new purely thermal-wave photopyroelectric interferometry · 2016. 4. 28. · purely thermal-wave...

JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 12 15 JUNE 1999

Purely thermal-wave photopyroelectric interferometryChinhua Wang and Andreas Mandelisa)Photothermal and Optoelectronic Diagnostics Laboratories (PODL), Department of Mechanical andIndustrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, M5S 3G8 Canada

~Received 22 October 1998; accepted for publication 16 February 1999!

A thermal-wave interferometric technique based on pure thermal-wave interference usingtwo-cavity photopyroelectric~PPE! detection has been developed. The salient features of thisinterferometry are investigated with a general theory, which describes the dependence of the PPEsignal on the optical, thermal, and geometric parameters of the thermal-wave cavity configuration.Preliminary experimental results are presented. A major feature of the technique is the efficientsuppression of the background PPE signal base line. The physical origins of the interferometry arealso discussed. ©1999 American Institute of Physics.@S0021-8979~99!02811-X#

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I. INTRODUCTION

The single-beam photopyroelectric~PPE! technique is awell-established photothermal method used for spectroscand thermal characterization of various materials as welfor studies of thermophysical properties of gases.1–9 The ba-sic principle of the single-beam PPE technique is that wheperiodically modulated energy source impinges on the sface of a sample, the sample will absorb some of the incidenergy and will, in turn, produce a localized temperatincrease following a nonradiative deexcitation process. Tperiodic temperature variation in the sample can be detewith a pyroelectric transducer made of a thin-film pyroeletric [email protected]., polyvinylidene fluoride~PVDF!#. The PPEsignal from the pyroelectric transducer is due totemperature-dependent change in polarization of the pelectric material.1,2 If the sample does not contact the tranducer, the air gap formed in the medial region amounts tthermal-wave cavity. This cavity confines the thermal-waoscillation and has been shown to give rise to a standwave-equivalent structure, the length of which can be tuto resonant antinode and node patterns.5 Recently, suchthermal-wave resonant cavities have been used for highcision measurements of thermophysical properties of thetracavity gases.5,8,9,10

In all the conventional embodiments of the PPtechnique,1,4,7,11a single excitation source is employed withe radiation impinging on either the front-~PPE!, or therear-surface@inverse PPE~IPPE!# of a PVDF transducer. Thesensitivity and dynamic range, however, of the PPE msurement scheme can be compromised in the study of stransparent materials, due to the substantial base line sfrom direct transmission of the incident light onto the detetor. In addition, in the case of IPPE, the restriction imposby the requirement for thermally thin PVDF film and fosample-detector contact further compromises the dynarange of photopyroelectric sensor devices. Some earlierforts were made to lower the instrumental base line of P

a!Electronic mail: [email protected]

8360021-8979/99/85(12)/8366/12/$15.00

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detection systems: differential front-surface polarizatimodulation12 and out-of-phase sample-surface modulatio6

PPE schemes involve the formation of destructive thermwave interference inside the body of a solid sample. Thhave been used with considerable enhancement of the sigto-noise ratio owing to base line suppression. Neverthelthe specific thin-surface-film preparation, vacuum enviroment, and optical excitation wavelength requirementsposed on the former technique cannot be met by a lanumber of geometries and materials, which do not exhdiscrete polarization states. Furthermore, bulk thermal inmogeneities and the strong thermal-wave dispersion insidsolid, photothermally excited by out-of-phase surface molation of two laser beams clearly limit the dynamic rangethe latter technique, which usually results in an incomplbase line suppression, exacerbated with increasing samthickness.

In this work, the theory and preliminary experimentmethodology of a thermal-wave interferometric techniquedeveloped. The technique is based on the spatial interfereof thermal waveswithin the body of the pyroelectric transducer, independently of the sample. It has been developeorder to overcome the shortcomings of the earlier interfemetric methodologies and may be regarded as the evoluof the method presented in Ref. 6, in that it creates outphase destructive interference patterns between higaligned thermal waves in the transducer itself, resultingcomplete base line suppression. Unlike other prior~‘‘con-ventional’’! photothermal interferometric schemes,12–15 thenew technique isnot based on monitoring thermal waveresulting from direct optical interference patterns, suchthose generated by two appropriately modulated laser be~e.g., intensity, phase or polarization modulation!. In thepresent coherence scheme, thermal waves are inducetwo intensity-modulated beams, split off a single laser souand with a fixed phase shift relationship between them. Tusually large instrumental PPE base line signal and a sigcant portion of the noise can be efficiently suppressed witthe PVDF detector if the two laser beams are collineaincident on opposite surfaces of the thin pyroelectric fil

6 © 1999 American Institute of Physics

license or copyright; see http://jap.aip.org/jap/copyright.jsp

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8367J. Appl. Phys., Vol. 85, No. 12, 15 June 1999 C. Wang and A. Mandelis

and with 180° relative phase shift. In this fashion, muhigher signal sensitivity and dynamic range PPE measments than with the conventional single-beam PPE confirations are expected and have been confirmed very recein this laboratory. In order to demonstrate the physical prciple of this technique, in view of our preliminary expermental results, a general theory is presented, which descthe dependence of the PPE signal from a thin-film PVdetector on the optical, thermal, and geometric parametethe dual thermal-wave-cavity configuration, formed by tnoncontacting sample, the PVDF and the reference soSeveral special cases of the theory corresponding to varpossible experimental configurations are also presentedits advantages over the single-beam noninterferometricmethod are discussed.

