Optothermal depth profiling by neural network infrared radiometry

JOURNAL OF APPLIED PHYSICS 97, 014701 (2005)

Optothermal depth profiling by neural network infrared radiometry signal recognition Jyotsna Ravi, Yuekai Lu, and Stéphane Longuemart Laboratorium voor Akoestiek en Thermische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

Stefano Paoloni Dipartimento di Energetica, Università di Roma “La Sapienza,” Instituto Nazionale perla Fisica della Materia, Via Scarpa 16, 00161 Roma, Italy

Helge Pfeiffer Laboratorium voor chemische en biologische dynamica (LCBD), Departement Chemie, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

Jan Thoen and Christ Glorieuxa) Laboratorium voor Akoestiek en Thermische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

(Received 6 July 2004; accepted 27 September 2004; published online 16 December 2004) The feasibility of a neural network radiometric photothermal depth profiling method is verified using well-defined artificial samples with varying optical properties across the layers. The signal calculation model is shown to be accurate and the neural network approach to solve the inverse problem is shown to be feasible. Both from simulated and experimental radiometric signals, accurate reconstructions are obtained for heat source and optical-absorption coefficient profiles. © 2005 American Institute of Physics. [DOI: 10.1063/1.1821635] I. INTRODUCTION

Photothermal radiometry (PTR) is an established nondestructive, noncontact technique for characterization of thermal, optical, and electronic properties of materials.1–4 In the recent past PTR was also employed as a tool for mapping subsurface defects,5 for correlating thermal properties with microstructure and mechanical hardness in the case of hardened steel,6 as well as in biomedical studies.7,8 Important biomedical applications include characterization of the distribution of topically applied drugs in tissues and determination of the depth distribution of pigments in skin, i.e., hemoglobin and melanin for skin topology and disease diagnostics. The basic principle involving the PTR technique is the detection via black body infrared (IR) radiation of temperature variations of a sample induced by dynamically heating its surface region by absorbed pulsed or modulated laser light. For functionally graded materials, the time evolution of the depth dependence of temperature variations in the sample is closely related with the profile of the thermal properties of the sample via the thermal diffusion equation. This can be understood by looking at thermal diffusion in terms of the action radius of “thermal waves.” The so-called thermal diffusion length, which is proportional with the square root of diffusion time and inversely proportional with the square root of the modulation frequency, can be controlled and allows probing a material till a given depth. This feature has already been exploited to perform photothermal depth profiling of the thermal conductivity.6,9 On the other hand, for semitransparent samples, the tema)

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perature distribution in the sample is also related to the depth profile of its source, the deposited heat source distribution, which in turn is uniquely determined by the depth profile of the optical-absorption coefficient (and the optical reflectivity) in the sample. Hence, the depth profile of the concentration of the optically absorbing substance (e.g., pigment or dissolved fluid absorber) can be obtained directly by profiling the optical-absorption coefficient or the heat source distribution via the time or frequency dependence of the photothermal signal. This interesting potential has led to a growing interest in the development of inverse theories to reconstruct a map or profile of the underlying thermal or optical depth structure of the material from photothermal signals.10–15 In the present work, we describe a theoretical model for calculating the IR radiometric signal of the optically excited samples with varying optical-absorption profile and homogeneous thermal properties. To validate the model, we use least-squares fitting of experimental data obtained from photothermal radiometry measurements on artificial samples with known optical-absorption profiles. We also investigate the feasibility of neural network recognition of infrared radiometry signals. With a minimum of a priori information, we train a neural network to reconstruct an optical absorption or heat source profile from the frequency dependence of a radiometric signal. II. INFRARED RADIOMETRY SIGNAL OF AN OPTICALLY EXCITED SEMITRANSPARENT SAMPLE

In photothermal radiometry temperature variations in a sample, excited by a dynamical heat input caused by optical absorption, are detected via the related changes in black body radiation. The theory of photothermal radiometry with

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modulated and pulsed excitation has been discussed by several authors.16–18 Besides its dependence on the temperature profile in the sample, the radiometric signal has a complicated dependence on the spectral properties of the sample, both for optical absorption of the heating beam wavelength as well as for IR emissivity in the wavelength range of the detected infrared radiation. The influence of various factors such as detector sensitivity and spectral variation of the infrared-absorption coefficient on the radiometric signal has been detailed in the past19,20 and was usually circumvented by using a narrow detection range with the help of appropriate filters. In the present work, we are combining different aspects to define a general model for the optically inhomogeneous (multilayered) samples under investigation, taking into account the full detection range 共2 – 12 ␮m兲 of the used HgCdTe (MCT) infrared detector. Consider a Gaussian excitation beam of wavelength ⌳ and intensity modulated at a frequency f. Let ␤共⌳兲 be the absorption coefficient of the sample at the excitation wavelength and ␤IR共␭兲 represent the infrared spectral variation of the absorption coefficient of the sample. The locally emitted radiation intensity per unit wavelength ␭, IBB,␭ 共W m−3兲 resulting from a black body at temperature T is given by Planck’s law of radiation, IBB,␭共␭,T兲 =

