Photodeflection signal formation in photothermal measurements: comparison of the complex ray theory, the ray theory, the wave theory, and experimental results

Photodeflection signal formation in photothermalmeasurements: comparison of the complex ray theory,

the ray theory, the wave theory, and experimental results

Dorota Korte Kobylinska,* Roman J. Bukowski, Bogusław Burak, Jerzy Bodzenta,and Stanisław Kochowski

Institute of Physics, Silesian University of Technology, Krzywoustego 2, 44-100 Gliwice, Poland

*Corresponding author: [email protected]

Received 1 December 2006; revised 13 April 2007; accepted 14 April 2007;posted 17 April 2007 (Doc. ID 77674); published 9 July 2007

A comparison is made of three methods for modeling the interaction of a laser probe beam with thetemperature field of a thermal wave. The three methods include: (1) a new method based on complex raytheory, which allows us to take into account the disturbance of the amplitude and phase of the electricfield of the probe beam, (2) the ray deflection averaging theory of Aamodt and Murphy, and (3) the wavetheory (WT) of Glazov and Muratikov. To carry out this comparison, it is necessary to reformulate thedescription of the photodeflection signal in either the WT or the ray deflection averaging theory. It isshown that the differences between calculated signals using the different theories are most pronouncedwhen the radius of the probe beam is comparable with the length of the thermal wave in the region of theirinteraction. Predictions of the theories are compared with experimental results. A few parameters ofthe experimental setup are determined through multiparameter fitting of the theoretical curves to theexperimental data. A least-squares procedure was chosen as a fitting method. The conclusion is that thecalculation of the photodeflection signal in the framework of the complex ray theory is a more accurateapproach than the ray deflection averaging theory or the wave one. © 2007 Optical Society of America

OCIS codes: 080.0080, 080.2710, 080.2720, 120.0120, 120.4290.

1. Introduction

The photodeflection method [1,2] is a useful tool forthe investigation of thermal and optical parametersof solid-state objects [3–6] and thin-layered struc-tures [7–9] as well as for spectroscopy [10–12] andthermal wave microscopy [13,14]. In this method, thepump beam heats the sample periodically (Fig. 1)while the second beam probes the temperature fielddisturbance in the gases surrounding the sample.The probe beam has an optical wavelength � and aGaussian profile with radius a at its waist. The probebeam propagates through the gas surrounding thetest sample. The mechanism arising from the signalis as follows. As a result of periodic heating and heatdiffusion in both the sample of a given thickness andthe gas over it, periodic temperature disturbancetakes place. That causes changes in the gas refractive

index in the adjacent medium close to the samplesurface—the thermal lens (TL) is produced. Theprobe beam is deflected by this periodic gradient ofthe index of refraction (this is so called the “mirage”effect).

One of the theories of photodeflection signal cre-ation is the ray deflection theory (RDT) proposed byAamodt and Murphy [15]. According to the RDT, theprobe beam is assumed to be an infinitely thin ray,and its deflection at the TL is proportional to thetemperature gradient in the TL. The development ofthe RDT is the ray deflection averaging theory(RDAT) [16–18]. It lets us calculate the photodeflec-tion signal for a probe beam of finite size. The beamis then considered to be a bundle of infinitely thinrays. Each of these rays is deflected on the TL in adifferent way. The photodeflection signal is ob-tained by averaging their contribution in accor-dance with the undisturbed probe beam intensityprofile. Such an approach is a phenomenological oneand is not satisfying on theoretical grounds.

0003-6935/07/225216-12$15.00/0© 2007 Optical Society of America

5216 APPLIED OPTICS � Vol. 46, No. 22 � 1 August 2007

Glazov and Muratikov proposed a theory of form-ing the photodeflection signal based on the solution ofthe wave equation for the probe beam propagating inthe thermal wave field [19,20]. This is called the wavetheory (WT). It takes into account only the change inphase of the electric field in the probe beam after itpasses through the TL.

The RDAT and the WT methods do not alwaysagree with the experiment, especially for high mod-ulation frequency of the temperature field and wideGaussian beams (such as the probe beam radius abeing comparable with the length of the thermalwave �g). Thus another approach, based on the com-plex geometrical optics equations [the complex raytheory (CRT)], was proposed [21–24].

The calculation of the photodeflection signal usingthe RDAT method and its comparison with the CRTmethod has been done [24]. In this paper, the photo-deflection signal in the framework of the WT is re-calculated for the assumed experimental setup, andthe obtained analytical formulas are presented. Fi-nally, the comparison of CRT to both RDAT and WT,as well as with some experimental results, is shown.

