ACKNOWLEDGMENTS

This material is based upon work supported under a NationalScience Foundation Graduate Research Fellowship. This work wassponsored by the Department of the Air Force under Air ForceContract F19628-00-C-0002, and the ONR under ContractN00014-01-1-0713. Opinions, interpretations, conclusions, andrecommendations are those of the authors and are not necessarilyendorsed by the United States Government.

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4. X. Chen, T.M. Grzegorczyk, B.-I. Wu, J.J. Pacheco, and J.A. Kong,Robust method to retrieve the constitutive effective parameters ofmetamaterials, Phys Rev E 70 (2004), 016608.

5. R.A. Shelby, D.R. Smith, and S. Schultz, Experimental verification ofa negative index of refraction, Sci 292 (2001), 77–79.

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7. Z.M. Thomas, T.M. Grzegorczyk, B.-I. Wu, X. Chen, and J.A. Kong,Design and measurement of a four-port device using metamaterials,Optics Express 13 (2005), 4737–4744.

8. T.M. Grzegorczyk, Z.M. Thomas, and J.A. Kong, Inversion of criticalangle and Brewster angle in anisotropic left-handed metamaterials,Appl Phy Lett 86 (2005), 251909.

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© 2006 Wiley Periodicals, Inc.

IMPACT OF A LIQUID-JET PRODUCEDBY THE COLLAPSE OF LASER-INDUCED BUBBLES AGAINST A SOLIDBOUNDARY

Xiao Chen,1 Rongqing Xu,2 Zhonghua Shen,2 Jian Lu,2 andXiaowu Ni21 State Key Laboratory for Macroscopic PhysicsDepartment of Physics, Peking University,Beijing, 100871, P. R. China2 Department of Applied PhysicsNanjing University of Science & TechnologyNanjing, 210096, P. R. China

Received 2 February 2006

ABSTRACT: The mechanical effect induced by the collapse of cavita-tion bubbles in the neighborhood of a solid boundary is investigated byfocusing a Q-switched laser pulse on a metal target in water. By meansof a fiber-coupling optical beam deflection technique, the displacementgenerated by liquid-jet impact at the final stage of the bubble collapse isdetected at the epicenter of the rear metal surface. Furthermore, com-bining a widely used laser-ablation model with the detection principlesof this detector, the transient impact-force value loading on the targetmaterial can be easily estimated. Besides, according to experimentalresults and the modified Rayleigh theory, the maximum bubble radiusand the liquid-jet pressure are also obtained, which are in good agree-ment with the results of other studies. © 2006 Wiley Periodicals, Inc.Microwave Opt Technol Lett 48: 1525–1528, 2006; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21702

Key words: liquid-jet impact; bubble collapse; laser ablation model;optical deflection technique; laser

1. INTRODUCTION

The interest in cavitation bubbles in liquids mainly arose fromtheir destructive action on solid surfaces, which was first reportedon ship propellers in the last century [1]. In general, cavitation canbe initiated by either setting up a tension in a liquid or by depos-iting energy into it [2]. Tension appears in fluid flows, such as withship propellers, hydrofoils, pipes, and pumps. It also occurs insound fields in the under-pressure cycle of sound waves, forexample, in shock-wave lithotripsy and sonochemistry. Local dep-osition of energy is brought by heat transfer in pipes or bydumping hot bodies into liquids. Nowadays, for the experimentalinvestigation of bubble dynamics, two methods of single-bubblegeneration are often used: spark generation [3, 4], where an elec-trical discharge produces a bubble, and optic cavitation [5–7],where a bubble is induced by a laser. The latter method hasadvantages over the spark generation, as a single bubble can bemade highly spherical, is free from mechanical distortions, and thecavity inception is easy to control. When a bubble collapses in thevicinity of a rigid boundary, a high-speed wall-directed reentrantjet begins to form, which generally causes serious destruction tothe nearby solid [8, 9].

