pdf1_tcm1021-48829.pdf

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 72, NUMBER 4 APRIL 2001

REVIEW ARTICLE

Application of acoustic resonators in photoacoustic trace gas analysisand metrology

Andras Miklos and Peter Hessa)

Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253,69120 Heidelberg, Germany

Zoltan BozokiResearch Group on Laser Physics, Hungarian Academy of Sciences, Do´m ter 9, H-6701 Szeged, Hungary

~Received 25 August 2000; accepted for publication 15 January 2001!

The application of different types of acoustic resonators such as pipes, cylinders, and spheres inphotoacoustics is considered. This includes a discussion of the fundamental properties of theseresonant cavities. Modulated and pulsed laser excitation of acoustic modes is discussed. Thetheoretical and practical aspects of high-Q and low-Q resonators and their integration into completephotoacoustic detection systems for trace gas monitoring and metrology are covered in detail. Thecharacteristics of the available laser sources and the performance of the photoacoustic resonators,such as signal amplification, are discussed. Setup properties and noise features are considered indetail. This review is intended to give newcomers the information needed to design and constructstate-of-the-art photoacoustic detectors for specific purposes such as trace gas analysis,spectroscopy, and metrology. ©2001 American Institute of Physics.@DOI: 10.1063/1.1353198#

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I. INTRODUCTION

The photoacoustic~PA! effect in solids1 was discoveredby Bell in 1880. It was soon realized that the same effexists in liquids and gases as well. Also in these first expments, resonant amplification of the photoacoustic sigwas discovered.2 The possibility of using a cavity as aacoustic amplifier of the signal was very important for deonstrating the effect. However, due to the lack of propinstrumentation~such as light sources, microphones, aelectronics!, the PA effect was almost completely forgottefor more than half a century.

Finally, in 1938, Viengerov introduced a PA systebased on a blackbody infrared source and a microphoneanalysis of gas mixtures.3 In the 1960s, an important breakthrough was achieved by the first use of a laser light souin PA gas detection.4 Compared to conventional lighsources, lasers have superior beam quality and spectrarity, and can provide high power radiation. Trace gas anasis was targeted first by the pioneering work of Kreuzer.5 APA cell, operated in a resonant mode by tuning the lamodulation frequency to one of the acoustic resonancethe cylindrical gas cell, was introduced by Deweyet al.6 andby Kamm.7

In the 1970s and 1980s photoacoustic gas detecboomed. High sensitivities were achieved by PA systeusing midinfrared gas lasers such as CO and CO2 lasers.8–12

Because of their high output power in the watt range atheir line tunability to strong fundamental vibrational tran

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tions, these lasers turned out to be ideal sources to pushsensitivity of PA gas detection into the ppbV ~parts perbillion volume! concentration range or even below. Initiresults using line-tunable CO2 and CO gas lasers, with modest power of about 1 W, were reported as early as 1972.13 Incurrent state-of-the-art experiments CO2 and CO lasers withmuch higher power (;10 W) are employed. The signal cabe further enhanced by placing the PA cell in the opticavity of the laser resonator, where a laser power of ab100 W can be reached~intracavity photoacousticspectroscopy!.11 Although line-tunable CO2 and CO gas la-sers are relatively large, complex, expensive systems,have already been used in mobile stations forin situ envi-ronmental gas detection.14

The outstanding features of the PA cell, most impotantly its small size, its simplicity, and robustness, can obe fully exploited when it is combined with a suitable lassource. Therefore, recent progress made in the developmof diode lasers has had an increasing influence on the acation of compact PA gas analyzers.15,16The currently avail-able power of near infrared~NIR! diode lasers, operatingnear room temperature, is small compared with that of C2

and CO lasers, and therefore the sensitivity is limited mosto ppm, or for certain gases to the sub-ppm range. Howethey may offer an alternative solution where small size, reability, low costs, and long lifetime are required. A specadvantage of diode lasers is their electronic modulation fture, where both intensity and wavelength modulationpossible. Diode lasers are continuously tunable, but theirability range is very limited~usually less than one wav

7 © 2001 American Institute of Physics

IP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp

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1938 Rev. Sci. Instrum., Vol. 72, No. 4, April 2001 Miklos, Hess, and Bozoki

number!.17 A larger tunability range~20–50 nm! can be ob-tained with external cavity diode laser~ECDL! systems.18,19

Beginning with the renaissance of photoacoustics aro1970, progress in this field always has been closely cnected with the development of laser technology. A maimpact on the field of trace gas detection can be expefrom new extensively tunable solid-state laser systems wing in the fundamental IR spectral region. In this respectrecent realization and further improvement of periodicapoled lithium niobate~PPLN! optical parametric oscillators20

~OPOs! and quantum cascade diode lasers21 could be an im-portant breakthrough in the practical application of photcoustic spectroscopy~PAS! in trace gas monitoring. Theslaser sources deliver continuous wave~cw! as well as pulsedbroadly tunable radiation with reasonable power and bawidth.

A crucial part of a PA gas detection setup is the cellwhich the PA signal is generated and detected. The‘‘gas-microphone cells’’ were small cylindrical cavities wita transparent window.22 The microphone was connectedthe cavity by a thin hole in one of the side walls of the ceThese PA sensors could be manufactured very easily; mover, they were very cheap when miniature electret micphones were used. Since the PA signal is inversely protional to the cell volume and the modulation frequenchigh PA signal levels can be obtained by taking a small cvolume (,10 cm3) and low modulation frequencies(,100 Hz). However, noise sources~intrinsic noise of themicrophone, amplifier noise, external acoustic noise! show acharacteristic 1/f frequency dependence, and thereforesignal-to-noise ratio~SNR! of such a gas-microphone cellusually quite small. Moreover, light absorbed in the windoand in the wall material generates a coherent backgrosignal, which is practically impossible to separate fromPA signal generated by the gas absorption itself. Therefsmall gas-microphone cells, which are still the most suitaPA detectors for solid and liquid samples, are hardly usegas phase photoacoustics anymore.

The SNR of a PA cell can be increased by applyihigher modulation frequencies~in the kHz region! andacoustic amplification of the PA signal. For this resonantcells operating on longitudinal, azimuthal, radial, or Helmholtz resonances have been developed.23–26 Furthermore,resonant cells can be designed for multipass or intracaoperation.11,12,27,28The influence of the window signal habeen minimized by introducing acoustic baffles29 or by thedevelopment of a ‘‘windowless’’ cell.30 Recently, a differen-tial cell, specially designed to suppress flow and windnoise, was introduced.17

In the following the requirements that need to be ffilled by resonant PA cells in gas measurements will be dcussed in detail, considering both theoretical and practissues. First, the characteristics of the PA effect in gasesbe briefly surveyed.

II. THE PHOTOACOUSTIC EFFECT IN GASES

The PA effect in gases can be divided into three msteps:15


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~1! localized heat release in the sample gas due to relaxaof absorbed light energy through molecular collisions

~2! acoustic and thermal wave generation due to localitransient heating and expansion;

~3! detection of the acoustic signal in the PA cell withmicrophone.

Molecular absorption of photons results in the excitatiof molecular energy levels~rotational, vibrational, elec-tronic!. The excited state loses its energy by radiation pcesses, such as spontaneous or stimulated emission, aby collisional relaxation, in which the energy is transforminto translational energy. In the case of vibrational excition, radiative emission and chemical reactions do not pan important role, because the radiative lifetimes of vibtional levels are long compared with the time neededcollisional deactivation at pressures used in photoacous(;1 bar) and the photon energy is too small to indu

FIG. 1. Schematic of the physical processes occurring after optical extion of molecules. Modulated or pulsed laser radiation leads to the poption of rotational, vibrational, and electronic states. Collisional deactivatby R–T, V–R, T, and E–V, R, T processes leads to localized transieheating. The resulting expansion launches standing or pulsed acowaves, which are detected with a microphone.


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1939Rev. Sci. Instrum., Vol. 72, No. 4, April 2001 Acoustic resonators

chemical reactions.31–33 Thus, in practice the absorbed eergy is completely released as heat, appearing as trational ~kinetic! energy of the gas molecules. The deposiheat power density is proportional to the absorption coecient and the incident light intensity. A detailed schemaillustrating the series processes occurring during PA siggeneration is presented in Fig. 1.

The laws of fluid mechanics and thermodynamics canused to model the acoustic and thermal wave generatiogases.34 The governing physical equations are the consertion laws for the energy, the momentum, the mass, andthermodynamic equation of state. The physical quanticharacterizing the acoustic and thermal processes aretemperatureT, pressurep, densityr, and the three components of the particle velocity vectorv. By eliminating thevariablesT, r, andv a linear wave equation can be derivefor the sound pressure:

] t2p~r ,t !2c2¹2p~r ,t !5~s21!] tH~r ,t !, ~1!

wherec, s, andH are the sound velocity, the adiabatic cefficient of the gas, and the heat density deposited in theby light absorption, respectively.

This wave equation always has two independsolutions:35 a weakly damped propagating acoustic wawith wavelengths in the centimeter range, and a headamped thermal wave. Since the latter wave has a very swavelength~in the submillimeter region! and does not propagate beyond the distance of a few wavelengths, it canobserved only in the vicinity of the exciting light beamTherefore, thermal and acoustic waves are separated in sand can be investigated independently.

Photoacoustic gas analysis is based on the detectiothe acoustic signal. The source term of the acoustic wequation is proportional to the deposited heat power denThe spatial size and shape of the source volume depenthe light-beam geometry and on the absorption length ingas, while the time dependence of heat deposition is ctrolled by the time dependence of laser excitation. Thuslaser pulse generates a heat pulse, which excites an aco

FIG. 2. Resonant acoustic cell designs.~a! Simple pipe~tube! for excitationof longitudinal modes,~b! pipe with two buffer volumes,~c! Helmholzresonator with separate sample and detection chambers for solid sam~d! coaxial excitation~e.g., of radial modes! in a cylinder,~e! asymmetricmultipass arrangement for excitation of azimuthal modes in a cylinder,~f! cylindrical cell suitable for excitation of the first radial mode with supression of the window noise.