In the following, we first review the basis of thetheoretical model for the calculation of the photode-flection signal on the grounds of the CRT, the RDAT,and present the calculation of it in the framework ofthe WT. We identify normal signal dependencies on afew parameters of the experimental setup. Next wediscuss the differences between all presented theories.Finally, we present experimental results obtained for asample with known thermal and geometrical parame-ters in order to validate the CRT and to point out thelimitation of the methods of description the probebeam interaction with the thermal waves used so far.

2. Theory

A. Geometry of the Experiment and the TemperatureField

The sample is assumed to be homogeneous and iso-tropic and surrounded by air (Fig. 1). Its surface is

illuminated uniformly by a modulated light beam ofangular frequency �. The surrounding gas does notabsorb any radiation from the incident light while theopaque sample absorbs energy from the beam.

The temperature field in the air, described as thethermal waves, can be obtained by solving the heatdiffusion equation with the appropriate boundaryconditions (the temperature and the heat flux conti-nuity at the interfaces). For the illumination spotmuch wider than the width of the probe beam, thesolution for the time-dependent part of the tempera-ture in air is one dimensional (1D) [25]:

�g�x, t� � T�x, t� � T0 � bg exp��

2�g�x � h��

� cos��t � � �

2�g�x � h� � g�.

(1)

Here bg is the amplitude of the temperature distur-bance, g is the phase shift between the sample’ssurface temperature and the exciting beam, �g is thegas thermal diffusivity, � is the angular modulationfrequency of the pump beam, and h is the distancebetween the probe beam axis and the sample surface.

The thermal waves change the gas refractive index[21–23]:

n�T� � n0 �dndT�T0

�T � T0� � n0 � n0sT�g�x, t�, (2)

Here n0 is the index of refraction in ambient temper-ature T0, sT � n0

�1�dn�dT�|T0is the refractive index

thermal sensitivity.

B. Complex Ray Theory

1. Propagation of a Gaussian Beam in anOptically Homogeneous MediumIn the frame of the complex geometrical optics, theprobe beam is considered to be a bundle of rays thatpropagate in a complex space [26]. The coordinates ofsuch a ray r�� x��, y��, z�� are described by thisset of equations [21–24]:

x�� 1 � in0

zRC�, (3a)

y�� 1 � in0

zRC�, (3b)

z�� n0�1 ��2 � �2

zRC2 , (3c)

where � and � are the ray’s coordinates in the inputplane of the experimental setup �z � 0�, � is therunning coordinate along the ray, zRC � zR � iL isthe complex Rayleigh’s length, zR � ka2n0 is theFig. 1. Schematic of the experimental configuration.

1 August 2007 � Vol. 46, No. 22 � APPLIED OPTICS 5217

Rayleigh’s length, L is the probe beam waist position,a is its radius in the waist, k � 2 �� is the wave-number, and � is the wavelength of the probe beam ina vacuum.

2. Propagation of a Gaussian Beam in aThermally Disturbed MediumThe Gaussian probe beam interaction with the field ofa thermal wave consists of two effects [21–24,26].One of them is the deflection of the beam, and thesecond one is the phase change of it. The deflectionresults from the gradient of the refractive index andcauses the ray trajectory change in the temperaturefield. Considering that the field of the thermal wavein the gas above the sample is 1D, that means onlythe x coordinate of the ray trajectory is changed. Thecorrection to the ray x coordinate can be determinedusing the perturbation calculus [22–24,26]:

x1��, � � n02sT

0

� � ��g

�x d�

� P��n0 � zs��zp � zl�, (4a)

P�� n0sTbg��

�gexp��

2�g�1 � i

zs

zRC��

� sin��t � ��

2�g�1 � i

zs

zRC�� g �

4�,

(4b)

where zl and zp are the position of the left and rightedges of the sample (Fig. 1), zs � �zp � zl��2 is themiddle of the sample. The integral in Eq. (4a) is anelementary one, but in this perturbation calculus, weuse it to estimate the simplified middle point method.