Although cavitation bubble phenomena have been the subjectof extensive research for more than a century, fundamental mech-anisms underlying the connected phenomena, particularly erosivedamage, are not well established. In the development of under-standing of bubble-boundary interaction, Lauterborn first utilizedhigh-speed photography to investigate this transient process and,nowadays, high-speed photography has been playing an importantrole due to its high spatial resolution [2, 5, 6, 9]. However, thismethod cannot provide a good temporal development of a transient

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 8, August 2006 1525

impact. Alcock even presented high-speed optical transducers formeasuring acoustic transients propagating in water associated withlaser cavitation [10]. These devices are based upon the measure-ment of reflectance at a solid-water interface using a probe beam,with perturbations in reflected intensity being brought by thechanges in pressure from an acoustic transient traversing theboundary. In this paper, we utilize a detection technique based onfiber-coupling optical beam deflection (OBD) [10]. Compared withother detection techniques, this detection method has many advan-tages, such as low cost, simple structure, and high-frequencyresponse. It can present the temporal development of transientforce acting on the target, especially a point-force loading. By thisoptical detector, we investigate mechanical effects of the bubble-boundary interaction. Then, combining the detection principles ofthis detector with a widely-used laser ablation model, the value ofloading forces, including the laser ablation force and liquid-jetimpact force, can be easily estimated. Besides, based on the bubblecollapse duration and the modified Rayleigh theory, the maximumbubble radius and the jet-induced impact can also be calculated.These results help to better understand the cavitation erosion andoptimize laser parameters in the fields of laser ophthalmology andunderwater laser processing.

2. EXPERIMENT

The experimental arrangement based on OBD is outlined in Figure1, which was first reported in the literature in detail [11]. A TEM00

Q-switched Nd: YAG laser 1 (wavelength 1.064 �m, pulse dura-tion 30 ns) is focused on the iron-plate surface 6 (6-mm diameter,0.25 mm-thick) with the iron rim stuck tightly on the inner wall ofa cuvette 5. The cuvette (100 � 100 � 150 mm3) is filled withdistilled water. In the detection region, a part of the glass cuvetteis removed, so that the probe beam can be directly focused on thepolished rear surface. The probe beam emitted by a He–Ne laser 8(power 5 mW, wavelength 0.6328 �m) is used to detect theepicenter displacement at the rear surface that is caused by arrivalwaves excited by a normal applied force. The reflected beam isfocused into a single-mode optical fiber 12 mounted on a five-dimensional fiber-regulating stand with 0.1-�m spatial resolution.The light from the optical fiber is then fed into a photomultiplier 13(Hamamatsu H5773 with 2-ns rise time) and recorded with a

digital oscilloscope 14 (Tektronix TDS340). A part of the scatteredlaser is fed into a photodiode 15 to generate the trigger signal. Thedetector parts numbered 7 to 13 in Figure 1 build up a subsystem,called OBD test system.

The fiber-coupling OBD setup as a position sensitive detectorobtains the information by detecting the reflected beam deflectioncaused by acoustic or thermal waves. The whole setup is verysimple, effective, and easy to adjust.

3. MODEL

During the process of a laser-induced cavitation bubble collapse,due to the radial water flow retarded by the solid boundary, thepressure at the lower bubble wall is smaller than that at the upperwall, and the bubble becomes elongated perpendicular to theboundary. Therefore, the velocity of the bubble center has toincrease, that is, the velocity of the upper bubble wall exceeds thatof the lower. Consequently, the fluid volume above the bubble isaccelerated and focused during the collapse, leading to the forma-tion of a liquid jet directed towards the boundary. This jet hits thelower bubble wall, causing a funnel-shaped protrusion. The impactforce against the boundary caused by a liquid-jet and its momen-tum transfer can be represented by a time varying force actingnormal to the surface. As the duration of this liquid-jet impact isvery short [12, 13], it can be considered as a point force withamplitude fa and the time dependence of a Dirac function �(t). Theaction of this transient force will excite multiple waves in thetarget material, for example, a longitudinal wave, a shear wave,and a head wave. Upon the arrival of these ultrasonic waves, themetallic epicenter will displace.

Here we adopt the laser ablation model and combine it with theOBD detection principles to estimate the force causing the dis-placement. This ablation model was first proposed by Miklowitz[14] and then has been successfully applied in laser ultrasonics[15]. The model takes both the loading force and the motionequation of the acted material into consideration. The displacementcaused by a normal force F is obtained by using Laplace trans-forms (transient conditions), Hankel transforms (axial symmetry),and by inverting the resulting transform solution by the Cagniard–DeHoop method [14]. Then, according to Miklowitz’s work, Sin-clair [16] further wrote out the analytic forms of a special case fora point force source with the receiver at the epicenter correspond-ing to pulse conditions �(t):