II. ONE-DIMENSIONAL GENERAL MODEL OFPURELY THERMAL-WAVE PPE INTERFEROMETRY

The most general configuration diagram for purethermal-wave PPE interferometry using a one-dimensioheat transfer model is shown in Fig. 1. Two laser beamsintensitiesI 1 and I 2 , respectively, are split off of a lasesource and are modulated at the same angular frequency~v!.They have a fixed, adjustable phase shift~Dw!, and are inci-dent onto the front and rear surfaces of a PVDF detecpassing through noncontacting optically transparent samand reference media, which, along with the PVDF sensothe middle, form the thermal-wave cavitiesg2 and g3 asshown in Fig. 1. The incident beams are assumed to illunate the PVDF sensor uniformly with spot sizes much larthan the thermal diffusion length in PVDF, so that the ondimensionality of the heat transfer model is assured. Tthickness of the sample, the reference, the PVDF deteand the lengths of cavitiesg2,g3 are l , m, d, L and L1 ,respectively. The sample and the reference have opticasorption coefficientbs , b r , respectively. Light absorptionby the sample—PVDF transducer—reference systemnonradiative energy conversion to heat increases theperature of the PVDF sensor, which results in a potendifference between the two surfaces of the transducer duthe photopyroelectric effect. The photopyroelectric sigfrom the PVDF detector is proportional to the averagetemperature of the PVDF film detector.7 It is governed bycoupled one-dimensional heat diffusion equations subjecappropriate boundary conditions of thermal-wave field aflux continuity across each interface (g1-S, S-g2, g2-P,

FIG. 1. General PPE geometry for purely thermal-wave interferometr


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P-g3, g3-R, andR-g4) of Fig. 1. In the theoretical treatment of the most general case, gaseous media of diffethermophysical properties are assumed to exist in regionsg1,g2, g3, andg4. Assuming negligible radiative heat transfto the sensor from the heated sample and reference, ecially in the case of transparent materials,9 the appropriatethermal-wave equations have the form

d2Ti~x!

dx22s i

2Ti~x!50,

i 51,2,3,4 for regionsg1, g2, g3 and g4, ~1a!

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8368 J. Appl. Phys., Vol. 85, No. 12, 15 June 1999 C. Wang and A. Mandelis

coupled boundary-value problems. In particular, the heatboundary conditions at the front and rear surfaces ofPVDF sensor can be written as

kgdT2~x!

dx2kp

dTp~x!

dx5~12Rp!I 1

~12Rs!2

12Rs2e22bsl

e2bsl ,

x5 l 1L, ~3a!

kpdTp~x!

dx2kg

dT3~x!

dx5~12Rp!I 2e

j Dw

3~12Rr !

2

12Rr2e22brm

e2brm,

x5 l 1L1d, ~3b!

where the right-hand sides of Eqs.~3a! and~3b! represent theincident beam intensities at the front and rear surfaces ofPVDF thin film, respectively, with due consideration of th

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infinite set of reflections of the incident beam within thsample and the reference media.3 The PPE signal from thePVDF detector, assuming a short thermal time constant cpared to the inverse of the modulation frequency, canwritten as7

V~v!5S~v!El 1L

l 1L1d

Tp~x,v!dx. ~4!

HereS(v) is the instrumental transfer function, a frequencdependent factor related to the associated PVDF-signalplification and processing electronics.16 It is usually normal-ized out by means of frequency scans of samples with wknown frequency responses.Tp(x,v) is the local complexthermal-wave field in the PVDF, which must be calculatfrom the above coupled boundary-value problems.

After a considerable amount of algebraic manipulatiothe output photopyroelectric signal~voltage mode! for thegeometry of Fig. 1 can be finally expressed as follows:

V~v!5S~v!

sp~11b2p!~11b3p!

3H1~11b3p!@G1~11W21e

22s2L!12b2pG3e2s2L#1H2~11b2p!@G2~11V34e

22s3L1!12b3pG4e2s3L1#

@espd~11g3pV34e22s3L1!~11g2pW21e

22s2L)2e2spd~g2p1W21e22s2L!~g3p1V34e

22s3L1!#, ~5!

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where

bi j 5kiAa j /kjAa i , ~6a!

g i j 5~12bi j !/~11bi j !, ~6b!

W2152g2se

ssl2g1se2ssl

essl2g2sg1se2ssl

, ~6c!

V3452g3re

srm2g4re2srm

esrm2g3rg4re2srm

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G15~12Rp!

kpsp3

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G25~12Rp!

kpsp3

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2

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e2brm. ~6f!

Expressions forH1 , H2 andG3 , G4 are given in the Appen-dix. From the structure of the PPE output voltage of Eq.~5!,it can be seen that the first term in the numerator represthe contribution from the front beam, through the sample athe intracavity gas layerg2. The second term originates fromthe rear incident beam, through the reference layer andintracavity gas layerg3. Therefore, the overall output signis the result of the complex~vectorial! superposition of two

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thermal wave fields within the PVDF detector generatedtwo incident laser beams of equal fluences modulated wifixed phase shiftDw.

III. SPECIAL CASES

Equation ~5! explicitly demonstrates that the generPPE output signal from the system shown in Fig. 1 isfunction of the solid sample and the reference, as well asconstituents of the two thermal-wave cavities formed btween the PVDF film and the two solids on either side. Tcomplicated dependence of the signal on the parameterthis system, however, makes any specific physical insinto Eq. ~5! very difficult in the general case. Therefore,this work it is instructive to consider several special casestheoretical and practical importance, according to sevepossible configurations among the incident beams,sample and the reference.

A. Single laser beam; no reference medium;one „front … thermal-wave cavity

If the rear laser beam and the reference medium aresent, or if the rear beam is absent and the reference solplaced far away from the PVDF compared to the thermdiffusion length in the gaseous cavityg3, then in terms of theforegoing model we must setI 250 and eitherm50 ~in theabsence of a reference solid!, or e2s3L1→0 ~if the referenceis placed far away from the detector!. In both cases Eq.~5!reduces to the following simplified expression:



V~v!5S~v!

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22s2L!12b2pG3e2s2L!]

espd~11g2pW21e22s2L!2g3pe

2spd~g2p1W21e22s2L!

. ~7!

