## 3 Theory  
Earlier in [17; 20], calculations of the power of the time-varying component of the probe radiation were performed for short pulse durations of heating radiation, when the heat diffusion effect can be neglected. In this section, we calculate the power of the time-varying component of the probe radiation with an allowance for heat diffusion. When solving this problem, we considered the deformation of the temperature profile \(\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})\) due to the heat conductivity effect described using the equation:  
\[\mathrm{C}\rho\frac{\partial\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z })}{\partial\mathrm{t}}-\Lambda\bigg{(}\frac{\partial^{2}}{\partial\mathrm{r }^{2}}+\frac{1}{\mathrm{r}}\frac{\partial}{\partial\mathrm{r}}\bigg{)}\Delta \mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})-\Lambda\bigg{(}\frac{\partial^{2 }}{\partial\mathrm{z}^{2}}\bigg{)}\Delta\mathrm{T}(\mathrm{t},\mathrm{r}, \mathrm{z})=\alpha_{\mathrm{h}}\mathrm{I}(\mathrm{t},\mathrm{r},\mathrm{z}). \tag{1}\]  
Here, \(\alpha_{\mathrm{h}}\) is the absorption coefficient of the heating wave (\(\mathrm{m}^{-1}\)) and \(\mathrm{I}(\mathrm{t},\mathrm{r},\mathrm{z})\) is the spatiotemporal distribution of the power density of the heating radiation (\(\mathrm{W}/\mathrm{cm}^{2}\)) with a profile corresponding to the Gaussian beam: \(\mathrm{I}(\mathrm{t},\mathrm{r},\mathrm{z})=\frac{\mathrm{I}_{0}\exp[- \alpha_{\mathrm{h}}(\mathrm{r}-\mathrm{z}_{0})]}{1+(\mathrm{z}/\mathrm{z}_{ \mathrm{d,h}})^{2}}\exp\left[-(\mathrm{r}/\mathrm{a}_{\mathrm{h}})^{2}\bigg{(} \frac{1}{1+(\mathrm{z}/\mathrm{z}_{\mathrm{d,h}})^{2}}\bigg{)}\right]\), where \(\mathrm{z}_{\mathrm{d,h}}=2\pi\mathrm{ma}_{\mathrm{h}}^{2}/\lambda_{\mathrm{h}}\), \(\lambda_{\mathrm{h}}\) is the wavelength of the heating radiation and \(\mathrm{z}_{0}\) is the coordinate of the entrance face of the sample. We transform the equation to dimensionless form using the replacements \(\mathrm{r}=\mathrm{r}^{\mathrm{old}}/\mathrm{a}_{\mathrm{h}}\), \(\mathrm{z}=\mathrm{z}^{\mathrm{old}}/\mathrm{z}_{\mathrm{d,h}}\), and \(\mathrm{t}=\mathrm{t}^{\mathrm{old}}/\tau_{\mathrm{diff}}\) (\"old\" refers to the old variables of Equation (1)). Then, we obtain an equation in dimensionless independent variables:  
\[\frac{\partial\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})}{\partial \mathrm{t}}-\bigg{(}\frac{\partial^{2}}{\partial\mathrm{r}^{2}}+\frac{1}{ \mathrm{r}}\frac{\partial}{\partial\mathrm{r}}\bigg{)}\Delta\mathrm{T}( \mathrm{t},\mathrm{r},\mathrm{z})-\mu\bigg{(}\frac{\partial^{2}}{\partial \mathrm{z}^{2}}\bigg{)}\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})= \delta\mathrm{T}_{0}\frac{\mathrm{I}(\mathrm{t},\mathrm{r},\mathrm{z})}{ \mathrm{I}_{0}}. \tag{2}\]  
Here, \(\delta\mathrm{T}_{0}=\tau_{\mathrm{diff}}\alpha_{\mathrm{h}}\mathrm{I}_{0}/ \mathrm{C}\rho\) and \(\mu=\big{(}\mathrm{a}_{\mathrm{h}}/\big{(}\mathrm{z}^{\mathrm{old}}/\mathrm{z }_{\mathrm{d,h}}\big{)}\big{)}^{2}\) is a small parameter, the value of which in the experiment was \(\sim\)\(4\times 10^{-6}\). The smallness of the parameter is related to the slow change in the temperature distribution \(\Delta\mathrm{T}(\mathrm{t},\mathrm{r},\mathrm{z})\) along the \(z\) coordinate in a focused single-mode laser heating beam compared with the corresponding transverse change. With an allowance for the value of the parameter, the term \(\mu\big{(}\partial^{2}/\partial\mathrm{z}^{2}\big{)}\Delta\mathrm{T}( \mathrm{t},\mathrm{r},\mathrm{z})\) in Equation (2) can be neglected. Then, in further expressions, all functions will depend parametrically. The right-hand side of Equation (2) has a Gaussian dependence on normalized coordinates, which corresponds to a real heating beam, \(\mathrm{I}(\mathrm{t},\mathrm{r},\mathrm{z})=\mathrm{I}_{0}\exp\left[-\mathrm{ r}^{2}\big{(}1/\big{(}1+\mathrm{z}^{2}\big{)}\big{)}\right]/\big{(}1+ \mathrm{z}^{2}\big{)}\).