Numerical Modeling of Photothermal Experiments on Layered Samples with MirageEffect Signal Detection
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Abstract
Modeling of the probe beam deflection caused by temperature gradients for layered sample was realized in COMSOL Multiphysics, which utilizes finite element method to analyze heat transport. The sample consisted of a 100nmthick layer on a 500\(\upmu \)mthick substrate. It was also assumed that the sample was illuminated with either a Gaussian or a flat top beam of harmonically modulated intensity. To obtain the probe beam deflection signal, the normal and tangential components of the temperature gradient in the air above the sample were integrated over the probe beam path. The numerical model of the experiment gave insight into the various parameter dependencies, e.g., the thermal and optical properties of the substrate and the layer, and the geometry of the experiment. These insights are used in the analysis of experimental data and in the planning of future measurements.
Keywords
Layered samples Mirage effect Numerical modeling Photodeflection1 Introduction
The detection of temperature disturbances by the use of the mirage effect is the basis of various optical techniques known as photothermal deflection measurements. Illuminating a sample with an intensitymodulated light beam generates a nonstationary temperature disturbance connected with temperature gradients. These temperature gradients induce changes in the refractive index through the thermooptic effect, which is essential to the optical detection of temperature gradients. A probe light beam passing through the disturbed region can be deflected, changing the intensity distribution in its cross section. This effect is called the mirage effect. Application of the mirage effect for signal detection in photothermal measurements was proposed by Boccara and Fournier [1], and Murphy and Aamodt [2].
Qualitative descriptions of the mirage effect are simple. However, its quantitative model—especially when the real probe beam and 3D temperature field are taken into account—is quite complicated. The ray model is the simplest theoretical model. It is based on the principles of geometrical optics and the assumption that the probe beam can be treated as a single ray [3]. This ray is deflected on temperature gradients, and the sum of the deflections along a ray path is proportional to the measured signal. This model was recently improved to take into account the finite cross section of the probe beam [4]. The probe beam was considered as a bundle of rays. The signal was the weighted sum of the rays’ deflections over the beam cross section. However, this model does not take into account the differences in the phase shifts for rays using different pathways and having varying interference effects. The wave nature of the probe beam was taken into account in the model proposed by Glazov and Muratikov [5]. The propagation of probe beam through the region with a perturbed refractive index distribution—called the thermal lens—was analyzed. The thermal lens was treated as a single phase, so the influence on the distribution of the probe beam amplitude was not considered. A model considering both the amplitude and the phase change of light in the probe beam caused by the thermal lens was proposed in Ref. [6, 7, 8, 9]. It is based on the complex geometrical optics equations, but is restricted to probe beams with Gaussian profiles.
This short review of existing models of photothermal deflection measurements contains only the main proposals for theoretically modeling. However, even in the simplest geometrical optics models, an analytical solution for probe beam deflection can only be obtained in simple cases, e.g., 1D temperature fields.
2 Numerical Model
2.1 Temperature Field
The first step in the modeling of the photothermal deflection measurement is the determination of the temperature field in the sample and its surrounding. Consider a sample with a 100nmthin layer on 500\(\upmu \)mthick substrate. The sample was surrounded by air modeled by two 3mmthick layers above and below the sample. To maintain axial symmetry, all layers were enclosed in a cylinder 5 mm in radius. The geometry of the system is shown in Fig. 1. In such geometry, the symmetry axis coincides with the zaxis, the temperature field is a function of two variables r and z, and the solution must be found at the halfplane \(r > 0\).
 the Gaussian beam with the light intensity at \(z = 0\)$$\begin{aligned} I\left( {r,t} \right) =I_0 \hbox {exp}\left( {\frac{r^{2}}{2R^{2}}} \right) \hbox {sin}\left( {2\pi ft} \right) , \end{aligned}$$(1)
 and the flat top beam with the light intensity at z = 0where \(I_{0}\) is the maximum light intensity in the power beam, R is a parameter characterizing its radius, u is a parameter describing the width of the flap top beam edge, and f is the light modulation frequency.$$\begin{aligned} I\left( {r,t} \right) =\frac{I_0 }{\hbox {exp}\left( {\frac{rR}{u}} \right) +1}\hbox {sin}\left( {2\pi ft} \right) , \end{aligned}$$(2)
Exemplary temperature and temperature gradient distributions obtained for the Gaussian beam with \(R = 88\, \upmu \hbox {m}\), and homogeneous polyethylene (PET) sample (thermal conductivity \(\kappa = 0.052\,\hbox {W}{\cdot }\hbox {m}^{1}\hbox {K}^{1}\), density \(\rho = 950\,\hbox {kg}{\cdot }\hbox {m}^{3}\), specific heat \(c = 2300\,\hbox {J}{\cdot }\hbox {kg}^{1}\hbox {K}^{1}\), and the optical absorption coefficient \(\beta _{1} = \beta _{2} = 80\, \hbox {m}^{1})\)) are shown in Fig. 2. These calculations were carried out for the modulation frequency \(f = 500\hbox { Hz}\), with maximum light intensity \(I_{0} = 3{\times }10^{7}\hbox { W}{\cdot }\hbox { m}^{2}\), and the distributions shown correspond to \(t = 0.059\,\hbox {s}\).
