## 3 Theoretical Models  
As shown in Figure 5 we used two geometries: cross and in-plane. In-plane geometry is achieved by significantly reducing the width of the excitation beam. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Thicknesses**} & \\  & **Estimated from** & & **Description** & **Metalic Layer on** \\  & **Reflectivity** & & **the Top** \\  & **Measurement**[24] & & & \\ \hline Sample \#1 & - & Undoped GaAs used & \\  & & as substrate; & Ti 50 nm \\ Sample \#2 & 1980 nm & Thin Al\({}_{0.33}\)Ga\({}_{0.67}\)As alloy & Au 100 nm \\ Sample \#3 & 520 nm & AlAs/GaAs superlattice with & Ti 50 nm \\  & & period thickness of 52 nm & \\ Sample \#4 & 520 nm & AlAs/GaAs superlattice with & \\  & & period thickness of 52 nm & Au 100 nm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the sample properties. Figure 4: SEM image of superlattice sample (samples #3 and #4). Although Figure 5 suggests that the lower the frequency the better the measurement of in-plane thermal conductivity, it still depends on the thermal parameters (the better the parameter determination sensitivity, range shifts toward higher frequencies, see Figure 6). The three-dimensional signal can be written based on the models already developed and published literature in [23]:  
\[\Theta(\omega,0)=-\frac{\mathrm{Q}}{2\pi}\int_{0}^{\infty}\frac{\mathrm{D}( \lambda)}{\mathrm{C}(\lambda)}\mathrm{e}^{\frac{-(\lambda_{\mathrm{WW}})^{2}}{ \mathrm{B}}\lambda}\mathrm{d}\lambda \tag{2}\]  
where \(\mathrm{Q}\) is the laser power and w0 is the laser spot \(1/\mathrm{e}2\). The coefficient \(\mathrm{C}\) and \(\mathrm{D}\) can be calculated using:  
\[\left[ \begin{array}{cc}\mathrm{A}(\lambda)&\mathrm{B}(\lambda)\\ \mathrm{C}(\lambda)&\mathrm{D}(\lambda)\end{array} \right]=\mathrm{A}_{3}\; \mathrm{M}_{32}\mathrm{A}_{2}\mathrm{M}_{21}\mathrm{A}_{1} \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Three Layer** \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{T}}\mathrm{(W/mK)}\) at \(-50\;^{\circ}\mathrm{C}\) & 22.9 \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{T}}\mathrm{(W/mK)}\) at \(0\;^{\circ}\mathrm{C}\) & 21.5 \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{T}}\mathrm{(W/mK)}\) at \(50\;^{\circ}\mathrm{C}\) & 21.0 \\ \hline \(\alpha_{1}=\alpha_{\mathrm{T}}\mathrm{(m^{2}/s)}\) for all temperatures & \(8\times 10^{-6}\) \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{au}}\mathrm{(W/mK)}\) at \(-50\;^{\circ}\mathrm{C}\) & 321 \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{Au}}\mathrm{(W/mK)}\) at \(0\;^{\circ}\mathrm{C}\) & 318 \\ \hline \(\mathrm{k}_{1}=\mathrm{k}_{\mathrm{Au}}\mathrm{(W/mK)}\) at \(50\;^{\circ}\mathrm{C}\) & 315 \\ \hline \(\alpha_{1}=\alpha_{\mathrm{Au}}\mathrm{(m^{2}/s)}\) for all temperatures & \(127\times 10^{-6}\) \\ \hline \(\mathrm{k}_{\mathrm{T}/\mathrm{GaAs}}\mathrm{(R_{12})}\) & \(2.0\times 10^{-8}\mathrm{(m^{2}\;KW^{-1})}\) from reference measurement \\ \hline \(\mathrm{k}_{3}=\mathrm{k}_{\mathrm{GaAs}}\mathrm{(W/mK)}\) at \(-50\;^{\circ}\mathrm{C}\) & 70 \\ \hline \(\alpha_{3}=\alpha_{\mathrm{GaAs}}\mathrm{(m^{2}/s)}\) at \(-50\;^{\circ}\mathrm{C}\) & \(4.2\times 10^{-5}\) \\ \hline \(\mathrm{k}_{3}=\mathrm{k}_{\mathrm{GaAs}}\mathrm{(W/mK)}\) at \(0\;^{\circ}\mathrm{C}\) & 60 \\ \hline \(\alpha_{3}=\alpha_{\mathrm{GaAs}}\mathrm{(m^{2}/s)}\) at \(0\;^{\circ}\mathrm{C}\) & \(3.2\times 10^{-5}\) \\ \hline \(\mathrm{k}_{3}=\mathrm{k}_{\mathrm{GaAs}}\mathrm{(W/mK)}\) at \(50\;^{\circ}\mathrm{C}\) & 30 \\ \hline \(\alpha_{3}=\alpha_{\mathrm{GaAs}}\mathrm{(m^{2}/s)}\) at \(50\;^{\circ}\mathrm{C}\) & \(1.7\times 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Constant parameters used for calculation. Figure 5: Cross-plane (**left**) and in-plane (**right**) geometry in frequency-domain methods. Each layer in the model is described by matrices A3, A2, A1 specified by:  
\[\mathrm{A_{n}}=\left[