# Silicon Integrated Nanophotonic Devices for On-Chip Multi-Mode Interconnets
## 3 Silicon Multi-Mode Routing Devices
### Silicon Multi-Mode Crossings
** \\ \hline
[167] & 2012 & 78.8 & 4 & / & \(<\)3 & / & / & / & / \\
[166] & 2014 & 10 & 8 & \(<\)0.1 & / & \(<\)\(-\)20 & / & / & / \\
[169] & 2017 & 5 & 2 & \(<\)0.1 & \(<\)0.2 & \(-\)23 & \(-\)22 & 100 & 100 \\
[170] & 2 such a direct crossing is not applicable for practical uses unless the taper length can be effectively shrunk to \(\approx\)10 \(\upmu\)m.  
A multi-mode crossing with a reduced size was firstly reported in 2016 [197], as shown in Figure 15b. The proposed scheme is still based on the MMI structure, but the focal length is chosen to be the common multiple of beat lengths for TM\({}_{0}\) and TM\({}_{1}\):  
\[\text{L}_{\text{f}}=\text{p-L}_{\text{B},0}=\text{q-L}_{\text{B},1} \tag{8}\]  
\[\text{L}_{\text{B},0}=\lambda/(\text{n}_{\text{eff},\text{TM}0}-\text{n}_{ \text{eff},\text{TM}2}) \tag{9}\]  
\[\text{L}_{\text{B},1}=\lambda/(\text{n}_{\text{eff},\text{TM}1}-\text{n}_{ \text{eff},\text{TM}3}) \tag{10}\]  
where p and q are two integers, L\({}_{\text{f}}\) is the focal length, L\({}_{\text{B},\text{i}}\) is the beat length for TM\({}_{\text{i}}\), n\({}_{\text{eff},\text{TM}i}\) is the TM\({}_{\text{i}}\) effective index. In this way, light focusing can be simultaneously achieved for TM\({}_{0}\) and TM\({}_{1}\). The measurement results exhibit insertion losses of IL \(\approx\) 1.5 dB and crosstalk of XT \(<\) -18 dB. The working performances can be further improved by cascading several multi-sectional taper structures (IL \(\approx\) 0.1 dB, XT \(<\) -30 dB) [198]. However, there are two major drawbacks for the MMI-based crossings: first, the device size is still relatively large (\(\approx\)30 \(\times\) 30 \(\upmu\)m\({}^{2}\)) since the focal length is multiplied; second, the capacity is restricted to two modes, because it is tough to find the common multiple of beat lengths for a large number of higher-order modes. Another type of multi-mode crossing is based on mode converters. In [199], a two-mode crossing was proposed by using a 2 \(\times\) 2 single-mode waveguide crossing matrix cascaded with four symmetric Y-junctions, as illustrated in Figure 15c. For such a scheme, the input TE\({}_{0}\)/TE\({}_{1}\) modes are converted  
Figure 14: (**a**) The calculated beat lengths (L\({}_{\text{B}}\)) for TE\({}_{0}\) and TE\({}_{1}\) modes with varied waveguide width (w\({}_{\text{wg}}\)). (**b**) The calculated light propagation profiles for MMI-based crossings when TE\({}_{0}\) (left panel) and TE\({}_{1}\) (right panel) are launched. The beat lengths are different between TE\({}_{0}\) and TE\({}_{1}\). Figure 15: The schematics for different silicon multi-mode crossings: (**a**) direct crossing [194], (**b**) MMI-based crossing [197,198], (**c**) mode converter-assisted crossing [199,200], (**d**) Maxwell’s fisheye lens (MFL)-based crossing [201,202]. to the symmetric/asymmetric super-modes (i.e., \(\mathrm{TE_{S}/TE_{AS}}\)) in the Y-junction, so that all the crossing structures only need to work with \(\mathrm{TE_{0}}\). The footprint is quite small (\(\approx\)21 \(\times\) 21 \(\mathrm{\SIUnitSymbolMicro m^{2}}\)), but the insertion losses are significant (\(\mathrm{IL}\approx\) 1.82 dB) owing to the intrinsically strong scattering at the junction tip. In [200], the similar idea was implemented to realize a three-mode crossing by using a \(3\times 3\) crossing matrix cascaded with four asymmetric Y-junctions. The insertion losses and crosstalk were measured to be \(\mathrm{IL}<0.9\) dB, \(\mathrm{XT}<-24\) dB, respectively. Nevertheless, the mode capacity is still quite limited for this type of multi-mode crossing since higher-order mode conversion is difficult to realize. [PERSON]'s fisheye lens (MFL) is a perfect imaging instrument with a gradient index map that focuses rays emanating from a point on the boundary to a conjugate point on the opposite side [203]. In [201], an MFL-based multi-mode crossing was proposed and demonstrated, as shown in Figure 15d.