2.2 Photodeflection Signal
3 Analysis
The numerical model of photothermal deflection measurements allows for the analysis of the influence of various parameters of the model on measured signals. Two main goals of this analysis are follows: (1) Does the variation of selected sample parameters influence measured dependencies and can it be determined? And (2) does the simplified method for analysis of experimental data give correct values of sample parameters? It is important to carry out the analysis for realistic distributions of the light intensity in the power beam cross section. The general assumption is that the power beam is Gaussian. Nowadays, the laser diodes are used in many photothermal experiments to generate the disturbance of the temperature field. The light from the source is guided to the sample through an optical fiber. The light intensity distribution along the diameter of a light spot is depicted in Fig. 5. The spot is obtained by focusing light from optical fiber on the sample surface. The experimental distribution is fitted by two theoretical distributions: the Gaussian one described by Eq. 1, and the flat top beam described by Eq. 2.

Gaussian beam: \(R = 88.4\,\upmu \hbox {m}\), \(I_{0} = 4.9\,\hbox {a.u.}\)

flat top beam: \(R = 114\,\upmu \hbox {m}\), \(u = 10\,\upmu \hbox {m}\), \(I_{0} = 4.4\,\hbox {a.u.}\)
The main conclusion built on a comparison of Figs. 6 and 7 is that the influence of the power beam shape on analyzed dependencies is more pronounced in the transparent sample with low thermal diffusivity. In the sample with high thermal diffusivity and relatively long thermal diffusion length, the temperature disturbance propagates along the sample surface. This leads to broadening of calculated dependencies, but also homogenizes the temperature distribution. Therefore, differences between dependencies for Gaussian and flat top beams are small. In the PET sample, these dependencies are narrower and the influence of power beam shape is more pronounced.
As the power beam used in experiments can be better described as the flat top beam, further analysis was carried out for this type of beam.
3.1 Influence of Transparent Thin Layer on Thick Substrate
As is mentioned in the previous section, an demonstrative analysis presented in the paper was carried out for two sample substrates—PET and Si. The purpose of this analysis described in this section is to determine how a thin layer can be deposited onto a \(500\,\upmu \hbox {m}\) substrate to still be “visible” in photothermal deflection measurements. Calculations were carried out for semitransparent layers \((\beta _{1}= 80\,\hbox {m}^{1})\) of various thicknesses (\(10\,\upmu \hbox {m}, 1\,\upmu \hbox {m}, and\;0.1\,\upmu \hbox {m}\)). In the case of a Si substrate, layers with high (\(\kappa _{1} = 500\,\hbox {W}{\cdot }\hbox { m}^{1}\hbox {K}^{1})\) and low (\(\kappa _{1} = 10 \hbox { W}{\cdot }\hbox { m}^{1}\hbox {K}^{1})\) thermal conductivities were considered. In both cases, the layer density was \(\rho = 2250\,\hbox {kg}{\cdot }\hbox { m}^{3}\), and the heat capacity \(c = 707\hbox { J}{\cdot }\hbox { kg}^{1}\hbox {K}^{1}\). These parameters correspond to the thermal diffusivities \(3.14\times 10^{4}\,\hbox { m}^{2}{\cdot }\hbox { s}^{1}\) and \(6.29 \times 10^{6}\,\hbox { m}^{2}{\cdot }\hbox { s}^{1}\), respectively. In the case of a PET substrate only, the layer with high thermal diffusivity was considered.
The normal and the tangential phases of the probe beam deflection calculated for a pure Si substrate sample and a layered sample with various layer thicknesses are shown in Fig. 8. Calculations were carried out for the layer with relatively high thermal conductivity (\(\kappa _{1} = 500\,\hbox {W}{\cdot }\hbox {m}^{1}\hbox {K}^{1})\), and \(h = 100\, \upmu \hbox {m}\). The influence of the layer on the photodeflection signal is visible only for the 10\(\upmu \hbox {m}\)thick layer. The influence of thinner layers is practically insignificant. Moreover, similar graphs obtained for the layer with lower thermal conductivity (\(\kappa _{1} = 10\,\hbox {W}{\cdot }\hbox { m}^{1}\hbox {K}^{1})\) revealed that even a 10\(\upmu \hbox {m}\)thick layer on Si does not cause noticeable changes in analyzed dependencies.
