physical acoustics - fundamentals and applications - laszlo adler
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Physical Acoustics - Fundamentals and Applications - Laszlo AdlerTRANSCRIPT
PHYSICAL ACOUSTICS Fundamentals and Applications
PHYSICAL ACOUSTICS Fundamentals and Applications
Edited by
Oswald Leroy Katholieke Universiteit Leuven Campus Kortrijk Kortrijk, Belgium
and Mack A. Breazeale The National Center Jor Physical Acoustics University oj Mississippi University, Mississippi
PLENUM PRESS • NEW YORK AND LONDON
LIbrary of Congress Cataloglng-In-Publlcation Data
International Symposiuo, on Physical Acoustics (1990 Kortrijk, Belgium)
Physical acoustics fundamentals and applications / edIted by Oswald Leroy and Mack A. Breazeale.
p. cm. "Proceedings of an International Symposium on Physical AcoustIcs,
held June 19-22. 1990, in KortriJk, Belgium"--Includes bibliographical references and index. lSBN-13, 978-1-4615-9575-5 e-lSBN-13, 978-1-4615-9573-1 DOl, 10.1007/978-1-4615-9573-1 1. Acoustics--Congresses. I. Leroy. Oswald. II. Breazeale, Mack
A. III. Title. OC221. 158 1990 534--dc20 91-16452
CIP
Proceedings of an International Symposium on Physical Acoustics, held June 19-22, 1990, in Kortrijk, Belgium
ISBN-13: 978-1-4615-9575-5
(c) 1991 Plenum Press, New York Softcover reprint of the hardcover 1st edition 1991 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic. mechanical. photocopying, microfilming, recording, or otherwise, without written permission from the Publisher
ORGANIZING COMMITTEE
O. LEROY (Chairman)
Katholieke Universiteit Leuven Campus Kortrijk, Kortrijk, Belgium
J. THOEN
Katholieke Universiteit Leuven, Leuven, Belgium
P. BOONE
Rijksuniversiteit Gent, Gent, Belgium
G. QUENTIN
Universite Paris VII, Paris, France
B. POIREE
Ministere de la Defense, Paris, France
R. REIBOLD
Physikalisch-Technische Bundesanstalt, Braunschweig, Germany
HONORARY COMMITTEE
Mgr. G. MAERTENS, Rector of KULAK
Prof O. VANNESTE, Governor of West-Flanders
SPONSORSHIP AND SUPPORT
Catholic University Leuven Campus Kortrijk
National Foundation for Scientific Research
Research Council of the Catholic University of Leuven
N.V. Bekaert S.A.
N.V. Devlonics S.A.
CERA
Picanol
TECHNICAL ASSISTANCE
Interdisciplinar Research Center
Faculty of Science of the Catholic University Leuven Campus Kortrijk
v
PREFACE
This book contains 17 invited papers and 80 communicated papers presented at
the International Symposium on Physical Acoustics, held at the University Campus of
Kortrijk, Belgium, from 19-22 June 1990.
The twenty-fifth anniversary of the Campus was celebrated with special activi
ties such as concerts, exhibitions and scientific meetings. This symposium was a part
of the celebration. The 120 participants came from 18 different countries. Among the
largest groups we mention 32 French contributions and 19 contributions from the
U.S.S.R. We especially thank Prof. V.V. Proklov from Moscow and Prof. S.V.
Kulakov from Leningrad who helped us with the distribution of invitations in the
U.S.S.R. We also thank Prof. G. Quentin and Ir B. Poiree from Paris who endeav
oured to inform all French acousticians. We thank all the lecturers for their effort in
producing the material for the book in time. The invited lectures have been collected
and retyped by Prof. M. Breazeale (U.S.A.), while the contributed papers were collec
ted by Prof. O. Leroy and retyped in Belgium.
The first 200 pages of the book comprise the invited lectures, not classified by
topic, but are in alphabetical order with reference to the first author. The second part
of the book contains the contributed papers and posters also classified in alphabetical
order according to the first author.
vii
CONTENTS
Section I: INVITED PAPERS
The Role of Surface Acoustic Waves in Scanning Acoustic Microscopy ............................................ 3 L. Adler
Ultrasonic Waves in Monodimensional Periodical Composites ........................................................ 13 A. Alippi, A. Bettucci and F. Crac:iun
What To Do When Your World Turns Nonlinear .......................................................................... 21 M.A. Breazeale
Thermal Waves for Material Inspection .......................................................................................... 31 G. Busse
Ultrasonic Backscattering: Fundamentals and Applications ........................................................... .41 B. Fay
Lasers in Acoustics ........................................................................................................................... 55 L.M. Lyamshev
Optical Generation and Detection of Ultrasound ............................................................................ 65 J.P. Monchalin
Acousto-Optical Interaction in a Medium with Regulated Anisotropy ........................................... 77 V.N. Parygin
Surface and Subsurface Waves for Characterization of Weakly and Highly Anisotropic Materials ....................................................................................................................... 87
A.Pilarski
Complex Harmonic Plane Waves ..................................................................................................... 99 B. Poire'e
Use of Short Pulses and Ultrasonic Spectroscopy in Scattering Studies ........................................ 119 G. Quentin
Optical Nearfield of Ultrasonic Light Diffraction ........................................................................... 129 R. Reibold and P. Kwiek
Resonance Scattering Spectroscopy by the M.I.I.R ........................................................................ 143 J. Ripoche
Reflection and Refraction of Heterogeneous Waves at Plane Interfaces ........................................ 155 J. Roux
ix
Modulation Effects in Light Diffraction by Two Ultrasonic Beams and Applications in Signal Processing ........................................................................................... 165
A. Sliwinski
Photoacoustics Applied to Liquid Crystals .................................................................................... 179 J. Thoen, E. Schoubs and V. Fagard
Recent Development of Ultrasonic Motors ..................................................................................... 189 S. Ueha
Section II: COMMUNICATIONS and POSTERS
Photoacoustic and Photothermal Characterization of Amorphous Semiconductors Thin Films ..................................................................................................................................... 199
G. Amato, G. Benedetto, L. Boarino and R. Spagnolo
Two-Beam Bragg Diffraction ......................................................................................................... 205 S.N. Antonov, V.N. Kotov and V.N. Sotnikov
High-Frequency Modulation of the White Light with Acoustooptic Modulator ............................ 209 S.N. Antonov and V.N. Sotnikov
Leaky Waves in Solid-Liquid-Solid Systems. Acoustoelectronic Microanalysis of Viscous-Elastic Properties for Liquids of Biological Nature ...................................................... 213
A.V. Arapov, V.S. Goncharov, S.V. Ruchko and I.B. Yakovkin
Acoustic Wave Propagation in Media Containing Two-Dimensional Periodically Spaced Elastic Inclusions ............................................................................................................... 219
C. Audolyand G. Dum&y
Scattering of Shear Horizontal Waves by Massive Metallic Electrodes in SAW Devices ............... 225 A.R. Baghai-Wadji
Acoustooptic Interaction Application to Optical Wavefront Analysis ........................................... 231 V. Balakshy and L.V. Balakin
Determination of Parameters for the Simulation of Surface Acoustic Wave Devices with Finite Elements ......................................................................................................... 237
P. Bauerschmidt, R. Lerch, J. Machui, W. Ruile and G. VlSintini
Analysis of the Nearfield of Laser Light Diffracted by a Plane Ultrasonic Wave .......................... 243 E. Biomme, R. Briers and O. Leroy
Magnetoelastic Mode Effect on Spin-Wave Instability Threshold ................................................. 249 A.S. Bugaev, V.B. Gorsky and A.V. Pomyalov
Methods of Acoustic Microscopy in Investigation of High-Temperature Superconductors ............ 253 M. Bukhny, L.A. Chemosatonsky and R.G. Maev
On the Extension of Horn Theory to Non-Uniform Visco-Elastic Rods ....................................... 261 L.M.B.C. Campos
Enhanced Propagation in a Foamy Medium .................................................................................. 271 A. Cowley, A. Baird, C. Harrison and T. Gedrich
Analysis of Surface Acoustic Wave in Layered Structure with Periodic Delamination .................. 281 E. Danicki
x
Surface Acoustic Wave Scattering by Elliptic Metal Disk on Anisotropic Piezoelectric Halfspace ........................................................................................................................................ 287
E. Danicki
Optical Detection and Analysis of Non-Linear Optoacoustic Waves ............................................. 291 J. Diaci and J. Mozina
Experimental Study of Guided Waves Propagating at the Interface Between a Fluid Cylinder and a Liquid Medium ......................................................................................... 297
J.M. Drean J.M. and M. de Billy
Surface Acoustic Wave Reception by an Interdigital Transducer .................................................. 307 J. Duclos and M. Leduc
Underwater Sound Scattering by Surface Gravity Waves .............................................................. 313 D. Euvrard and O. Mechiche Alarui
Finite Amplitude Acoustic Waves Radiating from a Non-Resonant Vibrating Plate .................... 319 M.A. Foda
Scholte-Stoneley Waves in a Multilayered Medium with Elastic Bond Conditions at an Interface ................................................................................................................................ 327
H. Franklin, M. Rousseau and Ph. Gatignol
Sound Intensification in Foam ....................................................................................................... 335 1.1. Goldfarb and I.R. Shreiber
The Influence of Heat Transfer and Liquid Flow on Sound Propagation in Foam ........................ 341 1.1. Goldfarb, I.R. Shreiber and F.1. Vafina
The Peculiarity of Non-Linear Waves Evolution in Bubbly Liquids ............................................. 347 A. Gubaidullin
Boundary Element Method Analysis of Surface Acoustic Wave Devices ....................................... 353 K. Hashimoto and M. Yamaguchi
Application of the Finite Element Method to Analyze the Scattering of a Plane Acoustic Wave from Doubly Periodic Structures ........................................................................... 359
A.C. Hennion, R. Bossut, J.N. Decarpigny and C. Audoly
Sound Propagation in Glass-Ceramic ............................................................................................ 365 Z. Hong, Z. Ming-Zhou, X. Yong-Chen and F. Cui-Ying
Visualization of the Resonances of a Fluid-Filled Cylindrical Shell Using a Low Frequency Schlieren System ........................................................................................................... 371
V.F. Humphrey, S.M. Knapp and C. Beckett
Light Scattering on Magnetostatic Waves in Thin-Film Gyrotropic Waveguide .......................... 377 LA. Ignatiev, V.G. Plekhanov and A.F. Popkov
Rayleigh Wave Tomography .......................................................................................................... 381 D.P. Jansen and D.A. Hutchins
Optical Interferometric Detection of Plate Waves on Circular Piezoelectric Transducers ............. 385 X.P. Jia, P. Mantel, J. Berger and G. Quentin
Experimental Study of Reflected Beam Profile by Two-Layer Plate Systems Immersed in Water ......................................................................................................................................... 391
A. Jungman, Ph. Guy, G. Quentin and J.C. Le Flour
xi
Study of Phase Transitions by Frequency Dependent Photoacoustic Measurements ..................... 399 S. Kojima
Multichannel Acoustooptical Modulators and their Applications in the Devices of Signal Processing ....................................................................................... ' ............................... .405
S.V. Kulakov, O.B. GUIleV, D.V. Tigin and V.V. Molotok
An Acoustic Ray Analysis of Wave Dispersion in Layered Structures .......................................... .413 J. Laperre and W. Thys
The Angular Resolution of Acousto-Optical Lamb Mode Detection ............................................ .417 J. Laperre, W. Thys and H. Notebaert
Inhomogeneous Plane Waves in Layered Media ........................................................................... .425 W. Lauriks, J.F. Allard and J. Cops
Depth Profiling by Fourier Analysis of Photoacoustic Signals ...................................................... .433 W. Lauriks, C. Glorieux and J. Thoen
A New Method for the Detection of Viscoelastic Solid Resonances: The Im/Re Spectrum .......... .441 O. Lenoir, P. Rembert, J.L. Izbicki and P. Pareige
The Mode Method in the Theory of Acoustic Wave Diffraction on Division Boundaries Between Different Structures ...................................................................................... .451
O. Leroy and G.N. Shkerdin
Acoustic Waves in Two-Phase Media .......................................................................................... .457 J. Lewandowski
Propagation Velocity and Attenuation Coefficient of Rayleigh-Type Waves on Rough Solid Surfaces ............................................................................................................... .461
J.Lewandowski
Localized Wave Transport of Pulsed Beam Energy ...................................................................... .467 D.K. Lewis, R.W. Ziolkowski and B.D. Cook
Imaging Vertically Oriented Defects with Multi-SAFT ................................................................ .473 M. LoreIllll, U. Stelwagen and A.J. Berkhout
Temperature Dependence of Optical Energy Gap in (As)x(Se}t-x and (Ge)x(Se}t-x Glasses ....... .479 K.N. Madhusoodanan and Jacob Philip
Investigations of Photoacoustic Signals in Powders ...................................................................... .485 U. Madvaliev, V.V. Proklov and A.M. Ashurov
Photothermal-Wave Diffraction and Interference in Condensed Media: Experimental Evidence in Aluminum ........................................................................................... .489
A. Mandelis and K.F. Leung
Evaluation of the Thickness of Shells by the M.I.I.R. . ................................................................. .497 G. Maze, H. Cahingt, F. Lecroq and J. Ripoche
The Nth Order Approximation Method in Acousto-Optics and the Condition for "Pure" Bragg Reflection ................................................................................................................ 505
R.A. Mertens, W. Hereman and J.P. Ottoy
An Improved Theory of Photoacoustic Signal Generation in Gases and Liquids ........................... 511 A. Miklos, Z. Bozoki and A. Lorincz
Theory of Photoacoustic Effect in Linear and Nonlinear Gyrotropic Piezoelectric Crystals .......... 517 G.S. Mityurich, V.P. Zelyony and A.N. Serdyukov
xii
Effects of Self-Action - Unexplored Field of Nonlinear Acoustics of Solid Surfaces ..................... 523 V.G. MOBhaev
A Novel Technique for Interface Wave Generation ....................................................................... 529 P .B. Nagy and L. Adler
Parametric Mixing Effects in Surface Acoustic Waves Caused by Gas Bubbles in Liquids ........... 537 Y. Nakagawa
Photoacoustic Investigation of Optical Energy Gap in As-Se-Te Glasses ..................................... 545 K. Nandakumar and Jacob Philip
Interferometric Probing of Optically Excited Surface Acoustic Wave Pulses for Thin Film Characterization ...................................................................................................... 551
A. Neubrand, L. Konstantinov and P. Hess
Investigation of Thermal Wave Interference in CdGa2S4 by the Photoacoustic Method .............. 557 P.M. Nikolic, D.M. Todorovic and Z.D. Ristovski
Pressure Waves Propagation in Gas-Liquid Foam ........................................................................ 563 Z.M. Orenbakh, I.R. Shreiber and G.A. Shushkov
Thermosensitivity of Generalized Rayleigh Waves for Rotated Y-Cuts in a-Quartz .................... 569 N.S. Pashchin and I.B. Yakovkin
Compression Elastic Wave Velocity and Absorption Measured by Electrical Impedancemetry ..... 573 J. Pouliquen and J.M. DurieB
Isolation of Resonances of a Cylindrical Target Immersed in Water by Means of a New Method Using Phase Information ........................................................................................ 581
P. Rembert, P. Pareige, O. Lenoir, J.L. IBbicki and G. Maze
Acoustic Emission Characteristics of Schists and Sandstones ........................................................ 591 M.C. Reymond, J.Fr. Thimus and Ph. Linse
Optoacoustic Raman Gain Spectroscopy of Binary Mixtures ........................................................ 599 M. Rohr, G.M. Bilmes and S.E. Braslavsky
Light-induced Vortex Current Generation as a New Mechanism of Photoacoustic Phenomena in Semiconductors ....................................................................................................... 605
V.A. Sablikov
Some Aspects of Lateral Waves Generation and Detection by Acoustic Microscopy Using V(z) Technique .................................................................................................................... 613
A. Saied, H. Coelho-Mandes, K. Alami, C. Amaudric du Chaffaut, J .M. Saurel and J. Attal
Study of Inhomogeneous and Heterogeneous Ultrasonic Waves in Kidney Stones ......................... 621 V.R. Singh and Agarwal Ravinder
Scattering of Acoustic Waves in Randomly Inhomogeneous Media by means of the Small Perturbations Method .................................................................................................... 625
E. Soczkiewicz
Electroacoustic Effect in Capillars Containing an Electrolyte ........................................................ 629 N. Tankovsky and J. Pelsl
Scholte Wave Diffraction by a Periodically Rough Surface ........................................................... 635 A. Tinel, J. Duclos and M. Leduc
xiii
Quantitative Determination of Absorption Using Coupled Amplitude and Phase Photoacoustic Spectra .................................................................................................................... 641
D.M. Todorovi~, P.M. Nikoli~ and Z.D. Ristovski
Acoustooptic Reflection Coefficient for Bounded Beams on Plates Using Inhomogeneous Wave Description .................................................................................................. 647
K. Van Den Abeele and O. Leroy
Reflection and Transmission Characteristics of an Alberich-Type Acoustic Barrier ...................... 657 P. Vidoret
Acousto-Optical Filtration of Electromagnetic Radiation in Ultraviolet Region ........................... 665 V.B. Voloshinov
On the Theory of Acoustooptic Interaction in Fabry-Perot Cavities ............................................ 671 A.!, Voronco, Ye.G. lsajanian and G.N. Shkerdin
Secluded Supersonic Surface Wave in the Transversely Isotropic Materials .................................. 677 L. Wang, S.A. Gundersen and J. Lothe
The Inverse Acoustic Scattering Problems for One-dimensional Lossy Media ............................... 687 N. Wang and S. Ueha
High Resolution Laser Picosecond Acoustics in Thin Films ........................................................... 695 O.B. Wright, T. Matsumoto, T. Hyoguchi and K. Kawashima
Photoacoustic Simultaneous Studies of Thermal Conductivity, Diffusivity and Heat Capacity Over the Smectic A-Nematic Phase Transition in Alkylcyanobiphenyls with Varying Nematic Range ............................................................................................................................... 703
U. Zammit, M. Marinelli, R. Pizzoferrato, S. Marlellucci and F. Scudieri
Properties of Surface Acoustic Wave Devices under Strong External Fields ................................. 709 J. Zelenka and M. Kosek
Acoustooptic Nonreciprocity .......................................................................................................... 715 G.E. Zilberman, L.F. Kupchenko and V.V. Proklov
Author index ............................................................................................................................... 721
Subject index .............................................................................................................................. 725
xiv
Section I:
INVITED PAPERS
THE ROLE OF SURFACE ACOUSTIC WA YES IN SCANNING ACOUSTIC MICROSCOPY
Laszlo Adler
The Ohio State University 190 W. 19th Avenue Columbus, Ohio 43210 USA
IN1RODUCTION
The reflection type acoustic microscope with water couplant has been used successfully in the last 10 years to image grain boundaries, solid-state bonds, or integrated circuits. The reflection coefficient of acoustic waves for most metals in water is in the order of 0.9; therefore the acoustic contrast even of different metals is quite weak when the transducer is focused to the surface of the sample. Strong contrast is observed even in the grain structure when the transducer is focused below the surface, provided that the angle of the aperture is large enough to generate Rayleigh type surface waves. Figure 1 shows the geometrical configuration of the scanning acoustic microscope. The rays (B) generate leaky Rayleigh waves which interfere with the specularly reflected wave (A) thereby producing contrast between points of constructive and destructive interference. In studying the contrast mechanisms it is found that the output voltage (V) of the microscope is sensitive to the vertical distance (z) of the lens from the sample, and a series of maxima and minima is observed. The characteristic peripdicity of V(z), often called acoustic materials signature (AMS), is related to the Rayleigh velocity of the sample and hence to near-surface material properties. Two equivalent explan~tions of the AMS using either ray theory or Fourier optics were given in the literature .1-1 According to the latter approach, the output of the transducer can be written as 9
(1)
where R(9, <1» is the reflectivity of the sample, P(9) is the pupil function, a. is the viewing angle of the focusing lens, 9 and <I> denote the polar and azimuthal angles, respectively, and k is the wave number in the coupling liquid. In a recent paper, 11 an alternative approach to V(z) was suggested: since the defocusing (z) is measured in terms of the acoustic wavelength in a couplant, therefore it always appears as a product with (k) or subsequently with frequency (f). It was pointed out that only V(fz) can be regarded as a system independent material parameter and should replace V(z) on the left side of Eq. 1. V(z) is simply a particular cross section of V (fz) at a certain frequency. In a similar way we can consider V(f) i.e. the cross section of V (fz) at a certain defocusing depth. On Fig. 2, the detected V(fz) is shown for glass as a two dimensional distribution over the range z = -1 to o mm and f = 20 to 40 MHz. The Rayleigh wave velocity can be measured then from the periodicity of either V(f) or V(z). In addition to V(z), V(f) can also be used to optimize the contrast in the acoustical image.
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 3
4
ZnO transducer
B
Sapphire rod
Water droplet
A
I
, I I
_ ',t' _
B
Focal plane - --Fig. 1. Geometrical configuration of the reflection-type acoustic microscope.
o
-0.2
e-0 .4 .§ Ul :::) o o u.. ~ - 0 .6
- 0 .8
- 1.0
FREQUENCY (MHz)
Fig. 2. Amplitude of the detected ultrasonic signal from glass as a function of defocusing distance and frequency.
The presence of the leaky Rayleigh waves also explains the strong contrast obtained at various interfaces such as grain boundaries, cracks, solid-state bonded metals, or layered materials. These boundaries reflect and scatter the Rayleigh surface wave and therefore alter the V (fz) dependence as the microscope is scanned over the boundaries.
The role of the surface wave in the acoustic microscope is to increase sensitivity to material and interface property measurements. We will describe two inspection techniques taking advantage of surface waves to maximize the sensitivity of the acoustic microscope for interface studies. Figure 3 shows the geometrical configurations for normal and parallel interface studies.
NORMAL INTERFACE
When a Rayleigh wave is incident on a normal discontinuity such as a crack or boundary between grains, it is partially reflected and partially transmitted. One may distinguish between two contrast mechanisms according to whether or not the insonified spot lies directly on the material discontinuity. 12, 13 On Fig. 4 the two cases are shown on a schematic diagram. The first case is called short range contrast and results in the boundary line brighter or darker than the neighboring grains, depending on the defocusing. The second effect is referred to as long range contrast and it results in periodic fringes running parallel to the reflecting discontinuity.
Short Ran~e Contrast
Grain Boundary. V(z). Figure 5 shows the origin of both grain and short range grain boundary contrast of a polycrystalline material. Consider two neighboring grains and the boundary between them. Since the periodicity of the corresponding Vl(Z) and V2(Z) curves is different, their relative contrast is alternatingly positive or negative as we increase the defocusing. There are points where the grain contrast is maximum in either direction, but grain boundaries are not visualized. On the other hand, there are the intersection points of the V(z) curves where the grains appear to have the same brightness and the grain
NORMAL:
Sample
Interface
PARALLEL:
Interface
Sample
Fig. 3. Interface inspection techniques with acoustic microscope. (a - normal, b - parallel).
5
6
A B
Lens
Specimen
B B
leaky Rayleigh wave
reflective discontinuity
A
A axial ray
" OR: OR :' ·I~.·
, . , . , , . " , '" '.'
focus (0,0)
Discontinuity
liquid
specimen
Fig. 4. Schematic diagram of Rayleigh wave interaction with normal interface. (a - short range, b - long range contrast).
VIz)
-z o
Fig. 5. Grain and grain boundary contrast in acoustic microscopy.
boundary between them shows up as brighter or darker regions depending on whether the intersection point is below or above the specular reflection. This is because the grain boundary scatters the surface wave component and the detected interference signal becomes offset toward the specular components. On Fig. 6, is shown the acoustic micrograph of a titanium sample taken at four different defocusing levels at 1.6 GHz. It is clear that both grain and grain boundary contrast can be enhanced by taking advantage of interference between the specularly reflected and the leaky wave components.
Solid State Bond. VW. The interchangeabilityll of z and f in obtaining both material and interface parameters is illustrated on Fig. 7. On this figure an experimental V(f) curve is shown on a stainless steel-aluminum solid-state bond at z = -1.1 mm defocusing. The periodicity of V(f) curves is 4.64 MHz and 4.98 MHz for steel and aluminum respectively giving their respective Rayleigh velocities at 2 875 mls and 2970 mls. The intersection points can be used to visualize the boundary between the two metals. Figure 8 shows an acoustic micrograph of the steel-aluminum weld taken at a, 32.3 MHz; b, 33.5 MHz; and c, 34.5 MHz demonstrating both material contrast at a and c, and interface contrast at b.
~~~~c~o~n~tra~s~t at different aetocllSHlg titanium taken with a frequency of 1.6 GHz.
mm; b, z=-3.8 mm; c, z=-4.2 mm; d, z=-4.8 mm).
7
8
5
4 • • ••.• St eel -- Aluminum
3
iii 2 .. .-
~ . .
w 0 ::> ..... 0 ~ Q. :t: ~
- 1
-2
- 3
- 4
- 5 0 5 10 15 20 25 30 35 40 45 50
FREQUENCY (MHz)
Fig. 7. VCf) curves of aluminum and stainless steel at z = -1.52 mm.
a
b
c
Fig. 8. Acoustic microscopic material and interface contrast of an aluminum-steel solid-state bond at z=-1.52 mm. Ca, 32.3 MHz; b, 33.5 MHz; and c, 34.5 MHz).
Long Range Contrast
The long range contrast (see Fig. 4b) is demonstrated on Fig. 9 where the acoustic micrograph of a nickel bicrystal at 1.6 GHz is shown. The darker and brighter domains are the two large grains of different crystallographic orientations while the multiple fringes running parallel to the grain boundary are caused by the long range contrast of reflecting interfaces. Because of alternating constructive and destructive interference between the incident and reflected surface waves, a standing wave pattern appears, resulting in fringes running parallel to the interface. The separation of fringes can be used to measure surface wave velocity.
Another example of long range contrast is shown on Fig. 10 where the acoustic image of a graphite-fiber-reinforced composite is shown which was taken at 1.7 GHz. The fiber diameter is 8 - 10 1.1., approximately 10 times higher than the image resolution. The long range contrast causing the Rayleigh wave reflection at the fiber-matrix interface produces interference fringes as seen on Fig. 10. The period of these standing wave patterns can be used to measure the surface wave velocity in the graphite fiber. In this case the measured average separation between bright fringes is approximately 1.5 1.1.. Since the periodicity corresponds to a half wavelength at 1.7 GHz, the leaky Rayleigh velocity is 1760 mls. Even more importantly, the amplitude of these fringes is proportional to the surface wave reflection coefficient in the fiber-matrix interface. Continuity of these fringes is a clear qualitative indication of bond integrity.
PARALLEL INTERFACE
A layered structure can support an infinite number of so-called generalized Lamb modes which are strongly dispersive. For the special case of a thin (much less than the wavelength) layer on a substrate, the principal mode of practical interest is the so-called dispersive Rayleigh mode. At very low frequencies, the thin layer has negligible effect on the surface wave propagation and the principal mode behaves like a Rayleigh wave on the free surface of the substrate. At very high frequencies, the substrate has an effect on the layer and the principal mode degenerates into a Rayleigh mode on the free surface of the layer. As an example,14 Fig. 11 shows theoretical and experimental phase velocity versus frequency-times-thickness for a stainless-steel layer bonded to an aluminum substrate. The solid line represents theoretical calculation for an ideal bond condition and the dashed curve is for a complete misbond. The theory agrees well with experiments for these two extreme cases. The other poor bonds. so-called kissing bonds, fall somewhere between these two extremes. In another paper, 15 we have suggested an approximate method to calculate the
Fig. 9. Acoustic micrograph of a nickel bicrystal taken at 1.6 GHz.
9
Fig. 10. Acoustic micrograph of graphite-fiber reinforced epoxy composite taken at 1.7 GHz.
phase velocity of the dispersive Ra(jleigh wave for a poorly bonded layered substrate using a finite boundary stiffness model. 1 The actual stiffness constants were chosen to cover a large range and the result is given on Fig. 12. In order to estimate the resulting contrast produced by such interface imperfections under acoustic microscopy inspection, V(z) curves were calculated for different bond qualities. Figure 13 shows the calculated V(z) curves and frequency-times-Iayer-thickness equal to one (f x d = 1) for different bond conditions. As can be seen from Fig. 13a, the highest inspection sensitivity is in the range of fz - 4 to 14 MHz mm. While the actual contrast changes from positive to negative values
-OJ
E ~ > ~ (3 0 -' w > W ffI ...: :J: ~
3.0..--- --------- --- ---...,
2.0
o
o ;' .. // ;'
-" -"
-"..--
... y o / .. ,,/ .. ..,
• • -II
0/
o I
• I 0,1
o I
f * /
1.0
EXPERIMENT : • good bond • poor bond 1Io free plate
THEORY: _ rigid
--- Iree
2.0
FREQUENCY X THICKNESS (MHz x mm)
Fig. 11. Theoretical and experimental phase velocity versus frequency x thickness curves for stainless-steel layer on aluminum substrate.
10
3,---------------------------------~
-;; E ~ 2.5
rilid bond .Ironl Idsslnl bond medium klssin, bond wuk kluinl bond rree p'"le
1.5+-...... ~---_--...,....--~---...... --_f o .5 1.5 2 2.5 3
Frequency l[ Thickness (MHz mm)
Fig. 12. Calculated dispersion curves of the modified Rayleigh mode for steel layer on aluminum substrate for different interface qualities.
at different defocusing depths, the highest obtainable contrast is dependent on the velocity change caused by a g ,iven boundary imperfection with respect to the ideal rigidly bonded layer. Figure 14 shows this acoustic contrast for medium kissing bond as a function of fd. The sharp maximum at fd :::; 0.4 indicates that there exists an optimal frequency where the interface imperfectioll produces the strongest defect signal on the acoustic micrograph. These results show that acoustic microscopy has a quite unique contrast mechanism since conventional bulk inspection always offers higher sensitivity at higher frequencies.
CONCLUSIONS
Leaky Rayleigh waves generated by the transducer of the scanning acoustic microscope are shown to play an important role in the generation of image contrast. Different techniques were shown to increase this contrast by optimizing the effect of normal and parallel material interfaces on the V(z) or Vef) curves .
.. .. . 8
~ .6 o :> .. .: .. ;; ;;; = ,2
""uk kwinC bond ~ medium kiss in& bond ",one ki .. in~ bond~
0T-~~~~_.~,_~--...... _r_.~,__4 - 20 -18 -16 -14 -12 -10 - 8 - 6 · 4 o
rz (MHz mm)
.. .. .. .8
;:: ,8 o
:>
,2
(,oop'"'e ~
rI&idbond~
O~-r~~~~_.~,_~-~,_~~ - 20 -18 -16 -14 -,2 -10 - 8 - 6 - 4 - 2 0 2
rz (MHz mm)
Fig. 13. Calculated V(z) curves at fd = 1 for different bond qualities.
11
500
'" 400 E .. u
300 c .. .. ~ "-
is 200
~ U 0 100 .. >
0 0 .5 1.5 2 2.5 3
Frequency" Thickness (MHz mm)
Fig. 14. Acoustic contrast for medium kissing bond as a function of fel .
ACKNOWLEDGEMENT
The author would like to thank Dr. Peter B. Nagy for his many contributions. This work was supported by the U. S. Department of Energy Basic Energy Science Grant No. DE-FG02-84ER45057 .AOOO5.
REFERENCES
1. A. Atalar, C. F. Quate, and H. K. Wickramasinghe, App\. Phys. Lett. 31:791 (1977).
2. H. K. Wickramasinghe, Electron. Lett. 14:305 (1978). 3. R. D. Weglein and R. G. Wilson, Electron Lett. 14:352 (1978). 4. A. Atalar, J. App\. Phys. 49:5130 (1978). 5. R. D. Weglein, App\. Phys. Lett. 34:179 (1979). 6 . H. K. Wickramasinghe, J. App\. Phys 50:664 (1979). 7 . W. Parmon and H. L. Beroni, Electron. Lett. 15:684 (1979). 8. A. Atalar, J. App\. Phys. 50:8237 (1979). 9 . C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38:858 (1981). 10. H. L. Bertoni, IEEE Trans. Son. Ultrason. SU-31:105 (1984). 11. P. B. Nagy and L. Adler, J. App\. Phys. 67:3876 (1990). 12. M. G. Somekh, G. A. Briggs and C. lIett, Phil. Mag. 49: 179 (1984). 13. H. L. Bertoni, "Wave Phenomena" V2, E. A. Ash and E. G. S. Paige, eds.,
Springer-Verlag (1985). 14. L. Adler, M. deBiIIy, G. Quentin, M. Talmant, and P. B. Nagy, "Rev. Frog.
QNDE" Vol. 8B, D. O. Thompson and D. E. Chimenti, eds., Plenum Press, New York (1989).
15. P. B. Nagy and L. Adler, CoIIoque de Physique Suppl. Jour. de Physique, Fasc. 2 C-2:1273 (1990).
16. N. F. Haines, Report R. D./B/N4744 C.E.G.B. Berkley Nuclear Laboratories (1980).
12
ULTRASONIC WA YES IN MONODIMENSIONAL PERIODIC COMPOSITES
A. Alippi*, A. Bettucci, and F. Craciun
C. N. R., Istituto di Acustica "0. M. Corbino" Via Cassia 1216 1-00189, Rome Italy
*also at·
Universita "La Sapienza" Dipartimento di Energetica Via A. Scarpa 14 1-00191, Rome Italy
INTRODUCTION
Composites is a name commonly reserved for a large class of materials, where two or more different constituents, or phases, combine, and while remaining distinct, give rise to a new material with macroscopic properties sensibly different from those of the original ones. l The presence of discontinuity surfaces between different phases is the origin of new characteristics of the acoustic propagation; namely, the propagation becomes dispersive, because of the multiple resonances produced by the bounded middle scale structures inside the material. Features are largely different according to whether the distribution of one phase within the composite is random or is ordered. In this latter case, dispersion curves can be predicted and investigated at ease, while in the former one, a statistical approach should be followed. In addition, the dimensionality of the spatial distribution is a conditioning parameter for the success of any theoretical approach.
Presently we give a model for the interpretation of dispersion characteristics of acoustic waves that propagate in monodimensional composites, that can be applied to any ordered or disordered monodimensional structures. Study was stimulated by the interest in composite piezoelectric plates, widely used in ultrasonic applications for improving matching and efficiency conditions in underwater propagation. Some experimental results are presented for comparison, relative to composite piezoelectric plates; theoretical predictions also are given in case of acoustic propagation along strings locally loaded by pointlike masses.
ELASTIC W AYE PROPAGATION IN COMPOSITES
Properties of composites strongly depend upon the properties of the constituents, and upon their relative distribution, size, and shape. General approximations describe composite geometries in terms of the reinforcement (the structural constituent) which may be in the shape of cylinders, rectangular prisms, needles, spheres, plates, etc. A branching classification of composites in terms of the reinforcement geometrical features2 is given in Table 1.
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 13
Table 1. Classification of composite materials.2
FIBER REINFORCED
COMPOSITE MATERIALS
COMPOSITES A PARTICLE REINFORCED COMPOSITES
RANDOM A PREFERRED ORIENTATION ORIENTATION
SINGLE - LAYER COMPOSITES MULTILAYERED COMPOSITES
LAMINATES A HYBRIDS
CONTINUOUS FIBERS DISCONTINUOUS FIBERS
UNIDIRECTIONAL REINFORCEMENT
RANDOM A PREFERRED ORIENTATION ORIENTATION
BIDIRECTIONAL REINFORCEMENT
The theoretical model that will be given here is well adapted to monodimensional structures, as they are single - or multi-layered composites, and laminates. Most generally, the elastic propagation in inhomogeneous materials can be simply treated by considering that the physical properties of interest, mass density p and elastic constants CIJ, are functions of the ~atial coordinates. Newton's equation for displacement u (in reduced subscript notation) then becomes
(1)
where i, j = 1, ... 3, J, J = 1, ... 6, co is the angular frequency of the propagating wave, and p = p(x,y,z), CIJ = CIJ(x,y,z).
For homogeneous and isotropic constituents, p and c are two-value constants, whose spatial distributions depend upon the considered materials, and can be expanded in Fourier series for regularly, or quasi regularly, structured composites. 1 However, the typical feature of elastic propagation in composites being interfacial reflection and transmission of the wave, it is convenient to approach the problem in this way and to consider a monodimensional structure as a sequence of several segments; in each one of them the velocity of propagation and then the wavevector are given, for any frequency. At each segment boundary a transmission and a reflection coefficient [t] and [r] will be defined respectively, that may generally assume a matrix notation in the case where several propagation modes are allowed in each segment. If only two constituent materials are present, matrices [t] and [r] are simply defined through the following equation (see Fig. 1)
(2)
[a+] [r][a-]
where [a'-], [a-] are n' and n dimensional column vectors, respectively, describing the wave amplitudes of the n' backward propagating modes allowed in the primed medium, and of the n modes allowed in the unprimed one. Analogously [a'+] is a n column vector describing the forwardly propagating modes in the primed medium. Matrix [t] is, therefore, a n x n' rectangular matrix, whose general term tij is the ratio of the transmitted mode j in medium 1 (primed) to the incident mode i in medium 2 (unprimed), in the case
14
2 2
. + + +
• • • .. • • .- -a a • • .. ..
t' t
r ~ c;:r
Fig. 1. Definition of propagation parameters through the interface between two semi-infinite media.
where the media are infinitely extended so that no stationary conditions establish because of finite dimensions of the segments. The transmission matrices [t] and [t'] are related by
, [Z'] [t] = t z . (3)
where Z', Z are the mode impedances in the two media. Matrix [r], conversely, is a n x n square matrix, whose general term lld defines the ratio of the reflected mode 1 to the incident mode k in medium 2, and analogously for r'kl in medium 1. It is, obviously:
kJ = - [r] . (4)
Due to the finite dimensions of each segment, a stationary wave condition developes in the composite, so that at each interface between adjacent segments a set of equations can be written among the four vectors that describe forward and backward propagating waves on Faqt side of the interface. In a multiple structure, it is convenient to redefine wave vectors La±J as the wave vectors at one end of each segment; therefore phase factors Sf, S'± for each component have to be defined:
S± = exp( +ikd) (5)
S'± = exp( +ik'd')
that take into account the phase lag of the propagation inside of each segment, when writing the boundary conditions at the interfaces. The lengths d', d are the lengths of segments in medium 1 and 2, respectively, and are supposed to be equal for all segments of the same material.
In this case, the boundary conditions are:
where
[a'+] = [L][a+] + [M][a-]
[a'-] = [P][a+]+ [Q][a-]
[L] = [n-1[S'-]
[M] = [t']-I[r][S'-]
[P] = [r'][t']-l[S'+]
[Q] = ([t] - [r'][t']-l[r])[S'+]
(6)
(7)
15
The overall transmittivity and reflectivity coefficients of a finite element structure can thus be evaluated by simply computing the ratios of the amplitudes
Tjj = a+j las!
a\ firs!
(8) Rkl = a'l firs!
a+k frrs!
in the assumption that a-j las! = 0, for every L In case the system allows only one mode to propagate in each segment, or that no
coupling is really efficient at the interfaces between modes save for one alone in each segment, Eqs. (6) simplify inasmuch as the matrix notation reduces to a scalar representation. That is the case that we consider in the following, for the two possible practical conditions: propagation of Lamb modes in composite plates and waves along a string, where the cross section alternately changes between two different values.
RESULTS
Comvosite plate
In a two-phase structure of infinite extent, Eqs. (8) represent the sequence of pass bands and stopbands that are predicted by Kronig-Penney model, if one introduces the periodicity conditions into Eqs. (6). Before considering the composite periodical structure, however, it is interesting to note how the transmission coefficient of a two-phase structure evolves toward the generation of stopbands, as the number of the constituent material segments increases4. Figure 2 represents the transmission function vs. frequency in case of structures having equal values of the acoustical paths in the two component media and a transmission coefficient t of 0.99 between them4. It can be observed that the stopband conditions are better satisfied in a sample with a greater number of elements contributing to Bragg scattering of the acoustic wave.
Figure 3 represents the transmission function vs . frequency for a structure of 5 elements for different values of transmission coefficient t = 0.75,0.91 and 0.99 between the two media4. The additional stopbands are due to the finite number of elements and they are more evident for lower values of t.
16
0 .5
N=31 o ~ ____________ ~~ ______________ ~
o n/2
'" (d Ie. d l c")
Fig. 2. Transmission function T vs. frequency in a two phase periodic structure, for different total number of elements.
~ 0.5
o L_ ______ ______ L_ ____ £_ ______
o
Cd (d l e+ d '/e")
Fig. 3. Transmission function T vs. frequency in a two phase periodic structure for different values of the transmission coefficient t between the two media.
The propagation of elastic waves in a two-phase composite was experimentally verified in composite plates. The plates were made by filling with melted epoxy resin the interspaces between parallel strips of piezoelectric material sliced from a ceramic plate. Thin plates thus were obtained, in which segments of given lengths d and d' alternate regularly along the propagation direction. The plates were metallized on both surfaces and the frequency response of the structure was measured with an impedance bridge meter. In this way, the resonant acoustic modes were determined in place of the transmission function. Lamb waves were excited in each segment and coupled together at the interfaces. Below the first thickness resonance of the slowest medium, however, only the first modes llo and So in each medium could be excited and freely propagate in the structure. Because of the excitation geometry, in addition, no energy could be fed into the antisymmetrical mode llo. Working, thus, at such conditions, the propagation of the Lamb modes in composites plates would only couple So modes between each other in the two different media and Eqs. (6) would then reduce to scalar relations for one propagating mode in each segment. Figure 4 represents the frequency response of a piezoelectric plate made of segments of thickness h = 0.4 mm and lengths d = 4.1 mm (epoxy) and d' = 1 mm
..., o z ~ !: 2 a
'" I
0 .8 il • z en
0 ·8 ~
~ 0 .4 <:
=i -<
0 .2
l '. L---~~--'-~~--~----~~--~----~L-~~10
0 .8 0·8
FREQUENCY (MHz)
Fig. 4. Frequency response of a piezoelectric plate made of eight segments (four ceramic and four epoxy). Broken curve is the calculated transmission vs. frequency for the same plate with v = 1950 mis, v' = 3200 mis, p = 1/17 x 103 kg/m3, p' = 7.65 x 103 kg/m3,
d = 4.1 mm, d' = 1 mm, N = 8.
17
(ceramic), together with the theoretical transmission calculated with the use of the following values for the propagation velocities of So modes and for the densities of the two media: v = 1950 mis, v' = 3200 mis, p = 1.17xlO3 kg/m3, p'= 7.65xlO kg/m3 (the geometrical parameters are: d = 4.1 mm, d' = 1 mm, total number of elements N = 8).
It is worthwhile to note that the experimental conditions of excitation of the plate are such that only those frequency modes are resonant which vibrate with the same phase in each one of the same medium segmentS: these are stopband edge frequency modes of the structure, as the correspondence with the theoretical predictions suggests. The experimental peaks are broader than the calculated ones, probably due to the mechanical losses.
Mass loaded strin~
As a second example of the monodimensional structured material, a vibrating string is considered, where pointlike masses are located at definite points along the axis. This example is considered, because it explicitly shows the influence of discontinuities in the propagation, that are scattering centers for the wave, but don't separate different media.
First, we shall consider what a mass m is producing along an infinite string of linear density p, that is stretched with a tension 'to The continuity conditions of the displacement through the point where the mass is located, together with the Newton equation for the mass forced by the restoring tension of the string, give the. following values of the scalar transmission and reflection coefficients t = Itle'«Pt and r = Irle'«Pr, to be used in Eqs. (6):
It! = (1 +y2k2r 1/2
Irl = )'k(I+y2k2rl/2
4>t = tan-l(-)'k)
4>r = tan-l(l/)'k)
(9)
where 'Y = f;, and k is the wave number. It is interesting to note that rand tare frequency dependent and their ratio is: ~ = - i ~ (the transmitted and the reflected waves are n/2 out of phase with respect to each other) . Using these values for t and r in Eqs. 6, together with the periodicity conditions, the following equation is obtained:
18
cos Kd = cos kd + )'k sin kd
2400 r = 2 .8 em
..... 2000 N
l:
1&00 >-U Z 1200 III :I a
800 III a:: ...
400
r = 7.1 em
0 .4 0 .8
K (:tId)
r = 8.5 em
0 .4 0 .8
Fig. 5. Calculated dispersion curves for mass loaded string. The parameters used in calculations are: d = 15 cm, c = 230 mls.
(10)
where K is the effective wave number and d the distance between two consecutive pointlike masses along the string.
Eq. 10 is the dispersion relation co(K) which relates the circular frequency of the wave co to its effective wave number. The real solutions are the well known acoustic branches, separated by stop bands where no frequency can propagate. Calculated dispersion curves are presented in Fig. 5 for a few values of y parameter. A peculiar feature is to be noted for this structure; namely: the lowest acoustic branch does not originate at zero above a certain y value. This critical value can be obtained easily by studying the behavior of Eq. lOin the vicinity of the origin (co = 0). It is Ycritical = d/2 which means that mcritical = pd. This makes sense when one remembers the geometry of the studied structure: masses m between stretched strings of lengths d. The masses filter low frequency vibrations if they are too heavy.
CONCLUSIONS
A model has been presented for the calculation of wave fields propagating in monodimensional periodical composites. A matrix representation has been used which reduces to a scalar representation when no coupling occurs between different propagating modes at the interfaces. The model has been applied to Lamb wave propagation in piezoelectric composite plates and some experimental results are presented for comparison. Theoretical predictions also are given for the dispersion characteristics of acoustic waves propagating along strings locally loaded by pointlike masses.
REFERENCES
1. B. A. Auld, Three-dimensional composites, in "Ultrasonic Methods in Evaluation of Inhomogeneous Materials," A. Alippi and W. G. Mayer, eds., Nato Adv. Study Inst. Series 126, Martinus NijhoffPublishers, DordrechtIBostonlLancaster (1987).
2. J. A. Gallego-Juarez, "Physical and elastic characteristics of fiber reinforced composites," ibidem.
3. B. A. Auld, "Acoustic Fields and Waves in Solids," Vol. I, Wiley-Interscience, New York (1973).
4. A. Alippi, "Propagation of elastic waves in one-dimensional composites," Materials Science and Engineering A122:71 (1989).
5. A. Alippi, F. Craciun, and E. Molinari, 1. Appl. Phys. 66:2828 (1989).
19
WHAT TO DO WHEN YOUR WORLD TURNS NONLINEAR
M. A. Breazeale
National Center for Physical Acoustics University, MS 38677
University of Tennessee Knoxville, TN 37919, USA
IN1RODUCTION
The research I intend to describe is a summary of the efforts of a number of students and postdocs. In order to produce these results they faced choices, and their choices not only have determined the results I will discuss, they also determined the progress of the student. When a student faces a difficult problem he can choose to give up. Such a student never will become a physicist. When a student faces a difficult problem he also can attack the problem and solve it. Such a student needs a more difficult problem, and usually gets one. Then there is the third reaction to a difficult problem. Some students have the unique ability to attack an exceptionally difficult problem, partially solve it, then gain insight that might contribute to the general solution by analyzing the partial solution. These are the physicists - and I feel very fortunate to have been able to work with several of them. These are the people who respond in a very special way. When life hands them a lemon, they don't complain. They make lemonade.
This is exactly the situation any physicist finds himself in when he begins a study of the physical properties of condensed matter. The lemon is the fact that all of nature is nonlinear. The human desire to consider only linear processes is thwarted by mother nature herself. One always encounters nonlinearity if one subjects any theory to experimental confirmation, so the solution never is a complete one. On the other hand, the lemonade is the fact that the linear approximation works well as it does. There actually are scientists who believe that they can simply ignore nonlinear effects as being insignificant, then they are offended when they discover that the real physical system has unavoidable nonlinearities in it and they must modify their approach if they are to progress further.
This situation has existed at least since the time of Hooke, but I really think that finally we are beginning to understand nonlinear behavior well enough that we can live with it without wanting to ignore it so it will go away. We finally are beginning to understand in a more complete way the fact that we should look for useful aspects of nonlinear behavior, not bemoan its inevitability. Physics is becoming more sophisticated, so we really need to know what to do when nonlinear behavior is the only thing that keeps our mathematical theory from being absolutely correct. We need to know what to do when our world turns nonlinear.
For many years, in my laboratory we have made measurements of the nonlinear properties of solids. We have accumulated data on a number of crystals, but one set of crystalline solids, the diamond lattice solids, had proved to be describable on the basis of a relatively simple lattice dynamical model. Other solids, even other cubic solids, are more complicated. As a basis of categorizing the behavior of the complicated solids, I choose to
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New \' ork, 1991 21
remind you of the behavior of the diamond lattice solids silicon and germanium we can measure, then contrast the behavior of NaCl, an ionic crystal with a very large nonlinearity parameter, with it. Thus, I hope to provide insight on the complicated nonlinear behavior of NaCl by comparing it with the behavior of silicon whose nonlinear behavior is more comprehensible.
THEORETICAL CONSIDERATIONS
One form of the equation describing propagation of an ultrasonic wave in a solid (including nonlinear terms) along the a, direction is:
(1)
where Jik is the Jacobian matrix and <P(ll) is the strain energy. By inserting the definition of the strain energy as an expansion in the strains
<P(ll) = i, L Cijklllijllkl + 3\ L Cijklmnllijllklllmn + ... , . ijkl . ijklmn
(2)
one can evaluate the nonlinear wave equation in terms of linear combinations of second order elastic (SOE) constants K2 and third order elastic (TOE) constants K3. For pure mode propagation directions in crystalline solids the nonlinear wave equation is found to have the form
(3)
The exact values of K2 and K3 depend upon crystalline symmetry as well as the propagation direction, but they are linear combinations of SOE and TOE constants. For the principal directions in a cubic lattice the expression for K2 and K3 are given in Table I.
Table I. K2 and K3 for [100], [110], and [111] directions in a cubic lattice
Direction
[100]
[110]
Cll
CII ! + 3CII2 + 12C!66 4
[111] C111 + 6CII2 + 12C144 + 24C166 + 2C123 + 16C456
9
An initially sinusoidal ultrasonic wave is described by assuming that
Sa,! = A sin (ka-rot) at a = o. (4)
The solution ofEq. 3 then takes the form
. ~A2k2a Sa,! = A SIn (ka-rot) + 8 cos 2(ka-rot) + .... (5)
22
where the nonlinearity parameter
(6)
is specifically identified. The measurement of the nonlinearity parameter, and hence the measurement of the TOE constants, is accomplished by measuring the amplitudes of the fundamental and the second harmonic after the wave has propagated a distance a. A modification of the capacitive microphone1 has proved to be especially useful in making these measurements. The use of this device has made possible measurement of the nonlinearity of a wide variety of samples? A modification for keeping the spacing constant at different temperatures has made possible measurement at temperatures as low as 3°K and as high as 500°K. With an accumulation of data has come the possibility to correlate nonlinear behavior with other physical properties. Generally speaking, the diamond lattice solids (zincblend structure) with covalent bonding exhibit the smallest magnitudes of nonlinearity parameter and the theory relating nonlinearity parameter with interatomic anharmonicity is most straightforward. NaCI structure with ionic bonding exhibits the largest magnitudes of nonlinearity parameter observed to date, so it may be expected that the theoretical explanation would be more complicated. A comparison of the data on NaCI with that on the diamond lattice solid silicon is quite informative.
NONLINEARITY OF DIAMOND LATTICE SOLIDS
By measuring the amplitudes of the fundamental and the second harmonic of an initially sinusoidal ultrasonic wave we have found that the predictions of Eq. 5, that the second harmonic amplitude is proportional to the square of the fundamental amplitude, is very well satisfied for both silicon and germanium. An example is given in Fig. 1, in which we plot measured A2 versus A12 for silicon. The linearity of the curves assures us of the accuracy of the predictions of Eq. 5, and the slopes of the curves is proportional to the nonlinearity parameter. We can see immediately that in silicon the nonlinearity parameter is largest in the [110] direction and smallest in the [100] direction. This is true of germanium as well. Such information can be correlated with the anharmonicity of the interatomic potential function, as we have shown. To make the correlation we measure as a function of temperature and plot the temperature variation of the nonlinearity parameters for the three
18
17
16
IS 14
13
12
e l l !:! 10
'0 9
8
'" «
11001
11101
o 10 20 30 40 SO 60
A~ (10.20 m2 )
Fig. 1. Plot of A2 as a function of AT for silicon.
23
principal directions in the diamond lattice solid silicon. This produces data such as shown in Fig. 2. Since the nonlinearity parameter is related to the combination of third order elastic constants K3 by
~=fK~:K3), (7)
and since K2 = pv2 is known, one has immediately a curve of the temperature dependence of K3 such as shown in Fig. 3. Table I, shows that the expressions for K3 all contain the TOE constant Clli. This quantity, and another one, can be subtracted from K3 [110] and K2 [111] to give the simplest combinations of TOE constants available from our data. The results are given in Fig. 4. For comparison, Fig. 5 presents corresponding results for germanium. The fact that these two curves have the same functional form assures us of the validity of the data, because silicon and germanium are both diamond lattice solids. The corresponding curves for a metal such as single crystal copper3 are fundamentally different as shown in Fig. 6. We have used the Keating4 theoretical model of diamond lattice solids to evaluate the temperature dependence of all six of the third order elastic constants5; however, for present comparisons, we will quote the result of assuming that central forces and nearest neighbor interactions predominate in defining the anharmonicity of the potential function in diamond lattice solids.
If central forces and nearest neighbor interactions are solely responsible for the interaction potential, then, of necessity, (Cauchy relations)
(8)
and for TOE constants
C111 = 2C112 = 2CI66 and (9)
In terms of the simplest combinations of TOE constants plotted in Figs. 4 and 5, these relationships would require that
CI12 + 4Cl66 = 5/2 ClIl
C123 + 6Cl44 + 8C456= o. (10)
These relationships arise from specific assumptions about the interatomic forces. The extent to which these assumptions are satisfied for silicon and germanium can be seen in Figs. 4 and 5 in which we plotted 5/2 CIII for reference. Examination of Figs. 4 and 5 leads one to the general statement that for diamond lattice solids the Cauchy relations
>-
~ 2.0"'--
~ 1.0
z o 50 100 150 200 250 300
TE MPERATURE (K)
Fig. 2. Silicon nonlinearity parameters as a function of temperature.
24
~ _°00<>0:>00000 OO000'-.:O0;~] 000 00000
10
19
0~-4~0~~8~0--~12~0~I~G~0~20~0~2~4~0~2~8~0 TEMPERATURE ( K )
Fig. 3. Silicon K3 values as a function of temperature.
(of 2: u <f) w z >-0
C\J ~o
<f)
z Q ~ z iD L
4
0
- 4
-8
8 - 12 IZ ;!: !;£ -16 o u
, TEMPERATURE (OK)
\ 50 100 150 200 250 3CX: \ \
\ , , "-
..... " LC,23+6Cw4+8~6 --------
;Cm
SILICON
/~C111 ----;---
-24 ~~--~--~--~--~--~
Fig. 4. Temperature variation of Silicon TOE constants.
25
26
E -2 u
~ 0 >. ;' - 2 g - - 4 I/l ~ Z -6 ~
TEMPERATURE (K) 50 100 150 200 250 300
~ -81 __ """"':=====-....::::::::--o U - l0 w ~ -12 ~ ::> -14 Z ~ - 16 0: w - 18 (!)
-20
em
~ 5 __ ]':II!. ____ - - - - - - -
Fig. 5. Germanium TOE constant temperature variation.
TEMPERATURE (·K) 50 100 150 200 250 300
-2
o - -.,.- - - -- 2
-4 ..
I/l -6
. . .. o 0 0 0
~ :11~I ...... z 0000 o 0 0 0
8 -16 ~ -34 I-a: - 36 w &: -38
8 - 40
.... q,
o 00 0
0 00 0
o-o-....R~o_-.. (C123 +6 C144+ 8C4 5S )
C111 . . 00 0 000000
000000 o 00 0 0
I
Fig. 6. Copper TOE constant temperature variation.
among the TOE constants appear to be satisfied at OOK, but are less well satisfied at higher temperatures. Let us compare these results with those from measurement of the nonlinearity parameters of NaCl.
NONUNEARITY OF NaCI CRYSTALS
To provide an extreme contrast with the data on diamogd lattice solids, we recently measured the nonlinearity parameters of NaCI single crystals. In comparing the two types of crystals one finds that the nonlinearity parameters of N aCI are approximately five times as large as those of diamond lattice solids. With such large nonlinearity parameters, it should not be surprising that the simple models used with diamond lattice solids no longer work; however, the comparison of behaviors still can be informative. Furthermore, NaCI has the largest thermal expansion coefficient at room temperature among the crystals measured in our laboratory. This has provided special problems for the experimenter. The results of the experiment, however, are of primary concern to the comparison.
Measurement of the amplitudes of the fundamental and second harmonic of an initially sinusoidal ultrasonic wave results in a plot of the second harmonic amplitude as a function of the square of the fundamental amplitude as shown in Fig. 7. Since the slope of the curves is proportional to the nonlinearity parameters, one can observe that in N aCI the largest nonlinearity parameter is observed in the [100] direction whereas it was smallest for the [100] in silicon (and germanium). This difference appears to be of fundamental importance. It can best be seen by comparing a plot of the temperature variation of the nonlinearity parameter of NaCI, Fig. 8 with the corresponding plot, for silicon, Fig. 2. The ordering of the magnitudes of the nonlinearity parameters in the two graphs is different. This means that the anharmonicity in the interionic potential function in NaCl is fundamentally different from that in silicon, as one might expect.
The nonlinearity parameters can be used to calculate the K3 values of NaCI, as shown in Fig. 9. In Fig. 10 are presented the simplest combinations of TOE constants of NaCI available from the data in Fig. 9.
DISCUSSION AND SUMMARY
Since the data on NaCI presented in Figs. 7 through 10 are analogous to Figs. 1 through 4 for silicon, the figures can be compared directly. First, the ordering of the nonlinearity parameters for the [100], the [110] and the [111] directions is different for NaCI compared with silicon. To the extent that there is a one-to-one correspondence between the nonlinearity parameter and the anharmonicity of the interatomic potential function, the information suggests that the anisotropy of the interatomic (or interionic) potential functions is fundamentally different for the two crystals. Considering the difference in the magnitudes of the nonlinearity parameters, this is not a surprising result. The temperature dependence of the K3 curve for silicon is unambiguous, even though third order elastic constants have been assumed to be temperature independent in order to test a theory.? This observation suggests that the assumption of temperature independence of the third order elastic constants doesn't even work for the relatively simple diamond lattice solid silicon. Thus, it probably would be less accurate to assume temperature independence of the TOE constants of N aCI. .
It is tempting to suggest that in the absence of measured values (thermal expansion of NaCI has caused experimental difficulty at low temperatures) one could assume a central forces, nearest-neighbor model is valid at OOK and extrapolate CIII linearly to zero temperature, extrapolate Cl12 + 4CI66 to 5/2 CIII and extrapolate Cl 23 + 6CI44 + 8C456 to zero magnitude at OOK as indicated by the solid lines in Fig. 10. Experience indicates that this probably is a good extrapolation of CUI; however, it probably is not good for the other curves.
An alternative, used to evaluate the "bare" constant, without zero-point motion and thermal contributions in KMnF3,8 is a linear extrapolation of the curves in Fig. 10. This would be equivalent to assuming that many-body forces are of great importance in the behavior of the TOE constants of NaCI. The behavior of the data suggests that many-body forces probably are very significant; however, this does not prove that the linear extrapolation is a correct one. More likely, a value somewhere between the two extremes is
27
20.------,-----------------------,
15
e 10
5
(Ill - 1
2 3 4 5 2 -20 2
Al (10 m)
Fig. 7. Plot of A2 as a function of Ai for NaCl.
15
CcDo D r::Fl:PCD,gc a I[s:J
(100)
5
CJl 0 CDO 0 OoQXOO QXlO o o amm 0 0 I)
(Ill ]
amtb QSItIa) OOCClD) Q)O CD cJtD 0 (I) 00 000
(111]-1 0+-----~-----.----~----_.----~~--__4
o 100 200 300 TEMPERATURE (K)
Fig. 8. NaCl nonlinearity parameters as a function of temperature.
28
::
0,-------------------------------------,
(111)-1
·20 .....,. ......... -.00 oeD ......
(Ill) -.-.... - .. -... ~- .. ·40
(1 10) .. _ ............ 6 ... _ ..... 6 .........
·60'
. . .... . ..... . ... . ...... ~ ..... ·80 (100)
. 1 00 +-----~------._----~------._----~----~ o 100 200 300
T EMPERATURE (K)
Fig. 9. NaCI K3 values as a function of temperature.
E
1 o
50 - -- -- --... ~~( 111}. 1 " C +6C +8C
o
-50
-100
" 123 144 456 , __ , t. _ _ ;';:' ..2 - .. 8eaw e.o e8 0 , ~~ , ~~
~ ~
80 "" ''0 (Ill }
___ _ _ ___ _ ; .r .... "_ .. "'_ .......... " ... ., ... ,,~,, ... "..,"o___"H"....," ... " ....... 0 0
" C 112 + 4C 166 I
I
___ _ I- _ --oo4£L ~CCf5 I
I
I I
se e 0 8 &
C II I
. 1 50+-~1--~----~----~---,.---~~---,----~ o (00 200 300
TEMPERATURE (K)
Fig. 10. Temperature variation of NaCI OTE constants.
29
the correct one. One can introduce more and more sophisticated theories to account for thermal motion and many other physical processes. Our ultimate goal is a fundamental understanding of the nonlinear behavior of crystalline solids and how this information can be used in modern technology.
ACKNOWLEDGEMENT
We gratefully acknowledge the support of the US Office of Naval Research and the Science Alliance, A State of Tennessee Center of Excellence.
REFERENCES
1. W. B. Gauster and M. A. Breazeale, Rev. Sci. Instrum. 37:1544-1548 (1966). 2. M. A. Breazeale and Jacob Philip, in Physical Acoustics, edited by W. P. Mason and
R. N. Thurston, Vol. XVII, Academic Press, New York, pp. 1-60 (1984). 3. W. T. Yost, John Cantrell, Jr. and M. A. Breazeale, J. Appl. Phys., 52:126 (1981). 4. P. N. Keating, Phys. Rev. 145:637 (1966); Phys Rev. 149:674 (1966). 5. Jacob Philip and M. A. Breazeale, J. Appl. Phys. 54:752-757 (1983). 6. Wenhwa Jiang and M. A. Breazeale, Submitted to J. Appl. Phys. 7. K. Brugger and T. C. Fritz, Phys. Rev. 157:524 (1967). 8. W. Cao, G. R. Barsch, W. Jiang and M. A. Breazeale, Phys. Rev. B 38:10244
(1988).
30
THERMAL WAVES FOR MATERIAL INSPECfION
G. Busse
Institut fiir Kunststoffpriifung und Kunststoffkunde Universitiit Stuttgart Federal Republic of Gennany
IN1RODUCfION
A thennal wave is the description of how a temperature modulation propagates as a function of time and coordinate. Compared to light (or generally electromagnetic) waves and even to sound waves, thennal waves are very slow. They are also strongly damped. Within one wavelength their amplitude is reduced to O. 2 %. The final reason for both the low velocity and the attenuation is the diffusion process that describes heat propagation. The parabolic differential equation for this process has only one parameter, which is thennal diffusivity A.. If a sinusoidal temperature modulation is generated at a frequency co, then one finds 1 that group velocity v g of the thennal wave produced this way is
Vg = 12 A. CO • (1)
As an example, if one modulates the temperature at the surface of a solid at 10 Hz, the thennal wave has a velocity of several mm/s in metals and an order of magnitude less in polymers. Obviously there is a strong dispersion that makes waves move faster at higher frequencies. The consequence of this dispersion is that group velocity is twice the phase velocity.
Though thermal waves differ quite a bit from other waves, a common feature of all waves is reflection at discontinuities. The physical quantity of interest is the change of impedance at a boundary. If the materials on both sides of a boundary have similar thennal properties, the reflection coefficient may be small (e.g. in the case of paint on a polymer substrate). It can be close to 1 if the materials are as different as air and metal. But even in that case the "thermal wave echo" superposed to the original wave may have a very small effect due to the strong attenuation. The distance where one can barely detect this echo is roughly given by the thennal diffusion length 1l:2-5
Il=V2A./co (2)
To detect a boundary by its thermal wave echo one needs low frequencies if the boundary is far away from the surface; e.g., 10 Hz or less if the boundary is at a depth of 1 mm in metals or 0.1 mm in a polymer. From this, it is evident that thennal waves provide a good method to probe in a remote way near-surface boundaries, to measure layer thicknesses and to locate faults. As an example Fig. 1 shows how the temperature modulation at the
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 31
4 I
I :> N 90 At • air .s
10 Hz c.
..: OJ
~ Lo.J a 60 9-:::J >- Lo.J
Z ~ --' t.:J
t.:J I ~ z ..: I -0--::>--0- ..: l:
Lo.J --' 30
V1 ..: ..: z I °A :r: t.:J a.. Vi fli
0 I
0 4
SAMPLE THICKNESS (mml -
Fig. 1. Magnitude A and phase <p offront surface thennal wave signal with wedged aluminium sample. Data points4 compared to model calculation.2
surface of aluminum is modified if the reflection from the rear surface (at a variable distance, the thickness of a wedge) is superposed.2. 4 The dashed line indicates the thickness at which thickness equals thennal diffusion length.
Thennal waves are not new,6-8 but they became popular only after the laser was used as a powerful tool to generate them by absorption of modulated radiation.9• 10 After the photoacoustic detection had largely been replaced by photo-thennal,ll· 12 they could also be analyzed in a remote way. The principle of photo-thennal radiometryli is illustrated in Fig. 2. The laser beam is modulated and then focused to the surface of an absorbing sample. The temperature modulation in the thennal wave causes a modulation of emitted infrared thennal radiation which is monitored by an infrared detector. Using infrared optics one can confine the thennal wave area observed. A spectral filter keeps the excitation light of the laser away from the infrared detector. 1 1 The minimum detectable temperature amplitude is between about 10-4 K and 10-5 K at a 1 Hz detection bandwidth.
By moving the sample one can measure amplitude and phase of the thennal wave as a function of sample coordinate. Therefore this is an arrangement for local dynamic heat transport measurement. However, such a one-dimensional (or two-dimensional) plot displays local changes of several sample properties. The phase angle plot is not sensitive to optical or infrared surface properties of the sample; 13 it therefore provides infonnation that cannot be obtained otherwise. That is why most of the examples presented in this article are phase angle plots.
TRANSLATION STAGE
Fig. 2. Experimental arrangement for scanned photothennal radiometry. 1 1
32
10
:;:- (3
.s w 0
< ~ UJ 0 w ::> ...J l- t:) ::::; Z 0. < ::li: < w
Vl < I 0.
0.1 0.2 0.3 0.4
AL - THICKNESS (em!
Fig. 3. Photothennal transmission signal of wedged aluminium sample at various modulation frequencies. 14
INSPECITON OF METALS
Many early thennal wave measurements were perfonned on metals. One reason is that the thennal diffusion length is large enough to be comparable with structures that can be machined easily, so one perfonns tests on range and resolution and compares the results with theory. The other reason is that one does not need to worry about the effect of laser radiation on the sample.
The following examples were obtained with thermal wave transmission: the laser focus is on the front surface of the sample while the detector observes the thermal wave when it arrives at the rear surface. In such an arrangement one expects that the phase angle is linear in the distance z that the wave has travelled,
<p =z/Il . (3)
This relation is well confIrmed by the experiment. 14 With a phase angle resolution of better than 0.1 degree it is possible to observe in a remote way where a 1 mm aluminum plate has a thickness variation of several 11m. Instead of the geometrical wedge of Fig. 3 one can also monitor a thennal wedge. This sample has been made by shaping a surface-hardened steel sample in such a way that the overall geometry was a plate with parallel surfaces while the hardened area with the carbon atoms in it had a wedge-like cross-section (Fig. 4). For a thermal wave propagating through this sample, one part of the phase lag is produced in
33
the hardened area, the other in the non-hardened layer where there is a different concentration of carbon atoms. The phase angle plot in Fig. 4 displays the crosssection of the thermal wedge. IS The smooth transition from the horizontal part to the sloping part indicates that the concentration profile is not a step-function. A more careful analysis of such a shape should also include corrections for lateral resolution.
Investigations of resolution were performed on a sample provided with two holes next to each other. The question was how well the holes could be resolved by using both signal phase and magnitude at various frequencies. The result wasI6 that the phase scan had a better resolution which could be improved by modulating at higher frequencies (Fig. 5). Besides depth range and independence of surface absorption and surface infrared emission, this better resolution is one more argument to favor phase angle scans. An application where thermal wave transmission was used to monitor a defect in welded steel is shown in Fig. 6.
POLYMERS
Polymers have a smaller thermal diffusion length than metals. Also, their threshold for radiation damage is much lower. Both facts limit the sample thickness for transmission measurements to about 150 11m. As most samples are thicker, one has to use the singleended experimental arrangement of Fig. 2.
There are essentially two kinds of subsurface features in polymers that are of interest for thermal wave inspection. One is the boundary in sandwich !itructures. Delamination or imbedded material may give rise to thermal wave reflection. I I However, the maximum depth that can be probed is essentially the boundary under the first laminate layer. The depth-range at acceptable modulation frequencies is too small for the inspection of realistic
34
side view:
case-hardened
coordinate x----...,
hardened--+-_oft--~ .. .. .. c
_ soft
~ __ ~~~~~rr77~~r777r-tj~
<-L.4-L.4.<~'-"-~~~~~f--t-;;; ~
(u- coating
c: o u
grain structure: low- carbon
18r-----------~----------~
.. c. :i 12 .. .. 10 ~ 0. 9
h~rdened - soft soft
6+-----.--0000"""T~~ ....... ~_._-~-....,I o 16 20 mm
coordinate x -
Fig. 4. Top: Preparation of steel sample with a wedge-shaped hardened region. Bottom: Scan with photothermal radiometry in transmission. IS
1.0 Hz
f 30Hz
20DEG.
j 20Hz
15Hz
A
l 2.3mm J_
- ' .J~ --0.16mm
Fig. 5. Transmission photothermal radiometry of an alumiium sample with two subsurface holes at several modulation frequencies. 16
multi-layer laminates. The other subsurface features are imbedded fibers that are supposed to increase material strength. This increase depends on the content and the orientation of fibers, as well as on the fiber/matrix boundary. It has been shown previously that an increase of fiber content reduces the thermal diffusion length. I8 The orientation of long carbon fibers has been monitored with a front-surface offset arranl)ement where the detector spot moved on a circle around the laser spot (or vice versa). 9 The phase lag comes to a minimum when the detector monitors thermal wave propagation along the fibers. For an unknown orientation the minima in Fig. 7 would indicate the fiber direction. As the diameter of the circle is about 0.5 mm, this arrangement allows a determination of the local direction. As an example, Fig. 8 shows how signal phase changes during a twodimensional scan across an injection-molded sample with short fibers. Fiber orientation changes in the weld line area where two opposite meltflows join each other. Scans with offset along the weld line (top) or perpendicular to it (bottom) reveal the local change of fiber direction. Investigations of this kind could be of relevance to reveal areas of reduced strength.
OPT. A
Fig. 6. Weld seam in a stainless steel plate. Gray-scale (top) and perspective line drawings (bottom) of raster scan results with optical reflection (left),
photothermal signal magnitude A (middle) and phase <I> (right).
35
COATINGS
The limitation of thennal waves to the near-surface area and their sensitivity to thennal boundaries makes them suited for coating inspection. The general rule for this application is that the thennal diffusion length should not be smaller than the coating thickness. For most organic paints and their nonnal thickness one needs modulation frequencies around 10 Hz or less. For thin metal layers deposited by evaporation, the frequency could be much higher.
If instead of the aluminum wedge in Fig. I, one probes a wedged sample of paint deposited on a substrate, one finds that the phase angle depend not only on the local thickness of the wedge, but also on the kind of substrate (Fig. 9). From the remarks on impedance this is obvious. Though curve (a) looks attractive, there has not been much response to the idea of using thennal waves to measure the thickness of paint on metal. The reason is that there are competing well-established techniques based on induction or capacity measurements. Curve (b) is very flat due to the similarity between paint and the polymer substrate underneath. The phase angle change is only 2° for 10 11m thickness change. However, there is no competing method. Fig. 10 shows how the curves depend on frequency.22 It is evident that the frequency can be optimized to give a maximum sensitivity for a given thickness range. At low frequencies one can monitor thick layers, but the time required for phase angle measurement increases.
It should be kept in mind that the reflection coefficient, depending on both materials affects the "calibration curves" in Fig. 10. Also the surface pretreatment or contamination has an effect which may correspond to a thickness change of up to 10 Ilm.23 However, thennal wave inspection can readily be applied to monitor local changes. As an example, Fig. 11 shows a two-dimensional scan across a spin-coated polymer sample. The rotational symmetry of thickness distribution is clearly revealed.23
B
FISER ORIENTATION
IJ~
1]0
t)
126 e. 9--<l
122 w (/) ~ :x: Cl.
118
11~
l ___ ~ ____ ~ ____ ~ ____ ~ ____ ~ __ ~~ __ ~ ____ -J 110
o 90 180 270 )60
ORIENTATION a (OEG.)
Fig. 7. Carbon fiber orientation observed with anisotropy of phase shift at a given offset. 19
36
ci w e. : w (J)
< :r a..
x·COORDINATE (mm)
Fig. 8. Local fiber orientation in the seam of an injection-molded component. Offset direction between laser spot and detector spot is perpendicular to seam
direction (top) or along seam direction (bottom).
CONCLUSION
Thermal waves, with their high attenuation, allow for remote near-surface probing of boundaries in materials at a range that can be tuned with the modulation frequency. Fibers and layers are features of interest for inspection. Therefore thermal waves fit well into the gap of existing conventional methods: ultrasonic methods are not applicable in the nearsurface region and low reflection coefficient (like polyurethane coating on polyurethane substrate). Also, they require mechanical contact. On the other hand, x-rays allow for remote inspection but they are not sensitive to boundaries. There is no competition with optical methods which depend on transparency, since the thermal wave generation is based on optical absorption.
o
9--20 <l III
'" '" .&:; c..
p~ @jO ®
/ 40 80 pa int th ickness [~ml
Fig. 9. Phase angle change during scan across a wedged paint sprayed on metal b polymer substrate.2r
37
92
88 ~2.25 Hz
_ 9Hz
_ 18Hz
--36 Hz
86L-~~-L--~--L-~~~---:: 20 40 60)lm 80
THICKNESS d
Fig. 10. Scan across paint wedge on polymer substrate at several modulation frequencies.22
, .. ..... ...... .. . :~ ::::: ::: :~~:':: : ;:
:--.... .. """ ..... _,t",,' : ::::~ ; :: : ::::: ;. , ........ . - -_ ...... . ....... - .. ... ... .... .
~n~~":..:::~~~ .' :: :::::::::::::: . . ::. :::: : -:: ....... . ~ : ~~:~ ~: :~:::: ::: : :: :-.:.':' ::- ' : .. :::::;:: ~: .' . ~ :::: ;::~!,:",:::"' .' ~~:::: ~ . ::: :" .... .... . ..... ::; ~ ~~~:~ ~ ~: ~~~ ~f:~~ ~ ~~:~ ~~: : . -:::
Fig. 11. Raster scan across polymer sample with paint deposited by spin-coating.23
The drawback of thermal waves is the long time they require for accurate phase angle measurements at low modulation frequencies. To determine a 50 Ilm thickness polymer coating with an accuracy of ±2 Ilm, one needs about 2 seconds. This would be just one data point in a line scan. A two dimensional scan like the one in Fig. 11 is too slow to be integrated into a production process. The present field of laboratory application seems to be analysis of production parameters and of gradual changes correlated with pretreatment and service-life considerations.
REFERENCES
1. H.S. Carslaw and J.e. Jaeger, "Conduction of Heat in Solids," Clarendon, Oxford (1959).
2. G. Busse, Appl. Phys. Lett. 35:1979 (1979).
38
3. R. L. Thomas, J. J. Pouch, Y. H. Wong, L. D. Favro, P. K. Kuo and A. Rosencwaig, J. Appl. Phys. 51:1152 (1980).
4. A Lehto, M. Jokinen, J. Jaarinen, T. Tiosanen, and M. Luukkala Electr. Lett 17:364 (1981).
5. C.A Bennett and R. R. Patty, Appl. Opt. 21:49 (1982). 6. J. Fourier, Mem. de l'Acad. des Sciences 4:185 (1824). 7. M.A. J. Angstrom: Phil. Mag. 25:130 (1863). 8. A.G. Bell, Phils. Mag. 11:510 (1881). 9. Y.-H. Pao, "Optaoacoustic spectroscopy and detection," Academic Press, New York
(1977). 10. A. Rosencwaig, "Photoacoustics and phtocacoustic spectros-copy," John Wiley &
Sons, New York, (1980). 11. P.E. Nordal and S.O. Kanstad, Phys. Scripta 20:659 (1979). 12. AC. Boccara, D. Fournier, and J. Badoz, Appl. Phys. Lett 36:130 (1980). 13. A Rosencwaig and G. Busse, Appl. Phys. Lett 36:725 (1980). 14. G. Busse, Infrared Phys. 20:429 (1980). 15. G. Busse, "Photoacoustic, photothermal and photochemical processes at surfaces and
in thin films, (Hrsg. P. Hess) Springer, Berlin, (1989) p. 251. 16. R.L. Thomas, L.D. Favro, K.R. Grice, L.J. Inglehart, P.K. Kuo, J. Lohta, and G.
Busse, Proc. IEEE Ultrasonics Symp., (1982) p. 586 . 17. B. Rief, G. Busse and P. Eyerer, Proc. 6th Int. Conf. on Composite Mat., Elsevier
Sc. Pub., London, 1:349 (1987). 18. P. Eyerer and G. Busse, Kunststoffe 73:547 (1983). 19. G. Busse, B. Rief and P. Eyerer, Polymer Composites 8:283 (1987). 20. B. Rief, "Zerstorungsfreie Charakterisierung von kohlenstoffaserverstarkten
Kunststoffen mittels Warmewellenanalyse," Fortschrittberichte VDI Reihe 5, Nr. 145. VDI, Dusseldorf (1988).
21. G. Busse, D. Vergne, and B. Wetzel, Photothermal nondestructive inspection of paint and coatings in "Photoacoustic and Photothermal Phenomena," (P. Hess and J. Pelzl, eds.) Springer, Berlin (1988) p. 427.
22. G. Busse and W. Karpen, "Neue Entwicklungen bei der Lackierung von Kunststoffen," DFO, Dusseldorf, in press.
23. G. Busse and D. Vergne, "Neue Entwicklungen in der Lackiertechnik - Perspektiven fur die Zukunft, " Vesper and V. Stange, eds., DFO, Dusseldorf, (1989) p. 362.
39
ULTRASONIC BACKSCA TIERING: FUNDAMENTALS AND APPLICA nONS
B.Fay
Physikalisch-Technische Bundesanstalt Braunschweig Germany
INlRODUCTION
Ultrasonics denotes mechanical vibrations at frequencies of approximately 16 kHz to 1 GHz. Because of its quasi-optical propagation behavior, in medicine and technical applications, ultrasonics is chiefly used to image or detect objects. The two acoustic parameters determining the transparency or the ability of the ultrasonic wave to penetrate matter are absorption which indicates the quantity of ultrasonic energy transformed into heat in the specimen, and scattering. Scattering means that an ultrasonic wave when striking an obstacle radiates part of its energy in all spatial directions.
When the diameter d of the scattering center is very small compared with the ultrasonic wavelength A, the so-called Rayleigh scattering (Fig. 1) occurs. Rayleigh scattering increases very strongly with frequency f, i.e. with the 4th power of the frequency. This frequency dependence is observed, for example, when blood is exposed to ultrasonic waves. The blood corpuscles give rise to scattering. An example from optics, where basically the same scattering laws hold, is our atmosphere. Its gas molecules scatter sunlight, particularly the high-frequency components of the visible spectrum, and thus produce the blue color of the sky. When the diameter of the scattering center is of the order of magnitude of the wavelength, stochastic scattering is encountered. Such a marked directional dependence is caused, for example, by a single thread stretched in a water tank. Stochastic scattering increases only with the 2nd power of the frequency. In stochastic scattering, the scattering centers are great compared with the wavelength but distinctly smaller than the diameter of the incident sound beam. Scattering is then produced by the diffuse reflection of the incident sound wave in the scattering centers, for example, in the shrinkage cavities of a metal specimen. This so-called diffuse scattering is independent of frequency.
BACKSCA TIER METHOD
The backscattered signal can readily be detected by experiment by the well-known pulse echo technique. In this method, an ultrasonic transducer radiates a short ultrasonic burst into the specimen to be investigated, for example, a steel plate. As the pulse hits the specimen surface at an angle of 90·, the transducer first receives the echo from the front and a little later also the echo from the back face of the specimen. All additional echoes occurring between these surface echoes are provoked by structural defects such as shrinkage cavities or slag inclusions. In order that very small defects might be revealed, the received signal is often amplified until the background between the echoes distinctly increases to values above zero. The background is produced by the scattered waves which
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 41
42
Rayleigh scaliering
d«A. ~_(4
Siochaslic scattering
d = A. !XS _(2
Diffuse scattering
d>A
~'-' --
--~(:"(;'-'-=l' ~\ '/ " ~ ,-. .'-- -... "
Fig. 1. Ultrasonic scattering. l
transducer specimen q ·······..--P-u-Is-e---.
/ ....... "' 0<, scattering cen tre
. /
water ' ~v.
sound pressure
'1' scattering
wave /'
, backseat tered
/ ;19001
- I_t-· ... ' ___ +-... _~ __ ~
time 1---
position X ---<-
Fig. 2. Backscatter method.
o
5
-10
L5 20lgL
Po
-20
-25
-
/ A Ibackscattered signal)
,......- 8 lecho pulse)
o 25 xlc •
Fig. 3. Ultrasonic measurements on a steel prism p: sound pressure, PO: reference sound pressure,
x: pulse propagation path, c: sound velocity.
have formed due to the inhomogeneities inside the specimen such as, for example, the grains in a steel specimen.
Just as the scattering background has a disturbing influence on ultrasonic imaging, the surface echoes of the specimen complicate the evaluation of the backscattered signal. This is why the method of ultrasonic backscatter (Fig. 2) uses in most cases a delay path in the form of water through which the pulse is obliquely introduced into the specimen to be tested. Except for spurious echoes which can form due to rough specimen surfaces, only the scattered waves from inside the specimen reach the transducer. The backscattered signal is a result of the superposition of all waves scattered in the specimen. For specimens with position-independent structure which will be considered fIrst, the scattering coeffIcient <Xs, the absorption coeffIcient <Xa and also the attenuation coeffIcient which is the sum of the other two coeffIcients, have the same value in every point of the specimen. In this special case, due to the locally constant attenuation, the back scattered signal might be a simple exponential function of the coordinate x.
This conjecture is confIrmed by the measurements represented in Fig. 3. The continuously drawn curve represents the signal, measured at 5 MHz, which is backscattered from a forged prism-shaped steel specimen with position-independent structure. Here the logarithm of the sound pressure is plotted against the pulse propagation path x. The backscattered signal therefore decreases linearly with this path. It is a specific advantage of this specimen form that it allows both attenuation measurements to be carried out on it by the above-mentioned pulse echo method and also the propagation path of the pulse to be varied at will by displacement of the transducer parallel to the irradiation surface. This means that an entire series of back-face reflections can be measured. As the attenuation of the echoes and that of the scattered waves are identical when the propagation paths in the specimen are equal, the slope of the echo envelope conforms as expected to that of the back scattered signal. In the case of substances \vith position-independent structure,
43
it is therefore possible to detennine the attenuation coefficient from the gradient of the backscattered signal. As the attenuation of the scattered waves which are produced below and at a short distance from the irradiation surface is negligible due to the short propagation paths of these waves in the specimen, at x = 0, the sound pressure of the backscattered signal is a function only of the scattering coefficient. Thus the scattering coefficient of the specimen can also be calculated from the initial value of the backscattered signal. The absorption coefficient of the specimen is obtained as the difference between attenuation and scattering coefficient.
Figure 4 shows an application of the backscatter method4. The curves represented refer to backscatter measurements at 19 MHz on five different steel specimens with locally constant structures. As has just been mentioned, the scattering coefficient can be detennined from the initial value of the curves. When, in addition, the scattering law is known as here where as = d (diameter of the grain) because stochastic scattering is produced in these specimens, the second ordinate axis can serve to directly detennine the grain size of the specimens? The grain sizes which have been determined by the backscatter measurements are in very good agreement with the grain sizes which were detennined by metallography by the MPI for Iron Research in Dusseldorf.3 This method of non-destructive grain size detennination is meanwhile increasingly used by industry.
Nonnally, however, the structure of the specimens is not position-independent but changes along the propagation path of the irradiated ultrasonic pulse. But in this case, too, it is possible to determine the fosition-dependent acoustic characteristics of the specimen by a graphic evaluation method. Let us examine a model specimen consisting of two layers (Fig. 5). In the first layer, the scattering coefficient is assumed to be locally constant but in the second layer, to increase gradually. As in the measurements just dealt with, the first section of the backscattered signal 1 decreases linearly with time. But in contrast to this, despite increasing attenuation, the second section of the backscattered signal even increases again slightly due to the increase in the scattering coefficient in the second layer. This backscatteredsignall does not allow a statement to be made on the position dependence of
44
3
5 4
12
- 10
- 12
-14 o
grain size'
IMP!)
0.5
12,11,10 11, 10,12 8,7, 9,10 4,5,3.2.6 3,4.5.6
em x-
+ 6.11 " 0.12 o 111 x 3.12 • 3.13
1.5
Fig. 4. Backscatter measurements on specimens with position-independent structure.
measurement 1
Ig L Po
<X , . , : Increases :
x----
2 (ref lected)
: -Ig ~ ; $.0
Fig. 5. Detennination of the position -dependent scattering coefficient as(x).
the acoustic characteristics of the specimen; this would require at least one additional measurement. The evaluation will be particularly simple if in the second measurement, the pulse follows the same path through the specimen but in the opposite direction. In this case, the back scattered signal obtained will naturally be different. For evaluation purposes, the second backscattered signal is reflected from an axis parallel to the ordinate. Reflection ensures that the signals coming from the same areas of the specimen, for example, from the interface of the two layers, lie at the same abscissae after reflection. When the mean value of the two measuremfnts is now fonned- an operation by which attenuation is eliminated-, the scattering coefficient is obtained directly as a function of position, that is to say, 19 a s/ us,Q where as,Q is a reference value, here the scattering coefficient in the left specimen surface. From the difference between the two curves, the attenuation coefficient can be detennined in a similar way as a function of the position.
Figure 6 shows an application of this evaluation procedure, namely the determination of the internal structures of a hardened steel roller. The backscattered signal 1 is obtained when the pulse is irradiated from above through the shell surface of the roller. Curve 2 represents the reflected backscattered signal which is obtained when the pulse is irradiated through the roller's shell surface from below. As I have just demonstrated, the mean value of the two curves; i.e. the curve of points, gives the variation of the scattering coefficient along the pulse propagation path x. As hardening of the roller yields a cylindrically symmetrical structure, the curve of the scattering coefficient also has a symmetrical shape. In the center, i.e. on the cylinder axis, the scattering coefficient is maximum, meaning that here the structure is still relatively coarse-grained. Next to the shell surfaces, the smallest values of the scattering coefficient are to be found because here a fine-grained structure which scatters only weakly, has fonned due to rapid cooling during the hardening operation.
BRAGG BACKSCA TIER METHOD
As the spacings between the scattering centers in biological tissues are in the millimeter range, and thus of the order of the ultrasonic ' wavelength, ultrasonic backscatter methods can also be used to determine the spacings between the scattering centers.
45
20 19~
-6
-12
x
101g~ a~
Fig. 6. Backscatter measurement on a hardened steel roller.
When the scattering centers are arranged at regular intervals as in the one-dimensional target composed of three scattering centers in Fig. 7, the Bragg backscatter method developed by HillS for this application can be used. For this, the Bragg reflections occurring at different irradiation angles are determined- a procedure similar to that used to determine the lattice constant with x-rays. Bragg reflections occur when the individual scattering pulses interfere constructively as in Fig. 7 on the left where all scattering centers lie at the same distance from the transducer. When the specimen is rotated through an angle <p, further Bragg reflections occur whenever the differences between the scattered pulse propagation paths is a multiple of the wavelength. The upper backscattered signal, for example, is obtained computationally for a target consisting of ten scattering centers arranged at equal intervals of 1 mm exposed to a 5 MHz wave at different angles <po The closer together the Bragg reflections, the greater the spacing between the scattering centers. When the scattering centers are regularly arranged, the Bragg backscattered signals thus allow the spacing between the scattering centers to be determined.
Figure 8 shows Bragg backscatter measurements carried out by Nicholas6 on healthy human liver and on human liver indurated due to cirrhosis. Even for a non-physician it is not difficult to see the difference in the scattering behavior of the healthy and of the diseased hepatic tissue, and this method is suggested for use in medicine for tissue differentiation. That the Bragg reflections occurring here are not so sharply defined as with the one-dimensional target is due to the fact that the scattering centers in the tissue are not arranged as regularly. The extent to which this irregularity of arrangement changes the backscattered signal can be demonstrated by again referring to the curves in Fig. 7 . The upper curve results from uniform spacing of ten scatterers. If the individual scattering center spacings differ from the mean spacing by 7%, then uniform Bragg reflections are much less perceptible, as shown in the lower curve. In medicine there are scarcely any substances with a completely regular structure. In most medical applications, then, this method, probably would not be completely suitable for unambiguously determining the spacing between scattering centers.
SYNTHETIC FOCUSING
With synthetic focusing the backscattered signals also are measured at different irradiation angles and stored. From the stored data, the position of the scattering centers in the irradiated specimen is reconstructed with a computer. The principle of the method, which also is valid for scattering centers which are not regularly arranged, is shown in Fig. (9).
46
transducer ~ ttt
d =1mm ) 5MHz
<p ----
O.93mm oS d oS 1.07mm
'P - - -
~-------
Fig. 7. Bragg backscatter method .
. r.. f\J\/\ (a) J 'vJ V V I.r~
/"-J\ A. 1\" /" sound tissue
(b) t' . VV V V'
(C)~~ ~J. . I Aft'. / diseased tissue (cirrhosis)
!d) '~~
rp-
Fig. 8. Bragg backscattered signals from hepatic tissues at 2, 5 MHz.
47
sca t tered
[$ pul~e " ....... -.. ...... -.-~- . --- . -. ---:-: ..•... .
- f'00CVv ;?, , ~:. __ sca lt ering .... .. ...... .. ... ; ....... : ............... -.. - centre
: time window
~ I I
l ime 1----
transducer \
~ ......•.•.... ~ .••.••• ';~("
: ! _____ backscatt ered ~ signal
t I
time/----
Fig. 9. Synthetic focussing procedure.
For reasons of simplicity, let us consider first a definite position inside the specimen, for example, the position of the scattering center marked by the open circle. As we know the coordinates of the space point considered, we also know at which interval of time the measured backscattered pulse is influenced by this scatterer. When the irradiation angle is changed, the spacing between space point and transducer changes as well. As we specify, however, the angle of rotation E:, we are able to calculate the time window in which the respective scattered pulse lies. This pulse thus lies always exactly within this time window, whereas all other scattered pulses shift relative to this window when the specimen is rotated. Due to the in-phase superposition of the scattered pulses considered, the summation of the back scattered signals measured at different irradiation angles yields a large characteristic value. If there had not been a scattering center in the space point considered, the summation of the back scattered signals would have yielded only a small characteristic value.
In Fig. 10, 60 high-frequency backscattered signals from a thread specimen are represented at different irradiation angles. On the right, the arrangement of the threads in the sound field of the transducer is shown. The mean spacing between the threads is 1.5 mm and corresponds approximately to the fibre spacing of human muscular tissue. This is why such specimens are used as model specimens of human muscular tissue.
The ultrasonic image of the thread specimen reconstructed by the synthetic focusing procedure described is shown in Fig. 11; the line density in the individual fields is proportional to the calculated characteristic value. On the right, Fig. 11 shows a transmitted-light photograph of one of the two plates with the ten holes through which the threads have been drawn. A comparison of the two images shows that the ultrasonic image correctly describes the thread arrangement in the sound field of the transducer. The resolution is approximately 0.3 mm and thus corresponds approximately to the length of the 5 MHz ultrasonic wave introduced.
IMPROVEMENT OF ULTRASONIC IMAGING IN TIlE CASE OF STRONG SCATTERING
Until now, the backscattered signal was regarded as a useful signal. For a physician or materials testing engineer who wants to see an object inside a body but is impeded in doing so by the strong bulk scattering, however, this will not be so. His situation can be compared with that of a motorist who encounters dense fog at night. The visibility worsens because light from the vehicle's headlights is strongly attenuated and the road in
48
P K i _ or.: Po 0
_ i_
o
-4 -2 0 /--
Fig. 10. High-frequency back scattered signals from a thread specimen. The time window illustrated belongs to the upper right thread of the model specimen.
front is much less illuminated than under normal conditions. But what is worse, the motorist is dazzled by the light scattered by the fog and can therefore identify to a still lesser extent the course of the road, which is lighted only weakly. It will now be shown by the example of austentic steels how ultrasonic imaging can be improved under such conditions.
For this purpose, the sound pressure PE of a back-face echo of a specimen is compared with the sound pressure Pb of the backscattered signal which has been produced in the vicinity of this back-face. As both ultrasonic signals have covered the same distance inside the specimen, the ratio pFfpb is independent of the attenuation in the specimen. This is why for the ratio of echo height to backscattered signal a simple relation is obtained: pEiPb = Grrr;:8, where Us is the scattering coefficient and () the length of the ultrasonic pulse used. The factor G is a function of the sound diffraction losses of the echo and of the
! I
o 2 mm 3
®
• • • • • • • @
• • • I I I
800 0 mm 4
Fig. 11. Reconstructed ultrasonic image of the thread specimen.1
49
backscattered signal. As to the root W it appears in the relation because the sound pressure of the backscattered signal is proportional to this quantity. In order that the echo might be distinguished from the scattering background, the scattering coefficient and the pulse length must become as small as possible. A small scattering coefficient is obtained at low ultrasonic frequency and the pulse length is adjustable.
Figure 12 shows that the echo from a cylindrical hole which is intended to simulate a structural defect in a strongly scattering austentic steel specimen actually exceeds the backscattered signal amplitude only at frequencies below approximately 2 MHz. Very low frequencies have, however, the disadvantage that the wavelength can become much greater than the delay time. As a result, the echo can fall below the detection limit of the testing system. This means that an optimum frequency must be found for each individual application. The optimum frequency is a function of the defect, of its depth in the specimen and of the scattering coefficient.
As was already pointed out, the relative echo height can also be improved when short pulse lengths are used. Figure 13 shows a practical example, namely the ultrasonic signal from an austenitic cast steel speciment at 2.5 MHz, which was measured by the pulse echo method. Only because the pulse length is very short - only 1.5 wavelengths - do the echos from a shrinkage cavity, the defect echo DE, and the back-face echo BFE project from the scattering background. Nevertheless, in this case too, the sound pressure of the back scattered signal is in some places almost the size of the defect echo. Only by averaging a very great number of individual measurements - in this case, 512 - with the transducer in different positions can it be clearly seen that the defect echo and the back-face echo do not belong to the backscattered signal. This means that the relative echo height also is improved by averaging.
It is not possible to reduce the pulse length arbitrarily, as this would result in changes of the bandwidth as is shown in Fig. 14. The ultrasonic burst on the right whose center frequency is 2 MHz and which is approximately 5 wavelengths in length still is a relatively narrow-band pulse. By shortening the pulse, the spectral bandwidth increases until the frequency components in the range from 2 MHz to 6 MHz are almost equal as in the case of the almost needle-shaped pulse on the left side of Fig. 14. As a greater scattering coefficient must be assigned to the greater frequency components, it becomes more difficult to recognize the object when the pulses are very small. What is to be considered very short is naturally a relative matter. This means that in this case, too, the optimum pulse length depends on the specific application. Here it particularly depends on the depth at which the object is placed in the specimen and on the value of the scattering coefficient.
Further possibilities of improving ultrasonic imaging are focusing of the the incident wave by which the echo of small objects can be increased with the scattering background remaining unchanged, and the use of longitudinal in the place of the usual transversal waves by which the backscattered signal can be reduced by the factor 3 to 4 even if the wavelength is the same.
50
o ----10
f -20 /
-30 I
201g..£ Po -1,0
-50
ecl'O of 0 cylinder bore (03mm)
/
\ backsCallered signol
depth zone 35 -t.Omm
-6 0 +--.----.-----r---,----,---r--,o 6 MHz 7
Fig. 12. Echo of a cylindrical hole and backscattered siral from a strongly scattering austenitic steel specimen.
DE BFE
A-Scan p
o 20 mm 40
DE
8FE averaged
p 512'
0 20 mm 40 x
Fig. 13. Backscattered signals from an austenitic cast speciment which contains a shrinkage cavity.9
Another important measure to improve ultrasonic imaging is the elimination of multiple scattering. So far, the echo of the object; e.g. , a defect, was only compared with single scattering, i.e. , with the back scattered signal which is ~roduced in the vicinity of the defect and returns, together with the echo, to the transducer. 0 On this condition, the object thus will be recognized if its echo is greater than this so-called single scattering. This is no longer the case for strong scattering because the energy of the waves scattered several times in the specimen is then no longer negligible. The waves scattered several times then are superposed on the waves scattered only once. The greatest part of the multiply scattered waves proceed from the near-surface area in the vicinity of the transducer. Multiple scattering can be eliminated by using separate transducers for emitting and receiving the ultrasound because most of the multiply scattered waves scattered to and fro in front of the transmitter transducer cannot reach the receiver.
By analogy to this, the motorist of our example will see much more in the fog when he comes to a locality where the road is predominantly illuminated by lanterns. For in this case, too, the transmitter, i.e. the light source, and the receiver, i.e. the motorist's eyes are at such a distance from one another that only a relatively small percentage of the scattered light reaches the motorist's eyes.
8 8 B
6 6
4 ~ 6
P P P 2
0 2 6 MHz 8 4 6 MHz B 0 2 6 MHz B
Fig. 14. Spectra of various ultrasonic pulses.S
51
ullrasonlc transoucer
~ scattering :entres model specimen L--=-f -... • • / 1.. • ... : ••• I· .... : ... : •••.• o¢ •
2t • I scaltenng centre denSity
n lLltALLU.lLUhtllLJJJluwJNJLJuUJlUUL
0)
b)
received Signal d)
I ,
100 150 x---
Fig. 15. Calculation of the scattering center density from the received signal.
e)
FURTHER DEVELOPMENT OF THE BACKSCA TIER METIIOD
In conclusion, an example illustrates how the methods hitherto known can be further developed. Figure 15 illustrates a model specimen consisting of homogeneous scattering centers which are distributed on the transducer axis. The transducer sends the transmitted pulse (c) into this specimen. On the assumption that a scattered pulse emanates from every scattering center, which differs from the transmitted pulse only by its smaller amplitude, the transducer receives the backscattered signal (d) from this model specimen. As the transmitted pulse (c) is much longer than the mean spacing between the scattering centers, the dependence of the back scattered signal on the given scattering center distribution cannot be recognized. It is , however, possible to calculate the scattering center distribution (e) in the specimen from the back scattered signal (d) and the transmitted pulse (c). In analogy to computer tomography, a linear system of equations must be solved. In practice, the results obtained by this method are unfortunately inadequate. This is chiefly because the frequency dependent scattering and absorption causes the difference between the scattered pulses and the transmitted pulse to increase with propagation path in the specimen. For this method to succeed, the change of the spectrum of each individual scattered pulse along its propagation path in the specimen would have to be allowed for in the calculation. This work would certainly be justified, for then a single backscatter measurement would be sufficient to obtain the scattering center distribution in the sound field of the transducer with a very high resolution.
REFERENCES
1. D. Nicholas, Orientation and frequency dependence of back scattered energy. In: "Recent advances in Ultrasound in Medicine," D.N. White, Ed. (Research Studies Press, Forest Grove) pp. 29-54 (1977).
2 . B. Fay, Ermittlung der Korngr6Be von Stahl mit dem Verfahren der Ultraschallruckstreuung. Arch. Eisenhtittenwes. 47, pp. 119-126 (1976).
3 . H.P. Hougardy, Vergleich der metallographisch und durch Ultraschallrtickstreuung errnittelten Korngr6Be von Stahl. Arch. Eisenhtitenwes. 47, pp. 127-130 (1976).
52
4. B. Fay, Ultrasonic back-scattering, a method for non-destructive structure testing. IEEE-Ultrasonics Symp. Proc., pp. 51-53 (1976).
5. C.R. Hill, R.C. Chivers, R. W. Huggins and D. Nicholas, Scattering of ultrasound by human tissue. Ultrasound: Its application in medicine and biology, Part 1, Elsevier scientific publishing company, pp. 441-493 (1977).
6. D. Nicholas, Interference effects in the backscattered signals from human tumors. Ultrasonic tissue characterization. Staflen's scientific publishing company, Brussels, pp. 197-202 (1980).
7. B. Fay, High resolution ultrasonic imaging by controlled averaging of back scattered signals. Ultrasound Med. BioI. 9, pp. 467-472 (1983).
8. A. Hecht, E. Mundry and E. Neumann, EinfluB der Bandbreite auf die Priifbarkeit streuender Werkstoffe. Berichtsband der DGZfP-Fachdiskusion "Ultraschallpriifung grobkorniger Werkstoffe," pp. 85-94 (1980).
9. S. Kraus and K. Goebbels, Grundlagen der Signalmittelungsverfahren. Berichtsband der DGZfP-Fachdiskussion "Ultraschallpriifung grobkomiger Werkstoffe," pp. 121-126 (1980).
10. B. Fay, Ausbreitung von Ultraschall in streuenden Substanzen, Acustica 48, pp. 218-227 (1981).
53
LASERS IN ACOUSTICS
Leonid M. Lyamshev
N. N. Andreev Acoustical Institute Academy of Sciences of the USSR Moscow, USSR
INTRODUCTION
The thirtieth anniversary of the discovery of the laser is celebrated in 1990. Modem lasers emit in a wide range of wavelengths and major progress has been made in the development of tunable dye lasers, as well as of gas and solid-state lasers. Semiconductor lasers are now mass-produced and represent the smallest was well as the most reliable components of quantum electronics. 1 In the last 30 years lasers have become generally accepted in science and technology, as well as in industry, medicine, and protection of the environment. In practically all applications the introduction of lasers has led to or will lead to revolutionary changes.2
The last decade has seen publication of the results of many theoretical and experimental investigations of the generation of sound as a result of interaction of coherent optical radiation with condensed media. There have been many papers on the applications of lasers in contactless remote optical methods for the investigation and detection of acoustic fields and vibrations. The construction of lasers and fiber lightguides has made it possible to study and develop new acoustic detectors in the form of fiber-optic sound detectors. The first papers appeared in the second half of the last decade. Various configurations of such detectors have been considered in numerous studies and the results have been reported. Coherent optical computing devices are being used more for the acquisition, storage and processing of data. Modem acoustic data systems are employing more and more channels, and the use of coherent optical computing devices in such systems is a pressing task.
The use of lasers and the progress made in fiber and integrated optics in the development of new sources and detectors of sound, and use of coherent optical processors for the analysis of multichannel acoustic data are opening up new opportunities in technology. In some cases it is possible to combine the progress made in laser technology with the traditional ultrasonic technology in the development of new methods and devices for nondestructive testing and improvements in physicochemical properties or materials.
Lasers in acoustics represent unique sources and detectors of sound and they also are used in coherent optical analysis of signals in multi-channel acoustic systems as well as in laser-acoustic technology. The results of research at the interfaces between acoustics, quantum and physical electronics, and fiber and integrated optics provide opportunities for developing new experimental methods and technological techniques for tasks which cannot be tackled by traditional methods.3
This introduction shows how extensive are the opportunities for the use of lasers in acoustics.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 55
LASER EXCITA nON OF SOUND
In 1880 A. Bell first observed the optoacoustic effect in the fonn of pulsations of pressure in a closed gas-filled chamber when it was exposed to a modulated infrared radiation flux.4 In the early sixties A. M. Prokhorov and his colleagues observed fonnation of shock waves due to the interaction of a laser beam with water.5 In recent years many papers have been published on the laser generation of sound both in the Soviet Union and abroad so that it now is possible to speak of optical or more usually optoacoustic sources of sound.
Optoacoustic sources have a number of advantages over the older acoustic radiators: sound can be generated remotely; there is no direct contact with the medium along which sound is propagating; it is possible to alter easily the geometric parameters of an optoacoustic antenna and the range of the emitted frequencies; sources of sound moving at practically any (subsonic, sonic, or supersonic) velocity can be constructed and these sources do not suffer from the effects of flow of a medium past the radiator in the traditional sense. Optical methods can be used to generate sound in a very wide frequency range - from very low acoustic to hypersonic frequencies.6-10
We shall now consider the characteristics of optoacoustic sources of sound. We shall assume that a laser beam is incident on the surface of a liquid or a solid (Fig. 1). The action of light on matter creates perturbations of the medium which are accompanied by the emission sound. There are many mechanisms of this effect and they depend primarily on the volume density of the energy dissipated in matter and on the way in which this energy is emitted. The mechanisms of generation of sound include thennal expansion, surface evaporation, explosive boiling, and optical breakdown.
In light-absorbing media at low densities of the dissipated energy the main role is played by the thennal mechanism of the generation of sound which is usually called the thennooptic excitation process. In this case there is no change in the aggregate state of matter in the region of absorption of light, and sound is generated by thennal expansion of the parts of the medium heated by optical radiation. An increase in the energy density dissipated in a medium enhances the effects associated with an increase in the rate of expansion of the heated part of the medium and the changes in the thennodynamic parameters of the medium in the course of its interaction with laser radiation. A further increase in the energy density gives rise to more complex processes of the generation of sound involving phase transitions and optical breakdown.
Thennooptic Excitation of Sound
We shall now assume that the intensity of laser radiation is varied periodically (modulated) at the frequency of sound and that the density of the energy which is evolved is low. Then, in a surface layer of a liquid (for simplicity, we shall consider a liquid, although the analysis applies also to a solid) a pulsating region is fonned and this region emits the acoustic wavelength, depending on the diameter of the illuminated spot on the surface of the investigated liquid. The acoustic pressure in the far-field zone in the liquid is
56
p( } _ romreAIoa2 exp(ikr) ~kcos8 ( k2a2 . 28) r - exp --- sm 2Cp r ~2 + k2cos28 4 .
,
Figure 1. Optical excitation of sound: (1) laser beam of variable intensity; (2) air-liquid interface; (3) optoacoustic source of sound (region of absorption of light in a liquid: (4) acoustic waves in the liquid.
(1)
Figure 2. Angular of the radiation from an optoacoustic sources: (a) dipole radiation; (b) radiation oriented along the surface of a liquid; (c) radiation oriented along the direction of propagation of a laser beam in a liquid
Here, p is the acoustic pressure; re is the volume thermal expansion coefficient; Cn is the specific heat of the liquid; k = ro/c is the wave number of sound; c is the velocity of sound in the liquid; 10 is the intensity of light in the incident laser beam; m is the modulation index, ro is the angular frequency of sound (modulation frequency of light); Il is the absorption coefficient of light in the liquid; A is the transmission coefficient of light at the boundary of the liquid; a is the radius of the illuminated spot on the surface of the liquid; e is the angle between the direction of the laser beam and the direction of the line from the point of observation to the origin of the coordinate system; r is the distance from the point observation to the origin of the coordinates.
It follows from Eq. (1) that the ~plitude of the acoustic pressure rises on increase in the laser power proportionally to loa and it also increases on increase in the frequency and modulation index. The directionality of the acoustic radiation depends on the parameters ka and kw i .
If kW I « 1 and ka « 1, the emission of sound is a dipole process, because under these conditions a monopole source appears on the free surface of a liquid and the radiation field of this source represents a field of a dipole because of the influence of the free surface (Fig.2a).
If kWI » 1 and ka« 1, sound is emitted mainly along the surface. A set of volume sources forms a thin (in the transverse direction) and long (compared with the acoustic wavelength) vertical antenna directed along the laser beam (Fig. 2b).
If kWI « 1 and ka» 1, the antenna is in the form of a disk with a diameter much greater than the acoustic wavelength. The sound is emitted mainly along the direction of the laser beam (Fig. 2c).
An analysis of Eq. (1) shows also that the optimal conditions for the generation of sound by laser excitation are observed when k '" Il. This imposes certain requirements on the frequency (wavelength of light) emitted by a laser.
The distance traveled by laser radiation (quantity WI) in a liquid (or in matter generally) depends on the radiation frequency (wavelength of light). For example, the distance traveled by infrared radiation (from a C02 laser) in water is approximately 1O-5m, whereas blue-green light (from a vapor laser) penetrates to a depth of tens of meters. Variation of the laser emission frequency, focusing and defocusing of the laser beam on the surface of a liquid, and variation of the frequency of light modulation can be used for remote adjustment of the characteristics of the acoustic field in a liquid. If a laser beam scans the surface of a liquid, a moving optoacoustic source can be constructed and the velocity of motion of this source can be subsonic, sonic, or supersonic. . 1?e<;>retical relationships c~ll1"lI:cterizing the process of th~rmooptic ¥eneration of soun~ In a hqUId are supported convIncIngly by numerous expenments. 11- 3 For example, It follows from the theory of such generation that the amplitude of the acoustic pressure rises linearly on increase in the optical radiation power. This has been confirmed experimentally. A solid line in Fig. 3 is the theoretical dependence and the circles are the experimental results. 13 The ordinate gives the acoustic pressure on the axis of an optoacoustic source; i.e., in the direction of propagation of a laser beam. The pressure is reduced to a distance of 1 m and is normalized to 10-6 pa. The abscissa gives the change in the optical radiation power in kilowatts. The experiments reported in Ref. 13 were carried
57
p,dB
110
0.8 1.2 2.0 4.0 W,k'll
Figure 3. Dependence of the acoustic pressure on the axis of an optoacoustic source on the laser power. The continuous line is theoretical and the circles are the experimental results.
out in a lake. A neodymium laser (A. = 1.06 !lm) was operated in the pulsed regime and intra-pulse modulation of the intensity of optical radiation was imposed. The modulation frequency was such that the generation of sound in water was a quasimonochromatic process.
Figure 4 shows the theoretical (continuous curve) and exp-erimental (points) angular dependences of the acoustic field generated by laser excitation. 13 The frequency of sound was f = 50 x 103 Hz; the condition ka « 1 was observed; measurements were made at a distance of 16.8 m, and the absorption coefficient of light in water was ~L = 15.7 m-1•
Sound through Vaporization
The sound-generation effect of vaporization of a substance becomes significant when laser action, such as a laser pulse, brings its temperature close to the boiling point. If this happens at the end of the pulse, the acoustic signal triggered by vaporization takes the form of an additional pressure peak in the 'tail end' of the signal produced by thermal expansion of the medium. With an increase in the intensity of the energy released, the maximum rises and lasts throughout the acoustic signal. When the laser action on the surface of the liquid is intensified, this produces a rapid boiling of the surface layer, from which a jet of vapor breaks loose and meets the laser beam. Rushing at high speed into the air, the vapor jet causes an intense shock wave, while the recoil impulse acting upon the surface of the liquid produces a compression wave in it.
On completion of the laser pulse, and as a result of reflection of the compression wave from the free surface region of the liquid, a rarefaction wave is set up. The latter causes cavitation in the surface of the liquid, producing clearly observable bubbles. Such is the picture so long as the volume density of the optical energy imparted to the substance does not reach a critical threshold at which an optical breakdown occurs in the vapors of the vaporizing matter. This happens, in particular, where there is interaction between the
58
p,dB
-10
Figure 4. Angular distribution of the acoustic field generated by laser excitation in water. The continuous curve is theoretical and the circles are the experimental results.
radiation of a carbon-dioxide laser at an intensity of 108 watts per square centimeter and the surface of a non-conducting liquid, and also where optical radiation at an intensity of 106 to 107 W/cm acts upon the surface of a metal. The optical breakdown in the vapors of the vaporizing matter leads to the formation of a plasma, which partly absorbs the optical radiation and screens the substance, whereupon the amplitude of the acoustic compression wave ceases to match the increased intensity of the light in the laser beam.
Three Modes of Laser-Induced Sound Generation
We may thus arbitrarily distinguish three modes of laser generation of sound as a result of vaporization of a substance with increased intensity of light: (a) weak vaporization, where the intensity of the energy released in the matter is comparable to the latent heat of vaporization; (b) intense vaporization (violent boiling), where the intensity of the released energy substantially exceeds the latent heat of vaporization but no optical breakdown of vapor occurs; (c) the plasma mode, there the intensity of the light is so great that an optical breakdown of the vaporization products takes place and gives rise to plasma, which absorbs the laser radiation and screens the surface of the substance.
This division of the modes of laser sound generation is of course quite arbitrary, because the process underlying the vaporization mechanism - the transition from a condensed to a gaseous state under laser action - is generally marked by a combination of complex nonlinear phenomena. However, this arbitrary division enables us in a number of cases to construct a theory regarding the phenomenon and, in particular, to gauge the conversion efficiency of optical into acoustic energy. It turns out to be almost four times as great as when a thermal mechanism is used and may exceed 1 percent. In the plasma mode, efficiency drops as a result of the screening of the substance.
The conversion of optical into acoustic energy has been found to be most efficient where there is an optical breakdown in the liquid or substance involved. The general picture of the phenomenon is this. When a particular threshold intensity in the focal area is exceeded, micro-explosions take place and produce cavities filled with luminescent plasma. The laser radiation is absorbed in the dense plasma, imparting additional energy to the cavity. The cavity expands under the increased pressure, emitting a shock wave. After completion of the laser pulse and the cessation of energy release into the plasma cavity, the gas cools, the luminescence fades out and a small bubble is formed which goes through a number of pulsations. Estimates indicate that the efficiency of conversion from optical to acoustic energy with optical breakdown may attain double-figure percentage points.
A few figures for illustration purposes may now be helpful to characterize the optoacoustic sound sources operating under various modes. A neodymium-doped glass laser, for instance, functioning at 1 MHz in water can be used to induce a 10 Pa acoustic field 0.1 m from the surface if the optical radiation output is 100 watts. With vaporization of a metal or liquid under carbon-dioxide laser action, for instance, sound pulse amplitudes of up to over 1Q6 Pa can be obtained at a distance of 0.1 m where the duration of the laser pulse is 10-5, the radius ~f the optical spot on the surface 1 cm and the light intensity of the beam around 108 W/cm . With optical breakdown of water, for example, by a ruby laser pulse at an output of 0.1 joule lasting 1O-9s, shock waves are set up at a pressure of about 4 MPa (or 40 atmospheres) admittedly over distances of no more than about 1 cm. It will be recalled that the standard threshold for human sound perception is 2.10-5 Pa or 2.10-10
atmospheres. It is clear from the above figures that modem lasers can be used to generate tremendously high-amplitude ultrasonic waves.
OPTICAL-FIBRE SOUND DETECI'ORS
Some fifteen years ago fibre and integrated optics, emerging from a number of disciplines comprising optics, electrodynamics, materials technology and semiconductor and quantum electronics, began its development as an independent field of applied research. Fibre optics has owned its rapid development to the advent of lasers and the exciting prospects for their application in optical communication systems following marked progress in the development of light guides with low optical losses.
A typical fibre guide is a dielectric fibre consisting of a core (usually cylindrical) and a sheath. For light to be transmitted along it, the refractive index of the core must be greater than that of the sheath. Light travels along the guide without escaping through the
59
sheathing. A finite number of optical wave-guide modes can be carried in the light guide. The diameter of the guide core is generally 5 - 10 microns for single-mode guides and from a few tens to a few hundred microns for multimode guides. The relative difference in refractive indices of materials constituting the core and the sheath amounts as a rule to 1-2 percent for multimode guides and a few tenths of 1 percent for single-mode guides. The basic material used for light guides is glass, such as germanium-doped quartz glass. Total optical losses in such a guide prove to be less than 1 dB/km for optical radiation in the 1-1.8 micron range of the spectrum.
The advent of low-loss optical fibres spurred the development in integrated systems consisting of miniature optical components, foremost among which are laser sources and photodetectors. The active medium for lasers in integrated optics may be a film containing a dye or an ion-doped semiconductor or insulator. Miniature gas lasers are being developed in which the light guide itself serves as the discharge tube. Short capillary lasers just a few centimeters long already exist. Semiconductor sources of radiation in heterostructures, light guides emitting through the surface or the end face, superluminescent emitters and semiconductor lasers have been the most widely used in fibre and integrated optics. Photodetectors in optical-fibre devices need to possess high sensitivity and low remanence. These conditions are met for instance, by photodiodes.
Progress in the production of low-absorption optical fibres with set parameters and in the development of laser sources and photodetectors has paved the way for the use of light guides both in communication systems and in physics research. Prospects are also good regarding the development of sound detectors.!
The idea underlying optical-fibre sound detectors is as follows. When sound acts upon a medium in which light in conducted, a change takes place in the length of the light path, which in tum modifies the phase of the light wave. This phase modification may be recorded with an interferometer. The acoustic field generally has a complex effect on the light wave, modulating its amplitude, polarization, frequency and phase. Sound generally has much the same effect, whatever the medium traversed by the light. Light guides are particularly suitable for sound recording, however, since the fact that optical losses are slight means that acoustooptic interaction over great length can be obtained.
The simplest optical-fibre sound detector is an interferometric device (Fig.5) in one arm of which is a signal light guide while the other contains a sound reference-beam guide. The reference and signal light beams form on the photoelectric cathode of the photodetector an interference pattern created by the action of the acoustic field on the signal-beam guide. At the output of the photodectector an electrical signal is observed at the sound frequency.
The optical-fibre sound detector shown in Fig. 6 works on the basis of amplitude modulation of the light in the guide. The amplitude modulation is generally a result of the production by acoustic oscillations of additional losses in the fibre. Losses occurring in the curves and microcurves of the light guide play the main role. The sensitive element of this kind of sound detector is provided by a multimode optical fibre. The microcurves of the fibre caused by sonic pressure give rise to mode interaction, resulting in a redistribution of energy among the curves. Part of the energy of the modes in the core of the guide is transferred to its sheathing, and this results in amplitude modulation of the light wave.
In a straight, single-mode fibre, ideally round in cross section, and in the absence of mechanical tension, two degenerate modes of optical radiation may be emitted. The state of
60
I
I
I 6 I
Figure 5. Optical-fiber sound detector. I-laser; 2-beam splitting plate; 3-signal wavebeam guide; 4-reference wavebeam guide; 5-photodetector; 6-airliquid interface.
Figure 6. Optical-fiber sound detector using wavebeam guide microcurves. I-laser; 2-wavebeam guide; 3 and 4-grooved plates; 5- photodetector
polarization of the light travelling along such a fibre remains unchanged. It does change, however, under the effect of an external acoustic field, since the light guide is distorted by the sound, and an exchange of energy takes place between the modes. If a photodetector sensitive to polarization is placed at the output of the guide, the polarization modulation produces in the detector current variable components with the modulation frequency; i.e., the sound frequency. The optical-fibre sound detector based on polarization modulation makes use of this phenomenon.
ADVANTAGES OF THE NEW DETECfORS
Great interest is being taken in optical-fibre sound detectors because of their advantage over traditional detectors (e.g. piezoelectric acoustic detectors): simplicity, smaller mass and the possibility of utilizing a great length of optoacoustic interaction, which permits high sensitivity and predetermined directivity. For example optical-fibre detectors using phase modulation (Fig. 5) offer exceptionally high sensitivity, outclassing in that respect the best piezoceramic detectors. The flexibility of the sensitive element - the light guide - means that optical-fibre detectors of the most varied configurations can be used. Such detectors are practically insensitive to electromagnetic interference and are more corrosionresistent. Furthermore, they permit a simpler and more compact system of optical processing of multichannel acoustic information.
Research and development findings on optical-fibre sound detectors were first published in 1977. Since then considerable progress has been made. These advantages were reported at the Eleventh International Congress held in Paris in July 1983. It is still too early to say whether optical-fibre detectors will shortly be replacing traditional sound detectors. Undoubtedly, however, they will in the future be used primarily in multichannel acoustic systems, where information will be processed by means of optical computers and coherent optical processors. Optical-fibre detectors, in conjunction with optical-fibre transmission systems, integrated optics and opto-electronic components, can be expected to influence the development of acoustic systems.
LASER-ACOUSTIC TECHNOLOGY
Industrial technology allover the world now seems poised for a new leap forward, in terms of quality, owing to the widespread applications of lasers, and production efficiency will to a considerable extent depend on how fast laser engineering develops. At the same time, increased production efficiency and product quality continue to lean heavily on ultrasonics, already to some extent a traditional branch of engineering.
New prospects are opening up for a combination of laser engineering and ultrasonics, using lasers to generate and receive ultrasonic oscillation, particularly for non-destructive product-ijuality control and in order to act on the structure and physical properties of matter. IS
For the sake of brevity, we shall take just two examples concerning product quality control. One relates to a special new field: laser-acoustic microscopy. It consists in scanning the surface of the object or specimen point-by-point with a focused laser beam
61
whose intensity is modulated by a sound frequency. The specimen is placed in a closed chamber to the wall of which an acoustic detector is attached. Depending on the structural characteristics of the specimen, the light-absorption coefficient changes from point to point, as a result of which there is a change in sound-pressure amplitude in the chamber and hence of signal amplitude on the output of the sound detector. This makes it possible to obtain, using an electronic system, a television image of the specimen.
If the requirement is to inspect a large area of the surface of the specimen rapidly and in high resolution, the specialist equipped with an ordinary microscope may find the assignment highly arduous and error-prone. A laser scanning microscope will help to remove this drawback. But it can detect only extemal defects in the specimen - those lying literally on the surface. An acoustic laser scanning microscope makes it possible to examine the layers near the surface and check their structure. This important quality may make it irreplaceable for product-quality control in micro-electronics, for example, when testing the quality of integrated systems and of components in integrated and fibre optics. 16
The second example concerns the possibility of acousto-optic exploration of a heterogeneous condensed medium. 12 Its essence is that, unlike traditional acoustic or laser exploration, optical radiation of modulated intensity is used to set up an acoustic signal by remote control in the medium under investigation. Travelling in a set direction, the signal scatters in accordance with the irregularities of the medium. The acoustic signal thus dispersed is recorded by remote control (in the return direction) by optical methods using, for instance, a laser speed gauge based on the Doppler effect or an optical 'read-out' of the surface relief, which is extensively used in acoustic holography. Alternatively, it may use a method based on Bragg's law of ultrasonic diffraction of light in the immediate subsurface layer of the medium, if it is sufficiently transparent optically. As in the previous example of acoustic laser scanning microscopy, it is possible with suitable electronic devices to scan laser beams emitting ultrasonic waves and receiving an acoustic signal, and to obtain a television image.
To take a further possibility in acoustic laser engineering, if laser pulses of enormous intensity are directed at a condensed medium, acoustic waves of finite amplitude may be set up, which in the course of their propagation turn into shock waves. These shock waves alter the structure of the substance and my affect is physical properties and strength. 18
COHERENT OPTICAL ANALYSIS OF SIGNALS IN ACOUSTIC DATA SYSTEMS
The coherent optical processing of signals is an independent subject and it is part of an extremely wide field of optical processing of data and optical computing. Optical processing methods have made major progress in the last 25 years. Optical spectrum analyzers of electrical signals and images, apparatus for the formation of images with aperture synthesis, correlators, devices for calculation of convolutions, optical character readers, and other systems have been developed. There is a fairly extensive literature on the optical processing of data and its applications (see, for example, special issues of the Proceedings of the IEEE and monographs).
We shall not consider details but simply point out recent suggestions of the use of lasers, fibre-optic sound detectors, fibre-optic communication lines, and optical processing methods not only in acoustic nondestructive quality control, ultrasonic tomography, etc., but also in h~droacoustics. For example, a passive hydroacoustic system ('optical sonar') is described. 6 In this system the components of a linear hydroacoustic antenna are fibreoptic sound detectors; use is also made of acoustooptic devices for the formation of the directionality characteristic of the antenna and for scanning; finally, the signals are processed optically.
CONCLUSIONS
We shall conclude by noting that optoacoustic sources and fibre-optic sound detectors, or laser methods for remote determination of vibrations and the reception of sound do not replace completely the traditional sources and detectors, just as coherent optical systems for data processing have not superceded computers completely. However, there is no doubt that the use of the lasers will have a major influence on the future development of physical and technical acoustics.
62
REFERENCES
1. N. G. Basov, P. G. Eliseev, and Yu. M. Popov, Usp. Phys. Nauk, 148:35 (1986) [Sov. Phys. Usp. 29:20 (1986)].
2. AM. Prokhorov, Usp. Phys. Nauk, 1984:3 (1986) [Sov. Phys. Usp. 29:1 (1986)]. 3. L. M. Lyamshev, Lasers in Acoustics, Sov. Phys. Usp., 30:252 (1987). 4. A G. Bell, Paper presented to National Academy of Sciences, USA, April 21,
(1881). 5. G. A Askar'yan, A. M. Prokhorov, G. F. Chanturiya, and G. P. Shilnikov, Zh.
Eksp. Teor. Fiz. 44:2180 (1963) [Sov. Phys. JETP 17:1463 (1963)]. 6. L. M. Lyamshev, Usp. Fis. Nauk, 135:637 (1981) [Sov. Phys. Usp. 24:977 (1981)]. 7. A I. Bozhkov, F. v. Bunkin, AI. A. Kilomenskii, A. I. Mallyarovskii and V. G.
Mikhalevich, Tr. Fiz. Inst. Akad. Nauk USSR, 156:123 (1984). 8. F. V. Bunkin and V. M. Komissarov, Akust. Zh., 19:305 (1973) [Sov. Phys.
Acoust. 19:203 (1973)]. 9. L. M. Lyamshev and L. V. Sedov, Akust. Zh., 27:5 (1981) [Sov. Phys. Acoust.,
27:4 (1981)]. 10. L. M. Lyamshev and K. A. Naugol'nykh, Acoust. Zh., 27:641 (1981) [Sov. Phys.
Acoust. 27:357 (1981)]. 11. F. V. Bunkin, V. G. Mikhalevich and G. P. Shipulo, Uvantovaya Elektron.(Moscow)
3:441 (1976) [Sov. J. Quantum Electon. 6:238 (1976)]. 12. L. Hutcheson, D. Roth and F. S. Barnes, Record of the Eleventh Symposium on
Electron, Ion, and Laser Beam Technology, Boulder, Colorado, 1971 (ed. by R. F. M. Thornley), San Francisco Press (1971), p. 413.
13. T. G. Muir, C. R. Culbertson and J. R. Clynch, J. Acoust. Soc. Am., 59:735 (1976). 14. L. M. Lyamshev and Y. Y. Smirnov, "Volokonno-opticeskie priemniki zvuka (okzor)
[Fibre-optic Sound Detectors (A Servey)]", Akust. Zh., 24:289 (1983). 15. L. M. Lyamshev, "Ultrazvukovaja i lazernaja technologija [Ultrasonic and laser
technology], Report to the Plenary Session of the All-Union Conference "Osnovnye napravlenija razvitija ultrazvukovoj tekhniki i tekhnologii na period 1981-1990 g." [Basic trends in the Development of Ultrasonic Technology and Engineering in the Period 1981-1990]. Suzdal, 1982.
16. A. I. Morozov, and V. Yu. Raevskii, Zarubezh. Electron. Tekh. No. 2(248), 46(1982).
17. L. M. Lyamshev, Optiko-akusticeskoe zondirovanie neodnorodnoj kondensirovannoj sredy [Acousto-optic Sounding of a Condensed Heterogeneous Medium]. Reports of the USSR Academy of Sciences, 1979.
18. L. I. Ivanov, N. A Litvinova and V. A Yanushkevich, Glubina obrazovanija udarnoj volny pri vozdefstvii lazernogo izlucenija na poverhnost monokristalliceskogo molibdena [Depth of the Formation of the Shock Wave Caused by Laser Action on the Surface of Monocrystalline Molybdenum], Kvant. Electron. 4:204 (1977).
19. Optical Computing (special issue), Proc. IEEE 65, No.1 (1977). 20. Optical Computing (special issue), Proc. IEEE 72, No.7 (1984). 21. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New York, 1968
[Russ. transI., Mir, m. 1970]. 22. V. A. Zverev and E. V. Orlot, Optical Analysers [in Russian], Sovetskoe Radio
Moscow (1971). 23. W. Munk and C. Wunsch, Deep Sea Res. Part A, 26:123 (1979). 24. G. A. Askariyan, B. A Dolgoshein, A N. Kalinovskii and N. V. Mokhov, Nuci.
Instrum. Methods, 164:267 (1979). 25. Acoustic Imaging: Cameras, Microscopes, Phased Arrays, and Holographic Systems
(Lectures presented at the University of California on Acoustic Holography and Imaging, Santa Barbara, 1975, ed. by G. Wade), Plenum Press, New York (1976), p. 379 [sic.].
26. G. Hetland, C. M. Davis and R. E. Einzig, EASCON '79:Record IEEE Electronics and Aerospace Systems Conf., Arlington, VA, 1979, Part III, pubi. by Institute of Electrical and Electronics Engineers, New York, 1979, p. 602.
63
OPTICAL GENERATION AND DETECTION OF ULTRASOUND
Jean-Pierre Monchalin
Industrial Materials Research Institute National Research Council Canada 75 De Mortagne Blvd. Boucherville, Quebec J4B 6Y4 Canada
INlRODUCTION
The optical generation and detection of ultrasound, often called laser-ultrasonics, has attracted considerable interest and presents numerous advantages for scientific and industrial applications, compared to classical piezoelectric generation and detection. These advantages follow essentially from generation without contact and at a distance by using lasers for generation and detection. Therefore, ultrasonic measurements and ultrasonic inspection on specimens at elevated temperature are readily feasible and the conventional uses of buffer rods, special bonds or momentary contact are eliminated. Such a feature is very important for industrial applications since materials like metals and ceramics are processed at elevated temperature. Also important for industrial applications is the ease of inspecting curved parts. Unlike conventional techniques which require precise transducer orientation to follow a complex surface contour, transduction of ultrasound with laserultrasonics is performed by the surface itself. A third limitation of conventional piezoelectric technology, unless special techniques are used, is its limited bandwidth at emission and reception. In contrast, lasers enable the production of broadband ultrasonic pulses extending from zero frequency to 50 MHz and even more, and interferometric receivers can be made with a bandwidth extending from a minimum value of 10 to 50 KHz (limited by ambient vibrations) to a maximum value given by the cutoff frequency of the detector (50, 100 MHz and above).
However, laser-ultrasonics is not generally as sensitive as conventional piezoelectric techniques, essentially because of detection. Nevertheless, as demonstrated by the results shown below, it often is possible to reach adequate sensitivity to perform various laboratory measurements and industrial inspections.
In this paper, we first outline the various mechanisms used for generating ultrasound, briefly review the optical methods used for detection and then give examples of the use of laser-ultrasonics taken from work performed in our laboratory. These examples illustrate the application to ultrasonic metrology, to thickness gauging and ultrasonic velocity determination, flaw detection, ultrasonic attenuation and microstructure characterization.
LASER GENERATION OF ULTRASOUND
The generation of ultrasound, following the absorption of a high power laser pulse, can proceed essentially from two mechanisms. 1-3 At low laser power density, there is no
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 65
phase change at the surface and only transient surface heating which produces essentially tangential stresses (thermoelastic regime).4 At higher laser power density there is surface melting and surface vaporization giving near the surface a hot expanding plasma (ablation regime). The ultrasonic stress in this case is produced by the recoil effect following material ejection and is essentially normal. A longitudinal wave is then emitted normally to the surface. Ultrasonic stresses in this case are comparable in magnitude to the ones produced by conventional piezoelectric transducers using peak voltage excitation of a few hundred volts, whereas they are weaker in the thermoelastic regime. The ablation regime has the drawback of vaporizing a small quantity of material at the surface, but in many cases, this small surface damage of the order of a micron per hundred shots is acceptable. It is also acceptable on steel products at elevated temperature on a production line. Strong longitudinal pulses are also produced in the damage-free thermoelastic regime by covering the surface with a transparent layer.5 The acoustic source in this case is distributed underneath the surface and the stresses are essentially normal to the surface as in the ablation regime. The same effect also is observed without additional coating when the material weakly absorbs laser light.6
One important advantage of the generation of ultrasound with lasers is the generation of shear waves as well, simultaneously to longitudinal waves. The amplitude and characteristics of the displacements associated with these two kinds of waves (steplike, monopolar or bipolar pulse) depend on the generation mechanism (thermoelastic or ablation), on the penetration of light through the material, on the size of the illuminated zone and whether detection is performed on-axis (at epicenter) or off-axis. 1-4 In all cases, the generated ultrasonic wavefront follows the surface curvature, thus permitting ready to probing of parts of complex shape.
Rayleigh surface waves and plate waves also can be generated with magnitude comparable or exceeding that of traditional means. Good directivity can be obtained by focusing the beam with a cylindrical lens in order to obtain a line source.? Large signal magnification has been demonstrated by generating a circular wave with a conical lens (axicon) and detecting with an interferometer at the center of convergence.S This setup minimizes heat loading on the surface, which is important for some materials.
OPTICAL DETECTION OF ULTRASOUND
Concerning the optical detection of ultrasound, the various methods recently have been reviewed,9 except for the reflection/ortical sidebands stripping technique lO and the technique using a phase-conjugating mirror. 1 Most of these methods are based on optical interferometry, except the knife edge technique which detects the deflection of an optical beam caused by the surface ripple produced by ultrasound.9, 12 For detecting normal displacement, which is generally sufficient in the case of laser generated ultrasound, two interferometric methods can be used.
The first one, which we call optical heterodyning or simple interferometric detection, consists in making the wave scattered by the surface interfere with a reference wave directly derived from the laser9 and is sketched in Fig. 1. Such a technique is sensitive to optical speckle and the best sensitivity is obtained when one speckle is effectively detected. This means that the mean speckle size 0n the focusing lens has to be about the size of the incoming beam and that this beam should be focused onto the surface. Therefore, this technique generally permits the measurement of the ultrasonic displacement over a very small spot, which, except at high ultrasonic frequencies, can be considered as giving point-like detection. Compensation for vibrations can be performed by an electromechanical feedback loop which uses a piezoelectric pusher for path length compensation. For more severe vibration environments, a heterodyne configuration is preferred. In this scheme, the frequency in one arm is shifted by an RF frequency and the detector receives a signal at this shifted frequency, phase modulated by ultrasound and vibrations. Electronic circuits can be devised to retrieve the ultrasonic displacement independently of vibrations.
The second detection method, called velocity interferometry or time-delay interferometry,9 is based on the Doppler frequency shift produced by the surface motion and its demodulation by an interferometer having a filter-like response (see Fig. 2). This technique is sensitive primarily to the velocity of the surface and is therefore insensitive to low frequencies. The filter-like response is obtained by giving a path delay between the
66
Mirror
""F ,.._1 .. , rT\ ! i Frequency shifter 'CJ r.. •• .A
Ultrasound
Detector
Fig. 1: Basic configuration for optical heterodyning (simple interferometric detection). A frequency shifter (e.g. Bragg cell) can be introduced in either arm (heterodyne Michelson interferometer). The inserts in circles indicate
the optical frequency spectrum at various locations.
interfering waves within the interferometer. Although two-wave interferometers (Michelson, Mach-Zehnder) can be used, a more compact design for frequencies 1 - 100 MHz is obtained with a multiple-wave interferometer (Fabry-Perot) . Unlike optical heterodyning, this method permits the reception of many speckles and allows a large detecting spot (several mm and even more), especially when known modifications which increase throughput and field of view are used (field-widened Michelson interferometer, confocal Fabry-Perot interferometer).9
Velocity interferometry, being based on a filter-like principle, does not have a flat detection response, but has generally a large etendue (or throughput) corresponding to its ability to detect over a large spot. Optical heterodyning or simple interferometric detection, on the other hand, has a flat response, limited by the detector cut-off frequency or the Bragg frequency, but a small eteridue (=1.2, A. being the optical wavelength), corresponding to its restriction to small spot detection. This limitation has been circumvented by two recent variants of the technique. One of them makes use of nonlinear optics to generate a phase-conjugating mirror with beams derived from the detection laser. This mirror causes light to retrace its path back onto the surface and then into the interferometer as a speckle-free wave which interferes with the reference wave. I I The second variant we have developed makes use of a confocal Fabry-Perot interferometer to strip the scattered light from its sidebands, thus generating a reference beam with an optical wavefront substantially matching the incident wavefront. A simple implementation of this scheme, sketched in Fig. 3, is obtained by noting that in the confocal Fabry-Perot interferometer the reflected light actually includes the interference of a beam with sidebands directl1' reflected from the surface and a beam stripped from its sidebands leaked by the cavity. 0
~ ~ Ultrasound - Response ~"
1jJ{~~ Laser ~ Optical
frequency.... frequency
Fig. 2. Ultrasound detection with a velocity or time-delay interferometer. The insert indicates the principle of detection.
67
Sample Confocal Fabry·Perot
Ultrasound Detector
Fig. 3. Optical heterodyning by sideband stripping using a confocal cavity in reflection mode.
Finally, an optical technique similar to differential Doppler anemometry allows the detection and the measurement of in-plane ultrasonic motion. The principle of this technique is sketched in Fig. 4 and consists in making two light beams, issued from the same laser, intersect on the surface, symmetrically with respect to the normal. A detector is used to receive the two speckle fields, the interference of which provides a signal representative of the in-plane motion. This detection method can be combined with optical heterodyning into a probe which measures at the same location on the surface in-plane and out-plane displacements.13
APPLICATION TO ULTRASONIC METROLOGY
Optical probes, being contactless sensors of ultrasound, are particularly useful for measuring the displacement produced by ultrasonic transducers. Problems associated with couplant thickness and applied pressure are thus eliminated. This application generally requires a small probing spot and easy and reliable calibration and therefore probes based ?n ~tical heter<X!yning (hom~yne or heterodyne int~rferometers~ are perfect~y suita?le for It. 1 ·18 UltraSOnIC field mappmg can be performed eIther br>f movmg the speCImen WIth the transducer attached to it15 or by scanning the optical beam. 6 This application is illustrated in Fig. 5, which shows the displacement variation along the circumference of a thin wall pipe made by Electrical Resistance Welding, in which ultrasonic waves are launched by the combination of an acoustic lens and a water film. By scanning the inside and the outside of the pipe, it is possible to assess the performance of the ultrasonic technique for defect detection in the weld zone.
68
Fig. 4. Schematic of the principle for in-plane motion detection. Introducing a frequency shifter (e.g. Bragg cell) along one of the incident
beams makes the probe heterodyne.
.012 Probe Transducer ; Water gap /
~
./"',...-.,--- .. ;;-.... -.. ~ ........... -... -. -----=._ ... .".. ',,::,:'" o "2 o 20 40
, 60 80 100
Position (away from transducer centre. rrvn)
Fig,S. Plot of maximum normal displacement measured inside a pipe (16 in. in diameter, 0.228 in. thick). The displacement is produced by a focusing transducer
water gap coupled to the pipe. The insert indicates the ultrasonic configuration. Each curve corresponds to different offsets in the transverse direction
(separated by 4 mm).15
Immersion transducers can be tested readily by using a standard water tank and its associated scanning and imaging system with a partially immersed plate specimen. Displacement is measured from the top surface while the transducer is mechanically scanned underneath. I8 More complete information on an ultrasonic field is obtained by using the probe which allows the measurement of two components of the ultrasonic displacement. 13
APPLICATION TO THICKNESS GAUGING AND VELOCITY DETERMINATION
Ultrasound is widely used in industry for thickness gauging by measuring the time of flight between consecutive echos. The same application can be performed at a distance by laser-ultrasonics even on parts at elevated temperature and of complex shape. This application is the same as velocity determination, since both are based on time-of-flight measurement; the former requires knowledge of velocity while the latter assumes that thickness is known.
Laser-ultrasonics has the same limitation as classical ultrasonics: the systematic error introduced by ultrasonic diffraction. In practice, since laser beams rarely have a well defined intensity distribution, two regimes can be considered, corresponding respectively to point source6Point receiver and to large source/large receiver generating/detecting configurations. 1 Both give negligible corrections to the time delay. To measure precisely time-of-flight independently of pulse shape, data is digitally sampled and the time interval between selected echoes is determined by a cross-correlation technique. Such a technique, which uses the whole shape of the signal, is very precise and errors down to 1 ns can be obtained with sufficient signal-to-noise ratio and sufficient signal bandwidth .20
Laser-ultrasonics has an additional source of error compared to classical ultrasonics: the fact that the emitting and the receiving systems are different, thus giving raise to possible misalignment between the generating and detecting spots. When operating in the point source/point receiver regime, it often is found necessary to readjust from time to time the relative position of the generating and detecting spots by observing on a scope the arrival of ultrasound. The proper position is the one which gives the minimum delay after generation. The large source/large receiver regime is less sensitive to misalignment and is consequently more suitable to industrial implementation.
Using the point source / point receiver approach, we have performed many experiments using a transmission configuration20-21 as have other laboratories.22-24 In all
69
these experiments, the detecting beam is focussed onto the surface and optical heterodyning is used. We have measured as a function of temperature the velocity of many specimens made of steel or ceramic, some of them being non-planar. The shear wave arrival was observed at the same time as the longitudinal wave arrival and shear velocity was also determined. A typical signal observed on a ceramic-metal composite is shown in Fig. 6. Determination of both velocities allows the determination of the elastic constants of the material. To enhance longitudinal features and to permit data averaging for improved signal-to-noise ratio, all the measurements were performed in the slight ablation regime. Data which shows the limitations of signal-to-noise ratio and alignment is presented in Fig. 7, which also shows how interpolation eliminates the limitation introduced by the sampling interval.20 In this figure the deviations from a smooth variation of velocity versus temperature are caused by the finite signal-to-noise ratio and alignment errors. Each data point was taken after proceeding to a new beam adjustment.
We are currently exploring the use of the large source / large receiver approach for industrial thickness gauging of steel products such as rolled sheets and seamless pipes. Thickness determination requires the measurement of ultrasonic velocity as a function of temperature which can be performed in the laboratory using specimens of known thickness. The experimental system being developed is sketched in Fig.8.25 This system permits single-side inspection and is made up of two units linked by optical fibers. In this system, the three beams: generating, illuminating for detection and the received beam, are all collinear. The generating unit includes the generating laser (frequency-doubled Qswitched Nd-Y AG laser or an excimer laser) and the beam mixing and light collecting optics (15 cm in diameter), which are located at 1.5 m from the surface of the workpiece. The receiving unit includes the receiving Nd-Y AG laser and the confocal Fabry-Perot
70
AI . Oj AI ·SiC Ceramic· metal compOSite
c OJ E OJ
L u os Ci Vl
'i5 n; E 35 0 Z
o 2
Time ( Ils)
Fig. 6. Experimental displacement measured at epicenter on a 3.25 mm thick AI203-Al-SiC ceramic-metal composite plate. The laser pulse has 10 ns
duration and the spot size is - 0.5 mm in diameter.20
AI 2 03-AI -SiC composite
• • •• o 00000
• •
• •• • 000
o 5ns (sampling time) resolution
•• •
• 0.05ns cubic spline interpolation
6000~~-+--r-~-+--r-~-+--r-~
o 500 1000
Temperature (0C)
Fig. 7. Enhancement of the precision of the time delay measurement by interpolation. The sample is a 2.360 mm thick AI203-AISiC ceramic-metal composite.20
r-------------------, Digital sampling
I -1 Receiving laser I 1 and signal
Optical fiber
I 1 processing
1 ID I E « »ffi) +-I Confocal Fabry-Perot I L ________________ ,-__ ..J
------.., I I I I
.----------,1 I
I Generating laser 1
I I L _________________________ ~
Beams mixing and light
collecting optics
Fig. 8. System in development for high temperature gauging in the steel industry.
71
receiver which allows detection over a large spot (::;; 10 mm in diameter). This unit is followed by digital sampling and digital processing electronics.
The accuracy of the method was tested on several low carbon steel plates of different thicknesses by using the system of Fig. 8 with generating and detecting spots of -9 mm diameter. 19 Generation was performed in the slight ablation regime, and no averaging was performed to increase the signal-to-noise ratio. A standard deviation of 2 ns on the round-trip time was obtained. This gives a resolution of 6 11m or 0.1 % on the thickness. The systematic error was evaluated to be smaller in this case. 19
APPLICATION TO HAW DETECTION
Laser-ultrasonics, like conventional ultrasonics, allows the detection of flaws inside and at the surface of materials. This has been illustrated by the detection of various artificial defects (flat-bottomed or side-drilled holes, slots).26-28 The technique is particularly useful to detect defects in samples of complex geometry. This is illustrated in Fig. 9, which shows the detection of an artificial delamination in the corner region of a graphite epoxy specimen.26 By using an optical scanner to scan the generating and detecting beams over the surface of the sample, large and curved areas can be inspected.
APPLICATION TO ULTRASONIC ATTENUATION MEASUREMENT AND MICROSTRUCTURE CHARACTERIZATION
Ultrasonic attenuation can also be measured by laser-ultrasonics and could provide information on the microstructure of the material. Like velocity measurement, two regimes in practice can be considered corresPQnding to a point source / point receiver and to a large source / large receiver configuration.29 In the first case, the ultrasonic amplitude decreases
72
1.0
t INNER SURFACE
3.0 5.0 7.0 9 .0
MICROSECONDS
DELAMINATION
1.0 3 .0 5 .0 7 .0 9.0
MICROSECONDS
GENERATION AND
RECEPTION
GENERATION AND
RECEPTION
Fig. 9. Inspection of a comer-shaped graphite-epoxy specimen. Above: zone free of delamination, below: delamination introduced at mid-thickness.26
100 T-------------------------------,
/ I
, , , ,
/ /
• •
/ - piezoelectr ic
/ I
/ transducer
- diffracl ion corrected
0.1 +-----~--~-+~-+~~+-----~--~ 10
frequency (MHz)
Fig. 10. Longitudinal ultrasonic attenuation measured by laser ultrasonics on a PZT ceramic sample illustrating the point source / point receiver case.29
as liz, where z is the distance from the generating to the detecting spot and the diffraction correction in dB is 20 log (Zl/Z2) for two echoes corresponding to the distances Zl and Z2. The diffraction correction is therefore equal to - 9.54 dB for a measurement in transmission (Z2 = 3Z1) and - 6.02 dB for a measurement made in reflection (Z2 = 2Z1). This regime is more easily applicable to thick samples and low frequencies. It is illustrated by the data taken on a 6.38 mm thick lead zirconate titanate (PZT) ceramic sample shown in Fig. 10.29 Ultrasound was generated in the ablation regime by a Q-switched Nd: Y AG laser focused to a spot of the order of 0.5.mm in diameter. An argon ion laser, coupled to a heterodyne displacement interferometer was used for detection and was focussed at epicenter (on the opposite side of the sample in front of the generating spot). After applying the 9.54 dB frequency-independent correction, we obtained the curve shown in Fig. 10. This curve is compared with results obtained by conventional piezoelectric transducers (only the fitted smooth curve is shown for the sake of clarity). It can be seen that the agreement is good (a variation in f2 is observed), except above 13 MHz where the signal-to-noise ratio is low and below 2 MHz because of the error introduced by data truncation.
In the second regime where the sizes of the source and the receiver are lar~e, the amplitude variation with z is small, especially if the sizes are about the same.2 This regime is more easily applicable to thin samples and high frequencies. It is illustrated by the data taken on a 4. 19-mm-thick hot-rolled carbon-steel plate with 14-~m average grain size shown in Fig. 11. Ultrasound was generated in the slight ablation regime with a frequency-doubled Nd:YAG laser giving a generating spot on the surface estimated to be 8 mm in diameter. The receiving laser is a Nd:YAG long-pulse laser focused onto the surface to a spot estimated to be 10 mm in diameter on top of the generating zone. The scattered light was coupled to a Fabry-Perot velocity interferometer. The results uncorrected for diffraction are plotted in Fig. 11 and it can be seen that they are in good agreement with a diffraction-corrected measurement obtained with a piezoelectric transducer and that they roughly follow an t4 variation, which is characteristic of Rayleigh scattering.
73
; I
I I
10.0 I I
~ I
E I
laser-ultrasonics ,
~ I
a ~ 1 mm. b ~ Smm I CD I :s!.. (uncorrected) I
I c: I .2 I
J iii I ::> I c: I ~ I
;( laser-ultrasonics / I
a ~ 4mm. b ~ Smm I (uncorrected) I
.... I I I I I I I I
I ' I piezoelectric transducer I
I (diffraction corrected) I I I
1.0 I
10 20 30 40 50 60
Frequency (MHz)
Fig. 11. Longitudinal ultrasonic attenuation of a hot-rolled steel plate measured in transmission. a and b are the radii of the generation and
reception spots, respectively.29
Figure 11 also shows the uncorrected attenuation obtained with the generation laser focused to a diameter of - 1.5 mm. It is clear that, in this case, the diffraction correction is not negligible any more. It is larger than 10 dB/cm near 20 MHz, which shows the importance of having beams of large and approximately equal size.
When the microstructure of the material is smaller, a higher range of ultrasonic frequencies should be used. The velocity / time delay Fabry-Perot receiver can then be replaced by the reflection/sideband stripping Fabry-Perot system. More recently, using such a system, we were able to obtain an excellent correlation between measured attenuation and grain size, for sizes ranging from 4 !lm to 15 !lm and using frequencies extending up to 60 MHz.
SUMMARY AND CONCLUSION
We have outlined the principles of laser generation of ultrasound as well as the various optical methods for detection. These techniques have found applications in ultrasonic metrology, in particular for mapping the emission pattern of ultrasonic transducers. Like classical techniques, they can be used for thickness gauging, flaw detection and material characterization with the advantage of not being restricted by the temperature of the specimen or its shape. Sensitivity and cost appear to limit their widespread use, which so far has been essentially limited to the laboratory. On-going developments, in particular in our laboratory, should make the technology sufficiently mature so its use in industry, in particular as a process control sensor, could begin.
74
REFERENCES
1. C. B. Scruby, R. J. Dewhurst, D. A. Hutchins and S. B. Palmer, Laser generation of ultrasound in metals in "Research Techniques in Nondestructive Testing," Vol. V, R. S. Sharpe ed., Academic Press, New York (1982), pp. 281-327 .
2. D. A. Hutchins, "Mechanisms of pulsed photoacoustic generation", Can., J. Physics, 64:1247-1264 (1986).
3. D. A. Hutchins, Ultrasonic generation by pulsed lasers, in "Physical Acoustics," Vol. XVIII, W. P. Mason and R. N. Thurston eds., Academic Press, New York, (1988), pp. 21-123.
4. U. Schleichert, K. J. Langenberg, W. Arnold, S. Fassbender, A quantitative theory of laser-generated ultrasound, in "Review of Progress in Quantitative Nondestructive Evaluation," Vol. 8A, D. O. Thompson and D. E. Chimenti eds., Plenum Press, New York, (1989), pp. 489-496.
5. R. J. Von Gutfeld, 20 MHz acoustic waves from pulse thermoelastic expansions of constrained surfaces, App. Phys. Lett., 30: 257-259 (1977).
6. R. J. Conant and K. L. Telschow, Longitudinal wave precursor signal from an optically penetrating thermoelastic laser source, in "Review of Progress in Quantitative Nondestructive Evaluation," Vol. 8A, D. O. Thompson and D. E. Chimenti eds., Plenum Press, New York, (1989), pp. 497-504.
7. A. M. Aindow, R. J. Dewhurst and S. B. Palmer, "Laser-generation of directional surface acoustic wave pulses in metals", Optics Com. 42:116-120 (1982).
8. P. Cielo, F. Nadeau, M. Lamontagne, "Laser generation of convergent acoustic waves for material inspection", Ultrasonics, 23:55-62 (1985).
9. J.-P. Monchalin, "Optical detection of ultrasound", IEEE Trans. on Ultrasonics, Ferr. and Frequency Control, 33:485-499 (1986).
10. J.-P. Monchalin, R. Heon, P. Bouchard and C. Padioleau, "Broadband optical detection of ultrasound by optical sideband stripping with a confocal Fabry-Perot", Appl. Phys. Lett., 55:1612-1614 (1989).
11. M. Paul, B. Betz and W. Amold, "Interferometric detection of ultrasound at rough surfaces using optical phase conjugation", Appl. Phys. Lett., 50:1569-1571 (1987).
12. L. W. Kesler and D. E. Yuhas, "Acoustic Microscopy - 1979", Proc. IEEE, 67:526-536 (1979) ..
13. J.-P. Monchalin, J.-D. Aussell, R. Heon, C. K. Jen, A. Boudreault and R. Bernier, "Measurement of in-plane and out-of-plane ultrasonic displacement by optical heterodyne interferometry", J. of Nondestructive Evaluation, 8:21-133 (1989).
14. J.-P. Monchalin, "Heterodyne interferometric laser probe to measure continuous ultrasonic displacements", Rev. Scient. Instr., 56:543-545 (1985).
15. J.-P. Monchalin, R. Heon and N. Muzak, "Evaluation of ultrasonic inspection procedures by field mapping with an optical probe," Canadian Metallurgical Quarterly, 25:247-252 (1986).
16. B. C. Moss and C. B. Scruby, "Investigation of ultrasonic transducers using optical techniques", Ultrasonics, 26: 179-188 (1988).
17. J. C. Baboux, "Interferometric measurements of transient ultrasonic fields: application to hydrophone calibration", IEEE Ultrasonic Symposium, IEEE Press, Chicago, 857-861 (1988).
18. J.-P. Monchalin, R. Heon, unpublished; see also the commercial brochure on OP-350 and OP-35I/O probes of Ultra Optec, available from Ultra Optec, 27 Lauzon Street, Boucherville, Quebec J4B 1E7, Canada.
19. J.-P. Monchalin and J.-D. Aussel, "Ultrasonic velocity and attenuation determination by laser-ultrasonics", J. Nondestructive Evaluation, 9:211-221 (1990).
20. J.-D. Aussel and I.-P. Monchalin, "Precision laser-ultrasonic velocity measurement and elastic constant determination", Ultrasonics, 27: 165-177 (1989).
21. J.-P. Monchalin, R. Heon, J. F. Bussiere and Farahbakhsh, B., Laser-ultrasonic determination of elastic constants at ambient and elevated temperatures, in Nondestructive Characterization of Materials II, J. F. Bussiere, J.-P. Monchalin, C. O. Ruud and R. E. Green, Jr., eds, Plenum Press, New York, (1987) pp. 717-723.
75
22. R. J. Dewhurst, C. Edwards, A. D. W. Mckie and S. B. Palmer, "A remote laser system for ultrasonic velocity measurement at high temperatures", J. Appl. Phys. 63:1225-1227 (1988).
23. L. F. Bresse, D. A. Hutchins and K. Lundgren, Elastic constants determination using ultrasonic generation by pulsed lasers, in "Review of Progress in Quantititative Nondestructive Evaluation," Vol. 7B., D. O. Thompson and D. E. Chimenti eds., Plenum Press, New York, (1988), pp. 1219-1226.
24. B. Pouet and N. J. P. Rasolofosaon, "Ultrasonic intrinsic attenuation measurement using laser techniques", Ultrasonics Symposium Proceedings IEEE Press, Chicago, pp. 545-549 (1989).
25. J. -Po Monchalin, J.-D. Aussel, R. Heon, J. F. Bussiere, P. Bouchard. J. Guevremont, C. Padioleau, "Laser-ultrasonic developments towards industrial applications", Ultrasonic Symposium Proceedings, IEEE Press, Chicago, pp. 1041-1043 (1988).
26. J. -Po Monchalin, J.-D. Aussel, P. Bouchard and R. Heon, "Laser-ultrasonics for industrial applications" Review of Progress in Quantitative Nondestructive Evaluation, Vol. 7B, D. O. Thompson and D. E. Chimenti, eds., Plenum Press, New York, (1988), pp. 1607-1614.
27. A. M. Aindow, R. J. Dewhurst, S. B. Plamer and C. B. Scruby, "Laser-based nondestructive testing techniques for the ultrasonic characterization of subsurface flaws", NDT International, 17:329-335 (1984).
28. D. A. Hutchins, F. Nadeau and P. Cielo, "A pulsed photoacoustic investigation of ultrasonic mode conversion", Can. J. Phys., 64:1334-1340 (1986).
29. J. -D. Aussel and J. -Po Monchalin, "Measurement of ultrasound attenuation by laser ultrasonics", J. Appl. Phys., 65: 2918-2922 (1989).
76
ACOUSTO-OPTICAL INTERACITON IN MEDIA WIlli REGULA lED ANISOTROPY
V. N. Parygin
Department of Physics Moscow State University 119899 Moscow USSR
INTRODUCITON
Anisotropic Bragg diffraction of light has significant advantages in comparison with isotropic diffraction. These advantages are defined by the complicated character of Bragg angle dependence upon acoustic frequency. Devices like light modulators, light deflectors, tunable filters and etc. based on anisotropic diffraction have better parameters than apparatus based on isotropic diffraction. However there is a group of acousto-optical materials, such as cubic crystals and glasses, that have no optical anisotropy at all.
ANISOTROPIC BRAGG DIFFRACITON IN ISOTROPIC MEDIA
In order to achieve anisotropic diffraction in isotropic media it is possible to create anisotropy artificially. It may be done by application of external mechanical pressure as well as by using an external electrical field in case of electro-optical media.
The analysis is performed for cubic crystals belonging to 23 or m3 classes with dielectric permeability Eo. Let the static mechanical pressure be applied to the crystal and directed at an angle ~ with the x axis (see Fig. 1). This pressure is characterized by a longitudinal static deformation vector:
z z'
Fig. 1. Coordinate system used in theory.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 77
Y = y [ COS2p; 0; sin2p; 0; sin 2P; 0]
Components of the dielectric penneability tensor are in this case:
- 2 ( 2 ) Ell = Eo + "fEo P11 cos2p + P13sin p
E22 = EO +;YC5 (P12sin2p + P13cos2P)
E33 = EO +;YC5 (Pl1sin2p + P12COS2P)
-2 E13 = "fEo P44sin 2P; E12 = E23 = 0
(1)
(2)
Here Pij are photoelastic tensor components of the medium. The characteristic axes of the tensor (Eqs. 2) do not coincide with crystallograph axes
of the initial crystal. It is necessary to rotate the coordinate system around the y axis through an angle a which can be found from the following equation:
(3)
It is interesting to mention that the angle a does not depend on the value of static pressure but it depends on the direction of this pressure and photoelastic constants of the medium only.
Let a longitudinal acoustic wave propagate in the direction Oz in this medium. This wave changes the third component of the three first equations of tensor (Eqs. 2)
El1 = YOE5 P13cOS (Kz - 0 t)
E22 = YOE5 P12COS (Kz - 0 t)
E33 = YOE6Pl1CO~Kz - Ot} .
(2a)
Here Yo is the longitudinal defonnation created by the acoustic wave, 0 is the frequency and K the wave vector of sound.
If incident light propagates along the y axis, then a component of diffracted light with polarization orthogonal to polarization of incident light is equal to
(4)
where EO is the amplitude and A. the wave length of light, I the interaction length, and Peff is the effective component of the photoelastic tensor:
Pl1 - P13 Peff = -----;=i::::;::::==============~]:;:::::2=
2 1 (Pl1 - P13)sin2p + (P12 - P11};:os2p .
2P44sin 2P (5)
In order to simplify the analysis ofPeffdependence upon p, we can define PI = (P11 - P13)/P44 and P2 = (P12 - P11)/P44· If PIP2 < 0, then P effmax = 0.5(P13 - P11)
P13 - Pll forp = arctan 1 -P?/Pl· If PIP2 > 0, then P effmax = 211 + P1P2 for p = arctan V PiP1 .
Analogous results may be received for other directions of incident light and the same direction of longitudinal acoustic wave. If a transverse acoustic wave propagates in the Oz direction then Peffmax is equal to P44 for;y orthogonal simultaneously to light and acoustic wave direction.
These results are valid also for cubic crystals belonging to 43m, 432 and m3m classes. In these classes Pl2 = Pl3 and as a result P2 = -PI, P opt = 1t and Peffmax = 0.5(PI2 - Pl1)· It is evident that in isotropic media like glasses the directioffOz is arbitrary and besides this, P44 = 0.5(P12 - Pll).
78
In an isotropic medium we have a maximum efficiency of anisotropic diffraction if the longitudinal static deformation is applied at a 45° angle to the longitudinal acoustic wave direction. The constant Peff = 0.5(PI2 - Pll) in this case. The same value of Peff we have for a transverse acoustic wave. But in this case the static deformation may be orthogonal to the wave direction. This variant is more convenient.
Acousto-optical figures of merit M are defined by the relationship
2 6 M=~.
pv3 (6)
In Table 1 are given figures of merit for some glasses and cubic crystals with anisotropic diffraction.
