## 3 Results  
### Wideband Polarization-Insensitive Waveguides  
For the implementation of polarization-insensitive Ge-rich SiGe waveguides, the following epitaxial configuration was chosen (Figure 2a): 2 \(\upmu\)m of constant Si\({}_{0.2}\)Ge\({}_{0.8}\) on an 11-\(\upmu\)m-thick SiGe graded buffer layer (GB), where the germanium concentration increased linearly in the vertical direction until the terminal composition of Si\({}_{0.21}\)Ge\({}_{0.79}\). The GB provides three major advantageous features: (i) it guarantees a good material quality through the accommodation of a gradual lattice [48], (ii) it isolates the optical mode from the silicon-rich region and the silicon substrate, avoiding loss via multi-phonon absorption at wavelengths higher than 7 \(\upmu\)m [49], and (iii) it allows broadband and low-loss operation by means of combining low MIDIR material dispersion in Si and Ge, and the mode size self-adaptation effect [34; 35]. Noticeably, low propagation losses and a wideband Mach Zehnder interferometer (MZI) operation were previously demonstrated based on this epitaxial layer [35; 36]. The inspected wavelength was limited by the available spectral range in our experimental set-up, which covered the wavelengths between 5 \(\upmu\)m and 8.5 \(\upmu\)m. Flat low-loss conditions were obtained for both TE and TM polarizations. On the basis of the above features, such an epitaxial layer is a promising candidate for achieving a polarization-insensitive waveguide targeting both MIDIR atmospheric transparency windows with a single-material platform. However, despite all the above-mentioned works on SiGe platforms, polarization sensitivity is yet to be evaluated, and consequently, became the main subject of interest in the presented work. Firstly, the waveguide geometry was optimized to achieve polarization-independent behavior in the MWIR range. The waveguide birefringence, i.e., the difference between effective mode indexes for TE (neff\({}_{\text{TE}}\)) and TM (neff\({}_{\text{TM}}\)) polarizations (\(\,\)neff\({}_{\text{TE}}\)-neff\({}_{\text{TM}}\)1), was studied as a function of the width (W) and etching depth (D) for 10 different wavelengths between 3 \(\upmu\)m and 5 \(\upmu\)m. As an example, the maps for the wavelengths of 3.5 \(\upmu\)m, 4 \(\upmu\)m, and 5 \(\upmu\)m are shown in Figure 2b-d, respectively. These maps clearly illustrate the wavelength dependence of the birefringence, and give insight into the zone where polarization-insensitive geometries can be localized. One can observe the evolution of the zero-birefringence line as a function of the wavelength. When the wavelength increases, the zero-birefringence line shifts toward wider and deeper-etched waveguide geometries. Figure 1: Refractive index evolution of the Si\({}_{0.2}\)Ge\({}_{0.8}\) alloy in the mid-infrared (MIDIR) spectral region. Remarkably, the bottom zone including the zero-birefringence line (black dashed rectangle in Figure 2b-d) kept low birefringence values despite the wavelength change. Hence, wideband polarization-insensitive waveguide configurations were expected to be localized in this region. In order to finely optimize waveguide dimensions for broadband polarization-insensitive operation, we studied the birefringence variation as a function of the wavelength. Firstly, we defined the beating length \(\mathrm{L_{\pi}}\) as  
\[\mathrm{L_{\pi}}=\frac{\lambda}{2\big{|}\big{(}\mathrm{n_{eff_{TE}}-\mathrm{n_ {eff_{TM}}}}\big{)}\big{|}}. \tag{2}\]  
At a given wavelength, the polarization-insensitive waveguide maximizes the \(\mathrm{L_{\pi}}\) value. However, if a broadband polarization-insensitive operation is envisioned, maximizing \(\mathrm{L_{\pi}}\) at a given wavelength is not sufficient. Hence, we defined the following figure of merit:  
\[\mathrm{L_{geom}}=\left(\prod_{i=1}^{\mathrm{N}}\mathrm{L_{\pi_{i}}}\right) \quad\text{,} \tag{3}\]  
Figure 2: **(a)** Schematic view of the waveguide cross-section.