Effect of the multiple reflections of a light beam on the thermal wave field of a sample of finite thickness

Abstract

The thermal wave field in a sample of thickness L and optical absorption coefficient β is derived and analyzed, by considering the effects of the multiple reflections of a modulated light beam within the sample. By comparing the corresponding amplitude and phase spectra, in presence and absence of those reflections, it is shown that their effects are strongly determined by the product βL. When the thermal wave field is measured at the illuminated surface, those effects appear remarkably on the amplitude and phase in the thermally thin and thermally thick regimes, respectively, and for βL≲2. The deviation of the amplitude spectrum due to the multiple reflections of incident light beam can be as large as 25%. In contrast, when the thermal wave field is measured at the non-illuminated surface, the contribution of the multiple reflections shows up on both the amplitude and phase in the thermally thick regimes, and for βL≲5. These spectra exhibit deviations of about 20% from their corresponding values without the effects of the light reflections. The obtained results show that the accurate measurement of the thermal and/or optical properties of the sample material, based on the amplitude and phase spectra, requires the consideration of those multiple reflections, especially for the cases where βL≈1. The predictions of the proposed approach for the optical absorption coefficient are in reasonable agreement with the results obtained through an independent technique based on a spectrometer.

Photothermal characterization of the thermal properties of materials using four characteristic modulation frequencies in two-layer systems

“…The values of a 1 and e 1 obtained using this two-parameter fitting procedure of the normalized amplitude, normalized phase, and of these PTR signals simultaneously; are also summarized in Table I. Note that in Table I, the values of the thermal diffusivity and thermal effusivity obtained from the four characteristic frequencies f 1 , f M , f 0 , and f m are in good agreement among them, with the values determined by the twoparameter fitting procedures and are consistent with the values reported in the literature, 20,21 within a deviation of about 8% or smaller. This underestimation, though small, may be due to the fact that our samples perhaps were not totally opaque to the red laser beam used with a wavelength of 635 nm, 22 as is usually the case for glassy carbon samples. Concerning the different degrees of accuracy, the following remarks can be pointed out: (1) The best accuracy for the thermal diffusivity and thermal effusivity with respect to the their corresponding values obtained by means of the simultaneous fitting of A and /, and the ones reported in the literature; are provided by the frequency f 0 at which the normalized phase vanishes.…”

Experimental photoacoustic observation of the photogenerated excess carrier influence on the thermoelastic response of n-type silicon

“…23 Here, we must pay attention to the possible important effects of the multiple reflections of the modulated light beam on the temperature field of the sample. These effects are strongly determined by the product βl between the absorption coefficient (β) and thickness (l) of the sample [29][30][31] such that they are significant at the illuminated (non-illuminated) sample surface for βl 2 (βl 5) only. 31 In our case, the absorption coefficient of our Si samples is β = 2.58 × 10 5 m −1 at the wavelength of 660 nm (red) of the used light source.…”

“…These effects are strongly determined by the product βl between the absorption coefficient (β) and thickness (l) of the sample [29][30][31] such that they are significant at the illuminated (non-illuminated) sample surface for βl 2 (βl 5) only. 31 In our case, the absorption coefficient of our Si samples is β = 2.58 × 10 5 m −1 at the wavelength of 660 nm (red) of the used light source. This yields values βl > 5, for the three samples considered in our experiments (βl 1 % 39, βl 2 % 22 and βl 3 % 8), and, therefore, the semi-infinite sample approximation used in the present work is well justified.…”

Photothermal Techniques in Material Characterization

“…This discrepancy could be due to the simplified theoretical model which ignores multiple reflections in the thin film. According to Arjona et.al [36] this is more significant for phase channel in the higher modulation frequencies while for the amplitude channel is more significant for the lower modulation frequencies. The calculated thermal parameters are within the range of values reported in the literature [30] for polymer carbon-nanotube composites.…”