# An \(N\)-Port Universal Multimode Optical Router Supporting Mode-Division Multiplexing
## 2 Materials and Methods
Figure 5d is the single-mode bending; when the fundamental mode is input from the Input port, the power output from the Output port can be calculated by Equation (10). \(L_{sc}\) is the crosstalk noise coefficient of each PSE in the ON 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 PSE in the ON state, and \(C_{p,\rho f}\) is the crosstalk noise coefficient of each PSE in the OFF state [30]. \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]1 (**a**) Single-mode CSE in the OFF state. (**b**) Single-mode waveguide crossing. (**c**) Single-mode CSE in the ON state. (**d**) Single-mode bending. be established as Equation (16), where \(L_{sb}\) describes single-mode bending loss coefficient. Crosstalk noise will be produced when two optical signals pass through a waveguide crossing at the same time. When the optical signal enters the waveguide crossing from the Input port, the output power of Port 1 and Port 3 can be described as Equation (17). \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]2 \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]3 (16) \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]4  
Through the above analysis of insertion loss and crosstalk noise sources, we can obtain the insertion loss and crosstalk noise between port pairs of the multimode switching element. Since the optical devices are used to transmit multiple modes, the insertion loss will vary with the input mode. When the TE\({}_{k}\) mode is input from Port I of this multimode switching element and finally restored to the TE\({}_{k}\) mode output from the Port III, the insertion loss generated by this process can be obtained by Equation (18), \(k\in\{1,2,\ldots,N-1\}\). As can be seen from Figure 3, when the input port is Port III, and the corresponding output port is Port I, the insertion loss can also be calculated by Equation (18). In addition, due to the symmetry of the device, when the input port is Port II, the corresponding output port is Port IV, and when the input port is Port IV, the corresponding output port is Port II; their insertion losses are equal to Equation (18). Similarly, when the TE\({}_{k}\) mode is input from Port III of this multimode switching element, finally restored to the TE\({}_{k}\) mode and output from Port IV, the insertion loss generated by this process can be established as Equation (20). For TE\({}_{0}\) mode, when it is input from Port I, Port III, Port II, and Port IV of the multimode switching element, the corresponding output port is Port III, Port I, Port IV, and Port II, respectively; the insertion loss can be calculated using Equation (19). Moreover, when the TE\({}_{0}\) mode is input from the Port I and the corresponding output from the Port IV, the corresponding insertion loss can be obtained by Equation (21). \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]5  
\[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]6  
\[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]7  
\[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]8  
Compared with the insertion loss, the crosstalk noise analysis of multimode switching element is more complicated, and it is mainly divided into 6 situations shown in Figures 6 and 7. Figure 6 shows the four communication situations where two modes are output from the same port of the multimode switching element. Figure 7 describes the two communication situations where two modes are output from different ports of the multimode switching element. Among them, Figure 6a-c show the situations that the two modes enter from the same port of the multimode switching element and then output from the same port.