## 3 Model  
The nearly 1 intrinsic DoP determined by compressively strained In\({}_{x}\)Ga\({}_{1-x}\)As QW material can be reduced by external mechanical stresses induced by chip fabrication (dielectrics, metals) and chip mounting on the carrier, through the photo-elastic effect [7; 8; 9; 10]. Mechanical stresses are enhanced by chip morphology and can be particularly high on waveguide edges (see Figure 3 in which the distribution of \(xy\) stress component is calculated using a FEM simulation [10] performed with a COMSOL Multiphysics(r) tool). External stress effects on the polarization state of laser diode beams are well known; in the case of bulk diode lasers, in which both \(e-hh\) and \(e-lh\) optical transitions have the same energy gap, these effects were exploited to demonstrate polarization switching at different injection currents [11]. It is also worth noting that, according to our simulations and the literature [12; 13; 14] on compressively strained QW, external mechanical stress induces additional strain, affecting QW optical properties. However, the induced strain contribution to the QW, for the typical external stress induced by device fabrication and mounting, is significantly lower than the 1.1% epitaxial strain [12] and produces negligible effects on the DoP. The mechanical stress translates into an anisotropic refractive index change [15] through the photo-elastic tensor, which in turn determines the polarization change in the optical field within the active cavity, causing a DoP reduction [9]. The anisotropic refractive index of the semiconductor structure is due to both the quantum well active material and the optical waveguide. Starting from the material composition of each epitaxial layer and solving the Helmholtz equation, we can quantify the refractive index split between the TE mode, with electric field oscillating in the QW plane, \(n_{xx}^{0}\), and the TM mode, with the electric field oscillating along the epitaxy growth direction, \(n_{yy}^{0}\). Figure 2: (**a**) Refractive index profile n(x), effective index value n\({}_{\text{eff}}\) and optical mode \(\mid\)Y(x) \(\mid\) 2. (**b**) TE and TM spectral optical gain calculated at different carrier density N ranging from \(1\times 10^{17}\) cm\({}^{-3}\) to \(3\times 10^{18}\) cm\({}^{-3}\). It is useful to define the intrinsic anisotropy \(\Delta n_{ii}^{0}\) due to the material structure [16]:  
\[\Delta n_{ii}^{0}=n_{xx}^{0}-n_{yy}^{0} \tag{2}\]  
The laser cavity along the \(z\)-axis is defined through semi-reflecting mirrors at cavity facets. The TE-polarized light propagating along this direction is not affected by the intrinsic anisotropy \(\Delta n_{ii}^{0}\). Furthermore, the small refractive index changes associated with the photo-elastic effect do not produce any significant change to facet reflectivity. The anisotropic mechanical stress, due to the technological process and the mounting process, changes the refractive index, which can be conveniently represented as a tensor, through the photo-elastic effect. Refractive index components are related to the stress components through the following equation [15]:  
\[\left( \begin{array}{c}n_{xx}\\ n_{yy}\\ n_{zz}\\ n_{xy}\\ n_{xz}\\ n_{zz}\end{array} \right)=\left( \begin{array}{c}n_{xx}^{0}\\ n_{yy}^{0}\\ n_{zz}^{0}\\ n_{xy}^{0}\\ n_{zz}^{0}\\ n_{zz}^{0}\end{array} \right)-\left( \begin{array}{cccccc}C_{1}&C_{2}&C_{2} &0&0&0\\ C_{2}&C_{1}&C_{2}&0&0&0\\ C_{2}&C_{2}&C_{1}&0&0&0\\ 0&0&0&C_{3}&0&0\\ 0&0&0&0&C_{3}&0\\ 0&0&0&0&0&C_{3}\end{array} \right)\times\left( \begin{array}{c}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \sigma_{xy}\\ \sigma_{xz}\\ \sigma_{zz}\end{array} \right)=\left( \begin{array}{c}n_{xx}^{0}+\Delta n_{ xx}^{s}\\ n_{yy}^{0}+\Delta n_{yy}^{s}\\ n_{zz}^{0}+\Delta n_{zz}^{s}\\ n_{xy}^{0}+\Delta n_{xy}^{s}\\ n_{zz}^{0}+\Delta n_{zz}^{s}\end{array}