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Equation~7! gives the general expression correspondingthe single-beam PPE technique.3 In the present case, however, the finite thickness of the PVDF sensor is accountIn the earlier theoretical treatment3 the PVDF was assumeto be semi-infinite. Figure 2 shows the effect of the finthickness of the PVDF element on the output signal fofixed cavity length,L5200mm ~see Fig. 1!. It was assumedthat the transparent sample is a Ti-sapphire laser crystalwith a thickness of 0.1295 cm3 and the gaseous media oboth sides of the PVDF detector are the same~air!. Theparameters which were used in Eq.~7! for the simulationare: kp51.3310

23 W/cm K, ap55.431024 cm2/s;15 ks

50.33 W/cm K, as50.106 cm2/s;3 k25k35kair52.62

31024 W/cm K, a25a35aair50.22 cm2/s;5 modulation

frequencyf 510 Hz. It can be seen from Fig. 2 that the PPoutput signal depends on the thickness of the PVDF detein the thermally thin range.7 With increased thickness, ththermally thick condition prevails:7 the amplitude of the out-put signal exhibits the expected oscillatory behavior withglobal maximum at thickness;40mm, before reaching aconstant~thermally thick! value; and the phase decreasmonotonically before assuming a constant~thermally thick!value. Figure 2 shows that for the foregoing parameters,PVDF film may be assumed ‘‘semi-infinite’’ if its thicknesis greater than;200mm. Figure 3 shows the output sign~amplitude and phase! as a function of the front cavitylength, L, formed between a Ti-sapphire sample and

FIG. 2. Effect of the PVDF detector thickness on the output PPE signal~a!amplitude;~b! phase.


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PVDF detector, with the sensor thickness as a parameThe output PPE signal amplitude peak position shows soshift to longer cavity lengths with increasing PVDF thicness, while the overall phase lag decreases. This is physias expected, because thinner transducers confine the comite thermal-wave centroid closer to the far wall of the cav~origin!, as witnessed by the amplitude shift toward the ogin, and the decreased phase lag. Therefore, for quantitaanalysis it is of practical importance to take into accountthickness of the pyroelectric transducer for PVDF films thner than ;200mm, as is usually the case with PPmeasurements.1–10 If the thickness of the PVDF is assumeto be semi-infinite, Eq.~7! reduces, as expected, to thsingle-beam expression given as Eq.~11! in Ref. 3.

B. Two laser beams; no sample or reference;no thermal-wave cavity

If both sample and reference are absent@or both sampleand reference are placed in the optical transmission~OT!mode (L,L1→`)#, the following limiting values must be sein Fig. 1: l 50, m50, ~or e2s2L→0, e2s3L1→0), andRs50, Rr50. In both cases, Eq.~5! is simplified to

FIG. 3. The output PPE signal vs cavity lengthL, using PVDF detectors ofdifferent thickness;~a! amplitude,~b! phase. The parameters used are tsame as those of Fig. 2.


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V~v!5S~v!~12Rp!

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I 1~11b3p!j11I 2ej Dw~11b2p!j2

~11b2p!~11b3p!~espd2g2pg3pe

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where

j1[espd211g3p2g3pe

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and

j2[espd211g2p2g2pe

2spd.

If the experimental system is exposed to the same ambgas medium~usually, air! on both sides of the transducethen subscripts 2535g in Eq. ~8a!, which further reduces to

V~v!5S~v!~12Rp!

kpsp2~11bgp!

•

@ I 11I 2ej Dw#~espd211ggp2ggpe

2spd!

espd2ggp2 e2spd

. ~8b!

This is the simplest case of coherent interference of two thmal waves of different amplitudes, which are generatedrectly within the PVDF detector by two oppositely incidelaser beams of arbitrary fluence and arbitrary phase sEquation~8b! shows that constructive or destructive interfeence patterns will appear when the relative phase shiftDw50° or 180°, respectively. Figure 4 shows the amplituand phase of the interference-generated PPE signal, Eq.~8b!,as functions of the phase shiftDw at different fluence ratiosof the two laser beams. The parameters used in the simtion are:I 151.0 W/m

2, d552mm. Other parameters are thsame as those used in Fig. 2. It is observed in Fig. 4 thatfully destructive interference pattern occurs in the amplituchannel whenI 2 /I 151 and only for odd integral multiplesof 180°, curve b, while the phase increases linearly wincreasing phase shift. A large constant base line PPE siappears in the case ofI 2 /I 150 ~i.e., the single-beam case!,curve a, independent ofDw, characteristic of complete absence of thermal-wave interference, as expected. Compathe two foregoing patterns, the advantage of the situadepicted by curve b at the nodes becomes immediatelyparent in terms of the large coherent base line signal suppsion, theoretically to zero level. In this mode, the thermwave interferometer acts as a sensitive detector of puoptical radiation variations between the intensities of the tincident beams. In practice, one is left with the residualperimental system noise under full destructive interferenexperimentally, we found that if we use a 5 mWlaser beammodulated at 10 Hz, we obtain an output PPE signalV0;3 mV by blocking the rear incident beam. Under the saexperimental conditions, we can easily obtain a miniminterference output signalVi;1 mV for fully destructive in-terference. This represents a figure-of-merit (FOM5V0 /Vi)of the PPE thermal-wave interferometer of;3000. Underthese conditions, a very small variation in output signalthe order of 1mV caused by an intensity change of the fro


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laser beam as a result of an optical or photothermal fluction can be readily sensed by a fully destructive thermwave interferometer with practical resolution of 0.1mV.17

By contrast, the same variation in the output signal is impsible to detect with the single-beam method with practiresolution of 0.01 mV. It should be noted that conventionoptical detectors such as photodiodes, cannot exhibit shigh FOM, because the protective transparent overlayer aor the nonnormal incidence of two laser beams on the acarea~front surface! of the detector make it harder to producprecise spatial symmetries, coincidence and equal spot sTherefore, optical detectors are capable of only partialence~and thus base line! cancellation, as equivalently showby the intermediate case of partially destructive interferein Fig. 4~a!, curve c. This is not a problem using the simpnormal incidence arrangement from both sides of the Pinterferometer depicted in Fig. 1. Thus, unlike conventionoptical detectors, the PPE fully destructive interferomehas the potential to be an optimum FOM instrument for otical beam power/fluence characterization.

