## 3 Results
### Wideband Polarization-Insensitive Waveguides
Depending on the design, the etching depth (D) can stop in the graded buffer layer, or in the Si\({}_{0.2}\)Ge\({}_{0.8}\) top layer. Birefringence evolution as a function of waveguide dimensions for wavelengths of **(b)** 3.5 μm, **(c)** 4 μm, and **(d)** 5 μm. In **(b–d)**, the gray line represents the zero-birefringence line, i.e., no difference between effective mode indexes for transverse electric (TE; \(\mathrm{neff_{TE}}\)) and transverse magnetic (TM; \(\mathrm{neff_{TM}}\)) polarizations, \(\mathrm{|\,neff_{TE}-\mathrm{neff_{TM}}|=0}\). The black dashed rectangle corresponds to a set of dimensions where low birefringence was obtained for the various wavelengths. where L\({}_{\text{geom}}\) is the geometric mean value of L\({}_{\pi}\) over N different wavelengths. The waveguide geometry was optimized to maximize L\({}_{\text{geom}}\), thus minimizing the birefringence over the selected spectral range. Figure 3a shows L\({}_{\text{geom}}\) as a function of waveguide width (W) and etching depth (D), calculated for 10 wavelengths within the MWIR range. As expected from the birefringence maps shown in Figure 2, L\({}_{\text{geom}}\) was maximized in the bottom zone close to the zero-birefringence line (black dashed rectangle). L\({}_{\text{geom}}\) was also maximized in the zone comprising waveguides with widths between 4 \(\upmu\)m and 5 \(\upmu\)m, and etching depths between 2.5 \(\upmu\)m and 5 \(\upmu\)m. However, these waveguides are highly multimode, which is not desirable for the implementation of complex circuits. Therefore, we chose W = 3.4 \(\upmu\)m and D = 1.9 \(\upmu\)m as the optimal design. The field-intensity profiles for the optimized waveguide geometry are depicted in Figure 3b,c for TE and TM modes, respectively. As can be observed, similar mode profiles and confinements were obtained for both polarizations. Figure 3d shows the birefringence as a function of the wavelength for the optimized waveguide. Interestingly, the proposed design ensured birefringence below 2 \(\times\) 10\({}^{-4}\) across wavelengths of 3 \(\upmu\)m and 4.8 \(\upmu\)m, almost entirely covering the atmospheric transparency MWIR window. In other words, the TE and TM modes were \(\uppi\)-shifted only after a propagation distance of 1.2 cm at a wavelength of 4.8 \(\upmu\)m.  
Figure 3: (**a**) L\({}_{\text{geom}}\) (L\({}_{\pi}\) geometric mean value calculated over 10 different wavelengths) as a function of the waveguide dimensions; the L\({}_{\pi}\) values are given in cm. Waveguide geometries optimized for mid-wavelength IR (MWIR) and field-intensity profiles for (**b**) TE polarization and (**c**) TM polarization at a 4-\(\upmu\)m wavelength. The white dotted line delimits the frontier between the graded buffer and the constant composition layer. The shown field profiles are cropped zoomed images from the bigger simulation window. (**d**) Birefringence evolution as a function of the wavelength for the optimized cross-section. The blue line marks the chosen limit for the birefringence at \(\Delta\)m\({}_{\text{eff}}\) = 2 \(\times\) 10\({}^{-4}\). A similar approach was used to design the wideband polarization-insensitive waveguide in the LWIR. The optimized dimensions and the TE-mode field-intensity profile are presented in Figure 4a; the waveguide was 5.4 \(\upmu\)m wide with a 3.1-\(\upmu\)m etching depth. Its birefringence, which is shown in Figure 4b, was below 2 \(\times\) 10\({}^{-4}\) across the wavelengths spanning 8.22 \(\upmu\)m to 10.36 \(\upmu\)m (i.e., a polarization-insensitive bandwidth larger than 2 \(\upmu\)m). These low birefringence values correspond to L\({}_{\pi}\) values higher than 2.5 cm on the full operational wavelength range.