Table 1. Figures of Merit M for Selected Glasses and Cubic Crystals
Sound Sound M Medium Peff polarization ve~ity
10 mls 1O-15s/kg
FusedSi02 0.63 L 5.95 0.12 3.76 0.47
KRS-5 0.63 L 2.08 5.2 S 0.87 1300
KRS-6 0.63 L 2.32 0.27 S 1.07 130.0
GaP 0.63 S 4.13 25.0
GaAs 1.15 S 3.32 39.0
A~Se3 1.15 L 2.25 6.3 S 1.23 40.0
Ge 10.6 10.6 S 3.51 93.5
The efficiency of anisotropic diffraction in an isotropic medium does not depend on the static deformation value. This deformation creates the possibility of anisotropic diffraction and determines the Bragg angle dependence upon acoustic frequency:
sin OB = 2A. (f ± 2nv28nlA?r). nv
(7)
Here 8n is the anisotropy birefringence value induced in the medium by the static deformation. For an isotropic medium
(8)
Experiment
Experimental investigation of anisotropic diffraction of light in the medium with artificial anisotropy was performed using isotropic materials fused Si02 and cubic crystal GaP. The experimental arrangement is shown in Fig. 2. A longitudinal acoustic wave
79
a b
Fig. 2. Scheme of anisotropic diffraction observation in isotropic media. (a) Longitudinal waves in Si02. (b) Transverse waves in GaP.
propagates in fused Si02 at the angle 45° to the static deformation direction. In GaP the transverse acoustic wave propagates in the direction orthogonal to y .
Dependence of experimental Bragg angles eB upon acoustic frequency f are present in Fig. 3. Polarization of incident light is made parallel to the axis of artificial anisotropy. When the output polarizer is parallel to the input, then eB(f) dependence is linear (dotted line). In this case eB does not depend on external pressure and on the polarization direction of the incident light. The observed scattering corresponds to the isotropic diffraction of light. If the output polarizer is orthogonal to the first one, then eB (f) dependence is different from the previous case. (Solid lines in Fig. 3).
All the curves above the dotted line correspond to the same polarization of incident light. The curves below the dotted line correspond to orthogonal polarization of the incident light. The greater the external pressure the greater the shift observed between the curves corresponding to the orthogonal polarization of incident light. At the smne time, the frequency fo corresponding to the minimum Bragg angle value increases. We can find the value of artificial anisotropy on by measuring the difference between Bragg angles for isotropic and anisotropic diffraction for a fixed frequency f:
(9)
In our experiments with GaP the artificial anisotropy is equal to 10-4 when () ",1O-7N/m2. In this case fO is equal to 200 MHz. The efficiency of anisotropic diffraction is 40% when the acoustic power is about I watt. The anisotropic diffraction is fused Si02 is considerably less efficient.
80
e13 mrad
12
8
4-
0
-4-
-8
Fig. 3. Bragg angle dependence on acoustic frequency. 1. Isotropic diffraction P2I1Pl. 2, 3, 4. Anisotropic diffraction P2..l P1.
Fig. 4. Geometry of Raman-Nath diffraction. (a) Anisotropic case; (b) Istropic case.
ANISOTROPIC RAMAN-NATH DIFFRACTION
In a medium with artificial anisotropy anisotropic Raman-Nath diffraction can be observed. This phenomenon consists of simultaneous diffraction in a few maxima with different directions of polarization. The necessary condition for simultaneous existence of a number of maxima is llpl < 1 for all of the maxima:
(10)
The geometry of interaction during anisotropic Raman-Nath diffraction isrepresented in Fig. 4a. In Fig. 4b is given geometry of isotropic Raman-Nath diffraction. As a result, besides the necessary condition 2nAI« A 2n in isotropic medium, an additional condition 2noni < A must be satisfied in the anisotropic medium. This supplementary condition may be satisfied if 1 or On is sufficiently small. In a typical anisotropic medium the values of On are fixed (10-2 - 10-3 as a rule). However in the medium with artificial anisotropy we can vary the value of On as necessary.
In our experiment the value of On is equal to 10-5. The inequality 2nOnl/A. < 1 is satisfied for A = 633 nm with I = 5mm. We could observe 8 to 10 diffraction maxima at the acoustic frequency f = 36 MHz in GaP cubic crystals. The external pressure is directed along the [001] axis and transverse acoustic waves propagate along the [100] axis. The corresponding experimental arrangement is shown in Fig. 5. Incident light polarization is determined by the input polarizer. It is collinear to one of the axes of artificial anisotropy. When a drive electric power of about 2 watts is applied then we could see 8 to 10 diffractional maxima symmetrically located near the transmitted light. Polarization of light in the neighboring maxima is orthogonal. When the output polarizer is parallel to the input one we observe even maxima only. When the second polarizer is orthogonal to the first one we register only odd maxima. These pictures are shown in Fig. 6. In the diffractional picture weak isotropic diffraction may be distinguished besides anisotropic diffraction. Its efficiency is ten times less than the efficiency of anisotropic diffraction.
s hear sound
s c
Light r e
GaP e 1'1 n
P2
Fig. 5. Scheme of anisotropic Raman-Nath diffraction in a cubic crystal GaP. The output polarizer is not necessary.
81
• • • • • • • • 0_. 0- - 0 • • • • • • • • • without P2 P2 1P1
P2 " P1
Fig. 6. Anisotropic Raman-Nath diffraction. Pictures on the screen.
EXACf SATISFACTION OF BRAGG CONDmON IN A WIDE FREQUENCY BAND
Optical anisotropy regulation by external influence in an anisotropic medium may be used for satisfaction of Bragg conditions in a wide band of acoustic frequencies. Acoustic waves in an acousto-optic cell are generated successively at different frequencies. This regulation is very effective when the acoustic cell is used as a light deflector. It is well known that the change of diffraction angle in an acousto-optic deflector is caused by the variation of acoustic wave frequency.
Change of the frequency may cause the Bragg condition for a given incident angle no longer to be satisfied. This distortion of the Bragg condition results in diffraction efficiency decrease. If simultaneously with the changes of sound frequency we regulate the refractive index of the medium, then it is possible to satisfy the Bragg angle conditions over the entire frequency band.
If the sides of a deflector are orthogonal to the acoustic wavefront, then incident 8 j
and diffracted 8d angle values satisfy the conditions (see Fig. 7):
82
sin 8d = A-[f - v2 (n7 - nd2)/fA?] 2v 1 •
incident light
t
diffractional light
transmitted light
Fig. 7. Conditions satisfied by 8 j 8d when sides of the deflector are orthogonal to the acoustic wavefront.
(11)
It can be seen in Eq. 5 that for isotropic diffraction (nj = nd), the 9j angle value does not depend on the value of n. As a result, the correction of 9j is impossible. However it is possible during anisotropic diffraction. It is necessary to change nj or nd in such a way that the angle 9j is the same over the entire band of frequencies. We have to change, for example, the value nd by the external electric field in the following manner:
2 ~nd = Ild - ndo = A (f - fo) (ffo - £1)
2ndv2fo (12)
In this equation fo is a frequency corresponding to exact satisfaction of the Bragg condition in the absence of the electric field when ndo = nd (fo); f} is the frequency corresponding to the minimum value of the Bragg angle:
(13)
The dependence ~nd on acoustic frequency f is represented in Fig. 8. This figure helps to determine the frequency band in which an exact satisfaction of the Bragg condition is possible. If nd can be changed within the limits of ndo ± ~ndmax then the frequency band ~f achieves a maximum for
(14)
In this case the incident angle value is close to the minimum Bragg angle value 9j "" Af}/v. The Bragg condition is exactly satisfied in the optimum case in a frequency band
~f = 4v1 ndo~ndmax fA· If we use the electro-optic effect for regulation of the refractive index, then
(15)
where r is an effective component of the electro-optic tensor of the medium, and E the electric field.
Experiment
A light deflector with regulated anisotropy is investigated in our experiments. It is fabricated from a LiNb03 crystal (Fig. 9). The interaction length of light and sound is 1 to 4 cm. A light beam is propagated in the direction characterized by the angle ~ to the z axis. This angle determined the f} frequency value:
f1 = Y1n~ - nt sin~. (16) A
The index of refraction is changed by the electric field directed along the y axis. A slow transverse acoustic wave propagates along the x axis. The experimental Bragg angle dependence upon acoustic frequency is represented in Fig. 10 for three different values of the external electric field:
O ~--~--------~----f
Fig. 8. Refractive index frequency dependence in optimum case.
83
84
light IX sound
~~=,,£=~'~=~-~ ~'~'~w~tl ~ !I LiNb0 3 p
Fig. 9. Schematic for experiment.
8 B mrad
100
90 3 1
8O~::~' 70
300 400 500 600 700 f,t1Hz
Fig. 10. Frequency dependences of Bragg incidence angles. 1. E = 0; 2. E = 9.6kV/cm; 3. E = -9.6 kV/cm.
E k V em
12
8 3 4
0
-4
-8
-1 2
Fig. 11. Frequency dependences of electric field. 1. 8 j = 78mrad; 2. Hj = 80mrad; 3. 8j = 82mrad
(17)
The Bragg condition can be exactly satisfied in the region between the curves 2 and 3. Each point in this region has a corresponding electric field E value for which 9j = 9B.
It can be seen that the maximum ~f value corresponds to 9j = 82mrad and ~f=380 MHz in this case. The change of E in the limits from -9.6kV/cm to 9.6kV/cm makes it possible to correct the Bragg angle value in the frequency band from 340 to 720 MHz. If we change the incident angle, then the band of frequencies increases. The experimental dependence E(f) is shown in Fig. 11 for different values of incident angles and f l =500 MHz. For 9j=78mrad the correcting field is positive and M = 270 MHz. For 9j=80mrad the correcting field changes its sign. For 9j=82mrad the band of frequency is maximum for the given limits of the electric field.
Bragg condition regulation by means of electro-optical variation of medium refractive index may be used also for light amplitude modulation. In case the deflector with electrooptical Bragg angle correction is used for scanning purposes then there is no need for an additional modulator. Besides a correcting signal, additional voltage may be applied in order to modulate light intensity due to Bragg condition distortion. The calculation shows that 100% diffracted light modulation is achieved in case an electric field is approximately twice as large as V Al2 for a corresponding electro-optical device.
REFERENCES
1. V. I. Balakshy, V. N. Parygin and L. E. Chirkov, "Physical Foundation of Acousto-Optics," Radio and Communication Moscow (1985) in Russian.
2. V. I. Balakshy, E. I. Zotov, V. N. Parygin, Anisotropic Light Diffraction in the Medium with Artificial Anisotropy, Sov. Journ. Quant. Electr., 3:2187 (1976).
3. V. I. Balakshy, V. N. Parygin, Electric Control of Bragg angle in Acousto-Optic Devices, Sov. Journ, Radiotechn. and Elect., 18:115 (1973).
4. V. B. Voloshinov, V. N. Parygin, Raman-Nath Diffraction in Optical Anisotropic Media, Sov. Joum. Techn. Physical Letters, 7:145 (1981).
85
SURFACE AND SUBSURFACE WAVES FOR CHARACfERlZATION OF WEAKLY AND HIGm.. Y ANISOTROPIC MATERIALS
Aleksander Pilarski
Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw, Poland
INTRODUCTION
Of recent concern in Nondestructive Evaluation (NDE) is the importance of inspecting anisotropic materials. In addition to inspecting advanced high-strength, high-modulus composite materials and layered structures, there is a need to evaluate the anisotropic character of such planar materials as rolled polycrystalline metals, paper or even wooden products. The anisotropy can be described in terms of acoustic birefringence a. the ratio of the difference in velocities of two transverse waves propagating through the thickness and polarized into both principal perpendicular directions to their average value. In anisotropic materials the range of a. values can be from a few percent to several tens of percent.
Characterization of anisotropic materials based on an effective elastic constant determination is utilized for at least two reasons. Firstly, it is necessary to know material characteristics for suitable transducer selection and appropriate signal analysis before examining waveform characteristics and arrival time for defect detection, location, classification and/or sizing, residual stress measurements etc. Secondly, a careful study of the elements of the matrix of elastic constants could be used to determine material integrity and potential global deterioration from a particular reference state. For instance, one can estimate, using Hashin's theory of the prediction of the effective elastic constants of heterogeneous materials, that 1 % increase of porosity degree (PC) in a unidirectional graphite-epoxy composite causes almost 50% decrease of off-diagonal terms in the stiffness matrix and at the same time keeps almost unchanged stiffness in the fiber direction. Hence, the stiffness coefficients could be considered as actual feature values which can be used to carry out NDE analysis of the anisotropic materials.
Our objective was the development of an NDE technique which could be used to characterize the local mechanical behavior of anisotropic materials in field conditions; i.e., with access to one side of the inspected material. Furthermore, for some particular cases like composites in hydrothermal environments with the accompanying swelling phenomenon, we were looking for a thickness independent elastic constant determination technique, even though the overall philosophy is the same as that presented in reference 1.
BACKGROUND
Ultrasonic techniques have been widely used for dynamic elastic property characterization for an anisotropic materials. They are based on velocity measurements in known directions in the specially prepared test specimens that are cut along symmetry and off-symmetry planes.2-6 For fiber reinforced composites or rolled materials, the significant
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 87
case with respect to gross anisotropy is orthotropy with nine independent effective elastic constants, and transversely isotropic with five independent constants. The six (for orthotropic case) or four (for transversely isotropic case) diagonal terms of the stiffness matrix can be obtained from on-axis velocity measurements. Even for this step one needs to prepare specimens in which the desired propagation direction (i.e. normal to the surface) can be varied in a controlled manner. This requires a relatively thick specimen and is destructive in nature.
The remaining three or two off-diagonal terms (Cij, i=j), could be obtained from velocity measurements on the specimens cut along off-symmetry planes. Several attempts have been made to determine the off-diagonal terms using through - transmission pulse techniques utilizing oblique incidence of longitudinal waves in either an immersion or contact mode and their subsequent refraction into quasilongitudinal and quasitransverse waves. This bulk wave velocity measurement technique re9,uires access to the opposite sides of inspected plane-parallel materials.7-S The authors -S have employed a digitalultrasonic spectrointerferometer with a correlation algorithm. In through-transmission a fixed transmitter is used with a receiver with mechanically controlled position, and specimens mounted in a rotating fixture which permits the angle of incidence to be varied. The proposed method has some limitation due to a temporal and/or spatial mixing of the different modes. Another limitation is that mode conversion and multiple incidence angles in two planes of incidence coinciding with two accessible (without cutting) planes of symmetry allows one to determine a full set of five elastic constants for a transversely isotropic sample, but one cannot determine a full set for an orthotropic one. A variation upon this method with controlled transducer angles on one side of a fixed specimen has been proposed.9
An attempt to avoid the inconveniences of destructive examination and two sided inspection was recently made.lO The proposed method is based on the application of two angle probes placed on the same surface (contact technique) in front of each other, one as a transmitter and the other as a receiver. By changing the incidence and receiving angles, one can generate and receive quasilongitudinal waves propagating in an oblique manner in the inspected thick plate. By measurement of group velocity and the different angles of skip distance, the characteristics of that particular material can be obtained. Comparing such characteristics with those obtained from numerical calculations for different material properties, one can indirectly evaluate the parameters of elasticity. There is another possibility based on conversion of the group velocity profile to phase velocity information. The conventional Cristoffel IS coefficients can be calculated, but again as in the previous technique some terms will be missing.
A search for another practical solution has led to guided wave considerations. For thin layers the plate waves of known modes can be generated to obtain a measure of the in-plane properties. By critical anglell or resonance frequency measurements,12 phase velocity determination, and hence material characterization, seems possible. For a thick layer, a layer with thickness much larger than the wavelength, the critical angle technique13 and the related subsurface waves14 were proposed as altemative solutions which are not only one sided but also thickness independent. Additional wave modes which bring us nearer to a one sided measurement technique for a full elastic material characterization are the surface wave modes. l,15 The main contributions of this paper are brief descriptions of three techniques investigated recently by us based on critical angle and subsurface and surface wave velocity measurements and the demonstration of their combine feasibility for full one sided thickness independent elastic constant determination for the materials with transversely isotropic symmetry.
CRmCAL ANGLE TECHNIQUE
The ultrasonic critical angle technique is a quasi-local one based on an ultrasonic oblique incidence and critical angle measurements. Such angles generally are defined 16 as the angle of incidence of longitudinal waves from a liquid onto the interface with a solid such that an energy flux vector of refracted longitudinal or transverse waves is directed along the interface. These critical angles are determined by the relevant maxima for the "thick" layer (assumed as a semi-space) and by the minima for the thin layer. For a thick orthotropic layer up to three critical angles may be observed, corresponding to the quasilongitudinal and fast or slow quasi transverse waves. For a rolled thick metal plate
88
there also can be deep minima corresponding to the surface Rayleigh wave. Knowing the velocity in the liquid and determining the critical angles one can use Snell's law for phase veloci~ calculation. This also is possible for some anisotropic situations, as we have shown. 4 For orthotropic materials with the plane of symmetry in the interface, as is the case for most rolled or two-dimensional composite materials, the energy flux vector and the wave propagation vector are in the plane of the interface for critical angle incidence. Generally at the critical angle the energy flux vector can be within an off-symmetry plane out of the plane of incidence.
In Fig. 1 the results of measurements for a 12-mm thick graphite-epoxy (GrIEp) unidirectional composite for different angles of deviation between the incident plane and the fiber direction are shown. The measurements have been made using an ultrasonic refractometer. I7 A rotating specimen is fixed at the center of a cylindrical vessel in such a manner that one transducer operating in the pulse-echo mode produces double-reflection. The critical angle can be determined with a resolution of 1'. Because of diffractional effects the signal disappears for incident angles below ±3·. The first critical angle for the plane of incidence coinciding with the fiber direction is less pronounced than for the plane of incidence perpendicular to the fiber. This is understandable because the magnitude of energy flow for refracted longitudinal waves at the first critical angle (subsurface longitudinal waves) is larger in the fiber direction than in the perpendicular one. For the range of the incident angles shown the second maximum corresponding to the faster quasitransverse wave also is visible for 600 angle of deviation (Fig. 1.).
Using the values of phase velocities determined for four angles of deviation (0·, 30·, 60·' and 90·) corresponding to the first critical angles (Table 1) and applying the formulas given in the Appendix of reference 13 four elastic constants (C22=Cll' C33, C44 = Css, and C23 = CI3) were calculated and are given in Table 2. For this case, transversely isotropic symmetry with axis of symmetry x3 in the fiber direction was assumed. The ambiguity in calculating C23 was eliminated by applying one of the positive definite constrains on the elastic constants,I8 which for a transversely isotropic solid is:
(1)
The missing value of the C l2 constant was determined using surface waves (see SURFACE WAVES chapter).
1st/;
Figure 1. Examples of angular characteristics of reflection factor in the range of incidence angle from -50· to +50· for three angles of deviation from the fiber direction.
89
Table 1. Phase velocities of quasi longitudinal waves in rn/s for four angles of deviation (critical angle measurements without parentheses, subsurface measurements in parentheses)
Angle of deviation
Velocities [rn/s]
o·
9425 (9463)
30·
8282 (8279)
60·
5159 (5160)
Table 2. Calculated elastic constants in GPa
12.96 137.22 9.99 6.06
90·
2896 (2885)
A nonlinear least-squares technique also can be used for data fitting. Theoretically, one can utilize the second critical angles corresponding to the faster quasi transverse waves, but even then only the same four elastic constants could be obtained. The slow transverse waves have velocity below or close that in the water used as a liquid in our experimental set up. Therefore, the critical angle corresponding to them does not exist or is not visible in Fig. 1.
SUBSURFACE WA YES
In his famous paper on elastic solids19 Lamb made a distinction between "minor tremor" and "main shock." The first "tremor" contains quasilongitudinal or quasi transversal displacements as plane bulk waves in an unbounded medium and is known in literature as surface skimming bulk waves (SSBW) or shallow bulk acoustic waves (SBA W). The main shock corresponds to the classical Rayleigh waves (SAW). Such a situation occurs under general conditions of normal and shear loads applied to the surface of the half-space. Here the term of subsurface waves is used as a term describing the field of waves excited in a solid half-space by an angle-beam transducer with angle of incidence close to one of the critical angles. If the longitudinal waves are used for subsurface wave generation it can be subsurface longitudinal waves (SSL) at the first critical angle or subsurface transverse waves with vertical polarization (SSSV) at the second critical angle. Using a transverse wave horizontally polarized for generation one can obtain a subsurface transverse wave polarized horizontally (SSSH) at the relevant critical angle.
Subsurface longitudinal waves were extensively investigated, both theoretically and experimentally for an isotropic materials20-21 and recently for anisotropic composite materials. 13 It was found that two waves coexist co-operatively fulfilling the boundary conditions on the free surface: SSL and head waves. A disturbance on a free surface moves with a velocity equal to the velocity of longitudinal waves in the solid. The amplitude of this displacement decreases with depth exponentially. This means that the SSL waves close to the free surface are strongly attenuated. Characteristic for this kind of wave is that the largest amplitude of acoustic pressure occurs at an angle of 10 to 20 degrees from the free surface. Hence, the name chosen for SSL waves is very appropriate. They can be considered for the detection of subsurface defects, especially since SSL waves show a relatively small sensitivity to surface roughness.
90
Even for highly anisotropic materials, such as unidirectional composites, the possibility of generation and reception of SSL waves in the symmetry and off-symmetry planes is observed when the free surface of the specimen coincides with the plane of symmetry. With the transmitting and receiving variable angle probes situated face to face along a line in the chosen direction, one can use the through transmission mode to measure the phase velocity by evaluating the differential arrival time. This was done only after both transmission and reception angles had been adjusted for maximum received amplitude. To enhance the signal to noise ratio, signal averaging procedures were utilized. Arrival time measurements have been made with 1 ns resolution. In Table 1 in the brackets the results of measurements of phase velocity of SSL waves in the four directions with respect to the fiber direction are given.
The results obtained can be used again for elastic constant determination, or indirectly for other purposes. For instance, one can carry out SSL wave velocity measurements in the fiber and in the normal to the fiber directions for simultaneous evaluation of fiber fraction (FF) and porosity of composite (PC). The diagram given in Fig. 2 was obtained through calculations which have been done for the elastic material properties and densities of the constituents (matrix - epoxy resin, fiber - graphite) used in reference 1. The effects of the void and fiber content on the material properties of the composite were based on Rashin's results and Smith's expressions for the elastic constants of unidirectionally reinforced composite with transversely isotropic fibers.
The contact one sided approach used needs some comments. First of all, the arrival times are measured in the phase direction. Secondly, we assume that the shape of the transmitted pulse is the same at both positions of the receiving probe. It can be proved that this assumption is valid for lossless materials, which means that the form of the wave packet remains constant in the phase direction. Composites with a very small fiber diameter or rolled steel plates with small grains for a narrow band pulse and low frequency can be considered as nondispersive media.
The attenuation of SSSL waves and the skew angles for these waves propagating in off-symmetry directions also were measured. The attenuation of SSSL waves is very substantial, as expected. In the fiber direction a distance of 50 mm is related to more than 10 dB amplitude decrease, across the fibers the losses at the same distance are almost 30 dB. This is a confirmation of our observations from the CRITICAL ANGLE TECHNIQUE chapter.
The measured values of the skew angles are of the same sign for every angle of phase propagation for unidirectionally reinforced composite, reaching the value more than 50· with respect to the fiber direction. For the cross-ply (0-90) composite the distribution of the skew angle is very interesting, having two extreme values between 20-30 and 70-80 degrees from the one chosen symmetry axis direction. It reaches the values of about 30·-35" with opposite signs.
Utilization of SSSV or SSSR waves in the contact technique for polymeric fibrous composite materials is difficult if not impossible, because of their low values of both fast
Figure 2.
c o =fl .,
3100.-------(6-0-.0-) ---:(:::"70:-.0"'"")--'
:5 2700
o .... 2300 C E o c (50.3) 1900~~~~~~~~~~
8500 9000 9500 1 0000 10500 fiber direction
Changes in velocities of SSL waves propagating in parallel and perpendicular directions with respect to fibers caused by changes in porosity (PC) and fiber fraction (FF) for unidirectional graphite-epoxy composite.
91
Table 3. Orientation distribution coefficients detennined ultrasonically with SSSH waves
0.001183 -0.0001792 0.0001802
and slow transverse (quasitransverse) wave velocities compared with the characteristic velocity for a wedge material used for an angle-beam probe.
Applying the SSSV waves for metals one should be conscious of the problem of separating a weak signal corresponding to SSSV waves from a strong one characteristic of Rayleigh waves. These surface waves can be generated even with a wedge angle of exactly the same value as the second critical angle. The same problem occurs in the case of line source generation of surface waves (see SURFACE WAVES chapter) propagating in the fiber direction along the free surface of unidirectional composites. The difference between SV velocity in this direction and the relevant surface wave velocity does not exceed 20 m/s.
As an example of material characterization by utilizing a subsurface transverse wave polarized horizontally (SSSH), an ultrasonic detennination of texture in a rolled steel plate has been made. Rolled plates typically exhibit slightly orthotropic symmetry (a < 3%) described by nine independent elastic constants Cij by three orientation distribution coefficients (ODC) W 4mo (m = 0,2,4) or by a set of elastic constants characteristic of a cubic metal (Cll,C12 and C44), averaged using one of the Voigt-Reuss-Hill procedures. To evaluate these three ODCs in a rolled steel plate 20 mm thick we have made use of the longitudinal and transverse waves polarized along and nonnal to the rolling direction (RD) propagating through the thickness and SSSH waves propagating along the surface in RD direction and in 45" direction. SSSH waves were generated and received using a special set of probeheads previously described.22 Using five measured velocities and applying published fonnulas,23 the three ODCs were obtained and are given in Table 3. The Lame constants and the anisotropy factor needed for the calculation were computed using Hill averaging procedure for a-Fe single-crystal constants given elsewhere.
To verify results obtained the angular distribution of the velocities of SSL and SSSV waves on the free surface were measured. The results are shown in Fig. 3 together with the results of computation using the previously evaluated set of ODCs. The agreement of relative velocity changes for all three waves obtained both experimentally and numerically is excellent. The differences in absolute values are mainly because of arbitrarily assumed values of density and possibly because of inaccuracy of the chosen Hill's averaging procedure.
SURFACE WAVES
For anisotropic material characterization of thick layers with the one-sided access technique, a thorough understanding of surface wave propagation is required. Surface waves are another type of guided waves discussed in the present paper.
Waves on the plane surface of a crystal were investigated very extensively many years ago.24 It is well known, that for the case in which the wave front and the surface of the crystal are parallel to the principal axis of elastic symmetry, the defonnation constitutes a plane strain state. There is no component of displacement nonnal to the sagittal plane (the plane perpendicular to the wave front and the free surface). Since discovery of Rayleigh waves, surface wave propagation has been analyzed theoretically for propagation in the sagittal plane for almost all crystal classes.25 When the sagittal plane is not a plane of elastic symmetry, however, waves of plane strain are not possible. Since the displacements have to satisfy three coupled equations of motion and three boundary conditions, the solution to the problem becomes more difficult. Therefore many solutions are based on the reduction of the situation to a case where Rayleigh (surface) waves are
92
5960
'Vi' 5950 ....... ! 5940
'"" "0 5930
!' '0 0 ';j > 5910
59000
3270
...... 3265 ~
........ Exceriment ! 3260
-- CIl culation &i "0 3255 I>.. ~ 3250 0 ,g
...... ~riment " > 3245 -- C culation
10 20 30 40 50 60 70 eo 90 32400 10 20 30 40 50 60 70 eo 90 Angle [deg] Angle [deg]
3270 r----------,
~ 3260 e
';;' 3250 fIl
"0 3240
t-'0 3230 ,g
~ 3220
32100 10203040 50 60 70 80 90 Angle [deg]
Figure 3. Velocities of subsurface: (a) longitudinal, (b) SV and (c) SH waves versus direction of propagation for a steel plate with weak texture.
polarized in the sagittal plane by transformation of the constants qjkl from the crystallographic coordinate system to a specially chosen Cartesian one.
The most frequently used approach to the problem is a so-called three dimensional approach26 which considers the wave equation and the boundary condition as two separate algebraic conditions. For successively chosen values of phase velocity in the secular equation we are looking for admissible decay constants and amplitude vectors until one is found for which tfe boundary condition determinant vanishes. This technique was used in our earlier work for numerical analysis of the surface wave for off-axis propagation in both a unidirectional and a more general orthotropic thick composite. Some of the conclusions are presented below. Another recently utilized six-dimensional formalism which combines the equation of motion and the stress-free boundary condition into an eigenvalue problem is presented elsewhere.27-28 This formalism was also used to prove existence and uniqueness theorems for surface waves, thereby closing the previous discussion about so-called "forbidden" directions.29 In our recent paper30 we describe the propagation of surface waves on a medium consisting of a transversely-isotropic layer in rigid contact with a transversely isotropic half space substratum of a different material. The total system is in contact with a fluid space. This work was done as a fIrst step toward modeling a layered anisotropic structure for material characterization with surface waves.
For further analysis in our present paper some of the conclusions from our earlier work! are given here. For a graphite-epoxy unidirectional composite, treated as a transversely isotropic homogeneous medium, the wave velocity for surface waves propagating along the off-axis direction on the free surface (xl=O) were calculated. Unlike the fast transverse waves, the velocities of surface waves along two perpendicular directions of the free surface (fIber and normal to fIber directions) are not equal. The differences between them, however, are not large and the velocities change smoothly over the entire range of propagation directions. The value of the velocity of the faster surface
93
waves in the fiber direction is very close to the velocity of the vertically polarized transverse wave in the same direction. The changes of the absolute values of the relative amplitudes of the three components of the displacement vector are similar to those for Rayleigh waves in an isotropic medium. There are, however, some differences worth noting. First of all, for surface waves propagating in the fiber direction, a transverse displacement dominates over the longitudinal one to a larger degree than is the case of propagation normal to the fibers. This explains why the value of the surface wave velocity is so close to the value of the transverse wave velocity in the fiber direction. Also, the depth of penetration of the surface wave along this direction is large, reaching a depth of 2-3 wavelengths. For an offaxis direction, the additional third component of the displacement vector appears. The existence of this horizontal displacement confmns the expected skewness of the energy flux vector for the surface wave propagating in an off-axis direction. In general we can conclude that, unlike other waves such as quasilongitudinal, quasi transverse, or even plate modes in a transversely isotropic strongly anisotropic medium, the skew angles in the surface waves are relatively small. They do not exceed 10· compared to almost 80· for some cases of bulk or plate waves. Since the differences between the phase and group velocity of the surface waves are smaller than 1 %, we can consider the measurements of the surface wave velocities along any direction in a meridian plane as an approximate phase velocity measurement.
The most popular NDT method for generation and reception of surface waves is the wedge transducer which utilizes oblique incidence of longitudinal waves at a critical angle from a liquid or a solid. Utilization of this very convenient contact technique, used conventionally for metal inspection, is almost impossible for the generation of surface waves in polymeric composites with surface wave velocities smaller than the velocity of longitudinal waves in a perspex wedge. Therefore in our experimental measurements, a modified line source method 1 with a differential arrival time measuring technique was employed for composites. Our measurements can be called quasi-point (local) because of a relatively small measurement base of 10 mm. Such a small base was chosen not only because of large attenuation (0.335 dB/mm in the fiber direction and 0.354 dB/mm in the normal to fiber direction), but also to avoid difficulties with the interpretation of the received signals due to the finite thickness of the composite layer and the possible occurrence of reflected waves (weak longitudinal or transverse waves) from the opposite side. The measuring technique employed with arrival time resolution of 1 ns, enabled us to monitor the changes of surface wave velocity in graphite/epoxy composite of 1 m/s even for a measurement base of only 10 mm ..
To evaluate the accuracy of surface wave velocity measurements needed for composite material characterization, a simulation modelling the changes of fiber fraction and porosity of unidirectional graphite/epoxy composite was computed (Fig. 4). The material property data were taken from Fig. 2. The results presented also suggest a practical application: surface wave velocity measurements along the fiber direction and normal to it.
As was mentioned in the INTRODUCTION and BACKGROUND chapters,one of the purposes of employing surface waves for anisotropic material characterization is to develop a thickness-indeperident elastic constant determination technique. The surface wave velocity measurements can be used for calculation of the off-diagonal terms of the stiffness matrix. In our earlier work} formulas are given for C12 and C13 computation for an orthotropic case based on the assumption that the other seven terms of the stiffness matrix are known from previous ultrasonic measurements. Here, if we do not want to use measurements through the thickness of the composite layer, we have to determine the value of the fifth missing constant C12 through the computation of C66 from the equation as follows
(2)
where
(3)
94
" o
2160c------------,.~~_.
2110
~2060 :0
(FF,PC)
1960~~~~~~~~~~
1580 1630 1680 1730 1780 1830 normol to fiber direction
Figure 4. Effect of changes in porosity (PC) and fiber fraction (FF) on velocities of surface waves propagating along and nonnal to fibers in a unidirectional graphite--epoxy composite ..
(4)
Here C23R is the velocity of the surface wave propagating on the surface xrx3 in the nonnal to the fiber direction (x2). A root of equation (2) was searched for numerically in the range 0<C66<C22/2 to fulfil the positive definite constraints. For unidirectional graphite/epoxy composite samples as used for critical angle and subsurface longitudinal wave velocity tests, the relevant surface wave velocity was measured. The result obtained (C23R=1452 m/s) gave C66 = 3.79 GPa. Because a transversely isotropic symmetry was assumed, the missing fifth effective elastic constant C12 is as follows
C12 = C22-2C66 = 5.37 GPa. (5)
Hence, employing subsurface longitudinal waves whose velocities for four directions were measured together with surface wave velocities, one can obtain a full set of stiffness matrix components for a unidirectional composite without using the thickness of the sample.
The velocity of Rayleigh waves propagating in polycrystalline materials with weak orthotropic symmetry was discussed elsewhere. It was revealed that the ODC coefficient W 420 is proportional to the difference between velocities of surface waves propagating in the rolling (x3) direction and the transverse (x2) one. Measuring these two surface wave velocities on the 20 mm thick steel block, the same as for SSSH measurements, and applying fonnulas23 a value OfW420 = -0.000182 was obtained.
The choice of the proposed ultrasonic modes (subsurface and surface) has a practical meaning in that they are accessible by cutting the inspected material on only one side and they do not use the far side, hence, are thickness independent. Such an approach is especially useful when a nonunifonn material property distribution in the thickness direction is to be expected.
ACKNOWLEDGEMENTS
In embryonic fonn, this work was initiated while I was a visitor to the Mechanical Engineering Department at Drexel University in Philadelphia, USA. I wish to acknowledge with thanks the stimulating discussions at that time with my helpful and generous host, Professor J. L. Rose. I would like also to thank A. Brokowski, J. Deputat and J. Szelazek - my colleagues from the Institute of Fundamental Technological Research of the Polish Academy of Sciences, for their instrumental support and many useful conversations.
95
REFERENCES
1. J. L. Rose, A. Pilarski, and Yimei Huang, Surface Wave Utility in Composite Material Characterization, Res Nondestr Evall:247-265 (1990).
2. R. Truell, C. Elbaum, and B. Chick, "Ultrasonic Methods in Solid State Physics", Academic Press, New York (1969), Ch. 2.
3. J. E. Zimmer, and J. R. Cost, Determination of the Elastic Constants of a Unidirectional Fiber Composite Using Ultrasonic Velocity Measurements, J Acoust Soc Am 47:795-803 (1970).
4. M. F. Markham, Measurement of the elastic constants of fiber composites by ultrasonics, Composites 5: 145-149 (1970).
5. R. E. Smith, Ultrasonic Elastic Constants of Carbon Fibers and Their Composites, J Appl Phys. 43:2555-2561 (1972).
6. W. Sachse, Measurement of the Elastic Moduli of Continuous-Filament and Eutectic Composite Materials, J Comp Mat. 8:378-390 (1974).
7. B. Hosten, M. Deschamps, and B. R. Tittmann, Inhomogeneous wave generation and propagation in lossy anisotropic solids. Application to the characterization of viscoelastic composite materials, J Acoust Soc Am 82: 1763-1770 (1987).
8. B. Castagnede, J. Roux, and B. Hosten, Correlation method for normal mode tracking in anisotropic media using an ultrasonic immersion system, Ultrasonics 27:280-287 (1989).
9. R. A. Kline, Wave Propagation in Fiber Reinforced Composites for Oblique Incidence, J Comp Mat. 22:287-303 (1988).
10. J. L. Rose and A. Tverdokhlebov, Ultrasonic testing considerations for metals with mild anisotropy, British Journal ofNDT 31:71-77 (1989).
11. A. Pilarski, J. L. Rose, K. Balasubramaniam, and D. Lecuru, Angular characteristics for layered structures with imperfections, in: "Ultrasonics International 87 Conf. Proc.", Butterworth Scientific Ltd., London (1987), 114-119.
12. D. E. Chimenti, and A. H. Nayfeh, Leaky Lamb waves in fibrous composite laminates, J Appl Phys 58:4533-4536 (1985).
13. A. Brokowski and A. Pilarski, Critical angle technique for composite material characterization, in: Proc. XV National NDT Conf, Jadwisin (1986), 55-60.
14. A. Pilarski and J. L. Rose, Utility of subsurface longitudinal waves in composite material characterization, Ultrasonics 27:226-233 (1989).
15. V. Bucur and F. Rocaboy, Surface wave propagation in wood: prospective method for the determination of wood off-diagonal terms of stiffness matrix, Ultrasonics 26:344-347 (1988).
16. E. G. Henneke and G. L Jones, Critical angle for reflection at a liquid-solid interface in single crystals, J Acoust Soc Am 59:204-205 (1976).
17. A Brokowski and J. Deputat, Experience in Applying the Critical Angle Reflectivity in NDT, in: Proc. 9th WCNDT, Melbourne 1979, 4-EDD1.
18. F. I. Fedorov, "Theory of Elastic Waves in Crystals", Plenum Press, New York (1968).
19. H. Lamb, On the propagation of tremors over the surface of an elastic solids, Phil Trans Royal Soc, London A203, 1 (1904).
20. L. V. Basatskaya and I. N. Ermolov, Theoretical study of ultrasonic longitudinal subsurface waves in solid media, Defektoskopiya 7:58-65 (1981).
21. L. A. Nikiforov and A. V. Kharitonov, Parameters of longitudinal subsurface waves excited by angle-beam transducers, Defektoskopiya 6:80-85 (1980).
22. J. Szelazek, Ultrasonic probeheads for measurements of SSSH waves velocities in steel product, in: Proc. 12th WCNDT, Elsevier, Amsterdam (1989), 977-979.
23. M. Hirao, K. Aoki, and H. Fukuoka, Texture of Polycrystalline Metals Characterized by Ultrasonic Velocity Measurements, J Acoust Soc Am 81:1434-1440 (1987).
24. H. Deresiewicz and R. D. Midlin, Waves on the Surface of a Crystal, J Appl Phys 28:669-671 (1957).
25. D. Royer and E. Dieulesaint, Rayleigh wave velocity and displacement in orthotropic, tetragonal, hexagonal, and cubic crystals, J Acoust Soc Am 76:1438-1444 (1984).
26. G. W. Farnell, Properties of Elastic Surface Waves, in: "Physical Acoustics", W. P. Mason and R. N. Thurston, ed., Academic Press, New York 1970, 6:109-166.
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27. G. T. Mase, Rayleigh wave speeds in transversely isotropic materials, J Acoust Soc Am 81:1441-1446 (1987).
28. P. Chadwick, Wave propagation in transversely isotropic elastic media. II. Surface waves, Proc Roy Soc Lond A 422:67-101 (1989).
29. D. M. Barnett and J. Lothe, Consideration of the existence of surface (Rayleigh) wave solutions in anisotropic elastic crystals. J Phys F 4:671-686 (1974).
30. J. L. Rose, A. Nayfeh, and A. Pilarski, Surface Waves for Material Characterization, J Appl Mech 57:7-11 (1990).
97
COMPLEX HARMONIC PLANE WA YES
B. Poirce
DRET/SDR/G63 26 bd Victor 00460 Annces France
1 INTRODUCTION
Complex harmonic plane waves, which are characterized by a complex wave-vector and a complex frequency, may be divided into homogeneous plane waves (having parallel propagation and attenuation vectors) and nonhomogeneous or inhomogeneous plane waves. The last ones may be evanescent plane waves (having perpendicular propagation and attenuation vectors) or heterogeneous plane waves (the propagation and attenuation vectors being neither parallel nor perpendicular). All these waves may be either permanent ones or transient ones.
Complex harmonic plane waves only exist locally both in space and time, but there are indeed many reasons to study them. Here are the main ones: (1) complex harmonic plane waves are the most simple solutions of the propagation equations for nonlimited homogeneous media; (2) the plane wave decomposition of an acoustic beam describes complex harmonic plane waves as well as classical ones l -9; and (3) interface waves, which exist in the vicinity of a plane interface, are locally a linear combination of complex harmonic plane waves3,1O-25.
The most studied complex harmonic plane waves in the literature are the permanent ones. Evanescent plane waves have been studied both in optics and electromagnetics26-37 (the first ~arers are from Fresnel35, Cauchy (1836), Green28 (1838» as well as in acoustics. ' 7,38-41 Heterogeneous plane waves have been analyzed in optics and electromagnetics ~cf. references in Ref. 28), in acoustics42-50, in viscoelasticity and in geophysics51-57,7 , and in thermoviscoelasticity.58 But only a few studies are concerned with transient harmonic planes waves.52, 59, 60
Following is the organization of the present paper. A qualitative analysis of complex harmonic plane waves is presented in Section 2. A few consequences of the fact that the wave-vector and the frequency are complex valued are discussed. A classification of the different harmonic plane waves is proposed. In the literature, one may find different names for each of these waves, so we have tried to choose what we thought the best name for them. Only a small amount of information about polarization and about the consequences of the complex amplitude vector is give in this section. One will find a very good study of these in Ref. 38. Section 3 is a quantitative analysis of complex harmonic plane waves in thermoviscous fluids and in inviscid fluids. The different velocities associated with harmonic plane waves are calculated in Section 4: phase, amplitude, group and energy velocities. It is shown that these velocities are usually different for this kind of wave.
Complex quantities z wiIl be denoted z' + iz", with z' and z" real: the 1\ sign designates a unit real or complex vector.
Physical Acoustics, Edited by 0, Leroy and M.A. Breazeale Plenum Press, New York, 1991 99
2 ANATOMY OF COMPLEX HARMONIC PLANE WA YES
In this section, complex hannonic plane waves are qualitatively described and some of their properties analyzed. In particular, the consequences of the fact that the wave-vector and the frequency are complex valued are inves~gated. Complex hannonic plane waves are defined from their characteristics: amplitude W, wave-vector k and frequency ro, all of them being complex constants.
First, k and ro are considered as elements of C3and C, respectively; then, in order to get an easier geometrical interpretation, they are considered as elements of R3EtlR3 and REtlR.
2.1 Definition of Complex Hannonic Plane Waves
Any instantaneous real vectorial field w associated with complex harmonic plane waves may be written as:
~ ~, t) = Re { W, exp i (k . ~ - ro t)}
= Re W = ~ ~, t; W, k, ro). (2.1)
, The complex constants W, k, and ro are respectively: (1) the complex amplitude vector
at the origin ~ = 0, t = 0; (2) the complex wave-vector or complex spatial frequency; (3) the complex pulsation or complex angular frequency. Re stands for "real part of;" k . ~ - rot is the complex phase; and W is the instantaneous complex vectorial,field. Complex hannonic plane waves are fully determined by the two complex vectors Wand k, and by the complex scalar roo
The physical interpretation of complex hannonic plane waves is not easy when one uses the complex entities k and roo It is similar to the interpretation of complex rays61-66 in which space and time variables are complex quantities. Actually, as the velocity d~dt has the same dimension as klro, ~ and t have to be complex when k and ro are. This is the reason we shall consider from now on k and ro as elements of R3$R3 and REtlR, respectively. The derived velocities will then be real quantities.
2.2 General Description of Complex Hannonic Plane Waves
In this section, we introduce the complex polarization vector and we decompose the k vector and the frequency ro into their real and imaginary parts. Since y is a complex vector, we define the unit vector'y that satisfies.Y . Y..* = 1, or
(2.2)
II y II ~ 0 is the modulus of the vector y and is defend as II y 112 = Y . y* (where y* is the conjugate vector of y and the dot indicates the scalar product in C3).
Relation (2.2) may be applied to the complex amplitude vector VI • 0 /;'\
~=IIWIIW; (2.3)
IIWII is the modulus of the complex ampl~de vector VI at the space-and-time origin or the real scalar amplitude at the origin, while W IS the polarization vector.
Let:
k = k' + i k" and ro = ro' + ro" , (2.4)
100
where k', k", ro' and ro" are real quantities (ro' > 0 in order to restrict our study); k' is the propagation vector of the phase; k" is the attenuation or damping vector, or the propagation vector of the amplitude; ro' is the real classical angular frequency or the permanent state frequency; i.e. the number of oscillations per unit time; ro" is the extinction coefficient (ro" < 0) or the switching on of the source coefficient (ro" > 0); ro" describes the timedependent exponentially transient part of the wave.
Putting (2.3) and (2.4) into (2.1), the instantaneous complex field describing the complex hannonic plane waves becomes:
• A ~ <A, t) = II ~ 1I.w: exp (-k" . ~ + ro" t) exp i (k.' . ~ - ro' t) . (2.5)
II~VIl ~xp (_k,." . !,. + ro "t),iS the real scalar amplitude at point (!,., t); IIWlI Wexp (-k" . ~ + ro"t) = W exp (- k" . ~ + ro"t) is the complex amplitude vector at point <A, t); k'· ~ - ro't is the real phase at point <A, t) (more or less an additive constant).
The equiphase planes, defined by k' . ~ - ro't = constant, are the planes perpendicular to k' at time t fixed, and the equiamplitude planes, defined by IIWII exp (- k" . ~ + ro"t) = constant, are the planes perpendicular to k" at time t fixed (Fig. 1).
2.3 Classification of Complex Harmonic Plane Waves
Depending on the relative positions of vectors k' and k", the complex harmonic plane waves are called: (1) damped (k." "# 0) or undamped (k." = 0) homogeneous harmonic plane waves when k' II k", the classical plane wave corresponding to the case k" = 0, ro" = 0; (2) evanescent harmonic plane waves when k' -L k"; (3) heterogeneous harmonic plane waves when k' and k" are neither parallel nor perpendicular (Fig. 2, Tables 1 and 2).
Evanescent and heterogeneous plane waves are also called inhomogeneous or nonhomogeneous waves (Table 2). Depending on whether ro" = 0 or ro" "# 0, all these waves are specified to be permanent or transient ones. Quantitative details about these waves are given in Section 3.
In order to complete this classification, two different notions of polarization must be introduced. The first one is related to the shape of the extremity of the real vector ~ locus, and the second one to the relative position of this locus, when compared to vectors k' and k". The first one states that a complex harmonic plane waves is elliptically, circularly or linearly polarized, depending on whether the extremity of the real vector ~ describes, at a given point and as time progresses, either an ellipse, a circle, or a line segment.
Fig. 1. Equiphase plane, equiamplitude plane, and exponential variation of the amplitude along the phase propagation direction.
101
(a)
k" .. k' .. (b)
k"t ... _____ k' .... ~ (c)
k' .. Fig. 2. Classification of complex hannonic plane waves into (a) homogeneous plane
waves, (b) evanscent plane waves, and (c) heterogeneous plane waves.
The second notion of polarization ~tates that complex harI?0nic plane waves may be divided into (1) lamellar plane waves O:YL, kL' ro) for which WL x kL = 0, longitudinal homogeneous plane waves ~,kz, ro) being a particular case of them with WI x kL = 0; (2) torsional plane waves (WT, kT' ro) for which WT · kT = 0, transverse homogeneous plane waves <Y::!..t, kt, ro) being a particular case of them with ~ . k t = 0. Lamellar waves are irrotational waves as V x wL = 0, and torsional waves are solenoidal waves as V . ~T = 0.25 A complete discussion of these polarization concepts is given in Reference 38. In the next section important definitions are given for plane waves propagating in a physical space endowed with a coordinate system.
2.4 Complex Hannonic Plane Waves in Different Coordinate Systems
2.4.1 Plane Waves in a General Coordinate System Let Vj' where j = 1,2,3, be the components of any vector .Y. in the coordinate system (R, y, £) = (gl, g2' g3). Then (2.5) may be written as:
W (K, t) = W exp (- k"m Xm +ro"t) exp i (k'n Xn - ro't) , (2.6)
with summation over m, n = 1,2, 3. We define Ik'jl = Ik' . gjl the wave number in the gj direction, or number of oscillations per unit length in the]j direction; Ik"·1 = Ik" . gjl the attenuation or daI?ping in the. Kj directi~n; Aj ~ 2~/lk'j~ the w.av.e length in the gj dir~ction; Ik'FI the attenuatIon or dampmg length m the Kj dIrectIOn (thIS IS the length after whIch the amplitude is either decreased or increased oy a factor of lie). These definitions are important in the experimental study of interface waves along the Xj = ° interface.*
2.4.2 Plane Waves in the Oblique Coordinate System In the oblique coordinate system ~, k"), Eq. (2.5) may be written as:
W(K,t) = W exp (- IIk"II~' . K + ro"t) exp i (II k' II g'. K - ro't), (2.7)
where: II k' II is the wave number (along g') and II k" II is the attenuation or damping (along k .... '); k' . K is the abscissa in the phase propagation direction, and k" . K is the abscissa in the amplitude propagation direction; A = 21[/ II k' II is the wave length (along g' ) and II k" U- l is the attenuating or da~ping length (along ~'); f ~ ro' / 21[ is the p~rmane~t state frequency and T = f-l = 21[/ro' IS the permanent state penod; f'-l = ro"-1 IS the tIme constant, and % is the heterogeneity angle.
2.4.3 Plane Waves in an Orthogonal Coordinate System Related to k' and k" In an orthogonal system ( X, 2, 2) related to vectors k' and lL, as is shown in Fig. 3, 1 k'x 1 = Ill{ II is the wave number; Ik"x 1 is the heterogeneity; i.e., attenuation or damping
*kj, k'j and k"j are respectively the components in the fj direction of the complex wave-vector k, the phase propagation vector k', of the attenuation vector k"; kj and k'j are sometimes referred to as wave numbers.
102
A
X
o~ _ _ .....
Fig. 3. The heterogeneous plane wave in a coordinate system related to !s' and !s".
along the X. phas-x propagation direction; I k"z I is the evanscence; i.e., attenuation or damping along the Z, direction which is nonnal to the phase propagation direction. *
In the ( X, .Y:. 2) system, the complex harmonic plane wave may be written as:
W (A, t) = W exp (-kx" X - kz" Z + (0" t) exp i (kx' X - (O't). (2.8)
Fi ure 4 represents the real amplitude cos (kx' X) exp - (kx" X + kz" Z), with kx' = 1 + k"Z2 ,for the different values of kx" and kZ" given in the Table 1.
3. QUANTITATIVE ANALYSIS OF COMPLEX HARMONIC PLANE W A YES IN A THERMOVISCOUS FLUID
In this section we present the complex harmonic acoustic plane waves in a thermoviscous fluid, and we study a few particular cases.
3.1 Setting up the Equations
The fluid is supposed to be a classical viscous one, heat-conducting, nonlimited, homogeneous, initially at rest, with no external forces and no internal sources. The acoustic displacement d satifies:49,50,67-69
a. d tt d - (C02 + bdt) ~d = Q (3.1)
b. \7xd=Q; \7·d*O,
where Co and b are two real thermodynamic constants; Co has the same dimension as a velocity and b is the dissipation coefficient. The equation governing the acoustic energy is obtained by multiplying Eq. (3.1a) by Po d t d, and by taking into account Eq. (3.1b). This gives:
dd~ Po dtd· dtd + ~ Po co2 (\7 . dP}
(3.2)
* In the optics literature27,28 these quantities Ik'XI, Ik"XI and Ik"ZI are respectively associated with the refractive index, extinction index and the heterogeneity coefficient (in the pennanent case). It must be kept in mind that in the transient case the extinction index is caused by both the thennoviscous dissipation and the transient evolution (what we have called extinction), and that these two quantities are linearly added in the particular case that each one of them is small.
103
k"
Fig. 4. The amplitude of cos (k'X X ) exp - (k"x + k"z Z), where (k'x = ~ 1 + k"z2),for different complex harmonic plane waves given in Table 1.
Table 1. Attenuation vector k" of the complex harmonic plane waves appearing in Fig. 4
~ 0 0.05 1 0.1
0 classical homogeneous damped homogeneous
plane waves plane waves
0.01
0.05 evanescent helerogeneous
0.1 plane waves plane waves
0.5
104
3.2 The Characteristic Equation.
We look for a solution of the differential system, Eq. (3.1), corresponding to a complex harmonic plane wave:
d = Re U2 exp i CK . ~ - m t) } = Re D . (3.3)
We then get the algebraic system:
a. {m2 - C02 (1 - i m b co-2) E . E} 6. = 0
b. kx)2=O; '£·)2#0, (3.4)
where 6., D, 1.. are contained in C3, m is contained in C .
From equations Eq. (3.4b) it can be seen that the acoustic plane waves in thermoviscous fluids are lamellar ones.
In order for the system (3.4) to have a non trivial solution, the characteristic equation:
(3.5)
must be satisfied. This equation is also called the dispersion equation or the fundamental equation. Let K and n = Co K be respectively a rea wave number and a real frequency taken as a reference. With the help of the nondimensional quantities:
OJ = m / n, k = .k. / K = Co k / n,
Eq. (3.5) may be written in the nondimensional form
OJ2 - (l - i£OJ)k . k = 0 , (3.6)
with £ = n b co-2 a positive real constant. The complex Eq. (3.6) gives rise to two real
equationili,2 _ OJ"2 - (1 +£ OJ"X k'2 - k"2) - 2£OJ' .k' . k" = 0
OJ' OJ" - (1 + £OJ") + k' . k" + t £OJ' (k'2 - k"2) = 0 (3.7)
where k'2 = k' . k' and k"2 = k" . k". This algebraic system may be solved by finding either k (if the complex frequency is given: case of transducer radiation), or OJ (a few components of k are given: case of propagation in a wave-guide). In the next section we shall search for a k solution.
3.3 Complex Harmonic Plane Waves in a Thermoviscous Fluid
In order to find k, we use the (X, Z> coordinate system related to k' and k" as shown in Fig. 3 and we suppose OJ', OJ" and k"z to be known. We first look for (k,2 - k"2) and k' . k", then for k'x2 and k"x2. The result is:
±[ {k"z2 + (OJ,2 - OJ"2) _ £OJ" (OJ,2 + OJ"2)}2 (1 + £OJ"f + £2 OJ'2
+ 4 {OJ'OJ" + t £OJ' (OJ,2 + OJ"2 W)]} . ((1 + £OJ"f + £2OJ'2)2
(3.8)
105
In the next two sections, a few particular solutions of Eq. (3.8) are discussed: first in the case of a weakly thermoviscous fluid, then in the case of an inviscid fluid. In each case, the weakly transient waves are seen first, then the permanent ones. For each situation, the heterogeneous waves are described, then the evanescent waves, then the homogeneous waves. We shall require k'x and 0)' to be positive. In Tables 2 to 4 is a classification of complex harmonic plane waves successively into homogeneous plane waves, evanescent waves and heterogeneous waves.
3.4 Complex Harmonic Acoustic Plane Waves in a Weakly Thermoviscous Fluid (10« 1)
3.4.1 Weakly Transient Complex Harmonic Acoustic Plane Waves in a Weakly Thermoviscous Fluid We assume that: . 10« 1,0)" « 1,0)' = 0(1). 'In this case of weakly transient heterogeneous waves, Eqs. (3.8) reduce to:
k'x = ,.,fk"z2 + 0),2 + 0(10, 0)")
(0)'0)" + ~ 10 0)'3) k"x = --'---r=~2,===-,- + 0(10, 0)")
,.,fk"z2 + 0)'2 (3.9)
From the second of Eqs. (3.9) one can see that the heterogeneity of weakly transient waves in a weakly dissipative fluid is due to two contributions which add up algebraically (Fig. 5). The first one is related to the time evolution:
(k"X)evol = 0)' 0)" / ,.,fk"z2 + 0),2 ,
and the second one to the thermoviscous dissipation:
(k" \.. . = ~ 10' 0),3 /,.,fk" 2 + 0)'2 X)dISSlp. 2 z .
There are five particular cases of the solution, Eqs. (3.9):
a. Very weakly dissipative fluid (10« 0)" < < 1). In this case Eqs. (3.9) become
b. For very weakly transient waves (0)" « 10 « 1), Eqs. (3.9) become
k' =.fk" 2+0),2. k" =kO),3/. fk" 2 +0)'2 x'Vz , x 2 'Vz .
(3.10)
(3.11)
c. The weakly transient evanescent plane wave in a dissipative fluid: If the heterogeneity which is due to the time evolution is opposed to the one due to dissipation,
0)'0)" = - ~ 100),3 or 0)" = - -.L 10 0),2 22' (3.12)
then the heterogeneous wave turns out to be evanescent: k"x = 0 (Fig. 6). This occurs when 0)" < 0, that is to say when a source is switched off, and only when the frequencies 0)' and 0)" satisfy Eq. (3.12).
d. With a weakly transient damped harmonic homogeneous plane wave in a weakly thermoviscous fluid (k"z = 0), Eqs. (3.9) become
k'x = 0)' ; k"x = 0)" + i 100),2 . (3.13)
e. The weakly transient undamped (k"x = 0) harmonic homogeneous plane wave in a weakly thermoviscous fluid gives a more restrictive solution. As in the case of transient evanescent waves in a thermoviscous fluid, the undamped homogeneous plane wave exists only when
0)" - 1. eO),2. (3.14) - -2 '
106
Table 2. Classification of the different complex hannonic plane waves
Complex harmonic plane waves Eq. (3.8)
homogeneous plane waves non homogeneous or inhomogeneous plane waves
undamped damped evanescent heterogeneous
k' k" k' kIlt k' ~ k' --=-- .. .. .. .. table 3 table 4
Table 3. Homogeneous hannonic acoustic plane waves in fluids (crossed squares indicate the lack of a solution)
undamped homogeneous plane waves damped homogeneous plane waves
k' k" =0 k" k' k' /lk" .. .. .. classical damped
classical pennanent pennanent
ro" =0 homogeneous homogeneous plane waves plane waves
pennanent state in an inviscid fluid ina weakly Eq. (3.22) thermoviscous fluid
Eq. (3.17)
the IUldamped transient damped homogeneous and
homogeneous and homogeneous
ro" -:to weakly transient (Eq. (3.18» and plane wave
weakly transient weakly transient plane
exponentially in a weakly plane waves
waves (Eq. (3.14» in an inviscid fluid
transient state thermoviscous fluid Eq. (3.21)
in a weakly Eq. (3.13) thennoviscous fluid
propagation inviscid fluid weakly inviscid fluid weakly medium thennoviscous fluid thennoviscous fluid
Table 4. Evanscent and heterogeneous hannonic acoustic plane waves in fluids
ro" =0
pennanent state
ro" -:to exponentially transient state
propagation medium
evanescent plane waves
k'lk" k" tL. ____ .::k:...' ...... ~
pennanent evanescent
plane waves in an inviscid fluid
Eq. (3.20)
inviscid fluid
the evanescent weakly transient
plane wave in a weakly
thermoviscous fluid Eq. (3.9) et (3.12)
weakly thennoviscous fluid
heterogeneous plane waves
heterogeneous weakly transient
plane waves in an inviscid fluid
Eq. (3.19)
inviscid fluid
k' .. permanent
heterogeneous plane waves in a weakly
thermoviscous fluid Eq. (3.15)
heterogeneous weakly transient
plane waves in a weakly
thermoviscous fluid Eq.(3.9)
weakly thermoviscous fluid
107
a
Fig. 5.
A
Z b A
Z
k"
k" A
X A
X
k' (k'~ )evol k'
The k" vector for heterogeneous waves in the case of (a) a source which is switched off ({J)" < 0 and k"x «} £ (J),2) (b) a source which is switched on ({J)" > 0 and k"x ~} £ (J),2). Dashed vectors represent the heterogeneity due to time evolution and to thennoviscous dissipation.
i.e., when the damping due to time evolution is opposed to damping due to thennoviscous dissipation.
3.4.2 Permanent Complex Harmonic Plane Acoustic Waves in a Weakly Thermoviscous Fluid These waves have been studied49,50 A summary of the results follows. One more condition, apart from the hypothesis stated at the beginning of the last section, is (J)" = O. For pennanent heterogeneous plane waves, Eqs. (3.9) become:
k'x = -Jk"Z2 + (J),2 + 0(£)
k"x = k {J)'3 / -Jk"Z2 + (J),2 + 0(£) 2
Two particular cases are as follows:
1. With 0"# k"z « 1, one has Alais's waves:42-45 ,49,50
k' - ,.,' . k" _1 <".,,2 - 1 "k' 2. X - UI, X - 2 CUI - 2 <- X
2. One can have damped pennanent homogeneous plane waves (k"z = 0). Then
a A
Z b A
Z
A
X
k' (k'~ ).,01
(3.15)
(3.16)
A
X
k'
Fig. 6. The transient (a) evanscent plane wave and (b) the undamped homogeneous plane wave in a thennoviscous fluid when (J)" = ~ . (J) ,2
108
(3.17)
These waves are the classical homogeneous plane waves in weakly thennoviscous fluids.
3.4.3 Damped Transient Homogeneous Hannonic Acoustic Plane Waves in a Weakly Thennoviscous Fluid The hypotheses are: CO"::F- 0 ; e « 1 ; k"z = O. From Eqs. (3.8), one gets:
k'x = co' (1 - eco") + 0 (e)
k"x = co" + :!t{CO'2 - CO"2) + O(e) 2
3.5 Complex Harmonic Acoustic Plane Waves in an Inviscid Fluid (e - Q)
(3.18)
3.5.1 Weakly Transient Complex Harmonic Acoustic Plane Waves in an Inviscid Fluid (e - 0: co"«I: co' - 0(1)). Setting e = 0 in Eqs. (3.9) one gets:
k' = '" k"z2 + co'2; k"x = co'co" / '" k"z2 + co'2 . (3.19)
The waves tend toward the pennanent evanescent waves in an inviscid fluid as co" tends toward zero. Then,
k'x = '" k"z2 + co,2; k"x = 0 . (3.20)
If k"z = 0, the solution in Eqs. (3.19) describes weakly transient homogeneous plane waves in an inviscid fluid satisfying:
k'X = co'; k"x = co" . (3.21)
This solution also is valid for every co" associated with transient homogeneous plane waves in an inviscid fluid.
3.5.2 Pennanent Complex Hannonic Acoustic Plane Waves in an Inviscid Fluid (e = 0; co" = Q) These waves have been studied.39,40 It is shown that the only possible permanent plane waves in an inviscid fluid are evanscent waves (Eq. 3.20) and the classical homogeneous plane waves for which k'x = co' ; k"x = 0 ; k"z = 0 .
4 MAIN VELOCITIES.
In this chapter, we give the expressions for different velocities or propagation speeds associated with complex hannonic plane waves, and we give a few results about acoustic waves in fluids.
4.1 Phase Velocities
Phase velocities are defined from the real phase K . A - co't. They are:
1. the phase velocity in the .& direction: N _ co' ,., ~ --,-,.,N , (4.1)
,.,k . N ,., ,., where I k' . Nil is the wave number in the N direction (Fig. 7), and N is a unit real vector.
2. the phase velocity (when N = [0 ): co' ,.,
Ccp = 11)&'11 k' = Ccpk' (4.2)
One has:
Ccp = ~ . k' and.c.cp. ~ = C~
109
As shown in Fig. 7, it is necessary to make a distinction between C~ and C<p because, when one experimentally studies interface waves, it is often the C~ velocity which is measured, as ~ lies along the interface. Depending on whether thed'ispersion equation is expressed in ro or in k, the phase velocities are functions of either K or ro', ro" and k" being constant:
C<p = C<p(k', ro", IIk"lI) or C<p = C<p (ro', ro", Ilk"lI) (4.3)
The evolution of C)p(k'x, ro"=O, 1 k"z I) versus k'x for permanent evanescent plane waves propagating along X in an inviscid fluid for different values of 1 k" z 1 is given as figure 8 in reference 40. Forthese waves, the phase velocity C<p is derived from Eqs. (4.2) and (3.20):
Op (k'x) = ,..jk'x2 - k"Z2 / k'x (4.4)
It can be seen from Eqs. (4.3) that complex harmonic plane waves are usually spatially dispersive waves, so that k<p depends on II k" II. Such waves are also time dispersive ones, so that .c<p depends on ro" (Eq. 3.8). Actually, the very space-and-time structure of such waves inplies their dispersive nature, whether or not the fluid in which they propagate is a dispersive one. For example, permanent evanescent plane waves in an inviscid fluid are spatially dispersive, though the medium is not (Eq. 4.4). Also, damped transient homogenous plane waves in a thermoviscous fluid are time-dispersive, even when the fluid is not dispersive (Eq. 3.18).
For one-dimensional propagation in space, the study of the dispersion curve
ro' vs. k'(ro')
can be made from:
Clk' / Clro' = 1/ Cg
where cg is the group velocity (see section 4.3 below).
4.2 Amplitude Velocities
(4.5)
(4.6)
These are defined from the real amplitude or from k" . X - ro"t. As for the phase velocities, there are:
• -N -
110
Fig. 7. The phase velocities ~ and ~.
equiphase planes
0)"> 0 0)"< 0
• k'
Fig. 8. Amplitude velocity £a for (a) switching on a source (w" > 0) and for (b) switching off (w" < 0) in an inviscid fluid.
1. the amplitude velocity in the & direction: ~
d!=~N -a - -k"· N
where I k" . N I represents the attenuation in the N direction;
2. the amplitude velocity (when.& = k" ): 1'0"" -UI k" k" £a = IIk"lI - = ca - .
• k'
(4.7)
(4.8)
Values of ca cover the range _00 < ca < 00. For switching on a source, 0)" > 0 and ca > 0; for switching off, w" < 0 and ca < o. For an inviscid fluid £a . ~q> = cos2 % ~ 0 (Fig. 8).
The same relations as those shown in Fig. 7 for the phase velocities still hold for the different amplitude velocities mutatis mutandis.
4.3 Group Velocity
Let us consider a wave-packet composed of complex harmonic plane waves which all satisfy the complex dispersion equation J9(w,k) = 0, which all have the same WOO and k" but different 0)' and k', so that all the k' vectors are parallel and of not-too-different lengths (Fig. 9). The group velocity £g of this wave-packet of complex harmonic plane waves is:
£g = dW' / dk.: . (4.9)
It is not necessary to solve the dispersion equation in W to calculate the group velocity. Let us write the two parts of the complex dispersion equation J9 (w,k) = 0 as:
J9' (0)', w", k', k") = 0
and J9"(w', w", k', k") = 0 , (4.10)
where J9' and J9" are the real and imaginary parts of J9. straightforward calculation, one gets:
Then, after a long but
dW' cg = dk'
fdJ9' dJ9" dJ9' dJ9"} fdJ9" dJ9' dJ9' dJ9"} \----- -----
dW" dW' dW' dW" \ dW" d.k' dW" d.k'
(dJ9')2 (dJ9")2 + (dJ9')2 (dJ9")2_ dW' dW" dW" dW'
dJ9' dJ9' dJ9" dJ9" 2------
dW' dW" dO)' dW"
(4.11 )
This result may be applied to the dispersion Eq.(3.6) with the result that Eqs. (4.10) become Eqs. (3.7) and one can deduce the expression for the group velocity of a wavepacket of complex weakly transient harmonic plane waves in a weakly thennoviscous fluid:
111
(ro'k: + ro"k") + 1 £ro,2k" £ = 2 + 0(£ ro") g 2 ' ro'
(4.12)
A few particular cases can be considered:
1. For a weakly transient heterogeneous plane wave in an inviscid fluid £ = 0, and
£g = (ro'k' + ro"k") / ro'2 + O(ro") . (4.13)
2. For permanent evanscent plane waves in an inviscid fluid £ = 0 and ro" = 0, and
£g = k' / ro' . (4.14)
3. For permanent damped homogeneous plane waves in a weakly thermoviscous fluid, ro" = 0 and k" = 0(£), and
'" £g = k' ; lI£gll = 1 (4.15)
4.4 Eneq~y Velocity
The energy mean velocity or energy velocity ~ of a complex harmonic plane wave is defined from the energy equation. We distinguish two cases: the inviscid fluid and the thermoviscous one.
4.4.l The Inviscid Fluid The acoustic energy equation is a conservation equation in the form:
(4.16)
where e stands for the instantaneous density of acoustic energy and ~ for the instantaneous flux of acoustic energy. Values for these quantities may be obtained from Eq. (3.2) with b = O. The energy velocity ~ is then69:
(4.17)
where < > denotes the mean value over a time period. For complex weakly transient harmonic plane waves in inviscid fluids, Eq. (4.17) becomes:
£e= k ·k! + roro*
rok* + ro*k rok' + ro"k" + O(ro") , (4.18)
where the asterisk denotes a complex conjugate (Fig. 10). This result already has been obtained for electromagnetic waves in vacuum.59 For pennanent evanescent plane waves,
.. .. .. k'
~. extremity of the k' vector
Fig. 9. Group velocity £g of a wave packet.
112
Tab
le 5
. T
he m
ain
velo
citi
es f
or c
ompl
ex h
arm
onic
pla
ne w
aves
(no
ndim
ensi
onal
val
ues)
~ W
eakl
y tr
ansc
ient
com
plex
pla
ne
Wea
kly
tran
sien
t com
plex
P
erm
anen
t eva
nsce
nt p
lane
P
erm
anen
t w
aves
w
aves
in w
eakl
y th
erm
ovis
cous
pl
ane
wav
es in
invi
scid
w
ave
in in
visc
id fl
uids
ho
mog
eneo
us p
lane
fl
uids
fl
uids
E
= 0
; c
o"«
l E
= 0
; c
o"«
l w
aves
V
eloc
ity
Pha
se v
eloc
ity
co'
co'
co'
Ccp
= Cc
p =
Ccp
= ro
l 1\
, ...;
co'
2+k"
z2
...; c
o'2+
k"Z2
...;
co'
2+k"
Z2
1 .cc
p =
II t'
II K
= ~k
'x2
-k"
Z2
/ k'
x
Am
plit
ude
velo
city
CO
""'; c
o'2+
k"z2
co
" co
" C
a=
Ca =
k"z
0
Und
eter
min
ed v
alue
kil
, ...
; EC
O'4
co"+
k"Z2
(co'
2+k"
z2)
~=lIt" 1
1-
Gro
up v
eloc
ity
(co'
k' +
co "
k")
+ 1
ECO'
2k"
c =
(co'
k' +
co"
k")
/ co
'2 -
-2
-£g
= 15
.' / c
o' 1
£g =
oco
' / o
k'
C g =
-g
-
--
co'2
Ene
rgy
velo
city
£e
= (
co'k
' + c
o"k"
) / 1
5.'2
To
be c
alcu
late
d £e
= £<
1' 1
£e=<~>/<e>
w
0)"> 0 0)"< 0
• k'
k"
Fig. 10. The velocities ~, ~, !<e in an inviscid fluid.
(4.19)
4.4.2 The Thermoviscous Fluid For a thermoviscous fluid the acoustic energy equation is no longer conservative. From Eq. (3.2), it may be written as:
(4.20)
which may be turned into a conservation equation by setting ~ = o. To evaluate the energy velocity from Eq. (4.20), let
e = e e' and ~ = e ~' ,with
e =exp f ~/e The energy velocity then is:
~ = <~ '> / <e '> .
The explicit calculation of this velocity has yet to be made.
(4.21)
(4.22)
(4.23)
Table 5 presents a summary of the main velocities for complex harmonic plane waves. As a conclusion of this chapter, let us insist on the fact that all the velocities we defined are usually different (Fig. 10).
CONCLUSION
A systematic study of complex plane waves has been presented in the general case (analyzed qualitatively), and in the particular case of inviscid and thermoviscous fluids (These were analyzed quantitatively.) This study may be extended to other geometries (cylindrical, spherical, or any other),1°-72 other media (elastic or viscoelastic solids, dielectrics, ... ), other modes than the acoustic mode in thermoviscouss fluids (en tropic and rotational modes).46, 47
Due to the very many possible ~plications of complex harmonic plane waves, we shall conclude as did O. de Beauregard: complex harmonic plane waves are a treasure.
ACKNOWLEDGEMENTS
We wish to thank B. Duchene and Th. Godefroid for having done the figures.
114
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43. Alais, P., and Hennion, P.-Y., Interaction non lineaire parametrique de deux modes acoustiques d'ondes planes attenuees inhomogenes, C. R. Acad. Sc. Paris, 282 B:385-387 (1976).
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media, Wave Motion, 12:143-159 (1990).
117
USE OF SHORT PULSES AND ULTRASONIC SPECTROSCOPY IN SCATTERING
STUDIES
Gerard J. Quentin
G.P.S., Tour 23 Universite Paris 7 2 place Jussieu 75251 Paris Cedex 05 France
INTRODUCTION
In the early days of Ultrasonics, most transducers were piezoelectric moncrystals. The main advantage of most of these transducers is that they can be used to make very high quality resonators which means narrow band devices.
In order to carry out precise measurements of time delay or sound velocity, the pulseecho method was initiated more than fifty years ago. It used "pulses" generally consisting of a train of sinusoids. In the beginning of the sixties, O. Gericke l invented the so-called Ultrasonic Spectroscopy and he explained that a short ultrasonic pulse is very similar to white light and that it would be possible to see the ultrasonic "color" of objects. Nevertheless, at this time the state of the art was such that wide-band transducers, wideband amplifiers and fast frequency analyzers did not exist or were very expensive in the high frequency range. A real development of ultrasonic spectroscopy began onl,?' at the beginning of the seventies2 and the use of this technique has continuously grown.3,
It is now quite easy to use a short pulse generator, a wide-band transducer, good wide-band amplifiers, and either an analog spectrum analyzer or a Fast Fourier Transform computer program. These now conventional techniques do not allow exceeding of a 100% bandwidth because transducers with higher bandwidths deliver very little power. A new technique arose in the last decade: laser generation of ultrasound and laser detection of ultrasonic displacement. For laser generation the upper frequency limit is determined by the laser pulse duration and can be very high. For laser detection the high frequency cutoff depends on the Bragg cell frequency.
After explaining the main advantages and disadvantages of the use of short pulses, we shall present some examples of applications of Ultrasonic Spectroscopy. We devote some of this paper to Resonance Scattering and finally explain why the group velocity concept is, in most cases, more useful that the phase velocity one in such experiments.
ADVANTAGES AND DISADVANTAGES OF SHORT PULSE INSONIFICA TION
Time Domain
The use of very short pulses allows a very good measurement of the time delay of echoes. But in the classical pulse-echo techniques using longer pulses, the pulse superposition technique is more accurate for measurement of phase shift and phase velocity. The main advantage of short pulse insonification seems to be the possibility to eliminate spurious signals due to scattering from objects having nothing to do with the phenomenon under study (limits of the tank for example). With short pulses, it is very
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 119
easy to gate out these signals. For scattering studies it also is very easy to gate the interesting part of the scattered signal for spectroscopic analysis.
FTeCJllency Domain and Time-FreQ.uency Analysis
The ftrst point to notice is that frequency domain analysis of a signal has very little interest if the phenomenon under study exhibits no frequency dependence. In scattering studies, frequency dependence is a common feature whatever the scatterer: voids, inclusions, cracks, interface, plate, roughness, porosity, target in water. Some of the waves involved are dispersive, and frequency dependence always is observed when the size of the scatterer is close to the ultrasonic wavelength.
The main advantage of short pulse insoniftcation is the wide band nature of short pulses. This means that a single signal can be used to study the whole frequency dependence of the process. The widest bands are obtained with noncontact low frequency ultrasonics. They may extend from some kilohertz to 70 Megahertz. In addition such signals are very suitable for time-frequency analysis. Without adding any information to the signal one can obtain very good results for dispersive wave studies.
Fourier Transform and Discrete Fourier Transform
In Ultrasonic Spectroscopy, the features of the object under study are deduced from the analysis of the signal in the frequency domain. The signal is converted from time domain through Fourier analysis. In the beginning of Ultrasonic Spectroscopy, spectrum analysis was mainly performed using spectrum analyzers and analog processing. Such an analysis has the advantage of being able to give a very good accuracy in the frequency domain, but such apparatus is quite expensive and precise spectrum analysis is a rather slow process. Progressively, digital spectrum analysis has become more and more popular with the increasing use of computers. For digital Fourier analysis, the Fast Fourier Transform algorithm is used in most procedures. At the opposite of analog spectrum analysis which gives the power spectrum, discrete Fourier Analysis provides both the real and imaginary parts of the amplitude spectrum. But this procedure involves some dangers of misinterpretation. The Discrete Fourier Transform of a signal is not its Fourier Transform, in general. The only case where they are equivalent is that of a band-limited periodic signal digitized in accordance with the Shannon criterion and analyzed using a window equal to an integer number of periods.S This is never the case in ultrasonic spectroscopy studies and many people forget that the Discrete Fourier Transform implies a periodization in both the time and frequency domain. Not being the case, many deformations of the spectrum with respect to the real Fourier Transform occur (aliasing, leakage, etc.). In addition, in order to obtain good accuracy in the frequency domain, one must use a very large window in the time domain. Let us take a very simple example.
Suppose that the transmitted signal has a bandwidth extending from 2 to 10 MHz and that we want to analrze the resonances of a scatterer insonifted by this signal with a 5 kHz accuracy using a 21 (1024 pts) FFT, is it possible? In order to have a 5 kHz accuracy the width T of the window must be
T = 5 tkz = 0.2 ms = 200 Jls (1)
which means that the samplin:1 frequency fis 210rr -5 MHz and aliasing will occur. The analysis can be made with 21 (4096 pts) FFT, but only if one uses before the digitizer an analog low pass ftlter with a very sharp cut-off at 10 MHz. In addition, for low Q resonances, the scattered signal is not very much longer than the incident one and for Q = 5, it will not be signiftcant for a longer duration than 5 Jls. This means that a significant part of the time signal sent to the digitizer must appear in the ftrst 5 Jls, then during the remaining 195 Jls the signal ideally would be zero. Such a procedure almost never is implemented in the usual FFT programs or FFT hardware.
Enerc;y Considerations
For a given transducer, both the voltage applied and the energy that it can receive must be limited to avoid destruction. If a given electrical energy is applied to a transducer with a
120
given electromechanical coupling in short pulses, the acoustical energy is spread out in a wide frequency range. As a consequence, the amount of energy per unit of frequency is better for narrow band excitation than for wide band, and the signal-to-noise ratio also is better. With laser excitation the situation is even worse than for conventional wide band transducers because the ultrasonic displacement generated is lower than when a classical transducer is used. This bad signal to noise ratio of wide band systems reduces their wide band efficiency.
EXAMPLES OF APPLICATION
Scattering from Bulk and Surface Defects (Voids. Inclusions. Surface Cracks. Porosity. Delaminations, etc.)
Our first examples are concerned with the field of Nondestructive Evaluation and Materials Characterization. Ultrasonic Spectroscopy is specifically useful when the size of the scatterer is of the same order of magnitude as the ultrasonic wavelength. This is the case for many kinds of defects listed below:
voids and inclusions: surface cracks: pores: roughness:
from microns to millimeters from some microns to centimeters from less than one micron to some hundreds of microns from zero to hundreds of microns.
This is to be compared with the wavelength of ultrasound in a typical metal which is between 30 11m at 100 MHz and 6 mm at 1 MHz. Ultrasonic spectroscopy is a very useful means of Nondestructive Evaluation and has been successfully used in this field for twenty years. Our team has been one of the first in Europe to use this technique.
Surface. Interface, and Plate Wave Generation
Ultrasonic Spectroscopy is useful when the physical phenomenon under study is a function of the frequency. This is the case for many surface, interface and plate waves which exhibit a dispersive behavior. The conventional method of generation of such waves is to launch from a fluid or solid wedge an ultrasonic wave onto the interface in order to match, along the interface, the velocity Cw of the wave which is to be generated. This means an incidence at an angle e such that
sin e = £1 Cw (2)
where cl is the phase velocity in the incident medium. If narrow band insonification is used and the velocity of the wave Cw is deduced from the measurement of the angle e, this gives only one point in the dispersion curve (Fig. 1), and only if Cw corresponds to a propagating wave at this frequency. Wide-band insonification at a given angle allows the measurement of the velocities of all waves generated at this angle inside the frequency range of the insonification. This has been used by M. de Billy et. a1.6 for studying dispersion of generalized Lamb waves in plates and multilayered structures.
Another very efficient technique is to use mode conversion at a grating engraved on the interface. With such a method and even at normal incidence A. Jungman has been able to generate waves having a velocity smaller than the sound velocity inside the incident medium'? Such a generation is impossible using conventional techniques.
Scattering from Targets
In Resonance Scattering, even for very simple targets, the number of resonances in a given frequency range may be very large and their width relatively small. The team of Ripoche developed a very clever method of Isolation and Identification of Resonances, the RIIM,8 using narrow band insonification. We applied pulse insonification9 and Ultrasonic Spectroscopy to the study of Scattering from cylinders and spheres. The most recent results have been obtained with pulse insonification and will be addressed later in this paper.
121
o 5 r---,.---- r----r---...----, 17.5°
4. 5 +----.::IJIl---.....JIr---~.__--~,......__l1 9. 5°
~ 4+-----.. ----'t'----~---~22 .00
E
"'" ~ 3. 5 +----->Jr----......::iIIM:::-----......::~25. 4 ° u o ~ 3+------~~~--------_; 30°
QJ VI
~ 2.5 +-----1'-----------------.....-1
2+-+-----------------;
1. 5 ~__r-_r____,-_,_-_r___r-_r__,r____r____l o 2 3 4 5 6 8 9 10
Frequency x Thickness (MHz x mm)
Fig. 1. Measurement of Lamb wave dispersion curves in a stainless-steel plate using using Ultrasonic Spectroscopy: with a 2-10 MHz wide band insonification and angles of incidence 30°, 25.4°, 22°, 19.5" 17.5", it is possible to determine the points marked by (*) on the dispersion curves.
ISOLATION AND IDENTIFICATION OF RESONANCES
One Simple Example
In order to explain the difference between narrow band and wide band techniques and the equivalence of the results obtained, let us choose a very simple example. Picture a lossless, isotropic, homogeneous infinite solid plate (medium 2) in contact with two fluids, 1 (incident) and 3 (transmitted) and an infinite harmonic plane wave at normal incidence. Details of this study are available. 10 The amplitude An of the reflected wave may be written as
(3)
where 13 and "( are defined in reference 10. Two terms appear inside the parentheses: one corresponds to the specularly reflected
wave and the second one is a geometrical series whose sum in the stationary state is
A=A{I+ k ) n ~ 1-"(
Narrow Band Insonification and the RIM
(4)
In Ultrasonics one can use continuous waves or such a signal replaced by a long train of sinusoids. The latter means approximately the product of a continuous harmonic wave by a rectangular gate. The reflected signal is easy to calculate analytically but a more straightforward explanation is based on Fig. 2 where we have plotted the result obtained at resonances and at antiresonances with 13 = -0.1, "( = 0.8 and a duration T of the gate equal to 20 times the duration of a roundtrip inside the plate. At a resonance, the amplitude in the
122
- - - - --~
Forced reg ime Free regime
Fig. 2. Reflected signal from a plate insonified by a train of sinusoids having a duration T.
"forced regime" decreases from the specular values and tends asymptotically toward the stationary value. In the free regime, the plate continues to oscillate and the amplitude decreases from AO py/(l-y) to zero. Both decreases are achieved by steps, each step having a duration equal to that of a round trip inside the plate. For antiresonance, in the forced regime the amplitude oscillates with a slight variation from the specular value, and in the free regime it decreases very rapidly from a very small value to zero.
In order to analyze the frequency behavior of resonances we may choose a time 'tl in the forced regime or a time 't2 in the free regime. For the first choice we obtain the "form function" with minima at each resonance and for the second choice the "resonance spectrum" with maxima at each resonance (Fig. 3a). This method is a very efficient way to isolate resonances.
Wide Band Insonification and the PRIM
We may first choose an infinite bandwidth insonification which means a 8 distribution signal in the time domain. The calculation of the reflected signal from the same plate as before is straightforwardlO and leads to the results displayed in Fig. 3b. The first echo is
a
time 1;
timeT,
Fig. 3.
Form function
!I<
Resonance spectrum I
b
spec.ula r 1-1
~ whole signal
1 AAA~A. I
signal without specular '_1
Spectral analysis of reflected signals from a plate using narrow band insonification (a) or 8 impulse insonification (b).
123
the specular one and the following ones are all in phase opposition with it. Two different frequency analyses may be performed with this signal. One can Fourier Transform the whole signal and obtain the "form function" or one can Fourier Transform the Series of echoes without the specular one and obtain the "resonance spectrum."
In real life, 0 impulses do not exist. The signal always has a finite bandwidth. But a convolution in the time domain means a simple multiplication in the frequency domain and it is very easy to deconvolute (divide) the experimental spectrum using the spectrum of the incident pulse (Fig. 4).
One can conclude from this discussion that both narrow band and wide band insonification lead to rather easy isolation of resonances. What about the identification of resonances in situations much more complicated than plate thickness resonances?
Consider the scattering from a cylindrical target of an ultrasonic beam with an axis perpendicular to the cylinder axis. Resonances are related to surface waves circumnavigating the cylinder, and the resonance condition is simply that the circumference is equal to an integer number n of wavelengths. Another integer number 1 labels the resonances and is related to the nature of the surface wave. It has been shown in Le Havre that it is easy to measure the value of n using a receiving transducer rotating around the target. A polar plot of the amplitude at resonance exhibits 2n maxima over a complete 3600 round trip. They' called the method that they invented the Resonance Isolation and Identification Method.S More recently both this team and our team adapted this technique to pulse insonification and we published simultaneously our first resuIts. ll ,12 We called this new technique the Pulsed Resonance Identification Method and shall explain it briefly now. The experimental set up is shown schematically in Fig. 5. The transmitter 1 and the axisymetrical target are kept steady while the receiver 2 is rotated by steps Ll9 around the axis of the axisymetrical target at a constant distance from it. For each step the signal is recorded and its frequency spectrum stored in the computer controlling the rotation. After carrying out the largest rotation range available, the computer checks the resonances appearing inside the frequency spectrum in backscattering, and for each resonance plots the amplitude versus the angle. From the number 2n of maxima, one can deduce the integer n for the corresponding resonance. Typical polar plots are displayed in Figs. 6 and 7. The first one corresponds to the (6,1) resonance of a copper wire (Fig. 6) and the second one to the (21,2) resonance of an aluminum cylindrical shell.
Fig. 4.
124
a
b
c
• I , I I \
f,equ"ncy domain f
Results obtained using wide band insonification both in the time and in the frequency domain: incident signal (a), whole reflected signal spectral analysis (b), spectral analysis without the specularly reflected signal (c).
WATER
1 TRANSMIT TER
2 RECEIVE R
Fig. 5. Experimental configuration for Resonance Identification.
Fig. 6. Polar plot of the amplitude of the (6, 1) resonance of a cylindrical copper wire (experimental ka = 10.34; theoretical ka = 10.36)
Fig. 7. Polar plot of the amplitude of the (21, 2) resonance of a cylindrical duralurninum shell (Inner radius b = 10 mm; outer radius a = 10040 mm). PRIM technique used.
125
GROUP VELOCITY AND PULSE INSONIFICA TION
With continuous wave or narrow band insonification, the concept of phase velocity is straightforward and very useful. This is no longer true with very short pulse insonification and especially with laser generation of ultrasound. Our first example is very short pulse excitation of a dispersive wave. The expected signal is shown in Fig. 8 for both situations: velocity increasing or decreasing versus frequency. In such experiments, the meaningful concept is group time delay and, consequently, group velocity. If the velocity is an increasing function of frequency (AO Lamb wave in a plate, for example) the high frequency components arrive at the receiver location before the low frequency ones. The reverse is true if the velocity is a decreasing function of frequency. This has already been proved experimentally by Monchalin, et al.
But this is a case where the usefulness of group velocity is related to the nature of the insonification. We shall see later that group velocity has its own meaning in Resonance Scattering, independently of the nature of the insonification.
We have shown in a recent paperl4 that if in the Regge trajectories commonly used in the Resonance Scattering Theory the slope of the line connecting a given point to the origin of coordinates is related to the phase velocity Cp through the very simple relationship
(5)
valid only for a cylinder, then the local slope p is related to the group velocity cg through a similar equation (Fig.9). This relation is valid for cylindrical, spherical, and, probably, most usual kinds of symmetries. This result is very similar to the usual interpretation of phase and group velocities from w(k) dispersion curves. As a consequence, the connection between the width of resonances of a surface wave and the decay y of this wave involves also the group velocity.
We return now to the interpretation of experimental results in terms of wave velocity when pulse insonification is used. From Figs. 10 and 11 one can see that from time domain signals it is difficult to deduce the grouJ' velocity using the time delay between echoes. On the other hand, it is easy to show l that in the frequency spectrum we can correlate the spacing in frequency between two successive resonances of a given wave fif with an approximate group velocity (fiw/fik) which can be very accurate if fik is sufficiently small:
126
21taM = fiw -7 dw - c fik k-->() dk - g
1 - Vg / wi th frequency
Excitation "fl f\ !\ A +_· .. ·· ··· .. -vvVVV \It ~Group time delay --- \J
2 - vg"'with frequency
Excitation (\ ('\ f\" A
~""""" VVV~t Fig. 8. Signal observed after very short pulse excitation of a
dispersive wave and its propagation.
(6)
n = Re( ,,)
o
Regge trajectory_
--t--= 19- 1(C/Cg) M(n 1x1)
Xl = Kl ex
Fig. 9. Phase and group velocity concepts related to Regge trajectories for cylindrical symmetry.
10 20 30 40 50 60 70 1(" .)
Fig. 10. Typical signal scattered from a thin shell immersed in a fluid and filled with air. The specular echo has been omitted and pulse insonification has been used.
c.u.
60 80 100 120 140 k , c
Fig. 11. Typical frequency spectrum of the signal scattered from a thin shell. In the chosed situation, there is only one surface wave circumnavigating the shell.
127
CONCLUSION
The use of pulse insonification in Ultrasonic Scattering Studies is now of major importance and this field is rapidly growing. The advantage of probing a wide frequency range and the rapidity of processing of such signals represents a big advantage to be compared with the usually small disadvantage of a bad signal to noise ratio obtained for very narrow resonances.
ACKNOWLEDGEMENT
Pulse insonification and Ultrasonic Spectroscopy have been used in our team since the beginning of the 70's and I want to acknowledge all my coworkers in this field: M. de Billy, F. Cohen-Tenoudji, A. Jungman, M. Fekih, F. Luppe, I. Molinero, M. Talmant, B. Poiree, A. Hayman, A. Cand, P. Guy, H. Batard, and also people invited to participate in our work: B. Tittman, L. Adler, K. Lewis, and A. Derem. Without them, the progress we have made would not have been possible.
REFERENCES
A complete set of references about Ultrasonic Spectroscopy in Scattering Studies is a whole book in itself. Most references given below are concerned with the work of our team. For more complete references about the early years, the reader may use Ref. 2; for more recent applications in NDE, Ref. 3; and in Resonance Scattering Ref. 4.
1. O. Gericke, "Spectrum and Contour Analysis of Ultrasonic Pulses for Improved Nondestructive Testing - Watertown Arsenal Laboratories Technical Report" WAL TR 830.5/1 (1960).
2. D. W. Fitting and L. Adler, "Ultrasonic Spectral Analysis for Nondestructive Evaluation", Plenum (1981).
3. O. Thomson and D. Chimenti, "Review of Progress in Quantitative Nondestructive Evaluation", Vol. 1-9, Plenum, (1981).
4. N. Gespa, "La Diffusion Acoustique," CEDOCAR, Paris (1987). 5. O. Brigam, "The Fast Fourier Transform," Prentice Hall (1982). 6. M. deBilly, I. Monlinero and G. Quentin, "Etude experimentale de effects non
geometriques lies a 1 interadion d'un faisceau acoustique borne avec un plaque immergee," Acustica 64:40-45 (1987).
7. A. Jungman, L. Adler, and G. Quentin, "Ultrasonic Anomalies in the Spectrum of Waves Diffracted by Periodic Interfaces," J. Appl. Phys. 53:4673-4680 (1982).
8. G. Maze and J. Ripoche, "Study of the Propagation of a Franz Type Wave on Cylinders," Phys. Lett. A 75:216 (1980).
9. P. J. Welton, M. de Billy, A Hayman and G. Quentin, "Backscattering of Short Ultrasonic Pulses by Solid Elastic Cylinders at Large Ka," J. Acoust. Soc. Am 67:670-676 (1986).
10. G. Quentin, Resonance Spectroscopy, in "Ultrasonic Signal Processing", World Scientific Edition (1989).
11. G. Quentin and A. Cand, "Pulsed Resonances Identification Method," Electronics Lett., 25:353-354 (1989).
12. P. Pareige, P. Rembert, J. L. Izlicki, G. Maze, J. Repoche, Phys. Lett. 135A:143-146 (1980).
13. M. Talmant and G. Quentin, "Backscattering of Short Ultrasonic Pulse from Thin Shells, " Journal Appl. Phys. 63:1857-1863 (1988).
14. G. Quentin, B. Poiree, and A. Derem, "Group Velocity and Acoustical Resonances," J. Phys. France 50:1943-1952 (1989).
128
OPTICAL NEARFIELD OF ULTRASONIC LIGHT DIFFRACflON
R. Reibold and P. Kwiek*
Ph ysikalisch -Technische Bundesanstalt, Braunschweig Germany
*Inst. of Experimental Physics University of Gdansk Poland
HISTORICAL SURVEY
Contrary to the widely held view which assumes 1932 as the year of the birth of ultrasonic light diffraction, the origin of this field dates well back into the 19th century.
In 1986, A. Toepler had already observed the propagation of spherical waves from an electrical spark discharge, using a technique which is now well known as the Schlieren method. Ernst Mach improved the experimental technique and in 1887 he published the first Schlieren photographs ever taken showing the shock waves from bullets. From today's point of view, this visualization technique can be considered a special case of the use of ultrasonic light diffraction and that in terms of optical nearfield investigations.
Nevertheless, modem ultrasound research is closely connected with the sinking of the Titanic in 1912 and the outbreak of World War I in 1914. These events initiated a chain of developments in the field of underwater acoustics. The main representative of the period that followed is Paul Langevin, known as the father of ultrasound.
The landmark in acousto-optics was without doubt the discovery of light diffraction by ultrasonic waves by Debye and Sears, and Lucas and Biquard in 1932, and the analytical description of the phenomenon by Raman and Nath in 1935. Subsequently, extensive theoretical and experimental research work was started and is still in progress.
Interest in this field has been chiefly focused on optical farfield investigations; that is, on individual intensities in the Fraunhofer diffraction pattern. A number of authors, especially Mertens and his collaborators, introduced various approximate solutions in order to extend the range of validity of the theory beyond the Raman-Nath regime. Extensive experimental research work was performed by Hiedemann and his group.
Disregarding Toepler's powerful visualizing technique, just one year after the discovery of light diffraction by ultrasonic waves, Hiedemann and his group directed their interest towards the optical nearfield. In 1933 they published a paper dealing with the investigation of sound field characteristics using the optical convergence lines predicted by Lucas and Biquard.
Hiedemann was also the first to point out, in 1935, the periodicity effect in the ultrasonically-produced optical Fresnel region. This effect had been well-known from optical diffraction gratings since Talbot's discovery in 1836 and Lord Rayleigh's theoretical explanation of it in 1881. An analytical description of the corresponding ultrasonic processes was given by N ath in 1936. A discussion of these early findings and their
Physical Acoustics. Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 129
controversial interpretation worth reading can be found in Hiedemann's "Grundlagen und Ergebnisse der Ultraschallforschung" published in 1939.
Investigations into the phenomena of light diffraction by ultrasonic waves carried out in the thirties for the Fresnel region were not pursued so intensively as the investigation of farfield effects. This lack of interest in the Fresnel region can be explained by difficulties which occurred in the experimental work due to the complex phase relations involved. Despite the development of laser technology, little interest was shown in this field and there have been very few papers published in the last fifty years. It was in 1987 that Belgium scientists working with Leroy rediscovered the ultrasonically-produced optical nearfield. Their analytical findings were experimentally verified by Kwiek and Reibold.
PROPERTIES OF THE FRESNEL REGION OF LIGHT DIFFRACfED BY OPTICAL AND ULTRASONIC GRATINGS
In 1836, Talbot} investigated diffraction effects produced by an amplitude grating which was illuminated by a plane light wave. He observed several images of the grating along the direction of light propagation. The experimental results he obtained were analytically explained by Lord Rayleigh2 in 1881. Rayleigh showed that an amplitude grating with the period A and illuminated by parallel monochromatic light of wavelength A produces its own images which are separated by the distance
For A I A « 1 this formula can be expressed as
2 D=2A fA .
(1)
(2)
In the literature various names for these images can be found. In the field of optics they are termed images or self-images, and the planes on which the images are located are Talbot planes or self-imaging planes. The corresponding denotations in acousto-optics are visibility patterns or secondary interferences and Nomoto planes.
Figure 1 shows the self-images of an optical amplitude grating. At half distance between the self-imaging planes exactly the same patterns occur but they are shifted by half the period Al2 in relation to the amplitude grating. Thus in the literature they are frequently referred to as positive (zn+ = N A 2 / A, N even) and negative (zn- = N A 2 / A, N odd) self-images. It should be mentioned that here the planes of maximum visibility (maximum intensity variation), zL\, coincide with the self-imaging planes, which means that zL\ = zn. Figure 1 also clearly reveals that for a limited entrance aperture a nearfield (interference region) and a farfield (spatial separation of diffraction orders) can be distinguished.
130
w w
Fig. 1. Schematic representation of the self-imaging planes and the planes of maximum visibility of an optical amplitude grating.
Fig. 2. Schematic representation of the self-imaging planes and the planes of maximum visibility of an optical phase grating.
In the case of a pure optical phase grating (Fig. 2) it is clear that neither in the positive nor in the negative self-imaging planes is a spatial light intensity distribution to be observed. The planes of maximum visibility, however, lie at half distance between the self-imaging planes at
Z~ = (N + r) .1\:, N = 0, 1, 2, ... (3)
For a combined optical amplitude / phase grating the self-imaging planes are the same as for the individual gratings. The planes of maximum visibility, however, are shifted depending on the amount of phase shift in the phase grating component3.
A p'ropagating ultrasonic wave can be considered a combination of both types of gratings4. The main differences from the optical gratings mentioned are:
1) The grating moves at the velocity of sound. 2) The diffraction orders are Doppler shifted by the ultrasonic frequency. 3) The ultrasonic field behaves like a volume grating.
The early papers dealing with the properties of the optical nearfield produced by ultrasound waves were published by Hiedemann5 , Pisharoty,6 Nath,7 Nomot08,9 and ParathasarthyIO. A comprehensive analytical treatise on the subiect was given in Blomme's Ph.D. thesis II in 1987 and in the paper by Blomme and Leroy. r2
In the case of a harmonic ultrasonic wave the self-imaging planes (Nomoto planes) are separated by
2 21lo A
D= , A (4)
where 110 is the refractive index of the undisturbed medium and A the ultrasonic wavelength. Introducing the new spatial variable Z = (z - L)/D (Fig. 3), the light intensity can be written 11, 12
with
I (x, Z, t) = L e2im2n Z Cm(Z) eim (Kx - n t) m
Cm (Z) = L Em+p(L) E;(L) e4imp7tZ , p
(5a)
(5b)
where Em(L) is the amplitude of the diffracted light in the exit plane z = L. K and Q are the wave number and the angular frequency of ultrasound. Equations (5) clearly reveal that the light intensity is a periodic function of x, Z and t.
131
In the particular case of a pure phase grating (Raman-Nath regime), the light amplitudes can be expressed by Bessel functions. Eqs. (5) then read
00
I ( x, Z, t) = 1 + 2 L Jm(Wm) cos m (Kx - Q t + 1t /2) m= 1
(6a)
with
<Om = 2 v sin (2m1t Z) (6b)
Here, v = kill L is the Raman-N ath parameter, k is the wave number of the light and III the maximum variation of the refractive index in the ultrasonic wave.
It follows from Eqs. (6) that the light intensity in the exit plane is independent of the spatial coordinate x and time t when a pure phase grating is considered. This is also the case when Z = 1/2, 1, ... meaning for multiples of D/2. In the general case (Eqs. (5)), the amplitudes of the diffracted light Em can be found either numerically using the MOA methodll ,12 or analytically using the theory of Kuliasko, Mertens and Leroy. 13
DEFINITION OF RN AND KML REGIMES USING OPTICAL NEARFIELD CONDmONS
Theory of Kuliasko. Mertens and Leroy
Light diffraction by progressive harmonic waves is well described by the Raman-Nath system of difference-differential equations, the solution of which represent the normalized amplitude En of the n-th diffraction order:
2 dEn -En-l + En+l = i n2 pEn, n = ... -2, -1, 0, +1, +2, d~
where
2 and p = -----'A-"----_
2 110111 A
and z is the direction of light propagation.
(7)
The mathematical treatment requires a number of simplifying assumptions concerning the ultrasound field which, however, can be approximately realized by carefully selected experimental conditions. The main restrictions to this are parallel sound beam of width L and constant sound pressure amplitude and phase along the light path through the sound field.
132
x
------... Incident Light
z t------,---~----~-.
Diffracted Light
-.- ---1--'-----·
I I I
L+O
I I I I I I I I I I ____ 1 __ _
L+20 1
Fig. 3. System of co-ordinates including the normalized spatial variable Z.11.12
With the method of generating functions for a collimated monochromatic light beam and normal incidence, the theory of Kuliasko, Mertens and Leroy13 (in the following termed the KML theory) yields the amplitude of the diffracted light wave of order n providedl4 that p3 v8 S; 15:
i p [ 2 v3 ] Eo(v} = Jo(v) + 8 v (10-1 - Jo+l) + 300-2+ 2Jo + Jo+2)
p2 rv3 y4 17 17 -16 L3 (10-1 - Jo+l) + 4 (6 Jo-2+ Jo + 6 Jo+2) (8)
+ ;~ (J0-4 + 4Jo-2 + 6Jo + 4Jo+2 + Jo+4)]
where Jo is the Besssel function of order n, and argument v. The corresponding approach for the Raman-Nath solution (RN solution) reads:
Eo(v} = Jo(v) . (9)
Diffraction Orders O. + 1. -1
Let us consider the amplitude of the diffracted light waves of the order 0, + 1 and -1 in the exit plane Z = O. The technique for verifying this experimentally will be described below. The resulting amplitude then reads:
EO. ±I = Eo + E+I ei (0 t - Kx) + E-l e-i (0 t - Kx) , (10)
where x is the direction of propagation of the ultrasound wave. The corresponding light intensity is obtainedl3 by multiplying Eq. (10) with its complex conjugate and taking into account that E.o = (-1 )°Eo :
10. ±I = (ao2 + ~02) + 2(0.12 + ~12) + 4(o.l~0 - ao~l) sin (Q t - Kx)
- 2(0.12 + ~12) cos 2 (Q t - Kx) .
(11)
0.0,0.1 and ~O, and ~1 represent the real and imaginary parts of the amplitude in the respective diffraction orders (Eq. (8».
If p ~ 0 or v ~ 0; that is. for a pure phase grating, the imaginary parts ~O and ~1 can be neglected and the light intensity for the RN solution then reads:
10.±1 = ao2 + 20.12 - 20.12 cos 2(Q t - Kx)
= J02 + 2J12- 2JI2 cos2(Qt-Kx) .
From Eq. (11) for the Q component, it follows that
dldv) = p v = 21t L}.. for v «I . dv Il<> A2
(12)
(13)
Equation (13) clearly reveals that the slope of the Q component for small v values is a dimensionless quantity which completely characterizes the geometrical properties of the diffraction problem. It depends on the width of the sound field and is proportional to the square of the ultrasonic frequency.
Experimental Investigation
To investigate the light modulation in the exit plane z = L, this plane must be imaged to the observation plane outside the water tank ( Fig. 4). Furthermore, measurement of the
133
time wavefonn and frequency components requires the exit plane to be magnified on the image plane and a photo-diode to be used, the aperture of which is smaller than the wavelength of the ultrasound « NI0).
The focal plane of the lens makes spatial frequency filtering possible. When low pass filtering is used with the 0, + 1 and -1 diffraction order passing the slit aperture, experimental conditions are obtained, the theoretical description of which is given in Eq. (11).
Figure S shows the shapes of the nonnalized light intensity for the dc, the n and the 2n component of Eq. (11) as a function of the Raman-Nath parameter for three frequencies, 2.067 MHz, 4.101 MHz and 6.201 MHz together with the experimental results. It is easy to see that the n component which is responsible for the fonnation of the amplitude grating increases considerably with frequency. This curve increases linearly from its origin and its slope seems to be a suitable criterion for the definition of the RamanNath regime. For f = 2.067 MHz (Fig. Sa) and f = 4.101 MHz (Fig. Sb) there is fairly good agreement between theory and experiment over the entire range studied up to v = S. Furthennore, this figure clearly shows that an amplitude grating component is always present in the sound field, and to an extent that increases with the square of the frequency. The range of validity of the Raman-Nath solution can therefore only be given by defining an upper limit below which the amplitude grating component can be assumed to be negligibly small.
In this investigation, in addition to the frequency components in the exit plane of the light beam, time wavefonns in this plane were also measured and compared with the predictions of the KML theory. As an example, Fig. 6 shows the theoretical and experimental time wavefonns of the light intensity for f = 4.101 MHz and for Raman-Nath parameters in the range from v = o.s to v = 4.9. Disregarding minor discrepancies, theory and experiment are in convincing agreement. The contribution due to the n component and the change in the modulation amplitude due to the decreasing dc component are clearly recognizable.
Definition of RN and KML regimes
For ease of comparison, throughout the following discussion the Q parameter of Klein and Cook15 and Raman-Nath parameter v will be used. It should be noted that the slope of the n component (Eq. (13)) is 21t L ')J( J.I<> A2) = P v = Q, which is a constant for a given experimental system. For f = 2.067 MHz the value of Q is 0.149 and the maximum value of the nonnalized light intensity of the n component turns out to be about O.OS ( see Fig. Sa). If a light intensity of 10 % for the maximum value of the n component is chosen as a criterion for the Raman-Nath regime, the upper limit proves to be Q = 0.2. For higher values of the Q parameter, the Qv product can be used as an additional criterion. In order not to exceed the given 10 % limitation, this product also must not exceed Q v = 0.2.
The upper limits of the Raman-Nath regime defined using the amplitude grating content in the sound field are more restricted than the limits given by Extennann and Wannier,16 Willard 17 and Klein and Cook 15 (see Table 1). The upper value of v = 6, also given by Klein and Cook, is reliable for avoiding thennal fluctuations and non-linear distortions of the ultrasound wave.
134
Light Beam
x
r -- _ I -- --------------
Spatial Filter
Image Plane
Fig. 4. Diagram of the optical arrangement. The exit plane L is imaged on the observation plane with an enlarged scale.
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4
:~ -- ------@
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,[ ----------------------------, @
:~ ____ mnn _____ n_n _______ 0
o 1 2 ] ~
Ul trasonlc Per iod -----c:-
Fig. 6. Theoretical (left) and experimental (right) time waveforms of the light intensity4 for f = 4.101 MHz. a) v = 0.5, b) v = 1.0, c) v = 2.0, d) v = 4.0, e) v = 4.9.
As in the case of the Raman-Nath regime, the limits of validity of the KML theory can be expressed in terms ofQ and v. As already mentioned for f= 2.067 MHz and f= 4.101 MHz (Figs. 5a and b), there is satisfactory agreement between experiment and theory. The corresponding values of the Q parameter are 0.149 and 0.58. For Q = 1.33 (Fig. 5c) the limit of validity of this theory seems to be exceeded. For this Q value the KML theory no longer represents the amplitude grating component with acceptable accuracy. The upper limit of the KML theory therefore is defined as Q = 0.5, Q v = 2.5 and v = 6 . These values are less than the corresponding quantities given by Klein and Hiedemann 18 (in our notation) Q v = 8 and Q = 1.25. The results presented here are given in more detail in the paper by Reibold and Kwiek.4
Table 1. Survey of the Limits of the Raman-Nath Regime Obtained from Literature
Extermann Wannier [16]
Q 2
136
Willard [17] Klein Cook [15]
Q 0 . 5
v 6
Reibold Kwiek [4]
Q 0.2 Q v 0.2
v 6
ULTRASONIC PHASE-AMPLITUDE DIFFRACTION GRATING IN THE OPTICAL NEARFIELD
In the previous section, optical nearfield investigations have been described in which only three diffraction orders were taken into account by spatial frequency filtering. This section deals with the light intensity modulation in the optical nearfield when all diffraction orders (of non-negligible amplitude) are present. The ana9'tical treatment of his problem has been performed by Blommell and Blomme and Leroy I using the method of Mth order approximation (MOA).1l,19
The experimental investigations were performed20 in the same way as already described in connection with Fig. 4 with the only exception that the spatial filter is now absent. Figure 7 shows the theoretical results of the time waveform in the exit plane of the light beam obtained by Blomme and Leroyll,12 for a Raman-Nath parameter v = 1. It can easily be seen that no modulation exists for p = 0, which means that the sound field behaves like a pure phase grating (Raman-Nath regime). For p > 0 and p < 12 a remarkable modulation of the light intensity is obvious with a modulation maximum for p = 4. This light modulation is due to the amplitude-grating content in the sound field4.
Figure 8 shows the measured time waveforms for p = 0.025, 0.25, 0.50, 1.0, 2.0 and 4.0. The Raman-Nath parameter was kept constant, v = 1. The comparison of the numerical and the experimental results reveals good agreement for the sound field parameters studied. The experimental condition for p = 0 could only be approximately verified for v = 1.
In Fig. 9 (top) the predicted time waveforms taken from Blomme's Ph.D. thesisll are presented for the normalized distances Z = 0, Z = 1/8, Z = 2/8 and Z = 3/8, P = 1 and v = 1. The figure shows a pronounced change in the intensity distribution with distance.
The corresponding experimental results are shown in Fig. 9 (bottom) using the same p, v and Z values. In this example the same good agreement between theory and experiment also was obtained.
4 @
4 @
4
3 3 3 @ 2 2 2
1 ~
I 0 0 0 0 2 0 2 0 2
4 4 4
0>- 3 ® 3 (0 3 C8
~
Vl 2 2 2 C OJ
C
0 I 0 0 r: 01 0 2 0 2 0 2 -'
4 :l;u0 4
3 (0 3 CD 2 2 {\ 2
1 \
0 (3 - [ ..J 0 0 2 0 2 0 2
Ultrasonic Period ---t::>
Fig. 7. Theoretical time waveforms of the light intensity for v = 1 in the plane z = L (Z = 0) at increasing values of p according to references 11, 12. P parameter: a) 0, b) 0.25, c) 0.5, d) 1, e) 2, f) 4, g) 8, h) 10, i) 12.
137
SOUND FIELD INVESTIGATIONS USING OPTICAL NEARFIELD TECHNIQUES
In contrast to the assumption on sound field properties used in Sections 3 and 4, quantitative sound field investigations require that some restrictions be made on the field parameters and at the same time some generalizations about acoustic wave propagation. Only a brief outline can be given here; for more details the reader is referred to the original publications. 21,22,23
Let us consider a real progressive ultrasonic wave. The pressure, p, at any arbitrary point in the sound field can be described by
p(x,y,z,t) = p(x,y,z) sin (0 t - Kx + <l>(x,y,z)) . (14)
Under Raman-Nath conditions (see Sections 3,4) , the optical phase retardation (RamanNath parameter) in the plane x = xo is
v(y,t) = k ~:f p(y,z) sin (0 t + <l>(y,z)) dz . (15)
The derivative an / ap is the adiabatic piezooptic coefficient. After some slight modifications, the final expression for the generalized Raman-Nath parameter is obtained:
v(y) exp (i ~ (y)) = k ~: f p(y,z) exp (i <l> (y,z)) dz . (16)
On the left-hand side, the projection data to be measured are given. On the right-hand side, the local sound field parameters to be calculated by means of computer tomography are given.
If only the zero and first positive diffraction order pass the spatial filter (Fig. 10), an expression is obtained which is very similar to Eq. (12):
(17)
4 @
4 @ 3 3
2 2
1 1
f 0 0
0 2 0 2
4 4
.c 3 C0 3 0 -Vi
2 2 c 2 1 .s ~ 0 0 L 01 0 ~
2 0 2
4 (0
4
3 3 CD 2 2
1 1
0 0 0 2 0 2
Ul trasonic Period -------c>
Fig. 8. Experimental time waveforms of the light intensity for v = 1 in the plane z = L (Z = 0). p values as in Fig. 7, only in a) p = 0.025.20
138
t !~, iLJv\ ?: 'Vi a 2 a 2 c: ~ 4
0 4 .s
3 @ - 3 .c::. 2 0\ 2 :::;
I
a a a 2 a 2
Ul t rasonlc Period -<>-
4 @
4 ® 3 3
t 2 2
I I
13 13 ?: 13 2 13 2 'in c: ~ 4 4 .s 0 3 - 3 .c::. 2 0\ 2 :::;
I
a 13
13 2 a 2
Ultrasonic Period -<>-
Fig. 9. Theoretical (top )11 , 12 and experimental (bottom)20 time waveforms of the light intensity for v = 1 and p = 1 at increasing values of the normalized
distance. a) Z = 0, b) Z = 1/8, c) Z = 2/8, d) Z = 3/8.
139
Image plane
Fig. 10. Optical arrangement used for data acquisition of the projection values.
180 0.25
deg
90 0.20
£ '" '" :G " .<:: 0.15 ~
Q. '" Q.
'" 0 £: !5'
~ 0.10 '" "F "'" ffi
-90 iil 0,05 ""
-180 0 -15 - 10 -5 0 mm 15
Position
Fig. 11. Projection of the sound field in terms of the average phase and the Raman-N ath parameter at a distance y = a2/ A from the transducer face.
140
Fig. 12. Cross-sectional pressure profile obtained by means of computer tomography from a set of projection data as shown in Fig. 11.
The projection values in Eq. (16) can be obtained by measuring the time-dependent term of the light intensity. The acquisition of the required data set is performed by shifting the transducer horizontally (y direction in Fig. 10) and turning it around its axis by a small angle after each scan.
Without going into details of the rather complex data acquisition system, only one typical example of application will be given here. A circular transducer a = 5 mm in radius was operated at a frequency of 3 MHz. The data were sampled at a distance of y = a2 / A = 50 mm (nearfield length) with a sampling interval of 0.2 mm and angle increments of 3.60 •
Figure 11 shows one individual projection of the sound field in terms of the averaged phase <I> and the amplitude of the ac term (2Jo(V) J 1 (v) - v) in Eq. (17). A significant variation of both phase and amplitude is obvious. After tomographic back projection, the threedimensional pressure distribution shown in Fig. 12 is obtained. This example demonstrates that optical nearfield investigations are not only of interest for basic research but also for routine work in the field of applied metrology.
REFERENCES
1. Talbot, F., Phil. Mag. 9:401 (1836). 2. Rayleigh, Lord, Phil. Mag. 11:196 (1881). 3. Patorski, K., Parafjanowicz, G., Self-imaging phenomenon of a sinuosidal complex
object, Optica Acta 28:357 (1981). 4. Reibold, R. and Kwiek, P., Optical nearfield investigation into the Raman-Nath and
KML regimes of diffraction by ultrasonic waves, Acustica 70:223 (1990). 5. Riedemann, E., Ultraschall, Ergeb. d. exakt. Naturwiss. 14:201 (1935). 6. Pisharoty, P.R., On the visibility of ultrasonic waves, Ind. Acad. Sci. 4:27 (1936). 7. Nath, N.S., The visibility of ultrasonic waves and its periodic variations, Proc. Ind.
Acad. Sci. 4:262 (1936). 8. Nomoto, 0., Uber eine neue Sichtbarmachungsmethode stehender Ultraschallwellen
in Fliissigkeit~n, Proc. Phys. Math. Soc., Japan 18:402 (1936). 9. Nomoto, 0., Uber eine neue Sichtbarmachungsmethode stehender Ultraschallwellen
in Fliissigkeiten, Proc. Phys. Math, Soc. Japan, 19:337 (1937). 10. Parathasarthy, S., Diffraction of light by ultrasonic waves. Proc. Ind. Acad. Sci.
3A:442, 594 (1936). 11. Blomme, E., Theoretical study of light diffraction by one or more ultrasonic waves in
the MHz region. Ph. D. thesis, Katholieke Universiteit te Leuven 1987. 12. Bloome, E. and Leroy, 0., On the nearfield of light diffracted by a plane periodic
ultrasonic wave, Proc. Fourth Spring School on Acoustooptics and Application, Gdansk 1989. A. Sliwinski, P. Kwiek, A. Markiewicz, eds., World Scientific, Singapore 1990, p.47.
13. Kuliasko, F., Mertens, R. and Leroy, 0., Diffraction of light by supersonic waves: The solution ofthe Raman-Nath equations. Proc. Ind. Acad. Sci., 67A:295 (1968).
14. Leroy, O. and Claeys, J. M., Light diffraction by one ultrasonic wave: Laplace transform method, Wave Motion 6:33 (1984).
15. Klein, W. R. and Cook, B. D., Unified approach to ualtrasonic light diffraction, IEEE Trans. Sonics Ultras. SU-14:123 (1967).
16. Extermann, R. and Wannier, G., Theorie de la diffraction de la lumiere par les ultrasons., Relv. Phys. Acta 9:520 (1936).
17. Willard, G. W., Criteria for normal and abnormal ultrasonic light diffraction effects, J. Acoust. Soc. Amer. 21:101 (1949).
18. Klein, W. R., and Riedemann, E. A., An investigation of light diffracted by wide, high-frequency ultrasonic beams, Physic a 29:981 (1963).
19. Mertens, R., Diffraction of light by two parallel superposed supersonic waves, one being the n-th harmonic of the other: A critical study of the methods leading to approximate solutions in finite form, Proc. Ind. Acad. Sci. 55A:63 (1962); Mertens, R. and Rereman, W., The Raman-Nath equation revisited. Proc. Ultrasonics International '85, Butterworth, Guildford 1985, p.422.
20. Kwiek, P. and Reibold, R., Experimental investigation of the ultrasonic phaseamplitude diffraction grating in the optical nearfield, Acustica 71:69 (1990).
141
21. Reibold, R. and Molkenstruck, W., Light diffraction tomography applied to the investigation of ultrasonic fields, Part I: Continuous waves, Acusta 56:180 (1984).
22. Reibold, R., Light diffraction tomography applied to the investigation of ultrasonic fields" Part II: Standing waves, Acustica 63:283 (1987).
23. Reibold, R., Field reconstruction using tomography and Fourier-optical techniques, in "Third Course on Ultrasonic Signal Processing," A. Alippi, ed., World Scientific, Singapore 1989, p. 45.
142
RESONANCE SCATTERING SPECTROSCOPY BY THE MIIR
Jean Ripoche
Laboratoire d'Acoustique Ultrasonore et d'Electronique LAUE, URA CNRS 1373 Universite du Havre Place Robert Schuman 76610 Le Havre, France
IN1RODUCTION
The "Resonance Scattering Theory," discovered in 1977 and 1978 by Uberall l -3 was illustrated for the first time in 1981 by the researchers of the "LEAH Ultrasons" (now called: LAUE, URA CNRS 1373), Maze et al.4-7 who obtained the first "Resonance Spectrum" of a cylindrical target immersed in water by means of the MIIR (Method of Isolation and Identification of Resonances). This "quasi-harmonic MIIR" gave rise to derived methods: the "short-pulse MIIR" (1986-1989) and, more recently, the "Im/Re MIR" (1989). These methods are applied to different studies (external or internal excitation) of infinite or finite length targets and inclusions in elastic materials.
Before 1981, the acoustic signatures of elastic targets immersed in water were "time signatures" (total response to an incident short pulse) and "form functions" (back scattered pressure versus frequency). The computed far-field form-functions were compared to experimental form-functions obtained by means of the FFT techniqu~.s applied to backscattered time signatures8,9 in the geometry of Fig. 1. At that time, Uberall,I-3 by means of the "Resonance Scattering Theory" (RST), knew how to calculate the modulus of the partial wave scattering amplitude with the rigid background subtracted, but the experimentation didn't give representations different from total form functions because there were no methods giving the spectra of the elastic scattering amplitude.
In 1981, the first "resonance spectrum" was published by Maze et a1.4 and the resonances were identified by experimental determination of the mode number. The method is called RIIM (Resonance Identification and Isolation Method) or, as it is now usually called, MIIR (Method of Isolation and Identification of Resonances).5-7 In my lecture, I describe the MIIR and derived pulse methods. I give results obtained for various (passive and active) targets and inclusions in elastic materials. I shall add a brief description of a new Method of Isolation or Resonance: the Im/Re MIR (this Symposium: P. Rembert et al.; O. Lenoir et al.) and a short presentation of recent results for finite length cylindrical shells.
METHODS
Ouasi-hannonic MIIR
Resonance spectrum Cmonostatic method), A cylindrical target, very long in relation to its diameter, is insonified in the geometry of Fig. 1, by ultrasound bursts made up of
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 143
phlSe fr on ts
WATER
transmitting and receiving i nci dent Wi ve transducer
bur st
M
cy l i nder o r cyl indr icil s hell
Fig. 1. Geometry of the scattering problem.
cycles numerous enough so that a permanent vibration is reached during which the amplitude of the echo may be considered as constant. A single transducer acts as transmitter and receiver. The quasi-steady state amplitude (at point A, Fig. 2) is recorded versus normalized frequency when the frequency of the incident signal varies slowly between one burst and the next (Figs. 3a and 4a: aluminum solid cylinder; air-filled copper cylindrical shell, bla = 0.90). These spectra are compared to the far-field form function given by the classical scattering theory when the target is insonified by an infinite plane acoustic wave: 10-13
F ~(Xl'CP) = (2/V1txd L. En(d~llDn) cosncp (1)
n=O
where Xl '7 kl a is the normalized frequency; kl = co/C I ; a is the outer, b is the inner radius; n is the mode number; C1 is the sound speed in water; and cp is the azimuthal angle. The theory agrees with the experiment. 12,13 The resonances are located at each minimum or anomaly, but, often, their accurate location is difficult in this spectrum consisting of background, peaks, and valleys.
In 1981, Maze et al.4 recorded a "quasi-line spectrum" by measuring the amplitude versus normalized frequency in the transient stage after the end of the excitation (at point B in Fig. 2 which represents a backscattered echo when the excitation frequency is equal to a resonance frequency: the "ringing" of the resonance is very visible). In the "Resonance Spectrum" obtained, a line appears for each resonance of the target. Figures 3b and 4b
A B Y Y
Fig. 2. Backscattered echo at a resonance.
144
. ... '" a. E
""
( iI)
'-JVI'--, ...... _( b )
• ... ,.
•
Fig. 3. Experimental spectra (solid aluminum cylinder). (a) backscattered signal spectrum at point A (steady state); (b) resonance spectrum at point B (second transient state) .
10 20 30
Fig. 4. Experimental spectra normalized by taking the frequency response of the transducers into account (copper cylindrical shell; b/a = 0.90),
(a) backscattered signal spectrum at point A (steady state); (b) resonance spectrum at point B (second transient state).
145
represent the "Resonance Spectra" of two previous cylindrical targets. The RST gives the expression of the resonant contribution:2,1
(2)
where T~(Xl) is the partial resonant amplitude of the vibration mode n; x~ the resonance normalized frequency; and r the resonance width. The experimental results were compared to theoretical studies. They agree perfectly. 14
Resonance identification Cbistatic method). In the "Resonance Scattering Theory" developed by Uberall,2 a relation has been discovered between the circumferential waves and the normal modes of vibration of an immersed target. If we assume that circumferential waves travel around the shell in two opposite directions and form a standing wave at a resonance frequency, it is possible to "see" nodal and antinodal points of displacement on the target. For the observation, we use a bistatic method: a receiver-transducer turns around the shell at a constant distance. At a resonance frequency, the pressure amplitude of the echo at point B is measured, versus azimuthal angle, in the second transient stage during the "ringing" of the resonance. The number of nodes or anti nodes divided by two gives the n mode number (Figs. Sa and Sb give examples of experimental "daisy" patterns). Thus, after the ISOLATION OF RESONANCES in a quasi-line spectrum, the IDENTIFICA TION OF RESONANCES gives the n mode number and allows a distribution of resonances observed in spectra among series indexed by the symbol L (Rayleigh, Scholte, Whispering Gallery waves ... ) in "Regge trajectories." Besides, the knowledge of n for each resonance gives the experimental phase velocity Cs of the circumferential waves (Cs = (k1 a/n) C1)· The group velocity C~ has been related to the separation between successive resonances by Talmant et al. 14 and ~uentin et al. 16
Stroboscopic visualization allowed us to examine, at one and the same time, a part of the steady state and the transient state after the end of the excitation.6,17 It corroborates the previous electronic measurements.
Short-pulse MIIR
Resonance spectrum Cmonostatic method). Several studies have been developed independently by three teams. The pulse method of Dragonette et al. 8,9 gives the total form functions of targets after processing the time signatures by means of an FFT algorithm when the incident signal is a short pulse. Numrich et al. 18 completed this method: only a selected portion of the entire time record of the echo (from cylindrical bodies with hemis&herical endcaps) is transformed, using an FFT algorithm. Independently, de Billy, in February 1986, isolated a part of the time signature from a cylinder by a gate and
146
90 C
D~ _______ ~~ ________ ~
Fig. S. Identification patterns (cylindrical shell) Quasi-Harmonic MIIR (a) n = 3 (b) n = 6
Short-Pulse MIIR (c) n = 3 (d) n = 6.
Sl
S2 Cb)
• ~ A2 Al M
~~
A2
S2
Ce) Cd)
Sl -2:'::~70 ----:5~IOO=-----::8';:;OO:-- N (kHz!
Fig. 6. Experimental spectra (aluminum plate) (a) backscattered spectrum (QH MIIR) (b) resonance spectrum (QH MIIR) (c) resonance spectrum (SP MIIR) (d) resonance spectrum (Im/Re MIR).
used a spectrum analyzer, and our researchers, Delestre et al.,20 in July 1986, using a single incident signal obtained the frequency response of an aluminum plate after temporal filtering and processing of a part of the echo by means of an FFf algorithm (Fig. 6c). This frequency response is to compare to the resonance spectrum (Fig. 6b) of the quasiharmonic MIIR. They are obtained by processing the elastic part ("2") of the time signature after subtracting the specular echo ("I") (Fig. 7). When part "2" is translated along the time axis, resonances have a higher relative amplitude in the spectra, as in the resonance spectra obtained by the quasi-harmonic MIIR when point B is translated on the time axis, according to the coefficient of reradiation in water of waves which generate the resonances. We quote Deprez et al.21 who studied impulse responses from targets in air by pulse methods.
Resonance Identification (bistatic method). In 1986, I announced22 the identification by the short-pulse method and Pareige et al.,23 in February 1989, published the first identification patterns of pure resonances when the target is insonified by short pulses. In the geometry of Fig. I, a receiver-transducer is rotated around the target as in the quasiharmonic MIIR. The time-signatures are processed every fifth degree. The resonance spectra are stored and it is possible to extract the "daisy" patterns, for example, for n = 3, 6 in Figs. 5c and 5d. Independently, Quentin et al.24 also obtained, in March 1989, identification diagrams by means of the short-pulse method.
ImlReMIR.
The Method of Isolation of Resonances called Im/Re MIR is a pulse method; it allows detection of the resonances of targets using the entire backscattered signal, including the rigid echo and the elastic echos from a target, by means of the ratio of the FFf imaginary
147
2
I' rrr '
.. o 20 t (PO s ) ....
Fig. 7. Time signature of a target. (1) specular response (2) elastic response.
part spectrum and the FFT real part spectrum.25 This method is described by Rembert et al. and used by Lenoir et al. (this symposium). I give only an example for an aluminum plate and I compare four spectra in Fig. 6: (a) backscattered signal spectrum; (b) resonance spectrum by the quasi-harmonic MIIR; (c) resonance spectrum by the short-pulse MIIR; (d) ImlRe spectrum by the ImlRe MIR. The method has been tested on plane structures, then on cylindrical targets. Rembert will say that (this symposium): (a) the ImlRe MIR doesn't require the isolation of the elastic response of the target, (b) the ImlRe spectrum seems to contain as much information as both backscattered and resonance spectra, (c) the ImlRe MIR is efficient when a metal plate is coated with a very absorptive material; when the absorptive material reduces the amplitude of the first echo by about 15 dB, for instance, the resonance spectra obtained by the previous MIIR don't clearly allow detection of the resonance peaks; however, the ImlRe MIR is very efficient for detection of the resonances of an aluminum plate behind the absorptive material.
INDUCED STUDIES
Guided waves along the axes (helical waves)
The MIR gives the mode number n necessary to graph the experimental "Regge trajectories," grouping the resonances into a series as in Fig. 8 for an air-filled aluminum cylindrical shell b/a = 0.90 with three series: L = 0; L = 2; P = 1. Some resonances which have a weak amplitude when the insonification is normal to the axis appear very clearly when the incidence angle is equal to a few degrees. The calculation of eigen frequencies of an infinite cylinder or cylindrical she1l26,27 on the basis of studies of guided waves shows that resonances are due to guided waves with a direction of propagation parallel to the axis of the target. Using a second type of bistatic observation, with the geometry of Fig. 9a, the resonances of guided waves are observed (Fig. 9b) and their identification by the MIIR permits one to assign them to the family p = 1 (Fig. 8b). (These waves are generated because the transmitting-transducer is not perfectly directional and a part of the incident energy is exciting the shell with an oblique incidence =3°) (Cf. recent studies by Li et al.,28 and Bao et al.29).
148
. ... ;! Go
E ..:
•
10
• &
4
2
o o 5 10 15 2D
Fig. 8. Regge trajectories (air filled cylindrical shell; b/a = 0.90) (a) whispering gallery wave resonances
(b) guided wave resonances (c) Scholte wave resonances.
(l .11 ~~ _ ~_-_::i'. II)
\4 .11
12.11 (5 .11
I~I
(i .11
D 5 10 IS
Fig. 9. Guided waves (air-filled cylindrical shell; b/a = 0.90) (a) geometry for the study and the identication of guided wave resonances;
(b) resonance spectrum observed by transducer R.
149
Scholte txpe waves
In Fig. 8, we see several Regge curves: (a) the 1. = 2 series of resonances of circumferential Whispering Gallery waves, (b) the p = 1 series due to guided waves and (c) a third series ofresonances identified for the first time in 1986 by Izbicki et al.30 and having high experimental n mode numbers. They are assigned to interface waves: Scholte waves. Theoretical studies related to those waves have been developed especially by Breitenbach et al,31 Rousselot,32 and Veksler.33 With Gerard and Rousselot, we have examined,34 theoretically and experimentally, the resonances of these interface waves for various air-filled shells (Fig. 10) with 0.67 < b/a < 0.99. The determination of mode number, phase velocities, line-width, and observation windows in the frequency domain show that these resonances are due to waves of the 1. = 0 family which behave as the antisymmetrical Scholte waves As. These results have been supplemented by the measurements of group velocities by Talmant et al. 35
Liquid-filled cylindrical shells
The resonance spectra of liquid-filled cylindrical shells can be interpreted after identification of resonances by the MITR. 36 Many resonances appear. They are plotted in Regge curves in which we also notice the Scholte wave resonances. It was possible to examine the influence of the normal modes of the liquid column in light of theoretical studies.
Liquid cylindrical inclusions
Pareige et al.37 have examined the scattering from liquid cylindrical inclusions in an aluminum block. Only the short-pulse MIIR is used and the identification diagrams show that modes n = 0 and n = 1 occur.
Active tar&ets
The internal excitation of air-filled elastic shells immersed in water generates a radiation of elastic waves in water. The experimental approaches to this subject by Pareige et al. 38 have shown that the methods used for the study of acoustic scattering from shells insonified in water apply perfectly to this problem. The resonance spectra and identification are obtained by the quasi-harmonic and short-pulse MITR. For air-filled stainless-steel
150
'/I..
6 / 40 ;Y
7 ( b ) 20 fl3 ~ 0.11
o 0.61 0 20 40 &0
k,1
Fig. 10. Regge trajectories of the Scholte wave resonances for aluminum shells (0.67 < b/a < 0.98).
n ssg 7 6~ k,a
30
25
20
15
m
10 4
3
5 2
5 10 20 a·
Fig. 11. Resonances identified for a finite length target (b/a = 0.89; L/2a = 1.66).
cylindrical shells (b/a = 0.94 and 0.97), two series of resonances are observed: L = 0 (Scholte waves) and L = 2 (Whispering Gallery waves). The frequency signatures are very similar, whether the excitation is internal or external.
Finite len~h tar&ets
The problem of the scattering by finite length targets is more difficult. Nevertheless, some results have been obtained, by means of the MIIR, for solid cylindrical targets with hemispherical ends18,39 and air-filled cylindrical shells with plate ends.40 For the latter targets, it is possible to explain resonances due to guided waves and circumferential waves and their splitting.41 ,42 In a first approach possible, except for interface waves, the calculation of natural frequencies of the infinite cylindrical shells by the formulation of normal modes is used. In the case of oblique insonification, F. Lecroq graphs a system of curves (k1a versus incident angle a for the helical waves, and for various n values in each series (p = 1, 2 ... ; L = 1,2 ... ». In addition to a standing wave in the circumfe..rence, a new condition is imposed: kz = mn:/L (kz component along the axis of the shell of kl wave vector; there is an integer multiple m of half wave lengths in the length L of the shell.) This condition leads to a relation which gives a second system of curves. The identification of resonances (a < 10") by the MIIR allows one to place them in the systems of curves for 7 < kl a < 30. Figure 11 is an example for the L = 2 series resonances of an aluminum target (b/a = 0.89; L/2a = 1.66). I quote some recent studies related to elastic spheroids and other finite-length targets by Uberall et al.,43 Werby et al.,44,45 Hackman et al.,46 Muzychenko et al.,47 Miller et al.,4'8 Brill et al.,49 and Madigosky et al.50
151
CONCLUSION
By using the previous techniques with temporal filtering, FFT algorithm ... implemented for the MIIR, other problems have been solved especially the determination of physical characteristics of the layers of plane multilayered structures immersed in water (for instance of Polystyrene/Water/Aluminum).51 ,52 Sessarego et al.53 and Rousselot et al.54 obtained resonance spectra from spheres by spectral analysis of the second transient state after the end of the forced vibration. Molinero et al.55 identified "breathing" modes for a cylindrical wire. MIIR gives rise to other studies; results obtained by MIIR are examined in various papers.28,29,33,21'8,56-61
This method permits, in a first stage, the ISOLA nON of resonances and, in its second stage, their IDENTIFICA nON by determination of mode number. Quasi-harmonic MIIR and Short-pulse MIIR introduce the RESONANCE SPECTRA, scattering spectra without "background" and new measurements: phase and group velocities, width of resonance lines, coefficient of "reradiation" of circumferential waves. It corroborates and illustrates the "Resonance Scattering Theory" discovered in 1978 by H. tiberall. It allowed the first experimental identification of circumferential wave resonances, especially Scholte type wave resonances on cylindrical shells insonified or excited by an internal source, and the first experimental identification of guided waves in a direction parallel to the axis of cylindrical targets.
MIIR is used for more complex targets: prolate spheroids, cylindrical shells with hemispherical end-caps ... and I hope that the understanding of physical phenomena will make easier the study of the inverse problem related to the underwater detection and nondestructive evaluation.
ACKNOWLEDGEMENTS
I thank Professor G. Maze who implemented the MIIR in 1981 and the other researchers of the LAUE URA CNRS 1373, especially Drs. 1.L. Izbicki, P. Pareige, F. Lecroq, O. Lenoir, and P. Rembert for their numerous studies.
I also acknowledge the financial support of the "Direction des Recherches Etudes et Techniques" (DGA, Paris) under contracts 86/045 and 88/245.
REFERENCES
1. H. tiberall, L.R. Dragonette, and~. Flax, 1. Acoust. Soc. Am. 61,711-715 (1977). 2. L. Flax, L.R. Drago~l?tte, and H. Uberall, 1. Acoust. Soc. Am. 63, 723-TH (1978). 3. 1.W. Dickey and H. Uberall, 1. Acoust. Soc. Am. 63, 319-320 (1978). 4. G. Maze, B. Taconet, and 1. Ripoche, Phys. Lett. 84A, 309-312 (1981). 5. G. Maze,B. Taconet, and 1. Ripoche, Rev. Cethedec, 72, 103-119 (1982). 6. G. Maze and 1. Ripoche, 1. Acoust. Soc. Am. 73, 41-43 (1983). 7. G. Maze and 1. Ripoche, Rev. Phys. Appl. 18,319-326 (1983). 8. L.R. Dragonette, S.K. Numrich, and L. Frank, 1. Acoust. Soc. Am. 69, 1186-1189
(1981). 9. S.K. Numrich, L. Frank, and L.R. Dragonette, ICASSP '82 Conference, Paris, 3-5
May 1982, Proceedings pp. 327-33Q .. 10. L. Flax, G.C. Gaunaurd, and H. Uberall, in Physical Acoustics, edited by W.P.
Mason and R.N. Thurston, Vol. XV, Academic Press, New York, pp. 191-293 (1981).
11. A. Derem, Rev. Cethedec 58, 43-79 (1979). 12. A. Derem, 1.L. Rousselot, G. Maze, 1. Ripoche, and A. Faure, Acustica 50, 39-50
(1982). 13. G. Maze, 1. Ripoche, A. Derem, and 1.L. Rousselot, Acustica 55,69-85 (1984). 14. R. Burvingt, 1.L. Rousselot, A. Derem, G. Maze, and 1. Ripoche, Rev. Cethedec 78,
73-93 (1984); see also: R. Burvingt, Wave Motion 8, 346-369 (1986). 15. M. Talmant, and G. Quentin, 1. Aprl. Phys. 63, 1857-1863 (1988). 16. G. Quentin, A. Derem, and B. Poiree, 1. Phys. 50, 1943-1952 (1989). 17. 1. Ripoche, G. Maze, and 1.L. Izbicki, 1. Nondestructive Evaluation 5, 69-79 (1985).
152
18. S.K. Numrich, N. Dale, and L.R. Dragonette, in Advances in Fluid-Structure Interaction, PVP VI. 78/AMD Vol. 64, Am. Soc.Mech.Eng., pp. 59-74 (1984).
19. M. de Billy, J. Acoust. Soc. Am. 79, 219-221 (1986). 20 P. Delestre, J.L. Izbicki, G. Maze, and J. Ripoche, Acustica 61,83-85 (1986). 21. G. Deprez and R. Hazebrouck, Acustica, 45, 96-102 (1980); Rev. Cethedec 72, 73-90
(1982). 22. J. Ripoche, G. Maze, and 1.L. Izbicki, Symp'osium "Vibrations-Chaos," Lyon, 10-12
June 1986, Proceedings in: Mecanique, Matenaux, Electricite 416,47-51 (1986). 23. P. Pareige, P. Rembert, J.L. Izbicki, G. Maze, and J. Ripoche, Phys. Lett. 135A,
143-146 (1989). 24. G. Quentin and A. Cand, Electr. Lett. 25, 353-354 (1989). 25. P. Rembert, O. Lenoir, J.L. Izbicki, and G. Maze, Phys. Lett. 143A, 467-472
(1990). 26. G. Maze, J.L. Izbicki, and J. Ripoche, J. Acoust. Soc. Am. 77, 1352-1357 (1985). 27. 1.L. Izbicki, G. Maze, and J. Ripoche, 1. Acoust. Soc. Am. 80, 1215-1219 (1986). 28. T. Li and M. Ueda, J. Acoust. Soc. Am. 86,2363-2368 (1989). 29. X.L. Bao, H. Cao, and H. Uberall, J. Acoust. Soc. Am. 87, 106-110 (1990). 30. 1.L. Izbicki, G. Maze, !l.nd J. Ripoche, Acustica 61, 137-139 (1986). 31. E.D. Breitenbach, H. Uberall, and K.B. Yoo, J. Acoust. Soc. Am. 74, 1267-1273
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23, 289-299 (1988). 35. M. Talmant, G. Quentin, J.L. Rousselot, J.V. Subrahmanyam, and H. Uberall, J.
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(1989). 39. G. Maze, F. Lecroq, P. Pareige, J.L. Izbicki, and J. Ripoche, J. Acoustique 1, 171-
175 (1988). 40. J.L. Izbicki, G. Maze, and J. Ripoche, Acustica 64,50-53 (1987). 41. F. Lecroq, J.L. Izbicki, G. Maze, and J. Ripoche, Phys. Lett. 146A, 101-106 (1990). 42. F. Lecroq, G. Maze, and 1. Ripoche, First French Conference on Acoustics, Lyon,
lO-p April 1990, Proceedings in: 1. Phys. 51C2, 395-398 (1990). 43. H. Uberall, Y.J. Stoyanov, A. Nagl, M.F. Werby, S.H. Brown, J.W. Dickey, S.K.
Numrich, and 1.M. d'Archangelo, J. Acoust. Soc. Am. 81, 312-316 (1987). 44. M.F. Werby and G..C. Gaunaurd, J. Acoust. Soc. Am. 1369-1377 (1987). 45. M.F. Werby, H. Uberall, A. Nagl, S.H. Brown, and J.W. Dickey, J. Acoust. Soc.
Am. 84, 1425-1436 (1988). 46. R.H. Hackman, G.S. Sammelmann, K.L. Williams, and D.H.Trivett, J. Acoust. Soc.
Am. 83, 1255-1266 (1988). 47. V.V. Muzychenko and S.A. Rybak, Sov. Phys. AC~l!st. 34, 325-333 (1988). 48. R.D. Miller, S.K. Numrich, M. Talmant, and H. Uberall, 13th ICA, Belgrade, 24
August - 1 September 1989, Proceedings, Vol. 1, pp. 229-232. 49. D. Brill and G.C. Gaunaurd, First French Conference on Acoustics, Lyon, 10-13
April 1990, Proceedings in: 1. Phys .. ?IC2, 383-386 (1990). 50. W. Madigosky, X.L. Bao, and H. Uberall, First French Conference on Acoustics,
Lyon, 10-13 April 1990, Proceedings in: 1. Phys. 51C2, 424-430 (1990). 51. J.L. Izbicki, O. Lenoir, P. Rembert, G. Maze, and J. Ripoche, J. Acoust. Soc. Am.
85(S 1), 35 (1989). 52. O. Lenoir, P. Rembert, J.L. Izbicki, G. Maze, and J. Ripoche, First French
Conference on Acoustics, Lyon, 10-13 April 1990, Proceedings in: J. Phys. 51C2, 455-458 (1990).
53. J.P. Sessarego, J. Sageloli, and C. Gazanhes, Acustica 64, 206-210 (1987). 54. J.L. Rousselot, A. Gerard, J.P. Sessarego, and J. Sageloli, Acustica 66, 203-213
(1988). 55. I. Molinero, M. de Billy, and G. Quentin, Appl. Phys. Lett. 49, 693-695 (1986).
153
56. C.Y. Tsui, G.N. Reid, and G.C. Gaunaurd, 12th ICA, Toronto, 24-31 July 1986, Proceedings Hl-2; see also: J. Acoust. Soc. Am. 80, 382-390 (1986); 83, 1946-1951 (1988).
57. N.D. Veksler, Acoustic Spectroscopy, Valgus, Tallinn, 1989. 58. N.D. Veksler and V.M. Korsunskii, Acoust. Lett. 10, 38-42 (1986); J. Acoust. Soc.
Am. 87,943-962 (1990). 59. W.E. Howell, S.K. Numrich, and H. Uberall, J. Acoust. Soc. Am. 78, 1125-1127
(1985). 60. S.K. Numrich, W.E. Howell, J.V. Subrahmanyam, and H. Uberall, J. Acoust. Soc.
Am. 80, 1161-1169 (1986). 61. W.E. Howell, S.K. Numrich, and H. Uberall, IEEE Trans. UFFC 34,22-28 (1987).
154
REFLECTION AND REFRACTION OF HETEROGENEOUS WA YES AT PLANE INTERFACES
Jose Roux
Mecanique Physique URA CNRS 867 Universite de Bordeaux I 33405 Talence France
INTRODUCTION
It is well known, from a long time, that simple monochromatic plane waves in ideal non-dispersive media are insufficient tools for modelling the propagation of space-time bounded beams and their reflection-refraction through layered realistic media. All these simple concepts must be considered only as asymptotic values of a more general form in agreement with the wave equations and the discontinuity in physical properties at the interface.
General Snell-Descartes laws for such heterogeneous waves may be derived from the continuity equations of displacement and stress fields. In terms of reflection and transmission coefficients, the general reflection-refraction process may be expressed as a simple condensed linear operator. Discussion about its regularity or singularity versus physical parameters enables a synthetic classification of solutions in regular (bulk) and free (interface) modes. Special features in media and incident waves leads to surprising results in view of the so-called "physical sense". However, the reader must always keep in mind that, as ordinary plane waves, heterogeneous waves have no physical meaning by themselves.
Any experiment involves beams bounded in time and space in lossy media. The heterogeneous wave is only the kernel of a generalized integral representation always including a weighting function - say the angular spectrum in the Fourier sense - that fits in with the regularity at infinity and a finite energy balance.
HETEROGENEOUS WAVES IN FREE SPACE
The more general form for such monochromatic plane waves is usually written in terms of displacement field by:
u (X,t) = 9\ [*a *p exp i (rot - *K.X)] (1)
where left starred symbols refer to complex quantities and bold faced ones to vectors. So *a is the complex amplitude, X the real position vector.
The complex polarization vector *P, and the complex wave vector *K are well know as bivectors. 1 Such complex vectors may be split in their real and imaginary parts which are true real vectors. Polarization vectors are normalized by *p = P' - iP" such that the scalar product p'.p" = p.2 - p"2 = 1.
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 155
Several descriptions may be used for the wave bivector *K in regard with computation and for physical meaning. The first of one is the form *K = K' - iK" where K' is the real wave vector and K" a real damping vector.
When K' and K" are collinear the wave is said to be homogeneous and damped, the sign being chosen so that it vanishes at infinity in the direction of propagation K'. If K' and K" are orthogonal the wave is said to be pure evanescent. A second form, displayed in the Fig. 1, is widely used in optics and electromagnetism. The wave bivector is split in local coordinates (M, n, h) so that:
*K = kn - ian - ihh (2)
As in optics2 the real numbers k, a, h are respectively the refraction and vanishing indexes and heterogeneity coefficient. The signs in Eq. (2) are chosen so that when k and a are positive, the wave is damped in the direction of propagation n normal to constant phase planes. Constant amplitude planes have the normal h.
The displacement field described in relation (1) must describe the wave equation in the free space. This leads to the general dispersion equation:
*K. *K = (ko - iaof (3)
where real positive numbers leo aO are functions of the angular frequency (J) in close relation with the mechanical behavior laws of the medium and their polarization. This equation has been widely described for linear isotropic elastic or viscoelastic bodies, 1 linear anisotropic elastic, 1 linear anisotropic weakly absorbing in principal planes3 perfect thermoviscous gas,4 general thermoviscous fluids5 and for thermoviscoelastic isotropic solids.6 Combining relations (2) and (3), wave indexes are related to the medium properties by:
(K - iaf - h2 = (ko - iaof (4)
Looking at the imaginary part, we see that any lossless medium, aO = 0, may propagate only waves with a = 0, i.e. ordinary plane waves with K" = 0 or evanescent ones: K" J.. K'. Such evanescent waves have been extensively described along plane interfaces between lossless isotropic media,7,8 and at a fluid-anisotropic solid plane interface.9 These interface waves - including pseudo-Rayleigh, Stoneley, Scholte waves etc. - are often called free waves or free modes of plane interfaces because they satisfy mathematically the boundary conditions by themselves in the absence of any incident wave source.
However a bulk evanescent wave in a nondissipative liquid may exist as experimentally shown by M. Deschamps and B. Hosten. JO Let us consider, in Fig. 2, two successive refractions of a plane longitudinal wave at the edges of a PVC prism, with a
X2
ph;se / plane amplitude plane
K·
Fig. 1. Heterogeneous wave vectors in local coordinates.
156
t ~ I S;:I ,
- ---- -... Fr= 3.5 Mhz Q)
" " .... E w
500mm --Water
Fig. 2. Experimental set-up for observing double refraction through a dissipative plastic wedge.
high damping factor, say <Xo = 80 m- I at 3.5 MHz, immersed in water which is assumed to be lossless at this frequency.
The fIrst refraction at normal incidence converts the homogeneous plane incident wave into a damped wave in the same direction according to relation (2) with h = 0; a = ao. Anticipating upon the next paragraph on generalized Snell-Descartes laws, the second refraction create in water an undamped wave a =0 but this time with h '* 0 because of the equiprojectivity of wave bivectors at this interface. So the outgoing wave is an evanescent one with K" .1 K' not confIned along the interface. The spatial structure is experimentally verifIed, I I inside the area around the beam axis where the incident wave is locally quite homogeneous. The results obtained by mechanically scanning the receiver along constant phase planes with normal K' are shown in the Fig. 3. In logarithmic scale, the exponential decay leads to the straight line with the same negative slope h = 40 m-I at any distance from the source.
REFLECfION-REFRACTION PROCESS
In this paper, the following discussion will be limited to a plane interface between isotropic media, whether fluids or solids, absorbing or not. Generalized Snell-Descartes laws are deduced from the continuity equations at the interface between the two media.
Let Xl be the unit vector normal to the interface whose points lie in the plane XI = O.
Denoting the total stress fIeld and the total displacement field
where L is the sum over all elementary waves at the interfa<;:e and [f]x I = 0 = f(O+) - f(O-) the jump of any quantity at the interface. The continuity equations are then:
[9\(*U0]xl=0 = 0
[9\ (*T1: xI)]xl = 0 = 0
(5)
(6)
The direction of propagation OJ of the incident wave and the perpendicular to the interface XI define the incidence plane. Regarding the continuity equation (Eq. 5) we now consider only in-plane waves whose polarization bivectors *p lie in the plane of incidence. Out-of-plane transverse waves do not interact and their reflection-refraction, is a one-to-one elementary process. For the in-plane process, the continuity equations (Eqs. 5 and 6) form a set of four scalar equations. So in the more general case, shown in Fig. 4, an incident wave (subscript i) will generate four waves, one longitudinal (subscript 1) and one transverse (subscript t) in each medium numbered 1 to 4.
In order to simplify and compact the calculations further, let us define for a momentary purpose, a complex reflection/refraction angle for each wave. In the (xl> x2) coordinates a pseudo-complex normalization of the wave bivectors may be written:
157
Without Prism
..................... ~ ..... • • ••
-10
•
_w~-----L--~--~--~--~-----L~----~------~--~~------~ -60 -so -40 -30 -w -10 o
Scanning Position in mm
Fig. 3. Experimental amplitudes at two distances. Arrows indicate the axis of the beam according to Snell's law.
*K = (kO - ino) [xl cos *8 + x2 sin *8]
10 w
(7)
The complex angle *8 is referring to the true angle 8, through equations (2) and (3) by the set
158
cos *8 = (kO -ino) -1 . [ (k - in) cos 8 - ih sin 8] sin *8 = (kO -ino) -1 . [ (k - in) sin 8 + ih cos 8]
" incident
'\
medium 1
medium 2
t
I reflected 1 I
2/
Fig. 4. General sketch with five waves.
(8)
The polarization bivectors are now in the fonn:
*p = xl cos *9 + x2 sin *9 *p = - Xl sin *9 + X2 cos *9
forI waves for t waves.
(9)
Writing the sum of the displacements *Ul: for the five waves (Fig. 4) and the continuity equation (Eq. 5) which are invariant in time and space coordinate x2 results in the generalized Snell-Descartes law: '
*Kmx2 = (kOm - ia. Om) sin *9 = cste = (kOi - ia.ol) sin *9 m = 1,2,3,4
(10)
It has been demonstratedl2 that according to Eq. (10), continuity equations (Eqs. 5 and 6) may be written in the full complex fonn since the discontinuity of the imaginary part is also null on the interface. Finally, writing the continuity equations in their full complex fonn leads to a linear set of equations in the complex matrix fonn:
L.A = *ao. B (11)
where A= {*a 1, *a2, *a3, *14, } T is the column vector of the four complex amplitudes, B a column vector playing in the role of source via the amplitude *ao of the incident wave and L a 4x4 complex matrix. Nonnalizing the vectors by the amplitude of the incident wave leads to a more condensed notation:
L.R=B, (12)
where R = {*r1> *r2, *t1> *t2, }T is the vector whose two first and two last tenns are respectively the complex reflection and transmission coefficients.
Particular solutions arise in conjunction with the singularity of L; i.e., det L = O. For particular values of the initial set of material parameters 7 solution of the homogeneous equation L.R = 0 are known to be the free waves at the interface: pseudo-Rayleigh, Stoneley and Scholte waves. Further investigations are necessary to demonstrate our idea that for any incident heterogeneous wave and for two lossy media, the operator L will never be singular; i.e., det L "# O.
Assuming the regularity of L, a direct inversion has been performed by M. Deschamps.13 Due to the complexity of expressions, a numerical solution is more suitable through a computer package. The R vector (12) is the output while the input is made of the whole set of 12 parameters: (p, kah kat, a.o1o a.ot>for each medium, the incidence angle 9j and the incidence heterogeneity coefficient hi'
In this work, the computation of reflection and transmission appears to be a one-to-one process, through the generalized Snell-Descartes laws (Eq. 10) and the dispersion equation (Eq. 4). This is illustrated in Fig. 5.
The incident wave known by the set of parameters (data):
generates by reflection or refraction the "m" wave (m = 1,2,3,4) defined by the set:
inside which, only the last four tenns are the unknown, computed from a set of four equations. Separating the real and imaginary parts, respectively, in the Snell-Descartes law and the dispersion equation leads to the following set of four real equations:
km sin 9m = ki sin 9i
a.m sin 9m - hm cos 9m = a.i sin 9i - hi cos 9i
k~ - ~ -h~ = k5m - a.5m
kma.m = kama.Om . (13)
159
Ai K ' = k D r r r
Fig, 5, Equiprojectivity of the wave bivectors according to Snell's law,
The last relation shows that km and am have the same sign so that the angle between the real K' and imaginary Kit parts of the wave bivector is always less than 90·, Note that km and am may be either positive (refracted waves) or negative (reflected waves), However, no hypothesis may be made a priori on the sign of hm' As we will see later, its determination is one of the main problems,
There is such a wide variety of situations, combining media: fluids, solids, absorbing or not, the incident waves related to the initial twelve-parameter data set, that it is quite impossible to draw a general sketch of the whole process, When limiting the computation to the regular solutions of R= L -I ,B the general flow chart of the computer package has the appearance of Fig, 6,
Pseudo-3 dimensional charts may be drawn from a two variables loop returning to the initial set of parameters, Large possibilities are allowed for these two variables to illustrate the behavior of one reflection or transmission coefficient versus experimental conditions, Before further discussion of some particular results, it should be noticed that all regular solutions of the reflection-refraction process are in agreement with the conservation laws and energy balance throu!)h the interface, These considerations, which exceed the aim of this paper, may be found, 3 Energy flows for singular solutions (free modes) have been discussed,7,8
SOME THEORETICAL RESULTS
All the situations presented now are chosen to be close to some physical cases, Data sets correspond to realistic fluid/solid and solid/solid interfaces, The incident wave may be heterogeneous when coming form a previous interface as in a layered medium, or homogeneous damped as near the axis of a finite beam generated by a transducer. In most pictures the incidence angle will always be one of the two variables because of its main importance in any experiment and in theory.
Ruid-Solid Interfaces
Figures 7 (a), (b), and (c) are related to an interface between an absorbing liquid and a lossy solid. The heterogeneous damped incident wave comes from the liquid. Its incidence angle Si and heterogeneity coefficient hi are the variables. The initial data setting was as follows:
kol aol kot a ot p m- I m- I m- I m- I kg,m-3
Ruid 4000 100 00 1000 1000 Solid 1000 50 2000 150 2700
160
Data ini tialization
Media: first and second
I
Incident Wave
p. kol> uo!. ket. Uot ai. hi
I I
I Dispersion Equation
First medium
1-J
Incident wave set Source vector
I Ai : kei. uoi. hi. Uj. ai B
[ ___ L_ Snell - Descartes Laws
I Dispersion Equations
AI' A2• reflected 1-- Waves sets-]
A3• A4. transmitted
I complex angles 'am ] I L (4x4matrix)
I 1
L_ R=L-l B I
---------- I I
I Rm 3D Chart drawing I
Fig. 6. Flow chart of the whole computation process in regular cases.
Values were chosen to be representative of the water-aluminum interface in a frequency range> 50 Mhz.
Some classical results are shown in Fig. 7. First, for a heterogeneous incident wave in the liquid, the reflection coefficient may be greater than unity. Particular situations of this kind are described.16, 17 Secondly, the "Rayleigh hole" near the critical Rayleigh angle, is always present in this reflection coefficient. Its sharpness is in conjunction with the damping factor of the two media even for a homogeneous damped incident wave.
For some particular values, the viscosities of the two media may cancel each other leading to an ideal reflection as between two lossless media. I8 Figure 8 shows this variation in reflectivity near the Rayleigh critical angle for a homogeneous (hi = 0) incident wave versus the damping factor of the liquid and the incidence angle.
CONCLUSION
In acoustics as well as in optics and electromagnetism, heterogeneous waves appear now to be the more suitable tool in describing realistic wave fields. Computations of beams bounded in time and space, reflected or transmitted through plane interfaces, requires a complete knowledge of the complex reflection or transmission coefficient of such waves to be inserted in a subsequent generalized integral as in the Claeys-Leroy seriesI9 or in some Fourier space-time integral.
Reflection-refraction process at a single interface has been solved in the general case in terms of a linear operator. Regular solutions are now available through a computer package of moderate size runable in any micro-computer. However, one should be especially careful in applications involving lossless media and/or homogeneous undamped incident waves. In such cases, singularities of the linear operator may rise and free modes must be taken in account in addition with the suitable particular solution involving the right sign of the heterogeneity coefficient in the quadratic Eq. (13). Referring to the Sommerfeld radiation condition; i.e., vanishing at infinity in this occurrence, is not pertinent because
161
162
L reflection coefficient
incidence angle
L lransmission coefficiem
T lransmission coefliciem
incidence angle
incidence angle
... <>
... '"
a
incidence helerogeneily
b
c
Fig. 7. Reflection and transmission moduli versus the incidence angle and the heterogeneity of the incident wave for an absorbing liquid-absorbing
solid interface. a. reflected longitudinal wave. b. refracted longitudinal wave. c. refracted transverse wave.
Lreflection coefficient
p
incidence angle
damping factor
Fig. 8. Reflection coefficient of a homogeneous incident wave versus the damping factor of the liquid for an absorbing liquid-absorbing solid interface. Arrow indicates the Rayleigh hole disappearing
for a suitable value of the damping factor.
plane waves of infinite extent in time and space are not realistic physical waves. In such cases, branch points in determinations must be considered in the limit of the general regular form in accordance with some specific inequalities.14
However, some spurious results as refraction angles greater than 90' or discontinuities in reflection coefficients are remaining problems that are in an apparent contradiction with common physical sense. Energy flows through and along the interface are to be carefully discussed at a later time.
REFERENCES
1. M. Hayes, Inhomogeneous Plane Waves, in Arch. Rational Mech. Anal. , 85:41 (1984) .
2. P. Cuvelier and J. Billard, Quelques proprietes des ondes electromagnetiques heterogenes planes et uniformes. Nouv. Rev. Opt., 4:23-26 (1973).
3 . B. Hosten, M. Deschamps and B.R. Tiumann, Inhomogeneous wave generation and propagation in lossy anisotropic solids. J. Acoust. Soc. Am. 82:1763-1770 (1987).
4. B. Poiree, Les ondes planes heterogenes dans un fluide thermovisqueux. Acustica 64:73-79 (1987).
5. M. Deschamps and C. Changlin, Ondes heterogenes et reflexion-refraction Ii l'interface liquide non-absorbant, liquide thermovisqueux. Acustica 68:96-103 (1989).
6. M. Deschamps and C. Cheng, Liquid thermoviscoelastic solids interface. Ultrasonics 27:308-313 (1989).
7. L. Sebbag, Les lois de la reflexion-refraction des ondes planes harmoniques evanescentes. in: Thesis, University of Paris 7 (1987).
8. B. Poiree and L. Sebbag, Les lois de la reflexion-refraction des ondes planes evanescentes. I, Mise en equation. Jal. Acoust. (to be published).
9. A. Braga and G. Herrman, Free waves at a fluid-layered composite interface. in: Elastic waves and Ultrasonic Non Destructive Evaluation". S.K. Datta et al. eds. Elsevier (1990).
10. M. Deschamps and B. Hosten, Bulk heterogeneous wave generation in a non viscous liquid. Acustica 68:92-95 (1989).
11. B. Hosten and M. Deschamps, Generation and propagation of bulk heterogeneous waves, in: Proceedings VITAM Symp. Boulder 1989. Elsevier (1990) p. 421.
12. M.Deschamps and C. Changlin, Reflexion-refraction de I'onde plane heterogen~: Lois de Snell-Descartes et continuite de I'energie. Jal. Acoust. 2:229-240 (1989).