The ray trajectory change results in the probebeam amplitude change caused by the change of thedivergence of the bundle of rays [22–25]:

A�z� � A� � 0��x��y��z��|�0

��x��y��z�� 1�2

� E0

zR

zRC�1 � i

zzRC

��1�1 �12

�P��

zs

n0��zp

n0�

zl

n0�

� �1 � iz

zRC��1�, (5)

where E0 is the electric field intensity in the middle ofthe probe Gaussian beam waist. The field of the ther-mal wave also influences the Gaussian beam phase.The corrected eikonal can be written as [21–24]

� � �0 � n02 � n0

2sT 0

�g�x��d�, (6)

where �0 is the eikonal of the undisturbed beam inthe input plane �z � 0�. The integration in this for-mula is carried out along the corrected ray path. Itmeans that the change of the eikonal is caused by thechange of the optical path of the ray after undergoing

the thermal lens (because of the refractive indexvalue change) and the change of the geometric path ofthe ray (because of the refractive index gradient). Insuch a situation, the correction to the eikonal consistsof two components:

�1 � �1f � �1d, (7a)

�1f � kn0sTbg�zp � zl�exp��kgxs�cos��t � kgxs � g�,(7b)

�1d �zn0

x

zRC2 P��z � zs��zp � zl��1 � i

zzRC

��2

. (7c)

The first is due to the change of the index of refractionvalue ��1f� and the second from the gradient of it ��1d�.

Because of the 1D character of the temperaturefield, the transverse component of the photodeflectionsignal equals zero, and only the normal componentneeds to be considered. The normal photodeflectionsignal is measured by the quadrant photodiode andresults in a difference in illumination between upperand lower photodiode halves. It is given by the fol-lowing [21–24]:

Sn � Kd ��

��

dyD� 0

��

� �h

0 �dxDI�r�D�

� Snd � Snf � At cos��t � g � t�. (8)

Here Kd is the photodetector constant, I�r�D� is theprobe beam intensity in the detector plane, Snd �Ad cos��t � g � d� is the deflected component of thesignal, and Snf � Af cos��t � g � f� is the phasecomponent of the signal. The amplitude of the totalphotodeflection signal At, the phase t, the ampli-tudes and phases of the deflected and phase compo-nents of the total signal Ad, Af, d, f, respectively, alldepend on experimental setup parameters. These in-clude the following: the height of the probe beam overthe sample h, the angular modulation frequency ofthe pump beam �, the probe beam radius a, thebeam-waist position L, the detector zD, the sample zl

positions [8–11], and also the thermal properties ofthe sample and the air (see Appendix A).

C. Ray Deflection Averaging Theory

The angle of normal deflection of the single ray prop-agating through the TL can be found from the expres-sion [15,17]

�n � ��1n

dndT��T0

zl

zp��g�x, t�

�x dz. (9)

The photodeflection signal results from the displace-ment of the ray in the direction perpendicular to thesample surface. For the probe beam modeled as abundle of rays, the photodeflection signal is treated as


a weighted sum of their contribution [15,17]:

Snr � yD��

�� xD��h

��

P�xD, yD��n�xD, yD�dxDdyD

� Ar cos��t � g � r �

4�, (10)

where P�x, y� is the normalized power distribution inthe probe Gaussian beam. The quantities Ar and r

are functions of parameters of the experimental setup[24] (see Appendix B).

D. Wave Theory

According to the WT, the TL is considered to be duesolely to phase changes (this is so called the thin lensapproximation). This means that only the phasechanges in the electric field of the probe beam causedby the probe beam interaction with the thermal wavefield are taken into account. Distortions of amplitudedistribution caused by deflection on refractive indexgradients are not considered. Therefore, the changeof the distribution of the electric field in the probebeam due to its interaction with the temperature fieldis [19,20]

�u�x, y, z� � i��u0�x, y, z�, (11)

where u0�x, y, z� is the distribution of the field in theabsence of the TL. The change of the phase �� due topassing through the temperature field is described bythe formula

�� 2

�

dndT�T0

zl

zp

�g�x, t�dz. (12)

The distribution of the electric field in the detectorplane on the basis of WT is found by the use of theFresnel–Kirchhoff integral. Here it is assumed thatthe distance between the TL and the detector planeexceeds the probe beam diameter in the region ofthermal wave interaction. In such a situation, theFresnel–Kirchhoff integral can be written in a form

u�xD, yD, zD� �i

��zD � zs� ��

��

��

u0�xs, ys, zs�

� �u�xs, ys, zs�exp��i

��zD � zs�

� �xD � xs�2 � �yD � ys�2�dxDdyD.

(13)

Here u�xs, ys, zs� is the electric field of the probe beamin the zone of the thermal wave’s action.

