## 3 Results
### Wideband Polarization-Insensitive MMI  
Following the optimization of the waveguide cross-section for wideband polarization-insensitive operation, a wideband polarization-insensitive MMI was designed in the MIDIR LWIR range. The MMI was designed according to the principles exposed in References [50; 51]. To design such a device, it was necessary to introduce the MMI beating length (L\({}_{\pi_{\text{MMI}}}\)), defined as  
\[\text{L}_{\pi_{\text{MMI}}}=\frac{\lambda}{2\big{|}\big{(}\text{n}_{\text{eff}_ {\text{mode 1}}}-\text{n}_{\text{eff}_{\text{mode 2}}}\big{)}\big{|}}, \tag{4}\]  
where the mode effective indexes correspond to the lowest-order modes in the multimode region for a given polarization. The length of the 1 \(\times\) 2 MMI is related to the beat length by \(\frac{3}{8}\text{L}_{\pi_{\text{MMI}}}\)[50]. To design an MMI operating for both polarizations in a wideband range, it is important to consider the evolution of \(\frac{3}{8}\text{L}_{\pi_{\text{MMI}}}\) for both TE and TM polarizations. More specifically, Figure 5a shows the influence of the MMI width on the difference between optimal MMI lengths for TE and TM as a function of the wavelength. The etching depth was chosen as 3.1 \(\upmu\)m, as it was shown to provide a polarization-insensitive waveguide in the LWIR range. Figure 5a shows that the difference between MMI lengths in TE and TM polarizations was minimized to a width of 20 \(\upmu\)m in the spectral range between 7.5 and 13 \(\upmu\)m. Increasing MMI width resulted in an increase in the difference between TE and TM MMI optimal lengths. For a width of 20 \(\upmu\)m, at a wavelength of 13 \(\upmu\)m in TE polarization, the MMI contains three modes which is the limit for obtaining MMI behavior.  
Figure 4: **(a)** The waveguide geometry optimized for long-wavelength IR (LWIR), and the field-intensity profile for TE polarization at a wavelength of 9.75 \(\upmu\)m. The field-intensity profile is a cropped zoomed image from the bigger simulation window. The white dotted line delimits the frontier between the graded buffer and the constant composition layer. **(b)** Birefringence evolution as a function of the wavelength for the optimized cross-section. The blue line marks the chosen limit for the birefringence at \(\Delta\text{n}_{\text{eff}}=2\times 10^{-4}\).  
The optimized MMI design is presented in Figure 5b, whereby a 20-\(\upmu\)m-wide and 84-\(\upmu\)m-long MMI was chosen with input and output tapering of 80 \(\upmu\)m to avoid any bandwidth limitation from the tapers. The optimized MMI structure induces a phase shift between TE and TM modes of 3.9\({}^{\circ}\) at the output of the MMI at a wavelength of 9.75 \(\upmu\)m (i.e., more than 40 MMIs are needed to realize a \(\uppi\)-shift between the TE and TM modes). The field-intensity profiles extracted from the propagation simulations in TE and TM polarizations are shown in Figure 5c. Both polarizations exhibited similar propagation profiles. Moreover, as shown in Figure 5d, TE and TM polarizations yielded very similar performances, with losses lower than 1 dB in the 7.5 \(\upmu\)m to 12.65 \(\upmu\)m and 7.5 \(\upmu\)m to 13 \(\upmu\)m wavelength ranges for TE and TM polarizations, respectively. Consequently, the MIDIR LWIR transparency window is successfully covered with a single MMI device.