163
13. M. Deschamps, Reflexion-refraction de l'onde plane heterogene. Repartition de l'energie. Jal. Acoust. (to be published in Vol. 3, 1990).
14. M. Deschamps and J. Roux, Some considerations about the evanescent surface wave. (submitted for publication in Ultrasonics 1990).
15. O. Leroy, B. Poiree, L. Sebbag and G. Quentin, On the reflection coefficient of acoustic beams. Acustica 68:96-103 (1989).
16. P. Cuvelier and J. Billard, Refraction des ondes planes uniformement heterogenes. Jal. Optics 9:9-14 (1978).
17. V.A. Vasilev, Reflection coefficient for transverse waves in an absorbing solid. Sov. Phys. Acoust. 23:127-129 (1977).
18. F. L. Becker and R. L. Richardson, Influence of material properties on Rayleigh critical angle reflectivity. J. Acoust. Soc. Am. 51: 1609-1617 (1972).
19. 1. M. Claeys and O. Leroy, Reflection and transmission of bounded sound beams on half-spaces and through plates. J. Acoust. Soc. Am. 72:585-590 (1982).
164
MODULATION EFFECTS IN LIGHT DIFFRACTION BY TWO ULTRASONIC BEAMS AND APPLICATION IN SIGNAL PROCESSING
Antoni Sliwinski
Institute of Experimental Physics University of Gdansk 80 - 952 Gdansk Poland
INTRODUCTION
Among the light-ultrasound interaction phenomena, the modulation oflight amplitude, phase, and intensity is a. knownt>roblem since the beginning of acousto-optics (Debye, .and Sears, l Lucas, and Blquard, Raman and Nath3 and others.4-8 In the case of hght diffraction by a single ultrasonic beam in the Raman-Nath region4-8 the light amplitude is spatially distributed in diffraction orders in the far field - Fraunhofer-region. In contrast, in the near field - Fresnel - region they form an interference pattem9-11 and a given component of every order (except the O-order) is shifted in frequency due to the phase modulation inorder. In the Fraunhofer region every order is shifted in frequency due to the phase modulations introduced by the Doppler effect according to the known formula
(1)
where Vo is the frequency of the incident light, f the frequency of the ultrasonic beam and p is an integer.
This frequency shift (equivalent to the phase modulation) in diffraction orders in the case of progressive ultrasonic waves is not evident in the intensities of diffracted light beams, which are proportional to J~(a) and time independent. The Jp are the p-order Bessel functions of the first kind with an argument
a = 2nOnL A
(2)
The argument a is the Raman-Nath parameter, L is the width of the ultrasonic beam, A the wave-length of light out of the medium and 011 is the amplitude of the variation in refractive index 11,
L11l = 11- 110 = Ollsin (Qt - Kx), (3)
Here Q = 2nf and K = 2n/L, A is the wavelength of an ultrasonic wave and x its propagation direction.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 165
However, in the case of light diffraction by a stationary ultrasonic wave the time modulation is evident. Then the Raman-Nath theory provides the following formula for the temporal modulation. 4,8
Ip(t) = Ii J~ (acos ilt) . (4)
The interaction of light with the stationary wave may be seen as a peculiar case of light diffraction by a system of two ultrasonic beams progressing in opposite directions and overlapped when the mutual phase shift & = ±1t.
In general, the diffraction of light by two ultrasonic waves is considered in many aspects (Fig. 1). Namely:
1) as the case of two ultrasonic waves progressing in the same direction and superposed (overlapped) (Fig. la), partly superposed (Fig. Ib), adjacent (Fig. lc) or separated (Fig. Id).
2) as the case of two ultrasonic waves progressing in the opposite directions, in the same configurations (Fig. la'-d').
The case of superposed ultrasonic waves may be ~eneralized for a complex ultrasonic signal re.f,resenting a spectrum of components6, , 11 (for short pulse signals, for instance2 ). The theoretical approach of Blomme11 has shown that such generalization describes many particular physical situations and includes more simple theories elaborated for diffraction of light by two ultrasonic waves, for instance.
It was shown in many papers9-23 that for the systems of two ultrasonic beams where the frequency ratio between the beams were harmonic f}: f2 = 1: m or n: m (n,m being integers) the phase shift & between two ultrasonic beams progressing in the same direction has played an essential role in the dependence of diffracted light intensity distributions. Then the regular modulation of light in diffraction orders as a function of & has been observed. For the case of the two beams propagating in opposite directions the light intensity modulation is time dependent and & plays not so essential role: & determines only the starting phase difference between the beams. In the case of separated beams (Fig. Id
Ii I
0) b) c) d )
I I
(, : I I,
-H--H I
I I T, I Tz
t--L,,~ ,~ r-L~ z
;- L,--+-l.~ Lz--:
Q') b') d)
I,
I I
I ~' __ It r-'If, I I I
r, I I
Fig. 1. Different geometries of light diffraction by two ultrasonic beams progressing the same (a-d) and in opposite directions (a' - d'); a, a' - superposed,
b, b' - partly superposed, c, c' - adjacent, d, d' - separated.
166
)(
,<1m ,u"
!!!J!>.+fl.~m( f4~~ } ] + 1 order light a order
J -/ order
I ,
:.- L,---":'-Lz+rear field -+-- ror fo"eld I I I I
Fig. 2. Diffraction of light by two adjacent ultrasonic beams with a phase shift 8. The nearfield and the farfield of light diffraction is illustrated.
and 1d') there exists an additional modulation as a function of the distance of the separation L, a subject of the very detailed study (theoretical and experimental) perfonned by P. Kwiek.9
In all these cases the modulation effects are of a great interest both for the far field and for the near field, as shown in the thesis of K wiek9 and in the thesis of Blomme 11 as well as in a paper. I2 All problems are discussed6-11, 17 not onl~ for the Raman-Nath regime (8 < 1) but also for the Bragg regime (8 > 1) (8 = A 2/1l01lA ) where A, A wavelength of light and ultrasound, respectively, 110, 11 refractive indices for non-disturbed and disturbed medium, respectively.
In this review we shall limit our considerations only to the case of adjacent* parallel and antiparallel beams in the Raman-Nath regime taking into account some examples of results obtained in our laboratory during the last decade. Also some possibilities for application in signal processing will be discussed.
MODULA nON OF LIGHT INTENSITY IN DIFFRACfION ORDERS IN THE CASE OF TWO ULTRASONIC BEAMS OF HARMONIC FREQUENCY RATIO m:n
In Fig. 2 the scheme of interaction of light with two ultrasonic beams of frequency ratio m:n is presented in the case of perpendicular incidence. Two beams are shifted in phase 8. In the first step we shall consider the case
m = 1, n = 1,2,3, ....
Two adjacent parallel ultrasonic beams of frequency ratio 1:n progressing in the same direction.
The expressions for diffracted light amplitudes (nonnalized to the incident amplitude Eo) at the output plane z = LI + ~ (Fig. 2 for m = 1) for p < 1 are as follows I4• 18:
-t<><>
<l>± (a) = L J±r.nq (a-a!) Jq (<Xn(a-a!)]e-iqli , (5)
q=~
wh~re r is the number of a given order, Jrq ( ... ), Bessel functions of the first kind; r, n, q are mtegers,
*It is known9.12.17 that for /) « 1 the cases of adjqcent and superposed beams give the same diffraction pattern and the same formula Eq. (5) for amplitude distribution is valid.
167
The light intensity distribution throughout diffraction orders in the far field can be calculated as
* l±r(a) = <I>±r(a) <I>±r(a) (6)
* where <I>±r represents the conjugate quantities. Taking into account only the terms for q = 0, + 1 in Eq. (S) and calculating intensities
from Eq. (6) for r S; n one gets
For small arguments of the Bessel functions using the approximations one can replace Jr[a'O(a-al)]/Jo[un(a-al)] by 1/2 sin[un(a-al)] and put J5[an (a-al)] "" 1.
Furthermore Jl (al) represents the light intensity in the r-order after interacting with the first ultrasonic beam only. Denoting that by I? (al) one can write
(8)
The expressions for light amplitudes Eq. (S), as well as for intensities Eq. (6), do not contain any time modulation factor; however, the periodic dependence on 0, has appeared.
It can be shown l7 from Eq. (7) and Eq. (8) that the diffraction pattern caused by two ultrasonic beams, depending on phase shift 0, is symmetric for n odd (the second beam is an odd harmonic of the first beam) and it is asymmetric for n even. In the latter case the diffraction pattern can be made symmetric by choosing a phase shift between ultrasonic beams of 0 = n12.
The theoretical dependences Eq. (7) and Eq. (8) were experimentally proved in our laboratory in many experiments and the results were compared with numerical calculationsl6-20 for different values of Raman-Nath parameters; i.e., different values of an against 0 for various frequency ratios.
Some examples of numerical calculation results are presented in the Fig. 3 (for n = 2) and Fig. 4 (for n = 3), for 0, ± 1 orders according to the formula derived from Eq. (7) and
Eq. (8) for n = 2: I±I (a) = t? (al)[l±sin (U2al)COS 0] (9) rt> • J2(al) ~
and for n = 3: I±l (a) = II (al)[I+sm (U3al)Jl(al) cos u] (10)
For a comparison some experimental results27 are given in Fig. S (for n = 2) and Fig. 6 (for n = 3) for ±1 orders. The agreement between the experimental results and the numerical calculation is quite good. The modulation factor (depth of modulation) against 0, depending on the Raman-Nath parameter, can be 90-100 per cent. The fact of 1800 of phase sifting of the optical signals in + 1 and -1 orders for the case of n even can be used in signal processing (see below).
168
~aoa ~ 007
;; 0.06 ..., 005
004 0.03 o~--~~--~----~~--~~
Fig. 3. Light intensity in diffraction orders ± 1, a I = 0.5, u 2 = 1 ana O.S for 1 :2, as a function of 0 (reference 17).
Fig. 4. Light intensity in diffraction order 1, al = 0.8, u3 = 0.5 and 0.25 for l:n = 1 :3, as a function of 0 (reference 17).
3.,
Fig. 5. Light intensity in diffraction + 1 orders as a function of 0 for n = 2; al = a2 = 1, a2 = 1, fl = 0.8 MHz, f2 = 1.6 MHz (reference 27).
Fig. 6. Light intensity in diffraction orders + 1 for n = 3 as a function of 0; al = 2, a2 =0.2; f = 0.5 MHz, f = 1.5 MHz; L = 28 mm, L = 24 mm' (reference 19).
169
In the near field due to the interference among components having different phases, time modulation takes place though it is not directly seen from (5) and (6). The formula for light distribution in the near field taking into account the time dependence, is as follows12:
-t<>o -t<>o
'¥(x,y,t) = L L Jp(al)Jq(a2)ei(2q+p)~1lz·ei([ro+(2q+p)nlt.[kZ+(2q+p)Kxl+2qS) (11)
p=_oo q=_oo
From Eq. (9) the intensity of light at any particular point (x, z) may be calculated as
I(x,z,t) = E~'¥ (x,y ,t) '¥. (x,y ,t) , (12)
where Eo is the amplitude of the incident light. The intensity depends on the distance z· showing characteristic spatially periodic self
imaging phenomena (in Nomoto planes) behind the ultrasonic beams. The time modulation as well as the dependence on d disappear in the planes at the distances z· = MJlA2JA, and proved for the case of frequency ratio 1: n=l: 1. Recently P. K wiek and A. Markiewicz12 continued the examinations and experimentally confirmed theoretical predictions for the case of frequency ratio 1:2 with good agreement, too.
Two adjacent antiparallel beams progressing in opposite directions.
Figure 7 represents the geometry of the case of light diffraction by two adjacent ultrasonic waves propagating in opposite directions for frequency ratio l:n. This case was theoretically studied by Leroy and Blomme,25.1l Calligaris, Ciuti, and Gabrielli26 and experimentally proved in our laboratory by Kwiek, Markiewicz, and Sliwinski19 for frequency ratio 1: 1 and 1:2.
In the Raman-Nath regime the light amplitude of the r-th order is given by
(13) q=_oo
and the light intensity
Ir = <Dr <D; = L jIrq cos k (2n nt-B) (14) k=O
where n = 27tf is the angular frequency of the first (fundamental) beam, B the phase shift between ultrasonic beams,
170
incident /iqhll
} .2·nd oraer ® time mOdulation with frequenCl.j
® ® ®
Fig. 7. The geometry of light diffraction by two adjacent ultrasonic beams propagating in opposite directions.
+1
03 -1
0.2
01
o ~--------~------~~--~~ 1f/2
Fig. 8. Theoretical (solid line) and experimental results (dashed line) for light diffraction by two adjacent ultrasonic beams propatating in opposite directions, as a function of nt for frequency ratio n = 1 (al = a2 = 1.2, fl = f2 = 1.6 MHz) (reference 19).
(15)
Considering the frequency ratio of ultrasonic beams n as a parameter Eq. 14 one can examine several properties of the light intensity depending on n. In the case of n-odd the diffraction pattern is symmetric25; Le.,
Lr(E) = Ir (E) for n = 21 + 1 (16)
and E = 2n:n n t - 8 .
When n is even the time-dependent intensity of light in the opposite diffraction orders is shifted in phase25:
L r (E) = Ir (E ± fri") for n = 21 (17)
In addition, the coefficients Irq [Eqs.(14) and (15)] for nand q even become 0 and the intensity of the O-th diffraction oraer is modulated with frequency twice as great as the other orde!:P and then the Eq. (14) becomes
10 = I 10 k' cos 2k'(2nnt-8n) for n = 21 (18) k'=O
In Fig. 8 (for n = 1) and Fig. 9 (for n=2) some experimental and calculated results l9 are compared. It is seen that the agreement between the theoretical prediction [Eqs. (14) and (15)] and the experiment is good, even for rather large values of the RamanNath parameters. Also, the phase relation described by Eq. (16) for odd and Eq. (17) for even n are such as predicted. When n is even the phase shift between the opposite orders is equal to n:/2n and moreover changes of intensity in these orders are opposite in phase (see also Fig. 15, below).
Modulation of light intensity in diffraction orders in the case of two ultrasonic beams of frequency ratio m:n. m;en.
The theory of diffraction of light by adjacent ultrasonic beams which have frequencies fl = mf and f2 = nf (where m, n are integers) (Fig. 2) were presented by Hargrove,
171
Heideman and Mertens 12 in 1962 and later by Gabrielli and Ciuti.22 Interesting possibilities of light modulation in comparison to those occurring for the case when m = 1, appear for m ~ 1, n ~ 1 and m ~ n. In that case there appears in the diffraction spectrum additional orders at spots between those which originally existed for each ultrasonic beam independently, as opposed to the case of fre!1uency ratio 1:n. The phenomena were recently examined and experimentally verified28. 29 for m:n = 2:3 and 3:4.
The appearance of new orders in the diffraction pattern is described by the following28 [It is more general than Eqs. (5) and (6)]:
+~
(19)
q=.~
(20)
where 1 = pm + qn and p, q are integer. Summation in CPL is carried out only for those values of the index 1 - nq/m which are integer numbers. The other notation is the same as for Eqs. (5) and (6). The <l>L are normalized to the amplitude of the incident light Ea. For 1= ... , ±7, ±5, ±1 one gets the additional diffraction orders except for the cases when every beam is acting separately or when m = 1 (the case 1 :n). In these orders the regular dependence on () takes place.
As an example for m:n = 3:4, some results for L=±L orders29 are presented in Fig. 10 where the experimental records (B) are compared with the calculated curves (A). The frequencies used were fl = 1.53 MHz, f2 = 2.04 MHz and Raman-Nath parameters were al = a2 = 2.4. For this value of a, the Bessel function 1(2.41) = 0 and consequently one gets a simpler formula for intensities. For ±1 orders it is:
Equation (21) was used for the numerical calculations in Fig. 10 A. The modulation factor for these orders has reached 100 percent.
APPLICATION OF TWO ULTRASONIC BEAMS TO LIGHT DIFFRACTION FOR SIGNAL PROCESSING.
There are many devices applying Raman-Nath or Bragg acousto-optical cells for signal processing described in the Literature6,7,30.38 like dispersive filters, spectrum
172
1 O~
---- expo -- ilJeor.
Fig. 9. Theoretical (solid line) and experimental (dashed line) results for light diffraction by two adjacent ultrasonic beams propagating in opposite directions as a function of Qt for frequency ration n = 2 (al = a2 = 1.2,
fl = 0.8 MHz and f2 = 1.6 MHz) (reference 19).
I 8.15 ••
8.J 8.85
a
8, 1811_
i ,... -.J -
8.85
b
I 8.15 - 1
a
1 8. 1S .='
E& t-=== :~~:--:!
8. 18' -
8.85
3. 14 6.28 6 3.14 6.28 6 3. 11 6.28 6 b 3. 14 6.28 6
Fig. 10. The modulation of light intensity as a function of 8 for ±1 orders, (reference 29); a -theoretical, b - experimental.
analysers,32,33,35,37 bistable systems, heterodyne system convolvers36and others. We shall limit ourselves to some possibilities which provide two ultrasonic beam light diffraction systems based on the phenomena discussed above.
A match-filtering principle
Let us consider application of an acoustioopitcal system for such operations as signal compression with match-filtering which are widely used in electronics, and also applied in ultrasonic spectroscopy.39 The operation of such a filter is equivalent to the correlation process accordin,g.to the expression (Fig. 11).
R12 (t) = L g(t-t) Kg(H,)dt =KRlI (He) (22)
The match-filtering process is very often used in ultrasonic sending-receiving systems which in block diagram is presented in the Fig. 12a. An electric signal of frequency ffi is generated by (1) then modulated in (4), amplified in (6) and transmitted by the ultrasonic transducer (8) into the medium as g(t). After scattering or reflection on an object (e.g. material flaw) (10) an attenuated signal approaches the receiving tranducer (9) as a replica of the transmitted signal Kg(-t), where K is an attenuation coefficient. The received signal is amplified in (7), demodulated in (5) and in (3) correlated with the generating signal being delayed in (2) by a determined time t. With the delay line (2) and using an additional summation processor (Fig. 11) one can get the output signal compressed in time and make its amplitude greater. The process is called match-filtering and its principle is explained in the Fig. 11.
Let the transmitted signal g(t) be frequency modulated as a chirp pulse Fig. 11 (a) having the spectrum G(t) of the frequency band - with M = f} - f2, Fig. 11 (b). The received signal Kg(-t) detected after some time T (transmitting time through the medium) and demodulated, consists of determined fragments corresponding to time intervals ta, tb, te, ... , tn· If one uses an adequate delay line and summation processor (Fig. llc) to overlap the consecutive maximal values of those fragments, the resulting signal will be built up and compressed as shown in the Fig. lId.
Signal phase quadrature procedure
The system shown in Fig. 12a is very often more developed in the detecting part. The received signal, amplified (13), is divided into two channels I and Q (Fig. 12b) in such a
173
9(t) a
16ml b input c
f,
,...-I-a-'d~deJ.r-'-- output
t ransmillea signal
~------ T--------~ m :
9 a , 0 c a~----------------~--d
I~ --------- T ------1
9(·/ )1 received siqna/ T-----:=.....j ~ ia::::J t- I
r- tb-1 : r-- 1r--j T~
I T'/'itc~
T"a·tb~ T.td·ta~
I I
h(/)f
~ Fig. II . Principle of the match-filtering: (a) chirp signal, b) frequency band,
c) delay line and adder, d) summation process and the result.
way that the phase of the I(t) signal is not changed but the phase of the Q(t) signal is shifted by 900 (by the Hilbert transform, see below) due to the phase quadrature processing.39
This way of detection provides the possibility to measure separately changes in time of an amplitude (envelope) and phase of a signal. In both channels the signals are mixed (5) with the transmitted signal of the carrier frequency wand multiplied by cos wt (6) and sin wt (14), next filtered (7) (15) to get rid of slowly varying components. Then the signals in separate channels are processed with matched filters; i.e., compres~ed (del&,ed (16), correlated (8), (17) and added in fragments) . Next, the output signals R II and Rl1 of I and Q channels after correlation are squared (9) (18), summed (10) and finally the r.m.s. value (19) of that sum is obtained as the resulting envelope:
(23)
Acousto-optical match filtering
The match-filtering process may be performed with the two ultrasonic beam light diffraction system as is demonstrated in Fig. (13). The signals g(t) and Kg(-t) are applied to drive the ultrasonic transducers. The light beam passing through the two signals interact with them and in the diffraction pattern one has information about the convolution of the two signals. In the configuration of Fig. (13), when the two beams are progressing in opposite directions the optical signal is compressed as is seen in the Fig. (14). The compression may be controlled by the time delay matched by moving the incident light beam with a system of mirrors.
174
a
kR,,(t -tJ
Fig. 12. A block scheme of an ultrasonic sending-receiving system using match-filtering (a) and phase quatrature procedure (b).
9/t )
-,- • t
light l 0
-f
92 · 9, (-t)
Fig. 13. Two ultrasonic beam light diffraction cell for autocorrelation processing.
175
- wv vvJ
%Iv
Fig. 14. Compressing of two pulse signals progressing in opposite directions.
Acousto-optical signal Quadrature
The distribution of light intensity at the output plane of the system is a Fourier transform in the frequency domain of the convolution between the light signal and the resulting ultrasonic signal (a product of two beams) S(t) in the time domain. Simultaneously, in the time domain, the plus and minus orders of the spectrum are shifted in phase to correspond to the Hilbert transform40 of the signal S(t) which is the convolution
H[S(t)] = l S(t) * 1 1t t
(24)
where t = x/c, c is the ultrasound velocity, and x = 0 corresponds to the main optical axis i.e. to the spot where the incident light beam meets the system. The Fourier transform for the function 11m gives the function:
(
.1t -I-e 2
(-isgnf) = .1t
e+l 2"
for t = K.> 0 c fort =~ 0 c
which corresponds to the situation presented in the Fig. (15).
(25)
Thus, the transformation describes the mutual phase shift in the intensities of ±1 orders as has been presented in Figs. 3-6 and 8-9. Since the optical signals in the + 1 and -1 orders are separated and shifted in phase by +900 with respect to the 0 order, one can treat the photo signals from these orders (Fig. 16) as, respectively, corresponding to I(t) and Q(t) signals in the phase quadrature procedure described above (Fig. 11 b).
176
x • I
Sri) , 9,(0 '9, (1)
Fig. 15. Fourier transform of the function -i sin f.
received signal
= -c}---- :' outpu{ )
r v-"1- output L _"
Fig. 16. A scheme of a two beam acousto-optical processor in a sending-receiving system.
Figure (16) represents a scheme of the two beam acousto-optical signal processor applied in a sending-receiving system as a match-filter with the phase quadrature signal processing in I and Q channels. The acousto-opitcal characteristic of such systems have not been specially examined yet; however they can be deduced from the results of experiments on light diffraction by two ultrasonic beams described above.
CONCLUSIONS
Modulation of light in diffraction orders caused by two ultrasonic beams has a pecular character depending on the mutual phase shift and geometrical relations between the beams. Some examples of theoretical and experimental results of examinations of the topic were presented here. The application for acousto-optical match-filtering and phase quadrature signal processing has been described as promising possibilities.
REFERENCES
1. P. Debye, and F. W. Sears, Proc. Nat. Ac. Sci. US., 18:410 (1932). 2. P. Biquard, and C. R. Lucas, A. C. Sci., Paris, 195: 121 (1932); J. Phys. Rad.,
3:464 (1932). 3. C. V. Raman and N. S. N. Nath, Proc. Ind. Ac. Sci., A2: 406 (1935); A2:414
(1935); A3:119 (1936); A3: 459 (1936). 4. L. Bergmann, "Der Ultraschall und Seine Anwendung in Wissenschaft und
Technik," Zurich, (1954). 5. M. V. Berry, "The diffraction of light by ultrasound," Academic Press, London,
(1966). 6. A. Korpel, "Acousto-optics," Marcel Dekker, Inc. New York, Basel, (1988). 7. W. I. Balakshii, V. N. Parygin, and L. E. Chirkov, Fizicheskiye Osnovi
Akustooptiki, Radyo i Zvyaz, Moskow, 1985. 8. I. Gabrielli, in the Proc. 4-th Spring School on Acousto-optics, Gdansk, (1989), pp.
69-91. 9. P. Kwiek, "Near and far field in light diffraction by two ultrasonic beams,"
University of Gdansk, D. Sc. Thesis, (1985),1-78 (in Polish); Light diffraction by two spatially separated ultrasonic waves, J. Acoust. Soc. Am 86:2261-2272 (1989).
10. K. Patorski, Ultrasonics 19: 169 (1981); Acustica 52:246 (1983); 53: 1 (1983). 11. E. Bloome, "Theoretical study of light diffraction by one or more ultrasonic waves in
the MHz region," D. Thesis, KU Leuven, (1987). 12. P. Kwiek, A. Markiewicz, in the Proc. 4th Spring School on Acousto-optics,
Gdansk, (1989); World Scientific, Singapore (1990), pp. 129-151. 13. R. Mertens, Z. f. Phys. 160:291 (1960). 14. L. E. Hargrove, E. A. Hiedemann, and R. Mertens, Z. F. Phys., 167:326 (1962). 15. O. Leroy, 1. Sound, Vibr. 26:289 (1973); 32:241 (1974); Acustica 29:303 (1973). 16. O. Leroy, E. Bloome, P. Kwiek, A. Markiewicz, and A. Sliwinski, Proc. ultr.
Intern. 81, Brighton IPS, Sc. Techn. Press Ltd, (1981), pp. 98-102. 17. O. Leroy, and E. Bloome, Ultrasonics 19:172 (1981); Ultrasonics 22:125 (1984). 18. O.-teroy, A. Sliwinski, P. Kwiek, and A. Markiewicz, Ultrasonics 20:135 (1982).
177
19. E. Blomme, P. Kwiek, O. Leroy, A. Markiewicz, and A. Sliwinski, Proc. Utr. Intern. 85, London, Butterworth (1985) p. 533.
20. F. Calgaris, P. Ciuti, and I. Gabrielli, Acustica, 38:37 (1977). 21. I. Gabrielli, and P. Ciuti, Proc. 1st Spring School on Acousto-optics, Gdansk,
(1980) p. 120. 22. I. Gabrielli. and P. Ciuti, Proc. 2nd Spring School on Acousto-optics, Gdnask,
(1983) p. 84. 23. A. Sliwinski, "Acousto-optics and its perspectives in research and applications,"
R.W.B. Stephens Lecture, Proc. Inst. Acoust., Portsmouth 87; 9: 15-60 (1987), Ultrasonics, 28(1990) in press.
24. T. H. Neighbors, and W. G. Mayer, J. Acoust. Soc. Am. 74: 146 (1983). 25. O. Leroy, and E. Bloome, Proc. of the 2nd Spring School on Acousto-optics;
Gdansk, (1983) pp. 125-157. 26. F. Calligaris, P. Ciuti, and I. Gabrielli, Acustica 38: 37-43 (1977). 27. P. Kwiek, A. Markiewicz, and A. Sliwinski, Acoust. Lett, 3:16-22 (1979); 3: 164-
167 (1980). 28. I. Gabrielli, P. Kwiek, A. Markiewicz, and A. Sliwinski, Acustica 66:281-285
(1988) . 29. I. Gabrielli, P. Kwiek, A. Markiewicz, and A. Sliwinski, Proc. Ultr. Intern. 89,
Madrid, Butterworth (1989) pp. 230-235. 30. A. Sliwinski, P. Kwiek, and A. Markiewicz, eds., Application of light diffraction by
two ultrasonicc beams for singal processing, in "Acousto-optics and Applications," Proc. 4th Spring School on Acousto-optics, Gdansk-Sobieszewo, 1989, World Scientific, Singapore, 1989, pp. 407-418.
31. N. J. Berg, and J. N. Lee, Editors, "Acoustooptic Signal Processing, Theory and Implementation," Marcel Dekker Inc., New York, (1983).
32. N. J. Berg, Proc. Ultrasonics Intern. 85 London, Butterworth, (1985), pp. 183-188. 33. C. Garvin, N. J. Berg, and R. Felock, ibid., 429-434. 34. I. C. Chang, ibid., 175-182. 35. V. N. Parygin, and W. I. Balakshii, Optitcheskaya orbrabotka informacyi, izd.
Moskovskovo Universiteta, (1987). 36. T. C. Poon, in the "Proc. 4th Spring School on Acousto-optics, Gdansk, 1989,"
World Scientific, Singapore (1990), pp. 213-228. 37. J. Koziowski, ibid., 327-334. 38. V. B. Voloshinov, ibid., 335-349. 39. F. K. Lam, Ultrasonics 25:166-171 (1987). 40. N. Thrane, The Hilbert Tranform, Techn. Rev. B. K., 3:3-15, (1984) .
178
PHOTOACOUSTICS APPLIED TO LIQUID CRYSTALS
1. Thoen, E. Schoubs, V. Fagard
Laboratorium voor Akoestiek en Warmtegeleiding Departement Natuurkunde Katholieke Universiteit Leuven Celestijnenlaan 200D B - 300 1 Leuven Belgium
INTRODUCTION
Liquid crystals are composed of organic molecules of asymmetric sha~e which do not melt in a single stage from the crystalline state to an isotropic liquid. I, For the most common type of liquid crystals the shape anisotropy is prolate. In this case the liquid crystalline phases are characterized by orientational order of the long molecular axes of the rod-like molecules. A variety of mesophases with symmetries and properties intermediate between those of a crystal and a normal isotropic liquid can be present. Differences in the orientational and spatial ordering of the molecules define the mesophases. The nematic (N) phase has the translational symmetry of a fluid but a broken rotational symmetry characterized by long-range orientational order produced by the alignment of the long molecular axes along a unit vector called the director. In the nematic phase the centers of mass of the molecules are, however, still randomly distributed. This is not the case anymore for the many different kinds of smectic phases which show layered structures described by a one-dimensional density modulation.3 One of the most common smectic phases is the smectic A phase which has a layered structure with layer planes perpendicular to the director. Within the layers there is no long-range order in the position of the centers of mass of the molecules. In the related smectic C phase there exists a tilt angle between the direction of the normal to the layers and the director. Here also in the layers there is no long-range positional order of the centers of mass of the molecules. For other types of smectics, however, positional order within the layers can be present.3
The measurements of thermal quantities of liquid crystals playa significant role for locating the different phases,and phase transitions. Although differential scanning calorimetry is rather well established and of great practical importance, it is not very reliable for detailed studies of pretransitional behavior. High resolution calorimetric measurements, in particular near phase transitions are usually carried out by adiabatic scanning calorimetry4 or a.c. calorimetric techniques.5,6 These methods give information only on the static quantities enthalpy and heat capacity. In a more complete thermal characterization one would also like to get information on thermal transport properties such as the thermal conductivity and thermal diffusivity. This dynamic thermal behavior of liquid crystals has not been studied in much detail. Conventional steady state gradient and transient techniques have been used in a number of cases to arrive at thermal conductivity results.7-9 These methods have the disadvantage of requiring large samples and large temperature gradients, which makes them unsuitable for phase transition investigations. Some high resolution a.c. techniques; e.g., forced Rayleigh light scattering, can be used to obtain the
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 179
thennal diffusivity or the thennal conductivity.lO,11 Ideally one would like to use a high resolution technique which allows for simultaneous measurements of both the heat capacity and the thennal conductivity.
Although some other possibilities exist,12,13 the fact that in a photoacoustic experiment the signal depends on the heat capacity C and the thennal conductivity of the sample,14 make it a very attractive alternative. In a photoacoustic experiment for a given liquid crystal sample, the signal originates from a surface layer with a thickness characterized by the thennal diffusion length given by 112 = 2ajro where ex = KlpCp' is the thennal diffusivity, p is the density and ro the modulation frequency of the absorbea optical radiation. Even for quite low (audible) frequencies, 11 is usually in the micrometer range for liquid crystals, and samples of small size can be used (typically lO-lOOmg). By simultaneously measuring the amplitude and the phase of the photoacoustic signals and proper analysis of the data, one can separate the K and C contributions of the sample.
We have carried our photoacoustics investigations of several liquid crystal compounds using a completely automatic, Personal Computer controlled, photoacoustic setup with microphone detection. 15,16 In this paper we present new data for octylcyanobiphenyl (8CB) and for mixtures of this compound with nonylcyanobiphenyl (9CB) the next compound in the same homologous series. Our experimental set-up also has been modified. Previously, a sample orienting magnetic field could only be applied parallel to the sample surface. In the modified experimental configuration a field direction perpendicular to the surface of the sample also is possible.
ME1HOD AND EXPERIMENTAL SET-UP
The photoacoustic effect is based on the periodic heating of the sample induced by the absorption of modulated or chopped (electromagnetic) radiation. In the gas microphone detection configuration the sample is contained in a gas-tight cell. The thennal wave produced in the sample by the absorbed radiation couples back to the gas above the sample and periodically changes the temperature of a thin gas layer above the sample surface. This results in a periodic pressure change which can be detected by a microphone. The theory for the photoacoustic effect has been deyeloped by Rosencwaig and Gersho. l4 For the photoacoustic microphone signal Q = qe-I\V, with amplitude q and phase \jf (with respect to the radiation modulation) the following equation holds:
Q= l1~loYgPO .[(r-IXb+I)ecrl-(r+IXb-I)e-crl+2(b-r)e-~l] (1)
2a ToKIgag (~2 _ cr2) (g + I)(b + I)ecrl - (g - IXb - 1) e-crl
In this equation 10, PO, TO, Yg' are respectively the radiation power density, the cell ambient pressure and temperature and the ratio of the specific heats at a constant pressure and volume of the gas. l:l is the (optical) absorption coefficient of the sample, 1 is the sample thickness, Ig is the thickness of the gas column above the sample in the cell and cr = (1 + i)a, with a = 1111 the thennal diffusion coefficient. 11 is the thennal diffusion length,l1 = 1 - R is the photothennal conversion efficiency, with R the reflectivity. One further has: b = Kb ab I1/K, g = Kg ag I1/K, r= (I-i)~11/2. Here, and also in Eq. (1), quantities without subscript refer to the sample and the subscripts g and b refer to the cell gas and the sample backing material. When the sample is optically and thennally thick (ecr1»e-crl , e-~l), it was shownl7 that in Eq. (1) the contribution of the backing material disappears and one has the following simplified results:
q = 11ygPolot {2t2 + 2t + IT 1/2
2a TOlg (1 + s) Kgag2
tan \jf = 1 + lit
(2)
(3)
In equation (2) and (3) one has t = 11~/2 and s = aK(Kg agrl . A schematic diagram of our experimental setup is given in Fig. 1. The actual
photoacoustic measuring cell is given in Fig. 2. The sample and the reference material are contained in disklike slots (with a diameter of 8mm and a depth of O.3mm) in the same
180
-=~ I I
Fig. 1. Schematic diagram of the photoacoustic set-up. s refers to the liquid crystal sample and r to the reference material. The arrow B gives the direction of the magnetic field parallel to the sample surface.
removable gold plated copper sample holder. A He-Ne laser operating at a wavelength A= 3.39 11m was chosen because it coincides with the strong absorption band of the C-H groups of the liquid crystal compounds. For the generation of a photoacoustic signal form the reference material for which we used a different liquid crystal in the isotropic phase, the same laser could be used. The switching from the sample to reference, or from one laser beam configuration to the other, can be done automatically and is under the control of a personal computer.
Temperature sensing and electric heaters have been incorporated in the brass body of the photoacoustic measuring cell of Fig. 2. The temperature is measured and controlled via digital multimeters and programmable power sources properly interfaced with the Pc. An algorithm adjusts every five seconds the heating power in a proportional integrated manner. Stabilization at fixed temperatures as well as scanning rates as low as a few mK/min can be programmed. The laser beam is modulated mechanically with a chopper. Modulation frequencies between a few hertz and 5 kHz can be set via the computer program. The amplitude and the phase of the PA signals are measured with a dual phase lock-in amplifier in connection with the PC where they are collected and stored for further analysis. At each
Fig. 2. Photoacoustic measuring cell with microphone detection a. brass cell body, b. sample, c. sample holder, d. reference material, e. quartz window, f. electret microphone, g. microphone holder, h. epoxy,
i. microphone leads.
181
measuring point five data values for the sam~le and one of the reference material are usually collected. In our previous experiments lS, I we could only apply a magnetic field parallel to the surface of the sample (see Fig. 1). Recently we have modified our set-up and a different configuration with the magnetic field of a small electromagnet perpendicular to the surface of the liquid crystal sample is also possible.
RESULTS AND DISCUSSION
A photoacoustic investigation was carried out for octylcyanobiphenyl (8CB) and nonylcyanobiphenyl (9CB) and for some of their mixtures. Both 9CB and 8CB have a smectic A as well as a nematic phase below the isotropic liquid phase. The results which we report here were obtained for a modulation frequency of 80Hz and for samples with thicknesses of about O.5mm. For this thickness, modulation frequency, and the laser wave length A = 3.39 /lm, the conditions for thermal and optical thickness were largely satisfied.
Both the direct amplitude and phase results have been corrected for the temperature dependence on the cell characteristics on the basis of calibration measurements with hexylcyanobiphenyl (6CB) in the isotropic phase. This is necessary because the sensitivity of the electret microphone, incorporated in the cell body, to reduce the gas volume and ensure high sensitivity, is temperature (and frequency) dependent. The normalization for the T -dependence of the system sensitivity fsys and the system phase shift 'V sys can be obtained from qref and 'Vref for the reference material, which can (within a few percent) be considered as temperature independent for the temperature ranges involved. The following relations are then applicable:
~~~T) = fsys(T) . qref
~~T) = 'Vsys(T) + 'Vref
q~T) = fsys(T) . qs (T)
'l'T(T) = 'V sys(T) . 'V s (T)
(4)
(5)
(6)
(7)
The superscript m refers to the direct experimental signal values. The index s explicitly refers to the sample. Combining the above equations, we obtain the correct (not influenced by the temperature dependence of the measuring system) photoacoustic signal parameters for the sample:
qs(T) = qref . qT (T) / ~~f (T)
'Vs (T) = 'Vref +'Ifs" (T) - ~f(T)
(8)
(9)
In Eqs. (8) and (9) one still needs values for qref and 'Vref. In principle these should be obtainable from the material properties of the reference material and different parameters (e.g. the laser light intensity 10 ) of the experiment. It is, however, much easier to use a reference point fcir the sample, e.g. in the isotropic phase, and calculate ~ef and 'Vref which are assumed to be temperature independent, at that temperature from the known (or assumed) sample properties. In this case the absolute accuracy of the derived quantities depend on the accuracy of the sample quantities at the chosen reference point. The temperature dependence of the measured sample quantities is, however, not affected by it. For the data displayed in the figures the highest temperature in the isotropic phase was chosen as reference point.
In Fig. 3 results for the amplitude q and the phase 'V are given for pure 8CB and 9CB and for three different mixtures of the compounds of the same homologous series. These results have been obtained without a special treatment of the sample holder and in the absence of a magnetic field. The two phase transitions (nematic - isotropic at high temperatures and nematic - smectic A at lower temperatures) can be clearly seen as anomalies at the expected temperatures. 18 The results for the temperatures of the anomalies can be used to construct the phase diagram for this binary liquid system. The phase d~agram is given ~n. Fig.~. Th~took1surement of th~ data for each of the samples with different cOmpOSitIOn given 111 • ,g. 3, only a couple of hours and shows that
182
1.4
1.3 -
1.2 -
1.1 -
" :l lIS '-'
til 0.9 0 ;:)
f-< 0 .8 oJ 0.. ~ 0.7 «
0.6
0.5
7
6
5
til 4 rn « X 0. 3
2
30 32 34 36 38
o o
40 42
TEMPERATURE (C)
44 46 48 50
Fig. 3. Temperature dependence of the amplitude and the phase of the photoacoustic signal for different mixtures of the liquid crystal compounds 8CB and 9CB. The crosses and the squares are, respectively, the pure compounds 8CB and 9CB. For the other symbols the following percentages of 8CB apply: 70.0% for the pluses,
33.8% for the diamonds and 16.6% for the triangles.
183
51
50
49
41'1
47
46
~ 15 u .:... 44
~ 43
~ 42 « 0:: 4 1 [,J
"" 40 ::< W 39 ...
38
37
36
35
34
33
32
0 20 40 60 80
MASS PERCENTAGE OF BCB
Fig. 4. The phase cliagram for mixtures of 8CB and 9CB derived from the photoacoustic results of Fig. 3.
100
photoacoustics can be a valuable alternative for locating phase transitions in liquid crystal systems.
The temperature dependence of the heat capacity Cp and of the thermal conductivity can be deduced from the amplitude q and the phase \jf VIa Eqs. (2) and (3).15,16 In Fig. 5 the results for Cp and lC for 8CB in the absence of a magnetic field are compared with results in the presence of a magnetic field B parallel to the sample surface. For the B = 0 case one should normally expect a polycrystalline-type distribution of orientationally and positionally (for the smectic A phase) ordered blocks in the saIBple. For a random distribution of domains one expects an average thermal conductivity lC = (lCll + 2lC.d / 3 for uniaxial mesophases. 1,2 In Fig. 5 one notes a substantial increase of the thermal conductivity in going from the isotropic phase to the nematic phase. The difference is much larger here than typically observed between K and lCis (the thennal conductivity for the isotropic phase) in the bulk of large size samples,1O and points in the direction of lCll for the N phase, suggesting a surface induced perpendicular alignment of the long molecular axes at the liquid-gas (and the liquid-solid) interface. For parallel to the sample surface, we observe in the N phase a smaller lC (B *" 0) value than lC (B = 0). For this field configuration, one would expect a parallel alignment of the molecules and a lC.L value for completely oriented samples. For this small Il == 17 Ilm value, B (of the order of 0.1 T) is apparently not strong enough to impose a complete parallel orientation of the molecules and arrive at the true lC.L values. That the parallel alignment is better realized away from the surface, could be concluded for 7CB on the basis of measurements as a function of modulation frequency.16 Indeed, for lower co values (larger Il value) smaller lC (B *" 0) values (closer to lC.L) were obtained. For the heat capacity one does not expect a field effect and the differences observed between Cp (B = 0) and ~ (B *" 0) are a consequence of the analysis assuming an homogeneously aligned sample, which is, as pointed out above, apparently not the case. For the SAthe magnetic field B is not large enough to change the surface induced orientation and produce a sample with layer planes perpendicular to the surface of the liquid-gas interface (at the top) and the liquid-solid interface (at the bottom).
184
2.6 ,---------------------------------------------------------------,
2.7
2.6
1.9 2l I - - ------- .---.-.- ------------ ---- --/
0.26
0 .25
~ 'Il 0.24 E "-2.. 0 .2 3
)0-
r 0.22 ;; ~ () ::> o z o U
0.2 1
0 .2
..l 0.19 < ~ 0.18 t<l
~ 0.17
0.16
0.15
0.14
I
29 31 33 35 37 39 41 43
TEMPERATURE ('C)
Fig. 5. Temperature dependence of the heat capacity and the thennal conductivity of an 8CB sample as derived from photoacoustic measurements. The squares are the results without a magnetic field. The pluses represent data for a magnetic field
parallel to the sample surface.
45
185
2.8 ~------------------------------------------------~
2.7
2 6
Q Of 2.5 -;.: \ ~ 2.4
~ 2.3
~ 0. 2.2 « u E- 2.1 « w :c 2
1.9
0. 27
0. 26
QO .25 -~
E 0 .24 -\ .3 0 .23 -
~ 0 .22
> i= 0 .21 U ;J 0 .2 o ~ 0 .19 u
0 . 18 .J « ::l 0. 17 a: W 0.16 X !- 0. 15
+ +
[IIJ +
o + +0
o d +
o
+ o
+
o
Q.
0 . 14 ~---.---.---.---r---.--.---.---'---'---'---'---'---'---'---'---~
29 3 1 33 35 37 39 41 43 45
TEMPERAT URE ee)
Fig. 6. Temperature dependence of the heat capacity and the thennal conductivity of a 8CB sample as derived from photoacoustic measurements. The squares are the results
without a magnetic field. The pluses represent data for a magnetic field perpendicular to the sample surface.
186
To arrive at completely homogeneous parallel samples one would need much larger magnetic fields. It would thus be much easier to arrive at completely homogeneously oriented samples in a field direction perpendicular to the surface of the sample. Recently we have modified our experimental set-up in order to arrive at such a magnetic field configuration. In Fig. 6 the results for c;, and I( for B = 0 and B #; 0 perpendicular to the sample surface are compared. Within the experimental uncertainty there are no differences between the two cases, indicating that the surface alignment alone produces a nearly homogeneous sample with the director perpendicular to the sample surface. Measurements at different modulation frequencies (different J.I. values) gave the same results. One should also note that results of Figs. 5 and 6 were obtained for different samples. The sample used for the measurements of Fig. 5 was less pure than the one for Fig. 6, which is indicated by the lower transition temperatures and rounded heat capacity peaks. From the results of Fig. 6 one can conclude that, with the given configuration, one obtains values for 1(//. In order to arrive at the the true 1(1. values (from B parallel alignment) one would need larger magnetic field strengths. Our results also show that from detailed measurements as a function of the modulation frequency (probing at different J.I.) interesting information on surface effects can be obtained.
REFERENCES
1. P.G. de Gennes, "The Physics of Liquid Crystals," Clarendon Press, Oxford (1974). 2. G. Vertogen and W. H. de Jeu, "Thermotropic Liquid Crystals, Fundamentals,"
Springer Verlag, Berlin (1988). 3. G. W. Gray and J. W. Goodby, "Smectic Liquid Crystals," Leonard Hill, London
(1974). 4. J. Thoen, Phys. Rev. A, 37:1754 (1988). 5. K. J. Stine and C. W. Garland, Phys. Rev. A, 39:3148 (1989). 6. C. C. Huang, G. Nounesis, R. Geer, J. W. Goodby and D. Guillon, Phys. Rev. A,
39:3741 (1989). 7. V. S. V. Rajan and J. J. C. Picot, Mol Cryst. Liq. Cryst., 20:55 (1973). 8. R. Vilanove, E. Guyon, C. Mitescu and P. Pieranski, J. Phys. (Paris), 35:153
(1974). 9. T. Akahane, M. Kondoh, K. Hashimoto and M. Nagakawa, Japan, J. Appl. Phys.,
26:1000 (1987). 10. W. Urbach, H. Hervet and F. Rondelez, Mol. Cryst. Liq. Cryst., 46:209 (1978). 11. M. B. Salamon, P. R. Garnier, B. Golding and E. Buehler, J. Phys. Chern. Solids,
35:851 (1974). 12. C. C. Huang, J. M. Viner and J. C. Novak, Rev. Sci. Instrum., 56:1390 (1985). 13. G. Koren, Phys. Rev, 13:1177 (1976). 14. A. Rosencwaig and A. Gersho, J. Appl. Phys., 47:64 (1976). 15. C. Glorieux, E. Schoubs and J. Thoen, Mat. Sc. and Eng., A122:87 (1989). 16. J. Thoen, C. Glorieux, E. Schoubs and W. Lauriks, Mol. Cryst. Liq. Cryst.,191:29
(1990). 17. M. Marinelli, U. Zammit, F. Scudieri, S. Martellucci, J. Quartieri, F. Bloisi and L.
Vicari, Nuovo Cimento, D9:557 (1987). 18. J. Thoen, H. Marijnissen and W. Van Dael, Phys. Rev. Lett, 52:204 (1984).
187
RECENT DEVELOPMENT OF ULTRASONIC MOTORS
Sadayuki Ueha Tokyo Institute of Technology 4259 N agatsuta Midoriku, Yokohama 227, Japan
INTRODUCTION
A conventional ultrasonic motor consists of a rotor and a stator, the stator being ultrasonically exited by piezoelectric elements. The rotor is pressed to the stator and forced to rotate by the frictional force between them. Such an ultrasonic motor was first proposed by Barth in 1973, 1 and an improved model was made by Sashida in 1981.2 Many kinds of ultrasonic motors since have been proposed. In this paper, after brief description of the principle, several typical motors are described together with newly developed ones. The trends of research on ultrasonic motors also are reviewed.
FUNDAMENTAL PRINCIPLE AND FEATURES
If a rotor is pressed to a vibrating stator, the rotor contacts the stator only intermittently as shown in Fig. 1, because the time constant of the rotor is larger than the period at which the vibrating stator is driven. If the rotor is movable along the contacting surface and the stator has a displacement component parallel to the surface, the frictional force between the two makes the rotor move during some portion of a cycle. For the remainder of the cycle, the compressive force between the two parts is reduced, and the rotor continues the motion.3 This is the fundamental principle of the ultrasonic motor.
pre-Ioad (b) (8) Y
/rotor
"0 Uo I x i IIIIIIII II, /////1//111
stator I
locus of particle _time
Fig. 1. Schematic diagram for explaining the fundamental principle: (a) geometry; (b) time chart of displacement.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 189
Ultrasonic motors are expected to have the following features:
(a) High torque, high efficiency at low speed; (b) High torque-to-weight ratio; (c) Versatility of shapes; (d) Position maintained when the electric power source is turned off; (e) Good controllability due to high torque and low inertia; (f) Capability of direct drive without gear train; (g) No electromagnetic interference; (h) Low acoustic noise even at heavy load operation.
TYPICAL ULTRASONIC MOTORS
Standing Wave Ultrasonic Motor (SWUM)
Particles on a surface vibrating in a standing wave motion usually move linearly and the direction of motion depends on their position along the standing wave, as shown in Fig. 2. If the rotor contacts the stator only at certain positions whose Lissajous figures are coincident with each other, it must move in only one direction. Such contact can be accomplished by means of bumps, for example.4 It should be noted that the direction of movement is determined by the position of the bump. This type of motor is applicable to an axial flexural vibration of a ring or plate, and also to a longitudinal vibration of bar.5,6 If active bumps positioned at nodal points are employed, as shown in Fig. 3, and if the phase difference between the flexural and the longitudinal vibrations is properly adjusted, bi-directional motion of the motor becomes possible.? In this case the motor falls into the category of hybrid ultrasonic motor.
Traveling Wave Ultrasonic Motor (IWUM)
A wave traveling along a bounded solid body is recognized as a coupled vibration between longitudinal and transverse ones, and the particle on the surface moves elliptically. A slider or rotor pressed on the surface is forced to move by frictional forces as shown in Fig. 4. The direction of motion is determined by the motion of the particles. To excite a traveling wave in a straight waveguide, an impedance-matched load should be terminated at the end of the waveguide.8,9 A truck-shaped waveguide can also be used to obtain a traveling wave. 10
190
a
b
Fig. 2. The principle of SWUM: (a), standing wave excited in an elastic bar; (b), driving direction along the bar; (c) and (d), selection of
motion by using bumps.4
Multilayered -Piezoe/eclric _ Aclualor
Duralumin
Counler We/glll
PZT
Fig 3. An example of SWUM which has active bumps in order to obtain bi-directional motion.
z
wave
~- - - ---- .. --" .;" ~ matertaI
Fig. 4. Scheme of traveling wave ultrasonic motor.
Mode Conversion Ultrasonic Motor (MCUM)
This type of ultrasonic motor utilizes a mode conversion to obtain the two vibration modes from a single piezoelectric element. There are several kinds of MCUM. One utilizes a vibrating piece which is attached at the front end of a longitudinal ultrasonic vibrator as shown in Fig. 5.2 The piece touches the rotor at a slight angle between the normal to the rotor and the axis of the piece. To make an elliptic Lissajous figure, the tip of the piece is forced to bend along the motion direction of the rotor. The second one utilizes a torsion coupler which converts a longitudinal vibration to a torsional vibration. I I The tip of the coupler moves elliptically as a combination of torsional and longitudinal vibrations. To obtain a bending vibration from a longitudinal vibration, an eccentric vibration system is also used. 12 These motors are driven by a single power source and are difficult to reverse without a change of frequency.
Longi t udinal v i brator
Vi bratory piece
V
++
.Y
r a
x
Fig. 5. An example of MCUM with a vibratory piece.2
191
Concenlric shear mode
Poling direct ion
Radial mode
Fig. 6. Coupled vibration of concentric shear mode (a) and radial extensional mode (b): (c) and (d) respectively show poling direction and electrodes for
simultaneous excitation.8
Multi-Mode Ultrasonic Motor CMMUM)
This motor utilizes a coupled vibration or a multi-mode vibrator excited by one or two piezoelectric elements. Several vibration modes can be used; one employs a longitudinalflexural coupled vibration of a circular hollow cylinder.13 Figure 6 shows an example of coupled vibration of concentric shear and radial modes generated in a circular piezoelectric element.5 The particles of the outer surface move elliptically if the driving conditions are appropriate. Other ultrasonic motors use: a coufling of non-axisymmetric vibration with radial extensional vibration of a circular plate; I longitudinal and flexural vibrations of a cross-shaped vibrator;5 two bending vibrations of a plate or a tuning fork with arms of different lengths. 15 Figure 7 shows a newly developed 1t-shaped linear actuator. 16 The piezoelectric elements excite both longitudinal and flexural vibrations of the "legs" and this kind of motor now is successfully used for x-y positioners.
192
Fig. 7. 1t-shaped ultrasonic linear motor which utilizes a coupled bending and longitudinal vibration of "legs".16
- cos WI -- - . ... - sin wi
cos WI
t t
cos WI
PZT
- sin WI
Fluid
( I-----J?-Fig. 8. A motor employing extensional vibration of ring and liquid coupling.18
Mode Rotation Ultrasonic Motor (MRUM)
This motor utilizes doubly degenerate vibration modes where the modes are excited by a phase difference of 90°, resulting in mode rotation.3,1? Figure 8 shows a newly developed motor which employs an extensional vibration of a ring and uses liquid coupling.18 Division of the electrodes of the piezoelectric element to achieve the rotation also is illustrated.
Figure 9 shows an example of a motor which employs a degenerate non-axisymmetric vibration mode. 19 Many other vibration modes can be used for MRUM: flexural and extensional vibrations of rings20, 21 and flexural vibration of a hollow cylinder.22
(a)
(b)
Fig. 9. An example of motor which employs de~enerate non-axisymmetric vibration of an annular plate.1g-
193
'" Uultik1yered lStator PitIl«/ectTic kllMltor TOI'SiomI Villnltor
Fig. 10. An example of a HTUM.24
Hybrid Transducer Ultrasonic Motor CHTUM)
This motor utilizes two separate vibrations which are independently excited by two separate piezoelectric elements. There are several kinds of HTUM. A stator which is composed of a torsional vibrator and a longitudinal multi-layered cylindrical piezoelectric actuator are used for the motor as shown in Fig. 10.23. 24 The actuator is bonded on the flat surface of the vibrator. In this case, the thrust is generated by the vibrator, and the actuator acts as a clutch to control the frictional force. A sandwich transducer is used for the stator, which contains longitudinal and torsional piezoelectric elements.25. 26 The combination of two longitudinal vibrations whose polarizations are orthogonal is used for a linear motor.27 The combination of two orthogonal bimorph vibrators is also used for a linear ultrasonic motor. 10
TRENDS OF ULTRASONIC MOTOR RESEARCH
The following are the trends of motor research listed in random order.
Optimization of design by numerical analysis
Both for establishment of design procedure and for improvement of performance, precise theoretical treatment of the operation of the ultrasonic motor is necessary. These investigations are now made by using electrical equivalent circuits.28. 29 The precise vibration amplitude distribution of motors for optimum operation also are studied numerically and experimentally.
Limit estimation of specifications
Limits to performance shown by torque, efficiency and output power should be investigated. The mechanical output power, for example, now ranges from several watts to several tens of watts. But these values are not the minimum nor maximum ones, because a single ultrasonic vibrator, which is similar to a stator in an ultrasonic motor, has a maximum output power of a few KW30 and a maximum cumulative power of about 50 KW. The limits of other specifications should be studied experimentally and/or theoretically.
Study of controllability
A few studies on such dynamical properties as repetition accuracy, wow and flutter have been reported.31 Successful a££lications of ultrasonic motors to positioning apparatus have been reported recently. .32 This trend will continue.
194
Improvement of efficiency
The efficiency of TWUM has been studied analytically and experimentally.33 The analysis shows that both incorrect loading of the stator and slippage between rotor and stator are the reasons for relatively low efficiency. Since the stator can operate at a high efficiency of 96%,30 at least 90% efficiency should be possible ifthe loading is optimized, for example, by changing the construction or vibration amplitude distribution of the stator. Little theoretical study has been done on the efficiency of other motors, so analytic investigation should be made concurrently with an evaluation of the mechanism of vibratory frictional thrust.
Scale up and/or miniaturization of ultrasonic motors and adoption of new vibration modes
Appropriate applications of ultrasonic motors must be sought which utilize the special features listed,34 and optimization of these motors for special applications also is necessary.
Development of friction material with long wear
As every performance of an ultrasonic motor depends mainly on the frictional material used, effort should be made to find, or develop, material with a large frictional coefficient which affords long wear.35, 36 The coefficient of friction, which depends on the materials and the conditions of the contacting surfaces, must be considered in any evaluation of these factors. The highest values measured to date are from 0.2 to 0.3.
SUMMARY
The principle and newly developed ultrasonic motors are briefly described together with the research trends. The ultrasonic motor is now beginning to demonstrate its potential. In a few years the motor will be used in applications which take advantage of the special features of ultrasonic motors.
REFERENCES
1. H. V. Barth: IBM Technical Disclosure Bulletin, 16:2263 (1973). 2. T. Sashida: Ohyo Butsuri, 51:713 (1982) (in Japanese). 3. S. Ueha: 1. Acoust. Soc. Jpn., 44:519 (1988) (in Japanese); Proc. 1989 IEEE
Ultrasonics Symposium 2, 749 (1989). 4. T. Iijima, M. Wada, Y. Nakagawa and H. Itho: Jpn. J. Appl. Phys. 26, Suppl. 26-
1:191 (1987). 5. Y. Tomikawa, T. Ogasawara, S. Sugawara, M. Konno and T. Takano: Jpn. J.
Appl. Phys., 27, Suppl. 27-1:195 (1987). 6. Y. Tomikawa, T. Ogasawara, S. Sugawara, T. Takano, M. Konno and K. Toda:
Preprints of the Annual Meeting of the Acoustical Society of Japan, Tokyo, 1988, 1-6-18 (in Japanese).
7. K. Nishita, M. Kurosawa and S. Ueha: Reprint of Annual Meeting of the Acoustical Soc. of Jap., (May, 1990) pp. 675-676 (in Japanese).
8. M. Kuribayashi, S. Ueha and E. Mori: 1. Acoust. Soc. Am., 77:1431 (1985). 9. M. Kurosawa and S. Ueha: Ultrasonics, 27:39 (1989). 10. Y. Tomikawa, T. Kondo, T. Ogasawara, S. Sugawara and M. Konno: Jpn. J.
Appl. Phys., 26:194, Suppl. 26-1 (1987). 11. A. Kumada: Jpn. 1. Appl. Phys. 24:739, Suppl. 24-1 (1985). 12. M. Masuda, M. Kuribayashi and S. Ueha: Preprints of Annual Meeting of the
Acoustical Society of Japan, Akita, 1986, 1-2-17 (in Japanese). 13. A. Ukita and S. Ueha: J. Acoust. Soc. Jpn. 44:173 (1988) (in Japanese). 14. T. Takano, Y. Tomikawa and M. Masuda: Jpn. 1. Appl. Phys., 27, 192:Suppl. 27-
1 (1987). 15. K. Uchino, K. Kato, K. Imaizumi and M. Tohda: 1. Ceram. Soc. Jpn., 12:1131-
1136 (1988) (in Japanese).
195
16. K. Onishi and Y. Kimura: Preprint of Annual Spring Meeting of lEE Jap., 7:148, (1990).
17. M. Kuribayashi and S. Ueha: Preprints of Annual Meeting of the Acoustical Society of Japan, Akita, 1988, 1-2-15 (in Japanese).
18. K. Nakamura, T. Ito, M. Kurosawa and S. Ueha: Jpn. 1. Appl. Phys., 29;1:L160-L161, (1990).
19. T. Takano, Y. Tomikawa, T. Ogasawara and H., Hirata: Proc. IEEE Ultrasonics Symposium, 2:735 (1989).
20. Y. Ise: J. Acoust. Soc. Jpn. 43:184 (1987) (in Japanese). 21. S. Iwamatsu, S. Ueha and M. Kuribayashi: Jpn. J. Appl. Phys. 25, Suppl. 25-
1:174 (1985). 22. M. Kurosawa, K. Nakamura, T. Okamoto and S. Ueha: IEEE Trans. UFFC,
36:517, (1989). 23. M. Kurosawa and S. Ueha: Tech. Rep. IEICE SU87-31: 27 (1987) (in Japanese). 24. K. Nakamura, M. Kurosawa and S. Ueha: IEICE SU88-17:23, (1988) (in
Japanese). 25. S. Ueha, H. Nagashima and M. Masuda: Jpn. 1. Appl. Phys. 26, Suppl. 26-2:188
(1987). 26. O. Ohnishi, O. Myouga, T. Uchikawa, M. Tamegai, T. Inoue and S. Takahashi:
Proc. IEEE Ultrasonics Symposium, 2 :739(1989). 27. M. Kurosawa, H. Ymada and S. Ueha: Jpn. J. Appl. Phys., 28:158, Suppl. 28-1
(1989). 28. M. Kurosawa and S. Ueha: Proc. 13th ICA, Belgrade, Yugoslavia, 3:421, (1989). 29. K. Nakamura, M. Kurosawa and S. Ueha: Proc. 13th ICA, Belgrade, Yugoslavia,
3:425 (1989). 30. E. Mori, S. Ueha, Y. Tsuda, S. Kaneko, K. Okada, M. Masuda and K. Ohya:
Tech. Rep. IEICE (1981) SU81-37 (in Japanese). 31. T. Kamano, T. Suzuki and E. Otoi: Jpn. Appl. Phys., 27, Suppl. 27-1:192
(1988). 32. K. Nakamura, M. Kurosawa and S. Ueha: Preprints of Annual Meeting of the
Acoustical Society of Japan, Tokyo, 1990, 1-1-1, p. 669 (in Japanese). 33. M. Kurowasa and S. Ueha: J. Acoust. Soc. Jpn. 44:40 (1988) (in Japanese). 34. R. M. Moroney, R. M. White and R. T. Howe: Proc. of 1989 IEEE Ultrasonics
Symposium, 2:745 (1989). 35. A. Endo and N. Sasaki: Jpn. J. Appl. Phys., 26, Suppl. 27-1:197 (1986). 36. K. Nishita, M. Kurosawa and S. Ueha: Preprints of Annual Meeting of the
Acoustical Society of Japan, Tokyo, 1988, 1-6-12 (in Japanese).
196
Section II:
COMMUNICATIONS and POSTERS
PHOTOACOUSTIC AND PHOTOTHERMAL CHARACTERIZATION OF
AMORPHOUS SEMICONDUCTORS TIDN FILMS
G. Amato, G. Benedetto, L. Boarino, R. Spagnolo
Istituto Elettrotecnico Nazionale Galileo Ferraris Strada delle Cacce, 91 10135 Turin, Italy
INTRODUCTION
In recent years a great deal of work has been devoted to amorphous semiconductors, the main interest concerning not only their technological potentialities, but also the investigation of their physical properties. Amon~ the various techniques used for thin films characterization, photo acoustic (P A) and photothermal (PT) methods are now widely used. Their most prominent feature is the particular signal generation mechanism, which is directly connected to the non-radiative recombination of photo excited carriers and therefore gives direct information on non-radiative de--excitation energy channels.
One of the main applications of P A and PT techniques in this field is the measurement of subgap optical absorption spectra, which in particular allows to evaluate the density of states of these materials. In general, in amorphous semiconductors the absorption spectrum can be divided into three regions: the high absorption region, which determines the optical gap, the exponential edge, which is related to structural randomness and the weak absorption tail, mostly related to the defect structure. This low-absorption region is directly related to the gap-state profile, which gives important information about transport properties and is strongly related to different parameters, as deposition conditions, impurity levels and post-deposition treatments. Generally, the absorption coefficient a values under consideration are lower than 102 cm -1 and the films are deposited with a thickness f. of 10-4 cm. Therefore absorptions lower than 10-2 have to be measured, which is an unattainable limit for conventional techniques, such as transmission spectroscopy, which are limited to optical absorption measurements of a· f. tv 1.
In this paper we present and discuss some results of photoacoustic spectroscopy (PAS) and photothermal deflection spectroscopy (PDS) measurements of subgap optical absorption coefficient on thin films of hydrogenated amorphous silicon, silicon carbide and alloys a-(Si-Ge):H, a-(C-Si-Ge):H, produced by different techniques.
MEASUREMENTS OF OPTICAL ABSORPTION COEFFICIENT
The measurements are performed by means of an experimental apparatus which has already been described in detail elsewhere [1]. A beam from a 1000 W xenon lamp is filtered by a monochromator, mechanically chopped and focused on the sample surface. A fixed modulation frequency of 10 Hz is used, which allows to consider all the films tested as thermally thin. For P A
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 199
105 100 E -a " 104
:r ::::- .. .. .. .... .. c 103 5 ., ~ V 102 ~ .. 0 .. " ... c ..
101 .. ~ ~ a. ... 0
100 0 til .!l 0.5 1 1.5 2 2 .5 3 ..:
Energy (eV)
Figure 1. Comparison between experimental and calculated spectra for an a-SiC:H film deposited on microscope glass. Upper curve, experimental. Middle, calculated through the model in Ref. 3. Lower, calculated on the basis of 45' phase shift. The phase shift curve is also reported .
measurements, a cell of small dimensions (volume N 500 mm3) is used, which has been designed in order to minimize stray-light effects. These are mainly due to reflections of light on the cell walls and can originate a background P A noise. The cell consists of a cylindrical aluminum body delimited by two quartz windows, the film being deposited on the rear window.
For PDS measurements, the sample is immersed in CC14. The probe beam is a 2 mW He-Ne laser. A lateral position sensor is used to determine the beam deflection. The signals are measured in amplitude and phase by a two-channel lock-in analyzer and referred to the lamp spectrum as it had been obtained on a carbon black sample. Both in PAS and PDS measurements, the signal Q( a) is proportional to the surface temperature of the sample. If the substrate is transparent, this only depends on the mutual relationship between the sample thickness e, the optical absorption length a-I and the thermal
diffusion length It = (fJ / rl) lh, where fJ is the thermal diffusivity and f the modulation frequency . At high a values (a- 1 « e « It), the signal saturates to a value Qs, which does not depend any more on the absorption coefficient of the film and is only related to its thermal parameters and to the measurements set-up. In the range of low a values (e« a-I, It), it is possible to define a normalized signal q( a) = Q( a) / Qs from which, taking into account the multiple reflections of the light in the film, the absorption coefficient can be calculated, according to the relationship:
a _ 1/ e en _r...t2'--_1----=-+_..I..J_->(-.!r ::,,-=---,1 ),-2_-_4+(q~(c:.:.aL) -_1 )"(Lr-"-2-q---"-'.(,,,,,a )~r l""r"-L2) 2(r2 - q(a) rlr2) (1)
where rl and r2 are the reflectance at the interfaces air-sample (CC1 4-
sample in PDS) and sample-substrate. At low a values the substrate contribution can be non negligible. It has
been demonstrated r21 that, under some simplifying hypotheses, there is a constant 45' phase (litference between surface and bulk absorption to the total signal. This allows to separate the bulk and surface contribution. For this purpose, the phase of the surface signal is assumed to be that of the signal in optical saturation conditions . For all the signal values which depend on the absorption coefficient, the total signal is decomposed in two components, one in phase with the saturation signal and the other one shifted of 45'. The latter component is then vectorially deducted from the total signal. This procedure is also valid for very low values of the surface to substrate absorption coefficient ratio, only if the thermal effusivities of film and substrate are equal. In case of some thermal mismatch occurring at the interface, this approximation can give
200
Figure 2.
E ()
" ~ C ., ~ Q; 0 ()
c :3 Q. I-0 (/)
.D <
105
104
103
102
101
100
10- 1 1.25 1.5 1.75 2
Energy (eV)
Absorption spectra obtained by means of PAS and CMS on an a-Si:H film.
an underestimation of the film absorption and a more precise evaluation of this parameter should be based on a complete two-layer model, as the Mandelis' model. Figure 1 reports the absorption spectra of a hydrogenated silicon carbide film, 0.98 /tm thick, deposited by Glow Discharge on a microscope &lass substrate. The details of the application of this model are already described [3] .
APPLICATIONS
Absorption properties of a-Si:H samples deposited at 420· C by Low Pressure Chemical Vapour Deposition (LPCVD) have been investigated by PAS in comparison with Constant Photocurrent Method (CPM). The results obtained by the two different techniques provide information about the density of surface states. The standard explanation of this fact is that , while PAS is related to the energy absorbed through the whole sample, CPM is dominated by the bulk density of states. The absorption spectra determined by means of the two techniques are shown in Figure 2. The authors have shown [4] that , besides different sensitivity to surface states, this difference can reveal inhomogeneities in the material along the growth axis, which are probably due to a change in growth kinetics during the film deposition procedure.
A comparison between amorphous hydrogenated silicon carbide films deposited respectively by Glow Discharge (GD) and reactive Sputtering (SP) technique was performed [5]. The substrate deposition temperature was varied between 100 and 300· C. The absorption coefficient values were obtained from PAS measurements. Figure 3 shows the absorption spectra of SP and GD
a
E ()
" -... c ., ~ ., 0 ()
c 0 ~ Q. I-0 II)
.D <
Figure 3.
b 104 104
103 103
102 102
101 101
100 10° 0 0 .5 1 1.5 2 2.5 3 0 0.5 I 1.5 2 2 .5 3
Energy (eV) Energy (eV)
PDS absorption spectra for Sputtering (a) and Glow Discharge (b) a-CSi:H films, at different substrate temperatures.
201
a
E <.> "--... c ., ~ Q; 0 <.>
C
:§ Q. ... 0 UJ .n ..:
Figure 4.
104 b 104
103 103
102 102
101 101
100 100 0 0 .5 1 1.5 2 2.5 3 0 0 .5 1 1.5 2 2.5 3
Energy (eV) Energy (eV)
PDS absorption spectra for Sputtering (a) and Glow Discharge (b) a-CSi:H films , before and after annealing at 250" C.
deposited samples at two different substrate temperatures (100 and 250' C) . The spectra have a long exponential tail expressed by a = ao exp(hv lEo) where hv is the photon energy and Eo the characteristic Urbach energy. The widening of Urbach edge in SP samples with respect to GD samples can be attributed to a greater compositional disorder of sputtered films.
In order to better understand the motion of hydrogen and its influence on the electronic properties of amorphous silicon carbide, the GD and PS samples were annealed at different temperatures in the range of 100-400' C. Figure 4 shows the optical absorption spectra for samples with a substrate temperature of 100' C, before and after the isochronal annealing at 250' C. For GD samples the characteristic energy Eo shows a sharp decrease after the annealing. The gap energy Eg remains almost constant at values similar to those of as-deposited films . As tar as SP samples are concerned, an increase of Eg and a decrease of Eo have been observed as the annealing temperature increases up to 250' C. The results of PAS were compared to those obtained by different techniques, as electron spin resonance (ESR) , infrared spectroscopy, dark conductivity measurements, and it was possible to infer that annealing is responsible for the possible reorganization of the amorphous network and thus for a reduction in the number of incomplete bonds .
Measurements of optical absorption were made by PDS techniques also on samples of the binary alloy a-SiXGel-X:H for different x values. It was
Figure 5.
202
~
" Ie ~ .. z
Defects density Ns determined from ESR measurements as a function of the parameter R for a-SiGe:H films, where R is the ratio between Ns and the integral of the residual absorption coefficient down to the low energy limit of the spectrum.
Figure 6.
E ()
"--C II>
~ 4; o ()
c: .2 0. ... o '" .0 « 100
0 .5 0 .75 I 1.25 1.5 1.75 2 Energy (eV)
PDS spectra of an a-Si:H film, illuminated from the surface side (lower curve) and from the substrate side (upper curve) .
observed that the silicon content has a large influence on the optical absorption edge. The sub-band-gap absorption indicates a decrease of the amount of defects when increasing x [6]. For a-Si:H it is possible to relate the integral of residual absorption coefficient (after subtraction of the exponential component) down to the low energy limit of the spectrum, with the defects number determined from ESR measurements. A proportionality factor 7.9 10 15
cm-2 eV-I has been fixed for un-alloyed a-Si:H7. From the measurements on a-SiGe:H samples it has been possible to observe a non-linearity between the defects number determined by PDS and the dangling bonds measured by ESR (see Figure 5), what is probably due to the fact that Ge dangling bonds states are not optically active in the same energy range of Si dangling bonds .
An alternative method for the determination of surface states in amorphous thin films from PDS spectra has been suggested [81. The method is based on the observation of enhancement of the interference fringe amplitude due to the presence of surface states. Figure 6 shows the PDS spectra of an a-Si:H sample, obtained for illumination from the film surface side and from the substrate side. The fringe amplitude enhancement in frontside spectrum is due to the presence of a strong defective overlayer on the surface.
CONCLUSIONS
As it is evident from the growing number of publications, photoacoustic and photothermal methods are extensively used for the characterization of amorphous semiconductors. Some improvement is still necessary in order to make photothermal techniques a conventional method for film characterization. However, their particular features (being non-invasive, not requiring particular sample preparation, allowing remote mOnitoring) can justify effort in this direction.
REFERENCES
[1]
[2]
[3]
G. Amato, G. Benedetto, and R. Spagnolo, Photoacoustic spectroscopy of LPCVD hydrogenated amorphous silicon, Proc. 13th Int. Congr. on Acoustics 4:357, Belgrade 1989. B. Mongeau, G. Rousset, and L. Bertrand, Separation of surface and volume absorption in photothermal spectroscopy, Can. J . Phys. 64 :1056 (1986) . G. Amato, G. Benedetto, L. Boarino, M. Maringelli, and R. Spagnolo, Substrate influence in photothermal measurement of thin films absorption, to be published.
203
[4]
[5].
[6]
[7]
[8]
204
G. Amato, G. Benedetto, F. Fizzotti, C. Manfredotti, and R. Spagnolo, Photothermal and photoconductive measurements of LPCVD amorphous silicon, Phys. Stat. Sol. (a) 119 (1990). F. Demichelis, C. Pirri, E. Tresso, and G. Benedetto, Influence of hydrogen on substrate and annealing temperature dependence of a-Si:H films properties, to appear on Phil. Mag. B. F. Demichelis, C.F. Pirri, E. Tresso, and G. Amato, Electron Spin Resonance and Photoacoustic Spectroscopy of a-CSi:H and a-SiGe:H alloys, to appear on Thin Solid Films. W.B. Jackson, and N.M. Amer, Direct measurement of gap-state absorption in hydrogenated amorphous silicon by PDS, Phys. Rev. B, 25:5559 (1982). G. Amato, G. Benedetto, L. Boarino, and R. Spagnolo, Photothermal detection of surface states in amorphous silicon films, Appl. Phys. A 50:503 (1990).
TWO-BEAM BRAGG DIFFRACTION
S.N. Antonov, V.M. Kotov and V.N. Sotnikov
Institute of Radio Engineering and Electronics U.S.S.R. Academy of Sciences Moscow, U.S.S.R.
MONOCHROMATIC POLARIZATION SPLITTING
The Bragg anisotropic acoustooptic (AO) diffraction is accompanied by change of the optical radiation mode, with the interaction parameters (in particular, the Bragg angle) depending on the initial light polarization [1]. We shall examine the conditions under which the Bragg synchronism is satisfied simultaneously for two eigenmodes of the Te02 crystal on one acoustic wave (AW) . First we shall consider a situation when monochromatic light is incident on the crystal, then develop methodology for two light wavelengths each of which corresponds to solution.
Proceedings from the pulse conservation law we write a set of three equations:
[To )( Te] = 0, Ko:!: q = K~, Ke T q = KJ (1)
where To and Te are wave vectors of the incident light in the air for ordinary and extraordinary modes, respectively, Ko and Ke the same vectors
for beams in the crystal, K~ and KJ are wave vectors of the diffracted radiation in the crystal and q is a wave vector of sound. The "+" in Eq.(l) means raising of the diffracted beam frequency by a value of sound frequency and "_" its lowering by the same value. Figure 1 illustrates fulfilling of equation set (1), reflecting the experimental geometry of interaction in the Te02
[0011
28
Figure 1. The vector scheme of simultaneous satisfaction for two eigenmodes of the crystal.
Physical Acoustics. Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 205
crystal, i.e. slow shear A W propagates along [110] directions with deformation
along [flO] axis and the light beam propagates in the plane containing rOOI] and [110]. Here no and ne are the major refraction indexes at a light wavelength of A, 0 is the gyration constant 0 = p(A) A /2rno and p( A) is the optical rotation. Satisfaction of Eq. (1) leads to an observation that depending on the polarization of the light incident on the crystal diffraction is either +1 or -1 orders (for light of random polarization into the two simultaneously).
The magnitude of the sound frequency at which Eq. (1) may be solved we shall search with taking into account the following approximation [2] for refraction indexes in optically active crystal (Te02) :
no = no (1 - 0)
no (1 + 0) (2)
nl = ---------------------------------n~ (1 + 0)2 1
[ 1 - sin 2 01 (1 - --n-~ -- ) ] /2
Dependence of the incidence angle 01 of an ordinary light beam in crystal on the sound frequency f we may from Eqs. (1) and (2) as
sin 01 = q [ 1 ± M {I - (M2 - 1) 40.Jg- } ]1/2 K (M2 - 1) q
(3)
where M = ne/no, K = 27rno/A, q = 2rl/v, and v is the velocity of light. The signs "+" and "_" the two diffraction modes in the crystal far-off axis and off axis. For angles of incidence of light from air al lwith vector To) and a2 (with vector T e ) we write [2]
sin al = no sin 01
sin a2 = ne [ 4 0 + sin2 01 ]1/2
(4)
(5)
The plot of Eqs. (4) and (5) is given in Figure 2, where al(f) is a solid and a2(f) a dashed line. The curve's character is well known, but we are interested in their intersection at f = fo. For the monochromatic light this intersection is singular and appears in the region of minimum of the Bragg diffraction angles . In fact such diffraction may permit creation of "regulated Wallaston prism" in which when sound is switched the polarization components will decline in +1 or -1 orders.
Table 1 comprises calculated results of the fo values in Te02 for several laser lines together with minimum lengths L of transducers at which the Bragg regime is achieved. A criterion for the latter is 0 = 211 AL/ A 2 > 411, where A is the acoustic wavelength [3].
206
'" '"
f
Figure 2. Quality figure of the Bragg angles vs sound frequency.
Table 1
A, nm lase r t.ype f" d MHz L. s m
440 He - Cd 3 0 ,1 0,19 488 Ar 23, 4 0 ,28 514 Ar 21 ,1 0 ,33 633 He - Ne 13 ,4 0 ,67
1060 Y3A15012: 5 ,5 2 , 4
:Nd+3
TWO-COLOUR AND TWO-COORDINATE SPLITTING
Consecutive switching (relative to the light direction) of two monochromatic waves splitters operating at two waves permits to create principally new device two-{;olor splitter with symmetrical relative to the optical axis defraction of the light beam from the A 1 and A 2 axis in the orthogonal planes (Figure 3), with the AO modulators being oriented to which other.
The device created by us was designed for development of a laser Doppler anemometer having the Ar laser with wavelengths AI = 488 nm and A2 = 514 nm. The first modulator operated at frequency of sound 15.4 MHz and second at 16.1 MHz.
The measured splitter parameters were as follows: diffraction efficiency not lower than 90%, optical losses 10%, light leaking with 514 nm wavelength into the "blue" spectrum orders did not exceed 16 dB and that of 488 nm into "green" was less than 25 dB .
TWO-COLOUR SINGLE-FREQUENCY POLARIZED SPLITTING
A rather interesting property of this regime under study consists in simultaneous satisfaction of the Bragg conditions at one A W for the light containinl$ two wavelengths. In this case there appears two points of cross-section of al(f) and a2(f) corresponding to off and far-off axis mode. It should be also noted that due to optical activity of the Te02 crystal these modes are significantly differenced according to the necessary polarization of the incident light (and respectively diffracted), in the first case the polarization is elliptical and in the second is in principle linear.
To find values of acoustic frequencies we must define the dependencies of al(f) for light wavelength AI from (1) - (4) where magnitudes no, ne and 5 should be taken at AI ; a2(f) vs A2 is found from (1) - (3), (5) similarly.
We have manufactured AO beam splitter designed for a Doppler laser anemometer built around an Ar laser wi th wavelengths A I = 488 nm and A 2 = 514 nm for which the condition al = a2 was satisfied at f = 43 and 25 MHz. We used the mode of large incidence angles (43 MHz) .
J ,)
Figure 3. Optical diagram of the two-{;oordinate beam splitter.
207
Table 2
~ 1 3 4 2
488 nm 496 nm 501 nm 514 nm
1 0 -14,3 -18,9 -30
2 -26.2 -28 -24 0
The splitter optical system shown in Figure 4 contains only two elements, i.e. selective rotator of polarization (1) and AOM (2) . Beam Kh2 of two wavelength ,\ 1 and ,\ 2 with parallel polarization vectors is incident on the selective rotator after output of which the polarization of one of the beams rotates by 90° . At the modulator output the light with wavelength 488 nm declines by +(31 angle relative to the transmitted beam and with wavelength 514 nm by angle -(32. The selective rotator is made of a Te02 plate 3.4 mm thick cut normally the optical axis.
The AO crystal had dimensions 7 x 7 x 10 mm along the [001], [110] and [1l0] axes, respectively, with a LiNbO 3 piezotransducer.
The diffraction efficiency for the both wavelength was more then 90% at a electric power 0.4 W. The angle (31 was 1.93° and (32 2.02°, respectively.
For our splitter we measured the intensity I) of each spectral component of the AI laser (,\1 = 488 nm, '\2 = 514 nm, '\3 = 496 nm, and '\4 = 501 nm) in the corresponding diffraction directions. Here i = 1,2 is the number of the main diffraction order (1 - blue and 2 - green) , j = 1,2,3 and 4 are the numbers of the light wavelengths. The results of measurements, the ratio of intensities of the major diffraction components to the intensity of leaking, in dB, are shown in Table 2.
Thus realization of the Bragg conditions on one sound wave for two orthogonally polarized light beams (in general case of different wavelength) permits to develop new AO devices, i.e. polarization splitters. This is promising for systems of laser Doppler anemometry when such devices comprise both function of spatial separation of blue and green spectrum components of Ar laser and shift of frequency of light in the same AO element.
Figure 4.
2
Optical scheme of the two colour single frequency polarization splitter.
REFERENCES
[1]
[2]
[3]
208
R. W. Dixon, "Acoustic diffraction of light in anisotropic media", IEEE Trans., v.QE-3, 2:85 (1967) . S.N. Antonov, Yu.V. Gulyaev, V.M. Kotov etc., "On the problem of developing acoustooptical switches of optical channels", Radiotechnika i Electronika, v.32, 3:623 (1987) . W.R. Klein, B.D. Cook, "Unified Approach to Ultrasonic Light Diffraction", IEEE Trans., v.SU-14, 3:123 (1967).
ffiGH-FREQUENCY MODULATION OF THE WffiTE LIGHT WITH
ACOUSTOOPTIC MODULATOR
S.N. Antonov, V.N. Sotnikov
Institute of Radio Engineering and Electrons U.S.S.R. Academy of Sciences Moscow, U.S.S.R.
INTRODUCTION
Amplitude modulation of optical radiation is widely used in various laboratory and measuring installations. The acoustooptic (AO) method of modulation is as a rule used for controlling monochromatic radiation which is due to a strong dependence of major parameters of AO interaction on the wavelength of light.
The present paper deals with feasibility of AO modulator usage for highfrequency (several tens of MHz) modulation of a high-band light (approximately an octave). The method is based on applications of zero order ot the RamanN ath diffraction.
THEORY
Let a standing acoustic wave be excited in an AO crystal which may be represented in the form of two mutually contrary directed waves having frequency fl with wave vectors KI and K2• In the Raman-Nath region the modulator produces on output of a spatial spectrum of an order 0, I, 2 etc. with frequency [1]:
IIIlm = III + (1 + 2m) fl (1)
where III is the incident optical radiation frequency, 1 is the ordinal number of the diffraction and m is the harmonic order. The 1-th order amplitude has the following form:
w
Al = Aexp[il(:r - 0.5 KL tgO))L,J I+rn ((3) Jrn (a{3) exp [i(l +2m) fit) (2) m= -00
where A is the incident light amplitude, L is the length of the piezoelectric transducer, K is the sound wave vector, J(x) is the Bessel function, 0 is the Bragg angle and {3 is the phase modulation index:
(3 = 2 Ank sin(0.5 KL tgO) nKsinO
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
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209
rz at
(JOB
005
I I
/i 1/' I ' I I
I ,. , I
,/
I I ,.
... ,,_ .---- .- .... ..
/"~ 2
Figure 1. Diffraction efficiency of the zero modulated order as the function of the light wavelength at different powers of the acoustic wave: 1. for P = 0.26 W, 2. for P = 0.4 W, 3. for P = 0.63 W.
where n is the refraction index of the crystal, an is the amplitude of changes of the refraction index induced by the AO wave and k is the light wave vector. Equation (2) shows that the zero number of diffraction contains a harmonic of 2fi frequency. Of substantial importance is the fact that the direc- -tion of the zero order wave has no association with the light wavelength or with the acoustic wave frequency.
Figure 1 demonstrates calculation results for diffraction efficiency of a
Te02 crystal of zero order (17 = A~/A2) with the modulation frequency 2fi as the function of the light wavelength ,\ for different powers P of the acoustic wave for L = 0.2 sm, and a = 0.9 . It is seen that there exists a range of wavelengths within which the efficiency changes rather weakly, by sound power it may be regulated over spectrum.
EXPERIMENT
The method of white light modulation suggested was used for measuring of material dispersion in multimode fiber light guide (FLG). The experimental installation is shown in Figure 2. It includes wide-band light source (1) (xenon high-pressure tube), collimator (2), acoustooptic modulator (3) (AOM), highfrequency oscillator (4), monochromator (5), beam splitter (6), objective for input of light into FLG (7), coil of the FLG (8), lens (9), the first photodetector (10), output objective (11), the second photodetector (12), signal processing unit (13) and computer for controlling the monochromator (14).
AOM built on a Te02 crystal in which a slow shear wave at a frequency
Figure 2. The experimental installation flowing diagram.
210
40
30
20
fO
Figure 3.
, ............. ~ .... \ , \
" \ ~/ \
" \ \ \
o -~
-fStJ
7tItJ ..1.-
Amplitude and phase characteristics of the output light vs the wavelength: the voltage at the photodetector (1) and the change of phase (2) .
of 13 MHz modulated light with double frequency. In measurements of amplitude and characteristics of the light out coming from the FLG influence of additional change of J?hase was observed in the AOM which was dependent on the diffraction region [2].
Figure 3 shows it clearly. In this figure experimental dependencies of the voltage U at the second photodetector which is proportional to the output light intensity and the change of phase as the function of the light wavelength. In a rather wide region of wavelengths (560 to 680 nm) the diffraction efficiency changes but weakly. Simultaneously a substantial phase change is observed. To overcome this difficulty disadvantage we used two photodetectors. In experiment quartz FLG sections of 390 and 2000 m length were used. Figure 4 shows the dependence of the phase difference !J.~ between the signals received by two photodetectors vs A.
The dispersion dependence M(A) may be calculated from the following equation [3]:
Figure 4.
, , , " , , , , ,
''; , ,
The phase shift of the light at the output vs the length of the fibers.
21 1
M(A) = _1_-M 471'%
(4)
where b is the light waveguide length.
REFERENCES
[1]
[2]
[3]
212
V.N. Mahajan, "Diffraction of light by sound waves of arbitrary standing-wave ratio", JAP, v.46, 9:3707 (1975). S.N. Antonov and V.V. Proklov, "Osobiennosti prokhozhdeniya svieta cherez ultrazvukovoy puchok pri silnom akustoopticheskom vzaimodeistvii", Zhurnal Technicheskoi Phiziki, v.53, 2:306 (1983). L.G. Kohen, C. Lin, "Pulse delay measurements in the zero material dispersion wavelength region for optical fibers", Appl. Opt., v.16, 12:3136 (1977).
LEAKY WAVES IN SOLID-LIQUID-SOLID SYSTEMS.
ACOUSTOELECTRONIC MICROANALYSIS OF VISCOUS-ELASTIC
PROPERTIES FOR LIQUIDS OF BIOLOGICAL NATURE
A.V. Arapov, V.S. Goncharov, S.V. Ruchko, LB. Yakovkin
Institute of Semiconductor Physics Siberian Branch of the U.S.S.R. Academy of Sciences 630090, Novosibirsk, U.S.S.R.
INTRODUCTION
Recently developed methods for acoustic microanalysis of liquids utilize the distinctive attributes of wave propagation in solid-liquid-solid tSLS) structures [1]. Further reductions in the volume of the investigated liquid are of major interest for biological and medical applications. The normal-mode formalism is used to describe the acoustical processes in the structure in the case of a layer whose thickness is much smaller then its length. It is important to analyze the possibility of determining the properties of a liquid from data on parameters of normal modes in a layered structure. Such parameters are the phase and group velocities. The objective of the present study is to find an approximate analytical relation between the measured parameters and acoustical properties of the liquid. For this aim we use a representation on leaky waves.
An approximate dispersion relation can be obtained from the description of a normal mode as a plane longitudinal wave, which is reflected at the SLS interfaces. The balance of the phases in the process can be expressed as follows:
2~r + 2~f = ~ + 2?ITl,
where $f = Hkr/sinO is the phase shift of the plane wave in the liquid layer, ~r is the phase shift of the wave in the reflection, ~ = kl is the phase shift of normal mode after passing the distance in the waveguide 1 = 2H/tanO, at which the plane wave undergoes double reflection (H is the thickness of the liquid layer, k = w/V and kf = W/Vf are the wave numbers of the normal mode and the plane wave in the liquid, respectively, w is the frequency, o = cos-1(k/kf) is the angle of incidence of the plane wave). We find an approximate solution of the equation for angles of incidence of the plane wave close to the Rayleigh angle Or = cos-1 (kR/kr), where kR = w/V R is the
Rayleigh wave number. We use the expression for the phase of the reflection coefficient in the form [2]:
when 6. = k-KR, fr-space rate decay parameter of leaky wave. The
approximate dispersion relation can be rewritten in the form:
HktanO - 26./ a = ?r(n-l), n = 1,2, ....
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
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213
It is evident that the normal-mode velocity is equal to the Rayleigh wave velocity at structure thicknesses such that Hn = n{n-1)/krsinOR. We solve the
solution of Eq.(l) in the form of the dependence of the quantity V on the parameter h = H - Hn. We retain only first-order terms in h. The phase velocity is described by the relation
To calculate the group velocity, we consider the motion of the energy fluxes. The dependence of the phase of the reflection coefficient on the angle of incidence at 0 ~ Or is manifested in a shift of energy flux lines by the amount
in reflection [2]. The velocity of propagation of the energy flux in liquid is equal to the sound velocity in the liquid, and its velocity in solid is equal to the mode phase velocity. The equation for the group velocity has the form
V(l + (Hn + h)/5tanO) V = ---------------------
g 1 + (Hn + h)/6sinOcosO (2)
The results of calculation of V g according to Eq.(2) and the numerical solution of the exact dispersion relation are shown in Figure 1. The discrepancy between the exact and approximate values of V and V g does not exceed 0.16% and 0.6%, respectively, at Ihkfl < 1 for the first seven modes.
It is evident that the mode phase and group velocities are determined by the quantities (l and OR at thicknesses of the liquid layer H ~ Rn. The
approximate solution of the dispersion relation for a leaky wave under the condition (l/kR « 1 shows that
where f3 is a combination of parameters of the solid, and Pf is the density of the liquid. At Hn = 0 and a small h the values of the phase and group velocities
depend only on the density of the fluid.
Figure 1.
214
3300 1 ...)'-0.0.0 .... 0..
"/ °'°'03
0·0-0'°'0 ~ ~ '05
27 "~ L-__ ~~~ ____ L--hk!/2~
-0.2 0 0.2
3000
Group velocities for modes versus normalized layer thickness. 1,2 - n = 1,4; 4,6 - accurate; 3,5 - approximate.
Figure 2. Block diagram of measuring system using pulse recirculation: 1 - a SLS sensor, 2 - videopulse generator, 3 - comparator, 4 - counter, 5 - frequency counter.
Let us consider the situation when the channels of energy dissipation exist in the solid and the liquid and both solids are assumed identical. Let Ef and Es be energy flows, af and as are the plane wave loss parameters in the liquid and solid, respectively. Then the loss factor for the normal wave is given by:
O:s + 0: f * E dEscos ° 1 + EriEs
Suppose that ratio Et/Es is not a function of loss parameters. This fact means that energy dissipation does not lead to the additional dispersion of normal waves. It can be shown on the basis of the ray model that Et/Es = HI8 . tan 0, where 8 is the beam displacement. Let us find quantities O:s, O:f such that the change in dispersion dependence for normal waves becomes negligible. It is known that the sound velocity in a liquid does not depend on O:f in the first approximation. It follows that the phase shift change in the liquid can be neglected. The incidence angles close to the Rayleigh value are examined to analyze the behavior of the reflection coefficient phase. Here we use the approximate expression for the reflection coefficient to the form:
Hence, the reflection coefficient phase can be written as follows:
f3n - O:s + 0: f3n - O:s - 0: ~r = tan -1 - tan -1 -----
kx- ka kx- ka (3)
The examination of Eq.(3) reveals that under the condition I f3n-O:s I « 0: the energy dissipation does not produce an effect on the reflection coefficient phase. This unequality holds true without any restrictions on the layer thickness in case when a = Io:t/coseal« 0:. An arbitrary quantity a requires
H « 2tanOa/a. The approach suggested in this paper can be generalized to the
case when the solids are different. For their operation the surface acoustic wave (SAW) sensors, including SLS
sensors, depend upon the measurements of the SAW propagation parameters at the surface of a half-space liquid state produce the SAW amplitude and phase variations registered through the corresponding changes in the value of :propagation time T between the input and output interdigital transducers (IDT). A number of techniques for measuring T has been developed. We use a method of frequency measurements employing a SAW delay line oscillator with a SLS sensor in the feedback loop. An oscillator operating on a pulse
215
recirculation principle is a variety of such approach. The measuring system works as follows (Figure 2). A short front videopulse applied from a controlled generator to the input IDT launches a packet of acoustic waves at the substrate surface. Then this packet propagates through the SLS sensor and is finally detected by the output transducer. With the aid of a threshold circuit, as a comparator, from the signal received which is the sensor impulse response, a new pulse is formed to enable the videopulse generator. It is obvious that the generation process of the SAW packet is periodic. The basic parameter to measure is either period or frequency of the SAW packet recirculation.
A resolution of the method discussed is determined by the way how an initiating pulse is synchronized with the impulse response of the SLS sensor. The solution of this problem can be reduced to synchronizing this pulse either with envelope or the high frequency carrier of the impulse response. Of the two ways, the latter is more preferable for achieving the higher frequency stability of a pulse generator. It is evident that for small amplitude variations of the impulse response with a fixed nonzero threshold level of the comparator, the change in the frequency of the SAW packet recirculation which is not related to the delay time takes place. Whereas, for the significant variations there occurs an abrupt frequency change due to the disappearance of the H.F. halfperiod by which the synchronization of the generator triggering is performed. The only chance to eliminate the influence of insertion losses upon the period of pulse recirculation is the synchronization of initiating pulse front with the impulse response at the zero level of the comparator.
However, since the -comparator operation level can not be lower than that of noise and spurious signals, this fact sets the substantial limit on the dynamic range and does not allow us to get rid of the connection between the impulse response amplitude and the period of pulse recirculation.
In order to extend the dynamic range it is offered to set the comparator level above the noise level and gain the difference between the amplitudes of neighboring half-periods at a time by using phase coded IDT's.
Thus, we have worked out the SAW oscillator using pulse recirculation and operating in the dynamic range of 30dB and frequency region 10-100 MHz. The study of the SAW oscillator stability has been carried out on the delay line of 7.04 /18, fundamental frequency of 43.6 MHz, the number of electrode pairs - 5. R.m.s. deviation of oscillator period - 5.7 10-8.
The SLS structure shown in Figure 3 is chosen as a tool for microanalysis of liquids. An investigated liquid is contained in the gap between two planeparallel plates of LiNbO 3 with the gap thickness of 3425 11m and length of 6100 11m. Method of measuring described above is applied to measure the delay time T of the SLS sensor. First experiments on the liquids of biological nature have been carried out with a suspension of human erythrocyte shadows and were aimed at observing phase transformations in biological membranes. The investigated erythrocyte suspension has a concentration of 2 mg/ml in buffer solution.
The measurement results on the fractional delay time tH / T for the SLS sensor containing buffer are plotted in Figure 4.a. An approximation for buffer by least squares fitting has the form:
tH/T = -2·69·10-311T + 4.74·10-511T 2
The same relation for distilled water ignoring the T - T dependence for the
Figure 3. Geometry of the SLS sensor.
216
a
b
- 3 t:ir:jr. 10
-2
32
t:..'I:/rr:: -]
-2·1 0
-3 - 1010
3S
A
34 36 38
37 39
Figure 4. Fractional delay time of the SLS sensor versus temperature. The liquids under investigation are: 4.a - buffer solution, 4.b - human erythrocyte suspension in buffer.
SA W propagation in the free surface can be written as follows:
It is obvious that the buffer affects mainly the nonlinear part of this relationship. In the limited temperature range t. T ~ 5' C it is linear within accuracy of 4%.
Figure 4.b displays the t.r/r relationship against temperature for the SLS sensor with erythrocyte suspension. It is clearly seen that the curve shows a bend at T = 37.6 'C. In the narrow temperature intervals 36' C < T < 37.6' C and 37.6' C < T < 39' C straight lines can be drown through the experiment dots. the accuracy of measurements makes it possible to distinguish the different slopes of this lines at T < 37.6' C and T > 37.6' C.
REFERENCES
[1]
[2]
A.V. Arapov, V.S. Goncharov, I.B. Yakovkin, Leaky waves in layered system, Sov. Phys. Acoust ., 31:439, (1985) . H.L. Bertoni, T . Tamir, Unified theory of Rayleigh-angle ,Phenomena for acoustic beams at liquid-solid interfaces, Appl.Phys., 2:157, 11973).
217
ACOUSTIC WAVE PROPAGATION IN MEDIA CONTAINING
TWO-DIMENSIONAL PERIODICALLY SPACED ELASTIC INCLUSIONS
Christian Audoly and Guy Dumery*
Groupe d'Etudes et de Recherches de Detection Sous-Marine Le Bruse, 83140 Six-Fours, France *Universite de Toulon et du Var 83130 La Garde, France
INTRODUCTION
The propagation of plane harmonic waves in heterogeneous media is a problem of continuing interest, for example to study underwater acoustic materials, porous or fibrous [1] materials, and has been treated by many authors. At low frequencies, when the wavelength is much larger than the size of the inclusions, it is possible to use quasi-static theories such as those of Berryman [2] and Kuster-Toksoz r3]. At higher frequencies, when the size of the inclusions is comparable to the wavelength, the problem is more complicated because of the resonances of the inclusions and the multiple scattering effects. Some authors have studied the case of randomly distributed inclusions using the multiple scattering theory [4]. However, the actual structure of some media of interest is periodic rather than random (underwater acoustic materials with resonant inclusions or bundles of tubes for heat exchangers in power plants for example). In the case of periodically spaced inclusions, it is possible to use the finite element method [5], but a mesh generation is required for each numerical case. We propose here to use a multiple scattering model that applies to cylindrical inclusions placed in a fluid or to the propagation of SH waves in a porous medium.
2d 2a
"&. o o o
2d I
~ -$-:- -e- - -0-- ~x 0:: 6 fY~ 0 X Y
o o o o cb
I o o
reticular plane T Figure 1. Definition of the geometry.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 219
THEORETICAL ANALYSIS
We consider an acoustic medium defined by its volumetric mass PI and speed of sound CI , containing cylindrical inclusions periodically spaced in the x and y directions. The grating spacings are 2d and 2d' (see Figure 1). It is also assumed that the problem is twCHlimensional (independent to z-axis). In all what follows, complex notation with implicit time dependence exp( -i"'t) will be used. The starting point of the method is that the doubly periodic structure can be decomposed as an infinite superposition of reticular planes consisting of infinite plane gratings of cylinders.
Reflection and Transmission Coefficients of a Periodic Plane Grating
We consider now a single reticular plane in free-field. This grating, when submitted to a harmonic plane wave, is characterized by reflected and transmitted fields having the forms:
+00 +00
L Rn exp(i(-knx+any)) L Tn exp(i(knx+any)) n = -00 n = -00
with an = n7rjd + sin 00 , kn = (k2 - an2/ h, 00 is the incident angle and k = ",jcI. When the grating spacing 2d is smaller than the wavelength and if the fluid is not lossy, all the partial waves are evanescent (i.e. kn is a pure complex number) and the energy propagates in the zeroth order terms. The grating is then characterized by the zeroth order coefficients Rand T. These coefficients can be computed using the multiple scattering method, presented in an earlier paper [6]. Basically, the reflection and transmission coefficients have the form (for normal incidence):
+00 +00
R = Ja L i m Cm T = 1 + Ja L (-i)m Cm
m= -00 m= -00
where the coefficients Cm are the coefficients of the cylindrical waves scattered by the elements placed in the grating. The coefficients Cm are found by solving a linear system involving the transition matrix of an individual scatterer in free-field conditions, and Schlomilch series.
Transfer Matrix of a Reticular Plane
We suppose now that the reticular plane is symmetric relative to y-axis and that it is placed among the others, as on Figure 1. On both sides of the grating, waves can propagate in the positive and negative x--direction. The transfer matrix A relating the acoustic field of the two sides is:
Using the symmetry of the grating and taking x origins, A takes the form [7]:
± d' as phase
[ A] = [ Q 2 (T-R2T -I) -RT-I
RT-I ] T-IQ-2
where Q = exp(ikd').
Propagation in the Heterogeneous Medium
The final step is to look for the natural plane waves propagating in the
220
medium, whose properties were detailed by Brillouin [8J. They verify: y± = exp(±2ikeqd / ) X± (1)
where keq setting e give:
is the equivalent wavenumber in the heterogeneous medium. By exp(2ikeqd / ), Eq.(l) and the definition of the transfer matrix
Thus, the speed of sound Ceq = w/ke in the heterogeneous periodic medium is determined by the eigenvalues of fue transfer matrix of a reticular plane.
EXAMPLES
All the examples that follow are given for circular cylinder obstacles of radius a, equally spaced in both directions (d = d / ).
Elastic Obstacles in Water
In this section, we consider the case of elastic obstacles in water (PI = 1000 kg/m 3, CI = 1500m/s) and dispersion curves of the normalized equivalent wavenumber keq -d/1f (real and imaginary parts) versus reduced frequency k.d/1f are presented. Figure 2 gives the results for steel circular cylinders (p = 7800 kgjm 3, E = 2.1 x 10 11Pa, /J = 0.3) in water for different grating spacings d/a. As predicted by Brillouin, pass-bands related to real wavenumbers and stop-bands related to imaginary wavenumbers (evanescent
D."'-0 cT v
0.40
60.20 v ct:
a:: "'u
cT v
0.15
60.08 ..E
.' / .... /
..... / .... /
... ;/ ... ;; .. '/
.. y
.. .,j';'
0.00 +--,--,.----,----.-.,------.-----r~i__'L,___r-'r_'_+_.-.__.___r-+__r__L,___'L.j
0.00
Figure 2.
0.10 0.20 0.30 0.40 0.50 0.60
kd/Pi
0.70 0.80 0.90
Steel cylinders in water - Dispersion curves. (--) d/a = 1.5; (- - - -) d/a = 2.0; (- - - -) d/a
toO
3.0
221
0.40
0:: '-.,.
" ci-v ':<:
'J[ 0.20
0:: '-.,.
" ci-v
0.16
~0.08 1; ~
II ,1 I I I I
0.00 -t---.--i----'-.----.-..--,---r--.---.--',---'----+----.--'---,--'t--'r---,-.--,---.--j 0.00 0.10 0.20 0.30 0.40 0.50 0.60
kd/Pi
0.70 0.80 0.90
Steel tubes in water - Dispersion curves. (--) d/a = 1.1; ( .... ) d/a = 1.2; (- - - -) d/a
Figure 3.
too
1.5 .
waves) appear. The low-frequency branch of the curves corresponds to the first Brillouin zone. These branches can be compared to the dispersion curve of water without inclusion, which is a straight line of slope unity. In that case, the sound propagates slower in the heterogeneous medium than in free-field, and the effect increases with higher concentration of inclusions. Figure 3 shows similar results for steel tubes of thickness h/a = 0.1. On the first branches, we see that the speed of sound is larger than in free field and that some perturbations due to the shell resonance modes appear. On Figure 4, when the elastic inclusion is lossy, as for PVC tubes (h/a = 0.12, E = 4.1 x 10 9Pa, v = 0.4, tgo = 6%), the pass-bands and stop-bands cannot be defined because the equivalent wavenumber is complex for all frequencies.
Comparison to Quasi-Static Approximations
For low frequencies, the previous results show that the dispersion curves are straight lines and the slope gives the equivalent speed of sound. KusterToks6z's theory can be used to evaluate the equivalent bulk modulus. Then the speed of sound ceg = Keq/ Peq1/2 in the heterogeneous medium. For inclusions of a material of VOlumetric mass P2 and bulk modulus K2 , we have:
1/Keq = (1 - ¢)/K 1 + ¢/K 2
Peq/(Pl+Peq) = (1-¢) + ¢ P2/(Pl+P2) (including added mass effect),
where ¢ is the concentration of inclusions and Kl = Pl-C12. Table 1 gives a comparison between the multiple scattering model and the
quasi-static approximation. The good agreement between the results show the coherence of the two approaches.
222
......
0.40
.....
0.00 ~-r---.----.---.----. __ '--.---.--'-----r-----,------.-~:::;:===:::::::::::;i::::;:==r=~~
0:: , -u rr .,
0.40
iO.20
................... ...-/ ..... . ............ \ .....
0.00 +--,---r-9"'''''''T=.,..---.---r-.-......,---,~-,---.---,-,.--,--.--..,.......-.-
0.00 0.10 0.20
Figure 4.
0.30 0.40 0.50
kd/Pi 0.60 0.70 O.BO
PVC tubes in water - Dispersion curves. (--) d/a = 1.2; ( .... ) d/a = 1.5 .
1.00.,----------------------------,
0.90
0,80
0.90 1.00
0.70 (----l G /G eq
-------------0.60
0.50
........ " " \
\ \
0.40+--~__r-.,---._--r-.__-~__r-~-r___r-.._-~__.-.,--__i
0,00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequence kd/Pi
Figure 5. (--) ceq/c; (- - - -) Geq/G
223
Table 1. Comparison of quasi-static and multiple scattering theories for steel cylinders in water
Quasi-static model Mult. Scat. d/a Peq/ Pl Keq/Kl ceq/c 1 ceq/c1
1.5 1. 7388 1. 5252 0.9365 0.9334 2.0 1. 3577 1.2402 0.9557 0.9548 3.0 1.1446 1. 0942 0.977'7 0.9776
2.3 SH elastic waves in a porous medium
SH Elastic Waves in a Porous Medium
The formalism for SH elastic waves in a porous medium is equivalent to rigid inclusions in a fluid. Figure 5 shows the evolution of the equivalent shear modulus Geq and shear velocity for a 19.63% porosity. The results using the multiple scattering formalism are in good agreement with ref. [5).
CONCLUSIONS
This paper has shown that the multiple scattering formalism, combined with a reticular plane transfer matrix approach, can be a useful tool for determining the equivalent speed of sound in periodic heterogeneous media. Some results· for water containing elastic inclusions were presented, and we find the phenomena described by Brillouin. In the very low frequency limit, the results agree with quasi-static approximation. The same model can be used for elastic SH waves in a porous medium.
REFERENCES
[1)
[2)
[3)
[4)
[5)
[6)
[7)
[8)
224
G.A. Hegemier, G.A. Gurtman and A.H. Neyfeh: A continuum mixture theory of wave propagation in laminated and fiber reinforced composites., Int. J. Solids. Structures, Vo1.9, p.395, (1973). J.G. Berryman: Long-wavelength propagation in composite elastic media I- Spherical inclusions., J. Acoust. Soc. Am., Vo1.68, p.1809, (1980). G.T. Kuster and M.N. Toksoz: Velocity and attenuation of seismic waves in two-phase media., Geophysics, Vo1.39, p.587, (1974). V.K. Varadan, Y.Ma and V.V. Varadan: A multiple scattering theory for elastic wave propagation in discrete random media., J. Acoust. Soc. Am., Vol. 77, p.375, (1985). T.-C. Ma, R.A. Scott and W.H. Yang: Harmonic wave propagation in an infinite elastic medium with a periodic array of cylindrical pores., Journal of Sound and Vibration, Vol. 71, p.473, (1980). C. Audoly and G. Dumery: Etude d'ecrans sous-marins constitues de reseaux de tubes elastiques., Acustica, Vo1.69, p.263, (1989). G. Dumery: Propagation d'ondes sonores dans un milieu contenant des inclusions periodiques., Revue du Cethedec-ondes et signal, N' 78, (1984). L. Brillouin et M. Parodi: Propagation des ondes dans les milieux periodiques., Masson et Dunod ed., Paris, (1956).
SCATTERING OF SHEAR HORIZONTAL WAVES BY MASSIVE METALLIC
ELECTRODES IN SAW DEVICES
A.R. Bahgai-Wadji
Inst. f. Allgem. Elekt. u. Elek. TU-Wien Gusshausstr. 27-29, A-1040 Wien, Austria
INTRODUCTION
In surface acoustic wave (SAW) devices the excitation and detection of acoustic waves is conveniently controlled b:y a set of metallic electrodes on the plane surface of a piezoelectric substrate ll]. An input interdigital transducer (IDT) consisting of a number of electrodes converts the electric input signal into the propagating acoustic waves, and conversely, an output IDT converts the signal carried by the acoustic waves back into the electric signal. The fundamental process behind the generation and detection of acoustic waves is the interaction of the acoustic waves with the electrodes. In many applications the electrodes are modeled as being ideal conductive, massless thin electrodes, leading to excellent simulation results compared with carefully obtained experimental data [2,3]. On the contrary, in some other applications, the above idealizations seem to be merely vague assumptions which, possibly hand-in-hand with other second- [4] or higher order effects [5], cause considerable discrepancy with experimental results. Our intention in this work is focused on the description of the massloading effect in SAW devices. In SAW devices like reflective arrays and unidirectional transducers in general, and single phase unidirectional transducers (SPUDTs) [6] in particular, the massloading effect due to the mass of the electrodes, plays a fundamental role in the functional behavior of the devices, and in this sense, it is rather primary effect.
The theoretical framework presented here is capable of describing the interaction of any type of acoustic waves with the electrodes. However, for simplicity sake and for the purpose of concentrating on the essentials of the method, we treat only the shear horizontal waves.
,,(i)
piezoelectric substrate ~ ~ ~
Figure 1. Geometry of the problem and specifications.
Physical Acoustics. Edited by O. Leroy and M. A. Brea,eale Plenum Press, New York, 1991 225
THEORETICAL BACKGROUND
Consider a set of N metallic massive electrodes on the plane surface of a piezoelectric substrate, Figure 1. In this configuration the substrate is assumed to support only shear horizontal waves (STWs) characterized by the transverse mechanical displacement U2(X,Z) and the electrical potential ¥I(X,Z). The electrodes are allowed to possess individual material constitution; i.e ., p ( i), q p and finite thickness h( i) . The p( i) and qp, respectively, denote the mass
density and the elastic constant of ith electrode. In Figure 1 ¥I( i) denotes the given electrical potential of i th electrode. Furthermore, the electrodes are assumed to be electrically ideal conductive. Our purpose is to describe the interaction of a SHW propagating along the substrate surface with the electrodes.
Instead of the terms excitation, scattering and detection we prefer to use the term interaction, because as we will see below, our formalism for the interaction allows the analysis of the excitation, scattering and detection as special cases . Let us now concentrate on one of the electrodes, say, ith one. As we previously have shown [7], from the ideal conductivity condition the following model for the electrodes can be proposed in Figure 2.
Figure 2 suggests treating the problem as consisting of three parts: a mechanical part, an electrical part, and the interaction between these two.
Mechanical part: A modal (eigenfunction) expansion [8] for the mechanical displacement U2( x,z) as given by
ID
= E (n) U(n)p(x) cos(n1r x - Xe) cos{,),(n)(z - h)} n=O Xe- Xb
(1)
fulfills the stress-free condition on the free part of electrode bounding surface (r' = r(l)+r(2)+r(3)), as is seen from the resulting expression for Tdx,z) ; i.e.,
where up to this stage the U( n) are unknown expansion coefficients, which have to be determined from the interface conditions on r( 4), Figure 2.b. Here, P(x) is a pulse function being equal to unity if x passes the electrode region, and zero otherwise. Xb and Xe, respectively, are the start and end point
coordinate of the electrode. ,( n) is defined as (,( n))2 = (-~)2 _ ( n1r )2, c Xe-Xb
Figure 2.
226
massi ve metallic electrode
a)
piezoelectric substrate
b)
t @
•
(non. piezoelectric) elastic ridge with constitulive parameters of lhe electrode
interaction at the electrode/ su bstrale interCace
massless melallic hollow pipe filled with £0
Proposed model for the ideal conductive, massive metallic electrodes.
where c = rC~4]\(n) is Neumann's factor. We now interpret the function
T(X) = T23(X,Z~ I z=O: T(X) represents the mechanical force distribution actin~ on the substrate surface due to the presence of the massive electrode. T(X) in conjunction with u( x,z) I z=o uniquely describes the mechanical loading effect of the electrode on the substrate surface in electrode region. Let us now replace T(X) by a collection of infinite number line forces parallel to y-axis. Because of the linearity of the problem, the superposition of the line force actions on the substrate surface is an alternative way of obtaining the resulting deformation due to T(X). In this way':, we reduce the mechanical part of the problem to a fundamental line force (Dirac's delta function) excitation of the medium. The response of the medium to a stress distribution T(X) on the surface manifests itself by two independent variables U~1) (x,z) and ¥I(1) (x,z) in the substrate;
here, ¥I (1) (x,z) denotes the electrical potential distribution. To distinguish the uL and ¥I- responses due to an isolated line force located on the surface at (x',O) we use the symbols G(U,1) (x,zlx',O) and G(\p,1) (x,zlx',O) (Green's
functions). For numerical calculations we should restrict ourselves to a finite
number of eigenfunctions (truncation), to obtain an approximation U2(X,Z) for
the mechanical displacement and from that an approximation T23( x,z) for the stress distribution in the electrode.
Electrical part: Let us now concentrate on the electrical part. The acoustic wave interacting with the electrode, Figure 2.a, leads to an electric charge distribution p(x,z) on the bounding surface of the metallic electrode (f = f( 1) +f( 2) +f( 3) +f( 4»), Figure 2.c, such that the tangential component of the resulting electric field on f vanishes. The response of the piezoelectric substrate to p(x,z) may now be denoted by U~Il) (x,z) and ¥I( (1) (x,z). Analogous to the preceding case we now regard p( x,z) as an infinite number of isolated line charge sources located on f. The reaction of the medium to a line char1!je source with the coordinates (x' ,z') may now be denoted G( U,Il) (X,zl x' ,z') and G( \P,Il) (x,zl x' ,z'). Note that p( x,z) is a priori
unknown and thus for the numerical calculations we must look for a suitable
approximation [7,9,10].
p(x,z), which contains a finite number of unknown coefficients
Interaction between the electrical and mechanical part: Referring to Figure 2 we want now to discuss what is meant by the interaction of the two cases sketched in Figure 2.b and Figure 2.c. For this purpose assume that the distributed mechanical and electrical sources T(X) and p(x,z) act simultaneously on the piezoelectric substrate. T(X) and p(x,z) can be thought of as being associated with one or with a set of electrodes. As we have already mentioned, the response of the medium to T(X) and p(x,z) is uniquely determined by the resulting U2(X,Z) and ¥I(X,Z). Thus, because of the linearity of the problem (validity of the superposition principle), we can write
U2(X,Z) = U~T)(X,Z) + u~P)(x,z), ¥I(X,Z) = ¥I(T)(X,Z) + rp(P)(x,z),
which with the above discussed Green's functions read
+00 +00 +00
(3a)
(3b)
U2(X,Z) = J G(U,1) (x,zlx',O)T(x')dx' + J J G(U,Il) (x,zlx',z')p(x',z')dx'dz', -00 -00 -00
(4.a and 4.b) +00 +00 +00
¥I(X,Z) = J G(\p,1) (x,zl x' ,0)T(x')dx' + f f G(\p,Il) (x,zlx',z')p(x',z')dx'dz'. -00
227
The term interaction merely means that the above two integral equations relating U2(X,Z) and V'(x,z) with r(x) and p(x,z) are coupled. Alternatively, we can state that the resulting system of equations for the determina-
tion of the unknown coefficients entering into r(x) and p(x,z), as approximations for r(x) and p(x,z), are coupled. A similar statement, in this sense, is valid, if we consider an infinite number of electrodes [11]. The method and the details concerning the derivation of G<'I',g) (x,zix',z') and G<U,g) (x,zix',z') closely
parallel the electrostatic approximation we have discussed elsewhere [12]. In the special case where the line charge resides on the substrate surface, the corresponding expressions G< u,g) (x,zi x' ,0) and G< 'I',g) (x,zi x' ,0) have already been
discussed [9,10]. The next section is devoted to the application of (4) to a scattering problem.
SCATTERING OF SHW WITH A SINGLE ELASTIC RIDGE
To demonstrate the application of (4) we now treat the scattering of SHW excited by an isolated line charge source located on the surface of a piezoelectric substrate (quartz) with a large amplitude elastic ridge on the substrate surface. The specifications are as given in the inset of Figure 3.a. In this example the scattered mechanical displacement field is of special interest.
The line charge source is assumed to be located at the origin of the coordinate and can be described by a Dirac's delta function p(x,O) = 8(x). The line charge excites a SHW which will interact with the ridge. Now because the scatterer is assumed to be pure elastic (non-metallic), only the induced mechanical stress at the electrode/substrate has to be calculated. Therefore the relevant equation for this problem is (4a). With regard to p(x,O) = 8(x) we obtain
fro fro f G<U,1) (x,zix',O)r(x')dx' + f G<u,g)(x,zix',0)8(x')dx'. (5a)
ro -ro
To interpret this equation assume the elastic ridge to be absent. In this case we would not have any induced stress on the surface (r( x) = 0), and we would obtain the outgoing SHW (incident wave) excited by the line charge. Thus with
fro u~nc(x,z) f G<u,g) (x,zix',0)8(x')dx' G< u,g) (x,zi 0,0) (5b)
ro
Eq. 5a can be written as
fro
U2(X,Z) = f G<u,1)(x,zix',0)r(x')dx' + u~nc(x,z). (5c) -ro
We conclude that the integral on the right hand side of (5a) represents the scattering process of u~nc(x,z) with the ridge. Now how to calculate the
scattered SHW? (Based on the assumption that the substrate supports only shear horizontal waves, there is no mode conversion into Rayleigh wave.) Using an approximation for r(x), as discussed above, and employing Fourier-Technique [7,8,9,10]' (5a) can be converted into a system of equations for the deter-
mination of U< n). Having determined U< n), substituting back into r(x) and
insertion of r(x) in the integral in (5c) we obtain an approximation for u~cat(x,z) and therefore for the total field U2(X,Z):
228
U2( X,Z) = u~cat( X, Z) + u~nc( X, Z), M
u~cat(x,z) =n~o(u~cat)(nl (X,Z),
(u~cat)(nl(x,z) = C44 [(nlU(nl'l(nlsin(-'l(nlh)
xe x'-x f G( li,ll (x,zl x' ,O)cos(ll7r -_e)dx' . Xb Xe-Xb
(5d)
(5e)
(5f)
Note that in (5e) the order of summation and integration has been interchanged
and M + 1 eigenfunctions are used to construct the approximation 7'(x) . From (5e) it is seen that the contribution of each expansion mode to the scattered wave individually can be calculated. This fact is of prime importance for numerical calculations, if we as usually is the case, are interested in the frequency dependence of the scattered wave (the number of the modes necessary to achieve a required accuracy depends on the frequency). Figure 3 shows the calculated results for the problem sketched in Figure 3.a .
Figure 3.
. 02 r-------~----~~----------------, incident SHW excited by an isolated line charge source at the origin of the coordinate
- - - - imaginary part - real part
;: O. E Il .. ~ 'ii . _ . 02 p.+J4-'+-L+'-++I--'--lf..L
_ .002 ..--- ---- o
. 002 . 02
o.
- .02 .020
. 015
. 010
. 005
o. 0 50
.. 0 o .; ..
~ ~Nyv~'VY'~""7'""-;-""""''-''r-1--T--r-."..=t " - 5
100 150 200
Scattering of SHW excited by a line charge source with a single elastic ridge. Calculated results for the incident-, scattered- and the total-field.
229
For the cases where the electrode profiles are not rectangular, the proposed formalism remains valid. However for the description of the electrodes, instead of Fourier-Technique, we would have to employ either the Boundary Element Method, the Finite Element Method or the Method of Least Square on the Boundary.
CONCLUSION
It is shown that Green's function concept in conjunction with FourierTechnique (and the Method of Weighted Residuals [7]) is amenable to calculating the interaction of acoustic waves with large amplitude electrodes on the piezoelectric substrates. Numerical results are presented for the scattering of a shear horizontal wave with a single elastic ridge.
ACKNOWLEDGEMENT
The author wish to thank Prof. F. Seifert and Dr. C. Ruppel for their continuous encouragement and support.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[S]
[9]
[10]
[11]
[12]
230
D.P. Morgan, Studies in Electrical and Electronic Engineering 19, Surface-Wave Devices for Signal Processing, Elsevier (19S5). C. Ruppel, E. Ehrmann-Falkenau, and H.R. Stocker, Compensation Algorithm for SAW Second Order Effects in Multistrip Coupler Filters, Proc. Ultrason. Symp., 7-10, (19S5). G. Visintini, W. Till, and E. Ehrmann-Falkenau, Diffraction Analysis with Extended Angular Spectrum of Waves Formalism, Proc. Ultrason. Symp. 145-14S, (19S7). W.S. Jones, C.S. Hartmann, and T.D. Sturdivant, Second Order Effects in Surface Wave Devices, IEEE Trans. Sonics and Ultrasonics, vol. SU-19, W 3, 36S-377, (1972). A.R. Baghai-Wadji, O. Manner, and R. Ganf3-Puchstein, Analysis and Measurement of Transducer End Radiation in SAW Filters on Strongly Coupling Substrates, IEEE Trans. Microwave Theory and Techniques, vol. MTT-37, 150-15S, (19S9). C.S. Hartmann, P.V. Wright, R.J. Kansy, and E.M. Garber, An Analysis of SAW Interdigital Transducers with Internal Reflections and the Application to the Design of Single-Phase Unidirectional Transducers, Proc. Ultrason. Symp., 40-45, (19S2). A.R. Baghai-Wadji, F. Seifert, and K. Anemogiannis, Rigorous Analysis of STWs in Nonperiodic Arrays Including Mechanical and Electrical Interactions, Proc. Ultrason. Symp., 303-306, (19SS). A.A. Maradudin, Surface Acoustic Waves on Rough Surfaces, Proc. of European Mechanics Colloquium 226, University of Nottingham, U.K., 100-12S, (19S7). A.R. Baghai-Wadji, Analysis of Surface Transverse Waves in Nonperiodic Arrays, to be published in Proc. ISSW AS 'S9 Symp., (19S9). A.R. Baghai-Wadji, Scattering of Piezoelectric Surface Transverse Waves
From Electrodes with Arbitrary ~, Proc. Ultrason. Symp., 377-310, (19S9).
A.R. Baghai-Wadji, and A.A. Maradudin, Rigorous Analysis of Surface Transverse Waves in Periodic Arrays with Arbitrary Electrode Profiles, Submitted to Ultrason. Symp., (1990). A.R. Baghai-Wadji, Electrostatic Field Analysis of Arbitrary Shaped Cylindrical and Spherical Metallic Structures in Anisotropic Multilayered Media, Proc. IGTE Symp., 195-202, (19SS).
ACOUSTOOPTIC INTERACTION APPLICATION TO OPTICAL
WAVEFRONT ANALYSIS
Vladimir I. Balakshy and Leonid V. Balakin
Department of Physics Moscow State University 119899, Moscow, U.S.S.R.
INTRODUCTION
The problem of optical wavefront analysis arises very often in experimental physics. Wavefront probing devices are necessary for optical elements quality control, in laser physics, holography and adaptive optics. One of the ways of solving the problem is based on acoustooptic interaction application [1-3]. This paper presents theoretical and experimental results of investigations of an acoustooptic system intended for light field phase structure probing. The system allows in real time and with high spatial and angular resolution to measure wave normal local directions along the wavefront and also to visualize the wave front structure in stereoscopic projection on a TV monitor screen.
PRINCIPLE OF OPERATION OF THE ACOUSTOOPTIC PROBING SYSTEM
In this system (Figure 1) the optical wavefront probing is carried out from point to point successively by an acoustic pulse (1) which propagates in an acoustooptic cell (2). The acoustic pulse is generated by a transducer (3) excited by a short radiopulse (4). An optical wave to be analyzed is projected by an input lens (5) into the cell. When the wave passes through the cell it diffracts on the acoustic pulse. An output lens (6) collects the scattered light (7) on a photodetector (8). So the photodetector output signal i(t) contains information about local directions of the optical wave vector.
In a common case a phase-modulated light field may be written as
U(y,z) = Uo exp[jr(y,z)] (1)
where Uo is the amplitude and a function 'Y(y,z) describes a spatial phase modulation. In this field the wave normal direction varies in the xz plane as
B = (>'/27r) Uhl8z) (2)
where >. is the optical wave length. There are two varieties of acoustooptic probing systems which differ in
physical mechanism of wave vector direction measuring. The system with a narrow aperture photodetector (NAP system) has been analyzed in [2].In this case only a part of scattered light is received by the photodetector. The acoustic pulse may be considered here as "a moving wave window II which picks out suc-
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 231
Figure 1. One-dimensional acoustooptic probing system.
cessively different sectors of the optical field. The output lens focuses diffracted light in a spot whose position depends on local directions of optical wave normal. When the acoustic pulse propagates across the cell (along the z axis) a diffracted spot moves in the lens focal plane in accordance with (2). These displacements are registered by narrow aperture photodetector. In pure form this method of wavefront analysis operates only in the Raman-N ath diffraction regime.
In the Bragg regime of diffraction the most important factor is an angular selectivity, i.e. the diffraction efficiency dependence on the optical beam incident angle. When the diffraction efficiency is small, this dependence can be expressed as
(3)
where 10 and Id are the intensities of incident and scattered beams, q is the parameter being proportional to the acoustic wave amplitude, I the acoustooptic interaction length, >. the acoustic wave length, On the Bragg
angle. In diffraction process, due to angular selectivity, spatial phase modulation of the optical wave is transformed to intensity modulation of the scattered beam. The acoustic pulse operates here as lIa selective moving windowll, its transparency depends on local values of phase gradient. In this case it is possible to use a broad aperture photodetector (BAP system) which receives all diffracted light [3].
BASIC RELATIONS
The time which is necessary to analyze one line if the light field is defined by transit time of the acoustic wave r = A/v where A is the acoustooptic cell aperture along the z axis, v is the sound velocity. For the frame scan it can be used a scanner of any type (mechanical, acoustooptic etc.) placed in front of the cell.
An important characteristic of the analyzing system is a number of resolvable elements in a line given by N = A/dmin where dmin is the resolvable element size. Spatial resolution of the system depends in the acoustic pulse length h or, in limit, on the transducer bandwidth .6.f:
N = R A/h = R M r. (4)
The parameter R is defined by the optical and acoustic wavelengths, the acoustooptic interaction length, the incident angle and the acoustooptic diffraction type. It is about 0.5 in optimum case.
Information possibilities of the analyzing system is determined also by a number of wave vector directions M which can be measured in an angular range .6. 0 :
232
M = t,.0/80 (5)
where 80 is the angular resolution. In the NAP system the range t,. 0 is defined by the diffracted beam
spread which is equal approximately CPd = )"/h. Thus, varying the acoustic pulse length we can change very easily angular range. In this case the photodetector must be placed in the direction
(6)
where 00 is the angle of incidence of the light on the acoustooptic cell. The angle 00 is chosen equal to Bragg angle 0B'
In the BAP system case it is necessary to choose the operating point in the middle of the slope of the characteristic curve (2), thus, 00 = 0B ± ncps/2
where CPs = A/I is the acoustic beam spread, n the index of refraction. The value CPs determines the angular range t,.0. It can be changed only by small variations of the acoustic frequency in the transducer band.
Angular resolution 80 depends on noise characteristics of the photodetector used in the system. If a photoelectric multiplier is employed as photodetector, then one can yield for the NAP system
(7)
where m is the signal-to-noise ratio, K the photodetector sensitivity, E the power density of incident light, ( the diffraction efficiency, X the optical transmittance of the system, e the electronic charge, CPo the photodetector angular aperture. The expression (7) connects three basic characteristics M,N,r which determine angular and spatial resolution and operation speed. For the
BAP system the value M differs by the multiplier (CPO/CPd)t:
(8)
Since CPo has to be much less than CPd the BAP system allows to obtain bigger angular and spatial resolution. This gain in resolution is achieved because of more completer usage of diffracted light.
The following example shows information possibilities of the acoustooptic analyzing system. The best acoustooptic material for the systems operating in visible range is paratellurite Te02. In an often used cut of the crystal an acoustic shear wave is excited alon& the axis [110] and an optical wave passes through the cell near the axis [001]. If the cell has an aperture A = 4 cm, then one line scanning time is r = 65 f.LS • For real values K = 10 mA/W, E = 20 mW/cm 2 , R = 0.6 m = 1 , ( = X = 1 and CPO/CPd = 0.1 one can obtain from (7): M(1)N 3!2 = 24.104 . Thus, if the acoustic pulse length is 0.025 cm (t,.f = 1.5 MHz), then the system is capable to analyze an optical
field in N = 160 resolvable positions along a line with M(l) = 120. At ).. = 0.6 /-Lm the system has angular range t,.0 = 1.5 mrad and angular reso-lution 80 = 1.2.10-2 mrad. For the BAP system we have M(2) N3h = 72.104.
EXPERIMENT AL RESULTS
In our experimental arrangement both variants of the acoustooptic analyzing system were realized: the BAP system was employed for line scan and the NAP system for frame scan.
233
(a)
\\ - r\ \
v '"'
flY v
Figure 2.
I
1mm ~
7! t---; 1.6/,s ..-'
\ A /
(\\ / v
\ b( y---'-"'
Oscillographic images of one-dimensional (a) and two-dimensional (b) optical fields.
One-dimensional analyzing system
The acoustooptic cell with the aperture A = 4 cm was fabricated of a Te02 crystal of the above-mentioned cut. The LiNB0 3 transducer with the central frequency 40 MHz had the width 1 = 0,23 cm. Phase objects to have been analyzed were placed in a collimated laser beam (A = 0.63 /-Lm) in front of the cell. After the object the phase-modulated wave passed through the cell close to the optical axis . In this case acoustooptic interaction was in a form of anisotropic diffraction. Due to orthogonal polarization the diffracted light could be separated from the undiffracted one.
The transducer supply system included a HF generator and an electronic modulator which formed radiopulses with pulse duration varying from 5 to 0.05 f-LS . It allowed to change the spatial resolution in the range 2.3-110 elements/cm. The maximum resolution was limited by the transducer bandwidth M. Thus, in the limit case information processing speed was N/r = 6.8.10 6
elements/so The angular range /::, 0 defined by acoustic beam spread was about
9.2 mrad. Measurements of angular resolution carried out with a test object gave the value 80 = 0.2 mrad (at m = 1 and N = 28). It is l.5 times worse than the theoretical value which follows from (8) for next experimental parameters of the system: (X = 0.02, E = 0.08 mW/cm 2 and K = 8.2 mA/W. As it follows from (8) the spatial and angular resolution can be considerably increased due to more powerful light source and optical loss decreasing.
Figure 2.a illustrates system operation in the one-dimensional scanning regime when the object represented a spherical mirror with a sinusoidal relief upon its surface. Three oscillograms of photomultiplier signal i(t) are put each on other. They are received for the same sector of the mirror but at different incident angles 00. These signals contain all necessary information about optical field structure and allow to study wavefront form even in the presence of spatial amplitude modulation. Results of these data processing are presented in Figure 3. The curve (1) shows light intensity distribution in the cell along the axis Z. This curve was used as reference to obtain wave normal distribution that is shown by the curve (2) . As the signal i(t) is proportional to phase gradient (h/8z, it is necessary to integrate i(t) in order to receive the function 'Y(z). The curve 3 yields this integration result showing the wavefront form.
Figure 3.
234
2
1
Intensity (1), phase gradient (2) and phase (3) distributions of light field formed by a spherical mirror.
Two-dimensional analyzing system
For analysis of two-dimensional optical fields a mechanical frame scanner was applied additionally. It had a form of rectangular prism rotated with angular velocity 587 clm that allowed to scan 64 image lines for 51.2 ms. A wavefront visualisation was realized on an oscillograph screen by frame and line scanning of electronic beam. Separate lines of scanning were shifted discretely in horizontal and vertical directions to create a spatial picture of light field. Phase gradient o'Y/ By was measured by an additional narrow aperture photodetector which registered a diffracted spot shift in the xy plane. To obtain the function t{y,z) the photodetector signals were integrated, then summed and sent to the oscillograph input. Figure 2.b presents an oscillographic picture of visualized wavefront formed by a spherical lens. For correct estimation of the picture it is necessary to take into consideration the presence of dc component in the photodetector signal which gives a linear growth of voltage after integration.
Therefore, our investigations show that acoustooptic interaction allows to construct effective wavefront probing devices which combine high angular and spatial resolution with high information processing speed.
REFERENCES
[1]
[2]
[3]
V.l. Balakshy, V.N. Parygin and L.E. Chirkov. "Physical principles of acoustooptics", Radio i sviaz, Moscow (1985). V.l. Balakshy, V.N. Parygin and H.A. Upasena. About a possibility of light field phase structure registration by acoustooptic method, Quant.El.(USSR) , 8:865 (1981). V.l. Balakshy, A.G. Kukushkin and M.Yu. Torgovkin, Light field phase structure registration with use of anisotropic Bragg diffraction selectivity, Radiotechniques and Electronics (USSR) , 32:724 (1987).
235
DETERMINATION OF PARAMETERS FOR THE SIMULATION OF
SURFACE ACOUSTIC WAVE DEVICES WITH FINITE ELEMENTS
P. Bauerschmidt*, R. Lerch, J. Machui, W. Ruile, G. Visintini
Siemens AG Corporate Research and Development Erlangen and Munich, FRG. • Institut fUr Hochfrequenztechnik Universitat Erlangen, FRG
INTRODUCTION
Surface acoustic wave (SAW) filters are widely used in the frequency range between 30 MHz up to 3 GHz due to their outstanding filter characteristics and their reproducible production at low costs. This frequency range and thus also SA W filters will become even more important with the growth of new communication techniques in the near future.
Due to increasing requirements on quality and low-loss accurate simulations of SAW filters have become indispensable for the computer aided design of these devices.
The simulation software we employ needs input parameters such as reflection and transmission coefficients of a SAWin an array of periodically arranged metal electrodes perpendicular to the direction of propagation.
Measurements of these parameters are very time consuming because of the broad variety of different materials, cuts and parameters. Additionally measurements suffer from technology tolerances and unwanted effects in experimental test-setups. Therefore great efforts have been made to calculate these data analytically [1,2]. In the past, however, the complexity of the problem still required some Simplifications and the results were limited in accuracy.
In contrast to the analytic approach the finite element method (FEM), which has been proven to be a powerful and flexible tool, avoids these restrictions thus leading to a reliable determination of the above mentioned parameters. The used FEM is described in detail in [3,4] and is therefore not discussed in this paper.
Figure 1. Surface acoustic wave resonator.
Physical Acoustics. Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 237
Figure 2.
be ___ l ___ ____ _ ----1--------- ,;
o 0 Tr/p 2Tr/p wave number -
Diagram of dispersion of a SAW on unperturbed, free surface (straight line) in a periodic array (line with "band gap").
EIGENFREQUENCIES AND THEIR INTERPRETATION
In most cases, SA W devices consist of several elements (transducers, reflectors) made up of a periodic array of metal electrodes On a piezoelectric substrate (Figure 1) . If the physical behavior of a single electrode in a periodic array is known, the whole element can be calculated using the coupling of modes theory [5] or a P-matrix formalism [6]. Then the elements are cascaded to give the performance of the whole SAW device.
The influence of a periodic array On the propagation of a SAW can be shown in a dispersion diagram w({J) (Figure 2) . The straight line represents the unperturbed propagation of the SAW which exhibits no dispersion, which means phase velocity Vph = wi {J and group velocity Vgr = dw/d{J are equal. The addition of an array of period p changes the unperturbed line in two aspects : the slope decreases which means a decrease of the SAW velocity. This effect can be seen as a frequency shift of 80 at {J = 7r I p. On the other hand there is a "band gap" at {J = '!rIp with the width 81, which is proportional to the reflectivity of a single electrode in an infinite array [1]. In this gap between WI and W2 the group velocity is zero, hence the SAW cannot propagate.
The numerical values of 80 and 81, were determined by an eigenfre-quency analysis performed with the finite element method. One of the FEMmeshes used for modeling electrodes On piezoelectric substrates is shown in Figure 3. It had a width of p/2 and a depth of up to 10 p in total. The electrode height was from 0.6 % up to 4 % of 2 p. The boundary conditions are periodic on both sides and free On top. Care was taken, that the boundary condition at the bottom had no influence on the results . As the result of the
Figure 3.
238
P ------------~. I
~--
W. ·:I·::·--l.~a + .. .. ... -. . ".-- -. _.-- --. .. . ,
~ . _. -.· . . Jd Finite element mesh of One single metal electrode in a periodic array. The lower boundary is at 10 p.
Figure 4.
002
~ 0.01
~ ;g wiT' = 0.5 0; 3 0 ~--------------------~~~~--~~~ c: .g '"' ~ ~
... (lOl
... 0.02 L-... __ ...J-__ ----L ____ .1...-__ ......l... __ ---''--~.....L...-----L----'
o 0.02
Ir /2p -
Rayleigh wave reflection coefficient of a metal electrode in a periodic array of aluminum electrodes on 128'rotYX-LiNb0 3 with electrode shape as parameter (FEM-results).
FEM--calculations two eigenfrequencies WI and W2 for stancling waves in a periodic array were obtained. On the free, unperturbed surface these eigenfrequencies degenerate to one single eigenfrequency Wo. From these eigenfrequencies the values for velocity perturbation /::, V /vo as well as the magnitude of the reflection coefficient of the single metal electrode 1,.1 p can be derived [1] :
Figure 5.
/
(1)
(2)
t 0.04
C4
Cl"'90°
w/p = 0.6 /
/ /
/
/ /
/
II /
" / /
" " " /
" / /
/ /
" " ,,/
/,,"
0.02 0,04
hl 2p _
Rayleigh wave reflection coefficient of a metal electrode in a periodic arral of aluminum electrodes on AT-Quartz FEM-results loctogons) interpolation of FEM-results (--) extrapolation of data from [5] (- - - -) .
239
RESULTS
We used Al-electrodes on 128'rotYX-LiNb0 3 for our FEM-calculations and proceeded as outlined above. For this choice of materials the Rayleigh wave reflection coefficient of a single metal electrode in a periodic array as a function of the normalized electrode height h/2p with the electrode angle a as parameter is shown in Figure 4. The electrode angle a cannot be chosen arbitrarily, but is determined by the photolithographic process. For a given electrode angle a a suitable height h can be chosen e.g. in order to obtain a zero reflection coefficient . This allows the design of transducers without internal reflections . In contrast to the well-known splitfinger technique this method is suitable even in the GHz range to achieve broadband suppression of reflections. At these frequencies, however , the electrode shape is not rectangular and the electrode angle a has to be taken into account.
On the other hand , large reflection coefficients are desired in the case of SA W resonators on AT-Quartz, for example. In Figure 5 the increase of the reflection coefficient with the electrode height h is shown. The broken line is an extrapolation of experimental data found by Wright [5] . The full line interpolates the results of the FEM analysis. It can be seen that deviations are significant for higher metal heights. The effect of these different reflection coefficients on the transfer function of a resonator with normalized metalization height of 2 % is shown in Figure 6. The upper picture (a) represents the
240
CD -20 .., <I> .., :l = -40 c: CD
~
CD -20 .., ~
<I> ..., :l = -40 c co ~
CD -20 ..., ~
<I> ..., :l = -40 c: CD
~
Figure 6.
223 224 225 226 227
f r equency [MHz]
Transfer function of a resonator on AT-Quartz:
tal measurement . b simulation with FEM-results. c simulation with data from [5].
measurement with transverse modes on the right side of the resonator peak. I~ is in excellent agreement with Figure 6.b, a P-Matrix calculation with the reflection coefficient obtained from the FEM analysis as input parameter. The transverse modes were not taken into account. For comparison, Figure 6c shows the same calculation but this time the extrapolated value for the reflection coefficient taken from Wright [5] was used as input parameter. It is evident, that with our FEM results the stopband can be described more correctly.
CONCLUSION
The finite element method has proven a flexible and powerful tool for the determination of input parameters for the simulation of surface acoustic wave devices. The calculated input parameters are reflection and transmission coefficients of SAWin periodic arrays of metal electrodes. Full anisotropy of the piezoelectric material, as well as an arbitrary geometrical shape of the electrodes are considered. There is excellent agreement between measurement and calculation if input parameters obtained by the FEM are used in the simulation process.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6)
D.P. Chen, H.A. Haus, "Analysis of metal strip SA W gratings and transducers", IEEE Trans. Sonics Ultraso., vol. SU-32, p.395-408, 1985. H. Robinson, Y. Hahn, "Analysis of SAW Reflectors", IEEE Ultrason. Symp. Proc., p.131-134, 1988. R. Lerch, "Finite Element Analysis of Piezoelectric Transducers", IEEE Ultrason. Symp. Proc., p.643--654, 1988. R. Lerch, "Simulation of Piezoelectric Devices by Two- and Three-Dimensional Finite Elements", IEEE Trans. Sonics Ultrason., vol. SU-37, May 1990. P.V. Wright, "Analysis and Design of Low-Loss SAW Devices with Internal Reflections Using Coupling-of-Modes Theory", IEEE Ultrason. Symp. Proc., p.141-152, 1989. G. Tobolka, "Mixed Matrix Representation of SAW Transducers", IEEE Trans. Sonics Ultrason. SU-26, vol. 6, p426-428, 1979.
241
ANALYSIS OF THE NEARFIELD OF LASER LIGHT DIFFRACTED BY A
PLANE ULTRASONIC WAVE
Erik Blomme, Rudy Briers* and Oswald Leroy*
V.H.T.!. Campus CATHO, Doorniksesteenweg 148 8500 Kortrijk, Belgium *K.U.L. Campus Kortrijk, Universitaire Campus 8500 Kortrijk, Belgium
INTRODUCTION
As for light diffraction by fixed gratings, one can distinguish between the Fraunhofer region (farfield) and the Fresnel region (nearfield) of the diffracted light. Whereas the individual farfield diffraction orders have constant light intensity, at least if the ultrasonic wave is a progressive one, the intensity in the nearfield of the diffracted light is modulated in both time and space. This intensity modulation is due to the mutual interferences of the various diffracted light waves and may be very complex, even for moderate ultrasonic amplitudes. Moreover, the intensity distribution in the nearfield of the diffracted light appears to be much more sensitive to variations of the ultrasonic power than the individual Fraunhofer intensities, which provides a useful technique for observing weak ultrasonic waves. In addition, the intensity measurement in the nearfield requires a simple experimental arrangement.
The purpose of the present paper is to give some general results, together with some specific properties and a sound-reconstruction algorithm in the case of Raman-N ath type diffraction.
GENERAL THEORY
Consider a plane monochromatic and linearly polarized light wave entering parallel to the Z-axis, a rectangular, homogeneous, isotropic and transparent medium (e.g. a liquid cell), which is disturbed by a plane ultrasonic wave propagating in the X-direction (Figure 1). The variation of the refractive index of the medium, adopting the periodicity of the ultrasonic wave through the fluctuations in pressure and density, can be approximated by a Fourier series of the form
N /L(x,t) = /Lo + ~ /Lrsin[r( w* t-k*x) + or] ,
r=i (1)
where /Lo represents the refractive index of the undisturbed medium, and J.Lr the maximum variation of /L caused by the harmonic of order r; w* = 21T'1I* is the frequency of the lowest order component (fundamental frequency), k* = 21T'/ A* the corresponding wave number and Or a phase constant. The restriction to N frequency components is not a real limitation of theory and is only of practical interest.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 243
After leaving the ultrasonically disturbed medium, the light intensity distribution in a plane parallel to the exit plane z = L is described by the Fourier series [1,2]
I(x,Z,t) too Cm(Z)exp[-im(w*t-k*x)] (2a) m= -00
where +00 = exp(2im27rZ) E 1/im+p(L)iJ!p(L)exp(4mpmZ) (2b)
p = -00
Z is a normalized spatial variable defined by
Z = z - L D '
where D is the distance
D
A being the wavelength of light in vacuum; 1/ip(L)
the diffracted light component with frequency conjugate.
(3)
(4)
represents the amplitude of
w-pw*, iJ!p(L) its complex
Note that the intensity distribution of the light, which was assumed to be equal to unity before reaching the entrance plane z = 0, has now adopted the periodicity of the ultrasonic wave and is repeated in the Z-direction with period Z = 1.
THE CASE OF RAMAN-NATH TYPE DIFFRACTION
For Raman-N ath type diffraction, the amplitude of the ptLorder diffracted light wave, evaluated in the exit plane z = L, is given by [1,3]
N x exp {-i E [nr(or-ro1)] } (5)
r=2
244
where
2'ffJ.Lr L vr = ~ , r = 1, ... ,N (6)
is the Raman-Nath parameter associated with the rth--order Fourier component of the sound si~nal (1); E is a summation from r = 2 to r = N. Substituting (5) into (2), rearranging terms and repeatedly applying properties of Bessel functions, the following expression for the intensity distribution inside the diffracted light beam is obtained [1]:
+00 +00 +00 I(x,Z,t) = 1 + 2 E E... E J m-Erq (wl m)Jq (w2 m)'"
m = 1 q2 = -00 qN = -00 r' 2 '
N
... JqN(wN,M)cos{m(w*t-k*x + oc ~) + r~2qr[(or-rol) + (r-1)~]) (7)
where
r = 1, ... ,N . (8)
Note that in this specific case, the diffracted light wave front has constant intensity in the plane z = L (no amplitude modulation) and that this state is repeated with period D/2. In other terms:
I(x,O,t) = I(x,f.~,t) = 1 f = 0,1,2, ... (9)
This property is related to the fact that in the case of Raman-Nath type diffraction, the ultrasonic beam only acts as a phase-grating: as far as the exit plane, the light is not affected in its amplitude.
It should be mentioned here that the early experiments of Colbert and Zankel [1,4] in the case of a harmonic ultrasonic wave, and the recent experiments of Kwiek and Markiewicz [5] in the case of an ultrasonic beam consisting of two harmonics of frequency ratio 1:2 give a very good evidence of the reliability of the theory.
In the last quoted reference, K wiek and Markiewicz also showed that the influence of the harmonic of order 2 disappears in the planes Z = 1/4 and 3/4. This phenomenon is an example of the more general property, which can easily be derived from (8). Indeed, in the planes Z = f/2r (f = 0,1,2, ... ), Wr,m = 0 ,and the influence of the harmonic of order r is canceled. An example of this effect is shown in Figure 2. The sawtoothed ultrasonic signal (a) produces in the plane Z = 1/4 the light intensity signal (d). Adding 'ff to the phases of the even order harmonics, the direction of the sawtooth-profile is inverted (b), or dropping the even order harmonics, the ultrasonic signal becomes rectangular (c). In each case however, the light intensity signal remains unaffected. The similarity between (c) and (d) is not coincidence as well, as can be explained from the theory [2].
A RECONSTRUCTION METHOD
Choosing Z $ 1 in such a way that I Wr,m I = 12v rSin( 2rm 'ffZ) I $ 1 for all values of rand m appearing in the non-negligible terms of (9), the latter expression can be approximated by [1,2]
I(x,Z,t) ~ 1 + Co "'=":~'-=""'r sin [m(w*t-k*x) + oJ (10)
245
1
o f'
-1 a
o
b -1
~------------------------------------------------------~
lI1Jl~j -1 C
2.0
1.5
1.0 I
0.5
d O.D~ ________________ -. __________________ .-________________ ~
o
Figure 2.
246
2
ULTRASONIC TIME PERIODS ~
(a) Sawtoothed ultrasonic signal normalized with respect to the fundamental frequency component; Vr = 0.5/r, Dr = 0, r = 1, ... ,30
(b) As in (a), but with Dr = 7r if r is even. (c) Rectangular ultrasonic signal obtained from (a) by
canceling all even order harmonics, i.e. Vr = 0 is even. (d) Intensity signal of light diffracted by the ultrasonic waves
(a), (b) or (c), registered in the plane Z = 1/4 (normalized with respect to the incident light intensity).
3
a
0.5
e
Figure 3.
b
1.5 d
-2
ULTRASONIC TIME PERIODS ~
(a) Ultrasonic signal with Vr = I sin [ 1f(r-8)/2]I for 1f( r=8)
r = 1, ... ,25 (r :f 8), VB 0.5; Or = 0 if r is even and Or = 1f if r is odd. The signal is normalized with respect to the 8-t h-order frequency component.
(b) As in (a), but with Or = 1f for r = 3,7,9,13,17,21,25 and Or = 0 otherwise.
(c) and (d) Intensity signals of the light diffracted bX the ultrasonic pulses (a) and (b) as calculated from (7) in the plane Z = 0.0002.
(e) and (f) Ultrasonic signals reconstructed from (c) and (d) using (10).
where Co is some proportional constant. Hence, applying a FFT-procedure to the light intensity signal registered in a point close to the exit plane Z = 0, one can reconstruct the time-waveform of the original sound signal (1) by multiplying the mtLorder Fourier component of (10) with the inverse ratio sin(27rZJ/sin(2m27rZ). Note that this procedure not only provides the sound frequency components and their relative amplitudes, but also their phases. For small Raman-Nath parameters vr , this nearfield method is comparable to the farfield method proposed by Van Den Abeele and Leroy [6], a method based on ultrasonic diffraction of a conical laser beam.
As any ultrasonic pulse can be approximated by a discrete Fourier series of the form (1), the present nearfield method is also applicable to pulsed ultrasonic waves. A computer simulation of the method is shown in Figure 3 for the case
247
of two pulses both having a repetition frequency w* (lowest frequency component) and a modulation frequency of 8w* (strongest frequency component). For w* = 250 KHZ and green argon laser light (,\ = 0.5145 ·10-6m), the plane of observation Z = 0.0002 corresponds to a real distance of 3.7 cm from the exit plane. It can be seen that a perfect reconstruction is established. Although both pulses have the same Fourier components, only differing by their phases, the method can distinguish them.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
248
E. Blomme, 11 Theoretical study of light diffraction by one or more ultrasonic waves in the MHz-region", Doctoral Thesis, K.U.L. Leuven (1987). E. Blomme and O. Leroy, On the nearfield of light diffracted by a plane periodic ultrasonic wave, Proc. 4th Spring School on Acousto-Optics, 47-68, Gdansk 1989, Ed. A. Sliwinski, University of Gdansk. O. Leroy, Theory of the diffraction of light by ultrasonic waves consisting of a fundamental tone and its first n-1 harmonics, Ultrasonics, 10:182-186 (1972). H. Colbert and K. Zankel, Light diffraction by ultrasonic waves - Fresnel region, J. Acoust. Soc. Am. 35:359-363 (1963). P. K wiek and A. Markiewicz, Investigation of the nearfield of light in the diffraction system consisting of two ultrasonic beams, Proc. 4th Spring School on Acousto-optics, 129-151, Gdansk 1989, Ed. A. Sliwinski, University of Gdansk. K. Van Den Abeele and O. Leroy, Optical measurement of plane superposed ultrasonic waves, Proc. 4th Spring School on Acousto-Optics, 291-300, Gdansk 1989, Ed. A. Sliwinski, University of Gdansk.
MAGNETOELASTIC MODE EFFECT ON SPIN-WAVE INSTABILITY
THRESHOLD
A. S. Bugaev, V. B. Gorsky, A. V. Pomyalov
Moscow Institute of Physics and Technology Dolgoprudny, Moscow, 141700, U.S.S.R.
INTRODUCTION
A new attraction has been given to nonlinear effects and ferromagnetic resonance (FMR) phenomena in single crystal ferrites in recent years, motivated by the availability of high-quality epitaxial thin films of yttrium iron garnet (YIG) and its possible use for microwave signal processing. This work presents experimental investigation of the influence of magnetoelastic interaction on nonlinear FMR behavior in normally magnetized YIG films. It is well known that dipole magnetostatic modes (MSM) representing natural oscillations of magnetization vector in bounded ferrite medium become unstable at high microwave power level [1]. Magnetoelastic interaction was previously shown to exert its influence on the first order spin wave instability threshold in bulk ferrite crystals [2]. Planar YIG film - gadolinium gallium garnet (GGG) substrate structures are high quality resonators and the magnetoelastic excitation of elastic modes must be especially strong in the structures and therefore is expected to influence significantly nonlinear behavior in YIG films. In normally magnetized ferrite films MSM instability is caused by the second order parametric processes. Parametric spin-wave (SW) generation is known to occur above the critical level of exciting microwave power P s. In this work the second order SW instability of dipole MSM is investigated for resonance interaction of secondary SW with elastic system. Strong magnetoelastic interaction of secondary SW with elastic system takes place provided frequencies and wavenumbers of the secondary SW correspond to dispersion curve crossover region of elastic wave (EW) and spin wave.
MSM INSTABILITY IN FERRITE FILMS
First of all let us consider the main features of the second order SW instability of dipole MSM in normally magnetized ferrite films in the absence of magnetoelastic interaction. Dipole MSM energy is determined by the energy of dipole spin-spin magnetic interaction. Dipole MSM in ferrite films are known to have quasi homogeneous distribution of rf magnetization over the film thickness. For a bulk ferrite crystals it was shown by Suhl [1], under condition of MSM second order SW instability those SW have a minimum parametric excitation threshold with wavevector km parallel to magnetic field direction. In normall~ magnetized ferrite films such a SW are spin-wave resonance (SWR) modes l3], so secondary SW modes generated at the dipole MSM instability have to be SWR modes. To check this hypothesis the MSM instability thres-
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, J 991 249
hold in thin ferrite films was investigated. The characteristic peculiarity of the thin films is a discrete SWR modes spectrum. So, if SWR modes had minimum excitation threshold the MSM instability threshold Ps would be a periodic function of MSM resonance magnetic field Hmsm. Besides, Ps would be minimum provided Hmsm were equal to resonance magnetic field Hswr of one of the SWR modes. This phenomenon was actually realized in the experiment. The decrease of dipole MSM instability threshold more than two times was observed provided its resonance magnetic field coincided with resonance magnetic field of one of the SWR modes. Under the experiment Ps denoted the microwave power level corresponding to the drop of microwave magnetic susceptibility of the dipole MSM. Thus, in normally magnetized ferrite films the dipole MSM instability is accompanied by the parametric generation of short exchange SW modes. SWR modes have the smallest parametric excitation threshold.
MAGNETOELASTIC MODE EFFECT ON DIPOLE MSM INSTABILITY THRESHOLD
Let us now consider the influence of magnetoelastic interaction on dipole MSM instability threshold. The magnetoelastic interaction was revealed in the instability threshold change of a group of MSM. The instability threshold Ps of dipole MSM vs its resonance magnetic field Hmsm was measured. Ps (Hmsm) dependencies were measured at several microwave frequencies f. In each case the selected f was equal to one of the Lamb mode cut-off frequencies fi for magnetoelastic interaction effect on the SW instability threshold to be maximum. In fact the efficiency of elastic wave excitation in planar film-substrate structure must be maximum provided microwave frequency f coincides with one of fi. fi frequencies are resonant frequencies of standing EW across the thickness of the film. The experimental dependence Ps (Hmsm) at f = 4235.4 MHz is shown in Figure La. It is evident that the MSM instability threshold at resonance magnetic field Hmsm ~ Hs surpasses P s of other MSM. f frequency being equal to one of fi ,the increase of MSM P s with Hmsm ~ Hs was maximum. Calculations reveal that dipole MSM with increased Ps have f frequency and magnetic field H degenerated with f and H of SWR modes having wavenumbers km equal to transverse EW wavenumber
ps•
(lrb .... tI.
3
2
1
Figure 1.
250
a b Hs.
/\ Oe h
h 20 /,
h ,/;
h
.-¥+ : ~+-+ (0 h .....:::
I I fa Hs 30 ~ 2 .3 Ii -!,GGe 1-t"""',Oe
a) Dipole MSM instability threshold Ps vs its resonance magnetic fields Hmsm. Hmsm were counted from the bottom of the MSM spectrum in ferrite film. The sample was a (111) YIG film 10 p.m thick grown on GGG substrate 450 p.m thick.
b) Magnetic field Hs where the increase of P s was observed vs microwave frequency f. Hs was counted from the bottom of the MSM spectrum. The dotted line is the experimental dependence. The solid line is calculated
dependence: Hs = 4 'if M . a· k~ , where ke is defined by the frequency ke = 2rl/V.
ke. In fact, as it was shown in the previous section secondary SW excited at the dipole MSM instability are SWR modes. SWR modes with km::i ke efficiently excite elastic modes in planar film-substrate structures. Such excitation increases relaxation parameter AHk of the SWR modes. Increasing AHk makes dipole MSM instability threshold P s larger since according to Suhl [1] P s is proportional to the secondary SW relaxation parameter AHk. Ps (Hmsm) dependencies obtained at other frequencies reveal frequency dependence of the characteristic magnetic field Hs at which the increase of Psis observed (Figure lob). The dependence Hs(f) is sure to be observed in the experiment. Really EW wavenumber ke depends on f: ke = 2rl/V , where V is EW velocity. SWR mode resonance magnetic field Hmsm, counted from the bottom of the magnetostatic wave (MSM) spectrum of the film depends on SWR mode wavenumber km as Hswr = 41rM· a· k~ , where a is exchange constant, M is saturation magnetization of the film. The resonant interaction of SWR mode with elastic system will take place provided km is equal to ke . Thus Hs must have quadratic frequency dependence. The comparison of experimental and calculated dependencies Hs (f) reveals their good agreement (Figure lob). Thus magnetoelastic interaction of secondary SW accompanying dipole MSM instability cause the instability threshold to increase if the MSM involved is degenerated in frequency and magnetic field with SWR modes having wavenumber equal to EW wavenumber.
It should be emphasized that the direct interaction of dipole MSM with an elastic system would have also increased the instability threshold of dipole MSM. Actually dipole MSM excites elastic modes in planar film-substrate structure if f = fi. Such excitation increases dipole MSM relaxation parameter AH and consequently could increase its instability threshold. However, as it was shown in [4J dipole MSM excites EW much weaker than SWR modes with km ::i ke. Thus according to [4] EW amplitude excited by dipole MSM should be 5.10 3 times smaller than EW amplitude excited by SWR modes in the experimental conditions. Thus the influence of direct interaction of dipole MSM and elastic system on P s would be negligible and was not found in the experiment. The dipole MSM instability threshold increase could not be induced by direct magnetoelastic interaction of dipole MSM. Has it been the case the increase of instability threshold of all dipole MSM would have been observed. But this effect was not observed and P s increased only for a certain group of dipole MSM degenerated with SWR modes defined above. Thus it was confirmed that it was secondary SW magnetoelastic interaction that increased dipole MSM instability threshold.
MAGNETOELASTIC MODE EFFECT IN THIN FERRITE FILMS
Included in this section is the frequency dependence of instability threshold of the dipole MSM generating at the instability SW resonantly coupled with elastic system. It will be shown that magnetoelastic interaction may decrease the instability threshold of the dipole MSM in thin ferrite films with discrete SWR mode spectrum. Experimental dependence of the instability threshold for one of dipole MSM is shown in Figure 2.a. Secondary SW generated at the MSM instability was resonantly coupled with elastic system. From section 2 if microwave frequency f was equal to one of fi the Ps was maximum. But it is evident that the P s at some f~ frequencies is less than the value of instability threshold p£ observed without coupling of secondary SW and elastic system. p£ was measured in another frequency range lest EW wave numbers should be equal to secondary SW wavenumbers. The decrease of Ps is explained by the coincidence of resonance magnetiC fields of the dipole MSM Hmsm and one of SWR modes Hswr at f frequencies equal to fi·. Really, in general, resonance fields Hmsm and Hswr do not coincide because of the discrete character of SWR mode spectrum. From section 1 instability threshold in this case is not minimum and should surpass the threshold which would have been observed if Hmsm had been equal to Hswr. However, in addition to
251
1-1, b I '
(k ': ,
" /
, I;
/ I , 3126 / / , -r -" -r , , -. . r--- /~ , : ( 37f8 :/ , 1/
- ;! I I
'-{(3182 ~i 3185" I -I,->J. Ii ... , {,M(;j~
Figure 2. a) Instability threshold frequency dependence of dipole MSM, generating at the instability SW with km = ke, where km, ke are spin and elastic wave wavenumbers respectively. YIG film 3 f1.m thick was used. fi are the Lamb mode cut-off frequencies.
b) Resonance magnetic fields H vs microwave frequency. Solid line is Hmsm(f) for dipole MSM. Dotted line is Hswr(f) for SWR mode with krn ~ ke. Dash-dotted line is Hmsm( f) for SWR mode in the absence of magnetoelastic interaction.
the increase of hHk magnetoelastic interaction causes anomalous frequency dependence of resonance field Hswr of SWR mode with km ~ ke. As a result there may be a c}'incidence of SWR mode and dipole MSM resonant magnetic fields at some fi frequencies (Figure 2.bl. From section 1 the dipole MSM stability threshold should be minimum at fi frequencies and this has actually been observed in the experiment. Thus the magnetoelastic interaction causes the periodic decrease of P s at ft frequencies.