The normal photodeflection signal is proportional tothe difference between a probe beam intensity distri-

bution at the photodetector (quadrant photodiode) af-ter its interaction with the thermal lens Is�xD, yD, zD�and that without it I0�xD, yD, zD�:

Snw � 2Kd yD��

�� xD��h

��

Is�xD, yD, zD�

� I0�xD, yD, zD�dxDdyD

� Aw cos��t � g � w�, (14)

where Kd is the detector constant, Is�xD, yD, zD� �|u�xD, yD, zD�|2, I0�xD, yD, zD��|u0�xD, yD, zD�|2. As inthe case of the CRT and the RDAT, the photodeflec-tion signal depends on the parameters of the experi-mental setup (see Appendix C).

E. Data Analysis

The ray averaging theory takes into account only thedeflection of the beam in the field of thermal wave,whereas the WT deals only with the influence of theTL on the phase of the electric field. That is why noneof these theories are an accurate approach for de-scribing the light beam interaction with the temper-ature field. The CRT is introduced [21–24,26] toaddress this deficiency by including both the deflec-tion and the phase change of the probe beam after itpasses through the temperature field of the thermalwave. To show the differences in predictions of a be-havior of the photodeflection signal, calculations of itsamplitude and phase are carried out using three mod-els: (1) the RDAT, which takes into account the finitevalue of the probe beam radius [Eq. (10)], (2) the WT[Eq. (14)], and (3) the CRT [Eq. (8)]. The results ofcalculation are presented in Figs. 2–5 for assumeddifferent values of experimental setup parameters:the height of the probe beam over the sample h, theprobe beam-waist position L, the angular modulationfrequency of the temperature field �, the detectorposition zD.

In Fig. 2, the photodeflection signal dependence ona��g is shown for the probe beam-waist position overthe sample. The photothermal measurements areusually performed at this configuration. The photo-deflection signals calculated using WT and CRT havesimilar characters. The main difference in ampli-tudes is the rapidity of their changes for small valuesa��g � 0.07. There are also significant differences inphases for a��g � 0.15. It can be explained by the factthat the CRT takes into account phase changescaused either by the change in refractive index or thechange in the probe beam path, while in the WT, onlythe first effect is considered. For a��g � 0.15, predic-tions of both mentioned theories converge for ampli-tude and phase dependencies. Over the full range ofa��g values, the calculated values from the RDATdiffer from these of the CRT.

When moving the probe beam waist behind thedetector, the character of the photodeflection signaldependence on a��g changes, as seen in Fig. 3. Fora��g � 0.03, amplitudes calculated by the use of theRDAT and the CRT are in good agreement but differ


from the calculations based on the WT. The phase ofthe photodeflection signal received on the basis of thecomplex geometrical optics equations depends ona��g only for small values of it (a��g � 0.02) whereasfor the RDAT it is not a function of a��g in the wholerange of its changes. In the case of the WT, the pho-todeflection signal changes in the same way versusa��g as it was for the probe beam waist over thesample. In this case, different behavior of the photo-deflection signal is predicted by the CRT and the WT.

The RDAT is based on the geometrical optics ac-cording to which the photodeflection signal increaseswith the increase of the distance between the inter-action region and the detector position (Fig. 4). Itresults from the fact that with the increase of zD, thedeflection of the ray increases. In the case of the CRT,the amplitude of the signal does not increase with theincrease of zD in all analyzed range of it. For largervalues of zD, the amplitude remains practically con-stant. Such a situation is also observed in the case ofthe WT (Fig. 4). That is why significant differences

between the RDAT and the CRT can be noticed forthe whole range of changes the detector position, es-pecially in the amplitude dependence on zD. For thecase of the WT, the agreement with the CRT can beseen for large zD �zD � 1.4 m� (Fig. 4).

Figure 5 presents the photodeflection signal depen-dence on the probe beam-waist position L. It is evi-dent from this graph that in all cases a drop in thesignal is observed when the probe beam waist is overthe detector. According to the RDAT, the photodeflec-tion signal increases sharply with the increase of thedistance between zD and L for small values of it. Withfurther increase of that distance, this dependenceremains the same in character but becomes less sig-nificant. The increase in the signal with the increaseof the distance between zD and L results from the factthat the larger it is, the larger the probe beam in thedetector plane is, which increases the deflection andfinally the photodeflection signal. In the case of theCRT and the WT, and for L a bit before or afterthe detector plane, an increase in the amplitude of

Fig. 2. (a) Calculated amplitude and (b) phase of photodeflec-tion signal changes versus the probe beam radius a to the lengthof the TW in air �g ratio for different theories: CRT, the complexray theory; RDAT, the ray deflection averaging theory; WT, thewave theory. The height of the probe beam over the sampleh � 800 �m, the angular modulation frequency of the tempera-ture field � � 60 rad�s ��g � 5130 �m�, the detector positionzD � 1.5 m, the sample position zl � 0.5 m.