C. Two laser beams; no reference medium;single thermal-wave cavity

If the reference medium is absent~or if it is fixed at avery large distance compared to the thermal diffusion len

FIG. 4. Interferometric PPE signal vs phase shift between two incidbeams at three different fluence ratios~a! I 2 /I 150 (I 250), ~b! I 2 /I 151.0, ~c! I 2 /I 152.0. The off-line phase points in curve~b! at Dw5180°and 540° are due to the 0/0 operation~destructive interference!.


ame


in the gas!, one must setm50, ~or e2s3L1→0) in Eq. ~5!. Assuming that the experimental setup is exposed to the sambient gas~air! on both sides, the subscripts 15253545g, result in the following simplified expression:

V~v!5S~v!

sp~11bgp!•

~espd211ggp2ggpe2spd!@G1~11W21e

22sgL!12bgpG3e2sgL#1G28H28

espd~11ggpW21e22sgL!2ggpe

2spd~ggp1W21e22sgL!

, ~9!

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where G1 is given by Eq.~6e!, G28[(12Rp)/kpsp I 2ej Dw

and

H28[~espd21!~11ggpW21e

22sgL!

1~12e2spd!~ggp1W21e22sgL!. ~10!

It is easy to see that the difference between this case ancase discussed in Sec. III A is that the interference te(G28H28) appears in Eq.~9!. This is an experimentally simpleand useful case, because only the properties and the georic parameters of the sample are involved. This situationencountered in the study of optically transparent sampleswhich a large base line signal from the direct optical tramission of the incident beam passing through the samplusually the dominant factor limiting the sensitivity of thPPE measurement.18 Fully destructive thermal-wave interferometry in the single-cavity mode can suppress the transsion base line and enhance the thermal component oftotal PPE signal compared to case III B, where the interometer in the absence of a sample can only detect op~fluence! variations between the two incident laser beams

A simple application of this PPE interferometric detetion mode in the scanning imaging of a transparent optmaterial using the experimental setup illustrated in Fig. 5shown in Fig. 6. Two laser beams, which are split from olaser and modulated at the same frequency, are incidentthe front and rear surfaces of a PVDF detector. The relaintensities of the two beams can be adjusted by a lineartensity attenuator~neutral density filter!, and the phase shifbetween the two beams is precisely controlled by adjusan optical chopper. The reference is placed onto a single-micro stage for the purpose of scanning the cavity (g3)length between the reference and the PVDF element.sample is mounted onto a three-axes micro stage, whichlows for changing of the cavity (g2) length and the scanninimaging of the sample. A PVDF circular film, 52mm thickand 1.2 cm in diameter, is mounted on an aluminum supwith a hole in its base and acts as the PPE signal transdu

FIG. 5. Experimental setup for PPE thermal-wave interferometry.sample, P: PVDF detector; R: reference.


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The thermal-wave signal from the PVDF is then fed intolock-in amplifier ~EG&G model 5210! without further am-plification.

The PPE image shown in Fig. 6 is that of a Ti-sapphlaser disk with a diameter of 1.5 cm and thickness of 0.12cm. The experiment was conducted in the purely optimode, with the Ti-sapphire disk having been placed far frthe PVDF [email protected]., L→` in Eq. ~9!#. Thus, a verysimple formula only involving the optical properties of thsample can be obtained

V~v!5S~v!~12Rp!

kpsp2~11bgp!

•

~espd211ggp2ggpe2spd!

espd2ggp2 e2spd

3F I 1 ~12Rs!2e2bsl12Rs2e22bsl 1I 2ej DwG . ~11!

:FIG. 6. Comparison of PPE scanned images of a Ti-sapphire laser rod u~a! the single-beam method, case A,~b! the optical-beam PPE thermal-wavinterferometer, case C.


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Once the PVDF detector and the measuring system aresen, the first two ratios in Eq.~11! become fixed constants. Ithe phase shiftDw5180° and the relative fluences of the twbeams (I 1 ,I 2) are adjusted such that the output signal is zat the onset of the scanning, this effectively zeros the termbrackets in Eq.~11!. In principle, the ratioI 1 /I 2 resultingfrom this zeroing operation yields an accurate value ofoptical absorption coefficientbs of the transparent materiaif its surface reflectanceRs is known or remains constanthroughout the scan. After the single-point zeroing corrtion, any optical defects and/or surface or bulk nonuniformties will introduce changes inRs and/orbs in Eq. ~11!, whichcan be sensitively monitored as a signal departure fromnoise-limited base line. Moreover, another advantage ofscanning imaging technique is that the noise componentto intensity fluctuations or drift of the common laser-beasource is also coherently suppressed. Figure 6 showsPPE images of a 636 mm2 area at the center of the lasdisk. Both the single-beam method and the thermal-winterferometric method yield the same basic image of tarea. In the region labeledx54 – 6 mm, both images showhigher transmission but less uniformity than the regionbeled x51 – 4 mm. The highest optical uniformity of thimaterial is located in the region bounded betweenx51 – 4 mm andy51 – 4 mm, as deduced from the interfermetric image, Fig. 6~b!. The single-beam image, Fig. 6~a!,shows almost perfect flatness in this region caused by lacdynamic range, a result of the dominating optical transmsion signal base line. The thermal-wave interferometricage, however, shows the surface morphology amplified bfactor of approx. 300. The most significant difference btween the two images occurs in the region bounded bx51 – 3 mm andy54 – 6 mm. The thermal-wave-interferencimage shows a gradual and continuous increase in the amtude from ;0 – 12mV with the practical resolution of;1 mV, while the single-beam image only exhibits threlevels ~labeled 0, 1, and 2! limited by lack of signal quanti-zation. In conclusion, Fig. 6 shows that optical imaging wPPE interferometric thermal waves is superior to singbeam imaging with respect to sensitivity, resolution and snal dynamic range. Therefore, this technique is a promisnew tool for optical scanning imaging.