SUMMARY
A dipole MSM has mllllmum instability threshold if degenerated in frequency and magnetic field with one of the SWR modes in normally magnetized ferri te films.
The instability threshold is increased by a factor of 3 for the dipole MSM generation due to instability SW resonantly coupled with elastic system.
Dipole MSM instability threshold is a frequency oscillating value because of discretization of magnetic and elastic spectra in planar film-substrate structures. Oscillation period is equal to frequency separation of the elastic Lamb modes of the structure. The maxima correspond to the Lamb mode cut-off frequencies, the minima being less than the instability threshold in the absence of magnetoelastic interaction.
REFERENCES
[lJ
[2J
[3J
[4J
2~2
H. Suhl, The Nonlinear Behavior of Ferrites at High Microwave Signal Level, Journ. Phys. Chern. SoL, 1:209, (1957). E.H. Turner, Interaction of phonons and spin waves in yttrium iron garnet, Phys. Rev. Lett., 5:100, (1960). C. Kittel, Excitation of spin waves in ferromagnet by a uniform of field, Phys. Rev., 110:1295, (1958). A.S. Bugaev, V.B. Gorsky, A.V. Pomyalov, Effect of exchange and spins pinning on acoustic wave excitation in YIG films, in Proceedings of ISSW AS '89, Varna, 205, (1989).
METHODS OF ACOUSTIC MICROSCOPY IN INVESTIGATION OF
IDGH-TEMPERATURE SUPERCONDUCTORS
Bukhny M.A., Chernosatonsky L.A., Maev R.G.
Center of Acoustic Microscopy Institute of Chemical Physics Kosygina str., 4 Moscow, 117334, U.S.S.R.
INTRODUCTION
We demonstrate the use of acoustic microscopy in investigation of both local acoustic properties of individual crystallites and various defects in HTSC-materials. It is important that the acoustic microscopy provides for: (a) viewing the defects in the depth of a non-transparent sample, unlike the optical methods; (b) determining the Rayleigh wave velocity on a small (N10 11m) section of surface (for example, on an individual crystallite); (c) detection of defects in the presurface layer by attenuation of such waves. In our work we used a commercial acoustic microscope ELSAM (Ernst Leitz Scanning Acoustic Microscope) [1].
In the acoustic image the brightness of each point is proportional to the Signal amplitude V(x,y,z) at the respective position of the lens in the scanning plane (x,y), z being the distance between the lens focus and the subject surface. The amplitude of the acoustic wave reflected from the subject and, consequently, V(z), depends both on the distance z, and the local acoustic properties of the subject. The greater the difference between the acoustic properties in the given point and the properties of the immersion liquid, the higher the reflection, the brighter the respective point in the subject image; V(z) is maximum, as a rule, when z = 0 (see Figure 1).
fl9rm
Figure 1. Output signal V of scanning acoustic microscope versus distance Z between lens focus and sample surface. V(z)-curve obtained on a BiCaSrCuO single crystal.
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 2~)3
Figure 2.
v Ray interpretation of SAM output signal generation with Z < 0 : A. mirror-reflected ray; B. ray transformed out of longitudinal wave in sample reflected from subsurface defect; C. ray transformed out of transverse wave; D. outflowing Rayleigh wave.
THEORETICAL CONSIDERATIONS
At image forming with Z < 0, in a ray approximation one may consider that only the rays going as if from the lens focus are received effectively. For example (see Figure 2), the contribution of ray C is much higher than that of ray A. Therefore upon focusing onto the subject the subsurface defects (discontinuities) will occur not in the focus, and they will be either not sharp in the image, or will be absent at all; such defects will get in the focus with the lens approaching the subject.
For practical application determination of the depth of occurrence of the defect is important. As was shown in [2], if the rays reflections between the surface and defect be neglected, the depth of occurrence of the defect equals approximately
(1)
where the maximum signal from the defect is attained at a shift of the lens from the sUbi'ect by oz, Bm - lens aperture C1 - sound velocity in immersion liquid (water, C - sound velocity in sample. To determine d, V(Z) - dependencies can e used measured on plane-parallel samples calibrated in thickness also.
It should be noted that even in an isotropic solid two types of waves (longitudinal and transverse) propagate. Numerical simulation and quantitative estimation show that the subsurface images in the majority of perovskite samples under investigation are created mainly by transverse waves [2].
The output signal of the SAM very greatly depends on distance z, hence for irregular subjects only their topography can be determined.
As mentioned above, the SAM can be used to measure the local velocity of the Rayleigh wave. It is possible to show [3] that for a homogeneous isotropic subject the expression for the Rayleigh wave velocity is used:
(2)
where !J.Z - the period of oscillations on V(Z) - curve, ,\ 1 - wave length in immersion. An experimental criterion of smoothness of the subject and its homogeneity in depth is the homogeneity of surface and subsurface image.
254
EXPERIMENTS
Ceramics
We have investigated 123-ceramics ABa2CusOx (A = Y, Yo, Ho), BiSrCaCuO, both hot-pressed up to 70 kbar and ordinary ones. Water heated up to 333K or methanol at a room temperature was used as an immersion liquid. Regions of crystallites are seen as bright sections, dark sections - are pores.
The focus being brought onto the bottom of the pores by the maximum of the acoustic wave reflected from them, makes it possible to determine their depth accurately to 0.05 p.m. Usually it lies within 1 to 3 p.m.
The value of Rayleigh velocity V n. was determined at the frequency of
1.7 GHz according to V(z) - dependence with variation of Z within 0 to 20 p.m; consequently, the maximum distance covered by the Rayleigh wave made up d = Zxtg0n. N 10 p.m [4], therefore measured V n. characterizes the
surface section with a diameter of about 10 p.m. Of course, V n. is an average
Rayleigh velocity in all directions in the plane of the sample. Measurements of V R. were taken on sufficiently large crystallites, so that their boundaries did
not influence the propagation of the Rayleigh wave. The velocity V n. was
found to be the same on different crystallites of the same sample. Since it is known that in the perovskite structures under investigation there is a considerable difference between the sound velocity along the axis c and the sound velocities in the plane (a,b) : V[100] ~ V[OlD] < V[OOl] [6], and the crystallites in ceramics are plates with dimensions in directions [100] and [OlD] considerably exceeding their thickness in direction [001], one may conclude that the axis c of the crystallites under examination is perpendicular to the ceramics surface.
For ceramics we obtained the following values of Rayleigh velocity: 3.01 km/s for BiCaSrCuO, 2.80 km/s, for YBaCuO, 2.86 km/s for HoBaCuO and YbBaCuO with the accuracy of 0.05 km/s, which made up 0.8 - 0.9 of the Ct values in the plane [001] for the respective materials.
The Rayleigh wave absorption an. is determined by the slope to the side
maxima of V(z) - curve (Figure 1). This makes it possible to determine the value of an. for every crystallite. The accuracy of determining the magnitude
was about 50%, since the magnitude of the side maxima determining the angle f3 greatly depends on the instrument radio engineering parameters. High magnitudes of absorption were observed on microcrystallites of many ceramics, especially on high-density ones. Such a high attenuation was caused by a great number of microcracks, sometimes the value of their exposure was far beyond the range of the microscope resolving power (0.4 p.m). This is associated with the fact that water does not penetrate into microcracks, and the hypersound of the used gigahertz frequencies practically gets fully attenuated in air at the wavelength distance. Such an estimation of an. characterizes the integral
quantity of attenuation of the Rayleigh wave on a section with the diameter of 10 p.m, and is rather sensitive to the magnitude of density of crack distribution on the crystallite.
Different ceramic phases feature different acoustic properties and, consequently, different factors of reflection from them. For instance, for phases YBa2CuS06'5 and YBa2CuS07, whose longitudinal velocities are equal to 4.9 km/s and 4.4 km/s, respectively [5], we obtain the difference in the reflection factors equal only to 2%. Thus, for intensively reflecting subjects the difference in the reflection factors for different phases is low, therefore, for visualization of different phases it is reasonable to use so-called quasi-three-dimensional image, at which the oscilloscope beam while scanning on the screen, besides variation in brightness, deflects from the scanning line proportionally with the magnitude of the output signal of the SAM. Figure 3 illustrates a quasi-three-dimensional image of high-density 123-ceramics, in which phase inclusions look like peaks.
255
Figure 3.
Figure 4.
Figure 5.
256
Quasi-three-dimensional image of surface of hot-pressed ceramics YBaCuO, 200x130 j1m. Phase 1236 is separated due to higher sound velocity. Frequency, 1.7 GHz.
Hot-pressed ceramics YBaCuO, 300x200 j1m, Frequency 1.7 GHz . Subsurface (Z = -5.5 j1m). Twins are visible.
Film YBaCuO applied by thermal evaporation on substrate Si. A multi phase film is obtained. Darker spots are inclusions of "green" phase 211 . Light spots are peelings.
We also observed a twin structure on some crystallites, which, generally speaking, should not be observed with the aid of a spherical lens featuring an axial symmetry. Apparently, such a picture is associated with formation of strained regions near the twinning boundaries. This is confirmed by the fact that the twin structure is invisible with Z = 0, and it becomes visible only when Z < 0 (Figure 4), when the lens starts receiving the outflowing Rayleigh wave. The probable mechanism of visualization of the twins boundaries could be as follows. While propagating over the monocrystallite surface, the Rayleigh wave is reflected from the twinning boundary because of the discontinuity of the latter, the wave outflow near the boundary increases. As a result, the twinning boundary is outlined by a brighter line and becomes visible in the image, even if the cross-sectional dimensions of the boundary are much less than the wavelength; a similar effect of outlining occurs for any boundaries with Z < O. To improve the contrast of the picture of the twin structure, use should be made of an asymmetric lens, for instance, with a shifted or elliptic transducer. It will allow to distinguish the components of the twin structure, different in orientation, according to the difference of the reflection factor.
Films
Films obtained by successive thermal deposition of Y, Ba and Cu on a silicon substrate at 720 K in the oxygen atmosphere were investigated. Investigation of the obtained films has shown that they are multi-phase ones ~Figure 5). Along with the YBaCuO phase there are sections of Y2BaCu05 - I green II phase (dark spots) and inclusions of CuO, Y20 3 and BaCu02 (dark points). As to the kind of V(z) - dependencies, phases 123 and 211 are identified unambiguously. In Figure 5 one can also see light spots with interference rings, peelings, caused by an insufficient adhesion and by the difference of the thermal coefficients of expansion at thermal cycling.
Crystals
Besides ceramic samples, we investigated also separate crystals YBaCuO and BiSrCaCuO. We revealed on them both surface (Figure 6) and subsurface (Figure 7) defects of structure. Twinning with a period of about 1 /Lm was observed on YBa2Cu307 samples. The values of V R measured in the planes
(a, b) of monocrystals were closed to the V R measured on non-strained crys
tallites of the respective ceramics.
STUDIES ON CHEMICAL DEGRADATION OF HTSC-MATERIALS
We investigated the degradation of single crystals and ceramics YBaCuO under the effect of water.
We have investigated the degradation of the microstructure of YBaCuO single crystals, prepared by the zone method, under the effect of water heated to 60· C, used as an immersion liquid. At the be&inning of the experiment the surface of single crystals was optically smooth lirregularities did not exceed 0.1 /Lm); acoustic images were recorded every several minutes. Degradation of the crystal started both with the interfaces between individual blocks and with the sites where cracks were initially observed (Figure 8). Degradation of the crystal was not uniform: thus, individual areas with noticeable cracks became destroyed to the depth of 1 - 1.5 /Lm during the first minutes, whereas some areas were destroyed to the same depth only after a period of 3 - 4 hours.
Similar experiments carried out on ceramics of the type 1-2-3 revealed a considerably higher stability of hot-pressed samples to the water influence, compared with ceramics produced by the usual sintering method. If the latter were destroyed under the water influence approximately with the same speed as monocrystals, then the not-pressed samples were destroyed to the 1 /Lm depth, as a rule, in a 1-2 hours. The surface of the hot-pressed samples was very homogeneous.
257
Figure 6. Surface of BiCaSrCuO monocrystal, 60x40 p.m.
Figure 8.
258
Figure 7. Same as in Figure 6. Subsurface (-6.6 p.m) . Subsurface defects are visible.
YBaCuO single crystal. Surface. 300x200 p.m. Degradation started with the cracks and the interfaces between blocks.
Figure 9. Copper sample holder. T 300x200 p.m.
88K, immersion liquid: propane.
ADDITIONAL CRYOGENIC EQUIPMENT
For examination of specimens in an acoustic microscope within a temperature range from 85 to 240 K we created a cryogenic thermostatted chamber providing an accuracy of 0.1K in the specimen area.
The chamber operates on the principle of a circulating cryostat with a two-step temperature control. The working space of the chamber is filled with liquid propane, which is used as an immersion. The microscope acoustic lens is inserted into the chamber installed on the microscopic stage after the optical adjusting of the latter. Figure 9 shows the surface of a copper holder on which the sample is secured. The temperature of the holder was 88K. The first experiments demonstrated the principle possibility of obtaining acoustic images and carrying out local measurements in the entire temperature range of 85 - 240K. At present the microscope scanning system is being modified to fit the cryogenic chamber.
REFERENCES
[3)
Iii
M. Hoppe & J. Bereiter - Hahn, IEEE, SU-32, no.2, pp. 289-301, 1985. Bukhny M.A., Chernosatonsky L.A., Khodan A.N., Maev R. G., Soifer Ya.M., Solid State Comm., Vol.72, No.12, pp.1177-1181, 1989. W. Parmon & H.L. Bertoni., Electron. Lett., voLl5, pp.684-686, Oct. 1979. A. Atalar, IEEE, SU-32, no.2, pp.164-167, 1985. T.J. Kim et. al., J. of Magn. and Magn. Mat. 76 & 77, pp.604-606, 1988. H.C. Gupta., Sol. St. Com. 65(6), 495-496, 1988.
259
ON THE EXTENSION OF HORN THEORY TO NON UNIFORM
VISCO-ELASTIC RODS
L.M.B.C. Campos
Instituto Superior Tecnico Av. Rovisco Pais, 1096 Lisboa Codex, Portugal
SUMMARY
We extend the classical theory of horns, which applies (Eisner 1966, Campos 1986) to the longitudinal vibrations of elastic bars of non-uniform cross-section, to include viscosity effects. The present viscoelastic horn theory is (§1) relevant to the damping of oscillations in tapered rods, and addresses the [ollowin/? topics: (§2) the distinct wave equations for the displacement and strain; l§3) a transformation to account for the change from elastic to viscoelastic material properties; (§4) extension of the duality principle, in the original (Pyle 1965) and two other (Campos 1987) alternative forms; (§5) the only selfdual shape, viz. the exponential tapered bar; (§6) the bars with constant cut-off frequency for the displacement or strain, viz. respectively the catenoidal and inverse catenoidal shapes; (§7) displacement and strain, viz. respectively the sinusoidal and inverse sinusoidal shapes; (§8) the cases of elementary exact solution of the visco-elastic horn equation, viz. those stated in §5-7; (§9) a non-elementary solution, viz. vibrations of visco-elastic rods with power-law cross-section, including simple degenerate cases, e.g. spherical waves in a conical rod; (§10) another non-elementary solution, viz. damping of vibrations of a Gaussian rod, which would have uniform stress in the elastic case (Bies 1962); (§11) plots of the filtering-damping function applying in cases (§5-6) of constant cut-off frequency; (§12) plots of the transparency-damping function applying in cases (§7) when there is no cut-off frequency.
§1 - INTRODUCTION
Bars of non-uniform cross-section are used as displacement amplifiers (Merkulov 1957, Eisner 1963), e.g. in power tools. The latter often operate near resonance, so that elastic models are not adequate, and must be replaced theories including damping. This is the motivation for the present theory of longitudinal vibration of visco-elastic rods of varying cross-section.
§2 - VISCO-ELASTIC HORN WAVE EQUATIONS
In a visco-elastic one-dimensional body the stress T is proportional to the strain u = auat, where ~ is the displacement, through the Young modulus E:
T = E(u + (au/at), U:: auax, Physical Acouslics. Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991
(la,b)
261
and the viscous effect appears as an additional term proportional to the timerate strain 8u/8t through Ef, where f is the relaxation time. The balance of forces for a bar states that the transverse inertia is the product of the acceleration 82U 8t 2, by the mass per unit length pS, which is the product of the mass density p by the cross-sectional area S:
p S 82U8t 2 = 8(ST)/8x, (2)
and is balanced by the rate of change, along the bar, of the stress force ST on a section of area S. Substituting (1a,b) into (2) leads to the wave equations for visco-elastic horns, using as variable (i) the displacement:
82U8t 2 - (pS)-1 8{SE(8U8x + f 82U8x8t)}/8x = 0; (3a)
(ii) the strain:
82u/8t2 - 8{(pS)-1 8{SE(u + f 8u/8t)}/8x}/8x = 0. (3b)
The wave equations (3a,b) do not generally coincide, either for non-homogeneous materials (p,E,f not all constant) or in the case of non-uniform cross-section S(x), and they generalize (Campos 1985, 1986) the usual horn equations (Rayleigh 1916, Webster 1919).
§3 - TRANSFORMATION WITH COMPLEX W AVE SPEED
If the properties of the material do not vary with time, and neither does the cross-section (i.e. the horn is not collapsible), we may use Fourier analysis:
f +00
f,U(X,t) = r,A(x,w) e-iLOt dw, -00
where the spectra of the displacement r and strain frequency w, at station x, satisfy respectively:
r H + L-l r' + {(w/c)2/(1-i8)} r = 0,
A' + L-l A' + {(1/L)1 + (w/c)2/(I-i8)) A = 0,
(4a,b)
A, for a wave of
(5a)
(5b)
where: (i) f,E are assumed to be constant in equation (5a), and p as well in equation (5b); (ii) prime denotes derivative with regard to x, e.g. r' == dr/dx, A" == d2A/dx2; (iii) both equations involve three parameters, namely:
L == S/S', c == ~, 8 == Wf, (6a,b,c)
the lengthscale L for changes in cross-section (6a), the elastic (6b) wave speed c, and the viscous damping 8, which equals the product of the wave frequency W by the relaxation time f. It is clear from (5a,b) that the viscous effect corresponds to:
c2 == E/p ---I c2/(1-i8) == c2/(1-iwf), (7)
replacing a real elastic wave speed (6b), by a complex viscoelastic one (7).
§4 - ORIGINAL AND ALTERNATIVE DUALITY PRINCIPLES
In a bar of constant cross-section L = w the visco-elastic wave equation is the same for all variables, e.g. displacement r and strain A spectra:
262
S' = 0: {d2/dx2 + (w/c)2/(1-io)} r, A(x;w) = O. (8a,b)
In the case of non-uniform cross-section S' * m the wave equations (5a,b) are distinct, but it is possible to find a relationship between the vibrations of dual horns, defined as those with inverse cross-section:
Horn Primal Dual
cross-section S(x) l/S(x)
lengthscale L = SIS' -L (1/S)/(l/S' )
duality principle-Driginal form (Pyle 1965)
variable ~(x,t ) S(x) u(x,t)
duality principle-alternative forms (Campos 1987)
for strain u(x,t) 8{S(x) u(x,t)}/8x
for displacement ~(x,t) S(x) 8~(x,t)/8x. (9)
The original form of the duality principle (Pyle 1965) states that the displacement ~ in a primal horn of cross-section S, equals the volume strain Su in the dual horn of cross-section l/S. This kind of duality relates different variables in dual horns. An alternative formulation (Campos 1987) relates the same variable in dual horns, either for strain (or displacement), viz. if u(O is the strain (displacement) in the primal horn of cross-section S, then (Su)'(S~') is the strain (displacement) in the dual horn of cross-section l/S. The proof of these results (Campos 1986) extends without modification from elastic to viscoelastic horns, bearing in mind the transformation (7).
§5 - THE SELF-DUAL OR EXPONENTIAL VISCO-ELASTIC BAR
The only self--dual horn shape is the exponential (lOa):
(10a,b)
which is also the only case of constant lengthscale (lOb), and hence also the only non-uniform bar for which the displacement and strain satisfy the same (5a)=( 5b) viscoelastic wave equation, whose solution is:
(Ha,b)
where F is the fil tering--damping function, defined by:
(12)
here A,B are arbitrary constants of integration, and:
2// :: {1 - (2wejc)2/(1-iO)} 1/2, (13)
in the visco-elastic case; in the elastic case this simplifies to:
(14a,b)
263
where w* is the cut-off frequency, because for w < w' we have real I/o and evanescent wave field in (12), and for w > w* we have imaginary I/o and propagating wave fields in (12). In the viscous case (13) the wavefield (12) are always complex, giving rise to damping effects (Campos 1988).
§6 - CASES OF CONSTANT CUT-OFF FREQUENCY
The transformation (Ha, b) is a particular instance, for the exponential horn (lOa), of the change of dependent variable:
r,A(x;w) = {S(x)} -1/ 2 D,R(x;w), (15a,b)
which eliminates the middle term in (5a,b), so that the reduced displacement D and strain R spectra satisfy a Schrodinger-type differential equation:
D" + {(w/c)2/(l-io) - 1/4L2 - (1/2L)'} D = 0,
R" + {(w/c)2/(1-ib) - 1/4L2 - (1/2L)'} R = 0,
(16a)
(l6b)
In the elastic case b = 0, this leads (14b) to a constant cut-off frequency (W*/C)2 = 1/4f), if the lengthscale satisfies:
(17a,b)
The solutions of (17a,b) specify the lengthscales:
L2(X) = e tanh, coth(x/U) = -L3(X), (18a,b)
corresponding, by (6a), respecti vely to catenoidal (19a) and inverse catenoidal (19b) horns:
S2(X) N cosh2, sinh2(x/U), S3(X) N sech 2, csch2(x/U). (19a,b)
Thus the displacement (16a) in the catenoidal (16a, 17a, 18a) horns:
r2(X) N F(x) sech, csch(x/U), (20)
and strain in the inverse catenoidal horns:
A3(X) N F(x) cosh, sinh(x/U). (21)
are specified by the same filtering--dampin~ function (12), which applies to the exponential horn (§5), and simplifies from 113) in the visco--elastic to (14a,b) in the elastic case.
§7 - CASES OF TRANSPARENCY TO ALL FREQUENCIES
The horn will be transparent, i.e. allow propagation of all frequencies, if the cut-off frequency (14b) is imaginary; thus the transformation e -I ie leads from the filtering--damping (12) to the transparency--damping function:
(22)
where A,B are arbitrary constants of integration, and:
(23)
264
In the elastic case:
(24)
the wave fields (22) propagate for all frequencies. The same transformation f. ---! if. shows that the displacement and strain:
r 4(X) N G(x) sec, csc(x/U),
A5(X) N G(x) cos, sin(x/U),
(25)
(26)
are specified by the transparency-damping function (22, 23, 24), respectively for the sinusoidal and inverse sinusoidal horns:
S4(X) N G(x) cos 2, sin2(x/U), S5(X) N sec2, csc 2(x/U);
the lengthscales for these:
satisfy the differential equations:
(27a,b)
(28a,b)
(29a,b)
which arise from the Schrodinger equations (16a,b) as before, replacing f. by if.
§8 - EXISTENCE OF EXACT ELEMENTARY SOLUTIONS
The schrodinger equation (16a, b) for the reduced variables has elementary exact solutions if the term in curly brackets is a constant:
{d2/dx2 + (w/c)2/(1-i8) ± 1/4f. 2} D,R(x;w) = 0, (30)
so that we have only the following cases: (i) the + sign corresponds (23a,b) to the transparent horns (§7); (ii) the - sign corresponds (17a,b) to the horns with constant cut--off frequency (§6), either for displacement or strain; (iii) a sub-case of (ii) is the exponential horn (§5). This shows that the visco--elastic horn equations (5a, b) have exact elementary solutions only for five shapes: exponential, catenoidal, sinusoidal and inverses. This result extends from elastic (Campos 1984) to visco--elastic horns, in view of the transformation (7). For all other shapes the solution involves special functions, i.e. series expansions, which may reduce to a finite number of terms in particular cases.
§9 - CASE OF BARS WITH POWER-LAW CROSS-SECTION
The horns having a cross-section varying with distance according to a power-law:
(31a,b)
lead to a visco--elastic horn equation (5a) which can be solved in terms of Bessel functions:
Zn_! ((wx/c)/IHO), (32)
of complex variable and real order:
265
(0, (33)
where A,B are arbitrary constants of integration, we can use either the Bessel
J and Neumann Y functions for standing waves, or Hankel H(1,2) functions for propagating waves. In the elastic case b = 0 the Bessel functions have real variable (Ballantine 1927). The Bessel functions reduce from series to polynomials in the case of spherical Bessel functions, i.e. n an integer.The simplest case is n = I, viz. the conical visco--elastic horn:
(34)
which propagates a spherical wave with viscous damping.
§10 - DECAY OF STRESS IN GAUSSIAN HORNS
In the case of a gaussian horn:
Ss(x) rv exp(-x2/ e2), Ls(x) = -e 2/2x, (35a,b)
the visco--elastic wave equations (5a, b) have solutions III terms of Hermite functions:
fs(x) rv H/1(x/e), As(x) rv H/1_1(x/e),
of complex order:
/1 == (1tIe/c)2/{2(I-ib)} = (1tI/1tI*)2/{8(I-ib)}.
In the elastic case the order is real, and if it is an integer /1
b = 0 : 1tIn = 2/ill 1tI* = (c/ /!) /ill,
(36a,b)
(37)
n:
(38)
we obtain a sequence of frequencies for which the vibrations are specified by
Hermite polynomials. In particular, at the 'design frequency' 1tI1 = 12 c/ e, the displacement is linear (36a), the strain is a constant, and hence the stress is uniform (Bies 1962); the presence of viscosity causes a decay of these states (Campos 1988).
§ll - PLOTS OF THE FILTERING-DAMPING FUNCTION
We demonstrate the damping effects first for horns with a constant cut-off frequency, viz. the catenoidal, inverse catenoidal and exponential shapes. The filtering--damping function (12) depends on the parameter v (13), which involves:
a + ib == v = {1-!l2/(I-iMl)} 1/2, (39)
the wave frequency 1tI and viscous relaxation time E, made dimensionless by using the cut-off frequency:
fi == 1tI/1tI* = 21t1/!jc, 11 == 1tIH = EC/2/!. (40a,b)
We plot in Figure 1 on the l.h.s. the decay a = Re(v) and on the r.h.s. the phase b = Im(v). Both increase strongly with frequency, and also as relaxation time reduces, for frequencies away from the cut-off. Near the cut-off frequency fi = 1 the phase becomes small, and the decay increases, corresponding to evanescent waves.
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§12 - PLOTS OF TRANSPARENCY-DAMPING FUNCTION
The latter effect is not present in the case of the transparency-dampinl$ function (22), which applies to sinusoidal and inverse horns. The factor (23) specifies:
b + ia = p, = {I + fi2/(1-iO)} 1/2,
specifies the damping a = Im(p,) and phase b = Re(p,):
eip,x/ e = e-ax/ e+ibx/ e ,
( 41)
( 42)
which are plotted respectively on the l.h.s. and r.h.s. of Figure 2. The damping and phase both increase as frequency increases, and as the relaxation time reduces.
REFERENCES
1 2 3 4 5 6 7
~l 10 11 12 13
S. Ballantine, J. Franklin Institute 203, 85-101, (1927). D.A. Bies, J. Acoust. Soc. Am. 34, 1567-1569, (1962). L.M.B.C. Campos, J. Sound Vib. 95, 177-201, (1984). L.M.B.C. Campos, Progress Ae.rosp. Sci. 22, 1-27, (1985). L.M.B.C. Campos, J. Sound Vlb. 110, 41-57, (1986). L.M.B.C. Campos, Rev. Mod. Phys. 58, 117-182, (1987). L.M.B.C. Campos and A. J. P. Santos, J. Sound Vib. 126, 109-115, (1988). E. E~sner, J. Acoust. Soc. Am. 35, 1367-1377, (1963). E. Elsner, J. Acoust. Soc. Am. 41, 1126-1146, (1966). L.G. Merkulov, Sov. Phys.-Acoustics 3, 246-255, (1957). R.W. Pyle, J. Acoust. Soc. Am. 37, 1178A, (1965). J.W.S. Rayleigh, Phil. Mag. 31, 89-96 (Papers 5, 375-382), (1916). A.G. Webster, Proc. Nat. Acad. Sci. 5, 275-282, (1919).
269
ENHANCED PROPAGATION IN A FOAMY MEDIUM
A. Cowley, A. Baird, C. Harrison*, T. Gedrich*
Admiralty Research Establishment Portland, Dorset DT5 2JS, U.K. * YARD, 233 High Holborn
London WC IV 7DJ, U.K.
INTRODUCTION
Historically, acoustic propagation modeling in the ocean has been largely restricted to highly stylized environments: typically, the problem is reduced from three dimensions to two by imposing cylindrical symmetry about the vertical axis through the source; the propagation medium is usually assumed to be stratified, and fully described by a frequency independent (more often than not range independent) sound speed profile. Even simplified in this manner, it is not always easy to see at a glance what transmission behavior to expect in a given environment.
The problem is vastly more intractable if the water contains a localized cloud of bubbles (generated in the ocean by breaking waves or by the passage of vessels, for example), for then none of the above approximations apply. Even if the basic mechanisms concerned can be modeled relatively simply, extracting predictions is far from trivial, and visualizing what will happen in any given case is next to impossible. Wildt [1] reproduces the following measured propagation anomalies (dB per unit distance above spherical spreading) at 5 and 25 kHz:
i) Along wake: 10 - 80 dB /kyd ii) Across wake: 300 - 6000 dB/kyd
Wildt remarks: IIThese observations are rather puzzling II . In this paper we describe a prediction program, SWARM, which makes
perfect sense of these results. Its functions are represented diagrammatically in Figure 1: from environmental data such as bubble populations at points within a coordinate grid, the program calculates local acoustic properties (absorption and volume scattering coefficients, sound speeds and sound speed gradients). These may of course be examined directly, but the stored parameters are also necessary for subsequent transmission loss predictions, reverberation simulations and ray plots. SWARM's two stages are represented for transmission loss across a wake, (an example of a dense bubble cloud) in Figure 2.
The first picture depicts the environmental input: the wake has been partitioned into small cuboidal volumes, within each of which a bubble population (as a distribution of bubble number against size) is assumed known. In the second, local attenuation coefficients, volume scattering strengths and sound speeds have been calculated at a lattice of points throughout the wake, and stored, and are then applied to a specific sonar problem.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 271
CALCULATIONS OF ACOUSTIC PARAMETERS
Within the volume of the sea, large bubbles are very short lived, nSlllg quickly to the surface and either contributing to the scattering properties of the interface or escaping. As a result, at frequencies below ultrasound it has been found adequate to treat the remaining bubbles as lumped acoustic systems, but with one, spherically symmetric, radial, mode of oscillation. The behavior of bubbles within this regime depends upon its size as a function of a wavelength. At sufficiently low frequenCies, the presence of bubbles will scarcely affect absorption and reverberation. They may considerably lower the speed of sound, however, by changing the bulk density and compressibility of the medium; the magnitude of this effect is, of course, governed by the volume fraction they constitute. Nearer to resonance, bubbles are driven into large amplitude oscillations, and a much greater proportion of the incident acoustic energy is absorbed and scattered.
Figure 1
272
n(8) RADIAL DISTRIBUTION FUNCTION -----.-~:-:--
a Sv c: TRANSMITTER
Figure 2. Acoustic Environment.
SWARM follows the single bubble analysis of R. Y. Nishi ([2] in the case ka « 1), in which he gives expressions for the resonant frequency and for the damping factor as a sum of terms due to reradiation of sound, viscosity, and thermal conduction. From these it is easy to calculate the scattering and extinction cross-sections of a bubble.
Us = ------------------[ [ff - 1 ] \ b2
b = brad + bVis + bth
These are functions of frequency and bubble radius. To calculate the bulk properties of bubble laden water it is necessary to make two, apparently rather contradictory assumptions . On the one hand, bubbles of any represented size should be so sparse that one may treat them in isolation, as above (Multiple scattering, for example, is not modeled); on the other, they should be numerous enough to permit treatment as a continuum. A development along these lines can be found in Brekhovskikh and Lysanov [3], and elsewhere; our void fractions, typically of the order of 10-6, lie within the window in which both of these approximations are reasonably applicable. Given a bubble population at a "point" lor elemental volume) in the cloud of bubbles, local acoustic parameters are calculated by integration over the bubble density function (Figure 3). Typical plots of acoustic attenuation, volume scattering strength and sound speed as functions of frequency are shown in Figures 4-6 .
RAY TRACING
Ray plots do not contain much quantitative information, but wherever the eikonal equation is applicable (broadly, wherever the environment does not change too markedly over the distance of a wavelength) they are very suggestive and provide helpful images of sound propagation. SWARM generates
273
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1000
100
10L-----L-----L-----L-----~----~----~--~
o 60 100 160 200 2150 300 350 Bubble Radius (microns)
Figure 3. Bubble Distribution.
polygonal rays (for a single frequency, of course, since the bubble cloud is highly dispersive) based on the finite difference formula:
d(ne) = dl . grad n
where 1 = path length, n = col c and e = dxl dl. Below resonance (actually a band of resonances, since there are bubbles of many sizes), the speed of sound is considerably lowered; at frequencies above, increased (Figure 6).
Orthogonal projections of these three-dimensional ray paths, strikingly refracted within the bubble cloud offer a very immediate representation of the physics.
Figures 7-11 show typical ray paths through a convected twin lobe wake. The ranges involved are much smaller than is usual in propagation studies, so
Allenuellcn dB/m 15 .--------------------------------------------,
12.5
10
7.5
5
2.5
o~~--~--~--~====~==~~ o 25 50 75 100 125 150 175 200
Frequency (kHz.)
Figure 4. Acoustic Attenuation.
274
Volume scattering strength Or-------------------------------------------~
_60L----------L----------J---------~L---------~
o 50 100
Frequency (kHz.)
150
Figure 5. Volume Scattering Strength.
200
the curvatures visible really are extraordinary: any bubble cloud as localized as a wake can give rise to extreme sound speed gradients. In these plots, a constant sound speed was assumed outside the wake, but a sound velocity profile may be defined (as well as a temperature distribution within the wake).
TRANSMISSION LOSS
In regions, and at frequencies of highest absorption, sound is attenuated beyond detection in the space of a few meters. In this situation, only short range propagation loss predictions are likely to be interest, and a simple spherical spreading plus absorption formula - in effect, ignoring the sound speed variation is often perfectly adequate. Low absorption and considerable ray
Sound Speed m/s 1600,-------------------------------------------,
1550
1500
1450
1400
1350
1300L---~-----L----J-----~--~-----L-----L--~
o 25 50 75 100 125 150 175 200
Frequency (kHz.)
Figure 6. Acoustic Sound Speed.
275
0
10
en 20 '" 1il §.
.<: 30 0..
'" 0
40
50
60
0 500 1000 1500 Distance Astern (metres)
Figure 7. Along the wake - Below resonance.
curvature can coexist at certain frequencies, however, and over longer distances the simple formula may be seriously misleading. Returning to Figure 7, for example, a shadow zone is perfectly evident, even without benefit of the third dimension; and there is ducting in Figures 9-11, and between twin bubble trails in Figure 8. These plots also suggest the possibility of focusing effects.
SW ARM estimates the true transmission loss (taking account of refraction) by projecting a number of rays from the source, each representing a proportion
50
50
o 500 1000 1500 Distance Astern (metres)
Figure 8. Along the wake - Below resonance.
276
a
10
'iii ~
20 a; .s .r:=
30 15. Q)
0
40
50
60
0 500 1000 1500
Distance Astern (metres)
Figure 9. Along the wake - Above resonance.
of the transmitted power, and summing their contributions, appropriately weighted, at the receiver.
The power a given ray represents initially depends on the angular spacing between it and its neighbors at the source - the area on the unit sphere it represents, in fact - and the directivity index applicable to it . At the receiver, this is diminished by the absorption suffered en route. About the receiver, a target area, more or less orthogonal to the incident rays, is defined. Each ray
50
:'l 0 c:
'" ;;; Ci E
'" ~ ;;; ,u, ...... - .. -----~ -----en o o
50
o 500 1000 Distance Astern (metres)
Figure 10. Along the wake - Above resonance.
1500
277
0
10
V> 20 ~ 0; .s .s::.
30 Ii OJ Cl
40
50
60 4-______ -. __________ -. __________ .-______ ~
·40 o 40 Cross·stream Direction (metres)
Figure 11. Across the wake - Below resonance.
passing through this area is deemed to make a contribution to the received signal based on the power it IIbearsll on arrival, the area the target presents to its direction on arrival, and the receiver's directional response. At present the total is calculated as a power sum, but coherent processing would also be possible.
It is in this area that SWARM makes its most striking, often counterintuitive predictions, and provides an illuminating theoretical model for the many discordant observations in this area. Ducting effects, in particular, can account completely for the discrepant losses reported by Wildt for propagation along and across a wake.
REVERBERATION
SW ARM calculates monostatic reverberation as a power sum of surface reverberation, IIbackground ll volume reverberation (that not due to the bubble cloud), and bubble scattering. Bottom scattering is not considered. The first two terms of the sum are estimated in a completely standard way (see [4] for example) . Bubble scattering is treated volumetrically: the wake is divided into elemental volumes, each of which makes a contribution based upon the local volume scattering coefficient, transmission loss, transducer directionality, pulse length, source level and so on. Transmission loss is estimated using the straight ray algorithm, where four round-trip paths (combinations of direct and surface reflected one way paths) between the source/receiver and the scattering element are considered.
A typical situation in which this algorithm may be exercised, and representative output, are shown in Figure 12. Three phases are clearly visible in the reverberation level time-history. At first only the background volume reverberation is detected, then we see the gradual onset of bubble reverberation, and finally surface reverberation, beginning with a sharp glint, and falling off unevenly as the returns pass through the sidelobes of the beam.
278
SHIP .... -..
l~
a; > Ql
...J
c: .2 iV a; n Q; > Q>
cr
. . " WARE: '"
NARROW BEAM UPWARD LOOKING
SONAR
80
60
40
20
0
-20
-40
-60 0 20 40
Time
·.SOUND SCATTERED .. AND "ABSORBED
CTl2
60 (ms)
Figure 12. Reverberation of a Bubble Layer.
279
CONCLUSIONS
The foamy medium prediction program, SW ARM, is still under development, but even in its current state, on a relatively simple theoretical basis, it can model, with considerable success, propagation and scattering within bubble clouds. Already it has greatly improved our understanding of these phenomena, and provided a convincing explanation of the disparity observed between propagation anomalies along and across a wake.
This work has been carried out with the support of the Procurement Executive, Ministry of Defense.
REFERENCES
[1]
[2]
280
Wildt R., Physics of Sound in the Sea - Part IV: Acoustic Properties of Wakes, Gordon and Breach Science Publishers. Nishi R.Y., The Scattering and Absorption of Sound Waves by a Gas Bubble in a Viscous Liquid, Acustica, Vo1.33, No.2, (1975). Brekhovskikh L., and Lysanon, Yu., Fundamentals of Ocean Acoustics. Urick R.J., Principles of Underwater Sound, McGraw - Hill, (1983).
ANALYSIS OF SURFACE ACOUSTIC WAVE IN LAYERED STRUCTURE
WITH PERIODIC DELAMINATION
E. Danicki
Institute of Fundamental Technological Research Polish Academy of Sciences 21 Swietokrzyska Str., 00-049 Warsaw, Poland
INTRODUCTION
The aim of this paper is to present a method for analyzing elastic waves in layered structure with periodic debonding or cracks between the layers [1]. This is a boundary problem with periodic mixed boundary conditions: some areas are stress-free, and they interlace with areas where the difference between particle displacement vectors on both sides of the boundary plane between two adjacent layers vanishes.
The method exploits known identity for Legendre polynomials [2]
n = 0
{ f'f (sint. )II( cos B-c 0 st.rll- t {O
o ; for t. < () < 7r
(1)
fo r 0 ~ () < t. < t. < 7r ; Re{lI} < 1/2}
We easily recognize the periodic function of () = Kx in the left-hand side. (K = 27r/ A is the wave-number of periodic cracks), as well as the required behavior of the right-hand side function, which is zero in one, and nonzero in the other domain of (). The function exhibits singularity at the edge of this domain, which is matched to the considered problem, provided that II is properly chosen. Here II = 0 (in this case P - Legendre polynomial).
In conclusion we see that (1) define a proper function for modeling acoustic field in the considered system. Similar approach was applied for solving certain 2--dimensional electrostatic problems [3] - l5].
Figure 1.
# slab # # # # # # # # # # # # # ~w ~# #
:r~~~~:;~::;~ %%1://;;;~;;;//~I~X1x x2 =y
An elastic slab is bonded to elastic halfspace periodically (dotted lines), where perfect mechanical contact exists, in remaining area (double lines) there is no mechanical contact.
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 281
GREEN'S FUNCTION FOR ELASTIC HALFSPACE
We consider a plane harmonic wave, propagating along the :J)-axis. On the y=O plane it has the form:
exp(jwt - jrx) (2)
where w is the angular frequency and r is the wave-number of the wave. Complex amplitudes of particle displacement vector components J! on the surface of the body depends on the stress 1. applied to it
u ~ = :w (jstk~T ~2 + jr(k~ - 2r2 - 2s1st)T 22]
u:i = :w [-jr(k~ - 2r2 - 2sjst)T~2 + jSjk~T22] (3)
where J.L, A are the Lame constants, p the mass density, and k~ = w2p/J.L, 1 .
k~ = W2p/(A + J.L), sp = (k~ - r2)2 for r > kp and sp = -J I sp I otherwise,
p = t, 1, w = (k~ - 2r2)2 + 4r2sjst.
GREEN'S FUNCTION FOR ELASTIC SLAB
Consider an elastic slab h thick (Figure 1), with its upper surface stressfree, and its bottom surface subjected to traction T 2p. The relations between field amplitudes on y=O plane, analogous to (3) are:
u l = jStk~ [W(1-VT2) + (X2-Z)(T2_L2)] [J.L[(1-L2)(1-T2)W 2+4X2Z(L-T)2]] -ITI2
W(X-2stsj)(1+L2T2)-(X2-Z) (X + 2s j St ) (T2_L 2 )-4LT(Z-2sts1X)X + jr T-
J.L[ (1-L 2 )(1-T2) W 2 + 4X2Z(L-T)2] 22
W(X-2stSj)(1+VT2)-(X2-Z)(X + 2s j S t ) (T2_L 2 )-4LT(Z-2s tsjX)X u2 = -jr T-
J.L[ (1-L2 )(1-T2) W 2 + 4X2Z(L-T)2] 21
+ jStknW(1-VT2)+(X2-Z)(T2-L2)] [J.L[(1-L2)(1-T2)W2+4X2Z(L-T)2 ]] -IT22
where (4)
FORMULATION OF THE BOUNDARY PROBLEM
Both the elastic slab and the elastic layer (Figure 1) are characterized by the Green's function given above in spatial representation. Boundary conditions, at y=O for one period of () are ( ()=Kx, /::'=pw / A, p=1,2,3 )
282
SOLUTION OF THE EIGENVALUE PROBLEM
(5)
(6)
(6')
We need nontrivial solution of equations (3)-(6). Following Floquet theorem, the solution has a form of a series of partial harmonic waves
with corresponding complex amplitudes having index n: u\n), T~Y), etc. These amplitudes are involved in Eqs.(3), (4), taken for rn instead of r.
Making use of (1) we can apply the following representation for them: M+
T -(n) - T+(n) - T(n) _ ~ ",(P) P ( h) p2 - p2 - p2 - LJ U m n-m COSu m=M-
M+ -J'rn[up+(n)- up-(n)] = ~ (.l(P) S P (COSh)
LJ f'm n-m n-m U
(7)
(8)
where Sv = 1 for l/ ~ 0, and = -1 otherwise, am and f1m are unknowns
yet, however they are mutually related. Indeed, Taking large value of n ¢ [Nl,N2] we easily notice, that the solution (8) holds if:
(9)
where matrix A ( p.q) is determined by the asymptotic values of coefficients involved in the right-hand sides of (3) and (4), for I r I --I 00, and provided that (for ° < r < K):
M- = Nl M+ = N2 + 1
The conditions (9) and (10) are sufficient for fulfillin~ the relations the solution (7), (8) for every n outside the limits lNl,N2].
(10)
(3)-(6) by
However, the above relations must be fulfilled for every n, also for n E [Nl,N2]. The equations resulting from these relations deliver triple (N2-Nl+l) conditions for a vector of unknowns a, that is for triple (N2-Nl+2) unknowns. The remaining three equations result from (6'). As we see, we obtained a complete set of homogeneous linear equations. The nontrivial solution exists provided that the determinant of the above set is equal to zero. This is the condition for r, the wavenumber dependent on w.
SAMPLE PROBLEM FOR LOVE WAVE
This is the simplest case, where we put up=O, T p2=0, a( P) =0, p=I,2,
and a(3) = a, A(3.3) = -j (1/J1-+1/J1-'), where J1-' and p' characterize the
material of the plate, correspondingly St(n) = (kt2_r~)lh = s~ in (4) and
(11) below.
283
The relations (3)-(8) results in the following equations for every nE[Nl,N2] separately.
[ -jrn _ ~ ct9(S'h)-[ ! + 1 ]s ]p ( !::,.) /lm f.LS n j.t' S ~ n Jl j.t' n -m n -00 cos o (11)
while the relation (6') yields (summation over m as above)
(_1)ffi /lm P _r/K(-cos!::,.) = 0 (12)
In case k t < r < k t « K »1/h we can put N1=N2=0, hence the
above yields
(13)
This is the dispersion relation for r dependent on w.
w AVE SCATTERING BY PLANAR SYSTEM OF PERIODIC CRACKS
SH z-polarized wave u 0 exp( -jrx-jsy) propagates from y=-oo onto cracks in the plane y=O of the isotropic body. The wave is scattered by them.
The stress T=T 23 in the plane of cracks is
(14)
Proceeding as previously we obtain equation analogous to (11)
(15)
where 0 is the Kronecker delta (Eq.(12) remains unchanged). In an example one can obtain for COS!::" =0 and a resonant case of
r = K/2, that the maximum value of backscattered wave takes place for r r:J .21s. In this case the backscattered wave takes about 20% of the incident wave power, while the transmitted wave takes about 30%.
CONCLUDING REMARKS
The above approach can be easily generalized to cases of anisotropic media, as well as to mUltiple layered media with cracks in contact planes between layers. They can have different period on and even different orientation on different planes [6]. Another generalization is possible for periodic system of group of cracks (multi-periodic system) [7].
284
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Y.C. Angel, J.D. Achenbach, Harmonic waves on elastic solid containing a doubly periodic array of cracks, Wave Motion, 9, 377-385, (1987). H. Bateman, A. Erdelyi, Higher Transcendental Functions, Mc Graw-Hill, NY, (1953). S.G. Joshi, R.M. White, Dispersion of surface elastic waves produced by a conducting grating on a piezoelectric crystal, J.Appl.Phys., 39, 5819-5827, (1968). K. Biotekjaer, K.A. Ingebrigtsen, H. Skeie, A method for analyzing waves in structures consisting of metal strips on dispersive media, IEEE Trans., ED-29, 1133-1138, (1973). E. Danicki, Theory of surface acoustic wave slant propagation in the :periodic electrode system, J.Tech.Phys. (Pol.Acad.ScL), 19, 1, 69-77, l1978). E. Danicki, Spectral theory of EM wave scattering by periodic metal strips, to be published. E. Danicki, D. Gafka, Propagation, generation and detection of SAWin a multi periodic system of metal strips on a piezoelectric substrate, JASA, (submitted for publication).
285
SURFACE ACOUSTIC WAVE SCATTERING BY ELLIPTIC METAL DISK
ON ANISOTROPIC PIEZOELECTRIC HALFSPACE
E. Danicki
Institute of Fundamental Technological Research Polish Academy of Sciences 21 Swietokrzyska Str., 00-049 Warsaw, Poland
INTRODUCTION
Surface acoustic wave propagating in a piezoelectric halfspace is accompanied with a wave of electric potential on the halfspace surface. Perturbation of the potential allows scattering SAW. In this paper we perfectly consider the case, where the perturbation of the potential is introduced by perfectly conducting elliptic disk. The orientation of the disk axes are assumed arbitrary, as well as the crystallographic axes of the anisotropic elastic media. The disk is considered weightless.
There are similar problems solved in acoustics and electromagnetics [1]. The problem considered in this paper differs from the above ones at least with the substrate anisotropy. The dielectric anisotropy of the substrate directly effects the electric charge distribution on the disk induced by the incident SAW. This, as well as the angular dependence of SAW velocity and piezoelectric coupling, influences on the angular dependence of the scattered SAW far-field (diffraction pattern).
INTEGRAL FORMULATION OF THE SCATTERING PROBLEM
The theory presented is a perturbation one. It neglects piezoelectric interaction when the electric charge distribution on the disk is evaluated. This simplification can be applied under the assumption of either weak piezoelectricity of the substrate or small disk diameter. Under this assumption, the Green's function expressing electric potential resulting from the point electric charge, both on the substrate surface, is
(1)
t is the so-called effective surface permittivity [2], r is the distance to the electric charge, e describes the orientation of r in the crystal frame [3]. .
The problem for elliptic disk can be led to the problem for circular one by suitable scaling and subsequent transformation of the coordinate system. The following considerations concern the transformed spatial coordinates where we have to do with circular disk of unit radius (note that t changes after the transformation) .
Let us consider the grounded disk which electric potential is zero. With help of (1) this can be expressed in polar coordinates r, e as follows
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
287
(k - wave-number of the incident wave) where r and r' belong to
211" R( 8) - ~ Jm(kr)ejun'l = ~ j[E(O - 11"/2]-1 dO j p(r',8') dr (2)
o 0
the disk area r < 1. The left-hand side represents the incident potential wave of unitary amplitude.
In a case of free disk, disk potential can have nonzero value. Corresponding scattering problem can be solved in few steps, provided that the above problem for grounded disk is solved first. After the charge on the disk is evaluated, the scattered acoustic field far from the disk can be found with help of the asymptotic form of Green's function [5].
AUXILIARY ELECTROSTATIC PROBLEM
Let us introduce a polynomial T(m) which is a Tchebyshev one truncated from the lower side. The only components left are these with power of r equal or greater than Iml tm and n have the same parity). Consider a charge distribution over the circular disk in form (a - arbitrary constants, n ~ I m I, summation symbol suppressed)
(3)
It can be shown that the electric potential ~ resulting in the area r < 1 from the above charge is described by relation (Xm is the Fourier expansion of 1/ E( 8), I m I ~ n < 00, -00 < m < 00, summation over p in limits dependent on m and n)
1 0 . for l=O and m t- 0
00 (k,n) , 1 . ~ O:p
~ = 2" akn Xrn-k eJrn-tl '- (_1)1 E -- Jrn(1I"lr) ; for NO 1 = -00 p (11" 1 ) p
1I"(_I)n/2 (l_n 2)-1 ; for l=O and k =0
(4)
where O:p - certain coefficients. Note, that components with p < I m I in (4) can be neglected, as their sum over 1 results in zero [4] (J - Bessel function).
CHARGE DISTRIBUTION INDUCED ON DISK BY THE SCATTERED SAW
In (2) the incident potential wave is represented in polar coordinates. Applying certain approximation, this representation can be further expanded into (summation over m)
ION
(_1)1. E p~v+. rn. (2)
00
for l=O and mt- 0 (i~m) -- Jrn( 1I"/r) (11" l)P
It-O (5) '- ~o = - ~ ejrn-tl L 1 = -00
for l=O and m=O
where {im are coefficients (1I=2 for m=O, otherwise 0). To obtain good approximation one has to apply large value of N.
288
A comparison of (4) and (5) makes evident that i) electric charge distribution is correctly expanded into a series (3),
(otherwise, there would have appeared terms +1/(7r0p in (4)), ii) the charge density expansion coefficients a can be evaluated on the
strength of equality of corresponding components in (4) and (5). A comparison of (4) and (5) is taken for every m and p, and separately
for terms with l=O. The later gives directly the total electric charge induced on the grounded disk.
(6)
In the case of free disk, where total charge over the disk is zero, the disk potential induced is V =sink/k.
ANGULAR DEPENDENCE OF THE SCATTERED FAR-FIELD
An asymptotic Green's function for large distance r from disk can be expanded into angular Fourier series (provided that the curvature of the slowness curve is finite [5])
(7)
where Cojj is an angular dependent parameter resulting from SAW velocity
w/kojj and SAW piezoelectric coupling in the propagation direction ()ojj, for
which the Pointing vector is oriented to the observation point ( r ---! 00 , () ).
Far-field potential wave amplitude is
(8)
The integral can be evaluated on a similar way as in Section 3:
where 'Y can be evaluated from a (see (4), except 'Y1 for even m. To obtain SA W scattering angular dependence, one has to take into
account that SAW and potential amplitudes are mutually related.
CONCLUDING REMARKS
An anisotropic diffraction problem is not expected to be solved explicitly, even in form of series analogous to Mie ones. The computations are necessary. The theory presented above in this paper allows to perform some steps of com-
putations only once. For example a~m,n) can be stored and applied in all
cases, independently on material, disk orientation with respect to wave, and a wavelength. What's more, these important coefficients are integer numbers, which can be calculated with large accuracy (they grow very fast with nand m).
289
SAw
U
2
Figure 1. Sample scattering patterns: upper row - circ., lower row - 4:1, 45 ° rotated, ellip. disk , -- ground., - - - free, k as marked in figs. Right column - radiation pattern, k=l,2,4.
NUMERICAL EXAMPLES
All results shown below were obtained for either circular disk of unitary radius or for elliptic disk with main axes l/a and a. Isotropic substrate was assumed, however the latter case was first transformed as to obtain circular disk, what results in great anisotropy of the equivalent substrate, and next the results were transformed back to the original system of coordinates. This way the theory developed for anisotropic medium was fully exploited .
In the calculations performed we applied I m I =9 and n= 15. These were
the largest numbers allowing exact representation of a~m)n) (18 digits in
double precision Fortran numbers for IBM PC). In this circumstance we were able to calculate examples for k up to about 27r and for 4:1 elongated disks . The accuracy of the computations was verified by applying smaller n.
REFERENCES
[1]
[4]
[5]
290
J.J. Bowman, T .B.A. Senior, P.L.E. Uslenghi , Electromagnetic and Acoustic Scattering by Simple Shapes, Nord-H . Amsterdam, (1969). B.A. Auld, Acoustic Fields and Waves in Solids, A Wiley, 1973 [3] E. Danicki, Green's Function For Anisotropic Dielectric Halfspace, IEEE Trans., UFFC-35, 5, p.643 , (1988) . A.P . Prudnikov, J .A. Marichev, Integrals and Series , Nauka, Moscow 1986, v.2 (in Russian) . E. Dieulesaint, D. Royer, Ondes Elastiques Dans Les Solides, Masson, Paris, (1974).
OPTICAL DETECTION AND ANALYSIS OF NON-LINEAR
OPTOACOUSTIC WAVES
J . Diaci, J. Mozina
University of Ljubljana Faculty of Mechanical Engineering 61000 Ljubljana, POB 394, Yugoslavia
INTRODUCTION
Detection and analysis of optoacoustic waves, generated by interaction of high intensity pulsed laser beams with solid surfaces, have opened new possibilities in monitoring of laser induced damage ~-41 and laser manufacturing processes [5-7] . Several experimental studies 1-3] have pointed out the nonlinear nature of these waves. It has been s own recently [4], that some nonlinear features of laser induced OA waves in air can be explained using the point explosion model. This contribution discusses the results of the time and frequency domain analysis of finite amplitude OA waves, detected by a knife edge laser probe in air at normal atmospheric conditions. considering OA wave generation and propagation as a laser induced point explosion, acoustic energy and transfer function of the detection system are determined.
H&-Ne laser
I R FILTER
DIGITAL PC OSCI L LOS COPE
Figure 1. Experimental set-up.
Physical ACOUSlics, Edited by O. Leroy and M. A. Breazea le Plenum Press, New York, 199 1
He-Ne laser
291
0.1
0.05
o
30
Figure 2.
'\ 1-.
1\" ~ .- -
"\.. V
I---""'""
J' '-'
40 50 60 70
10-8 =
10-7 L L
7
a t-V
10-8
r-- -l
p,L Il 10-
10·
t-
-
- 'S: -.- --- 3::'i -_., - '\. ,"\.
'\ .\ -/ \ \N
- -
..
108
frequency [Hz]
Examples of finite amplitude OA signals (left) and their spectra (right).
EXPERIMENTS
The schematic of our experimental arrangement is shown in Figure 1. A Q-switched Nd:YAG laser, providing pulses of 20 mJ energy and 16 ns FWHM duration at 1.06 pm, is used as the excitation source. In order to vary laser pulse energy and fluence on the surface of the sample, the excitation beam is passed through an optical system, consisting of a beam expander, a neutral density attenuator and a focusing lens. OA waves in air are detected by a He-Ne laser probe. The probe beam is aligned parallel to the surface of the sample and perpendicular to the excitation beam. Deflection of the probe beam due to the transition of OA wave is detected by a knife edge and a fast photodiode. The photodiode response is amplified, digitized by a sampling oscilloscope and fed into computer for signal analysis and storage. The output of an other photodiode, receiving a part of the excitation beam, is used as a triggering signal.
In the present experiments OA responses of .bare and painted metal samples are studied. Each site of the sample surface is irradiated by 25 subsequent laser
50r-----or-----,~----. 0 .06 ,------or-----,,-------,
J'I' E 0.04 --1------1"='- - 1 . .."..-----i 0 .02 1-'<"'- ><="""".-:::=*,=---It---
35L-----~----~------J
o 10 20 o ~----~----~~----~
o 30 10 20 30
pulse number pulse number
Figure 3. OA signal transit time (left) and amplitude (right) sequences at three pulse energies: *...20 mJ, 0 ... 10 mJ, x ... 5 mJ.
292
pulses with fluencies exceeding the threshold for permanent surface changes. The distance between the surface and the probe beam is constant throughout the experiment.
Typical detected OA signals (Figure 2) consist of compression followed by rarefaction. Frequency spectra of the detected signals exhibit typical bandpass amplitude characteristics with one central peak and approximately constant percentage bandwidth.
Evolution of OA signals generated by a series of constant energy laser pulses suggests that they depend not only on laser pulse energy but also on the state of the surface before irradiation. The changes of the surface affect some parameters of OA signals in the same way as the changes of the incoming laser pulse energy leading e.g. to the decrease of amplitude Sma x and to the increase of transit time ts during subsequent irradiation of the sample (Figure 3).
THEORY
Theoretical model is employed, assuming an instant release of finite amount of energy E in a small volume of a perfect gas [9, 10]. As a result, blast wave with a spherical-shaped shock wavefront is formed which propagates with a supersonic velocity. The model has two energy dependent scaling parameters:
re = (E/PO)1/3 and te = ro/co (1)
where Po and Co are the pressure and the sound velocity of the undisturbed gas. The solution is given in terms of dimensionless space ~ = r/re and time T = t/te coordinates [10, 12] .
Propagation of the shock wavefront is considered first, using published numerical solutions [12]. Since the source energy is generally not known and cannot be determined from laser pulse parameters, a new energy independent function of the shock wavefront is introduced:
!J.ps/po = 11(1 - T/~s(T)),
with T/~s(T) = ts/to
(2)
(3)
where !J.ps , is the shock overpressure (i.e. peak pressure of the blast waveform) ~s( T) is the shock position, ts is the shock wave transit time and to is the sound wave transit time.
Theoretical pressure waveforms behind the shock front are calculated next. Weak shock approximate solution of the point explosion model is given by [11]
t = ti + T - Ti + /1;+1 !J.Pi Ti log ~ s s ~ Po Ti
Figure 4.
o ~
0.1 ,---,--.------,,..---, •
';;- 0 .05 ... <I
o ~-~---~-~--~
0.05 0 . 1 0 .15 0 .2 0 .26
~t.e/to
Normalized peak pressure vs normalized transit time: Model: solid line, Exp. data: L.20 mJ, 0 .. . 10 mJ, x ... S mJ .
(4)
283
-- _ .. ---- --'""c-';--10-· L_ ____ -L ____ ~L_ ____ ~
o 10 20 30
pulse number
Figure 5. Blast wave energy of the AO signal sequences from Figure 3: Model: solid line, Exp. data: *...20 mJ, 0 . . . 10 mJ, x . . . 5 mJ.
T' ~p = ~Pi -.!
T (5)
where the solution of Okhotsimskii and Vlasova [12] is taken as the initial pressure profile (~Pi, ~ i) at T i·
DISCUSSION
Comparing theoretical and experimental results, we assume linear relationship between optical probe voltage peak Smax and blast wave amplitude ~ps. Using least squares method for data in Figure 3 and Eq. (2), we get the value 5.3 j.tV /Pa for the sensitivity of laser probe assembly. Most of the measured data fit very well into the theoretical function n (Figure 4) .
10 10-1 .:- - ----
~ --i-
V . to.
/\ "-~
o
. - === -- - --- ,-- - ..
Z -V
L I- '" -- V'- -~ '-~~
10-2
01 ., ~
~ b--._- ---
f-- - b \
/ 1\ f::= 7"'- . - .. _-
-
c ~
10- $
"- ~ - --- - -
-- -- _._ -------c------ r---- - -- -
30 50 60 70 10-'
10' 106 10'
Figure 6 .
294
time [ps] frequency [Hz]
Theoretical pressure waveforms corresponding to the OA signals of Figure 2.
o
Figure 7.
'";' Jl,..
~
10 20 30 40
time [ps]
10-~ !==
-=- · V -
~ -i-I- I-H+Ht----ji-1-
,----- -
- '-1--' -
10- 5 =:::=1:7= _ _ -.A-
-----, ~ -- : - _ __ !li;
=.z f= r'"I
/---r-- - HH-H; I---i -!-,!--V¥1\hl -1---
frequency [Hz]
Transfer functions of the detection system, determined by deconvolution of the waveforms of Figures 2 and 6.
Using Eq. (3), dimensionless shock wave transit time rs is determined . Then, Eq. (1) is employed to determine source energy of each pulse. Since we have a hemispherical blast wave due to half space geometry, the actual source energy (Figure 5) is half of the calculated one [13) .
Theoretical pressure waveforms (Figure 6) agree with the measured OA signals very well. Several predictions of the model are consistent with the measured data, for example: decrease of source energy leads to:
- decrease of the compression and rarefaction amplitude, the decrease of the former being larger than the decrease of the latter;
- decrease of the total waveform duration, with the decrease of the rarefaction duration being larger than the decrease of the compression duration;
- increase of the transit time; - increase of the central frequency of the OAS spectrum; - increase of the total frequency bandwidth; - decrease of the spectral magnitude;
Applying deconvolution of measured signal and theoretical pressure waveforms it is possible to determine transfer function of the detection system. The results of this procedure are shown on Figure 7 for several pairs of waveforms. Transfer functions are found to be r~producible with standard deviation of amplitudes and pulse widths less than 10%.
CONCLUSION
OA signals, detected by optical probe, are found to be in a good agreement with theoretically determined pressure waveforms, predicted by point explosion model. On this ground , sensitivity and transfer function of optical probe can be evaluated. Energy of OA wave and optical to acoustic energy conversion efficiency is determined. Several characteristics of nonlinear propagation of laser induced acoustic transients are then explained.
295
REFERENCES
I~I fil f~l [8]
[9]
Il~1 [13]
A. Rosencwaig and J.B. Willis, Appl. Phys. Lett., 36:667, (1980). J.A.M. Jeen and J.M. Green, J. Phys. E.: Sci. Instrum., 17:191, (1984). J. Diaci and J. Mozina, in: "Ultrasonics Int. 87 Conf. Proc.", Butterworth, Guildford, 96, (1987). S. Petzoldt et al., Appl. Phys. Lett., 52:2005, (1988) M.A. Saifi and S.J. Vahaviolos, IEEE J. Quantum Electron, 12:129, (1976). C.E~ Yeack et al., Appl. Phys. Lett., 41:1043, (1982). R. Sribar, J. Mozina and 1. Grabec, in: "Ultrasonics Int. 89 Conf. Proc.", Butterworth, Guildford, 211, (1989). J. Diaci and J. Mozina, in: "Ultrasonics Int. 89 Conf. Proc.", Butterworth, Guildford, 724, (1989). L.1. Sedov, "Metody podobya i razmernosti v mekhanike", Nauka, Moskva, (1972) . G. Taylor, Proc. Rov. Soc., A 201:159, (1950). S.A. Hristianovich, Prikl. Mat. Mekh, 20:599, (1959). D.E. Okhotsimskii and Z.P. Vlasova, Zh. Vychisl. Matematiki Matern. Fiziki, 2:107, (1962). N. Ferriter et al., AIAA Journal, 15:1597, (1977).
EXPERIMENTAL STUDY OF GUIDED WAVES PROPAGATING AT THE
INTERFACE BETWEEN A FLUID CYLINDER AND A LIQUID MEDIUM
J. M. Drean and M. de Billy
Groupe de Physique des Soli des Universite Paris 7 and Paris 6 Tour 23, 2 place Jussieu 75251 Paris Cedex 05, France
INTRODUCTION
It was shown [1-5] that different guided waves arise at the interface between a cylinder and the surrounding medium: the Rayleigh and the whispering gallery modes supported by the elastic cylinder and the Stoneley or Franz waves which are propagating in the external fluid. These different surface waves are characterized by their speed of propagation and their attenuation. Until now most of the studies involved an elastic solid cylinder immersed in water [6-91, but in the last few years some theoretical investigations were also devoted to the fluid cylinder [10-12].
In this experimental work we examine and discuss the existence of surface waves at the interface between rhodersil (Rubson Company) cylinders and a liquid. The case of high and low values of the parameter ka are studied.
EXPERIMENTAL PROCEDURE
The experiments are achieved with the set-up described in detail in Ref. 13; it includes two possibilities: a bistatic configuration and a Inonostatic arrangement. The probes and the fluid cylinder are immersed in a liquid used as a coupling medium. The physical parameters of the different media are given in Table 1. The techniques used for the present study are based on time measurements and frequency analysis. For these purposes, both long and short pulses were used. The nominal central frequencies are 0.9, 2.2, 5.0 and 7.5 Mhz. An analog gate is also available for isolating the part of the detected signal for which the frequency spectrum is desired.
Table 1. Physical parameters of the studied media
Water Rhodersil Ethanol Aluminium
1490 ~ 5 1017 ~ 10 1207 ~ 5 6640 + 7
Physical Acoustics. Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991
3125 ~ 6
3 f (kg/m )
1000 1040
790 2700
297
Figure 1.
~ .B
0 v <Il
L --C T =500ms"
~ ~ .6
-CT=Oms"
>-L 0
1= .is L .4 -S
w 0 :J
:::i .2 0..
L <t
0 0° 20° 40° 60° BO° H
Variations of the amplitude of the specularly reflected signal versus the angle of incidence: - - - - theory with CL = 1017 mls
theory with CL = 1017 mls
• experimental data .
500 m/s. o m/s.
As fluid cylinder we used rhodersil cylinders of 120 cm length and various diameters. With usual techniques, we did not succeed in measuring any shear speed in this material. In Figure 1 are plotted the experimental variations (.) with incident angle of the amplitude of the signal specularly reflected by the interface between water and a plane surface of rhodersil material. For comparison we also plotted the theoretical curves calculated for a fluid-fluid interface ( continuous line) and fluid-solid interface ( dotted line). We notice that the agreement is better with continuous line than with dotted line. This is the argument which lets us to consider the cylindrical target as a fluid cylinder. Another characteristic of this material is the "filter effect". This is illustrated in Figure 2 where the frequency spectra of the signals (1) and (2) are plotted -see Figure 2-a. A comparison between the two normalized spectra points out a 4 Mhz shift towards the low frequency indicating that the high frequencies are strongly absorbed by this kind of material.
MEASUREMENTS AT HIGH VALUES OF ka
We investigated first the existence of surface waves propagating at the interface between a rhodersil cylinder of 4.6 cm diameter and a liquid at large ka (k is the wavenumber in the outside medium and a designates the radius of the cylinder). The ka values ranged from 190 to 960. The experimental arrangement is sketched in Figure 3. Narrow band transducer are employed for these measurements and they are excited with large pulses. The transmitter (T) and the receiver (R) are in the same plane and located at 9.4 cm (TI = RI ') from the cylinder fluid. The experiment proceeds by directing an acoustic beam whose half angle opening is 4°, tangentially at the rhodersil cylinder in a plane normal to the cylinder axis. The mean axis of the receiver is tangentially orientated at the cylinder also.
Depending on the position of the receiver, the R.F. output voltage recorded by the receiver shows different signals, resulting from the bulk waves which transits through the cylinder (TKK I R) and from the reemission of a surface
298
/I .. , I , , I I
~ I I
~ , I J I a
j , ,
b I , I I I I
>- , , <-0 I I <-
:B I I , I <- I 2 I
W o' ,
0 " :::) I-
~ l: ~
024 6 8 10 12 14 F(MHz)
Figure 2. Frequency spectrum of the back reflected signals: a) by the front part of the cylinder. b) by the back part of the cylinder.
wave which propagates at the interface between the liquid and the cylinder. Because of the spreading of the beam, bulk waves can be produced in the cylinder. Four regions can be selected (Figure 4). The specular region (Rt) is limited by the beam tangentially incident at the cylinder and the speculady reflected ray of the beam. The region R2 covers both zona the specular region and the direct beam. R3 indicates the angular limitations of the region in which bulk waves have transited through the cylinder. Finally we have indicated a region R4 in which the surface wave leaks back into the fluid. Drawings are not scaled and the region R3 is smaller.
Specific experiments are then performed to measure the speed of the interface wave and its attenuation and to identify the medium in which it propagates.
Figure 3.
T 1-==::::::::::=::==71" x
Schematic diagram for the experimental measurements at large ka.
299
I
R3
Figure 4. Schematic diagram of the different regions.
Speed Measurement
For two positions of the receiver R located in the region R4 - Ra and marked by the angles 81 and 82 ,corresponding arrival times tl and t2 of the signal due to the reemission of the surface wave are measured. The speed of the interface wave is then calculated by the relationship:
C = a 81-82 t l-t2 (8 1 and 82 are expressed in radians).
From measurements performed at different frequencies, we obtained: C = 1470 ± 10 mls for speed of the surface wave on cylinder immersed in water whose the propagation velocity was measured: 1480 ± 5 m/s. We noticed that no dispersion of the velocity was observed in the range of ka.
Attenuation Measurements
From the plot showing the variations of the amplitude of the signal caused by the existence of the surface wave versus the angle of reception, it is possible to evaluate the attenuation of the surface wave as it propagates at the interface between the fluid cylinder and the liquid. Measurements are made at 5 and 7.5 Mhz and are plotted in Figure 5a and b respectively. For comparison we have plotted the results obtained for a brass cylinder immersed in the same liquid at f = 7.5 Mhz. We measured the attenuation values with an error less than 10%:
arhod = 0.20 Np/deg at ka = 480 arhod = 0.28 Npjdeg at ka = 720 abrass = 0.075 Np/deg at ka = 780
Supporting Medium. of the Interface Wave
For identifying the medium which supports this type of surface wave, we repeated the measurement described in the first part of this section by using ethanol as coupling medium. We obtained for speed C' = 1202 :!: 8 m/s. From these experimental observations we conclude about the existence of a creeping fluid-borne wave: a Franz wave as it was observed with elastic cylinders. The measured value of the speed of the propagation of the acoustic wave in ethanol is found to be: Cethanol 1207 ± 6 m/s. This result shows that the speed of the wave is only affected by the external medium.
300
o
Figure 5.
"-"-"--"----"-c
a
2 4 6 8 10 12 14 16 H
Experimental attenuation measurements: al case of a rhodersil cylinder at 5 MHz. b case of a rhodersil cylinder at 7.5 MHz. c case of a brass cylinder at 7.5 MHz.
RESONANCES INVESTIGATION-MEASUREMENTS AT LOW VALUES OF ka
For low values of ka, it was demonstrated that resonances can be excited in elastic cylinders [14-16]. In the present study we used the method based on the frequency analysis which was checked earlier on metallic cylinders immersed in water [17, 18].
Experimental Procedure
Broadband transducers are excited with short pulses (N 200 ns) and the response of the probes in the frequency domain is such that the values of ka cover the range from 6.8 to 143. The diameters of the probes are large enough to cover the whole cylindrical target of 3.2 and 6.8 mm diameter and part of the incident energy is tangentially guided. In Figure 6 is sketched the configuration that we used for experiments.
Figure 6.
T
p' p x
Schematic diagram of the experimental measurements for low values of ka.
301
Figure 7.
2
II 3
Ji. 4 5 6
!~ ' . 11.
I 'I
'"
'I'
I ' -10 ~s
• I , , I I I
Typical time record observed in backscattering with a rhodersil cylinder of 6.8 mm diameter.
Typical Records and Identification of the Echoes
An example of the time structure is recorded and given in Figure 7 for backscattering geometry (ka = 28.7) . The distance between the transmitter and the receiver is 11.8 cm. The echo 1 is identified as the specular reflection from the front face of the cylinder. The time structure shows a different "history" in comparison with the one observed in the same experimental conditions for an aluminum wire (Figure 8). The amplitude of the second echo which is higher than the amplitude of the first signal is interpreted as resulting from the constructive interference between two echoes which are very close in time domain because the "acoustical path" TPP I PT and TIP I I I T are very close.
The echoes labeled 3 to 6 in Figure 7 seem to be caused by the same scattering mechanism and have to be considered altogether. The time distance which exists between two consecutive echoes is constant (8t = 19.2 f1-s) and cannot be explained by multi pIe reflections into the target . For this reason, we considered that the periodic series of echoes is due to a creeping wave which is excited by the part of the incident beam which tangentially hits the cylindrical
Figure 8.
302
I i . ; ;
~. It I 'I' " I'
I -10 us I
I a b
Typical time record observed in backscattering with an aluminum cylinder of 6.8 mm diameter.
Figure 9.
-I
" ,r ~ p' J --' p~o
~ , .. I
l....-
I • J Schematic diagram of the experimental measurements for low values of ka (bistatic geometry) .
sample and which radiates out each time it goes through the points I or I I (see Figure 6) after one, two, .. . , and n circumnavigations.
Characteristics of the Creeping Wave
This circumferential wave can be characterized by its speed and by the medium in which its energy is concentrated. If we label ti the delay of the ith echo, the group velocity of the surface wave is given by C = 2nn/(ti.l - q where n is the number of circumnavigations achieved by the creeping wave. The experimental measurements yield to a mean value of C = 1110 ± 15 m/s. Note that no significant dispersion was observed with ka and all the measurements were included within the experimental error. Measurements were repeated in ethanol used as external medium. Similar time "histories" were obtained and the group velocity was estimated to be 1070 ± 25 m/s.
Extra measurements were achieved with a bistatic configuration including two probes immersed in water resonatin~ at the same central frequency and orientated perpendicularly to each other lFigure 9) . The presence of two series of echoes (Figure 10) separated by an interval of time (lH) which corresponds to a half path in time domain, is explained by the fact that the reemission of
I .h '(
-10 .. -
1 2--4--6--8 , 3--6--1 II
Figure 10. Typical record observed in bistatic geometry.
303
Figure 11. Experimental frequency spectrum of the signal observed in backscattering with a rhodersil cylinder of 6.8 mm in diameter.
the creeping waves which propagate in two opposite directions (Figure 9) are detected each time they go through P for series II and P I for series I. The speed measured in this configuration with ethanol as ambient medium is found to be 1100 ± 15 mfs, which is close to the previous experimental evaluations.
These results point out that the speed of the interface wave does not depend on the outside liquid and so that the creeping wave can be defined as a whispering gallery type wave structure borne wave.
Normal Modes of Vibration
As was done on metallic wires [17-18] a frequency analysis of the reflected signal was carried out. The curve given in Figure 11 represents a typical frequency spectrum obtained with a rhodersil cylinder of 6.8 mm diameter. It corresponds to the frequency spectrum of the whole signal plotted in Figure 7 except the echo 1. This frequency spectrum which is not corrected by the frequency response of the transducer points out minima quite regularly spaced which correspond to the normal modes of vibration of the rhodersil cylinder immersed in water.
CONCLUSION
In this paper we related experimental results on acoustic scattering by fluid cylindrical target immersed in water. Two types of waves were preferentially observed according to the value of the parameter ka: for low values of ka, whispering gallery type waves were pointed out and for high values of ka, Franz type surface waves are excited. Interesting information and additional confirmation of these guided waves should be obtained by comparison with theoretical calculations and with records obtained on metallic cylinders of the same diameter.
REFERENCES
1 2 3 4 5
304
R.D. Doolittle, H. Uberall and P. Vgincius, JASA 43, 1-14, (1968). G.V. Frisk, J .W. Dickney and H. Uberall, JASA 58, 996-1008, (1975). O.D.: Grace and R.R. Goodman, JASA 39, 173-174, (1966). H. Uberall and L. Flax, JASA 61, 711-715 , (1977). G. Derem, La diffusion acoustique par des cibles geometriques simples, Chap .9, N. Gespa (Editeur Cedocar). G. Maze, A. Faure and J. llipoche, JASA 77, 1352-1357, (1985). G. Maze, J.L. Izbicki and J. Ripoche, JASA 77, 1352-1357, (1985).
[8]
[9]
10 11 12 13 14 15 16 17 18 19
W.G. Neubauer, JASA 44, 298-299 (1968) and JASA 45, 1134-1144, (1969). .. A.J. Stoyanov, H. Uberall, F. Luppe and G. Quentin, JASA 85, 137-140, (1989). J .L. Rousselot, Acustica 64, 216-218, (1987). J.M. Conoir, These de Doctorat, Universite Paris 6, (1987). J.D. Alemar, P.P. Delsanto and E. Rosario, Acustica 61, 9-20, (1986). J.M. Drean, Rapport de Stage de DEA Paris, (1988). J.L. Izbicki, G. Maze and J. Ripoche, Acustica 55, 27-29, (1983). G. Maze and J. Ripoche, Rev. Phs. Appl. 18, 319-322, (1983). M. de Billy, JASA 79, 219-221, (1986). I. Molinero, These de Doctorat, Universite Paris 7, (1987). I. Molinero and M. de Billy, JASA 83, 1280-1283, (1988). P. Pareige, G. Maze, J.L. Izbicki and J. Ripoche, J. Acoustique 1, 165-169, (1988).
305
SURFACE ACOUSTIC WAVE RECEPTION BY AN INTERDIGITAL
TRANSDUCER
J. Duclos, M. Leduc
Laboratoire d' Acoustique Ultrasonore et d'Electronique URA-CNRS 1373, Universite Le Havre Place Robert Schuman, Le Havre, France
INTRODUCTION
Interdigital transducers are classically described by the crossed field model [1]; in this analysis the transducer is represented by an equivalent electric circuit, originally developed for bulk wave transducers, with one electric and two acoustic ports. But, in emission as in reception, the bulk acoustic waves generated by the interdigital array are not taken in account and the physical values are not determined.
A physical model [2], more recent, using the exact equations of the :problem and the Fourier transform, sUI2plies a good description of a delay line (two transducers on the same substrate). But a large amount of computation is necessary.
We present, for receiving transducers, an easier method, extending the already one presented for an emitting transducer [3]. A Rayleigh wave is incident upon the interdigited electrodes; its energy is partly transmitted to the electric load while bulk and surface waves are regenerated by the array. In our study, we neglect resistivity, mass and thickness of metallic electrodes.
BASIC EQUATIONS
An interdigital transducer (Figure 1) is settled on the z=O surface of a piezoelectric substrate (LiNb0 3, Y-Z cut), centred at x=O. It has a spatial period A and Np pairs of aAJ2 wide electrodes (a being the metallisation ratio).
Let us consider an harmonic surface wave incident on the transducer described by its electric potential ¢i(X,z,t). It induces reflected and transmitted wave potential ¢r(x,z,t) and ¢t(x,z,t).When Ixl -! 00 these waves are free waves on non metallised surface, with phase velocity Co and wave vector ko at frequency F. Such as we can express the surface potential ¢(x) by:
¢(x) = ¢iexp(jkox)U(-x) + ¢texp(jkox)U(x)
+ ¢l'exp(-jkox)U(-x) + ¢o(x) (1)
In this expression, we omit the time harmonic dependence exp( -j27l'Ft). U(x) is the Heaviside step function and ¢o(x) must vanish if Ixl -! 00.
Using dielectric function c(k) [2] [3], potential ¢(x) and surface charge density o-(x) are related through their Fourier transform by:
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 307
Figure 1.
I.
Receiving transducer with load Ye. Incident, reflected and transmitted surface waves and regenerated bulk waves.
~(k) = Ikl (J~N) = j sgn(k) ;t~j
where q(x) = f (J(x) dx vanishes outside the transducer.
Using Fourier transform of (1) supplies:
~(k) = ~i[l/z 6(k-ko) + 27i{Lko)l + ~t[t/z 6(k-ko) - 27rd-k o)l
+ ~r[l/z 6(k+k o) + 21rd+ko)l + ~o(k)
Using (2) with (3) we obtain:
~(x) = f j sgn(k);t~j exp (jkx) dk + l/z(~i+~t)exp(jkox) v.p.
(2)
(3)
(4)
So the poles (k= ±ko) of E (k) have a clear contribution. The calcul induces also important relations:
,j,.~ = 21r q(k o) 'f' 1--¥ t £lKOJ and ,j, ~ko 'f'r = 21r
E' 0
MODELISATION AND NUMERIC EVALUATION
(5)
For numerical evaluation, we proceed by linear approximation of q(x). Each finger of the transducer is divided in Ns parts (unequal for fastest computation); Zn, 6xn and 6qn are respectively, the middle, the half-width of the nth subdivision and 6qn is the variation of q upon it. So the transform of q(x) is approximated by:
Figure 2. Equivalent electrical circuit.
308
• N 6 q(k) = ~ E Q<ln sin(k6xn) exp(-jkzn) ~ n=! OX;;
(6)
denoting N = 2· N p' N s the total number of subdivisions of the transducer. Potential ~(x) expression is then:
If dk N 6q () ~(x) = 7r k2 c(k) n~! oi sin(k6xn) cos(k X-Zn) v.p.
N
+ ~i exp(jkox) + ka ~/(ko) n~! ~ sin(kooxn) cos(ko(x-zn))
We can express the voltage Y i1 = ~i-<P1 between two middles Zi and Zl
+ 2j~i expOjk o( Zi+Z1) )sinOko( Zi-Z1))
N
+ 2 k3 ((ko) n ~! ~ sin(ko6xn)sin(ko( Zn-HZi+Z1) )sinOko( Zi-Z1))
To determine the N unknown values 6qn, we must write N equations, describing:
a) potential equality upon Ns segments of the same finger; this supplies 2· Np ' (Ns-l) equations.
b) potential equality upon Np interconnected fingers; we obtain 2· (Np-l) equations.
c) Ohm law for the electric load of the transducer (Figure 1) I. = L =Yc'Yc (two equations).
In these equations it is convenient to give the same importance to each fin~er and subdivision. After solving the linear system, we obtain successively q(k) by Eq. (6), surface wave potential (Eq. (5)) then transmission and reflection coefficients (~t/~i, ~r/~i) and, at last, the voltage across the load.
In order to verify the computation accuracy, we make a balance of the active power absorbed by the transducer. The incident wave supplies the power [2]:
Pi = - t 7r F ko c/(ko) l~il2,
which induces transmitted power
reflected power
P r = - t 7r F ko c/(ko) l~rl2,
power absorbed by the load' termination
Pc = - t IYcl 2Gc,
and P y , the power dissipated by bulk waves generated in the substrate. We always verify the inequality:
< I~d 2
309
RESULTS
a) First we verify the equivalent circuit of a recelVlng transducer. In the crossed field model, the transducer is described by a current-generator Io with an internal admittance Yi equal to the input admittance Ye of the same transducer used as an emitter (Figure 2).
As an example, a transducer with the parameters A = 34.88 J1IIl, a == 0.5, Np = 5 has, at Fo = col A = 100 MHz, an input admittance:
310
Ye = 0.4228 -j 1.117 Sim
Or-------------------------------------------------~O
dB
-20 /'
, , , , , , , - 40 :
-60
o
;---~ ....
bulk ---- ---.- --- ---_ ...... ---
load
F (Mil»
100 2110
alin the load and bulk waves
Or=-~--~-7 __ ~·~ __ ~ __ 7. __ ~~--------------------------------_,O ----- .. _.... ,.- ......... ,-----_ .. -------_ ..... \ ,'---------
R dB
-20
-60
o
. 1 \ 1 ,
\ I' ,. .....
100
transmission \ .. ,/ dB T
riln~~ ~~n~ -5
reflect ion
F (MHr)
20U
bltransmitted and reflected surface waves
Figure 3. Insertion losses as a function of frequency.
Then this transducer, as a receiver loaded by Yc=Y: has an optimal power transmission at the same frequency. Our computation gives the following insertion losses:
3.5 dB in the load 5.1 dB in transmitted wave 7 dB in reflected wave
b) Dependence on frequency (Figure 3). The studied transducer has a load adapted at Fo= 100 MHz. We examine
how insertion losses depends on frequency without modifying the components of the load. We note a maximum of received power near Fo. with a large bandwidth (2Fo/Np = 40 MHz).
0.01 0.1 10
dB o
... -------
-to
bulk
-20
-- our computation. ---- crossed field model.
a)in the load and bulk waves
0.01 0.1 10
dB o ------ __
-10
-20
I I
I
, , ,
, , , I
, , I
, ,
, , ,
.....................
, v , ,
, ,
, , ,
, , , , ,
, ,
reflection
, \ ,
, , , \
\ \
b) transmitted and reflected surface waves
, \ , ,
, \ ,
\
100 b
100 b
Figure 4. Insertion losses as a function of load conductance.
311
The bulk wave level is rather constant (-15 dB) with an increase (-11 dB) near Fo. The surface wave is transmitted at high level (0.2 dB) except in frequencies neighbouring Fo (-5.1 dB) and 2Fo (-0.8 dB). On the contrary, reflection is important in the same areas and we notice secondary lobes, spaced by 100 MHz, which agree with a simple theory of reflection grating.
c) Load conductance variations (Figure 4) At a given frequency (F = 100 MHz), the transducer adaptation is
modified when Ge vary (Be constant) and our results can be compared with the crossed field one. Let us write b = Gc/0.4228, with this model the insertion losses, as a function of bare:
-20 lOg~ 1 + b) for reflection, 20 log b/(1+b)) for transmission, 10 log 2b/(1+b)2) in the load,
-00 for the bulk waves.
The agreement is good for the load 102). But, unfortunately, the crossed discordance with experimental values coefficients of surface waves.
power in a large scale of b (10 -2 to field model often supply values in [1] for transmission and reflection
CONCLUSION
A better knowledge of receiving transducers is supplied by our calculation, whose accuracy is greatly depending on computation time. We prove the large dependence on the electric load of physical process such as reflection and transmission of surface waves and generation of bulk waves. Our model must be adjusted to more complex transducers and to an array of metallic fingers such as multistrip couplers.
REFERENCES
[1]
[2]
[3]
312
WR. Smith, HM. Gerard, JH. Collins, TM. Reeder, HJ. Shaw, "Analysis of interdigital surface wave transducers by use of an equivalent circuit model", I.E.E.E. Trans., MTT17/11:856, (1969). RF. Milson, NH. Reilly, M. Redwood, "Analysis of generation and detection of surface and bulk acoustic waves by interdigital transducers", I.E.E.E.Trans., SU 24/3:147, (1977). J. Duclos, M. Leduc, "Calcul de l'admittance d'un transducteur interdigite depose a la surface d'un monocristal piezoelectrique", Revue d' Acoustique, 78/3-4:33, (1986).
UNDERWATER SOUND SCATTERING BY SURFACE GRAVITY WAVES
D. Euvrard, O. Mechiche Alami
Groupe Hydrodynamic Navale, URA D 0853 du CNRS associe a l'Universite Paris VI (UPMC) Laboratoire de Mecanique et Energetique, ENST A Centre de l'Yvette, Chemin de la Huniere 91120 Palaiseau, France
INTRODUCTION - HYPOTHESIS
This study concerns the acoustical radiation of an underwater motionless body with known vibrations, taking into account harmonic surface gravity waves of small amplitude.
We are interested in the case when both acoustical and hydrodynamic wavelengths, Aac and AG, are of the same order. It corresponds to a very low
sound frequency l/Tac of a few Hertz and to gravity waves periods T G of
some seconds. The depth of the ocean is infinite and the body is deep enough to allow us
to neglect the scattered gravity waves. We suppose water to be an ideal fluid and its motion to be isentropic. It
follows that the Brunt-Vaisal1i frequency is zero so that internal gravity waves are precluded.
We consider for the water a classical equation of state as the Tait modified equation. It can be written:
c2 = C6 -L-[p IlO] n-l (1)
where n = 7 for water, c is the sound speed in water, Co is a reference sound speed in water, P the water density and Po the value of p at the free-surface.
We also assume the flow to be irrotational and the free-surface to be isobaric at the constant atmospheric pressure Patm.
ASYMPTOTIC TREATMENT
Characteristic Dimensions
Before introduCing power series expansions, we must a-dimension the conservation equations. We introduce the following characteristic dimensions: Aac , AG, Po and Tac or TG whether we want to study an acoustical
problem or a hydrodynamic problem.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 313
According to the IIleast degeneracy principlell , the most interesting case is the one where Aac and AG are of the same order A.
Small Parameters
We introduce small adimensional parameters traducing the fact that the considered phenomena are of small amplitude:
A = h/2A
where h is the amplitude of the surface gravity wave,
e: = Pac/Pref
where Pref is a reference pressure and Pac the sound pressure. A third small adimensional parameter characterizing water compressibility
must also be taken into account:
o = gA/c~
where g is the gravitational acceleration. We are not in seek of a complete expanding in power series of 0, A and
e: but only to exhibit the necessary terms to study the influence of gravity waves on sound waves.
We can notice that Tac/TG is of the order of .[5 and that A.[5 is
nothing else but the Mach number of the gravity wave flow.
Free-Surface Condition
The following results concern the bidimensional problem in the Cartesian coordinate system (x,z) fixed relative to the undisturbed position of the free-surface z = O,with the z-axis pointing upwards. z = TJ is the equation of the free-surface in presence of surface gravity waves. We can precise that TJ is of order of magnitude A-
We have assumed that p = Patm on z = TJ, so
Pac = 0 on z = TJ (2)
Postulating the existence of the perturbation series expansion for the pressure p, we make a Taylor expansion of (2) around the calm-water position z = o. We shall write the sound pressure Pal at the order e:, Pa2 at the
order e: A and Pa3 at the order e: A.[5. We obtain on z = 0:
and
Pal = 0
Pa3 = 0
A i( ZlGX-IIIGt ) -i( ZlGX-IIIGt ) a [ + j. ~zal Pa2 = - "2". e e ------oz
(3) (4)
(5)
1112
where IIIG is the surface gravity waves pulsation and ZIG = ~ Because of (5), there will be two time-harmonic problems at the order
e:.A; one with the pulsation iliac + IIIG where iliac is the sound pulsation, the
other with the pulsation iliac - IIIG, which is typical in second order problems.
314
Successive Boundary-Value Problems
We expand the pressure p, the water density p and the velocity U in power series and we inject them into the conservation equations:
- at the order A we obtain the classical linearized gravity waves problem which solution is a regular plane progressive wave,
- at the order E we obtain the classical linearized acoustical problem without gravity waves. Pal satisfies the homogeneous wave equation in the fluid domain z < 0 and (3) on z = O.
- at the order E A we obtain the first acoustical term influenced by gravity waves. Pa2 satisfies the homogeneous wave equation for z < 0 with the same wave number as at the order E, and (5) on z = O. It is the problem of the acoustical scattering by surface gravity waves.
- at the order E A /5 we obtain the second acoustical term influenced by gravity waves. Pa3 satisfies a non-homogeneous wave equation for z < 0 with the same wave number as at the order E and depending on the upper orders, and (4) on z = O. It is the problem of the acoustical propagation in a moving medium. It is not a typical underwater sound problem, so we shall rather focus on the problem at the order E A.
We show that at the considered orders, it is not necessary to take the equation of state (1). We can take a constant sound speed c = co.
CASE OF A POINT-SOURCE
We suppose that gravity waves and acoustical waves are both time-harmonic:
Pal = Im{ Pal e-iwact } (6)
and
We shall consider here an acoustical point-source located at S(O, -a), with a > O.
At the Order E
At the first order we have a theorem of existence and uniqueness because of the Sommerfeld radiation condition. This condition precludes the possibility that there might be an incoming wave generated at infinity. We solve the problem via the classical image method: we bring the problem from the half plane z < 0 with the boundary condition (3) into the whole plane without
boundary condition, with a source located at S and a sink at S which is symmetrical of S with respect to z = O. The solution is:
where k = Wac is the wave number, rs = I MS I Co
At the Order E A
and rs IMSI·
The time-harmonic problem. Unfortunately we have no radiation condition at infinity for the second order that would ensure the uniqueness of the
315
solution. For the moment we can just exhibit a solution of the problem. By doing a Fourier transform on x, solving the corresponding problem and
inverting the Fourier transform we find (see [1]):
pB)=p(~)1 * -2~ a z=O X
1,2 . (8)
where pB) I is given by (5) and (7), and H is the elementary solution of z=O
the Helmholtz equation.
The initial-value problem. We shall use the "Limiting Amplitude Principle" in order to show that (8) is the solution of our problem: we formulate and solve an appropriate initial-value problem and then find the solution of the time-harmonic problem by allowing the time t to tend to infinity. This process will give us the physically acceptable solution.
Initially we consider a regular plane progressive gravity wave persisting for all time. The acoustical source starts pulsating from rest with strength
Im{e-iwact}. The imaginary part has been chosen not to create a shock-wave. We have a theorem of uniqueness (see [2]) in the '6 2([0, +00[; Y ') space.
Its conditions are satisfied here. For the existence we find a solution by making a Fourier transform on x and a Laplace transform on t. This solution is:
Pa2 = Pa21 z=O *
x, t 2 BE
- Oz
where E is the elementary solution of the wave equation.
(9)
When t tends to infinity in the solution (9), we find the solution (8) of the time-harmonic problem.
Numerical computation of the solution. To compute the solution (8), we make a Fourier transform on x and then an inverse Fast Fourier Transform. This is possible because the solution which is analytically known in the Fourier plane, has an exponential decay.
The following figures correspond to the time-harmonic problem j = 1. The time-harmonic problem j = 2 is symmetrical of the first one with respect to the z-axis.
The second order sound pressure is maximal on the free-surface z = 0 while the first order sound pressure is zero on it. The presence of surface gravity waves creates a kind of resonance and allows us to hear the underwater point-source. Most of this sound is concentrated above the point-source.
1 2
~IA J IIA :vv ~ IV Y
Figures 1 & 2. Re{ p~~) (x)} and Im{ p~~) (x)} with
--6000 m ~ x ~ 10000 m; a = 300 m and z
316
-150 m.
3 4
Figures 3 & 4. Re{ p~~) (x)} and Im{ p~~) (x)} with
-6000 m ~ x ~ 10000 m; a = 300 m and z O.
GENERAL CASE OF A RADIATING BODY
At the Order E
The problem with a non-homogeneous Neumann condition on the body boundary can be solved using a finite element-integral representation coupling method (see [3]) . It consists on bringing the problem in the unbounded domain to a problem in a bounded domain n by introducing a regular fictive surface ~ surrounding the body boundary. We use on ~ an integral representation of the solution involving sources and normal doublets on the body boundary. Both problems in unbounded and bounded domains are equivalent. The choice of ~ being arbitrary, we can choose it as small as possible to limit the finite element computation in n.
At the Order E J6
pH) can be split by linearity into pH), 1 and pH ), 2'
pH), 1 must satisfy a homogeneous Helmholtz equation in the whole half
space z < 0 ignoring the body and a non-homogeneous free-surface condition on z = O. Its solution is obtained by numerically computing a doublet distribution on z = 0 via a FFT as in the point-source problem.
pH), 2 must satisfy a homogeneous Helmholtz equation for z < 0 and
outside the body, a homogeneous free-surface condition on z = 0 and a Neumann condition on the body boundary . It is solved using the same coupling method that at the order E , which involves the standard radiation condition.
A similar procedure has already been used successfully for a tidal wave propagation problem (see [4]).
ACKNOWLEDGEMENTS
This study has been supported by DRET under contract n° 88/1009.
31 J
REFERENCES
[1]
[2]
[3]
[4]
318
G. Pot, Diffraction d'une on de acoustique sous-marine par une houle plane simple, Rapport ENSTA n° 233, 1989. R. Dautray, J.L. Lions, Analyse mathematique et calcul numerique pour les sciences et les techniques, Volume 3, Masson, 1985. M. Lenoir, Methodes de couplage en hydrodynamique navale et application a la resistance de vague bidimensionnelle, Rapport ENSTA n° 164, 1982. M. Verriere, Calcul numerique de champs de vagues line aires , en regime transitoire, en presence d'un obstacle tridimensionnel, These et Rapport ENSTA n° 235, 1989.
FINITE AMPLITUDE ACOUSTIC WAVES RADIATING FROM A
NON-RESONANT VIBRATING PLATE*
M. A. Foda
(on leave from Mansoura University, Department of Industrial Production Engineering, Mansoura, Egypt) Institut fiir Technische Akustik der TU Berlin, FRG
INTRODUCTION
The interaction between an acoustical fluid and a harmonically excited plate is examined first [1] using a dual renormalization method. A direct renormalization procedure [2], [3] is applied to the resonant excitation of the same problem. In these investigations one has to solve two coupled transcendental equations for the co-ordinate straining transformations before calculating the response at a certain location. Subsequent investigations [4] - [7] recast the problem into a superposition of two sets of uniform planar waves.
The following investigation describes the non-resonant response of the plate system described above using the method of average combined with the renormalization method. Series representations of the signals in terms of the physical co-ordinates are derived using a Fourier analysis. This results in explicit forms which are straightforward and efficient for computation and predict harmonic contents and the shock formation distance. A quantitative example is cited that shows the spatial and temporal waveforms enabling one to visualize the distortion phenomena.
VELOCITY POTENTIAL AND PLATE DISPLACEMENT
A simply supported large plate (idealized as infinite), located in the plane z = 0 and undergoing transverse vibration which is harmonic in time with frequency fi and sinusoidal in space, forms the boundary for an acoustic fluid within the half space z > O. Let tIn be the time scale and Lx and Lz are the Cartesian co-ordinates tangent and normal to the plate, where L is the constant spacing between the supports of the plate. Further, the particle velocity components and the pressure in the fluid are denoted as Lnvx, Lfivz and Po(p+l), respectively, where Po is the ambient pressure. The equation describing the displacement Lw of the plate is
wxxxx + owwxxxxt + W2Wtt - 6(~)2wxxf lw~dx = o
- p ~ L3 ( 4;0 exp(it) [i exp(iux) + cc] + pi zoo} + cc
* Dedicated to Professor M. Heckl on his 60 th Birthday.
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
(1)
319
where F is the amplitude of the excitation, h is the thickness of the plate, 8 is the nondimensional damping coefficient and
1
W = fiL2(mh/D)2 (2)
where D is the plate rigidity and m is the mass density for the plate, cc denotes the complex conjugate of the preceding terms. The boundary and the initial conditions are such that
w(x,t) = 0, at x = 0, ±1, ±2, ...
w(x,t +271') = w(x,t);
w(1/2n,O) = E, wt(1/2n,O) == 0 (3)
where the perturbation parameter E« 1 and n refers to the n-th vibration mode of the plate. The non-linear wave equations governing the nondimensional potential are
(4)
(5)
where fJo is the non-linearity coefficient and ,\ = fiL/co is the reduced frequency involving the speed of sound Co of an infinitesimal planar wave.
The continuity condition for the velocity component normal to the plate requires that
(6)
The transverse displacement of the plate may be expressed as
w(x,t) = E u(t) [2\ exp(inu) + cc] + cc (7)
Considering that POL3/D, (h/L)2, F/po and 8 will be O(E), the substitution of Eq.(6) into Eq.(l) leads to the equation governing u(t)
n471' 4u + w2 U ;~~ exp (it) - [8wn471'4 + (2fJo-1) ,\2~~L3] U
- 3 E2 (~)2 n471'4(u3+3u2u) (8)
where dots are the time derivates and the bar denotes complex conjugate. It is assumed a priori that only the outgoing wave which depends on t and z through the combination t-kz will be considered. The wave number k will be determined in the course of the analysis.
Following the Struble'S version of the method of average [8], the solution of Eq.(8) is sought to have the form
u(t) = % exp[i(t-O)] + EU1(t) + cc (9)
where a and 0 are slowly varying functions of time. Substitution of Eq.(9) into Eq.(8), equating the coefficients of exp(it) through O(E) in both sides, then separating the result into real and imaginary parts and finding the stationary solution of the resulting equations that satisfies the boundary conditions leads to the frequency-response relations
320
tanO = [ b'wn 47r4 + (2/30 - 1),\2p oL3/kD] / [ n 47r4 - w2 + ~1;:2(~)2 n 47r 4],
I;: = F L3 cosO/D [ n47r 4 - w2 + ~1;:2(~)2 n 47r 4]. . (10)
Equations (10) are the same as Ginsberg's [1] except that he neglected O( 1;:) terms since In 47r L W21 » 0(1;:). This indicates that the non-linearity caused by the mid-plane stretching due to the restraint of the supports has little significance for non-resonant motion. After redefining the dimensional time as (HO)/ll, the displacement of the plate is given by
w(x,t) = ~i { exp [i(t-n7rX)] - exp [i(Hn7rX)]} + 0(1;:2) + cc (11)
A solution of Eq.( 4) that satisfies the boundary conditions as well as the radiation condition at large distances from the plate is obtained by expandin~ ¢ in a perturbation series and solving the equations corresponding to 0(1;:) and 0(1;:2). When the solution of these equations are combined the result is
. 2,\ 4/3 ¢ = - ~~ [exp(iW+)-exp(iW-)] - I;: 32k~z[exp(2iW+)+exp(2iW-)]+NST+cc (12)
where
k2 = ,\2 - n 27r 2 (13)
W± = t 'f ll7rX - kz (14)
NST stands for all 0(1;:2) terms that are bounded and do not contribute to the distortion process.
EXPRESSIONS FOR ACOUSTIC SIGNALS
Differentiating Eq.(12) with respect to t, x, and z and decomposing each of the resulting expressions into two oblique waves gives
p = p+ + p- vx = v~ + v~ vz = v+ + v-z z
± p
± vx
± v z
( ) ,\ 2 ± [ l;:i/3 0'\ 4z (±)] 'f I;: 2/30-1 4k exp(iW) 1 'f ~ exp iW + NST + cc
± ll7rP± /(2/30-1),\2
kp ± /(2/30-1)'\ 2
(15)
(16)
(17)
(18)
The presence of the factor z in the second order term in the above equations indicates that these expressions lack uniform accuracy. This can be corrected by the renormalization method [8] using a different strain co-ordinate for each wave. The result of the renormalization procedure is
± ,\ 2 ± p = 'f I;: (2/30-1) 4k exp(ia ) + cc + 0(1;:2)
w± = a± 'f ~ c [exp(ia±) + cc]
where
c
(19)
(20)
(21)
Calculation of the acoustic pressure and the velocity components at set of values (x,z,t) requires only solution of each of the transcendental equations
321
given by (20) independently for the strained co-ordinates a+ and a-. To circumvent the difficulty of solving these transcendental equations, the Fourier series is invoked to express the acoustic pressure in terms of the physical co-ordinates. The procedures are similar to that used in [7]. The result in real form is
(IJ
p -e:(2,80-1)¥:L m1cJrn(mC)[Sin m(W++7r/2)+(-1)msin m(W-+7f/2))+0(e: 2)
m=l
(22) (IJ
vx = - T:L m1cJrn(mc)[sin m(W++7f/2)-(-1)msin m(W-+7f/2))+0(e: 2) (23)
m=l
vz = kp/(2,80-1),P + 0(e: 2) (24)
where Jrn is the Bessel function of order m. One may note that retaining only the first order terms the linearized signals are recovered. Specifically:
pLin -e:(2,80-1 )"k2 sin( t-kz) sin nn (25)
Lin vx = e:ll7f ( ) - T cos t-kz cos n7rX (26)
Lin vz kpLin /(2,80-1)" 2 (27)
Equations (22)-(24) are uniformly valid first approximations to the non-linear process until a shock forms . At shock, the transformations given by Eq.(20) cease to be single-valued. The smallest value z* where the discontinuity occurs corresponds to setting I c I = 1. Thus
2k 2 2k2 z* = £fJOJ4 = -=-e:(J7TO'("k2r+-n"27fo2"')2
This result is reminiscent of the result for the one-dimensional planar wave.
Figure 1.
322
O.t ~----------------------------~
_ 0.0
~.1~------------~ ____________ ~ 0.0 1.5 3.0
TIME ( NONJIM.1 / 21t
Temporal waveforms for pressure signal at z and at z = 2.81; x = 0.5 .
1.4
(28)
Figure 2.
0.1 r----------------------------,
0.0
~.1+-------------~--------------~ 0.0 1.5 3.0
TIME C MOtClIM.1 I 21t
Temporal waveforms for pressure signal at z = 1.4 and at z = 2.81; x = 0.25 .
EXAMPLE AND DISCUSSION
In the following, the non-resonant excitation of the n-th mode of a steel plate whose Young's modulus E = 2.068 x 1011 N/m2, Poisson's ratio /J = 0.3 and the mass density m = 7827 kg/m3 is considered. The material damping ratio 0 = 0.005. The ambient medium is water at 20· C for which flo = 3.6, Po = 1000 kg/m 3, Po = 5 x 105 Pa and the speed of sound Co = 1488 m/s. The independent parameters are L/h = 30, n = 1, £ = 0.001 and IJ} = 40/f2. The value of F can be calculated from Eqs.(10). Corresponding to these values are F /Po = 218 .5, k = 13.3928, ). = 13.7563 and the shock formation distance calculated from Eq.(28) z* = 2.8622. Only the diagrams for the pressure signal will be shown due to space limitation and the dashed lines in all the figures represent the linearized signals . Figures 1 and 2 show the temporal waveforms along the lines z =' 1.4 and z = 2.81, which are three wavelengths (,\z = 2/f/k) apart, and at the locations x = 0.5 and x = 0.25. The line x = 0.5' is a nodal line for the linear and the non-linear velocity component vx . In comparison to the linearized signal, the distortion becomes stronger as z increases . The spatial profiles when t = 0 along the lines of constant x, which are normal to the plate, are shown in Figures 3 and 4. In Figure 3, the
!
I
0.1 ~----------------------------,
0.0
~.1D+.-D------------~1-.4~----------~2.8 Z C NOtIJIN.)
Figure 3. Spatial variations of pressure signal along x = 0; t = O.
323
IL
0.1 __ ----------------------------,
0.0
-o.10~.-0--------------1~.~4------------~2.8 Z ( NONDIM.)
Figure 4. Spatial variations of pressure signal along x = 0.5; O.
pressure profile along x = 0, which is a nodal line for the linearized pressure signal, is shown. The non-linear signal grows with increasing z and the profile is distorted as the wave propagates. The profile along x = 0.5, .which is a nodal line for Vx in both linear and non-linear analysis is depicted in Figure 4. The distortion resembles that of the one-dimensional planar wave.
A different set of profiles appear in Figures 5 and 6, for t = 0 along lines of constant z which are parallel to the plate. At z = 2.6976 the linearized Vx vanishes and Figure 5 shows that the distortion of the portion of the waveform in the compression phase is identical to that in the rarefaction phase. While at z = 2.86, Figure 6 shows that the pressure profile exhibits negative spikes . Comparing the results presented here with those in [7], one can conclude that the qualitative behavior of the distortion process of the waves radiated from a non-resonant excitation of a plate is similar to that in the case of its resonant excitation. This is also noted by Ginsberg [1], and it is attributed to the result that the plate displacement given by Eq.(ll) has the same form in both cases .
In conclusion, the simple analytical results presented drastically reduce the computational effort. That can be used to generate different waveforms for the acoustic variables to provide a greater physical insight into the distortion phenomena.
IL
0.1 ~----------------------------,
0.0
I' , \
, , , , , "
r, , \
I I
-O.1~--------------~------------~ 0.0 3.0 6.0
x ( NCHJIM.)
Figure 5. Spatial variations of pressure signal along z = 2.6976; O.
324
0 . 1 r-----------------------------~
_ 0.0
, ... , \ \
~. 1 ~------------~--------------~ 0.0 3.0 &. 0 X ( NOtmIN.)
Figure 6. Spatial variations of pressure signal along z = 2.86; t O.
ACKNOWLEDGEMENT
This work is supported by The Alexander von Humboldt Foundation.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
J .H. Ginsberg, Multi-dimensional non-linear acoustic wave propagation, Part II: The non-linear interaction of an acoustic fluid and plate under harmonic excitation, J . Sound and Vib. 40, 375:379, (1975). A.H . N ayfeh and S. G. Kelly, Non-linear interactions of acoustic fluids with plate under harmonic excitations, J. Sound and Vib. 60, 371:377, (1978) . J .H. Ginsberg, A re-examination of the non-linear interaction between an acoustic fluid and a flat plate undergoing harmonic excitation, J. Sound and Vib. 60, 449:458, (1978). J .H. Ginsberg, A new viewpoint for the two-dimensional non-linear acoustic wave radiating from a harmonically vibrating plate, J. Sound and Vib. 63, 151:154, (1979) . A. Kluwick, On the non-linear distortion of waves generated by flat plates under harmonic excitations, J. Sound and Vib. 73, 601:604, (1980). A.H . Nayfeh, Non- linear propagation of waves induced by general vibrations of plates, J . Sound and Vib. 79, 429:437, (1981) . M.A. Foda, Analysis of non-linear propagation of waves induced by a vibrating flat plate, Acustica (accepted for publication) . A.H . Nayfeh, Perturbation Methods. Wiley-Interscience, New York, ch. 5, (1973) .
325
SCHOLTE-STONELEY WAVES IN A MULTILAYERED MEDIUM WITH
ELASTIC BOND CONDITIONS AT AN INTERFACE
H. Franklin, M. Rousseau, P. Gatignol
Laboratoire de Modelisation en Mecanique (URA 229) Universite Pierre et Marie Curie, Tour 66 4, Place Jussieu, 75252 Paris Cedex OS, France
INTRODUCTION
The aim of this study is to show the influence of less restrictive boundary conditions than those used traditionally (welded or smooth contact type along solid-solid interfaces), on the Scholte-Stoneley waves propagating at a liquidsolid interface or on the Stoneley waves propagating at solid-solid interfaces of a plane multilayered medium.
Welded and smooth contact are ideal conditions for adhesively bonded structures. They do not really describe one of the critical defects, besides debonding on poor cohesive strength, that is interfacial weakness caused by an imperfection in the adhesion process, between the adhesive and the solids.
In the case where the adhesive thickness is small compared with the wavelength, one can consider that the bond is characterized by a limiting surface where a discontinuity in displacements occurs, caused by the current stress (J
in the interface, which is continuous across the interface. The next section shows how a linear condition can be obtained between
stresses and displacement discontinuity including welded and smooth contact conditions, in the case of isotropy and elasticity of the bond structure.
The subsequent section is devoted to a theoretical study of the Stoneley wave range of existence, following Scholte (1947), in the case of two semiinfinite elastic media adhesively bonded. The influence of the bond conditions on the wave propagation is investigated.
In the last section, we study the Scholte-Stoneley wave propagation on the case of a multilayered liquid-solid-solid medium with this kind of interface.
LINEAR CONDITIONS BETWEEN STRESS AND DISPLACEMENT DISCO NTINUITY
In the case of two semi-infinite solid media in welded contact, Scholte has shown that Stoneley waves only exist for elastic parameters in a certain range. The addition of an elastic, adhesive bond to Stoneley's problem introduce interesting effects. In order to model these effects, let us consider two semi- infinite media joined together by an adhesive layer of very small thickness d. It is implicit in this description that the wavelength is large compared to the bond thickness.
7J2 and 7J 3 are two semi-infinite media 7Jc is the adhesive layer between 7J2 and 7J 3, with thickness d A cartesian coordinate system (x,y,z) is chosen so that the (x,y) plane divides 7Jc in two symmetrical parts.
Physical ACOUSlics, Edited by O. leroy and M.A. Brea7eale Plenum Press, New York, 1991 327
E2 and E3 are the intersection between the boundaries of the three media, Ec is the surface containing the (x,y) plane.
In a two dimensional description of the adhesive bond, an assumption concerning the displacement field in 11c is made. The equilibrium equations are then derived by using the principle of virtual work (with the assumption that external volumic and surface forces are ineffective):
3
or 11 alj ul j dv + 111caIj UIj dv = 0 (i,j = x,y,z) 1 = 2 111
where U\ and U1 are the virtual displacements, atj stress tensors, with the kinematically admissible field:
d
Cad = {(U2,U3,UC)j Ur = U~ (x,y) '2 : z + U~ (x,y)
(1)
being the
By using Green's theorem to transform the integrals (1), with the kinematical field (2), and assuming that the adhesive is a thin elastic isotropic medium, one must obtain the linear relationship between stress and displacement discontinuity vectors:
where
x = [K'6 K~ ~l o 0 KN
In our case, we use the two dimensional form of X (in the x-z plane). The coefficients KT and KN are related to Lame constants of the layer by:
KT = /.tjd ; KN = (A+2J.t)jd
The relation (3) is given by several authors: Jones (1967), Schoenberg (1980), Pilarsky (1988). X is the boundary stiffness matrix, in the case of plane displacements. The adhesive layer is reduced to an interface (d IV 0). Explicitly, one can write:
a~x (z=O+)
a~z (z=O+)
KT{U~(z=O+) - U~(z=O-)}
KN{U~(z=O+) - U~(z=O-)} (4)
SURFACE WAVE PROPAGATION AT THE ADHESIVELY BONDED INTERFACE
The stress and displacement vectors in the media can be written as functions of the potentials describing plane monochromatic and inhomogeneous waves which are solutions of the Helmholtz wave equation. In each medium, the waves are represented by:
~2 = Ai exp(-ikz21 z)"E(x,t)
III 2 = B i exp( -ikz2t z)" E( x, t )
328
~3 = A3 exp(ikz31 z).E(x,t)
IV3 = B3 exp(-ikz3t z) . E(x,t)
with (j = 2,3) :
k . - (kZ _ k2)1/2 w () ( ) zJI,t - J l , t x , kx = -, E x,t = exp ikxx - w t Cx
and where Cx is the phase velocity in the x direction of the interface. Equations (4) then lead to a set of four linear homogeneous equations for the four unknown constants A2, B2, Aj, Bj. Setting the determinant of the system
equal to zero yields the following dispersion relation:
7)=
1 (l-ci)1 [ 1 [ - 2 V 3) ] 1 [(1-C i 1 1 1 -2 1 ] - 1+2~ l-cx 7J 1J)2-a:--( Cx 1J -2)
Cx
-2 1 -1 [[ -2 V3] t 1 -2 1 ] [1+2,8 ~(I-ci ~) 1] -( l-c x v2) - l-cx T -~( Cx 1J -2) Cx Cx
V 1 _ 1
-(ci-2) &[ -2 3]' ~(ci ~ -2) 2 (1-civ2)2 2 l-c x 7J JL2 JL2
-(ci-2) 2(1-ci)t &(c 2 1 -2) JL2 x 1J
~[ -2 V 3J1 2 l-cx 7J JL2
o ..;
0 .. . .
"! , , , .. .. ... -- - -
_ _ _ ... .. ___ 00 _ _
.. ' . . ' 0 .,;
0 .0 1.0 2 .0 t .O
'------ - - - ------- - - 8 = C; t l C~t 3. 0 5 . 0
Figure 1. Stoneley boundary curves. 1/2 = 1/3 = 1/3. The shaded area corresponds to bond coefficients: a = 3, ,8 = 1.5 .
329
where all the terms In the determinant have been adimensioned, and where:
The following cases immediately appear:
tal if KT -i 0 and KN -i 00 , Q -i 00 {3 -i 0 and 1) = 1)smooth contact b if KT -i 00 and KN -i 00 , Q -i 0 {3 -i 0 and 1) = 7Jwelded contact c if KT -i 0 and KN -i 0 , Q --; 00 {3 --; 00 case of total debonding
between media.
One must note that the dispersion equation depends upon the wavelength only in combination with Q and {3 from dimensional considerations. For short wavelengths, the adhesive is weak, while for long wavelengths even a very flexible adhesive will appear to be stiff (Jones, 1967).
The question is now whether a real-valued solution of the dispersion equation is possible for every conceivable combination of 1/2, I/a, 0, Q and {3. In the case where Q=O and {3=O, Scholte has shown existence conditions for Stoneley waves. Achenbach (1967) has shown similar existence conditions for the case Q -i 00 and {3 = 0 corresponding to a smooth contact between solids. For other values of Q and {3, corresponding to certain bond stiffness, the curves plotted in Figures 1-2, show enlargement or contraction of the existence's range.
The Stoneley wave velocities are represented on Figure 3 for different bond coefficients Q and {3.
WAVES IN MULTILAYERED MEDIUM
We now consider the problem of a stratified medium composed of a semi-infinite liquid L1, an elastic layer S2 and an elastic semi-infinite
Figure 2.
330
r ~ , II'
r
o .;.
o ,.;
o ...
- ~ - - -- -- .. -- -- ... -- ... . -_ .. -------o
o+-----L-~----~------_r------,_----__, 0.0 1. 0 2.0 3. 0 1 .0 5.0
'------_________________________ g = C~t I C~ t
Stoneley boundary curves . 1/2 = 113 = 1/3 . The shaded area corresponds to bond coefficients : a = 10, {3 = 5 .
u .... u'
Figure 3.
~
II: Q
S Q
III Q
I Q
III Q
a; Q
... ... Q
III d
Ii!
. . .
· · · · · ·
--- --- ---\------------~ ... , ~ .. ) ----- --.
.. .... -... ~ -------------£. _~~~~I_~:~ -----___ _ //-____ :::: :::::::::::::::::::::: ::~ ~ ~~~~':r:~:::::::::
Q+-------.-----~-------r------,_----__, 0. 0 1.0 2. 0 3.0 1.0 5 .0
...,/..,
Stoneley wave velocities for different bond coefficients with
1/2 = 1/3 = 1/3, C~t/c~t = 1.
(a=O,,B=O) : welded contact; (a=oo ,,B=O) : smooth contact .
medium S3. Imperfections of bonding at the S2-S3 interface are taken into account by application of the preceding model. The equation relating phase velocities and wave numbers is derived by using the wave functions that were introduced in the previous section and by adding the wave function for the liquid Ll. The boundary conditions are then:
- at the L1-S2 interface: smooth condition - at the S2-S3 interface: bond condition (3)
The study of the dispersion equation allows to show the variations of the liquid-solid interface wave, also called the Scholte-Stoneley wave, as a function of the boundary stiffness matrix elements through a and {3. These waves have real phase velocities: the motion in the liquid L1 decreases with decreasing z (z < 0) . As a consequence the phase velocity must be less than the velocity of liquid sound wave c, (cx<C,< C2 and C3t). In all cases the liquid wave velocity is an upper bound for the phase velocities of ScholteStoneley waves in the multilayered medium. Only one mode of propagation with real velocity does not exist.
In the limit where the layer thickness is infinite, the dispersion equation reduces to:
1) is the equation of the preceding section, which has no roots in the relevant range of phase velocity; S2 represents the Scholte-Stoneley equation of the L1-S2 interface:
331
332
uft ~ -u'
m ci
II ci
.... '" ci
HI ci
~ ci
C1i ci
I>l ci
&! ci
C;; ci
Ii! .;
0 .0 0 .2 0.1 0 .6 O.B 1.0
~ ci
" ( .. ~o, ~ •• ) ... .;
~' '' I r'o )
~ .;
l.l.A, ~.o,.)
~ .;
( • • .t., ~~t)
~ 1"'--- lot. S'I r,J-,·) u ... ... .;
~ .;
~ .;
0.0 1.0 2.0 3.0 1 .0 5.0 k2t.Md
Figure 4.a Layered water-steel-aluminum medium. Scholte-Stoneley wave phase velocities versus wave number in the case a variable, (J = O. (a=O,(J=O) : welded contact; (a=oo,(J=O) : smooth contact.
Figure 4.b Layered w ater-steel-aluminum medium. Scholte-Stoneley wave phase velocities when a and (J vary.
S! ~ o+-----~r_----_,------_.------_.----__.
0.0
Figure 5.
1.0 2. 0 3.0 i.O 5 . 0
Layered water-aluminum-steel medium. Scholte-Stoneley wave phase velocities .
which has a real root Cx < CI .
o
In the limit of a very thin layer, the expression giving phase velocities is somewhat more complicated depending strongly on the adhesive bond. If /3=0, the dispersion equation reduces to the Scholte-Stoneley equation of the L1-S3 interface. If /3 * 0 debonding effects appear between the layer S2 and the semi-infinite medium S3: for very thin layers the interface wave velocity then tends to zero with increasing coefficients O! and /3; In the limit where O! and /3 -I 00, corresponding to total debonding between solids, the liquid-solid interface wave is the antisymmetrical mode of a layer with liquid on one face. For the calculus, we have chosen two combinations of materials : in Figure 4 are represented phase velocities for the water-steel-aluminum medium, in Figure 5 for the water-aluminum-steel.
Materials p (g/cm 3) CI (km/s) Ct (km/s)
Aluminum 2.7 6.45 3.04
Steel 7.7 5.73 3.10
Water 1.0 1.48 -
333
CONCLUSION
The propagation of interface waves in layered media have been examined for a range of solid-solid interface adhesive bonds. Dispersion curves have been presented for various combinations of material properties of the medium. In the case of two semi-infinite elastic media in contact, the range of existence of Stoneley waves is investigated, and Stoneley velocity curves have been obtained.
In the case of a multilayered medium, the influence of the bond on the Scholte-Stoneley wave is shown. We conclude from this study that the bond defects at the solid-solid interface in the multilayered medium could be tested by inspection of this liquid-solid interface wave. In fact, we show that this wave exists for all values of the bond parameters (independently of the existence conditions of the Stoneley wave at the bonded interface).
REFERENCES
[1]
[2]
[3]
[4J
[5]
334
Achenbach, J.D., Dynamic interaction of a layer and a half space, J. Eng. Mec. Divis., EM 5., (1967). Jones J.P., Whittier J.S., Waves at a flexibly bonded interface, J. Appl. Mec., Transactions of the ASME., (1967). Pilarsky A., and Rose J.L., A transverse-wave ultrasonic oblique incidence technique for interfacial weakness detection in adhesive bonds, J. Appl. Phys., 63 (2), (1988). Schoenberg M., Elastic wave behavior across linear slip interfaces, J. Acoust. Soc. Am., 68 (5), (1980). Scholte J.G., The range of existence of Rayleigh and Stoneley waves, Mon. Not. R. Astr. Soc. Geophs., Suppl 5, (1947).
SOUND INTENSIFICATION IN FOAM
Goldfarb 1.1., Schreiber 1.R.
Institute of Northern Development Siberian Branch U.S.S.R. Academy of Sciences 625003, Tyumen, box 2774, U.S.S.R.
INTRODUCTION
The paper considers sound propagation in non-conservative foam, i.e. under the condition when liquid can flow out and be back into the foam. It is shown that while performance of this condition there is a regime of acoustic perturbation propagation strengthening in gas-liquid foam. It is investigated acoustic characteristics of the secondary flows mode.
The process of acoustic perturbation propagation in foam as one can imagine, is accompanied by bubbles' size changing, Gibbs channels [1]. The liquid flow along the system of chaotically oriented capillaries is one of the hydrodynamics effects influencing on the acoustic signal evolution.
Using the elementary relations of two-phase medium [2] quasi homogeneous model and varying the expression for the volume moisture content (11 one can get a relation, connecting the value Ll (11 with gas volume perturbation Ll V 2
and liquid Ll V 1:
(1)
For the model under review an assumption on the difference between Ll V 1
and the identical zero means the possibility for liquid to leave control volume, i.e. to flow out and to be back into the foam freely. In the language of multi phase mechanics media it means that there is double-velocity effect [3]. Such situation appears, for instance, in the problem connected with foam combustion, foam destruction, disturbance or break down of its structure. Following the generally accepted point of view one can examine a conservative change when liquid has no possibility to leak from foam freely. In this case one should put Ll V 1=0 because of liquid incompressibility and make all the following operations in accordance with the standard algorithm, worked out for the twophase media [2,3]. Such assumption, for instance, is used in [4], therefore in this statement we limit ourselves to analyze the signal intensifying effect in the non-conservative foam.
Let us assume that the bubble volume is varying adiabatically. In this case Ll V2/V2 = -LlP2!rP 2, where 'Y is an adiabatic exponent, P 2 is gas pressure in a bubble. The heat transfer effects in this statement are neglected. Their influence comes to the determination of the polytropic exponent value. Let us also assume that the liquid volume change takes place only with the channel section
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 335
change for the value b.S. After that b. V tlV 2 N b.S/S. Substituting these relations into the expression for foam density disturbance b.Pf, we get a distinctive equation of the foam condition - the connection between b.Pf and the disturbances of the channel section area b.E and gas pressure in a bubble b.P 2•
b.Pf = b.P 2 + P a a [1 - f!1] b.S C~ I I 2 PI S
(2)
where PhP2 are the liquid and gas density respectively, kg/m 3; C4 is the sound velocity in foam, depending on the gas phase compressibility, m/s.
The dependence of b.Pf on two variables in (2), but not on gas pressure, as in case of classic bubble mixture, is stipulated by the accepted assumption on a non-eonservativeness of the considered foam structure.
The description of the liquid motion along the Plateau-Gibbs channels is the basic part of carried out analysis. The choice of model is determined by the specificity of liquid flow: a channel has a section of Plateau's triangle form like; the channel walls represent the boundary of gas-liquid phase division and deform easily. Numerous experiments [1] have shown that a creeping flow is realizing in Plateau-Gibbs channels, therefore as zero approximation one can use an assumption on liquid adherence on the interphase boundary. In particular, it means that the equation of liquid motion along the channel should include the forth of friction at the boundary. Moreover the channel should be considered as a kind of capillary filled with viscous fluid and having a Plateau's triangle form like section. In assumption that liquid in a channel is II heavy II and that it is realized Poiseuille flow regime the motion equation can be written down in approximation of one-dimensional hydraulics in the following form:
au = _ 1..- OP l _ fhlliu + g Of PI Ox S
(3)
where u is an average liquid velocity in channel, m/s; III is the kinematic liquid viscosity, m2/S; PI is the liquid pressure in channel, Pa; g the acceleration due to gra~{ity, m/s2; /32 is a numerical coefficient.
Authors [5] have made a numerical integration of N avier-Stokes equation for liquid motion in the pipe with the section like Plateau's triangle and have gotten the value /32 = 49.1. We should remind you that for the pipe with the
equilateral triangle form section /32 = 20y'3, and for the round section /32 = 871".
The restriction on the range of acoustic perturbation frequencies, the propagation of which can be described by the suggested model, is fully determined analogically to [4].
The Plateau-Gibbs channels, in accordance with the polyhedron model [1], have cylindrical walls. The channel wall surface curvature brings to an excess of gas in a bubble P 2 over the liquid pressure in channel Pl. All the dissipation effects in this formulation can be neglected, as it is set an object to investigate the effect of signal strengthening in foam. The connection between the radius curvature of the channel's wall lie and its section is determined by
simple geometrical dependence lIe=/31.[S" where /312= y'3 - 71"/2 [1]. It should
be underlined that the channel surface curvature radius differs from the radius of foam bubble. The relation of these values is determined by the polyhedron model and foam moisture content al. Taking into account the above-stated one can consider that pressure in gas and liquid is connected by Laplace's law.
(4)
where (J is a coefficient of surface tension, Pa.m. In the capacity of the continuity equation one can use the relation
336
connecting the liquid velocity in channel with the channel section itself.
~ + g£Su) = 0 (5)
For completing a system of equations we shall use a standard linear motion equation [2,3J, but for the full pressure in foam we'll use the expression which is analogous to the relation for
(6)
It should be mentioned here, that the pressure in foam is really determined generally by the gas pressure in bubbles, just as the pressure in the bubble suspension is generally determined by the pressure in liquid.
Let us examine the propagation in the linear approximation of the plane monochromatizational wave, i.e. let us seek the solution in the form Yi=Yioexp(iwt-ikx). Substituting such unknowns into the equations system, describing the suggested model, and setting the obtained determinant equal to zero, we obtain a dispersion equation in the following form:
w* =-- (7)
The analysis of the dispersion relation (7) should be started with the case, when the gravity is absent. In this case the equation (7) becomes biquadratic and assumes relatively simple analytical solution. It has two pairs of roots corresponding to two modes of acoustic oscillations: the acoustic mode and the secondary flows mode conditioned by the liquid motion along the Plateau-Gibbs channels. the existence of roots' pairs, as usually, means an equivalence of perturbation propagation in different sides (the absence of the marked direction). In the considered frequency diapason an acoustic branch practically doesn't feel dispersion the wave propagation velocity is determined by the value of C4; the other branch - the secondary flows mode increases its velocity
proportionally to rw in the sphere of the low frequency; the imaginary part of the secondary flows mode wave vector describes the attenuation of these motions, and it is a decrement in this field of frequencies (where w ~ 0
S'm(k(w)) N .jww* for the secondary flows mode). An acoustic mode tests the intensification at the expense of the energy transfer from the liquid motions and external medium motions (there is a change by mass and energy with the external medium in the non--conservative medium); an appropriate part of the wave vector is an increment in the considered field of frequencies (where w ~ 0, ~m(k(w) N w2/w*).
A solution of the full equation (7) with gravity consideration doesn't make changes in the dependencies of phase velocity and intensification increment for the acoustic mode, but brings to the splitting of the slow mode into two unequivalent waves. It is connected with the appearance of the marked direction - a vector of acceleration due to gravity in the problem. There is a dependence in Figure 1 of the phase velocity on the frequency for the secondary flows mode in the low frequency field. With the increase of the roots of equation (7) merge, that testifies to the fall of gravitation all influence on the perturbation propagation with the growth of main frequencies. It can be confirmed by the analytical investigation of the wave number k odd level influence on the solution (6). The analysis also shows that while w ~ 0, one of the roots (7) relating to the secondary flows mode, tends to a constant value, defined by the liquid viscosity 111 by the initial channel section S, and by the acceleration
337
1.02 (,,)(S-11
Figure 1. Phase velocity dependence on frequency for the secondary flows mode with gravity consideration. Numeral designation: 1 - is the brunch corresponding to kinematic wave of liquid leakage; 2 - another brunch of secondary flows mode.
due to gravity. This value corresponds to the motion velocity of the liquid leakage kinematical wave, which is in the gravity field. An equation describing this wave can be obtained in the following way. As all the characteristic signal propagation velocities in foam are considerably less than the value of C4, one can pass to the limit C4 --I 00 in the foam condition equation. Varying also the Laplace's law (4) and assuming with the same range of accuracy APr-'O, we get a connection:
(J 1
2{h .[S" AS
S (8)
The motion equation for liquid in channel one can somewhat simplify, taking into account a low frequency range under study. In this case without any loss of accuracy one can set of fJt = 0 and use the equation (3) in the following form:
u = ~ [<g> - <g~l>] {hI'-
(9)
where French quotes mean an averaging on the liquid motion direction in the Plateau-Gibbs channels system.
Substituting the relations (8) and (9) after the averaging into the equation of the continuity for the liquid motion in channel (5), we get the following expression:
(10) X -~ 1 -
3{hv
The equation (10) describes a microscale liquid motion in foam under the influence of capillary and gravity forces. It has a solition - like solution, extending with the constant velocity A;lS0. The formation and motion of kinematic wave are considered in detail in [6].
Let us return to the dispersion relation [7]. Analyzing it at the full diapason of frequency 0 < W < 00, one can make a conclusion that the charac-
338
ter of the solution will be principally changed in the frequencies field w N W*. It follows from that the sum 1 + iw* / w while w« w* is practically pure imaginary (the real part is much less than the imaginary one),but while w »W* with the same degree of accuracy it is valid, i.e. on the boundary of w N w* a considerable change of its behavior is taken place. The other coefficients (7), real and imaginary, don't play such role, as they are only growing with the increasing of w, without changing of the growth character. Physically it is consequence of the restriction, imposed by the taken assumption on the liquid motion regime. Reall;:, from the boundary frequency determined in [4], characteristic value w* in (7) differs by the multiplier (32 which is a geometrical factor and is determined by the shape of the channel section.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
J.J. Bikerman, Foams, Berlin-Heidelberg-N ew-York. Springer-Verlag, 337 p., 1973. V.E. Nakoryakov, B.G. Pokusaev, I.R. Schreiber, Waves propagation in gas- and vapour - liquid media. Novosibirsk, ITPh Siberian Branch USSR Academy of Siences, 238 p., 1983. R.I. Nigmatulin, Fundamentals of heterogeneous media. Moscow, Nauka, 336 p., 1978. 1.1. Goldfarb, I.R. Schreiber, F.I. Vafina, The influence of heat transfer and liquid flow on sound propagation in foam, Proceedings of Physical Acoustics: Fundamental and Applied, Kortrijk Belgium, 1990. R.A. Leonard, R. Lemlich, Laminar longitudinal flow between close packed cylinders, Chern. Engng. Sci., v.20, N 8, p. 790-795, 1965. 1.1. Goldfarb, K.B. Kann, I.R. Schreiber, Liquid flow in foam, Izvestiya AN SSSR, ser. MZhG, n 2, p. 103-108, 1988.
339
THE INFLUENCE OF HEAT TRANSFER AND LIQUID FLOW ON SOUND
PROPAGATION IN FOAM
1.1. Goldfarb, I.R. Schreiber, F.1. Vafina
Institute of Northern Development Siberian Branch U.S.S.R. Academy of Sciences 625003, Tyumen, box 2774, U.S.S.R.
INTRODUCTION
The paper considers the heat transfer and liquid motion influence on the propagation of acoustic disturbances in gas-liquid foam. It has been shown that the attenuation decrement consists of two parts: Thermal and viscous, which relative intensity is defined by foam dispersion ao (size of foam cell). When
ao < a~, the signal dissipation, to a great degree is caused by liquid motion, it
is possible to neglect the heat transfer. When ao > a~ the thermal processes define attenuation and one can ignore the liquid motion.
Gas-liquid foams represent a variety of two-phase disperse system differing from wide-spread and relatively well studied gas-liquid media by a hi~h void fraction close to unity and the presence of some quasi ordered structure [1]. The structure of foam is defined to a large extent by the ratio of gas and liquid phase volumes, i.e. volumetric moisture content Cl!. To describe various physico-ehemical properties of the foam in which Cl! does not exceed 0.05, there exists a generally accepted polyhedral model of the foam structure [1], according to which an elementary foam cell represents an irregular polyhedron. It follows from experimental Plateau's data [1,2] and general physical considerations that dodecahedron is the nearest correct approximation to the real foam cell. Three films forming equal angles of 21(/3 between each other converge in each rib of polyhedron. Thickenings which are called Plateau-Gibbs channels form at the joint of these films. Externally such channel resembles a straight cylinder having Plateau's triangle as guideway (plane figure bounded by three pairwise tangential circumferences of the same radius). The peculiarity of polyhedral foams is that practically the entire liquid is contained in the ribs of polyhedron, i.e. Plateau-Gibbs channels. One should also note that according to the same experimental Plateau's data [1] four Plateau-Gibbs channels converge at the vertices of polyhedron, i.e. foam nodes, three of which are the ribs of this polyhedron but the fourth one is oriented normally to the surface of elementary foam cell under consideration, if such expression is acceptable as applied to polyhedron.
The propagation of small perturbations in foam as in ordinary two-phase medium [2,3] is accompanied by various physical phenomena, viscous effects at the interface, effects of added mass, effect of the surface tension mechanism, effect of interphase heat transfer processes. Besides the presence of pseudostructure in foam results in the appearance of one more hydrodynamic effect affec-
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 341
ting essentially the evolution of initial signal, i.e. the liquid motion in the set of chaotically oriented microcapillaries, i.e. Plateau-Gibbs channels. This motion is caused by changing the sizes of gas bubbles while imposing the external pressure perturbation. The present paper shows that in the domain of the values of foam parameters, which is of interest, the mechanism of thermal dissipation and liquid flow Plateau-Gibbs channels produces a determining effect on the propagation of low-frequency sound.
The problem statement is reduced to the formulation of interphase heat transfer conditions and deriving the equations of liquid motion through the channels. We shall make a number of general physical assumptions to specify each problem separately. We consider the propagation of perturbations in the approximation of linear acoustics. We shall restrict ourselves to the consideration of only such foam motion, at which no destruction of the foam structure occurs. We shall neglect the effect of gravity force. We shall consider that liquid is incompressible but the homobaric condition is valid for gas in a bubble [2,3].
In the general form the problem of heat transfer between carrying and disperse phase is formulated, in particular, in [3] on the basis of cellular approximation of two-phase medium. In [4] the general statement is specified as applied to foam structure. We shall use the statement of heat transfer problem preferred in [4].
In the general case the system of equations describing the foam motion as a whole should be constructed taking into account the abnormal viscosity of foam structure [1]. Let us assume that the foam volume under study is in a cylindrical vessel of radius Rf, then in the frame of homogeneous approximation [2,3] the motion equation can be written in the following
aw _ _ l ap f + _2_ . T at - PfaX PfRf f
(1)
where w - is foam velocity, m/s; Tf - is shear stress at the boundary of the vessel under study, Pa; P - pressure, Pa; P - density, kg/m3; here and further subscript f is related to foam as a whole.
The shear stress Tf depends essentially on the velocity profile in the foam volume. Due to inertia that take place certain time is required for the profile formation, which is not allowed sometimes to be ignored. The criterion of such allowance is the relation between the vessel radius Rf and the depth Or of the penetration of viscous boundary layer which is determined as
Or = ..; 2vr/ w ,where w - is the characteristic frequency of imposed perturbation and v - is kinematic viscosity [5]. It is evident that
w » w* = ..; 2Vf/R~ the thickness of viscous sublayer amounts a small portion
of the vessel radius but since the value w~ is rather small (one can
approximation [1] Vf IV 10 VI for estimates) for Tf one can use expression for unsteady shear which is obtained following Landau [5] converting the corresponding integral relation
t
- j ~ f aw ----=;::a~T== Tf - Pf 7r tfi r o ..; t - T
It is expedient to write the discontinuity equation in the general form
use
the by
(2)
(3)
After linearization and elementary substitutions the set of Eqs.(I)-(3) can be reduced to the relation of the following form
342
~ = 82Pr _ 1..- § ft 82 pr dr 8t 2 8x 2 Rr ~ 1f -00 8r2 .j t - r
(4)
Since the transversal size of the vessel, in which foam is contained, is present in convolution in (4), this equation can be used for experimental processing of data on the propagation of acoustic perturbations in foam. With more general approach one can be abstracted from the specific experimental setup, put Rr --! ID and use only the part (4) representing the standard linear equation of motion as the motion equation
Pftt = P fxx (5)
For closing the complete system of equations it is necessary to obtain one more relation. One can readily understand that it should relate pressure in gas and liquid and radius of the gas bubble, i.e. play the role of the analogue of Rayleigh equation for foam. To derive it rather correctly, we shall analyze the liquid motion near the gas bubble in real foam. All four Plateau-Gibbs channels converging at each vertex of the cell under consideration can be divided as follows: three of them formed by ribs of the given polyhedron are lI own", one belonging to neighbouring polyhedrons is IIforeign li . While chan~'ng the average size of the cell (specifically while decreasing external pressure liquid as if is pushed in a II foreign II channel form the centre of fixed gas bu ble whereas the lIownli channels with liquid contained in them play the role of added mass for this cell. Hence, the dynamics of liquid motion in a IIforeign li channel is actually defined by the dependence of bubble radius on time and freedom of liquid motion in the foam. Hydroconductivity Kf i.e. the function of physical parameters of the foam (dispersion ao and void composition Ill) is the mathematical equivalent of the last notion [6).
Proceedings from the above we shall write a system of equations for liquid phase of the form
8(r 21lIu) _ 0 ar -
~ = - PI1 81rl + VllllKuf - Vl[ r\ ~ [ r2~ ] - ;~ ] (6)
where u is the velocity of liquid motion, m/s; index 1 corresponds here and further to liquid; r - microcoordinate, m; the discontinuity equation is written in the general form [2,3); the equation of momentum transfer is chosen in Brinkman's form that is peculiar superposition of Darcy and N avier-Stokes equation [7], in this case the term with Kf is responsible for dissipation when IIpushingli the liquid in IIforeignli channels, but the classical viscous term is responsible for ordinary losses [2,3).
Note the fact that writing the motion equation in the form of Brinkman's equation imposes restrictions on the range of frequencies of acoustic perturbations which can be described by the suggested model. The point is that the presence of friction in (6) which is linear in terms of velocity means the presence of Poiseuille flow regime in the channel. This is, in its turn, testifies to the fact that the viscous sublayer in Plateau-Gibbs channel increases and covers the entire volume of the channel, i.e. the sublayer thickness 01 is larger than
the characteristic cross size of the channel which can be denoted as .jS (s is the channel section). Thus the form (6) implies the quasistationary liquid flow in the channel, restricting from above the frequency of the model applicability
~ S
(7)
343
The system of equations (6) should be integrated with respect to r from ao to ao+b o under the conditions on the bubble surface
u = a (8)
where a is the radius of foam gas bubble in cell model, [3,4], m; b - liquid layer thickness surrounding gas bubble, m; index 2 here and further corresponds to gas; index 0 corresponds to unperturbed state. After integrating (6) under the conditions (8) we shall obtain the following equation
(9)
which is the analogue of Rayleigh equation for foam in the frame of the suggested model.
And finally the last equation relating pressure P I in liquid, gas pressure P2 in a bubble and total pressure Pf in a foam is taken in the form suggested in [3] and resembles the expression for density of two-phase mixture.
(10)
We shall seek the solution of the given system in the form of travelling plane monochromatic wave.
The dispersion relation characterizing the suggested model represents a complex transcendental expression with a large number of functions from complex variables and is of the form
I1 2(x) = 3(x·cthx - 1)x-2 (11)
F[ F[ YI -lWT1 Y2 = -1W- ; T2
xI x 2 1 -I a 20P 2 0 WTI WT2 EI = 1 +~ E2
B2 a 2 'fIlal al~p I + a2OP2O 0 0
PIC I Al A2 'fI1 =
P20 C P2 , xI =
C IPI x 2 =
P20 C P2
In this case the relations for temperature distribution III a cell coincide with formulas obtained in [4].
344
We shall choose the function Kf = (atO,ao) in the form of KozeniKarman relation [3] with numerical coefficient given in [6]
(12)
While analyzing the dispersion relation (11) it may be shown that in the frame of the suggested model we may neglect the influence of surface tension, effect of added mass and viscous deformable boundary. The main influence on signal evolution in foam is exerted by heat transfer and liquid motion along
Plateau-Gibbs channels. One can determine also the value of critical radius ab, at which the dominant mechanism of sound wave attenuation changes. Hence at
ao < ab the viscous mechanism of sound dissipation will predominate
essentially, but at ao > ab the mechanism due to the processes of interphase
heat transfer will prevail. The value of ab is defined by
Under normal condition and usual parameters of the foam (at = 10-2) the
value of ab amounts approximately 5.10-4 m that is in the centre of the spectrum of applied dispersion values.
Figure 1 shows the numerical modeling results. The analysis of the curves proves the fact that in foam with dispersion ao = 10-4 m the attenuation due to liquid motion essentially prevails on heat transfer mechanism.
Figure 1.
t A i A
0.2
t t A A
0.6
t
Signal evolution in foam with parameters: at=10-2j 1,2 - ao=10-3 mj 3,4,5 - ao=10-4 m. 1,3 - initial profiles; 2 - on the distance x = 6 m taking into account only heat transferj 5 - on the distance x = 0,2 m taking into account heat transfer, liquid motion.
345
REFERENCES
[1)
[2)
[3)
[4)
[~l [7)
346
J.J. Bikerman, Foams, Berlin-Heidelberg-New-York. Springer-Verlag., 337 p., 1973. V.E. Nakoryakov, B.G. Pokusaev, T.R. Schreiber, Wave propagation in gas and liquid-vapour media, Novosibirsk, ITPh Siberian Branch USSR Academy of Sciences, 238 p., 1983. R.T. Nigmatulin, Fundamentals of heterogeneous media, M. N auka, 336 p., 1978. V.Sh. Shagapov, Effect of heat transfer processes between phases on propagation of small perturbations in foam, TVT, 23, N 1, p. 126-132, 1985. L.D. Landau, E.M. Lifshitz, Hydrodynamics, M. Nauka, 1986. K.B. Kann, Capillary hydrodynamics of foams, Novosibirsk, Nauka, Siberian Branch, 167 p., 1989. J.S. Slettery, Theory of momentum. Energy and mass transfer in continuum, M. Energiya, 446 p., 1978.
THE PECULIARITY OF NON-LINEAR WAVES EVOLUTION IN BUBBLY
LIQUIDS
A. Gubaidullin
Department of Multiphase Systems Mechanics Siberian Branch U.S.S.R. Academy of Sciences Tayrnirskaya 74 Tyumen 625026 U.S.S.R.
INTRODUCTION
The theory of bubble liquids wave movements is based on a number of mathematical models differing by the degree of detailing the description of the passing processes of power, mass and heat interaction of disperse bubbles and surrounding liquid from balance relationship at the jump (Campbell and Pitcher, 1958) to the model taking into account non-coincidence of phasal pressures and temperatures (Gubaidullin, Ivandaev and Nigmatulin, 1976).
BASIC EQUATIONS OF MOTION FOR A BUBBLY LIQUID
In the present work the results of the research carried out on the basis of one - velocity two - temperature scheme with two pressures as well as schemes of viscous - elastic and ideal compressible liquids are presented. The first of the mentioned schemes has this form:
QE.'2 a3 = 0 at '
dU2 a2P2 + P2(IT = P'2 nq
PI = alP'l'
al + a2 = 1,
a2 !1ra3n 3 '
P2 = p'2 R 2T 2,
P dv + 8p - 0 at Ox- (i = 1,2)
da at = w
P2 = a2P'2,
p = alPl +
\01 = l.l ai/3 ,
U2 Cv2T 2,
Prp-2};/a PI
P =
a2(P2 - 2};/a)
\02 =
PI + P2
1.47aih
P'l = P'10)
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
(1)
T I = T lo
347
Nu = {10 , Pe ~ 100 /Pe, Pe > 100
Here and henceforth p, p, v are, respectively, the mean density, the reduced pressure, and the velocity of the mixture; T i, pi, Pi, Pi ai are the tempera-
ture, the true and the mean densities, the pressure, and the volume content of the i-th phase; the subscripts 1 and 2 refer, respectively, to parameters of liquid and gas; a, ware the radius and the radial velocity of the bubbles; \91>2 are corrections on the gas content, VI, ~ are the coefficients of viscosity and the surface tension; CY 2, Cp 2, " a2,'\2 are the specific heat capacities at constant volume and pressure, respectively, the adiabatic exponent, and the coefficients of the thermal diffusivity and the thermal conductivity of the gas; the initial values of parameters are indicated with subscript O.
When the model presented is computerized, one should go to Lagrangian coordinates r and perform some manipulations, see for example Gubaidullin et.al. (1976).
SHOCK-WAVE EVOLUTION DUE TO INTERPHASE HEAT TRANSFER IN A BUBBLY LIQUID
Here we produce results of numerical calculations and analysis of nonstationary wave processes in liquids with low viscosity (water or 50% waterglycerine solution) containing gas bubbles (air, carbon dioxide, helium) with a characteristic size of the order of 1 mm. When performing numerical simulation, we used set (1).
The calculations have shown that the structure of sufficiently weak waves (Pe/Po < ,) develops from oscillatory to monotonic one in bubbly mixtures
with low viscosity of liquid (Vi = /1-1 / ao C* pOl) « I, C* = (PO/pol/h , such
as in air-water mixtures with a bubbly size of the order of 1 mm (Po ~ 0.1 MPa). Structure of more strong waves (Pe/Po > ,) tends in their evolution to the limiting oscillatory configuration. Growth in the number of oscillations is the distinctive feature of the wave evolution. The latter is due, in the mixtures indicated, to effects of interphase heat transfer and transport of kinetic energy of radial motion to the neighbour volumes of the mixture at the expense of pressure disturbance (and not due to effects of viscosity at a relative phase motion). By way of example let us consider pressure profiles in Figure 1 at different instants after increasing pressure at x = 0 from Po to Pe and then remaining it unchanged.
The wave of moderate intensity is seen to have a pronounced oscillatory structure with characteristic peaks of oscillations during compression. Because of thermal dissipation the oscillation amplitude decreases as the wave propagates, while the number of waves increases.
Analysis of thermal dissipation has shown that adiabatic and isothermal modes of behavior of gas in the bubbles are not the limiting modes. The dissipation maximum is observed at a finite (nonzero) value of Nu. Since
realized values of Nu are functions of thermal diffusivity v~T) of a gas, evolution of a wave structure and attenuation of short pulses in a bubbly mixture must depend on a sort of gas in the bubbles, to be exact, on the
thermal diffusivity v~T) of the gas. In the context of this we compare three gases, namely, carbon dioxide, air and helium. At Po = 0.1 MPa and
T = 293 K, v~T) for helium is practically by an order of magnitude greater than that for air and almost 20 times more than that for CO 2. Therefore, attenuation of oscillations in the mixture containing helium bubbles and
348
Figure 1.
-/ '-,
i ..... ,.... i
-.-._. i - '- '-- ---i , , --/
[.ill --1 2. 3
'UUW - -J , ~
5
1 o 0.5 <,M
Shock wave evolution in water with air bubbles (Po = 0.1 MPa, To = 293 K, ao = 1 mm, a20 = 0.Ql) . Figures I, 2, 3 near the curves correspond to time (in ms).
formation of a monotonic wave configuration, all other factors being equal, must occur faster than in the mixture with bubbles of air and CO 2• This conclusion is confirmed by the results of calculation and their comparison with experiments. By way of example, we can consider calculated and experimental oscillograms presented in the same scale in Figure 2 for the pressure in shock waves propagating in the water-glycerine mixture with bubbles of air, helium, and carbon dioxide.
The oscillograms were measured at a fixed depth x = 1.6,m. It can be easily seen that a sort of gas influences significantly the structure of a shock wave in a bubbly mixture. In particular, shock waves in mixtures with bubbles of carbon dioxide and air are oscillatory, while in mixtures with helium bubbles they are monotonic. The calculated oscillograms are in good agreement with the experimental ones.
Paradoxicalness of wave properties of bubbly mixtures with inert gas is that variations, over a wide range, in such thermophysical properties of a liquid
Figure 2.
p~~;a~ cO2
0.10 o Z 4/;,",5
:.:~a tJm~ lie
aID ~ o 2 4
f,m.r
Predicted and experimental pressure oscillograms in shock waves with the fixed intensity (Pe = 1.18) in 50% water - glycerine mixture (Po = 0.118 MPa) with a bubble radius ao = 1 mm at void fraction a20 = 0.95% for different gases in the bubbles, carbon dioxide, nitrogen (air), and helium. (Experiment by Kuznetsov et al. (1977)).
349
phase (occupying almost the whole volume in the mixture) as thermal conductivity and viscosity do not practically influence the propagation of the waves, their structure and attenuation. At the same time, the variation in gas properties in a bubble (adiabatic exponent, thermal diffusivity) occupyin~ very small volume in the mixture (not to mention its negligible mass fraction) has significant effect on wave processes and their attenuation.
NUMERICAL SIMULATION OF SHOCK WAVES IN HIGH-VISCOSITY BUBBLY LIQUIDS
The objective of the study was to determine the specific features of shock wave propagation and attenuation when a carrier liquid is highly viscous, that is the liquid viscosity and the wave parameters are such that we can neglect the local deformation inertia of the small-scale motion of the liquid around bubbles. The main dissipative mechanism in this case is friction losses during bubble compression.
The numerical simulation was performed using the viscoelastic and viscous schemes for describing the motion of a bubbly liquid. To determine the contribution of appropriate nonequilibrium effects and the range of applicability of the above models we used the two-temperature scheme and the ideal compressible liquid scheme.
The equations of motion of a bubbly mixture have the following form:
ttL + flEi = 0 at ox: ' dv + op - 0 Pat Ox-
In the ideal liquid scheme they are closed by the equation of state:
p _ a20
Po - PO!P-alO
The closing relations in the viscous liquid scheme have the form:
da a (0 ) at = 4all1J P2 - P , 2);
P'2 = P2 - a
The carrier liquid in the viscoelastic liquid scheme is assumed to be compressible, the equation
is used instead of the condition pOl = pOlO'
It should be noted that the following estimate for the parameter ranges hold when the above schemes are used:
1
The ideal liquid scheme if (P/poIP·a'pol/jjl «1, r» jj!/p
(r is the characteristic duration of pulse disturbance). 1
The viscous liquid scheme if (p/poJ'i.a.pol/jjl« 1.
The carrier liquid in all the schemes can be considered incompressible if
D2 :: -p- « q. a2· P
The evolution and attenuation of strong pulse disturbances were studied depending on the intensity, duration, and shape of a wave, void fraction and
350
bubble size, initial medium pressure, and the density and viscosity of the carrier liquid. A pulse of higher intensity but less duration is shown to attenuate faster, other conditions being equal. A pulse attenuates more intensely and at shorter distances with increasing initial void friction, which is due mainly to increasing nonlinearity of the medium. Besides, in this case the wave velocity drops. The triangle disturbance with a steep leading edge attenuates faster than a sinusoidal wave with the same initial pulse since the latter only starts attenuating after its leading edge becomes steep enough. A rise in the initial pressure of the bubbly mixture (or a diminishing of the carrier liquid density) entails an increase in the wave velocity leaving the medium nonlinearity unchanged. Therefore a wave of the same dimensionless intensity attenuates at correspondingly larger distances. A decrease in bubble size affects but slightly the attenuation of pulse disturbances but leads to a small rise in the shock wave velocity since the bubbly medium compression diagram changes. As the viscosity of the carrier liquid increases, the width of blurring in shock transition grows too.
The presence of a high-viscosity carrier liquid brings about two different pressures to be observed in the bubbly system, viz. the liquid pressure and that of gas in bubbles, and this in turn gives rise to nonequilibrium effects in wave processes under consideration. These effects lead to a more intense wave attenuation and its leading edge is strongly blurred since dispersion and dissipation effects prevail over nonlinearity.
The simulation data were compared with other author's experimental results, which demonstrated an adequacy of the theory developed.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
L.1. Campbell, and A.S. Pitsher, Shock waves in a liquid; Shock waves in a liquid containing gas bubbles, Proc. Roy. Soc. London, A 234, N 1235:534, (1958). A.A. Gubaidullin, A.!. Ivandaev and R.I. Nigmatulin, Nonstationary waves in a liquid with gas bubbles, Soviet Phys. Dokl. 21, N 2:68, (1976). A.A. Gubaidullin, Pulse Disturbance attenuation in bubbly liquid, Fluid Mechanics, Soviet ResearchjScriptatechnica, 14, N. 3:7, (1985). R.1. Nigmatulin, "Dynamics of Multiphase Media", Hemisphere Publishing Corp., Washington (1990). A.A. Gubaidullin and R.I. Nigmatulin, Amplification of shock waves in Nonequilibrium gas-liquid systems, in: "Research Reports in Physics. Nonlinear waves in Active Media", J. Engelbrecht, ed., Springer-Verlag, Berlin, Heidel berg (1989). V.V. Kuznetsov, V.E. Nakoryakov, B.G. Pokusaev and LR. Shreiber, Experimental study of disturbance propagation in a liquid with gas bubbles, in: II Nonlinear Wave Processes in Two-Phase Fluids", S. S. Kutateladze, ed., Institute of Thermophysics, Novosibirsk (1977).
351
BOUNDARY ELEMENT METHOD ANALYSIS OF SURFACE ACOUSTIC
WAVE DEVICES
Ken-ya Hashimoto and Masatsune Yamaguchi
Department of Electrical and Electronics Engineering Chiba University Chiba-shi, Japan 260
INTRODUCTION
The paper describes boundary element method (BEM) analysis of surface acoustic wave (SAW) devices, and aims at developing an effective SAW device simulation technique [11. Precise device simulation, which can take account of various second-order effects, would be one of the most vital techniques for the design and development of sophisticated SAW devices [2].
When applying the BEM analysis to SAW device simulation, one should be most careful of choosing expansion functions in order that the charge distribution on the electrodes of an interdigital transducer (IDT) is properly and rapidly approximated. Commonly employed polynomial and piece-wise functions do not seem suitable for the approximation of the actual charge distribution: the charge concentrates and diverges at the edges of IDT electrodes. Unless proper expansion functions are employed, BEM-solutions become extremely illconvergent.
From this point of view, we have attempted to employ the expansion functions proposed by Skeie and R0nnekleiv for the analysis of static charge distribution on IDT electrodes [3]. The functions are simple but most effective in approximating the actual charge distribution: in particular, they are effective in taking account of the effect of the charge concentration at the edges of IDT electrodes.
First, IDT static capacitances and their dependence upon the electrode width relative to the electrode periodicity were calculated. In the calculation, the problem was assumed to be purely static. The result showed that the solution rapidly converges with an unexpectedly small number of variables and with considerable accuracy.
Secondly, by taking account of the effect of SAW radiation, the input admittance was analyzed for a floating electrode unidirectional transducer (FEUDT) [4] with 40 electrode-pairs. The result was in good agreement with experimental results.
The result allows conclusion that BEM employing the expansion functions proposed by Skeie and R0nnekleiv could be used as an effective tool for precise SA W device simulation.
BOUNDARY ELEMENT METHOD ANALYSIS
Figure 1 shows the space coordinate system used in the analysis where acoustic waves propagate on a plane surface of a semi-infinite piezoelectric
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 353
substrate. For simplicity, the resistivity of IDT electrodes and field variation along the Y -direction are ignored.
By using the Green function G(X), the electrical potential ¢(X) and charge q(X) on the surface are related to each other by [5],
+00
¢(X) f G(X-X')q(X')dX'. (1)
In Eq.(l), G(X) is defined by the effective permittivity E(S) [5],
+00
G(X) = (211")-1 f exp(-jf3X)/I,B1 E(,B/w)d,B. (2)
The charge qm(X) on the m-th electrode of width w is expressed by a series of expansion functions fn(X), i.e.,
(3)
where Amn are unknown weighting coefficients and Xm denotes the center of the m-the electrode. The charge q(X) in a gap between two adjacent electrodes is zero.
Substituting Eq.(3) into Eq.(l), one obtains,
M xm+w/2 N
¢(X) = m~1 f G(X-X' )n~lAmnfn(X'-Xm)dX' Xrn-w/2
In Eq. (4), M is the total number of IDT electrodes, and
+1
FrJX) = f G(X-wt/2)fn(wt/2)dt. -1
(5)
In Eq.(4), ¢(X) on each electrode is constant, and is given by the IDTapplied voltage. So, by sampling ¢(X) at M· N discrete points (conventionally N points on each electrode), Eq.(4) is rewritten in the form of simultaneous linear equations determining Arnn.
Some types of IDTs often employ floating electrodes [4], where ¢(X) is unknown. However, this unknown ¢(X) is simultaneously determined by ta-
354
IDT
+ - + - + -------I I w
p
air
substrate
Figure 1. Coordinate system used in the analysis.
Table 1. Co for single-- and split-type IDTs and FEUDT
Type of Transducer M=20 M=40 M->oo
Single-type IDT 0.97522 0.98765 1 6
Split-type IDT 1.42138 1.41812 1.414 7
FEUDT 0.60917 0.61632 0.625 4
king into account the additional requirement for Eq.(4) that the total charge on the floating electrode is zero.
Milsom et al. showed [5] that G(X) is expressed as the sum of contribu-tions from three terms associated with electrostatic coupling Ge(X), SA W radiation Gs(X) and bulk acoustic wave (BAW) radiation Gb(X).
Both Ge(X) and Gs(X) are given in an analytical form:
-l/7rt(O)"logIXI, (6)
and
(7)
In Eq.(7), a=k 2/2t(w), k2 is the electromechanical coupling factor for SAW, and (Jf is the SAW propagation constant on a free surface.