2␲hc2 , ␭5关exp共hc/␭kBT兲 − 1兴

共1兲

where c is the velocity of light and kB is the Boltzmann constant. The total emitted radiation intensity, Itot,BB 共W m−2兲 is approximately given by the well-known Stefan– Boltzmann law, Itot,BB共T兲 =

冕

IBB,␭共␭,T兲d␭ ⬵

12␲共kBT兲4 ⬅ ␴T4 , h 3c 2

共2兲

␴ = 5.67⫻ 10−8 W m−2 K−4 is the Stefan–Boltzmann constant. For limited temperature variations ⌬T, Eq. (1) can be linearized around the dc part of the temperature, To, and emitted intensity changes are given by ⌬IBB,␭共␭,⌬T兲 =

冏 冏 ⳵IBB,␭ ⳵T

⌬T.

共3兲

To

In our photothermal experiment, ⌬T is the ac component of the temperature distribution and corresponds to the depthdependent temperature profile determined uniquely by the absorption of the excitation wavelength by the sample. Due to the finite penetration depth of the emitted IR radiation, the radiometric signal comprises attenuated contributions from all depths z. For radiation detection in a range d␭ around a center wavelength ␭, combining the above factors, the radiometric intensity variation ⌬I 共W m−2兲 emitted by the whole cross section of the sample can be expressed as ⌬I共␭兲 =

冏 冏 ⳵IBB,␭ ⳵T

␤IR共␭兲d␭ To

冕

0

⬁

⌬T共z兲exp关− ␤IR共␭兲z兴dz. 共3a兲

The factor ␤IR共␭兲 takes into account the limited IR emissivity of real materials. In our experiment, broadband detection

is utilized and hence the infrared spectral variation of the sample for the wavelength range of detection should be taken into account in the calculation of the radiometric signal S 共volts兲. This is done by integrating the spectral components ␤IR 共␭兲 within the detection band, weighted by the wavelength-dependent detector sensitivity obtained from the data sheet by the provider. The generalized form of the radiometric signal S 共V兲 is given by S共⌬T兲 =

冕冕冕

⫻

dxdydz

冏 冏 ⳵IBB,␭ ⳵T

冕

⌬T共x,y,z兲R共␭兲

␤IR共␭兲exp共− ␤IRz兲d␭,

共4兲

To

where R 共V / W兲 is the wavelength-dependent responsivity of the detector. For a black body at room temperature with ⌬T = 1 K and emissivity= 1, the total radiated IR intensity change is given by, ⌬IBB =

冕

⌬IBB,␭共␭兲d␭ =

冕冏 冏 ⳵IBB,␭ ⳵T

⌬T共z = 0兲d␭ To

⬵ 4␴T3o⌬T共z = 0兲 = 6.5 W m−2 . In the case of a typical HgCdTe infrared detector with a detection range between 2 and 12 ␮m (peak responsivity R = 60000 V / W), for a detection area A = 0.05⫻ 0.05 mm2, this results in a signal S共⌬T兲 =

冏 冏 冕冕冕 冕 冕 冏 冏 dxdy

= A⌬T共z = 0兲

⌬T共x,y,z兲R共␭兲

R共␭兲

⳵IBB,␭ ⳵T

⳵IBB,␭ ⳵T

d␭ To

d␭ ⬵ 0.4 mV.

To

In general, the ac component of the temperature distribution ⌬T共z兲 can be determined by solving the one-dimensional21 or three-dimensional (3D) heat diffusion equations,22 approximating an inhomogeneous sample by a multilayer. If the spot size of the excitation beam is large compared with the thermal diffusion length ␮ = 共␣ / ␲ f兲1/2, where ␣ is the thermal diffusivity of the material, then a one-dimensional treatment is sufficient, if not, the 3D model has to be used. The onedimensional model is also valid when the IR radiation from the whole heated area is uniformly collected. III. INVERSE PROBLEM: NEURAL NETWORK PROFILE RECONSTRUCTION METHOD