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D. Two laser beams; two thermal-wave cavities

In view of the structure of Eq.~5!, it can be seen that thefirst term in the numerator is due to the front incident beawith the G1 term representing the contribution from the drect transmission of the incident light and theG3 term beingthe thermal contribution due to the optical-to-thermal coversion process in the sample. The same situation ocwith the rear beam and the reference sample: hereG2 andG4represent the direct transmission of the rear beam andoptical-to-thermal conversion process at the referesample, respectively. If both cavity walls are opaque,G1 andG2 are equal to zero, with only the contributions fromG3andG4 appearing in Eq.~5!. By contrast, if both cavity wallsare highly transparent, only contributions fromG1 and G2have to be taken into account due to the comparatively nligible contributions from theG3 and G4 terms. Therefore,two different operational mechanisms and configuratiowith either transparent or opaque walls can be employedwill be discussed separately.

1. Highly transparent cavity walls;spectroscopic mode

Due to the very weak absorption coefficient of thsample and the reference, the thermal contributions tooverall PPE signal resulting from the optical-to-thermal coversion processes after the optical absorption are very smTheoretical simulations of Eq.~5! show that the contribu-tions from the direct transmissions of the front and rear beonto the PVDF detector (G1 andG2 terms! are 3–4 orders ofmagnitude greater than contributions from the opticalsorption processes (G3 andG4 terms! for all cavity lengths.Therefore, the overall PPE signal can be simplified byglecting G3 and G4 terms in Eq.~5!. Owing to this fact, amajor application corresponding to this case can be expein the measurements of the optical properties of the samwhich are included in theG1 term if the reference is weldefined. For this application, the entire measurement sysis assumed to be exposed to the same gaseous ambiepractice. By setting subscripts 15253545g in Eq. ~5!, theoverall PPE signal reduces to

V~v!5S~v!

sp~11bgp!

H1G1~11W21e22sgL!1H2G2~11V34e

22sgL1!

espd~11ggpV34e22sgL1!~11ggpW21e

22sgL)2e2spd~ggp1W21e22sgL!~ggp1V34e

22sgL1!. ~12!

aretheionting

he

onhe

The symbols used in Eq.~12! have the same meaningsthose in Eq.~5!. As expected, if both sample and referenare placed far away from the PVDF detector, Eq.~12! re-duces to case III B above, withI 1 and I 2 being replaced byG1 and G2 , respectively. There is no advantage of tpresent geometry over that of case III B in the purely optimode. In fact, the simplicity of that configuration, withothe complicating presence of the reference sample, musnoted. The advantage of the double thermal-wave cavity

l

bep-

pears in the near field, where both sample and referenceplaced in thermal contact with the PVDF detector, i.e.,cavity lengths have to be smaller than one thermal diffuslength in the gas. Figure 7 shows the PPE signal resulfrom scanning the sample cavity lengthL from the purelyoptical mode to the thermally contacting mode, while treference cavity length is fixed at a thermally contactingL150.02 cm. To simulate the effect of the optical absorpticoefficient (bs) of the sample on the PPE output signal, t


sarw

uanti

eris-t

onc

owmmtie

n

ce,itydenel.

r-

is-isop-

on

in-ingof

icald, if

re-f thethetheare

aa

re-ity

r a

th

r at

f


sample and the reference were assumed to have thethickness and thermal properties but different optical absotion coefficients. Here parameters of Schott BK7 windoglass19 are used: l 5m50.3 cm, Rs5Rr50.07, ks5kr51.11431022 W/cm k, as5a r55.173310

23 cm2/s, andb r50.04 cm

21. The modulation frequency is fixed atf526.6 Hz and the entire system operates in open air. Fig7 shows that the theoretical outputs in both amplitudephase channels are very sensitive to the change in the opabsorption coefficient of the sample (bs). As expected fromsymmetry, the output in amplitude has a sharp dip to zwhen scanning the sample-PVDF cavity length to the dtanceL5L1 in the case ofbs5b r . Under those same conditions, the phase being the ratio of the quadrature andin-phase signal~both zero! becomes undetermined@a spikein Fig. 7~b!#; compare curve~b! in Fig. 4~b!. If bs is slightlydifferent from b r , the amplitude minimum is not equal tzero, and its position depends sensitively on the differebetween the values of these two parameters@Fig. 7~a!#. In theinsets of Fig. 7, experimental results are shown to demstrate the capability of the method. In the experiment, t‘‘identical’’ commercial Schott BK7 window glasses, 0.3 cthick were used. The experimental conditions were the saas those assumed in the theoretical simulation. The relaintensities and the phase shift between the two incidbeams were adjusted, such that the output equals zero i

FIG. 7. Theoretical behavior of a differential PPE interferometric sensovarious cavity lengthsL and f 510 Hz. ~a! Amplitude, ~b! phase. Insets:experimental results from Schott BK7 glass sample and reference, andoretical fits.


mep-

redcal

o-

he

e

n-o

eventthe

absence of both the sample and the reference:I 15I 2 andDw5180°. Then the reference and sample were put in plain thermal contact with the PVDF, and the sample cavlengthL was scanned. Fitting experimental data to amplituis preferable, due to the sharp minimum in this PPE chanBy fitting the amplitude of the PPE signal to Eq.~12!, adifference in optical absorption coefficient ofDb5(b r2bs)59.7310

23 cm21 between the sample and the refeence was found. Then, the PPE phase was fitted using theDbvalue obtained from the amplitude fit to ascertain constency and validity. Numerical fitting further showed that threlative measurement is independent of the value of thetical absorption coefficient of the reference (b r), which mustbe used in the fitting, in a wide range of optical absorpticoefficients from 1023 to 1021 cm21, which is the case formost optical glasses. Therefore, thermal-wave destructiveterferometry provides a very useful method for measurvery small differences in the optical absorption coefficienttwo samples quantitatively. The absolute value of the optabsorption coefficient of the sample can also be measureb r is known.