In general, Gb(X) is not analytical and should be evaluated numerically. However, the substrates such as 128' YX-LiNb0 3, X-112' Y-LiTa03, etc. are weakly coupled with BA Ws, and the effect of Gb(X) upon Gs(X) is negligible. In the paper, therefore, G(X) is assumed to be the sum of Ge(X) and Gs(X).
In order to reduce computational time, fn(X) should be properly chosen so that the solution may rapidly converge with N; (i) fn(X) should be suitable for the approximation of the actual charge distribution, automatically taking account of the effect of the charge concentration at the edges of IDT electrodes, and (ii) F n(X) should be either analytical or fast to calculate.
The following expansion functions have previously been proposed by Skeie and R0nnekleiv [3],
(8)
where
Y = X/(w/2) (-w/2 ~ X ~ +w/2). (9)
The appendix gives F n(X), where the contribution of Gb(X) is neglected. By using Eq.(8), the total charge on the m-th electrode, and the IDT
capacitance and input admittance are obtained. For comparison in the following numerical calculation, the IDT capacitance
is also estimated by the conventional BEM, where qm(X) (see Eq.(3)) is assumed to be
L
qm(X) = E Amn[U{X-w/L*(n-l-L/2)}-U{X-w/L*(n-L/2)}] (-w/2~X~+w/2) (10) n = 1
In the Eq.(10), U(X) is the step function.
355
Figure 2.
~ w '-
10
9.5
u 9.0
8.5
~ N=3 (Present method)
'. • • • •
• T o 1710 l)'S 3ho 2)'S 112
1/L
C j E(ro) calculated by the conventional BEM (.) and by the present method (0).
RESULT OF NUMERICAL ANALYSIS
First, in order to show how the present method using the expansion functions· in Eq.(8) is effective, the static capacitance C of a single-type IDT was calculated and compared with the result obtained by the conventional BEM (see Eq.(10)). In the calculation, the contribution of Gs(X) to G(X) was neglected to make the problem static, and N was set at 3.
Figure 2 shows Cj E(ro) for a single-type IDT consisting of 10 electrodepairs (M=20), where the electrode width to periodicity ratio w jp is 0.25 (see Figure 1). The result obtained by the conventional BEM is shown as a function of L-1 (L is the number of sampling points on each electrode in Eq.(10)), while the result by the present method with an open-circle on the Y-axis. As can be seen, with an increase in L (with a decrease in L-1), Cj E(ro) calculated by the conventional BEM converges towards the value obtained by the present method. However, the convergence is rather poor even when L is increased up to 20. Note that the conventional BEM should solve a set of simultaneous linear equations with M·L (=20·L) unknowns, whereas the total number of unknowns is only M· N (=60· N) in the present method. This implies that the present method is very effective in reducing both the computational time and required memory size.
Table 1 shows the capacitance electrode-pair, Co ( = Cj f(ro)jMj2 ), for single- and split-type IDTs and a floating electrode unidirectional transducer (FEUDT) [4], where M = 20 or M = 40. In the table, Co estimated for an infinitely large M [4,6,7) is also shown, which is a little different from the present results. Further analysis suggested that the difference is mainly caused by the end effect.
Figure 3.
356
1
10 -1
!-< 10- 2 0 !-< !-< <1l
10-3
-4 10 0.0 0.1 0.2 0.3 0.4 O.S
w/p
Relative error between Qrn on the center electrode and its numerical value.
Figure 4.
30
~ 20
C1J 10 fJ ....
. ~ 0 ¥:=::"~~::::L._-\.~= ___ """",<:
~
frequency, MHz
Input admittance of FEUDT with 40 electrode-pairs. Calculated: Experimental: - - - - - - - - Ref.4
The dependence of the total charge Qm on each electrode upon w /p was calculated for a single-type IDT with 10 electrode-pairs. Figure 3 shows the relative error between Qm on the center electrode and its analytical value [6]: on the center electrode, the end-effect is expected to be negligible. The error remains within a few percent for w /p of less than about 0.45. This suggests that the present method could also be applied to electrode-width weighted and narrow-gap IDTs, because Qm is proportional to Co. However, N may have to be increased for further accuracy.
Figure 4 shows the input admittance of an FEUDT on 128'YX-LiNbO a. Since the FEUDT consists of 40 (active) electrode-pairs (M=80) 120 floatin~ electrodes, the total number of variables becomes 600. In the calculation, G(X) is assumed to be the sum of Ge(X) and Gs(X). The result is in very good agreement with experimental data, which suggests that the proposed BEM is effectively applied to SAW device simulation.
The calculation was also carried out for a Narrow-Gap-FEUDT [8]. Nevertheless, there were some discrepancies between the theoretical and experimental directivities. In the actual device, the directivity, in particular, for the 2nd harmonics seems more dependent upon mechanical reflection, which has not yet been taken into consideration.
CONCLUSION
The paper has attempted to employ the expansion functions proposed by Skeie and R0nnekleiv for BEM analysis of IDTs, where the effect of SAW radiation is simultaneously taken into consideration.
First, accuracy and extremely rapid convergence of the analysis were shown by the calculation of static capacitances for several types of IDTs. Secondly, the analysis was applied to the characterization of floating electrode unidirectional transducer (FEUDT) with a large number of electrodes. This confirmed that BEM mentioned in the present paper could be used as an effective tool for SA W device simulation.
ACKNOWLEDGEMENTS
The authors would like to express their sincere appreciation to Professors K. Yamanouchi and M. Takeuchi of Tohoku University for their valuable suggestions and permission to use experimental result shown in Figure 4, and Mr. Y. Koseki for his assistance in numerical calculation. The numerical calculation was carried out by the HITAC M-280H system (using an integrated array processor) of the Information Processing Centre, Chiba University.
357
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
K. Hashimoto and M. Yamaguchi, IDT behavior for highly piezoelectric leaky-SAW on 36"YX-LiTa03, Trans. IECE Japan, J71-C, 3:479-485, in Japanese, (1988). R. Ganss-Puchstein, C. Ruppel and H.R. Stocker, Spectrum shaping SAW filters for high-bit-rate digital radio, IEEE Trans., UFFC-35, 6:673--684, (1988). H. Skeie and A. R0nnekleiv, Electrostatic neighbour- and end effects in weighted surface wave transducers, Proc. IEEE Ultrason. Symp., 540-542, (1976). M. Takeuchi and K. Yamanouchi, Field analysis of SAW single-phase unidirectional transducers using internal floating electrodes, Proc. IEEE Ultrason. Symp., 57--61, (1988). R.F. Milsom, N.H.C. Reilly and M. Redwood, Analysis of generation and detection of surface and bulk acoustic waves by interdigital transducers, IEEE Trans., SU-24, 3:147-166, (1977). H. Engan, Excitation of elastic surface waves by spatial harmonics of interdigital transducers, IEEE Trans., ED-16, 12:1014-1017, (1969). H. Engan, Surface acoustic wave multielectrode transducers, IEEE Trans., SU-22, 6:395-401, (1975). K. Yamanouchi, T. Meguro, Z.H. Chen and K. Matsumoto, New surface acoustic wave interdigital transducers with narrow electrode gaps, Proc. IEEE Ultrason. Symp., 63--66, (1988).
APPENDIX
= { E(Or1log(2) + a {j7rJ O(,Bfw/2) + To(,Bfw)}
E(Or1log(2X') + j7raexp(-j.BfsX )JoCBfw/2)
(X=O)
(Xi 0)
(X=O)
(Xi 0)
F,(X) ~ 1 E(Orl{log(2) /2-1 / 4}
+ 7r [ja{Jo(.Bfw/2)-J2(.BfW/2)}/2+T2(.BfW)]
E(Or1{log(2X') /2+X' 2/4} + 'If j aexp(-j.BfSX){J O(.BfW/2)-Jd .Bfw/2)}
(X=O)
(Xi 0)
where X, = 2sX/w-.jf.(2X/w)L1}, and s = sign(X). To(O) and T 2(O) are given by,
too
358
APPLICATION OF THE FINITE ELEMENT METHOD TO ANALYZE THE SCATTERING OF A PLANE ACOUSTIC WAVE FROM DOUBLY PERIODIC STRUCTURES
A.-C. Hennion, R. Bossut, J.-N. Decarpigny, C. Audoly*
ISEN Recherche Laboratoire d' Acoustique 41 Boulevard Vauban 59046 Lille Cedex. France
*GERDSM DCAN-Toulon, Le Brusc 83140 Six Fours Les Plages, France
INTRODUCTION
The scattering of a plane acoustic wave from immersed periodic structures has raised up a great deal of interest in underwater acoustics because they can be used as low frequency, wide bandwidth reflecting or absorbing baffles. These periodic structures are divided into two groups. The first one contains single periodic structures, such as single or double layered compliant tubes gratings, embedded or not in a viscoelastic layer. They are considered as reflecting baffles, in which the incident plane wave excites a resonance mode of the whole structure. The other group contains doubly periodic structures such as gradual tank lining or Alberich anechoic layers. They behave as a sound absorber, in which the incident plane acoustic wave excites a resonance mode of cylindrical or spherical inclusions, for instance. In order to explain their physical behavior and to help their design, several authors have built accurate mathematical models, which provided insertion loss values in nice agreement with measured values. Thus, Burke et al [1], Brigham et al [2], Dumery [3] and Audoly et al [4] have analyzed gratings of circular cylinders as well as of arbitrarily oriented, elliptically shaped compliant tubes, using a multiple scattering approach. From another point of view, Vovk et al [5], Radlinski et al [6] have used a waveguide approach to describe the behavior of tubes which have elongated rectangular section. Though powerful, all these methods require in every case a lot of specific algebraic developments, which restrict their use to a small number of given geometries. Moreover, they often rely upon simplifying assumptions for the structure or its displacement field, assumptions which rule out the resort to complex structures or materials. Thus, the use of the finite element method, with the help of ATILA code [7], to tackle these problems can strongly broaden the designer's possibilities.
In this paper, the mathematical model is first described for a tridimensional structure, in the case of a double periodicity. Then, several tests are presented which allows a first validation of the method. Finally, complex structures such as compliant tubes gratings or material with periodic inclusions are considered and demonstrate the accuracy of the method for a plane wave at any incidence.
* Associated to the CNRS, URA 253.
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 359
THEORETICAL FORMULATION
General Mathematical Model
The mathematical formulation for the description of periodic structures is presented for a tridimensional domain, in the case of a double periodicity (Figure 1). The whole domain is split into three successive regions by two planes which are parallel to the scattering structure plane, noted S, and S _ . The first and the third regions are semi-infinite fluid domains, in which the pressure field is expanded as a sum of propagating and evanescent plane waves. In the first region, one of the propagating waves is the incident wave. The second region includes the scattering structures and a small part of the surrounding fluid domain. In this region, the double periodicity of the problem is taken into account with the help of Bloch-Floquet theorem i.e. a specific phase relation between points separated by the grating spacing. This allows us to reduce the model for this region to a unit cell, which is then meshed using finite element. Bloch-Floquet relations between the displacement or the pressure values for nodes which are separated by one period, are used to define the boundary conditions between adjacent cells, on the Sl, S2, S3 and S4 planes. Moreover, continuity equations for the pressure field and its normal derivative are written at the boundary between regions I and II on the one hand, regions II and IlIon the other hand and allow to take into account the effects of external fluid domains.
Applications of the Finite Element Method
Bloch-Floquet relations allows to reduce the modeling of the region II to an unit cell, which is then meshed using the finite elements [8]. Classically, the equations of the system give the displacement field and the pressure field. Bloch-Floquet relations allow linear combinations between lines and columns of the system. In terms of finite elements, this operation is the static condensation of degrees of freedom belonging to faces S3 and S4, limiting the unit cell. Taking into account the effects of external fluid domains introduces matrices between the nodal values of the reflected (respectively transmitted) pressure normal derivative and the nodal values of the reflected (respectively transmitted) pressure. When the structure is infinite in one direction i.e. single periodic structure, the problem is bidimensional and evident simplifications can be done [9].
RESULTS
Using the finite element method described above, frequency variations of the transmission coefficient can be investigated for different types of periodic
region J s.
region 2
region 1
Figure 1. Doubly periodic structure.
360
, ,
-1
\ ,.:
-. ~
>Q
0 '0
~ - 6 -= ., -8 .~
.~.
0 :::: ., -10 0
" = 0 -12
-0 '" '" a -1'
'" = '" ... -16 ....
P ~ N: LD'I.ca ox. fiM '~II-~
ANALYT ICAL - i!:iriRl"-~NTAC- "' . \
-18
- 10 0 8 10 \2 14 16 18 10
Frequency (k Hz)
Figure 2. Plastic circular cylinders gratings .
structures. The transmission coefficient, as used here, is given by
20loglpTr/pil where pTr is the transmitted pressure and pi is the incident pressure. For the configurations which are considered hereafter and in the frequency range of interest, only one propagating wave is generally considered, with the same direction as the incident wave, the other waves being assumed to be evanescent with no contribution in the nearfield. Thus, the transmission coefficient is calculated by using the term of zero order in the expansion in series of the pressure.
. ,
, , , -s ,
,
~
0 '0
f:: -I. - d " ...
- ~
0 " -IS 0
" " .2 ..
0 ..
-2. 8 .. " • FB M r.bedded. ..
P! W DOli •• badGe • ... f-< - - ixi'Exl.teNTIC ·
- 15
I 10 J2 U 16 11 20 Frequency (kHz)
Figure 3. Encapsulated plastic circular cylinders grating.
361
Plastic Circular Cylinders
In this section a plastic circular cylinders grating made of PVC is considered, excited by a plane acoustic wave at normal incidence. The internal losses in the PVC are chosen equal to 6%. Results obtained with the finite element model, including and excluding losses in the PVC, are compared with measurements on Figure 2. A nice agreement is observed between the curves. Then the previous elastic tubes have been encapsulated in polyurethane. The finite element results are compared with experimental data on Figure 3. The finite element method provides results which agree fairly well with the experiments. The discrepancy near 13.5 kHz is probably due to the inaccuracy of the elastomer physical constants, because its characteristics depend on the temperature and on the frequency.
362
a
W --uu--_ ______ 0 __ - 0 D -------H---0 ______ 0 __
b
- -u----------Tr - ----------------H-- 0 - -----H--0 ________________ J[] __
,
- 5
iQ -I. '" f::; ;; -IS
" ~ "
-2. 0
2d "
" .2 - 1$ ., ., 8 WI -30
" .. .. f-<
-35
-40 0
o "
-5
" U -15
~ ~ -10
2d "
" -; -25 .. ·s It) -]0
" .. ~ -"
,
, "
FD N
-, / '\ ,
6 8 10 12 Frequency (k,l%)
ANALYTICAL
6- 8 10 11 14
Figure 4.
Frequency (10Hz) .
a. Doubly-layer compliant tube grating, d/W = Full line (FEM), dashed line (Analytic~.
b. Doubly-layer compliant tube grating, d W = Full line (FEM), dashed line (Analytic ).
6 .
0.85
14
.. -- -- .. , , . , , , , -' '-' -' -'
YL .- .-, .-, .-. , , , , -' ' -' -' -'
"
"
Figure 5. Alberich anechoic layer.
Double Layer Gratings with Different Separations
There is a considerable interest in the behavior of multilayer compliant tube gratings. The wide separation of the array center resonance frequencies for various single layer gratings suggests that combining the panels into double layer gratings can result in a transmission coefficient having a wider bandwidth. The grating is shown on Figure 4. The tubes have a rectangular cross section and are considered as two plates structurally coupled. These tubes, infinite in one direction, take part of a 2d wide grating.
The plane wave, at normal incidence encounters two gratings, the second, is parallel to the first, and made of tubes that are two times smaller and two times closer. The distance between the two gratings is noted W. Figure 4 shows results provided by an analytical model , using a wave guide approach [41 and by the finite element method, for two different separations: d/W = 6 and d/W = 0.85. There is a good agreement between the curves obtained by these two methods. As the separation distance is decreased, the pressure field in the nearfield of the tubes is less uniform and this nearfield interaction tends to modify the broadband performance.
Alberich Anechoic Layers
In this section, a plate contains air cylindrical inclusions, which form a doubly periodic structure (Figure 5). Properties of the material, which can be either polyurethane or silicone, depend on the experimental temperature and on the frequency. They were read on curves, provided by the manufacturer. The Figure 6 compares transmission coefficients obtained with the finite element
'OJ .. - s
~ " S -1 0 d o .;; .~ -1 5 B .. " ..
a
/' "
• H.:-l EXPERIHE~TAl
~-,oo~----~----~~--~~--~ 1 0 1 S 1 0
Frequency ( kHz )
0
b
-10
-1 0
- 30
-40 0 11 IS
Frequ eocy (kHz )
Figure 6. Alberich anechoic layers. a. Polyurethane, b. Silicone.
II
363
method on the one hand, and measured, on the other hand, for the two types of materials. There is a good agreement between the curves. When the material is the polyurethane, the discrepancy is certainly due to the difficulty to know exactly the properties of the elastomer.
CONCLUSION
This paper has presented a detailed analysis of the acoustical behavior of various periodic structures. Results calculated with the ATILA code have demonstrated the ability of the finite element method to predict these behaviors and good agreement has been found between results obtained by the finite element models and by measurements or other theoretical methods. Moreover, the efficiency and the versatility of this finite element approach have been demonstrated. Now, one of our greatest cares is to extend this technique to analyze periodic structures containing active materials and to triply periodic structure. One of the possible applications of this method could be to know the homogenized properties of these materials.
REFERENCES
[1)
[2)
[3]
[4]
[5]
[6]
[7]
[8]
[9]
364
E. Burke, V. Twersky, liOn scattering of waves by an infinite grating of elliptic cylinders II , IEEE Trans. Antennas Propag., AP-14:465, (1966). G.A. Brigham, J.J. Libuha, and R.P. Radlinski, II Analysis of scattering from large planar gratings of compliant cylindrical shells II , J. Acoust. Soc. Am., 61:48, (1977). G. Dumery, IISur la diffraction des ondes sonores par des grilles ou des reseaux d'obstacles ll , Acustica, 18:334, (1967). C. Audoly, G. Dumery, IIEtude d'ecrans sous-marins constitues de tubes eIastiques ll , Acustica, 69:263, (1989). V. Vovk, V.T. Grinchenko and L.A. Kononuchenko, IIDiffraction of a sound wave by a plane grating formed by hollow elastic bars II , Akust. Zh., 22:201, (1976), [Sov. Phys. Acoust., 22:113 (1976)). R.P. Radlinsk, M.M. Simon, IIScattering by multiple gratings of compliant tubes ll , J. Acoust. Soc. Am., 72:607, (1982). J.N. Decarpigny, J.C. Debus, P. Tierce, B. Tocquet, D. Boucher, IIIn-air analysis of piezoelectric Tonpilz transducers in a wide frequency band using a mixed finite element-plane wave method II , J. Acoust. Soc. Am., 78:1499, (1985). O.C. Zienkiewicz, liThe finite element method II , 3rd Edition, Mc Graw Hill Ed., New York (1977). A.C. Hennion, R. Bossut, J.N. Decarpigny, C. Audoly, IIAnalysis of the scattering of a plane acoustic wave by a periodic elastic structure using the finite element method: application to compliant tube gratings II , J. Acoust. Soc. Am., 87:1861, (1990).
SOUND PROPAGATION IN GLASS-CERAMIC
ABSTRACT
Zheng Hong, Zhao Ming-Zhou, Xu Yong-Chen, Feng Cui-Ying
Pohl Institute of Solid State Physics Tongji University Shanghai, PR CHINA
The acoustic theory of the suspensions is applied to study acoustic properties of glass--ceramic. The relations between the sound speed and the sound attenuation with the crystallinity in glass--ceramic are derived. The theoretical results are verified by the ultrasonic experiments.
INTRODUCTION
During the last sixty years many acousticians worked at the sound propagation in suspensions which is a liquid mixed with many solid particles. The works were base on the theory of Ament and Ahuja et. al. [1-2]. Recently the acoustic theory of suspensions has been improved gradually.
The glass--ceramic is an isotropic solid which is composed of the disorder glassy phase and many tiny crystal grains. It is similar to the suspensions in several respects. The glassy phase and the crystal grains can be considered as liquid and suspension particles respectively. Therefore, we can make use of the methods used in suspensions to study the acoustic properties of glass--ceramic. Meanwhile, the difference between them must be regarded. The suspension is a fluid, only the compressed longitudinal wave propagates in it. But the glassceramic is a solid in which the elastic waves go through (both longitudinal and transverse wave).
EFFECT OF CRYSTALLINITY ON SOUND PROPAGATION
The crystal grains of a glass--ceramic are distributed randomly and uniformly in a glassy J;?hase. They are neutral and hard tiny particles with diameters about 102-10 3A, much smaller than the sound wavelength. The interaction between grains can be neglected, so the microstructure of a glass--ceramic is isotropic. According to these assumption, we derived the expression of the sound speed and the sound attenuation in a glass--ceramic.
The elastic wave equation in a solid can be written as [3]:
(1)
... where u is the vibration displacement, p is the density, c is the tensor of
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 365
the elastic constant. For an isotropic material c is
cll C12 C12 0 C12 c11 C12
C = C12 C12 c11
C44
0
the differential operator Vs is
%x 0 0 0 o/Oy 0
V - 0 0 of oz s -0 %z o/Oy
of oz 0 o/ax o/Oy o/ax 0
We assume that the acoustic wave is a plane wave which can travel in any
direction. The vibration displacement is given by ~ IX exp[i(wt-k. ;)], where k is the wave vector. Hence the equation (1) can be rewritten as:
1 - .. 0" lw V . (c . Vs . v) = at pv (2)
.. .. where v=du/dt. Then the relations introduced by Ament's are quoted as follows
.. v =
.. .. .. pv = (l-VO)PVI + VOV2 P2
(3)
(4)
where PI, VI and P2 ,V2 are the densities and the vibration velocities of the glass and the crystal grains respectively, Vo is the volume fraction of the crystal grains, i.e. the crystallinity. Substituting Eq.(3) and (4) into Eq.(20), we get
For convenience' sake, let t .... A = [(l-VO)Vl + Vov2J/iw, .. .. .. B = (l-V O)VIPI + VOV2P2,
and then Eq.(5) comes into three scalar equations,
oBx/Ot = (C 1102/ax2 + C4402/Oy2 + C4402/8z2)Ax + (c I2 02/axOy + C44 02/axOy)Ay + (C I2 02/ax8z + C4402/8x8z)Az (6)
OBy/Ot = (C 4402/axOy + C1202/axOy)Ax + (C 44 02/ax 2 + CU02/Oy2 + C4402/oz2)Ay +
(C 1202/OyOZ + c44 02/Oy8z)Az (7)
366
EJBz/at = (c4482/{)x.& + CI282/{)x.8y)Ax + (C44 82/8y8z + c282/8y8z)Ay
+ (C 4482/{)x.2 + C4482/8y2 + cU82/8z 2)Az
.. (8)
Because of the viscosity between the glass and the grains, VI is different .. .. .. from V2 in both phase and amplitude, supposing V2 = f3v Ie 10, we get
where
R = 1 + ((3 e10-1)V 0 ,
Since the glass-ceramic is isotropic, we may assume that the wave propagates .. in x direction, Eq.(6), (7) and (8) become
(9)
(10)
(11)
respectively. From these three equations we can get three waves: a longitudinal wave and two transverse waves the polarized orientations of which are perpendicular each other. For the longitudinal wave:
w_lRc1i_ J [1+(/ie~~-l)VolCl1 K - r-SS-- - Pl+(/i elOp - p)Vo
For the transverse wave:
w _ J RC14 - J [1+(/i e~~- l)Vol C44 K - -S- - Pl+({j elOp - p)Vo
(12)
(13)
Using k/ w = l/c + i/ wand ignoring the terms of second or higher orders of
smallness such as V~, V~, etc., we get the sound speed and the attenuation
of the longitudinal wave as:
I .; [l+Vo( 2-2(3cosB) + V ~ ((32+1-2(3cos B)]C ll /PI
c = ----------------------------------------L 1 + Vo [(3(1+7J)cosB-2]/2 + V~(32[27]-(1+7J2)cosB]/8
w 4Vo ( 71-1) (3sinB - V5(32(7J2- 1)sin2B
a=g-;::========================:--L j [1+Vo(2-2(3cosB) + V5((32+1-2(3cosB)Jcu/p
(14)
(15)
367
where 'TJ is P21 PI· The transverse wave speed and attenuation can be expressed by replacing C11 by C44 in Eq.(14) and (15). It must be noticed that by the assumptions above (3 and 0 are the constants which are not dependent on Vo [4]. So Eq.(14) and (15) express the relation of the sound speed and the sound attenuation to the crystallinity Vo. Eq.(14) and (15) show that CTL increases with crystallinity and aTL increases at low crystallinity,
but decreases when V 0 reaches about 60%. If Vo is equal to zero, 1 1
cL = (c111 PIP, c = (c441 PI)' and aTL = O. In fact, the sound attenuation
does not become zero because of many other factors, so aTL is zero
reasonably when V 0 become zero. From Eq.(15) we find that the sound attenuation will be zero when 0 is
zero, i.e. if the vibration phase of glass is in agreement with that of crystal grains there exist no attenuation. Therefore, we draw the conclusions that the viscosity is the major cause of the sound attenuation. Besides, the higher the frequency, the larger the sound attenuation.
EXPERIMENTS AND DISCUSSION
The sound speed and the sound attenuation in a glass-ceramic with various crystallinity are measured by the echo method. The block diagram of the experiment is shown in Figure 1. The samples are the Li 20-A1 20 3-Si0 2 system glass-ceramic with two kinds of crystal phases: Eucryptite and Spodumene. They are made into 15 mm long cylinders. PVDF film transducer is used to produce ultrasonic wave at the frequency range 10-50 MHz.
The variation of the longitudinal wave speed and the attenuation with the crystallinity are shown in (Figure 2), where cLO =6055 mls is the sound speed
with Vo=O. The solid line in (Figure 2) is a theoretical fitting corresponding to Eq.(14) with 0=8', {3=0.9 and 'TJ=1.2 from (Figure 2) it can be concluded that the sound speed increases as the crystallinity in the glass-ceramic rises. In comparison with suspensions, we calculated 0 and {3 in suspensions and got: 0=2'-3', {3=0.9-1. The experimental results show that the transverse wave speed is about half of that of the longitudinal wave.
368
Figure 3 shows that aT' L increases quickly at first, and slower later.
FflEQvPAI(y
'''tiN rER p,., tc7:J
D WIIJ€8.1oND
051LLOSCOPG
p~ ~2'8
Figure 1. Block diagram of experimental device.
I_OeD
I. abc
" ~ I. CIS '-'
I. ~20
1.000 ZO ~O 60 V,(%J
Figure 2. Relation between longitudinal wave speed and crystallinity.
According to Eq.(lS), the sound attenuation should reduce at high crystallinity mainly because the crystal grains are isolated in the glass at low crystallinity. The increase in crystallinity means more or bigger crystal grains in glass. In this case the sound wave is scattered dominantly by crystal phase, therefore, the sound attenuation rises with crystallinity. When crystallinity is high enough, the crystal phase in the glass-ceramic become a continual phase and the glassy phase changes from a continual phase into a separated phase (similar to the crystal grains at low crystallinity). In this way, the sound waves are scattered mainly by the glassy phases instead of crystal phase. As a result, the sound attenuation from the scattering decreases with the increase of the crystallinity. Of course, the scattering here is caused by the viscosity between the crystal grains and the glassy phase.
The relations of the sound speed and the sound attenuation to the sound frequency are also studied. The results show that there is not noticeable dispersion and the sound attenuation is sensitive to the frequency (Figure 4).
d~",
0./6
o,~
oJ
~ o 15
0.10
o 08
006 20 qo 60 Vd%)
Figure 3. Relation between longitudinal wave attenuation and crystallinity.
369
o 2~
-J o 18 ~
o 12
0.06
o 00 /0 26 30 110 5TJ 60 f(f1H)
Figure 4. Relation of longitudinal wave attenuation versus sound frequency.
CONCLUSION
From the experiments we find that there are several common points between the glass-ceramic and the suspensions, one of which is the maximum in the curve of the sound attenuation versus the crystallinity etc .. However, there exists a minimum sound for suspensions at a certain fractional volume concentration while the sound in a glass-ceramic always increases with the crystallinity. In the theoretical part we introduced two parameters Band (3, they, in fact are the function of the sound frequency and the size of the crystal grains etc. . The analytical expressions of these function as well as the microscopic interaction between the crystal phases and the glassy phase will be studied in the future.
The authors are grateful to Mr. Zhang Gangyong and Ms. Zhang Weiqin for their help on the experiments.
REFERENCES
1 Ament, W.S., J. Acoust. Soc. Am., 25, 638, (1953). 2 Ahuja, A.S., J. Acoust. Soc. Am., 51, 182, (1972). 3 Sun Kang et. al, II Piezoelectricity II , 276-279, Beijing, (1984). 4 Ahuja, A.S., J.Appl. phys. 44, 4863, (1973). 5 Tang Y.W., Acta Acustica, 3, 181, (1981).
370
VISUALIZATION OF THE RESONANCES OF A FLUID-FILLED
CYLINDRICAL SHELL USING A LOW FREQUENCY SCHLIEREN SYSTEM
ABSTRACT
Victor F. Humphrey, Sharon M. Knapp and Carolyn Beckett*
School of Physics, University of Bath Bath BA2 7AY, U.K. *Current Address: Yard Ltd. Bridport Road Dorchester, Dorset DT1 1TL, U.K.
A low frequency Schlieren system designed to visualize ultrasonic waves over the frequency range 100 - 500 kHz has been used to observe the resonances of a submerged fluid-filled brass cylindrical shell with inner and outer radii of 8.25 mm and 9.5 mm. Under resonance conditions the incident acoustic field generated a standing wave pattern in the fluid column that was observed using the Schlieren technique. With this arrangement all of the resonances of the shell and of the fluid column between 180 kHz and 500 kHz were identified as were the Stoneley type wave resonances between 50 kHz and 220 kHz. The experimental images of the standing wave patterns are compared with theoretical predictions and found to be in good agreement. The frequency of each resonance was also determined experimentally using the Schlieren system. These frequencies are found to agree with the theoretical predictions to within 1% overall.
INTRODUCTION
Recent studies of acoustic scattering from discrete objects have concentrated on interpreting the frequency dependence of the scattered field in terms of the resonances of the scattering body [1). This has led to the development of a number of experimental techniques [2, 3] to identify the resonances of scattering objects by studying the external scattered field. For the particular case of a thin fluid-filled cylindrical shell Maze, Izbicki & Ripoche [41 experimentally identified three different types of resonance. The majority of the resonances were attributed to the fluid column contained within the shell. Others were observed to be due to circumferential waves in the shell and Stoneley type waves propagating on the shell.
This paper describes the use of a Schlieren system to locate and identify the resonances of a fluid-filled cylindrical shell, and to study the resulting field in the interior fluid. It is shown that under resonance conditions, the acoustic standing wave which exists in the fluid column of a cylindrical shell can be easily visualized using the Schlieren technique. This allows the frequency of each resonance to be accurately located, and the symmetry of the resonance to be determined.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 371
SCHLIEREN SYSTEM
The experimental results were obtained using a Schlieren system that has been developed to investigate the scattering of ultrasound in the frequency range 100 kHz - 500 kHz [5]. The system uses a standard Z configuration (Figure 1) with an ultrabright light emitting diode (Toshiba TLRA l50/C) radiating at 660 nm as a light source. Light from this source is collected and focused onto a pinhole aperture by a pair of condensing lenses. The light transmitted by the pinhole is then collimated by the first parabolic mirror, of 0.3 m diameter and 1.8 m focal length, and passed through a water tank containing an ultrasonic transducer and the cylindrical shell under observation. The axis of the acoustic field is arranged to lie perpendicular to the light beam, while the axis of the shell lies parallel to the beam. The light transmitted through the tank is brought to a focus by the second parabolic mirror to form a diffraction pattern which corresponds to the refractive index variations in the ultrasonic field. In order to visualise this field a spatial filter is introduced at the focus of the second mirror which removes the zero order of the diffraction pattern. The light from the remaining orders is then used to form an image of the acoustic field which is observed using a video or still camera.
At frequencies below 250 kHz the separation of the diffraction orders in the spatial filter plane is less than 200 /tm. For these orders to be resolved, a pinhole smaller than 200 /tm in diameter must be used. In practice, a multiple pinhole array [5] with eight 100 /tm pinholes was used to provide a suitable light level, whilst still achieving the required resolution.
The experimental results were obtained using a 100 kHz transducer with an active area 60 mm in diameter driven by a continuous sine wave. The results are presented for a water filled brass cylindrical shell, of 110 mm in length, with an inner radius b of 8.25 mm and an outer radius a of 9.50 mm, having an inner to outer radius ratio b/a of 0.868.
THEORY
The relationship between the image obtained with a Schlieren system and the acoustic field depends on both the acoustic pressure and the spatial filtering used. For zero order filtering and low acoustic pressures the results for a plane wave indicate that the intensity in the Schlieren image is proportional to the square of the pressure in the acoustic field [6]. This result also applies to low amplitude standing waves for which the bright areas in the Schlieren image correspond to the standing wave antinodes. In this study therefore, the experimental results are compared with the square of the theoretically calculated acoustic pressure distribution.
The pressure distribution was calculated assuming an elastic, fluid-filled, cylindrical shell insonified at normal incidence by an infinite plane wave. The
J-~~PI:nh~O:le~=========;:=~~~~~ Mirror 1
~~====T:ranSduce~ Mirror 2 \~
Spatial Filter Plane
Figure 1. Diagram illustrating the main components of the Schlieren system.
372
Figure 2.
,,-=,-,
hcQ)~ ~©,)~ v_e~\'j,
;@~ •• ~@ -6-~ o . ~ ~'~ ~
r..!. ~
Schlieren image and theoretical prediction for the (4,4) fluid column resonance.
theory uses the standard normal mode approach, similar to that used to evaluate the scattering of a plane wave from a cylindrical shell [7] except that in this case an expression for the pressure distribution in the interior fluid was derived and evaluated [8].
RESULTS AND DISCUSSION
For the low acoustic drive levels used, the Schlieren image of the water column within the shell was almost completely dark at most frequencies. As the transducer drive frequency was slowly and continuously varied, bright symmetric patterns were observed to appear within the shell at each frequency where a standing wave pattern existed. Each of these patterns could only be observed for a narrow frequency band (typically 2 - 4 kHz) due to the high Q of each resonance, enabling each resonant frequency to be accurately located. The resonance patterns were easily recorded using a stable synthesized oscillator to drive the transducer at the required frequency .
For the water filled cylindrical shell considered, it was possible to visualise and identify a large number of resonances in the frequency range 50 kHz to 500 kHz corresponding to ka values (wavenumber x outer radius) between 2 and 20. These resonances were of the three types identified by Maze, Izbicki and Ripoche [4]. The majority of the resonances are due to the fluid column within the cylinder, and occur when a standing wave pattern is created in the fluid . Other resonances are associated with resonances of the shell and Stoneley type waves propagating on the interface between the shell and the fluid . Experimentally it was possible to visualise all the shell and fluid column resonances between 180 kHz and 500 kHz, and the Stoneley wave resonances between 50 kHz and 220 kHz.
For a resonance of the fluid column it is possible to observe '2n' acoustic antinodes or 'n' wavelengths around the cylinder circumference. It is also possible to identify 'm' acoustic antinodes across the radius of the cylinder, and we have chosen to denote the fluid column resonances shown here by the notation (n,m). The other types of resonances discussed in this paper have only been assigned an 'n' value, according to the number of antinodes (2n) existing around the shell.
Figure 2 shows the Schlieren image of a typical fluid column resonance, in this case the (4,4) resonance observed at a frequency of 450.4 kHz. For comparison the theoretical contour plot of the same resonance is also shown in Figure 2. The agreement with the Schlieren image is very good. An example of the image obtained for a Stoneley wave resonance (n = 7) is compared with the corresponding theoretical prediction in Figure 3. The theoretical plot shows only the part of the Schlieren image within the shell. Other resonances, due to
373
Figure 3. Schlieren image and theoretical prediction for the n = 7 Stoneley wave resonance.
circumferential waves in the shell, can be identified by their effect on the inner fluid . Figure 4 shows the Schlieren image and the associated contour plot for the only third order (n = 3) shell resonance in the ka range 0 - 20 . The pattern seen is quite similar to that for the (3,2) fluid column resonance although there are differences especially near to the shell. The two resonances do, however, occur at significantly different frequencies.
The frequencies at which the resonances occur can be located accurately by noting the frequency of the maximum intensity of the Schlieren image. The frequencies of 'kb' (wavenumber x inner radius) values at which the fluid column resonances were observed with the Schlieren system are compared with those predicted theoretically in Table 1. Column 1 shows the identity of the resonance. The frequency and resultant kb value observed with the Schlieren system are shown in columns 2 and 3 respectively. Column 4 lists the kb value for the theoretical contour plots (i.e. the kb value for which the maximum pressure is seen). The values shown is columns 3 and 4 show excellent agreement for all of the resonances with agreement to within 0.3% (i .e. 0.7 kHz) . It should be noted that the theoretical values are mainly sensitive to the parameters of the inner fluid , although the (n,2) resonances show a slight dependence on the velocities in the shell. It should also be noted that the resonant frequencies could be located experimentally with a repeatability of better than 0.1 kHz, the largest source of experimental error being due to temperature drift in the water tank.
A similar comparison of the experimentally observed and theoretically predicted frequencies of the Stoneley wave resonances is made in Table 2. The
Figure 4. Schlieren image and theoretical prediction for the n = 3 shell resonance.
374
Table 1. Comparison of experimental and theoretical kb values for fluid column resonances
Fluid f ,," kb kb column Schlieren Schlieren computed
resonance 1kHz
2,2 194.1 6.89 6.87
2,3 282.8 10.04 10.05
2,4 372.2 13.21 13.22
3,2 232.1 8.24 8.22
3,3 321.3 11.41 11.42
3,4 411.7 14.62 14.63
4,2 261.4 9.28 9.27
4,3 358.8 12.74 12.75
4,4 450.4 16.00 16.01
theoretical kb values for n = 5,6 and 7 were obtained by finding the kb value at which the computed pressure distribution within the cylinder had a maximum amplitude. For higher orders, the values were obtained from calculations of the backscattered form function, as it was difficult to locate the resonances accurately using the pressure distribution due to the proximity of other resonances. These results again show very good agreement with theory (to within 0.5%) except for the n =11 resonance which was difficult to locate experimentally.
The Stoneley waves on a cylindrical shell are created at glancin& incidence and are known to propagate with most of their energy in the fluid [9J and with a phase velocity that is dispersive and lower than that for the fluid. From the observed resonance frequency, order n , and shell outer diameter it is possible to calculate the Stoneley wave velocity as shown in column 4 of Table 2. This clearly illustrates the dispersive and relatively lower velocity of the Stoneley waves.
Initial comparisons, based on a shear wave velocity of 2100 ms -1 showed considerable discrepancies between the predicted and observed frequencies for the shell and Stoneley wave resonances. For these resonances increasing the shear wave velocity by 100 ms -1 to 2200 ms -1 gave the good agreement shown in Table 2.
Table 2. Comparison of experimental and theoretical kb values for Stoneley wave resonances
Panial kb kb Velocity mode (n) Schlieren computed Ims·!
5 1.85 1.86 625
6 2.67 2.68 754
7 3.60 3.61 870
8 4.62 4.61 978
9 5.65 5.67 1063
10 6.78 6.78 1147
II 7.81 7.92 1202
375
CONCLUSIONS
A Schlieren system has been used to identify the resonances of a brass cylindrical shell by means of the standing wave patterns generated in the inner fluid. Within the range 180 - 500 kHz it was possible to visualise all of the resonances due to the fluid column and the shell. The stoneley wave resonances, however, were only visible between 50 kHz and 220 kHz. The visual agreement of the standing wave patterns at resonance with theoretical predictions is very good, indicating the usefulness of the Schlieren system for immediate identification of resonance modes. The accuracy with which resonances can be located experimentally has been shown to be very good, especially if the uncertainties in the shear and compressional wave velocities are taken into account. Overall the agreement is found to be within 1%.
ACKNOWLEDGEMENTS
The authors acknowledge the support of the Procurement Executive of the Ministry of Defence and the advice of D.H. Follett, Bristol General Hospital.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
376
G.C. Gaunaurd, Elastic and acoustic resonance wave scattering, Appl. Mech. Rev., 42:143, (1989). G. Maze and J. Ripoche, Methode d'isolement et d'identification des resonances (MIIR) de cylindres et de tubes soumis a une onde acoustique plane dans hau, Revue Physique Appl., 18:319, (1983). C.Y. Tsui, G.N. Reid and G.C. Gaunaurd, Resonant scattering by elastic cylinders and their experimental verification, J. Acoust. Soc. Am., 80:382, (1986) . G. Maze, J.L. Izbicki and J. Ripoche, Acoustic scattering from cylindrical shells: guided waves and resonances of the liquid column, Ultrasonics, 24:354, (1986). S.M. Knapp and V.F. Humphrey, Schlieren visualisation of low frequency ultrasonic fields, in: "Ultrasonics International 89 Conference Proceedings II , Guildford, Butterworths (1989). J.A. Bucaro, L. Flax, H.D. Dardy and W.E. Moore, Image profiles in Schlieren observations of acoustic wave fronts, J. Acoust. Soc. Am., 60:1079, (1976). .. R.D. Doolittle and H. Uberall, Sound scattering by elastic cylindrical shells, J. Acoust. Soc. Am., 39:272, (1966). S.M. Knapp, C. Beckett and V.F. Humphrey, Schlieren observation of the resonances of a fluid filled cylindrical shell, submitted to: J. Acoust. Soc. Am. H. Uberall, Surface waves in acoustics, in: "Physical Acoustics X", W.P. Mason and R.N. Thurston, eds., Academic Press, New York, (1973).
LIGHT SCATTERING ON MAGNETOSTATIC WAVES IN THIN-FILM
GYRO TROPIC WA VEGUlDE
LA. Ignatiev, V.G. Plekhanov, A.F. Popkov
Physical Department Krasnoyarsk State University Krasnoyarsk, U.S.S.R.
INTRODUCTION
The development of the microwave technique of the MSW basis stimulated the interest to the investigation on the effects, appearing under interaction between MSW and other types of waves, which propagate in magnetic dielectric films, including optic ones. In this case the use of magnetostatic waves gives us the possibility of both the modulation of light and of carrying out a special analysis of microwave signals: an instant spectrum Fourier analysis, disappearing and so on.
Earlier we studied the effects of collinear optical waveguide modes conversion in gyrotropic waveguide under non-elastic light scattering on magnetostatic waves with Faraday effect only [1). But magnetic birefringence greatly effects the whole picture of conversion, especially in case of backward volume MSW.
The light propagation in a gyrotropic waveguide is usually described by zero approximation equations of correlation modes: TE and TM. If the travelling MSW is absent, correlation equations define gyrotropic waveguide modes structure. The exact solution of the Maxwell's equations for gyrotropic waveguide was received earlier [2]. It was shown that their own modes have an elliptical polarization and their elliptical coefficients are rather small. The value of each elliptical coefficient is proportional to fiji f.f « 1, (i, j = x,y,z ifj).
If magnetostatic wave propagates, the dielectric film tensor f. acquires space and time modulations. So we may observe the resonance modes conversion.
Let us consider the case when the film is situated in the region O<x<d, the substrate and the cover are optically isotropic and have f.=f.s or f.=f.c respectively. The light propagates in Oz direction. Let us consider, for example, the propagation of the forward volume magnetostatic wave when external magnetic field is HI/Ox. The structure and dispersion of this wave are found from the magnetostatic equations and Landau'S equation. Under the real conditions of experiment Kd« 7r/2, 0 --I 0H (K is the wave number of
MSFVW, 0 is its frequency and 'Y is the gyromagnetic ratio) and mz = imy
= m~ exp[i(Kz-Ot)), mx ~ M = Const, the dielectric tensor f.f is expressed as:
(~f)xx = f.f + a2,
(~f)XY = -igoO::z + 1/3(2ao+ a2)o::y,
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 377
(~f)xz = igoO:y + 1/3(2ao+ a 2)o:z,
(~f)YY = Ef + v'2/3(aO-a2)(O:zcos3rp-0:;;in3<p),
(~f)ZY = igo + v'2/3(ao-a2)(o:ycos3<p+O:zsin3<p),
(~f)zz = Ef - v'2/3( aO-a2)( O:zcos3rp-o:ysin3<p),
. . * . . * . . * (Ef)yx = (Ef)xy , (Ef)yz = (Ef)zy , (Ef)zx = (Er)xz
Here <p is the angle in the plane (111) between the axis (211) and the direction of light propagation, 0:, are the directing cosines of magnetization M, go is the module of the gyration vector and ao, a2 are the constants of the magnetic birefringence.
During the calculations we'll neglect the modes conversion at the constant part of gyration vector go. And we'll consider only TE-TM type of the modes conversion.
We shall look for the solutions of the disturbing wave equations for the film region in the form of
where Pi(x)« E~ (i = x,y,z), E~ = ATE(z)e~(x) and EJ = BTM(z)eJ(x).
Here e~(x) is the transversal structure of a field in corresponding non-disturbing mode.
On preserving the terms of the first order of approximation in the wave equation
grad div E - L1E - k6 E E = 0
we may obtain the following system of equations for the Pi'
I I I I I
(Pz)xx - if3TM(pJx + k6 EfP z = (E~)xz - EzyE~k~exp(-iL1f3z)
f3~M(pJ + if3TM(PZ): - k~ErPx = 2if3TM(E~)~ - (E~):~ + ExyE~k~exp(-iL1f3z)
Here f3TE ,TM are the wave numbers of TE and TM modes in propagation di
rection, ko = 27r/ A, A is the wave length of light in vacuum. The general solution of this system includes the solutions analogous to the
structures of the own TE and TM modes, but the right hand side terms of this system are singular. The following connection equations
I
(BTM) z = O:z {v'2/3( aO-a2)w zycos3<p+i[W xy(go+2/3ao+a2/3)
+ v'2/3( aO-a2)w zysin3<p]} ATEexp[i( L1fJ-K)z],
378
f
(ATE) z = az hl2/3( aO-a2)1lr yzcos3cp-i[1lr yx(go+2/3ao+a2/3)
+ y'2/3( aO-a2)1lr yzsin3 rp]}BTMexp [-i( .6.,B--K)z]
are the consequence of the elimination of this singularity. The coefficients Ilr ij mean the connection coefficients between TE and TM modes.
The solution of the connection equations is well known. Under the initial conditions BTM(O) = Bo and ATE(O) = 0 the amplitudes of the modes may
be written in the form
1 1
ATE = Bo 1lr(.6. 21 4+ 1lr2(~ sin[(.6. 2/4+ Ilr 2) 2" z]exp( -i.6.z/2),
BTM Bo{cos[(.6.2/4+1lr 2)1z]-
-i(.6. 12)(.6. 2/4+ 1lr 2t 1 sin[(.6. 2/4+ 1lr 2) 1 z ]}exp( -i.6.z/2).
Here .6. = .6.,B--K and
1lr 2 = a;{llrxyllryx(go+2/3ao+a2/3)2 + 2IlrzyIlrYZ(aO-a2)2/g
+ 2y'2/3( aO-a2)1lr zy Ilr yx(go+2/3ao+a2/3 )sin3 rp}.
The maximum of efficiency is realized when .6.=0. When we take into account the dumping of the light, the maximum of the conversion efficiency will be
~ = [exp(-a&/Ilr arctg Ilr/a&)sin(arctg 1lr/a&)J2
at the length of interaction
I = Ilr -larctg Ilr 1 aa.
If we connect the amplitude of the MSFW with its power, we receive the following expression for Ilr
1lr 2 = p~SFVW /[2LOl71Md)2] {llrxyllryx(go+2/3ao+a2/3)2
+ 21lr zy Ilr yz( aO-a2)2 /9+2y'2/3( aO-a2)1lr zy Ilr yx(go+2/3ao+a2/3 )sin3 rp}
The light scattering on MSSW and MSBVW is considered to be analogous. In Figure 1-3 the connection coefficients for MSFW, MSSW and MSBVW are
presented. Here the structure is YIG film - GGG substrate, ao = -1.66 10-4,
a2 = -2.35 10-4, p~SW = 1 mW.
The analysis made shows that mode conversion in general is defined both by linear and by quadratic magnetooptic effect. When we observed light scattering on forward volume MSW, conversion is mostly contributed by gyrotropic
part of tensor ff. In case of backward volume MSW a more important role is played by birefringence. Light scattering on surface MSW is to a great extent a superposition of scattering processes on backward and forward magnetostatic waves. Modes conversion maximum is achieved in case of one number modes for all the three types of MSW in a film. Quadratic magnetooptic effect effect leads to space dependence of mode conversions. In Figure 4 one can see the angular dependence of conversion coefficient of optical modes TEo-TMo for the three
379
.1 ............. ..... : ........ , ........ : ....... ..
~.r~::: ~ : L ........ ; ...... , ......... : ...... ; ....... .
: : ~.~ :
.~~J 5 7 9 11
4,r
Figure 1
.......................... ", ......... ; ....... ";
: ItiIW :
1, ........ , ........ , ......... : ......... : ........ :
~ .. . . : 1£..TlI, : : : . . . . .
01 ....... : ............... . : ......... ; ........ ; , . . , . .
: [,11\ : : : e.al · ·~
3 5 1 9 11 d,r
Figure 3
u~ ·· · · .. ... y .... :· .... ~ .. : sn'
l : \~.~ .
1.0 ...... : ...... ~ .... §D ....... : ... : . ' ' . , . ' . , ,
~~.11\~ 0, 1 ···:·····~
Y'I~
3 5 7 9 11 4,r
Figure 2
7~,lBk: · :::::: ·2~<::~:>: 9,51"\7··· .. ·:····\:7 .. ·:
::!b .: .... ~< ',8r··· · ·: ··~·:
:,:tyy . . , . , . . . . . . .
30 69 ~ 1211 ~.t
Figure 4
types types of MSW in the film 3 /Lm thick. The strongest space dependence of the effect is observed in case of scattering on MSBVW, also characterized by the least mean effect magnitude in comparison with the scattering on forward volume surface MSW under the same conditions.
A similar analysis of mode conversion of the same polarization (TEi-TE·, TMi-TMj, ifj) shows as a rule that its magnitude is one-order less than mo~e conversion TE-TM. The case of resonance conversion of modes TMi-TMi+! in light scattering on MSFVW presents an exception. The efficiency of such conversion is comparable on the order of magnitude with mode conversion TEi-TMi+1'
When MSW power is about 1 W the conversion efficiency for YIG film doesn 't exceed several percents [3J. But in the film Bio.89Lu2.11Fe50 12 the achieved efficiency was 47% with MSW power 0.8 W [4J . It well corresponds to our numerical estimates of 30% conversion with MSW power 100 mW in such materials.
REFERENCES
[IJ
[2]
380
Yu .V. Gulyaev, LA. Ignatiev, V.G. Plekhanov, A.F. Popkov, Radioengineering and Electronics (SU), 8:1522, (1985). LA. Ignatiev, V.G. Plekhanov, A.F . Popkov, in "Phpical Phenomena in Electronic Devices and Lasers", MPhTI, Moscow (1982) . A. Fisher, J . Lee, E. Gaynor, A. Tveten, Appl.Phys.Lett ., 9:779, (1982). H. Tamada, M. Kaneko, T . Okamoto, J .Appl.Phys ., 2 :554, (1988).
RAYLEIGH WAVE TOMOGRAPHY
D. P. Jansen and D. A. Hutchins*
Department of Physics Queen's University Kingston, Ontario, Canada K7L 3N6.
*Department of Engineering University of Warwick Coventry CV 4 7 AL, England
INTRODUCTION
Tomographic imaging involves the collection of data from a series of projections through an object, and the reconstruction of variations in a chosen parameter across a two-dimensional section. It is now an established method for the reconstruction of images from ultrasonic data, and has been used widely in such areas as medical ultrasound and non-destructive evaluation [1-3].
In the latter application, the technique has not been widely used because of several problems. The first of these is the difficulty in coupling ultrasonic energy into the sample. The most obvious technique is to use an immersion method, where a water coupling medium is used. This allows easy scanning of the transducer around the sample under test, but if the sample has a curved surface, the effects of refraction will cause the beam to deviate from the chosen path. Such effects become more severe as the difference in acoustic velocity between the sample and the water coupling medium increases. The second problem is ray-bending, due to the changes in acoustic velocity throughout the sample. As these are often the variations of interest, it is important that methods be available to counteract such effects if they are a significant factor.
In the present work, the aim was to investigate the use of tomographic reconstruction for the imaging of surface defects in aluminum samples using Rayleigh waves. An immersion technique was chosen, as the flat surface of the sample removed potential problems due to sample curvature. Defects were to be in the form of drilled cylindrical holes, and it was thought that a straight ray image reconstruction technique was sufficient. The algorithm chosen was based on filtered back-projection, one example of a class of techniques known collectively as transform methods [4]. This algorithm is well-suited to the geometries that would be encountered in Rayleigh wave experiments, and is fast and efficient to implement on a lab-based microcomputer.
APP ARATUS AND EXPERIMENT
A schematic diagram of the apparatus is shown in Figure 1. 5 MHz Panametrics immersion transducers were used for generation and reception, both inclined at the critical angle for Rayleigh waves at an aluminum/water interface. The leaky Rayleigh wave travelled along the sample surface, and inter-
Physical Acoustics, Edited by O. Leroy anJ M.A. Urea/_calc Plenum Press, Nc\"., York, 1991 381
IBM PS/2
microcomputer
To motion
control system
IEEE Interfece . =======* Tektronix 2.30A
Panametrlcs
pulser/recelver
5 MHz Immersion transducers
digital
OSCilloscope
Figure l. Schematic diagram of apparatus.
acted with a series of artificial defects which were machined into the surface of the thick aluminum blocks. The transducer pair were initially aligned, and thereafter scanned in unison across the object surface at a fixed distance apart . Waveforms were recorded at fixed intervals along the scan path, using a Tektronix 2430A digital oscilloscope. The transducers were then rotated by a preselected angle, and the linear scan repeated for a set of angles to form a total rotation of 180'. All motion was performed by stepping motor drive systems, under the direct control of an IBM PS\2 microcomputer. The waveform from the transducers at each location of the transducers was also stored on the same computer, which controlled the operation of the oscilloscope and the motion control system via an IEEE interface.
An example of a typical waveform recorded using this apparatus is shown in Figure 2. Note that the Rayleigh wave (R) had a reasonable bandwidth and signal to noise ratio. At each transducer location, the amplitude and arrival time of the Rayleigh wave transient were estimated. As this had to be performed for each waveform, an automated procedure was developed. For amplitude measurements, which would lead to an attenuation image, the integrated energy within a time window was estimated, and its amplitude compared to a windowed reference signal on a flat smooth surface. To determine the times of arrival, which would lead to an image of slowness variations (the inverse of velocity), a cross-correlation was performed with the same reference waveform.
C> J f\ ..J\
"0 V
~ R Q. E <
o 10 20 30 40 50 80 Time (us)
Figure 2. Typical received Rayleigh waveform.
382
1 . El"
0' 35
'1 'l"
Figure 3.
0'03
-0 ·ea
DoC]) -0 ·e3
Reconstruction of (a) slowness and (b) attenuation following Rayleigh wave interaction with a 2 mm diameter hole.
DoC])
RESULTS AND DISCUSSION
The reconstructions obtained from a sample surface into which a 2 mm diameter hole was drilled are presented in Figure 3, where (a) is the slowness image and (b) that of attenuation. There are several features of interest . The slowness image derived from time of flight data contains a single prominent peak, corresponding to a decrease in velocity. This is caused by the delay encountered by the Rayleigh wave in passing around the defect . Note the interesting phenomenon of an apparent increase in velocity (i .e. a negative slowness) surrounding the single peak. The reconstructed variations in attenuation in Figure 3.b are more extended spatially, with a minimum at the location of the defect being a characteristic feature.
Experiments were also conducted on a sample containing two cylindrical holes, of 10 mm and 5 mm diameter, separated by 25 mm. The reconstructed slowness variations for this case are presented in Figure 4, where a double peak is visible, with the centres of the two features being separated by the expected distance.
There are some interesting factors involved in the interpretation of the
Figure 4. Slowness variations following Rayleigh wave interaction with cylindrical holes of 10 mm and 5 mm diameter, separated by 25 mm.
383
reconstructed variations in slowness and attenuation. In the former case, the maximum of each peak locates the centre of the defect, with the peak height being proportional to the diameter of the defect (which dictates the time delay of the Rayleigh wave). The width of the peak is influenced by the width of the Rayleigh wave beam, and other factors. This is more evident in the attenuation data, where the resultant variations arise from forward scattering of the Rayleigh wave from a wide beam. This forward scattering causes the attenuation to apparently decrease at the position of the defect centre. The width of the reconstructed attenuation peak is dependent upon the defect size, and because of diffraction effects the width increases with a decrease in defect diameter. This accounts for the extended width of the reconstructed attenuation data for the 2 mm defect in Figures 3.a and b. Further work is under way to deconvolve the effect of a wide Rayleigh wave beam, and to use the effects of diffraction to obtain a true image of the defect from amplitude data.
REFERENCES
[1]
[2]
[3]
[4J
384
R.K. Mueller, M. Kaveh and G. Wade, Reconstructive tomography and applications to ultrasound, IEEE Proc., 67:567, (1979). D. Hiller and H. Ermert, Ultrasound computerized tomography using transmission and reflection mode: application to medical diagnosis, Acoustical Imaging, Plenum Press, New York, 12:553, (1982). J.F. Greenleaf and R.C. Banh, Clinical imaging with transmissive ultrasonic computerized tomography, IEEE Trans. Biomed. Eng. 28:177, (1981). R.M. Lewitt, Reconstruction algorithms: transform methods, Proc. IEEE, 71:390, (1983).
OPTICAL INTERFEROMETRIC DETECTION OF PLATE WAVES ON
CIRCULAR PIEZOELECTRIC TRANSDUCERS
X. Jia, P. Mantel, J. Berger and G. Quentin
Groupe de Physique des Solides Universite Paris VII Tour 23, 2 place Jussieu 75251 Paris Cedex 05, France
INTRODUCTION
Ultrasonic pulse--echo techniques are increasingly used in nondestructive evaluation. A good knowledge of the influence of transducers on the radiated ultrasonic fields becomes of major importance to interpret correctly the information received by transducer. Actually the transient fields radiated by transducers are generally described by the piston model. It predicts quite well the radiation patterns in the far fields. However some experiments made with stroboscopic Schlieren system and hydrophone detection showed that it is impossible to explain certain experimental results obtained in near fields only with the piston theory. The origin of these anomalous behaviors, which are also called "head waves", are related to the acoustic radiation of the vibration modes of transducer other than the pistonlike one [1-3].
In the previous experiments l4] performed with an optical heterodyne interferometric probe, we have observed, in addition to the thickness mode, one wave propagating across the transducers from their edj2es. On thick plates this wave exhibits the characteristics of a surface wave lRayleigh type), while in thin plates does that of a plate wave (Lamb type).
The purpose of this paper is to report the quantitative study of the plate waves on circular piezoelectric transducers with the optical interferometric detection and to show, by the hydrophone measurements, the relationship between the plate waves and the head waves radiated into water.
EXPERIMENT AL PRO CED URE
In order to understand the characteristics of plate waves travelling laterally across the transducer, air-backed circular piezoelectric plates of PZT (lead zirconate titanate) ceramic are used in our experiments. These piezoelectric transducer are driven by two kinds of electric generators: one provides a short unipolar pulse of amplitude 150 V and 0.1 p,s duration (wide band); the other a gated sine pulse of the same amplitude and central frequency 650 kHz (narrow band). The transient displacements of transducer are detected by the heterodyne interferometer. This optical probe, associated with a broadband electric circuit, allows to measure the absolute normal components of displacements with a
sensitivity 1O-4A/,fHz, i.e. 0.5 A for a bandwidth 30 MHz [5]. The received signals at the output of the probe are sampled and averaged by a digital oscilloscope, then recorded and plotted.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 385
Figure 1.
8
E 4
.:. I-Z W ~ W U <:( ....I Q. II>
is
15
_ 10 E c: I- 5 z ~ 0 w ~ ....I
fh Q -10
-1 5
- 20
l
T
b u-____ ~ ____ ~ ______ ~ ____ ~ ____ ~
o 10 20 30 40 50 TIME 'jJS'
Transient displacements at the centre of PZT ceramic of 40 mm diameter and 1 mm thickness, excited respectively by a short unipolar pulse (a) and gated sine one (b).
INTERFEROMETRIC MEASUREMENTS
Figure 1.a and 1.b show the waveforms of the displacements at the centre of a PZT ceramic of 40 mm diameter and 1 mm thickness, excited respectively by the electric short unipolar pulse and the gated sine one. In addition to the first signal (T) in Figure 1.a, which corresponds to a thickness vibration of frequency 1.6 MHz, multiple echoes (L, Ll, L2, ... ) are detected. These L-pulses become even more important in Figure 1.b, where the displacement due to the thickness vibration is, on the contrary, much depressed. This is explained by the fact that the central frequency of electric excitation is near that of L-pulses (see next paragraph) but far less than that of thickness resonance.
In fact the L-pulses concern the cylindrical Lamb waves ori~inating at the edge of transducer and travelling laterally across the transducer lFigure 2). The maximum signal is obtained at the centre (focus) of amplitude about 10 nm (Figure 1) . When the optical probe beam is moved away from the centre, L-pulses are divided into two parts. Figure 2.a displays the signal detected at a point about 4 mm away from the centre. The thickness vibration remains constant as expected, while the first part of pulse L( 1) corresponds to the cylindrical plate wave converging from the edge and the second (2) to the cylindrical plate wave diverging from the centre after passing through it. By comparing the two calculated pulses in Figure 2.b, it appears clearly that the
386
Figure 2.
3 T
2 ,
E 0 :
ffi - 1 :::E w ~ - 2 ...J 0-
~ -3
PZT CERAMIC ,
o 2
LASER PROBE
2
4
r - - - - - - - - - --- - -,
, :8 1
i 2
,
, , , b ' L. ____ _ _ _ _ __ __ _ ..I
6 8 10
TIME 'j.lS>
Measured displacements of cylindrical Lamb waves at 4 mm away from the centre (a) . Pulse 2 which has passed through the focus is a phase shift of 'If /2 relating to pulse 1 as confirmed by the calculated pulses (b) .
pulse shapes have been changed from unipolar to bipolar. Such shape variation can be understood by a phase shift of 'If/2 which pulse 2 experiences on passing through the centre. This phenomena related to the focused waves has been reported in optics as well as in acoustics [6 - 8] . Up to now the discussion of the origin of the plate waves generation is still open, it is generally thought to be caused by the discontinuity of piezoelectric stress at the edge of plate produced during the polarization process.
Figure 3.
(:' 1
o 0.2 0 .4 0.6 0.8 1.0
FREQUENCY (MHz)
Resonance spectrum of multiple echoes in Figure 1 with the profile (dotted line) corresponding to the spectrum of pulse L.
387
In order to characterize quantitatively the plate waves, a spectral analysis method currently used in acoustic resonance scattering [9, 10), is applied to ' determine the group velocities. Figure 3 shows the Fourier transformation of the multiple echoes in Figure 1. The profile (dotted line) of the resonance spectrum which shows the interference between the multiple pulses corresponds to the spectrum of pulse L. The maximum positions in the frequency spectrum correspond to the different order n of resonance of the plates waves. If n, n + 1 are the two successive resonance modes , then the condition of constructive interference requires:
flk - k _ k _ 2(n+I)1I" 2n 11" _ 211" - n<i n - D - D - D (1)
where kn is the wave number of nth modal resonance and D the diameter of the plate. Suppose fn is the nth resonance frequency, we deduce from (1) the following formula:
flU) _ 211"(fn' 1 - fn) _ D Af V IT - 6k - Ll 6k--!O I g (2)
which approaches the group velocity , if the chan~e in the wave number is small between two successive resonances. The formula l2) allows therefore to calculate the group velocity from the frequency difference M. The experimental values V g, deduced from this method, are plotted versus the product of frequency by plate thickness (FD) in Figure 4. The small decrease of the group velocity with frequency , which is also marked in the wave trains of echoes of Figure 1 with low-frequency components arriving first, verifies the dispersion curve of the symmetric Lamb mode So in the low frequency range [11). The group velocity curve of Lamb wave So, which is calculated from the average values of longitudinal (VI = 3950 ± 50 m/s) and vertically polarized shear (Vt = 1925 ± 50 m/s) wave velocities in the sagittal plane of the piezoelectric plate, confirms qualitatively this dispersion feature. The discrepancy from the calculated curve arises probably from the uncertainty in the velocity determination of PZT ceramics .
Figure 4.
388
3.5
~ :. 3.0
>-I-C3 o iil 2.5 > Q. ::> o IX (!) 2.0
1.5
o
• • • ..
0.2 0.4
• ..
0.6
• • .. .
0.8
• • .. ..
..
1.0 1,2
FREQUENCY x THICKNESS (MHz x mm)
1.4
Calculated and experimental group velocities of the Lamb wave So plotted versus FD (MHz x mm) . (e) and ( ... ) are the experimental values, obtained on the PZT ceramics of 40 mm and 20 mm diameters, respectively.
Figure 5.
a r WArnR H
D .. ~_A:~ Ll
- - - - --PIEZOELECfRIC PLATE
b
? E '2 I :> p
~ ~ :e ~
UJ H' 0:0: :J '" '" UJ 0:0: a. H
0 10 20 30 40 50
TIME (ps)
Radiation paths of the piston model and that of the Lamb waves (a). The acoustic pressure is measured by a 0.6 mm wide band hydrophone at the distance Z = 24 mm, on the axis of the PZT ceramic of D = 40 mm diameter (b).
HYDROPHONE MEASUREMENT
As mentioned previously, the radiation of Lamb waves into the surrounding medium (usually water) can not be predicted by the piston model as in the case of thickness vibration. These waves radiate out the acoustic energy into water following the Snell-Descartes law (Figure 5.a), as propagating across the surface of transducer. Figure 5.b displays the acoustic pressure measured by a 0.6 mm diameter broadband hydrophone at Z = 24 mm on the axis of PZT ceramic plate of 40 mm diameter.
In Figure 5.b the direct plane wave (P), centred at l.6 MHz, is followed by the wave pulses (H, H'). According to the radiation path of Lamb waves indicated schematically in Figure 4.a, the arrival of the first head wave is given by:
D - 2 Z tgOL Z
2 V L + ""V-o -c-os"""O'--L (3)
where V L is the phase velocity of Lamb wave, V 0 = 1480 mls that of
sound wave in water and 0L = arcsin(Vo/V L) the radiation angle. For a
frequency of 750 kHz, close to the central frequency of the pulse H, the group velocity obtained from the experimental dispersion curves (Figure 4) is 2560 mls (phase velocity is a little greater than group one [11]) . The correspondent delay time T L is equal to 21 its, in good agreement with the
measured value 23 its (Figure 5.b). The temporal interval ~T = 13 its between the successive echoes (H and H') corresponds the mean travel time of Lamb waves through the face of transducer (Figure 1) . The slight perturbation marked in the waveform of head wave H is probably due to the interference
389
of a diffracted edge wave (E), which, according to the piston model, is an inverted replica of the direct wave and arrives at 21.1 /1s [1 - 2].
CONCLUSION
In addition to the classical thickness mode, cylindrical Lamb waves So have been investigated optically on pulsed circular PZT ceramic plates by using an optical heterodyne interferometer. With a spectral analysis technique, the group velocities of these waves were deduced quantitatively. A phase shift of 1f /2 was noticed when the Lamb waves passing through the centre of plate (focus). The radiations of these waves into water, which create head waves not predicted by the piston model, were detected by a hydrophone. The good agreement between the optical and hydrophone probe measurements confirms experimentally that the head waves originate from the radiation of Lamb waves into water.
ACKNOWLEDGMENTS
The authors would like to thank M. Talmant for the computation programs and A. Hayman for helpful discussions.
REFERENCES
f~l [3]
[4]
[5]
[6]
[7]
[8]
[9] [10]
[11]
390
J.P. Weight and A. Hayman, J. Acoust. Soc. Am. 63, 396-404, (1978). M.R. Layton, E.F. Carome, H.D. Hardy and J.A. Bucaro, J. Acoust. Soc. Am. 64, 250-256, (1978). J.C. Baboux, F. Lakestani and M. Perdrix, J. Acoust. Soc. Am. 75, 1722-1731, (1984). X. Jia, J. Berger and G. Quentin, in Colloque de Phrsique, (Proceeding of the First French Conference on Acoustics, Lyon, 1990), pp. 567-570, 1990. D. Royer and E. Dieulesaint, in Proceedings of 1986 Ultrasonics Symposium, IEEE, New York, pp. 527-530, 1986. M. Born and E. Wolf, Principles of Optics, (Pergaman, New York), p. 446, (1959). C.K. Jen, P. Cielo, J. Bussiere, F. Nadeau and G.W. Farnell, Appl. Phys. Lett., 46, 241-243, (1986). D. Royer, E. Dieulesaint, X. Jia and Y. Shui, Appl. Phys. Lett., 52, 706-708, (1988). M. Talmant and G. Quentin, J. Appl. Phys. 63, 1857-1863, (1988). G. Quentin, A. Derem and B. Poiree, J. Phys. France 50, 1943-1952, (1989). LA. Viktorov, Rayleigh and Lamb Waves (Plenum, New York, 1967), p.73.
EXPERIMENTAL STUDY OF REFLECTED BEAM PROFILE BY
TWO-LAYER PLATE SYSTEMS IMMERSED IN WATER
A. Jungman, Ph. Guy, G. Quentin and J. C. Le Flour*
Groupe de Physique des Soli des Universite de Paris VII Tour 23, 2 place Jussieu Paris, 75005 France *RNUR 8-10 Av. Emile Zola 92209 Boulogne Billancourt, France
INTRODUCTION
Ultrasonic examination of layered media may improve the design of these components. One of the major problems of these materials is related to the prediction of the adhesive strength of a joint between plate-like structures. The background of this approach is based on the idea that measurable changes in the elastic properties of the plate-to-adhesive interface should be associated with changes in the adhesive strength. This means that the reflection coefficient can be used to characterize the interface properties of the system. A number of methods have been proposed to detect disbonds, such as resonance [1], pulse echo and through transmission [2], ultrasonic spectroscopy [3, 4, 5], interface waves [61 and leaky Lamb waves [7, 8, 9]. A review of these techniques is given by Bar Cohen and al. [10].
In this present paper, we examine the reflection of an ultrasonic finite beam from a double layered system made of a steel plate loaded with an adhesive one. In order to investigate the properties of such a bonding agent, comparisons are carried out for a single plate and for a double layered system with good and bad bonding. Measurements are based on the modification of the dispersion curves of leaky Lamb modes, the reflection coefficients and the shape of the beam profile. Comparison with available theory will be presented.
THEORETICAL BACKGROUND
The derivation of the reflection coefficient is based upon a two-dimensional wave propagation, model in fluid loaded layered structure. The layers are assumed to be isotropic and stacked normal to the z direction of a cartesian coordinate system (x,y,z) such that the x-y plane coincides with the plane of each layer. The wave is supposed to be incident on the solid system from a fluid at an arbitrary angle from the normal direction z.Without loss of generality, the wave will propagate [5] along the x-direction. For this system of isotropic layers, the propagation process will be independent of the y-axis.
The analysis proceeds by solving the field equation in each layer in terms of wave amplitude. For each layer 1 or 2, the field variables will consist of the displacement components u and w in the x and z directions, respec-
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 391
tively and the stress components Uzz and Uxz. By satisfying appropriate continuity conditions at layer interfaces and at the fluid layered system interfaces, we obtain an analytical expression for both the reflection and transmission coefficients. Two types of inter-layer boundary conditions are treated. The first concerns situations where rigid bonding condition exists and the second concerns smooth bonds (often referred to as the kissing condition). Here rigid bonding requires all field variables, namely u, w, U zz and u xz to be continuous across a given interface namely, UI = U2, WI = W2, uzzl = uxz 2. For the smooth interface condition, on the other hand, continuity is required only for the normal displacement and stress: WI = W2, uzzl = u xz2. The remaining two conditions are invoked by setting Uxz = 0 for both constituents sharing the interface. Hence the u displacement component is not involved in satisfying interface conditions for the smooth bonding case.
By investigating the behavior of either the reflection or transmission coefficients, the propagation characteristics will be readily available. In particular, all frequency dependent critical angles can be easily identified. The reflection coefficient for plane harmonic waves incident from a fluid onto a solid surface is determined by writing the Fourier transforms of the wave potentials with respect to the x coordinate, and assuming exponential solutions in the z coordinate. These lead to the expression [12]: R = <1>' rI<I>f where <1>' f and <I>r are the reflected and the incident amplitudes of the potential respectively.
Upon reflection of the incident beam into the fluid, the beam profile is modulated by the reflection coefficient R(O where ~ is the component of the wave vector along the x direction. It gives, for the beam profile [12]:
·00
~'f = h f ~f(O R(O exp(iex - iefz) de -00
EXPERIMENT AL METHODS
The experimental study is performed in order to evaluate the modifications in the leaky Lamb waves when the characteristics of the interface between a steel plate and an epoxy layer are changed. Three different approaches are carried out, such there all based on the generation, the propagation and the leaking of guided waves in fluid-coupled double layered systems. In the first one, we concentrate our attention on the analysis of the phase velocity dispersion curves. In the second one, we measure the reflection coefficient R( 0) for a fixed number of the product F.d (F is the ultrasonic wave frequency, d the thickness of the steel plate). And in the last one, we consider the distortion of the reflected beam profile, when a Lamb mode is excited. All these different methods are related to the measurement of the phase velocity of a given leaky guided wave. In the first approach, dispersion curves are obtained either by the critical angle technique or by spectral analysis. The critical angle technique gives plates modes by identifying the angles at which a minimum occurs in the reflection coefficient, at a fixed frequency. With spectral analysis, the modes are obtained by measuring the position of minima in the specularly reflected spectrum, at a fixed angle. In both techniques, the results are presented in the form of varying the phase velocity with the product of the frequency F by the thickness of the plate d.
In the second approach, the reflection coefficient R( 0) is plotted as a function of the angle of incidence O. Information about the mode can be obtained not onl;, from the angular position of the minimum, but also from the general shape t width, depth) of the dip in the reflected profile.
Finally, in the third approach, we are looking at the energy distribution of the beam profile, which exhibits, under certain conditions, a null zone between two lobes [11]. Angular position of the mode (i.e. phase velocity of the mode, according to Snell's law) is measured when the reflected beam profile has the deepest minimum. But in addition information about the interface can be extracted from the general shape of the reflected beam.
392
Receiver
L-__________ ~------ X
Figure 1
EXPERIMENTAL SET-UP
The experimental configuration for these measurements is illustrated in Figure 1. A double-layered specimen, immersed in water can turn around a vertical axis. Two similar broadband transducers (central frequency Fe = 2.25 MHz, diameter = 12.5 mm) are oriented under oblique incidence. The transmitter is fixed while the receiver has two degrees of freedom. One allows it to turn around the y axis to fit in the geometry of the specular reflection for any incidence. The other allows it to translate in a direction perpendicular to the axis of the reflected beam, in order to scan the field. The mechanical positioning system affords close control of the three degrees of freedom (2 rotations and 1 translation). All of the movements are computerassisted, with a precision of 0.01 degree and 0.01 mm. Data acquisition of time signal and processing (average, normalization, Fourier transform, deconvolution) are achieved by an on-line microcomputer. Storage and further processing of the data allow to plot the dispersion curves, the reflection coefficient as well as the beam profile.
Sample characteristics are presented in Table 1. They are made up of a steel plate loaded by an epoxy layer. Bad bonding is obtained by adding a thin layer of oil between a galvanized steel plate and an epoxy layer. In all our experiments, the product F.d is maintained constant. This is achieved by adjusting the frequency of the excitation of the broadband ultrasonic transducer. Changes in the interface conditions are detected by measuring the shift of the angle 0 at which a minimum occurs in the reflection coefficient compared to the angular position of the mode for a reference sample (i.e. single plate or perfectly good bond).
EXPERIMENTAL RESULTS AND DISCUSSION
On Figure 2, we have plotted the theoretical velocity dispersion curves for a single steel plate immersed in water. The slight discrepancy with the experimental points for the mode So in the frequency range 2 MHz· mm < Fd < 3 MHz.mm where the dispersion is high, may be due to the anisotropy of the plate. This problem has been discussed recently elsewhere [13]. The experimental results on Figure 3 for the double layer system C, considered as a good bonding, are also compared to the theoretical prediction
Table 1. Samples characteristics
Sample Sleel lh.ckncss (mm) Epoxy Ih,ckness (mm P 0.94 B U.'.I4 .46 B' U.'.I'.I U.4 L ).94 0.70 C 1.9'.1 u.75
393
394
8
III 6 ::I. .....
H
.... . ....... . . ' .. . . ...... ..
• • • • •••• ••• • K. ". ' ,. , ... . ~ .... .... ' ..... ~
>-fo< ~
U 4 0 ...;! tal > tal til .c :z: 2 0.
o o
Figure 2.
8
~
III :1 6 .... a >-.... ~
U 0 4 -...;! tal >
I-tal til .c :z:
2 I-0.
..,..
0 0
Figure 3.
. ~ , ,M
.. .. ~ .. .. " "
. " ~'
. .. • .• : :: :~:~::: :: tI::: I 1.111 •••••• • •• • •• •• . ....... . • • • • ..... " M
If' • .. "' ..... ~ . '.,. .,.
I
2 4 6 8
Fd (MHz . mm)
Dispersion curves ( .. , theoretical, * * * experimental for sample P).
•• •• •• •••• 101
." .. .. . . . ' ., "
.. : .. ... .. ... ..... . ...
.... .. ...... ....
• ,,' M" M ",.tII""
N •••• •
....... . . . .. :: :: :: :: :: : : : :1 111 ...... • . . .. . . .... . ..... .. ....
2 4 6 Fd (MHz . mm)
8
Dispersion curves ( , .. theoretical, * * * experimental for sample C).
calculated for the single plate. Experimental points do not exhibit any significant difference either. And hence it demonstrates that the existence of an epoxy layer on the back of the steel plate let the position of the modes unchanged.
If we consider now the reflection coefficient R( 0) of Figure 4, we observe almost no shift between a single plate and double layer systems with good (sample C) and bad (sample C') bond. The difference, less than 4%, is close to the precision of the measurements. Comparison between the single metallic plate and the hard bonding system C demonstrates as with the dispersion curves, that non significant change in the reflection coefficient can be noted. In addition, the minimum in the reflection coefficient for a bad adhesion is obtained for the same angle. Theoretical computation [14] of the phase velocity of the So mode confirms the experimental observation about the fact that no difference exists between a steel plate and a loaded plate with either good or bad bonding.
Finally, we present on Figure 5 the beam profile of samples Band B'. Again all three plots look rather similar, and particularly the position of the minimum between the two lobes.
These results confirm that neither good nor bad adhesion can be detected on a double layer steel-epoxy sample. Although the impedance of adhesive (zL = 3.111 106 SI, zT = 1.483 106 SI) is significantly different from that of
water (zL = 1.480 106 SI, zT = 0 10 6 SI), plate modes are not affected by
such a change in the system which behaves as a single plate. Computations are in progress to predict for which value of the parameters of the adhesion, the modes exhibit a different behavior. As a consequence, no reliable ultrasonic test based on leaky Lamb waves appears to be useful for nondestructive testing of adhesive properties of a double layer system. Fortunately, this simplified double layer model is not realistic, whereas an adhesive engineering structure involves at least three layers (two metallic plates with a glue layer in between).
1.0
< x+
0 .8 o )( .. , xx ' .
0 0000000
Col 0.6 Q
~ H ...l Ilo.
~ 0 .4
0.2
18 21 24 27 30 ANGLE (0)
Figure 4. Reflection coefficients (F = 2.91 MHz, for samples P (0), C (*), Co (x)) .
395
Figure 5.
TRANSVERSAL DISPLACEMENT (mm)
Reflected beam profiles at Fd = 2.72 MHz·mm.
36
for samples P (_._._._), B (- - - -), B' (_ .. - .. _)
Preliminary results show for such a structure a more complex shape of the reflection coefficient which involves the modes of the two plates.
CONCLUSIONS
The problem of characterization of a steel--epoxy interface has been addressed . Several sets of experiments have been carried out to emphasize a possible change in the speculady reflected beam at a Lamb critical angle. Results have been presented using different characteristic properties of the leaky Lamb waves to discriminate between a single steel plate and a steel plate loaded with an epoxy layer , and also to distinguish a good adhesive bond from a bad one. Phase velocity dispersion curves have been plotted as well as reflection coefficients and beam profiles. None of these exhibits significant features which may lead to an interpretation of the properties of bonding. Theoretical calculations, which have been undertaken to verify these unexpected results, agree quite well with the experimental results .
ACKNOWLEDGEMENTS
The authors are greatly indebted to Professor Adnan Nayfeh for the fruitful discussions and for assistance with theoretical computations. This work was supported by the Regie Nationale des Usines Renault (Direction des LaboratOires) under grant VI 6965.
396
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
C.V. Cagle, Ultrasomc testing of adhesive bonds using the Fokker Bondtester, Materials Evaluation, 26, 362, (1966). J. Krautkramer, and H. Krautkramer, Ultrasomc testing of materials, Springer-Verlag, 23-29, (1969). F.G. Chang et al., Principles and Application of Ultrasomc Spectroscopy in NDE of adhesive bonds, IEEE Transactions on Sonics and Ultrasonics, SU-23, W 5, 334, (1976). A.K. Mal, C.C. Yim and Y. Bar-Cohen, The influence of material dissipation and imperfect bonding on acoustic wave reflection from layered solids, Review of Progress in Quantitative NDE, 7, D.O. Thomson and D.E. Chimenti edit.,Plenum Press, New York (1987). C.H. Guyott and P. Cawley, Measurement of the ultrasonic vibration of adhesive joints, Eurotech 227, St. Etienne (1987). S.l. Rokhlin, M. Hefets and M. Rosen, An ultrasonic interface-wave method for predicting the strength of adhesive bonds, J. Appl. Phys. 52(4), 2847, (1981). Y. Bar-Cohen and D.E. Chimenti, NDE of Composites laminates by leaky Lamb waves, Review of Progress in Quantitative NDE, 5B, D.O. Thomson and D.E. Chimenti Edit., Plenum Press, New York, 1199, (1968). D.E. Chimenti and A.H. Nayfeh, Leaky Lamb waves in fibrous composite laminates, J. Appl. Phys., 58(12), 4531, (1985). F. Leomy, M. de Billy et G. Quentin, Milieux en couches et ondes dispersives, Revue Phys. Appl., 23, 1547, (1988). Y. Bar-Cohen, A.K. Mal and C.C. Yin, Ultrasonic evaluation of adhesive bonding, to appear in ASNT Handbook, vol.6, K.E. Green Edit., (1990). L.E. Pitts, T.J. Plona and W.G. Mayer, Theory of non specular reflection effects for an ultrasomc beam incident on a solid plate in liquid, IEEE Transactions on Somcs and Ultrasonics, SU-2U, W 2, (1977). A.H. Nayfeh, D.E. Chimenti, L. Adler and R.L. Crane, Ultrasonic leaky Lamb waves in the presence of a thin layer, J. Appl. Phys., 52(8), 4985, (1981). Ph. Guy, A. Jungman, G. Quentin and J.C. Le Flour, Ondes de Lamb dans les milieux multicouches, Congres Franc;ais d' Acoustique, Colloque de Physique, Supplement C2, (51), Les Editions de Physique, 1249, (1990). A.H. Nayfeh, Private communication, (1990).
397
STUDY OF PHASE TRANSITIONS BY FREQUENCY DEPENDENT
PHOTOACOUSTIC MEASUREMENTS
Seiji Kojima
Institute of Applied Physics University of Tsukuba Tsukuba, Ibaraki 305, Japan
INTRODUCTION
Currently the photoacoustic effect has been widely used in the study of various kind of phase transitions. The photoacoustic signals highly reflect the thermal properties of a specimen, and it has been frequently used to investigate the change of thermal properties in the neighbourhood of a transition temperature. In the early stage, it has been often applied to determine a transition temperature accurately, since Florian et al. reported the first observation of the remarkable anomaly at the melting of a gallium specimen [1]. However, in fact, such an information can be also obtained by the conventional instruments of thermal analysis, for example, DTA, DSC, etc.
The recent new trend is the application of the photoacoustic technique to the study of dynamic properties of various transitions. Due to the non-equilibrium statistical physics, heat capacity is a dynamic susceptibility. According to Kubo formula, it is given by the Fourier transformation of the correlation function of the fluctuation of enthalpy. In the gas microphone method, photoacoustic signals strongly connected to the heat capacity at the modulation frequency of an incident beam [21. Therefore the frequency dependent photoacoustic measurements become to be a powerful tool to study relaxational processes through thermal dispersion [3].
In the present work, the gas microphone photoacoustic technique has been applied to investigate the thermal dispersion of the solid-liquid phase transitions of pure Sn and Sn-Bi alloy, and the glass transition of glycerol.
EXPERIMENT AL
In the present work, all photoacoustic measurements were done by the original handmade setup as shown in Figure 1. The photoacoustic signals are excited by 780 nm visible light from a semiconductor laser, whose intensity is electrically modulated in the frequency range from DC to 1 MHz. The output diverging beam from a laser is slightly focused on the surface of a specimen with a spot size of 1 mm to prevent local heating. The photoacoustic signals are detected by a condenser microphone with diameter of 1/2 inch, which covers the frequency range from 1 Hz to 200 kHz. The microphone is connected to the temperature variable photo acoustic cell by the use of a probe tube with a length of 150 mm. The output signals from the microphone are always amplified to more than 1 mV by utilizing both a pre-amplifier and a main amplifier, and finally analyzed by the use of a two phase lock-in amplifier.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 399
Figure 1. Experimental setup for frequency dependent photoacoustic measurements.
THERMAL DISPERSION IN THE VICINITY OF TRANSITIONS
Liquid-Solid Phase Transitions of Pure Sn and Sn-Bi Alloy
Melting is well-known general phenomena in material science and also one of the technologically important problems. Various materials have been studied experimentally and also theoretically. Although many phenomenological theories have been developed, they do not correctly interpret a lot of experimental results. The physical mechanism of the melting has thus been unknown up to now. Especially, the experimental approaches on opaque material like metal were scarcely investigated.
With regard to photoacoustic investigation, Florian et al. first studied about the melting and freezing of a gallium specimen by the gas microphone method [2]. In their experiment, the photoacoustic amplitude suddenly decreased on heating in the vicinity of the melting temperature, whereas on cooling the amplitude never showed such a dramatic decrease. They interpreted the mechanism by the endothermal and exothermal processes.
In contrast, Korpiun and Tilgner proposed a theoretical model (KT 4 model) on the photoacoustic anomaly at a first order phase transition. In their model, the oscillation of an interface between a high temperature and a low temperature phases is considered under the cyclic illumination to a surface. From the analysis of the thermal diffusion equation, they concluded that the photoacoustic behavior is the same for passing the phase transition from below as above.
Kojima studied both the melting and freezing of Sn specimens carefully, and confirmed that the photoacoustic amplitude and phase show the remarkable change in the immediate vicinity of both a melting and a freezing temperature r5]. In the present study, the frequency dependence of the anomaly has been turther investigated for pure Sn and Sn-Bi alloy.
Figure 2 shows the temperature dependence of the photoacoustic amplitude and phase in the neighbourhood of a melting point in a pure Sn specimen of 99.999% purity. The behavior is qualitatively explained by the KT model. It is well-known in metallurgy that the interface between a solid and a liquid phases of pure Sn is smooth, so in such a case, one dimensional model like the KT model can be consistent with the experimental results.
Next, the case in which a rough interface exists was investigated. It is also known that when Bi is mixed to Sn in several atomic percents the interface shows a drastic change from a smooth to a rough surface. Another change due to the mixing of Bi is that a melting temperature decreases and spreads into the finite temperature region between solids and liquids. Therefore in this region solid and liquid phases always coexist, and the photoacoustic
400
Figure 2.
UJD ·~ :;:)'"
..... " - .0 ere; ~«
99.9999% Sn
f\ 950Hz
-J"--~
A
225 230 235 2LO TEMPERA T URE t'Cl
Temperature dependencies of the photoacoustic amplitude A and phase ¢ in the neighbourhood of a melting temperature of a Sn specimen of 99.9999% purity.
anomaly due to the interface oscillation is expected to occur in this temperature range.
Figure 3 shows the temperature dependence of the photoacoustic amplitude A and the phase ¢ around the solids and the liquids in Sn-Bi alloy. Both the amplitude and the phase show anomaly in the wide temperature range between 225' C and 180' C, where the two phases coexist. From the results it is concluded that the photoacoustic anomaly is caused by the interface oscillation as predicted by the KT model.
The frequency dependence of the decrease of the amplitude in melting was also investigated, because this quantity is strongly connected to the flatness of
Figure 3.
'" .. 35
0. .g 18 UJ III « 5: 0
50 100 150 200 250 300 TEMPERATURE (cOl
Temperature dependence of the photoacoustic amplitude A and phase ¢ in the neighbourhood of the melting region of a Sn-Bi specimen of 99 .999% purity.
401
Figure 4.
« <I
1.0 .------.---.-----r--r---r--r-I-----.-.---I -~-Ir--l
0.8
0.5
"0 . -a. 0.4 '0 .
0' .
0.2 0
0 I
1 10
0 0 o··p
' P
ci .. 0, .!J
o . I '.!.' •
100 lk FREOUENCY (Hz)
o 99.9999%Sn [J Sn-S%Bi
c> , c ,
c'" '. I . .C? .1.
10k lOOk
Frequency dependence of the normalized decrease of the amplitude f1A on melting in pure Sn and Sn-5%Bi specimens.
the interface. To normalize the maximum decrease on melting, the quantit,zr f1A is introduced by the equation f1A = (As - Amin)/ As, where As, Amin 1S the amplitude far below the melting temperature, the minimum amplitude on melting, respectively. Figure 4 shows the frequency dependence of the quantity f1A for a pure Sn and a Sn-5%Bi specimens. It is found that f1A of Sn-5%Bi is less than that of pure Sn in all the frequencies. The mechanism of the decrease of A is explained as follows. First the thermal waves are generated at the surface of a specimen due to the periodic illumination. Then waves reached the interface and cause the interface oscillation, which accompanies the absorption and the release of the latent heat alternatively. Consequently the waves from the surface are partially canceled by the waves generated by the oscillation. From the consideration it is concluded that the difference between two samples reflects the change of the flatness of interface. On the other hand, it is found in both cases that f1A decreases gradually as the frequency increases. This fact is considered to be due to the friction against the movement of an interface as discussed in [3]. It strongly suggests that a multi-relaxational mechanism exists in both dispersions.
glyce rol
180 190 200 210 220 230 240 TEMPERATURE(K)
Figure 5. Temperature dependence of A -2 and ¢ of glycerol.
402
1000 -. glycerol
-. 'eo
g > 100 u z
'"0 .. .. .~.
w ::> 0 w 0::
... LL 10 e .
4.4 4 .6 4.8 5.0 10001T
Figure 6. Vogel-Tammann-Fulcher law of glycerol.
Thermal Dispersion of the Glass Transition of Glycerol
The viscosity of supercooled liquid increases on cooling and finally goes to infinity at a glass transition temperature. The temperature dependencies of viscosity are classified into two types. Strong liquid shows the Arrhenius type behavior and fragile liquid obeys the Vogel-Tammann-Fulcher law (VTF) [6]. The Arrhenius behavior is deduced from an activation process, whereas VTF has been considered to be an empirical formula. Currently some theoretical derivations of VTF are discussed in the connection with the relaxational process of a glass transition, for example the model of the hierarchically constrained dynamics [7] . However experimental results of VTF and relaxational properties are not enough. Therefore the glass transition of glycerol, which is a typical material of fragile liquid, was studied.
Figure 5 shows the temperature dependencies of A -2 and ¢ of glycerol around the glass transition temperature, where a photoacoustic signal is inverse proportion to a square root of specific heat . From the peak of ¢, VTF was determined as shown in Figure 6. Next, the relaxational behavior was studied by the Davidson-Cole formula (DCF) [8] as shown in Figure 7. From these results parameters of VTF and DCF were determined as shown in Table 1. As to the VTF parameters, the values of photoacoustics are similar as that of dielectric and elastic measurements. In contrast, as to the value of DCF of photoacoustics is found to be less than the values of other methods.
The dielectric and the elastic dispersions are caused by the relaxation of dielectric polarization and strain, respectively. Whereas the photoacoustic dispersion corresponds to the thermal dispersion, which includes the degree of freedom of both polarization and strain. Thus the distribution of relaxation time of the thermal dispersion is expected to be broader than that of the dielectric
Figure 7.
glycerol
- fJ= 0.38 217K
Cole-Cole plot of the complex specific heat of glycerol at 217 K. The solid line shows the calculated values of DCF at f3 = 0.38.
403
Table 1. The values of parameters appeared in VTF and DCF
VTFa DeFb
method log fo To(K) A(K) S
photoacoustic 11.4 120 2011 0.38 ± 0.02 (217K)
dielectric9 13.8 137 ± 10 1740 0.S8±0.03
elastic10 16.7 129 2310 0.42±0.OS
a : f = foexp{-A/(T-To)}, b c(w) = (l+iuJT) -13
and the elastic dispersions. This means that the parameter of DCF of the thermal one can be the least. So the present experimental result is interpreted by the above consideration.
In conclusion, the photoacoustic measurements by the gas-microphone method is confirmed to be a powerful tool to study the relaxational properties of a glass transition through the thermal dispersion. This method can apply to any relaxational process even if a relaxational process has no interaction with dielectric polarization and strain.
ACKNOWLEDGEMENT
This work is partially supported by Grants-in-Aid from both the Ministry of Education, Science and Culture of Japan, and the University of Tsukuba Special Project Research. The author is grateful to Mr. H. Hoya, Mr. S. Doi, Mr. T. Kashiwada and Mr. S. Nakahira, of the University of Tsukuba, for the fabrication of the experimental setup and the technical assistance. The author is thankful to Prof. T. Suzuki for the valuable discussion.
REFERENCES
[1]
2 3 4 5 6 7
404
R Florian, J. Pelzl, M. Rosenberg, H. Vergas and R Wernhardt, Phys. Stat. Sol. (a) 48:K35, (1978). A. Rosencwaig and A. Gausho, J. Appl. Phys., 47:64, (1976). S. Kojima, Jpn. J. Appl. Phys., 28-1:228, (1989). T. Ko~piun and R. Tilgner, J. Appl. .Phys., 51:6115, (1980). S. KOJIma, Jpn. J. Appl. Phys., 27-1.226, ~1988). K.L. Ngai, J. Non-Cryst., Solids 95 & 96:969, (1987). R.G. Palmer, D.L. Stein, E. Abrahams and P.W. Anderson, Phys. Rev. Lett., 53:958, (1984). D.W. Davidson and RH. Cole, J. Chern. Phys., 19:1484, (1951). N.O. Birge, Phys. Rev. B, 34:1631, (1986). YH. Jeong, S.R Nagel and S. Bhattachya, Phys. Rev. A, 34:602, (1986).
MULTICHANNEL ACOUSTOOPTICAL MODULATORS AND THEIR
APPLICATIONS IN THE DEVICES OF SIGNAL PROCESSING
ABSTRACTS
S.V. Kulakov, O.B. Gusev, D.V. Tigin, V.V. Molotok
Leningrad Institute of Aviation Instrument Making 190000, Leningrad, Hertzen st. 67, U.S.S.R.
The authors are the representatives of a not great research group, which is engaged in the acoustooptical (AO) methods applications for information (signal) processing from the early sixties. Within the great number of AO-devices the more perspective are the space-time electrical signal processors. The basis of such processors are the multichannel AO space-time modulators. The problems of design and manufacturing of the bulk multichannel AO-modulators are under consideration. Some results of their use in the information processing devices are presented.
MULTICHANNEL BULK AO-MODULATORS, PROJECTION AND ELABORATION
The wideband multichannel bulk AO-modulators elaboration is connected with the solution of a number of complex problems. The main problems are: the achievement in one channel of AO-modulator of a big value of BT-product (BT ~ 1000); the creation of a large number of parallel channels; the ensuring of the minimum quality of the interchannel crosstalks (acoustical and electrical); the ensuring of the high electrooptical efficiency and high optical quality, the providing of stability, etc. Making a close study of the world experience, and also being based on our own tests and errors, we came to the satisfactory solution of the main problems of the AO-multichannel modulators construction in the scientific laboratory conditions.
Materials for Piezotransducers and media of interaction
In the case of designing AO-modulators for radio wave meter band signals (30 - 300 MHz) the plate half-acoustical wave piezotransducers are used from monocrystal LiNbO 3 (X-cut for shift acoustical wave excitation and y + 36° - cut for longitudinal waves). Different monocrystals can be used as an interaction medium of AO-modulator and the best one is the monocrystal Te02 for shift acoustical waves excited in [110] direction (fmax < 150 MHz) and for longitudinal waves in [001] direction (fmax < 500 MHz). The maximum usable frequencies are limited by the acoustical waves attenuation. The piezoelectrical plate is attached to the body of AO-modulator by means of Cu-In layer. The layer must have a sufficient thickness to compensate the effects, which appear on account of the temperature joint patterns expansion.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 405
The AO-interaction efficiency in Te02 modulators can be reached. (30 - 100)%/W in normalized frequency range M/fo < (0.3 - 0.5).
Plate LiNb0 3 piezotransducers for VHF AO-modulators (300 - 3000 MHz) should have too small thickness (a few units or parts of micron). The piezoplate polishing up to such thickness is a very complex technological problem. In this case the piezotransducers are manufactured from the vacuum deposited halfwave ZnO films. As a rule the longitudinal acoustic waves are used in VHF AO-modulators on account of their small attenuation in comparison with the shift waves. The monocrystals LiNb0 3 (X-<:ut) or GaP (Z-<:ut) are used as the media of interaction. In the normalized frequency band &f/fo = (0.3 - 0.5) it can be reached to the AO-interaction efficiency about (0.5 - 5.0)%/W. The delay time T of acoustic waves in the active part of the interaction medium is limited by the attenuation or by the diffraction spreading of acoustic waves. The second reason is particular essential when there is strong anisotropy of acoustic properties of AO-interaction medium near the direction of acoustic beam propagation (for example, near [110] direction in Te02 for shift type of acoustic waves). To compensate this effect it is necessary to increase significantly the surface of the eradiating piezotransducer. In AO-modulators of meter-band radio wave signals parameter T can achieve a few tens of microseconds, and in AO-modulators of decimetric ban - a few microseconds.
Wide-Band AO-Modulators Construction Peculiarities
In wide-band AO-modulators which act in a Bragg's isotropic regime (M/fo ~ 0.3 - 0.5), the permitted length of piezotransducer is limited by the value of 11£ (the more frequency bandwidth is, the less is the piezotransducer's size). The AO interaction effectivity decreases with the decrease of the length of piezotransducer. And for a normal AO-modulator functioning it is necessary to increase the driving electrical power density. This is not a good thing on a number of reasons (the electrical destruction danger, heating effects, etc.). Furthermore the restriction of the piezotransducer's sizes leads to the limitation of the piston (Fresnell's) piezotransducer's zone D = VT, where V is the acoustic waves velocity. However, this factor is not a limiting one in VHF frequency range in comparison with the acoustic waves attenuation.
In wide band OHF and VHF ranges AO-modulators the high AO interaction efficiency can be reached by two methods: by performing of anisotropic Bragg diffraction mode conditions or by using a multielement piezotransducer such as antenna array with autoadjusting by the Bragg's angle. The first heuristic method can be used only for input electrical signal frequencies which correspond to some conditions. These conditions are determined by the happy combination of acoustic and optic properties of the interaction medium. For example, it can be used for low part meter band radiosignals in the case of shift acoustic waves in [110]-<:ut Te02 [1] or in a frequency range (2-3) GHz in Y+120' - cut LiNb0 3. However, in a decimetric wave range this method is not accomplished because the longitudinal acoustic waves are usually used to maintain the small waves attenuation.
Multielement antenna array type piezotransducers are constructively enough complicated and hard to execute. These difficulties are becoming especially sensitive when one tries to make a multichannel AO-modulator.
The technological simplicity of making piezotransducers from ZnO films on non-flat surfaces discovers one more simple method of creating wide band VHF Bragg's AO-modulators. The crystal surface on which one deposits the ZnO film is made cylindrical. It leads to the expansion of a piezotransducers eradiation angle spectrum, and as a result, to the expansion of a frequency range of AO-interaction. The precise calculation of AO-frequency interaction characteristic of AO-modulator with cylindrical piezotransducer is suffiCiently complicated. However, even a rough estimation shows that in this case we can get the significant gain in the width of the range. When the length of the flat piezotransducer L = 3 mm, the central frequency fo = 1.5 GHz and the velocity of acoustic waves is 6.5.10 6 mm/sec, the 11£ ~ 40 MHz AO-inter-
406
action frequency range is maintained. The angular spectrum of :piezotransducer eradiation in this case is evaluated by the equation of A¢ = V ItfL) ~ 1.5.10-3
rad. Piezotransducer having the same length but deposited on a cylindrical surface with radius R = 150 mm maintains the angular spectrum of eradiation A¢ = L/R ~ 20 . 10-3 rad. The increase of the angular spectrum leads to the proportional expansion of AO-interactive frequency range, which in this case has to form the meaning M ~ 500 MHz. Such approximate estimation gets good enough confirmation in experimental patterns of AO-modulators with cylindrical shape piezotransducers. Of course the expansion of frequency range is accompanied by the corresponding decrease of AO-interaction efficiency. But this idea is a key one in solving the problem of electrical power density reduction inside the piezotransducer.
Wide Band Electrical Matching of Piezotransducer
The electrical properties of piezotransducer in the operating frequency range can be described quite exactly by an electrical equivalent scheme of a fourelement two pole network [2]. The scheme consists of a parallel connected capacity Co and a consequent LCR network. In this scheme Co is the static capacity of a squeezed piezotransducer. The consequent LCR network characterizes the reaction of a piezotransducer at the electrical input signal caused by the acoustic resonance. Resistor R reflects the active losses of an electrical energy in a piezotransducer - useful and dissipative. In this case the losses of electrical energy, which is transformed to the acoustic wave energy, are considered as useful losses. The parameters of the electrical equivalent scheme elements are usually determined by electrical measurements of frequency characteristics of the input piezotransducer's impedance.
In AO-devices of a wide band electrical signal processing it is necessary to maintain the input signal transformation to an acoustic wave and then - to the optical signal with small enough distortions in the cited frequency band. The frequency distortion of the signal are determined by the frequency characteristics of an electro-acoustic transformation, by the frequency dependence of an acoustic wave attenuation and other reasons. The problem of a frequency distortion correction in AO-modulator is being solved by the insertion of an electrical matching network. In OHF and VHF frequency bands a good qualit:y of matching can be reached in the normalized bandwidth M/fo = (0.3 - 0.5) by using the two stages of bandpass networks. Constructively the matching network can be accomplished by use of concentrated electrical elements with a quater-wave transformer or by use of microstrip circuits. For example the two-stage matching network for ZnO thin film piezotransducer provides the standing wave coefficient SWC = 2 in a normalized frequency range Mlfo = 0.3 with the maximum flat type of frequency characteristics of an equivalent attenuation.
The construction of an electrical excitation network with the electrical matching elements is an especially complex problem in the realization of wide band multichannel modulators. Here besides to the difficulties of a constructive placing of matching network elements of different channels in a very limited space it is necessary to solve the problem of a reliable electrical separation of the neighbouring channels of an AO-modulator.
AO-modulators of OHF - VHF frequency ranges, especially - multichannel AO-modulators have a perspective construction of matching networks assembly. It is a set of special flat modules on the base of a microstrip rectangular coaxial waveguide. The excitation of a piezoplate or a thin film ZnO piezotransducer is achieved with the open end of a rectangular waveguide (the excitation of IIcapacitive joint II type). All other elements of matching network are included into the microstrip geometry of a waveguide. Alongside with a good electrical separation of neighbouring channels (> 40 dB) this construction provides effective removing of heat from the piezotransducer. Such construction of electrical excitation network permitted to make the experimental patterns of a multichannel AO-modulator of a VHF range with the step 0.3 mm between the channels.
407
MULTICHANNEL AO-SIGNAL PROCESSORS
In a coherent multichannel AO-processors the output optical signal is formed by the coherent summation of output optical beams from the separate channels. It causes strong demands to the identity of the separate channels characteristics and to the quality of their electrical and acoustical crosstalks as well. Furthermore, there appears the problem of effective heat rejection from the piezotransducer and from the media of interaction. This problem becomes complicated due to the fact that the crystals usually used as media of interaction have a very small heat conductivity. The heating effects lead to the necessity of average input electrical power limitation. To provide the sufficient level of the optical power on the surface of photodetector we must increase the output laser power.
The first publications on the multichannel AO-signal processors were devoted to the description of different systems for the parallel antenna array signal processors [3, 4] . Many specialists are still persistently working at the problems of processing algorithms perfection . The new perspective application of multichannel acoustooptics appeared in the last few years. These are the algebraic AO-processors, which are intended for the high speed linear mathematical transformation of the input matrix data (analog and digital) [5]. AO-algebraic processors have a potential superiority compared to the electronic analogs if their joint characteristics are taken into account : high calculation speed, gabarits and energetic requirements.
This article presents the results of investigations, which were obtained by the authors when the multichannel AO-signal processors were under developmen t.
AO-Spectrum Analyzer with a High Resolving Power
In a on~hannel AO-space-integrating spectrum analyzer the frequency resolving power AfR is limited by the duration T of the acoustic wave
process, which is in the aperture of the optical system: AfR = liT
AO-spectrum analyzer with the multichannel AO-modulator and multielement delay line has a more higher resolving power (see Figure 1).
A delay line permits to make the raster-type insertion of a signal into an AO-modulator. The summary duration of a processed signal is equal to T ~ = NT, where N is the number of channels. T - is the time aperture of
a one channel. When the input signal is a sinusoidal one, i.e . U(t) = Uosin(wt), the light distribution in a Bragg's diffracted light beam can be described by the equation [6]:
S(wx,Wy) ~ Uo sinc[0.5(wx - Dw/vd)d] sinc(0.5wyd)·
sin[0.5N (Wy - Dw/Vd)] I sin [0.5 (wy - Dw/Vd)]
where Wx and Wy - space frequency coordinates connected with the space
Figure 1. AO-spectrum analyzer with a high resolving power.
408
a HUIiU b . 8 Figure 2. The photographs of the diffracted light distributions in the
front (a) and in the back (b) focal planes of the Fourier lens.
coordinates X and Y by the equations: Wx = 21rXI AF; Wy = 21rY I AF; V -the acoustic wave velocity within the medium of interaction. D - the aperture size of an AO-modulator along the X-direction. d - aperture size of one channel along the Y -direction, F - the focal distance of the lens and A - the light wave length.
From the cited equation it follows that when the frequency of a signal w is changed linearly, the central maximum of the focused diffracted light beam moves the raster type scanning in the back focal plane of the lens. In this case the frequency resolving power is equal to afR = liNT. Figure 2 shows the
diffracted light distribution in the front and the back focal planes of the Fourier lens .
Figure 2.a was made when the input signal was a sinusoidal one. Figure 2.b was made when the input signal was a chirp one. The chirp rate during the time interval T was a insignificantly small.
The identity of the amplitude-frequency and the phase-frequency characteristics of different AO-modulator channels is quite easily provided by the tuning
. ~oo .~~ ~~'~ '·" )·· r . ........ "': ... .., , •
b
Figure 3.
c
The phase shift correction in different channels of AO-spectrum analyzer: a) the outward appearance of phase corrector; b) and c) the output diffracted beam light distri butions before and after the phase correction.
409
Figure 4. The scheme of AO-convolver.
elements in electronic excitation networks . However, one serious problem still remains. The problem consists in the presence of accidental phase signal shifts in individual channels. These phase shifts lead to the disfocusing of the output optical distributions. The electronic wideband phase correctors can not provide the necessary stability in time. The successful problem solution was found in using the optical phase correctors, which were made in each channel as a flat glass transparent plate, placed close to AO-modulator . The incline angle of each plate to the optical axis can be smoothly changed. It leads to changes of effective thickness of the plates, and as a result to the independent phase correction of optical signals in each channel. This phase corrector has a linear phase-frequency characteristic in a practically infinite frequency range of the input electrical signals. The sensitivity and phase tuning range are determined by the thickness of plates, its refractive index and the range of angles. It is easy to adjust the phase corrector parameters to the requirements of the optical system. Figure 3 presents the outward appearance of the 8-channel optical phase corrector and the results of its use in a real optical system.
The raster-type (folded) spectrum analyzer is a base for realization of many interesting signal processing algorithms. For example, we can design the matched filtering systems for signals with a huge (10 5 - 10 6) product MT . The signal processors for phased array antenna signals and other devices on its basis.
Figure 5.
410
The input and output signals of binary numbers AO-convolver-multiplier: a) 12-bit phase manipulated codes of two equal factors +1, -1, +1, ... , -1; b) convolution of input coding sequences; c) results of coherent output signal detection (analog-digital sign-variable code).
[AJ
Figure 6. Binary number matrix AO-multiplier [A]- [B] = [C].
AO-Convolvers and Matrix Digital Multipliers
AO-convolvers are also one variety of coherent AO-signal processing devices. The scheme of an AO-convolver (Figure 4) contains a laser, two AO-modulators, four Fourier lenses and a photodetector with a bandpass electrical filter. If input electrical signals U1lt) and U2(t) have a time duration not more than AO-modulator time aperture, the output signal is proportional to the input signal convolution, which is represented in a twice compressed time scale.
The convolution operation finds wide applications indifferent (including adaptive) signal processing algorithms. In this article we shall concentrate our attention only on one of the most popular in recent years application of AO-convolvers - on its use for performing mathematical operations - multiplication of the numbers in binary codes [5]. When the input signals of AO-convolver represent two numbers indigital code, the output signal is their product in the mixed analog - digital code. It is illustrated by Figure 5. Two identical factors are represented in a sign-variable binary code l which permits to produce the counting with sign-variable numbers).
The use of multi-channel AO-modulators in AO-convolver makes it possible to build different schemes of binary number matrix multipliers in one optical system. One of the high speed variants of a matrix multiplier is presented in Figure 6. The full time for getting the resulting matrix [cl (in the mixed binary-analog code) in this AO-multiplier is equal to the time representation of matrices [A] or [B].
This AO-matrix processors can solve (on the whole complex of parameters: high processing speed, gabarits, power use) certain problems which appear when creating special high speed digital processors.
CONCLUSION
In the last part of this review article we can come to a few short conclusions. Undoubtedly, the potential possibilities of AO-information processing methods are confirmed now both theoretically and experimentally. The use of AO-methods open up possibilities to achieve a very high speed of wide band signal processing. The AO-processors can provide the solution of a number of narrow places in digital electronics, especially if one has limitations in mass-gabarits and energetic resources. However, the designing of technically perfect AO-signal processing systems is now still a very complex and an expensive task.
REFERENCES
J. Sapriel, 1'Acousto-Optique, Masson, Paris, p.126, (1976) . O.B. Gusev and V.V. Kludzin, Acousto-optical measurements, Leningrad State University Press, (1987).
411
[3]
[4]
[5]
[6]
412
L.B. Lambert, M. Arm and A. Aimette, Electrooptical Processors for Phased Array Antennas, in Optical and Electro-Optical Information Processing, J.T. Tippett, et al. Eds., (MIT Press, Cambridge, 1965) p.715. M. King, Acousto-Optic Reconstruction of Microwave and Acoustic Holograms, in Acoustic surface Wave and Acousto-Optic Devices, T. Kallard Ed., (Optosonic Press, N°.4, 1972), p.163 - 169. Optical Computing, The Special Issue, Proc. of the IEEE, v.72, N"7, July 1984. O.B. Gusev, S.V. Kulakov, B.P. Razjivin and D.V. Tigin, Optical signal processing in real time, Radio and Communications, Moscow, 1989.
AN ACOUSTIC RAY ANALYSIS OF WAVE DISPERSION IN LAYERED
STRUCTURES
J. Laperre and W. Thys
Interdisciplinary Research Center Katholieke Universiteit Leuven Campus Kortrijk 8500 Kortrijk, Belgium
INTRODUCTION
In acoustics, a ray is defined as the traject followed by the energy in an ultrasonic beam of very small section. The basic equation which describes this trajectory is the eikonal equation which was first derived by A. Sommerfeld in 1911, starting from the scalar wave equation in the limit A .... 0 [1]. Ray theory has been successfully used to describe the propagation of electromagnetic waves in optical fibers [2] and to study ultrasonic wave propagation in oceans [3].
In this paper we will apply ray-theory to acoustic surface waves on elastic structures. More specifically, we will derive the dispersion relation from a constructive interference condition, and point out how the ray model can be used to find the resonances of a multilayer.
THE DISPERSION RELATION IN THE RAY MODEL
In this paragraph we derive, using ray theory, the dispersion relation of a thin liquid layer of thickness 2d sandwiched between two solid half spaces. Solid(i) has a density Pi and characteristic velocities (Vid, Vis); the liquid layer has a density PI and a sound velocity VI. A schematic overview of these parameters together with the frame of reference, is given in Figure 1.
The dispersion relation can be found by using a consistency condition on the phase of the plane wave constituting the ray path [4]. This condition arises because of the translation invariance along the layer. It requires that the change in phase of the plane wave progressing from A to D (Figure 1) along the layer axis, is equal to the phase change along the path ABCD within an integer multiple of 27r:
4 co~(B) k + 2 0 1 + 2 O2 = 4 d k tan(O) sin(O) + 2n7r (1)
Oi represents half the phase change at total reflection of a plane wave striking the liquid/solid(i) boundary at an angle 0:
[ (kr + s~) 2 - 4 kr si r i 1
o· = arctan 1 r 4 / Pi k is r i k3
Physical Acoustics, Edired by o. Leroy and "-1.A. Brealeale Plenum Pre". New York, 1991
(2)
413
with: pi = pd Pi
J k~ - W2/V~d and
After rearranging terms, Eq.(1) can be written as:
(3)
This equation is identical with the dispersion relation found m the normal mode theory [4, 5, 6].
The following special cases are interesting to consider:
When d approaches zero and the two solids are identical, then the denominator of the argument of arctan in Eq.(2) must be zero, leading to the equation of Rayleigh. This result is easy to understand since when d = 0, we have a free boundary between two solids and the surface wave is a Rayleigh wave.
When d -; (J) and k3 is real, then k3 has to be zero because ° is always finite. The phase velocity then equals the sound velocity in water.
When d -; (J) and k3 is imaginary, then arctan changes to arctanh. This function becomes infinite when its argument is equal to one. This leads to the equation of Stoneley, describing an interface wave on a liquid/solid structure. With non identical solids, this condition leads to two stoneley equations, one for each boundary.
THE DISPERSION RELATION OF A MULTILAYERED STRUCTURES
Consider a stack of n layers confined between two semi-infinite isotropic media. Each layer is labeled by an index running from 1 to n. Assuming that at least one layer is a liquid (suppose it has index i), we can use the condition on the phase to write down the dispersion relation of the stack. If we concentrate on the i-th layer, we get using Eq.(3), the following condition:
(4)
k3i = J w2/vL - kl ,and 0i,i-l and 0i'i+l are half the phasechanges at total reflection on the (i-I) th, respecti vely the (i + 1) th layer.
Figure 1.
414
B
2d x,
Agreements on the reference system and symbols, and the traject of a ray travelling in the liquid layer at an angle O.
a r r r z
E E Ul Ul W Z
r tP tP
...... . ..... .... .... . .... . ',,', .... . ........ '.
••••••••••• I, " ", .... ..... '"
..... ..... . .............. .
............. . . '.
" ............ . '. . ... ". ..... . .................... .
~ 1
.............
". ...............
.... . .................. . .......... ". ...... . . . .......... . • I, ••••••••••••••••••••••••••••
r l-
I
I
Figure 2.
.. 'I •••••••••••••••
.............. ". ........ . . .............. '" ................
' . .......... . ...... . ". ........ .......
FRE QUENCY I M Hz)
Grafical solution of the dispersion relation of a aluminum/water/plexiglass trilayer.
'.
We illustrate the graphical procedure to solve this equation for a stack of three layers in air: an aluminum plate/ a water layer / a plexiglass plate. The resonance frequencies of this trilayer have been studied experimentally as a function of the thickness of the water layer by M. Cherif et al. [7]. At normal incidence (k3i = W/VI), equation (4) reads as follows:
(5)
In this equation 2d is the thickness, and VI the sound velocity of the water layer. Oland O2 are half the phasejumps at total reflection on respectively the water/plexiglass- and the water/aluminum interface.
We can solve this equation grafically by plotting the family of curves
(6)
on a thickness 2d versus frequency coordinate system. This is done is Figure 2 for n=1 to 15.
Solutions of equation (5) can then be found grafically at the intersections of these curves with the horizontal straight line Y' = 2d.
For a given thickness 2d of the water layer, each intersection gives the corresponding resonance frequency of the trilayer. It is interesting to notice that these curves show steep parts near the resonance frequencies of the isolated Al plate (indicated with an A) and of the plexiglass plate (indicated with a P). This is comprehensible by means of Eq.( 5) and Figure 3 where we plotted the phasejumps 0 1 and O2 as a function of the frequency of the sound wave. We ascertain that the phasejumps change abruptly at the resonance frequencies of the plates. This means that in order to maintain constructive interference, expressed by Eq.(5), the term 2d(w/VI) representing the phase change due to the distance travelled by the ray, will have to change too. Since VI is a constant for a given liquid, the only change can come from d, causing the steep parts in the curves of Figure 3.
Another way of stating this is: near a resonance of one of the plates, the resonance tuning of the stack is critical. This is more pronounced at lower frequencies because of the W in the numerator of the phasejump 2d( W/VI).
415
:3 r ( ( r r r I
i 2 I
I I I
U I I
2 I I
W I VI I « I I ~
D-- .. 0 --'
-1
0 tA tA t/ FREQUENCY I MHz) -
Figure 3. Phasejump as a function of frequency at the AI/water inter-face (dashed lines) and at the plexiglass/water interface (solid lines) .
CONCLUSIONS
We have illustrated the use of a ray model to study wave dispersion in elastic structures containing at least one liquid layer. A graphical solution of the dispersion relation is presented and applied to a trilayer.
REFERENCES
[3]
[4]
[5]
[6]
[7]
416
Born and Wolf, "Principles of Optics". A.W Snyder and J.D. Love, II Optical waveguide theory", Chapman and Hall, London 1983. C.T. Tindle and D.E. Weston, Connection of beam displacement, cycle distances and attenuations for rays and normal modes, J.A.S.A., 67:1614, (1980). J. Laperre and W. Thys, Analysis of an acoustic ray model for solid/liquid/solid structures, Acustica, 67:86, (1988). P.W. Staecker and W.C. Wang, Propagation of elastic waves bound to a fluid layer between two solids, J.A.S.A., 53:65 (1973). J. Pouliquen, A. Defebvre, L.M. Moukala, ~vIode d'interface pour une couche liquide entre deux soli des semi infinis, Colloque de physique tome, 51:969, (1990). M. Cherif, J.L. Izbicki, G. Maze, J. Ripoche, Structure multicouche plane immergee dans l'eau. Influence des resonances sur la diffusion acoustique, Acustica, 64:180, (1987).
THE ANGULAR RESOLUTION OF ACOUSTO-OPTICAL LAMB MODE
DETECTION
J. Laperre, W. Thys and H. Notebaert
IRC Katholieke Universiteit Leuven Campus Kortrijk 8500 Kortrijk, Belgium
INTRODUCTION
Since the work of H. Lamb [1] in 1917, Lamb waves refer to elliptically polarized normal propagation modes of an elastic solid plate with free boundaries. For a given plate thickness and wave frequency, there exists a finite number of such resonances, specified by their phase velocity. Conventionally they are called ao, so, aJ, SJ, etc.; the an's denoting the asymmetric and the sn's the symmetric modes, respectively.
The most effective way of generating Lamb waves in a plate, consists of insonifying the plate, immersed in a liquid, at the Lamb wave critical angle, or shortly the Lamb angle [2-4]. Because the particle displacement of the Lamb waves has a component normal to the plate, energy continuously leaks from the plate into the surrounding liquid, setting up the so called "leaky" Lamb wave. As a consequence, the reflection and the transmission of the incident ultrasonic beam is no longer geometrical [5, 6]. Because the "leaky'! Lamb wave is present only near or at the Lamb angle, the plate modes can be studied by monitoring the reflected or transmitted field as a functltm of the angle of incidence.
In this paper we investigate how the angular resolution of Lamb mode detection, can be optimized by carefully positioning the detector in the "leaky field!!. The detection technique used in this study is a simplified version of the light diffraction tomograph developed by R. Reibold [7, 8].
ACOUSTO-OPTICAL LAMB MODE DETECTION
The optical detection of Lamb modes is done with an expanded laser beam and a small aperture photodiode which can be finely positioned in the acoustic field transmitted by the plate.
An ultrasonic burst approximately 100 cycles long, is generated in a watertank by means of a commercial broadband transducer with a diameter of 0.75 inch and a central frequency of 3.5 MHz. It can be finely adjusted with respect to a thin plate (140 mm x 200 mm) which is positioned in the farfield and which can rotate about an horizontal axis. The ultrasonic wave transmitted through the plate is illuminated normally by an expanded and collimated HeNe laser beam. The light, after being diffracted by the transmitted soundfield, is focused by means of a cylindrical lens on a Schlieren-stop which is adjusted to select the zero-th and the first diffraction order. These two orders are then let to interfere, giving rise to a light intensity beating with the sound frequency [7]. For small acoustic peak pressures and for frequencies less than 5 MHz, the
Physical ACOliSlics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 417
Transducer
y'
z·
Plate ' _3~ation axis
=-=b~~z/'~Yh~z~a~a~v.~2~a~?!~a~;Z~Z~1Z~?Z~2~?2~2~d~' t=::~ I m Expander Sl it
Absorber
Figure 1. Schematic diagram of the acousto~ptical set-up.
ac-{;omponent of the light intensity is given by:
lac N 2 Vrn cos(wt - kz + f)
Photodiode
(1)
where wand k are respectively the frequency and the wavenumber of the ultrasonic wave. Since the Raman-Nath parameter Vrn is directly proportional to the acoustic peak pressure [9], equation (1) allows the mapping of an acoustic field by simply measuring the modulation amplitude. This can be done by means of a digital oscilloscope and a small aperture photodiode (1 mm2) mounted on a xy-table for fine positioning in the image plane. The small aperture is needed to obtain sufficient spatial resolution, and to avoid spatially averaging the cosine in Eq.(l) . We used this set-up to measure the soundfield transmitted by a plate as a function of the angle of incidence, for different positions of the detector. The collected data can be plotted either as a function of the angle of incidence for a given detector position, or as a function of the detector position for a given angle.
EXPERIMENTAL RESULTS
The main objective of the experiments described in this paragraph, is to show that the angular resolution of Lamb mode detection strongly depends on the position of the detector in the "leaky" field.
We studied a one mm thick aluminum plate (140 mm x 200 mm), which has two pairs of closely spaced resonances in the frequency region between 3 and 4 MHz. The plane wave transmission at 3.16 MHz, plotted in Figure 2 (the solid curve) as a function of the angle of incidence B, clearly shows these modes. When a gaussian incident beam with a half-width of 4 mm is used, the integrated transmission (the dashed curve) no longer resolves the s n So- and ao- modes. This numerical result indicates , that when a relatively narrow incident beam is used and the whole transmitted field is integrated, which can be done experimentally for instance by means of a very large detector, that the obtained angular resolution is poor. It can be improved considerably by using a small detector and by positioning it properly in the transmitted field. This is shown in Figure 3 where we plotted the Lamb spectrum for seven different
418
.. 'a
.~ Ci E 0
o
1.3
1 . 2
1.1
1 . 0
0 . 9
0 . 8
0.7
0 .6
0.'
O . ~
0 . 3
0 . 2
0 . 1
0.0
0
· · · · · · · · · · · · · · · · · · · · · . .
Figure 2.
· · · , ,
Figure 3.
. ' 10
, ,
, , ,
, , , , , , , , , , ,
. . , , . , . , . . , · · · · , \ ~'
... .. ...... _ ... .-- ....... '
" ..... ~ " ..... " , , , , , . , . , .
!O
· · · · · ,
· · · · · · · · · · · · · , · · · · · · · , . , ' .
(J(d~I'ft$' ---
The plane wave transmission (solid line) and the integrated transmission of a Gaussian beam (dashed line) through a 1 mm thick Al plate at 3.16 MHz.
~o
K:18mm
10 15 20 25 30 35
(J (degrees) ---
The Lamb spectrum of a 1 mm thick Al plate at 3.16 MHz, for different positions of the photodetector.
40
419
positions of the photodetector. To untangle the curves, they have been shifted in respect to one another along the vertical axis. The position of the detector is measured relative to the center of the incident beam, and in the direction of Lamb wave propagation.
We ascertain that with the photodetector close to the center of the incident beam, the Lamb modes are not resolved. By shifting the detector away from the center, the angular resolution improves progressively up to a certain distance and then starts to deteriorate again. The best detector position to resolve the (so, ao) pair is at 9 mm from the center of the incident beam, while SI and a1 are well separated at 12 mm.
To explain these observations, we recorded the profile of the transmitted acoustic field with the angle of incidence 0 fixed in the interval 10" to 15°, where the SI- and a1 modes are observed. The results are plotted in Figures 4 and 5 with on the vertical axis the amplitude of lac (Eq.(l)), and the position of the photodetector measured in mm, on the horizontal axis. Inspection of these profiles leads to the following conclusions. When the angle of incidence is 9° (Figure 4), the transmitted profile is low and it reaches up to approximately 8 mm. With increasing angle, the transmitted profile grows both in height and in width, until the Lamb angle (11.1°) of SI is reached, where it extends beyond 20 mm. The same overall trend is observed when 0 approaches the Lamb angle (14.0") of the resonance a1 (Figure 5). Notice that when the angle of incidence is in between SI and ab both resonances are excited and a beating is set up causing large fluctuations at the end of the profile.
These profiles show that the onset of a resonance peak in the Lamb spectrum and thus the peak width depend on the detector position. For instance, with the detector at 3 mm, we already detect a relatively large signal for o = 9.5°, whereas at 9 mm, the signal is still zero. In other words: close to the center of the incident beam, there is transmission even if the angle of incidence is relatively different from the Lamb angle. This leads to a broad resonance on the measured Lamb spectrum. With the detector further into the "leaky" field, the recorded signal will be large only close to the Lamb angle, leading to a narrow resonance.
100
I ~ GO 'l:I
~ 0. 80 E o
40
20
Figure 4.
420
o 5
, , " '"
"~----------.~~------.... " .. , .. K"--~_
10 18 20
x(mm)---
Transmitted profiles for angles of incidence 0, close to the Lamb angle (11.1°) of the resonance s1.
GI '0 .z
100 ,,.-- ........ , , , , , ,
:' \ , . , . , . , . , . , . , . " \ , ,., \
, . , . , . :.a eo E . '
, . ,.,/ .,'.,---
o
40
20
Figure 5.
o 10 15 20 lC(mml---~
Transmission profiles for a 1 mm thick Al plate at 3.16 MHz, for angles (j close to the Lamb angle (14·) of the resonance al .
25
This means in fact that we can artificially narrow the resonance peaks by placing the detector far enough into the "leaky" field, at the position where the amplitude change as a function of (j, is as large as possible. Two closely spaced resonances like Sl and al which are not resolved with the detector close to the incident beam, can be narrowed artificially up to the point where they are separated.
This procedure works optimally if the "leaky" field of the two resonances to be separated, has approximately the same extension . Then there will be one best detector position where both resonances show up on the Lamb spectrum as well separated peaks of sufficient height and of almost equal width. When on the contrary, a narrow peak is close to a much broader one, then the much broader resonance will according to the resonance theory [10], radiate its energy over a much smaller region, it is heavily damped and does not propagate as far along the plate. Consequently, the best detector position for the broad resonance is close to the center of the incident beam, whereas the narrow mode is better resolved with the detector further away. A compromise will necessarily lower the angular resolution of the measurement.
A second example of how this procedure works, is shown in Figure 6 where we plotted the amplitude of the acoustic field transmitted by a 1 mm thick steel plate, as a function of the angle of incidence. We ascertain that, in order to resolve the (al, Sl) resonance pair, the detector must be displaced as far as 23 mm relative to the incident beam. At this position we miss the transmitted fi eld near normal incidence. This should urge us on being cautious. Indeed, with the detector far into the "leaky" field , there is always a chance to miss, either the transmission at normal incidence or even a resonance in some other part of the Lamb spectrum; especially broad and thus heavily damped resonances, with a less extensive "leaky" field, are at risk.
We wish to draw the attention to the fact that the measuring procedure outlined here, although ideal for mode separation and for determining the Lamb angles, does not give any reliable information about the width nor about the amplitude of the resonances.
421
1 .. ."
~ Ci. E o
o
Figure 6.
x; 9mm
x;23mm
10 20 30
Oldegrees) - - ..
Lambspectrum of a 1 mm thick steel plate at 3.16 MHz for two positions of the photodetector.
40
It is also worthwhile mentioning that when a transducer is used to detect the transmitted sound field, its position in the leaky field will also determine whether modes are resolved or not. The fact that a transducer integrates the transmitted field over a much larger area than does the diode, is not such a great disadvantage as is made clear by the extensive amount of data available in literature.
CONCLUSION
We have shown that experimentally, one can strongly enhance the angular resolution by properly positioning the detector in the "leaky" field of the plate. From the recorded acoustic profiles transmitted by the plate, we learned that resonances are artificially narrowed when the detector is shifted away from the center of the incident beam. This explains why broad and closely spaced resonances in the Lamb spectrum, which are not resolved when the whole transmitted acoustic field is measured, can nevertheless be separated.
ACKNOWLEDGMENT
The authors especially thank Dr. R. Reibold for his appreciated advice in building the experimental set-up, and for giving J. L. the opportunity to spend some time in his lab. They also thank Prof. O. Leroy for his scientific advice.
This research was supported by the Research council of the University of Leuven and by the Belgian N.F.W.O.
REFERENCES
422
H. Lamb, Proc Royal Soc. (London), 93, 114-128, (1917) . F.R. Sanders, Can. J. Res. A17, 179, (1939).
3 H. Reissner, Helv. Physica Acta, 11, 140, (1938). 4 A. Schoch, Acustica, 2, I, (1952). 5 H. Bertoni and T. Tamir, J. Appl. Phys., 2, 157-172, (1973). 6 T. Plona, L. Pitts and W. Mayer, J. Acoust. Soc. Am., 59, 1324-1328,
(1976). 7~1 R. Reibold and W. Molkenstruck, Acustica, 56, 180-192, (1984).
W. Riley, J. Acoust. Soc. Am., 65, 82-85, (1979). W. Klein and E. Hiedemann, Physica ~.9, 981-986, (1963).
10] R. Fiorito, W. Madigosky and H. Uberall, J. Acoust. Soc. Am., 66, 1857-1866, (1979).
423
INHOMOGENEOUS PLANE WA YES IN LAYERED MEDIA
W. Lauriks, J.F. Allard*, A. Cops
Laboratorium voor Akoestiek en Warmtegeleiding K.U.Leuven, Departement Natuurkunde Celestijnenlaan 200 D, 3030 Heverlee, Belgium
* Laboratoire d'acoustique Universite du Maine Route de Laval BP 535, Le Mans, France
INTRODUCTION
In this paper we are concerned with the sound propagation through plastic foams. Even though the porosity of such foams is often close to 1, it has been proved that the three Biot waves are necessary to explain the acoustic behavior of such foams. In the next pages, it is explained how the Biot theory can be used to calculate the acoustic reflection factor and the transmission coefficient of layered porous and solid systems.
SOUND PROPAGATION IN A POROUS MATERIAL
Let U be the average displacement of the fluid in a porous material and u be the average displacement of the frame of a porous material. Following Biot, the equations of movement of the porous material are:
82 8 PV(V·u) - QV(V·U) - N VxVxu = - (Puu + P12U) + b4U - U) 8t 2 8t
(la and Ib)
82 8 RV(V· U) - QV(V·u) = - (P12u + P22U) + b4U - U) 8t2 at
In these equations, Pu, P12 and P22 are given by:
PH = Ps + Pa, P22 = (J Pf + Pa, P12 = -Pa, (2)
where Ps is the density of the frame, Pf is the density of the fluid in the pores, (J is the volume porosity and Pa is the mass coupling term. This mass coupling term can be approximated by:
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
(3)
425
In this equation, ks is the tortuosity of the material. The coefficient b is a frequency Ciependent viscous coupling term, which is related to the flow resistivity 0". In its simplest form, b can be approximated by:
0" [ 4 i k ~ lIPfW 1/2 iksPf~ b = {p - (I - ) - -- ,
11 A 2 0" 2 {]2 0" {3 (4)
where A is the characteristic dimension of the pore as defined by Johnson [2):
(5)
in which C is a factor of order 1. The parameters P, Q, R and N are the elastic coefficients of Biot. TheX are defined by the stress strain-relations for the solid (s) part and the fluid If) part of the porous material:
T~j = [(P - 2N) V·u + Q V.u)8ij + N[~ + ~], (6a and 6b)
The coefficients P, Q, Rand N can be determined from the equations 6a and 6b and three "Gedanken" experiments. In the special case when a gas fills the pores, the compressibility of the frame material Kb and the compressibility of the gas Kf are very different and the elastic coefficients can be approximated by:
(1-{3) 2 4 P = -p- Kf + Kb + -gN,
Q = (1-{3) Kf ,
R = {3 K f ·
(7c)
N is the shear modulus of the frame.
(7a)
(7b)
At low frequencies, the compression of the gas is isothermal, due to the good thermal contact between frame and air. At high frequencies, the compression of the gas is adiabatic. The frequency dependent compressibility of a gas which takes this transition into account can be proved to be:
(8)
where >. = A W Pr!lI.
Equation 1a and 1b indicate that 3 waves can propagate in a porous material: two compressional waves and a shear wave. The propagation constant of the two compressional waves are:
k~'2 = w2 [(PuR + P22P - 2 PI2Q) :I: {!;:]. 2(PR-Q2)
(9)
t,. = (PuR + P22P - 2P 12Q)2 - 4 (PR_Q2) (PIIP22 - P~2)'
426
and the propagation constant of the shear wave is:
2 rv N N2
W PUP22 - P12
N "'2 P22
(10)
with Pu = Pll + i~,
Each wave propagates in the two phases of the porous material. The ratio of the amplitudes of each wave in the two phases is given by:
J1.i = pllR - P12Q - (PR - Q2)kIf w2
P 2 2Q - P 12R
'" I'" P,3 = -P12 Pn .
= 1,2
THE ACOUSTIC TRANSFER MATRlX FOR A LAYER
(11)
In a layer of finite thickness, incident and reflected waves can propagate and six waves contribute to the sound propagation. The amplitudes of these six waves are known if six independent velocity components and stresses are known. Let V be the vector:
The axis are represented in Figure 1. A 6x6 matrix [T] which relates V4 and V3 in the points M4 and
M3 can be calculated from the equations 6a,6b,9,lO and 11 [4J:
V4 = [T] V3 . (12)
For a material made of several layers, a similar matrix can be calculated from the matrix of each layer. The elements of the matrix [T] are given in [4].
Figure 1.
x
z
CD
A layer of porous material in contact with two layers of elasti c soli d. 1- porous material, 2 - elastic solid
427
PROPAGATION IN ALTERNATE POROUS AND SOLID LAYERS
A layer of porous material in contact with two layers of elastic solid are represented by the vector W:
A 4x4 matrix [aJ, relating WI to W 2 can be calculated [7]. Applying the continuity equations on the planes of contact F and R between the solid and the porous layers allows us to determine the transition matrices for each [~ and [,8]. The transfer matrix of the system is given by:
The elements of the matrices [~ and [,8] are given in [5]. This method can be used for an arbitrary number of layers.
PROPAGATION IN ALTERNATE POROUS AND FLUID LAYERS.
A layer of porous material in contact with two layers of fluid is represented in Figure 2. The acoustic field in the fluid is defined by the vector X:
X = [p,Uz]T, with p the sound pressure in the fluid.
A 2x2 matrix [AJ, relating X4 and X3 can be calculated J7]. Applying the equations of continuity on the planes of contact between the uid and the porous layer makes it possible to define transition matrices [II] and [IL] for each interface. The transfer matrix for the system is than given by:
Xs = [II] [T] [IL] [A] Xl
The elements of the matrices [II] and [IL] are given in [5]. The method can be used for an arbitrary number of layers.
Figure 2.
428
o CD 0
z
A layer of porous material in contact with two fluid layers. 1 - porous material, 2 - fluid layers
1000r-----,_----~------,_----~--~~~----_,
.... 1. E
A
• RcZ \ \ . ~
~ 0 \
\ .... N \
~\ " .. .... -
-1000';:::==::==~==:==~====~===. 1000
.... -; .. E 0 ,
Q,
N
B
-1000~ ____ -L ______ ~ ____ ~~ ____ -L ______ ~ ____ ~
Figure 3.
234 5 6 FREOUENCY (kHz)
Acoustic impedance of two layers of porous material at normal incidence (0=0 3.a) and oblique (0=60 3.b) sound incidence -- prediction real part, - - - prediction imaginary part. The dots and triangles are measuring results for the real resp. imaginary part of the impedance.
APPLICATION TO THE ACOUSTIC IMPEDANCE AND TRANSMISSION OF LAYERED SYSTEMS
The matrix formalism for layered systems makes it possible to calculate the acoustic impedance of layered systems, stuck on a hard surface [4,6]' the acoustic impedance of ground surfaces [5] and the acoustic transmission through layered systems in a fairly straightforward way. Figure 3 shows the impedance of a system of two layers of porous material, stuck on a hard surface. The material parameters have been measured independently. The agreement between theory and experiment is excellent, both at normal and oblique incidence.
Figure 4 shows the transmission loss factor of two different plate/porous material/plate systems. The parameters of the plates and the porous layers have also been measured independently. Measurements and calculations have been performed for random sound incidence. This condition is hard to establish experimentally at audio frequencies, which accounts for the deviations between theory and experiments.
429
100 A
R • (dB) • • •
• • • • • • • •
o~----~----~----~----~----~
l00+-----~----~----~----~------T
B R
(dB)
• • • • •
2 3 4 5
Frequency (kHz)
Figure 4. Transmission loss factor of two different plate/porous material/plate systems Figure 4.a are the results for a system consisting of a 1 mm thick steel plate, a 5 cm thick foam layer and a second steel plate. Figure 4.b are the results for a system consisting of a 8 mm thick wood plate, a 5 cm thick foam layer and a second wood plate. The continuous line is the theory, the dots are measuring results.
CONCLUSIONS
A matrix formalism for the sound propagation in porous media has been presented using the Biot theory. This formalism is easily generalized with the introduction of fluid and solid layers. The acoustic impedance and transmission of layered systems can be calculated in a straightforward way.
REFERENCES
I~I (4)
(5)
430
M.A. Biot, J.Acoust.Soc.Am. vol 28 nr 1, p.168-191, (1956). D.L. Johnson, J.Koplik, R. Dashen, J.Fluid.Mech 176, p.379-402, (1987). J.F. Allard, C. Depollier, W. Lauriks, J.Sound.Vib. 132(1), p.51-60, (1989). J.F. Allard, C. Depollier, P. Rebillard, W. Lauriks, J.Appl.Phys. 66(6) 15, p.2278-2284, sept. 1989. W. Lauriks, J.F. Allard, C. Depollier, A. Cops, II Inhomogeneous plane waves in layered materials including fluid, solid and porous layers II , accepted for publication in Wave Motion.
[6]
[7]
W. Lauriks, A. Cops, J.F. Allard, C. Depollier, P. Rebillard, J.Acoust.Soc.Am 87(3), p.1200-1206, March 1990. L.M. Brekhovskikh, "Waves in layered media", Academic Press New York 1980.
431
DEPTH PROFILING BY FOURIER ANALYSIS OF PHOTOACOUSTIC
SIGNALS
W. Lauriks, C. Glorieux, J. Thoen
Laboratorium voor Akoestiek en Warmtegeleiding K.U.Leuven, Departement Natuurkunde Celestijnenlaan 200D, B-3030 Heverlee, Belgium
INTRODUCTION
The photoacoustic (P A) technique has been used obtain depth profiles of solids by different authors in the past [1]. It can be very time consuming to detect discontinuities in solids, since an X-Y scan has to be performed at different chopper frequencies. In this paper, we shall point out that useful result for up to five different frequencies can be obtained simultaneously by calculating the Fourier transform of the P A signal.
THE PHOTOACOUSTIC EFFECT
The photoacoustic signal is generated by the periodic heating of the sample caused by the absorption of modulated light. The thermal wave in the sample generated by the modulated light, will be scattered by discontinuities in the sample, thus making the P A technique suitable for the detection of material defects which can not be located with optical techniques.
The photoacoustic signal is determined by the thermal properties of the sample within about one thermal diffusion length j.L, which depends on the modulation frequency of the incident light f [2,3]:
(1)
In this equation K is the thermal conductivity, C the heat capacity and p the density of the sample. This frequency dependence of the thermal diffusion length makes it possible to provide information about the depth of discontinuities in the sample.
THE FOURIER SPECTRUM OF THE P A SIGNAL
When the incident light is modulated with a mechanical chopper, the time signal (TS) is a square function. This time signal can be written as a Fourier series:
TS N ! L ~ sin(nwt) .
n odd
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
(2)
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The sample receives an excitation at the frequencies f, 3f, 5f, 7f, 9f ... and should give a response at those frequencies simultaneously. An ordinary lock-in amplifier can only measure the response at one of those frequencies. By using a two channel Fast-Fourier analyzer, the response of the sample can be measured at all those frequencies simultaneously. The experimental set-up was almost the same as in [2]. Light, generated by a HeNe laser is modulated by a mechanical chopper with a frequency f and is incident on the sample. The sample is mounted in a cell with dimensions much smaller than the wavelength of the acoustic signal, but greater than the thermal wavelength in air. The lock-in amplifier has been replaced by a Hewlett-Packard two channel Fast-Fourier analyzer. The entire PA cell was mounted on a X-Y positioning system. One channel of the analyzer has been connected to the chopper signal, while the other channel received the microphone signal from the P A cell. Phase and amplitude of the P A signal can be improved considerably by time averaging the P A signal, triggered on the chopper signal. In this way, reliable results for up to five different frequencies could be obtained simultaneously.
To demonstrate the power of the Fourier transform method, a brass sample has been constructed. Two holes have been drilled in the sample at a distance of 0.1 and 0.2 millimeter below the surface. Since the P A signal is proportional to the amount of absorbed energy, the surface of the sample has been covered with a thin layer of black paint. The influence of the layer of paint can be estimated with the two layer model of Mandelis, Teng and Royce [4]. They calculate the P A signal for a sample consisting of two layers of solid material:
[ 7J2.B2Ioblexp(-.Bl1l) PA N ------[(r2-1)(b2+1)exp(0"212)-
k2(.B~ - O"~)
(r2+1)(b r 1) exp(-0"212) + 2(b2-r2)exp(-.B212)] +
7J 1.B 110 ----[((r c 1)(b 1+1)exp( 0"111+0"212) -2k 1 (.B~ - O"i)
(r 1+ 1 )(b c 1 )(b 2+ 1 )exp( -0" 111+0" 212) +
(rc1)(b c 1)(b2-1)exp(0"111-0"212) -
(r 1+ 1 )(b1+ 1 )(b2-1 )exp( 0" 111+0" 212) +
2{(b c 1 )(b2+ 1 )exp( 0"212) + (b 1+r 1)(b2-1 )exp( -0" 212)} exp( -.BIll)]]
x
[(bl+1)(b2+1)exP(0"111+0"212) + (b c 1)(b2+1)exp(-0"111+0"212) +
(b 1+1)(b2-1)exp(-0"11C0"212) + (b c 1)(b2-1)exp(0"11C0"212) ] -1.
(3)
where .B is the optical absorption coeffiCient, 7J is the efficiency for nonraditive processes, 10 is the intensity of the incident light and I is the thickness of each layer. The properties of the paint layer are indicated with subscript 1, the properties of the brass sample are indicated with subscript 2 and the properties of the backing are indicated with a subscript b. Furthermore:
k2a2 kbab .Bl .B2 b1 = :K.a:" b2 = ~ r1 =
0"1 r2 =
0"2 l a l 2a2
O"i (1 + i)/ fl-i a i ( w/2ai) ai (K,J PiCJ i=1,2,b
438
Since the coating is optically thick exp( -,sIll) ~ 0 and r I » 1. If the coatin& is thin compared to the thermal diffusion length, then exp(±(Jlll) ~ (l±(Jllt). Equation 3 simplifies to the equation from the Rosencwaig and Gersho theory in the optically thick approximation:
(5)
Hence, the P A signal depends solely on the thermal properties of the sample and the backing material.
THE RESULTS
The Figures 1 and 2 show the results of the amplitude and the phase of the P A signal as a function of position on the sample, measured with the Fourier Technique. The chopper frequency was 10 Hz. Results have been obtained for the first five harmonics simultaneously.
CONCLUSION
Calculating the Fourier transform of the P A signal can give good results for up to five different frequencies simultaneously and can reduce the measuring time for photoacoustic depth analysis considerably.
REFERENCES
I~I [4]
G. Busse, A. Ograbeck, J.Appl.Phys., 51(7), p.3576-3578, July 1980. A. Rosencwaig, A. Gersho, J.Appl.Phys., 47, p.64, (1976). J. Thoen et al., "Photoacoustics applied to liquid crystals", paper published in this issue. A. Mandelis, Y.C.Teng, B.S.H. Royce, J.Appl.Phys., 50 (11), p.7138-7146, nov 1979.
439
A NEW METHOD FOR THE DETECTION OF VISCOELASTIC SOLID
RESONANCES: THE miRe SPECTRUM
ABSTRACT
O. Lenoir, P. Rembert, J. L. Izbicki, P. Pareige
L.A.U.E., URA CNRS 1373 Universite du Havre place R. Schuman, 76610 Le Havre, France
This paper deals with the theoretical basis of a new experimental method allowing the detection of the resonances of targets including viscous effects. Two types of plane structures have been investigated: a viscoelastic plate made of PVC and a bilayered plane structure composed of an elastic aluminum plate coated with an absorbing layer, called absonic 1. It appears, in both cases, that the modulus of the transmission coefficient, plotted versus frequency, for an angle of incidence of 10
, exhibits attenuated peaks (peaks of small dynamics with regard to the non resonant background). It lets us suppose that the spectral amplitude of the free backscattered elastic response of such targets may not allow their resonances to be detected. On the contrary, when we study the ratio of the imaginary part to the real part of their reflection coefficient, transitions of larger dynamics remain observable when resonances are excited.
INTRODUCTION
Viscoelastic plane, cylindrical targets or elastic targets coated with viscous layers have been the subject of numerous papers (see e.g. [1-3]). The influence of viscosity has principally been shown in the study of the transmission T and reflection R coefficient moduli. It has been shown that viscosity manifests itself in a reduction of the amplitudes of the resonance peaks, a broadening of their half-widths and a shift of their position towards lower frequencies. For homogeneous materials, at small incidence angle, viscosity is generally taken into account by considering the phase velocities of longitudinal and transverse waves as complex, their imaginary part depending on the absorption coefficients. Experimentally, if the viscous layers studied have large longitudinal and transverse absorption coefficients, it is difficult or even impossible to detect resonance peaks in the spectral amplitude of the free elastic response reflected by this type of structure. So, we have looked for an other method of detection of the resonances of viscoelastic solids. We have been interested in a parameter which is not very much exploited, i.e. the phase of the reflection coefficient [4]. A priori, it gives as much information as the modulus on the resonant character of the acoustic scattering. Indeed, considering the case of an elastic plate, the phase of the reflection coefficient, plotted versus frequency, exhibits transitions between 7r/2 and -7r/2, when a resonance is excited. The detection of the resonances is still more obvious when we study the tangent of the phase, namely the ratio of the imaginary part (1m) to the real part (Re) of the
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 441
l.er a.al..
0.8
e.7 a.s a.s 0.4
B.3
a.a a.l 0.2 a.3 a.4 0 . 5 a . 6 B.7 0 . 0 0 . S 1.0 ~(M-Iz)
Figure 1. Modulus of the reflection coefficient of the PVC plate, aL = aT = 0, 0 = 10 .
reflection coefficient, since, in this case, the transitions related to the excitation of resonances are not limited to a finite range. The purpose of this paper is to show the interest of the study of the ImlRe ratio of the reflection coefficient of plane structures in which viscous effects occur.
THEORETICAL RESULTS
We have first considered the case of a single viscoelastic plate of thickness d, density p, with phase velocities of longitudinal and transverse waves CL
442
I.
')Ji'-(\VI~ /"'~ I I . r I I I
I ! I
t .. 1",,0 I 1. O J " .. "" ,, '".!'' ' "".,, 11 I .. , ' "t"" ' "" It "It!l"t ''' '" ,,,l
a ' O a .1 0.2 0.3 0.4 a.s 0 . S 0.7 0.8 0 . 9 i.O
rnmIDlCY (MHz)
Figure 2. Modulus of the transmission coefficient of the PVC piate, aL = aT = 0, 0 = 10.
a.a e.! 1il . 2 0.3 13.4 0 . 5 e.s e. :> 0 . 8 0.9 I.e FRmJE}lCY (Kiz)
Figure 3. Im/Re ratio of the reflection coefficient of the PVC plate, aL = aT = 0, 0 = 1°.
and CT, embedded in water, whose density and sound phase velocity are noted
PI and C I. Let a plane wave of frequency N be incident on the plate, at the incidence angle O!, and refracted into longitudinal and transverse waves at angles a and p. The exact expressions of Rand T can be written [1] :
R = (CACS-r2)/(CA+CS)(1/(CA+jr)+1/(Cs-jr))
T = jr(1/(CA+jr)+l/(Cs-jr))
~.Sr
,·,f /\ r f\ L I 1\ \! \ f\ A (\ r I~'
a . 31 1 . \ I \ f \ I' \ I\,\ I II \ J \ ' I I \ / \ / \ i \ I
'·'ti \ I \ I \! I \ I \ I \ I j~~VVVVJ e.l ",,,,,,, ., . ,,,',,,,,,. 10,·", " "" ... .!" . ""Or",,,,,," ',,·,,,, I, .. . .. 1,, '
B.B 0. I 8.2 B. 3 0.' 0,5 B. 6 O,7 O,8 0.9 1.8 FREX:!UllICY (MHz)
Figure 4. Modulus of the reflection coefficient of the PVC plate, aL = 5.10-5 N, aT = 2.6.10-4 N, 0 = 1° .
443
13.a
0. 7~
I a.b~1 ~~~~~~~~~~~~~~~~~~~~
a.e a.i 0.2 il.3 <1 . 4 <l . S <i .o il . 7 0 . 8 il.3 1.8
FRmJENCY' (MHz)
Figure 5. Modulus of the transmission coefficient of the PVC plate, a = 5· 10 -5 N a = 2 6· 10 -4 N 0 = 1'.
L ' T' ,
where C A = cos 22,B.tan( jfN d/CL cosa )+( CT/CL)2sin2a.sin2,B. tan( jfNd/CTcos,B)
Cs = cos 22,B.cot( jfN d/CL cosa )+( CT/CL)2sin2a.sin2,B.cot( jfN d/ CTcos,B)
T = PICt/CL.cosa/cosOI
The viscous effects are included in these expressions by writing the phase velocities in the following complex form:
444
2
- 2~~~~~~~~~~~~~~~~~~~~ 0.0 a . I 0 . 2 0.1 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 ~ . 9 I.e
Figure 6.
~(MHz)
Im/Re ratio of the reflection coefficient of the PVC plate, aL = 5.10-5 N, aT = 2.6.10-4 N, 0 = I'.
aL T are the longitudinal and transverse absorption coefficients. So, the angles
of ' refraction a and {J become complex, as well as C A' CS and 7.
We have investigated the case of a 1-cm-thick PVC plate, whose parameters, experimentally evaluated are as follows: CL = 2629 mis, CT = 1207
mis, p = 910 kg/m 3. The longitudinal absorption coefficient has been found to approximately vary linearly with frequency N. aL is about 5.10-5.N Np/m in the
frequency range 200-800 kHz. The transverse absorption coefficient aT is
deduced from aL by means of the relation aT = aL /2.(CL /CT )3, which is
valid for polymers where ultrasonic losses are entirely due to viscous damping process [2]. The moduli of Rand T and the Im/Re ratio of the reflection coefficient R (Figure 1 to 6) are plotted in the frequency range 0-1 MHz, at an incidence angle of 1'. With no absorption, the modulus of R (Figure 1) exhibits, on the one hand, wide minima, regularly spaced every 130 kHz, due to resonances related to longitudinal waves, and on the other hand sharper transitions, about every 60 kHz, due to shear waves. The modulus of T (Figure 2) exhibits wide peaks at frequencies corresponding to those of the minima of the modulus of R and transitions due to shear waves. The Im/Re ratio of the reflection coefficient of the PVC plate (Figure 3) shows large transitions, in the vicinity of the frequencies at which the resonances due to longitudinal waves are excited, and other smaller ones, which correspond to the excitation of shear waves. When the longitudinal and shear absorptions are taken into account, the maxima and the depth of the reflection coefficient modulus (Figure 4) decrease when the frequency increases. Moreover, the transmission peaks (Figure 5) are also attenuated. Since the shear absorption coefficient is about 5 times larger than the longitudinal one, we notice, in both cases, that no more transitions due to shear waves are detectable. We also observe that the dynamics of the transmission peaks with respect to the non resonant background decrease when the frequency increases, and that it is always inferior to 0.1. When we study the Im/Re ratio of R (Figure 6), transitions related to resonances corresponding to longitudinal waves are" detectable. Their amplitudes range from 4 to 1 in the frequency range studied. It is not possible, for the chosen value of aT' to detect transitions due to shear waves,
but at greater incidence angle (0 = 5'), it remains possible to detect some of them at low frequency.
In the case of a viscoelastic plate, the Im/Re ratio of the reflection coefficient exhibits transitions of large amplitude compared to the dynamics of the transmission peaks, so it appears more efficient to study experimentally the Im/Re spectrum than the resonance spectrum in order to achieve the isolation of its resonances.
In a second time, we have investigated the case of an elastic plate coated with an absorbing layer. For the calculations of the reflection and transmission coefficients, we have assumed that the absorbent can be modeled by a homogeneous absorbing fluid material, which is completely defined by the thickness d, the density p, the longitudinal phase velocity CL and the longitudinal
absorption coefficient aL . According to the methodology described by Brekhov
skikh [5], the expressions of the reflection and transmission coefficients of a fluid absorbing layer/elastic layer structure embedded in water derive from the continuity conditions imposed upon the normal components of the particle velocity and the stress at the interfaces separating adjacent media. The viscosity is taken into account by the introduction of a complex longitudinal phase velocity in the absorbing layer.
The parameters of the absorbent are: d = 1 cm, p = 1083 kg/m 3,
CL = 1665 m/s and aL = 5.10-4.N Np.m (200 kHz < N < 800 kHz); those
of the elastic plate, made of aluminum, are d = 8 mm, p = 2616 kg/m 3,
445
1.,.-~ !..----t--. __ ~.3t- ( \ ~ ( \ (
0.ah \ 1\ I \ / ' 0 . 7~ 1/ ' 0.SH \ r I i \! ( 0 - 1' )
0.SW 111' \ I :::1 I I 1/
I I ill,' ~.2 I
0. !
~.~~~~~uw~~~~~~~~~~~~~~~~
a.B B.; 0 . 2 0.1 0.4 0. 5 0 . 6 0 . 7 FRFOJENCY ( Ifiz)
.... " 0 . 0 a.s I. e
Figure 7. Modulus of the reflection coefficient of the 8 mm thick aluminum plate, 0 = 1'.
CL = 6560 m/s and CT = 3300 m/s . The frequency range studied is 0-1 MHz
and the incidence angle is l' for the study of the moduli of the reflection and transmission coefficients and of the Im/Re ratio of the reflection coefficient.
When we study the single aluminum plate, the modulus of R (Figure 7) exhibits minima located at 206, 405, 420, 610 and 815 kHz, which are linked with the Lamb modes A1, SI, A2 and A3 and the modulus of T (Figure 8) corresponding peaks. The modes S2 and A3 are related to longitudinal waves and the others to shear waves. The Im/Re ratio shows sharp transitions at corresponding frequencies (Figure 9). In the case of the absorbent/aluminum bilayer, the modulus of R (Figure 10) shows, up to 400 kHz, smooth decreasing
l.
0.5
B.~
ij.3
0.2
I~
I \ I \
~ 0. I
0.~~~~~~~~~~~~ww~I·~~~wW~ww~~ 0.0 0. ) 0.2 0.3 0.4 0. 5 0.6 0 . 7 0 . S 0.9 1.0
F'REl:lUENCY (MHz)
Figure 8. Modulus of the transmission coefficient of the 8 mm thick aluminum plate, 0 = 1'.
446
51l r-
41:1r-
3tlr-
20f-
10f-
1
-Ie
-2~ I i -3e I I :: J" .. " ''' ''''' '' '''''' .,," ", ..... "" l.. .... , ......... " .. ,,, .. ,, , ....... " . .l. ..... , ..... "" ,
B.a e.l 0 .2 ~.3 e.4 Il .~ e .6 0.7 \l .8 \l .S I . ~ ffiEI1JENCY (MHz)
Figure 9. Im/Re ratio of the reflection coefficient of the 8 mm thick aluminum plate, e = 1'.
oscillations which can be assigned to the absorbent, then the amplitude tends to 0.1 without clearly displaying minima related to Lamb modes of the aluminum.· In the transmission coefficient modulus (Figure ll), it is still possible to detect peaks related to the modes AI, Sl and S2, but their heights are reduced by more than 20 dB with respect to the measured ones in the case of the single aluminum plate. Particularly, we notice that the amplitudes of the peaks corresponding to the modes Sl and S2 are less than 0.09 . Now, when we analyze the Im/Re ratio of the bilayered structure (Figure 12), variations, whose amplitudes are about 0.15, can be observed in the close vicinity of the frequencies at which the peaks, corresponding to the modes Sl and S2, are located. Thus, the dynamics of these variations is greater than the absolute amplitudes of the transmission peaks, so the study of the Im/Re spectrum may allow an easier experimental detection of the resonances of an elastic plate
.~~ .8
.7
'Sr
:;~\~ -
.~g \ (\ r \; .V(~~--------.11 0''''' I . , .• 1"""" 0I" "" " .! "'''''''!.'' '' '''' I.""" " I." "" I, I. """ "I ." " " "I ." ",," I 6.0 6. I B.2 0.3 0.4 0.5 0.6 0 . 7 0.9 0. 9 I.e
Figure 10.
~(MHz)
Modulus of the reflection coefficient of the absorbent/aluminum bilayer, Q L = 5.10-4 N, 0 l' .
447
448
a.l~
0.09 e'0S~ 0.07
fil . filS!
<i.el5~ 0.0~r
e.03~ \
0. 02t \ :::J .. " "" . I" " "".I"""".I"!"""!."""" I~!"""I.",,!;:~
0.0 5. I e.2 0.3 ~.4 0.5 0.6 0.7 0 . 6 0.S
Figure 11.
FRmJEK:'i ( ttiz)
Modulus of the transmission coefficient of the absorbent/aluminum bilayer, aL = 5.10-4 N, B
i'l.48
FRmJEK:'i (MHz)
Figure 12. Im/Re ratio of the reflection coefficient of the
1.0
1 ° .
absorbent/aluminum bilayer, aL = 5.10-4 N, 0 1°.
coated with an absorptive material than the classical method based on the spectral amplitude analysis of the free elastic response.
CONCLUSION
It has been verified in this work, for two specific cases, that viscosity results in a decrease of the maximum amplitudes of the modulus of the reflection coefficient and of the depth of its minima when the frequency increases, as well as a reduction of the transmission peak heights, which are due to the excitation of Lamb modes in the plane configurations. The larger the longitudinal or transverse absorption (transverse absorption being always greater than longitudinal one), the more important the effects recalled above. From the point of view of experimentators, whose interest is to enhance the resonant character of the acoustic scattering of all types of targets, every technique permitting a better detection of resonances than the classical method, which may fail to achieve it, is welcome. It appears that the study of the phase information contained in the reflection coefficient and more precisely the tangent of the phase, that the authors call the Im/Re ratio, may be more adequate to isolate the resonances of targets including viscous properties. It is experimentally confirmed, as shown in other papers [6].
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
R. Fiorito, W. Madigosky, H. Uberall, Theory of ultrasonic resonances in a viscoelastic layer, J. Acoust. Soc. Am., 77:489, (1985). W. Madigosky, R. Fiorito, Modal resonance analysis of acoustic transmission and reflection losses in viscoelastic plates, J. Acoust. Soc. Am., 65:1105, (1979). P.D. Jackins, G. Gaunaurd, Resonance reflection of acoustic waves by a perforated bilaminar rubber coating model, J. Acoust. Soc. Am., 73:1456, (1983). see e.g. N.F. Haines, J.C. Bell, P.J. Mcintyre, The application of broadband ultrasonic spectroscopy to the study of layered media, J. Acoust. Soc. Am., 64:1645, (1978). N. Mercier, J.F. De Belleval, Use of the phase of the signal in ultrasonic spectral analysis to evaluate flaws, Ultrasonic International (1981). L.M. Brekhovskikh, "Waves in layered media", Academic Press, New York (1960). P. Rembert, O. Lenoir, J.L. Izbicki, G. Maze, Experimental analysis of phase spectrum of cylindrical or plane targets: a new global method of isolation of resonances, Phys. Lett., 143:467, (1990).