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pulse,33 while a modulated laser beam generates periosound due to the periodic localized heating of the gas.36

The sound signal is generated in those locations, whlight is absorbed, for example, by the species to be motored. Interestingly, even the PA signal generated by a labeam propagating in free air has been observed.37 However,nearly all PA trace gas measurements are carried out inenclosure, called a ‘‘PA cell’’ or ‘‘PA detector.’’ Both termsare used in the literature to describe the device in whichPA signal is generated and monitored. In the following tterm PA resonator will be used for the cavity in which thresonant amplification of the PA signal takes place. Tname PA cell is reserved for the entire acoustic unit, incluing the resonator, buffer volumes, acoustic filters, windowgas inlets and outlets, and microphone~s!. Finally, PA sensoror PA detector stands for a complete instrument, includPA cell~s!, light source, gas handling system, and electronused for gas detection.

From an acoustic point of view the PA cell is a lineacoustic system, which responds as a whole to the disbance generated by light absorption. Since one of the mimportant parts of the PA cell is the acoustic resonator itsits properties will be discussed first. The influences ofother parts of the PA cell are taken into account onlyboundary conditions.

III. ACOUSTIC RESONATORS

The cylindrical acoustic waves generated by the absotion of a periodically modulated laser beam traveling throuopen air have a relatively small amplitude. However, whegas sample is confined within a closed cell the amplitumay increase considerably because of the constructive inference caused by the boundaries. A simple acoustic restor consists of a well defined regular cavity~ideally closed,but in reality equipped with one or more openings for tmicrophone and gas transport!. Three types of acoustic resonators have found widespread use in PA detection: Heholtz resonators, one-dimensional cylindrical resonators,cavity resonators. Figure 2 displays a series of fundameresonator types including a simple pipe or tube, a pipe wbuffers, a Helmholtz resonator, and three different confirations used for laser excitation of specific modes in cyldrical resonators.32

A. Helmholtz resonator

The Helmholtz resonator consists of a cavity and anjoining neck, which opens to open air. It is the acousequivalent of a simple mechanical oscillator composed omass and a spring. The air in the cavity plays the role ofspring, while the air mass in the neck corresponds tomoving mass of the mechanical oscillator. When the air pin the neck moves outward the pressure inside the cadecreases, resulting in an inward-directed force, which tto restore the original state. An oscillation will develop withe following resonance frequency:35

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where c, S, V, and l are the sound velocity, the crossectional area of the neck, the volume of the cavity, andlength of the neck, respectively. The amplification at tresonance frequency depends on the losses~viscous and ther-mal losses at the wall of the neck and radiation losthrough the opening! of the Helmholtz resonator.35 Theachievable amplification usually does not exceed a valu10.

A Helmholtz resonator can be represented by an equlent electric circuit using an analogy between acousticelectric networks.38 In this analogy the voltage and currecorrespond to sound pressure and volume flow~flow velocitytimes the cross-sectional area!, respectively. The equivalencircuit of a Helmholtz resonator is a serially connectacoustic capacitance and inductance withCA5V/rc2 andLA5r l /S. Since a PA cell has to be isolated against outsnoise, such a simple cavity with a neck cannot be usedPA cell. Therefore, the simplest, most practical applicaPA Helmholtz resonator consists of two cavities connecby a tube@see Fig. 2~c!#. In this case the volumeV in Eq. ~2!should be replaced by the effective volume defined asVeff

51/(1/V111/V2).Such coupled cavities have been used as PA cells,25 es-

pecially for the measurement of solid-phase samples. Hever, since the optical window is situated directly on twalls of the resonator, the window signal and the PA sigdue to gas absorption are generated in the same cavity,cannot in practice be separated. The PA sensitivity achable with Helmholtz resonators is relatively small,39 andtherefore they have not had widespread use in traceanalysis.

B. One-dimensional acoustic resonator

If the cross-sectional dimensions of a resonator are msmaller than the acoustic wavelength, the excited sounddevelops a spatial variation only along the length of the renator, i.e., a one-dimensional acoustic field is generatednarrow pipe~or tube! can be regarded as a one-dimensioacoustic resonator. A pressure wave propagating in thewill be reflected by an open/closed end with the oppossame phase.38 Through multiple reflections a standing wavwill be formed. Therefore, open–open@see Fig. 2~a!# andclosed–closed pipes should have resonances when thelength is equal to an integer multiple of the half wavelengwhile an open–closed pipe should resonate when its lengequal to an odd integer multiple of the quarter wavelengthreality the wavelengths at the resonances are somelarger.35,38 The corresponding resonance frequencies canobtained from the following expressions:

f n5nc

2~ l 1D l !n51, 2, 3,....,

~3!

f 2m215~2m21!c

4~ l 1D l !m51, 2, 3,....,

wherel is the length of the pipe. Heref n corresponds to theopen–open or closed–closed ends, whilef 2m21 correspondsto the open–closed end. The quantityD l is the so-called endcorrection, which should be added to the length of the p


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for each open end.35 The end correction for a closed endnaturally zero. Physically the end correction can be undstood as an effect of the mismatch between the odimensional acoustic field inside the pipe and the thrdimensional field outside that is radiated by the open en

The end correction can be approximated by the folloing expression:D l >0.6r , wherer is the radius of the pipeMore precisely, the end correction decreases slightly wfrequency,35 therefore the resonance frequencies of an opipe are not harmonically related but slightly stretched.the other hand, the resonance frequencies of a closed–clpipe are exact multiples of the fundamental frequency. Tstanding wave patterns are different in open–openclosed–closed pipes: the former has pressure nodes, thepressure antinodes at the ends.

Incompletely closed pipes or pipes with rounded holike openings naturally also have acoustic resonances, buresonance conditions cannot be expressed in such simforms as Eqs.~3! The end of a pipe that opens to a largdiameter volume@see Fig. 2~b!# behaves similarly to an ideaopen end, while the properties of a closed pipe end withopening having a much smaller diameter than the pipe itare close to that of an ideal closed end. Pipe resonatorsfound widespread application in gas phase photoacousTherefore, their properties will be discussed in more delater.

C. Cavity resonators

If the dimensions of a cavity are comparable with tacoustic wavelength then several distinct resonances cagenerated. The standing wave patterns and resonancequencies depend on the shape and size of the cavity. Alytical expressions can be given only for certain regushapes, such as spheres, cylinders, cubes, and rectanprisms. Although spherical resonators have been usephotoacoustics,40 the most frequently applied cavity resontor is the cylinder@see Figs. 2~d!–2~f!#, with symmetry thatcoincides well with that of a laser beam propagating alothe cylinder axis.33,36The resonance frequencies of a losslecylindrical resonator are as follows:

f jmq5c

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1S q

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, ~4!

whereR andL are the radius and length of the cylinder, thjmq indices~non-negative integers! refer to the eigenvaluesof the radial, azimuthal, and longitudinal modes, resptively, anda jm is the j th zero of the derivative of themthBessel function divided byp. The characteristic features othese different eigenmodes of a cylindrical resonator arelustrated in Fig. 3.

The properties of cavity resonators will be discussedSec. IV using a theoretical treatment discussed elsewhe41

This treatment can be applied also to one-dimensional presonators.


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IV. PHOTOACOUSTIC SIGNAL IN A CAVITYRESONATOR

A. General theory

All cavities used as photoacoustic cells have acouresonances. The acoustic eigenmodes of a closed cavitthe solutions of the homogeneous wave equation:35

] t2p~r ,t !2c2¹2p~r ,t !50. ~5!

It can be shown that in the lossless case the solutions oabove equation are orthogonal.35 In the case of an acoustiresonator with small openings the solutions can be writtena series expansion in the eigenfunctions of the closed lossresonator. Indeed, small holes disturb the acoustic fieldtribution only slightly and can be considered as sourcesloss of acoustic energy radiated out from the resonator. Tthe solution of the inhomogeneous wave equation,35 given inEq. ~1!, can be written as33,41

p~r ,t !5A0~ t !1(n

An~ t !pn~r !, ~6!

where An and pn are the amplitude of thenth eigenmodecomponent and the dimensionless eigenmode distributrespectively.33 Notice that the dimensions ofA0 andAn arethe same as those of the sound pressurep. As an example,the dimensionless eigenmode distribution function of alindrical resonator is given as

pn~r !5pjmq~r ,w,z!5Jm~pa jm!cos~mw!cosS qpz

L D . ~7!

The meanings of the indicesj , m, andq are explained above@see Eq.~4!#. The eigenmode indexn used in Eq.~6! and inthe following considerations represents a given (jmq) trialindex group. The series expansion of Eq.~6! can be used forboth pulsed and modulated excitation of acoustic resonanIn the modulated case the time dependence exp(ivt) can beassumed. Taking into account thatpn(r ) is a solution of thehomogeneous wave equation, the following expressionbe derived:

2v2A01(n

~vn22v2!Anpn~r !5 iv~s21!H~r !. ~8!

The sound pressure amplitudeA0 can be determined byintegrating the above equation over the volume of the re

FIG. 3. Schematic of the acoustic longitudinal, azimuthal, and radial moin a cylindrical resonator. Only oscillations at the resonant frequenciethese eigenmodes are amplified and they will significantly gain energy


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A05~s21!*H~r !dV

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TheAn amplitude can be determined by multiplying E~8! by pm(r ) and integrating over the volume. Since thepn

andpm eigenmodes are orthogonal,An can be expressed a

An5iv~s21!*H~r !pn~r !dV

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vvn

QnD E upn~r !u2dV

. ~10!

In order to take into account in a formal way the effeof losses~see Sec. VI C! an additional term containingQn ,the quality factor~Q factor! of the resonance, has been itroduced in the denominator of Eq.~10!. If the modulationfrequency is equal to one of the acoustical eigenfrequenof the cavity, the energy from many modulation cyclesaccumulated in a standing wave and the system works aacoustic amplifier. The final signal amplification is detemined by the total losses of the resonator. After an inittransient state, during which energy is accumulated instanding acoustic wave, a steady state is reached in whichenergy lost per cycle by various dissipation processesequal to the energy gained per cycle by the absorptionphotons. At resonance the amplitude isQn times larger thanthe amplitude far from the resonance frequency, i.e., theplification is equal to the value of theQ factor. The physicaldefinition of theQ factor is

Q52p accumulated energy

energy lost over one period. ~11!