To optimize the measurement, different modulation fquencies can be considered. Figure 8 shows the effect omodulation frequency on the PPE output signal, in whichcavity length at the reference side is fixed at 0.02 cm, andabsorption coefficients of the sample and the referenceassumed to beb r50.04 andbs50.03 cm

21, respectively.Figure 8~a! shows that higher modulation frequency givesgreater positional shift from the reference position andgreater deviation of the minimum output from zero. Thefore, higher modulation frequency provides higher sensitiv

t

e-

FIG. 8. Theoretical behavior of a destructive interferometric PPE sensothree modulation frequenciesf 510, 40, and 90 Hz.~a! Amplitude, ~b!phase. The number pairs shown in~a! indicate the coordinate positions othe minimum points for each curve.


vec

ro

r

ano-

he

igthgte

isav

oipinqvnth

d

d

ua

trin

sc

fi-he

raxe

i-

twond-

aslinetruenal

theste at

is aityrnsThee to

en-

r-de-thslthe

s be-eddif-y at

sthe

om

st

theaveen-gede-thectthesingvityr isn of


of the measurement in the amplitude channel. Howehigher frequency will produce a smaller output signal, henlower measurement signal-to-noise ratio, and will also pduce smaller phase variation@Fig. 8~b!#. As a result, an op-timal trade-off operation frequency should be chosen fogiven experiment.

2. Opaque cavity walls; thermophysical mode

In the case of entirely opaque walls, no direct optictransmission of the incident light is possible. The PPE sigis due to interfering thermal waves originating in optical-tthermal power conversion of the saturated7 optical absorp-tion of the incident light on the exterior surfaces of tsample and the reference. These thermal waves are~mainly!conductively transmitted to the PVDF. The demodulated snal should be zero if the gases in the two cavities aresame, the opaque sample and reference are identicalmetrically, and thermally, and the two beams are adjussuch thatI 15I 2 , with Dw5180° ~fully destructive interfer-ence!. If the gaseous ambient in one of the cavitieschanged, the symmetry between the two thermal wave cties is broken, and thus a net output signal is expectedappear. Therefore, a highly sensitive differential thermphysical gas sensor can be designed based on this princFor the purposes of designing this sensor, the foregotheory of the PPE interferometer suggests that totally opamaterials must be used for both sample and reference cawalls to reduce the complexity of the theoretical analysis aexperimental operation. It can be further assumed thatPVDF is thermally thick, the gas ambient in regionsg1, g2,andg4 is the same in the simplest case, while the gas uninvestigation is confined in regiong3 ~Fig. 1!. Under theseconditions, the following values are inserted in Eq.~5!: sub-scripts 15452 (Þ3) andr 5s; totally opaque sample anreference cavity ‘‘walls’’ givee2bsl→0, e2brm→0, whichleads toG150 andG250. Equation~5! reduces to

V~v!52S~v!e2ssl

kpsp2~11b2s!

•

1

12g2pg2se22s2L

•

1

12g3pg3se22s3L1

3F I 1b2s~12g3pg3se22s3L1!e2s2L~11b2p!~11b2s!~12g2s2 e22ssl !1

I 2ej Dwb3s~12g2pg2se

22s2L!e2s3L1

~11b3p!~11b3s!~12g2sg3se22ssl !G . ~13!

The output signal is a complex function of the thermal copling coefficients between the intracavity gases and the city walls. However, once the cavity walls and the geomeparameters of the system are fixed, the signal is only a fution of the intracavity gases, the thermophysical propertiewhich are included very sensitively in the exponential funtions in Eq.~13!, and less so in the thermal coupling coefcientsbi j andg i j . Figure 9 shows the characteristics of toutput signal versus the cavity lengthL on the sample sidefor several thermal diffusivities of the test gas in that intcavity space. The cavity length at the reference side is fi


r,e-

a

lal

-e

eo-d

i-to-le.g

ueityde

er

-v-cc-of-

-d

at L150.03 cm. In the simulation, the parameters of alumnum foil walls were used: as50.98 cm

2/s, ks52.37 W/cm K,20 l 515mm; regionsg1, g2 andg4 are ex-posed to air ~i.e., a250.22 cm

2/s, k252.6231024

W/cm K), and I 15I 2 , Dw5180°. The modulation fre-quency is 30 Hz. As can be seen, if the gases in thecavities are different, the amplitude minimum line shape aits position vary sensitively with the relative thermal diffusivities of the two gases: lower diffusivity of the test gresults in a sharper and deeper amplitude minimumshape positioned closer to the detector. The opposite isfor higher test-gas diffusivity. As expected, the output sigdecreases to zero when the scanned cavity length onsample side is equal toL1 when the gases in the two cavitieare identical, and the phase also becomes indeterminathe point ofL5L1 @see also Figs 4~b!, curve~b!, and 7~b!#.The signal phase past the indeterminate point, however,very sensitive function of the test-gas thermal diffusiv@Fig. 9~b!#, and switches from lead to lag as the test gas tufrom a poorer to a better conductor than the reference.dual-cavity gas sensor is expected to be more sensitivtest-gas thermophysical properties~diffusivity and conduc-tivity ! than its single-cavity counterpart,5 owing to the sup-pression of the thermal base line and the real-time differtial measurement effected by this novel cavity design.

