## 2 Theoretical Aspect and Design of the 1 \(\times\) 2 Demultiplexer  
Figure 1a shows the cross-sectional view at \(Z=0\) in the XY plane; yellow- and light blue-colored regions indicate the SiN and SiO2 covers, respectively. H\({}_{\text{strip}}\) represents the SiN layer height, and W\({}_{\text{strip}}\) indicates its width. The refractive indexes of the SiN strip and the SiO2 cover are n\({}_{\text{strip}}\) and n\({}_{\text{cover}}\), respectively, and they are 1.99 and 1.444. Figure 1b illustrates the two-channel MMI coupler created for the 1 \(\times\) 2 demultiplexer device in the XZ plane cross-sectional view at Y = 0. The MMI coupler has a width of W\({}_{\text{MMI}}\) and length of L\({}_{\text{MMI}}\). The input waveguide segment width and height were set at 500 nm and 320 nm, respectively. For the MMI coupler, the input taper length was specified at 40 \(\upmu\)m, and its width ranged from 0.5-0.75 \(\upmu\)m. The output taper length was specified to be 35 \(\upmu\)m, and its width ranged from 0.9-0.5 \(\upmu\)m. \(\text{G}_{\text{t}}\) and \(\text{G}_{\text{s}}\) are the gaps between the output tapers and the S-bends and were set to 0.78 \(\upmu\)m and 11.5 \(\upmu\)m, respectively. The lengths of the S-bends were set to 80 \(\upmu\)m.  
The two wavelengths that the MMI coupler was designed to divide are 1540 nm (\(\lambda\)1) and 1560 nm (\(\lambda\)2). Figure 1b illustrates how the pair (\(\lambda\)1/\(\lambda\)2) propagates into an MMI coupler before splitting into two channels. The length of the MMI coupler (L\({}_{\text{MMI}}\)) is 2067 \(\upmu\)m and the width of the MMI (W\({}_{\text{MMI}}\)) is 3.5 \(\upmu\)m.  
The width of the S-bends was set at 0.5 \(\upmu\)m to match the width of the output tapers, with a length of 80 \(\upmu\)m. The distance between the MMI S-bend outputs (Gs) was set at 11.5 \(\upmu\)m. The total length of the device is 2.272 mm. Figure 1: An illustration of the two-channel MMI demultiplexer. (**a**) The refractive index profile at the XY plane; (**b**) The device profile at the XZ plane. The self-imaging effect states that every wavelength reaching the device's multimode area periodically generates a direct or mirrored picture of itself. The beat length (L\({}_{\pi}\)) is the distance from the entry to the location of the first image and is provided by [21]:  
\[\text{L}_{\pi}\ \approx\ \frac{4n_{\text{eff}}W_{\text{eff}}^{2}}{3\lambda_{ \text{n}}}\ ;\ n=1,\ 2 \tag{1}\]  
where the operational wavelengths for \(n=1,2\) are given by \(\lambda_{\text{n}}\). The SiN effective refractive index for the fundamental electrical mode is known as n\({}_{\text{eff}}\). BPM mode solver was used to determine this parameter. Each fundamental mode has its own \"effective\" width, also known as W\({}_{\text{eff}}\), which takes into account the horizontal penetration depth due to the Goos-Hanchen shift. The Goos-Hanchen effect is a lateral displacement of a finite cross-sectional wave that is totally internally reflected at the interface between two different refractive index materials. The lateral shift results from the propagation of an evanescent wave parallel to the interface. The W\({}_{\text{eff}}\) size for the transverse electric (TE) mode is provided by [21]:  
\[\text{W}_{\text{eff}}\text{=W}_{\text{MMI}}+\frac{\lambda_{\text{n}}}{\pi\cdot \sqrt{n_{\text{eff}}^{2}-n_{\text{cover}}^{2}}} \tag{2}\]  
where W\({}_{\text{MMI}}\) is the MMI coupler's actual width, as illustrated in Figure 1b. This requirement must be satisfied for the MMI coupler to divide different wavelengths:  
\[\text{L}_{\text{MMI}}\text{= pL}_{\pi}^{\lambda_{1}}\text{= (p+q) L}_{\pi}^{\lambda_{2}} \tag{3}\]  
where p is an integer and q is an odd number. By canceling the third mode from inside the MMI, the propagation distance can be reduced by a factor of three. To achieve this, the MMI input taper needs to be moved out from its center by an offset of (1/6) W\({}_{\text{eff}}\)[21]. Additionally, by employing BPM simulations, these conditions can be optimized to obtain improved performances.