## 2 Design and Simulation  
Figure 1 shows the design architecture of our proposed wideband and channel switchable MDM optical power divider. It consists of two parts. The first part is a cascaded MZI for switching the data from their original TE\({}_{0}\), TE\({}_{1}\) and TE\({}_{2}\) modes to different modes among themselves. After the target modes are identified, mode up-conversion and Y-branch are utilized in the second part for the MDM power division. The insets in Figure 1 illustrate the cross sections of different orders of MDM modes. Before discussing the operation mechanism of the MDM optical power divider, we first discuss how to use the ADC for combining the TE\({}_{0}\), TE\({}_{1}\) and TE\({}_{2}\) modes in a single bus waveguide. Figure 2a shows the structure schematic of the ADC-based MDM Mux/Demux. As discussed in [19], an ADC structure consists of a narrower access waveguide (for the TE  
Figure 1: Design architecture of our proposed wideband and channel switchable MDM optical power divider. Insets: cross sections of different orders of MDM modes. mode transmission) and a wider bus waveguide (for the high order modes transmissions). When the access waveguide and bus waveguide are close to each other, the fundamental TE\({}_{0}\) mode can be converted to higher order modes, or vice versa when the phase-matching condition is satisfied, as discussed in the couple mode theory. As the whole proposed MDM power divider is on an SOI platform, the ADC structure is no exception. The dimensions of the silicon access waveguide are 0.45 \(\upmu\)m \(\times\) 0.22 \(\upmu\)m. The buried oxide layer (BOX) is 2 \(\upmu\)m, and the ADC gap in the coupling region is 0.15 \(\upmu\)m. Here, we used an Ansys Lumerical(r) finite-difference time-domain (FDTD) technique to simulate the ADC-based MDM Mux/Demux. As discussed before, the access waveguide has a width of 0.45 \(\upmu\)m supporting the TE\({}_{0}\) mode. In order to meet the phase matching condition, the bus waveguide widths are 0.932 \(\upmu\)m and 1.416 \(\upmu\)m, supporting the TE\({}_{1}\) and TE\({}_{2}\) modes, respectively. Furthermore, the coupling lengths for the TE\({}_{0}\)-TE\({}_{1}\) and TE\({}_{0}\)-TE\({}_{2}\) modes ADC are 17 \(\upmu\)m and 22.5 \(\upmu\)m, respectively. Figure 2b-d show the FDTD simulation results when light at the fundamental TE\({}_{0}\) mode is inputted from the left-hand side and outputted at the right-hand side. As shown in Figure 2b, the MDM Mux/Demux can maintain the TE\({}_{0}\) mode at the device output. Figure 2c,d illustrate that the fundamental TE\({}_{0}\) mode can be up-converted to TE\({}_{1}\) and TE\({}_{2}\), and then back to the TE\({}_{0}\) mode again at the output. The simulation results also reveal that the coupling efficiencies of ADC in all the channels are more than 95%. In the 3 \(\times\) 3 mode switch, the data carried in the TE\({}_{0}\), TE\({}_{1}\), and TE\({}_{2}\) modes can be converted among one other. It has a similar structure to that reported in [33]. As shown in Figure 1, it consists of three cascaded 2 \(\times\) 2 MZIs. Each MZI consists of two symmetrical arms with thermal-optic phase shifters and two multimode interferometers (MMI). In order to provide wideband operation, the path difference between the two arms of the MZI is zero. The thermo-optic phase shifter is based on a p-doped waveguide, which is used as a resistive heater to provide the phase shifting. Its characteristics will be discussed in the next section. In order to balance the doping-induced optical loss, the thermo-optic phase shifters are utilized in both arms of the MZI, but only one arm has the electrical contact for applying the external electrical bias. Finally, we discuss the operation mechanism of the MDM optical power divider. Figure 3a shows the schematic of the SOI-based MDM 3-dB power splitter structure. The operation principle is based on the mode up-conversions using ADCs, and then the higher order modes are split by the Y-bench into two TM\({}_{2n-1}\) modes, where n is an integer.