A quantitative way to monitor minute changes in themophysical properties of the intracavity gases can bevised. In this measurement method, the two cavity lengare fixed at the same value, i.e.,L5L1 , and the output signais calibrated to zero when the two cavities are exposed tosame gaseous medium. Thereafter, any relative changetween the two intracavity gases will be sensitively monitorby a nonzero output. Figures 10 and 11 show how thisferential gas sensor is expected to behave theoreticallseveral cavity lengthsL, such thatL5L1 and various modu-lation frequencies (f ) using Eq.~13!. The same parameteras those in Fig. 9 were used in the simulation. In Fig. 10,values of thermal diffusivity (a3) and conductivity (k3) ofthe intracavity gas were assumed to depart linearly frthose of the ambient~background air! with the forms [email protected]* 1.631023# cm2/s [email protected]* 8310

23#31024 W/cm K, wheren is an integer from 0 to 125. At thestarting point (n50) of Fig. 10, the parameters of the tegas in region 3 are the same as those in region 2~air!. There-fore, the output signal amplitude at this point is zero andphase is indeterminate due to fully destructive thermal-winterference inside the PVDF. Figure 10 shows that the ssitivity of the sensor in the amplitude channel to the chanin thermophysical properties of the gas is enhanced withcreasing cavity length, while the phase channel showsopposite trend feature from the amplitude. A similar effecan be seen in Fig. 11, which shows that the sensitivity ofsensor in the amplitude channel is enhanced with decreamodulation frequency, as expected, since the intracathermal-wave power reaching the surface of the sensoincreased. In practice, however, lower frequency operatiothe interferometer will result in a larger overall system (1/f )noise.


tripetu-

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hth

futio

banm

on

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-the

-the

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IV. DISCUSSION

The successful development of photopyroelecthermal-wave interferometry must be placed in the prophysical context, in order to understand the peculiar naof ‘‘thermal-wave interferometry.’’ A discussion of the nature of thermal standing-wave patterns leading to the sincavity sensor consistent with the heat-conduction continequation~otherwise known as Fourier’s law!

2k¹T~r,t !5F~r ,t! ~14!

has been given elsewhere.5 HereF is the thermal-wave fluxand k is the thermal conductivity of the medium in whicthermal waves propagate diffusively. The essence ofwavelike nature of thermal waves lies in the fact that difsive temperature-wave propagation cannot sustain reflecat the boundary walls of the cavity~or cavities!, where thethermophysical properties change abruptly. This is socause conduction heat transfer is unidirectional and onlytivated by existing temperature gradients, which generateheat fluxes in the appropriate direction of decreasing teperature in the cavity according to the continuity equatiEq. ~14!. This is unlike propagating wave fields~acoustic,optical!, where a definite mode structure exists, which stefrom a well-defined completeness relation linking temporaorthogonal eigenfunctions of the motion.21 In the thermal-wave field there are no orthogonal temporal eigenmodes,thus there is no completeness relation to assure the enconservation of a mode structure with well-defined ene

FIG. 9. Theoretical behavior of the destructive interferometric thermophcal PPE gas sensor operating in air~reference! and in a gaseous ambienwith three different thermal diffusivities.~a! Amplitude, ~b! phase.


cr

re

e-y

e-ns

e-c-et-,

s

ndrgyy

content~harmonic Fourier spectrum!, a prerequisite for con-ventional wave phenomena~e.g., reflection, refraction, dif-fraction and polarization!. Nevertheless, the intracavity spacenergy distribution can be sufficiently described by the ma

i-

FIG. 10. ~a! Theoretical amplitude, and~b! phase of the destructive interferometric thermophysical PPE gas sensor signal as a function ofintracavity-gas thermal diffusivity, with cavity length as a parameter.

FIG. 11. ~a! Theoretical amplitude, and~b! phase of the destructive interferometric thermophysical PPE gas sensor signal as a function ofintracavity-gas thermal diffusivity, with modulation frequency as a paraeter.


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12

thns,e

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thsa

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ly

the

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hea

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-

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ematical equivalent of a thermal standing-wave pattformed by the rate of coherent thermal oscillations transping an amount of net power across the cavity length. Tthermal transport rate is thus controlled by local phase rtions of the oscillating thermal energy, fixed by the souand the presence of the cavity wall~s!.5 This intracavitypower confinement can be mathematically described bstanding thermal-wave pattern as a function of the intracity coordinate. Based on this understanding of the waveture of temporal thermal oscillations as coherent phaseperpositions of unidirectionally transported power, Fig.helps describe the physical nature of~photo!thermal-wavedual-beam interferometry introduced in this work: only withe introduction of opposing temperature-wave gradie~thermal fluxes! in the form of two optothermal heat sourcecan propagation of thermal power be achieved in the dirtion opposite to an existing gradient~flux F j ; j 51,2), asshown in Fig. 12. In this situation, interferometric phenoena can be described as linear superpositions of the twodirectional thermal-wave fields launched photothermafrom opposite directions. It is to be noted that the role ofoptical beams is limited in the energy conversion procesthe modulated beams into thermal-wave surface sourcesthat no optical interferences occur anywhere in the confirations of Figs. 1 or 12~whence the name ‘‘purely thermawave interferometry’’!. Assuming equal laser fluences oboth sides of the PPE transducer, when both beams ar~in-phase operation!, thenet thermal flux into the PVDF de-tector is maximum, as theinward fluxes from both surfacesadd across the thickness of the transducer to generatemaximum possible temperature at that instant@Figs. 4~a! and4~b! and 12~a!#. Conversely, when both beams are off~stillin-phase operation!, the stored thermal energy conductive

FIG. 12. Physical origins of two-beam thermal-wave interferometry:~a!in-phase operation, maximum of fully constructive interference,~b! in-phaseoperation, minimum of fully constructive interference, and~c! out-of-phaseoperation, fully destructive interference.


nt-ea-e

av-a-u-

ts

c-

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the

diffuses out of the PVDF from both surfaces. This createsmaximum outward flux resulting in the minimum possibletemperature across the transducer at that instant of the olation cycle @Figs. 4~a! and 4~b! and 12~b!#. The linear su-perposition of the two thermal-wave fields~front and backsurface! yields the mathematical equivalent ofconstructiveinterferenceof the thermal waves generated in the bulkthe film, in the language of conventional propagating wafields.22,23 On the other hand, when the two laser beamsout of phase by exactly 180°, theinward thermal flux fromthe surface temperature gradient due to the incidence ofbeam is exactly balanced by theoutward flux from theopposite-surface temperature gradient due to the absenthe other beam at all instants of the oscillation [email protected]~c!#. The result is that the overall thermal content of ttransducer remains dynamically fixed at all times, yielding~theoretically! zero lock-in output. Again, the linear superpsition of the two fluxes across the film results in the maematical equivalent of fully destructive interference of ttwo out-of-phase thermal waves, in the language of progating fields. The operation of the present PPE interferomric device is based on the base line suppression affordethe dynamic steady state flux balance of configuration F12~c!.