Fig. 3. (a) Calculated amplitude and (b) phase of photodeflec-tion signal changes versus the probe beam radius a to the lengthof the TW in air �g ratio for different theories: CRT, the complexray theory; RDAT, the ray deflection averaging theory; WT, thewave theory. The height of the probe beam over the sampleh � 800 �m, the angular modulation frequency of the tempera-ture field � � 60 rad�s ��g � 5130 �m�, the detector positionzD � 1.5 m, the sample position was zl � 0.5 m, the probe beam-waist position L � 1.6 m.


the signal can be noticed after which a drop in it canbe observed.

The phase dependence on the probe beam-waistposition is the same in character in the case of thecomplex ray theory and the wave one. The phase ofthe photodeflection signal calculated by the use of thegeometrical optics depends on L only in the region ofthe detector plane.

3. Experimental Description

The presented literature data analysis shows that theWT satisfactorily describes the experimental data for awide range of the experimental setup parameterchanges [19]. We would like to show that the presentedCRT also agrees with experiment, and moreover, insome cases, it can explain the behavior of experi-mental dependencies that cannot be explained bythe WT. The block diagram of the experimentalsetup is shown in Fig. 6. We used a laser photodiodewith an output wavelength of �e � 830 �m for exci-tation and a He–Ne laser �� 633 nm� that provides

a Gaussian beam as the probe beam. The sample withknown thermal parameters was studied. It was a0.5 mm thick zinc sample (thermal diffusivity �s

� 0.42 cm2�s, thermal conductivity ls � 116 W�mK)placed on the support that allows variation in its po-sition. The change in light intensity distribution in theprobe beam after undergoing the field of the TW wasmeasured by a quadrant photodiode, which was placed

Fig. 4. (a) Calculated amplitude and (b) phase of photodeflec-tion signal changes versus the detector position zD for differenttheories: CRT, the complex ray theory; RDAT, the ray deflectionaveraging theory; WT, the wave theory. The height of the probebeam over the sample h � 1200 �m, the probe beam radius in thewaist a � 500 �m, the angular modulation frequency of the tem-perature field � � 600 rad�s ��g � 1622 �m�, the sample positionzl � 0.5 m, the probe beam-waist position L � 0.5 m.

Fig. 5. (a) Calculated amplitude and (b) phase of photodeflectionsignal changes versus the probe beam-waist position L for differenttheories: CRT, the complex ray theory; RDAT, the ray deflectionaveraging theory; WT, the wave theory. The height of the probebeam over the sample h � 800 �m, the probe beam radius in thewaist a � 150 �m, the angular modulation frequency of the tem-perature field � � 4500 rad�s ��g � 621 �m�, the sample positionzl � 0.5 m, the detector position zD � 1.5 m.

Fig. 6. Block diagram of the experimental setup.


zD � 77 cm beyond the input lens of the experimentalsetup. The resulting signal was analyzed by means ofa lock-in amplifier connected to a PC, which allowedus to record the amplitude and phase of the photode-flection signal as a function of (i) the height of theprobe beam over the sample h (Fig. 7), (ii) the mod-ulation frequency � (Fig. 8), (iii) the sample positionzl (Fig. 9). To fit the theoretical curves calculated onthe ground of the CRT into the experimental dataand to determine some parameters of the experi-mental setup, the least-squares procedure appliedto the complex quantities (Argand’s graph) wasused. The fitting accuracy was estimated by deter-mining the width of the fitting function at theheight equal to its double value in the determinedminimum. The fitted parameters were the radius ofthe probe beam a, the probe beam-waist position L,the probe beam height over the sample h (in case of

data presented in Figs. 8 and 9). Obtained values ofthe fitted parameters are shown in the graphs. Ad-ditionally, these graphs present the theoreticalcurves calculated on the grounds of the RDAT andWT for the fitted parameters.