For highQ values another definition is used:

Q5f 0

D f, ~12!

where f 0 andD f are the resonance frequency and the hawidth value of the resonance profile. The half width is mesured between the points where the amplitude is a 1/A2 valueof the peak amplitude~half-maximum values of the intensity!. Therefore,D f is also called the full width at half maximum ~FWHM!.

Equations~9! and~10! can be written in a more familiaform which takes into consideration that the deposited hdensity is equal to the product of the absorption coefficiand light intensity. The intensity can be written42 as follows:

I ~r !5WLg~r !,

whereWL denotes the light power andg(r ) is the normalizedintensity distribution. Its integral over the entire cross sectof the beam is normalized to unity. Now, the sound pressamplitudes can be written as follows:

A05~s21!aLWL

ivVres, ~13!

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The amplitude of the oscillation~the PA signal! will be pro-portional to the absorption coefficient, the absorption lengand the light power, but inversely proportional to the modlation frequency and the cell volume. More precisely, thesignal is inversely proportional to an effective cross sectdefined byVres/L.

The last factor of Eq.~14!, with the two integrals~in thefollowing denoted byFn!, describes the effect of the spatioverlap between the propagating laser beam and the predistribution of thenth acoustic eigenmode of the resonatThe numerator, the so-called overlap integral, describeseffectively thisnth eigenmode is excited by the spatial laslight distribution, while the denominator is a normalizatiofactor of thenth eigenmode. Therefore, the dimensionlenumberFn is called the ‘‘normalized overlap integral.’’ Byproper choice of the light path the predominant excitationa single resonance may be obtained, while the detectiounwanted resonances can be suppressed by makingoverlap integrals zero. Figure 4 shows, as an example,efficient excitation of a radial mode in a cylinder by thoverlap of the cylindrical spatial intensity distribution of thlaser beam propagating along the cylinder axis withmode pattern of the first radial mode.

The measured PA signal depends also on the exactsition of the microphone in the resonator. The signal detecby the microphone is proportional to the integral averagethe pressure over the microphone membrane. Since mominiature microphones are applied in photoacoustics, thetegral can be approximated by the value of the pressureplitude at the microphone location. If thenth eigenmode isexcited, and the microphone is located at the positionr M ,then the sound pressure atr M can be written as

FIG. 4. Schematic of the spatial overlap of the cylindrical intensity disbution of the laser radiation propagating along the cylinder axis withpressure distribution of the first radial mode. This configuration yieldlarge value of the corresponding overlap integral, and therefore efficexcitation.


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Substituting Eq.~14! into Eq. ~15! enables the sound pressure as a function of the modulation frequency to be writas

p~r M ,v!5~s21!L

Vres S 1

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n

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~16!

It can be seen from Eq.~16! that the suppression of unwanted resonance is possible also by placing the microphin a node of this particular eigenmode. In this case the vaof the corresponding eigenmode distribution functionpn(r M)is zero.

Notice that the right-hand side of Eq.~16! can be writtenas the product of an acoustic impedanceZ and a volume flowu:

p~r M ,v!5Z~r M ,v!u, ~17!

where

u5~s21!aLWL

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n

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Vres

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which corresponds to an electric network of serially conected parallel resonance circuits and one capacitancshown in Fig. 5. This means that each acoustic eigenmcan be modeled by a parallel resonance circuit, whilenonresonant part is represented by a capacitance. Thesource is described by a current sourceu, which does notdepend on the frequency or the eigenmode distributiondepends only on the quantityaLWL , which is the lightpower absorbed within the acoustic resonator.

The electric network described above can completrepresent the properties of an acoustic cavity resonator. San electric network model has been used successfullycomputer simulation of PA signal generation in cavresonators.43

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FIG. 5. Scheme of an electric network of serially connected parallel renance circuits, where each acoustic eigenmode is modeled by a paresonance circuit and one capacitance which represents the nonresonantribution to the acoustic signal.


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B. Nonresonant cell and resonant cell, high- Q andlow- Q resonators

In the photoacoustic literature the PA cells are oftcharacterized as ‘‘nonresonant’’ or ‘‘resonant.’’ This termnology is misleading, because all PA cells can be operatean acoustic resonance or far from their resonances. Thuspreferable to label the system according to the nonresoor resonant mode of operation.

If the modulation frequency is much smaller than tlowest resonance frequency, the cell is operated in a nonrnant mode. In this case the sound wavelength is much lathan the cell dimensions, thus sound cannot propagatestanding waves cannot form. The average pressure incavity will oscillate with the modulation frequency. The phtoacoustically generated pressure is given by the first termthe series presented in Eq.~6!, thus the sound pressuresimply equal toA0 in Eq. ~6!. Note that this term appliesonly for closed resonators, because in an open resonatopressure change simply drives gas in or out until the equrium pressure is restored. If an open resonator is built inclosed PA cell, then the pressure generated by the PA ewill be distributed over the entire closed volume. Therefothe total volume of the PA cell must be taken into accounEq. ~13!, instead of the volume of the resonator.

For a small closed cell and low modulation frequencthe signal can be relatively large, as mentioned before.fortunately, the noise also increases with decreasingquency and volume, which means that the SNR will usuadecrease substantially. It should be noted that the PA sihas a 90° phase lag with respect to the light power.

For resonant operation the modulation frequencytuned to one of the eigenresonances of the PA cell, i.ev5vm . As can be seen from Eq.~16! not only themth but alleigenmodes of the acoustic resonator will be excited. Tresonance amplitude is proportional to theQ factor Qm ,while the amplitudes of the other resonances are inverproportional to the quantityvn

22vm2 . Therefore, distant reso

nances will not be excited as effectively. As was discuspreviously, certain resonances can be suppressed sincoverlap integral will be zero for special symmetry condtions, e.g., azimuthal modes cannot be excited if the cydrical laser beam propagates exactly along the cylinder a

If the eigenresonances are well separated and theQ fac-tor of the particular resonance to be used is large enoughselected resonance can be excited much more effectithan the others. Therefore, the series expansion of the spressure in Eq.~6! can be approximated by one single terIn this ‘‘high-Q’’ case the sound pressure amplitudA(r M ,vn) of Eq. ~16! is given by the resonance amplitudAn(r M ,vn) of the nth resonance by settingv5vn in Eq.~14!:

An~r M ,vn!5~s21!LFnQn

VcellvnaWL . ~20!

If the Q factor of the selected resonance is not hienough, the contributions of the other resonances to thesignal cannot be neglected, and the full expression of~16! has to be used~low-Q case!:


att isnt

o-erndhe

of

the-act,

n

sn-e-yal

s

e

ly

dthe

-is.

helynd.

Aq.

p~r M ,vn!5~s21!L

Vres S 1

ivn1

Fnpn~r M !Qn

vn

1(j Þn

ivnF j pj~r M !

v j22vn

21 iv jvn

Qj

D aWL . ~21!

If v j /vn.1.2 or v j 3vn,0.8 andQj.20, the imaginaryparts in the denominators of the expression after the sumtion symbol can be neglected. In practical applications loorder eigenmodes are employed. Thus the ratiov j 3vn willbe larger than about 1.5, and therefore the ratio of the larterm behind the sum symbol to the second term, describthe selected resonance, will be smaller than 1/Qj . Since theother terms are much more separated, their contributionsbe even smaller. For these reasons,Qj,50 may be regardedas a limit for the low-Q case.

Since the quantities in the prefactor toaWL on the right-hand side of Eqs.~20! and ~21! are independent of the lighpower and the absorption coefficient, these factors canregarded as characteristic setup quantities for high-Q andlow-Q PA resonators, respectively. Denoting this quantityCn(vn) the PA signal amplitude can be written as

An5Cn~vn!aWL . ~22!

The quantityCn(vn) describes the sensitivity of the PA resnator at a given resonance frequency. It is usually called‘‘cell constant’’ in the photoacoustic literature. The cell costant depends on the size of the resonator, the frequencytheQ factor of the resonance selected for PA detection, hoever, it depends also on the spatial overlap of the laser band the standing acoustic wave pattern. Therefore, the ncell constant is misleading, because it characterizes the cplete measurement arrangement~including the acoustic resonator with a selected resonance, microphone position,laser beam profile with spatial location! rather than only thePA cell. Moreover, it depends on the frequency, and its vais different for the different eigenmodes. Therefore, it woube better to use the term ‘‘PA setup constant’’ instead of cconstant, despite the fact that the latter is already establisin the literature. The setup constant of a high-Q acousticresonator is a well-defined quantity@see Eq.~20!# that can bedetermined quite accurately by taking into account the gmetrical dimensions, acoustic constants, beam parameand eigenmode distributions. On the other hand, the seconstant of a low-Q resonator is a complicated function oseveral eigenmodes@see Eq.~21!#, and therefore it cannot bedetermined with sufficient accuracy by calculation. It hasbe determined by photoacoustic calibration measuremusing certified gas mixtures.

While in the overwhelming majority of photoacoustpapers excitation by modulated~periodic! radiation is ap-plied it is interesting to consider also the response ofacoustic resonator to pulsed laser excitation.