449
THE MODE METHOD IN THE THEORY OF ACOUSTIC WAVE DIFFRACTION ON DMSION BOUNDARIES BETWEEN DIFFERENT STRUCTURES
O. Leroy, G.N. Shkerdin*
Catholic University Leuven Campus Kortrijk Kortrijk, Belgium *Institute of Radio Engineering and Electronics U.S.S.R. Academy of Sciences, Moscow, U.S.S.R.
INTRODUCTION
Problems connected with propagation of acoustic waves (A W) in inhomogeneous structures have been intensively investigated [1]. Of special interest are problems of surface acoustic waves (SAW) propagation in structures inhomogeneous along SAW propagation direction. Such problems arise for example in structures with inhomogeneous surface or in structures consisting of different materials with gradual or abrupt transition between them (see Figure 1).
It is necessary to note that the existence of inhomogeneities in these structures in many cases leads to interaction of volume A W and SAW in inhomogeneity regions. It means that generally we must take into account both volume A Wand SAW when we consider A W propagation in such structures. In this work for the case of planar geometry we consider the mode met:l).od that can be used for solution of such problems.
Figure 1.
x medium 1 medium 3
incident. SA OW -----medium 2 medium 4
Geometry of abrupt division boundary (XY plane) between different structures. XZ plane is the plane of SAW propagation.
Physical Acoustics, Edited by O. L~roy and M.A. Breazeale Plenum Press, New York, 1991 451
THE BASIS OF THE MODE METHOD
The mode method is based on the construction of complete orthogonal set of acoustic modes for homogeneous planar structure (for example, for the structure consisting of media 1 and 2 divided by XZ plane in Figure 1). This set of A W modes consists of a discrete spectrum of localized near XZ plane eigenmodes (SA W or waveguide modes if acoustic waveguide exists in the structure) and continuous spectrum of acoustic radiation modes. Arbitrary volume A W in the structure can be expanded in these radiation modes.
To construct radiation modes it is necessary to introduce an A W source that can be placed at infinity when y -+ ± CD. In a planar geometry A W amplitudes excited by an A W source must be restricted. Then for every concrete situation we can calculate amplitudes of reflected and transmitted acoustic waves from and through the XZ plane. Combinations of incident, reflected and transmitted waves represent acoustic radiation modes. It's useful to note that the value of the y-component of the A W wave vector of radiation modes qy is real because of the restriction of radiation mode amplitudes when y -+ ± 00. Considering all possible types of incident waves and changing the value of qy for the incident wave in the interval [O,oo[ we can construct a complete set of all A W radiation modes of the structure [2]. Below we suppose that A W absorption in the structure is negligible and the Z axis is orthogonal to a plane of division between different structures. In this case it is necessary to take into account radiation modes with real and purely imaginary values of qz and with real values of qx. The total energy flux of A W radiation modes through XY plane is zero for radiation modes with purely imaginary values of qz. We'll call these modes local (damped) radiation modes.
The most important property of A W modes is the orthogonality between different modes. The orthogonality condition can be derived using the expression for A W energy flux and the equation of A W propagation. Below we'll consider only the case of isotropic media. Using the usual expression for stress tensor O'ik = >'Dik div(u) + 2f.£Uik (Uik is strain tensor, >., f.£ are Lame coefficients of a structure) and the equation of A W propagation, we obtain the following expression:
(1)
where numbers (1), (2) denote A W modes 1, 2 with values qzl and qz2, * means the operation of complex conjugation.
Then using (1) and Gauss theorem we obtain the following integral expression for A W modes with qzl * qz2:
+00
ff Sz(1,2)dxdy = 0, (2) -00
and in planar structures we have the following results.
a) For A W modes of discrete spectrum:
452
+00
f Sz(1,2)dy = -Pr/(iw) {3(qZI,qz2) {3(qXI,qx2) (3) -00
here P r is A W energy flux through XY plane per unit of length
along X axis, w is an A W angular frequency, i = .;=r.
b) For A W modes of continuous spectrum:
+00
f Sz(l,2)dy = D(l,l) o(qz\-qz2) ,8(qx\,qx2) (4)
here D(l,l) is a normalization constant, o(qz\-qZ2) is the 0
function, ,8( qx\,qX2) is the Kroneker symbol.
Expressons (2), (4) are conditions of orthogonality between all different A W modes for considered planar geometry. It is necessary to note that expression (4) is inconvenient to use for local radiation modes since for them D(l,l) = o. In this case it is better to use the following orthogonality condition:
+00
ff Sz(l,2)dxdy = 0 (5)
Then we can obtain expressions analogous to (3),(4). For this we must
make substitutions: Sz(l,2) => Sz(l,2), P r => P r/2 and here D(l,l) f 0 for
local radiation modes. It is useful to note that the same forward and backward propagating modes (when qz2 = -qz\, qx2= -qXl) are orthogonal according to (2) and not orthogonal according to (5).
Usually there are several different types of A W radiation modes with the same values of qz and qx because of the existence of different A W types. Generally these modes are not orthogonal and it is necessary to construct their orthogonal linear combinations.
The completeness of this set of modes gives us the possibility to expand an arbitrary A W in a homogeneous structure in these modes. These expansions for the case of A W propagation in a planar structure along Z axis are the following:
00
u(y,z) L Cnu< n) (y,z) + L f C< m) (qy)u< m) (qy;y,z)dqy (6) 0 n m
00
O"ik(Y'z) =L CnO"\k) (y,z) + L f C< m) (qy)O"\k) (qy;y,z)dqy 0
(7)
n m
where u is the vector of mechanical displacement, sums over n are sums over discrete spectrum of A W eigenmodes, integrals are taken over continuous spectrum of A W radiation modes and sums over m are sums over different types of radiation modes. Expansion coefficients Cn, C< ill) (qy) do not depend on the coordinates.
Mode can be used to calculate the amplitude of acoustic waves propagating in inhomogeneous structures along A W propagatin& direction structure. In this case if structure properties (elastic or geometric) change slowly along A W propagation direction expansion coefficients can be considered as slow functions of A W propagation coordinate and slow amplitude equations can be derived for them. The same procedure can be used for the case of gradual transition between different structures. In some cases it is possible to find approximate analytic solutions of such equations.[2].
For the case of abrupt transition between different structures (see Figure 1)
453
the slow amplitude method can not be used. Below we consider this case at greater length.
THE CASE OF ABRUPT TRANSITION BETWEEN STRUCTURES
Here we consider for simplicity the case of normal incidence of acoustic wave on X axis in the following structure of Figure 1: media 2, 4 are vacuum, media 1, 3 are different isotropic solids in contact. Let's consider the most interesting case of vertical polarization A W when surface (Rayleigh) waves exist.
In this case A W radiation modes are represented by combinations of longitudinal and transverse acoustic waves with the same values of qz. There are two different types of orthogonal radiation modes for the case being considered (expressions (8), (9) and (10), (11)):
. [2qytqy 1 . 1 uy = lAqz sm(qylY) - sin(qytY) exp(i(qzz-wt)) q~ - 2q~
(8)
(9)
(10)
(11)
where: A, B are arbitrary constants, q§t'l = qt,l - qi, qt = w2p/l-",
q~ = I-"(A+2I-"tlq~, P is the density. Expressions (8)-(11) can be used for real
qz if I qz I $ ql· For ql < I qz I $ qt we have the following expressions for u of A W modes:
(12)
(13)
here: a = arctg[4qylqytq~(q~- 2q~)-2], q~l = q~ - qr .
For real qz its value for radiation modes can not be greater then qt. For imaginary values of qz we have only expressions (8)-(11).
There are two different diffraction problems in this case: a) when A W incident on XY plane is a Rayleigh wave and b) when this incident A W is a volume wave. In both cases the acoustic wave reflected from and transmitted through the XY plane consists of both volume and surface waves.
To calculate amplitudes of reflected and transmitted waves we can expand unknown A W amplitudes to the left and to the right of the XY plane in
454
AW modes of appropriate homogeneous structures (see (6), (7)). Then using the following boundary conditions on XY plane: u(z=-O) = u(z=+O)), aiz(z=-O) = aiz(z=+O) and the above orthogonality conditions we obtain the system of integral equations for unknown expansion coefficients to the left and to the right of the dividing plane. Generally this system can be solved only numerically. The main difficulty of this solution is connected with correct calculation of interaction between A W radiation modes on both sides of XY plane. The following method of successive approximations can be used to find the solution of the system. In the first approximation we can find amplitudes of all A W modes neglecting the interaction between A W radiation modes for different structures. Then taking into account this interaction we can calculate corrections to these amplitudes in the framework of successive approximation method. In the real situation when difference of values A, J.L, P for media on different sides of XY plane is essential such calculation can be fulfilled only with the help of a computer. Below we give some approximate results for A W amplitudes that were obtained using this method of solution of the above system of integral equations.
SAW excitation by incident volume acoustic wave
If an incident volume A W represented by an expansion in radiation modes propagates from structure (1) through XY plane into structure (2) both reflected and transmitted surface waves are exited on this plane. For the case of small reflection from XY plane and assuming that incident A W is described for example by A W radiation mode given by (12), (13) we can obtain the following expressions for expansion coefficients of reflected (C r ) and transmitted (Ctr) SAW's:
where: cj is expansion coefficient of incident A W,
00
K = r [a( 2) (z=O)u * ( \) (z=O) - u ( 2) (z=O)a* ( I) (z=O)] dy 21 J o ZZ,Y z,r Y,Y YZ,r ,
values a( 2) u (2) are connected with transmitted volume A W in the lZ,V' 1,V
structure (2) and given by expressions (12), (13), values
connected with Rayleigh wave in the structure (1).
a( I) U (I) are lZ,r' l,f
Analogous expressions can be obtained for other types of incident acoustic waves. For model case when AI = J.L" A3 = J.L3, PI = P3 typical amplitudes of transmitted (utr ) and reflected (ur) Rayleigh waves can be estimated by the following expressions:
where 'Y = A3/ A I' ui is the amplitude of the incident volume wave, ftr.r( 8)
are functions of the angle of the volume A W incidence on XY plane. In the limit of normal incidence when 8 => 7r/2 values of ftr,r( 8) tend to zero. For
455
intermediate angles 8 these functions are of the order of 1, but values of fr( 8) are usually smaller in comparison with values of ftr( 8). For example for , ~ 0.83 ,8=1f/3 we have ft!' ~ 0.9, fr ~ 0.3.
Propagation of Rayleigh waves through XY plane
For the case of incident Rayleigh waves we can obtain expressions for transmitted and reflected surface and volume acoustic waves. But even in the first approximation these expressions are too cumbersome to give them here. As an example we give the following results for the relative power of transmitted Itr and reflected II' Rayleigh SAW's and for the relative power of transmitted P tr and reflected PI' volume A W's that were obtained for the above model case in the first approximation.
al I1-,1 « 1: I tr ~ 1; P t!, ~ 0.2(1-,)2 and PI" II' are of the order of 0.1 P tr . b ':: 1:5: It~ ~ 0.~5; P t:: ~ 0.?3; I~ ~ 0.0?1; P~ ~ O.OOL c , - 2. I tr _ 0.86, P tr - 0.08, II' - 0.004, PI' - 0.004.
Thus the mode method can be used for investigation of acoustic wave diffraction on division boundaries between different structures.
REFERENCES
[1]
[2]
456
L.M. Brekhovskikh and O.A. Godin, "Acoustics of Layered Media", Nauka, Moscow (1989~. D. Marcuse, 'Theory of Dielectric Optical Waveguides", Academic Press, New York (1974).
ACOUSTIC WAVES IN TWO-PHASE MEDIA
SUMMARY
Jozef Lewandowski
Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw, Poland
From the laws governing the flows of fluids in porous solids with interconnected pores, suspensions and emulsions, the system of differential equations is derived for the propagation of small time-harmonic fluctuations in the density, pressure and velocity fields. On putting the fluctuations in the form of a plane attenuated acoustic wave, the system of linear and homogeneous algebraic equations is derived for the amplitudes of the fluctuations. The frequency dependence of the propagation velocity and attenuation (dispersion laws) of the acoustic wave are established from the existence condition of the nontrivial solutions to the algebraic equations system.
INTRODUCTION
In this paper, the two-component media are described by using the Truesdell's [1] concept of replacing the components (phases) of the mixtures by fictitious continuous constituents. Ignoring the effect due to heat conductivity, the laws governing the adiabatic flows of the fluid components are derived on the basis of paper [2]. In these equations some terms are to be found from more detailed considerations as, e.g., in papers [3,4). In the approach presented, attenuation and dispersion are caused by the inability of the phases to follow each other in the changes of their mechanical state induced by the acoustic waves.
LINEAR ACOUSTIC EQUATIONS OF TWO-PHASE MEDIA
We consider a two-phase medium with a Newtonian fluid as one of the phases called the f-phase. The other phase is called the s-phase and is taken to have the form of a skeleton with a statistical distribution of interconnected pores. In the case of a suspension or emulsion the s-phase is taken to have the form of particles with the same or random shape and size. The particles are assumed to be made up of solid material or another Newtonian fluid which is non-soluble in the first one (f-phase) and chemically non-reacting with that. Throughout the paper all the abbreviations with the sub- or superscripts s and f denote quantities referred to the s and f continuous constituents, respectively. If a sub- or superscript a, a = s, f, is in parenthesiS then the quantity refers to phase a. The density, po,(r,t), pressure, Pa(r,t), and
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 457
.. .. velocity fields, vo.(r,t), a = s, f, are involved in the equations governing the
.. flows of continuous constituents as unknown functions, F(r,t), of the position .. vector, r, and time, t. These functions are assumed to be of the form
... - ...-F(r,t) = F(r,t) + AF(r,t), F = const. (2.1)
AF(;,t) denotes the local and instant adiabatic fluctuation in the quantity
F(; ,t) about its equilibrium value which is denoted by overbar. It is assumed that
.. -IAF(r,t)/FI « 1,
AF(;,t) = AFoexp[ - i(wt - k.;)), .. w .. k = (-c- + if.!)ek
(2.2)
(2.3)
where w denotes the angular frequency. Expressions (2.3) describe an attenuated plane wave with the amplitude ..
Aro propagating through a medium in the direction ek and with the velocity c. f.! denotes the attenuation coefficient.
Substituting (2.3) into the equations governing the flow of the two-phase .. medium yields, after the linearization with respect to AF(r,t), the system of 8
acoustic equations for 10 unknown functions: Aps, Apr, Aps, Apr, v~, Ui where
Ul• = v~ - v{ 1· 1 2 3 1 l' =".
The acoustic equations can be written in the following form:
a( Ap) + -Vs + - 0 ~ P jlj prUj.j
av~ _ 1 + _ aUi L p (Jf""" pr 7Jt = aij'j
a(v{ + Ui) - 1 FS _ Ps at - i - 0
a7j denotes the result of the linearization of the component
effective stress tensor of the mixture as a whole.
Po. = baP (0.) , P = Ps + pr, bs + br = 1, a = s,f
..
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
aij of the
(2.9)
Fs is the density of the viscous drag force experienced by the continuum s when it executes oscillations induced by the sound field. bOo' a = s,f, denote the
volume concentration of the phase a. To equate the number of acoustic equations with that of the acoustic disturbances we add to Eqs.(2.5) - (2.9) the relations
Apo. c~ = Apo.' a = s,f (2.10)
expressing the assumption that both the continuous constituents are disturbed adiabatically.
458
ACOUSTIC WAVES IN DILUTE SUSPENSIONS AND EMULSIONS
In this section, dilute suspensions and emulsions are considered and use is made of Eqs.(2.4) - (2.10) to derive the dispersion laws for the media under study. The dynamic and second viscosities, 1/( 0.) and e (0.)' a = s,f, of the
suspended and suspending fluid are assumed to be independent of the frequency III.
The assumption that the mixtures are dilute allow us both to neglect the forces with which the particles act on each other and take bfPf, bf1!f, bfef as the effective pressure and viscosities of the mixture as a whole.
As an example the propagation of acoustic waves in aqueous mercury emulsions is considered. Every suspended particle is assumed to maintain its spherical shape due to surface tension. The wave length is assumed to be much
greater than R. In order to calculate the interaction ps between the particle .. oscillating with the velocity vS and surrounding fluid which oscillates as a whole with the velocity ;f, it is necessary to find the local velocity and pressure fields in the suspending fluid (f) at the boundary of the particle. Solving this problem both the fluids are supposed to be viscous and incompressible. The local velocity and stresses fields in the fluids f at the boundary of the particle are required to fulfill the conditions of the tangent components continuity and normal components vanishing.
Although the applied method of calculation ps follows to a wide extent the Ahuja's method [5], it is more general. In paper [5], it is assumed that the dynamic viscosity of the dispersed fluid is much higher than the viscosity of the suspending fluid. Moreover, Ahuja omitted the time derivative and nonlinear terms (with respect to the fluid velocity) in the Navier-Stokes equation describing the viscous and incompressible flow of the fluid inside an oscillating particle. In this paper these constraints are released.
On finding the formula for ps and substituting its right-hand side to Eq. (2.8) we are able to obtain from Eqs.(2.5) - (2.8), (2.10) the secular equation which enable us to establish the frequency dependence of the propagation velocity and attenuation coefficient. Examples of results of such digital computations are presented in Table 1. More details will be presented in further publications.
Table 1: Frequency dependence of the propagation velocity and attenuation coefficient of ultrasonic wave in aqueous mercury suspensions
-1 -3 -3 Cr =1451ms P,r> =0. 997kgm , Peal =13. 595kgm
Y/'s) =0. 02103kgm" 5" . Y/'S) =0. 001kgiD." . be =0.05
-1 W [s 1 b
0.01 0.02 0.03 0.04 0.05 0.06
0.01 0.02 0.03 0.04 0.05 0.06
s
0.991 0.082 0.982 0.117 0.974 0.144 0.965 0.168 0.956 0.188 0.947 0.207
0.988 0.036 0.976 0.051 0.964 0.063 0.952 0.074 0.941 0.083 0.929 0.091
459
REFERENCES
[1]
[2]
[3]
[4]
[5]
460
I. Muller, A thermodynamic theory of mixtures of fluids, Arch. Rat. Mech. An., 29:1, (1968). W.B. Fulks, R.B. Guenther and E.L. Roetman, Equation of motion and continuity for fluid flow in a porous medium, Acta. Mech., 12:121, (1972). M.A. Biot, Generalized theory of acoustic propagation in porous dissipative media, J. Acous. Soc. Am., 34:1254, (1962). R.W. Morse, Acoustic propagation in granular media, J. Acoust. Soc. Am., 24:696, (1952). A.S. Ahuja, Effect of particle viscosity of propa&ation of sound in suspensions and emulsions, J. Acoust. Soc. Am., 51:1, (1972).
PROPAGATION VELOCITY AND ATTENUATION COEFFICIENT OF
RAYLEIGH-TYPE WAVES ON ROUGH SOLID SURFACES
SUMMARY
Jozef Lewandowski
Institute of Fundamental Technological Research Polish Academy of Sciences Warsaw, Poland
The propagation of Rayleigh waves on a rough free surface of a homogeneous and isotropic elastic half-space is considered, in terms of the average displacement field by using the Rayleigh method. The frequency dependence of the propagation velocity and attenuation under the assumption that the frequency is real and wave scattering on the surface roughness is the only considerable mechanism of attenuation.
INTRODUCTION
The interaction of a Rayleigh-type wave with a randomly rough boundary separating two different media is of considerable interest and utility in device application as well as in examining properties of solids in the near vicinity of their surfaces. Since the 1960's such problems have been studied by several authors, e.g. [1-10].
As a Rayleigh wave propagates over the planar free surface of a homogeneous solid, it is scattered by surface roughness and damped due to the inelasticity of the material. Consequently, processes are considerable, the wave vectors of the Rayleigh waves are complex and frequency dependent (dispersion phenomenon). Then the real and imaginary parts of the wave vector provide the propagation velocity and attenuation coefficient, respectively. It is assumed that the behavior of the homogeneous material is dominated by the elastic phenomena, so that the wave scattering on the surface roughness is the only considerable mechanism which results in the complexity of the wave vector of Rayleigh waves and its dependence on the frequency.
To analyze the dispersion laws for the Rayleigh waves we may make use of two appwaches. In the first approach called the Green function method, the problem for elastic half-space with conditions of vanishing stress on the solid boundary (free surface) is formulated as the problem for infinite heterogeneous space with the local elastic moduli and density being a step function of coordinate with respect to the reference axis perpendicular to the plane free surface. Such a problem can be studied by using the corresponding Green function as it was done, e.g., in papers [5,6,8,9]. On the contrary in the second approach called the Rayleigh method [11], the primary stress-free boundary problem for hal£-space is studied [7,10].
This work is strongly influenced by paper [7]. Studying the propagation of Rayleigh waves along a free rough surface of a semi-infinite homogeneous and elastic solid, the authors of paper [7] treated the wave frequency as a complex
Physical Acoustics. Edited by O. Leroy and M.A. Breazeale Plenum Press. New York. 1991 461
quantity. On the contrary to paper [7], the authors of paper [12] treated the wave frequency as a real quantity considering the same problem in the long wave approximation. In this approximation, the roughness induced attenuation is negligible small as compared with the changes in the propagation velocity caused by the same surface roughness.
In this paper, the same problem is discussed for a wide frequency range which was considered in paper [7J. In this discussion the use is made of paper [7] and the same modifications are introduced as in paper [12].
THE PROPAGATION VELOCITY AND ATTENUATION COEFFICIENT OF RAYLEIGH-TYPE WAVES
Suppose an isotropic elastic medium is of the form of a semi-infinite solid with the free surface
(2.1)
Suppose in the medium propagates a time harmonic displacement field, Ui(xlt), of the form
(2.2) -t -t ... -t -t .... 2-+
XII = x 1e 1 + x2e 2 ; qll = qle l + q2e2; d qll = dqldq2 i = 1,2,3
where ~ 1 and ~2 are the unit vectors in the directions of the axis OXI and OX2, respectively. Then the equations of motion for the elastic medium restricted to sufficiently small deformations without body forces can be written in the form:
{ 8ij { w2 + c~ [_ q2 + ~] } + (2.3)
(c~ - cD 2 [(1-8i3)iqi + 8i3~] . [(1-8j3) iqj + 8j3~] 1 uj(qIIWlx3) = 0
Solving this equation we obtain [7]
(2.4)
where
.. _-1 u 1(qIIWlx 3) - q [(a 1+a2)ql - a3q2],
.. -1 u2(qllwlx3) = q [(a 1+a2)q2 + a 3qJ, (2.5)
u3(QIIWlx3) = i[(allq)a 1 + (qlat )a2]
where
(2.6)
462
The unknown functions Ai = Ai(q" ,w) are to be determined by applying
the stress-free boundary conditions at the rough surface given by (2.1)
i,j = 1,2,3 (2.7)
.. .. n(x) is the unit vector normal to the surface (2.1) . ..
O"ij( q wi X3) (i,j = 1,2,3) denote the stress accompanying the displacement
field defined by (2.5), (2.6). We express these stresses in terms of Ul(q wi X3) (i,j,l = 1,2,3). The assumption that the surface roughness is small (I x31 « 1) .. .. enable us to expand the displacement u( qll wi X3) into the MacLaurin series
(about X3 = 0 ) and omit the terms of the third and higher orders.
In this way we arrive at an eigenvalue problem for Ai(qll'w). The integral
equations involved in this eigenvalue problem are valid to O(h2) where
(2.8)
The angle brackets < ... > denote averaging over the ensemble of all possible realizations of the surface roughness. On averaging we reduce the system of integral equations to much simpler system of linear homogeneous .. equations for <Ai(q" ,w». In turn, the solubility condition of the algebraic
equations can be expressed in the form of the following secular equation
(2.9)
where the elements of the matrix M( 0) are given by [7,(2.8)]' and
.. 2 [ (0.).. w)..] P(qlllw)ij = h Mij (qillw) + Mij (qillw) (2.10)
(0.) .. «(3) .. Mij (qillw) and Mij (qillw) are given by formulae [7,(2.37)] and [7,(2.38)]' respectively. Moreover, deriving Eq.(2.9) we have assumed that the incident Rayleigh wave propagates in the positive direction of the axis Ox! so that in
the absence of roughness (~(~,,) = 0) its wave vector would be given by
.. 0 0 w qll = (q ,0,0) = [c~,o,OJ (2.11)
where c~ is the propagation velocity of the flat-surface Rayleigh wave.
On introducing the surface roughness - induced perturbation ..
given by formula (2.11), we obtain the following expression for q
(2.12)
463
'TIa denotes the roughness - induced attenuation of the Rayleigh wave measured
in the positive direction of OXI. ca is the propagation velocity of the Rayleigh wave in the presence of rough-
ness. The component flqll' ~2 of the wave vector (2.12) is assumed to be neg
ligible small compared with flqll'~l (small roughness).
Eq.(2.9) yields the frequency dependency of flqll'
(2.13)
where flq~a) and flq~b) denote the real and imaginary part of flqll' In turns, Eqs.(2.9) and (2.13~ enable us to calculate ca and 'TIR as functions of
the frequency w,
[ 1 1 ]-1 C = -flq< a) + 'TJ = -flq< b) a w CO' 'fa a (2,14)
To perform the digital computations the two - dimensional Fourier transform of the correlation function of the surface profile
(2.15)
is taken to be of the form
(2.16)
as in paper [7]. The correlation length a can be considered as a measure of the average distance between successive peaks and valleys in the surface profile.
For this case we obtain for example that for a = 10 -4m, w = 2 '167rc~ / a
the ratio ca/c~ diminishes from 1 to about 0.9 with h/a increasing from 0
to 0.01 (small roughness) and the attenuation induced by surface roughness is negligible small as compared with the changes in ca caused by the roughness
(long waves). Some results of digital computations for such low frequencies and Al are presented in Table 1. There the following abbreviations are used:
Table 1
t ["z
o 0 0
9*10-· 10- 2
2.5*10-' 2.71*10- 2
t >0 4.9*10-' 5.18*10- 2
8.1*10-' 8.28*10- 2
10- 2 10- •
464
(2.17)
cn. and c~ denote the propagation velocity of the Rayleigh wave at time t I
and t=O, respectively, where t I >t and t=o is the beginning of the material life.!:::.z denotes the changes of h2/a2 during the time interval from t=O to t I. Material of the elements of constructions and devices continuously ages and degrades due to loading connected with the exploitation. The progress in such processes results in an increasing of the surface roughness of the material under study. In accordance with Table 1, the increasing of surface roughness results in with diminishing of the propagation velocity of the Rayleigh waves. Therefore, the results of this paper may be thought as a basis of Rayleigh waves diagnostics of solid surface smoothness.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
J.A. Viktorov and T.M. Kaekina, Scattering of ultrasonic Rayleigh waves by surface defects models (in Russian), Sov.Acous.Journ. 10:30, (1964). K. Sobczyk, Scattering of a plane elastic wave at random surface, Proc. Vibr. Probl. 6:1, (1965). K. Sobczyk, Scattering of Rayleigh wave at a random boundary of an elastic body, Proc. Vibr. Probl. 7:4, (1966). E.J. Urazakov and L.A. Fal'kovskii, On the propagation of Raylei~h wave over a rough surface (in Russian), Sov. Phys. JETP 63:2297, (1972). A.A. Maradudin and D.L. Mifls, The attenuation of Rayleigh surface waves by surface roughness, Ann. ·Phys. 100:262, (1976). A.G. Eguiluz and A.A. Maradudin, Effective boundary condition for a semi-infinite elastic medium bounded by a rough planar stress-free surface, Phys. Rev. B 28:711, (1983). A.G. Eguiluz and A.A, Maradudin, Frequency shift and attenuation length of a Rayleigh wave due to surface roughness, Phys. Rev. B 28:728, (1983). V.V. Kosacev, J.N. Lochov, and V.N. Cukov, On the damping theory of acoustic Rayleigh waves at a free solid surface with random roughness (in Russian), IFI Moscow Report 068-86:1, (1986). V.V. Kosacev, J.N. Lochov, and V.N. Cukov, On the damping theory of acoustic surface Rayleigh waves at a free solid surface and randomly rough surface (in Russian), Sov.Phys. JETP 94:162, (1988). M. de Billy, G. Quentin, and E. Baron, Attenuation measurements of an ultrasonic wave propagating along rough surfaces, J.Appl.Phys. 61:2140, (1987). J.W. Rayleigh, On waves propagated alon~ the plane surface of an elastic solid, Proc. Math. Soc. London 17:4, (1885). R.V. Goldstein and J. Lewandowski, Surface roughness induced attenuation and changes in the propagation velocity of long Rayleigh-type waves, Acta Mechanica, (to be published).
465
LOCALIZED WAVE TRANSPORT OF PULSED BEAM ENERGY
D. Kent Lewis, Richard W. Ziolkowski and Bill D. Cook*
Lawrence Livermore National Laboratory Livermore, CA 94550, U.S.A. *University of Houston, TX 77004, U.S.A.
INTRODUCTION
This paper describes the theory and experiments used to measure a new type of finite beam pulse, the localized wave pulse. The L W pulse is a linear superposition of Gaussian-like basis functions which are constructed from the focus wave modes of Brittingham [1] by means of the modified power spectrum [2].
Experiments to data [2,3] have made use of three techniques. Synthetic array experiments using acousto-{)ptic measurements were first undertaken to see if the localized wave pulses could be launched by normal acoustic devices. Next, synthetic array two transducer measurements were done to see if the beam generated would outperform a conventional continuous wave or pulsed array with both Gaussian shading and no shading. Finally, real array experiments were performed to see if the beam could be launched with actual acoustic arrays and still perform as well as in the synthetic array measurements.
In all cases, the L W pulse has outperformed comparable beams. The theory will be outlined and the methods of measurement sketched. Finally, comparison of beams produced by driving the array with different inputs will be presented.
THEORY
The basis functions of the L WT signals propagating along the z axis are the focus wave modes;
'" ( t) = ik(z+ct) e -kp2j(z 0 + i () 'l'k r, e 4:ri (zo+ i 0 (1)
with Zo a constant, k the wave number, z the axial distance, c the wave speed, t the time, and (= z - ct. The wave field at any position and time t can then be defined as
w
f(r,t) = f ~k(r,t) F(k) dk (2) o
The trick is to find a spectrum, F(k) for which f(r,t) is highly localized in the region desired. Using the focus wave basis function, the field becomes
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 467
f(r,t)
ID
= 1 f dk F(k) e-ks(p,z,t) 47ri[zo+ i (z-{;t)]
o (3)
where
s(p,z,t) (4)
The requirements on the spectrum, F(k), are that it lead to a finite energy in the wave field and that it introduce other parameters in order to allow the field to be tweaked for maximum localization in the region desired. The spectrum used is obtained from
(5)
by a scaling and truncation and is called the Modified Power Spectrum (MPS), defined as
F(k) = . ~~k_b)a-le-aCBk-b)
4n{3 ['(a) [k > *] 0 [0 $ k $ *] (6)
This choice fulfills the finite energy requirement and introduces the four new parameters, a, a, band (3, in addition to Zo contained in the focus wave mode expression. This yields a wave function on axis as
f( -0 - t) - [ cos(2bzL{3}- ~2zL~a}sin(2bzL~} ] 1 p- ,z-c - 1 + (2z (Ja) azo
(7a)
which behaves in the different regions along z as
1 [~ « 1 and ~ < 1] ~ azo (7b)
~ cos(2bzL~) [2Z < 1] azo a;J (7c)
and
~ ] ~ [2Z > a;J 1] (7d)
the three regions of different behavior for the wave function. Thus the amplitude on axis is constant for a distance, then begins to oscillate recovering its original amplitude every 7r{3/2b, then while still oscillating falls off as l/r as it must to obey the radiation condition.
EXPERIMENTAL PROCEDURE
Three experimental efforts have been completed to date. The first of these was a measurement of the field using an acousto-optical measurement system and synthetic array methods to determine the pressure over an area 20.48 microseconds long and 6 cm wide. A single transducer, a 5 MHz 3 mm radius commercial unit, was used to launch each of 11 waveforms, Lo through L IO , at 21 different positions 3 mm apart. The waveforms were computed and loaded into an arbitrary waveform generator, then recalled by name. A schematic of the experimental arrangement is shown in Figure 1.
Since acousto-optical detection is measurement over a line, reciprocity was
468
Figure 1.
Transducer
Sound Lens Diode Field
Immersion Tank
The experimental system schematic. The synchronization signal controls both the launching of the individual waveforms and, after a delay to compensate for propagation, for detection of the pressure field. In the acousto-optic experiments, a lens is used to project a virtual image of the photo-diode close to the sound beam.
used to interchange the role of transmitter and receiver. The theoretical model used for comparison consisted of a 21 by 21 element array with inter-element spacing of 3 mm. All of the elements in a row along x fire the same signal and each row fires a different signal in an arrangement along y symmetric about the center of the array. The excellent results of this first test of the theory, a measurement of the field along y and time at various z values, led us to the second set of experiments.
The second method was also a synthetic array model. The same transducer was used to generate the fields and a second transducer with similar properties replaced the laser and photo-diode. The receiving transducer was positioned at 441 different positions in x and y, separated again by 3 mm, and at each position the 11 LW waveforms were launched as well as continuous wave (CW) tone bursts of 0.5, 1.0 and 2.0 MHz. The tonebursts were 8 microseconds in duration. For field comparison purposes, the center portion of each CW field was extracted and processed to avoid the turn-on and turn-off effects from the power amplifiers.
The third series of experiments used a 25 element, highly damped PVDF acoustic array with a nominal resonant frequency of 5 MHz. The elements were arranged in a 5 by 5 pattern and were individually addressable. The element diameter is 0.5 mm and the elements are separated by a 2.5 mm center to center distance, making this a sparce array.
The signals were modified from the two previous experiments by a folding process, a method of mapping the driving functions of a larger array into the actual array. These signals were then further modified by a time gate which removed part of the resultant signals for efficiency. These were then used as the input signals to the array.
The experimental system was modified by using two waveforms generators and a demultiplexer unit which separated the signal from one of the generators into 5 different signals for the array elements. The signals were fed to the array symmetrically about the center so 6 waveforms were needed. The resultant sample interval for the source signals was still well within acceptable frequency limits for all waveforms.
The detection in the third series of experiments was again by the same transducer used for the previous experiments. The measurements were of pressure as a function of time in 0.5 mm steps along x at y = 0 and along y at x = 0 for -5 to +5 cm relative to the center of the array. For comparison purposes a 0.5 MHz piston beam was also launched.
469
DATA PROCESSING AND RESULTS
The first series of experiments was undertaken to see if the fields predicted by theory could really be created, so an analysis of this data involved only the comparison of measurement to prediction. Since this agreement was excellent, the two transducer experiments were begun immediately.
For the two transducer synthetic array, the tone bursts were assembled in the signal processing package VIEW. The resultant fields were assembled as both piston or uniform amplitude and Gaussian beams by proper weighting of the 6174 individual waveforms at each z value for comparison. The half width at half maximum for the CW cases and the L W beam are shown in Figure 2.
As expected, the Gaussian beams for all frequencies were much wider than either the LW or the piston CW beams for all cases. An indication of the complicated near field structure of the CW signals can be seen in the half widths for the 1 and 2 MHz CW signals.
Additionally, beams made up of the center driving function Lo were formed as both Gaussian and piston simulations. This signal has a spectrum that is the envelope of all the frequencies in the L W signals. This comparison was done to see if the broadband signals alone could account for the localization of the beam. We also Fourier analyzed the fields as magnitude versus frequency and beam position to look for frequency shedding, the progressive loss of low frequency content in the pulse.
For comparison purposes, we determined the equivalent frequency in the following way. All of the input signals were Fourier transformed, their magnitudes multiplied by the number of elements each fired, and the results summed. This magnitude was then integrated and the 1-e-2 point chosen for comparison. This point was well below 0.5 MHz.
The on axis intensities and energies are in all cases higher for the L W pulse than for any of the others. Some frequency shedding was observed in the Lo pulse in that the upper half power point remained fairly constant while the lower increased with increased distance travelled. The LW pulse did not exhibit this low frequency shedding leading us to a description of the pulse as a moving interference pattern. Apparently enough of just the right frequencies arrive on axis from the various sources at just the right time to reconstruct the pulse.
3.0
2.5
2.0 Half Width at
Half Maximum 1. 5 (centimeters)
1.0
0.5
:~.-~ ><
./ A
•
0.0 +-----+-----+-----+-----+-----+-----+-----; 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1. 75
Distance from array (meters)
Q. LW .• - CW@O. 5 MHz ••. CW@1. 0 MHz -A- CW@2. 0 MHz
Figure 2. Beam width for 441 element simulated array. The CW signals are obtained by driving all elements with the same amplitude. The simulated array is 6 cm by 6 cm.
470
N I.00E-02
0 r m a 1 I.00E-03 i z e d
1.00E-04 E n e r
9 1.00E-05 Y
Figure 3.
: .... , i ............. . ~ . • ,
". " ~
10 distance i n em
. ..:::.
~ • , .
•
-"Localized ··'Tone Burst
" 1' . '. .
1
100
The ener~y of the L W beam on axis compared to the energy of the 1/2 MHz CW toneburst. The normalization energy is a calculation of the total energy put into the water by the array elements assuming they are point sources. The experimental energy is obtained by integrating the intensity over time.
The third series of measurements using the 25 element array used only 6 signals. While not an ideal situation, the experiment allowed us to see if inter-element array effects would preclude actual launching of the pulse.
Figure 3 shows the on axis energy for the LW pulse compared to a 1/2 MHz CW toneburst. Both signals were launched from the array and detected by the transducer used before. The energy is determined for both cases by integrating the intensity of the on axis signal over the time of the pulse for the L W signal and over the chosen window for the toneburst. The normalization is calculated by determining the total energy generated in the water by an equivalent set of point sources.
The results for this case showed that the L W can indeed be generated by a real array. Further, these results show that the resultant beam, though generated by only 6 signals and 25 elements, will still beat a comparable CW beam by a factor of at least 4 in energy on axis. Simulation results were a full order of magnitude improvement .
SUMMARY
We have addressed three questions in this research: 1) can the LW signals be generated by real acoustic devices; 2) will these signals interfere with each other in the presence of noise and minor misalignment well enough to form a pulse which will behave as predicted; 3) can these signals be generated simultaneously by an actual array, or will the inter element interactions make the launching of an actual pulse impossible. The results to date have been overwhelmingly positive and indicate that this is a real, as opposed to a mathematical, effect.
Our research plan is currently focusing on three main areas: 1) propagation of the LW pulse through mode-<:onverting interfaces and in dispersive media; 2) generation of the L W pulse using broad bandwidth, high power-high efficiency transducer arrays; 3) resolution determination in imaging experiments.
471
ACKNOWLEDGEMENTS
The authors wish to thank Dr. Dale W. Fitting of N.I.S.T., Boulder, Colorado for the array used in the third series of experiments. Thanks also to Ken Waltjen an Denis Silva for the demultiplexer units. Work performed under the auspices of the U.S. Department of energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.
REFERENCES
[1]
[2]
[3]
[4]
472
J. N. Brittingham, "Focus Wave Modes in Homogeneous Maxwell's Equations: Transverse Electric Mode", J. Appl. Phys., 54, 1179, 1983. R. W. Ziolkowski, "Localized Transmission of Electromagnetic Energy", Phys. Rev.A, 39, 2005, 1989. R. W. Ziolkowski, D. Kent Lewis and B. D. Cook, "Evidence of Localized Wave Transmission", Physical Review Letters, 62, 147, 1989. R. W. Ziolkowski and D. Kent Lewis, "Verification of the Localized Wave Transmission Effect", submitted to J. Appl. Phys.
IMAGING VERTICALLY ORIENTED DEFECTS WITH MULTI-SAFT
M. Lorenz, U. Stelwagen and A.J. Berkhout*
TNO Institute of Applied Physics Instrumentation Department P.O. Box 155, 2600 AD Delft, The Netherlands
*Delft University of Technology Laboratory of Applied Physics P.O. Box 5046, 2600 GA Delft, The Netherlands
INTRODUCTION
Imaging vertically oriented defects using the Synthetic Aperture Focusing Technique (SAFT) requires special consideration. When the faces of the defect are reasonably smooth, diffraction energy is low and the indirect reflections
fwaves reflected at front and/or back-wall) must be used for proper imaging 1]. Conventional single-transducer measurements do not produce a correct
image of a vertically oriented defect after focusing (due to a lack of resolution) and one is forced to make use of separate sources and receivers. The tandem configuration, scanning with both source and receiver separated by some distance, is capable of producing useful measurements, but the use of two moving transducers makes the tandem set-up generally too complicated in practice. When measuring mode-converted waves, a single-transducer technique may be used (LLT-technique [2]). However, with LLT-techniques the imaging zone is limited.
In this paper we present a technique that makes use of one fixed source, illuminating a defined region of interest, and a scanning receiver. The position of the source and the receiver aperture have to be chosen carefully, as will be shown with the aid of modeled results. Taking into account direct and indirect wavepaths implies the use of multiple synthetic apertures. Therefore, the imaging method is referred to as Multi-SAFT. To avoid mode-conversion effects reducing the signal amplitudes, a simple modification in the set-up will be proposed. Finally, one of the first results of applying Multi-SAFT to a real dataset will be shown.
LATERAL AND AXIAL RESOLUTION
Good imaging requires optimal lateral and axial resolution. As will be shown, for the measured data this requires separation of responses from different point diffractors in each measured trace, as well as variation of insonification angles to those diffractors along different traces. The separation of responses depends on the difference in traveltime, i.e. the difference in pathlength to the different point diffractors. The variation of insonification angle depends on the available aperture. Making use of different wavepaths increases the range of insonification angles considerably.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 473
receiver source
Figure 1. The (1,0) wavepath.
For this section it is convenient to define a short hand notation to designate the different wavepaths that may exist in a plate. A wave may reflect one or more times at front- and/or back-wall of the plate before or after defect interaction. This will be denoted as "(n,m)", where the first number denotes the number of (front- + back-wall) reflections in the sourc~ffractor path and the second number the same for the diffractor-receiver path. Thus, the (1,0) wavepath implicates one reflection (at the back-wall) of the source-signal before defect interaction, and no reflection after defect interaction (see Figure 1).
To study the influence of traveltime differences consider a model which consists of two point diffractors, at 15 mm and 20 mm depth, in a 20 mm thick plate of steel (Figure 2). Transverse waves with a wave velocity of 3235 m/s will be used for direct and indirect insonification. Figure 3 shows the one-way traveltime differences, i.e. from a transducer to each of the point diffractors for the direct (0) and indirect (1) wavepaths. The two graphs have opposite signs as the two wavepaths encounter the two diffractors in opposite order .
From the graphs it can be seen, that the traveltime difference decreases for increasing x-offset. The net traveltime difference for any (0,1) or (1,0) wavepath can be reconstructed from the graphs in Figure 3. For a single-transducer (zero-offset) configuration the positions of source and receiver coincide and the net traveltime-difference curve is simply the sum of the two curves. This (dashed) curve is indicated in Figure 3. The source-signal, which will be used in the modeling experiments, has a Signal-length of approximately 0.4 J1.s. From Figure 3 we may thus conclude, that the zero-offset (0,1) or (1,0) ("roundtrip") wavepaths will never provide enough separation for two point diffractors to be imaged separately.
The variation of insonification angle is proportional to the change in traveltime difference. Thus, poor variation will be obtained when the traveltim~fference curve does not vary much, which is the case for the single-transducer roundtri p-signals .
Generally, there are four possible situations regarding the separation and variation of the echo-acoustic information from two vertically spaced point diffractors near a plane reflector:
1. no separation and no variation; 2. no separation but good variation; 3. good separation but no variation; 4. good separation and good variation.
474
Figure 3.
2.0
1.5
g 1.0
~ 0.5 !!
,E 0.0 ~ .~ -0.5 _ -1.0
~ -1.5 I-
-2.0 o 10 20 30 40 50 60 70 80 90 100
x-offset (mm)
Traveltime differences (transducer position - diffractor position, shear waves) for two 5 mm vertically spaced point diffractors as a function of position of the transducer for the (0,1) or (l,O)-wavepath.
The degree of separation and variation depends on the source--defect-receiver geometry. In Figure 4, for the two-point model, the four possibilities have been indicated for the (1,0) wavepath. The source position is 20 mm or 80 mm, the receiver aperture is 20 mm wide and centered around 20 mm or 80 mm. Using a simple ray-tracing algorithm and shear waves, these datasets were modeled and then focused with Multi-SAFT. The imaging results of the four different data-acquisition configurations are shown in Figure 5.
When there is neither separation nor variation (Fi&ure 5.a), only a part of an ellips (indicating the points with equal traveltimes) appears in the image, showing no lateral resolution (resolution in the direction of the ellipses, constructed by the source and receivers) and no axial resolution (resolution in the direction perpendicular to the ellipses). With no separation but . good variation (Figure 5. b) the lateral resolution improves, but the axial resolution is poor. Good separation but no variation (Figure 5.c) gives an image of two well-separated parts of ellipses, having good axial resolution, but no lateral resolution. At last, when there is good separation and good variation (Figure 5.d), both lateral and axial resolutions are satisfying.
LATERAL SPLIT DATA-ACQUISITION
The objects of interest in our research are all elastic media, which implies that compressional (P) as well as shear (S) waves may propagate and convert
Figure 4. The traveltime differences (shear waves) for the four different data-acquisition configurations using the (1 ,O)-wavepath (Figure 1) in the two-point model (Figure 2), for two source positions and two receiver apertures.
475
6-
4-
2-
0-... I .. 0 5 10 15 .E-3 ...
C L '" I I>l
8-
6-
4-
2-
0-... I .. 0 5 10 15 .E-3 ...
DL '" I I>l
8-
6-
4-
2-
0-I 0 5 10 15 .E-3
Figure 5. The images of two point diffractors designated in Figure 4 (region of interest lOx20 mm), for the four different data-aquisition configurations.
a. no separation and no variation; b. no separation but good variation; c. good separation but no variation; d. good separation and good variation.
476
Figure 6.
~ source
An indirect source-wavepath with 100% SS-reflection at the back-wall will result in almost 90% SP-conversion at the face of the vertical defect.
into each other. Whether or not, and what percentage of an incident wave will convert, depends on the angle of incidence and the characteristics of the interface. The interface may be a crack or a (free) boundary of the object. In practice, a number of wavepaths will not be possible due to mode-conversion effects. For instance, measurements are often made with shear waves (as they have a smaller wavelength, and therefore give a better resolution than compressional waves). But shear waves will convert into compressional waves for angles of incidence smaller than N 33' (with respect to the normal on a steel/air-interface; for steel: Cs = 3235 mis, c = 5960 m/s) . The interaction at the back-wall may give 100% SS-reflection (due to an angle of incidence larger than 33'), but the next interaction with a verticall'y oriented defect then may result in a dramatic mode-conversion of up to 90% lsee Figure 6).
It thus seems, that the wavepath of Figure 6, with the 60' insonification angle, cannot be used in practice because of the mode-conversion effects. However, when the insonification would not take place in the plane perpendicular to the vertical defect, the angle (3 with the face of the defect will increase while a, the insonification angle, remains 60' (Figure 7). From Figure 7 it may be deduced, that for increasing angle " which will be called the "lateral split an&le", the angle of incidence at the vertical defect (3 will become larger than 190-a). This implies, that the source has to be positioned out-of-plane, i.e. not on the line of the image-plane and the receivers. The image plane will also have an inclination , with the face of the vertical defect. This configuration will be called Lateral Split Data-acquisition: LSD.
Figure 7.
o 10 20 30 40 50 60 70 80 90
y(") ..
Angles of incidence as a function of split angle for different insonification angles a.
477
Figure 8. The image of a 10 mm vertically oriented defect as the result of applying Multi-SAFT and LSD to a real dataset.
Thus, to be able to insonificate a vertically oriented defect with a transducer under a certain angle (45', 60', 70'), the split angle has to be large enough so that 100% SS-reflection is ensured. The angles of incidence are plotted in the Figure 7 as a function of the split angle , .
One preliminary experimental result of Multi-SAFT using LSD is shown in Figure 8. It is the image of a 10 mm deep slot positioned at the back-wall of a 40 mm thick plate of steel. The defect's faces are smooth and well-reflecting. A split angle of 30' was used with the source at 45 mm (x = 0 at the center of the image) and the receiver aperture running from 0 mm to 150 mm (150 positions) . The (1,0) signals were used for imaging as the data-acquisition was optimized for that wavepath.
CONCLUSIONS
An improved imaging technique, Multi-SAFT, is proposed which utilizes direct as well as indirect wavepaths. The use of Multi-SAFT for one fixed source and one scanning receiver may provide excellent images, provided the data-acquisition configuration has been carefully optimized. For an optimized configuration the chosen source position and receiver aperture must allow separation of signals as well as variation of insonification angles during scanning. Furthermore, Lateral Split Data-acquisition may be used to avoid mode-conversion effects and allowing source and receiver transducers to be positioned approximately at the same distance from the defect . This proves to be very important in practice!
REFERENCES
[1]
[2]
478
L.F. van der Wal, M. Lorenz, and A.J . Berkhout, A unified approach towards inverse problems in ultrasonic NDT, in: "Elastic Waves and Ultrasonic Nondestructive Evaluation", S. K. Datta, J. D. Achenbach, y. S. Rajapakse, eds ., North-Holland, Amsterdam (1990) . F. Walte, W. Gebhardt, V. Schmitz, W. Miiller, K.J. Langenberg, and M. Berger, PC-SAFT mit LLT-Prufk6pfen, in: "Vortragen & Plakatberichten DGZfP Jahrestagung Trier 1990", to be published.
TEMPERATURE DEPENDENCE OF OPTICAL ENERGY GAP IN
ABSTRACT
K.N. Madhusoodanan and Jacob Philip
Department of Physics Cochin University of Science and Technology Cochin 682 022, India
We have recorded the PA spectra of ASxSel-x (0.10 ~ x ~ DAD) and GexSel-x (0.10 ~ x ~ 0.25) glasses in the absorption edge region at various fixed temperatures using a high temperature gas-microphone P A spectrometer. The spectra have been recorded for temperatures above glass transition temperature T g. It is observed that the exponential edge region of the spectrum broadens as the temperature is increased and the rate of this broadening is more for temperatures larger than T g. The optical energy gap Eg of the samples have been determined from tlie P A spectra. It is found that Eg decreases with increase in temperature and the rate of this decrease is larger for temperatures T > T g.
INTRODUCTION
A great deal of work has been done on the various physical properties of amorphous semiconducts in recent years owing to its technological applications as electronic and optoelectronic materials [1]. The chalcogenide glasses constitute an important class of amorphous semiconductors. As and Ge form good glasses with Se over a wide range of compositions. The addition of As and Ge in Se results in the crosslining of Se chain and gives rise to change in average coordination number and network dimensionality and thus the physical properties vary as the composition is changed. In earlier papers we have discussed· the composition dependence of optical energy gap Eg and thermal diffusivity in As-Se and Ge-Se glasses [2,3]. Here we present the temperature dependence of Eg and features of optical absorption in the absorption edge region. Such investigations on these materials will give information about the microscopic structures and defects because electronic properties depend largely on these features (4).
We have used photoacoustic (PA) technique (5), which depends upon the detection of acoustic signal produced when a sample kept in an enclosed chamber is illuminated by an intensity modulated light beam, to study the optical absorption in AsxSel-x (0.10 ~ x ~ 0.50) and GexSel-x (0.10 ~ x ~ 0.25) glasses in the absorption edge region, from room temperature to temperatures above glass transition temperature T g. Since the P A signal depends only on the absorbed power it is extremely useful in the study of
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 479
highly light scattering materials like amorphous semiconductors. Where conventional spectroscopic techniques do not give good results. Moreover the P A technique has been proved to be powerful tool to study optical absorption over a wide range of absorption coefficient. This is an additional advantage in the study of amorphous semiconductors because in most of these materials in the absorption edge region absorption coefficient drastically changes from about lO- lcm -1 to 10 5 cm -1.
EXPERIMENT AL
The P A spectrometer used for the present investigations consists of [6] 1 KW Xe lamp, monochromator, mechanical chopper, high temperature P A cell containing microphone and lock-in amplifier. The P A cell, which can be used up to 300° C, uses a Helmholtz type configuration with sample compartment and microphone compartment separated by a narrow stainless steel tube. The heater is wound on the sample compartment and the temperature can be controlled by a temperature controller. The P A spectra of the samples have been recorded by plotting the normalized P A signal amplitude as function of wavelength at a fixed chopping frequency. The normalization has been done with power spectrum of the lamp obtained using carbon block as the sample. The spectra have been recorded for AsxSel-x and GexSel-x glasses at various temperatures from room temperature ·to temperatures T > T g. The samples have been prepared using the conventional melt quenching technique. For this appropriate amount of constituent elements are taken in an evacuated quartz ampoule and kept at about lOOK for 24 hours. The ampoule is rotated periodically for homogeneous mixing. The ampoule is then quenched in ice water. The amorphous nature of the sample is checking using X-ray diffractometry.
Figure 1.
480
6
900
c b
-------~-a
a) T = 30°C
b) T = 50°C
c) T = 100°C
d) T = 130·C
e) T = 160·C
f) T = 200'C
600
P A spectra of AO.25 Seo. 75 samples recorded at different temperatures.
RESULTS AND DISCUSSION
The PA spectra of AsO.25SeO.75 and GeO.IOSeO.gO samples recorded at different temperatures are shown in Figure 1 and 2 respectively. Spectra have been recorded for other compositions also of As-Se and Ge-Se glasses. The P A spectra give information on optical absorption features. The signal increases from the lower energy side and reaches a saturation level for hv > Ez where Eg is the optical energy gap. As explained by Tauc [7] there are three different regions in the optical absorption edge for an amorphous semiconductor corresponding to three different ranges of the absorption coefficient {3. In the high absorption region where {3 ~ 104 cm -1 the absorption coefficient has an energy dependence given by
hv {3(hv) = B(hv - Eg) 2 (1)
where B is a constant. This equation can be used to define the optical energy gap Eg. In Figures 1 and 2 this region is characterized by the saturation of normalized P A Signal. The Rosencwaig-Gersho (RG) theory [8] predicts this type of behavior for optically opaque samples.
The absorption edge in these materials has an exponential part associated with disorder induced potential fluctuations which extends in the absorption coefficient range 1 cm -1 $ {3 $ 104 cm -1. The absorption in this region depends on the photon energy according to the relation
{3(hv) = (Joexp (hv lEo) (2)
where Eo characterizes the slope of the exponential region. In this region the normalized P A signal is proportional to the absorption coefficient and increases with photon energy as shown in Figures 1 and 2. The weak absorption region with {3 < 1 cm -I is characterized by an absorption higher than that expected from extrapolated exponential region. Its origin can be attributed to impurities,
Figure 2.
5 e d c
vi b
§4 a .ci ~
-.-J <i z a) T =30·C <.')3 tn b) T=50·C <i c) T = 100'C 0..
0 d) T = 150'C w elT =200'c N
=:i2 <i 2: a: 0 z
800 500
P A spectra of Geo.loSeo.90 sample recorded at different temperatures.
481
charged or neutral defects etc. and it is a measure of the states in the energy gap. The optical energy gap ~ of the samples investigated have been determined from the P A spectra. This has been done graphically by drawing the tangents to the wavelength vs. P A signal plots at the saturation region and the exponential edge region. The point where two tangents meet corresponds to Eg.
The value of Eg at various temperatures have been determined for As-Se and Ge-Se samples from the P A spectrum. The variations of E with temperatures for As-Se and Ge-Se glasses are shown in Figures 3 an~ 4 respectively. For all these samples Eg decreases with increase in temperature and the rate of this decrease is larger for T > T g. However there is no sharp change in Eg at T g' From the temperature dependence of P A spectra (Figures 1 and 2) it is observed that the exponential absorption edge region of the spectrum broadens as the temperatures is increased and the rate of this broadening is more for temperatures larger than T. Dow and Redfield [9] proposed that exponential edges are due to electric fiel~ induced ionization of excitons. The electric field can be caused by ionized impurities, phonons or by potential fluctuations. For large average electric fields the exponential slope of the edge is smaller and edge is broader. At higher temperatures phonon induced electric field will dominate and the slope of the edge will be temperature dependent. The electron-phonon interactions play a major role in determining the temperature variation of optical absorption.
Since noncrystalline solids lack long range order and their electronic properties are determined primarily by the nature of the short range order their properties can be explained on the basis of chemical bond approach. In chalcogenide glasses containing high concentration of chalcogen element the lone pair states associated with chalcogen atoms form the top of the valence band and the antibonding states form the conduction band [10]. Therefore the electronic properties can be related to the structural features like local bonding configurations, network flexibility and dimensionality etc. The average coordination number of bonds per atom z which represents the short range order of local atomic units is an important parameter in understanding the normal bonding structures. For chalcogenide glasses z is between 2 and 3 whereas for Si-type materials it is 4 [11]. Therefore one inherent property of chalcogenides lies in
Figure 3.
482
0.1
1.9
~ 18 0.3 ~ . 0.5
WOl 035
0.4
1.7
1.6
50 100 150 200 250 TEMPERATURE (·c )
Variation of optical energy gap Eg with temperature for AsxSel-x glasses. The x values are noted on the curves. Arrow indicates glass transition temperature.
Figure 4.
2·3
0.25
~ 0.15 Jf2.1
2.0
1.9
1.8 L---~50~~1~OO~~1~50~~20~O~-2~5~O~
TEMPERATURE (·C)
Variation of optical energy gap Eg with temperature for GexSel-x glasses. X values are noted on the curves. Arrow indicates glass transition temperature.
its small z which implies that the lattices are substantially flexible. Thus the atomic structures are likely to be distorted when electrons are excited. The strong electron-lattice coupling in these systems can be the reason for the temperature dependent exponential absorption. The decrease in Eg as the temperature increases may be due to the increased lone pair interaction, due to bond twisting and change in separation between chain or layers present in the glass system, which results in the widening of lone pair band. Therefore the change in Eg and exponential absorption edge as the temperature is varied can be attributed to the enhanced randomness of normal bonding structure associated with the flexibility of the glass network. Such effects are more predominant when T > T g.
CONCLUSION
The P A technique is found to be extremely useful to study the features of optical absorption and to determine optical energy gap is chalcogeriide glasses up to temperature T > T g. The broadening. of the exponential edge region and the decrease in Eg as the temperature is increased are explained on the basis of chemical bond approach and structural features of the glass network.
REFERENCES
I~I [~l [6]
D. Adler, Sci. American, 236:36, (1977). K.N. Madhusoodanan and Jacob Philip, Pramana J. Phy., 33:705, (1989). K.N. Madhusoodanan and Jacob Philip, Phys. Stat. Solidi(a), 108:775, (1988). S.R. Elliot, "Physics of Amorphous Materials", Longman, London, (1984). A. Rosencwaig, "Photoacoustics and Photoacoustic Spectroscopy", Wiley, New York, (1980). K.N. Madhusoodanan, J. Philip, G. Parthasarathy, S. Asokan and E.S. Rajhagopal, Philos. Mag.B., 58:123, (1988).
483
[7] J. Tauc, in: "Amorphous and Liquid Semiconductors", J. Tauc, ed., Plenum, New York, (1974).
89] A. Rosencwaig and A. Gersho, J. Appl. Phys., 47:64, (1976). J.D. Dow and D. Redfield, Phys. Rev., B5:594, (1972).
1°1 M. Kastner, Phys. Rev., B7:5237, (1973). 11 R. Zallen, "The Physics of Amorphous Solids", Wiley, New York, (1983).
484
INVESTIGATIONS OF PHOTOACOUSTIC SIGNALS IN POWDERS
U. Madvaliew, V.V. Proklov*, A.M. Ashurov
Physical Technical Institute of the Tadjik Academy of Sciences 734063, Dushanbe, U.S.S.R. *Institute of Radio Engineering and Electronics
Academy of Sciences of the U.S.S.R. K. Marx 18, 103907 Moscow, U.S.S.R.
Nowadays the processes underlying the photoacoustic signals (P A) are very well understood [1,2]. At the same time the P A signal in a porous material is a matter of great scientific interest because of its great promises for the applications and a poor knowledge of the physical concepts.
This paper is devoted to the mechanisms of the formation of the P A signal in powders in connection to the geometrical parameters of the samples. Using the single beam P A-spectrometer we investigated the properties of the several dyes (the malachite green, methyl violet, rhodamin 6G etc.) absorbed by macro porous silica powder (Sylochrome S-80, mean pore diameter is 50 nm) with dye concentrations in the range of 1 10-9 .;. 1 10-4 Mol/gr.
lt was shown earlier [4-6] that P A signals from porous materials in the case of gas-microphone detection are very sensitive to the thermal transmission through the boundary of the sample with the gas chamber and the periodic expansion of the "intrinsic gas" (the gas captured into the pores and the intergrain spaces). So first of all we studied experimentally the P A signals with a different filling factor of the gas chamber (when the ratio 19/1s was
U,I11V 06"'0
20 ltg#o
a c i.CJ , ,
JO ,
b 0.5
iO"9 iO"S 10"7 10"' C, mlgr
Figure 1. Dependence of P A signal (U) on concentration of the malachite green dye absorbed by macro porous silica S-80 (,\ = 620 nm, fr = 25 Hz) a) 19 * 0, b) 19 = O.
Physi("ai Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 485
changed, where 19 and Is are gas chamber and porous sample lengths, respectively). The typical results with the malachite green dye are shown on Figure 1. We clearly see the difference between PA signals U for the cases of the partially filled chamber (lg/ls * 0, see the curve (a) on Figure 1) and the fully filled chamber (1 /ls = U, the curve (b) on Figure 1). Its ratio is shown by the curve (c) on Filgure 1. The difference can be explained by the different processes responsible for the generation of the P A signal, for these cases - in the case ~b) there is just the "intrinsic gas expansion II and in the case (a) are both the 'intrinsic gas expansion II and the free (chamber) gas expansion as well. So the curve (c) shows that under small dye concentration e ~ 1 10-7 M/gr (and the small light absorption a) all light energy is distributed in all sample volume and in both cases the main mechanism is the same - intrinsic gas expansion. But when e ~ 1 10-7 M/gr we can see the growing of relative influence of the free gas in the chamber.
One of the matter of interest is the PA signal dependence on the grain sizes of the powders, what need to be appeared due to expected changes of the thermal and optical properties of the structure and its P A response. Figure 2 shows the PA spectra of GdSe powder with different grain sizes (solid lines) and the ratio of the spectra for minimum to maximum grain sizes (the dotted line). We can see an approximately constant PA signal (independently of the grain sizes A ~ 750 nm ) and the light absorption a is very small. When the light absorption a is growing (A S 750 nm) the P A signal is increasing with the remarkable influence of the grain sizes (the P A signal increased when the grain size decreased). As we have pointed out before, the dominant mechanism of the formation of the P A signal in the highly absorbent powders is the thermal transportation from the specimen to the free gas in the chamber. But it obviously must be enhanced when the grain size decreases because its effective square of the boundary between the absorbent body and the free gas of the chamber. When the light absorption decreases this effective square (and the grain size) is not so important compared with the value of the P A signal because in according to what has been mentioned before, at this condition the P A signal arises mainly due to the expansion of the "intrinsic gas".
Figure 3 shows the P A signal U as the function of the thickness Is of the silica powder with different concentration Sc of the malachite green dye. When e ~ 1 10-7 Mlgr (high absorption a) the P A signal is decreased with growing Is, because of the power absorbed near the surface, it must be distributed on the increased volume of the sample. But when e« 1 10-7 Mlgr (under small a) the P A signal is approximately independent of Is because at this condition the signal is just generated due to the energy absorbed quite far from the sample surface.
Figure 2.
486
U,mV
1.2
0.9
0.6
0.3
Photoacoustic spectra of edSe - powders with different grain sizes d (f = 25 Hz). d, 11m: (1) 50; (2) 50.;.63; (3) 63.;.100; (4) 100.;.200;
(5) 200.;.315; (6) 315.;.400; (7) ~ 400.
Figure 3.
U,mV u,n.V
SO aD
16 45
f2 40
.5 ----.
1. 2 .3 4 5 6 ],g,mm
Dependence of P A signal (U) on thickness of the macroporous silica with different concentrations of the absorbed malachite ~reen dye (C). ,.\ = 620 nm, f = 25 Hz. C, M/gr: {l) 2.5 10-6; (2) 2.5 10-7; (3) 2.5 10-9.
The investigated peculiarities of the P A signals in powders are very promising to ~et its quantitative description. For example the P A signal dependence Ut C) from Figure l.a it drawn on the linear scale of the axis is obviously nonlinear (see Figure 4.a). The question is: what is the nature of this nonlinearity? Now we know that under fixed grain sizes and thickness of the powder sample we need to take into account just one more argument, which is the light scattering as a function of the dye concentration. Following to Gurevich-Kobelka-Munck theory we can get in the case of a surface with diffused reflection
a (1-R)2 g=2R' (1)
where Sand R are the coefficients of the light scattering and reflection, respectively. To obtain characteristics of the material absorption a( C) we can use a proportionality of the P A signal to the absorbed light power i.e. U ~ Wo(l-R). From here we can rewrite equation (1) to obtain
a U2 g = 2(W o-U)Wo' (2)
Figure 4.
u:ya(i-U)
0.1
0.2
Qf
50
Dependence of PA signal (a) and normalized function a/S (b) on concentration of the methyl violet dye absorbed by macro porous silica S-80. ,.\ = 580 nm, fr = 25 Hz.
487
So, when Wo ~ const we can transform U(C) to a(C)/S, as it is shown on Figure 4 (curve b). As it is seen, a good linearization of a( C)/S took place over a wide region of C (up to C ~ 7 10-5 M/gr ), what a constant absorption mechanism on this region means. At higher concentrations (C > 7 10-5 M/gr) the curve (b) is sublinear, what has been explained as the appearance of dye aggregates on the powder surface (after additional experiments it was deflected another version due to II heat saturation" effect) and also we have directly observed at this conditions several specific features: broadening of the spectral line and arising of new spectral components.
REFERENCES
[1]
[~l
Iii
488
A. Rosencwaig, IIPhotoacoustics and photoacoustic spectroscopy II , J. Wiley and Sons, N.Y., Chichester, Brisbane, Toronto (1980). F.A. McDonald, G.C. Wetsel, J. Appl. Phys., 49:2313, (1978). A.M. Ashurov, U. Madvaliew, V.V. Proklov, J. Scientific Instruments (Sov.), 5:236, (1988). P. Helander, J. Appl. Phys., 54:3410, (1983). Z.A. Jasa, W.A. Jacson, N.M. Amer, Appl. Opt., 21:21, (1982). J.P. Monchalin, L. Bertrand, G. Roussez et al., J. Appl. Phys., 56:190, (1984).
PHOTOTHERMAL-WAVE DIFFRACTION AND INTERFERENCE IN
CONDENSED MEDIA: EXPERIMENTAL EVIDENCE IN ALUMINUM
Andreas Mandelis and K wan F. Leung
Photoacoustic and Photothermal Sciences Laboratory Ontario Laser and Lightwave Research Center Department of Mechanical Engineering University of Toronto Toronto, Ontario, Canada M5S 1A4
INTRODUCTION
The peculiar nature of thermal-waves as spatially heavily damped pseudowaves [1] is mathematically the result of a specific, Helmholtz wave-like form of the Fourier heat conduction equation. In a general three-dimensional (3-D) theoretical framework, such pseudowaves have been shown [2] to possess diffractive and interference characteristics. A diffraction integral has been established in the experimentally justifiable limit of a small-aperture (SA) approximation, which allows analytical treatments of thermal-wave fields generated by arbitrary aperture functions. In this work the theoretical prediction [2] for the diffraction field generated from a photothermal aperture function, as well as the predictions [2] for the interference fields resulting from two photothermal apertures operating coherently (in-phase) or anti-coherently (out-of-phase), have been tested experimentally.
THEORETICAL BACKGROUND AND NUMERICAL SIMULATIONS
Diffraction
The geometry of laser beam generated thermal-wave propagation in 3-D space is shown in Figure l.a. Under the condition of the small-aperture (SA) approximation, a convenient representation of the diffraction integral can be written encompassing all field locations on the observation plane, such that ro > p in Figure l.a, where (ro,wo) ....... (x,y) are observation plane variables, and (p, WI) ....... ((,11) are aperture plane variables. Experimentally, when focused laser beams are used, the SA approximation is valid for essentially all field positions outside the symmetry axis of the exciting laser beam [21.
For the TEMoo mode of a Gaussian laser-beam, the aperture profile of unit magnitude is given by
To(p) = exp(-p2/w2), (1)
where w is the beam waist. The diffraction integral then yields the complex photothermal-wave temperature field, which can be written out in terms of its experimentally relevant components, amplitude and phase [2].
Numerical simulations of the resulting expressions indicate the effect of the
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 489
Aperture Plane
Incident Optical
Field ------
Figure 1.
{
Aperture b Plane
(',,,.0)
a
,
x
Observation Plane
Photothermal-wave diffraction geometry; and Interference geometry on the aperture plane ((,1]) .
exciting laser-beam waist size is quite small in the 30 11m to 800 11m range. This result, in turn, indicates that the resolution of the diffraction field is only weakly dependent on the laser-beam size in the above range, and is a guide to the design of the optical part of the experimental set-up.
Interference
The geometry of thermal-wave interference fields generated by two laser-beams incident on a material surface, a distance 2d from each other along the 1] axis of the aperture plane, is shown in Figure Lb. Due to the fact that the circular symmetry is broken when two photothermal sources are considered, a Cartesian coordinate representation (x,y,z) of the superposition thermal-wave field becomes necessary [2], thus forcing one to abandon the simpler polar coordinate formulation (ro,z).
Assuming that two exciting laser-beams of unit irradiance have Gaussian TEM oo profiles of equal spatial spot sizes (WI = W2 :: w), and also assuming
490
in-phase operation (constructive interference indicated by the (+) s.ign), or out-of-phase operation (destructive interference indicated by the (-) sIgn), we obtain source functions (see Figure 1.b):
(2)
Now insertion of Eq.(2) into the diffraction integral [2) yields complex expressions for the temperature field in either case of photothermal-wave interference. These expressions can also be reduced to convenient amplitude and phase components and compared with experimental data.
EXPERIMENTAL AND RESULTS
The details of the instrumentation and experimental scheme for thermalwave diffraction detection have been presented elsewhere [3). The exciting beams (s) were supplied from a Hughes Aircraft He-Ne laser delivering ca. 10 mW at 632.8 nm. The detector element was a 28 /lm-thick polyvinylidene fluoride (PVDF) pyroelectric film with .an upper (grounded) electrode made of standard Pennwalt AI-Ni layers (200 A Ni covered with 600 A AI) [4). The lower surface of the PVDF film was not electroded and was in contact with a brass pin of 0.8 mm circular tip diameter . In this arrangement, the tip was able to monitor local charge changes on the PVDF surface due to the photopyroelectric (P2E) effect [3]. A cylindrical aluminum sample (diameter, D: 1 cm, thickness, L: 1.5 mm) was mounted on the upper flat surface of the PVDF P2E detector and intimate contact was assured. At the lowest modulation frequency (f = 18 Hz) of our experiments, the thermal diffusion length in aluminum was /ls(f = 18 Hz) = 1.2 mm « D so that the sample could be adequately approximated by a radially semi-infinite aluminum solid. The fact that the aluminum sample used in this work was very thick compared to the PVDF detector ensured [5] that the transducer would operate as a thermometer, producing a pyroelectric charge proportional to the PVDF thickness-averaged local temperature in the pyroelectric element, which is essentially equal to the local temperature change at the sample-transducer interface. Therefore, the P2E signals thus obtained were found to be proportional to the local values of the thermal-wave field at the probe pin position over the plane of the sample back surface.
Figure 2 shows the schematic diagram of the photothermal-wave interference experiment for dual laser-beam incidence on sample D and in-phase
l ~s:. ~I L.......-He_._NC_L_AS_ER---JIU Z : uT,u_n ~ Hc·NeLASER f· ....... · .. ·-.. ~ .. ·-~ , . . ,
CHOPPER [:>-<j B
•
Figure 2.
CHOPPER : M3 , LI '- /. : l-{-l·k :: U ~ ~ 0
[==:::J SAMPLE
b
LI T /. ~ /-1 .. ··/ ,, 17 M2 :: L2 I I 0
SAMPI..E~ Schematic diagram of optical circuit used for photothermalwave interference detection. C: mechanical chopper, BS: beamsplitter, L: lens, M: mirror;
(a) In-phase modulation. (b) Out-of-phase modulation.
491
A
B
Figure 3.
492
Amplitude (x 10-5 ) v. a
o .• ~Z
Z.ooo
Z·ooo:-Z.ooo
Z . OOO
Z· 000:-2.ooo
Ampl itude (x 10-4 ) (arb . units) a
~.oaz
3 . 1' Z.OOO
0'3051~§ -Z.ooo~
Z . ooo-Z.ooo
:11.000
A) Experimental diffraction patterns generated by a w=300 J1.m laser beam and a 1.5 mm-thick Al sample. Modulation frequency: 20 Hz; (a) Amplitude and (b) phase. B) Theoretical simulation of thermal-wave diffraction field generated by a w = 300 J1.m size laser-beam impinging on the surface (z = 0) of a semi-infinite aluminum sample (aA1
= 0.82 cm2/s. The temperature field is shown at z = 1.5 mm. Modulation frequency: 20 Hz; (a) Amplitude and (b) phase.
A
B
Figure 4.
Ampl i tude (x 10-5 )
1 . <1"
1 . 5 00
Phase
1 . 500
Ampl i tude (x 10-5 ) (arb . units)
Phase (x 102 ) deg .
A) Experimental constructive interference patterns generated by a two-laser-beam in-phase geometry. Modulation frequency : 18 Hz. Beam waists: 300 11m. 2d = 1.2 mm. B) Theoretical simulations of thermal-wave interference field generated by two in-phase modulated laser beams of equal irradiances, impinging on the surface (z = 0) of a semi-infinite Al sample. The beam waists are 300 11m each and the temperature field is shown at z = 1.5 mm.
493
A
B
Figure 5.
494
Amplitude (x 10-5 ) v. a
1.2""
•. 000
0 .• 6:1
-0 . 707
- L •• O - 01 . 2:10
Amp l itude (x 10-5 ) (arb. uni ts )
Phase (x 102) deg.
b o .
- 1 .
000
A) Experimental destructive interference patterns generated by a geometry identical to the one shown in Figure 2.b. Modulation frequency : 18 Hz. Beam waists: 300 pm. (a) 2d = 1.8 mm. B) Theoretical simulations of thermal-wave interference field generated by two out-of-phase modulated laser beams of equal irradiances, impinging on the surface (z = 0) of the semi-infinite Al sample. The beam waists are 300 pm each and the temperature field is shown at z = 1.5 mm and f = 18 Hz.
(Figure 2.a), or out-of-phase (Figure 2.b), irradiance modulation. The beamsplitter of Figure 2 was a variable absorbance neutral density filter mounted on a micrometer stage, so as to produce two laser-beams of approximately equal irradiance at the sample surface. The optical absorption coefficient of the aluminum sample surface was further increased by using a very thin black paint film on the surface, which minimized reflections and yielded maximum photothermal signals. Scanning of the laser beam(s) was performed, with the probe pin remaining stationary in contact with the unelectroded PVDF film surface at the center of the exposed film area.
Figure 3.A shows entire thermal-wave diffraction patterns (amplitude and phase) generated upon scanning a 0.3 mm-size laser-beam on the aluminum sample surface. The experimental parameters chosen were identical to those which generated the theoretical simulations of Figure 3.B and thus a direct comparison is possible. The qualitative agreement (overall spatial distribution profiles/morphologies of amplitude and phase images) between Figures 3.A and B is excellent and shows that the semi-finite solid approximation assumed in the theoretical formulation [2] is essentially adequate for analyzing diffraction results from our 1.5 mm-thick sample. .
Figure 4.A shows experimental results of constructive interference patterns in the geometry of Figures l.b and 2.a. For this experiment, 2d = 0.0 mm and 0.6 mm interference fields were obtained by a slight tilt of mirror M3, so as to render both beams co-incident or nearly so. No significant disturbance of the (measured) Gaussian profile of the beams results from this operation. The amplitude results show a monotonic maximum amplitude decrease with increasing separation, as predicted and discussed theoretically. Both amplitude and phase shapes display trends with increasing separation qualitatively similar to theory with the individual source contributions to the thermal-wave interference pattern becoming resolvable at a minimum distance of ca. 1.8 ± 0.2 mm. Figure 4.B shows theoretical results of constructive interference patterns, qualitatively similar to those of Figure 4.A. Figure 5 shows experimental and theoretical thermal-wave destructive interference patterns in the geometry of Figures 1.b and 2.b. The data, in Figure 5.A show clear evidence of destructive interference with spatially overlapping sources, becoming less effective with increasing beam separation distance, in excellent qualitative agreement with theoretical simulations, such as the images shown in Figure 5.B. This agreement extends to field maximum amplitude increases with increasing separation, a minimum halfway between the two beams, amplitude peaks somewhat beyond the actual laser-beam positions, and a step-like structure of the associated phases.
ACKNOWLEDGEMENTS
The support of the Ontario Laser and Lightwave Research Center and of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
REFERENCES
[1]
[2]
[3]
[4]
[5]
H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd. ed. Chap. 2.6, (Clarendon, Oxford, 1959). A. Mandelis, "Theory of photothermal-wave diffraction and interference in condensed media", J.O.S.A. A 6, 298-308, (1989). M. Mieszkowski, K.F. Leung and A. Mandelis, "Photopyroelectric thermal wave detection via contactless capacitive polyvinylidene fluoride (PVDF)-metal probe-tip coupling", Rev. Sci. Instrum. 60, 306-316, (1989). KYNAR Piezo Film Technical Manual, Pennwalt Corp., King of Prussia, PA, (1983). H.J. Coufal, R.K. Grygier, D.E. Horne, and J.E. Fromm, "Pyroelectric calorimeter for photothermal studies of thin films and adsorbates", J. Vac. Sci. Technol. A 5, 2875-2889, (1987).
495
EVALUATION OF THE THICKNESS OF SHELLS BY THE M.I.I.R.
Gerard Maze, Henri Cahingt, Florence Lecrocq, Jean Ripoche
Laboratoire d' Acoustique Ultrasonore et d'Electronique L.A.U.E. U.R.A. C.N.R.S. 1373 Universite du Havre Place Robert Schuman 76610 Le Havre, France
INTRODUCTION
The method of Isolation and Identification of Resonances (M.I.I.R.) was perfected by the authors in the eighties [1-3] in order to study the scattering of a plane acoustic wave from targets of simple shape (plates, cylinders, spheres, ... ) immersed in water. This method is founded on an experimental observation which shows that, at resonance frequencies, targets accumulate energy during the insonification by ultrasonic waves and reradiate it gradually into the surrounding liquid as soon as the excitation is stopped. This phenomenon appears on the chronograms of backscattered echoes when the target is insonified by a burst of long duration. The backscattered echo can be decomposed into three parts (Figure 1): an initial transient state, a quasi-steady state, and a second transient state after the end of the forced vibration, where the amplitude of the backscattered echo decreases exponentially.
The M.I.I.R. consists of two experimental steps: - the first step allows the isolation of resonances in quasilinear spectra, the "Resonance Spectra". We measure as a function of the dimensionless frequency k,a (k, being the wave vector modulus in water, and a the outer radius of the cylindrical shell) the amplitude of the backscattered echo during the second transient state, a few microseconds after the end of the excitation. For this experiment, a simple transducer is used as a transmitting and receiving transducer. One example is given in Figure 2 for an aluminum cylinder. This result represents a true "spectral signature". - the second step furnishes the experimental knowledge of the mode number n of each resonance. A receiver-transducer is rotated around the cylindrical target at a constant distance. It measures, versus azimuthal angle, the amplitude of the echo in the second transient state when a resonance is set up. The "reradiation" occurs in preferential directions. A maximum pressure appears in front of each antinodal point of displacement on the target. This phenomenon is related to the standing waves which are established around the target from the propagation of circumferential waves in two opposite directions. The number of nodes or antinodes divided by two gives the mode number n. This n is exactly the number of wavelengths of the circumferential wave on the cylindrical shell. In Figure 3, we give an example of the reradiation pattern for n = 4. The M.I.I.R. is very useful for studying [4,5] the influence of circumferential waves and guided waves on the scattering pattern of a target. It
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 497
BACKSCATTERING SPECTRUM
RESONANCE SPECTRUM
8
FREE REEMISSION
QUASI -STEADY STATE
INITIAL TRANSIENT STATE
Figure 1. Backscattered echoes, A: off resonance, B: at a resonance .
.... ....
A
20 k,a
Figure 2. Resonance spectra, A: solid aluminum cylinder, B: cylindrical shell (b/a = 0.9).
498
Figure 3.
b/a ~ 0.87
20
II ~(I i ~ ~ \ I B
I \ II ~~j~ ~ J \ i ·· i . i . i I· . !. 11
180
Distribution of pressure versus azimuthal angle around the cylinder.
A: (n,l) :: (2,3) . B. (n,l) - (5,4).
k,a
i " .. ..,
k,a
te.o I
Figure 4. Regge trajectories.
I
I
I , I I
it" +
l i'
/ -I
/ P:l I
/
k,1
499
allows to verify the Resonance Scattering Theory (R.S .T.) [6,7) . In this paper, this second experimental step is used to evaluate the thickness of elastic shells.
INFLUENCE OF THE THICKNESS ON THE RERADIATION PATTERN
For a given cylindrical shell, when the mode number n is known, it is possible to group the resonances among series indexed by 1 along Regge curves. Regge trajectories, as described in a previous paper [8J for cylindrical shells of radius ratio b/a (b = inner radius, a = outer radius) between 0.0, and 0.9 show that:
- for the I = 1 series (Rayleigh wave) , the trajectory shifts to low frequencies when the radius ratio increases.
- for the 1 = 2 series (Whispering Gallery wave n'l), the trajectory depends only weakly on the radius ratio.
- for 1 > 2, the trajectories are shifted to high frequencies.
This last observation is conspicuous because the shift is significant when the shell becomes very thin.
We now attempt identification of resonances chosen in the 1 = 3 family for a special aluminum shell with the axis of the cavity shifted with regard to the axis of the shell. The thickness varies continuall,Y between 0.92 mm and 1.12 mm (a = 10 mm); the average radius ratio bja is equal to 0.9. With the bistatic M.I.I.R. method, we plot, for example, the identifying angular pattern versus the azimuthal an~le of the (8,3) resonance (n = 8, I = 3) usin~ angular coordinates (Figure 5.A), and of the (9,3) resonance (n = 9, I = 3) using Cartesian coordinates (Figure 5.B). In Figure 5.A, we observe that there is no lobe at 0' in the direction opposite of the incidence. On these curves, it is easy to note that the lobes have various widths. If we take the previous
Figure 5.
500
B
I~~W~ I I I I i .,' •• 0
180
Distribution of pressure versus azimuthal angle for the cylindrical shell (b/a = 0.9). A: (8,3), B: (9,3).
n
k,a
Figure 6. Regge trajectories of 1 3 series for various b/a ratios .
remarks into account, we know that the wider the lobe, the smaller the thickness of the shell; indeed, for a same normalized frequency, a pipe whose shell is thinner gives fewer lobes in the identification pattern. Figure 6 shows Reg~e trajectories of the 1 = 3 series with k1a between 20 and 90 for various b/a ratios. If we plot vertical lines which pass through the integer modes n of the shell studied here (b/a = 0.9) (Fig. 6), these lines intersect the Regge trajectories of the shells which have other radius ratios. Each intersection gives a value of n which is not an integer. If we carry back these n values versus the radius ratio on a graph with the integer mode n of the b/a = 0.9 shell as parameter, we obtain Figure 7.
With the angular pattern of Figure 5, the measurement of the lobe width allows the calculation of the fictitious number n of wavelengths on the circumference of the cylindrical shell. This number is not necessarily an integer. By means of Figure 7, it is then possible to determine the corresponding b/a radius ratio and to know the thickness "e" of the shell adjacent to this lobe (e = a(l - b/a)) . It is possible to use the method for an angular pattern which
n
20
10 -
0.7 0.9 b/a
Figure 7. n versus b/a ratios for the 0.9 cylindrical shell.
501
Table 1. thickness of the cylindrical shell bja 0.9
half wavelength n b/a e mm mm
3.00 10.47 0.892 1. 08 2.86 10.99 0.890 1. 10 2.72 11.55 0.886 1. 14 2.86 10.99 0.890 1. 10 2.97 10.58 0.892 1. 08 3.21 9.79 0.895 1. 05 3.25 9.67 0.896 1. 04 4.19 7.50 0.906 0.94 4.82 6.52 0.910 0.90 5.41 5.81 0.913 0.87 4.89 6.43 0.911 0.89 4.26 7.38 0.906 0.94 3.49 9.00 0.900 1. 00 3.14 10.00 0.895 1. 05 3.04 10.34 0.893 1. 07 2.79 11.26 0.887 1. 13 2.65 11.86 0.884 1. 16 2.93 10.72 0.891 1. 09
is plotted in cartesian coordinates or angular coordinates. The resolution is better if the wavelength number is large. We display the results obtained with the described calculation in Table 1.
CONCLUSION
Measurements with the M.I.I.R. show that a quantitative valuation of shell thickness is possible. In particular, the method allows to determine if the thickness of a pipe is uniform over the whole circumference. This measurement procedure is usable for testing pipes of small cross section with a radius below 5 cm because of limitations of existing ultrasonic installations. The accuracy of this method can be improved by the calculation of Regge curves for predetermined width variations of the thickness.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
502
G. Maze, B. Taconet, J. Ripoche, Influence des ondes de galerie 11 echo sur la diffusion d'une onde ultrasonore plane par un cylindre, Physics Letters, 84A:309, (1981). G. Maze, J. Ripoche, Methode d'Isolement et d'Identification des Resonances (MIIR) de cylindres et de tubes soumis 11 une onde acoustique plane dans l'eau, Rev. Phys. Appl., 18:319, (1983). G. Maze, J. Ripoche, Visualization of acoustic scattering by elastic cylinders at low ka, J. Acoust. Soc. Am., 73:41, (1983). G. Maze, J.L. Izbicki, J. Ripoche, Resonances of plates and cylinders: Guided waves, J. Acoust. Soc. Am., 77:1352, (1985). J.L. Izbicki, G. Maze, J. Ripoche, Influence of the free modes of vibration on the acoustic scattering of a circular shell, J. Acoust. Soc. Am., 80:1215, (1986). _ L. Flax, L.R. Dragonette, H. Uberall, Theory of elastic resonance excitation by sound scattering, J. Acoust. Soc. Am., 63:723, (1978).
[7]
[8]
L. Flax, G.C. Gaunaurd, H. Uberall, Theory of resonance scattering in:"Physical Acoustic", W.P. Mason and R.N. Thurston, ed., Academic Press, New York (1981). G. Maze, J. Ripoche, A. Derem, J.L. Rousselot, Diffusion d'une onde ultrasonore par des tubes remplis d'air immerges dans l'eau, Acustica, 55:69, (1984).
503
THE NTH ORDER APPROXIMATION METHOD IN ACOUSTO-OPTICS
AND THE CONDITION FOR IIPUREII BRAGG REFLECTION
R. A. Mertens*, W. Heremant, J.-P . Ottoyt
*Instituut Theoretische Mechanica, Rijksuniversiteit Gent B-9000 Gent, Belgium
tDepartment of Mathematics, Colorado School of Mines Golden, CO 80401, U.S.A.
tSeminarie voor Toegepaste Wiskunde & Biometrie Rijksuniversiteit Gent, B-9000 Gent, Belgium
INTRODUCTION
It is well-known that Bragg diffraction in acousto-optics occurs if the incident light makes a Bragg angle with the ultrasonic wave planes and the diffraction spectrum only consists of the orders -1, 0 and + 1.
IIPure ll Bragg reflection arises if the diffraction results in a spectrum of orders 0 and +1, with evanescent order -1 (Figure 1) . Theoretically those phenomena were respectively described by Nagabhushana Rao [1] and Phariseau 12], approximating the Raman-Nath system for the amplitudes of the diffracted fight waves. Those results may be rederived from the NOA method [3] for N = 1 and from the MNOA method [4] for M = 0, N = 1 and treated as eigenvalue problems. We shall compare both solutions with the experimental data of Klein et al. [5]. Further we employ the lOA method to investigate the occurrence of "pure" Bragg reflection for large and increasing values of the Klein-Cook parameter Q and the Raman-Nath regime parameter p.
THE NOA AND MNOA METHODS
The starting point for those methods is the Raman-Nath system for the amplitudes of the diffracted light waves [6]
Oth ORDE.R
1st ORDER
Figure 1. Geometry of single-order Bragg diffraction ("pure" Bragg reflection).
Physical A coustics, Edited b y O. Leroy and M.A . Breazeale Plenum Press, New York , 1991 505
2dle - (¢n-I - ¢n+l) = ipn(n + {J)¢n, (1)
with boundary conditions
n = 0, ±1, ±2, .... (2)
In (I) , ( = 1fEIZ/Er),COS'P, p = 2Er),2/EI), *2, {J = -(2), */),)sin'P , where 'P is the angle of incidence of the light (the z-axis being parallel to the ultrasonic wave fronts), El the maximum variation of the relative permittivity Er, ), the wave length of the light in the medium, ),* the wave length of ultrasound. If {J = -p,
with p integer, then p()./2),*) = sin'PH), where 'P~i) is called the Bragg angle
of order p. We also introduce the Raman-Nath (RN) parameter v = (L/ z, L being the width of the ultrasonic field, and the Klein-Cook parameter Q = vp. In the NOA method [3] one neglects the energy in the diffraction orders higher than N and lower than -N. The truncated system can then be solved by an
eigenvalue method. For N = 1 we obtain for {J = -1 ('P = 'P~i))
with
Ll = 4 [SIS2S1 sin2(sl - S2) ~ + cycl.],
10 = 1 + 4 [SIS2SI(2p - sl)(2p - S2) sin2 (Sl - S2) ~ + cycl.],
1.1 = 4 [SI(2p - sl)(2p - S2) sin 2 (81 - S2) ~ + cycl.],
and where s1, S2, S3 are the single real roots of the characteristic equation
S3 - 2pS2 - 2s + 2p = O.
100~
\ 10=9.31 i \ i
80 I I
60 \ I :: ~ i I Vl w
0 >= t I iii
40 z w
\ ..... ~
20 ,
.~\~ , 10=9.31 P . i \
80 . ~ ll j /, \. . \ i b I \
60 i> \ ! /\\ /\.1 I \ ! I 'i ; ~\ r . ! I \. ,.
40 f \ I I ! i: b \ ;) '" \\: I \ :]. I \ 1-1 i} I . : / '. I \ I
20 r ./ if \ ./\ \ . to . . . 'f..
0 \ ..... ~ o l ............... JI \ i '"
0 2
Figure 2.
506
4 6 8 10 12 14 o 2 I. 6 8 10 12 II.
v v
10 versus v (left) for Q = 9.3 and {J = -1 calculated from Phariseau's formula (8) (_.-._) and from Nagabhushana Rao's formula (4) (-) compared with experimental data of Klein et al (x x x ) . II versus v (right) calculated from Equation (9) (-.- . _) and from (5) (- - -) compared with experimental results of Klein et al (0 0 0) and LI versus v from (3) for the same values of Q and {J .
(3)
(4)
(5)
(6)
(7)
~
<II W
~ z w l-
~
t <II ~ I-
iii z w l-
~
Figure 3.
~ <II W ~ iii z W I-
~
v v
;!
III W ~ iii z W I-
~
v v
10(--), 11(- - -) and Ll(- - -) versus v at Brag~ incidence ({3 = -1) respectively calculated from (4), (5), (3) for Q = 10, 20, 3D, 50.
Those results are in fact Nagabhushana Rao's formulae [1] written in a more explicit form, for (3 = -1. In the MNOA method [4] it is assumed that only M negative and N positive orders are present in the diffraction spectrum, with M ~ N for rp > O. Considering M = 0, N = 1 and using the
eigenvalue method we obtain, for perfect Bragg diffraction rp = rp~i) ({3 = -1),
Phariseau's well-known results [2],
10 = cos 2(v/2) 11 = sin2( v/2) .
NUMERICAL RESULTS
(8) (9)
In Figure 2 the intensities 10 (left) and 11 and Ll (right) versus v are shown. The various curves are calculated with formulae (8,9) and with (3), (4), (5) . Both sets of theoretical results are compared for Q = 9.3 with experimental data obtained by Klein et al. [5] . The fitting of the lOA curves with the experimental points is excellent. Unfortunately the data are restricted to the domain v E [0,4]. In this region for v there is a rather good
507
agreement with Phariseau's results, but it fails beyond v ~ 5, due to the fact, that from thereon Ll is no longer negligible. Hence, we can conclude that for Q = 9.3 there is only "pure" Bragg reflection up to v::oJ 5.
In Figure 3 we represent the curves for 10, I +1 and Ll versus v, calculated with the lOA method at Bragg incidence (Equations (3), (4), (5)), for increasing values of the Klein-Cook parameter, namely Q = 10, 20, 30, 50 . The larger the value of Q, the better the condition for "pure" Bragg reflection is satisfied. This is because the intensities Ll decrease with higher values of v. Incidentally, the deviation of the curves for 10 and II from Phariseau's theory becomes small with larger Q.
Finally, in Figure 4 we show 10, 1.1 and L1 versus v, computed from Equations (3), (4) and (5) at Bragg incidence, but now for increasing values of the regime parameter, i.e . p = 1, 5, 10 and 20. Similar calculations were performed for p = 30, 40 and 50, but the results were identical with those for p = 20. Observe that for p = 1 most values of Ll are too large, and second order intensities are not negligible, so that this case does not illustrate Bragg reflection very well. But for p ~ 5, the calculated values of L1 keep decreasing, practically vanishing for p. = 20. Furthermore, 10 and 1.1 are nearly represented by Phariseau's formulae (8) and (9). This shows that approximately for 5 ~ P ~ 20 there is near Bragg reflection, whereas for p ~ 20 we have
60
;"
'" 60
~ ..... iii z
~o w .... :;?;
20
0 0
t '" w ;:: v; z w .... ~
Figure 4.
508
, ...... I
;"
~ I 1
{
2
\ I '" \ w
I ;:::
\ v; z w .... :;?;
1\ \~ /f \\ f./
~ 6 6 10 12 14 v v
;"
'" '=' .... iii z w J-
~
v
10(--), 11(- - -) and Ll(- - -) at Bragg incidence ((3 = -1) calculated from (4), (5) , (3) for p = 1, 5, 10 and 20.
"pure" Bragg reflection. Hence, it is clear that both the parameters Q and p are relevant for determining the condition for Bragg reflection.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
K. Nagabhushana Rao, Diffraction of Light by Supersonic Waves - Part 1, Proc. Indian Acad. Sci., 9A:422, (1939). P. Phariseau, On the Diffraction of Light by Progressive Supersonic Waves. Oblique Incidence: Intensities in the Neighbourhood of the Bragg Angle, Proc. Indian Acad. Sci., 44A:165, (1965). R. Mertens, W. Hereman, and J.-P. Ottoy, The Raman-Nath Equations Revisited. II. Oblique Incidence of the Light - Bragg Reflection, in: "Ultrasonics International 87 Conference Proceedings", Butterworth, Guildford (1987). E. Blomme, and O. Leroy, Diffraction of Light by Ultrasound at Oblique Incidence: A MN-Order Approximation Method, Acustica, 63:83, (1987). W.R. Klein, C.B. Tipnis and E.A. Hiedemann, Experimental Study of Fraunhofer Light Diffraction by Ultrasonic Beams of Moderately High Frequency at Oblique Incidence, J. Acoust. Soc. Am., 38:229, (1965). C.V. Raman and N.S. Nagendra Nath, The Diffraction of Light by High Frequency Sound Waves. Part V: General Considerations. Oblique Incidence and Amplitude Changes, Proc. Indian Acad. Sci., 3A:459, (1936). G. Plancke-Schuyten and R. Mertens, The Diffraction of Light by Progressive Supersonic Waves. Oblique Incidence of the Light. II. Exact Solution of the Raman-Nath Equations, Physica, 62:600, (1972).
509
AN IMPROVED THEORY OF PHOTOACOUSTIC SIGNAL GENERATION
IN GASES AND LIQUIDS
Andras Mikl6s, Zoltan Boz6ki and Andras Lorincz
Institute of Isotopes Hungarian Academy of Sciences P.O. Box 77, H-1525 Budapest, Hungary
INTRODUCTION
The photoacoustic (PA) phenomenon, which was discovered by A.G. Bell in the last century, always was interpreted as a basically thermal process. The theoretical models of Kreutzer [1], of Amer [2] and of Rosencwaig and Gersho [3] are based on the assumption that the modulated light heats the sample causing heat expansion, and this process generates sound as a secondary effect. In other words there is a causal relation between heat and sound. The thermal process is described by the well-known heat diffusion equation of Fourier, while the sound generation may be determined using the Navier-Stokes equations with a source term containing the gradient of the temperature. This phenomenological model proved to be effective for characterizing different PA arrangements, but there are some discrepancies between theory and practice.
One may ask whether such a phenomenological model would be applied for physical processes starting with the absorption of photons by single molecules of the illuminated sample. The answer depends on the actual physical situation, namely on the density of absorbing molecules compared to the photon density and on the rate of energy transfer due to the collisions with the neighbouring particles. If the radiationless energy transfer is fast compared to the radiation processes and the number of excited molecules is small compared to the total number of them, the P A process may be described using the tools of statistical mechanics and thermodynamics. Nevertheless, there are few exceptions like the case of high light intensity together with very small concentration of the absorbing molecules when saturation occurs, and the case when the absorbing molecule excites another type of molecule by collision, but this other one decays by radiation, thus radiationless and radiation processes are involved simul taneously.
These cases may be included into the phenomenological P A theories by using quantummechanical description for the molecular absorption and decay processes and by calculating the statistical averages used by the phenomenological model. This method seems to be very good for extreme problems in optoacoustics, too.
Nevertheless accepting the classical phenomenological theories for describing P A processes another question arises, namely the correct form of the equations of thermodynamics and fluid dynamics when the physical quantities are very small. The photoacoustic signal may be extremely small, sometimes the detected sound pressure is smaller than 0 dB, which corresponds to 2 10-5 Pa, i.e. the relative pressure change may be about 2 10-10 Pa. This value is commensurable with the statistical pressure fluctuations of the sample gas at ambient pressure.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 511
In the case of such small changes there is no reason for regarding one thermodynamical quantity as constant and the others as free variables. The values of all the physical quantities which are involved into the photoacoustic signal generation and propagation are very-very small compared to the ambient values of the same quantities, therefore all of them have to be taken into account.
THEORY
The basic equations of a P A theory may be imported from thermodynamics and from hydrodynamics using the definitions of the thermodynamical variables. For getting a better insight of the problem, let's first investigate the experimental data of acoustics! The measurement of sound propagation in different liquids and gases proves that the sound velocity is independent of the frequency up to very high frequencies, and its value is very close to the value calculated assuming adiabatic processes. This fact may be understood by the fact that the characteristic distance or correlation length of the sound field is much larger than the thermal wavelength at the same frequency, therefore thermal processes cannot equalize the changes generated by the sound field. The sound and thermal wavelengths will be of the same order of magnitude only at about 125 MHz in air. Above this frequency thermal processes may be more or less regarded as isothermic ones, but in the usual range of PA experiments the assumption of adiabatic processes is valid.
As the changes of thermodynamical variables like density, pressure and temperature are very small compared to their ambient values, all quantities which are defined as partial derivatives of the variables, like compressibility, volume heat expansion coeffiCient, specific heats, thermal conductivity and diffusivity, sound velocity, viscosity, etc. may be regarded as constants, but their actual value may slightly differ from the initial value due to DC processes. Nevertheless, one may assume, that after switching the illuminating light on, the DC temperature will approximate an equilibrium value, thus after a certain time the DC temperature may be regarded as constant, and the numerical values of the above mentioned quantities will correspond to the equilibrium temperature.
Starting from the thermodynamic equations [4] and the Navier-Stokes equation of fluid mechanics [5], and using the equations of mass conservation and adiabatic equation of state one may get two equations of the following form:
¥t + c2 div v - /f,!J. Y = Q (1)
~ + grad Y - I grad div v - v!J. v = g (2)
where the sound velocity, thermal diffusivity, kinetic viscosity, acceleration of gravity and fluid velocity are denoted by c, /f" V, g and b respectively. The quantity denoted by Y may be written as
y=&e fJTa
(3)
where the adiabatic exponent, the specific heat of constant density, the volumetric expansion coefficient and the ambient temperature are denoted by 7, Cv, f3 and T a respectively. Here 0 denotes the fluctuating part of the temperature, i.e. T = T a + O. The effect of light absorption is included in Q, which is given as:
512
Q = -.L .!. 8wa {lTa P Ov (4)
where Wa is the light power absorbed by the medium. For small velocities the convective parts of the time derivatives may be
neglected. The effect of gravitation may be separated from (2) by assuming Z = z + gr and recognizing that gv is much smaller than c2div v both for thermal and sound waves.
The linear approximation of equations (1) and (2) may be written as follows:
¥t + c2div v - KAy = Q (5)
%r + grad y - I grad div v - vA v = 0 (6)
The plane wave solutions of (5) and (6) in the case of no heat sources may be substituted as:
y = Aei( kr-tOt) and v = Uei( kr-tOt)
The determinant will be:
(7)
The most interesting property of the coupled equations of (5) and (6) is that the pure thermal wave solution does not exist. If div v = 0 would be assumed and a pure thermal wave solution y of (5) would be determined, then this solution would lead to div v * 0 in equation (6). This is the mathematical consequence of our considerations about the impossibility of keeping the value of any thermodynamical variable constant during P A processes. On the other hand, there are transversal viscous waves which are not coupled to the temperature field, but the condition of decoupling may be fulfilled only if all the thermodynamical variables are homogeneous in the medium. However, in PA experiments one is always dealing with inhomogeneous thermodynamical variables.
The only relevant solution of (5) and (6) is a propagating extensional wave with wavenumber:
The relation between the velocity amplitude U and temperature amplitude 00 is:
00 ~ ~ I U I p
The sound pressure, which is the only measurable quantity of a sound field may be calculated by using the equation:
1 op 'Y - 1 oY c2 d' P or = 7 Of -? zv v (8)
The proposed equations (5) and (6) show, that the viscosity in (6) plays a similar role as the thermai diffusivity in (5), therefore one must not neglect only the viscous terms, as usually done.
CONVECTIVE TERMS IN PA PROCESSES
It was stated that the convective terms in equations (1) and (2) may be neglected because they are products of very small quantities. It is to be noted,
513
however, that there are very important exceptions, namely the case of gas flow of external origin and the case of strong DC heating. In both cases a time independent velocity field may be assumed, consequently the convective terms in (1) and (2) have to taken into account. Even at very low flow velocities the convective term may be greater than the diffusive one. That is, convective flows can influence the photoacoustic signal generation in the case of flow-thorough PA cells.
Similar problem may arise when the temperature of the sample considerably increases during the PA experiment. Nonuniform temperature distribution, according to Eq.(2), will generate steady flow in the medium for decreasing the temperature differences. The magnitude of flow velocity may be estimated as v2 ~ 2 C pdT, where dT is the temperature difference. Although this expression overestimates the value of the generated velocity, it may call one's attention to the influence of DC temperature fields on the photoacoustic signal generation. Strictly speaking, the AC and DC parts of equations (1-2) must not be separated because of the strong coupling between them, but the complete solution has to be determined, taking into account the boundary conditions, too. The DC temperature field and the flow velocity field are strongly influenced by the boundary conditions and even by the gravitation. As the nonuniformity of the temperature is the source of flow velocity, the intensity distribution of the light beam may influence the amplitude of the generated flow velocity, thus the PA signal generation may be changed by using laser beams of different power distributions.
Nevertheless, the absorbed light power in a common PA experiment is too small for observing the effect of DC temperature field on the signal generation, but this phenomenon may be investigated in photothermal deflection experiments, where the absorption of solvent may be considerable. Generally, the effect of solvent absorption on wavelengths which are different from the probe beam wavelength is neglected, because it doesn't influence the deflection of the probe beam in a direct fashion. But the temperature distribution generated by the pump beam depends on the total absorption of the solvent-sample mixture, thus the probe deflection may be influenced via the convective flow terms of equations (1-2). For quantitative results a very detailed theoretical and experimental investigation of the problem would be needed.
CONCLUSION
The photoacoustic and photothermal phenomena may be described in a common theoretical frame as adiabatic processes characterized by very small changes of the thermodynamical variables T, p and p, which are related to one another by the laws of thermodynamics. The assumption of adiabatic process may be based of the empirical fact that the sound velocities of gases and liquids under normal circumstances correspond to the adiabatic value. Using this assumption a very symmetric set of differential equations of the diffusion type may be introduced. In this model the thermal diffusivity and the viscosity play a similar role, therefore it is not allowed to neglect the viscosity, as usual in photoacoustics. Pure thermal wave cannot exists in this model, because temperature changes without density and pressure changes is impossible. A new type of propagating wave is predicted by the model, but it is attenuated such a great extent that its observation is almost impossible. The attenuation coefficient of this wave decreases with increasing viscosity and thermal diffusivity, therefore this type of wave, provided it exists at all, may be observed in liquids of high viscosity only.
The question of convective flows and DC temperature and velocity fields was discussed very briefly. A detailed theoretical and empirical investigation of photoacoustic and photothermal signal generation based on the above introduced model would bring new results of practical importance in photoacoustics.
514
REFERENCES
[1]
I!I [5)
L.B. Kreutzer, Optoacoustic Spectroscopy and Detection, ed. by Y.H. Pao, (Academic Press, New York, 1977). R. Gerlach and N.M. Amer, Appl. Phys. 23, 319-326, (1980). A. Rosencwaig and A. Gersho, J. Appl. Phys. 47, 64, (1976). D. Chandler, Introduction to Modern Statistical Mechanics, (Oxford Univ. Press, New York, 1987). P.M. Morse and K.U. Ingard, Theoretical Acoustics (McGraw Hill, New York, 1968).
515
THEORY OF PHOTO ACOUSTIC EFFECT IN LINEAR AND NONLINEAR
GYROTROPIC PIEZOELECTRIC CRYSTALS
G.S. Mityurich, V.P, Zelyony, A.N. Serdyukov
Department of Physics Gomel State University 246699 Gomel, U.S.S.R.
Amplitude-modulated optical radiation effect on piezocheramics caused, as shown in [1] the induction of low-frequency acoustic signal in the latter, and the specimen under examination was simultaneously a photoacoustic signal detector.
Piezoelectric characteristics are known to be found in exclusively noncentersymmetric crystals [2] many of which have the natural or induced gyrotropy [3]. Besides, thermooptical generation of signal may occur both in linear and nonlinear piezocrystals.
The given paper represents the consideration of photoacoustic transformation in gyrotropic linear and nonlinear piezoelectric crystals of different symmetry classes.
LINEAR PIEZOELECTRICS
In case of linear piezocrystals the consideration is based upon the use of Dyugamel-Neyman relations [4]
(1)
where C~jkl' e~jm are tensors of elasticity coefficient and piezoelectric
constants, P~ the pyroelectric coefficient, A ~j' Elj temperature voltages tensors
and dielectric permeability. While spreading the incident amplitude-modulated light wave of arbitrary
polarization along the direction [110] the correlations (1) in matrix expressions take the value
D T USE 3 e 14 6 + Ell 3
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York. 1991
(2)
517
The radiation upon the highest axis coinciding with the direction [001] of unaxial piezoelectric crystals of classes 3, 4, 6 from (1)
(3)
that in [1] are used for photoacoustic effect calculation in non~yrotropic piezoelectric materials. The periodic composite of temperature field in l2), (3) is determined from the solution of inhomogeneous equation of thermal conductivity for gyrotropic specimen [5]:
(4)
In (4), the following notations are introduced
A±a± cP n 21 E 12(1±r) 2 E± = ---, A± =
a; - u~ Ii Ksln+noI2(1±r)2
no = (n++n_)/2, where n± = Ii ± 1 are complex indexes of diffraction of
isonormal waves, 1 = l' +h" is the complex pseudoscalar gyration parameter, l' is the circular dichroism parameter, r the incident wave ellipticity, nits modulation frequency, U 0 the constant, determined by crystal parameters.
Using the equation of elastic medium movement as well as correlations (2)-(3) the expressions for potentials difference (see Table 1) occurring in piezoelectric samples at boundary conditions (free: u(O)=O, u(l =0; squeezed: U(O)=O, U(l)=O and alternatively loaded crystal boundaries: u 0)=0, U( f)=O or U(O)=O, u(l)=O ) are received. Here we take the rotations Vo = A~[exp(-a/)-l] - A'_[exp(-a_f)-l]' A~ =
h ,E Ie h TIS E "110, =e 14 f W CO =c 44 +
(e~4) 2 I f~l ,K is the wave number of elastic wave and f the specimen
length. It should be noted that in the expressions for crystals of classes 3, 4, 6 instead of the value ho will enter ho
Table 1
crystals of class 23
i ~(oJ=o /JV1 =-h.IV: -/d~ q(B) e-oI• f_ 1) - B_ (e-oU - i))] 6"(£) =0
ii <:>"(0)=0 V. h [V,'- t~d(B -B)- cos,d-f(A er/+e_ A e.t..~] 1.I.1t.) = 0 .c1 2::: -. • K 't - COSK/! + -
ij, <:>"(e) =0 AV3=-h. rv: -KtVllf (BJ·,.,·e_/ie.r· f ) ... c~~~~/ (A+ - k)] Wp; =0
/..V U(II) =0 4 V .. = 0 ure)=o
crystals of classes 3, 4, 6 , I
f - ,!i IJ Vi. = - t,vi T 4 V4 , i. = 1,'2,3
,ii A V~ = t: [J:'f (e- or. R_ 1) - 0/;'£ ... (e-o(~R - l' )] 3&
518
-3 ~O
Figure 1.
<i'V i ! ___ 2
r- 3 r 4 " 'I .I
'I Ii
ft " ii 'I i l II , 1 ' I
I · I \ ' \
I ' ~~~-~ SZ,MHz
roo 200 aoo 400 $00
The dependence of the difference of PA signals amplitude on the frequency of incident radiation for gyrotropous crystal LiJ0 3 having boundary conditions 11(£)=0, U(O)=O) 1. £=2,lcm; 2. £=1.8 cm; 3. £=1.5 cm; 4. £=1.2 cm.
The numerical analysis of the correlations received shows that within the kHz (>100 kHz) and MHz frequencies of incident radiation modulation the squeeze of crystal front boundary of classes 3, 4, 6 causes photoacoustic signal amplitude resonance increase (see Figure 1). The nature of resonance curves, as shown in Figure 1, depends largely upon the geometric dimensions of piezoelectrics. Having mechanically squeezed sample boundaries the P A response in cubic piezocrystals equals zero and in class 3, 4, 6 crystals the benefit in PA signal value is completely determined by pyroelectric coefficients.
NONLINEAR PIEZOELECTRICS
Piezoelectric crystal properties will be described on the basis of nonlinear crystalacoustics relations [6]
(5)
1 1 = emijUij + '2 emijklUijUkl + '2 fmnijEnUij
+ f~nEn + ~ f~npEpEn + ~ fmnklUklEn + P~B
including the expression for B(z,t) (4). In (5) the nonlinear properties are
described by tensors : C~jklmn piezoelectric coefficients, Elimp
electrostriction coefficients.
the nonlinear elastic constants, emijkl
the square dielectric permeability and
nonlinear
the
To make the calculations concrete, let's consider the case of amplitudemodulated light beam spread along the bisector of the angle formed by directions [010] and [110] of cubic piezocrystals of class 23, 43m. From (5) it follows, considering that 11 = 1112 + I1n .
519
Table 2
crystals of classes 23 , 43m, 432 (eJ, ... o)
Vi = Q,. (eu + e32 , l.L~) ~f v.l "'; ... (8) z E~ 01, I
I - Iii (l.Lf2eU6 + eJ,)B +£,42 a.. b v.l2/+ It
~ 2. e3 ""2' ~J .. ~ ~2,3
IV V .. .: Corni
crystals of classes J , 4, 6 , 3m , 4mm, 6mm
I -'I' hAD (~) . 1,23 (9) V :: - - V· ... V '=" c. C." 44 )
IV V. =: - ~['& (e-~ t_ f) - ~ (e....l· '- -() - &. (e~~.( J]"'COl'Ist 4 C, Cf. 0(, "'-
EEl E ( 2) E (I C22U2 + C66U6 + 2" C626 2U 23U6 + U6 - e 36 s
1 1 - 2" em E3(U 2 + U 6) - 2" f32E3E3 - )'10, (6)
Transforming the component e36 into zero from (6) we receive correlations satisfying crystals of class 432. In case of uniaxial crystals of classes 3, 4, 6, 3 m , 4 mm, 6 mm where radiation falls among the highest symmetry axis the equations (5) have the form
(7)
Having done the linearization of correlations (6), (7) according to [7] and using elastic medium movement we easily receive the value for potential difference occurring in piezoelectric crystal specimens various boundary conditions (see Table 2) .
Figure 2.
520
-6 /0
L
L---z ~
" ,I
" " " " "
j \
The dependence of P A signal amplitude on frequency of incident light modulation for crystal of class 432. 1 - ((1(0)=0, U(£)=O, or U(O)=O, (1(£)=0) 2 - ((1(0)=0, (1(£)=0) .
Let's take into account that for the crystals of class 3 m, 4 mm, 6 mm, 43 m in the expressions (8), (9). at = a_ = a = 47r/ A( EO /2n, E t = E_ = E.
The analysis of the expressions received , shows that the P A signal amplitude in cubic crystals is determined by the ratio of values of linear and nonlinear piezoeffect, and in uniaxial crystals depends greatly upon the value of pyroelectric coefficients. For the crystals of 432 class in which linear piezoeffect is forbidden by symmetry, the resulting P A signal is determined completely by the value of component of the nonlinear piezoeffect tensor em. A kHz frequencies of modulation (500 kHz) resonance phenomena are detected, determined by boundary conditions, piezoelectric properties and specimen thickness.
In conclusion it is necessary to point out the received results concerning resonance P A transformation may find use to increase the capabilities of P A spectroscopy as well as to define the tensor component of linear and nonlinear piezoeffect, absorption parameter E" and circular dichroism 'Y' of gyrotropic piezoelectrics.
REFERENCES
[6]
[7]
Wetsel G.C.Jr., J.Opt.Soc.Am., v.70, N" 5, p.471-474, 1980. Landau L.D., Liphsits E.M. Electrodinamica sploshnyh sred. M. Nauka, 622 p., 1982. Fyodorov F.I., Teoria gyrotropii. Minsk: "Nauka i technika", 456 p., 1976. Novatsky V., Teoria uprigosti. M. Mir, 872 p., 1975. Mityurich G.S., Zelyony V.P., Pisma v ZTF, v.14 issue 20, p. 1879-1883, 1988. Lyamov V.E., Polyarisatsionnie effecti i anyzotropnie vzaimodeistviay acustitcheskih woln v crystallah., MSU Publishevs, 223 p., 1983. Parton V.Z., Kudravtsev V.A., Elektromagnitouprugost' piezoelectricheskih i electroprovodnich tel. M. Nauka, 620 p., 1988.
521
EFFECTS OF SELF-ACTION - UNEXPLORED FIELD OF NONLINEAR
ACOUSTICS OF SOLID SURF ACES
Vladimir Mozhaev
Faculty of Physics Moscow State University, U.S.S.R.
INTRODUCTION
Nonlinear properties of surface acoustic waves have been investigated since Viktorov's theoretical paper of 1963 and Rischbieter's experimental paper of 1965. More than two hundred papers are devoted to this problem at present time, not including numerous publications on surface wave convolvers and similar nonlinear acoustoelectronic devices. However the most part of the available papers concern mainly the question of second and higher harmonic generation of surface wave and only few of them are devoted to the effects of self-action.
Self-action of fundamental frequency wave may appear as changes of its characteristics (in comparison with linear problem) in three orthogonal directions: along direction of propagation (x-axis in this paper), normal to surface (y-axis here) and transversal direction (z-axis). Self-action in z-direction results in self-focusing, or self-defocusing of surface wave beam. These effects are weakly investigated today. In particular, self-focusing of surface acoustic waves has been investigated in the only one paper [lJ by Nakagawa. Amplitude-frequency effect for surface acoustic wave resonators related to self-action in x-direction (direction of propagation) was investigated in the papers [2,3J. Problem of existence of surface acoustic solitons in nonlinear elastic media, related to such surface wave self-action, have been considered by a number of authors [4-71. And now some general, words about the main subject of the present paper--effects of self-action related to the changes of surface wave structure in depth. Similar effects have been widely investigated in plasma physics and optics, especially during the last ten years. In recent paper [8J of present author, it was shown that analogous effects can also be realized under certain conditions in the case of acoustic surface waves in solids. Present paper is a development of the paper [81 and its purpose is to reveal the possibilities of existence of a new class of nonlinear phenomena in acoustics of solid surfaces.
NEW SURFACE SHEAR WAVE DUE TO ELASTIC NONLINEARITY
Let us recall that the shear wave does not generate second harmonics of the same polarization in unbounded homogeneous isotropic media and this property is valid also for shear waves of horizontal polarization (SH-waves) in
Physical Acoustics, Edited by o. Leroy and M. A. Breazeale Plenum Press, New York, 1991 523
half-space (y ~ 0 here). So, the case of SH-waves is similar to problems of optics with cubic nonlinearity. It is also known that SH-waves, propagating along the plane free-traction surface of homogeneous linear isotropic media, are bulk waves, but are not stable in the sense that small changes of properties of medium, form of boundary, and boundary conditions may lead to transformation of the SH-bulk wave to the surface wave. The reason of such a transformation, as it is shown in paper [8], may be nonlinearity of the medium. The corresponding nonlinear equation of motion and boundary condition for this problem have the form [4,5]
(2)
where uy = aul ay, U x = aul ax, /l, a the linear and nonlinear elastic constants, p the mass density of the medium, u, T 4 the mechanical displacement and normal stress created by SH-waves. If a < 0 (case of a self-focusing of a shear wave), the solution of Eqs.(l), (2) in the form u = Uo U(y) exp(iwt-ikx) where Uo = I u(y=O) I, in case of neglecting interaction of fundamental frequency wave with higher harmonics, describes a surface wave, arising entirely due to nonlinearity of the medium
k2 = k~ - 3ak4u5/(8/l)
f u dU Y = U
l-E2U2+J(1-E;2U 2 )L27E4U2(1-U2) 2 (1-U2)
(3)
(4)
1 1
where Y = y(kLkD', E2 = -ak2ua/(4/l), k? = w2/Vt, Vt = (/lIp)' the velocity of bulk SH-wave for a linear problem. Transversal structure of the surface wave, U(y), is found as an inverse function from Eq.(4). In the small amplitude limit, E2 « I, this function has a very simple form U = sechy' Note that predicted surface acoustic wave exists in a free-boundary half-space simultaneously with bulk nonlinear SH-waves, possessing a dispersion relation
k2 = k? - 3ak4u5; (4/l). Generalizing the solution presented above for a case of orthorhombic
crystals we have the same relation (3), if we put there /l = C55. The structure is described by an expression similar to Equation (4)
Y = fU dU 1 U
l-E2U2+J(1-E2U2) 2-2 7 (a1//32)E4U2(1-U2) 2(1-U2) (5)
If coefficients a, 13, 1, involved in Equations (3), (5), are defined by the following combination of linear and nonlinear elastic constants
a (3Cll + 6C155 + C5555) I 6
13 = (C12 + 2C66 + C144 + C255 + 4C456 + C4455) I 2 (6)
1 (3cn + 6C244 + C4444) I 6
and
Y
524
SURFACE SH-WAVES DUE TO NONLINEARITY IN QUADRIC NONLINEAR ORTHORHOMBIC ELASTIC MEDIUM AND THEIR STATIC ACTION UPON A SURFACE
Nonlinear Hook's law has the following form
T ij = CijklUk,l + (1/2)CijklmnUk,lUm,n+ (1/6)CijklmnpqUk,1 Um,n Up,q (7)
The solution presented in preceding section corresponds to the case, when contribution of double interaction due to quadratic nonlinearity (second term in Equation (7)) to the effective cubic nonlinearity of SH-waves, is negligible in comparison with direct cubic nonlinearity (third term in Equation (7»). For simplicity of explanation here we include geometric nonlinearity to Equation (7), assuming nonlinear constants to be equal to their effective values (as in Equations (6)).
Consider an opposite case. In this case inhomogeneous in depth, SH-wave creates static fields of strain S2 and normal displacement U2, which are functions of depth and don't depend on other coordinates
(8)
Besides, SH-wave of fundamental frequency generates a second harmonic of a Rayleigh-wave type, as in similar problems considered earlier [9-11]. But this process is not effective due to lack of synchronism and so we may neglect it. Taking into account interaction of the initial SH-waves with the static strain S2 shows that this interaction can be a reason for surface localization of SH-waves. A solution for this case has the same form as Equations (3), (5), if coefficients u, {J, 1 are defined by
u - 2 (c 12+C255) 2 / (3C22)
(9)
In the small-amplitude approximation, when U ~ sechY, static displacement of a near-boundary layer is determined by
U2(y) = sgn(c12+Cm) VC44!C22 (l-tanhY) Uo (10)
Therefore the region which is occupied by the surface SH-wave pulse lowers or raises as a whole one. To account for simultaneous quadratic and cubic nonlinearity it is enough to use simply a sum of Equations (6), (9), as the coefficients u, {J, 1 involved in Equations (3), (5).
STRUCTURE AND SPECTRUM OF SURFACE SHEAR WAVES IN NONLINEAR ELASTIC HALF-SPACE WITH BOUNDARY LOADING
Up to this moment we have considered a half-space with a free surface. In this case, boundary condition (Equation (2)) is factorized and we have used one possible type of boundary condition coincided with the linear one. Under a loading, the degeneracy of the boundary condition is taken of and it becomes also nonlinear as the equation of motion. For our purposes the most convenient form of writing the boundary conditions is their impedance form: continuity of impedance Z = T 4/U at the boundary y = O. Impedance of nonlinear isotropic substrate may be written in a form
(11)
525
where
Q
Equating this value to the impedance of a layer with a free upper side Zj = ttlXltanXlh, where h is a thickness of layer, we find a dispersion relation of nonlinear Love waves. Here for simplicity we restrict our consideration to an isotropic linear layer on an isotropic nonlinear half-space with negligible quadratic nonlinearity. Equation (11) describes two families of dispersion branches of love waves. One of them corresponds to known (in linear theory) branches perturbed slightly by nonlinearity. Transversal structure of these modes in small-amplitude limit is determined by
(12)
Equation (12) has an interesting feature which will be discussed later. The second family of dispersion spectrum branches is absent in a linear case. The form of these curves are similar to a spectrum of SH-waves in a layer with rigid lower side. The question of physical realization of these new branches demands to investigate the structure of the waves in the nonlinear half-space. For the present this question is not solved finally.
Equation (11) allows us also to find amplitude-independent coefficient of reflection of SH-waves oblique incident on a nonlinear half-space from a linear half-space with material parameters marked here by subscripts L
1
where ZL = (f.tLPL)'cos(), 0 is angle of incidence. From condition R = CD, we
find dispersion relation of specific SH-waves guided by interface between nonlinear and linear media.
INVERSION OF CHARACTER OF SELF-ACTION UNDER THE CHANGE OF DEGREE OF INHOMOGENEITY OF SH-W AVES IN DEPTH
The nonlinear term in Equation (12) is proportional to the product of a and (3kL2X2k2-9X4). The same combination is involved in the nonlinear term of the evolution equation obtained for love waves by Hadouaj and Maugin [7]. If inhomogeneity of the wave in depth is weak (X« k), the sign of the nonlinear term is determined by the sign of nonlinear coefficient a. However
when X > k j 2.[7+1 / 3, the second factor changes its sign. So effective nonlinear coefficient behaves itself under the change of degree of inhomogeneity of SH-waves in depth so as we would have transition from self-focusing to defocusing or vice versa.
SURFACE LOCALIZATION AND NONLINEAR DISPERSION OF SHW AVES OF INFINITELY SMALL AMPLITUDE TRAPPED BY SURF ACE SH-WAVES OF FINITE AMPLITUDE
Consider now a propagation of SH-wave of infinitely small amplitude in half-space in the field of surface SH-wave of finite but enough small amplitude so that we may describe its transversal structure by the function U = sechY. We denote the surface amplitude and frequency of the wave mentioned first as U2 and W2 and the proper variables of the second wave as Ut, Wl. Because Ul »U2 self-action of U2-wave is negligible. Then it is a problem of shear
526
wave propagation in stratified normally half-space with shear-modulus inhomogeneity of Epstein type. This problem can be solved analytically and the result for the lowest mode is u = u2(sechy)m,
where m 2 = (k~-kh)/(k~-k£I)' ktl = wl/Vt, kt2 = W2/Vt, Y = yj k~-k~1 ,
with dispersion relation
(14)
This solution shows that bulk SH-waves of infinitely small amplitude under the influence of surface SH-waves of finite amplitude becomes surface waves possessing dispersion due to nonlinearity.
WHERE ARE EFFECTS OF SELF-ACTION OBSERVABLE?
In usual conditions, effects described above are as a rule very weak, through self-action of shear waves in quartz plate have been already observed experimentally [12]. Another situation may be expected in ferroelectrics at a region of structural phase transition where as it is known linear shear elastic modulus can turn to zero at the point of the transition. Strong nonlinearity is available also in magnetoelastic materials and in piezoelectric-semiconductor structures where we may expect that such or similar effects of self-action are detectable.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Y. Nakagawa, Self Focusing of Surface Acoustic Waves, in: II Proceedings of ~hird Symposium on Ultrasonic Electronics", Jap. J. Appl. Phys., 22-3.157, (1983). M. Planat, G. Theobald, and J. J. Gagnepain, Propagation Non-lineaire d'Ondes Elastiques dans un Solide Anisotrope. II Ondes de Surface, Onde Electrique, 60:61, (1980). M. Planat, and D. Hauden, Nonlinear Properties of Bulk and Surface Acoustic Waves in Piezoelectric Crystals, Ferroelectrics, 42: 117, (1982). K. Bataille, and F. Lund, Nonlinear Waves in Elastic Media, Physica, 6D:95, (1982). A. A. Maradudin, Surface Acoustic Waves on Real Surfaces, in: "Physics of Phonons", T. Paszkiewicz, ed., Springer, Berlin (1987). A.A. Maradudin, Nonlinear Surface Acoustic Waves and Their Associated Surface Acoustic Solitons, in: II Recent Developments in Surface Acoustic Waves II , D.F. Parker, and G.A. Maugin, eds., Springer, Berlin (1988). H. Hadouaj, and G.A. Maugin, Une Onde Solitaire se Propageant sur un Substrat Elastique Recouvert d'un Film Mince, Compo Rend. Acad. Sci. Paris, 309:1877, (1989). V.G. Mozhaev, A New Type of Surface Acoustic Waves in Solids due to Nonlinear Elasticity, Phys. Lett. A, 139:333, (1989). De Krishna, Finite Strain Theory of Love Waves, Pure and Appl. Geophys., 80:114, (1970). Y. Shui, and LYu. Solodov, Polarization effects accompanying nonlinear reflection of SH-acoustic waves, Acta Acoustica, 12:129, (1987). F.M. Severin, and LYu. Solodov, Polarization Effects in Nonlinear Acoustic SH-Wave Reflection, Moscow University Physics Bulletin, 43:92, (1988). M. Planat, and M. Hoummady, Observation of Soliton-like Envelope Modulations Generated in an Anisotropic Quartz Plate by Metallic Interdigital Transducers, Appl. Phys. Lett., 55:103, (1989).
527
A NOVEL TECHNIQUE FOR INTERFACE WAVE GENERATION
Peter B. Nagy and Laszlo Adler
The Ohio State University Department of Welding Engineering Columbus, Ohio 43210, U.S.A.
INTRODUCTION
Guided acoustic waves along interfaces are especially sensitive to specific properties associated with interface conditions and bond quality since their energy is effectively confined to the region of interest. On the other hand, this inherent advantage turns out to be a significant drawback for generation and detection of such guided waves. These are two basic types of propagating interface modes, which are shown schematically in Figure 1. First, there are leaky modes with higher phase velocity than at least one of the bulk velocities in the surrounding media. These modes "leak" their energy into one or more phase-matching bulk modes as they propagate along the interface and they can be readily generated by these mode-coupled bulk modes at the same incidence angle. In other words, the energy of leaky interface modes is not strictly
a)
adherend
adhesive
adherend
b)
adherend
adhesive
adherend
guided wave
evanescent wave
~ ~ guided wave
evanescent wave
Figure 1. Schematic diagram of leaky (a) and true (b) guided mode propagation along an adhesively bonded intertace.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 529
TRANSMITTER Material A RECEIVER
It' Rayleigh Wave Interface Rayleigh Wave
Wave
Material B
Figure 2. Guided interface wave inspection by Rayleigh wave coupling.
confined to the boundary region therefore they are relatively easy to generate and detect. Because of their relatively short propagation length, leaky interface modes provide localized information on boundary properties and possible imperfections, which can be taken advantage in ultrasonic NDE of bonded structures [1].
True guided modes of lower phase velocity than any of the bulk velocities in the surrounding media are much more difficult to generate and detect since they produce evanescent fields in the bounded materials as they propagate along the interface. Such guided modes are especially sensitive to overall boundary properties averaged along the interface to be inspected, but their NDE application is badly limited by their poor accessibility. Figure 2 shows the most commonly used geometrical arrangement for guided interface wave inspection of bond properties. Wedge transducers are used to generate Rayleigh waves on the free surface of one of the joining parts, which are coupled to one or more vertically polarized interface modes over the bonded area. This technique works not for true guided modes [2,3] as well as for slightly Leaky ones [4,5], but the awkward geometry required for positioning the surface wave transducers renders this technique useless in most NDE applications.
DIRECT EXCITATION OF INTERFACE WAVES
Figure 3 shows an alternative geometrical configuration for direct generation and detection of interface waves between two bonded half-spaces [6]. A contact ultrasonic transducer is placed directly over the boundary region so that it can generate both bulk and interface waves. A vertically polarized symmetric mode produces identical transverse, but opposite normal displacements on the two
530
I. Symmetric Rayleigh -Type
--------Longitudinal ___
--------III. Symmetric Love - Type
+ + +
Shear ISHI + + + +
II. Antisymmetric Rayleigh -Type
f f
Shear ISVI f f f t
Figure 3. Direct generation of guided interface modes.
Q Bulk Wave t --------------
Interface ,<.
~ Wave 1 --------------Q Bulk Wave
Figure 4. Energy partition between interface and bulk modes.
sides of the boundary, therefore it can be excited by a longitudinal transducer which generates only parallel vibration relative to the interface. In a similar way, a vertically polarized anti symmetric mode produces identical normal, but opposite transverse displacements on the two sides of the boundary, therefore it can be excited by a vertically polarized shear transducer which generates only normal vibration relative to the interface. Finally, a horizontally polarized symmetric mode produces identical transverse vibrations on the two sides of the boundary without any normal components, therefore it can be excited by a horizontally polarized shear transducer.
Figure 4 demonstrates the main concept of energy partition between the generated bulk and interface waves. That part of the transducer, which lies directly over the interface region within approximately one wavelength generates mostly interface wave while the remaining part radiates mostly into the bulk mode. Since the transducer diameter-to-wavelength ratio is proportional to frequency, the two frequency spectra are complementary with the low frequency components carried by the interface mode and the high-frequency ones by the bulk mode.
Figure 5 shows the geometrical arrangement used in direct interface excitation experiments. A contact ultrasonic transducer is used to generate the ultrasonic waves as well as to detect the reflected signals from the back wall of the specimen. If the sample is long enough and the interface mode is not too dispersive, two separate signals can be detected due to the interface and bulk
w
~ II: W I-!!: I() w U. II: w Q.
~
Figure 5. Geometrical arrangement for direct interface wave excitation.
531
modes, and they can be analyzed independently. In most cases, however, these signals are not fully separated and we have to use spectrum analysis to get velocity and amplitude information on the more interesting interface wave component [7].
EXPERIMENT AL RESULTS
In the following, we are going to present three typical examples of direct interface wave generation by contact shear transducers. First, we consider the simple case of dry contact between slightly rough aluminum counterparts under compressive pressure. At zero compressive pressure, two separate signals can be observed. The first arrival is a shear-type bulk wave while the second one is a Rayleigh-type interface wave. At higher compressive pressure, the two signals are not separated sufficiently to directly measure their respective time delays, but frequency analysis can still readily reveal the sought separation. Distinct minima occur as a result of destructive interference between the two principal modes, and the periodicity of the observed frequency modulation can be readily used to determine the sought interface wave velocity as a function of either frequency or compressive pressure.
Figure 6 shows the measured interface wave velocity as a function of compressive pressure between the counterparts. Lee and Corbly were the first to experimentally observe this kind of gradual transformation of a Rayleigh wave into a Stonely wave due to strong pressing together of two solids [4]. Between similar materials, the interface wave generates into a vertically polarized shear mode as the boundary stiffness increases with increasing compressive pressure. The slight discrepancy between the theoretical prediction and the measured experimental data is due to the limitations of Haines' model used to calculate the boundary stiffness of the interface [7].
The second example is a thin adhesive layer between aluminum adherents. Although, in this case, there exist an infinite number of both vertically and
Vi ....... E ~
>-:!:: <.)
~ '" >-
'" > '" ~ '" <.)
~ '-'" ..... ~
Figure 6.
532
3.2'...,------------------------,
3.1
3.1
• Experimental - Theoretical
• • • • • • • • • • •
2.91~-_r--.--r--._-_.-_r-_,--._-,_-~
o Compressive Interface Pressure [ksi]
Measured and calculated interface wave velocity as a function of compressive pressure between aluminum counterparts at 5 MHz.
Figure 7.
Frequency x Thickness [MHz mml
Theoretical and experimental dispersion curves for the zero order guided modes in an adhesive layer at different polarizations.
horizontally polarized guided modes [8], the strongest one is always the zero order mode which has the most even phase distribution along the face of the contact transducer. The 1/211-long sample was made of aluminum plates bonded with a 140 J1.m thick FM300 adhesive layer. Figure 7 shows the measured interface wave velocities at different frequencies along with theoretical predictions for the phase velocities of the two zero order modes at vertical and horizontal polarizations. The fairly good agreement between the experimental and theoretical results well demonstrates the accuracy of the suggested direct excitation technique.
The third example to be discussed is the simple case of nondispersive Stonely wave propagation along a liquid-solid interface. Figure 8 shows the continuous transformation of the Rayleigh-type surface mode into a Stonelytype interface mode on the surface of a 42 mm long plexiglass sample as it is gradually immersed in water. The shear pulse is very weak because of the high attenuation in plexiglass at high frequencies, and it is not affected by the immersion. The Stonely wave is slightly more attenuated than the Rayleigh mode, and it comes approximately 14 s later. The interface wave velocity obtained from this experiment was 1046 mls which agrees very well with the calculated value of 1042 m/s.
CONCLUSIONS
A new experimental technique was introduced to generate and detect interface waves along otherwise unaccessible plane boundaries. Theoretical and experimental results for the phase velocities of propagating guided modes along different solid-solid and liquid-solid interfaces were found to be in good agreement. The suggested simple technique may found numerous applications in ultrasonic assessment of bond properties requiring guide mode inspection.
533
">
~ o N
..
II.
-'- I L-malr -
I\.
J 25~ inw~ter =
...I 50~ In ~.ter ;:: .....
I"
--1 75i ini.ter L-r-
Ii
!
...I I~ .. at~r L.. . r--
TIME (5Ils/div I
Figure 8. Continuous transformation of the Rayleigh mode into Stonely mode on the surface of a plexiglass sample as it is gradually immersed in water.
ACKNOWLEDGEMENTS
This work was supported by the Office of Naval Research under Contract NOO 014-88-K0452 . Scientific officer: Dr. Y. Rajapakse.
REFERENCES
[1]
[2]
[3]
534
P.B. Nagy and L. Adler, Nondestructive evaluation of adhesive joints by guided waves, J. Appl. Phys., 66:4658, 1989. S. Rokhlin, M. Hefets, and M. Rosen, An elastic interface wave guided by a thin film between two solids, J. Appl. Phys., 51:3579, 1980. S. Rokhlin, M. Hefets, and M. Rosen, An ultrasonic interface-wave method for predicting the strength of adhesive bonds, J. Appl. Phys., 52:2847, 1981.
[4]
[5]
[6]
[7]
[8]
D.A. Lee and D.M. Corbly, Use of interface waves for nondestructive inspection, IEEE Trans. Sonics Ultrason., SU-24:206, 1977. A. Pilarski, Ultrasonic evaluation of the adhesion degree in layered joints, Mater. Eval., 43:765, 1985. P.B. Nagy and L. Adler, Guided wave generation by direct excitation, J. Acoust. Soc. Am., 86:594, 1989. P.B. Nagy and L. Adler, New ultrasonic techniques to evaluate interfaces, in: "Elastic Waves and Ultrasonic Nondestructive Evaluation", S. K. Data, J. D. Achenbach, and Y. S. Rajapakse, eds., North-Holland, Amsterdam, pp. 229-239, 1990. R.C.M. Li and K.H. Yen, Elastic waves guided by a solid layer between adjacent substrates, IEEE Trans. Microwave Theory Tech., MTT-20:477, 1972.
535
PARAMETRIC MIXING EFFECTS IN SURFACE ACOUSTIC WAVES
CAUSED BY GAS BUBBLES IN LIQUIDS
Yasuhiko Nakagawa
Faculty of Engineering Yamanashi University Takeda-4, Kofu 400, Japan
INTRODUCTION
A surface acoustic wave (SAW) is a wave that propagates along a surface of a semi-infinite elastic body, with the major portion of its energy being confined at a depth of one wavelength from the surface. Therefore, a nonlinear effect is readily obtained on the surface of the elastic body by relatively small input power [1). For example, when two SAWs are propagating on a waveguide in opposite directions, a SAW at the sum frequency of the two input SAWs is generated as a result of the parametric mixing effect [2).
This paper presents a new phenomenon in the parametric mixing effects of SA W. When two SAWs were perturbed by a very small amount of liquid above the waveguide surface as shown in Figure 1, the generation of the SAW at the sum frequency was increased in amplitude about 30 to 40 dB over that in a state of no perturbation. Here, we propose a qualitative model based on the nonlinear oscillation of gas bubbles in liquid to explain this phenomenon.
ENERGY STORE IN PERTURBATION MATERIAL
As shown in Figure 2, the perturbation material is assumed to be a semiinfinite nonviscous fluid above the substrate surface. It is well known that
Figure 1.
Output (WI, k. )
Input I
Illustration of the device and interaction geometry of the two input SAWs.
537
Radiated compressional wove
p,:=,MtHf11 I
Rayleigh wove
Pin
Figure 2. Fluid loading of a surface acoustic wave.
the power is radiated from the substrate surface into the fluid and that the SAW is attenuated by virtue of the power . lost in this way [4]. A nonviscous fluid supports only waves of the compressional type. For our waves with a propagation velocity of Ve, and a radiation angle of
(1)
Where V R is the propagation velocity of the SAW propagating on the sub
strate. If the attenuation is small, the attenuation constants of the SAW are given by [3],
a = (20 log (2)
Here, v Ry is the particle velocity in the normal direction to the surface, P R is
the SAW power flow per unit width and Pc is the mass density of the fluid.
Table 1 shows the calculated radiation angles and the attenuation constants for the. perturbation material used in the experiments. Numerical calculations have been done for the waveguide fabricated on a 128' YX-LiNb0 3 and the SAW frequency was 48 MHz. If the perturbation material is in the shape of a small sphere, the energy is stored in the perturbation material, and thus high power density is possible with only moderate total input power. This fact reveals several nonlinear effects of SAW caused by gas bubbles in liquid.
Table 1.
538
Compressional propagation velocity, mass density, radiation angle and attenuation constant for the perturbation materials.
Compressional Mass density Radiation Attenuation Material propagation [g/cml) angle[OI constant
velocity [m/ s) [dB/mml
Acetone 1170 0.79 17.5 5.8 Methanol 1280 0.78 19.2 6.4 Mercury 1450 13.6 21.9 130.0 Water 1500 1.0 22.7 9.6
VR =3888 m/s. 1=48 MHz.
LIQUID !iNCOMPRESSIBLE) p,p.
Figure 3. The bubble model used showing parameters.
NONLINEAR OSCILLATION OF GAS BUBBLES IN LIQUIDS
Here forced oscillatfons of a spherical gas bubble in an incompressible, viscous liquid (water) are calculated numerically. A sketch of the bubble model used is given in Figure 3. The bubble is supposed to remain spherical through out its motion (having a momentary radius R and a radius at rest Rn) and be surrounded by an infinitely extended incompressible liquid. The bubble will contain some gas and vapor. The gas in the bubble is compressed according to a polytropic gas law.
When this bubble model is given a mathematical form, the following nonlinear ordinary differential equation of second order is obtained [4].
pRR + ~ pR2 = po[R~] 3k 2 (j 4 J1. • + p - p - - R - p( t)
y stat If If (3)
Po = 2 (j + P P stat - y Rn
(4)
Where a dot denotes a derivative with respect to time. Physical parameters are: Rn the radius of the bubbles at rest, R momentary radius; p the density, (j
the surface tension of the liquid, J1. the viscosity, It the polytropic exponent of the gas, Pstat static pressure, P y hydrostatic pressure, P(t) the external pressure. This model is suggested to be called the RPNNP-bubble model. In the numerical calculations reported here p( t) was taken as sinusoidal of the form
p(t) = -P Asin(wt) (5)
As p( t) must be constant at the bubble surface, the wavelength of the sound wave must be large compared to the bubble radius. This holds true for frequencies up to far above the linear resonance frequency of the bubble.
Figure 4 shows an example of a steady-state solution of Eq.(3) for water as follow: Rn=179 nm, p=O.98 g/cm2, 1T=72.5 dyn/cm2 J1.=O.Ol g/cm, 1t=1.33, Pstat=l bar, Py=O.023 bar, P A =12 bar. The frequency of the input sound
pressure was 46 MHz. The upper curve is the driving sound pressure , the curve in lower curve is the resulting steady-state bubble oscillation (radius time curve). It has a strong component at higher frequency.
Figure 5 shows the coefficient of second harmonic component as a function of the input sound pressure. At lower input sound pressure the revel of the second harmonic component is proportional to the power of the input sound. There is a threshold value of the sound pressure amplitude where the bubble may burst. When two sounds of frequencies 46 MHz and 44 MHz are applied
539
~ 6 A A V V V PA= 12( bar)
3
o~--~----~--~--~
o 4.4 8.8 x 10.2
t (jLsec)
Figure 4. Example of the computer output.
to the bubbles, a component of sum frequency 90 MHz is generated by the nonlinear oscillation of gas bubble. Figure 6 shows this case. There are also the phenomena of the saturation and the bursting of gas bubble. When two or more bubbles oscillate at the same time, the interaction between them must be considered in Eq.(3). The numerical calculation for this case is very complicated.
PARAMETRIC MIXING EFFECTS
Figure 1 schematically illustrates the device used in this study. It consists of a 128 0 YX-LiNb0 3 substrate with one strip waveguide, two input interdigital transducers (IDT) for generation of the SA Ws and one output mercury is fabricated on the waveguide. The generate mechanism of the SAW frequency WI
and W2 and the wave vectors kl and -k2 propagate in opposite directions, a nonlinear interaction in the perturbation material. This becomes a point source
Figure 5.
540
is too ii'i
~ x W 0:: W ii: :::> 1t to ~ I-Z W U li: It o <..>
2w=92MHz
10 tOO
SOUND PRESSURE (bar)
The coefficient of the second harmonic component as a function of the input sound pressure.
6 100 (i'\
~ x W 0: W a: :J
~ ~ I-Z W U ;:;:: ... w o u
10
0.1
WI+ WZ " 90tv'Hz
10 r ~ (b o r)
0~5 (w. -46 MHz)
SOUND PRESSURE (barl (w l " 44 MHz)
10
Figure 6. The coefficient of the sum frequency as a function of the sound pressure at the frequency of 46 MHz.
and the SAW at the sum frequency is generated in all directions as a cylindrical wave.
EXPERIMENTAL RESULTS
A 128' YX-LiNb0 3 substrate was used in the experiments. Al was deposited to form IDT and a waveguide. In the first experiments, the frequencies of the two input SAWs were the same. The center frequency of the input and output IDTs is 46 MHz and 92 MHz, respectively. The output IDT was fabricated in the normal direction to the waveguide (0=0" in Figure 1) in this case. The width of a waveguide is 80 /-Lm and the length is 10 mm. Experiments were done for perturbation materials such as liquid (water I acetone, methyl alcohol and mercury). The size of the material was about 0.3 mm in diameter and 0.1 mm in height.
An rf pulse with a 46 MHz and a length of 0.8 f.LS was applied to each input IDT. The resultant auto correlation pulse for the water is shown in Figure 7.b. The signal at the (a) represents the SAW at the sum frequency of 92 MHz for the no liquid on the waveguide. When the perturbation material was not fabricated on the waveguide, the voltage of the generated SAW caused by the nonlinearity of the only substrate was very low as shown in cylindrical wave when a small amount of perturbation material was fabricated on the waveguide.
- . . . . .... ........ • t •. -
Figure 7. Voltage waveform of the generated SAW.
541
Figure 8. Variation of the level of the SAW at the sum frequency for water.
The conversion efficiency F -factor for the parametric mlxmg effect presented in this paper is defined by F = P Ou t/P jP 2; where P j and P 2 are the input SAW powers and Pout is the SAW power of the F-factor for several perturbation materials are -70 to 80 dBm which are about 30 to 40 dBm larger than those for crystal substrate and evaporated Al.
The SAW at the sum frequency showed a violent variation in the liquid perturbation materials of water, acetone, methylalcohol and mercury . This phenomenon is thought to have been caused by the instability of the chemical state of the liquid. Figure 8 and 9 show the variation of the level of the SAW at the sum frequency for the perturbation of water and methylalcohol respectively. The period of the variation for methanol is very shorter than the period of water. Methylalcohol disappeared soon by the SAW.
Figure 10 shows the output power of the SAW at the sum frequency as a function of the sound pressure at the frequency of 46 MHz. The pressure of the input sound was estimated from the total SAW power divided by the effective cross section as shown in Figure 2. The parameter is the pressure of the sound at the frequency of 44 MHz. The output level of SAW at the sum frequency is saturated at the input sound pressure of 1 to 10 bar. The result of Figure 10 agrees well with the calculated results in Figure 6.
CONCLUSION
In conclusion, a new phenomenon in parametric mlxmg effects of SAW has been studied experimentally. When two SAWs were perturbed by a very small amount of liquid above the waveguide surface, generation of the SAW at the sum frequency was increased in amplitude about 30 to 40 dB over with that in an unperturbed state. We have here proposed a model which can explain this
Figure 9.
542
- t METHANOL ~ 20 4.' c£ ~
, (sec I
Variation of the level of the SAW at the sum frequency for methylalcohol.
:3 4..
ai '0
I::> (L f::> o
slope I
50
SOUND PRESSURE (bar)
Figure 10. Out put power of the SAW at the sum frequency as a function of sound pressure at the frequency of 46MHz.
phenomenon. Here, we propose a qualitative model based on the nonlinear oscillation of gas bubbles in liquid to explain this phenomenon. The theoretical calculation showed the phenomena such as the saturation of the SAW and the explosion of the gas bubbles. They are approximately similar to the experimental results.
REFERENCES
[3]
[4]
IEEE Trans. Microwave Theory Tech. MTT-17, No. 11, 1969. Y. Nakagawa and M. Ono, Proc. 4th Symp. Ultrasonic Electronics, Tokyo 1983, Jpn. J. Appl. Phys. 23, Suppl. 23-1, 148, 1984. B.A. Auld, Acoustic Fields and Waves in Solid, John Wiley & Sons, New York, Vol. II, Chap.12, p. 283, 1973. W. Lauterborn, J. Acoust. Soc. Am., Vol. 59, 283, 1976.
543
PHOTOACOUSTIC INVESTIGATION OF OPTICAL, ENERGY GAP IN
As-Se-Te GLASSES
K. Nandakumar and Jacob Philip
Department of Physics Cochin University of Science and Technology Cochin - 682 022, Kerala, India
INTRODUCTION
Over the past few years the photoacoustic technique has developed as a powerful tool for studying the optical an thermal properties of solids [1]. Several researchers have reported the results of studies of the optical properties of semiconductors in crystalline and amorphous forms using this technique [2,3,4]. This technique has several distinct advantages over the conventional reflection and transmission techniques in the study of highly absorbing and powder materials where the scattering of light is a serious problem.
The photoacoustic (P A) effect is the generation of acoustic waves in the gas medium surrounding the sample when it is irradiated by an intensity modulated beam of radiation. The absorption of radiation by the sample and the subsequent non-radiative relaxation processes give rise to thermal waves having the same frequency as the modulation frequency in the sample. The thermal waves in the sample is then converted into periodic pressure variation or an acoustic wave in the gas medium. So the amplitude of the PA signal is proportional to the amount of energy absorbed [5].
Chalcogenide glasses are finding extensive applications as electronic and optoelectronic materials [6]. This class of materials, containing a large percentage of chalcogen elements, is widely used in switching and memory devices [7]. An understanding of the dependence of various properties of such chalcogenide glasses on composition is important, because the continuously variable composition of these glasses may be utilized to prepare materials for particular applications.
As amorphous materials lack long-range order, the chemical bond approach [8] which considers the local bond and short-range order is considered to be the most appropriate to explain the various properties exhibited by these materials. This approach has proved to be useful in the study of a covalent chalcogenide glasses [9]. In chalcogen rich chalcogenide glasses, the top of the valence band is formed from the lone pair non-bonding states and the bottom of the conduction band is formed from the anti bonding states of the chalgogen atom [10]. The energy difference between the top of the valence band and the bottom of the conduction band closely corresponds to the optical energy gap Eo.
Here we report the results of the measurements of the variation of the optical energy gap Eo with composition in glasses of the As-Se-Te system belonging to the A vBvi family using photo acoustic (P A) technique. The composition studied can be categorized into two groups. The composition of one group of glasses can be represented as ASxTe5Se95-x with x at 3D, 35, 40, 45 and 50 atomic %. The other group of glasses can be represented as AS40 (Se,Teho and
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 545
fall along the AS2Se3-As2Te3 pseudobinary tie line and constitute the so-called stoichiometric compositions of the As-Se-Te system.
EXPERIMENT AL
Bulk glasses of the As-Se-Te system have been prepared by the usual melt quenching technique. Appropriate quantities of the constituent elements of five nine purity sealed in an evacuated quartz ampoules were melted in a rotary furnace at about 950" C. The ampoules were continuously rotated at 10 rpm about 30 h to ensure homogenization of the melt which is then quenched in the ice water to obtain the glass samples. The amorphous nature of the samples were checked by X-ray diffractometry.
The spectrometer used to record the photoacoustic spectra of As-Se-Te glasses consists of a 1000 W Xe lamp, a monochromator, a variable-frequency optical chopper (PTI model DC 4000), a home-made small volume PA cell and a lock-in amplifier (Stanford model SR510) to analyze the signal detected by the high-sensitivity electret microphone (Knowles model BIT 1753) kept in the P A cell. The block diagram of the experimental set-up is shown in Figure 1.
The optical energy gap Eo is determined by measuring the variation in the normalized P A signal with incident wavelength (photon energy). The PA spectrum obtained for highly absorbing carbon-black sample is used to normalize the spectrum obtained for each sample. The chopping frequency used for the measurements was 31 Hz.
RESULTS AND DISCUSSION
The plots of normalized PA signal amplitude against wavelength for Asx Te5Se95-x with x = 30, 40 and 50 are shown in Figure 2. Similar plots have been obtained for all other samples investigated. As can be seen from the figure the PA signal increases with increasing photon energy and reaches a saturation level. The P A signal saturation corresponds to when the optical absorption coefficient becomes very high for photon energies greater than the optical energy gap Eo. The curves also give information about the optical absorption coefficient fJ. In amorphous semiconductors the optical absorption curve is found to have three regions. The first is a weak absorption region with fJ < 1 cm -1.
The second has an absorption coefficient in the range 1 cm -1 S fJ S 10 4 cm -1. In this region there is an exponential dependence of absorption on the incident photon energy given by
fJ(hv) = exp(hvjE)
MONO-
LAMP LENS CHROMATOR LENS PA CELL
LOCK IN AMPLIFIER
Figure 1. Block diagram of the experimental set-up.
546
6
~ 5 c :J
..6 L ",
....J « z l!) 4 iii
~ Cl W N
:J ~ 3 a: o z
Figure 2.
2
50
40
30
400 600 800 1000 WAVELENGTH (nm)
Normalized PA spectra of AsxTe5Se95-x glasses. The x values are noted against each curve.
For hll < Eo, the P A spectra follow such a dependence. The third region corresponds to a high absorption coefficient with fJ ~ 10 4 cm -1 which follows the equation.
hllfJ(hll) = B(hll - Eo)2
Figure 3 shows the variation of optical energy gap with composition parameter x for AsxTe5Se95-x glasses. As can be seen from Figure 3, the optical energy gap decreases with decrease in the chalcogen content for the system and shows a marked change in the decrease at x = 40, which corresponds to the stoichiometric composition of the As-Se-Te system. The rate of decrease is higher for compositions with x < 40 than for composition with x > 40. Figure 4 shows the variation of optical energy gap with Te content for the stoichiometric compositions. It can be seen that the optical energy gap decreases with increase in the Te content for the system.
The observed behavior can be explained on the basis of chemical bonding between atoms and the changes in the short range order that take place when the As content increases. According to the chemically ordered network (CON) model [12], the As-Se-Te glasses can be thought of as being made up of completely cross-linked three dimensional structural units of As2Se3 and As2Te3 with either As or Se in excess. Apart from As-Se, As-Te, Se-Te bonds, As-As, Se-Se and Te-Te bonds will be involved in forming the glass. For the AsxTe5Se95-x glasses, with increase in As content the average bond energy decreases since As-As bond energy is less than the As-Se bond energy, and a corresponding decrease in the optical energy gap. Beyond the stoichiometric
547
Figure 3.
1·76
1 f" > '" -0> 168 W
1·64
160'---'---__ --.J'-----__ ---' ___ ---' ___ --'---'
30 35 40 45 50 Atomic '/, of As
Variation of optical energy gap with composition for Asx Te3Se95-x samples.
composition, the number of Se-Se bonds decreases resulting in a decrease of average bond energy which in turn reduces the optical energy gap. For the stoichiometric compositions the optical energy gap decreases with increase in Te content. According to CON model an ideal glass is formed at the stoichiometric composition and only heteropolar bonds are present at this composition. So with increase in Te content the number of As-Te bonds increases resulting in a decrease of average bond energy and a corresponding decrease of optical energy gap.
CONCLUSION
The present work shows that the photoacoustic technique is an excellent tool to study the optical properties of amorphous solids. The technique has been
Figure 4.
548
1·8
W01 1·6
1·4
o 5 10 15 Atomic '/, of Te
Variation of optical energy gap with composition for As 40(Te,Se ho samples.
used to study the composition dependence of optical energy gap in As-Se-Te glasses. For AsxTe5Se95-x glasses with x = 30, 35, 40, 45 and 50, the optical energy gap decreases with increase in As content and shows a marked change in its rate of decrease at the stoichiometric composition. For the stoichiometric compositions, the optical energy gap decreases with increase in Te content. The observed behavior is explained on the basis of chemically ordered network structure and changes in the short range order with increase in the chalcogen content.
REFERENCES
[1]
6 7 8 9
1°1 11
[12]
A. Rosencwaig, Photoacoustics and Photoacoustic Spectroscopy, (New York: Wiley), (1980). K. Nagata et aI., Jap., J. Appl. Phys. 24, L858, (1985). K. Nandakumar and Jacob Philip, Bull. Mater. Sci., VoLll, N".4, p.297-301, (1988). K.N. Madhusoodanan et al., Phil. Mag.B, Vol.58, N" .1, p.123-132, (1988). A. Rosencwaig, Advances in Electronics and Electron Physics, (New York: Academic press), Vol.46, (1978). D. Adler, Sci. Amer, 236, 36, (1977). R. Zallen, The Physics_ of Amorphous solids, (Wilex: New York), (1983). A.F. loffe and A. Regal, Prog. Semicond. 4, 239, lI960). M. Lannoo ap.d M. Bensousan, Phy. Rev. B16, 3546, (1977). M. Kastner, Phy. Rev. Lett. 28, 355, (1972). N.F. Mott and E.A. Davis, Electronic Processes in Non--crystalline Materials, (Oxford: Clarendon), (1971). G. Lucovsky and T.M. Hayes, Amorphous semiconductors, (ed.) M.H. Brodsky, (Berlin: Springer-Verlag), p.215, (1979).
549
INTERFEROMETRlC PROBING OF OPTICALLY EXCITED SURFACE
ACOUSTIC WAVE PULSES FOR THIN FILM CHARACTERlZATION
A. Neubrand, L. Konstantinov, and P. Hess
Institute of Physical Chemistry University of Heidelberg 1m Neuenheimer Feld 253, D-6900 Heidelberg 1, FRG
INTRODUCTION
Surface acoustic waves (SAWs) have been widely employed for the nondestructive testing of materials and thin films due to their effective interaction with various inhomogeneities and defects. Such waves are usually generated and detected by piezoelectric transducers coupled to the surface to be inspected. However, the high sensitivity and the well-defined acoustic response of these transducers are combined with a relatively narrow frequency bandwidth and serious restrictions concerning the type and quality of the surface being studied, the contact between the sample and the transducer, and the experimental flexibility. That is why the efficient and contactless generation and detection of wide-band SAW transients in pulsed-laser photoacoustics have recently been given much attention [1,2]. Realized in an all-optical system (e.g. SAWs excited by a short laser pulse and detected by a focused interferometric probe) such schemes allow the above-mentioned restrictions to be overcome at the expense of a reasonable loss of sensitivity. The main problem in employing optically excited SA Ws seems to be the complexity of the resulting signals, which obscures in some cases their interpretation and requires a more sophisticated analysis. The aim of this paper is to describe a highly sensitive wide-band method for the optical generation and detection of SAW pulses of ns duration and to discuss its applicability for thin film characterization.
EXPERIMENT AL
The experimental setup used is shown schematically in Figure 1. The SAW pulses were excited by a 308 nm exciter laser EL (Lambda Physik LPX 105 i) of 20 ns (FWHM) pulse duration illuminating a slit aperture SL. The slit is projected onto the sample surface as a strip of about 25 mm length and 20 /lm width by a quartz cylindrical lens CL. In order to assure a thermoelastic regime of SAW generation, in all experiments the energy density was kept far below the threshold for damaging the surface (pulse energy of about 1 mJ on the sample resulting in a fluence of 0.2 Jjcm2). By precisely moving the cylindrical lens one could scan a length of 25 mm along the sample surface with a lateral resolution of about 5 /lm. After propagating a given distance, the SAW pulses were detected by a Michelson interferometer in which the inspected surface serves as one of the interferometer mirrors. For this purpose the beam of a polarized 5 mW HeNe laser PL (Uniphase ll05 P) is split into sample and
Physical Acoustics. Edited by O. Leroy and M.A. Brealealc Plenum Press, New York, 1991 551
Figure 1. Experimental setup.
reference beams of equal intensity by an optical system consisting of a polarizing (PBS) and a non-polarizing (BS) beamsplitter and a quarter-wave plate QW. The reference beam is then reflected by a dielectric mirror DM mounted on a piezoelectric vibrator PZT, while the sample beam is focused onto the sample surface to a spot of about 20 {Lm diameter by an achromatic lens ALI of 1 cm focal length. The light reflected by the sample is collected with the lens ALI and directed through the beamsplitter BS to interfere with the reference beam. After passing through the quarter-wave plate once again, the light reflected towards the laser is polarized perpendicularly to its initial direction of polarization and is directed out of the interferometer by the polarizer PBS. The other part of the light is focused by the lens AL2 onto a fast low-noise PIN photodiode PD (Hamamatsu S2839, risetime less than 1 ns) through an interference filter. The signal from this diode is amplified and recorded by means of a fast digital storage oscilloscope DSO (Tektronix 2440) triggered by a UV sensitive photodiode UD and connected to a computer.
As the interferometer sensitivity is a function of the phase difference between the sample and the reference beams, which are subject to thermal and mechanical noise, a stabilizing circuit was necessary to defeat ambient disturbances and to adjust the interferometer to its most sensitive operating point. This is achieved by a feedback system which automatically compensates for low-frequency deviations from this point, thus assuring the temporally constant sensitivity required for measuring short SAW pulses. In practice, the
552
~ '" " :E -a 0.5 ~
Or--C~------------~~
1.25 1.27 1.29 1.31
TIme (.us)
Figure 2. SAW pulse waveform for a dispersionless medium.