In Fig. 7(a), the study of the photodeflection signalchanges versus the height h of the probe beam overthe sample is shown. Experimental conditions re-spond to the case of the probe beam radius compara-ble with the length of thermal wave �g � 682 �min the zone of its interaction a�zl� � 215 �m [27,28]�zl � 22.5 cm�. It can be seen that experimentallydetermined amplitude dependence of the photodeflec-tion signal in the logarithmic scale and the phasedependence of it on h is not a linear one as predictedby the RDAT and the WT. But this behavior can beexplained in the frame of the CRT.

Figure 8 presents the photodeflection signal depen-dence on the angular modulation frequency �. With

Fig. 7. (a) Amplitude and (b) phase of photodeflection signal de-pendence on the height h of the probe beam over the sample.EXPERIMENT, the experimental results; the solid curve repre-sents the best fitting to the experimental data on the basis of theCRT. The obtained by fitting values of the parameters are shownin the figure. RDAT are the dependences calculated on the groundsof the ray averaging theory for the parameters obtained by fitting.WT are the dependences calculated on the grounds of the WT forthe parameters obtained by fitting. The probe beam radius in theplace of probe beam interaction with the thermal wave field a�zl�� 215 �m, the angular modulation frequency of temperature field� � 3391 rad�s ��g � 682 �m�, the sample position zl � 22.5 cm,the detector position zD � 77 cm.

Fig. 8. (a) Amplitude and the (b) phase of photodeflection signaldependence on modulation frequencies of the temperature field �.EXPERIMENT, the experimental results; the solid curve repre-sents the best fitting to the experimental data on the basis of theCRT. The obtained by fitting values of the parameters are shownin the figure. RDAT are the dependences calculated on the groundsof the ray averaging theory for the parameters obtained by fitting.WT are the dependences calculated on the grounds of the WTfor the parameters obtained by fitting. The sample positionzl � 7.5 cm, the detector position zD � 77 cm.


the increase of �, not only does the TW attenuationincrease, which causes the drop of the signal, but alsothe temperature field gradient increases, which re-

sults in the increase in the photodeflection signal.Such behavior cannot be reproduced with the RDATnor the WT approach, but can be reproduced usingthe CRT.

Figure 9 shows that the photodeflection signal de-pends on the sample position zl. It should be notedthat such a dependence cannot be explained solely onthe basis of the RDAT but can be reproduced by theuse of the CRT. Although the WT also predicts thephotodeflection signal amplitude and phase depen-dence on zl its agreement with the experimental re-sults is not satisfying.

4. Conclusions

A comparison between three theories describing theinteraction of the Gaussian beam with the TW hasbeen made; namely, the ray deflection averagingtheory (RDAT), the wave theory (WT), and the com-plex ray theory (CRT). Their predictions were alsocompared with experimental results. For some pa-rameters differences between experiments, CRT,RDAT, and WT occur. The differences become moreapparent in case of the RDAT and the CRT for theamplitude of the signal and the phase for high mod-ulation frequencies, when the probe beam radius iscomparable with the length of the TW in the zone ofinteraction.

The computed and experimental results show thatthe CRT can play an important role in the correctquantitative and qualitative interpretation of photo-deflection data, especially those related to measure-ments at high modulation frequencies (e.g., thedetermination of thin film thermal parameters byphotothermal methods). This is because it takes intoaccount some effects that cannot be described by ei-ther the RDAT or the WT.

Appendix A: Complex Ray Theory Signal

The amplitude of the total photodeflection signal At,the phase of it t, the amplitudes and phases of thedeflective and phasial components of the total signalAd, Af, d, f are expressed by the formulas

At � �Ad cos�g � d� � Af cos�g � f�2 � Ad sin�g � d� � Af sin�g � f�2, (A1)

tan t �Ad sin�g � d� � Af sin�g � f�Ad cos�g � d� � Af cos�g � f�

, (A2)

Ad � ��AdA cos dA � Adf cos�df �

4��2

� �AdA sin dA � Adf sin�df �

4��2

, (A3)

tan d �AdA sin dA � Adf sin�df � �4�AdA cos dA � Adf cos�df � �4�

, (A4)

AdA � 2Kd�Re�G1 � G2�2 � Im�G1 � G2�2, (A5)

Fig. 9. (a) Amplitude and the (b) phase of photodeflection signaldependence on the sample position zl. EXPERIMENT, the experi-mental results; the solid curve represents the best fitting to theexperimental data on the basis of the CRT. The obtained-by fittingvalues of the parameters are shown in the figure. RDAT are thedependences calculated on the grounds of the ray averaging theoryfor the parameters obtained by fitting. WT are the dependencescalculated on the grounds of the WT for the parameters obtained byfitting. The angular modulation frequency of the temperature field� � 1445 rad�s ��g � 1095 �m�, the detector position zD � 77 cm.