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C. Excitation of acoustic resonances with pulsedlasers

A resonator responds differently to pulsed and molated disturbances. In pulsed excitation the absorptionphotons ceases after the laser pulse terminates, but the ration processes will continue until full thermalization of thdeposited energy is reached. Therefore, the heat pulsesomewhat longer than the light pulse. Nevertheless, thepulse may be regarded as a source of instantaneous acoexcitation if the characteristic time of the acoustic procesunder consideration is much longer. Since the acousticsponse time is normally in the millisecond range~the lowestacoustic resonance of the PA cells used in practice is usuin the kilohertz range!, while the pulse duration of mospulsed lasers employed is in the nanosecond range, the acondition is generally fulfilled. The short heat pulse acts abroadband acoustic source, exciting all eigenmodes ofresonator simultaneously, similarly to a nonselective acouexcitation process. The acoustic resonator responds tosudden disturbance similarly to a ballistic pendulum or gvanometer reacting to a force or charge pulse: a sudden oof acoustic oscillations is followed by a slow characterisdecay of their amplitudes. The amplitude of the very fioscillation period will be proportional to the released heenergy. Figure 6 shows an example of the excitation oflindrical modes with a single laser pulse. The inset exhibthe decay of the photoacoustic signal measured in thedomain with the microphone. The corresponding frequespectrum is obtained by Fourier transformation of the micphone signal and indicates the excitation of the longitudimodes@~002! and~004!#, the radial modes@~100! and~200!#,and a combination mode~102! in the frequency range detected.

A short pulse excites a PA signal described by a seexpansion of decaying sine functions having the followiform:33,41

p~ t !5(n

Ane2bnt sin~Avn22bn

2t !

5(n

Ane2 ~vnt/2Qn! sinSA12S 1

2QnD 2

vnt D . ~23!

FIG. 6. Time dependent PA signal monitored with the microphone~inset!and the corresponding frequency spectrum of the cylindrical resonshowing the modes excited with a laser pulse.


-ofax-

stsatstics

e-

lly

oveae

ichisl-set

tt-sey-l

s

For Q.5 the term under the square root can be regardeunity. The spectrum of the sum of the decaying sine futions is a sum of Lorentzian profiles:

p~v!5(n

An ~vn2/4Qn

2!

~v2vn!21 ~vn2/4Qn

2!. ~24!

The amplitudeAn is given by

An5~s21!LFnpn~r M !

VresaEL , ~25!

whereEL denotes the laser pulse energy. This indicatesthe amplitudes of the decaying sine functions do not depon theQ factors of the resonances.33,41 The dependence onthe location of the laser beam and the microphone positiothe same as for the modulated case. When the frequeresponse of a resonator is measured by tuning the modulafrequency over a certain frequency range, a response gby Eq. ~16! will be obtained. On the other hand, when theffect of a short pulse is analyzed by Fourier transformatof the time response a spectrum corresponding to Eq.~24! isfound. Note that the frequency dependence of the resonacurve is different in these two cases. The properties of cydrical resonators have been discussed in detail in prevpapers33,41and their application in pulsed photoacoustics wdemonstrated in Ref. 44.

V. THEORETICAL DESCRIPTION OF ONE-DIMENSIONAL RESONATORS

A. General considerations

One-dimensional acoustic resonators can be descrby two different methods. The first method is very similarthe treatment applied to cavity resonators earlier in SIV A. In this case the eigensolutions of the lossless waequation@see Eq.~5!# for the particular resonator have to bused in the series expansion of the sound pressure. Thathe pn(r ) eigensolutions will be different for open–opeopen–closed, and closed–closed pipes. The PA signal cadetermined similarly, thus Eqs.~13! and~14! can be appliedto calculate the sound pressure amplitude at the microphposition.

Although this method seems to be appropriate, it tuout that the basic assumptions of the method are not alwjustified for one-dimensional pipe resonators. For optimizcavity resonators the real solutions do not differ considerafrom the eigensolutions of the lossless closed resonatorthis case the theoretical treatment of Sec. IV can be usedthe other hand, the frequency,Q factor, and mode shape ofpipe resonator depend strongly on the terminating impances, which are functions of the sizes and shapes ofadjoining parts of the PA cell. Therefore, proper theoretitreatment should take into account the acoustic propertiethe whole PA cell, not only those of the isolated resonato

The second method is based on the formal analogy ofone-dimensional acoustic equations to the electromagnequations of a waveguide, such as a coaxial cable.propagation of an acoustic wave in a pipe can then bescribed by a complex wave number, while the propertiesthe pipe are represented by a complex wave impedanc

or


rk

nd

red

c

ic

e,

e

tintin

oec

b

PAnly

pipe-

pipe

ghtipe.atl

e-

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unit


piece of pipe corresponds to an electric four-pole netwoand arbitrary arrangements of pipes can be investigatedthe well-known methods of electric network analysis asynthesis.

In the following, both methods will be described in modetail, and their advantages and disadvantages will becussed.

B. Serial expansion of the PA signal for one-dimensional resonators

As mentioned already, the treatment described abovebe applied to one-dimensional pipe resonators. The formthe eigensolutions in the ideal lossless case for a cylindrpipe of lengthl can be given as follows:open–open resonator:

pn~x!5sinS npx1D l 1

l 1D l 11D l 2D ; ~26!

closed–closed resonator:

pn~x!5cosS npx

l D ; ~27!

closed–open resonator:

p2m21~x!5cosS ~2m21!p

2

x

l 1D l 2D ; ~28!

wherex50 andx5 l are the left and right ends of the piprespectively. The end corrections are denoted byD l 1 andD l 2 for the left and right open ends, respectively. The indicn andm are positive integers.

The PA signal generated by a laser beam propagaparallel to the axis of a pipe can be calculated by substituthe above eigensolutions into Eq.~14!. If the light power isconstant along the beam, i.e., the attenuation due to abstion is negligible~which is the usual case in trace gas dettion!, the overlap integral in the numerator ofFn will beproportional to the integral of the mode distributionpn(x)over the length of the pipe. The value of this integral cangiven as follows for the three different cases:open–open resonator:

E0

l

pn~x!dx5l 1D l 11D l 2

np FcosS npD l 1

l 1D l 11D l 2D

2~21!ncosS npD l 2

l 1D l 11D l 2D G ; ~29!

closed–closed resonator:

E0

l

pn~x!dx50; ~30!

closed–open resonator:

E0

l

p2m21~x!dx52~ l 1D l 2!

~2m21!p~21!m

3cosS ~2m21!p

2

D l 2

l 1D l 2D . ~31!


,by

is-

anofal

s

gg

rp--

e

The expressions show that under these conditions asignal cannot be excited in a closed–closed pipe, and oodd eigenmodes can be generated in a simple open(D l 15D l 2). Moreover, the efficiency of excitation is inversely proportional to the eigenmode indexn or (2m21).Therefore, in most cases the fundamental resonance of ais used for photoacoustic detection.

Closed–closed pipes can also be employed, if the lipath in the resonator is shorter than the length of the pLet us assume that the light enters into the resonatorx5x1 and leaves atx5x2 . In this case the overlap integrawill be proportional to the following integral:

Ex1

x2pn~x!dx5

l

np FsinS npx2

l D2~21!n sinS npx1

l D G . ~32!

In the case of the ‘‘banana cell’’ described by Sigristet al.45

n52, x15 l /4, andx253l /4, thus the absolute value of thintegral is 1/p, which is about half of the corresponding integral of an open–open pipe operating in the fundamemode.

In practice, the resonant PA signal can be generatedin a closed pipe by a laser beam propagating along theof the pipe. However, this signal will no longer be propotional to the absorption coefficient. It may be excited eithby background effects~for example, a window signal or PAsignal caused by scattered light absorbed by the walls!, ordue to asymmetries of the acoustic properties of the resotor, or caused by attenuation of the radiation inside the renator. In this last case the PA signal shows a quadraticpendence on the absorption coefficient.

C. Equivalent four-pole network of pipe resonators

The acoustic equations of a one-dimensional losssystem can be written as

]p

]t1

rc2

S

]u

]x50, ~33!

r

S

]u

]t1

]p

]x50, ~34!

whereS is the cross-sectional area of the pipe. These eqtions are very similar to the corresponding transmission lequations:

]U

]t1

1

C*]I

]x50, ~35!

L*]I

]t1

]U

]x50, ~36!

whereL* andC* are the inductance and capacitance of ulength of the cable, respectively. Based on the similaritythe above pairs of equations sound propagation in pipesbe treated similarly to the propagation of electromagnewaves in cables. The sound pressurep and volume velocityu correspond to the voltageU and currentI , respectively.The acoustic inductance and capacitance of a pipe withlength can be expressed as


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the

ca

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he

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lete

thes.


Lac* 5r

S, Cac* 5

S

rc2 . ~37!

The wave impedance of the pipe can be defined the sas for electromagnetic waveguides:

Z05ALac*

Cac*5

rc

S. ~38!

The solutions of the above equations are composed of award and a backward propagating wave:

p5~Ae2 ikx1Beikx!eivt, ~39!

u51

Z0~Ae2 ikx2Beikx!eivt, ~40!

where the the wave numberk is given as

k5vALac* Cac* . ~41!

The losses of the pipe can be taken into account by introding loss terms into the above equations. In the case of etromagnetic waves the termsR* I andG* U are added to thefirst and second equations, respectively. These terms resent the ohmic resistance of unit length of the cable andloss per unit length due to the conductance of the dielecinsulator of the cable. The wave number and the wavepedance will be complex, becauseivL* and ivC* shouldbe replaced byivL* 1R* and ivC* 1G* , respectively.Using a similar substitution in the acoustic equationscomplex wave number and wave impedance can bepressed as

g5k1 ik5A~ ivLac* 1Rac* !~ ivCac* 1Gac* !, ~42!

Z05A ivLac* 1Rac*

ivCac* 1Gac*. ~43!

The dimensionless viscous and thermal loss factorsbe written as follows:

Rac*

vLac*5

1

rA2v

v,

Gac*

vCac*5~s21!

1

rA2m

v, ~44!

wherev, m, and r are the kinematic viscosity and thermdiffusivity of the gas and the radius of the pipe, respective

Since other loss components may also be importsuch as the volume losses in the pipe and the energy trferred to wall vibrations, the above expressions cannot demine the loss of the pipe accurately.

In the case of small losses their effect on the acouwave impedance can be neglected and the acoustic wimpedance can be approximated by the real quantity giveEq. ~38!. In order to describe the attenuation of the acouswave during propagation the acoustic wave number, hever, must be complex.

