## 2 Theory  
We calculate in this section the evolution of the surface temperature distribution produced by a rectangular heat source of area \(\Omega\) in a plane \(\Pi\) having an angle \(\varphi\) with the surface. We consider that the rectangle emits a homogeneous power density (flux) \(\varPhi\) that is constant during a given time interval [0,\(\tau\)] that we call burst. The rectangle has a width \(w\), parallel to the surface where the temperature is measured. The height of the rectangle is \(h\) and the depth of the shallowest side is \(d\). The maximum depth of the heat source is \(d_{max}=d+h\) sin\(\varphi\). The geometry of the problem is depicted in Figure 1.  
We start the calculation considering an instantaneous point-like heat source located at position \(\overrightarrow{r}\) that emits an energy \(H\) at a time \(t^{\prime}\) in an infinite homogeneous and isotropic medium of thermal conductivity and diffusivity \(K\) and \(D\), respectively. The temperature elevation above the ambient at any point \(\overrightarrow{r}\) and time \(t>t^{\prime}\) is given by the Green function [19]:  
\[T(\overrightarrow{r},t)=\frac{HD}{8K[\pi D(t-t^{\prime})]^{3/2}}e^{-\frac{ \overrightarrow{(r}-\overrightarrow{r}^{\prime})^{2}}{4D(t-t^{\prime})}} \tag{1}\]  
We consider now that the point-like heat source emits a burst of constant power \(P\) during a time interval [0,\(\tau\)]. The temperature elevation at any point in the infinite material can be calculated by integrating Equation (1) [19]:  
\[T(\overrightarrow{r},t)=\frac{P}{4\pi K|\overrightarrow{r}-\overrightarrow{ r^{\prime}}|}Erfc\left[\frac{|\overrightarrow{r}-\overrightarrow{r}^{\prime}|}{ \sqrt{4 Dt}}\right],0\leq t\leq\tau, \tag{2}\]  
\[T(\overrightarrow{r},t)=\frac{P}{4\pi K|\overrightarrow{r}-\overrightarrow{ r^{\prime}}|}\left\{Erfc\left[\frac{|\overrightarrow{r}-\overrightarrow{r}^{ \prime}|}{\sqrt{4 Dt}}\right]-Erfc\left[\frac{|\overrightarrow{r}- \overrightarrow{r}^{\prime}|}{\sqrt{4D(t-\tau)}}\right]\right\},\tau<t \tag{3}\]  
where \(Erfc\) is the complementary error function. In the next step, we take into account the spatial extension \(\Omega\) of the heat source. The temperature at any position and time can be calculated by integrating the contribution of point-like heat sources of an infinitesimal area \(ds^{\prime}\) within the region \(\Omega\), with an eventually position dependent flux \(\varPhi(\overrightarrow{r})\)\(\varPhi(\overrightarrow{r}):\)  
\[T(\overrightarrow{r},t)=\iint_{\Omega}\frac{\varPhi(\overrightarrow{r}^{ \prime})}{4\pi K|\overrightarrow{r}-\overrightarrow{r^{\prime}}|}Erfc \left[\frac{|\overrightarrow{r}-\overrightarrow{r}^{\prime}|}{\sqrt{4 Dt}} \right]ds^{\prime},0\leq t\leq\tau, \tag{4}\]  
Figure 1: Geometry of the problem. \[T(\overrightarrow{r},t)=\iint_{\Omega}\frac{\Phi(\overrightarrow{r}^{\prime})}{4 \pi K\overrightarrow{r}-\overrightarrow{r}^{\prime}}\bigg{\{}Erfc\bigg{[} \frac{\overrightarrow{r}-\overrightarrow{r}^{\prime}}{\sqrt{4 Dt}}\bigg{]}- Erfc\bigg{[}\frac{\overrightarrow{r}-\overrightarrow{r}^{\prime}}{\sqrt{4D(t- \tau)}}\bigg{]}\bigg{\}}ds^{\prime},\tau<t. \tag{5}\]  
Finally, we consider that the material is an infinite plate in the directions \(x\) and \(y\) but has a given thickness \(e\) in the \(z\) direction. The surface where data are collected is located in plane \(z=0\) (see Figure 1). If heat losses by convection and radiation can be neglected, the effect of the sample surfaces can be taken into account by applying the images method, i.e., considering the effect of reflected images of the heat source at the front and back surfaces.