## 2 Theory
The temperature elevation in the material is given by:  
\[T(\overrightarrow{r},t)=\iint_{\Omega+\Omega^{\prime}+\Omega^{\prime\prime}+ \Omega^{\prime\prime\prime}+-}\frac{\Phi(\overrightarrow{r}^{\prime})}{4\pi K \overrightarrow{r}-\overrightarrow{r}^{\prime}}Erfc\bigg{[}\frac{ \overrightarrow{r}-\overrightarrow{r}^{\prime}}{\sqrt{4 Dt}}\bigg{]}ds^{ \prime},0\leq t\leq\tau, \tag{6}\]  
\[T(\overrightarrow{r},t)=\iint_{\Omega+\Omega^{\prime}+\Omega^{\prime\prime \prime}+\Omega^{\prime\prime\prime}+-}\frac{\Phi(\overrightarrow{r}^{\prime} )}{4\pi K\overrightarrow{r}-\overrightarrow{r}^{\prime}}\bigg{\{}Erfc\bigg{[} \frac{\overrightarrow{r}-\overrightarrow{r}^{\prime}}{\sqrt{4 Dt}}\bigg{]}- Erfc\bigg{[}\frac{\overrightarrow{r}-\overrightarrow{r}^{\prime}}{\sqrt{4D(t- \tau)}}\bigg{]}\bigg{\}}ds^{\prime},\tau<t, \tag{7}\]  
where \(\Omega^{\prime}\) is the reflection of \(\Omega\) at the front surface of the sample (\(z=0\)), \(\Omega^{\prime\prime}\) is the reflection of \(\Omega\) at the rear surface of the sample (\(z=-e\)), \(\Omega^{\prime\prime\prime}\) is the reflection of \(\Omega\) twice, first at the front surface and then at the rear surface, and so forth. If the thickness of the sample \(e\) is much larger than the thermal diffusion length \(\mu_{\tau}=\sqrt{4D\tau}\) associated to the duration of the burst \(\tau\), then only \(\Omega^{\prime}\) contributes significantly in the summations in Equations (6) and (7). Taking into account that the temperature of interest is the distribution at the sample surface (plane \(z=0\)), the effect of a reflected image \(\Omega^{\prime}\) is equivalent to having a heat source of double the flux. The evolution of the temperature distribution at the surface then can be calculated as:  
\[T(\overrightarrow{r}_{z=0},t)=\iint_{\Omega}\frac{\Phi(\overrightarrow{r}^{ \prime})}{2\pi K\overrightarrow{r}_{z=0}-\overrightarrow{r}^{\prime}}Erfc \bigg{[}\frac{\overrightarrow{r}_{z=0}-\overrightarrow{r}^{\prime}}{\sqrt{4 Dt }}\bigg{]}ds^{\prime},0\leq t\leq\tau, \tag{8}\]  
\[T(\overrightarrow{r},t)=\frac{HD}{8K[\pi D(t-t^{\prime})]^{3/2}}e^{-\frac{ \overrightarrow{(r}-\overrightarrow{r}^{\prime})^{2}}{4D(t-t^{\prime})}} \tag{1}\]0  
In the particular case of the rectangular heat source shown in Figure 1 of width \(w\), height \(h\), and depth \(d\), making an angle \(\varphi\) with the surface of a semi-infinite material, and emitting a homogeneous flux \(\Phi\), the temperature at the surface results:  
\[ \begin{array}{c}T(x,y,0,t)=\frac{\Phi}{2\pi K}\int_{-\frac{\Psi}{2}}^{\frac{ \Psi}{2}}\int_{0}^{h\cos\varphi}\frac{\sqrt{1+m^{2}}}{\sqrt{\left(x-x^{\prime }\right)^{2}+\left(y-y^{\prime}\right)^{2}+\left(mx^{\prime}-d\right)^{2}}}\\ \\ \times Erfc\bigg{[}\frac{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime} \right)^{2}+\left(mx^{\prime}-d\right)^{2}}}{\sqrt{4 Dt}}\bigg{]}dx^{\prime}dy^ {\prime},0\leq t\leq\tau,\\ \\ T(x,y,0,t)=\frac{\Phi}{2\pi K}\int_{-\frac{\Psi}{2}}^{\frac{\Psi}{2}}\int_{0}^{h \cos\varphi}\frac{\sqrt{1+m^{2}}}{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y -y^{\prime}\right)^{2}+\left(mx^{\prime}-d\right)^{2}}}\\ \\ \times\bigg{[}Erfc\frac{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime} \right)^{2}+\left(mx^{\prime}-d\right)^{2}}}{\sqrt{4 Dt}}-Erfc\frac{\sqrt{\left( x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+\left(mx^{\prime}-d\right)^{2}}}{ \sqrt{4D(t-\tau)}}\bigg{]}dx^{\prime}dy^{\prime},\tau<t,\end{array}  \tag{10}\]  
where \(m=\tan\varphi\). In reference [20], we tackled the characterization (i.e., determination of width \(w\), height \(h\), and depth \(d\)) of rectangular vertical (\(\varphi=\pi/2\)) heat sources, assuming knowledge of the orientation of the heat source. Among all possible inclinations described by Equations (10) and (11), vertical heat sources are the most difficult to characterize as the depth of the elements emitting heat increases faster than for any other orientation. Accordingly, here we will focus on the impact of the angle of the heat source on the surface temperature.