0.5
- 0.5
1.1~m a- Si: H on (l00)Si
~----~~~~----~~ 4.4 4.5 4.6 4.7
rome (jJs)
Figure 3. SAW pulse waveform for a dispersive medium.
piezoelectric transducer PZT is oscillated with an amplitude of 50 A at 2 kHz by a sine voltage from a frequency synthesizer FS in order to modulate one of the interferometer arms . When the operating point of the interferometer deviates from the optimal position, corresponding to a phase difference of ),/8 between the two interfering beams, a weak component with the doubled modulation frequency arises in the photodiode output signal. This component is separated from the high-frequency signal by a frequency divider FD and detected by a lock- in analyzer LI. The "in-phase" signal is then processed by a differential amplifier and integrator DAI to form a correction voltage for the amplifier HV A supplying the piezoelectric ovibrator. The sensitivity of the arrangement described above amounts to 0.2 A in a 100 MHz bandwidth for a single shot and can be further improved by averaging.
RESULTS
Figure 2 presents the SAW pulse waveform excited on a (100) Si wafer and measured after propagating a distance of 6.3 mm along the surface. The waveform is typical for thermoelastic excitation of SAWs in a non-dispersive medium and is quite similar to the laser pulse shape. Figure 3 shows the waveform of a SAW pulse excited on the same wafer but after a distance of 22.3 mm, 14 mm of which was covered by a 1.1 J1m-thick film of amorphous hydrogenated Si (note that due to the greater pathlength the time scale is now four times larger) . It is evident that the film modifies the SAW pulse waveform substantially, transforming it into a bipolar pulse and introducing in its trailing edge a sequence of gradually contracting oscillations indicative of the presence of dispersion.
Figure 4 shows the dispersion curves for two aluminum films of thickness 1.0 J1m ± 3% (curve a) and 3.0 J1m ± 3% (curve b) deposited on quartz. The
Figure 4.
~ 3400
.5. ~ 3380 '8 J C> 3360 ., 0 .c b) 0..
:I' alljJm AI on fused quartz -< 3340 V'> b 3)Jm AI on fused quartz
0 20 40 60 80
Frequency (MHz)
SA W phase velocity versus frequency for Al films of thickness 1 J1m and 3 J1m on fused silica.
553
Figure 5.
~ 3395 -5 >-
"8 3390 1 .. 3385 ., 0
.c. <>. lJJm AI on fused quartz a) ~ 3380 < '"
a) as deposited b) annealed ot 58O·C
0 20 40 60 60
Frequency (101Hz)
SA W phase velocity versus frequency for an Al film of 1 f,lm thickness as prepared and after 2 h thermal annealing at 580' C.
curves are obtruned by averaging 1000 SAW pulses excited at two different distances x and x ' from the point of detection. The SAW phase velocity v and the attenuation a are then calculated by Fourier analysis of the SAW pulse waveforms using the corresponding spectra of the phase ¢ and the Fourier amplitude .A (see for example [3]):
v(w) w· (x ' - x) I [¢(x ' ,w) - ¢(x,w)] a(w) - In[ A{x ' ,w) I A{x,w)] I (x' - x)
The accuracy in determining the SAW velocity by this procedure is about 2 m/s in the whole frequency range studied, which allows this parameter to be specified very precisely for different films in order to characterize them. It can be seen that the film thickness affects the dispersion considerably, the ratio of the slopes of the two curves in the low-frequency region being equal to 3.0, in excellent agreement with that of the film thicknesses. At the same time no attenuation was observed up to 100 MHz even with a s ensitivity of about 0.2 Np/cm (see Figure 6, curve a) .
In order to study the influence of film treatment on the SAW propagation parameters the films were heated in air to different temperatures in the range 500' -£40' C. It was found that annealing up to 500' C does not produce any observable change in either the dispersion or attenuation curves, while above this temperature, when the melting point of the film was approached, considerable variations occurred. To illustrate this, Figure 5 presents the dispersion curves for an Al film of thickness 1 f1-m as deposited (curve a) and
Figure 6.
554
5 lJJm AI on fused quartz
~ 4 c) al as deposited b annealed at 58O"C
<>. 3 c annealed at 64O"C 3-
~ 2 0 b) j :;: a)
0 0 20 40 60 80
Frequency (MHz)
SA W attenuation versus frequency for an Al film of thickness 3 f,lm on fused silica as prepared, and after thermal annealing.
after heating to 580· C for 2 h (curve b). It is evident that the rise of the annealing temperature by about 80" C nearly halved the initial dispersion without changing the character of the curve. On the other hand, Figure 6 (curve b) shows the SAW attenuation as a function of frequency for a film of thickness 1 J.tm heated for 2 h at 580· C, while curve c presents the attenuation for a film heated for 2 h at 640· C. It is seen that in the latter case above 30 MHz the attenuation sharply increases to such an extent that measurement at higher frequencies becomes impossible, while in the former case it rises more slowly. It should be noted that these changes are accompanied by visible alterations in the film appearance, i.e. in its optical parameters and morphology, as well as in the formation of inhomogeneities of different size on the film surface. The temperatures mentioned above serve only as indications of the deviation of the film properties from the usual ones due to oxidation, recrystallization, etc.
The same kind of experiment were carried out on a sample consisting of an approximately 1.1 J.tm-thick film of amorphous hydrogenated Si prepared b;r laser-induced chemical vapor deposition [4) at 300· C (hydrogen content ~ 17%) on a polished (100) silicon wafer of 2.1 mm thickness. In this case the SAW pulses were generated on the substrate and detected either on the film surface or on the substrate after passing through the coated area to avoid film damage. If the substrate thickness was less than 1.5 mm, the acoustical response observed was quite complicated due to multiple reflections of various bulk and surface modes, but for greater thicknesses it was dominated by a Rayleigh-type dispersive SAW whose phase velocity is shown in Figure 7 as a function of frequency. In this case, the dispersion is stronger than that measured for Al films due to the larger difference in the acoustical parameters of the film and the substrate. At the same time, similarly to the case of Al films, no SAW attenuation was observed for Si films in the frequency range studied.
DISCUSSION
The characteristics of SAW propagation can be modified by a thin film deposited on the substrate surface. Despite the mathematical complexity of this problem there are methods available to treat the matter. The most dominant effects produced by the film are a shift of the SAW phase velocity and a dispersion effect (the dependence of SAW velocity on frequency).
The film, assumed to be isotropic and thin with respect to the wavelength, is usually characterized by its mass density p and the velocities for shear and longitudinal waves ~ and 1/, respectively. All three parameters influence the SA W velocity in different ways, which in principle allows one to determine them by fitting the measured SAW dispersion curves to ones calculated on the basis of relevant theoretical models. The best three-parameter fit of both curves in Figure 4 using the theory proposed by Tiersten for the description of SAWs
Figure 7.
~ 4890 ..§. ", ~
4860 " " 1 • 4830 ., " .c CL
~ 4800
'" 0 20 40
1.111m a-51: H on (100) 51
60 80
Frequency (MHz)
SA W phase velocity versus frequency for an a-Si:H film of thickness 1.2 J.tm on a (100) Si wafer.
555
guided by thin isotropic films on isotropic substrates [5] was obtained with p = 2.77 kg/cm 3, e = 3110 mis, 'fJ = 6600 mls (Rayleigh SAW velocity of 2933 m/s), values close to those known for the bulk material [6,7] (p = 2.7 kg/cm 3, e = 3110 mis, 'fJ = 6420 m/s). It should be noted that the model function for the SAW velocity in the case of Al films of a thickness of about 3 11m on fused silica is sensitive to changes of the order of 2% in both p and e, and of 6-7% in 'fJ. The procedure, however, is not unequivocal and additional criteria are necessary to distinguish the actual set of parameters, e.g. the sign and the magnitude of deviations from the corresponding bulk material parameters. Since the chan~e in SAW velocity caused by the film is constant for a given value of kh (k is the SAW wavevector and h the film thickness), one can use dispersion curves as those in Figure 4 for measuring h provided that a film of known thickness serves as the reference standard. Such an approach would be especially effective for in situ control of the film growth, when no other means such as microbalances, optical probes, etc. can be employed.
The thermal treatment of Al films at temperatures close to the melting point produces considerable changes in their acoustical and optical parameters as well as their morphology and homogeneity. The observed sharp increase in SAW attenuation for frequencies above 30 MHz (Figure 6, curve c) and the features present in curve b could be related to some specific mean grain size or to a given mechanism of SAW decay in the film. The discussion of this problem, however, is beyond the scope of the present study and will be the subject of a forthcoming paper.
ACKNOWLEDGEMENTS
L.K. is indebted to the Alexander von Humboldt Foundation for a fellowship. We wish to thank Dr. H. Oetzmann from ABB, Heidelberg, for providing us with the aluminum films. Financial support of this work by the German Ministry of Research and Technology (BMFT) under contract N'. 13N 5363 8 and by the Fonds der Chemischen Industrie is also gratefully appreciated.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
556
L. Konstantinov, A. Neubrand, and P. Hess, Surface Acoustic Waves in Solid State Investigations, in "Topics of Current Physics", Vo1.47: "Photoacoustic, Photothermal and Photochemical Processes at Surfaces and in Thin Films", P. Hess, ed., Springer, Berlin, Heidelberg (1989). A.A. Karabutov, Laser excitation of surface acoustic waves: a new direction in opto-acoustic spectroscopy of a solid, Sov. Phys.-Usp., 28:1042, (1985). A. Neubrand and P. Hess, Study of Attenuation and Dispersion of Optically Excited Surface Acoustic Waves Employing Small PVDF Foil Transducers, Mater. Sci. & Eng., A122:33, (1989). D. Metzger, K. Hesch, and P. Hess, Process Characterization and Mechanism for Laser-Induced Chemical Vapor Deposition of a-Si:H from SiH4, Appl. Phys., A45:345, (1988). H.F. Tiersten, Elastic Surface Waves Guided by Thin Films, J. Appl. Phys., 40:770, (1969). "Handbook of Chemistry and Physics", R.C. Weast, ed., Chemical Rubber, Cleveland (1971). H.M. Ledbetter and J.C. Moulder, Laser-induced Rayleigh waves in aluminum, J. Acoust. Soc. Am., 65:840, (1979).
INVESTIGATION OF THERMAL WAVE INTERFERENCE IN CdGa2S4 BY
THE PHOTOACOUSTIC METHOD
P.M. Nikolic, D.M. Todorovic, Z.D. llistovski
Belgrade University 1100 Belgrade, P.O. Box 816 Yugoslavia
INTRODUCTION
The photoacoustic (PA) effect is a phenomenon of generation of acoustic waves in a material which absorbs modulated electromagnetic radiation or a modulated energy beam (electrons, ions, etc). The generated acoustic signal depends on optical, thermal and elastic properties of the material.
The sample, which absorbed the modulated radiation, locally and periodically heats (the photothermal effect). This thermal energy can directly produce acoustic signals (the thermoelastic effect) or can be transferred to the sample surface by means of diffusion (the thermodiffusion effect). Periodical heating of the sample-gas boundary surface can produce periodical expansion of the gas i.e, variation of pressure (the acoustic response). The generated acoustic signal has a frequency which is equal to the modulation frequency of the incident radiation.
Bennett and Patty [1] have investigated carbon using the PA method and they have noticed that the change of the PA signal with the frequency of modulation is not in agreement with theory anticipation. They have explained these differences as being due to interference of thermal waves which were generated in the sample whose thickness was smaller than the thermal diffusion length.
Mandelis et al. [2] investigated thin films of Si0 2 on a Si substrate. They have demonstrated the feasibility of quantitative PA spectroscopy as a simple and sensitive thermal interferometric method for thin film transmittance measurement.
Todorovic and Nikolic [3] have experimentally studied interference of thermal waves in semiconductor GaSe. The P A spectra were measured as a function of modulation frequency and the sample thickness. The thermal wave interference was clearly noticed. An optical interference, i.e. periodical change of the P A signal as a function of wave length was noticed. Optical interference was observed for a particular modulation frequency of incident radiation and sample thickness. This implies that the thermal phenomena are connected to the optical interference.
In this work we have studied interference of thermal and optical waves in thin film semiconductor - CdGa2S4.
This spectra showed simultaneous interference of optical and thermal waves for some frequencies of modulation of the light beam and sample thickness. These results showed that the photoacoustic spectra was complementary with the interference spectra.
Physical Acoustics, Edited by O. leroy and M. A. Breazeale Plenum Press, New York, 1991 557
EXPERIMENTAL RESULTS
The P A amplitude spectra were measured for thin film semiconducting CdGa2S4 using a standard PA spectrometer (GILFORD R-1500) in the range between 400 and 800 nm.
The thin film of CdGa2S4 was evaporated on a glass substrate and was about 2.8 micrometers thick. The amorphous nature of the samples thus prepared was confirmed by X-ray method using a Philips diffractometer . The composition of the samples was analyzed and confirmed by microscope analysis.
The PA amplitude spectrum for the thin film CdGa2S4 in the range of the absorption edge is given in Figure 1 with a full line. The transmission spectrum of the same sample obtained using UV-VIS spectrophotometer (Perkin-Elmer Lambda 5) is given with a dotted line in the same figure . The modulation frequency for the P A spectrum, given in this figure, was 43 Hz. Similar P A spectra for the same thin film CdGa2S4 sample, but for three modulation frequencies (35, 43 and 70 Hz) are given in Figure 2. In this figure is shown that an interference P A spectrum can be obtained only for a particular modulation frequency i.e . 43 Hz.
DISCUSSION
As previously mentioned, Bennett and Patty have derived relations for amplitude and phase of PA signals under conditions of thermal wave interference . They used the Rosencwaig-Gersho theory [4] introducing into the boundary conditions, coefficients of reflection and transmission of thermal waves. The thermal waves which are generated in a sample propagate through the boundary surfaces but they also partly reflect on them. If we apply the boundary conditions of continuous temperature and thermal flux it is possible to calculate the reflection Ri and the transmission T i coefficients of the thermal waves for various boundary surfaces ( i = b for the boundary surface of the sample background; i = g for the surface sample - gas) . These coefficients have the following forms:
I-b 2 _ .!....=...g . Rb = 1 + b; T b = lTD; Rg - 1 + g' T g
2 r+g
60 r---.,....---.-----r----r---,---.100
... 50
40
Vl if. 30
......... 50 .!(
20
10
~25 475 525 575 625 675 72~ >./ nm/
I-
Figure 1. The PA amplitude spectrum for the thin film CdGa2S4 in the range of the absorption edge is given with a full line and the transmission spectrum of the same sample obtained using UV-VIS spectrophotometer is given with a dotted line.
558
(1)
where band g are known constants
where p is density, Ci specific heat and Ki the thermal conductivity of materials.
If the sample thickness is small when it is compared with diffusion length of the thermal waves, many reflections in the boundary surfaces can produce thermal wave interference. The the following relation can be obtained for the normalized amplitude of the PA signal, using a thermodynamical formalism of the Rosencwaig-Gersho theory:
where
is the thickness of the sample and thermal diffusion length fJ, is:
~ fJ,=~~
'l'
~ a.
Figure 2.
80
70
60 60
30 50 50
70 40
50 40
.30 50 .30
20 40
20 10 f-35 Hz 30
0 20 10
f : 43 Hz
10 o f- 70 Hz
o j.illj""j""jllil i" "j""""'j .375 425 475 525 575 625 675 725
Vnm/
P A spectra for the thin film CdGa2S4 sample, for three modulation frequencies (35, 43 and 70 Hz).
(2)
(3)
559
It may be noticed that F has a similar expression to the relation for the analogous optical problem.
The P A signal should be normalized using a reference P A signal for a sample of the same material with a thickness much greater than the thermal diffusion length (f.L » 1).
The amplitude of the P A signal theoretically calculated as a function of the thermal thickness of the sample (f.Ll) has been given in the literature [5] for various values of thermal wave reflectivity coefficient on the boundary surface (Rb). There it is possible to see that for Rb > 0, the reflected thermal waves from the boundary surface sample-holder (background) arrives in phase with the direct wave to the boundary surface sample-gas. Then a constructive interference appears and the normalized amplitude of the P A signal is bigger than one.
Our experimental results given in Figure 2 and the theoretical considerations can be used to analyze thermal wave interference i.e. to analyze the thermal properties of our CdGa2S4 samples. The calculation of the change of amplitude of P A signal as a function of the frequency of modulation for frequencies in which the thermal diffusion length was much bigger than the sample thickness (f.L » 1) is in progress.
Using this method it is possible to examine thickness of thin film samples as well. Mandelis et al [2] have developed a theoretical model for photoacoustic interferometry . They confirmed that photoacoustic minima occur for values of (d, + ¢h2) where:
arctg-----k~ + n~ - ni
(4)
corresponding to optical maxima, or optical transmission numma, obtained from thin transparent films. The thickness L, of the thin film can be obtained from the two executive PA minima using a simple relation .
(5)
This equation can be simplified for the case when the substrate is much more transparent compared with the measured thin film sample. Using our experimental results given in Figure 1 we have calculated the thickness of our sample which was in reasonable agreement with the experimental results obtained using a Talystep. From Figure 1 it is possible to conclude that for the wave lengths bellow the absorption edge photoacoustic diagrams are much more informative compared with the classical optical transmission.
CONCLUSION
In this paper it has been shown that the optical interference can be studied by using the P A method which is especially useful when the samples are optically opaque. It was also shown that the interference conditions can be obtained for a particular frequency of modulation. Besides, optical and thermal properties can be analyzed by investigation of the interference P A spectra. It is also possible to obtain the thickness of various layers in a multilayer sample by using the P A method, practically with a nondestructive method.
REFERENCES
[1] C. Bennet, R. Patty, "Thermal Wave Interferometry: Application of the Photoacoustic Effect", Appl. Opt. 21, 49, (1983).
560
[2]
[3]
[4]
[5]
A. Mandelis, E. Sui, S. Ho, "Photoacoustic Spectroscopy of Thin Si0 2 Films Grown on (100) Crystalline Si Substrate", Appl. Phys. A, 33, 153-159, (1984). n.M. Todorovic, P.M. Nikolic, Pr. of Int. Con. on Ac. Belgrade, 4(10,11,12,13), 349, (1989). A. Rosencwaig, A. Gersho, "Theory of the Photoacoustic Effect with Solid", J. Appl. Phys. vol. 47. pp.64 (1976). P.M. Nikolic, n.M. Todorovic, Progress in Quantum Electronics, 13(2), 107-190, (1989).
561
PRESSURE WAVES PROPAGATION IN GAS-LIQIDD FOAM
Z.M. Orenbakh, LR. Shreiber, G.A. Shushkov
Institute of Northern Development Siberian Branch U.S.S.R. Academy of Sciences 625003, Tyumen, Box 2774, U.S.S.R.
INTRODUCTION
There are presented the experimental researches results of pressure waves propagation in gas-liquid foam. There have been defined velocity meanings and space coefficient of sound attenuation in different, close to a unity gas content. The comparison with theoretical data is given.
In experimental researches of acoustical disturbances propagation in gas-liquid mixtures of bubble structure [1,2], the media with small gas content volume have been researched, as a rule. It's explained, at first, by the difficulty of stable concentrated water-bubble suspensions receipt through the bubbles coagulation. The small addition of surface-active substance, without increasing of liquid viscosity, permits to stabilize mixture and define its acoustic characteristics in gas contents above 90%. Such medium consists of the majority of gas bubbles separated by thin liquid peUicles and is called foam [3]. Since the foam characteristics change considerably, during the small alterations of volumetric gas content, foams are characterized by the foam volume V f relation to the volume of liquid VI containing in the foam (K = VdVI). This parameter is connected with the volumetric gas content rp by the relation K = 1/(1 - rp) and changes from 1 to w. The average bubble size obtained at microphoto analysis compiles 0.15 mm.
The experimental plan presents the vertical shock tube (Figure I), where the pressure disturbance initiates during the gap of diaphragm, separating the high pressure chamber from the working area. The initial amplitude of pressure impulse is defined by a material of the diaphragm. The working area is made of organic glass and supplied with six piezoelectric sensors arranging through 0.05 m. The distance from the diaphragm to the first sensor and from the last sensor to the bottom of the working area is more than 1 m. This fact permits to exclude reflected waves from the consideration.
The researching medium is driven up into the working area from the special foam generator [4] permitting to set the expansion factor with the order of accuracy 5% in wide diapason. The transparency of the working area gives the opportunity of visual observation of the process and foam photographing. Depending on experimental conditions pressure impulses have been initiated in the medium with maximum amplitude from 3 to 10 kPa. The picture of disturbances evolution practically doesn't depend on its initial amplitude, which testifies the absence of nonlinear effects.
The signals from sensors after analog-to-digit conversion are put into the operation of computer, reflected on the screen of coloured monitor and written on the magnetic carrier for the long keeping. The software permits, in future,
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 563
Figure 1.
l
3
" 5
6
7
12 13
1--6 - pressure sensors; 7 - low pressure chamber (LPC); 8 - high pressure chamber; 9 - diaphragm; 10 - starting sensor; 11 - overlapped holes; 12 - multiplexor 6 to 1; 13 - analog-to-digit convertor; 14 control computer; 15 - color monitor; 16 - hard disk drive.
to read the information from magnetic disk, to print the wave structures on the dot matrix printer, and also to calculate the spreading velocity and wave attenuation coefficient. It is possible also to define average magnitudes for the given expansion factor.
According to the formula C-2 = 8p / 8P, sound velocity C in any medium is defined by the dependence of density p from pressure P. Without concretizing of this dependence, in other words, without observing of concrete mechanism of acoustic disturbances propagation, but coming only of the general relations of homogeneous model of two-phase mixture, one can get the expression [5) :
c (m/s)
110
90
70
1/C2
50 <:> o
Figure 2.
564
<:>
40
c (m/5)
100
80
60
40 B A
80 I20 K 0,95 0, '!7 0,99
Sound velocity dependence in gas-liquid foam on expansion factor (A) and on gas content (B) . Sheer lines: 1 - results of formula (1); 2 - low frequency phase velocity, results of paper [6). Experimental dots: 0, D. - accordingly one and two percent solution of sulphanol in water.
(1)
"
Figure 3. Pressure disturbances evolution in foam with expansion factor Sensors 1 - 6 data.
where, X is the mass gas content. The subscript 1 corresponds to liquid, 2 to gas.
The dependence of sound velocity on expansion factor (Figure 2.A) looks exotic for the specialists working in the sphere of small gas content, so Figure 2.B presents gas content as an argument.
During the treatment of experiments the sound velocity is defined accordingly the time intervals between arrivals of disturbance maximum in two neighbour sensors and the next averaging of obtained magnitudes . In spite of the considerable variation of measured meanings along the length of working area, the average velocities divergence is 1% for the foam of definite expansion factor.
The experiments show (Figure 2) the dependence of sound velocity in foam on the expansion factor (or gas content) , coinciding with the formula (1) with error less than 5%. The stable calculated velocity excess above experimental meanings testifies, possibly, about the systematic error of foam expansion factor defining. Nevertheless , the solution of the inverse problem is possible: the expansion factor defining according to the results of sound velocity measuring, i.e. acoustic diagnostics of foam currents .
The typical picture of pressure disturbances evolution (Figure 3) is evident of the absence of sound velocity dispersion in foam, at least, for the experimental conditions.
The space coefficient (f) was defined from the average ratio if disturbance maximums on the neighbour sensors (every other 0.05 m).
Table 1 shows the absence of evident dependence of attenuation coefficient on foam gas content , though there is a tendency to foam damping properties
Table 1.
K
f
Correlation between expansion factor and coefficient of attenuation
23 40 57 74 100 143
3,65 3,70 3,18 3 , 02 3 , 01 3,14
565
Figure 4.
. .-.;n------~:-----:-. "".-;::-----
I I ' , , I t I - t
Comparison of experimental pressure profiles with linear theory [6]. A without consideration of liquid motion along Plateau-Gibbs channels; B - full theory. Curves made of close dots- sensors indications; rare dots - calculation results.
reduction with the expansion factor increase. Comparison with the data of paper [2], presents that coefficient of sound attenuation in foam is roughly 5--6 times above, than that in two-phase bubble mixture in gas content 1%.
With control computer, shown in Figure 1, there was made comparison of experimental pressure profiles with linear theory [6] . The indications of the first sensors were taken as boundary conditions. After that using the fast Fourier transform algorithm method, one received solutions corresponding to sensors positions. The results of this comparison, shown in Figure 4, prove that theory, considering liquid motion along Plateau-Gibbs channels is more appropriate for the description of the experiment .
566
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
V.V. Kuznetsov, V.E. Nakoryakov, B.G. Pokusaev, LR. Shreiber, Experimental research of disturbances propagation in liquid with gas bubbles, "Non-linear wave processes in two-phase media", Novosibirsk, ITPh Siberian Branch USSR Academy of Sciences, p.32-44, (1977). N.A. Pribaturin, The influence of pressure on the disturbances propagation in vapour-liquid medium, "Non-equilibrium processes in one- and two-phase systems", Novosibirsk, ITPh Siberian Branch USSR Academy of Sciences, p.5-10, (1981). K.B. Kann, Capillar hydrodynamics of foams, Novosibirsk, Nauka, p.167, (1989). K.B. Kann, V.N. Feklistov, Experimental determination of flow regimes and hydraulic resistances of foam flows in tubes, "Researches in hydrodynamics and heat transfer". V.E. Nakoryakov, B.G. Pokusaev, LR. Shreiber, Waves propagation in gasand vapour-liquid media, Novosibirsk, ITPh Siberian Branch USSR Academy of Sciences, p.238, (1983). LL Goldfarb, LR. Shreiber, F.l. Vafina, The influence of heat transfer and liquid flow on sound propagation in foam, "Physical acoustics: fundamentals and applications", Kortrijk, Belgium, 1990. Z.M. Orenbakh, L.P. Smirnova, LR. Shreiber, Structure of linear waves in tubes filled in with viscous fluid, Acoust. Journal, Vo1.33, N4, p.729-735, (1986).
567
THERMO SENSITIVITY OF GENERALIZED RAYLEIGH WAVES FOR
ROTATED Y-CUTS IN a-QUARTZ
N.S. Pashchin, LB. Yakovkin
Institute of Semiconductor Physics Siberian branch of the U.S.S.R. Academy of Sciences 630090, Novosibirsk, U.S.S.R.
It is of practical importance to study the temperature sensitivity as the fractional change of the surface acoustic wave (SAW) characteristics due to the substrate temperature variations. So, the temperature coefficient of the delay time (TeD) is a major consideration for the selection of the substrate material in the design either of stable SAW-oscillators or temperature sensors [1-2].
SA W propagation characteristics in a anisotropic half-space which is arbitrary orientated in the crystallophysic coordinate system can be determined numerically from the simultaneous solution of the differential equation system. These equations are: motion equations, Maxwell equations and appropriate mechanical and electrical boundary conditions including constituent material relations. The consideration of this problem is being given in the spatial (Euler) coordinate system. The assumption of the infinitesimal strains produced by the SAW is the main argument to use this approach. It seems to be natural to determine the thermal sensitivity by calculating the SAW characteristics at different temperatures as is the case with the bulk waves, when the substrate material constants can be approximated as a power series in temperature [3]. Yet, even for the case of bulk wave devices the discrepancies between the calculated and experimental results [4,5] stimulated subsequent theoretical investigations. They were aimed at developing more precise numerical technique on the temperature sensitivity determination with due account of, for instance, thermally induced static strains which change essentially the dynamic behavior of the piezoelectric media. In this case the general problem of the determination of the SAW propagation characteristics demands the solution of the equations for nonlinear solid state mechanics, i.e. the introduction of material (Lagrange) coordinate system with the subsequent transformation from the material to space frame description. An initial nonlinear system was linearized [6,7], and the values of material constants introduced into the equation coefficients became'the functions of the static biasing field. In particular, thermally induced strains can serve as such field.
The main objective of the present work is the study of the SA W thermosensitivity for the rotated Y -cut quartz substrates, also to make a comparison of the results measured and calculated, and an identification of new SA W orientations having linear frequency/temperature behavior. An analysis of the potentialities inherent in the results obtained for practical use in the temperature sensor design is given.
An information about the experimental TeD is contained in either phase or frequency response of the SAW device. Measurements of the TeD have been carried out with an automatic system incorporating a SAW delay line oscillator for frequency control. The experimental data processing followed, provides the
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 569
frequency and TCD dependencies grafically in the convenient form. The experimental setup used works in the "CAMAC" standard. The substrate with a SA W delay line mounted on the massive copper holder is placed into a closed heated camera that allows us to keep the minimal temperature deviations all over the substrate surface. A precalibrated semiconductor diode serves for temperature registration in the 25" C to 120" C region. An average rate of the temperature variation is about 1" C/min.
Figure 1.a,b shows the experimental TCD values plotted for the two different propagation directions in ST --cut quartz. In the first step the verification of the experimental technique employed was made by measuring the material with the well known properties, such as ST,X-quartz (Figure La). Furthermore, a comparison of the measured and calculated results on the TCD was performed. The calculated values are given with and without taking into account the static thermally induced strains. A known Q-matrix method, described in detail [8] was applied when calculating SAW characteristics. It is important to note that the temperature deviations of the substrate were included by introducing "effective" material constants involving thermally induced static strains. As it is seen in view of this approach, Figure 1.a,b show a satisfactory fit of the experimental TCD values to calculated values with regard to thermally induced strains. The maximal deviation of the absolute values over the whole temperature range does not exceed 1.2 x 10-6(" C)-I. By virtue of the fact that thermosensitivity measurements were made at frequencies near 200 MHz it was expected to observe an effect of contribution on the part of the damaged subsurface layer and aluminum electroded regions, as also impurities and inclusions adsorbed on the substrate surface. It is likely that difference in the theoretical and experimental curve slopes up to 20% can be explained on the assumptions discussed above.
We investigated more carefully the SAW characteristics for ST-quartz in the range of propagation directions making angles , of 26" - 52" with the X-axis (Figure 2,3). The existence of particular propagation directions was revealed with zero first order temperature coefficient of the delay time (TCDl) at room temperature and second order coefficient (TCD2) less by half of that for commonly used ST,X-quartz. One such direction is at ,= 47.3" (Figure 3). Evidently, this fact is important and beneficial for the high-stable SA W-oscillator design. However, this orientation exhibits nonzero power flow angle and the electromechanical coupling coefficient (k2) about 16% less than that of ST,X-quartz. From TCDI = TCDl( ,) curve one can expect the existence of similar propagation directions in the , angle region from 15 to 20 degrees but calculations show that k2 value becomes very low at ,< 20".
Figure 1.
570
o T 100 ·c
_" ,. .......... ....... 5 .......... .. ........... .. , ............ . r-~~--------~----~-
Experimental (2,5) and calculated (1,3,4) TCD values in a-quartz: a) ST,X. 1 (1.2 x 10-6 + 57.5 x 1O-9T);
2 (0.6 x 10-6 + 42.5 x 1O-9T); b) ST,X + 4l.5". 3 (-16.2 x 10-6 + 73.7 x 1O-9T) without
thermally induced strains; 4 (-16.7 x 10-6 - 2.5 x 10-9T) with strains included; 5 (-15.5 x 10-6 - 3.4 x 10-9T).
Figure 2.
m/s V
33 50
4
B OO 2
.. 3250 T
30 40 SO deg
Phase velocity (V) and its fractional chan~e (DV IV) for ST-quartz versus SAW propagation direction (1) .
At angles 1 = 34.8' and 1 = 42.5' the TCD2 value becomes close to zero and, hence, a linear frequency/temperature behavior takes place which is important for the temperature sensor design. The TCD1 calculated value is equal to 21.3 x 10-6(' C) -I for the first direction and to 13.7 x 10-6(' C)-I for the second one (Figure 3) . The experimental values of the phase velocity, TCD1 and TCD2 are marked with dots in Figures 2, 3. A good agreement between experimental and computed data allowed us to calculate the temperature sensitivity in the wide range of rotated Y -cuts in quartz .
These results are shown in Figure 4. The TCD1(,8,1) and TCD2(,8,1) as functions of orientation (,8) and propagation directions (1) are represented here in the form of two three-dimensional loci . Also we have indicated the intersection line of the TCD2 loci and TCD2 = 0 plane and its projection on the TCDl loci. From examination of Figure 4 a set of angles ,8 and 1 best suitable for the selection of the cuts and propagation directions with a linear frequency /temperature relationship can be found .
It is evident that with ,8 decreasing, the SAW thermosensitivity becomes higher. At the same time the errors due to the deviation of the substrate orientation increase too. The error estimations are: for ST-cut an unaccuracy in ,8 PM 0.5' changes the TCD1 by 0.5 x 10-6(' C)-I at 1 = 34.8' and by 1.0 x 10-6 at 1 = 42.5' ; the 1 deviation of the same order changes TCD1 by 1.0 x 10-6(' C)-I and by 4.0 x 10-6( ' C)-I, respectively. The largest deviation of the TCD2 does not exceed 3.6 x10-9( ' C)-2 for 1 = 42.5' . The orientation revealed with ,8 = 20', 1 = 34.8' , TCD1 = 41.4 x 10-6(' C) -I and TCD2 = 0 requires more tighter angular tolerance. It has ATCD1 = 1.8 x 10 -6(' C)-I and ATCD1 = 6.0 x 10-6(' C)-I due to the orientation error in ,8 and 7, respecti vely.
Figure 3. TCDl , TCD2 for ST-quartz versus SAW propagation direction.
571
T(O ppm 20 TC02ppb
Figure 4. TCD1, TCD2 as functions of orientation ((3,1) for Y-cuts in quartz.
We have briefly reviewed our recent results on SAW temperature sensitivity for a number of new rotated Y-cuts in quartz with a high temperature resolution and a linear frequency/temperature behavior. In conclusion, we underline the importance of keeping in mind the thermally induced static strain effects when calculating SAW thermosensitivity.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
572
G.K. Montress, T.E. Parker, M.J. Loboda, J.A. Greer, Extremely low-phase-noise SAW resonators and oscillators : design and performance, IEEE Trans . on Ultrason., Ferroel. and Freq. Contr., UFFC-35 :657, (1988). D. Hauden, G. Jaillet, R. Coquerel, Temperature sensor using SAW delay line, 1981 IEEE Ultrason. Symp. Proc., Chicago, p.148 , (1981). P. Bechmann, A.D. Ballato, T.J. Lukaszek, Higher order temperature coefficients of the elastic stiffness and compliancies of alpha--quartz, Proc. IRE, 50:1812, (1962). A. Kahan, Elastic constants of quartz and their temperature coefficient , Proc. 36-th Ann. Symp. Frequency Control, p.159, (1982) . C.K . Hruska, On the locus of zeros of the first order temperature coefficient of frequency for thickness mode C of a--quartz plate, IEEE Trans. on Sonics and Ultrasonics, SU-30:324, (1983). B.K. Sinha, H.F. Tiersten, First temperature derivatives of the fundamental elastic constants, Journ. Appl. Phys., 50:2732, (1978) . B.K . Sinha, H.F. Tiersten, On the temperature dependence of the velocity of surface waves in quartz, Journ. Appl. Phys., 51:4659, (1980) . N.Z. Ljachov , R.M. Taziev, An improved algorithm for the SAW parameters calculation in the anisotropic media, Deponent paper, N 5235-84, Novosibirsk (in Russian) , (1984).
COMPRESSION ELASTIC WAVE VELOCITY AND ABSORPTION
MEASURED BY ELECTRlCAL IMPEDANCEMETRY
ABSTRACT
Jean Pouliquen, Jean-Michel Duriez
Faculte libre des sciences Laboratoire d'acoustique-ultrasons (U .A. 253) 13 rue de Toul 59046 Lille Cedex, France
We present a method to characterize acoustically absorbing media using only electrical z measurement in C.W. on a delay line consisting of a piezocerarnic bounded to the sample.
The impedance z is measured whether the end line is in air or in water. The compression wave velocity c and absorption coefficient (3 are given without any transducer hypothesis.
For a perfect transducer all the characteristics are measured. This method is verified by a simulation.
INTRODUCTION
Velocimetry measurements using pulse superposition methods are sometimes impossible: the phase displacement due to reflexion between the transducer and the sample cannot be found if the medium is dispersive. [1 tot 8]
The acoustic characterization (velocity c and absorption coefficient (3) of middle losses isotropic media ((3 ~ 30m -1) has previously [9] been carried out by an impedancemetry method. The transducer which stimulates the sample had
i
n: c
Figure 1. Equivalent circuit transducer around its resonance F o.
Physical Acoustics. Edited by O. Leroy and ~1.A. Breazeale Plenum Press. New York. 1991 573
( L )
Sample (c.{J)
Transducer
Figure 2. Acoustical delay line made with the sample.
to be "perfect" around its resonance. Moreover, its electromechanical coefficient N2 had to be known.
In this paper we propose another method derived from the former where no transducer hypothesis is necessary. It gives the N2 value and, if the transducer is "perfect", all its characteristics.
We illustrate this method by a simulation.
METHOD REPRESENTATION
Around its resonance frequency Fr a perfect transducer can be shown as in Figure 1. The admittance diagram of which is a circle (Kennely diagram). For an im1?erfect transducer the motional branch ab will be zt(F) instead of (lw - l/wc) and the admittance diagram will be any curve (w = 27fF) [9}
When the transducer is bounded on the acoustical line (Figure 2) its electric impedance is given by Figure 3 where Zl is the electrical impedance introduced by the acoustical line in the motional branch. [10]
Zl = SZl/2N2 (1)
where Zl is the line acoustical impedance and S the transducer surface. The line is either in air or in water this impedance is written by Zla or
Zlw. In the latter case, the acoustic wave must be totally absorbed in water. The motional impedance is either
Zma = Zt + Zla (2) or
(3)
( a)
(b)
Figure 3. Equivalent circuit transducer bounded on the sample.
574
by subtracting (3) from (2) we have:
(4)
Referring to [11] we show:
Zl = {b2 - a 2 + 2jab sin 2kL} / {b 2 + a2 + 2ab cos 2kL} (5)
if b = exp (2,BL) and a = (Zo - Zw)/(Zo + Zw) L is the line length, a the real part of reflexion coefficient (sample water) k the acoustical wave number, Zo and Zw the characteristic impedances either the sample or the water.
We suppose plane waves in the sample, so the transducer diameter must be sufficient versus the wavelength.
For the line in air Zw ~ 0 and a = 1; equation 5 gives Z la, if Zw is characteristic impedance of water ( poco: Po the water mass density, Co the compression wave velocity in water).
In the complex plane (X,Y) Zr (or Zr) draws a loop R (Figure 4) which is osculatory to the circle 'Tr in r I and r 2, which are two x axis points.
The co-ordinates of the center Care
(Xo = Zo {b 2S(I-a2)} / {N2(bL l)(b L a2)} , Ye 0) (6)
and its ray
r = Zo {b S (b 2+a) (I-a)} / {N2 (b L l)(b L a 2)} (7)
The wave velocity c is given by
c = 4 L OF pip (8)
where OF p is the frequency deviation between the two points for p half-round on the loop.
The damping coefficient
(3 = (1/2L) LOg{(T(l+a) + ,; (T2 (1+a)-4a)) / 2}
if T = r/Xe
r l and
(9)
At last the electromechanical N is derived from either (6) or (7) or Xrl' r l abscissa
N2 = Zo b S (a-I) I (Xrl(b+l)(b+a)) (10)
If the transducer is "perfect", the former method [9] could be used to calculate the other characteristics since N is known.
Table 1
Fmin Fmax of Fr Co 1 N Zo Zw B L kHz kHz Hz kHz nF "H N/V (*) (* ) m- 1 m
a 180 210 30 200 1 90 6,5 2.48 1.488 10 0.1
b 190 210 20 200 1 90 6,5 2.48 1.488 10 0.1
(*) kg.m-2.s-1.10B
575
y 100 <II IIWlI' "'0 , 00 '
80 60 ~0
213 o
-213
I"PCORNCC : ll
--R
, •• ..•.• \:" R
-40 KHI
-60 - 80
-100, =~""""".....,,I -- CSl~-<Sl~<Sl~X
I:C /II H'I , · S . · Q.OOI
= 'T C\J C\J~' <.n co ~ C\J I I I 1
Figure 4, The R loop (---) and the oscula tory circle (- - -) .
MEASUREMENT SIMULATION
From the values given in Table 1, equations (1) and (5) give Zilla and Zmw, Then, from Co, we calculate the impedance values Zca and Zcw corresponding to the end line either in air or in water. (Figure 5,a and Figure 5, b),
These impedance value Zilla and Zcw calculated versus the frequency around the resonance form the stimulated measures.
EXPLOITATION OF SIMULATED MEASURES
For a perfect transducer
In subtracting a value of Co from l/zca the spiral inclination (the at
a4 slope) changes. The exact value of Co renders ata4 vertical. It is the way to determine It (Figures 6,a and 6,b),
Then we calculate Zilla and Zmw and Eq,(4) gives Zr (Figure 7), At last, equations (8), (9) and (10) yield the results. Table 2 shows a
perfect restoration if the frequency deviation F between two neighboring measure points is small enough (Figure 7),
y 15 Ohms IMPEO RNCE : zea II (a) o{t-2IO , 000 kHl
7
3 - I
-5 - 9
-IJ -17 -2 1 c:t,. _ tOO , 000 kHl Ohms -2 5 (9 C\J ,,·-lo-cO~ N <T U) co CSl X
.- C\J ('\J I"\J N C'\J (T)
y Ohms (b)
-13 -17 -2 I
IMPEDRNCE: le w
210 , 000 kH l
Ohms 160 , 000 kHl
-25 CSl N OJ' lD co CSlNOJ'U)COC:ilX N N N N (\J (Tl
Figure 5, (a) The ceramic impedance for the end line in air. (b) The ceramic impedance for the end line in water.
576
Table 2
mode l Restored values 0.1% (a)
c(m/s) 2000 1992-2004 a(m- l.) 10 9.995- 10 . 008
Zo (* ) 2 . 48 2 . 47 - 2 . 485
N (N/V) 6.5 6 . 4958- q . 4982
f(kHz) 200 200 . 07
y I MPE DRNCE : z ma 15 Oh ms
': (~ ~ -7-- (.,----5 ~ -9
-13 -17 -21 -25 . ___ --.!..!2~0 kHz
c;> N .,. lO OJ ($j N ..,. cD -- - - - N N N N
Ohms
OJ iSl X N M
0 . 1% (b)
2000 10.0026-
10 . 0065 2.48
6 . 4955-6.4977
199.98
0.01% (b)
2000 10 . 0034-
10.0037 2 . 48
6 . 4975-6 . 4977
199 . 98
IMPEDR NCE :zmw
210 . 000 kHl
Figure 6. (a) The motional impedance for the end line in air. (b) The motional impedance for the end line in water .
y
5 Ohms iMPEDANC E :zr
--1---' /86C1 . ClOO k11 915 . 000
_<1 L Ohms
-5 ---1-----. X. vM N ~ ~ - N M V ~ W I I I I
kH l
Figure 7. The res tored loop from the simulated measures.
577
578
ROMITTAIK E
kHl
kHl
51 cmcf'\S • ,0G
~~~=-~~~~-UO-~-ci>U~
~- C\J ,., ~ ~ Ul ,... OJ m .-
Figure 8. An admittance curve of an imperfect transducer .
y
G. e Ohms IMPEDRNCE:zr
(0-0.85 n(
(oc-l . 05 n(
-2." -J . 6
- ~ . 8
-6 . 0~m~~~~~~N~'~N~~v~~~m~~ x . .. V"ifP) rv - I - N M .,. '" " " ,
Figure 9. The restored R curve is a spiral with an inexact Co value.
y Ohms IMPEDRNCE : Zr
~ r:IJ~cu_ , . ? n) 1 nl
3 (oc- I. 2020 nl
2 I I 8bO . 000 kHz
I -= ~I ~900 . 0,00 kHz
- ~ Ohms -5 "',., (\J _ CSl - C\J ,., " ~ I.D X
I I I I
Figure 10. R is a loop with an exact Co value.
Table 3
model Restored values 0.1% 0.01%
c(m/s) 2000 2000 2000 ~(m-l.) 10 9.998-10.01 10.002-10.009 Zo (*) 2.48 2.48 2.48
N (N/V) 6.5 6.492-6.498 6.492-6.496
(*) kg.m-2.s-l..10S
For an imperfect transducer
The measurement admittance l/zt(f) of a real transducer is given in Figure 8; it is not a circle.
We start the previous process again. Starting from an inexact Co value, the calculated Coe is different from the Co value and the R loop is a spiral (Figure 9). The spiral whorls change directions whether the Co value is either superior or inferior to the true value. When the true value is found, R is a loop (Figure 10) and the restored values are given in Table 3.
CONCLUSION
The method which is here explained is able to characterize absorbing media by only electrical measurement without knowing the transducer characteristics if fl s 30 m-1.
We have to remind that the compression wave must be absorbed at the water delay line end.
In the hypothesis of a perfect transducer ( Yt = l/zt is a circle), its characteristics can be calculated. They are obtained in its working conditions and not with an un-installed ceramic.
REFERENCES
1 2 3 4 5 6 7 8
[9]
i~l
Willams J. and Lamb J., J.A.S.A., 30, 308, 1958. Mac Skimin H.J., J.A.S.A., 22, 413, 1950. Mac Skimin H.J., J.A.S.A., 33, 12, 1961. Mac Skimin H.J., J.A.S.A., 34, 609, 1962. Mac Skimin H.J., J.A.S.A., 37, 325, 1965. Papadakis E.P., J.A.S.A., 42, 1045, 1967. Matec operating manuel. Defebvre A., Pouliquen J., Chastagner M., Revue d'acoustique, 45, 71-76, 1978. Pouliquen J., Defebvre A., 13th !CA, 4, 305-308, 1989. Warren P. Masson Physical acoustics Vol. 1 part A, 238-244. Mercier J., Acoustique P.U.F., 69-73.
579
ISOLATION OF RESONANCES OF A CYLINDRlCAL TARGET IMMERSED IN WATER BY MEANS OF A NEW METHOD USING PHASE INFORMATION
P. Rembert, P . Pareige, O. Lenoir, J .L. Zuber, J .L. Izbicki and G. Maze
Laboratoire d' Acoustique Ultrasonore et d'Electronique U.R.A. C.N.R.S . 1373, Universite du Havre Place Robert Schuman, 76610 Le Havre, France
INTRODUCTION
Every classical method which allows the isolation of resonances of cylindrical targets only involves the modulus of the complex number provided by the Fast Fourier Transform of a part of the backscattered signal. In this paper, the authors propose a novel experimental method which takes into account another aspect of this complex number. The ratio of the imaginary part (1m) to the real part (Re) of the F.F.T. processed on the whole backscattered signal , is used for the isolation of the target resonance frequencies. It is cal.led the Im/Re method [1,2].
TRANSMITTBI I RECEIVER TRANSDUCER
11I---(((eTARGET
PULSE GENERATOR
NUMERIC
trigg.r
WAVEfORM GENERATOR
OSCillOSCOPE trigg.r
BUS INTERfACE
NUMERIC PLOTTER
Short Pulse M.!.tR &
COMPUTER
Im/ Re me thod
Figure l. Experimental set-up .
Physical Acoustics, Edited b y O. Leroy a nd M. A. Breazeale Plenum Press, New York , 1991 581
full scale 20011.1
Figure 2. Aluminum tube (b/a = 0.9): backscattered signal.
The isolation of resonances of cylindrical targets can be performed by means of different methods such as the quasi-harmonic Method of Isolation and Identification of Resonances f31 (M.I.I.R.), or the short pulse M.I .I.R., involving a short emitting pulse [4,5. For several kinds of infinite cylindrical targets immersed in water, experimental results provided by means of the short pulse M.I.I .R. with those obtained with the help of the new ImlRe method.
EXPERIMENTAL CONDITIONS
In both the short pulse M.I.I.R. and the ImlRe method, the isolation of resonances of a cylindrical shell is provided by means of a monostatic method.
Shorl Pulse M.I.I.R
1:0
1:0
1:0
500 1000 kHz
Figure 3. Aluminum tube (b/a 0.9): backscattered power spectrum.
582
The same experimental set-up (Figure 1) is used for the two methods. A pulse generator sends a short pulse (less than 300 nanoseconds) to a wideband transducer which insonifies the target. The incident mean beam lies on the horizontal plane which is perpendicular to the shell axis. The signal backscattered from the target is received by the same transducer and transmitted to a numeric oscilloscope. The digitized signal can be sent to a numeric plotter and to a microcomputer via an I.E.E.E. interface bus . The oscilloscope can perform several waveform processings such as: sum averaging, arithmetics (for instance ratio), F.F.T. power and phase spectra, imaginary and real part spectra. Further processings are performed by the microcomputer.
COMP ARISON BETWEEN THE SHORT PULSE M.I.I.R. AND THE ImjRe METHOD
An Infinite Aluminum Tube
We first investigate an elastic cylindrical target. A 3.7 mm radius aluminum shell (bja == 0.9) is insonified by a wideband transducer of 1 MHz central frequency . The signal which is backscattered from the tube (Figure 2), is composed of two parts: the specular echo and a tail which is the elastic response of the target . The F.F.T. power spectrum of the whole signal exhibits minima (Figure 3). The wide minima are related to the propagation of interface waves of the Scholte-Stonely family [6] (1 == 0); the sharp minima are due to the propagation of circumferential waves of the Whispering Gallery family (1 == 2,3 .. ). The isolation of resonances of the shell is performed from the plot of the power spectrum of its free elastic temporal response (the rigid echo or more is eliminated and replaced by an average amplitude signal). Wide peaks mainly occur on the resonance spectrum (Figure 4), related to the Scholte family. Sharp peaks are also detected; they are associated to the reemission of Whispering Gallery waves.
Preliminary Considerations about the Im/Re Spectrum [7]. In order to obtain the ImjRe spectrum, the F.F.T. is performed on the whole signal and
ShOll Pulse M.I.I.R
1: 0
1: 0
500
Figure 4. Aluminum tube (bja
1000
0 : 5
1: 2
0.9): resonance power spectrum.
583
2) (2) (2)
(1)
u I~!: I,! 111 (
1m / Re method
500
Figure 5. Aluminum tube (b/a
1000 kHz
0.9): Im/Re spectrum.
the ratio of its imaginary to its real part is plotted . The resonances of the target are detected on the curve owing to more or less sharp transitions. When performing an Im/Re spectrum (Figure 5) on the whole backscattered signal, several transitions occur on the curve. Most of them are located at resonance frequencies of the target (1); other regularly spaced infinite transitions (2) cannot be attributed to the resonant response of this target. When we plot the Im/Re spectrum (Figure 7) of the isolated specular echo (Figure 6), the unexplained transitions only remain. In fact, if we note the time delay ~ t 0
between the beginning of the temporal window (t = 0) and the start of the head signal, and ~N 0 the average frequency interval between two unexplained transitions on the related Im/Re spectrum, a simple law binds these two variables:
584
~to = (1/2 ~N 0) + K,
Figure 6.
full scale 200,1I~
Aluminum tube (b/a = 0.9): isolated specular echo from the backscattered signal (Figure 2).
Iml Re method
~l (2) 121
(2)
500 1000 kHz
Figure 7. Aluminum tube (b/a = 0.9) : Im/Re spectrum of the specular echo (Figure 6).
where K is an arbitrary constant which depends on the frequency band of the used transducer.
Then, in order to reject these transitions out of the studied frequency domain, the whole temporal signal must be translated of -LHo-K. The oscilloscope LeCROY 9400 only processes on the displayed trace (t ~ 0), and it appears that a part of the specular echo clearly shifts toward the negative time domain, when the translation is performed. The truncation of the head-signal is avoided by reporting the part of the specular echo which should have been eliminated , at the end of the temporal window (Figure 8): in other words , a II circular permutation II is performed on the whole signal into the temporal window. Experimentally, we notice that the signal processing operation we have done in order to eliminate non-resonant transitions always leads to a quasi-symmetrisation of the specular echo in the case of elastic targets. The corresponding Im/Re spectrum (Figure 9) is non-resonant transitions free .
fu ll scale 200JI.I
Figure 8. Aluminum tube (b/a 0.9): permuted backscattered signal.
585
Figure 9.
Im / Re method
., o
500 1000 kHz
Aluminum tube (b/a = 0.9): Im/Re spectrum of the permuted signal (Figure 8).
In the following experiments , a circular permutation will be performed on the temporal signal when the non-resonant transitions disturb the isolation of resonance frequencies of the target.
On the Im/Re spectrum of the permuted entire signal backscattered from the tube (Figure 9), all the resonance frequencies detected with the help of the short pulse M.I.I.R. are isolated (there is a good agreement between resonance frequencies of the cylinder pointed out on the Im/Re spectrum and those isolated on the resonance spectrum). So the Im/Re spectrum also allows to isolate resonances of a cylindrical elastic target.
A Viscoelastic Cylindrical Target
In order to test the efficacy of the Im/Re method on a viscoelastic cylindrical shell [8], a PVC tube is investigated . The tube is insonified by a transducer of 500 kHz central frequency. The signal is composed of a specular
,- 0 .. , , iii ..
0
full scale 300,uJI
Figure 10. PVC tube: backscattered signal.
586
Short Pulse M.I.I.R
full scale 1 MHz
Figure 11 . PVC tube: backscattered spectrum.
echo, and the free elastic response (A) of the target with little energy reemitted. We can remark two small oscillations in the free elastic response target (Figure 10): the average frequency of each oscillation is respectively 320 kHz and 535 kHz. The backscattering spectrum (Figure 11) exhibits low amplitude undulations with different frequencies on both side of 500 kHz: the frequency interval between two minima is of 50 kHz on the left side of 500 kHz and of 27 kHz on the right side. The power spectrum of the (A) part of the signal (Figure 12) exhibits two main wide peaks. The central frequency of each peak corresponds to the average frequency of each oscillation on the temporal signal. This assumption is confirmed by the power spectrum of the only (B) part of the signal (Figure 13): only the right peak remains.
The Im/Re spectrum of the whole signal backscattered from the PVC tube exhibits two average levels (Figure 14). The central frequency of each level corresponds to the frequency of each wide peak of the (A) resonance spectrum. On the left level, transitions are pointed with a 50 kHz periodicity; on the right side level the frequency interval between two transitions is of 27 kHz; the same phenomena have been remarked on the backscattered spectrum. The Im/Re spectrum still provides as much information as both the backscattered and resonance spectra. This method is efficacious on viscoelastic targets.
Short Pulse MJ.l.R
full scale 1 MHz
Figure 12. PVC tube: power spectrum of the (A) part of the signal.
587
Short Pulse M.I.l.R 535 kHz
full stile 1 MHz
Figure 13. PVC tube: power spectrum of the (B) part of the signal.
'l7
50kHz
Im / Re method
full scale 1 MHz
Figure 14. PVC tube: Im/Re spectrum.
fu ll scale 200 14.1
Figure 15. Plaster cylinder: backscattered signal.
Short Pulse M.I.I.R
Figure 16. Plaster cylinder: backscattered spectrum.
588
full scale 1 MHz
Figure 17. Plaster cylinder: resonance spectrum.
Im/ Re method
Figure 18. Plaster cylinder: Im/Re spectrum.
A Non-resonant Cylindrical Target
Finally, a plaster cylinder has been investigated in order to verify the results obtained with this method on an "a priori non-resonant II cylindrical target. As the backscattered (Figure 15) signal exhibits no elastic response, the backscattered and resonance spectra allow no resonances to be clearly detected (Figure 16,17). The Im/Re spectrum (Figure 18) of the whole signal exhibits a flat level. This achieves to confirm that no resonances are excited in the cylinder. This method clearly shows the non-resonant aspect of a cylindrical shell built in a non-resonant material.
CONCLUSION
The new Im/Re method is as efficient as the classical ones to detect resonances of elastic or viscoelastic cylindrical targets. O. Lenoir et al. show [1,2] (these proceedings) that resonance frequencies of an elastic plate coated with an absorbent can be isolated by means of the Im/Re method. We base our hopes on this method to detect resonances of a cylindrical elastic shell coated with an absorbent, when the classical methods are bound to fail. In order to detect the resonances with the Im/Re method, the entire signal has to be processed. This may be interesting for the study of multilayered media. The problem of the temporal separation of the resonant contributions of the reflected signal ( where is the resonant tail for a multilayered medium reflection ? ) could be solved by the Im/Re method.
REFERENCES
[1]
[2]
[3]
P. Rembert, O. Lenoir, J.L. Izbicki, G. Maze, Experimental analysis of phase spectrum of cylindrical or plane targets: a new global method of isolation of resonances, Phys. Lett., 143A:467, (1990). O. Lenoir, J.L. Izbicki, P. Rembert, P. Pareige, The Im/Re method: A new experimental determination of resonances, J. Acoust. Soc. Am., 87 (Suppl. I), S86, (1990). J.L. Izbicki, G. Maze, J. Ripoche, Influence of the free modes of vibration on the acoustic scattering of a circular cylindrical shell, J. Acoust. Soc. Am., 80:1215, (1986).
589
[4]
[5]
[6]
[7]
[8]
590
P. Pareige, P. Rembert, J.L. Izbicki, G. Maze, J. Ripoche, Methode Impulsionnelle Numerisee (MIN) pour l'isolement et l'identification des resonances de tubes immerges, Phys. Lett., 135A:146, (1989). G. Quentin, A. Cand, Pulsed resonance identification method, Electronics Lett., 25:353, (1989). A. Gerard, J.L. Rousselot, J.L. Izbicki, G. Maze, J. Ripoche, Resonances d'ondes d'interface de coques cylindriques minces immergees: determination et interpretation, Rev. Phys. Appl., 23(3):289, (1988). N. Mercier, D. Lecuru, J.F. de Belleval, Probh~mes poses par l'utilisation de la phase en spectroscopie ultrasonore, Rev. Acoust., 64:38, (1983). V.M. Ayres, G.C. Gaunaurd, Acoustic scattering by viscoelastic objects, J. Acoust. Soc. Am., 81:301, (1987).
ACOUSTIC EMISSION CHARACTERISTICS OF SCHISTS AND
SANDSTONES
M. C. Reymond, J. Fr. Thimus*, Ph. Linze*
L.C.P.C . CNRS Paris, France *Universite Catholique de Louvain
Civil Engineering Department Place du Levant, 1 B-1348 Louvain-Ia-Neuve, Belgium
INTRODUCTION
Acoustic emission designates the spontaneous setting up of mechanical waves inside a stressed material. This phenomenon occurs during the discontinuous release of stored energy in the material due to the brutal and definitive change of this material in the microscopic scale (decohesion at the interface between grains) or in the macroscopic scale (crack propagation). The reception and analysis of acoustic emission are very efficient techniques for the study of rocks behavior under different kinds of sollicitations.
Very various materials are able to produce acoustic emission: metals, ceramics, composites, concrete, rocks, ... It is well known that the ultimate state of failure for a rock occurs with a sound . The release, at this moment, is very important and therefore, the noise emitted by the rock is clearly audible. Before this complete fai lure, the stressed rock can produce lower amplitude elastic waves only detected by very sensitive sensors.
Laboratory tests have been performed aiming the study of the possibilities of acoustic emission in geological research. The different phases of cracks evolution of rocks samples have been tested to the failure . Similar tests performed by other laboratories [1,2,3,4] have shown a possible behavior analogy between laboratory and on site.
To find out a good definition for this analogy forms the topic of the present study.
~»)~ •
0: sample a : piezoeleClric accelerometer b : preamplifier c : bandpass filter d: amplifier e : recorder f : magnetic tape 9 : frequency analysis
Figure 1. System for recording and analysis of emission data.
Physical A coustics, Edited b y o. Leroy and M.A. Breazea le Plenum Press, New York, 1991 591
EXPERIMENTAL DEVICES
The first step of the study consists is emphasizing by acoustic ermSSlOn the different stages of the evolution of a rock showing a brittle failure (sandstones, schists, ... ) under uniaxial stresses until the failure of the sample.
The experimental device to apply the vertical load on the sample is an hydraulic jack handy set in motion to reduce parasite sounds which could interfere with sample information.
The axial strain is measured by mechanical deformeter along two generating lines at 180 0 on the cylindrical sample.
For the acoustic emission, an accelerometer with a sensibility of 20 mvolt/g, with a linear response between 2 Hz and 20 kHz is connected to a preamplifier. After signal filtration and amplification, the signals amplitudes above a chosen door-step are computed at regular time intervals and stored versus sollicitation level (recorder).
A magnetic tape unit (20 Hz to 20 kHz) is connected to the amplifier, which allows to perform a deffered spectral analysis of signal versus sollicitation level (Figure 1).
The sollicitation is applied on the sample with a continuously increase of load (100-200 N/sec) until the failure. Regularly, stages of constant stress are maintained for a few minutes, allowing the data acquisition. With this procedure, the cracks can be taken form yet before the failure of the sample for a stress O"f.
RESULT ANALYSIS
The results of uniaxial compression tests (stress versus axial strain) are shown on the Figure 2 for schist, sandstone and siltstone. Different acoustic techniques (frequency analysis, summation and histogram of the number of recorded signals (N s) versus sollicitation values) have shown different steps of sample damage.
a (t.lPa)
60
50
40
30
20
10
0 0
592
0. 1 0.2
• SILTSTONE o SANOSTONE ~ SCHIST
Figure 2. Results of uniaxial compression test.
Relative amplilude
(dB)
20
10
o 0. 16
(dB)
20
10
o 0.2
10
o
Spectral Analysis
20% of 01
40% 01 0,
0.63 8
61% of Of
0.4 2 5 8 20 kHz
Figure 3. Spectral analysis for sandstone.
Opposite to the three types of rock (sandstone, siltstone and schist), an assumption can be made on cracking mechanisms (Figure 3).
- for u ~ 20% O"f: prevalence of acoustic emission at lower frequencies which are the expression of microscopic events with low energy distributed in all the sample volume but statistically more frequent for lower loadings
- for u ~ 40% 0"[: increase of acoustic emission at higher frequencies due to the extension of macroscopic events such cracks propagation between or through rocks grain : how higher the stress-level how greater the amount of ener,gy required. This observation has also been made by other scientists ll,2,3,4] on different materials (steel, concrete, composite, .. . ). We can thus assume an analogy between these different samples.
- for u ~ 60% 0"[, brutal decrease of frequency which can be assumed as a criterion of a next failure of the sample and, by extrapolation, a next failure on site.
593
Table 1
Type of rock Strergth % of strergth Predaninating (MPa) frequercies
(kHz)
Schist 78 16 0 .5 to l.2 33 20 66 0 .8
Siltst~ 32 5 20 95 0.5
Sardst~ 63 20 0.6 50 12.5 to 20 60 1
Depending on the rock type, different limits of cracking can be observed (Table 1):
for schists: spectral slipping of lower frequencies (16% ITf) to higher frequencies (between 25 to 41% ITf) , average 70% ITf, fall of predominating frequency
for siltstones: the samples were initially cracked; it is observed only a fall of higher frequencies just before the failure of the sample
for sandstones: for IT > 50% ITf, a spectral enrichment can be seen. The fall of higher frequencies appears for a load level between 60 and 62% of failure .
Analysis of Stress-Strain Curve according to the Number and Histogram of counted Events
A good correlation generally exists between these three parameters , appearing by change of curve slope and by spectral evolution.
C1 (l04P.l
SCHIST (C1,e: ) CC1,LNs )
(FreQUenCY analysis)
60 C1f : 78 MPa
50 ---<> --<> ~
M ~ ~kHz 0 40
ee)
30
---<> kHz 20
Cb) 10
Ca) '" 0 e:1'"
0.05 0. 10 0. 15 0.20 0.25
, I I " LN. 0 1000 2000 3000 4000
Figure 4. Stress-strain curve, number of events and spectral analysis .
594
Acoustic emission : number of events
Ns
5000
4000
3000
2000
1000
200
~ LOADING
n STAGE
5
Load (kN)
10 15
10 15
20
Time (min)
Time (min)
20
Figure 5. Histogram of events for a sandstone.
595
596
Acoustic emission : number of events
Ns
10000
8000
6000
4000
2000
Load (kN)
100
~ LOADING
n STAGE
20 Time (min)
Time (min)
5 10 15 20
Figure 6. Histogram of events for a siltstone.
Acoustic emission: number of events
LNs
6000 1 I I I
J • SILTSTONE
'~~ I 0 SANDSTONE
I ... SCHIST
I 4000
I i
3000
r 2000
1000
O~~-r---'----r---~--'----r--~r---.---.----r--~Load (kN) o 20 40 60 80 100 120 140 160 180 200
Figure 7. Acoustic emission: number of events versus load.
597
Four different phases can be determined (Figure 4):
- phase of sample put in place (a) - phase of microcracking (~ 30% O"f): increasing slope of the stress-strain
curve; it is assumed that decohesions appear at grain interface (b) - phase of macro cracking (> 40% (Jf): the phenomenon of plastic strain
and mechanical slipping becomes preponderant (c) - phase just before the failure (~ 70% (Jf): a fall of higher frequencies
appears with an increase of strain (change of slope for the stress-strain curve) and a deffered phenomenon which is shown in Figure 5 and 6.
These figures give the histogram of events for a sandstone or a siltstone versus time and the loading historic. A different behavior can be seen for the two types of rock:
for the sandstone, the number of events is low and the alone deffered event appears only just before the failure
for the siltstone, the number of events at each stage is more important and, for (J ~ 50% (Jf, at each stage, a deffered event can be seen (acoustic emission after the load increasing step).
Acoustic Power of Different Materials
The relation between the total number of events and the sollicitation level is shown at Figure 7: siltstones appear clearly to be more emission able than schists and sandstones.
CONCLUSION
In this study, the acoustic technique is able to predict the first step of damage. Nevertheless the laboratory study of the different types of rocks present in the rockmass to be checked is necessary to know the specific sound behavior of each material for the different loading phases.
The analogy existing between the different physical and mechanical parameters can be applied to study and predict the natural phenomenon.
REFERENCES
[1)
[2)
[3)
[4)
598
B.T. Brady, An investigation of the scale invariant properties of failure, Int. J. Rock. Mech. Min. Sci. and Geomesh, AbstL, Vo1.14, pp. 121-126, 1977. M.C. Reymond, H. Clergeot, B. Lumeau and F. Perrot, Acoustic emission in coal masonry tunnels, Proceedings third conference on Acoustic Emission Microseismic activity in geological structures and materials, Trans. Techn. Publ., Vol.8, pp. 117-130, 1984. C.M. Scholtz, The frequency magnitude relation of microfracturing in rock and its relation to Earthquakes, Bull. Seism. Soc. Am., Vo1.58, N° .1, 1968. M. Trombik and W. Zvberek, Microseismic research in polish coal mines, Proceedings first conference on Acoustic Emission Microseismic activity in geological structures and materials, Pennsylvania State University, June 75, Trans Techn. Publication, pp. 169-194, 1977.
OPTOACOUSTIC RAMAN GAIN SPECTROSCOPY OF BINARY MIXTURES
Mathias Rohr, Gabriel M. Bilmes, and Silvia E. Braslavsky
Max-Planck-Institut fiir Strahlenchemie D-4330 Miilheim a.d. Ruhr, F.R.G.
INTRODUCTION
In recent years optoacoustic and non-linear Raman techniques have become more and more popular. A combination of both, i.e. the optoacoustic detection of Raman transitions, however, has not yet been fullr explored. Few reports of so called photoacoustic Raman spectroscopy (PARS) in the gas phase show important advantages over optical detection [1,2]. Pioneer work' in the liquid phase was performed by Patel and Tam [3] but no further studies have been reported in this area. We now present new results on the application of optoacoustic Raman spectroscopy (OARS) to liquid systems.
The theory of the OAR effect is similar to that for stimulated Raman scattering (SRS) [4]. The sample is irradiated simultaneously by two lasers of different frequencies. The interaction of the so called pump beam (vp) and the Stokes beam (vs) with the molecules produces a SR effect. The raruationless relaxation of the molecules in the Raman level gives rise to a pressure signal which is detected by an optoacoustic detector. The signal (H) is described by equation 1.
!J.v H = k _R_ E~ g 1 + Hp + Hs
Vs (1)
Here E~ is the incident Stokes energy, 1 is the pathlength of both lasers in the sample, k is a proportionality constant, includin$ the thermoelastic properties of the sample and the geometrical parameters [5J. Hp and Hs are the optoacoustic absorption signals of each laser, !J.vR = vp - Vs is the
frequency of the Raman transition, g is the Raman gain coefficient and describes the nonlinear amplification of the Stokes beam as a function of the pump energy, Ep, Stokes frequency, and the imaginary part of the third-order susceptibility of the sample.
EXPERIMENT AL
The second harmonic of a Nd:Yag laser (532 nm) was used for simultaneously pumping of two dye lasers. The laser system was operated at 5 Hz with pulse widths of 8 ns and energies < 1 mJ. In order to guarantee the overlap in time in the sample the time synchronized beams were built up in a symmetric, opposite geometry [2]. The additional requirement of spatial overlap
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York. 1991 599
of pump and Stokes beams in the cuvette was easily monitored and adjusted by means of a polyvinylidenedifluoride (PVF2) film detector. Instead of the sample a calorimetric reference like CuCl2 or CoCl 2 was irradiated with the two lasers [5) . Because of the high time resolution (20 ns) an excellent space location of the evolved OA signal is obtained. Since the time difference between trigger and OA signal amplitude is proportional to the distance between laser beam and detector both beams were adjusted until only one OA signal with an amplitude equal to the sum of pump and Stokes signal is obtained . This procedure is repeated for a 90 ' turn of the detector in order to obtain a three dimensional adjustment. The OAR signal, H (equation I), was detected by a piezoelectric ceramic transducer. The data were transferred to a computer system
(PDP 11/23), where each signal was normalized to E8. For the measurement of the OAR spectrum the pump laser (vp) was fixed while the Stokes laser (vs) was tuned by a stepmotor through the Raman transition (LlvR = vp - vs)·
The Raman spectrum was calculated step by step using the difference between the positive and negative amplitudes of each signal (equation 1) . Further analysis and simulation of the data were done on a pc. For this purpose a fit program using a sum of Lorentzian functions was used to analyze the data [6) . Before use, all samples were carefully filtered through Millipore filters (0 .05 - 0.Q1 jtm).
RESULTS AND DISCUSSION
High resolution Raman spectra between 3089 and 214 cm -I were measured [7). Figure 1 shows the OAR spectrum of benzene. vp was fixed at 559 nm while Vs was tuned from 672 to 676 nm. The figure shows a main peak at LlvR = 3059 em-I. A simulation of the data with a sum of two Lorentzian
shows a second peak at Ll va = 3046 cm -I with a height of N 0.18 of the
main peak . Both are identified as CH valence stretchings. The 3059 cm -I peak
can be assigned to the v~H vibration (alg symmetry) of benzene while the
3046 cm peak is assigned to the v~~ vibration (e2g symmetry) [8). Within the range of used pump energies a linear dependence ot Raman gain on Ep was obtained.
Figure 1.
600
6
...., 5 :i ~ 4
({) 3 <{
o 2
3020 3040 3060 3080 3100
Observed OAR-spectrum of liquid benzene v~H (alg) :
VI : LlvR = 3059 cm-I, Hmax = 5.78 a .u., FWHM = 9.5 cm-I
V15 : Llva = 3045 em-I, Hmax = 1.01 a.u., FWHM = 20.3 cm-I
10 /
802 em-I 8
,.., ~ 6
" u 1267 em-I
(/) • « 0
2
0 780 800 820 1260 1280 1300
Figure 2. Observed OAR-spectrum of liquid cyclohexane: a: dVR = 802 cm-I, FWHM = 2 cm-I, SjN = 2.3
b: dVR = 1267 cm-I, FWHM = 13 cm-I, SjN = l.1
Figure 2 shows two CC-stretching bands of cyclohexane with different structures. Similar to the CC-stretchings at LlvR = 1267 cm -I with a signal to
noise ratio (SjN) of l.1, the stretching at dVR = 802 cm-I can be observed
even though the full width at half maximum (FWHM) is only 2 cm-I (Figure 2) . .
In order to determine the sensitivity we looked at a binary mixture of benzene and toluene in the region of the CH-stretching (dVR N 3060 cm-I). We
could see as little as 10-20% of benzene in toluene. Because of the broad bands no further increase of sensitivity was possible (Figure 3.1) . For that reason we
0.8 n
~ 0.6 III
u
(/) 0.4
« 0
02
0 3000 3020
Figure 3.
Benzene 1\' 0.8
0.6
0.4
0.2
0 3040 3060 3080 3100 970 986 1002 1018 1034
!;VR [cm- 1 J !;VR [cm-1J
VI = OAR spectrum (CH stretchi!lg) of benzene and toluene mixtures, a: 100% benzene, b: 80% benzene + 20% toluene, c: 60% benzene + 40% toluene, d: 40% benzene + 60% toluene, e: 20% benzene + 80% toluene, f: 100% toluene. V2 : OAR spectrum (CC stretching) of benzene and toluene mixtures, a: 10% benzene + 90% toluene, b: 5% benzene + 95% toluene, c: 1% benzene + 99% toluene.
1050
601
1.4 n .. 1.2 _ 0 --I 0 ..-
E ...-U /' u /
0 / 0.8 .... / 0
'-/ E 0.6 " . II) /
Q) 0.4 /
> /
.... ~ 2! 0.2 ...-Q) • ...-<.. ..-
0 0 20 40 60 80 100
concentra lion of cyclohexane [ mol %J
Figure 4. Relative shift of the v~H stretching of benzene as a function of cyclohexane molar fraction in a binary mixture.
shifted to the CC-transitions (benzene v~c( alg) = 991.6 cm -1, toluene
v~C(alg) = 1004 cm- I ) . The FWHM of each signal was about 2-3 cm- I . In this region we could increase the sensitivity by a factor of ca. 10 although the intensity of the signal decreased relative to that of the CH-region by a factor of about 3. Figure 3.II shows the results for a mixture of toluene and 10, 5, and 1% benzene. Further analysis of the data showed, for both the CC- and CH transition, a linear dependence of the Raman gain (rv peak maximum) with
concentration. This reflects the fact that the enthalpy of the mixture I1H~i~8 is very small [9J.
In order to investigate the influence of the interaction between molecules on the OAR-spectrum we conducted experiments with mixtures of benzene and
cyclohexane (I1H~~~12 = 836.8 Jmol- I [9]). We found an increasing shift of the CH hi f b f . f h I h f t ' v I stretc ng 0 enzene as a unctIOn 0 t e cyc 0 exane rac IOn
(Figure 4).Investigations of the v~C transition of benzene also showed a shift to larger wavenumbers . This shift was, however, smaller than the shift in the CH region. These measurements demonstrate the high spectral resolution of the OARS method, in addition to its sensitivity.
ACKNOWLEDGEMENTS
We are very grateful for the support and interest of Prof. K. Schaffner. G.M.B. was supported in part by CONICET (Consejo Nacional de Investigaciones Cientificas y Tecnicas, Argentina) and M.R. is a recipient of a fellowship from the Alfried Krupp von Bohlen und Halbach Foundation.
REFERENCES
602
J . Barrett, Appl. Spectr. Rev., 21, 419-426, (1985) . A.M. Brodnikovskii, B.P. Zharov and N.!. Koroteev, Sov. J. Quant . Electr. 15, 1600-1605, (1986) . C.K.N. Patel and A.C . Tam, Appl. Phys. Lett. 34, 760-763, (1979). J. Barrett, D.F. Heller , J. Opt. Soc . Am. 71, 1299-1308, (1981). S.E. Braslavsky, R.M . Ellul, R.G. Weiss, H. AI-Ekabi and K. Schaffner, Tetrahedron, 39, 1909-1913, (1983).
[6]
[7]
[8]
[9]
R.D.B. Fraser and E. Suzuki III Spectral Analysis: Methods and Techniques, Dekker, NY (1970). G.M. Bilmes, M. Rohr, S.E. Braslavsky in Photoacoustic and Photothermal Phenomena II, Springer-Verlag Heidelberg (1990). G. Herzberg, Molecular Spectra and Molecular Structure, II. Infrared and Raman Spectra of Polyatornic Molecules, (1968). K.S. Pitzer and L, Brewer Thermodynamics, McGraw-Hill book Company, Inc. (1961).
603
LIGHT-INDUCED VORTEX CURRENT GENERATION AS A NEW
MECHANISM OF PHOTO ACOUSTIC PHENOMENA IN SEMICONDUCTORS
V.A. Sablikov
Institute of Radio Engineering and Electronics Academy of Sciences of the U.S.S.R. Moscow, U.S.S.R.
INTRODUCTION
Photoacoustic and thermal wave phenomena in semiconductors are of interest in view of their application as a basis of non-destructive diagnostic methods in microelectronics [1). Such methods are to be local, so the investigation of photoacoustic phenomena under the local illumination is the urgent problem in this field. At present the most perspective are following local methods: photothermally modulated optical reflectance, laser beam deflection and IImirage ll
effect. In these methods the responses are determined by temperature field, deformation and electron-hole plasma density distribution around the beam. In previous paper [2) we have analyzed these factors for homogeneous semiconductor. Semiconductor structures (p-n junctions [3,4) and heterojunction [5)) were considered only in the case of one-dimensional geometry'.
Photoacoustic and thermal wave phenomena in semiconductor structures under the local illumination are investigated very weakly. The first problem in such investigation is the analysis of electron processes which are known to determine the thermal sources and mechanical strains. Recently we have shown [7) that in semiconductors with nonuniform potential shape there exists a new photoelectric phenomenon which consists in the vortex (closed-inside the sample) current generation. The vortex currents manifest in photoacoustic phenomena via their effect on the spatial distribution of carriers which results in the distribution of the recombination heat sources and electron-induced strains in lattice. Another effect of the vortex currents consists in an additional heat generation (or absorption) when the current passes the potential inhomogeneity.
The purpose of the present paper was to investigate the electron, thermal and deformation phenomena in semiconductor structure with p-n junction taking the vortex currents into account. We have shown that the vortex currents give rise to essentially new features of the thermal and deformation responses. In what follows we consider the planar pen junction (Figure 1) illuminated by Gaussian shape light beam in the centre of p+-region. The beam intensity is modulated
I(r,t) = (Poj211"a2)exp(-r2ja2)(1+coswt)
*A. Mandelis [6) made recently attempt to reexamine the photothermal response of p-n junction. Unfortunately the author ignores the cited results [3,4). In our opinion his work is erroneous.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 605
NONEQUILIBRIUM DENSITY
CARRIER CONCENTRATION AND CURRENT
Nonequilibrium charge carriers are generated by two sources: direct generation by light and injection due to nonequilibrium drop at the junction barrier. Let for simplicity the conductivity of p- and n-semiconductors be high enough for the photovoltage Y to be considered as spatially independent. The nonequilibrium carrier concentration at the junction edges are
i\po = Pn [exp(eY/kBT)-lJ, i\n = n [exp(eY/kBT)-l] (1)
for n- and p-regions respectively. Here p and n are the equilibriumminority carrier concentrations, kB is the Boltzman constant, T is the temperature.
For the sake of simplicity' we take the following assumptions: (i) geometry of the system is cylindrical, lii) light is absorbed near the surface in p +-region, (iii) the doping level of p+-region is much higher than that of n-region, (iv) the minority carrier lifetime Tn in p+-region is small compared with the carrier lifetime T p in n-region. These assumptions do not restrict the common character of the phenomenon under consideration.
The most important problem is the carrier spatial distribution in n-region. The nonequilibrium carrier concentration is determined by the ambipolar diffusion from p-n junction where it is given by Equation (1). The problem can be simplified when the fact is taken into account that the thickness d of p+-region is compared with the carrier diffusion length Ln = J(DnTn). Thus d « Lp, where Lp = J(DpTp) is the carrier diffusion length in n-region, DR being the hole diffusion coefficient. In this case the boundary of n-semiconductor can be considered to be plane.
The problem is solved using the Fourier-Bessel transform for the complex component which is proportional to exp(-iwt). Let us introduce the dimensionless variables
All functions of Rand Z are presented in the form
with
i\p(R,Z) is determined by the diffusion equation with boundary condition at the plane Z ~ 0:
i\p(R) = i\po, R ~ Ro
8i\P/8Z -(sLp/Dp) i\p = 0, R > Ro,
s being the surface recombination velocity. The solution gives:
(2)
where
606
g(0 (l-ifi)~ [8(0 + foo dR R Jo(~R) /lg(R)], o
/lg(R) is the function which takes the edge effect of the p-n junction into account and depends on the surface recombination velocity. When s« Dp/Lp,
~g(R) N (1/2) exp[(R-Ro) J(l-in)]
Nonequilibrium carrier concentration ~n(R,Z) in p+-region determined by the carrier diffusion from the light beam and the p-n junction surface. Straightforward calculations give
KU( CZ) = (P oLn/2hvDpL6){[sinh( 8-Z/,\)/ (cosh8+lT*sinht5)]exp( -~2A2/ 4)
+ No 8(0[cosh(Z/A)+lT*sinh(Z/A)]/(cosh8+lT*sinh8)}, (3)
where No is dimensionless concentration which is proportional to ~no, hv is the photon energy, 8 = d/Ln, ,\ = Ln/Lp, IT* = s*Ln/Dn , S* is the surface recombination velocity at Z = O. Equation (3) is written under the simplifying conditions: WT n «1 and Ln« a,Lp.
The boundary concentrations ~Po and ~no can be found taking into account of the fact that the total current through p-n junction is zero:
~Po = Po (/2hv Dp Lp J (I-in), No = 7(/ J(l-ifi), (4)
where
(5)
The current density through the p-n junction is given by
j(O = [ePo/2hvL~(cosh8+lT*sinh8)]{-exp(-~2A2/4) + 1 _
~7r-l(l-inr'[g(O + 7B(0(sinh8+lT*cosh8)/(cosh8+lT*sinh8)]} (6)
Here the first term in curly bracket corresponds to the electron current generated by the light beam. The second term describes the injection current
Figure 1.
LIGHT
r
n
Planar p+-junction structure. Dotted line shows the nonequilibrium carrier cloud. Full closed lines show the toroidal-type current density distribution.
607
caused by the photovoltage. The current density pattern is toroidal-type (see Figure 1).
The photovoltage at the p-n junction can be found from Equation (1) and (4) in which must be taken into account that 8p is the sum of stationary Ap(o) and non-stationary 8p(li») components:
(7)
Here 8PSli») is given by Equation (4) and (5), while 8PSO) is determined by
the same equation under the condition !l = O. The solution of Equation (7) has the form:
m V = V( 0) + E V( k) cos[k(wt-¢)]
k=l (8)
It can be shown that
V(k) N (_1)k 2knT/(ek)
under the condition 8p( 0) » Pn.
THE TEMPERATURE
The heat sources in system under consideration are as follows: 1) The thermalization of light-induced electrons and holes acts as a
surface source of power
2) where Eg is the gap energy. The carrier recombination at the surface (Z the surface source
W( s· reC) = Eg s*8n(r,z=O).
0) in p +-region acts as
3) The carrier recombination in p +-region is a bulk heat source. However, in present paper we suppose the frequency is not too high so that the thermal diffusivity length exceeds the p+-region thickness. Thus the bulk recombination in p +-region can be considered as the surface source
w(rec) s
c! Eg f (8n/rn )dz.
o
4) Generation or absorption of the Peltier heat is considered when the current, passing the junction barrier. In accordance with Equation (8) the power of this source is a series of harmonic components. However the only first harmonic, exp( -iwt), is of interest because we are considering the fundamental harmonic response. Thus the power of this source is
W(li») = -J'V pn n'
where V B ~ V k - V ( 0), V k being the build-in voltage.
The heat sources discussed can all be considered as an effective surface source
608
The bulk heat source is caused by the carrier recombination in n-region. Its power is
The heat conduction equation is easily solved giving
KT(~,Z) = (P oEg/2hvI'Lp){-wo([B(0+Kg(0]w1(Oexp[-Jt(OZ] +
[w jexp( -~2A2/ 4 )-( W2( -w3)B( 0-w2(Kg( 0]1)-1( Oexp [-1) ( OZ},
where 1)(0 = v'(~2-ill/llo), !Ie 1)(0 > 0, 110 = x/D p , I' conductivity, x is the thermal diffusivity,
Wo = 11 0/[11 o+ill(l-11 0)],
Wj = (hv/Eg)-(l-eV B/Eg)(coshtl+lT*sinhO)-l
W2 = (eV B/Eg)-wo,
THE DEFORMATION
is thermal
Let the sample be mechanically homogeneous and isotropic. We consider here two strain mechanisms: temperature strain and electronic strain caused by the carrier-lattice interaction via the deformation potential E. The stress tensor is
with
f = [atE/3(1-21T)]~T+E~p,
where E is Young's modulus, IT is Poison's coefficient, at is the thermal expansion coefficient (volumetric), Uik is .the .strain tensor.
Let us restrict our consideration to the region of sufficiently low frequency (It! « ro/v, v being the sound velocity). In that case the quasistatic approximation can be used. Thus we have the following equation for the strain
{}IT ik/ {}Xk = O.
The calculations, which are rather cumbersome, give the following result for the displacement vector of the free surface (external normal is supposed to be the positive direction)
u(R) = [P oEgaT (1+1T)/61fhvl'] U(R)
U(R) = (E-wo)([Il(R)+Hl(R)]-[(W2(-W3)I2(R)+W2(H2(R)]+W1I3(R), (10)
where
609
I 1(R) 2Ro fOO d~ Jo(~R) Jl(~Ro) / }t(0 [}t(o+n o
I2(R) = 2Ro fOO d~ JoUR) Jl(~Ro) / '1/(0 ['l/U)+~]' o
LHl(R) fOO d~ {~ Jo(~R) / }t(0 [}t(o+m Kg;(o, o
LH2(R) = fOO d~ {~ Jo(~R) / '1/(0 ['I/(o+m Kg;(O, o
I3(R) = fOO d~ H Jo(~R) / '1/(0 ['I/(O+~]} exp(-~2A2/4). o
All components in right-side of Equation (10) are qualitatively shown in Figure 2 as function of the radius. The line (1) corresponds to the first term. It shows the deformation under the p-n junction caused by the carrier recombination in n-region (its contribution is proportional to wo) and by carrier,lattice interaction (its contribution is proportional to E). The line (2) corresponds to the second term in Equation (10). It shows the deformation caused by heat generation due to the carrier recombination as well as heat absorption due to the Peltier effect. The line (3) shows the thermal deformation caused by heating the central region due to carrier thermalization, the Peltier effect and the recombination in p +-region.
Relative contribution of the components discussed to the whole deformation is dependent on the modulation frequency. When fi« fi o, the components (3) and (2) are predominating. In the case of high frequency (fi » fio) the compo-
nent (3) predominates in the region R < R*,
while at Ro ~ R > R* the greatest is the component (1) caused by vortex current. In that case the replacement of the deformation mechanisms occurs
when crossing the point R = R * which results in characteristic features of the displacement (its amplitude and phase). The critical parameter for those features is E (in the first place its sign being essential).
In experiment the surface deformation is measured by the probing beam reflection, i.e. the surface inclination is of interest as a function of the radius and the frequency. The inclination angle is determined by the derivative dU/dR. Its amplitude and phase are shown in Figure 3 as function of R. The amplitude reaches its maximum at the p-n junction edge. The maximum width
Figure 2.
610
lui
o R
Modulus of the components in Equation (10) as function of the radius
1i '2 <.<0
:;-6 lU 0
'" UJ :::l II)
!:: c:z: ::z:
...J a. :Ii a. -z ~ ..:
R- Ro R
Figure 3. The amplitude and phase of the surface inclination angle as function of the radius.
is of the order of the carrier diffusion length Lp. The phase behavior is determined by sign E. The case of positive E corresponds to lattice expansion under the carrier action, while negative E corresponds to the compression. Accordingly the phase for E > 0 differs from that for E < 0 in about 180·.
At R > R* the phase gives information about Tp and E.
CONCLUSION
In this paper we have developed the method of detail analysis of thermal and deformation phenomena in semiconductor structures under the local illumination. The electron transport processes, in particular the vortex currents, are shown to be of great importance. The vortex current effect consists in the redistribution of the carrier concentration as well as in the creation of the additional heat sources. The carrier-lattice interaction can generate a non-monotone surface profile when the deformation potential is negative (E < 0).
REFERENCES
[1]
[2]
[3]
[4]
(5)
(6)
(7)
"Photoacoustic and thermal wave phenomena in semiconductors", ed. by A. Mandelis, Elsevier, New York (1987). A.N. Vasil'ev, V.A. Sablikov, Detormation of semiconductor surface under the local illumination, Fiz. Techn. Poluprov. (in Russian), 23:33, 1989. A.N. Vasil'ev, V.A. Sablikov, V.B. Sandomirskii, Photothermal and photoacoustic effects in semiconductors and semiconductor structures, Izvestia Vysshikh Uchebnykh Zavedenii, Fizika (in RUSSian), 30:119, 1987. H. Flaisher, D. Cahen, Computer simulation of the photoacoustic cells, IEEE Trans. Ultrason., Ferroelec. and Freq. Contr., 33:622, 1986. A. Mandelis, Coupled ac photocurrent and photothermal reflectance theory of semiconducting p-n junctions, J. Appl. Phys., 66:5572, 1989. V.A. Sablikov, V.B. Sandomirskii, A.N. Vasil'ev, Electron phenomena in photoacoustics of semiconductors and semiconductor structures, Izvestia AN SSSR, Fizika (in Russian), 53:1162, 1989. V.A. Sablikov, V.B. Sandomirskii, Photomagnetic effects in semiconductors induced by local illumination, Pis'ma Zh. Eksp. Teor. Fiz. (in Russian), 49:548, 1989.
611
SOME ASPECTS OF LATERAL WAVES GENERATION AND DETECTION
BY ACOUSTIC MICROSCOPY USING V(z) TECHNIQUE
A. Saied, H. Coelho-Mandes, K. Alami, C. Amaudric du Chaffaut, J.M. Saurel, J. Attal
Laboratoire de Microacoustique de Montpellier Universite des Sciences et Techniques du Languedoc Place E. Bataillon, 34060 Montpellier, France
INTRODUCTION
In the non scanning configuration, the characteristic response V(z) of the acoustic microscope treated as a signature is a valuable method for measuring quantitatively on a microscopic scale the elastic properties of a sample and for explaining the contrast observed in surface images [1]. This signature is obtained by recording the reflected acoustic signal variation V as a function of the sample defocus z and arises from an interference between a bulk wave specularly reflected at the normal direction and generally a leaky Rayleigh wave generated near and at the critical angle OR at the liquid sample interface and
reradiated in the liquid medium at precisely OR' V(z) is periodic and its
periodicity is directly connected to the leaky Rayleigh wave velocity V R in
the material. In a recent work [2], we have pointed out that depending on both the material and the liquid loading, other modes in addition to the leaky Rayleigh mode can contribute with a significant efficiency to the interference mechanism in V(z) signature. These modes result in waves known as "longitudinal lateral waves" or "surface skimming bulk waves" generated at the longitudinal critical angle 0L and involving radiation of energy in the liquid.
They propagate along the interface with a velocity equal to that of the sample longitudinal bulk wave. Owing to the complex reflectance function R( 0) variation, we have shown that longitudinal lateral waves appear when a phase transition exist at 0L'
In this paper we study thoroughly the effect of the coupling liquid loading on the Rayleigh and lateral waves efficiency conversion and velocities when we change the density of the fluid to a large extent. This was achieved by using water and mercury. The experimental results reported here are obtained on a variety of candidate materials and are consistent with theoretical V(z) curves analysis.
ROLE OF THE COUPLING FLUID ON THE V(z) SIGNATURE
Material characterization and analysis require the determination of elastic constants which are dependant on acoustic velocities. V(z) technique is suitable for determining accurately the material acoustic surface velocity since it is directly proportional to the magnitude and phase of the sample complex
Physical Acoustics. Edited by O. l.eroy and M. A. Breazeale Plenum Press, New York, 1991 613
reflectance function information [3J.
R(O) which contains the material properties
V(z) signature using Water as Coupling Fluid
Extensive work on V(z) technique have been done with water and in that case most of investigators agree with the following comments on experimental V(z) curves.
1) When V R > V liq and operating with a lens having a large opening
semi angle Om (typically 50') to enable the Rayleigh wave excitation, V(z) exhibits oscillations whose periodicity characterizes the Rayleigh velocity V R.
With regard to the reflectance R( 0) variation, we notice a phase change at 0L and a 7r phase transition at OR allowing the determination of the leaky
surface wave velocity according to Snell's formula sinOR = Vliq/V R. A unity
magnitude is also observed at 0L and around OR if we don't take into
account of the acoustic wave attenuation in the material.
2) When VR < Vliq and VL > Vliq (VL being the longitudinal
wave velocity in the material), the Rayleigh wave can not be excited but we still observe oscillations in V( z) signature which are related to the longitudinal velocity V L. We have particularly observed this phenomenon with plexiglass for
which the phase transition at 0L reaches 7r /2. These waves known as
longitudinal lateral waves play a similar role as leaky Rayleigh waves: excited at longitudinal critical angle, they are guided along the liquid-solid interface and radiate energy in the liquid. Their presence was detected by some authors such as Neubauer [4J and Mayer [5J when a parallel bounded acoustic beam is incident on a solid liquid-interface at 0e In a previous work, we have
detected them as well as Chubachi and al. [6J by V(z) technique employing a focused beam. However, using scanning acoustic microscopy we are first to visualize the propagation of these lateral waves. This is displayed in Figure 1 where an epoxy-sapphire interface is imaged at 580 MHz with water as coupling fluid . Two different ranges of fringes are visible and are caused by interferences between normal rays and surface waves generated in sapphire (leaky Rayleigh waves) and in epoxy (lateral waves) and reflected both at the boundary between the two materials. Fringes spacing in epoxy gives a lateral
sapphire
Figure 1.
614
Visualization by acoustic microscopy of interferences due to reflection of Rayleigh and lateral waves at epoxy/sapphire interface. F = 580 MHz.
R +n OOI1'UT VOLTAGE
H2J) /PleIiglasl
o
-n ... :ro 640 00) 1200 Hm 1921l Z Cum\
20 9 0
R
AMPLmJDE +n VOLTAGE H20f.,11X2
0
PHASE
-fi . . 20 40 9 0 :ro 640 00) 1200 HID 1921l Z Cuml
R
00ll'tIT VOLTAGE H2O/AI
0
PHASE
-n 20 40 9
1921l Z fu ml 0 :ro 640 !a) 1200 11m
Figure 2. Reflectance function R( 0) variation and experimental curves plotted with water as coupling fluid at 135 MHz.
V(z)
wave velocity of about 2700 mls which is near the value V L = 2580 mls obtained by the other techniques . The Rayleigh wave velocity measured for sapphire is about 6200 mjs. Measurements by acoustic microscopy agree well with those obtained by V(z) and echography taking into account the difficulty for precise evaluation of the fringes spacing on the image.
Besides epoxy and plexiglass , we have noticed that lateral and Rayleigh waves can both contribute simultaneously to V(z) . In Figure 2, we report experimental V(z) of plexiglass , 8i02 and aluminum acquired at 135 MHz. These materials were chosen because of their significantly different densities for a range of velocities. We note that compared to Rayleigh waves, lateral waves have generally less amplitude but lower damping. They modulate the Rayleigh oscillations at the beginning of V(z) and distinctly appear at the end part of the curve.
Investigation of the material reflectance function set on the left side of V(z) curves shows that whereas the phase transition at OR is constant and
615
Table 1. Comparison between Rayleigh and lateral waves excitation efficiency calculated at 135 MHz
Ii : J..rleigb nn • lfateriu. 6'# EL :.LLteral nve FL - ER at. IL Efficiency excitation Efficiencr excitation
AI 10' 3,8 dB 27 dB 23 dB
Si02 ,. S,' dB 29,5 dB 24dB
funt,lsbn I," 14 dB 36 dB 22 dB
* referred to the amplitude at focus .
equal to 211", the phase transition at 0L varies according to each material: it
is of 6' for glass, 10' for aluminum and was found to be of 1.6 ' for tungsten (not shown here). As the phase transition increases, the reflectance magnitude decreases . On the other hand, as it can be seen on experimental V(z), the lateral waves amplitude increases with increasing phase transition at 0L: these
waves are not visible for tungsten but are significantly detected for aluminum and glass. It then becomes evident that the phase variation at 0 L is
indicative of the material-liquid coupling and is responsible for efficiency conversion of lateral waves as well as for their contribution to V(z) mechanism. Velocity measurements reported in Table 2 are achieved by FFT technique and are equal to within 0.3% to velocities obtained by echography.
With regard to Rayleigh wave excitation, we notice that the signal amplitude reflected by the material is directly related to its relative impedance (density x velocity). A relative low impedance such as of glass (16.10 5 gjcm 2s) results in efficient coupling to Rayleigh waves leading to an efficient emission of energy in the liquid displayed by a high V(z) amplitude and a rapid decay. The radiation of Rayleigh waves is less for tungsten since it has a high acoustic impedance of about 6 times that of glass.
Furthermore, we have computed V(z) response for glass, tungsten and aluminum using the following formula denoting the correlation with the complex reflectance function R( 0):
Om V(z) = f P2(0) R(O) e2jkzcosO cosO sinO dO
o where P( 0) and z are respectively, the pupil function and the defocus of
Table 2.
lfat.eriu.
Plui
Si02
Al
r
616
Experimental (exp) and simulated (sim) velocities expressed in mjs
2 -3 K ° I ' = 1500./5 p = Ig c.
lIq I ' " 1450 rols p= 13,6g cm3
p , R exp 'RSill
Y Lexp 'Lsi.
, Rexp
VRsill y Lexp \ sill
1,2 - - 2758 2751 - - 2752 2750
2,6 3447 3408 6009 5927 3754 - 5981 -2,7 2919 2930 6324 6299 - - - -19,3 2640 2639 - 5240 2849 2830 5249 5244
R HGISI02 AMPIlIUDE +n 00I1'tTI' VOLTAGE
o
-n 40 a 0 :r2O ~ 900 1200 un> I!alZ (uml
R
AllPUI'UDE +n OU1l'Ul' \'OLTAGE I!g/W
0
~
-n 31 40 a 0 :r2O ~ OOJ 1200 1001 I!mZ(uml
Figure 3. Reflectance functions R(O) variation and experimental V(z) curves plotted with mercury as coupling fluid at 135 MHz.
the lens and k the wave number in the liquid. Only the aluminum signature simulated at 135 MHz and showing the lateral and Rayleigh oscillations is illustrated in Figure 4.a. Theoretical V(z) simulation has revealed that when the phase transition at 0L does not exist, no longitudinal lateral waves
oscillations are displayed and V(z) is not sensitive to longitudinal waves. Otherwise, these waves are generated by the smallest phase variation. In fact, computed tungsten V(z) has shown that longitudinal lateral waves do exist but have a low amplitude that they can not be experimentally visualized. In Table I, we have reported the excitation efficiency of lateral and Rayleigh waves calculated from theoretical V(z) . We note that using water the excitation of Rayleigh waves is 10 times more efficient than that of lateral waves.
V(z) signature with Mercury
With intent to optimize a material characterization, it is necessary to choose an appropriate fluid yielding an efficient excitation of surface and longitudinal waves. Since many years, we successfully use mercury for imaging internal structures in materials [7] . This liquid has almost the same velocity as water (1450 m/s) but a d ensity 13.6 times larger. This latter assign to the liquid a higher loading and impedance l eading to a better matching with most of solids . Besides, mercury is four times less absorbent than water.
Experimental V(z) curves obtained with mercury at 135 MHz are reported in Figure 3. Comparing with V(z) plotted with water, first we see that mercury has significantly enhanced the oscillations amplitude and induced a higher emission of longitudinal lateral waves which improves the accuracy of the velocities measurements. Secondly and the most important is that while longitudinal velocities remain unchanged , Rayleigh waves velocities are raised to shear velocities . These results are summarized in Table 2.
To explain the effect of mercury loading, let's carefully examine the reflectance function which exhibits important singularities (Figure 3) . First, we
617
0U'lPUI' mLTAGE H20/AW OUl'Ptrr mLTAGE
(a) (b)
o :2l 640 !m 12m 11m 1\3) Z (um) o :2l 640 !m 1:Bl um 1\3) Z (11m)
Figure 4. Simulated V(z) curves at 135 MHz. (a): AI/H 20, (b): W /Hg.
notice that at 0L and for all the materials the phase transitions 6.¢ are
much more important compared to that obtained when we operate with water: 6.¢ = 'If for Si0 2 and 'If/4 for tungsten which justifies the higher longitudinal waves amplitude. Secondly, it is very important to look at the phase variation around OR i.e. at the vicinity of the shear critical angle 0T : whereas the
phase transition goes sharply to 'If when we operate with water, this 'If radian transition becomes less rapid when the material is loaded with mercury. Significantly starting at 0T' it occurs over about 3° angular range for Si02 and
tungsten. So this phase variation may well induce a high coupling between the liquid and the shear lateral wave in the solid which by radiating in the liquid can obscure the Rayleigh wave effect. In other words, the use of mercury makes the effect at shear critical angle much more Significant than that at Rayleigh angle. In Figure 4.b, simulated V(z) of tungsten/mercury at 135 MHz clearly shows two kinds of oscillations related to the bulk waves velocities in the material.
CONCLUSION
We have found that using a high density liquid such as mercury increases the liquid-material coupling and leads to a high efficiency excitation of longitudinal and shear lateral waves. This is of the highest importance since getting both bulk waves from the same V(z) signature, we can determine at a microscopic scale the Young and Poisson elastic modulus of the material . Moreover, referring to the reflectance function R( 0), we have firstly, demonstrated that phase transition at the critical angles is responsible for the excitation efficiency of the different modes propagating at the liquid-material interface. Secondly, the derivative of the angle dependent phase ¢'(O) near 0T identifies the phase velocity and indicates whether leaky Rayleigh wave or shear lateral wave propagates at the interface. In other words, as using a dense coupling liquid the leaky Rayleigh wave velocity is strongly affected, the regular location of OR at ¢ = 'If is no more valid to determine the surface velocity
since it does not take into account of the liquid loading.
REFERENCES
[1] R. Weglein, Acoustic Micro Metrology. IEEE trans . sonics ultras ., 3:225, 1985.
[2] J. Attal, C. Amaudric du Chaffaut, K. Alami, H. Coelho Mandes, A.
618
Saied, Role of the coupling fluid in acoustic signature V(z), Elect. Lett. 25(24):1625, 1989.
[3)
[4)
[5)
[6)
[7)
A. Atalar, A physical model for acoustic signature, J. Appl. Phys., 50:8237, 1979. W.G. Neubauer, Ultrasonic reflection of a bounded beam at Rayleigh and critical angles for a plane liquid-solid interface, J. Appl. Phys., 44(1):48, 1973. T.D.K. N'gok, W.G. Mayer: Ultrasonic non specular reflectivity near longitudinal critical angle., J. Appl. Phys., 50(12):7948, 1979. J. Kushibiki, K. Horii, N. Chubachi: Material characterization by L.F.B. acoustic microscope., IEEE trans. sonics ultras., SU32:189, 1985. A. Saied, C. Amaudric du Chaffaut, J.M. Saurel, J. Attal, Some aspects of the scanning acoustic microscope contributions in the evaluation of device reliability., J. de Phys. suppl., 9(49):801, 1988.
619
STUDY OF INHOMOGENEOUS AND HETEROGENEOUS ULTRASONIC
WAVES IN KIDNEY STONES
ABSTRACT
V.R. Singh and Ravinder Agarwal
National Physical Laboratory New Delhi - 110012, India
Renal calculi are inhomogeneous, heterogeneous and complex in nature. There is a large variation in their chemical composition due to food habits of patients and hence geography of the inhabitants in different parts of the world. Ultrasonic studies are made to investigate various physical and other constants of such renal calculi. A double-probe through transmission technique is used to study various propagation parameters. The ultrasonic waves displayed on cathode-ray oscilloscope are complex and complicated in nature. Complete analysis of such wave patterns is given in the present study. The data reported would help in the design of an ultrasonic lithoripter.
INTRODUCTION
The stones formation in the kidney has been concerned with longstanding disease. Many workers have analyzed chemical composition with several methods [1-3]. Some of the constituents like calcium oxalate calcium phosphate, are mainly present as major constituents, while others like magnesium ammonium phosphate, cystine, uric acid, etc. occur rarely. Invariably, renal stones have inhomogeneous, heterogeneous and complex nature. The present paper is limited to ultrasonic studies of such inhomogeneous and heterogeneous calculi.
MATERIALS AND METHOD
Twenty five kidney stones collected from local hospitals in Delhi (India) were used in this study. Five of the samples could not, however, be used due to their being broken during shaping. The samples were machined to make their opposite faces flat and parallel, as this was required to achieve better ultrasonic measurements and to facilitate contact with the front face of the transducers. [4].
A pulse-receiver (make Panametrics, model 5052 PR), contains a pulse generator producing 200 to 1000 short electrical pulses per second. The pulses of ultrasound emitted from the probe were partially reflected from the boundary between two media having different acoustic impedance (Z),
Z = pC
where p is density and C is the propagation velOcity.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 621
The reflected pulses were picked up by a second ultrasound transducer placed immediately near to the transmitter (double probe technique), and are transformed into electrical signals, amplified in the apparatus, rectified and displayed at cathode ray oscilloscope (model PM 3206, Philips, India). The attenuation coefficient a , of the renal stones was found by using the relationship:
A a = 20 log x-; / d
where Al and A2 are the amplitudes in terms of voltages of the reference signal with and without the sample of the renal calculi of thickness d.
RESULTS AND DISCUSSION
The ultrasonic parameters for seven samples are given in Table 1 at frequency 2.5 MHz at room temperature 27' C. A large variation in the results have been found due to its complex, inhomogeneous and heterogeneous nature of the calculi. In order to reduce the error and to have the best possible results the measurements have been made in different positions of the sample by rotating around their axes.
The ultrasonic propagation velocity of the renal stones has been found to vary from 1808 to 3015 m/s while the acoustic impedance from 1.803 x 10 6 to 4.763 x 10 6 Kg/m2/s . The propagation velocity has been found to be inversely proportional to the attenuation. The specimens having inhomogeneous structure have higher attenuation factor and low ultrasonic propagation velocity factor. The attenuation has been found to vary from 253 to 1037 dB/m respectively.
Figure 1.
622
Ultrasonic Wave Pattern for (a) Heterogeneous (b) Inhomogeneous Renal Calculi.
Table 1. Ultrasonic wave propagation parameters of renal calculi
S.No.
l.
2.
3.
4.
5.
6.
7.
Velocity m/sec
1808
1864
2010
2107
2200
2406
3015
Acous~ic Im~edance 2xlO Kg/m /Sec
1. 826
1. 803
2.030
2.380
2.712
3.240
4.763
Attenuation dB/m
103
1012
876
613
465
413
253
Figures l(a) and l(b) show two typical cases for heterogeneous and inhomogeneous ultrasonic wave patterns. In Figure l(a), for heterogeneous stone specimen, attenuation value has been found to be very low and for Figure l(b), that is inhomogeneous, attenuation factor is very high due to its porous structure. Results in Table 1 also show that the hard specimens with less pores have low value of attenuation (dB/m). Further, this systematic study would be helpful in the identification of the type of renal calculi that occur in the human ureter.
CONCLUSION
Ultrasonic properties with regard to complexity and porosity of renal calculi have been studied and analyzed.
REFERENCES
[1]
[2]
[3]
[4]
D.J. Suter and E.S. Wooley, "Composition of Urinary Calculi by X-ray Diffraction", Br. J. Urol. Vol. 44, pp. 287-291, (1972). Etruji Takashi, "Chronological Variation in the Chemical Composition of Upper Urinary Tract Calculi", J. Urol., Vol. 136, pp. 5-91, (1986). Ravinder Agarwal, V.R. Singh, "A Comparative Study of Fracture Strength, Ultrasonic Properties and Chemical Constituents of Kidney Stones", Ultrasonics, In Press, (1990). V.R. Singh and Ravinder Agarwal, "A Study of Ultrasonic Characteristics of Renal Calculi, In Vitro, "J. Acoust. Soc. Am.", Vol. 85, pp. 962-963, (1989).
623
SCATTERING OF ACOUSTIC WAVES IN RANDOMLY INHOMOGENEOUS
MEDIA BY MEANS OF THE SMALL PERTURBATIONS METHOD
Eugeniusz Soczkiewicz
Institute of Physics Silesian Technical University 44-100 Gliwice, Poland
INTRODUCTION
Various mathematical methods have been employed in studies of wave propagation in randomly inhomogeneous media [1-3]. The choice of the one which is actually used in a given situation depends on the scale of inhomogeneities in relation to the wavelength of radiation, on the mean square fluctuation in the medium refractive index for acoustic wave, as well as on the distance wave passed. The single scattering approximation in calculation of acoustic wave scattering coefficient is justified if the scattered energy is small in comparison with the energy of initial wave [4]. Acoustical properties of random media depend not only on the value of the mean square fluctuation of the medium refractive index for acoustic waves, but also on the correlation function of refractive index fluctuations [2]. The Gaussian form of the correlation function:
(1)
corresponds to continuous changes in the medium refractive index [2], where .. f(r) is the fluctuating part of the square of the medium refractive index for acoustic waves, < f2> denotes the mean square fluctuation of f, and a is the radius of correlation of fluctuations. If changes of the medium refractive index are discontinuous in space, the correlation function of medium inhomogeneities has an exponential form [2]:
(2)
In the case of inhomogeneities caused by turbulences, the correlation function of medium inhomogeneities is described by the von Karman function [1,5]:
K(r) = <f2> [.!.]V K [.!.] 2v- 1r( v) a va'
(3)
where Kv[~] is the Bessel function of second kind of imaginary argument,
r(v) denotes the Euler gamma function [6], and v is a number.
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 625
SMALL PERTURBATIONS METHOD IN CALCULATION OF SCATTERING COEFFICIENT
We look for the solution of the Helmholtz equation [4]:
\1 2'l1(;) + k2n 2'l1(;) = 6(;) (4)
(where 66) is the Dirac distribution, \1 2 denotes the operator of Laplace, k the wave number, n the refractive index) in the form of a series:
(5)
where j1 denotes a small parameter, 'l1 0(;) is the primary acoustic field,
'l1 1(;) the acoustic field generated by medium elements under the influence of
the primary wave, 'l1 2(;) the acoustic field generated in the medium under
influence of fields 'l1 o(h and 'l1 1(;) etc. In the Fraunhofer region, 'l1 1(;) is given by the formula [4]:
(6)
where Ro denotes the observation point, ki' ks are wave vectors of initial and scattered waves, V the volume of the inhomogeneous space. In the case of large scattering space in comparison with the correlation radius of inhomogeneities, it results from the above equation the following formula for the intensity of the scattered acoustic field [4]:
J = 7I"k 4 V (p(~), (7) 2R5
where x ki-ks, and (p(~) is the Fourier transform of the medium inhomogenei ties correlation function:
-> 1 foo.... ............ <\J(x) = -- K(q)exp(ixq)d 3q
(271")3 -ro (8)
For isotropic random media, these formulae lead to:
J = 7I"k 4 V <\J(2k sin~), 2R5
(9)
where () denotes the angle of scattering and k the wave number. From the above equation it results the following formula for the energy of singly scattered acoustic field:
(10)
while that of the initial plane wave is:
(11)
where V = L3, and one can assume that 1 'l1 0 12 1. From the formulae (10)
626
and (11) one obtains the following equation for the scattering coefficient T
(12)
DEPENDENCE OF SCATTERING COEFFICIENT AND VELOCITY DISPERSION OF ACOUSTIC WAVES IN RANDOM MEDIA ON THE FORM OF K(r)
The formula (8), after introducing spherical coordinates and integration over angles, takes the torm:
1 CD cp(x) = - f sin(xr)K(r)rdr,
21f 2x 0
(13)
and it is easily to obtain expressions for the scattering coefficient 1, for various forms of the medium inhomogeneities correlation function K(r). In the case of Gaussian form of K(r):
for the exponential form (2):
and for the von Karman correlation function (3):
Ji< E 2 >r( 1I+-23)k2a 13 = ------"-- [1 - (1 + 4k 2a2)-II-t]
( 2 11+ 1) r ( II )
(14)
(15)
(16)
The method of Green function gives the same as the above formulae if the Bourret approximation for the effective wave number operator is used [5]. We have calculated dispersion in velocity of acoustic waves in random media, by using the local Kramers Kronig relation derived by O'Donnell et al. [7]:
where c( w) denotes the velocity of acoustic wave at the cyclic frequency 1 and a = 2"1 . In the long wavelength limit ka< <1 we have obtained:
(17)
w,
(18)
where B = (12Ji)-1 for the Gaussian form of K(r), B = 2(31f)-1 for the
expression (2), and B = r(II+~)/(3Jir(lI)) in the case of the von Karman
correlation function. Co denotes velocity of waves in homogeneous medium. The single scattering approximation in calculation of the acoustic wave
scattering coefficient is justified if the scattered energy is much smaller than the
627
Table 1. Values of <c 2>L * a 1
2 and <~ >L~
ka I <E2> L* <~> t z L~ lez Q,--_1_-+-__ ~_ __
0.001 2.26 10~ 5.00 10" 4.51 0.01 2.26 10 8 5.00 107 4.51 0.02 'I· 1.41 107 3.13 106 4.50 0.05 3.61 105 8.08 104 4.47 0.1 2.27 104 5.20 103 4.36 0.2 I 1.44 103 3.62 102 3.97 0.5 4.08 10' 1.60 10' 2.55
energy of initial wave [4], and it results from equations (10) and (11) the following condition for the distance L passed by the wave:
(19)
f * * Using the above ormula, we have derived expressions for the lengths L 1, L 2,
L ~ respectively for the Gaussian, the simple exponential (2), and the von Karman correlation functions, obtaining the following formulae:
L* 3
(2v+1)r (v) (1+4k 2a 2t+ t [(1+4k2a2t+I/2 _ 1] -I. y'1f<c 2>r(v+3/ 2 )k 2a
L * decreases with increasing in ka. For the Gaussian form of medium
inhomogeneities correlation function L: is greater than L~ for the simple
exponential one, as one can see from the Table 1.
REFERENCES
[1]
[2]
[3]
[6]
[7]
628
B.J. Uscinski, The Elements of Wave Propagation in Random Media, McGraw-Hill, New York (1977). L.A. Czernov, Wolny w sluczajno neodnorodnych sredach, Nauka, Moskwa (1975). A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York (1978). K. Sobczyk, Fale stochastyczne, P.W.N., Warszawa (1982). E. Soczkiewicz and R.C. Chivers, Scattering of acoustic waves by turbulence, Proc.Inst. of Acoustics (G.B.) 8:20 (1986). E. Janke, F. Emde and F. Losch, Specjalnyje funkcji, Nauka, Moskwa (1977). M. O'Donnell, E.T. Jaynes and J.G. Miller, Kramers Kronig relationship between ultrasonic attenuation and phase velocity, J.Acoust.Soc. America 69:696 (1981).
ELECTROACOUSTIC EFFECT IN CAPILLARS CONTAINING AN
ELECTROLYTE
Nickolay Tankovsky' and Josef Pelzl
'On leave from: Sofia University Dept. Solid State Physics Blvd. A. Ivanov 5, Sofia-1126, Bulgaria
Ruhr-Universitaet Bochum Inst. Experimentalphysik VI Bochum, FRG
INTRODUCTION
Debye was the first in 1933 to make a prediction, based on analytical calculations, that the propagation of ultrasonic wave through an ionic solution gives rise to alternating potential differences within the solution [1]. The effect is due to the difference of the dynamic reactions of the cations and anions to the acoustic waves. In this paper the existence of a reverse effect, i.e. the excitation of acoustic wave into an electrolyte by applying an alternating electric field has been analyzed theoretically and proved experimentally. For reasons of efficiency it is necessary to apply the electric field within a small volume of the solution (length-size smaller than the acoustic wavelength). As far as the electric field can not be confined within a definite small volume it is easier to confine the electrolyte into a small vessel e.g. a thin glass capillar
Figure 1. Cross-section of the capillar.
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 629
tube. Thin metal-film electrodes are deposited over the outer walls of the capillar, as shown in Figure 1. The small diameter of the capillar (outer diameter 500 J.Lm, inner diameter 150 J.Lm) and the cylindrical geometry ensures a high value of the electric field in the central cavity containing the electrolyte. The cations and anions are accelerated in opposite directions by the a.c. electric field, but their friction with the solvent is different. As a result the molecules of the liquid are set into oscillating motion, transferred to the walls of the capillar as pressure variations. In this way the capillar, imbedded into a solid can serve as a line acoustic source.
THEORY
A simplified theoretical analysis can allow us to obtain a better insight into the physical phenomena. The electrolyte is considered to be univalent. The ions, together with the solvent molecules sticking to them can be treated as spheres with effective radii RI and R 2. The indices 1 and 2 indicate the two types of ions - positive and negative. The electric field is assumed homogeneous. Under these presumptions the equations of motion for the ions (1), (2) and for the solvent molecules (3) can be written as follows:
eEx;PI Ee 3x; eE - - -- - PIVI 61f1] 6DkT
eEx;P2 Ee 3 x; eE - - -- - P2V2 61f1] 6DkT
s dvo CIt
(1)
(2)
(3)
where ml and m2 are the masses of the ions, e is the fundamental electric charge, vI, V2 are the velocities and PI, P2 are the frictional coefficients of the ions in the solvent; n is the electrolyte concentration and s is the solvent density; va is the mean velocity of the solvent molecules, 1] is the coefficient of viscosity, x; -1 is the thickness of the ion atmosphere around a chosen central ion, defined by the relation:
(4)
Here D is the dielectric constant of the medium, T is the absolute temperature and k is Boltzmanns constant. The first term on the left side of equations (1) and (2) presents the force of the electric field upon the charged ion, the second and third term give the electrophoretic and relaxation forces for dilute solutions [2]. The last term expresses frictional forces. The system of three equations (1)-(3) can be solved easily relative to the velocities VI, V2 and Vo. As pointed out by Debye [1] the ions mass effects can be ignored in comparison with frictional effects i.e. wmj« Pj for frequencies w up to the infrared region. After these simplifications and having in mind harmonic time dependence we obtain the following result:
i e E x; ( ) = in e E"(RI-R2) va = ws 61f1] PI-P2 ws (5)
This simple analysis shows that the main role in the electroacoustic effect in dilute electrolytes is played by the electrophoretic forces realized through the friction of the ion complexes into the solvent. The effect is proportional to the difference of the frictional coefficients (effective radii) of the cations and anions.
630
EXPERIMENTAL RESULTS AND DISCUSSIONS
Experiments have been carried out with thin capillar tubes filled with electrolyte and rigidly imbedded into the surface of a parallelepiped plexiglass sample sized 6x5x2 cm 3. Different electrolytes have been used: NaCI, LiI, Na-acetate, CsF and others. Care was taken to avoid the presence of air bubbles in the electrolyte. The capillar openings are closed with the help of melted paraffin. The acoustic signal is detected by a wide-band piezoelectric transducer (ceramic or PVDF) pressed against the surface of the plexiglass sample, a few centimeters away from the capillar. The piezoelectric transducer is shielded by a special metal housing to eliminate the electric inductions. The electric signal is then fed to a lock-in amplifier and spectrum is readily obtained by automatic scanning of the frequency .
Dependence on the Difference between Cations and Anions
It is clear from (5), that a good efficiency of the electroacoustic conversion can be achieved by electrolytes whose ions of different charge are different in size. Having this in mind we have used water solution of Na-acetate. The acetate anions have mean molal volume equal to 50 .2 cm 3/mole and mass 52 g/mole giving a density equal to 1.04 g/cm3. The Na cations have, in turn a mean molal volume -7.4 cm 3/mole, mass 30 g/mole and density 4.05 g/cm3
[2] . An acoustic resonance spectrum, excited with the help of a 0.01 normal solution of Na-acetate is shown in Figure 2. The value of the a.c. electric voltage, applied upon an open circuit load is 35 Vp-p. In order to verify, that the obtained signal is due to an acoustic resonance the following experiment has been carried out. A second piece of plexiglass is brought into acoustic contact with the sample . Silicon oil has been used as a binding agent. Thus, an acoustic resonator, twice thicker is built. The resultant signal is shown in Figure 2 with dashed lines. The resonance line is distinguished. However, no resonance line could be discovered at the twice smaller frequency (about 6.5 kHz) . This fact might be explained by the assumption that the electrolyte can not produce an efficient acoustic signal in this frequency range. Finally, to exclude the possibility that the signal might be due to electromagnetic radiation by the electrodes a spectrum obtained at the same experimental conditions, but with an empty capillar has been recorded, as shown in Figure 2 with dotteddashed lines. Obviously this signal is much weaker.
Dependence on a d.c. Voltage
The ions in the electrolyte are distributed macroscopically uniform. The cations and anions are driven in opposite directions by the a.c. electric field
~ 1.512 OJ '0 ::> := 1.01 8 c. E o
0.524
0.030
.. . . .
.. . .. .
1.100 1.200 l300 1.400
frequency (Hz)
Figure 2. Acoustic resonance spectrum.
320
240 '0 :T
" '" '" 160 0: .. 2.
88
631
~ 0604 GI -0 :3 ~ D.406 Ci. E o
1.1 20 1.304 1.550 1.760 frequency (Hz)
Figure 3. Signal amplification by d.c. voltage.
and hence the oscillating velocity of the fluid will tend to zero, especially when the ions of opposite charge are similar in size and dynamic properties. The efficiency of electroacoustic conversion can be improved by applying additionally d.c. electric voltage to separate spatially the ions of different charge. In this way the local counter action of cations and anions can be partly removed. The signal gain due to a d.c. voltage should be better observed in electrolytes, whose ions of different charge are similar in size. Experiments have been carried out with water solution of NaC!. The results are shown in Figure 3, obtained for electrolyte concentration 0.02 and a.c . voltage 35 Vp-p . The spectrum, recorded when Ddc = 0 V is given with dashed lines, while the continuous lines represent the same spectrum at Udc = +750 V. It can be seen, that the signal gain reaches up to one order of magnitude under these conditions. Measurements have shown , that the signal dependence of Ud c is nonlinear and different for the different frequencies.
Dependence on the Electrolyte Concentration
Electrolytes have complicated structure, which depend on many factors: concentration, temperature , pressure, solvent etc. Around each ion there exists a complex of solvent molecules distributed into several zones [3] . In the inner zone, called ion zone the interactions between the ion and the solvent molecules are dominant. A definite number of solvent molecules are sticking to the central ion, forming dynamically stable structure. In the intermediate zone the solvent molecules are structurally changed by the influence of the central ion. In the
10-5
2.006
1.512 2: '" ."
E 1.01a -a. E 0 0.524
frequency (Hz)
Figure 4. Dependence on electrolyte concentration.
632
outer zone the solvent molecules behave as in a pure solvent. The zone model has a mean-statistical interpretation because molecules can change their places in the different zones governed by thermodynamic equilibrium rules. By increasing the electrolyte concentration or raising the temperature outer sheaths of the zones are successively destroyed and the effective diameter of the ions becomes smaller. That is why the efficiency of electro acoustic conversion should depend on the electrolyte concentration. Experiments with different concentrations of NaCI and Na-acetate have been carried out. In both electrolytes maximal signals have been obtained for concentrations lower than 0.01. Different concentrations of NaCI have been used and the results are shown in Figure 4 for Uac = 35 Vp-p and Udc = +750 V. It is seen, that the signal amplitude grows when diluting the solution. Experiments have not been performed for concentrations lower than 0.005, but saturation should occur for very dilute electrolytes. Two mechanisms of the signal dependence on electrolyte concentration can be deduced from formula (5): direct proportionality on n and indirect dependence through the change of the effective radii of the ions when the concentration is changed. The two mechanisms are opposite to one another, which ensures that an optimal concentration should exist.
CONCLUSIONS
The existence of electroacoustic effect in thin capillars containing an electrolyte has been experimentally verified. This effect might have different important applications as line acoustic source or as a means to study the electrolyte structure and its dynamic properties. However, we must say, that our knowledge about this effect is far from perfect. There are some confusing examples of poor reproducibility of the experiments. Other experimental methods and deeper theoretical analysis are needed for better understanding of this phenomena. Experiments with charged colloidal solutions seem promising, as well.
ACKNOWLEDGEMENT
One of us (Nickolay Tankovsky) has the pleasure to acknowledge the support by the Alexander von Humboldt foundation.
REFERENCES
[1]
[2]
[3]
P. Debye, A method for the determination of the mass of electrolyte ions, J. Chern. Phys. 1, 13, (1933). R. Zana and E. Yeager, Ultrasonic vibration potentials in the determination of ionic partial molal volumes, J. Phys. Chern. 71, 3, 521, (1967). "Ion solvation", G. Krestov ed., Nauka, Moscow (1987).
633
SCHOLTE WAVE DIFFRACTION BY A PERIODICALLY ROUGH
SURFACE
Alain Tinel, Jean Duclos and Michel Leduc
Laboratoire d' Acoustique Ultrasonore et d'Electronique CNRS 1373 Universite du Havre, France
INTRODUCTION
Bulk wave reflection and diffraction in an infinite medium or plate have been studied for a long time. In a liquid medium, a bulk wave impinging on a plate, with velocity Cf, may generate Rayleigh or Lamb waves. A wave with phase velocity c may be generated, under an angle (3 (with the normal to the interface) only if the equation (1) Cf == c· sin((3) is satisfied; so the Scholte-Stoneley wave whose velocity is slightly less then Cf for usual solid mediums, may not be generated. However if engraved by a periodical grating a solid substrate may generate waves not satisfying equation (1); on thick substrates [1 J and plates [2J part of the incident wave energy is considered to be converted in a Scholte-Stoneley wave, though such a wave has never been observed.
We will study the interaction between a Scholte-Stoneley wave and a periodical grating engraved on a thick substrate (duraluminium) and examine the inverse phenomenon (Scholte-Stoneley wave generation by a bulk wave impinging on the grating).
INTERFACE AND SCHOLTE-STONELEY WAVES
An interface wave is a plane wave characterized by its pulsation wand
its wave vector K generally imaginary; the 2-dimensional expression of the displacement of such an elementary wave in a homogeneous and isotropic medium is
Ui == Ui·exp(j(K,x,+K2X2-W·t)) for i == 1 to 2
Ox, denoting the propagation axis and OX2 the perpendicular one.
The components of K must satisfy the equation:
for m == f,l,t
Cf, Cj and Ct denoting the plane homogeneous wave velocity respectively in liquid (f) and solid (1 for longitudinal, t for transverse) mediums, K I and K I I
respectively K real and imaginary parts. Moreover the waves must satisfy the
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 635
Table 1
'-lava Scholte pseudo Rayleigh 1st critic::!l angle in i~aiG~uid Scholte waves
angle 13 90 90 30 . 327 14 . 433 13.618 in degrees
pl:.asa celer1 ty H8l.988 1484.995 2940.582 5953.926 6303.134 c (m/s)
K,' IK" 1,002032 1.0000035 0.'505002 0,249415 0,235597
/C2'/K~ 0 0 0,863274 0,969107 0,912537
K,"/Kf" 0 0 O. 014144 0.035941 0.035498
K,,"'IC F i 0,063786 0,002643 -0.008274 -0,00925 -0,008599
usual boundary conditions between liquid and solid mediums: continuity of the normal components of the displacement and stress for X2 = O.
Finally we obtain a linear homogeneous system of 3 equations whose determinant cancellation constitute Stoneley's equation f4]; the various solutions of the problem have been enumerated by Sebbag [5 . For each propagation direction, there are 16 solutions which can be associated by 2 or 4 (opposite or conjugate K); we gather the main values obtained for the studied interface (water/duralurninium) in Table l.
The parameters values we use, are for water: its density Pf = 0.9982 103 kg/m3 and velocity Cf = 1485 mls and for duralurninium: its density Ps = 2.799 kg/m3 and longitudinal Cl = 6355 mls and transverse Ct = 3138 mls velocity.
Let us recall the main features of Scholte-Stoneley waves upon usual solid bodies:
636
- velocity slightly less than that of the plane homogeneous wave- in the liquid medium;
I I I
- slow decay in the liquid medium, the decay coefficient a = K 2 IK, being related to its velocity Cs by the equation a2 = 1-( csl Cf)2
- quick decay in the solid medium, the wave becoming negligible about after one wave length;
- wave energy nearly totally (over 99%) located in the liquid medium; - elliptic displacement in both mediums, the eccentricity coefficient being
11 a along Ox, in liquid.
Figure 1
DIFFRACTION BY THE GRATING
We don't give the complete description of the diffraction of Scholte waves by a grating engraved in the solid part of the interface; we just indicate the equations supplying the sequence of angles under which bulk waves are emitted in both mediums, denoted f3n in the liquid medium and ,n in the solid (n E "0)
Let us assume that the grating saw-tooth profile is not too deep so that the waves diffracted by its various points don't cancel each other out; then denoting by A the spatial period of the grating, we obtain in the liquid
medium above the plate: sin(f3~) = n· Ad A - Ad As and inside the plate:
sin(/~) = -n·Arn/A + Arnj).s with m = 1 or t as represented in Figure 1.
After having crossed the plate, these waves are themselves refracted by its low face and emitted in the liquid medium under the plate; so we obtain the sequence of angles given by sin(f3n) = -n· Ad A + Ad As.
Finally the emission angles are the same on both sides of the plate; however the emission angles under the plate are bounded by the second critical angle sin -1( cr/ Ct).
EXPERIMENT AL APPARATUS AND RESULTS
Above a glass sided tank containing about 250 I water are horizontally settled two parallel rails along which vertical spindles may run; one of them carries an usual immersed receiving transducer; the other one carries the engraved plate together with the emitting transducer (an interdigital thin layered transducer on a glass substrate sticked at the end of the plate [6]).
Rotations and translations are automatically controlled by a system consisting in a HP330 microcomputer and a IEEE488 multi bus. For each angular position of the plate ( 0 = f3 ± 7r/2 ), the maximal amplitude of the received signal is automatically memorized (Figure 2) .
Our experimental results allow a quaJi tative verification of the theoretically expected phenomenons. The used samples were 20 stripes gratings of various spatial periods (200, 300, 400, 500 and 900 /tm) insonated by a Scholte wave of wavelength As) 300 /tm (about 5 MHz frequency) .
In each case, we observed the ultrasonic signals emitted in water either directly (0 > 0) or through the plate (0 < 0). They are identified according to their direction (see Figure 3 and 4) and denoted:
MICRO- COMPUTER
Figure 2
637
o
638
2. §v I :-<\ I I I 1 I I .1
I I I i I I 1 I I J I II I I I I I I i I I I , I I I I I I . I , I
I , I , I I , II I I I , I , " II R
[I , , I I I I I I ' p l 'I PI III' I I , I I I. , I lrY ,' 1\ 1 I I I
I I 't'-( . I IT I i' I J'-t-I.-l 0 90
,... 1 I I I I I I 1 I I I I II ! II , I I II I I II
I I I 1\ I I I I I , I I , I 1 I I , , I 1 I , I I , , ,
I _L 1 I I I I I II 1 I I I I I I I I I , I I , I I II ' ! I , , I I I I , I I 1 1 I I I I I 1 I I I I \I I 1 I 'p I I I I , ' I , 1 1 , 1 I , : ' .. I .. i I I I 1
I , o I , -188 .'II ..A~ 'I
R+D
Figure 3
50 o V
D 'I
I , 1\ '\ " lr.J I\" I I , ...... , ,
0 ~u o
50 mV ... I I I I I I i
I , I I , I I I ,
'0 , , , p i' ! I ! , I 11\1 1 I I ,
I I ""[ Yi' ! , I
180
I
, i o
18 o
IDO -90 0
R
.lIl ....
. y ....
Figure 4
- Dn the signal of order n diffracted by the grating, - R the Rayleigh waves emitted in water (under the 2nd critical angle), - P the interface waves near the 1st critical angle, - S the Scholte waves appearing near 0 = O.
Besides we present the oscillograms recorded for some angles, to show the various aspects of the diffracted signals and interface waves. For directly emitted waves, the various modes appear clearly and about under the expected angles. But when the spatial period and therefore the depth of the grating becomes greater than the Scholte wavelength, we observe that for increasing orders (for instance Ds with A = 900 I'm) the corresponding peaks flatten and broaden out; in such cases, the gratings widely obstruct Scholte wave propagation and reemission. However Rayleigh waves are always observable near o = 120·.
For the waves emitted through the plate, in accordance with Snell-Descartes formula, they are observable only for -120· < 0 < 60· and we may identify the waves diffracted by the grating and those emitted under the 1st critical angle.
At last, let us notice the reversibility of the phenomenon, which we observed in all our experiments: when a bulkwave impinges on the grating under one of the angles fln, a Scholte wave is generated and easily observable.
CONCLUSION
Our experimental study confirms Scholte wave diffraction by gratings which occurs under the expected angles. Interface waves also appear when the grating spatial period is not much longer than the Scholte wavelength. At last the phenomenon reversibility supplies a new process of generating Scholte waves.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
J.M. Claeys, O. Leroy, A. Jungman and L. Adler, "Diffraction of ultrasonic waves from periodically rough liquid-solid surface", J. Appl. Phys., 54(10):5657, (1983). X. Jia, A. Jungman and G. Quentin, "Generation and characterization of guided waves on a periodic corrugated plate", Ultrasonics International 89 Conference Proceedings, 1026. B. Poif(:~e "Les ondes evanescentes dans les fluides parfaits et les soli des elastiques", J. Acoustique, 2:205, (1989). L.M. Brekhoskikh, "Waves in layered media", Academic Press, New York (1980). L. Sebbag, "Les lois de la reflexion-refraction des ondes planes evanescentes et les ondes d'interface", these, Universite de Paris 7, (1987). S. Nasr, J. Duclos and M. Leduc, "Scholte wave characterization and its decay for various materials", J. Acoust. Soc. Am., 87(2):507, (1990).
639
QUANTITATNE DETERMINATION OF ABSORPTION USING COUPLED
AMPLITUDE AND PHASE PHOTOACOUSTIC SPECTRA
ABSTRACT
D.M. Todorovic, P.M. Nikolic, Z.D. llistovski
Faculty of Electrical Engineering Belgrade, P.O. Box 816, Yugoslavia
The procedure for the quantitative determination of absorption coefficient is a very important problem in the application of the photoacoustic method as a quantitative spectroscopic technique. In this work, a new procedure for the determination of the absorption coefficient using coupled amplitude and phase photoacoustic spectra, measured on two different modulation frequencies is demonstrated. The possibility of this procedure, using semiconducting sample, is analyzed. The photoacoustic results are compared with those obtained using a classical optical absorption method for the same sample.
INTRODUCTION
Quantitative determination of the absorption of the condensed materials can be performed by comparative analysis of the theoretical and experimental amplitude and phase photoacoustic (P A) spectra. Various authors have described a way of using the amplitude P A method to determine the low optical absorption coefficient [1-4]. For higher values of the absorption coefficient, the amplitude P A method gives the results which are quite different than those obtained by the classical optical method.
The main problem of quantitative determination of absorption coefficient on the basis of analyzing only the amplitude of P A spectra is a small linear range for the correlation between the P A amplitude and the optical absorption coefficient.
The phase P A method has been used by Roark et al. [5] for obtaining the quantitative absorption coefficient. They found that the practical limit of absorption using the PA phase was larger for about one order of magnitude compared with the values obtained using the amplitude P A method. On the other hand, the phase P A measurement suffer from large errors for a sample with small absorption due to the contribution of acoustic back ground and of extraneous absorption in the P A cell. In addition the fluctuations of the phase of P A signal depend on a more or less precise location of the modulator of incident light. Thus, Poulet and Chambron [6] have found that a 50 J.Lm displacement of the modulator correspond to a phase shift of 2 deg.
All these problems can be minimized by coupling both amplitude and phase spectra. Burggraf and Leyden [7] analyzed those problems in the case of intense light-scattering for thermally thick samples. In that case the phase measurements were used to make the linear amplitude, taking the advantage of
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 641
the greater sensitivity of the phase measurements. One should notice that small errors in determination of phase produce large errors in the relation for the absorption coefficient when the saturation limit is approached [7]. For a lot of materials, especially for single crystal semiconductors, this procedure does not appear satisfactory from the quantitative relations point of view.
In this work the amplitude and phase PA spectra have been analyzed at two different modulation frequencies of the excitation beam. The sample relation for both amplitude and phase PA response under above conditions, were derived the absorption coefficient, in regard to the spectral characteristic, was then obtained using those relation.
THEORY
Generally speaking, the generated P A signal, can be expressed as a complex function of optical, thermal, elastic and geometrical parameters of the sample, gas and backing materials. The PA response function may be recognized as a vector rotating in a complex plane with angular frequency, corresponding to the modulation frequency of the incident beam:
S(W,Il) = A(W,Il) e-j'P(W,Il) (1)
where A(W,Il) is the amplitude of the response function, 'P(W,Il) is the phase, w is the modulation frequency and II is the absorption coefficient.
It is possible to develop various simplified expressions corresponding to the limiting cases. The thermally thick limiting condition has general analytical apPlicabilit~ because that is not difficult to achieve experimentally for many materials 6-8]. In that case the sample thickness is much greater than the thermal di fusion length (1 » f.L). This limiting case is not very restrictive. The thermal diffusion length corresponding to the frequency of 100 Hz is for various materials usually between 20 and 200 f.LID. In this case the expression for the amplitude and phase are given:
A( w, ll) = C -;==Il::J1.===== j (1lf.L+1)2 + 1
'P(W,Il) = -arctg [ ~rt + 2 ] + 'Pc
(2)
(3)
The expressions (2) and (3) for the amplitude and phase of the P A signal depend on following parameters: the absorption coefficient II the thermal diffusion length f.L (which depends on frequency of modulation w), but does not depend on the thickness of the sample 1.
The constants C and 'Pc are parameters independent of the optical absorption coefficient, which can be eliminated from (2) and (3) if the amplitude and phase P A spectra are measured at two different modulation frequencies. It is assumed that for these two frequencies the optical absorption coefficient does not vary, this means that the sample is optically homogeneous. If the two frequencies used are fl and f2 with thermal diffusion lengths f.LI and f.L2, it can be written:
~~=m==m (4)
The ratio of the amplitudes at these two frequencies is then given by:
n (5)
642
Equation (4) can be solved for Ctf.Ll in terms of m and n:
Ctf.Ll = (m-n2) + j (m-n2 ) + 2(m2- n2)(n2_ 1)
n 2_ 1 (6)
where 1 < n < m. This equation is used for evaluation of Ctf.Ll using the obtained parameters m and n. This value is then substituted into equation (2) to obtain the constant C. Similarly to the amplitude PA signal, the phase correction factor 'Pc can be found by measurements at two different frequencies of modulation. The phase angle difference for these two frequencies is then:
[Ctf.L2+ 2 ] [Ctf.Ll+2]
'Pc 'P2 = arctg Ctf.L 2 - arctg Ctf.L 1 = d (7)
Equation (7) can be solved for Ctf.Ll in terms of m and d to give:
1[m-1 ] J 1[m-1 ]2 Ctf.Ll ="2 t g(d) - m - 1 + 4 t g(d) - m - 1 - 2m (8)
where 0 < tg(d) < (m-1)/(64m2 + m + 1). This equation is used for evaluation of Ctf.Ll using the experimentally obtained parameters m and d. These values are then substituted into equation (3) to obtain the phase correction factor 'Pc e.g. a phase angle difference from the experimental phase data and theoretically assumed value.
These corrected experimental amplitude and phase PA data can be coupled. The equation (2) can be then combined with equation (3) and in that case it is possible to obtain the relation for the absorption coefficient:
(9)
In this relation the experimentally determined parameter Ae and 'Pe, and calculated of parameters C and 'Pc are combined. The equation (9) describes actually a new procedure of quantitative determination of the absorption coefficient using coupled amplitude and phase PA data for two frequencies of modulation.
EXPERIMENT AL RESULTS
The experimental amplitude and phase P A spectra have been measured at two modulation frequencies for various semiconducting materials. In this work the experimental results for a II-VI semiconducting compound GaSe are given.
The PA spectra were obtained using a commercial system GILFORD, model R-1500. The GaSe samples for P A measurements were cleaved from large single crystals. The sample thicknesses were between 100-1000 f.J.ffi. Typical amplitude PA spectra of GaSe for two modulation frequencies (30 and 300 Hz) are shown in Figure 1. These spectra were obtained by dividing the response of the semiconducting specimen with the response of a reference sample (carbon black). The phase P A spectra measured on the same GaSe specimen are shown in Figure 2. The phase spectra were obtained by substituting the sample phase angle of the phase angle of carbon black.
The transmission and adsorption measurements were done using PERKIN ELMER UV /VIS double beam spectrophotometer - Lambda 3. The thicknesses of GaSe samples for these measurement were between 5 and 40 mm. The refec-
643
Figure 1.
f ~ 30 Hz
0,8
r ~ 300 Hz 0 .2
o.q ,1,..6-~~1~.8~~--:2.-l.0n--'----"2.J,;, 2--,--."J2,4
E / eV /
Photoacoustic amplitude spectrum of GaSe on two frequencies of modulation.
tivity was measured on a ZEISS SPM 3 photospectrometer (using an aluminum reference mirror). These experimental results were used to calculate the absorption coefficient . The absorption coefficient of GaSe, calculated from these experimental data for the sample L = 6,33 J.Lm thick, is shown in Figure 3.
DISCUSSION
The experimental PA data are related to the thermally thick sample. The thickness of GaSe sample was L = 791 J.Lm while the thermal diffusion length was J.L = 116 J.Lffi (f = 30 Hz) . For this case the requirement of the thermally thick condition (L » J.L) was met . In the literature it was shown that the thickness of the thermally' thick sample must be about six times larger than the thermal diffusion length l8].
Figure 2.
644
10
5
'-g' 0 v
~ -5 o .s:: a.
it -10
- 15
-20, ,6 1.8 2.0 2.2 2,4
E / eV /
Photoacoustic phase spectrum of GaSe on two frequencies of modulation.
Figure 3.
3000
2500
,2000
i
~ 1500 , tl
1000
500
0 1.4 1.6 1.8 2.0 2.2 2.4
E I eV I
Absorption coefficient of GaSe obtained from experimental transmission and reflection data.
The experimental P A data given in Figure 1 and Fi~ure 2 were used to calculate the absorption coefficient a using equation (9) . The result of this analysis, in regard to the calculated absorption coefficient is shown in Figure 4. The parameters Ae and CPe in equation (9) were taken from the experimental P A data (Figure 1 and Figure 2). The amplitude correction factor C was obtained using equation (2) at each value of the energy of incident light, using the experimentally obtained parameters m and n (equation (4) and (5)). Sometime it can be found that the value of n given by equation (4) outside the range 1 < n < m expected from theory. Under these circumstances, for each energy of incident light, when n lies within the expected range, the mean value of C can be used in evaluating aJtl.
The correction parameter CPc was obtained from equation (3). The value cp(w,a) was taken from the experimental data (Figure 2) and aJt was ob-
Figure 4.
3000
2500
,2000 .. § 1500 , <l
1000
500
?.4 1.6 1.8 2.0 2.2 2 .4
E I eV I
The spectral characteristic of GaSe obtained by analysis of the coupled amplitude and phase spectra on two frequencies of modulation.
645
tained from equation (8), using the experimental parameter d (equation (7) and the experimental data in Figure 2).
This described method gives better results comparing with either amplitude or phase PA methods.
CONCLUSION
A new method of determination of the absorption coefficient using coupled amplitude and phase PA spectra, measured on two different frequencies of modulation, is demonstrated. The comparison of the results obtained with this method and the results obtained with the classical optical method gives good agreement in the range of about 10%.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
646
H. Bennet and R. Forman, II Absorption coefficients of highly transparent solids: photoacoustic theory for cylindrical configurations II , Appl. Opt., 15(5), 1313, (1976). A. Hordvik and H. Schlossberg, IIPhotoacoustic technique for determining optical absorption coefficients in solids II , 16(1), 101, (1977). A. Mc Donald, IIPhotoacoustic determination of small absorption coefficients: extended theory II , 18(9), 1363, (1979). A. Rosencwaig and T. Hindley, "Photoacoustic measurement of low level absorptions in solids II , 20(4), 6060, (1981). J. Roark, R. Palmer and J. Hutchison, IIQuantitative absorption spectra via photoacoustic phase angle spectroscopy (~AS), Chern, Phys. Lett., 60, 112, (1978). P. Poulet and J. Chambron, II Quantitative photoacoustic spectroscopy applied to thermally thick samples II , J. Appl. Phys., 51 (3), 1738, (1980). L. Burggraf and D. Leyden, IIQuantitative photoacoustic Spectroscopy of intensely light-scattering thermally thick samples II , Anal. Chern. 53, 759, (1981). Y. Teng and B. Royce, II Absolute optical absorption coefficient measurements using photoacoustic spectroscopy II , J. Opt. Soc. Am., 70, 557, (1980).
ACOUSTOOPTIC REFLECTION COEFFICIENT FOR BOUNDED BEAMS ON
PLATES USING INHOMOGENEOUS WAVE DESCRIPTION
Van Den Abeele K. and Leroy O.
K.U.Leuven Campus Kortrijk Interdisciplinary Research Center Kortrijk, Belgium
INTRODUCTION
Basic work in the study of reflection of bounded beams on liquid-solid interfaces was done by Schoch [1], showing a displacement of the beam when the angle of incidence corresponds to what is known as the Rayleigh angle. As this simple model did not account completely for different observed phenomena, Bertoni and Tamir [2] developed a more successful analytical model to describe nonspecular reflectivity for liquid-solid at this critical angle of incidence. Pitts, Plona and Mayer [3] have extended Bertoni and Tamir's method to describe nonspecular reflectivity for a bounded beam incident at Lamb-mode angles on a plate immersed in a liquid. Although these results are qualitatively consistent with experimental observations, the method yields to considerable analytical problems when applied to non-Gaussian beam profiles and is only applicable when the resonant conditions prevail.
In order to overcome these inconveniences, Ngoc and Mayer [4] introduced a numerical integration method which uses spectral analysis to represent the profile of any bounded beam as a superposition of an infinite number of plane homogeneous waves with different amplitudes and different wave vectors. The reflected profile can be calculated as a superposition of the incident plane waves reflected from the system. This method can be applied for any layered media and for any range of incidence angles. Recently an alternative method using a repre~entation of the bounded beam profile by means of a finite number of plane inhomogeneous (evanescent) waves with different velocities but incident at the same angle, has been presented by Claeys and Leroy [5]. Also this method can be applied to calculate the reflected profile for any angle of incidence and for any layered structure.
As classical measurement techniques using transducers as receivers could no longer be relied on due to the complete alteration of the incident profile upon reflection, and as collecting data using small transducers is lengthy, investigation of ultrasonic fields and critical angle phenomena by light diffraction was suggested. Knowing that the sensitivity of light diffraction to amplitude and phase variations along its direction of propagation is very high, we defined an easy measurable acoustooptic reflectioncoefficient for bounded beams on plates using the inhomogeneous wave description technique. The same procedure can be applied for transmissioncoefficients and for any layered structure.
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 647
THEORY
The wave equation for a plane wave travelling in zi-direction of the ( Xi,zi)-plane
has two types of solutions: the homogeneous waves
o
¢(Xi,Zi) = A exp(i(kozi))
and the inhomogeneous or evanescent waves
¢(Xi,Zi) = A exp(f3oxi) exp(i(v' k2 + (32 oZi))
where f3 is the exponential decay and A exp(f3 0 Xi) the amplitudeo
For a bounded beam ¢(Xi,Zi) with profile f(xi) in the plane Zi 0 (Figure 1), the boundary problem:
+ k2 ¢ = 0
~i(Xi'O) ~ of (x i) (no big divergence while propagating)
can be solved by expressing the unknown amplitude distribution ¢(Xi,Zi) in terms of a finite number of plane inhomogeneous waves, having the same direction but different velocities v~ = 1.iJ2 / (k2 + f3~):
648
N
L An exp(f3noX i) exp(i(v' k2 + (3~ oZi))
n=O
Figure 10 Geometrical interpretation of the incident and reflected beam profile.
(1)
N
with L An exp(Pn 0 Xi) = f( Xi) for -m < Xi < (l)
n=O
This non-linear system can be reduced to a least-square problem applying the transformation Xi = poln(y) with p > 0 and an appropriate choice of Pn:
Pn = n / p
N
L An yn = f(p oln(y)) for 0 < y < (l)
n=O
or
N
L Bn Ln(Y) = f(poln(y)) for 0 < y < (l)
n=O
where Ln(Y) is the Laguerre Polynomial of order nand Bn is given by least-square theory:
(l)
Bn = f e-y f(poln(y)) Ln(Y) dy o
The unknown coefficients An can be found as a linear combination of Bn.
This procedure can be performed for any kind of profile provided that
- f(poln(y)) is piece wise continuous function
f (l) 2 - e-y f (poln(y)) dy < (l)
o
For a Gaussian profile we already obtain a good approximation by choosing N = 50 (Figure 2).
In an arbitrary coordinate system the representation of the bounded beam in terms of a finite number of plane inhomogeneous waves is given as follows:
N
¢(;) = L An exp( Pn ~.8";) exp( i (.j k2 + /3J ~ko;)) n=O
with ~k a unit vector in the direction of propagation and ~ po ~k = 0
Because of the linearity of the problem, the amplitude and phase distribution of the reflected profile can be found by multiplying each plane inhomogeneous wave component of the decomposition by its reflectioncoefficient R(kx;O), which is essentially of the same form as the one for homogeneous waves :- defined f.i. in [3-4) for half-spaces and plates -, but in which kx has to be taken complex now.
If R( (kn)x; 0) is the reflectioncoefficient for the n-th component with ex-
ponential decay Pn , and (kn)x = .j k2 + /32 0 sin( 0) - i 0 P 0 cos( 0) the re
flected beam propagating in the zr<iirection can be calculated from the following analytical expression:
649
,--------------------
"1 --- ------------------,
0.81 1 1 t 0.6j
~ 0.41
I 0.21
1
o.oC, '" I" '" "" I '" '" '" I~~n'" I " ~n~r;~~' ~, ~, '~I:;' ;:;n::;:--;-;::;~;::;:r:;::-~;::;-:;::;::;::;:;::;::;:::;::;::;J -4
Figure 2.
-3 -2 -1 o 2 3
x / width
Geometrical representation of the numerical approximation of a Gaussian profile by means of Eq.(l) with N = 50.
N L An R((krJx;8) eXP(fJnoxr) exp(i(v' k2 + fig oZr)) n=O
Numerical calculations, shown in Figures 3 and 4 clearly illustrate nonspecular reflection effects at various Lamb angles in the case of a brass plate immersed in water. However, we again emphasize that this method can be applied for any angle of incidence and for any layered structure.
The finite series describing the incident and reflected ultrasonic beam profiles near the interface enables us to define a reflectioncoefficient for bounded beams which can be easily measured acoustooptically. From calculations performed by J.M. Claeys and O. Leroy [6], we know that in the case of RamanNath type diffraction the averaged amplitude of the light diffracted by a profiled ultrasonic wave in the first order is proportional to the integral of the profile along the direction of propagation of the light. This theory applied to both the incident and reflected ultrasonic beam provides us two measurable values connected to the respective profiles.
For the incident bounded beam, the first order intensity corresponds to:
N b 7r L" J (JnY " (b-a) i..J An e dy
n=O a
and for the reflected beam the intensity of the same diffraction order leads to:
650
1.0 & a b c d e
0.8
9 h
~ 0 .6 ::J :::: -a. e '" 0.4
0.2
-4
Figure 3.
1.0 @ a
b c
0.8 d e f 9
~ 0.6 h
::J :::: i -a. e k '" 0.4
0.2
-4
Figure 4.
20 21.5
22 . B 23.5 24 . 05 24 . 3
25 26
- 3 - 2 - 1 o 2 3 4
l[ I width
Calculated amplitude distribution of the reflection of a bounded beam of 3 mm width on a 0.5 mm thick brass plate, considering different angles of incidence around the Lamb-angle 24.05° (frequency = 5 MHz).
41 42 43 44
44 . 6 45 . 2
46
46 . B 47 . 6
4B . 3 49
51
-3 -2 -1 o 2 3 4
l[ I widlh
Calculated amplitude distribution of the reflection of a bounded beam of 3 mm width on a 0.5 mm thick brass plate, considering different angles of incidence around the Lamb-angles 44.6° and 47.6° (frequency = 5 MHz).
651
a
~
..., c Ql
~ ... Ql 0 u I c 0
:;:; U Ql ;;:: Ql ...
1.0
II 0.8
0.6
0 . 4
0 .2
0 . 0
0
b
..., C Ql '0 :;: ... Ql o u
1.0
I c 0.5 o
:;:; U Ql
;;:: ~
( \
10 20 30 40 50 60
8
o . o ~ ______ ~ ______ ~ ______ ~ ______ ~~ ______ ~ __________ -+
o
Figure 5.
652
10 20 30 40 50 60
8
Acoustooptical reflection coeffi cient of a b ounded beam (f = 5 MHz and width = 3 mm), reflected from a 0.5 mm thick brass plate. (a) theoretical results, (b) experimental .
a 1.0
'" O.B .... c qJ
'u 0 . 6 I;: ....
qJ 0 CJ I c 0 0.4 :;:; (,,) qJ ;: ~
0.2
0.0
0 10 20 30 40 5 0 60
8
b
A!:
C 1.0 qJ
'u :;:: 'qj 0 u , 0:::: 0
0 . 5 :;:; u Q) ;: ~
O.O L-------------------------------------------------~--_+ o
Figure 6.
10 20 30 40 50 60
8
Acoustooptical reflection coefficient of a bounded beam (f = 5 MHz and width = 3 mm), reflected from a 1.0 mm thick stainless steel plate. (a) theoretical results, (b) experimental.
653
The ratio of these first 'Order diffraction intensities is certainly a value connected to the reflection effects occurring in the case of bounded beams.
=
Indeed, outside the critical angles, this coefficient will be unity because:
[a,bl = [ar- brl
R( (kn)x; 0) ~ 1 ==} Rbb( 0) ~ 1
which means that specular reflectivity occurs. For critical angles the interpretation gets more complicated. Because of the
phase difference between the two lobes in the profile, (I1)ref becomes smaller while [br - ad becomes larger which causes a minimum for the reflectioncoefficient and is an indication for nonspecular reflectivity. The position, depth and width of the minima in the reflectioncoefficient depends very much on the sample and on the frequency used to examine it.
COMPARISON WITH EXPERIMENTAL RESULTS
Through the years a lot of definitions of reflectioncoefficients for plates and half-spaces were given. Some experimentalists measured the intensity of the reflected beam profile in one specific point, others searched for the position of the minimum in the profile, and again others connected the coefficient to the normalized peak intensity of the first lobe. As no acoustooptic experiments in the line of our theory has been done we choose to compare the numerical results - obtained with the acoustooptic reflectioncoefficient - with the experimental results obtained in Paris (reflectioncoefficient obtained by measuring the amplitude of the reflected beam in one specific point.). As one can see from Figure 5 and 6, the agreement is very good. The advantage of this acoustooptical Rayleigh and Lamb-mode detection technique over the IIpeak intensityll and the llminimum location II coefficient is that only one measurement is needed at each angle of incidence.
WHY INHOMOGENEOUS WAVES DESCRIPTION
As inhomogeneous waves are characterized in space by their amplitude A·exp(p·x), we can interpret the reflectioncoefficient for one such a wave as follows:
If R(kx; 0) is smaller than one, the wave is displaced in the direction of Re(kx).
If R(kx; 0) is larger than one, the wave is displaced in the opposite direction.
Now, all inhomogeneous waves used in the description of a bounded beam have different velocities, depending on the value of Pn. This means that these components also have a different reflectioncoefficient. At plate-mode angles nearly all components are reflected in the same amount, except one specific inhomogeneous wave which has a reflectioncoefficient close to zero. The interpretation we gave to this peculiar fact is that this single component is shifted over a certain distance in the direction of Re(kx) and along this distance a plate-mode wave is excited. Indeed, we can come to this interpretation very quickly because the velocity of this specific IInon-reflectedll component is equal to the velocity of the excited surface wave, which is also inhomogeneous, at the considered Lamb-angle.
654
Between two plate-mode angles all components are totally reflected, so that no component is shifted over the interface and consequently no surface waves are excited. The reflected profile is the same as the incident one.
So by using the inhomogeneous wave description we obtain an easy formalism to explain the generation of the inhomogeneous surface waves at the Lambmodes of the plate.
CONCLUSION
We have shown that the acoustooptic reflectioncoefficient for bounded beams on plates is an efficient technique for the detection of Lamb-modes of plates. Its results can be put on the same accuracy level as for the other definitions of the reflectioncoefficient and it has this advantage that only a reduced number of measurements is needed. The description of the bounded beam profile by means of inhomogeneous wave enables us to give a reasonable explanation for the generation of surface waves on the interface.
REFERENCES
I~I [4]
A. Schoch, Ergeb. Exakten Naturwiss., 23, 127, (1950). H.L. Bertoni and T. Tamir, Appl. Phys., 2, 157-172, (1973). L.E. Pitts, T.J. Plona and W.G. Mayer, IEEE Trans. Sonics Ultrason., SU-24, 2, 101-109, (1977). T.D.K. Ngoc and W.G. Mayer, IEEE Trans. Sonics Ultrason., SU-27, 229-236, 5, (1980). J.M. Claeys and O. Leroy, J. Acoust. Soc. Am., 72, 2, 585-590, (1982). J.M. Claeys and O. Leroy, Acustica, 55, 21-26, (1984).
655
REFLECTION AND TRANSMISSION CHARACTERISTICS OF AN
ALBERICH-TYPE ACOUSTIC BARRIER
Vidoret Patrick
G.E.R.D.S.M. Le Brusc 83 140 Six-Fours, France
INTRO DUCTION
Acoustic barriers can be made with arrays of inclusions periodically spaced and embedded in a plate (Alberich). Their frequency behavior conditions their use as reflectors or absorbing coatings. Generally the study of reflection and transmission coefficients when the structure is insonified by an incident wave, allows to characterize the barrier.
These structures have been studied by multiple scattering methods in case of simply periodic arrays (with one period only). The result is given in an analytical form, not always easy to use because of the slow convergence of series of special functions [1-4], specially for doubly periodic arrays in 3-D or vectorial fields.
Our aim is to present a numerical method based on an integral representation of the field inside the Alberich. We study the acoustic pressure and use a Floquet mode decomposition outside the plate. One can find in the next paragraphs, the problem statement for doubly periodic array of inclusions. We first give the pressure representations in the different media, then the boundary equation formulation and the numerical resolution, and finally results concerning simply and doubly periodic array.
PROBLEM STATEMENT ..
A plane time-harmonic wave denoted Pinc( r) is incident on a doubly periodic array of spherical cavities in a slab (Figure 1). The cavities are of equal radius Ro. There are located in the plane z=O, every 2a along the x-axis and 2b along the y-axis. The slab thickness is 2h. The incident wave arrives on the upper side of the slab (z > h) and is written with a time dependence in e-itDt:
Pinc(;) = ei(kxx + kyy - kz(z-h)) (1)
where the incident pressure amplitude is assumed to be unity, without loss of
generality. kx, ky and kz are the components of the wavenumber k in the usual cartesian coordinate system.
We denote PR. and PT' the reflected and transmitted waves which are
respectively send back in the upper half space (z > h) and transmitted in the lower one (z < -h).
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 657
z
2h
x 2b
Figure 1. Periodic array of inclusion.
Periodicity of the structure and plane incident wave gives a special property to all fields satisfying the Helmholtz equation: the quasi-periodicity relation. It states that:
( ) ( ) i(k 2ma + k 2nb) p x + 2ma,y + 2nb,z = p x,y,z.e x y (2)
. () -i(k x + k y) DenotIng that p x,y,z . e x Y
represented by a Fourier series in the x and Application of this result to the reflected and the Sommerfeld radiating condition, gives:
is a periodic function, it can be y coordinates.
where:
658
transmitted waves, together with
m,n
~ . ei(am x + f3n y + 'Ymn4 (z+h)) -'-' tron
m,n
m E z
n E z
(Sx+)
x
z (Sz+)
(Sy+)
h
(SO) ''t--.....,'0 '. ---1-.( ;
\. I ~ ___ ~r_ I
• -h
y
Figure 2. Elementary cell representation.
(3)
(4)
(5)
(6)
and "12 - k2 a 2 (32 mnv - v - m - n (7) II is an integer used to pick out the different media and kv is the
wavenumber of medium II: kv = w/cv' Expressions (3) and (4) represent li
near combinations of planes waves. If 'Ymnv is real the corresponding plane
wave, called mode (m,n), propagates in the direction (am, (3n, 'Ymnv)' Else
'Ymnv is purely imaginary and the corresponding mode decays exponentially
from the surface of the Alberich. rmn and tmn are the reflection and transmission coefficients of the mode (m,n).
The following step of Our procedure consist in partitioning the slab in elementary cells containing only one inclusion (Figure 2). Addition of suitable boundary conditions on surface Sx., Sx-, Sy. and Sy_, allows to study the pressure in one cell only.
Volume V 2 (resp. V 3) of medium 2 (resp. 3) is delimited by the surface S2 (resp. S3), such that:
S2 = Sx. U Sx- U Sy. U Sy_ U Sz. U Sz- U So
and : S3 = So
.. In these domains, the pressure Pv( r), where 11=2,3, is expressed with an
Helmholtz Integral Representation, as:
(8)
.. .. where n' is the normal vector to the surface Sv at the point r' directed
toward the exterior of V v
gv is the free space Green function, given in three dimensions by:
The ev coefficient takes the value
Sv (and Sv is a regular surface) and 0
.. 1 if r is inside
elsewhere.