The relation between the input and output sound prsures and volume velocities for a pipe with uniform crosection can be represented by the impedance, admittaand transfer~chain! matrices46 as follows:


e

r-

c-c-

re-e

ic-

ex-

n

.t,s-r-

icveinc-

s-sce,

S p1

p2D5S Z0 cothg l

Z0

sinhg l

Z0

sinhg lZ0 cothg l

D S u1

2u2D , ~45!

S u1

2u2D5S cothg l

Z0

1

Z0 sinhg l

1

Z0 sinhg l

cothg l

Z0

D S p1

p2D , ~46!

S p1

u1D5S coshg l Z0 sinhg l

sinhg l

Z0coshg l D S p2

u2D , ~47!

where the indices 1 and 2 denote the input and output ofpipe, respectively. The direction of the volume flow was chsen in accordance with the convention of network theoi.e., the impedance and admittance matrices are definedflow directions into the pipe, while in the chain matrix thflow is directed inward at the input and outward at the oput, allowing the connection of several four poles~pipes! ina chain. In this particular case, which is very importantpractical applications, the chain matrix of the entire chaingiven by the product of the chain matrices of the elemenOn the other hand, impedance~admittance! matrices are pre-ferred, if the inputs of two or more elements are connecteseries~parallel!.

The equivalent electric circuits~p and T networks! of apipe with lengthl and cross sectionS can be seen in Fig. 7where Z05rc/S is the wave impedance of the pipe. Bocircuits can completely describe the pipe as an acouwaveguide. Thep network is preferred when the inputof two or more pipes are connected in parallel, while anetwork representation is better for serially connecwaveguides.

If the acoustic four-pole network is terminated at toutput by a load impedanceZT , the remaining two pole canbe represented by a single impedance, called input impance. The input impedance of a pipe of lengthl terminatedby ZT can be determined from Eq.~47! as

Zin5Z0

ZT coshg l 1Z0 sinhg l

Z0 coshg l 1ZT sinhg l. ~48!

~Note that the input impedance of a pipe of arbitrary lengterminated byZ0 always equalsZ0 .! This expression is also

FIG. 7. Two equivalent electric circuits that can be used for a compdescription of a pipe as an acoustic waveguide. Thep network is betterwhen the inputs of two or more pipes are connected in parallel, whileT-network representation is preferred for serially connected waveguide


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n

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suitable for calculating the input impedance of a chain coposed of pipes having different lengths and diameters.input impedance of the last pipe should be calculated fitaking into account the terminating impedance at the endthis pipe. Then this value should be used as the terminaimpedance to calculate the input impedance of the nexlast pipe and so on. Thus, the input impedance of a chcontaining an arbitrary number of pipe elements can beculated by using a simple loop in a computer program. Silarly, the sound signal appearing at the output of the chcan be transformed into the input or vice versa.

The input impedances of an open (ZT50) and a closed(ZT5`) pipe are given by Zin5Z0 tanhgl and Zin

5Z0 cothgl, respectively. Resonances occur when the vaof the imaginary part ofZin equals zero, i.e., at the roots othe functions tankl and cotkl, respectively. Sincek>v/c,the resonance frequencies for open and closed ends caexpressed as

f n5nc

2l, n51,2,3,....,

~49!

f 2m215~2m21!c

4l, m51,2,3,....

These expressions are similar to Eq.~3!, but no end correc-tion terms appear here.

According to Eq.~39! the sound field in the pipes igiven as the sum of two counterpropagating plane wavThese waves can be regarded as an incident wave areflected one. The reflection coefficient can be defined a

R5ZT2Z0

ZT1Z0. ~50!

For an open endZT!Z0 , thus the reflection coefficienis R'21, while for a closed end (ZT!Z0) the reflectioncoefficient isR'11. This means that the sound pressurereflected with the opposite phase from an open end andthe same phase from a closed end. The reflection coefficat the boundary of two adjoining semi-infinite pipes canwritten as

R5

rc

S22

rc

S1

rc

S21

rc

S1

5S12S2

S11S2, ~51!

which indicates that the sound is reflected by every sudcross section change. If the acoustic resonator tube is plbetween two wide cylinders~buffers!, the sound field will bereflected back to the resonator very effectively.

Unfortunately, the one-dimensional model cannot dscribe three-dimensional effects, such as sound radiatiothe pipe end into the buffer volume. Therefore, an end crection is not included in the distributed acoustic netwomodel and resonance frequencies and reflection coefficiof the pipe cannot be determined accurately by this moBetter results can be obtained using the following correctimpedance:35


-et,ofg

toinl-i-in

e

be

s.a

sthnt

e

ned

-byr-

tsl.n

DZ5rc

Sopening@a~kr !21 ib~kr !#, ~52!

whereSopening denotes the area of the actual opening atpipe end~it may differ from the inner cross sectionS of thepipe!, r is the equivalent radius of the pipe~r 523volume/area of the pipe wall for noncylindrical pipes!, anda'0.25 andb'0.6 are numerical parameters. Their valudepend on the geometry of the pipe opening, but the vagiven above ensure sufficient accuracy in practical appltions.

In this case the wave impedancerc/S2 should be re-placed byrc/S21DZ, or, more generally,DZ should beadded to the terminating impedanceZT . If S2 increases, theimpedance at the end of the pipe will approachDZ instead ofzero.

Note that the one-dimensional model can be used ofor frequencies below the cutoff frequency of the pipe, whiis defined as the lowest transversal resonance frequencthe pipe. Its value for a pipe of radiusr is defined by theexpression35

vcutoff

cr 51.84.

For a complex photoacoustic cell the equivalent radius ofbuffer element with the largest cross section should be uto determine the cutoff frequency. One-dimensional moding of the whole PA cell is possible only if the resonanfrequency of the resonator is smaller than the cutoff fquency.

The main advantage of the application of the equivalelectric network is that PA signal generation can be queasily included in the model. In this case a term describthe PA signal should be added to Eq.~33!,

]p

]t1

rc2

S

]u

]x5~s21!H. ~53!

The PA signal in a pipe can be determined by solviEqs.~34! and~53! simultaneously. This solution can then bused to determine the PA signal in a complex PA cell. Tmain steps of such a calculation can be explained witmodel containing two pipe elements~pipe 1 and pipe 2! withdifferent diameters. The goal is to determine the sound sigat the output of pipe 2. It is assumed that some kindacoustic signal~e.g., external noise, window signal, etc.! ispresent in the input of pipe 1. This input signal (p11) will betransformed to the output by the chain matrix of pipe 1. Tnext step is the calculation of the sound pressure (p12) at theoutput of pipe 1 as the sum of two components, namely,value of the solution of Eq.~53! at the output of pipe 1 andthe transformed value of the input signal. Nowp12 will beused as the input signal of pipe 2 and will be transformedthe output by the chain matrix of pipe 2. The resulting outppressurep22 will be obtained by adding the local value of thsolution of Eq.~53! for pipe 2.

The above method can also be applied to determinemicrophone signal in PA cells with complicated geometriThe calculation procedure should be started at the input~s! ofthe PA cell ~window or gas inlet, or both! and should be


ocnl

out

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le

nt

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rdle

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-ss

he

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-

ut

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ofn as

of

-ts.

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inar-

ed


repeated until the microphone has arrived. The same prdure must be carried out for the other half of the PA cell, aboth contributions should be added to obtain the PA signathe microphone position.

It is beyond the scope of this review to give all detailssuch a calculation. In fact, it is more complicated than olined above, because several sources~window absorption,external noise! and different losses have to be taken inconsideration. However, the calculation can be easily pformed on a PC. A PA signal calculation based on the etric circuit analogy was reported as early as 1990.10 Com-puter simulations of PA cells composed of pipe elemewith different cross sections were introduced in 1992.43

Although one-dimensional modeling of PA cells is veefficient and can be used for the analysis of complicatedgeometries, it is difficult to extract information on the sentivity of different PA resonators, because the sensitivity cnot be expressed in closed analytical form. Therefore,sensitivity of pipe resonators will be discussed and compawith other resonators by using the serial expansion moapplied before to cavity resonators.

VI. SENSITIVITY OF RESONANT PHOTOACOUSTICDETECTION

A. Setup constant

The PA amplitude generated in an acoustic resonatogiven by Eqs.~13!, ~20!, and~25! ~the last applies to pulseexcitation!. Although these expressions are valid in principonly for high-Q resonators, they may also be used to emate the sensitivity of PA resonators with medium quafactors (10,Q,50). It turns out that the sound pressurealways proportional to the absorption coefficienta and thelaser powerWL ~or pulse energyEL in pulsed photoacoustics!, and it is inversely proportional to an effective crosection defined by the ratioVcell /L. In the case of modulatedPAS the signal is in addition inversely proportional to tmodulation frequency and proportional to theQ factor of theresonance.

The sensitivity of a PA resonator to the pressure atresonance frequencyvn of the nth eigenmode can be represented by the setup constantCn (vn) defined above:

Cn~vn!5p~vn!

aWL5

~s21!LFnpn~r M !

Vcell

Qn

vn. ~54!

The units ofCn are given in Pa cm/W based on the usudimensions ofp, a, andWL . When the spatial dimensionare specified in centimeters and the frequency in hertz,setup constant~in units of Pa cm/W! can be calculated fromthe data of the resonator as follows:

Cn~vn!FPa cm

W G5106~s21!LFnpn~r M !

VcellDvn. ~55!

Here Dvn5vn /Qn is 2p times the half width~FWHM! ofthe resonance profile. Equation~55! clearly shows that thesetup constant of an acoustic resonator depends criticallthe effective resonator cross section and on the half widtthe resonance. The values ofFn andpn(r M) are usually closeto unity, and therefore the setup constant of a mediumQ


e-dat

f-

r-c-

s

ll--ed

el

is

i-

e

l

e

onof

resonator with 1 cm2 effective cross section and;20– 50 HzFWHM is expected to be in the 103 Pa cm/W range, while aresonator optimized for high-Q performance~effective crosssection;80 cm2, Q;1000! has a setup constant of abo200 Pa cm/W.

The setup constant of the same eigenmode for puexcitation can be defined as

Kn5p

aEL5

~s21!LFnpn~r M !

Vcell. ~56!

