# An \(N\)-Port Universal Multimode Optical Router Supporting Mode-Division Multiplexing
## 2 Materials and Methods
At last, TE\({}_{0}\) mode will be obtained at Port 1 and Port 2, and their corresponding output power can be given by Equations (5) and (6), respectively. The \(L_{dmxx}\) is the demultiplexing loss coefficient of the TE\({}_{x}\) mode restored to the TE\({}_{0}\) mode, and \(L_{dmx0}\) is the loss coefficient of the TE\({}_{0}\) mode passing through multimode taper waveguide during mode demultiplexing. \[P_{mxx} =P_{in}\cdot L_{mxx} \tag{3}\] \[P_{mx0} =P_{in}\cdot L_{mx0}\] (4) \[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]0 (5) \[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]1  
In addition, mode crosstalk noise inevitably occurs in mode-division (de)multiplexing. For example, when the TE\({}_{0}\) mode is input from Port 1 and converted to target TE\({}_{x}\) mode through the coupling region, a small part of TE\({}_{0}\) mode is converted to the undesired TE\({}_{j}\) mode, and the corresponding crosstalk noise can be illuminated as Equation (7). \(C_{0 jmx}\) is the crosstalk noise coefficient of mode-division multiplexing. Correspondingly, crosstalk noise will also occur when the mode is demultiplexed, that is, when the TE\({}_{n}\) mode is restored to TE\({}_{0}\) mode, part of the TE\({}_{n}\) mode is converted to TE\({}_{i}\) mode. The mode crosstalk noise can be illuminated as Equation (8), where \(C_{nidmx}\) is the mode crosstalk noise coefficient of mode-division demultiplexing. \[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]2  
\[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]3  
(2) The insertion loss and crosstalk noise generated by the basic optical switching elements (BOSEs), the single-mode waveguide crossings, and the single-mode bendings. As shown in Figure 5, they are the BOSEs, single-mode bending and single-mode waveguide crossing of the single-mode optical router. The BOSEs used in this paper is the basic \(1\times 2\) crossing switching element (CSE), as shown in Figure 5a,c. \[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]4 Mode (de)multiplexer for TE\({}_{x}\) mode. (**a**) The process of mode multiplexer. (**b**) The process of mode demultiplexer. As for the \(1\times 2\) CSE, it is composed of two single-mode waveguides and a microring resonator. When the microring resonator is in the OFF state, the fundamental mode is transmitted directly to the Through port, as shown in Figure 5a. Therefore, the output power of the Through port can be calculated by Equation (9). On the contrary, when the microring resonator is in the ON state, the fundamental mode is coupled into the microring and transmitted to the Drop port, as shown in Figure 5c; the output optical power from the Drop port can be calculated by Equation (10). \(L_{sp1}\) is the power loss of each parallel switching element (PSE) in the OFF state, and \(L_{sc2}\) is the power loss of each CSE in the ON state. For the \(1\times 2\) CSE, when the microring resonator is in the ON state, the output power of the Through port and the Add port can be calculated by Equations (11) and (12), respectively. \(K_{c}\) is the crosstalk noise coefficient of the waveguide crossing. When the microring resonator is in the OFF state, the output power of the Drop port and the Add port can be calculated by Equations (13) and (14), respectively. \(C_{p,\rho n}\) is the crosstalk noise coefficient of each \(1\times 2\) PSE in the ON state, and \(C_{p,\rho f}\) is the crosstalk noise coefficient of each PSE in the OFF state [30]. \[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]5  
\[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]6  
\[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]7  
\[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]8  
\[P_{mbx} = P_{in}\cdot L_{mbx} \tag{1}\]9  
\[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]0  
Figure 5b shows the single-mode waveguide crossing. When the fundamental mode enters the single-mode waveguide crossing from the Input port, it outputs directly from Port 2, and the power output can be established as Equation (15). \(L_{sc}\) is the single-mode waveguide crossing loss coefficient.