tan dA � �Im�G1 � G2�Re�G1 � G2�

, (A6)

Adf � 2Kd�Im�H1 � H2�2 � Re�H1 � H2�2, (A7)

tan df � �Re�H1 � H2�Im�H1 � H2�

, (A8)

Af � 2Kd�Im�F1 � F2�2 � Re�F2 � F1�2, (A9)

tan f � �Re�F2 � F1�Im�F1 � F2�

, (A10)

G1 � �m exp��i � 1�2C�

2

4a2

�g2��erf��i � 1�

C�

2a�g��

12 erf��i � 1�

C�

2a�g

�zR

�zR2 � �L � zD�2

ha��

12�, (A11)

G2 � �m exp��i � 1�2C�

2

4a2

�g2��erf��i � 1�

C�

2a�g��

12 erf��i � 1�

C�

2a�g

�zR

�zR2 � �L � zD�2

ha��

12�,

(A12)

H1 � �m�� i � 1�C�

2a�g

exp��i � 1�2C�

2

4a2

�g2��erf��i � 1�

C�

2a�g

�zR

�zR2 � �L � zD�2

ha�� erf��i � 1�

C�

2a�g��

� exp��2 �i � 1��1 � izs

zRC��1 � i

zD

zRC��1 h

�g�

zR

zR2 � �L � zD�2

h2

a2�� i � 1�C�

2a�g

� exp��i � 1�2C�

2

4a2

�g2��1 � erf��i � 1�

C�

2a�g��, (A13)

H2 � �m�� i � 1�C�

2a�g

exp��i � 1�2C�

2

4a2

�g2��1 � erf��i � 1�

C�

2a�g�� exp�2 �i � 1��1 � i

zs

zRC�

� �1 � izD

zRC��1 h

�g�

zR

zR2 � �L � zD�2

h2

a2�� i � 1�C�

2a�g

exp��i � 1�2C�

2

4a2

�g2��erf��i � 1�

C�

2a�g�

� erf��i � 1�C�

2a�g

�zR

�zR2 � �L � zD�2

ha��, (A14)

F1 � �m exp��i � 1�C�

2a�g�2��1 � 2 erf��i � 1�

C�

2a�g�� erf��i � 1�

C�

2a�g

�zR

�zR2 � �L � zD�2

ha��, (A15)

F2 � �m exp��i � 1�C�

2a�g�2��1 � 2 erf��i � 1�

C�

2a�g�� erf��i � 1�

C�

2a�g

�zR

�zR2 � �L � zD�2

ha��, (A16)

�m � 2 isTbgPl�zD � zs��zp � zl��g

2�1 � izD�zRC�2, (A17)

�m �sTbgkg

�2 azDPl�zD � zs��zp � zl��zR

2 � �L � zD�2�2izRC2�1 � i

zD

zRC�2��1

, (A18)

�m �14 n0sTbgkPl�zp � zl�, (A19)

C� �2

zR�1 � i

zs

zRC��1 � i

zD

zRC��1

�zR2 � �L � zD�2, (A20)


where erf�� is the error function.

Appendix B: RDAT Signal

Formulas describing the photodeflection signal re-ceived on the ground of the ray deflection averagingtheory have the form:

Ar � �Re�Q1 � Q3 � Q2 � Q4�2 � Im�Q1 � Q3 � Q2 � Q4�2,(B1)

tan r � �Im�Q1 � Q3 � Q2 � Q4�Re�Q1 � Q3 � Q2 � Q4�

, (B2)

Q1 � � exp� �1 � i�2a2

�g2��1 � erf� �1 � i�

a�g��,

(B3)

Q2 � � exp� �1 � i�2a2

�g2��1 � erf� �1 � i�

a�g��,

(B4)

Q3 � � exp� �1 � i�2a2

�g2��erf�� 1 � i�

a�g

�ha�

� erf� �1 � i�a�g��, (B5)

Q4 � � exp� �1 � i�2a2

�g2��erf� �1 � i�

a�g

�ha�

� erf� �1 � i�a�g��, (B6)

� ��2

4i a2sTbgkg�zp � zl�. (B7)

Appendix C: Wave Theory Signal

After calculating the integrals in Eqs. (13) and (14),the amplitude Aw and the phase w of the photode-flection signal received on the basis of the WT can bewritten as