The units ofKn are given in Pa cm/J. The setup constantthe same setup but for modulated excitation can be writte

Cn~vn!5Kn

Qn

vn5

Kn

Dvn. ~57!

The possibilities for improving the setup constantacoustic resonators are limited. Let us consider Eq.~54! inmore detail. The terms21 depends only on the gas involved, which is normally air in trace gas measuremenTherefore, we finds2150.4. The factorFn should be cal-culated for the specific arrangement, taking into accountlocation and intensity distribution of the laser beam andmode pattern of the particular eigenmode selected fordetection. This factor must be optimized, but the optimuvalues will be close to unity. The value ofpn(r M) is unity, ifthe microphone is placed at the antinode of an eigenmoThis, of course, is not always possible, but a value inrange between 0.5 and 1 can usually be obtained. Thusonly parameter that really can be changed in a larger rangthe effective cross section of the cell. A reduction of the cdiameter will increase the setup constant. A lower limit isby the diameter and divergence of the laser beam emploThe setup constant for modulated measurements is inverproportional to the FWHM value of the resonance profiUnfortunately, the half width cannot be reduced indefinitebecause it scales approximately with the surface-to-voluratio of the resonator. As the cross section of the celreduced, the surface-to-volume ratio increases. Thereforis impossible to realize small cross sections and a smbandwidth~high Q! simultaneously. The smallest diametused in practical PA systems is several millimeters, the laest about 10 cm.

If the eigenfrequencies and the acoustic eigenmodeterns can be calculated analytically, and the laser beamometry is also known,Kn can be calculated fairly accuratelyThe determination ofCn(vn) requires the measurement othe FWHM of the acoustic resonance. This can be doneacoustic excitation. Thus, the setup constants of thequently used acoustic resonator types can be determwithout using calibrated gas mixtures and/or calibratedcrophones for both pulsed and modulated excitation withprocedure.

Setup constants expressed in V cm/W units can befound in the PA literature. In this case the setup constantCn

is multiplied by the responsivity of the microphone givenV/Pa units. However, this kind of setup constant, as a chacteristic of the PA cell, can be used only if a calibratmicrophone is available.


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It can be stated that the PA resonators presently useseveral research groups are close to optimal. Differencesbe found; nevertheless, in the designs of the PA detecand in the methods used for measuring the PA signal.

B. Noise

Noise plays an important role in all photoacoustic mesurements and is of particular importance in the detectionultralow gas concentrations, because the noise level limthe ultimate sensitivity. In the photoacoustic literature tdetection level is usually defined by the SNR, wherenoise is given by the microphone signal measured withlaser light blocked. However, when light hits the PA celladditional background signal is generated which exists ewhen the absorbing species are not present in the deteThis background signal is often larger than the noise sigand therefore the detection limit or sensitivity has to befined by the signal-to-background ratio~SBR! in most ex-periments. Only the fluctuating part of the backgroushould be considered in the SBR, because the constantcan be subtracted. Unfortunately, it is common practiceconsider only the SNR. This procedure yields an extralated detection limit that may be considerably too small.

The noise level is determined by the microphone anddetection electronics employed. In addition external envirmental noise sources may play an important role, forample, the laser used for excitation can disturb the phocoustic measurement substantially.

The background signal is usually determined with a nabsorbing gas, such as nitrogen, in the PA detector. Iinfluenced by many system properties, such as the poinstability, the divergence, and the diameter of the laser beBesides the quality of the laser source the construction ofPA cell is another important factor. For example, the cleliness and optical absorption of the entrance and exit wdows can dramatically influence the background signal. Tdiameter of a pipe resonator cannot be reduced too msince it has to be selected according to the divergence olaser beam to avoid scattering of laser light at the cell wor even at the microphone. Impurities in the gas, whichsorb the laser radiation, also make a contribution to the baground signal. The background signal is expected to be mlarger for pulsed laser excitation than for modulated opetion.

Because of these dominating noise and backgroundnals the ultimate theoretical noise limit, which is due to tpressure transducer and stems from the Brownian motiothe sample gas molecules, cannot be reached.

C. Loss mechanisms

The energy accumulation attainable in the standing wof a resonant cavity is many times larger than the energyoccurring during a single period of an acoustic oscillatioThis acoustic amplification effect is limited, howeveby various dissipation processes. The first detailed discusof these dissipation effects observed in a closed cavitygiven in Ref. 7. In order to compare theory and expement the resonance frequency and half width were meas


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and compared with model calculations. Good agreementobtained for high quality spherical and cylindricresonators.32,47

The losses can be divided into surface effects and vometric effects. The surface losses are due to the interactiothe standing wave with the internal resonator surfacemay be subdivided into the following dissipation process

~1! viscous and thermal dissipation inside the boundary lers at the smooth internal surfaces;

~2! losses due to wave scattering at surface obstructionsas the gas inlet, microphone, and windows;

~3! compliance of the chamber walls;~4! dissipation at the microphone diaphragm.

In a carefully designed high quality resonator, the cotribution of the last three processes can be kept small.main contribution is caused by the viscous and thermboundary layer losses. Throughout the major portion ofresonator volume the expansion and contraction of theoccur adiabatically. Near the walls, however, the processcomes isothermal. This leads to heat conduction withintransition region~thermal boundary layer!, which is respon-sible for the thermal dissipation process. The viscous dipation can be explained by the boundary conditions impoby the walls. At the surface, the tangential component ofacoustic velocity is zero, whereas in the interior of the cavit is proportional to the gradient of the acoustic pressuThus, viscoelastic dissipation occurs in the transition regiwhich is called the viscous boundary layer.

In a spherical resonator no viscous surface losses apfor radial modes. This advantage and the favorable surfato-volume ratio make the sphere a superior resonator cpared to a cylinder. Very highQ factors~2000–10 000! canbe achieved for carefully constructed spheres.48,49Very goodagreement with measured half widths of radial modesbeen obtained by taking into account the thermal boundlosses and a small loss coming from free-space viscousthermal dissipation. On the other hand, the descriptionlaser excitation in a spherical resonator is more complicathan for one with cylindrical symmetry.40

The volumetric or bulk losses are caused by procesthat tend to establish equilibrium in the propagating wave35

These damping processes are

~1! free space viscous and thermal losses,~2! relaxation effects,~3! diffusion effects,~4! radiation effects.

Friction due to compressional motion and transformatof organized energy into heat due to temperature gradiare responsible for the free-space viscous and thermal losThese two processes are often called Stokes–Kirchlosses and play only a minor role compared to boundlayer losses. Diffusion effects are normally negligible, bmolecular relaxation processes can make significant cobutions in a limited pressure region which is given by theptvalue, wheret is the relaxation time. In this pressure regioan accurate determination of the corresponding relaxa


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time is possible from the resulting dispersion of the renance frequency and the strong broadening of the acouresonance profile.50

Radiation losses are negligible for completely closcavity resonators because of the nearly perfect reflectiothe sound at the walls. On the other hand, radiation losthrough openings, for example, pipes connecting the restor to buffer volumes, cannot be neglected. The radiatlosses can be reduced by increasing the acoustic inputpedance of the openings.30 This is achieved by terminatingthe cavity resonator at the openings with acoustic band-filters, which prevent the sound from escaping from the renator. In the case of one-dimensional acoustic resonatordiation is usually the dominating loss mechanism, whichways has to be taken into account.

VII. OPTIMUM DESIGN OF RESONANT DETECTIONSYSTEMS FOR TRACE GAS ANALYSIS

A. General requirements

In trace gas analysis usually small concentrationsbelow 100 ppm are detected. If it is assumed that absorpfollows the Beer–Lambert law the light intensity decreasin the PA detector asI 5I 0 exp(2ax), wherex is the opticalpath length. In the case of small absorption the fractioabsorption in the detector is given asaL, whereL is thelength of the optical path within the detector. Taking inaccount typical values for the absorption coefficients ofspecies to be measured (0.01– 10 cm21) and realistic diam-eters of the PA cells (;1 – 10 cm), the fractional absorptiois very small (;1023– 1027), which means less than 0.1%of the incident laser power. Often the windows of the Pdetector and the adsorbates at their surfaces may abmore light. Even the amount of light scattered by imperfetions of the windows and by the inner walls of the detecbecause of low beam quality may exceed the amountsorbed by the species to be monitored. Therefore, a cohebackground generated by window absorption and light stering is always present in the PA detector. Effective spression of this background is the most important requment for a sensitive PA detector. As shown in Eqs.~13! and~14!, the sound pressure generated by the modulated PAfect is proportional to the product of the absorption coecient and the laser power. Since the power of the availalaser sources is typically between 1 mW~diode lasers! and 1W ~CO2 and CO lasers!, the generated sound pressureusually smaller than 1 Pa. In the case of NIR diode laswhere overtones or combination bands are excited, the tcal values are 1023– 1026 Pa. By considering that the aveage noise amplitude@root mean square~rms!, broadband# ina laboratory corresponds to 1023– 1022 Pa, very goodacoustic isolation of the PA cell is required.

The measurement of such small sound pressures reqhigh standards for the microphones such as high responity, low noise characteristics, and a small size~and for prac-tical applications also a low price!. Since the responsivity ocondenser and electret microphones is about 10–100 mVPA signal levels in the nanovolt–microvolt range mustmeasured. Therefore, low noise microphones and mi


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phone preamplifiers are necessary and particular attenmust be paid to the connections between the cell andamplifier. Although high-quality condenser microphonesfer the best noise performance, they are rarely used in ptoacoustic gas detection because of their large size, lorobustness, and relatively high price. The most commoncrophones employed are miniature electret devices deoped originally as hearing aids~for example, different typessupplied by Knowles Electronics Inc. and SennheiGmbH!.

The general requirements for PA detectors can be smarized as follows:

~1! good background suppression;~2! good acoustic and vibrational isolation;~3! microphone with high responsivity;~4! low noise electronics;~5! good electronic isolation~no ground loops, prope

shielding!.

