## 3 Analysis and Discussion
The time interval was selected to match the excitation pulse duration, which reflects our experimental conditions. This pulse width corresponds to the excitation used in our PA setup and was chosen based on our real measurements on silicon membranes (see Refs. [22; 23]). In the analysis, it was assumed that the pulse duration is \(T=50\,\mathrm{ms}\). The normalized signal was observed over a time interval of \(t=2T\). The values of surface recombination velocity (Table 1) were chosen based on the analysis of frequency-domain PA measurements on semiconductors [18]. The parameters of the Si sample used in the calculations are provided in Table 1[18; 61]. Let us consider the thicknesses of the Si sample for which the assumptions of the derived model are satisfied (the absorption coefficient is significantly larger than the inverse of the sample thickness, justifying the assumption of strong absorption in the thin layer close to the illuminated region--practically surface absorption). The optical absorption coefficient of silicon is \(10^{5}\) m\({}^{-1}\) for light in the visible part of the electromagnetic spectrum [62]. Consequently, the approximation of an optically opaque sample is reasonable for silicon samples thicker than 10 \(\upmu\)m.  
Since the thermal diffusivity of silicon is 9\(\cdot\)10\({}^{-5}\) m\({}^{2}\)s\({}^{-1}\)[18], the cutoff frequency between thermally thin and thermally thick regimes for a sample thicker than 10 \(\upmu\)m can be calculated from Equation (41). It ranges from approximately 5 kHz for a 90 \(\upmu\)m thick sample to 300 kHz for a 10 \(\upmu\)m thick sample. For samples with a thickness of about 100 \(\upmu\)m, the cutoff frequency varies from 50 Hz for a 900 \(\upmu\)m thick sample to 3000 Hz for a 100 \(\upmu\)m thick sample. This implies that in the PA measurement range (from 50 Hz to a few kHz), silicon samples with a thickness of 10-150 \(\upmu\)m can be considered thermally thin. Finally, in our model, we considered two cases related to the ratio of the maximum diffusion length of minority carriers to the geometric thickness of the sample. These include the plasma-opaque case, where this ratio is less than one, and the plasma-transparent case, where the ratio is greater than one. Since the maximum diffusion length of minority carriers is given by the square root of the product of their diffusion coefficient and lifetime (see Appendix B), it can be estimated based on the parameters given in Table 1 that a p-doped silicon sample with a thickness greater than 135 \(\upmu\)m is plasma-opaque, while an n-doped silicon sample is plasma-opaque if it is thicker than 75 \(\upmu\)m. This means that Si samples, whether p- or n-doped, with a thickness below 75 \(\upmu\)m are plasma-transparent, whereas samples thicker than 135 \(\upmu\)m are plasma-opaque. In the further analysis, we considered silicon samples with a thickness of 20 \(\upmu\)m as representatives of plasma-transparent, thermally thin, and optically opaque semiconductor samples, and samples with a thickness of 140 \(\upmu\)m as representatives of plasma-opaque, thermally thin, and optically opaque semiconductor samples. Figure 5 shows the normalized time-domain temperature signal (evolution of temperature change on the non-illuminated side) for a thermally thin, plasma-opaque n-doped silicon sample (\(l_{s}\) = 140 \(\upmu\)m, \(D_{p}\) = 1.2 \(\times\) 10\({}^{-3}\) m\({}^{2}\)s\({}^{-1}\)). The red line represents the signal originating from lattice thermalization (fast thermal source), while the blue lines correspond to signals influenced by surface recombination for different values of \(s_{g}\). The green