In a frequency domain experiment, the photothermal inverse problem consists ofextracting the depth profile of the optical-absorption coefficient, ␤共z兲, from the frequency dependence of the IR radiometric signal, ⌬S共f兲. The neural network (NN) implementation of this problem consists in getting the desired NN output 兵Ok , k = 1 , 2 , . . . , NO其, which corresponds to the discretized profile, ␤共zk兲, as accurate as possible from an input vector 兵I j , j = 1 , 2 , . . . , NI其, here ⌬S共f j兲. This is realized by training the NN, i.e., to change the weight values or NN parameters in a series of iterative cycles until satisfying recognition is obtained. A well-trained NN

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FIG. 2. Microphotography of one of the multilayered samples under investigation showing clearly the three layers, namely, top transparent layer, central colored layer, and bottom transparent layer. FIG. 1. Schematic of the experimental setup.

can recognize similar patterns as the ones it has encountered during the training on the basis of the stored training information. In view of optimizing the calculation time and convergence rate, here we have chosen to use a simple oneoutput network for every ␤k value.9 It is a two-layer NN with two or more tangent hyperbolic units in the hidden layer and one linear output unit. The output of the NN is given by

冉

2

O = W2,1 0 +

兺 u=1

NI

1,u W2,1 u ⫻ tanh W0 +

W1,u 兺 j ⫻ Ij j=1

冊

,

共5兲

are weight values with j the NN input number, m where Wm,u j the layer number, and u the unit number. For training the NN, we used a Levenberg–Marquardt routine from Norgaard.23 In this method, the root-meansquare error between the NN predicted values of the jth profiling parameter, ␤ p and their real values, ␤r, given by

冑兺 N

␧ j共␤ ,I,W兲 = p

关␤rn共z j兲 − ␤np共z j,I,W兲兴2 ,

共6兲

the average rms error over the examples (index n) is minimized in an iterative way. In every iteration cycle, all the elements in the weight matrix W are simultaneously changed along the direction opposite to the gradient ⳵␧ / ⳵W to keep the error going down while adjusting the weight values. In every training, 800 random-generated profiles and their corresponding signals calculated by Eq. (4) in a logarithmically spaced frequency range between 0.1 and 100 Hz were used. In order to prevent the neural network from recognizing generally irrelevant properties of its input signals, 1% Gaussian noise was added to the real and imaginary part of the signals.9 IV. EXPERIMENT A. Sample preparation

The multilayered samples under investigation were prepared by a UV polymerization technique (EFOS Lite light source, 10 W / cm2, 320– 480 nm). The monomer, glycerol propoxylate 共1PO / OH兲 triacrylate, obtained from Aldrich was mixed with 1% by weight of photoinitiator, Irgacure 651

j=1

TABLE I. The description of the different multilayered samples under investigation and the thickness of the layers estimated for the different samples by optical microscope as well as the thickness of the top transparent layer (depth of absorber) determined by the direct problem are shown. Thickness of layers

Sample Sample 1

Sample 2

Sample 3

Sample 4

Sample description (in the order from top layer to bottom layer) Transparent–oil red O– Transparent Transparent–oil red–O Transparent Transparent– carbon– Transparent Transparent– carbon– Transparent

By optical microscope with a typical error of 20 ␮m Central Bottom colored Top layer layer layer 共␮m兲 共␮m兲 共␮m兲

Direct problem (numerical fit)

Top layer 共␮m兲

330

213

175

194± 3

173

213

152

135± 3

195

165

106

90± 3

314

214

196

215± 4

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FIG. 3. Variation of amplitude and phase of the PTR signal with frequency of samples having oil red O as the absorber but at different depths: 䊐 sample 1; 䊊 sample 2. The solid line represents the numerical fit and the dashed and dotted lines represent the numerical simulations for values of depth of absorber, i.e., the thickness of the transparent top layer, which are, respectively, ⫹ and ⫺ three standard deviations from the exact fit.

FIG. 4. Amplitude and phase variation of the PTR signal with frequency for the samples having carbon as the absorber at different depths: 䊊 sample 3; 䊐 sample 4. The solid line represents the theoretical fit. The dashed and dotted lines, respectively, represent the numerical simulations of ⫹ and ⫺ three standard deviations for the values of the depth of absorber obtained from the exact fit.

(Ciba). Three-layered samples, with a transparent top layer, a colored central layer, and a transparent bottom layer were prepared on a plexiglass substrate. Colorants with known percent by weight, namely, oil red O (0.21%) and carbon powder (0.05%) combined with the monomer and photoinitiator mixture were used to prepare samples with two different kinds of colored layers. The thickness of the top transparent layer was varied so as to have absorbers at different depths. Two samples with carbon as absorber and two with Oil Red O as absorber were prepared.

collected and focused to a liquid-nitrogen-cooled MCT detector (detector bandwidth 2 – 12 ␮m and sensing area 0.05 ⫻ 0.05 mm2) using two 90° off-axis gold-coated parabolic mirrors. The signal from the detector was amplified using a preamplifier and fed to a lock-in amplifier (SR 830).