In the case of a conductive film deposited on semitransparent substrate with very low thermal conductivity, different behavior is observed. The influence of the layer on both the normal and the tangential phases is well pronounced (Fig. 9). The same conclusion can be made for respective amplitudes. The analysis performed for the layer with lower thermal conductivity (\(\kappa _{1} = 10\,\hbox {W}{\cdot }\hbox {m}^{1}\hbox {K}^{1})\) showed that even 1\(\upmu \hbox {m}\)thick layer should be detectable in a photothermal deflection experiment.
3.2 Simplified Analysis Based on Linear Relations
The possibility of a simplified analysis of data from photothermal deflection measurements was analyzed by Salazar and SánchezLavega [13]. They concluded that in the case of a probe beam passing over a parallel sample surface, linear relations can be used in two aspects: the phase of normal deflection depends linearly on h for \(s = 0\), and the phase of tangential deflection depends linearly on s for \(h = 0\). In both cases, a pointlike heat source was assumed. In the case of \(\psi _{n}(h)\) dependence, it remains linear also for 1D temperature field (homogeneously illuminated sample surface).
The other expected linear dependence is the phase of tangential deflection as a function of the power–probe beam distance s. Two such dependencies obtained for a Si sample with power beam radii \(11.4\, \upmu \hbox {m}\) and \(114\,\upmu \hbox {m}\) are shown in Fig. 11. Both dependencies are practically linear for \(s > 200\, \upmu \hbox {m}\) and have almost the same slopes.
More detailed analysis revealed that the dependence calculated for a tightly focused beam is less steep. The slopes calculated for \(h = 100\,\upmu \hbox {m}\) and the s in range 1000 \(\upmu \hbox {m}\)–1500 \(\upmu \hbox {m}\) were \(4.59\times 10^{3}\hbox { m}^{1}\) and \(4.61\,\times \,10^{3}\hbox { m}^{1}\) for R equaled to \(11.4\, \upmu \hbox {m}\) and \(114\, \upmu \hbox {m}\), respectively. Calculated slopes depend also on h and the range of s taken. As a consequence, the thermal diffusivity value obtained from simplified analysis based on linear regression varies when h or s ranges are changed. Figure 12 shows dependencies of thermal diffusivities on the sample—probe beam distance h calculated for two s ranges: 600–900\(\upmu \hbox {m}\) and 1200–1500\(\upmu \hbox {m}\), and two diameters of the power beam: \(11.4 \upmu \hbox {m}\) and \(114 \upmu \hbox {m}\).
As it follows from the figure, the thermal diffusivity obtained from regression analysis carried out for s from 600 \(\upmu \hbox {m}\) to 900 \(\upmu \hbox {m}\) is underestimated of about 20 %. But analysis for 1200 \(\upmu \hbox {m}\)–1500 \(\upmu \hbox {m}\) gives an overestimated value of the thermal diffusivity. However, in this case, the error is smaller, and values obtained for small h give good estimation of the actual thermal diffusivity of the sample. It is worth mentioning that the influence of the power beam radius on obtained values is relatively small.
As it follows from Figs. 6 and 7, determination of the thermal diffusivity from the slope of the linear part of \(\psi _{\mathrm{t}}(s)\) dependence is possible only for opaque samples. For a transparent sample (Fig. 7), it is not possible to identify a linear portion in this behavior.
4 Conclusions
Numerical modeling of physical experiments has become more and more popular due to its flexibility and ability to consider complex models, which better convey real experiments. In photothermal experiments, this concerns the geometry of measurements (the distribution of the light intensity in power beam) and the thermal and optical properties of the sample. Pure analytical models are based on many simplifications because obtaining an analytical solution for more realistic models is not possible. As we have shown in this paper, the numerical modeling of photothermal deflection measurements allows for the analysis of the influence of various parameters of a model on measured signals, including the influences of power beam shape and the thin film deposited on the thick substrate. However, it is also possible to add further complexity, such as the influence of anisotropy in thermal properties of the layer and the influence of the thermal resistance at layer–substrate interface. It is also shown that numerical modeling allows for the analysis of the correctness of oftused simplified methods for experimental data analysis based on linear relations.
Another merit of numerical modeling is the possibility of preliminary analysis regarding the usefulness of selected measuring methods for the determination of defined sample properties. Results presented in this work illustrated the sensitivity of measurement to various parameters of a model and hint at possible modifications of experimental procedure to achieve defined goals.
Numerically modeling an experiment can be also used for the determination of physical properties of a sample based on best fitting procedures. In this case, calculations are more timeconsuming in comparison with curve fitting based on an analytical model, but it is still possible.
Notes
Acknowledgements
A. KaźmierczakBałata wishes to thank Dr. Kurt E. Harris for his assistance in the final version of the manuscript.
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