d�h, t� �fa

��hcS2 ��gL

E

�t�

�gSE

�t � ,

gLE �

�sL2 � cL

�2��2sL2 � cS

�2�2

��2sL2 � cS

�2�2 � 4sL2�sL

2 � cL�2�1/ 2�sL

2 � cS�2�1/ 2�2 H� t �

h

cL� ,

gSE �

�4sS2�sS

2 � cL�2��sS

2 � cS�2�

��2sS2 � cS

�2�2 � 4sS2�sS

2 � cL�2�1/ 2�sS

2 � cS�2�1/ 2�2 H�t �

h

cS�,

(1)

with sL2 � (t2/h2 � 1/cL

2) and sS2 � (t2/h2 � 1/cS

2), where d isthe displacement, corresponding to the impact force, F � fa�(t),at the epicenter. h is the target’s thickness, cL, cS are the acousticvelocities of the longitudinal wave L and shear wave S, respec-tively. � is the Lame constant and H(t � (h/cL)) and H(t �(h/cS)) are step functions.

For a 0.25-mm-thick iron plate as case, cL � 5850 m/s, cS �3230 m/s, � � 8.03 � 1010 Pa, and the arrival transfer function atthe epicenter is calculated as d/fa � 2.77 � 10�8 m/N. If a forcehas a different time-dependence, such as Gaussian or step time

Figure 1 Experimental scheme: 1. Q-switched Nd: YAG laser (wave-length 1.06 �m, duration 30 ns); 2. beam splitter; 3. attenuator group; 4.convex lens ( f � 147 mm); 5. glass cuvette; 6. 0.25 mm-thick iron plate;7. convex lens L1 ( f1 � 30 mm); 8. He–Ne laser (wavelength 0.6328 �m);9. microscope objective L2 ( f2 � 4 mm); 10. interference filter (centralwavelength 0.63 �m); 11. five-dimensional fiber-regulating stand; 12.single-mode optical-fiber; 13. photomultiplier (Hamamatsu H5773); 14.digital oscilloscope (Tektronix TDS340); 15. photodiode

1526 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 8, August 2006 DOI 10.1002/mop

function, not Dirac function, the response of the target plate to thisforce can be computed by convolution of Eq. (1) with the appro-priate function of time. Therefore, based on this ablation model,the relationship of the acting transient force and the displacementat the epicenter is obtained.

Furthermore, combining the detection principles of this tech-nique [11], the output voltage from the photomultiplier, U, and thenormal loading force, fa, are related by

U � �4I0Rf2

�hrscS2 � ��gL

E

�t�

�gSE

�t �� � fa, (2)

where I0 is the maximum intensity of the reflected probe beam, f2

is the focal length of the microscope objective, rs is the corre-sponding to the magnitude of longitudinal wavelength, R is theradius of fiber core, and is the conversion coefficient of thephotomultiplier.

From Eq. (2), it is obvious that the relation between outputelectrical signal U and the normal force fa with Dirac time functionis linear. By the detected signal U, the normal force loading on thematerial surface can be estimated.

4. RESULTS AND DISCUSSIONS

Figure 2 depicts a typical signal of the laser-iron interactionunderwater. The incident laser energy is 22 mJ with the focal spot50 �m in radius. Two peaks a and b are separated by a timeinterval of 136 �s. Peak a is actually induced by the expandinglaser plasma, called an ablation force. Peak b proves to be theliquid-jet impact, not shock waves or a combination of both at thefinal stage of a cavitation bubble collapse. This is mainly due totwo reasons: (i) This fiber-optic diagnostic technique is sensitive toa transient point-force loading, such as laser plasma ablation forceand liquid-jet impact, while is insensitive to the shock wave effectwith a relative “larger” applied surface; (ii) there is a 30-timesmismatch when acoustic wave transmits from the surroundingwater to iron plate. So the effect of shock waves is too weak todetect at the epicenter. Besides, the amplitude of peak b is largerthan that of peak a, which indicates that the mechanical effectinduced by the liquid-jet impact outweighs the well-known laser-plasma ablation force. Under the different laser energy radiation,the jet impact is monotonously increased with the laser energy asshown in Figure 3. Each data point is an average of five measuredvalues.

In order to estimate the value of liquid-jet impact against theboundary, in the following we take Figure 2 as an example tocalculate. The voltage amplitude of this liquid-jet signal is 0.069

V. We substitute the following physical parameters into Eq. (2).The conversion coefficient of the Hamamatsu (H5773) photo-multiplier used in this experiment is 4000 V/W and the He–Nelight intensity is 9422 W/m2. The radius of the optical fiber core,R, is 4 �m and the focal length, f2, is 4 mm. rs corresponding tothe magnitude of longitudinal wavelength is about 500 �m. So byEq. (2), the amplitude fa is calculated as 6.5 N.