(9)
if on
Then, we write classical boundary conditions in order to solve the problem and find the reflection and transmission coefficients.
The first ones are the continuity conditions. They state that acoustic pressure and normal velocity are continuous on interfaces between two different media, that is on surfaces Sx., Sx- (with m=l and n=O) and Sy., Sy_ (m=O, n=O).
Finally, we must connect integral representations to pressures on interfaces. They provide integral equations, the unknowns of which are the pressure and its normal derivative on the surface of the elementary cell. Irregular frequencies are avoided due to small dimension inclusions and low frequency range practically employed. We only write the integral equation concerning medium 3, the corresponding one for medium 2 being of analogous form:
(10)
659
RESOLUTION
In order to solve the preceding equations and to determine the reflection and transmission coefficients, we discretize the elementary cell surface (corresponding to S2) and break off the series which give the reflected and transmitted waves (expressions 3 and 4). We also take as extra-unknowns the pressures and their normal derivatives at each discretization point .
Continuity and quaSi-periodicity relations together with integral equations are written at these points. The boundary Sv is discretized with elementary
triangles, the tops of which represent the discretization points. On each triangle, the pressure and its normal derivative, are expressed in terms of their values at the tops, by means of linear shape functions . Thus integral equations like (10), result in linear equations with the pressure and its normal derivatives at discretization points as unknowns, and Green function integrals over each triangle as coefficients. These last ones are performed by a Gauss quadrature algorithm.
o,---~~--~----------------------~
-1)
-25
-35
o
Figure 3.
660
\ ~:':~ ... :.,;:~.,. l -r -w .... "
JI-' ....•. -':-. . " '., , , \, I \ \, I .... \. \
.... '. \
. - REFLEXIOH COEf H.LB.M.(48pW
2 4 6
"' \ \ \ \\ "' \ \ ~_"V \ '·1 r~' __ •
': II ~' \1 \ / . ~.# ' . \ I/"I
': \ (..... ; i'\ I / ' ! \\ I .. I :11 ... ; ",/ i \i I , .
:i f j
l:.' f ; i i .; i
\j ( ! t
i I j i . i I ! i ! ; ! i ! , ! i !i •• II f I
8 1) 12 14 16 18 20 .~
fREOWCY (Hz)
... - ReO) MOL TIPLE DlFFllACnON ME
- - MODCIl> EXJ'. HYDRO AT 20_
-.- MODCR> ZXP: HYDRO AT 40am
Air cylinders embedded in a rubber-like plate submerged in water: afRo = 5. Comparison with analytical and experimental results (array of cylinders) .
tOO'~~~------------------~,---r--------~
0.90
OJ!O
0.70
.-0..60
~ ~ 0.50 '-'
d > W ...J 0.4(1
o.JO
0.20
0.10
Figure 4.
o 2 4 6
I I
I I
I , I
I I
f
/ I
I
, \ , , , , \ \ \ \ \
" , , " ... ... ...
8 ~ n ~ ~ ~ ~
-10' FREOI..U-JCY (Hz)
- JUO)
_ •• 1l.Q)
-- TtO)
_ .- TCl)
Air cylinder array in a rubber-like plate submerged in water: afRo = 5. Absolute values of reflection and transmission coefficients versus frequency for a cylinder-array insonified by a normally incident plane wave.
Green's kernel (expression 9) becomes singular when the observation point (~o in (10)) is located on the integration triangle. To perform its integration one transforms each triangle to a square so as to regularized the Green kernel by multiplying it with the Jacobian of this transformation.
Thus all equations written previously bring a linear system which resolution gives the reflection and transmission coefficients for each mode (m,n) . These coefficients must satisfy the energy conservation law:
L [l r mnl 2 + Itmn l 2] •
(m,n)EP cos 00
= 1 (11)
where P is the set of couples (m,n) corresponding to propagating modes in the direction Omn such that: Imnl = k t • cos Omn (media 1 and 4 are supposed to be identical).
661
RESULTS
In our applications, the outside media (1 and 4) are filled with water (p=1000 kg·m-3, c=1500 m·s-1), and inclusions (medium 3) with air (p=1·3 kg·m-3, c=340 m,s-I).
We first present results concerning array of cylinders. In this case, medium 2 is filled with a rubber-like material in order to have results in the experimental conditions. The problem can be reduced to a two dimensional one and discretization holds on contours instead of surfaces. Circular cylinders of radius Ro=l cm, are spaced from each other with distance 2a=10cm. Reflected and transmitted pressures are expressed as Fourier series along one direction only. These series are broken off after 17 terms (8 modes apart from the zeroth). Seventeen is also the number of points taken on segments delimiting the elementary cell in order to obtain a square linear system of equations.
Our results are first compared with analytical (3-4) and experimental ones (Figure 3). One will note that they are in good agreement. Figure 4 shows the
Figure 5.
662
~I,---~~------------~--~------,
0.10
- JlCO.1)
- RQ,O)
-- RaJ)
--RQ.O>
.. . TtO,O)
o 5 10 15 20 25 30 35 40 .-q
fREOl£t.Cy (Hz)
Air-sphere array in water: afRo = 2.00, a=b=h=2.5 cm, #modes = 9x 9, normal incidence. Reflection and transmission coefficients versus frequency (array of spheres) Ro=1.25 cm.
o
Figure 6.
10 20 30 40 50 60
Ancja tetaO (Deg) 70 80 90
- MOD(KC o. 0)] MMAX"'4.NSO-Ut
- MOD{RC L 0)]
- - MOD{RC 1. 1)]
-- MOD{RC-l. 0>]
MOD[RC-l.-O]
-_. MOD{RC-2. 0)]
.... BILA.N NRJ OllDRB Co.O)
_. BIL.Uf NRJ TOTAL
Air-sphere 2-D array in water: P=O, F=30 kHz, a=b=h=2.5 cm, Mmax = Nmax, a/Ro=2.000. Reflection coefficients versus angle of incidence (array of spheres) .
absolute values of the reflection and transmission coefficients for modes 0 and 1, in the case of normal incidence. Cylinder resonances are found at about 11 kHz and 17 kHz. At the first cut off frequency of the grating (f1=15 kHz), modes 1 and -1 becomes propagating (at grazing incidence). The structure is very reflective beneath 10 kHz.
The two last figures concern an array of spheres submitted to normally incident plane wave. The array is not embedded in a plate. For this reason, medium 2 is considered as filled with water, like media 1 and 4. Each sphere has a radius of Ro=1.25 cm and is discretized with 216 triangles, the tops of which bring 110 points. Each of surfaces Sz+ and Sz - must have a number of points equal to the number of modes in the truncation of the reflected and transmitted pressures (expressions 3 and 4) . We have taken 9 modes for m and n, that is 81 points on each surface Sz+ and Sz-.
Figure 5 shows the most significant reflection coefficients (Roo,RIO,R ll and R20) and the zeroth transmitted one (Too). The first cut off frequency happens at 30 kHz and a sphere resonance takes place at about 9 kHz.
Figure 6 shows the reflection coefficients of various modes versus angle of incidence (0 with the z-axis) .
663
CONCLUSION
We have presented a numerical technique based upon integral equation discretization, in order to study the diffraction of a plane incident wave by a periodic array of inclusions. In the simply periodic case (grid problem), results are in good agreement with experimental ones and those obtained with multiple diffraction methods. For doubly periodic structures, results are still satisfactory, and specially with phenomenon that can be predicted by theory (sphere resonances, cut-off frequencies). In both cases, we have obtained a power balance (expression 11) equal to 1 with less than 1% fluctuation. Such a method could be applied to various forms of obstacles, needing only to discretize their corresponding elementary cell surface and no more theoretical developments. It can also be generalized to elastic media without great difficulties.
REFERENCES
[1]
[2]
[3]
[4]
664
Dumery G, IIContribution a la theorie des reseaux d'objets diffringentsll, These Doct. es Sciences, Universite d'Aix-Marseilles, 234 p., 1967. Bringham Gerald A., Libuha John J., Radlinski Ronald P., IIAnalysis of scattering from large planar gratings of compliant cylindrical shells II , JASA, Vol. 61 , n'l, pp. 48-59, 1977. Audoly C, IIEtude de barrieres acoustiques formees de reseaux d'obstacles resonnants ll , Nouvelle These: Universite de Toulon et du Var, 109 p., 1989. Achenbach J.D., Lu Y.C.,Kitahara M., 113-D reflection and transmision of sound by an array of rods II , J. of Sound and Vibration, Vol. 125, n'3, pp. 463-476, 1988.
ACOUSTO-OPTICAL FILTRATION OF ELECTROMAGNETIC RADIATION
IN ULTRAVIOLET REGION
Vitaly B. Voloshinov
Department of Physics Moscow State University 119899 Moscow U.S.S.R.
INTRODUCTION
The report is devoted to theoretical and experimental investigation of spectral filtration of electromagnetic radiation using acousto-optical interaction [1,2]. Acousto-optical devices of spectral analysis are successfully used in applied physics and technology. These devices are based on tunable acousto-optical filters. The filters have some advantages in comparison with spectral apparatuses of traditional types. The major advantages are rapid tuning in a wide band of optical wavelengths with a possibility of electronic regulation of the transmission. Acousto-optical filters provide with processing of signals in a narrow band of frequencies as well as in a number of spectral bands simultaneously. It is possible to achieve rapid switching and commutation of channels during the filtration. Therefore acousto-optical filters may be considered as reliable and universal devices.
ACOUSTO-OPTICAL FILTERS
The principle of operation of acousto-optical filters is based on diffraction of electromagnetic radiation on a diffractional grating. The grating is created by an acoustic wave propagating in a crystal medium. Spectral filtration of the radiation R = ).16), increases with the number of the grating periods, and spectral band of wavelengths 6), is in reverse dependence upon interaction length I of light. and sound. As a result higher resolution R will be achieved in filters with large in size crystals. .
Recently there has been considerable interest in filtration of radiation in ultraviolet region [3]. Quartz single-crystals due to high transparency in the ultraviolet and large sizes seem to be suitable for filtration purposes. A number of acousto-optical filters based on collinear diffraction in Quartz were fabricated and examined [3-5]. Interaction lengths in filters were defined by the dimensions of the crystals along X-axis L = 5 - 16 cm .• Corresponding spectral bands of wavelengths 6), were less than 6), = 10 A in visible region and up to 6), = 1 A in the ultraviolet. The only disadvantage of collinear filtration in Quartz crystals is a small value of acousto-optical figure of merit M2 of the material. Therefore transmission coefficients of the examined filters were low T < 4% with drive electric power levels P = 1 Watt.
A certain improvement of transmission parameters of Quartz filters was achieved in our experiments using light and sound interaction geometry examined by Kusters, Hammond and Wilson [6]. Corresponding wave-vector
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 665
Figure 1.
-=:::._-::!'-___ -JL--___ ............ [010]
Wave-vector diagrams of acousto-optical interaction in a - Si0 2 single-crystal. - - - Direction of optical energy flows; _.- direction of acoustical energy flow.
diagram is shown in Figure I, where t, kd' and K are wave-vectors of incident, diffracted light, and ultrasound. The main peculiarity of the interaction geometry is collinear direction of propagation of optical and acoustical energy flows. As shown in Figure I, phase velocities of interacting beams are not collinear in principle. However, selectivity of diffraction is the same as in the collinear case. Acousto-optical figure of merit M2 for this type of the filter is higher than for the previous one. Besides, interaction lengths of light and sound due to collinear energy flows can be made long enough, resulting in high spectral resolution.
Acousto-optic cell configuration was based on interaction geometry presented in Figure 1. Interaction length was equal to 1 = 8 cm, and piezoelectric transducer dimensions 0.2 x 0.2 cm2. Tuning curve of the filter '\(f) and frequency dependence of acoustic power of the transducer Pa(f) are shown in Figure 2. Preliminary investigation 9f the acousto-optical cell made with a He-Cd laser light source (,\ 0 = .;1416 A) showed that spectral band of the device was equal to b'\ = 2.7 A, transmission coefficient T = 5% (P = 1 Watt), and minimum quick-action of the cell r = 20 /Lsec. Tuning band of optical frequencies was defined by P a( f) curve of the transducer.
Figure 2.
666
6000 4-b
.i\, ~ Pa,arb. units
4-()00 2
100 14-0 180 f, MHz
Parameters of the filter. (a) Tuning curve; (b) frequency dependence of acoustic power in the cell.
Tuning of the filter to the ultraviolet region increased the transmission T and spectral resolution R values as (1/).) and (1/). 2) respectively.
SIGNAL-TO-NOISE RATIO DURING THE FILTRATIONS
Optical output signal of an acousto-optical filter, corresponding to diffracted light, is orthogonal to incident light. Usually output signal selection is made using a polarizer. It is evident that some defects of optical elements and the crystal of Quartz, birefringence of a-Si0 2 and etc. result in a background appearance together with the diffracted beam.
If spectral intensity distribution of the incident radiation is S(>.), and &). = (}.z--).1) is a photodetector spectral sensitivity band, usually equal to the tuning band of the filter, filtrated signal intensity corresponding to diffracted light is equal to
In the above equation T().) is a transmission function of the filter. The transmission function is different from zero in a band of wavelengths 8),.
Background signal intensity may be expressed in a similar form
where Kn is a parasitic transmission coefficient of the optical system. Usually a photoelectronic multiplier is used as a photodetector. Then the influence of In optical noise signal will generate fluctuation noise Sn = {lIn, where (l describes noise properties of the multiplier.
During the filtration of ultraviolet radiation spectral band of the filter was narrow 8),« &).. Parasitic transmission on each optical wavelength ). 0 was characterized by a low Kn value in comparison with T(>.o). However noise signal In exceeded Is due to difference in spectral bands &). and 8),. Thus rapid tuning of the filter all over the band of wavelengths &). = ().2-).1) was accompanied by a low signal-to-noise ratio SIN on the output terminals of the multiplier.
In order to achieve better selection of the useful signal it is necessary to use a low frequency amplitude modulation of ultrasonic intensity. By means of synchronous detection of the output voltage of the photomultiplier signal-to-noise ratio was increased
jl'A2 It (>.-)'o)S(>.)d)'
SIN = __ A ...... I ____ _
(l8F f A2Kn (>.)S()')d)' Al
where It is coupling coefficient, F and 8F - frequency and frequency band of synchronous detection. Synchronous detection band defines the real quick-action of the whole system of filtration.
Experimental investigation showed that transmission coefficient of the examined filter in the ultraviolet wa~ equal to T = 10 - 15%, spectral tllning band of wavelengths &). = 1400 A, and spectral resolution 8), = 1 A. In accordance with the above mentioned considerations satisfactory SIN values could be achieved in the system with a narrow 8F frequency band only. However, it is evident that narrow bands of synchronous detection correspond to
667
Figure 3. Block-scheme of the experimental system.
a dramatic increase in time of measurements allover the tuning band. Generally speaking, this is a disadvantage of the system.
BLOCK-SCHEME OF ACOUSTO-OPTICAL SYSTEM OF FILTRATION
Investigated peculiarities of operation of the filtration system were taken into consideration during the design of the device. Block-scheme of the device is presented in Figure 3. Optical scheme includes a light source I , a collimator 2, and a diaphragm 3. Acousto-{)ptic cell 4 is situated between a polarizer 5 and an analyzer 6. The diffracted light in a narrow band of wavelengths is directed to a photoelectronic multiplier 8 by a lens 7. Two Glann-Taylor prisms 5 and 6 are used as polarization selectors.
Electrical scheme of the system consists of high-frequency blocks for electric drive signal generation and a low-frequency part for analogous processing of electrical signals of the multiplier. A high-frequency generator 9, a low-frequency modulator 10 , and a system of regulation and control of electric drive power 11 as well as a wide-band amplifier 12 are used in the high-frequency scheme.
Signal processing part includes a synchronous detector 13, a low-frequency filter 14 with a cut-{)ff frequency t5F = 0.2 - 5 Hz, a logarithmic translator 15, and an electric signal generator 16 . Frequency of the generator 16 is equal to F = 10 kHz. Analogous output electrical signal is proportional to relative spectral intensity A of the optical signal . Output signal values in logarithmic scale are measured and displayed by a recording block 17.
EXPERIMENTAL RESULTS ON FILTRATION
In order to confirm the advantages and possibilities of acoustO-{)ptical filtration, control laboratory measurements were performed. Radiation spectra of mercury lamps were investigated. Experimental results are shown in Figure 4. Precision of the measuremeIJ.ts of discrete lines in the spectra was characterized by an error less than 0.5 A . Time of tohe recording of a single spectrum in a wavelength band ll'\ = 2600 - 4000 A was equal to t = 4 sec with a signal-to-noise ratio SIN == 60 dB .
The apparatuses were used for outdoor measurements of radiation spectra of sun in the ultraviolet region. The spectra were analyzed when the device was mounted on the surface of the earth . Corresponding experimental data are shown in Figure 4. Taking into consideration low intensities of direct sun
668
Figure 4.
o
- 20
A • dB
a
-60
3000
A. Z 3500
Measured spectra of radiation. (a) Spectrum of a mercury lamp; (b) spectrum of direct radiation of sun.
radiation, the time of measurement of the spectrum was increased to t = 80 sec. These losses in time were tolerated in order to achieve the desirable signal-tjl-noise ratio SIN = 60 dB. Spectral data obtained in the experiments were used for calculations of concentrations of gases in the atmosphere of the earth. For example, absolute concentrations of ozone in the atmospheric column were estimated as well as variations of the concentration with time could be investigated. These measurements gave information about local ecological condition of the atmosphere of the earth.
CONCLUSIONS
It can be stated as a resume that acousto-Dptical devices may be used for spectral filtration purposes. However, resolution and transmission parameters of the devices are coupled with processing time of the system. In order to get high values of resolution, each spectrum must be measured during a relatively long period of time.
ACKNOWLEDGEMENTS
The author wishes to express thanks to L.N. Magdich, I.B. Belikov and A.B. Kasjanov for support in experimental work and discussions.
REFERENCES
[1]
[2]
S.E . Harris, S.T.K. Nieh, and D.K. Wilson, Electronically Tunable Acousto-Optic Filter, Appl. Phys. Lett., 15:325, (1969) . I.C. Chang, Tunable Acousto-Optic Filters. An overview, SPIE 90, Acousto-Optics, 12, (1976).
669
[3]
[4]
[5]
[6]
[7]
670
P. Katzka and I.C. Chang, Noncollinear Acousto-Optic Filter for the Ultraviolet, SPIE 202, Active Optical Devices, 26, (1979). V.B. Voloshinov, I.V. Nickolayev, and V.N. Parygin, Collinear Acousto-Optical Filtration in Quartz Crystal, SOy. Moscow Univ. Bulletin, Phys. and Astron.) 21:42, (1980). V.B. Voloshinov, Tunable Photometer on a Base of an Acousto-Optical Filter, Proc. 4-th All-Union Conf. Photometry and Metrology Maintenance, USSR, Moscow. 137, (in Russian) (1982). J.A. Kusters, D.K. Wilson and D.L. Hammond, Optimum Crystal Orientation for Acoustically tuned Filters., J. Opt. Soc. Amer., 64:434, (1974). I.B. Belikov, V.B. Voloshinov, A.B. Kasjanov, and V.N. Parygin, Acousto-Optical Spectral Filtration of Radiation in Ultraviolet Region, SOy. Tech. Phys. Journ. Lett., 14:645, (1988).
ON THE THEORY OF ACOUSTOOPTIC INTERACTION IN
FABRY-PEROT CAVITIES
A.I. Voronco, Ye.G. Isajanian and G.N. Shkerdin
Institute of Radio Engineering and Electronics U.S.S.R. Academy of Sciences Moscow, U.S.S.R.
INTRODUCTION
Recently growing interest has been directed to investigation of Fabry-Perot cavities (FPC), which possesses important properties for developing systems for optical information processing. The intensity of the radiation that has passed through the FPC in the absence of electromagnetic absorption is defined by the initial linear phase detuning 'Po from the precise resonance conditions in the FPC. A maximum value of K, = 10/1, is reached at 'Po = 21m (n=O, ±1,
±2, .. ), where 10 and I, are the intensities of the transmitted and incident
radiation, respectively. When an acoustic wave (A W) propagates along a crystal it induces in the
latter a diffraction grating which moves at the velocity of the A Wand on which electromagnetic waves (EMW) are diffracted. If the crystal is placed in a FPC, then the A W modulates the phase 'Po (due to modulating EO) and it is clear that under certain conditions acoustooptic interaction (AOI) may produce a substantial influence upon the dynamics of propagation in the FPC. In the present paper such problem is theoretically solved for arbitrary AOI efficiency inside the FPC.
THE METHOD OF CALCULATION
We examine the case of collinear AOI in a FPC. Suppose that a travelling
A W having wave vector q and frequency [) propagates along the z axis
and a plane monochromatic EMW having wave vector k and frequency w is incident on one of the FPC facets from outside (z ~ -L) in the same direction. Here an assumption is made that the Bragg approximation of EMW diffraction on sound that is k ~ q/2 is fulfilled. The medium permittivity E(Z,t) in a general form taking into account the A W modulation is expressed by formula:
E( Z, t) (1)
Here EO is the medium permitivity in the absence of A W, E±l are the am
plitudes of the Fourier harmonics in the permittivity connected with sound, a
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press. New York, 1991 671
is the nonlinear optical coefficient. The EMW and A W absorption in the medium are neglected.
In the slow amplitude approximation the second-order wave equation is reduced to a set of two coupled differential equations of the first order that have the form of
fNo
i 8x + fN -1
-i-8x
(aU) + 1] F~L) V 0 = 11 exp(-iOt) V-1 1 + (aU) + 1] FNi) V- 1 = 11 exp(iOt) Vo
(2)
where VO,-1 are slow amplitudes of the forward (0) and backward (-1) waves in the FPC. The length L» A. x = z/L and inside the FPC changes from -1
j E' 1 <-1 to 0, 11 = - 4 kL is the parameter of efficiency of EMW diffraction on
EO
sound over the FPC length L; aU) = (E OW2/ c2 - q2/4) L/q is a parameter ex
pressing the de~ree of fulfillment of phase matching condition of the AOI in the FPC; E+I = I E+ll exp(i'l/J); for travelling AW we have already 'l/J = 'l/Jo-Ot;
1] ~ 0:/ 2E o x kL; F~L = 21V_112 + IVoI2; FNl = 21Vol2 + IV-d 2. The solu
tion of equation set (2) for slow amplitudes VO,-I is written in the form:
00
(3)
The boundary conditions have the form:
00
x = -1 (z -L) 1 0)
(4) n=O n=O Bo = -R Ao x = 0 (z
where T=l-R 2 is the EMW transmission coefficient at the boundaries z = -L, 0 (here Rlz=_L = Rlz=O is assumed).
For the EMW transmission coefficient K, taking into account the first
boundary condition of Eq.(4) we obtain
T2 K,
00 (5)
L (_l)n (An + R eiCPoB n)
n = 0
This is the principal expression which is solved through numerical calculations. The recurrent relations between the coefficients Aj and Bj have the form:
672
n
l.=.i.l [-iOt A ~ A ] An<! = n+ 1 /I e Bn - ali) n - 1/ i..J Hk n-k
k=o
k
where Hk=Ck+2Dk, Gk=2Ck+ Dk, Ck = LAm A~-m
A* d B~ j an J
m=O
are complex conjugate amplitudes Aj , Bj .
THE MAIN RESULTS
Linear Case
(6)
A general analytical expression for the EMW transmission coefficient in the FPC K~ under AOI is obtained:
K~ (7)
The dependencies of K~ as functions Ot that were calculated on the
basis of expression (7) are shown in Figure 1.
Nonlinear Case
Generally in this case the system (6) can be solved only numerically. The only exception is the case v « 1 when analytic solution of this system can be found (see below). The analogous dependencies of K~ for this case are shown
on Figure 2.
Figure 1.
Kr .t
o.s
~i
Dependencies of K~ of linear FPC as a function of Ot
for R=O, a=O and different values of v and CPo. a) v=2.944, CPo=7f; b) v=0.5, CPo=7f/lO; c) v=0.05, CPo=7f/lO.
673
Figure 2.
DISCUSSION
Linear Case
Nonlinea,. easo
K~ 1.
( B )
- U,)
o 3 6
Dependencies of K~ as functions of Ot for R=0.9;
L=O.lcm; a=10-7, 10=105 and different values of the following parameters: (A) a) v = 0 .05 , CPo = 7r/l00; b)v = 0 .5, CPo = 7r/l00. (B) a) v = 1.00, CPo = 7r/l0; b) v = 2.944, CPo = 7r/l00.
It is convenient to examine two particular cases : v < 1. In this case the contribution to EMW round trip phase inside FPC made by AOI is small and strong modulation of the transmitted radiation by sound is possible only at low values of 1 CPo 1 $ v. In the approximation 1 R 1 N 1; 1 CPo I, v «1 Eq.(7) takes the form :
1 1 K~ 1 = -----r--=-----.,-,;
1+(2F/7r)2 [~ _ v cos(Ot)] 2 (8)
where F = 7rR/(I-R2) . From this equation it may be seen that K~ reaches a
maximum at cos(Ot) = ICPol/2v (which is possible if v ~ Icpol/2 when K~ ~ 1). Note that strong modulation of the transmitted radiation produced by
the A W at 1 CPo 1 f. 0 occurs at v ~ 1 CPo 1/2 ~ (1-R2) which corresponds to
strong AOI over the length L N L/(I-R2), which is of the order of the effective length of EMW propagation within FPC taking into account multiple reflections from the cavity mirrors. At v > CPo/2 resonant EMW transmission through the FPC occurs twice during the sound period (when cos(Ot) = CPo/2), i.e. two high peaks having K~axN 1 and modulation depth of the order of
90% (Figure 1) appear.
v ~ 1. In this case AOI over the length L is effective. As a consequence, as the analysis of Eq.(7) shows, we have almost always K~« 1 at R N 1 for
674
any 'Po with the exception of 'Po ~ 7r(1+2n) [n=O, ±1, ±2, ±3 .. ] when cos2( 'Po/2) « 1. At 'Po = 7r (1+2n) the A W causes appearance of high peaks in the transmitted light at moments when cos('Po/2 - Ot) = -1. At 'Po = 7r the denominator in Eq.(7) acquires the form [chv (1+R2) + 2R shv sin(Ot)J2. If this occurs at Vrnax = 19[i+~] then at moments when sinOt = -I, we
have K!}lax= 1. Thus, in the present case the A W brings about appearance of
one transmitted light resonance peak during a single sound period, with the peak occurring in the region of maximum detuning from the FPC resonance frequencies when 'Po= (1+2n)7r (see Figure I, curve (a)).
Nonlinear Case
v (1. In this case analytical expression may be derived for the transmission coefficient K, which will have the form similar to that of Eq.(8)
with 'Po+2TJ V3(0) substituted for 'Po. There is a qualitative agreement
between the results of numerical calculations and analysis of Eq.(8) taking into
account optical nonlinearity. At I 'Pol> y'3(1-R2)/R = S and v« (1-R2) the AOI produces weak modulation of the radiation transmitted through the FPC (see Figure 2.A) with a small modulation depth being located during the sound period in the branch of weak or strong transmission. The system can be bistable when I 'Po-vi > S. This is a condition necessary for realization of optical bistability in the cavities of the type being considered. However when the AOI is effective in the FPC, i.e. 1 > v > (1-R2) and v ('oJ 'PNL which
corresponds, just as in the linear case, to strong AOI in the FPC with
L ('oJ L/(1-R2) and 11 ('oJ v/(1-R2) the A W produces strong modulation of K,(I,) (see Figure 2). As may be seen during the sound period the system has
time to transit from one stable state to the other and vice versa and the modulation depth reaches approximately 100% for K,('oJ 1 (Figure 2). Bista-
bility may be observed even at 'Po < S though provided that I 'Po-vi > S. Summarizing the results for the v« 1 case it may be said that provided v > 1-R2 the EMW phase modulation by sound produces strong effect on the dynamics of the system behavior if the conditions I 'Po-vi > S. Summarizing the results for the v« 1 case it may be said that provided v > 1-R2 the EMW phase modulation by sound produces strong effect on the dynamics of the system behavior if the conditions I 'Po-vi > S and I 'Po-vi ('oJ 'PNL = TJIin are satisfied simultaneously. Strong time modulation of the transmission coefficients K, and I, occur when, during the sound period, the dependence K,( t) goes
through the whole hysteresis loop of the dependence K" I, = f(Io) with the
modulation depth ('oJ 100%.
v ~ 1. In this case the problem being considered may be solved via numerical calculations only. Here besides finding a solution, it seems expedient to determine optimum values v; 'Po{L,R) for which the AOI effect on the FPC is maximum. As the numerical calculation results show at small initial phase detuning values 'Po« 7r and v = 1 the nonlinear FPC demonstrates bistability properties (see Figure 2.A). The A W produces time modulation at a deep modulation magnitude so that during the sound period K, = 1,/10 acquires
values from ('oJ 1 to K, « 1 and the dependence K,(t) goes through the all
hysteresis loop of dependence K,=f(Io). With the parameter v > 1 increasing,
first, a trend towards suppression of optical nonlinearity is observed and, second, the modulation depth decreases.
675
CONCLUSION
The results described above show that AOI may be effectively used for controlling the reflected and transmitted radiation of PPCs. The main conclusions may be formulated as follows.
1. A method of investigation of AOI in FPC has been developed which takes into account the dispersion optical nonlinearity.
2. The effect AOI on the EMW transmission through a FPC leads mainly to modulation of the cavity resonance frequencies and once ((jJo rv 7r) or twice ((jJo «7r) during the sound period coincides with resonance. At a constant input signal optical pulses are observed at the output whose duration depends on the value of finesse of the FPC.
3. The optical nonlinearity effect on AOI in the FPC has been studied. It has been shown that if the A01 in the cavity is effective at v $ 1 the A W affects the FPC bistability properties. During the sound period the system jumps from one stable state (high EMW transmission) to another (weak EMW transmission) and vice, versa, while the I~( t) and K~( t) dependencies go
through the all hysteresis loop of the dependence I~, K~ = f(lo)'
676
SECLUDED SUPERSONIC SURFACE WAVES IN THE TRANSVERSELY
ISOTROPIC MATERIALS
Litian Wang, Steinar A. Gundersen and Jens Lothe
Department of Physics, University of Oslo P.O. Box 1048 Blindern, 0316 Oslo 3, Norway
INTRODUCTION
The existence theorems for subsonic surface waves in anisotropic elastic media are well known [1,2,3J. The supersonic surface wave consists of two partial waves instead of three partial waves as is usual for the subsonic surface wave. Alshits and Lothe [4J predict that the supersonic surface waves would exist in a two-dimensional subspace, the so-called space of simple reflection.
We will apply the concept of a space of simple reflection to locate secluded supersonic surface waves. Existence criteria for various hexagonal classes are developed by means of regula falsi method.
GENERAL THEORY
Consider a half-infinite elastic medium, with inwards surface normal along II and an orthogonal triad m, ll, t (Figure 1), and the following displacement field function:
u(x) = Ao.exp[ik(m,x+po.n·x-vt)J (1)
Surface wave propagation then can be studied by solving the elastodynamic equation:
C 82uk - 0 ijkl aXj aX! - (2)
where CJ.jkl = Cijk1 - pv2milllbjk. Substituting (1) into (2), we get:
where (ab) is defined by (ab)jk = aiCJ.jklbl.
The traction force La associated with Ao. is:
The vectors (Ao.,Lo.) fulfill the orthogonality and the completeness relations:
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 677
Figure 1.
C<
crystal
surface
I
y' I
n
m
The geometry of the surface. (mO,llo,to) is the fixed orthogonal triad. The high symmetry axis is to and llO is inward normal, I(J is the angle between m and mo.
1 C<
(5)
The subsonic surface wave solution for can be expressed as a superposition of three partial waves with positive imaginary part in Pa. With appropriate
coefficients I a to satisfy the condition of free surface, we have:
3 3
U(x) L laAaexp[ik(m,x+p(lll'X - vt)], L(x) L laLa 0 (6) a=l a=l
The free--s urface condition (6) becomes I Lail 0, where 1·1 means
determinant, or:
3 3
B = 2i I LaiLaj , det B -Si II LaiLaj I 0 (7) a=l n=1
when B-matrix is a real matrix as follows from (5). This is Stroh's formulation [5].
THE SPACE OF SIMPLE REFLECTION
The space of simple reflection was introduced by Alshits and Lothe [4] in their discussion of the reflection problem in the supersonic region. Because there are two body waves and two partial waves involved, the displacement and the traction can be expressed as:
(S)
The free surface condition is then:
(9)
where the two terms (i) and (r) are incident wave and reflected wave.
678
If there exists a two--component surface wave with the free surface condition fulfilled, i.e., if La and Lf3 are linearly dependent, then [4):
La ± zLf3 = 0 (10)
Thus, La ® La + Lf3 ® Lf3 = 0, L: ® L: + L~ ® L~ = O. From the complete
ness relation (5.3), therefore, the two remaining body waves (Li,Lr) will obey:
(11)
Thus, simple reflection becomes a necessary condition for supersonic surface waves. The space of simple reflection forms a two-dimensional subspace of the four-dimensional space of velocity and orientation, and the supersonic surface wave will constitute a one-dimensional subspace of the space of simple reflection [4).
The existence criterion for supersonic surface wave will be in term of a reduced B-matrix:
2 2
2i L L"iL"j' det B -8i I L Lo.iLo.j I o (12) a=l 0.= 1
where a refers to two partial waves (a,(J) in (9). The supersonic surface wave can be determined by examining the behavior of Br along the space of simple reflection.
Alshits and Lothe [8) have carried out calculations for hexagonal materials, and gave a complete analysis of the transonic states, exceptional states and other related properties. The eigenvalues for the dynamic equation (2) can be expressed as:
(13)
where u = pV2 - c44sin2j?, W = pV2 - c33sin2j? and d = C44+C13, and the upper or lower sign in (13) takes the first subscript when Cl1 > C44 and the second when Cl1 < C44. Then:
-2 [ -C66X 2 L2 2C66P2COSj?
.j2P2U tC44si n2j?
a=1,3 -2 [ 1 C 6 6 Pac os j? 1
La ~ C66X2+C 44 Y"sin 2j? ( 14) .j2poX U+ C44 ygS1ll2j? -C44Po.( Ya -l)sinj?
where X2 Pt. - cos 2j?, Yo. = (Cl1Xa - u)j dsin2j? In the supersonic region, we
have that one pair of conjugated eigenvalues becomes real, (say, P, = -p;;), and
both vanish at the limiting velocity. This enables us to construct the space of simple reflection by setting:
1 = 1,3 (15)
leaving L, and L;; satisfying the condition for simple reflection:
679
L + iL* = 0 , , (16)
Here we regard the L, associated with P, < 0 as an incident wave, and
the L, associated with P, > 0 as a reflected wave. From the expressions of
Lo.'s, one can see that there is no way to have the simple reflection condition
fulfilled with L,( '"f = 2) except along high symmetry directions.
Hereafter, we will take C3 materials (for which C44 > Cl1) as an example. From (15), taking '"f = 3, we can have an analytical expression for the space of simple reflection as:
(17)
Thus, the space of simple reflection can be determined analytically by the equation:
(d+~ 14)(pv2)2+[-2c66(2d+~ 14)cos2)?+(I1-C44( dH 14) )sin 2)?]pv2 -c44I1sin4)?+4dcg6cos4)? - 2c66I1sin2)?cos2)? = 0 (IS)
where 11 = c13 - CllC33, ~ 14 = Cll - C44·
The space of simple reflection defined above is related to the slowness sheet
where the exceptional limiting wave is located. The vectors (L3, L~) are, in
fact, associated with the same sheet where the exceptional limiting wave is
reached when L3 = L~ = o. According to Chadwick's argument [7] (see next
section), there exists at most one exceptional wave on the Sa sheet in the axial sections of slowness surface. Therefore, we consider that the space of simple reflection is associated with the Sa sheet and that it crosses the Sa sheet at most once at the exceptional limiting wave. It should be emphasized that, in the supersonic region, the Br will only be real if the reflection (11) is fulfilled.
CLASSIFICATION OF SLOWNESS SURFACE AND SPACE OF SIMPLE REFLECTION
The slowness surface consists of three sheets denoted as Sa, Sb, Sc according to the polarization [7]. Sb is associated with a purely transversely polarized branch. The other two are polarized in axial planes with concave or convex slowness surface. The classification of the slowness surface and the space of simple reflection is then based upon the following facts.
Fact 1: In the axial planes, the exceptional limiting wave is always located at the Sa sheet and at most one exceptional state exists.
Fact 2: In the axial planes, the body waves associated with the Sb sheet (purely transversely polarized) are never involved in the forming of the space of simple reflection.
Fact 3: The space of simple reflection is associated only with the Sa sheet and will cross the Sa sheet only at a simple (Type 1) exceptional wave.
Fact 1 is based upon the discussion by Chadwick [7], the reasoning for Fact 2 and Fact 3 is in the preceding section.
Consequently, we classify the slowness surface according to the ordering of So. and Sb in forming the outer sheet of the slowness surface, 51:
680
Class 1: Class II: Class III: Class IV:
where \Gab is the direction of the acoustic axis along which Sa and Sb coincide.
Similarly, we can classify the space of simple reflection associated with the four classes of slowness surface configurations. Practically, the space of simple reflection is classified so that the regula falsi method can be adopted. Physically, we are most interested in the following four special points along the space of simple reflection (as shown in Figure 2), i.e., 'A', on axis; 'B ', in transverse axis; 'C', at 2nd transonic state; 'E', at exceptional state. Associated with the four points , we get three intervals where the reflection condition (9) is fulfilled (i.e., Po. and PI?> are complex, P"j is real) and the existence
condition of surface waves (12) can be applied, i.e., A-B interval rClass I,ll), E-B interval Class III), E-C interval Class I,IV).
Knowing the sign of the determinant of the Br-matrix at those four specific points , we can give three corresponding criteria which apply regardless of the possible non-monotonicity of det Br in the intervals . Some regions, like E-A interval, are never present in our consideration because they are forbidden by the stability of materials [7].
Figure 2.
80 40 (a) (b)
60 30
40 20
20 10
o / <p" 0
0
25
20
15
10
5
20 40 60 80 0 10 20 30 40
( ) 80
(d)
60
40
20
5 10 15 20 25 60 80
The slowness surface and the space of simple reflection of various hexagonal materials. The vertical axis is the symmetry axis, the horizontal axis is the transverse axis. The solid curves are slowness sections. The stapled curves are the subsonic surface (Rayleigh) waves, the thin stapled curves are the space of simple reflection. \Gex is direction of the exceptional limiting wave, 'Pss is the direction of the secluded supersonic surface wave. (a), Cd*; (b), Co; (c), Beryl [1]; (d), C3.
681
EXISTENCE OF SUPERSONIC SURFACE WAVES
Along the space of simple reflection using (12, 14, 17), Br is:
. 2 Al = ~ [C44Y3 Sin297 - 4 C66Pl cot 297]
Yl P2 C44(Yl-Y 3)
A3 - - J... [C44P 1 (Y 1-1)2 _ C44COS2f] - Yl (Yl-Y3) P2 Y3
(19)
The existence condition for a supersonic surface wave (12) can then be rewritten as:
det Br = A(A1A3 - 0) = 0 (20)
Noticing that A1A3 - 0 = 0, the sought for non-trivial solution for (20) is A = 0, i.e.,
(21)
We can get the eigenvalues for this Br-matrix (0, A, Al + A3). As A vanishes, we find that all the other three elements of the B,.--matrix (A!, A3, C) vanish simultaneously, implying that all eigenvalues of the B,.--matrix vanish where the supersonic surface wave exists.
In order to establish surface wave criteria in the above three typical intervals, we have to examine the behavior of A at the (A,B,C,E) points so that the regula falsi method can be employed. Using this method, we have to postulate that A in all those intervals of space of simple reflection is monotonic. This is a drawback and gives conclusions that could be wrong by a multi plum of pairs of supersonic waves. In fact, we do have such exceptions, e.g. TiB2.
A(E): At the simple (Type 1) exceptional limiting wave, the B-matrix and B,.--matrix are related to each other. Since the exceptional limiting wave is of Type 1, there is only one pair of conjugate eigenvalues P"( coalescing to zero.
Corresponding, the eigenvectors and collapse into a grazing
configuration with L"( = L~ =} L"( = o. At the Type 1 exceptional limiting
wave, there are two possibilities for eigenvalues Pu, "0' and ""() of B-matrix.
(a): ""( = 0 at the exceptional limiting wave. The other two vanish
simultaneously at the subsonic surface wave, and both become negative at the exceptional limiting wave; (b): ""( = 0 at the exceptional limiting wave, and
the other two remain positive at the exceptional limiting wave, leaving no Rayleigh waves in the subsonic region.
Because the space of simple reflection is associated with P"( and (L"(, L;), the eigenvalues of the B,.--matrix then are yielded from other two eigenvectors,
682
and exclude the eigenvector of L, and L~ in (12). The sign of the
eigenvalues of the Br-matrix is determined by the two remaining eigenvalues of the B-matrix. Thus, the eigenvalues of the Brmatrix will be negative (or positive), if there is (or not) a subsonic surface wave.
A(A): The point A is along the axis, or tp 1r /2. From equations (17, 19), therefore, we have (for C44 > Cll case):
A -i(C44/Pl) (for axial convex case),
A -i(C44/Pl)( Y I-Y3)/ YI (for axial concave case)
A(A) for the axial convex (on the Sa) case is always purely real and negative because PI is purely imae;inary and positive. However, for the axial concave (on the Sa) case, A(A) IS always purely imaginary, so we can exclude the axial concave case and put it into an (E-C) case because of the appearance of a C point.
A(B): At the point B, i.e., along the transverse axis, tp = o. We can deduce that only in the case of CII > CBB > C44 (Class III), is A real:
1 C 1 f(a) = (2-a)2 - 4(1-a)"2(1-a-...2.£)"2 Cll
where a = (CI3+C44)/(CI3+Cll). Consequently, if A(E) and f(a) have the same si~n, there will be a supersonic surface wave.
A(C): As the space of simple reflection approaches towards the second transonic state, the A becomes positive divergent because one pair of eigenvalues (Ph Pi, for the C44 > Cll case) in the equation (19) vanishes at
the cross point C. With the A behavior at all these four key points known, we can
summarize the three criteria under the reservation about monotonicity of A as follows: A-B criterion:
E-B criterion:
E-C criterion:
There will be a supersonic surface wave if f(a) < 0, when the axial re~ion is convex (in Class I, II). If there is (or isn't) a subsonic surface wave along tpex, and f( a) < 0 (or > 0), there will be a su personi c surface wave near of the transverse axis (in Class III). If there is a subsonic surface wave along tpex, there will also be a supersonic surface wave near the exceptional limiting wave (in Class I, IV).
NUMERICAL CALCULATIONS AND CONCLUSIONS
Table 1 summarizes the results for materials of the respective classes. In Class I, Cl materials exhibit an axial concavity in the Sa sheet and
thus the (E-C) criterion employed leaves no supersonic surface wave. Cd* has convex configuration near axis in the Sa sheet and we use (A-B) criterion which gives rise to a supersonic surface wave because f( a) < o.
For the materials of Class II, the Sa sheet lies in supersonic region. Along the space of simple reflection, Br will be complex in the supersonic region except along symmetry directions. Therefore, there will be no supersonic surface wave for the materials of Class II.
As for the material of Class III, we like to emphasize two materials, Beryl rl] and Beryl [2]. The elastic data for Beryl [1] (from [6]) is slightl~ different trom that of Beryl [2] which is from the Landolt-Bornstein (1979, IIIjll pp.39). For Beryl f1], .th~ e~ceptional limiting wa>:e is in the first transonic state, while for Beryl 2] It IS III the second transomc state. The supersonic surface wave exists near the transverse axis in both materials. In Class IV, we calculate three materials (C3, TiB2, TiBiJ We find a
683
Ol co
.j:>
Tab
le 1
. T
he n
umer
ical
cal
cula
tion
of
vari
ous
clas
ses
of m
ater
ials
. (T
he u
nit
for
the
elas
tic
cons
tant
s is
G
Pa,
de
nsit
y gi
ven
in
103
kg
jm3 ,
an
gles
ar
e in
de
gree
)
mate
r.
clas
s P
C
ll
c33
"4
4
Cl2
C
!3
'Pab
'P
ex
A(E
) A
(A)
A(B
) A
(C)
'Pss
C1
4
.0
35
6
8 15
8
28
.09
>
0
+00
Cd
· 8
.64
1
16
5
0.9
1
6
42
41
<
0
>0
5
4.7
8
Co
II
8
.9
29
5
33
5
71
15
9
111
<0
<
0
Cd
Te
III
6.2
6
2.2
6
8.9
1
1.6
3
5.9
29
.1
14
.39
7.
21
<0
<
0
Ber
yll
II
I 2
.75
2
87
.3
24
1.8
7
0.2
9
9.1
7
.3
31
.12
3
0.2
5
<0
>
0
14
.72
B
ery
F
III
2.7
5
29
0
25
7
67
.7
10
7
83
29
.41
31
.82
>
0
12
.75
C3
IV
4
.0
20
3
0
25
4 1
0
22.7
1 2
5.6
4
<0
+0
0 27
.61
TiB
2
IV
4.5
6
90
4
40
2
50
4
10
3
20
27
.65
46
.86
>
0
+00
TiB
; IV
4
.5
69
0
44
0
16
0
41
0
32
0
20
.78
4
3.4
6
>0
+0
0 6
9.5
4
62
.25
P
is
the
dens
ity,
j?
ab
is
the
dire
ctio
n of
aco
usti
c ax
is,
j?ex
is
th
e di
rect
ion
of t
he e
xcep
tion
al
lim
itin
g w
ave,
j?
ss
is
the
loca
tion
of
the
supe
rson
ic s
urfa
ce w
ave
wit
h ve
loci
ty
Vss·
Cd*
: A
n ar
tifi
cial
m
ater
ial
by
chan
ging
th
e el
asti
c co
nsta
nt
C44
of
Cd
from
19
.6
GP
a to
16.0
G
Pa.
TiB~
: A
n ar
tifi
cial
mat
eria
l by
ch
angi
ng
the
elas
tic
cons
tant
C4
4 of
T
iB2
from
25
0 G
Pa
to
160
GP
a.
supersonic surface wave near the exceptional limiting wave in C3 materials. The TiB2 has very similar slowness surface to that of C3 materials, but at l(Jex there is neither a supersonic surface wave nor a subsonic surface wave [7]. For the TiB2, A is not monotonic but keeps positive in the E-C interval, so that non supersonic surface wave is found. After changing the elastic constant C44
from 250 GPa to 160 GPa, namely, denoted as TiB2' without violating stability
of the material, in A appears a dimple below zero near the C point and this leads to the appearance of two zero points, i.e., two supersonic surface waves. Thus, the monotonicity assumption in the E-C interval eventually becomes a reservation for the applicability of the criteria.
The preceding results point towards a generalized theoretical framework, comprising subsonic as well as supersonic surface waves. The central role of the exceptional limiting wave in the subsonic surface wave theory is well known. This paper shows how the exceptional limiting wave points out a space of simple reflection where supersonic surface waves will be located if they exists at all. The secluded supersonic surface wave is deviating in direction from the exceptional limiting wave. The subsonic and the supersonic surface waves are linked together by a common exceptional limiting wave. The investigation leaves some questions open, e.g., lack of monotonici ty of A along the space of simple reflection and the supersonic exceptional limiting wave in the Beryl [2] are situations not fully covered by our criteria. The criteria are established under the assumptions of the monotonicity of A along the space of simple reflection. This leaves some exceptions only for materials of Class IV.
ACKNOWLEDGEMENTS
The authors would like to thank The Norwegian Natural Science and Humanities Foundation for support.
REFERENCES
I~I 4 5 6 7
D.M. Barnett and J. Lothe, J. Phys. F 4, 671~86, 1974. J. Lothe and D.M. Barnett, J. Appl. Phys. 47, 428-433, 1976. P. Chadwick and G.D. Smith, in: Mechanics of Solids, The Rodney Hill 60th anniversary volume (ed. H.G. Hopkings & M.J. Sewell, Pergamon, Oxford), 47-;-100, 1982. V.I. Aishits and J. Lothe, Wave Motion 3, 297-310, 1981. A.N. Stroh, J. Math. & Phys. A 402, 77-103, 1964. V.I. Aishits and J. Lothe, Sov. Phys. Crystallogr. 23, 509-515, 1978. P. Chadwick, Proc. Roy. Soc. London A 422, 23-121, 1989.
685
THE INVERSE ACOUSTIC SCATTERING PROBLEMS FOR
ONE-DIMENSIONAL LOSSY MEDIA
ABSTRACT
Ning VVang and Sadayuki Ueha
Research Laboratory of Precision Machinery and Electronics Tokyo Institute of Technology, Nagatsuta 4259 Midori-ku, Yokohama 227, Japan
The inverse acoustic scattering problems for one-dimensional lossy media, is theoretically studied using the first-order differential equations of motion. A coupled vector-Marchenko equation is derived in association with the causality, and the uniqueness of solution is shown. The constraints on the absorption models are relaxed, compared to the approach of beginning with the wave equation. As a result, the impedance and absorption operator are proved to be extracted separately by using the scattering data obtained for both positive and negative directions of incidence.
INTRODUCTION
The inverse acoustic scattering problems have been studied by many researchers. VVare and Aki showed that the impedance profile can be uniquely reconstructed as a function of traveltime from the longitudinal displacement impulse response of normal incidence [1]. Howard considered the problem to reconstruct density and velocity profiles of a layered fluid, and showed that the two profiles can be reconstructed from two impulse responses of separate incidence by' transforming the acoustic wave equation to a first-order equation of motion l2]. Jaulent considered the inverse s-wave scattering problem for a class of potential which may depend on energy, and showed that the potential can be reconstructed from the scattering data l3,4].
The purpose of this paper is to consider an inverse acoustic scattering problem for a layered medium in which acoustic wave may be absorbed due to some absorption mechanisms.
GOVERNING EQUATION AND THE OUTLINE OF APPROACH
Consider a layered acoustic medium in which the material properties, density p(z) and sound velocity C(z) are dependent on the depth z as shown in Figure 1. According with Howard [2], the equation of motion for a monochromatic acoustic wave at normal incidence is conveniently expressed:
[) T ~(W,T) = (iwA - V(T)Q) ~ (W,T) (1)
where ~(W,T) = [p,VV]T is a column vector, the other quantities are
Physical Acoustics, Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991 687
incident wave
1 f reflected wave
Parameter distribution
X Z, .. . ..
z,
f transmitted wave z z
Figure 1. Schematic illustration of the continuous scattering problem.
summarized in Table 1. For our purpose, this equation is to be extended to
layered medium with density p(z), sound velocity C(z) coefficient a( z).
The extended formula is assumed to be given by:
8 l' cj>(w,1') = (i(w+i[(1'))A - V'(1')Q) cj> (w,1')
where
1(1') = a(1') C(1')
the case of lossy and attenuation
(2)
and V'(z) may be a function of the frequency w, and is assumed to be satisfied by the following relation:
V'(z,w)* = V'(z,-w*) (3)
where asterisk denotes complex conjugated. Generally , C(z) and a(z) are also functions of the temporal frequency.
For simplicity, the case in which both C(z) and a(z) are independent of frequencies will be treated at first. Then, it is shown how the formulation proposed here can be extended to more general cases in association with the causality.
688
Table 1. List of Symbols
r 1 A= I
L 0
o 1 I
-1 J
r 0 Q= I
L 1
(z
1 1 I
o J
Z2 =c p, .. (z) = I c- I (z) dz ) 0
(+00 f'(w,z) = I f(t,z)exp(iwt) dt
) -00
P; Pressure 11'; displacement, p ; density C; Velocity
'( (z) ; Travel time
THE DIRECT PROBLEM
The direct scattering problem is to determine the reflection and transmission coefficients of acoustic wave when the distribution of material properties are given.
Consider the following four solutions of Eq.(2), which are the solutions of the Volterra integral equation corresponding to four different boundary conditions:
~+(W,T) = exp(iK(T))[l,O]T + J+oov'(x)exp(iK(T-x)A)Q~+(w,X)dX (4) T
which is a pure downgoing-wave at T = +00
~_(W,T) = exp(-iK(T))[O,l]T - iT V'(x)exp(iK(T-x)A)Q~_(w,x)dx o
which is a pure upgoing-wave at T = ° ~+in(W,T) = exp(-iK(T))[O,l]T + J +00 V'(x)exp(iK(T-x)A)Q~+in(W,x)dx
T
which is a pure upgoing-wave at T = +00
(5)
(6)
~-in(W,T) = exp(iK(T))[l,O]T - fT V'(x)exp(iK(T-x)A)Q~_in(w,x)dx (7) o
which is a pure downgoing-wave at T = 0, where function K(x) is defined by:
r(T) = fT 1(z)dz. o
The reflection and transmission coefficients for positive and negative directions are defined by using the asymptotic form of Equations (4 )-( 7) as showed in Figure 2,
~+ N lit +( w)exp(iK( T) )[l,O]T + r +( w) It +( w) exp( -iK( T) )[O,l]T T --! ° (8)
~_ N l/L(w)exp(-iK(T))[O,l]T + L(w)/L(w) exp(iK(T))[l,O]T T --! 00 (9)
where subscripts ± mean the direction of incidence.
Replacing the a and V'(z) in Eq.(3) by -a and V'(z)*, we have another equation of motion for an acoustic medium with ·attenuation -a, which never exists in any stable real physical problem. It can be shown that the corresponding four solutions X+, X-, X+in and X-in of this equation, are related with ~ .. , ~_, ~"in and ~-in by the following relations
[Qx-*] = ~-in' [QX+*] = ~+in' [QX-in *] = ~-, [QX+in *] = ~+, (10)
X-in = [Q~-*], X+in = [Q~+*]' X- = [Q~-in*], X+ = [Q~+in*] (ll)
689
Positive incIdenCe f r ('l (,)
t.. (,) ,L (:=.
Parameter cll$tribuUon
Figure 2. Schematic illustration of the Acoustic scattering.
Namely, this replacement means interchange of the outgoing-wave and incoming-wave between the two systems. The corresponding reflection and transmission coefficients can be defined in the forms:
x. N l/t~(w) exp(iK*( T))[I,O]T + r~(w)/t~(w)exp(-iK*( T))[O,I]T (12) T -I 0
x- N l/t'_(w) exp(-iK*(T))[O,I]T + r:(w)/t:(w)exp(iK*(T))[I ,O]T (13) T-Iw
Using the relations Eqs.(10) and (11), the asymptotic values of ~., ~_,
'" '" are rewritten : 'f-in) 't'+in
~. N exp(iK( T)1\)[I,O]T; N exp(iK(T)1\)[I/t., r./t.]T
T-Iw T -I 0 (14)
~_ N exp(iK(T)1\)[r-/L,I/tl;' N exp(iK( T)1\ )[O,1]T
T-Iw T -I 0 (15)
~-in N exp(iK(T)1\)[r :*/t '_*,I/t '_*f; N exp(iK( T)1\)[I,O]T
T-Iw T -I 0 (16)
~.in N exp(iK( T)1\)[O,I]T; N exp(iK(T)1\)[I/t~ *, r~*/t~*]T T-Iw T --I 0 (17)
Since the trace of the matrix i(w/C(z) + io{z))A -V ' (z)Q is zero, then we have the following identities by comparing the values of the Wronskian of any two solutions of ~., ~ _ , ~-in' ~.in at T = 0 and T = w
690
Lt~* + r.r~* = 1,
(c/t~)* = -(r./t.)
t~ = t~
Substitution of the Eq.(18) into the Eq.(19) yields the following relation
c* = -r-f(t 2 - u.)
(18)
(19)
(20)
(21)
As a result of Eq.(21), we have a representation of the reflection coefficient of non-physical scattering problem by using the scattering data of the physical scattering problem. This provides a set of complete scattering data for our approach.
THE INVERSE SCATTERING PROBLEM
The inverse problem is to determine characters V( T) and r( T) from the measured scattering data, reflection and transmission coefficients.
Under the assumption of
00 f xV, (x)dx < 00, o
exp(r(oo)) < 00. (a)
The equations (4)-(7) can be solved by iteration not only for real w but also for complex w with Imw ~ 0, and the following properties (P-1) and (P-2) can be shown to hold for Imw ~ 0:
(P-1) ~., ~_ are analytical functions of w, which are continuous up to
and including the real w-axis.
I ~.(x) exp(-iK(x)A)-[l,O]T I N O(l/w) for I wi -; 00 (b)
I ~~x) exp(-iK(x)A)-[O,l]T I N O(l/w) for I wi -; 00 (c)
(P-2) lit -; 1 for I wi -; 00, lit is analytic and lit * o. (d)
Then, according to the Paley-Wiener theorem [5], Eqs.( 4)-(7) can be rewritten in the following form
00
~.(W,T) = exp(iK(T)A)[l,O]T - J T exp(iwx) B(x,T) dx (22)
T ~_(W,T) = exp(iK(T)A)[O,l]T - J exp(iwx) A(X,T) dx (23)
-ro
00
X.(W,T) = exp(iK*(T)A)[l,O]T - J T exp(iwx) D(X,T) dx (24)
T
X_(W,T) = exp(iK*(T)A)[O,l]T - f exp(iwx) C(X,T) dx (25) -ro
691
where A(x,r), B(x,r), C(x,r) and D(x,r) are vectors and are called transforming kernel.
In the virtue of the definition of scattering data, we have the following identities:
t.(w) +. = +-in + r.(w) +_,
t~(w) x. = X-in + r~(w) X-,
(26)
(27)
In fact, a pair (+-in, +_) forms a fundamental system of Eq.(3) and Eq.(26)
can be proved by comparing the asymptotic value of both hands of Eq.(26) at r = 0.
Using the relations (10) and (11), Eqs.(26) and (27) may be rewritten:
(t.-l) (+.-€xp(iKA)[I,OlT) + exp(iKA)[I,Of (t.-l) + ~.
- exp(iKA)[I,OlT = [Qx-l* -exp(iKA)[I,OlT + r. exp(iKA)[0,11T
+ r. (+_-exp(iKA)[0,11T), (28)
(t~-I) (X.-exp(iKA)[I,O)T) + exp(iKA)[I,O)T (t~-I) + X.
- exp(iKA)[I,Of = [Q+-l* -exp(iKA)[I,O)T + r~ exp(iKA)[O,l)T
+ r~ (X_-exp(iKA)[O,I)T), (29)
Substituting Eqs.(21)-(24) into Eqs.(28) and (29) and multiplying by exp(-iwx) at the both hands of the above two equations and integrating with respect to w, we finally obtain the matrix' Marchenko equations:
T I r C(x,r) = R.(x+r) exp(r(r)) [1,0) - Q A(z,r) R.(z+x) dz -x
T fr A(x,r) = R~(x+r) exp(-r(r)) [1,0)..., Q C(z,r) R~(z+x) dz
The following relations:
2Al(X,X) = exp(r(x)) V I (x)
2C 1(x,x) = exp(-r(x)) V'(x)
-x
(30)
(31)
(32)
(33)
are also satisfied, where subscript 1 means the first components of the vectors. The uniqueness of the Eqs.(30) and (31) can be proved easily in association with the properties (P-l) and (P-2) mentioned before.
Generally, o{ r), V( r) are also functions of the temporal frequency as mentioned before. We write the wave number
K(r,w) = w/Cmax + H(w,r), K(w)* = -K(-w*),
instead of the forms given in Eq.(3).
692
The function
exp(± iH(w,r))
is analytical and definite for Im( w) ~ 0, if causality is satisfied [6]. It is clear that the approach proposed here can be extended to this general case.
CONCLUSION AND DISCUSSION
In this paper, authors showed that the characters V( r), f( r) can be reconstructed uniquely from the scattering data of both positive and negative direction at normal incidence. However, the treatment given here is theoretical, a stable discretization of the approach proposed here is necessary for any practical application.
REFERENCES
1 J.A. Ware and K. Aki, J. Acoust. Soc. Am., 45(4), p.911, 1969. 2 M.S. Howard, Geophysics, 48(2), p.163, 1983, and papers referenced there. 3 M. Jaulent and C. Jean, Commun. Math. Phys., V. 28, p.l77, 1972. 4 M. Jaulent, J. Math. Phys., 17(7), p.1351, 1976. 5 K. Yoshida, IIfunctional analysis II , Spring-Verlag, Berlin, 1965. 6 J.S. Toll, Physical Review, 104 (6), pp. 1760-1770, 1956.
APPENDIX
Inserting Eq.(23) into Eq.(5), yields
r r -f A(x,r)exp(-iwx)dx = -f V(x)exp(iK( r-x)A)Qexp(-iK(x))[O,l]Tdx -oJ -oJ
r r + f V(x)exp(iK( r-x)A)Q f A(y,x)exp( -iwy)dy
o -oJ
multiplying by exp( -iw')') at both hands of the equation and integrating with respect to w, we have
-Ab,r) = -1/2 V(( r + ')')/2) exp(f( ')'))[l,O]T
r + f V(x)exp(-f( r-x) A) Q Ab + r - x, x) dx
o
Putting r = ')' in this equation and noting A(x, r) = 0 for x > r, we have Eq.(32).
693
IDGH RESOLUTION LASER PICOSECOND ACOUSTICS IN TIDN FILMS
ABSTRACT
O.B. Wright, T. Matsumoto, T. Hyoguchi and K. Kawashima
Nippon Steel Corporation Electronics R&D Laboratories 5 - 10 - 1 Fuchinobe, Sagamihara Kanagawa 229, Japan
Using the pump-probe technique with laser noise compensation, laser picosecond acoustic measurements were carried out for the amorphous sputtered thin films a-Ge and a-InSb on sapphire substrates using front side pumping and probing. The measurements were extended in two new ways. Firstly rear side pumping with front side probing was used for the film of a-Ge on a sapphire substrate, allowing background signals from non-acoustic sources to be greatly reduced. Secondly measurements were made with the front side pump-probe geometry for a transparent film on an opaque substrate using a-Si02 on a-Ge. The detected echo pulse shapes were modelled theoretically.
INTRODUCTION
The use of picosecond light pulses to generate and detect very short stress pulses is a promising technique for the investigation of the thickness, adhesion, and elastic constants of thin films in the 100 - 10000 A thickness range [1,2]. In this paper we extend this technique to a rear side pump geometry in order to reduce or remove background signals and to the measurement of transparent films on opaque substrates. The latter research has important applications in the electronic device field where films such as passivation layers are transparent to the probing light.
The pump-probe detection method for stress pulses relies on the measurement of very small changes in reflectance or transmittance ('" 10-6 to 10-5) of a thin film, and the resulting change in probe power depends on the variation of the film refractive index and absorption coefficient with stress. For films with a weak variation it is important to obtain a high signal-to-noise ratio. We have developed a noise compensation scheme to help reduce the effect of laser power fluctuations.
EXPERIMENTAL
Figure 1 shows the optical system used. Light pulses of width 1 ps, energy ",0.1-0.3 nJ, wavelength A = 590 nm, repetition rate 76 MHz, and vertical polarization were generated with a mode-locked dye laser synchronously pumped
Physical Acoustics, Edited by o. Leroy and M.A. Breazeale Plenum Press, New York, 1991 695
by a Nd-YAG laser. The crossed polarization pump-probe method was used to measure the change oR in film reflectance R using spot diameters (half power criterion) of 10 or 50 j.Lm. The pump and probe angles of incidence on the samples were 6' and 50' respectively. The pump beam was chopped at frequencies in the 100-200 kHz range and lock-in detection was used. In order to compensate for laser power fluctuations, the probe beam was split into two separate beams, one a beam (N 10% of pump power) for sample reflectance measurement and one a reference beam, and their powers measured at two Si PIN photodiodes (bandwidth 10 MHz).The average d.c. levels of these two beams were adjusted to be equal using a neutral density filter placed in the reference beam path. The difference in output voltage between the photodiodes was determined with a differential amplifier, and noise from laser power fluctuations could be reduced by a factor of about ten giving a best resolution oR/R ~ 10-7 r.m.s ., similar to that obtained by other groups [1,3]. To ensure long term d.c. stability of the laser power, a low frequency « 10 kHz) acousto-optic stabilization system was incorporated. The delay line was scanned at rate corresponding to 6ps/min. The lock-in time constants of 10 sand 30 s used thus correspond to effective measurement time constants of 1 ps and 3 ps respectively.
Thin film samples were made by room temperature r J. magnetron sputtering with deJlosition rates for a-Ge, a-InSb, and a-Si0 2 films equal tp 240, 135 and 150 A/min respectively. X-ray Bragg diffraction studies on 5000 A thickness films with Cu Ka radiation showed, in addition to a substrate component, broad peaks typical of amorphous short range order but no evidence of any film crystallinity. Filmo thicknesses were determined with a needle profile measurement system to ± 50 A.
FRONT EXCITATION TECHNIQUE FOR OPAQUE FILMS
To evaluate the oset-up, experiments were carried out on a 1000 A film of a-Ge and a 2000 A film of a-lnSb, both with Z-cut sapphire substrates thickness 0.5 mm. Figure 2 shows the traces of reflectance variation against delay time. The signal near zero delay time contains contributions from thermoreflectance effects, electronic excitation, and also from the stress pulse.
STRESS
FILM
NEUTRA.L
DENSITY
CHOPPER
A-8 LOCK-IN
AMPLIFIER
CORNER
( + 10)
CHART
RECORDER
ACOUSTO
OPTIC
STABILIZATION
SYSTEM
PHOTO DIODES LASER
Figure 1.
696
Diagram of the pump-probe optical system showing the technique for compensation of laser noise . The initial propagation of the stress pulse is also indicated.
Figure 2.
a
BR/R I~,I 1 st echo
2nd echo 3rd echo
., ,,·1 01
" '" ;5 OJ
t l ps '" " ~ ... 0 100 200 ~ b ~ ., > . ., '" BR/R ~
1st echo 2nd echo
10·· r
t I ps
0 100 200 delay time
Variation of relative film reflectance (peak-to-peak value) with delay time .showing acoustic echoes in a film .of a) a-Ge thickness 1000 A and b) a-InSb thickness 2000 A, both on sapphire substrates. The laser pulse energy is 0.25 nJ with spot size 10 pm and measurement time constant 1 ps . The inset of a) shows the calculated first acoustic echo reflectance variation (with arbitrary units) for a-Ge, with the same time scale calibration as in the main figure.
Successive acoustic echoes are phase inverted because of the 7r /2 phase change of the stress pulse on reflection from the top surface, and their attenuation can be used to assess the film adhesion. From the round trip propagation time the longitudinal sound velocity· VJ. can be estimated as 4000±200 ms- I for a-Ge, in approximate agreement with previous data [II, the principal error coming from the uncertainty in, thickness. Published data {or the sound velocity in sputtered a-Ge varies by up to ±50% depending on the film preparation conditions [4,5]. For a-InSb Vi is 2950±150 ms- I. To our knowledge no other data on VJ. for a-InSb is available for comparison, but this value is smaller than the mean value [6,7] for crystalline InSb in accordance with the results found for other amorphous solids l4,5] .
REAR EXCITATION TECHNIQUE WITH TRANSPARENT SUBSTRATES
The experiment was then modified by pumping from the rear side on an opaque film through a transparent substrate with the probe beam incide.nt on the front side, as shown in Figure 3. The results obtained for the 1000 A film of a-Ge are shown in Figure 4 for two different measurement time constants. Use of a longer time constant allows the higher order echoes to be resolved more clearly. The signal from the initial transmitted acoustic pulse and three
697
Figure 3.
PUMP BEAM -_-+-----0 ..........
,"
AIR TRANSPARENT OPAQUE SUBSTRATE FILM
" " "If" "
PROBE ," BEAM
Diagram showing the rear excitation geometry for an opaque film on a transparent substrate. The stress pulse shape (unipolar) is also indicated for a-Si02 on a-Ge.
acoustic echoes are indicated. As before successive echoes are phase inverted. Because the region of probe beam absorption is not the same as that of the pump beam, the contribution to the signal from thermoreflectance and electronic excitation in the region of pump beam absorption, the step-like response in Figure 2, is much smaller because very little light is transmitted through the film to the top surface [8]. This allows the acoustic pulse shape to be resolved more easily. The results give VI = 4100±200 ms-1 in close agreement with the front side technique.
Figure 4.
698
o 100 t I ps
initial t t t acoustic pulse 1 st echo 2nd echo 3rd echo
o 100
delay time
t l ps
Variation of film reflectance with delay time for the rear excitation geomeotry, showing the response for a film of a-Ge thickness 1000 A on a sapphire substrate with measurement time constant a) Ips and b) 3ps. The laser pulse energy is 0.1 nJ with spot size 10 /-tm . The inset shows the calculated first acoustic echo reflectance variation.
Figure 5.
PROBE BEAM , , , [f] @] , , , , ---L-- A--, , ,
........................ PUMP BEAM -v---, ,
" ,
" ,
~ , , , ~
;(' ,
, , [hl , ,
AIR TRAN SPARENT OPAQUE FILM SUBSTRATE
Geometry of experiment with a transparent film on an opaque substrate, showing the acoustic stress pulse shape and propagation for a-Si0 2 on a-Ge: a) shows the stress pulse initially transmitted to the substrate, b) and c) show the stress pulses in the transparent film, and d) shows the stress pulse retransmitted to the substrate.
FRONT EXCITATION TECHNIQUE FOR TRANSPARENT FILMS ON OPAQUE SUBSTRATES
The third series of experiments were carried out using a transparent film on an opaque substrate, as shown in Figure 5. This differs from previous picosecond acoustics experiments with transparent films [1,2,9) in that light is incident from the transparent film side without the use of a potentially inconvenient external transducer film. Figure 6 shows the results for a 2000 A sputtered film of a-Si0 2 on a 5000 A film of a-Ge with a silica substrate. The increased noise level for these measurements is not fully understood, but may be caused by laser instability. In previous experiments with bulk transparent
Figure 6.
<II CI c: IV ~ v <II v c: ~ v <II
~ <II > . .-IV
~
8R1R
o
1st echo
100 delay t ime
8R(t')
t ime t'
t l ps
Variation of film reflectance with delay time for a film of a-Si0 2 thickness 2000 A on an a-Ge substrate. Two traces taken for the same point on the sample are shown displaced. The laser pulse energy is 0.3 nJ with spot size 50 Jl-m and measurement time constant 3 ps. The inset shows the calculated first echo reflectance variation oR( t') (arbitrary units), with time scale calibration the same as in the main figure.
699
samples [10,11] oscillations produced by interference beats with a Doppler shifted beam reflected from the moving stress pulse were detected. However, in the present experiments the film thickness is small enough for the stress pulse propagation time from the Si0 2/Ge interface to the free top surface to be smaller than the expected beat period. We therefore interpret the maximum at delay time t 1 = 81.0 ps as the first echo of the stress pulse generated in the a-Ge substrate and reflected from the free top surface. In order to estimate the sound velocity of a-Si0 2 a delay &t in the position of the echo maximuIJl, which depends on the optical absorption depth of the a-Ge film (( N 230 A) [12], must be taken into account, and the sound velocity 'I![' for a-Si0 2 calculated from VI' = 2d/(tl-LH) where d is the film thickness.
This delay can be estimated as follows: the acoustic pulse is generated in the opaque substrate and transmitted to the transparent film (Figure 5. b). On reflection from the top surface (Figure 5.c) the returning acoustic pulse re-enters the substrate (Figure 5.d) where it can be detected by the probe light beam. The acoustic stress pulse shape can be calculated from the elastic wave equation assuming a thermal expansion generation mechanism, taking into account the elastic boundary conditions at the Si0 2/Ge interface. Ignoring thermal diffusion (c.f. reference 1) the longitudinal stress in the substrate (z > 0) associated with the returning stress pulse is a unipolar pulse (Figure 5.d) given by
0"33 0" maxexp( (Z-vIt' ) / 0 o
for Z < 'l![t',
for Z > 'l![t',
where 'I![ is the substrate sound velocity and z is the axial coordinate. Position Z = 0 corresponds to the film/substrate interface and t' = 0 corresponds to the time when the acoustic pulse just reaches the interface (When delay t = 2d/'I![). The maximum stress O"max is given by
_ 2Z j 2Z2 3B,8( l-Ro)Q _ 2Z jZ2 l(l-Ro)Q O"max - Zl+Z2 x Zj+Z2 x 2AC( - (Zl+Z2)2 A(
where Zj (= pVl), Z2 are respectively the acoustic impedances of the substrate and film, B, ,8, p, C, and 1 are respectively the substrate bulk modulus, linear thermal expansion coefficient, density heat capacity per unit volume, and Griineisen parameter (1 = 3B,8/C), Q is the laser pulse energy incident on the substrate, A the area of irradiation, and Ro the reflection coefficient of the film/substrate interface. The acoustic pulse shape, a truncated exponential function, is the same as the spatial distribution of the light absorption in the substrate.
The reflectivity variation can be calculated from 8R = Jf(z)1/33dz with limits Z = 0 to CD, where 1/33 is the longitudinal strain and f a sensitivity function [1]; for a.morphous semiconductors f can be related to the energy band gap Eg and refractive index n through f = b(81nEg/81/33)cos(4:mz/A-~)exp(--z/O, where ~ and b are functions of n, (, Eg, '\, and, for the geometry of Figure 5, depend also on the transparent film refractive index and probe beam incidence angle. Using 1/33 = 0"33/ P'l![ 2 for the returning stress pulse the reflectance variation of the film/substrate interface can be calculated:
8R(t') = ~ [81nEg] ~ exp(-vlt' /0 x {sin(m'l![t' /( - ~) + sin~} m 1/ 3 3 PVr
where ~ = 1.02 rad, m = 4:m(/'\ = 2.4, b( = -0.49, Eg = 0.7 eV and n = 4.9 for a-Ge at ,\ = 590 nm [12]. The angle of light transmission in a-Ge (9") is assumed to be negligible, but the effect of the probe beam incident angle (50') and the a-Si0 2 film refractive index (1.5) is included in the calculation of ~ using the Fresnel equations for the probe beam reflection (polarized perpendicular to the plane of incidence). Accounting for the pro be
700
beam an&le and film refractive index actually turns out to have a small influence l < N 3%) on the value of ¢ for the present case.
The predicted oR( t ' ) variation is shown in the inset of Figure 6 and At = 0.78(/VI ~ 4.5 ps is obtained. This correction gives VI' = 5200±300 ms- I ,
in reasonable agreement with previous data fQr a-Si0 2 films [9,13,14]. Experiments with an a-Si0 2 film of thickness 3000 A were also carried out. The first echo was observed at t I = 120.6 ps, giving the same value for VI'. The predicted reflectance pulse is nearly unipolar with the expected sign (oR > 0 since for a-Ge (}lnEg/ aTJ33 N -1 [15] and O'max > 0 *), in agreement with experiment, but the predicted pulse width is approximately half that observed. This suggests that the value of VI' may be greater than calculated although some broadening may be attributed to the measurement time constant and finite light pulse width. Using estimates for the parameters in the equations the predicted pulse height is of the observed order of magnitude. The effect of the background variation observed in experiment has not been included in the model.
DISCUSSION
For comparison we have also calculated the echo shape for the front and rear excitation geometries for a film of a-Ge on a sapphire substrate. For front excitation the stress pulse has a bipolar antisymmetric shape (see [1] and Figure 1). The predicted first echo pulse shape for oR ignoring background variation is shown in the inset of Figure 2, using the above model for f. The echo shape is symmetric, as in experiment, owing to the combination of the antisymmetric stress pulse shape and the acoustic reflection coefficient of -1 at the top surface. For rear excitation the stress pulse shape depends on the acoustic impedances of the film and substrate, which can be estimated from the literature [13,16]. The shape is unipolar (see Figure 3) because the substrate has a higher acoustic impedance than the film. The predicted first echo for 8R is shown in the inset of Figure 4. The echo shape is bipolar but not symmetric, as observed in experiment. In contrast to the results for the a-Si0 2 film, for both front and rear excitation with the a-Ge film the pulse widths are similar to those observed, although the detailed form of the pulse shapes are somewhat different from those predicted, and (for front excitation) from previous measurements on an a-Ge film [1]. A similar discrepancy is also found for the a-InSb pulse shape. This may be due to the inaccuracy of the assumed model parameters which depend on the conditions of film preparation. Further measurements with a wide variety of amorphous semiconductor films should help elucidate these deviations from the theory.
CONCLUSION
In conclusion the laser picosecond acoustic technique for thin films has been extended by using rear side pumping with front side probing for opaque films on transparent substrates and by application to transparent films on opaque substrates. In the future we intend to measure the adhesion of transparent films, such as silicon nitride passivation layers, for integrated circuit applications. By working with higher pump beam chopping frequencies of order 5 MHz, for which the laser noise is smaller, we expect to obtain a resolution approaching 8R/R ~ 10-8 r.m.s. with our noise compensation scheme. This should allow materials with a very weak response, such as crystalline silicon, to be probed using laser picosecond acoustics.
*) For time scales of order picoseconds in semiconductors the Griineisen parameter "f in the equation for 0' max should be replaced by the quantity (Eg/EheI+{(E-E,)/E},.,Ph, where reI is the electron Griineisen parameter (reI ~ - (}lnEg aTJ33), rph is the phonon Griineisen parameter (rph ~ 3BfJ/C at room temperature), and E is the phonon energy. For a-Ge these contributions are both positive (see [1]).
701
REFERENCES
[1]
[2]
[3]
[4]
I~I t~l [10]
[11]
[12]
[13]
[14]
702
C. Thomsen, H.T. Grahn, H.J. Maris, and J. Tauc, Phys. Rev. B, 34, 4129, (1986). H.T. Grahn, H.J. Maris, and J. Tauc, IEEE J. Quantum. Electron., 25, 2562, (1989). G.L. Eesley, B.M. Clemens, and C.A. Paddock, App1. Phys. Lett., 50, 717, (1987). I.R. Cox Smith, H.C. Liang, and RO. Dillon, J. Vac. Techno1., A3, 674, (1985). L.R Testardi and J.J .. Hauser, Solid State Comm., 21, 1039, (1977). J.D.N. Cheeke and G.Madore, Solid State Comm., 41, 899, (1982). K.W. Nill and A.L. McWhorter, J. Phys. Soc. Japan Supp1., 21, 755, (1966). Z. Bozoki, A. Miklos and A. Lorincz, Mat. Sci. and Eng., B5, 147, (1990). C. Thomsen, J. Strait, Z. Vardeny, H.J. Maris, J. Tauc, and J.J. Hauser, Phys. Rev. Lett., 53, 989, (1984). C. Thomsen, H.T. Grahn, and H.J. Maris, and J. Tauc, Optics Comm., 60, 55, (1986). C. Thomsen, H.T. Grahn, D. Young, H.J. Maris, and J. Tauc, Phonon Scattering in Condensed Matter 5, Proceedings of the 5th International Conference, Springer-Verlag, Berlin, 392, (1986). G.A.N. Connell, R.J. Temkin, and W. Paul, Advances in PhYSics, 22, 643, (1973). J. Kushibiki, H. Maehara, and N. Chubachi, J. App1. Phys., 53, 5509, (1982). M. Rothenfusser, W. Dietsche and H. Kinder, in "Phonon Scattering in Condensed Matter", W. Eisenmenger, K. Lassmann, and S. Dottinger, ed., Springer, New York, p.419, (1984). G.A.N. Connell and W. Paul, J. Non-Cryst. SoL, 8-10, 215, (1972). G.W. Paul, A.N. Connell, and RJ. Temkin, Advances in Physics, 22, 531, (1973).
PHOTOACOUSTIC SIMULTANEOUS STUDIES OF THERMAL CONDUCTIVITY, DIFFUSIVITY AND HEAT CAPACITY OVER THE SMECTIC A-NEMATIC PHASE TRANSITION IN ALKYLCYANOBIPHENYLS WITH VARYING NEMATIC RANGE
U. Zammit, M. Marinelli, R. Pizzoferrato, S. Martelucci and F. Scudieri
Dipartimento Ingegneria Meccanica, II Universita di Roma "Tor Vergata" via E. Carnevale, 00173, Rome, Italy
INTRODUCTION
In thermotropic liquid crystals (1. C.) the thermal parameters may depend strongly on temperature in the vicinity of their phase transitions. An accurate study of their critical behavior thus requires a technique which introduces small temperature perturbations in the investigated sample. In the photoacoustic (PA) technique in the gas microphone configuration [1 J an adequate signal to noise ratio may be achieved while introducing temperature oscillations in the sample of only a few mK. Also, the d.c. temperature gradients within the probed depth introduced by the a.c. laser source was also negligible. Finally the technique allows the simultaneous determination of the specific heat c, thermal conductivity k and diffusivity D. In this work the smectic A-nematic (A-N) transition in liquid crystal samples with different nematic temperature range was studied. The second order A-N transition in liquid crystals was placed in the same universality class as a superfluid, namely the three dimensional XY like transition [2]. Experimentally [3]' observations carried out in alky1cyanobiphenyl compounds and mixtures having different nematic temperature range showed that the critical exponent a of the specific heat ranged between the tricritical value, 0.5 and -0.03, a value close to the XY model one (-0.007). This was interpreted as due to a crossover between tricritical and SY like critical behavior as the nematic range increased. Even in compounds whose value approached the one predicted by the XY model, however, the critical exponents I for the susceptibility and v for the correlation length remained inconsistent with the XY model predictions [4]. We have studied the critical behavior of k and D, as well as c, over the A-N transition in three samples of alky1cyanobiphenyl liquid crystals with different nematic ranges.
THEORY
When the sample is optically and thermally thick [5J, the expression for the PA signal amplitude A and phase ¢ become
1 I P,], 2egli' (p2+2p+2)' (1+q)
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991
'If _ [ p] - 4 - tan 1 1+'2 (1)
703
where
p = J1.s/J1.f3'
1
J1.s = (2D s/W)2 is the thermal diffusion length, J1.f3 = l/fJ is the optical
absorption depth, e is the thermal effusivity, subscript sand g refer to sample and coupling gas respectively, p is the density, fJ is the optical absorption coefficient. I is the thickness, I is the laser intensity, P is the initial cell gas pressure, T is the d.c. ambient temperature, w is the modulation frequency in rad/s.
The quantity in the large brackets is the temperature oscillation induced on the sample surface. From the phase data the sample diffusivity D = k/ pc can be derived provided fJ is known, while from the amplitude data the effusivity e and therefore c and k can also be worked out. The three thermal parameters can thus be simultaneously worked out from measurements performed at one value of the modulation frequency.
EXPERIMENTAL
The measurements were performed on 5 mm diameter 0.4 mm deep samples contained in a gold plated copper sample holder. The samples were illuminated with a He-Ne laser operating at 3.39 J1.m wavelength, at which there is strong absorption by the C-H bonds of the liquid crystal compounds. The value of fJ at such a wavelength is 1500 cm-1 (J1.f3 = 7 J1.m). The modulation frequency was
30 Hz. With a typical D value for liquid crystals of 0.0005 cm 2s-1
J1.s = 25 J1.m. The conditions for thermall~ and optically thick sample are thus fulfilled. The laser intensity was 1 m W / cm -2 and the temperature oscillations induced on the sample surface as estimated from Eq.(l) was only 0.5 mK. It is worthwhile pointing out that from Eq.(2) it is clear that p must no be » or « than 1 or the phase would then tend to -1r /2 or -31r / 4 respectively and would become insensitive to D. In our experimental conditions p = 3.5 and thus useful for the detection of D. Finally the dc temperature variation over the probed depth (J1.s) for an optically thick sample is given by [lJ.
dT = 2k!/3s [J1.s[1+fJlg~]/[lg~+lb~+ls]-l] subscript b refers to the sample backing material. Under our experimental conditions dT = 1.4 mK and therefore once again negligible.
The investigations were carried out on samples of 9CB, 8CB and a 0.76 8CB mole fraction mixture of 8CB and 7CB. The respective values of TAN/TN! were 0.994,0.967 and 0.962 respectively. TAN and TN! are the
smectic A-nematic and nematic-isotropic transition temperatures respectively. The temperature rate change during the measurements was less than 1 mK/min.
RESULTS AND DISCUSSION
The results for the amplitude and phase for the 9CB sample are shown in Figure 1. Both feature very sharp dips over the transition region. The behavior of c, D and k are reported for all three samples in Figure 2. In all the investigated samples the heat capacity and the thermal conductivity show peaks over the transition region. The thermal diffusivity data show a dip with a small peak structure in the vicinity of the transition temperature. The peak is due to a sharper rise in the thermal conductivity than in the heat capacity in the region right next to the transition temperature. The slower rise in c can be due to rounding of the data near the peak, as shown in the double log plot of
704
the singular term contribution to the c data f1c vs t (Figure 3), which may effect the heat capacity more thau the thermal conductivity. f1c has been obtained by fitting the c data w;th the power law expression c = A 1 t 1-0. + B + Et and then subtracting from the data the constant and linear terms: f1c = c - (Et + B). t = (TIT AN - 1) is the reduced temperature.
In order to determine the critical exponents of the thermal parameters, since the k data were calculated off the c and D data, a fit was first performed on the latter two quantities values, on either side of the transition temperature, using the above mentioned expression for c and the power law expression D = R Itl-b + S + Ut for D.
The critical exponent a for the thermal conductivity, which may also be described by a singular and background terms with an expression similar to the ones used for c and D, was then determined from the a and b values. The points closest to the transition temperature considered in the fit procedure were those for which the I t I > 1.2xlO -5. To within experimental error the
::::i -< w Cl :::J I-:J Il. ~ c(
w til
~ Il.
Figure 1.
120
100
80 \ ~ .' .' .: .
60 .' :-..
40L-~--~--~~~·~--~--~--~~
47.5 47 .6 47 .7 47.8 47.9 48 48 . 1 48.2 48.3
T ( 'c )
43.------------------------------,
~ /---'" 42
\ , 41 \ I
! I ! ;
40 I
· . · . · . : : · 39 ., (.
~
38 ~~--~~~--~--~~~~~~~
H .5 4 7. 6 47.7 47 .8 47.9 48 48. 1 48.2 48.3
T ( 'c )
PA amplitude A and phase ! over the smectic A-nematic phase transition in geB.
705
Figure 2.
706
3r------------------------------,
· · ,..... 2.5 :.: E v
"'~ ' S!
:.: co "-...,
u
2 ....... ~ c - --
· . .. :\ a
• __ JIA! .... . ...,....- - --'" • to
~~.--b .v ~ 1.5
- 0.3 - 0.2 - 0.1 0 0.1 0.2 0 .3 T- Tm ( 'C)
.. 10
.' '. · , · 8 · . · . · . · . i :
_:& -~ 6
4
--.. -----~-----------_. c 2
0 .3 0.2 -0. 1 0 0 . 1 0 .2 0 .3
T-T .. ( ' C)
10r---------------------
30.3 - 0.2 - 0 . 1 0 0 . 1 0.2 0.3 T-TAN ( 'C)
c, D and k data over the A-N transition in 9CB (a), 8CB (b) and 7CB+8CB mixture (c). The different sets of data have been shifted relative to each other to avoid overlapping.
Table 1.
Sample
TANITN/ a b a
Smectic A-Nematic to Nematic-Isotropic transition temperatures ratios and critical exponents of heat capacity a, thermal diffusivity b and thermal conductivity a of the investigated samples
0.24 7CB + 0.76 SCB mole SCB 9CB fraction
0.962 0.967 0.994 0.14±O.03 0.2S±O.03 0.52±O.04 -0.0I±O.03 -0.04±O.03 -O.OS±O.04 0.13±O.06 0.24±O.06 O.44±O.OS
critical exponents calculated for T > TAN and T < TAN were equal and
the values are reported in Table 1 together with the samples respective T AN/TN! values. The absolute values of the critical exponents of each thermal
parameter progressively increase as TAN/TN! increases. Regarding the heat
capacity data the critical exponent values for 9CB and 8CB are in agreement with the ones reported by Thoen et al. (0.5 and 0.3 respectively) [3] . As for the mixture value, Thoen .et al. reported a value of a = 0.1 for TAN/TN! = 0.961.
In the present case the value of a is slightly larger, consistent with the trend of increasing values of a as T AN/T NI increases . Employing the
hyperscaling relation dl/ = 2-a [6] where d = 3 dimensions, v may be determined from the experimental values of a. Using the dynamic scaling law for an asymmetric planar magnet model which describes the superfluid like
transitions [6], a = ~[E:+~] where E: = 4-d and a = max( a,O), the pre
dicted values of a in the case of a 3 dimensional XY model can be obtained. The calculated values for 9CB, 8CB and mixture are a = 0.5, 0.43 and 0.38 respectively. Although the same trend as the one obtained experimentally of the a values as a function of T AN/T NI is observed, the values predicted
by the scaling laws are substantially larger than the experimental ones and they would lead, in all three cases, to a stronger divergence in the thermal conductivity than in the heat capacity over the phase transition. This would
Figure 3.
10',-- ------ - __
5
b
.ql
0 .5
dc vs t data over the A- N transition for 9CB (a), 8CB (b) and 7CB+8CB mixture (c) .
707
cause a divergence of the thermal diffusivity as well, in contrast with the reported data. This behavior would also be found for a value of a consistent with the three dimensional XY model as a would be 0.34. It is most surprising that the discrepancy between predicted and experimental values increases as the TAN/TN! value progressively decreases and that is where the
value of a tends to approach the aXY value. The best agreement is in fact
found at the TCP. A similar result had been obtained for the Hexatic B -Smectic A transition in n-hexyl-4-n pentyloxybiphenyl-4-carboxylate (650BC) r7], another transition whose critical behavior had been predicted to be XY fike. There was a good agreement between the predicted and experimental values of a even though a value of a = 0.6, much larger than aXY' had
been obtained. In conclusion it has been shown that the critical exponents of the heat
capacity, thermal conductivity and thermal diffusivity obtained at the Smectic A-Nematic transition in samples with different Nematic ranges are inconsistent with the dynamic scaling law predicted by a model which applies to three dimensional XY like phase transitions. The discrepancy has been shown to increase the more the sample Nematic range increases even though the critical exponent of the heat capacity tends to approach uXY' This is consistent with
the picture that even in compounds with large Nematic ranges the critical exponents of the correlation length and susceptibility are also inconsistent with the ones predicted by the XY model.
REFERENCES
1 2 3 4
[5]
708
A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64, (1976). P.G. de Gennes, Solid State Commun., 10, 753, (1972). J. Thoen, H. Marynissen, W. Van Dael, Phys. Rev. Lett., 52, 204, (1984). C.W. Garland, M. Meichle, B.M. Ocko, A.R. Kortan, C.R. Safinye, L.J. Yu, J.D. Lister, R.J. Birgeneau, Phys. Rev.A, 27, 3234 and references therein, (1983). M. Marinelli, U. Zammit, F. Scudieri, S. Martellucci, F. Bloisi, 1. Vicari and J. Quartieri, Nuovo Cimento D9, 557, (1987). P.C. Hohenberg, B.!. Halperin, Rev. Mod. Phys., 49, 435, (1977). G. Nounesis, C.C. Huang, J.W. Goodby, Phys. Rev. Lett., 56, 1712, (1986).
PROPERTIES OF SURF ACE ACOUSTIC WAVE DEVICES UNDER
STRONG EXTERNAL FIELDS
Jiri Zelenka and Miloslav Kosek
Technical College of Mechanica and Textile Engineering 46117 Liberec I Czechoslovakia
INTRODUCTION
Due to material nonlinearities the parameters of the SAW propagating on the substrate will be changed somewhat when the substrate is subjected to strong external fields. The most important external fields are the acceleration, the electric field, the external stress or strain and the temperature change. The knowledge of the effect of these fields on the SAW parameters is important namely in the design of the SAW sensors and precise SAW devices, for example resonators and oscillators. In this work we consider a special case of the general problem. We suppose that SAW with a small amplitude are propagating on the surface of the medium, which is subjected to strong external fields. This means that the nonlinearities need to be considered are those nonlinearities that are connected with the external fields only.
The material nonlinearities have been studied extensively since 1970 and two basic methods have been developed: the nonlinear theory, see for example [1], and the perturbation theory [2]. The nonlinear theory is quite general, which has the consequence that its conclusions are in the relatively complicated relations. On the other hand the perturbation theory applied to the bulk wave resonators [2] includes only the elastic nonlinearities described by the third order elastic moduli. This theory does not take into account the other nonlinearities and the piezoelectric effect. In this paper we apply the basic principles of the perturbation theory in order to develop the equations for the propagation of the small amplitude SAWin the generally nonlinear piezoelectric medium in the presence of external fields. Therefore the basic equations will be written in the form suitable for the solution of the SAW problem and the principles of the perturbation theory will be applied to them.
PERTURBATION THEORY
The perturbation theory considers three states: natural, initial and final state. In the natural state no fields are applied and the medium is in rest. In the initial state the static external fields act on the substrate. This state is described by the elastic displacement U i and the electric potential ~. The following nonlinear equations can be written [2] for the initial state of the medium that occupies the halfspace
Physical Acoustics. Edited by O. Leroy and M. A. Breazeale Plenum Press, New York, 1991
(la)
709
1 7: CijklmnSklSmn - ekijmnEkSmn 1
- ekijEk - 7: HklijEkEl +
1 Di = eiklSkl + :riklmnSklSmn + HijklEjSkl
+ EijEj + ~ EijkEjEk + D10)
(Ib)
(Ic)
(Id)
(Ie)
(1£)
In the equations above the Tij, Sij, Ei, Di are the thermodynamic stress, elastic strain, the electric field intensIty and electric displacement, respectively.
T lJ) and D 1 0) are the stress and electric displacement due to the external
sources and Po is the medium density in the natural state. The equations (Ia) and (Ib) define the strain and the electric field intensity, respectively. The state equations (Ic) and (Id) include all the basic nonlinearities, see for example [3]. The Cijkl and Cijklmn are the elastic moduli of the second and the third rank, eijk and eiklmn are the linear and quadratic piezoelectric stress-tensor components , Hijkl are the electrostriction coefficients and Eij, Eijk are the linear and quadratic permitivities. The equation (Ie) of motion and electrostatic condition is given by (1£). Einstein summation rule is used and the space derivatives are given -by the indices following the comma. The point above the symbol is used for the time derivative.
In the vacuum above the medium only the Laplace equation for the electric potential is needed
(2)
The mechanical boundary conditions on the surface of the medium have the form
(3a)
where ni are the components of the unit vector normal to the surface.
Two extreme electric boundary conditions are usually considered: the free surface and the short-circuited surface. The free surface requires the continuity if the potential and the normal component of the electric displacement
(3b)
where symbols + and denote the vacuum and material side of the surface. The short-circuited surface requires the zero potential on it,
~ = 0 (3c)
The elastic displacement and the electric potential must vanish at infinity.
In the final state the SA W of small amplitude is superposed on the medium in the initial state, for which the equations (1) to (3) are valid. This final state is described by final variables
710
where Ui, <p, Sij, ei, ... are the quantities connected with the propagating SAW. For the final state quantities the same equations as (1) to (3) must be valid. By subtraction of these two sets of equations, some manipulations and neglecting all small terms the following linear equations for the SA W parameters are obtained.
(5a)
(5b)
<P,ii = 0 (5c)
( 5d)
(5e)
<p = 0 ( 5f)
where the effective material properties are given by the following formulae
(m) '" + 'T(O) cijkl = Cijkl Uli jk (6a)
cijkl cijkl + cijkn V l,n
cijkl Cijkl + cjnkl V i,n
Cijkl Cijkl + CijklmnSmn - enijklEn (6b)
(6c)
where EO is the permittivity of the free space and is the Kronecker symbol.
The equations (5) are the well-known linear equations for the SAW problem. They have the following meaning: the equations of motion (5a) and the electric condition (5b) in the medium, the Laplace equation (5c) in the free space above the medium, the mechanical boundary conditions (5d) and the electrical boundary conditions for the free surface and the short-circuited surface (5e) and (5f), respectively. The influence of the external fields is included into these equations by means of the effective second order elastic moduli (6a), the effective piezoelectric stress-tensor components (6b) and the effective permittivity (6c). By the inclusion of these effective material parameters the SA W equations remain linear so that the standard well-known methods can be used for their solution.
Computation of the effect of the external field on the parameters of the propagating SAW consists of two steps. In the first step the elastic and electric quantities Vi, ¢, Sij, Tij, Ei and Di are computed for given external fields by the use of the equations (6). In the second step the standard linear method for the SAW based on equations (5) is applied.
711
TEMPERATURE DEPENDENCE OF SAW VELOCITY
The described method is a general one but its general use requires the solution of the nonlinear static problem and the knowledge of the nonlinear material parameters in equation (1). We used this method for the computation of the temperature dependence of the SAW velocity according to [2]. We neglect all the nonlinear parameters with the exception of the third order elastic moduli. Further we suppose that no other external field acts,
T( 0) = 0 Ei = 0 lJ '
(7)
The strain is given by the equation (2)
(8)
where a ij is the thermal expansion. After substitution of the equations (7) and (8), the equations (6) have the form
C1Jk1 = C1Jk1 + [ (Ik + (Ok ( I) Cij 1 Cij 1mn a mn + c(gk a( I)
nJ 1 1 n + c(Ok all)] 1 J n n 0
+ [1 (2 k + (Ok (2) 2' c ij 1 C ij 1mn a mn + (I k (I) cnj 1 a in +
C( Ik a( I) nJ 1 1 n + cPk all) lJ n n + cPk all) lJ n n + ( Ok \ I) (I) cnj 1mo a n a mo
+ (Ok (I) (I)] c ij nmoain a mo 02
elm) ik1
e( 0) ik1 [el~ i + e(O)a(l)]
1 kn In o + [e(O) (1 2 )
lkn n + e\ I) a\ I)] lkn n 02
[\'!1) = [(0) + [(I) 0 (9) lJ lJ lJ
where the upper index denotes the order of the quantity in the temperature dependence according to the equations
( 0) ( I) 0 1 (2) 02 Cijk1 = Cij k1 + Cij k1 . + ~ijk1
(10)
a .. = a\ I) 0 + a\ 2) 02 lJ 1 J 1 J
and 0 is the parameter difference. The first equation in (9) is the same as the equation (20) in [2].
The temperature dependence of the saw velocity on the Y cut X propagating quartz was computed by the use of a personal computer. The contribution of different parts in the equations (10) to the velocity changes was investigated. The preliminary computations showed that the main effect was due to the linear temperature dependence of the second order elastic moduli
712
coefficients Cl]ltl in (10). The other members have a non-neglect able effect only in the case of greater temperature changes. The quadratic term cUltl improves somewhat the velocity dependence on temperature in the whole temperature range. The thermal expansion improves considerably the velocity dependence at low temperatures and disturbs somewhat the agreement with exact values for the high temperatures. The inclusion of the third order elastic moduli is needed to achieve a good description in the whole temperature range. The temperature dependence of the piezoelectric moduli and the permittivity have practically no effect.
CONCLUSION
We have adopted the special case of the perturbation theory to the more general case, which considers all the basic nonlinear effects and which can be applied to the piezoelectric medium. By this theory the modified equations for the propagation of the SAW under the strong external fields were prepared.
With an aid of these equations the temperature dependence of the SAW velocity was computed for the Y cut X propagating quartz. The preliminary computations show that the main temperature effect was due to the linear temperature dependence of the second order elastic moduli. Only this member can be used with a satisfactory accuracy in the relatively narrow temperature range. In order to achieve good agreement in the wide temperature range the thermal expansion and the third order elastic moduli should be considered. On the other hand the temperature dependence of the piezoelectric moduli and the permittivity does not have any effect.
In a similar way the obtained equations can be used for the computations of the effect of the other fields on the SAW velocity if the material parameters are known for these fields. As an example the effect of a strong electric field can be investigated . The results can be used in the design and analysis of the precise SAW devices.
REFERENCES
[1]
[2]
[3]
J.C. Baumhauer and H.F. Tiersten, Nonlinear electroelastic equations for small fields superposed on a bias, J. Acoust. Soc. Amer., 54:1017, (1973). P.C.Y. Lee and Y.K. Yong, Temperature behavior of quartz plates, J. Appl. Phys., 56:1514, (1984). J. Zelenka, "Piezoelectric Resonators and their Applications", Academia, Prague (1986).
713
ACOUSTOOPTIC NONRECIPROCITY
G.E. Zilberman, L.F. Kupchenko, V.V. Proklov
Institute of Radioengineering and Electronics Academy of Sciences of the U.S.S.R. K. Marx avo 18, GSP-3 103907, Moscow, U.S.S.R.
The effect of the acoustooptic (AO) nonreci\lrocity means the existence of the differences of the phase (or the amplitude) changes for two contrapropagated light beams under its AO interactions with the sound wave. Up to now it were two well known nonreciprocal optical phenomena - the phase nonreciprocity of the light propagation in the magnetics under magnetic field (discovered by M. Faraday more than 140 years ago) and in some moving media (by H. Fizeau more than 70 years ago).
The nature of the third (AO) nonreciprocal effect is a quite different one because it doesn't require any magnetic properties or movement of the media. It needs only the acoustic wave propagation (an elementary excitations) inside a static sample of some material.
Recently were developed several theoretical approaches of the AO nonreciprocity including the isotropic [1], the uniaxial [2J, the optically active [3] and the highly dispersive [4) materials. Finally a few applications for the laser gyroscope [7), the optical directional couplers (8) and other laser devices were also experimentally verified [5,6) and proposed.
First of all the existence of the drastic phase change A¢ of the O-order propagated light beam under strong AO interaction (when {i ~ AnKI/n cosO ~ 7r,
where An is the refractive index modulation, I is the AO interaction length, K is the light wave vector) and its proportionality to AO synchronism mismatching i.e. A¢ N (8-e B), where 8 and 8B are the angle the incidence and the
Figure 1. The amplitude I and the phase A¢ of O-order AO diffracted light as a function of {i (a) and A ¢ as a function of A O-synchronism mismatching 841 B (b)
Physical Acoustics, Edited by O. Leroy and M.A. Breazeale Plenum Press, New York, 1991 715
Bragg angle, respectively was shown theoretically and experimentally [9-11] (see also Figure 1). To explain this effect we need to note that the O-order light consists of both the undiffracted and the double diffracted light with different phase delay but the same frequency, what yields to the averaged phase shift depending on the mentioned AO phase mismatching.
On this background we can simply understand AO nonreciprocity. In fact it is well known that the Bragg condition for the resonant light scattering by the
static grating is: sinfl~ = -K o/2K (Ko = 27r / A, A is the grating period).
For progressive grating (sound wave) it /cields to two cases: sinfl" = -K o/2K + nD/C and sinfl' = K o/2K + nD C for so called antistockes and the stockes light incidence, respectively (see also Figure 2). So in case of AO diffraction we have the difference of the sound - light mismatch for two contrapropagated beams (1 and 2 on Figure 2), what might give a difference in both phase and amplitude sound induced nonreciprocities.
From the theoretical points of view we need to solve the wave equation in the second (or higher) order approximation
.. .. 1 82D grad div E - dE = ~ (Jt2 , (1)
where D £ E, £ = n 2 + 2a cos( wt - Ky), a is the index of the acousto-optic coupling.
Because Eq.(l) in our case is the differential equation with the periodic coefficients we can use Floquet's theorem to get a general solution as follows:
Ei = L A~l) ei[KijX + (K 2+nKo)y] - i(!l+nwo)t (2) n, j
where Ei is the amplitude of the i-component of the diffracted light. Nowadays the second order solution of Eq.(l) was developed by so called continuous fractions techniques for several cases: the isotropic media with shear or longitudinal sound waves, the uniaxis crystal with shear wave, the optically active crystal with shear wave etc. For example in the case of the isotropic media with longitudinal wave we can obtain
m = 0, ±1, ±2 .... , (3)
where Am = ;\-m2 + mZ; Z - 2K2/K o + 1/;
Figure 2. Explanation of the different Bragg angles for the contrapropagated light beams.
716
Figure 3.
I 1[1
~:
A sample of the calculation of !J. ¢ 12 as a function of (3 for uniaxial crystal
q = aK 2/Kd; v = 2n 2KU /(K C); K is the light wave vector in vacuo. In the
general case we can find from Eq.(3)
----"-q------ Am-f , (4) q2 Am - ---''-------
q2 Am +f - --"----
Am +2 -
which might be solved with parameter v to obtain the phases !J.¢12 = 1 !J.¢1 - !J.¢21 !J.I 12 ~ 111 - 121)·
desirable approximation concerning the small values of the AO-nonreciprocities of the light and the O-order light amplitudes (or intensities
In one of the realistic situation - the slow shear wave in an uniaxial crystal (like Te02) - we could find the phase AO-nonreciprocity in the first order approximation under 181 « 7r, ( 8 = 7r-(3 ) as follows (see also Figure 3):
where fc = 1 + (2Ksin8)/Ko-(K27) 2sin 28)/Ko; 7) = (n~-n~)/n~.
Figure 4.
fO
The scheme of the experimental setup: I-laser (A = 0.63 Itm); 2 - AO modulator (T ~ 5 its, f = 1 KHz); 3 - polarizer; 4,5,6,8 and 10 - mirrors; 7 - AO cell under investigation; 9 and 11 - photodetectors.
(5)
717
ACf'2' GO
degr.e 4D 20 0
-aD (O,Arh.un.
-40
-GO
Figure 5. The theoretical and the experimental dependence !J. ¢ 12((J)
The mostly improved experimental verification has been performed with Te02 Bragg cell installed in the optical ring interferometer for two contrapropagated waves as it is shown in Figure 4.
The measurements of !J.¢12 by this technique were independent of the intensities of the interfering light beams when it changed remarkably under experiments. The typical dependence of !J. ¢ 12 as the function of the sound amplitude (exactly of (J, which is the phase modulation index) is shown in Figure 5.
Fairly good qualitative accordance was found between experiments and the related theoretical calculation, what suggests all main features of the AO phase nonreciprocity effect. Here we have realized a quite big nonreciprocal phase difference (up to 60') when (J is not so far from 'Ir. Unfortunately at these conditions we have the loss of the propagated light intensity, so in the applications of this effect we need to find. the compromise of two important features - the phase nonreciprocity !J.¢12 and the light attenuation. We need to note that the difference of the light losses for the contrapropagated waves
fwhat is the AO amplitude nonreciprocity) are ordinarily very small, because 12,13]
In conclusion the authors hope to meet in the future much more applications of AO-nonreciprocity in the different fields of optics.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
718
G.E. Zilberman and L.F. Kupchenko, Propagation of light through ultrasonic beam in isotropic dielectrics, Journal Radioengineering and Electronics (Sov.), 20:2347, (1975). G.E. Zilberman, L.F. Kupchenko, G.F. Goltvyanskaya, To theory of light diffraction by shear waves in uniaxial crystals, Journal Radioengineering and Electronics (Sov.), 29:2095, (1984). G.E. Zilberman, L.F. Kupchenko, G.F. Goltvyanskaya, Light diffraction by ultrasound in uniaxial optically active crystals, Journal Radioengineering and Electronics (Sov.), 29:2449, (1984). G.E. Zilberman, V.V. Proklov, Nonreciprocal acoustooptic effect in materials with enhanced dispersion of dielectric permittivity, Journal Radioengineering and Electronics (Sov.), 30:156, (1985). S.N. Antonov, Yu.V. Gulyaev, V.V. Proklov etc., Effect of the optical nonreciprocity under strong acoustooptic interactions in Te02, in: "Digests of XII Allunion Conference on acoustoelectronics and quantum acoustics" (U.S.S.R.), Saratov, 1:277, (1983). G.E. Zilberman, L.F. Kupchenko, S.N. Antonov, V.V. Proklov, etc., Nonreciprocal acoustooptic effect, Preprint IRE (U.S.S.R.) Moscow, N' .28 (400), (1984).
[7]
[8]
[9]
[10]
[11]
[12]
[13]
A.A. Zadernovsky, Acoustooptic phase nonreciprocal cell, Journal of Quantum Electronics (Sov.), 12:1748, (1985). P.P. Holokoz, Yu.L. Oboznenko etc., Ring laser with acoustooptic nonreciprocal cell, in: "Digests of III Allunion Conference on laser optics", Leningrad (USSR), 230, (1981). G .E. Zilberman, L.F. K upchenko, Light propagation through ultrasonic beam in the uniform isotropic dielectrics, Journal Radioengineering and Electronics, 22:1592, (1977). G.E. Zilberman, L.F. Kupchenko and S.S. Zhukov, Experimental investigations of the light wave phase under propagation through ultrasonic beam, Journal Raclioengineering and Electronics (Sov.), 25:1991, (1980). S.N. Antonov, V.V. Proklov, Peculiarities of the light propagation through ultrasonic beam under strong acoustooptic interactions, Journal Technical Physics (Sov.), 53:306, (1983). P.P. Holokoz, Yu.L. Oboznenko, Amplitude nonreciprocity of the acoustooptic interaction in Ge, in: "Acoustooptic and acoustoelectronic devices for radioelectronic systems", S. V. Kulakov ed., Leningrad (U.S.S.R.), (1985). L.F. Kupchenko, Yu.V. Astashov etc., Journal Radioengineering and Electronics, 34:1341, (1989).
719
AUTHOR INDEX
A Coelho-Mandes H., France 613 Cook B.D., U.S.A. 467
Adler 1., U.S.A. 3;529 Cops J., Belgium 425 Agarwal Ravinder, India 621 Cowley A., U.K. 271 Alami K., France 613 Craciun F., Italy 13 Alippi A., Italy 13 Cui-Ying F., China 365 Allard J .F., France 425 Amato G., Italy 199 Amaudric du Chaffaut C., France 613 D Antonov S.N., U.S.S.R. 205;209 Arapov A.V., U.S.S.R. 213
Danicki E., Poland 281;287 Ashurov A.M., U.S.S.R. 485 Attal J., France 613
de Billy M., France 297
Audoly C., France 219;359 Decarpigny J.N., France 359 Diaci J., Yugoslavia 291 Drean J.M., France 297
B Duclos J., France 307;635 Dumery G., France 219
Baghai-Wadji A.R., Austria 225 Duriez J .M., France 573
Baird A., U.K. 271 Balakin L.V., U.S.S.R. 231 E Balakshy V., U.S.S.R. 231 Bauerschmidt P., FRG 237
Euvrard D., France 313 Beckett C., U.K. 371 Benedetto G., Italy 199 Berger J., France 385 F Berkhout A.J., The Netherlands 473 Bettucci A., Italy 13
Fagard V., Belgium 179 Bilmes G.M., FRG 599
Fay B., FRG 41 Blomme E., Belgium 243
Foda M.A., Egypt 319 Boarino L., Italy 199
Franklin H., France 327 Bossut R., France 359 Bozoki Z., Hungary 511 Braslavsky S.E., FRG 599
G Breazeale M.A., U.S.A. 21 Briers R., Belgium 243 Bugaev A.S., U.S.S.R. 249 Gatignol Ph., France 327 Bukhny M., U.S.S.R. 253 Gedrich T., U.K. 271 Busse G., FRG 31 Glorieux C., Belgium 433
Goldfarb 1.1., U.S.S.R. 335;341
C Goncharov V.S., U.S.S.R. 213 Gorsky V.B., U.S.S.R. 249 Gubaidullin A., U.S.S.R. 347
Cahingt H., France 497 Gundersen S.A., Norway 677 Campos L.M.B.C., Portugal 261 Gusev O.B., U.S.S.R. 405 Chernosatonsky L.A., U.S.S.R. 253 Guy Ph., France 391
721
H M
Harrison C., U.K. 271 Machui J., FRG 237 Hashimoto K., Japan 353 Madhusoodanan K.N., India 479 Hennion A.C., France 359 Madvaliev U., U.S.S.R. 485 Hereman W., U.S.A. 505 Maev R.G., U.S.S.R. 253 Hess P., FRG 551 Mandelis A., Canada 489 Hong Z., China 365 Mantel P., France 385 Humphrey V.F., U.K. 371 Marinelli M., Italy 703 Hutchins D.A., U.K. 381 Martellucci S., Italy 703 Hyoguchi T., Japan 695 Matsumoto T., Japan 695
Maze G., France 497;581 Mechiche Alami 0., France 313
I Mertens R.A., Belgium 505 Miklos A., Hungary 511
Ignatiev LA., U.S.S.R. 377 Ming-Zhou Z., China 365 Isajanian Ye.G., U.S.S.R. 671 Mityurich G.S., U.S.S.R. 517 Izbicki J .L., France 441;581 Molotok V.V., U.S.S.R. 405
Monchalin J.P., Canada 65 Mozhaev V.G., U.S.S.R. 523
J Mozina J., Yugoslavia 291
Jacob Philip, India 479;545 Jansen D.P., Canada 381 N Jia X.P., France 385 Jungman A., France 391
Nagy P.B., U.S.A. 529 Nakagawa Y., Japan 537
K Nandakumar K., India 545 Neubrand A., FRG 551
Kawashima K., Japan 695 Nikolic P.M., Yugoslavia 557;641
Knapp S.M., U.K. 371 Notebaert H., Belgium 417
Kojima S., Japan 399 Konstantinov L., FRG 551 0 Kosek M., C.S.S.R. 709 Kotov V.N., U.S.S.R. 205
Orenbakh Z.M., U.S.S.R. 563 Kulakov S.V., U.S.S.R. 405 Kupchenko L.F., U.S.S.R. 715 Ottoy J.P., Belgium 505
Kwiek P., FRG 129
L P Pareige P., France 441;581
Laperre J., Belgium 413;417 Parygin V.N., U.S.S.R. 77 Lauriks W., Belgium 425;433 Pashchin N.S., U.S.S.R. 569 Lecroq F., France 497 Pelzl J., FRG 629 Leduc M., France 307;635 Pilarski A., Poland 87 Le Flour J .C., France 391 Pizzoferrato R., Italy 703 Lenoir 0., France 441;581 Plekhanov V.G., U.S.S.R. 377 Lerch R., FRG 237 Poiree B., France 99 Leroy 0., Belgium 243,451,647 Pomyalov A.V., U.S.S.R. 249 Leung K.F., Canada 489 Popkov A.F., U.S.S.R. 377 Lewandowski J., Poland 457;461 Pouliquen J., France 573 Lewis D.K., U.S.A. 467 Proklov V.V., U.S.S.R. 485;715 Linze Ph., Belgium 591 Lorenz M., The Netherlands 473
Q Lorincz A., Hungary 511 Lothe J., Norway 677 Lyamshev L.M., U.S.S.R. 55 Quentin G., France 119;385;391
722
R Tinel A., France 635 Todorovic D.M., Yugoslavia 557;641
Reibold R., FRG 129 Rembert P., France 441;581
U Reymond M.C., France 591 Ripoche J., France 143;497 Ristovski Z.D., Yugoslavia 557;641 Ueha S., Japan 189;687 Rohr M., FRG 599 Rousseau M., France 327
V Roux J., France 155 Ruchko S.V., U.S.S.R. 213 Ruile W., FRG 237 Varma F.I., U.S.S.R. 341
Van Den Abeele Koen, Belgium 647
S Vidoret P., France 657 Visintini G., FRG 237 Voloshinov V.B., U.S.S.R. 665
Sablikov V.A., U.S.S.R. 605 Voronco A.I., U.S.S.R. 671 Saied A., France 613 Saurel J.M., France 613
W Schoubs E., Belgium 179 Scudieri F., Italy 703 Serdyukov A.N., U.S.S.R. 517 Wang L., Norway 677 Shkerdin G., U.S.S.R. 451;671 Wang N., Japan 687 Shreiber I.R., U.S.S.R. 335;341;563 Wright O.B., Japan 695 Shushkov G.A., U.S.S.R. 563 Singh V.R., India 621 y Sliwinski A., Poland 165 Spagnolo R., Italy 199 Soczkiewicz E., Poland 625 Yakovkin LB., U.S.S.R. 213;569 Sotnikov V.N., U.S.S.R. 205;209 Yamaguchi M., Japan 353 Stelwagen U., The Netherlands 473 Yong-Chen X., China 365
T Z Tankovsky N., Bulgaria 629 Zammit U., Italy 703 Thimus J.Fr., Belgium 591 Zelenka J., C.S.S.R. 709 Thoen J., Belgium 179;433 Zelyony V.P., U.S.S.R. 517 Thys W., Belgium 413;417 Zilberman G.E., U.S.S.R. 715 Tigin D.V., U.S.S.R. 405 Ziolkowski R.W., U.S.A. 467
723
SUBJECT INDEX
Absorbance, 495 Absorbed energy, 438 Absorbed frequency, 298 Absorbed power, 309 Absorbed radiation, 180 Absorbent, 445, 589, 617
absorbent powder, 486 Absorber
sound absorber, 359 Absorbing baffle, 359 Absorbing body, 156 Absorbing coating, 657 Absorbing layer, 441 Absorbing material, 545 Absorbing medium, 573
light-absorbing media, 56 Absorbing molecules, 511 Absorption, 32, 41, 65, 271, 402, 452,479, 545,
573, 641 absorption coefficient, 43, 180, 199, 441,
480, 495, 573, 641, 695 absorption depth, 704 absorption edge, 558 absorption model, 687 absorption of light, 56, 433 absorption of phonons, 511 absorption of radiation, 180 absorption operator, 687 absorption parameter, 521 absorption region, 199 acoustic wave absorption, 672 electromagnetic absorption, 671 heat absorption, 605 light absorption, 512 material absorption, 557 molecular absorption, 511 optical absorption, 37, 479
optical absorption coefficient, 438, 546, 704
optical absorption spectrum, 199 optoacoustic absorption signal, 599 Rayleigh wave absorption 255 surface absorption, 34
Absorptive material, 148
Acoustoelectronic device nonlinear acoustoelectronic device, 523
Acoustoelectronic microanalysis, 213 Acoustooptical cell, 666
Bragg acoustooptical cell, 172 Acoustooptical detection, 468
acoustooptical Rayleigh mode detection, 654 acoustooptical Lamb mode detection, 417,
654 Acoustooptical device, 665 Acoustooptical figure of merit, 79, 666 Acoustooptical filter, 665
acoustooptical match filtering, 174 Acoustooptical filtration of radiation, 665 Acoustooptical interaction, 77, 665 Acoustooptical material, 77 Acoustooptical method, 405 Acoustooptical modulator
multichannel acoustooptical modulators, 405 Acoustooptical processor
multichannel AO signal processor, 408 two beam acoustooptical processor, 177
Acoustooptical setup, 418 Acoustooptical signal
acoustooptical signal quadrature, 176 multichannel AO signal processor, 408
Acoustooptic analyzing system, 233 Acoustooptic cell, 82, 231 Acoustooptic coupling, 716 Acoustooptic deflector, 82 Acoustooptic device, 62 Acoustooptic diffraction, 232
Bragg anisotropic acoustooptic diffraction, 205
Acoustooptic experiment, 469, 654 Acoustooptic exploration, 62 Acoustooptic interaction, 60, 231, 671
acoustooptic interaction length, 232 Acoustooptic measurement, 467 Acoustooptic method, 209 Acoustooptic modulator, 209 Acoustooptic nonreciprocity, 715 Acoustooptic probing system, 231 Acoustooptic reflection coefficient, 647
725
Acoustooptics, 129, 165, 505 multichannel acoustooptics, 408
Acoustooptic stabilization system, 696 Amorphous film, 695 Amorphous form, 545 Amorphous hydrogenated Si, 553 Amorphous material, 545 Amorphous nature, 558 Amorphous semiconductor, 199, 479, 700 Amorphous solid, 691 Anisotropic birefrigence value, 79 Anisotropic composite materials, 90 Anisotropic diffraction, 77, 234, 289
anisotropic Bragg diffraction, 77 anisotropic Bragg diffraction mode, 406
anisotropic Raman-Nath diffraction, 81 Bragg anisotropic acoustooptic diffraction,
205 Anisotropic elastic body, 156 Anisotropic halfspace, 569
anisotropic piezoelectric halfspace, 287 Anisotropic materials, 87 Anisotropic medium, 81, 284, 677
anisotropic elastic medium 287 Anisotropic situation, 89 Anisotropic structure
layered anisotropic structure, 93 Anisotropy, 27, 36, 77, 87, 179, 241, 287, 393,
406 AO, 205,209 AO beam splitter, 207 AO cell, 717 AO convolver, 410 AO crystal, 208, 209 AO device, 208, 405 AO diffracted light, 715 AO diffraction, 716 AO element, 208 AO interaction, 209, 406, 715
AO interaction length, 715 AO method, 405 AO modulator, 207, 209, 717
AO space-time modulator, 405 bulk multichannel AO modulator, 405
AO nonreciprocity, 715 AO phase, 716 AO signal, 294 AO spectrum
AO space integrating spectrum, 408 AO spectrum analyzer, 408
AO synchronism, 715 AO wave, 210 Attenuated peak, 441 Attenuated signal, 173 Attenuated wave, 514
attenuated acoustic wave, 457 attenuated surface acoustic wave, 538
Attenuation, 31, 43, 91, 253, 297, 337, 341, 348, 457, 538, 554, 691
acoustical wave attenuation, 405 acoustic attenuation, 273
726
Attenuation (continued) attenuation coefficient, 43 173, 271, 458,
461, 514, 564, 622, 688 attenuation image, 382 attenuation measurement, 72, 73 attenuation vector, 99 light attenuation, 718 sound attenuation, 365 ,563 surface acoustic wave attenuation, 554 thermal attenuation, 341 ultrasonic attenuation, 65
Attenuator neutral density attenuator, 292
Backscattered echo, 144, 497 Backscattered form function, 375 Backscattered pressure, 143 Backscattered signal, 581
backscattered signal spectrum, 145 Backscattered spectrum, 587
backscattered signal spectrum, 145 Backscattered time signatures, 143 Backscattered wave, 284 Backscattering, 301
ultrasonic backscattering, 41 Backscatter method, 41 Backscatter response, 441 Backscatter signal, 41 Bicrystal, 9 Bilayer structure, 441 Bragg
Bragg angle, 79, 205, 209, 232, 406, 505, 716 Bragg AO modulator, 406 Bragg approximation, 671 Bragg backscatter method, 45 Bragg cell, 67, 718
Bragg acoustooptical cell, 172 Bragg condition, 82, 207, 716 Bragg diffracted light, 408 Bragg diffraction, 205, 505
anisotropic Bragg diffraction, 77 anisotropic Bragg diffraction mode, 406 Bragg anisotropic acoustooptic diffraction,
205 X-ray Bragg diffraction, 696
Bragg frequency, 67 Bragg cell frequency, 119
Bragg reflection, 46 pure Bragg reflection condition, 505
Bragg regime, 167, 206, 232, 406 Bragg scattering, 16 Bragg's law, 62 Bragg synchronism, 205
Bubble, 58, 271, 335, 563 air bubble, 631 foam bubble, 336 gas bubble, 342, 348, 537, 563 bubble layer, 279
Bubbly liquid, 347
Capillar, 629
Capillary, 335 Ceramic, 65, 255, 385, 591
ceramic impedance, 576 ceramic-metal composite, 70 ceramic plate, 17 ceramic transducer, 631 glass ceramic, 365 piezoceramic, 517, 573 piezoelectric ceramic transducer, 600
Complex harmonic plane waves, 99 Complex reflection/refraction angle, 157 Complex reflection/transmission coefficient, 159 Composite, 9, 13, 518, 591
ceramic-metal composite, 70 composite material, 87
anisotropic composite materials, 90 composite piezoelectric plates, 13 multilayered composite, 14
Conductance load conductance, 311
Conduction band, 482, 545 Conductivity, 606
conductivity condition, 226 conductivity measurement, 202 heat conductivity, 408 hydroconductivity, 343 thermal conductivity 179, 348, 433, 512,
559,609,703 inhomogeneous thermal conductivity
equation, 518 Constructive interference, 3 Critical angle, 88, 392, 613, 637, 647
critical Rayleigh angle, 161 Lamb critical angle, 396
Crystal, 21, 78, 205, 233, 253, 287, 365, 542, 665, 671
AO crystal 208, 209 crystal ferrite, 249 crystalline form, 545 crystalline solid, 21 crystallinity, 696 crystallite, 253 crystallographic axis, 287 crystallographic coordinate system, 93 crystallographic orientation, 9 crystallophysic coordinate system, 569 crystal semiconductor, 642 cubic crystal, 77 liquid crystals, 179 monocrystal, 405 noncrystalline solid, 482 orthorhombic crystal, 524 piezoelectric crystal, 517 piezoelectric monocrystal transducer, 119 thermotropic crystal, 703
Cylinder, 13, 45, 124, 145, 220, 341, 359, 373, 497, 586, 660
circular hollow cylinder, 192 cylinder grating, 361 fluid cylinder, 297 scattering from cylindes and spheres, 121
Cylindrical body, 200 Cylindrical core, 59 Cylindrical film, 406 Cylindrical hole, 381 Cylindrical inclusion, 219, 359
liquid cylindrical inclusions, 150 Cylindrically symmetrical structure, 45 Cylindrical piezoelectric actuator, 194 Cylindrical piezotransducer, 406 Cylindrical plate wave, 386 Cylindrical sample, 491, 592 Cylindrical shell, 124, 143, 371, 497, 582 Cylindrical target, 124, 143, 298, 441, 497, 581 Cylindrical vessel, 89 Cylindrical wave, 220, 541 Cylindrical wire, 125, 152
Defocus, 613 Defocusing, 3
defocusing depth, 3 defocusing distance, 4 defocusing of laser beam, 57 surface wave beam self-defocusing, 523
Destructive examination, 88 Destructive interference, 3, 491, 532 Detection
acoustooptical detection, 468 acoustoopticaJ Lamb mode detection, 417,
654 acoustooptical Rayleigh mode detection, 654 flaw detection, 65 photo acoustic detection, 32 underwater detection, 152
Diffracted light Bragg diffracted light, 408 focused diffracted light, 409
Diffracted radiation, 205 Diffraction
acoustooptic diffraction, 232 Bragg anisotropic acoustooptic diffraction,
205 anisotropic diffraction, 234, 289, 77
anisotropic Bragg diffraction, 77 anisotropic Bragg diffraction mode, 406
anisotropic Raman-Nath diffraction, 81 AO diffraction, 716 Bragg diffraction, 205, 505
Bragg anisotropic acoustooptic diffraction, 205
anisotropic Bragg diffraction, 77 anisotropic Bragg diffraction mode, 406
diffraction mode, 206 Fraunhofer diffraction pattern, 129 isotropic diffraction, 77
isotropic Raman-Nath diffraction, 81 photothermal wave diffraction, 489 Raman-Nath diffraction, 209, 232
anisotropic Raman-Nath diffraction, 81 Raman-Nath type diffraction, 243, 650
Scholte wave diffraction, 635 X-ray Bragg diffraction, 696
727
Diffusion diffusion length
thermal diffusion length, 31, 180, 200, 433, 491, 557, 608, 642, 704
heat diffusion equation, 511 thermal diffusion, 400, 700
thermal diffusion coefficient, 180 Diffusivity
thermal diffusivity, 31, 179, 200, 348, 479, 512, 609, 707
Disfocusing, 410 Dispersion
foam dispersion, 341 photoacoustic dispersion, 403 thermal dispersion, 399
Dissipation thermal dissipation, 348
Effusivity thermal effusivity, 200, 704
Eigenmode, 205, 452 Electroacoustic effect, 629 Electroacoustic transformation, 407 Electrode 192, 225, 237, 491
electrode pair 216 IDT electrode, 353 in terdigited electrodes, 307 thin film electrode, 630
Electroded region, 570 Electrodynamics, 59 Electrolyte, 629 Electromagnetic absorption, 671 Electromagnetical coupling, 121 Electromagnetic interference, 61, 190 Electromagnetic radiation, 557, 631, 665 Electromagnetics, 99 Electromagnetic wave, 31, 112, 413, 671 Electromagnetism, 156 Electromechanical coefficient, 574 Electromechanical coupling, 355
electromechanical coupling coefficient, 570 Electromechanical feedback, 66 Electron-lattice coupling, 483 Electron-phonon interaction, 482 Electron, 557
electron hole, 605 Electronic beam, 235 Electronic charge, 233 Electronic device, 62 Electronic excitation, 696 Electronic material, 479, 545 Electronic measurement, 146 Electronic modulator, 234 Electronic property, 202 Electronics, 173
digital electronics, 411 digital processing electronics, 72 quantum electronics, 55
Electrooptical device, 85 Electrooptical efficiency, 405 Electrooptical media, 77
728
Electrooptic effect, 83 Electrooptic tensor, 83 Electrophoretic force, 630 Electrostatic approximation, 228 Electrostatic condition, 710 Electrostatic coupling, 355 Electrostatic problem, 281, 288 Electrostriction coefficient, 519, 710 Emission, 65, 307
acoustic emission, 591 emission angle, 637 emission of sound, 57 emission pattern, 74 energy emission, 616 laser emission frequency, 57 surface infrared emission, 34
Eradiating piezotransducer, 406 Eradiation angle spectrum, 406 Evanescent field, 530 Evanescent wave 99, 156, 220, 264, 360, 647 Expansion mode, 229
Fabry-Perot cavity, 671 Fabry-·Perot interferometer, 67 Farfield 243, 417
farfield region, 165 optical farfield investigation, 129
Ferromagnetic resonance, 249 Film 60, 256, 341, 368, 406, 491,
amorphous film, 695 cylindrical film, 406 ferrite film, 249 film detector, 600 heated film, 554 isotropic film, 555 semiconductor film
photoacoustic characterization of semiconductor films, 199
photothermal characterization of semiconductor films, 199
thin film, 199, 249, 377, 551, 557, 695 thin film electrode, 630
water film, 68 Flaw
flaw detection, 65 material flaw, 173
Foam 335, 341, 563 foam bubble, 336 foam cell, 341 foam combustion, 335 foam density disturbance, 336 foam destruction, 335 foam dispersion, 341 foam generator, 563 foam mode, 341 foam motion, 342 foam velocity, 342 gas-liquid foam, 563 polyhedral foam, 341
Foamy medium, 271 Focus 372, 386
Focus (continued) focus wave mode, 467 laser focus, 33
Focused beam, 199, 614 focused laser beam, 32
Focused diffracted light, 409 Focused interferometric probe, 551 Focused laser, 399, 489
focused laser beam, 32 Focused light, 232, 372, 417 Focused transducer, 3 Focused wave, 387 Focusing
beam focusing, 66 focusing effect, 276 focusing lens, 3, 292 focusing of laser beam, 57 surface wave beam self-focusing, 523 Synthetic Aperture Focusing Technique
(SAFT), 473 synthetic focusing, 46
Fraunhofer diffraction pattern, 129 Fraunhofer intensity, 243 Fraunhofer region, 165, 243, 626 Free mode, 156 Fresnel equation, 700 Fresnel region, 165, 243
optical Fresnel region, 129
Fresnel zone, 406
Generation heat generation, 605 interface wave generation, 121 photoacoustic signal generation, 511 plate wave generation, 121 surface wave generation, 92, 121, 523, 655
surface acoustic wave generation, 551 thermooptical signal generation, 517
Group velocity, 31, 88, 110, 126, 213, 238, 388, Guide
multi-mode guide, 60 single-mode guide, 60
Guided mode propagation, 529 Guided wave, 88, 148, 297, 497, 529
guided wave resonances, 149
Heat, 41, 407, 511 dynamic heat, 32 effective heat rejection, 408 heat absorption 605 heat capacity 179, 399, 433, 700, 703 heat conducting fluid, 103 heat conduction, 489, 609 heat conductivity, 408, 457 heat diffusion equation, 511 heat exchanger, 219 heat expansion, 511 heat generation, 605 heat interaction, 347 heat loading, 66 heat saturation effect, 488
Heat (continued) heat source, 513, 605 heat transfer, 335, 341, 348 latent heat release, 402 specific heat, 57, 180, 403, 512, 559, 703
specific heat capacity, 348 Heated camera, 570 Heated film, 554 Heater, 480
electric heater, 181 Heating
DC heating, 514 heating effect, 406 periodic heating, 433, 557 surface heating, 66
Heterogeneity, 102 heterogeneity angle, 102 heterogeneity coefficient, 156
Heterogeneous calculi, 621 Heterogeneous materials, 87 Heterogeneous medium, 62, 219 Heterogeneous space, 461 Heterogeneous wave, 99, 155
heterogeneous plane wave, 99 heterogeneous utrasonic wave, 621
Horn theory, 261 Hydrothermal environment, 87
IDT electrode, 353 Im/Re method, 143, 581 Im/Re ratio, 442 Im/Re spectrum, 441, 583 Inhomogeneity 43, 201, 451, 526, 551, 605, 625 Inhomogeneous calculi, 621 Inhomogeneous materials, 14 Inhomogeneous medium, 625 Inhomogeneous structure, 451 Inhomogeneous surface, 451 Inhomogeneous thermal conductivity equation,
518 Inhomogeneous thermodynamical variable, 513 Inhomogeneous wave, 328, 425, 647
inhomogeneous plane waves, 99 inhomogeneous utrasonic wave, 621
Interdigital transducer, 215, 225, 307, 353, 540, 637
Interdigited electrodes, 307 Interface, 5, 15, 45, 55, 88, 120, 155, 184, 200,
213, 226, 257, 272, 279, 327, 341, 373, 381, 391, 400, 427, 445, 471, 477, 491, 526, 583, 591, 613, 635, 647, 659, 700
air-liquid interface, 56 fiber-matrix interface, 9 interface contrast, 7 interface imperfections, 10 interface inspection technique, 5 interface mode, 155, 529 interface parameter, 7 interface property measurements, 5 interface quantities, 11
729
Interface (continued) in terface studies, 5 interface wave 99, 150, 156, 299, 331, 414,
529, 583, 635 interface wave generation, 121
liquid-gas interface, 184 mode converting interface, 471 normal interface, 5 parallel interface, 9 plane interface, 99 vertically polarized interface mode, 530
Interfacial reflection and transmission, 14 In terference
constructive interference, 3 destructive interference, 3, 491, 532 electromagnetic interference, 61, 190 photothermal wave interference, 489 thermal wave interference, 557
Interferometer digital ultrasonic spectrointerferometer, 88 Fabry-Perot interferometer, 67 focused interferometric probe, 551
In terferometry photoacoustic interferometry, 560
Irradiated specimen, 46 Irradiated surface, 292 Irradiated ultrasonic pulse, 44 Irradiation 293, 545, 599, 700
irradiation angles, 46 irradiation surface, 43
Isotropic body, 284 isotropic elastic body, 156 isotropic viscoelastic body, 156
Isotropic constituents, 14 Isotropic diffraction, 77
isotropic Raman-Nath diffraction, 81 Isotropic fiber
transversely isotropic fibers, 91 Isotropic film, 555 Isotropic halfspace, 461
isotropic nonlinear halfspace, 526 transversely isotropic halfspace, 93
Isotropic layer, 391 isotropic linear layer, 526 transversely isotropic layer, 93
Isotropic liquid, 179, isotropic liquid phase, 182
Isotropic material, 79, 90, 366, 677, 715 Isotropic medium, 77, 156, 243, 328, 414, 452,
573, 626, 635, 716 homogeneous isotropic medium, 523 transversely isotropic homogeneous medium,
93 Isotropic plate, 122 Isotropic sample, 609 Isotropic subject
homogeneous isotropic subject, 254 Isotropic solid, 89, 254, 365 Isotropic substrate, 290, 556
nonlinear isotropic substrate, 525 optically isotropic substrate, 377
730
Isotropic transition nematic-isotropic transition, 704
Isotropy, 327 transverse isotropy, 88
Lamb angle, 417, 647 Lamb critical angle, 396
Lamb mode 9, 16, 250, 388, 417, 446, 647 acoustooptical Lamb mode detection, 417,
654 leaky Lamb mode, 391, 417
Lamb spectrum, 418 Lamb wave, 17, 121,417,635
Lamb type surface wave, 385 Laser
defocusing of laser beam, 57 focused laser, 399, 489 focusing of laser beam, 57 laser-acoustic microscopy, 61 laser acoustics, 695 laser emission frequency, 57 laser focus, 33 laser-induced sound generation mode, 59 laser radiation, 33, 56 laser scanning microscope, 62 semiconductor laser, 399 short capillary laser, 60
Layer, 9, 44, 88, 152, 179, 201, 213, 333, 342, 359, 405, 413, 438, 483, 525, 532, 560, 695
bubble layer, 279 fluid layer, 427 isotropic layer, 391
isotropic linear layer, 526 transversely isotropic layer, 93
layer thickness, 31 multilayer, 363, 413
multilayer actuator, 194 multilayered composite, 14 multilayered medium, 327, 589 multilayered structure, 121, 152, 414 multilayer sample, 560
overlayer, 203 solid layer, 427 surface layer, 56, 180
subsurface layer, 570 transparent layer, 66 two layer plate system, 391
Layered fluid, 687 Layered materials, 5 Layered medium, 155, 160, 391, 425, 647 Layered porous system, 425 Layered solid system, 425 Layered structure, 9, 87, 179, 213, 281, 413
layered anisotropic structure, 93 Layered transducer, 637
Magnetoelastic interaction, 249 Magnetoelastic material, 527 Magnetoelastic mode
magnetoelastic mode effect, 249
Magnetooptic effect, 379 magnetoelastic mode effect, 249
Magnetostatic mode, 249 Magnetostatic wave, 251, 377 Marcoporous powder, 485 Metrology
applied metrology, 141 ultrasonic metrology, 65
Microcapillary, 342 Microelectronics, 605 Microphone
photo acoustic microphone, 180 Microphoto analysis, 563 Microscale motion, 338 Microscope
acoustic microscope, 253 laser scanning microscope, 62 microscope analysis, 558 microscope glass, 200 scanning acoustic microscope, 3, 253
Microscopic events, 593 Microscopic interaction, 370 Microscopic scale, 591, 613 Microscopic structure, 479 Microscopy
acoustic microscopy, 613 laser-acoustic microscopy, 61 scanning acoustic microscopy, 614
Mode, 191, 497, 527, 613, 639 acoustic mode, 337, 452
acoustic radiation mode, 452 antisymmetrical mode, 333
vertically polarized antisymmetric mode, 531
bulk mode, 555 mode coupled bulk mode, 529 phase-matching bulk mode, 529
contact mode, 88 diffraction mode, 206
anisotropic Bragg diffraction mode, 406 eigenmode, 205, 452 entropic mode, 114 Floquet mode decomposition, 657 foam mode, 341 focus wave mode, 467 generation mode
laser-induced sound generation mode, 59 interface mode, 155, 529
vertically polarized interface mode, 530 isothermal mode, 348 Lamb mode, 9, 16, 250, 388, 417, 446,
647 acoustooptical Lamb mode detection, 417,
654 leaky Lamb mode, 391
leaky mode, 529 leaky Lamb mode, 391 leaky Rayleigh mode, 613
magnetoelastic mode effect, 249 magnetostatic mode, 249
Mode (continued) mode conversion, 121, 228, 378
mode conversion ultrasonic motor, 191 mode converted wave, 473 mode converting interface, 471 mode coupled bulk mode, 529 mode coupling, 238 mode method, 451 mode number, 143 mode propagation, 22
guided mode propagation, 529 mode rotation ultrasonic motor, 193 multimode guide, 60 multimode fiber light, 210 multimode vibrator, 192 normal mode 213, 373, 414
normal mode of vibration, 146, 303 normal propagation mode, 417
oscillation mode, 272 plate mode, 417, 654 polarized mode, 417 propagating mode, 14, 331, 453 pulse echo mode, 89 radiation mode, 452
acoustic radiation mode, 452 optical radiation mode, 60
Rayleigh mode, 9, 297, 533 acoustooptical Rayleigh mode detection,
654 reflection mode, 68 resonance mode, 222, 359, 376, 377, 388
spin-wave resonance mode, 249 resonant acoustic mode, 17 rotational mode, 114 SAW mode, 452 secondary flows mode, 335 single-mode fibre, 60 single-mode guide, 60 spin-wave resonance mode, 249 Stoneley mode, 534 surface mode, 533, 555
SAW mode, 452 symmetric mode, 417
horizontally polarized symmetric mode, 531
vertically polarized symmetric mode, 530 TEM mode, 489 thickness mode, 385 vibration mode, 146, 192, 320, 385
normal mode of vibration, 146, 303 waveguide mode, 60, 452
optical waveguide mode 377 Whispering Gallery mode, 297
Motor linear motor, 192 mode conversion ultrasonic motor, 191 mode rotation ultrasonic motor, 193 multimode ultrasonic motor, 192 ultrasonic motor, 189
Multilayer, 363, 413
731
Multilayer (continued) multilayer actuator, 194 multilayered composite, 14 multilayered medium, 327, 589 multilayered structure, 121, 152, 414 multilayer sample, 560
Multimode ultrasonic motor, 192
NDE/NDT, 94, 87, 128, 530 Nearfield, 129, 243, 361, 385, 470
nearfield region, 165 Nematic transition
A-nematic transition, 704 nematic-isotropic transition, 704
Nondestructive control nondestructive product-quality control, 61
Nondestructive determination nondestructive grain-size determination, 44
Nondestructive evaluation, 87, 121, 152, 381, 385
Nondestructive method, 560, 605 Nondestructive testing, 55, 395, 551 Nonradiative deexcitation, 199 Nonradiative recombination, 199 Nonradiative relaxation, 545 N on reciprocity
acoustooptic nonreciprocity, 715 Nonspecular reflection effects, 650
Optical absorption, 37, 479 optical absorption coefficient, 438, 546, 704 optical absorption spectrum, 199
Optical farfield optical farfield investigation, 129
Optical fibre, 60 Optical Fresnel region, 129 Optical mode
optical radiation mode, 60 optical waveguide mode, 377
Optical radiation, 56, 180, 209, 517 optical radiation mode, 60
Optical reflection, 35 Optical scanner, 72 Optical scanning
optical beam scanning, 68 Optical substrate
optically isotropic substrate, 377 Optoacoustic signal
optoacoustic absorption signal, 599 Optoacoustic spectroscopy
optoacoustic Raman gain spectroscopy, 599 Optoelectronic components, 61 Optoelectronic material, 479, 545
PA, 199 P A arrangement, 511
• PA cell, 438, 480, 514, 546, 641 P A effect, 545, 557 PA method, 557, 641 PA phenomenon, 511 PA process, 511
732
P A response, 519 P A signal, 181, 433, 480, 485, 514, 519, 545,
557, 641, 703 PA spectrometer, 479, 485 PA spectroscopy, 521, 557 PA spectrum, 479, 486, 546, 557, 641 P A technique, 433, 479, 480, 545, 703 Phase velocity, 10, 88, 109, 119, 146, 213, 238,
307, 329, 338, 375, 389, 392, 443, 529, 553, 564, 571, 617, 635
Photoacoustic cell, 399 photo acoustic measuring cell, 180
Photoacoustic characterization of semiconductor films, 199
Photoacoustic depth analysis, 439 Photoacoustic detection, 32 Photoacoustic dispersion, 403 Photoacoustic effect, 180, 433, 517, 545, 557 Photoacoustic interferometry, 560 Photoacoustic investigation, 545 Photoacoustic measurement, 399 Photoacoustic method, 557, 641 Photoacoustic microphone, 180 Photoacoustic phenomenon, 511, 605 Photoacoustics, 179, 403, 514, 551 Photoacoustic setup, 180 Photoacoustic signal, 180, 399, 433, 485, 517
photo acoustic signal generation, 511 Photoacoustic spectroscopy, 199
photoacoustic Raman spectroscopy, 599 Photo acoustic spectrum, 486, 546, 557, 641 Photoacoustic study, 703 Photoacoustic technique, 433, 479, 545, 545,
703 Photocurrent method
Constant Photo current Method, 201 Photodetector, 60, 210, 231, 408, 667, 717 Photodiode, 60, 134, 292, 417, 469, 552, 696 Photoelastic constant, 78 Photoelastic tensor, 78 Photoelectric cathode, 60 Photoelectric multiplier, 233 Photoelectric phenomenon, 605 Photoelectronic multiplier, 667 Photolithographic process, 240 Photomultiplier, 667 Photomultiplier signal, 234 Photopyroelectric effect, 491 Photospectrometer, 644 Photothermal aperture function, 489 Photothermal characterization of semiconductor
films, 199 Photothermal conversion efficiency, 180 Photothermal deflection, 514
photothermal deflection spectroscopy, 199 Photothermal effect, 557 Photothermally optical reflectance, 605 Photothermal phenomenon, 514 Photothermal radiometry, 32
scanned photothermal radiometry, 32 Photothermal signal, 495
Photothermal spectroscopy photothermal deflection spectroscopy, 199
Photothermal transmission signal, 33 Photothermal wave diffraction, 489 Photothermal wave interference, 489 Photovoltage, 608 Piezoceramic, 517, 573
piezoceramic detectors, 61 Piezoeffect, 521 Piezoelectric acoustic detectors, 61 Piezoelectric actuator
cylindrical piezoelectric actuator, 194 Piezoelectric coupling, 287 Piezoelectric crystal, 517 Piezoelectric effect, 709 Piezoelectric element, 189 Piezoelectric generation, 65 Piezoelectric halfspace, 287
anisotropic piezoelectric halfspace, 287 Piezoelectric interaction, 287 Piezoelectricity, 287 Piezoelectric material, 17, 241, 516 Piezoelectric medium, 569, 709 Piezoelectric plates, 385
piezoelectrical plate, 405 composite piezoelectric plates, 13
Piezoelec tries linear piezoelectrics, 517 nonlinear piezoelectrics, 519
Piezoelectric semiconductor structure, 527 Piezoelectric sensor, 563 Piezoelectric stress, 387
piezoelectric stress tensor, 71 0 Piezoelectric substrate, 225, 238, 307, 353 Piezoelectric technique, 65 Piezoelectric technology, 65 Piezoelectric transducer, 66, 209, 385, 551, 631,
666 piezoelectric ceramic transducer, 600 piezoelectric monocrystal transducer, 119
Piezoelectric vibrator, 552 Piezooptic coefficient
adiabatic piezooptic coefficient, 138 Piezotransducer, 208
cylindrical piezotransducer, 406 eradiating piezotransducer, 406
Plastic foam, 425 Plate mode, 392 Polarization, 60, 78, 90, 99, 155, 194, 205, 377,
454, 517, 52~ 680, 695 dielectric polarization, 403 polarization modulation, 61 polarization selector, 667 polarization splitting, 205
Polarized light, 243 Polarized mode, 417
horizontally polarized symmetric mode, 531 vertically polarized antisymmetric mode, 531 vertically polarized interface mode, 530 vertically polarized symmetric mode, 530
Polarized orientation, 36'7
Polarized transducer vertically polarized shear transducer, 531
Polarized wave, 284 polarized wave velocity, 388
Polarizer, 667, 717 Polarizing and nonpolarizing beamsplitter, 552 Polycrystalline material, 5, 95 Polycrystalline metal, 87 Polycrystalline type distribution, 184 Porous material, 219, 425, 485 Porous solid, 457 Porous structure, 623 Porous system
layered porous system, 425 Propagation mode, 14, 331, 453
guided mode propagation, 529 normal propagation mode, 417 surface wave propagation, 9, 92, 328
surface acoustic wave propagation, 451 underwater propagation, 13
Radiated power, 538 Radiated ultrasonic field, 385 Radiating acoustic wave, 319 Radiating beam, 303 Radiating body, 317 Radiating condition
Sommerfield radiating condition, 658 Radiating diode, 372 Radiating transducer
eradiating piezotransducer, 406 Radiation, 55, 150, 545, 625, 671, 696
absorbed radiation, 180 absorption of radiation, 180 acoustical radiation, 313 acoustic radiation, 57, 385
acoustic radiation mode, 452 bulk acoustic wave radiation, 355
acoustooptical filtration of radiation, 665 bulk acoustic wave radiation, 355 diffracted radiation, 205 dipole radiation, 57 electromagnetic radiation, 557, 631, 665 energy radiation, 421, 613 eradiation angle spectrum, 406 infrared radiation, 57
infrared thermal radiation, 32 irradiation surface, 43 laser radiation, 33, 56 modulated radiation, 32 nonradiative relaxation, 545 optical radiation, 56, 180, 209, 517
optical radiation mode, 60 radiation angle, 538 radiation condition, 321, 468
Sommerfeld radiation condition, 161 radiation damage, 34 radiation flux, 56 radiation frequency, 57 radiationless energy transfer, 511 radiation less relaxation, 599
733
Radiation (continued) radiation mode, 452
acoustic radiation mode, 452 optical radiation mode, 60
radiation modulation, 180 radiation of energy, 41 radiation pattern, 290, 385 reradiation, 147, 497 surface acoustic wave radiation, 353 thermal radiation
infrared thermal radiation, 32 transducer radiation, 105, 531
Radiator, 56 acoustic radiator, 56
Radiometry photothermal radiometry, 32 scanned photothermal radiometry, 32
Raman-Nath diffraction, 209, 232 anisotropic Raman-Nath diffraction, 81 isotropic Raman-Nath diffraction, 81 Raman-Nath type diffraction, 243, 650
Raman-Nath parameter, 132, 165, 245, 418 Raman-Nath regime, 129, 232, 505, 165, 209 Raman-Nath region, 165, 209 Raman-Nath system, 505 Raman-Nath theory, 166 Raman scattering, 599 Raman spectroscopy
optoacoustic Raman gain spectroscopy, 599 photoacoustic Raman spectroscopy, 599
Raman spectrum, 600 Raman technique, 599 Raman transition, 599 Rayleigh angle 213, 647
critical Rayleigh angle, 161 Rayleigh equation, 343, 414 Rayleigh hole, 161 Rayleigh mode, 9, 297, 533
acoustooptical Rayleigh mode detection, 654 leaky Rayleigh mode, 613
Rayleigh scattering, 41, 73 Rayleigh light scattering, 179 Rayleigh surface wave scattering, 5
Rayleigh velocity, 3, 556 Rayleigh (type) (surface) waves, 3, 66, 146, 213,
228, 307, 381, 385, 414, 454, 461, 500, 525, 555, 569, 635, 681
pseudo-Rayleigh wave, 156 Rayleigh surface acoustic wave velocity, 556 Rayleigh surface wave scattering, 5 Rayleigh wave absorption, 255 Rayleigh wave coupling, 530 Rayleigh wave reflection, 9
Rayleigh wave reflection coefficient, 239 Rayleigh wave tomography, 381 Rayleigh wave velocity, 95, 253, 613 reradiated Rayleigh wave, 613 scattered Rayleigh wave, 461 surface Rayleigh wave, 89
Reciprocity, 468 Reemission, 298, 583, 639
734
Reflection, 3, 31, 66, 155, 173, 213, 253, 357, 391, 413, 417, 455, 473, 487, 495, 545, 589, 610, 61~ 635, 645, 657, 677, 691,
acoustic reflection, 425 back-face reflection, 43 Bragg reflection, 46
pure Bragg reflection condition, 505 complex reflection/refraction angle, 157 complex reflection/transmission coefficient,
159 diffuse reflection, 41 double reflection, 89 interfacial reflection and transmission, 14 multiple reflection, 200, 555, 674 nonspecular reflection effects, 650 optical reflection, 35 Rayleigh wave reflection, 9 reflection coefficient, 3, 14, 31, 213, 391,
526, 647, 700 acoustooptic reflection coefficient, 647 Rayleigh wave reflection coefficient, 239 surface wave reflection coefficient, 9
reflection mode, 68 reflection of light, 200 reflection/transmission coefficient, 220, 237,
309, 392, 441, 558, 689 specular reflection, 7, 301, 301, 393 thermal wave reflection, 34
Refraction, 155, 206, 233, 276, 381, 445 complex reflection/refraction angle, 157 double refraction, 157 index of refraction, 83 refraction index, 210 subsequent refraction, 88
Reradiated Rayleigh wave, 613 Reradiation, 273, 497 Resonance, 122, 219, 261, 272, 300, 316, 371,
391, 413, 417, 441, 497, 519, 573, 581, 663
acoustic resonance, 407 acoustic resonance scattering, 388
antiresonance, 122 electron spin resonance, 202 ferromagnetic resonace, 249 guided wave resonances, 149 multiple resonance, 13 resonance condition, 124, 371, 671 resonance frequency, 88, 144, 539
nominal resonance frequency, 469 resonance identification
method of isolation and identification of resonances, 143
pulsed resonance identification method, 124
resonance interaction, 249 resonance isolatio~, 124
method of isolation and identification of resonances, 143
resonance magnetic field, 250 resonance mode, 222, 359, 376, 377, 388
Resonance (continued) resonance scattering, 119
resonance scattering spectroscopy, 143 resonance spectrum, 124, 143, 387, 497, 583,
631 Scholte wave resonances, 149 spin-wave resonance mode, 249 thickness resonance, 386 Whispering Gallery wave resonances, 149
Resonant plate nonresonant vibrating plate, 319
Resonant acoustic mode, 17 Resonant case, 284 Resonant condition, 647 Resonant excitation, 319 Resonant inclusion, 219 Resonant light scattering, 716 Resonator, 119, 249, 523, 709
acoustic resonator, 631 surface acoustic wave resonator, 237
Rough counterparts, 532 Rough interface, 400 Roughness, 120
surface roughness, 90, 461 Rough surface, 43, 400, 461, 635
SAW mode, 452 Scanned radiometry
scanned photothermal radiometry, 32 Scanner, 232
optical scanner, 72 Scanning 157, 381, 409, 473, 495, 631
adiabatic scanning, 179 differential scanning, 179 laser-scanning microscope, 62 non scanning configuration, 613 optical beam scanning, 68 scanning acoustic microscope, 3, 253, 614 scanning rate, 181 scanning receiver, 473 scanning regime, 234 scanning system, 69
Scatter, 62 Scattered energy, 625 Scattered field, 371 Scattered light, 67, 231, 433 Scattered pulse, 46 Scattered signal, 120 Scattered wave, 369, 626
scattered gravity wave, 313 scattered Rayleigh wave, 461
Scatterer, 120, 220, 228 Scattering, 41, 80, 173, 225, 359, 369, 384
acoustic scattering, 371, 441 acoustic resonance scattering, 388 acoustic wave scattering, 625 inverse acoustic scattering, 687 surface acoustic wave scattering, 287
Bragg scattering, 16 elastic scattering amplitude, 143
Scattering (continued) light scattering, 377, 480, 487, 641
Rayleigh light scattering, 179 resonant light scattering, 716
multiple scattering, 359, 657 multiple scattering effects, 219
Raman scattering, 599 Rayleigh scattering, 41, 73
Rayleigh light scattering, 179 Rayleigh surface wave scattering, 5
resonance scattering, 119 resonance scattering spectroscopy, 143 resonant light scattering, 716
scattering background, 43 scattering center, 18, 41 scattering coefficient, 43, 271 scattering from bulk and surface defects, 121 scattering from cylinders and spheres, 121 scattering from target, 121 scattering of acoustic wave, 497 scattering of light, 545 scattering studies, 119 surface acoustic wave scattering, 287 surface scattered wave, 66 wave scattering, 284, 461
Rayleigh surface wave scattering, 5 surface acoustic wave scattering, 287
Schlieren image, 372 Schlieren method, 129 Schlieren stop, 417 Schlieren system, 371, 385 Schlieren technique, 371 Scholte-Stoneley wave, 327, 583, 635 Scholte wave, 146, 156, 327, 635
Scholte wave diffraction, 635 Scholte wave resonances, 149
Semiconductor, 55, 545, 557, 605 amorphous semiconductor, 199, 479, 700 crystal semiconductors, 642 semiconducting sample, 641 semiconductor film
photoacoustic characterization of semiconductor films, 199
photothermal characterization of semiconductor films, 199
semiconductor laser, 399 semiconductor structure
piezoelectric semiconductor structure, 527 Spectrointerferometer
digital ultrasonic spectrointerferometer, 88 Spectrometer, 546
P A spectrometer, 479, 485 Spectrophotometer, 558, 643 Spectroscopic analysis, 120 Spectroscopic technique 480, 641 Spectroscopy
infrared spectroscopy, 202 photoacoustic spectroscopy, 199
photoacoustic Raman spectroscopy, 599 photothermal deflection spectroscopy, 199
735
Spectroscopy (continued) Raman spectroscopy
optoacoustic Raman gain spectroscopy, 599
photoacoustic Raman spectroscopy, 599 resonance scattering spectroscopy, 143 transmission spectroscopy, 199 ultrasonic spectroscopy, 119, 173, 391
Spectrum analyzer AO spectrum analyzer, 408
Specular reflection, 7, 301, 393 Spherical inclusion, 359 Stoneley equation, 414, 636 Stoneley mode, 534 Stoneley wave, 156, 297, 327, 371 Subsurface layer, 570 Subsurface waves, 87 Superconductor, 253 Surface
inhomogeneous surface, 451 irradiation surface, 43 rough surface, 43, 400, 461, 635 scattering from bulk and surface defects, 121 surface absorption, 34 surface acoustic solitons, 523 surface acoustic wave, 3, 215, 225, 237, 281,
287, 307, 353, 413, 537, 569, 709 (see also surface waves)
attenuated surface acoustic wave, 538 Rayleigh surface acoustic wave velocity,
556 surface acoustic wave attenuation, 554 surface acoustic wave device, 225, 709 surface acoustic wave generation, 551 surface acoustic wave propagation, 451 surface acoustic wave radiation, 353 surface acoustic wave resonator, 237 surface acoustic wave scattering, 287
surface heating, 66 surface infrared, emission 34 surface layer, 56, 180 surface mode, 533, 555 surface roughness, 90, 461 surface wave, 87, 297, 385, 614, 654
(see also surface acoustic waves) Rayleigh surface wave scattering, 5 Rayleigh (type) surface wave, 3, 66, 89,
555 subsonic surface wave, 677 supersonic surface wave, 677 surface scattered wave, 66 surface wave beam self-defocusing, 523 surface wave beam self-focusing, 523 surface wave convolver, 523 surface wave generation, 92, 121, 523, 655 surface wave propagation, 9, 92, 328 surface wave reflection coefficient, 9 surface wave velocity, 9, 92
waveguide surface, 537
Thermal annealing, 554
736
Thermal attenuation, 341 Thermal characterization, 179 Thermal conduction, 273 Thermal conductivity, 179, 348, 433, 512, 559,
609, 703 inhomogeneous thermal conductivity
equation, 518 Thermal contributions, 27 Thermal cycling, 257 Thermal deposition, 257 Thermal diffusion, 400, 700
thermal diffusion coefficient, 180 thermal diffusion length, 31, 180, 200, 433,
491, 557, 60~ 642, 704 Thermal diffusivity, 31, 179, 200, 348, 479, 512,
609, 707 Thermal dispersion, 399 Thermal dissipation, 342, 348 Thermal effusivity, 200, 704 Thermal energy, 557 Thermal evaporation, 256 Thermal expansion, 56, 700, 712
thermal expansion coefficient, 27, 609, 700 Thermal fluctuation, 134 Thermal gas
isothermal gas, 426 Thermalization, 608 Thermally induced static strains, 569 Thermally thick sample, 641 Thermal mechanism, 56 Thermal motion, 30 Thermal noise, 552 Thermal parameter, 642, 703 Thermal process, 341, 511
endothermal process, 400 exothermal process, 400
Thermal property, 31, 399, 433, 486, 545, 557 thermal transport property, 179
Thermal quantities, 179 Thermal radiation
infrared thermal radiation, 32 Thermal sensitivity, 569 Thermal transmission, 485 Thermal transportation, 486 Thermal transport property, 179 Thermal wave, 31, 180, 402, 433, 489, 513, 545
thermal wave echo, 31 thermal wave interference, 557 thermal wave phenomenon, 605 thermal wave reflection, 34
Thermal wavelength, 438, 512 Thermal wedge, 33 Thermodiffusion effect, 557 Thermodynamical formalism, 559 Thermodynamical quantity, 512 Thermodynamical variable
inhomogeneous thermodynamical variable, 513
Thermodynamic constant, 103 Thermodynamic equation, 512 Thermodynamic equilibrium, 632
Thermodynamic parameters, 56 Thermodynamics, 511 Thermodynamic stress, 710 Thermoelastic effect, 557 Thermoelastic excitation, 553 Thermoelastic property, 599 Thermoelastic regime, 66, 551 Thermooptical signal generation, 517 Thermooptic excitation process, 56 Thermophysical property, 349 Thermoreflectance effect, 696 Thermosensitivity, 569 Thermotropic crystal, 703 Thermoviscoelastic isotropic solid, 156 Thermoviscoelasticity, 99 Thermoviscous dissipation, 106 Thermoviscous fluid, 99, 156 Thermoviscous gas, 156 Transducer
ceramic transducer, 631 focused transducer, 3 interdigital transducer, 215, 225, 307, 353,
540, 637 layerer transducer, 637 piezoelectric transducer, 66, 209, 385, 551,
631, 666 piezoelectric ceramic transducer, 600 piezoelectric monocrystal transducer, 119
Transducer (continued) piezotransducer, 405
cylindrical piezotransducer, 406 eradiating piezotransducer, 406
shear transducer vertically polarized shear transducer, 531
transducer radiation, 105, 531 wedge transducer, 94, 530
Underwater acoustic material, 219 Underwater acoustics 129, 359 Underwater body, 313 Underwater detection, 152 Underwater propagation, 13
Waveguide, 190, 212, 213, 359, 377, 537 waveguide mode, 60, 452
optical waveguide mode, 377 waveguide surface, 537
Wedge, 32, 121, 157 thermal wedge, 33 wedge angle, 92 wedge material, 92 wedge transducer, 94, 530
Wedged sample, 32 Whispering Gallery mode, 297 Whispering Gallery wave, 146
Whispering Gallery wave resonances, 149
737