B. Photothermal radiometry setup

The experimental setup used for the measurement is as shown in Fig. 1. The excitation source was a diode-pumped solid-state laser (DPSS, ␭ = 532 nm). Intensity modulation of the laser beam was achieved using an acousto-optic modulator of IntraAction and was directed on the sample surface by a flat mirror. The infrared emission from the sample was

V. EXPERIMENTAL FREQUENCY SPECTRA AND RECONSTRUCTED PROFILES A. Direct problem

Figure 2 shows the cross section of one of the multilayered sample recorded using an optical microscope from which the thickness of the layers can be estimated (with an error of ±20 ␮m). The description of the different multilayered samples and the thickness of the layers estimated for the different samples are given in Table I. The variation of normalized amplitude and phase with frequency of the PTR signal for the samples having Oil Red O as absorber at different depths is shown in Fig. 3 and that

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FIG. 6. FTIR spectrum of the polymer sample. FIG. 5. Numerical simulations showing the influence of the spot size of the excitation beam on the ac temperature distribution and comparing it with the one-dimensional ac temperature distribution.

for samples having carbon as absorber at different depths is shown in Fig. 4. Typically, our signals were of the order of 1 mV at 1 Hz for the different samples. The symbols denote the experimental values and the solid line represents the theoretical fit, performed using ␹2 minimization. We have used a one-dimensional model to calculate the ac temperature distribution generating the radiometry signal. The numerical result is in good agreement with the experimental data. The frequency range of fitting is so chosen as to avoid the region where the signal-to-noise ratio is small. In the low-frequency region, there is no sign of three-dimensional effects. This is not surprising, since the pump beam spot size used in our experiment is large 共a ⬃ 5 mm兲 compared to the thermal diffusion length in the frequency range of experiment. To confirm this, numerical simulations (Fig. 5) were performed to compare one-dimensional and three-dimensional ac temperature distributions with the same thermal parameters of the multilayered samples under investigation. The threedimensional ac temperature distribution is calculated for different spot sizes of the beam (a = 0.5, 2, 4, and 5 mm). It is seen that as the spot size increases, the three-dimensional effect decreases and the signal finally coincides with the onedimensional case for the beam spot size 共a ⬵ 5 mm兲 used in the present experiment. For the precise calculation of ac temperature [Eq. (4)] for fitting and calculation of NN training signals, the absorption coefficient of the colored layer and the thermal diffusivity of the polymer sample were determined. The absorption coefficient of the two different colored polymer layers at the excitation wavelength was determined optically using Beer– Lambert’s law, I = I0 exp(−␤共⌳兲x), where I0 is the incident light intensity. I is the output light intensity after passing through the sample of thickness x. The values are found to be 13565 and 3500 m−1 for our polymer colored with oil red O and carbon, respectively. The thermal diffusivity of the sample was determined by the photopyroelectric technique in a standard (back) detection configuration24–26 using a 186-␮m-thick LiTaO3 single crystal as a pyroelectric sensor. The frequency behavior of the amplitude and phase of the pyroelectric signal was analyzed for obtaining thermal diffu-

sivity. In the model, the thermal diffusivity value used is 共1.8± 0.1兲 ⫻ 10−7 m2 s−1, which is the average of the values obtained from the amplitude 共1.9± 0.1兲 ⫻ 10−7 m2 s−1兲 and phase 共1.7± 0.1兲 ⫻ 10−7 m2 s−1兲 of the pyroelectric signal. Another parameter required in the numerical fitting is the IR-absorption coefficient, ␤IR, of the polymer sample in the detection wavelength range 共2 – 12 ␮m兲 of the HgCdTe detector used. The Fourier transform infrared (FTIR) spectrum of the sample was recorded using a FTIR (IFS 66) spectrophotometer and is shown in Fig. 6. From those data the infrared absorption coefficient (␤IR = absorbance/ thickness of the sample) was deduced. The thickness of the sample ␦ used for recording the spectrum, a crucial parameter in accurate determination of absorption coefficient, is difficult to obtain precisely. Hence in the present model, the thickness to be used in the calculation of the absorption coefficient was also left as an unknown parameter in numerical fit (Figs. 3 and 4) of the photothermal radiometry signal, which was found to be 75± 5 ␮m for all the multilayered samples. A close observation of the spectrum reveals that the value of absorbance of the polymer film varies considerably accounting for the large infrared penetration depth (=reciprocal of absorption coefficient) at certain wavelengths. This, together with the large variation of the infrared absorbance with wavelength in the detection range 共2 – 12 ␮m兲, increases the importance of contributions from different depths in the sample into the calculated radiometry signal, and the need for a weighted integration over the wavelength as in Eq. (4). In the FTIR spectrum, an absorbance peak corresponding to the wave number 共1700 cm−1兲 is saturated which makes it impossible to calculate the absolute value. Since the IR penetration depth 共dIR兲 at a particular wavelength (or wave number) is the reciprocal of the corresponding absorption coefficient, dIR would be very small. This implies that the IR emission at this wavelength can be treated as from the surface and has no contribution from the bulk of the polymer. The depth of the absorber in different samples, which is also equal to the thickness of the top transparent layer, determined from the fit, is given in Table I and is in good agreement with the value obtained using the optical microscope (typical error of 20 ␮m). It is also clear that the strength of the absorber (whether it is weak or strong) does

