## 3 Theory
Then, the solution for \(\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})/\delta\mathrm{T}_{0}\) under zero initial conditions has the following form:  
\[\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})=\delta\mathrm{T}_{0}\int_ {0}^{\mathrm{t}}\frac{1}{\mathrm{t}-\theta+(1+\mathrm{z}^{2})/4}\exp\bigg{[} \frac{-\mathrm{r}^{2}/4}{\mathrm{t}-\theta+(1+\mathrm{z}^{2})/4}\bigg{]} \mathrm{d}\theta. \tag{3}\]  
This solution describes the dependence of the temperature profile on the heating pulse duration. After the end of the heating pulse, the sample cooling process begins due to heat diffusion. The process is described by Equation (2) with a zero right-hand side and the initial condition in the form of the resulting solution for the temperature profile after heating. The solution \(\Delta\mathrm{T}_{\mathrm{free}}\) in this case has the following form:  
\[\Delta\mathrm{T}_{\mathrm{free}}\Big{(}\mathrm{t},\tau_{\mathrm{pulse}}, \mathrm{r},\mathrm{z}\Big{)}=\delta\mathrm{T}_{0}\int_{0}^{\tau_{\mathrm{ pulse}}}\frac{1}{\tau_{\mathrm{pulse}}-\theta+\mathrm{t}+(1+\mathrm{z}^{2})/4} \exp\Bigg{[}\frac{-\mathrm{r}^{2}/4}{\tau_{\mathrm{pulse}}-\theta+\mathrm{t}+( 1+\mathrm{z}^{2})/4}\Bigg{]}\mathrm{d}\theta. \tag{4}\]  
Further, this function describing the spatiotemporal temperature distribution should be substituted into the right-hand side of the parabolic equation (its derivation is given in [17] (Equation (2)) for the field \(\mathrm{E}_{\mathrm{1,pr}}\) of the time-varying component of the probe beam propagating through a region disturbed by inhomogeneous heating:  
\[\frac{\partial\mathrm{E}_{\mathrm{1,pr}}}{\partial\mathrm{z}}=-\frac{1}{2 \mathrm{i}}\Delta_{\perp}\mathrm{E}_{\mathrm{1,pr}}+\mathrm{i}\bigg{(}\frac{ \Delta\mathrm{T}(\mathrm{r},\mathrm{z},\mathrm{t})}{\delta\mathrm{T}_{0}} \bigg{)}\mathrm{E}_{\mathrm{0,pr}}(\mathrm{r},\mathrm{z}). \tag{5}\]  
In this equation, \(\mathrm{E}_{\mathrm{0,pr}}(\mathrm{r},\mathrm{z})\) is the field of an undisturbed Gaussian probe beam, and time \(\mathrm{t}\) is included in the equation as a parameter. When solving this equation, we first perform a direct Fourier transform over the transverse coordinate \(\mathrm{r}\). The result is a differentialequation with respect to the longitudinal coordinate z, which the inverse Fourier transform is performed after solving. Due to the Gaussian shape of the functional dependence of the right-hand side on the transverse coordinate, all the described transformations are trivial. Then, it is necessary to integrate over the diaphragm aperture the expression for the power density of the time-varying component \(\mathrm{L}_{\sim}=\mu\mathrm{cnR}\mathrm{e}\mathrm{R}\mathrm{e}\left(\mathrm{E} _{0.\mathrm{pr}^{*}}\times\mathrm{E}_{1.\mathrm{pr}}\right)/2\pi\) (c is the speed of light in vacuum, and n is the refractive index (\(\lambda=633\) nm) of the tested sample) in the cross section at a distance L from the exit face of the sample. The result is a formula for the time dependence of the power \(\widetilde{\mathrm{P}}\) of the time-varying component of the probing beam, which is incident on the photocathode:  
\[\widetilde{\mathrm{P}}\left(\mathrm{L},\mathrm{L}_{\mathrm{sample}},\mathrm{ R},\mathrm{t}\right)=\mathrm{K}\left(\mathrm{L},\mathrm{L}_{\mathrm{sample}}, \mathrm{R},\mathrm{t}\right)\mathrm{P}\frac{\mathrm{W}_{\mathrm{h}}(\mathrm{t })}{\pi\mathrm{a}_{\mathrm{h}}^{2}\mathrm{C}\rho}\alpha_{\mathrm{h}}, \tag{6}\]  
where L is the distance from the exit face of the sample to the section projected by the lens L\({}_{6}\) onto the photodiode diaphragm (see Figure 1); L\({}_{\mathrm{sample}}\) is the length of the tested sample; R is the radius of the diaphragm in front of the photodiode, which is normalized to a\({}_{\mathrm{h}}\) and taken with a coefficient corresponding to the projection into the section at a distance L from the exit face of the sample; and \(\mathrm{W}_{\mathrm{h}}(\mathrm{t})\) is the current energy of the heating pulse.