Since noise sources~intrinsic noise of the microphoneamplifier noise, external acoustic noise! have nearly a 1/ffrequency dependence, low modulation frequencies shobe avoided in PA trace detection. In order to minimize tnoise, modulation frequencies in the 1–5 kHz frequencygion are recommended. The signal loss due to the higmodulation frequency can be compensated for by using renant cells with a reasonableQ factor.

PA detectors generally can be used for both modulaand pulsed excitation, but the requirements for optimaltection are different in the two cases. A comprehensive dcussion of microphone selection criteria in conjunction wpulsed PAS can be found elsewhere.51

The sensitivity of the PA detector can be increasedemploying a microphone array instead of one single micphone device. This configuration has been demonstrateenhancing the detection sensitivity considerably.52

B. Design principles of PA detectors

The main question to be answered before the most sable design of a PA detector can be found is,Which specieshave to be detected in which concentration range?The nextquestion is,Which laser source is available to detect thgiven species in the given concentration region?

To answer these questions laser sources and absorbands of the species have to be selected. The product oavailable laser power and the minimum measurable abstion coefficient can be used to represent the sensitivity ofPA detector. This quantity can be determined by usingvalue of the setup constantC(vn) given in Eq.~54!:

aminWL5pmin

C~vn!, ~58!

wherepmin is the minimum detectable PA signal. Assuminthat the setup constantC(vn) of the resonator is alreadoptimized, the easiest way to improveamin is to select astronger laser line or a laser with higher power. If suchlaser is not available, further improvement can be achieonly by pushing down the detection limit of the PA signa


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This can be done by using PA cells, microphones, and etronics designed for low noise performance and by applymore sophisticated data acquisition and evaluation teniques. The noise of the measurement can be reduceusing state-of-the-art~and therefore very expensive! lock-inamplifiers, and/or by using longer time averaging~the noisedecreases with the square root of the averaging time!, withthe expense of longer measurement times. Effective nreduction can be achieved also by analog/digital converssynchronized to the modulation frequency.

The value ofpmin can be determined by dividing thminimum detectable microphone signalUmin by the respon-sivity Smic of the microphone. This means, the microphoresponsivity must be known to estimate the absorption ssitivity of the PA detector. Therefore, it is recommended tone applies calibrated microphones for PA trace gas msurements. Then the PA sensitivity can be calculated aslows:

aminWL5Umin

C~vn!Smic. ~59!

The minimum detectable microphone signalUmin can be de-termined experimentally. Both, the background signal athe noise should be measured, and the larger one shouregarded as the detection limit at SNR51 or SBR51. Amore realistic detection limit for gas detection can be definas SNR53 or SBR53.

The value ofUmin depends on several factors. In additioto the intrinsic noise of the microphone, preamplifier noigas flow noise, and acoustic and electronic noise fromenvironment, the background generated by the windowsscattered light may play an important role. The performaof the acoustic resonators applied in photoacoustics aremost optimal in many cases. On the other hand, howenoise and background levels of the PA detector could obe further decreased by proper detector design and sopcated methods of data collection and processing. Forample, the noise and background levels of a differentialdetector described previously17 are in the 100 nV–1mVrange~for a 1 stime constant!. These values can be reduceby an order of magnitude using state-of-the-art methodsmeasurement and data evaluation. If the average responsof 100 mV/Pa of the microphone is taken into account,minimum detectable sound pressurepmin is about1025– 1027 Pa. With a setup constant of 1750 Pa cm/W tvalue of the minimum detectable absorption coefficient cbe estimated asamin'531029– 5310211/WL . Assuming10 mW modulated light power a minimum detectable asorption coefficient ofamin'531027– 531029 cm21 can beobtained. If fundamental vibration bands are excited~for ex-ample, of hydrocarbons in the 3mm range!, sub-ppbV con-centrations can be detected.

VIII. APPLICATION OF RESONATORS INPHOTOACOUSTIC TRACE ANALYSIS

A. Complete photoacoustic gas detection system

The main parts of a complete photoacoustic systemshown schematically in Fig. 8. The laser radiation pas


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through the PA cell and enters a power monitor. Continuowave laser radiation is modulated by a mechanical chopan electro-optic device, or, as in the case of diode lasersdiode current is modulated directly. Pulsed lasers canused for broadband acoustic excitation. The excited sowaves are detected by an electret or condenser microphwith a responsivity in the 1–100 mV/Pa range. The micphone signal is measured by a lock-in amplifier or it is ditized and evaluated directly with a PC. In some cases aond, so-called ‘‘reference,’’ PA cell is used that containsrelatively high but known concentration of the gas compnent to be measured. The signal of this reference cell caused for tuning the laser wavelength to the maximum ofabsorption line. With this arrangement the effects of lapower fluctuations can be eliminated from the PA signalmeasuring the ratio of the signals of the detection and reence cells.

The PA cell itself consists of windows, gas inlets aoutlets, a microphone~s!, and an acoustic resonator~s!. Ad-vanced PA cells also contain an acoustic filter to suppressflow and window noise. Both high-Q and low-Q resonatorscan be applied. Since high-Q and low-Q resonators are usefor quite different purposes, they will be treated separate

B. High- Q resonators

Cavity resonators with cross-sectional dimensions inrange of several centimeters usually have high-Q eigen-modes. The value of theQ factor is normally around 100, bua value ofQ51000 has been reported41 for the first radialmode of a nearly perfect cylinder with carefully polisheinner surfaces. As mentioned before, even higherQ factorscan be realized with a spherical resonator~2000–10 000!.Such devices have been used for acoustic precision meaments of the universal gas constant.48,49

Since relaxation processes in the gas influence the actic losses and the sound velocity, the dispersion of the renance frequency and the broadening of the half width ofresonance profile can be used for accurate measuremenvibrational and rotational relaxation times in the frequendomain.50,53The same technique has been employed to stthe chemical relaxation of systems such as N2O452NO2,54

FIG. 8. Typical photoacoustic setup for modulated PA measurements.cw laser radiation is chopped and transmitted through the cylindrical renator to excite radial or longitudinal modes. The acoustic signal is measwith a microphone and processed by a lock-in amplifier and a PC.


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the chemical relaxation of hydrogen bonds and the dynamof hydrogen bond formation in dimeric formic acid,55 andother aliphatic carboxylic acids56 in the frequency domain.

High-Q acoustic resonators have also frequently beused for trace gas measurements. The first PA gas deteinvestigated were simple cylinders equipped with windoand gas inlets and outlets. But windows attached directlythe cylinder surface produce a significant background sigTherefore, more sophisticated PA detectors have beenstructed recently. In these never detectors the windowsseparated from the resonator by pipes and/or buffer volu~see Fig. 2!. This solution was applied, for example, in thmobile CO2 system.9,57

Different configurations for laser excitation of cylindrcal resonators have been applied in photoacoustic detecfor example, to excite the first~010! and second~020! azi-muthal modes, the first radial mode~100!, and the mixedradial-longitudinal~102! mode ~see Fig. 3!. Since the rela-tively large cylindrical cells have lower sensitivity, they wemostly used together with high-power gas lasers such as2lasers.

An open PA detector based on a cylindrical resonacombined with band-rejecting acoustic filters has beproposed.30 Since the acoustic filters have excellent nosuppression around the selected resonance, the outsidedoes not disturb the PA signal. This construction can ainclude windows,58 because the background due to the wdows can be efficiently eliminated by the filters.

A high Q factor can be achieved by keeping the lossesthe resonant cavity low. All loss sources should be takenconsideration, including surface and volume losses,losses through openings, microphones, and other senViscous and thermal losses at the walls can be decreassome extent by using smooth well polished surfaces. Acotic losses through openings can be minimized by usquarter-wavelength tubes to connect the resonator to extebuffers.

In the case of very highQ factors (Q.500) losses origi-nating from the microphone can even influence the valuethe Q factor.41 Therefore, miniature microphones with higacoustic impedance are strongly recommended for higQPA applications.

Unfortunately, high-Q resonators are very sensitivetemperature changes. Since the resonance frequency isportional to the sound velocity@see Eq.~3!#, the temperaturedependence of the sound velocity is directly mirrored byresonance frequency. The sound velocity of air has a tperature coefficient of about 0.18%/°C, thus a frequencyshift d f >0.0018f 03DT is expected for a temperaturchange ofDT °C. Therefore, a fixed modulation frequencmay deviate byd f from the true resonance frequency. Ththe PA signal will not be excited at the peak of the resnance, but slightly to one side. Since a detuning fromresonance peak by a quarter of the FWHM results in;10% drop of the PA signal, the detuning should be smathan f 0/4Q for 65% signal stability. This stability can bensured, if the conditionQ3DT<138 is fulfilled. The cor-responding condition for PA signal stability of61% can bewritten asQ3DT<56. These examples clearly show th


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high-Q photoacoustic resonators are difficult to use in modlated photoacoustics without temperature stabilization ortive tracking of the resonance to adjust the modulation fquency.

Several solutions to this problem have been reportThese include frequency scans of the modulation frequeover the acoustic resonance profile repeated at regularintervals using PA18,57 or acoustic17 excitation. Monitoringthe temperature and calculating the frequency shift by usthe known temperature dependence of the sound velocitysetting the modulation frequency to the calculated new vaat regular intervals are another possibility.10 Finally an activeresonance tracking method based on an acoustic osciloperating permanently on a higher eigenmode has bdeveloped.59 The last system also works well for suddechanges of temperature and/or gas composition.

The problem of the dependence of the resonancequency on sound velocity can be elegantly circumventedusing broadband excitation with pulsed laser radiation. HiQ resonators are very well qualified for pulsed operatiobecause, as mentioned above, the light pulse excites a wseries of decaying sine signals. The Fourier spectrum oftime response is composed of several Lorentzian resoncurves, whose peaks correspond to the actual resonancequencies, in agreement with the resonance spectra obseby point-by-point excitation in the frequency domain. Therfore, temperature shifts do not play a role in these pulmeasurements. It is also possible to design the resonatoa dominant mode by appropriate selection of the positionthe laser beam and the location of the microphone at noof acoustic modes not to be detected. In the ideal case thesignal is essentially a single decaying sine function whcan be easily distinguished from noise and backgrousignals.41

A high-Q resonator optimized for the detection of thfirst radial mode has been used for an absolute measureof trace concentrations of ethylene.41,44 Since the PA signaldid not depend on theQ factor of the resonance, the overlaintegral Kn and the eigenmode distributionpn(r M) of Eq.