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FIG. 7. Reconstructed profiles of absorption coefficients 共␤兲 of the four samples and corresponding heat source calculated by the reconstructed ␤ (circles); NN-predicted heat source of the four-samples (diamond). The inset shows the reconstructed images based on the predicted data by neural network.

not affect the accuracy in determination of the depth at which it is since we could determine the depth of a weak absorber 共3500 m−1兲 with the same accuracy as that for the relatively strong absorber 共13 565 m−1兲. B. Neural network-reconstructed optical-absorption coefficient depth profile from the infrared radiometric photothermal frequency spectrum

For depth profiling reconstruction problems, it is usually difficult to get a satisfying reconstructed result from signals corresponding to strongly varying profiles, especially the ones with step-shaped profiles. One can use various sets of randomly generated step profiles, and profiles consisting of a finite number of spatial Fourier harmonics, and calculate their corresponding signals to train the NN. The signal calculation is based on the theoretical model already explained. In order to optimize the recognition efficiency, here we have chosen as training profiles, Gaussian functions, which are quite simple and smooth in shape compared with step functions or profiles with a lot of Fourier harmonics. Our simulations have shown that Gaussian profile training sets are very suitable to train a NN to recognize the profile of sandwichlike samples, especially for predicting the position and

FIG. 8. (a) Reconstructed profile of sample 2. (b) Circles represent the recalculated amplitude signals by the reconstructed profile and the solid line represents the experimental data. (c) Circles represent the recalculated phase signals and the solid line represents the experimental data.

width of a step. We would like to point out that, in order to restrict all the NN output values in a physical allowed range, one can make a logarithmic transform on parameters (absorption coefficients or heat source) and then perform the inverse transform (exponent) on corresponding NN outputs. Moreover, it is worthy to mention that, in our case, the first training started from a random initialized weight matrix for the ␤1 network, and then used its final weights as the initial weights for training the ␤2 network, and so on. The best training for the ␤1 NN requires about 20–25 iterations and 10–15 for the rest. In this way, optimum advantage can be taken from the thus introduced correlation between recognized adjacent ␤ values and thus smoothing effect of the reconstructed profiles. The reconstructed ␤ profiles, ␤NN共zk兲, the heat source distribution deduced from the reconstructed ␤ profiles, Q关␤NN共zk兲兴 = ␤NN−1 exp共− ␤NNzk兲, and the heat source distribution QNN共zk兲 directly obtained using from a dedicated NN, recognized from the IR radio-

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metric signals of the four samples, are shown in Fig. 7. There is a good agreement between the true sample features (Table I) and the profiles predicted by the NN, both concerning the position and width of the colored layer. As an additional criterion to confirm the reconstructed profiles, we have used the NN-predicted profile to recalculate the IR radiometric signal. Figure 8 shows a good correspondence. VI. CONCLUSION

In the present work, we have validated a NN radiometric photothermal depth profiling method, using well-defined artificial samples with varying optical properties across the layers while the thermal properties are homogeneous. The signal calculation model is shown to be accurate and the NN approach to solve the inverse problem is shown to be feasible. The used theoretical model is also applicable to the case of multiple absorbers, gradient of optical absorption, etc. Various effects such as thermal wave reflection at the layer boundaries in thermally inhomogeneous samples, 3D effects, and several kinds of a priori information about the sample, can be taken into account in a straightforward way, both in the signal calculation and in the NN recognition approach. The method therefore seems very promising for challenging problems of optical depth profiling in technological, agricultural, and biomedical applications. At present, in many cases, the proposed method seems to be the only viable alternative. 1

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