As we know, during the bubble collapse near a rigid boundary,it is really not a spherical shape. As a matter of fact, the bubble iselongated normal to the boundary surface during the collapse witha liquid-jet through in center. Moreover, the nearby boundary alsocauses a prolongation of the collapse period over that of Ray-leigh’s model for a spherical cavitation bubble [17]. Nevertheless,according to Rattray theory [18], correction to the Rayleigh col-lapse duration is possible if the location of the bubble with respectto a nearby wall is available. Rattray derived an approximaterelationship between the prolonged collapsed TC and , given by

TC

T�C� k � 1 � 0.41

1

2, (3)

where T�C is the collapse time of a equivalent spherical bubble, TC

is the prolonged collapse time, k is the prolongation factor, and is the dimensionless parameter with its definition � L/Rmax

(with Rmax the maximum bubble radius and L the distance of acavity inception from a boundary). Equation (3), however, is notsuitable for very small , because it predicts k3 for 3 0. So,Godwin [19] evaluates a large number of high-speed photographi-cal series of the bubble dynamics, and then further pointed out byexperiment that at � 1 the experimental measurements are in fairagreement with the theoretical predictions, but at � 1 the actualcollapse times (or k values) are much smaller than predictions. Inour experiment, consider that laser pulses are directly focused onthe iron plate. So the distance between the solid boundary andbubble center is very small. When is in the range of 0.1, k isabout 1.14. Therefore, according to Godwin’s work associatedwith our detected results TC � 68 �s, the collapse duration of theequivalent spherical bubble is determined as

T�C � TC/k � 68 �s/1.14 � 60 �s. (4)

Then, according to the Rayleigh formula,

Figure 3 Signal amplitude of liquid-jet impact as a function of laserenergy

Figure 2 Typical signal detected by the optical beam deflection method

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 8, August 2006 1527

Rmax �T�C

0.915� /�Pstat � Pvap�, (5)

where is the density of liquids, Pstat is the static pressure, and Pv

is the vapor pressure of liquids (2330 Pa at 20°C). In this way, wecan calculate Rmax � 0.65 mm. Plesset [18] even indicated that thejet diameter is about one-tenth of the initial bubble diameter bynumerical simulation and this conclusion has been widely cited byothers [6, 8, 20]. So we also adopt it to estimate the acted spot ofa liquid-jet impact. Therefore, the corresponding liquid-jet pres-sure is expressed as follows:

P � � fa/�r2� � �6.5 N/3.14*�65 �m�2� � 488 MPa. (6)

Combined with the data of Figure 3, the pressure induced byliquid-jet impact as a function of laser energies shows in Figure 4.The calculated results show that when incident laser energies varyfrom 5.1 to 22 mJ, the pressures range between 322 to 488 MPa.The calculated results are in good agreement with the valuesobtained in others studies [21, 22]. Besides, due to the pressurevalues generally larger than the yield strength of several commonmetals, it is easy to judge whether this sample is damaged or not.

5. CONCLUSION

By means of the fiber-coupling OBD technique, we have detectedthe displacement at the metal’s epicenter induced by the laser-plasma ablation and the liquid-jet impact at the final stage of abubble collapse in a vicinity of a solid boundary. Combining thelaser ablation model with the detection principles of this technique,we calculated the value of this normal transient force acting on themetal surface. Furthermore, based on the bubble collapse durationand the modified Rayleigh theory, the maximum bubble radius andthe liquid-jet pressure were also estimated, and found to be in goodagreement with the results of other studies. The experimentalresults have shown that cavitation erosion caused by the liquid-jetimpulsive pressure is one of the main damage mechanisms inmaterial destruction and outweighs the well-known laser ablationeffect.

ACKNOWLEDGEMENTS

The project was funded by the National Natural Science Founda-tion of China under grant no. 60208004, the Natural ScienceFoundation of Jiangsu Province under grant no. BK2001056,

Teaching and Research Award Program for Outstanding YoungProfessor in Higher Education Institute, MOE, P. R. China.

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© 2006 Wiley Periodicals, Inc.

Figure 4 Pressure induced by liquid-jet impact on the solid boundary asa function of laser energy

1528 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 48, No. 8, August 2006 DOI 10.1002/mop