# Silicon Integrated Nanophotonic Devices for On-Chip Multi-Mode Interconnets
## 3 Silicon Multi-Mode Routing Devices
### Silicon Multi-Mode Crossings  
Waveguide crossing plays an important role in the photonic integrated circuit, with a complex pathway topology. Over the past few years, various single-mode waveguide crossings have been reported, including ones based on MMIs [178; 179; 180; 177; 178; 179; 181; 182; 183; 184; 185; 186], ridge expanders [187], inter-layer couplers [188; 189] and subwavelength structures [190; 191; 192; 193; 194; 195; 190; 191; 192; 193]. Among them, the MMI-based crossing is most commonly used due to its simple design, compact size (\(<\)10 \(\times\) 10 \(\upmu\)m\({}^{2}\)) and low insertion losses (IL \(<\) 0.2 dB). The MMI can serve as a waveguide-lens that focuses the incident light at the cross-over position, which effectively reduces the diffraction losses [178]. However, such a focusing process only works for a single fundamental mode. To explain this, we show a calculation example considering the first two TE modes (i.e., TE\({}_{0}\), TE\({}_{1}\)), as shown in Figure 14. The focal length of MMI is basically determined by interference-induced self-imaging, which can be formulated as [178]:  
\[\mathrm{L}_{\mathrm{f,0}}=\mathrm{L}_{\mathrm{B,0}}=\lambda/(\mathrm{r}_{ \mathrm{eff,TE0}}-\mathrm{n}_{\mathrm{eff,TE2}}) \tag{5}\]  
\[\mathrm{L}_{\mathrm{f,1}}=\mathrm{L}_{\mathrm{B,1}}=\lambda/(\mathrm{r}_{ \mathrm{eff,TE1}}-\mathrm{n}_{\mathrm{eff,TE3}}) \tag{6}\]  
where L\({}_{\mathrm{f,i}}\) is the focal length for TE\({}_{i}\), L\({}_{\mathrm{B,i}}\) is the beat length for TE\({}_{i}\), n\({}_{\mathrm{eff,TEi}}\) is the TE\({}_{i}\) effective index, \(\lambda\) is the working wavelength. We calculate the beat lengths for TE\({}_{0}\) and TE\({}_{1}\) with varied waveguide width (w\({}_{\mathrm{wg}}\)) using Equations (5) and (6), as shown in Figure 14a. Here, the waveguide thickness is chosen to be h\({}_{\mathrm{wg}}\) = 220 nm, while the working wavelength is set to be \(\lambda\) = 1.55 \(\upmu\)m. We also calculate the light propagation profiles for the MMI-based crossings with L\({}_{\mathrm{f}}\) optimized for TE\({}_{0}\) and TE\({}_{1}\), as shown in Figure 14b. It can be seen that the beat lengths are always different between TE\({}_{0}\) and TE\({}_{1}\) regardless of the variant waveguide widths. Consequently, it is challenging to focus multiple modes at the same cross-over position. Some works suggest that the multi-mode cross-over propagation could be simply realized by employing a direct-crossing structure [194], since the beam divergence is negligible with a wide waveguide, enabling a lossless multi-mode transport (see Figure 15a). To achieve this, the waveguide width (w\({}_{\mathrm{wg}}\)) should be:  
\[\mathrm{w}_{\mathrm{wg}}>>\lambda/\mathrm{n}_{\mathrm{s}} \tag{7}\]  
where n\({}_{\mathrm{s}}\) is the effective index for the slab mode, \(\lambda\) is the working wavelength. For instance, to support a four-mode cross-over (e.g., TE\({}_{0\text{-}3}\)), the waveguide width should be chosen as w\({}_{\mathrm{wg}}\) \(>\) 10 \(\upmu\)m. On the other hand, the multi-mode bus waveguide is usually quite narrow (e.g., w\({}_{\mathrm{wg}}\)\(<\) 2 \(\upmu\)m); thus, the long adiabatic tapers (typically \(>\) 100 \(\upmu\)m) should also be involved to convert the mode size [173; 174], leading to an extremely large total device size (\(>\)200 \(\upmu\)m). It is crucial that the crossing structures should have small footprints (\(<\)20 \(\upmu\)m) to fit in the dense cross-bar networks [195; 196]. In this regard,  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Ref. **} & \multirow{2}{*}{**Year**} & \multirow{2}{*}{**Radius (\(\upmu\)m)**} & \multirow{2}{*}{**Capacity**} & \multicolumn{2}{c}{**IL (dB)**} & \multicolumn{2}{c}{**IMC (dB)**} & \multicolumn{2}{c}{**BW (nm)**} \\ \cline{5-10}  & & & & **Sim. ** & **Exp. ** & **Sim. ** & **Exp. ** & **Sim. ** & **Exp.