Aw � 2Kd�Re�P1 � P2 � P3 � P4�2 � Im�P3 � P4 � P1 � P2�2,(C1)

tan w �Im�P1 � P2 � P3 � P4�Re�P1 � P2 � P3 � P4�

, (C2)

where

P1 � �mR�

2�R1R2��0 � i�1�cosk�zl � L�

� ��1 � i�0�sink�zl � L��exp��4 2a2

�g2��1 � i�2�

� ik�zl � L� � 2k2a2� a�g�2

��1 � i�2�2

� �zD � zs��2R1�1��1 � erf��i�R1h

� i kaa�g

��1 � i�2��zD � zs��1�R1�1��, (C3)

P2 � �mR�

2�R4R2��0 � i�1�cosk�zl � L� � ��1 � i�0�

� sink�zl � L��exp�4 2a2

�g2��3 � i�4�

� ik�zl � L� � 2k2a2� a�g�2

��3 � i�4�2

� �zD � zs��2R4�1��1 � erf��i�R4h � i ak

a�g

� ��3 � i�4��zD � zs��1�R4�1��, (C4)

P3 � �mR�

2�R3R2��0 � i�1�cosk�zl � L�

� ��1 � i�0�sink�zl � L��exp�4 2a2

�g2��3 � i�4�

� ik�zl � L� � 2k2a2� a�g�2

��3 � i�4�2

� �zD � zs��2R3�1��1 � erf��i�R3h � i ak

a�g

� ��3 � i�4��zD � zs��1�R3�1��, (C5)

P4 � �mR�

2�R5R2��0 � i�1�cosk�zl � L� � ��1 � i�0�

� sink�zl � L��exp��4 2a2

�g2��1 � i�2�

� ik�zl � L� � 2k2a2� a�g�2

��1 � i�2�2

� �zD � zs��2R5�1��1 � erf��i�R5h � i ak

a�g

� ��1 � i�2��zD � zs��1�R5�1��, (C6)

�m �kzR

2 a�Pl

zl � L � izR

zD � zszR

2 � L2 � 3�zR2 � L2��1zl

2L2

� �zR2 � L2��3zl

4L4�1, (C7)

�m �sTkbg

2 n02a

�Pl

zR3�zp � zl�

�zD � zs�2�zl2 � zR

2�

� �1 � �zl � LzR

�2�2�1 � �k2�zD � zs��1

� �1 � �zl � LzR

�2�� 2zl � Lkn0

�2��1

, (C8)


�0 � �m

�zR2 � L2�2

�zR2 � L2 � zlL�2 � zR

2zl2�2zR�zD � zs�

k

� �1 ��zl � L�2

zR2 ��1 � 2

zlL

zR2 � L2��

kn0

2zR�zl � L�

� �1 ��zl � L�2

zR2 ��1�, (C9)

�1 ��mk2 �1 � �zl � L

zR�2��1��zD � zs��1 �

n0

zR2�zl � L�2

� n0

�zR2 � L2�2

�zR2 � L2 � zlL�2 � zR

2zl2�, (C10)

R1 � �k

�zD � zs��zD � zs��1� zR

2n0� ka2�i�2 � �1��

�i2�1 �

�zl � L�zR

��, (C11)

R2 � �k

2�zD � zs�� zR

n0�zD � zs�� i�1 �

�zl � L�zR

��,

(C12)

R3 � �k


2n0� ka2��4 � i�3��

�i2�1 �

�zl � L�zR

��, (C13)

R4 � �k


2n0� ka2��4 � i�3��

�i2�1 �

�zl � L�zR

��, (C14)

R5 � �k


2n0� ka2��1 � i�2��

�i2�1 �

�zl � L�zR

��, (C15)

�1 �12�1 � �zl � L

zR�2��1 � � zR

n0�zD � zs��1 � �zl � LzR

�2��

�zl � L�zR

�2��1

, (C16)

�2 � �1� zR

n0�zD � zs�� 1 � �zl � L

zR�2��1�, (C17)

�3 �12�1 � �zl � L

zR�2��1 � � zR

n0�zD � zs��1 � �zl � LzR

�2��

zl � LzR

�2��1

, (C18)

�4 � �3�1 � �zl � LzR

�2�� zR

n0�zD � zs�

� �1 � �zl � LzR

�2��1�. (C19)

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Documents

Photodeflection signal formation in photothermal measurements: comparison of the complex ray theory, the ray theory, the wave theory, and experimental results