FIG. 9. Comparison of measured and calculated calibration curves.open squares points give the PA signal strength observed for the 10P14 lineof the CO2 laser and different C2H4 concentrations. The diamond marks thbackground signal in pure nitrogen and the open circles signal the maximdetectable signal. The line represents the calculated signal strength~calcu-lation of the setup constant!.


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~56! could be calculated accurately from the dimensionsthe resonator known with high accuracy. The calculationthe signal strength~setup constant! offers a new approach todetermining absolute gas concentrations. The comparisothe measured PA signals of certified ethylene–nitrogen mtures with the calculated values showed excellent agreemas can be seen in Fig. 9. The calibration curve is obvioulinear between the maximum signal detectable with thecrophone used and the nitrogen background signal of 0mV. It is also possible to determine the optical absorptcoefficient of the absorbing species if the concentrationknown, for example, by using a certified gas mixture.

In spite of these advantages, high-Q resonant PA detectors also have disadvantages if they are applied in traceanalysis. The reasons for this are that their size is larger, tsensitivity lower, and their weight higher than that of tmedium- and low-Q PA detectors.

C. Applications of medium- and low- Q resonators

Medium- and low-Q resonators are used frequentlyPA detectors built for ultrasensitive trace gas monitorinMostly one-dimensional pipe resonators are applied wresonance frequencies in the 500 Hz–5 kHz region. Tresonator itself is either a closed or an open pipe. Sinceopen pipe efficiently picks up and amplifies noise from tenvironment, it should be surrounded by an enclosureorder to ensure high acoustic reflections at the pipe endsudden change of the cross section is necessary. Therethe resonator pipe should open up into a larger volume obuffers with a much larger cross section. Such ‘‘organ piparrangements have found widespread use in phacoustics.60 The buffers can be optimized to minimize flonoise and/or window signals. An asymmetric buffer configration has been tested and successfully applied for supping the window signals.42 More sophisticated PA detectoruse adjoining acoustic filters for noise and backgrousuppression;61 a true differential PA detector has been devoped with two identical resonator pipes.17 This constructionwill be discussed in more detail below.

Due to their small size, medium-Q PA detectors are suitable also for intracavity operation. The best PA sensitivamin'1.8310210cm21, reported to date has been achievby an intracavity PA detector of organ pipe design. Tvalue corresponds to 6 pptV ethylene concentration in thgiven measurement setup.11

PA detectors based on one-dimensional pipe resonahave been optimized for special purposes. A special ‘‘hpipe’’ design has been used to investigate the PA signavapors originating from liquids with low volatility.62 Re-cently, a heatable PA detector has been applied for meaing pentachlorophenol~PCP! contamination of wood.63 Thisdetector was equipped with band-rejecting acoustic filtersadditional noise suppression. The microphone was separfrom the resonant cell by a 200 mm long thin stainless stube, thus allowing the PA cell to be heated up to 220 °C

The operation of a multipass PA detector, based onopen acoustic resonator pipe, has been demonstrrecently.57 The laser beam is reflected back and forth b


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tween two mirrors placed outside the PA detector. Tacoustic resonator of this detector was a broad pipe withQfactor of Q'70. In spite of the low sensitivity of the resonator itself, the overall sensitivity of the entire detector wamin'231029 cm21 ~corresponding to 70 pptV ethyleneconcentration in the given measurement setup57!, due to theincreased light absorption in the cell.

It should be mentioned that PA detectors can be ualso at reduced pressures. Since the absorption coefficincreases~due to the decreasing pressure broadening eff!and the microphone responsivity and theQ factor of theacoustic resonance decrease, a sensitivity maximum cafound somewhere.64 The exact pressure providing the maxmum sensitivity depends on the properties of the microphand on the excited molecular transition, thus no genervalid expression can be given for determining the maximsensitivity. It should be mentioned that the above considations are not valid for line tunable lasers. In that caseeffective absorption coefficient may significantly decreasereduced pressures because the overlap between the laseabsorption line may be degraded due to narrowing ofabsorption line.

Open pipes were introduced for PA detection as early1977,23 and the most sensitive PA detectors known todused are based on open resonant pipes.11,17 A closed pipeexcited in its second resonance was introduced in 19845

The banana cell~named after its shape! had two Brewsterwindows around the nodal points of the second longitudiacoustic resonance. Similar cell construction was usedcently for PA detection of aerosols.65

D. Differential photoacoustic detector

The differential PA detector was specifically designfor fast time response, low acoustic and electric noise chacteristics, and high sensitivity.17 In order to reduce the flownoise and external electromagnetic disturbances a fully smetrical design was developed. Two acoustic resona~5.5 mm diam tubes! were placed between two band-sto

FIG. 10. Differential PA cell with two resonator tubes, buffer volumes, al/4 filters. Two resonant acoustic pipes~5.5 mm diam tubes! were placedbetween two acoustic filters. The two similar microphones were placethe middle of each tube. The signal is amplified by a differential amplifi


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acoustic filters as shown in Fig. 10. The gas flow pasthrough both tubes, producing about the same flow noisboth resonators. However, the laser light irradiates onlyof them, thus the PA signal is generated in only one tuTwo microphones~Knowles EK 3029! are placed at themiddle of each tube flush with the wall and their signalamplified by a differential amplifier. The microphones havso-called ‘‘ski-slope’’ frequency characteristic, with resposivity decreasing rapidly below 2 kHz. Thus, they are nsensitive to the dominant low-frequency part of the enviromental noise. The selected pair of microphones has the sresponsivity around the acoustic resonance frequency otubes of 4 kHz. Thus all noise components that are cohein the two tubes are effectively suppressed. The microphoare isolated from the housing of the detector, which is cnected electrically to the chassis of the amplifier throughscreening of the cable. Three wires, including the commsignal ground, connect the microphones to the amplifiThus, ground loops are completely avoided. Figure 10 psents a schematic of this optimized differential cell.

In order to keep the flow noise at a sufficiently low levethe flow must be in the laminar regime. Another practicrequirement is the time response, which is determined byexchange rate of the gas sample in the resonant cell. Indition, a delay occurs because the gas flows from the inlethe resonant cell first through the acoustic filter. In the ctinuous flow mode a response time of,1 s and a delay of,10 s are possible. Taking into account the largest dimsion and the limiting value of the Reynolds number~Re<2300 for laminar flow! the flow velocity should not excee1.7 m/s. This value is far too large, since it would give a florate of about 4.8 l/min~large gas consumption!. As an oper-ating value a flow rate of up to 0.5 l/min has been used. Wthis value the maximum flow velocity was about 15 cm/s,Reynolds number Re,200, and the calculated response timt,0.7 s. The response time determined from the rise timthe PA signal is larger because of mixing of the gas inbuffer volumes of the cell. Nevertheless, the measuredsponse times are below 10 s for nonadsorbing gases. Adstion at the walls of the PA cell may increase the respotime significantly. The crucial problem of adsorption effecin photoacoustic detection is discussed in deelsewhere.66–68 Note that the adsorption effect can be effetively reduced by using appropriate wall materials andevated wall temperatures.68

The photoacoustic sensitivity of the differential cellabout aminWL51.031029 W cm21. The actual sensitivitydepends on the instrumentation applied and the data evation method. The above value is the lower limit of the Psensitivities of the differential cell obtained in measuremewith methane, ethylene, and ammonia using a CO2 laser,diode lasers, and an external cavity diode laser system.sensitivity of the PA cell applied in the intracavity measument in Ref. 11 wasaminWL51.831028 W cm21, muchlower than that of the differential cell. Thus, the parts-ptrillion concentration sensitivity reported for ethylene wachieved mainly by the high intracavity power and the stroabsorption of ethylene and not by an extremely sensitive cIt is worth mentioning that the dynamic range of the P


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method is considerably reduced by intracavity operation. Otical saturation may occur for molecules with high absotion cross section11 while uncontrollable signal changes mabe obtained at higher overall absorption in the PA cell, bcause the loss of light intensity influences the gain oflaser. This effect may cause erroneous results whensample concentration changes are large. Therefore, highsitivity single- and multipass extracavity PA detectors offesimpler alternative to intracavity devices.

E. Novel tunable light sources

While a state-of-the-art photoacoustic detector withelectret microphone and suitable electronics is relativcheap, high power laser sources, such as CO2 lasers, cur-rently needed to reach the sub-ppbV range, make the setutoo expensive and bulky for many practical applications. Tmore versatile single mode distributed feedback~DFB! ortunable ECDL diode lasers have, unfortunately, the disvantage of low power and narrow spectral range, andrestriction that room temperature diodes emit in the Nonly. Thus, the availability of suitable laser sources playfundamental role for the further development of the fieThey control the sensitivity~laser power!, the selectivity~tuning range!, and the practicability~ease of use, size, cosand reliability! that can be achieved with the photoacoustechnique.

Compared to the previous situation in which the specto be detected were selected according to accidental codence of their resonances with available laser lines, the sation is changing gradually to the point where the speciebe detected can be chosen according to scientific or pracinterest and a suitable laser source with enough powerbeam quality can be found for sensitive detection. Tunasolid-state laser devices especially may soon contributefurther development of the field. Since the power and tunrange of the currently available diode-laser sources, whare tunable in the fundamental IR spectral region~quantumcascade laser!, are still limited, it is very important to choosthe optimal design for the PA detector to reach state-of-tart performance. The recent realization and further improment of PPLN-OPO devices may provide an opportunityrealize widely tunable pulsed and modulated infrared radtion with high pulse energy or power. These devices willmore expensive but may find a niche, where high perfmance is needed.

ACKNOWLEDGMENTS

Financial support of this work by the bilateral GermanHungarian scientific cooperation project~HUN 97/016! andthe Fonds der Chemischen Industrie is gratefully acknoedged.

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