V. CONCLUSIONS

A novel thermal-wave interferometric device was deonstrated using photopyroelectric detection. A ondimensional theoretical approach to the PPE signal outputhe interferometer was developed and several operamodes were examined. The figure-of-merit~one-beam/two-beam signal amplitude ratio! of the new instrument was estimated to be on the order of 3000 if a 5 mW laser wasemployed. Both theoretically and experimentally, it wfound that optical-mode operation of the new PPE interfometer yields high-resolution imaging of optical structurewhich are usually limited by the base line of single-endPPE operation. A PPE measurement of the differential ocal absorption coefficient between nominally identicamanufactured transparent solids~Schott BK7 windowglasses! was performed in the thermally contacting mode.addition, high-resolution scanned images of optical defein a Ti-sapphire laser disk were obtained. The new devicexpected to be useful for optical studies and scanning iming of highly transparent media, as well as a high-resolutthermophysical gas sensor.

ACKNOWLEDGMENT

The support of a Research Grant from the Natural Sences and Engineering Research Council of Can~NSERC! is gratefully acknowledged.

APPENDIX

Complex quantities used in the general pyroelectric snal @Eq. ~5!#

H15~espd21!~11g3pV34e

22s3L1!

1~12e2spd!~g3p1V34e22s3L1!, ~A1!


.

,

,

ys.


H25~espd21!~11g2pW21e

22s2L!

1~12e2spd!~g2p1W21e22s2L!, ~A2!

G352Q12Q2~11b1s!e

ssl1Q3~12b1s!e2ssl

~11b1s!~11b2s!~essl2g1sg2se

2ssl !~A3!

with

Q15Es@b1s~N11N2e22bsl !1r s~N12N2e

22bsl !#,~A4!

Q25Ese2bsl@N11N21r s~N12N2!#, ~A5!

Q35Ese2bsl@N11N22r s~N12N2!#, ~A6!

Es5I 1hsbs

2ks~bs22ss

2!•

12Rs12Rs

2e22bsl, ~A7!

r s5bs /ss ~A8!

and

G452P11P2~11b4r !e

srm2P3~12b4r !e2srm

~11b3r !~11b4r !~esrm2g3rg4re

srm!~A9!

with

P15Er@b4r~N1r1N2re22brm!1r r~N1r2N2re

22brm!#,

~A10!

P252Ere2brm@N1r1N2r1r r~N1r2N2r !#, ~A11!

P352Ere2brm@N1r1N2r2r r~N1r2N2r !#, ~A12!

Er5I 2e

j Dwh rb r2kr~b r

22s r2!•

12Rr12Rr

2e22brm, ~A13!


r r5b r /s r . ~A14!

1A. Mandelis, Chem. Phys. Lett.108, 388 ~1984!; H. Coufal, Appl. Phys.Lett. 44, 59 ~1984!.

2H. Coufal and A. Mandelis, Ferroelectrics118, 379 ~1991!.3A. Mandelis, J. Vanniasinkam, and S. Budhudu, Phys. Rev. B48, 6808~1993!.

4M. Chirtoc and G. Mihailescu, Phys. Rev. B40, 9606~1989!.5J. Shen and A. Mandelis, Rev. Sci. Instrum.66, 4999~1995!.6A. Mandelis and K. F. Leung, J. Opt. Soc. Am. A8, 186 ~1991!.7A. Mandelis and M. M. Zver, J. Appl. Phys.57, 4421~1985!.8J. Shen, A. Mandelis, and B. D. Aloysius, Int. J. Thermophys.17, 1241~1996!.

9J. Shen, A. Mandelis, and H. Tsai, Rev. Sci. Instrum.69, 197 ~1998!.10M. Bertolotti, G. L. Liakhou, R. Li Voti, S. Paoloni, and C. Sibilia, Int. J

Thermophys.19, 603 ~1998!.11M. Chirtoc, D. Bicanic, and V. Tosa, Rev. Sci. Instrum.62, 2257~1991!.12H. Coufal, F. Trager, T. J. Chuang, and A. C. Tam, Surf. Sci.145, L504

~1984!.13Z. Sodnik and H. J. Tiziani, Opt. Commun.58, 295 ~1986!.14L. Wawrzyniuk and G. Dymny, Opt. Eng.~Bellingham! 36, 1602~1997!.15H. G. Walther, K. Friedrich, K. Haupt, K. Muratikov, and A. Glazov

Appl. Phys. Lett.57, 1600~1990!.16KynarTM Piezo Film Technical Manual~Pennwalt, King of Prussia, PA

1983!, p. 17.17C.-H. Wang and A. Mandelis, Rev. Sci. Instrum.~submitted!.18J. Vanniasinkam, A. Mandelis, S. Buddhudu, and M. Kokta, J. Appl. Ph

75, 8090~1994!.19Schott Optical Glass Catalog~Schott Glass, Duryea, PA, 1996!.20CRC Handbook of Chemistry and Physics, 74th ed., edited by D. R. Lide

~Chemical Rubber, Cleveland, OH, 1993!, pp. 12–134~Al !.21P. M. Morse and H. Feshbach,Methods of Theoretical Physics~McGraw–

Hill, New York, 1953!, Vol. I, Chap. 7.22M. Born and E. Wolf,Principles of Optics~Pergamon, New York, 1980!.23R. E. Berg and D. G. Stork,The Physics of Sound~Prentice–Hall, Engle-

wood Cliffs, NJ, 1995!, Chap. 2, p. 39.


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