## Appendix A Photogenerated Diffusion of Charge Carriers in Semiconductor  
The equation describing the transport of photogenerated charge carriers in a semiconductor, under the low-level injection approximation, is a linear, one-dimensional spatial diffusion equation for minority carriers. By applying the Laplace transform, this equation is reduced to the second-order differential equation in the complex domain [12]. In the case where optical absorption occurs in a thin layer near the illuminated semiconductor surface (i.e., when the sample is optically opaque and has a high optical absorption coefficient), the transport problem of photogenerated charge carriers is described by the following system of homogeneous differential equations (Equations (A1) and (A2)), with inhomogeneous boundary conditions (Equations (A3) and (A4)) [34]:  
\[\frac{d^{2}\overline{n}(x)}{dx^{2}}-\frac{1}{\overline{L}^{2}}\overline{n}(x)=0\] (A1)  
\[\overline{j}(x)=-D_{p/\epsilon}\frac{d\ \overline{n}(x)}{dx}\] (A2)  
\[\overline{j}(0)=\overline{G}(s)-s_{\overline{s}}\overline{n}(0)\] (A3)  
\[\overline{j}(l_{s})=s_{\overline{s}}\overline{n}(l_{s})\] (A4)In the above equations, \(\overline{G}(s)=(S_{0}/E)\overline{F}(s)\) denotes the spectral function of the optically generated source of excess carriers, under the assumption that the semiconductor is optically opaque (it is important to note that the model is derived under the assumption of shallow optical penetration depth, meaning that the generation of charge carriers occurs within a spatially confined region near the illuminated surface--practically at the surface itself) [34]. By solving the problem described by Equations (A1)-(A4), the following expression for the spectral function of concentration distribution of photogenerated minority carriers is obtained:  
\[\overline{n}(x)=\frac{S_{0}}{E}\overline{F}(s)\frac{\left(\frac{D_{e/p}}{L}-s_ {b}\right)e^{\frac{x-l_{b}}{L}}+\left(\frac{D_{e/p}}{L}+s_{b}\right)e^{-\frac{ x-l_{b}}{L}}}{\left(\frac{D_{e/p}}{L}+s_{g}\right)\left(\frac{D_{e/p}}{L}+s_{b} \right)e^{\frac{l_{b}}{L}}-\left(\frac{D_{e/p}}{L}-s_{g}\right)\left(\frac{D_{ e/p}}{L}-s_{b}\right)e^{-\frac{l_{b}}{L}}}\] (A5)  
Based on Equation (A5), expressions describing the carrier concentrations at the semiconductor surfaces can be readily derived [34]. \[\overline{n}(0)=\frac{S_{0}}{E}\overline{F}(s)\frac{\left(\frac{D_{e/p}}{L}-s _{b}\right)e^{-\frac{l_{b}}{L}}+\left(\frac{D_{e/p}}{L}+s_{b}\right)e^{\frac{ l_{b}}{L}}}{\left(\frac{D_{e/p}}{L}+s_{g}\right)\left(\frac{D_{e/p}}{L}+s_{b} \right)e^{\frac{l_{b}}{L}}-\left(\frac{D_{e/p}}{L}-s_{g}\right)\left(\frac{D_{ e/p}}{L}-s_{b}\right)e^{-\frac{l_{b}}{L}}}\] (A6)  
\[\overline{n}(l_{s})=\frac{S_{0}}{E}\overline{F}(s)\frac{2\frac{D_{e/p}}{L}}{ \left(\frac{D_{e/p}}{L}+s_{g}\right)\left(\frac{D_{e/p}}{L}+s_{b}\right)e^{ \frac{l_{b}}{L}}-\left(\frac{D_{e/p}}{L}-s_{g}\right)\left(\frac{D_{e/p}}{L}- s_{b}\right)e^{-\frac{l_{b}}{L}}}\] (A7)  
In this work, we analyzed two asymptotic cases--the plasma-opaque and the plasma-transparent semiconductor sample--using an analogy with propagation processes. If the complex frequency is written as \(s=j\omega\) (stationary propagation), where the symbol \(j\) denotes the imaginary unit, the real part of the inverse of the complex propagation coefficient (Equation (34)) corresponds to the diffusion length of charge carriers [34; 50]. \[\text{Re}\bigg{\{}\frac{1}{\overline{L}}\bigg{\}}=\mu(\omega)=\sqrt{\frac{2D _{e/p}\tau}{1+\sqrt{1+\omega^{2}\tau^{2}}}}\] (A8)  
As can be seen from Equation (A8), the carrier diffusion length depends on the frequency of the harmonic excitation. As the frequency increases, the diffusion length decreases to zero, while for a frequency of zero, it reaches its maximum value [34; 50]:  
\[\mu_{\text{max}}=\sqrt{D_{e/p}\tau}\] (A9)  
If the sample thickness is much greater than the maximum diffusion length (\(l_{s}/\mu(\omega)>>1\)), the sample is plasma opaque.