## 3 Model
\right) \tag{3}\]  
where \(n_{ij}^{0}\) are the unperturbed components, \(\Delta n_{ij}^{s}\) are the stress-induced changes, and  
\[C_{1}=\frac{n_{0}^{3}(p_{11}-2\
u p_{12})}{2Y},\;C_{2}=\frac{n_{0}^{3}[p_{12}- \
u(p_{11}+p_{12})]}{2Y},\;C_{3}=\frac{n_{0}^{3}p_{44}}{2G} \tag{4}\]  
Figure 3: Stress calculation for a HPLD. (**a**) Schematic of upside-down mounted HPLD in which index guiding is achieved through a ridge waveguide. (**b**) 2D distribution of the stress tensor component \(\sigma_{xy}\) obtained through a FEM calculation with COMSOL Multiphysics®. The parameters \(p_{ij}\) are the strain-optic constants; \(Y\), \(G\), and, \(v\) are, respectively, the Young's modulus, the shear modulus, and the Poisson's ratio of the material layers. From a mechanical point of view, we approximated the epitaxial structure with a single equivalent GaAs layer whose opto-mechanical properties are taken from literature and reported in Table 1. These properties are very different from all other materials considered in the simulations (dielectrics, metals) and similar for all the epitaxial layers, thus justifying the single equivalent layer approximation [17; 18] aimed to strong computational time reduction. Considering a simplified one-dimensional approximation with light propagating along the longitudinal axis \(z\), the Jones matrix formalism can be applied [22; 23]. The light propagation for a polarized field can be described as:  
\[\left( \begin{array}{c}\psi_{x}(z+dz)\\ \psi_{y}(z+dz)\end{array} \right)=e^{ik_{0}ntz}\left( \begin{array}{c}\psi_{x}( z)\\ \psi_{y}(z)\end{array} \right) \tag{5}\]  
where \(\psi_{x}(z)\) is the TE field component, \(\psi_{y}(z)\) is the TM field component, \(k_{0}\) is the wavenumber, and  
\[n=\left( \begin{array}{cc}n_{xx}-i\frac{\lambda}{4\pi}\delta_{xx}&n_{xy}\\ n_{yx}&n_{yy}-i\frac{\lambda}{4\pi}\delta_{yy}\end{array} \right) \tag{6}\]  
is \(\lambda\) the photon wavelength, \(g_{xx}\) is the TE optical gain, and \(g_{yy}\) is the TM optical gain, vanishing in our case (\(g_{yy}\cong 0\)). The DoP can be calculated as:  
\[DoP=\lim_{z\to 0}\frac{\left|\psi_{x}(z)\right|^{2}}{\left|\psi_{x}(z) \right|^{2}+\left|\psi_{y}(z)\right|^{2}} \tag{7}\]  
by propagating the electric field using (5). As shown in Figure 4, the DoP of the propagating field converges to a value defined by the stress-modified refractive index matrix, independently of the input polarization state, after a propagating distance  
\[z\gg v_{g}\tau=max\ (g_{xx})^{-1} \tag{8}\]  
where \(v_{g}\) is the group velocity and \(\tau\) is the photon lifetime within the laser cavity. A useful and, to the best of our knowledge, original representation of the DoP as a function of the material properties (refractive index anisotropy \(n_{xx}-n_{yy}\) and non-diagonal component \(n_{xy}\)) is reported in Figure 5. It is clearly visible that the DoP is maximized by increasing the refractive index anisotropy \(n_{xx}-n_{yy}\)[16; 24] and minimising the non-diagonal component \(n_{xy}\). A white contour line at DoP = 0.95 is reported in the figure highlighting the critical region above it (low anisotropy, high non-diagonal component). Refractive index anisotropy \(n_{xx}-n_{yy}\) can be enhanced by proper structure design, while \(n_{xy}\) can be minimized by optimizing chip morphology, technological process, and chip mounting. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constant** & **Value** & **Unit** & **Ref** \\ \hline \(p_{11}\) & \(-0.165\) & - & \\ \(p_{12}\) & \(-0.140\) & - & [19] \\ \(p_{44}\) & \(-0.072\) & - & \\ \hline \(Y\) & 85.5 & GPa & \\ \(G\) & 32.9 & GPa & \\ \(\gamma\) & 0.31 & - & [20; 21] \\ \(n_{0}\) & 3.47 & - & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Opto-mechanical properties of GaAs. An example of the model capability for the analysis and optimization of the DoP of HPLD is reported in Figure 6, in which the effects induced by Cu layers embedded within the HPLD carrier for thermal dissipation improvement (see Figure 7) are addressed.