# An \(N\)-Port Universal Multimode Optical Router Supporting Mode-Division Multiplexing
## 2 Materials and Methods
\begin{array}{cc}L_{dmx0}(C_ {pse\rho\rho ff}+L_{sp1}^{2}K_{c})&0\\ L_{dmxm}L_{sc}^{m}L_{sp1}L_{sb}K_{c}&0\\ 0&L_{dmx0}L_{sc}^{n}L_{sp1}L_{sb}K_{c}\\ 0&L_{dmxm}L_{sc}^{n+n}L_{sp1}L_{sb}^{2}K_{c}\\ 0&L_{dmxm}L_{sc}^{2m}L_{sb}^{2}(C_{pse\rho ff}+L_{sp1}^{2}K_{c})\end{array} \right] \tag{26}\]  
\[\cdot\left[ \begin{array}{c}L_{mx0}\\ L_{mxn}\end{array} \right]\quad a:m=0,n=0;b:m\
eq 0,n=0;c:m=0,n\
eq 0;\]  
\[d:m\
eq 0,n\
eq 0,m\
eq n;e:m\
eq 0,n\
eq 0,m=n\]\[\left[ \begin{array}{c}C_{3,2}^{a}\\ C_{3,2}^{b}\\ C_{3,2}^{c}\\ C_{3,2}^{d}\\ C_{3,2}^{e}\end{array} \right]=\left[ \begin{array}{cc}L_{dmxn}C_{sc}^{2(N-1)}(K_{ c}+L_{sc}^{2}C_{pse,off})&0\\ 0&L_{sc}^{2(N-1)-m}L_{sp1}L_{sb}K_{c}\\ 0&L_{dmxn}L_{sc}^{2(N-1)-n}L_{sp1}L_{sb}K_{c}\\ 0&L_{dmxn}L_{sc}^{2(N-1)-n-m}L_{sp1}L_{sb}^{2}K_{c}\\ 0&L_{dmxn}L_{sc}^{2(N-1-m)}L_{sb}^{2}(K_{c}+L_{sc}^{2}C_{pse,off})\end{array}  \right] \tag{27}\]  
\[\cdot\left[ \begin{array}{c}L_{mx0}\\ L_{mxm}\\ \end{array} \right]\cdot P_{in}\quad a:m=0,n=0;b:m\
eq 0,n=0;c:m=0,n\
eq 0;d:m\
eq 0,\]  
\[n\
eq 0,m\
eq n;e:m\
eq 0,n\
eq 0,m=n\]  
For the situation shown in Figure 7b, the two modes are input from the same port of the multimode switching element, but they are output from different ports. When \(n>m\), the two modes will produce crosstalk noise in the single-mode waveguide part of the mode switching element, and the corresponding crosstalk noise \(C_{4,1}\) and \(C_{4,2}\) can be illuminated as Equations (28) and (29). Equation (28) indicates the crosstalk noise generated from the TE\({}_{m}\) mode to the TE\({}_{n}\) mode, and Equation (29) indicates the crosstalk noise generated from the TE\({}_{n}\) mode to the TE\({}_{m}\) mode. \[C_{4,1}= \begin{cases}P_{in}L_{dmxn}C_{sc}^{n}L_{sp1}L_{sb}K_{c}L_{mxm}&m=0\\ 0&n<m\\ P_{in}L_{dmxn}L_{sc}^{m+n-m}L_{sp1}L_{sb}^{2}K_{c}L_{mxm}&m\
eq 0,n>m\end{cases}  \tag{28}\]  
\[C_{4,2}= \begin{cases}P_{in}L_{dmxn}L_{sc}^{N+n-2}L_{sc2}L_{sb}K_{c}L_{mx0}&m=0 \\ 0&n<m\\ P_{in}L_{dmxn}L_{sc}^{N+n-m-2}L_{sc2}L_{sb}^{2}K_{c}L_{mxm}&m\
eq 0,n>m\end{cases}  \tag{29}\]  
When a traditional single-mode optical router is working, there may be multiple optical signals inside. However, some specific microring resonators is only in the ON state on the link from a certain port to a certain port and in the OFF state on the other port-to-port link. Therefore, by adjusting the position of the multimode switching element, the design of the MDM-OR that supports MDM can be realized. Figure 8 shows a simplified model of an \(N\)-port optical router. The optical router has \(N\) bidirectional ports, namely Port 1, Port 2,..., Port \(N\). When the optical signal is transmitted from Port \(i\) to Port \(j\) of the MDM-OR, the optical signal power \(P_{ij}\) can be obtained by Equation (30). \(P_{in,i}\) is the power, which injects into Port \(i\) of the MDM-OR. \(L_{ij,(x,y)}\) is defined as the transmission loss coefficient from Port \(i\) to the Port \(j\) of the \(N\)-port MDM-OR. The process produces total crosstalk noise \(C_{ij,}(x,y)\), which can be calculated by Equation (31), where \(P_{k}(x,y)\) is defined as the signal power, which is injected into the MDM-OR from Port \(k\), and \(K_{i,j,k}(x,y)\) indicates the crosstalk noise coefficient of the optical signal introduced by \(P_{k}(x,y)\) into the MDM-OR. \[P_{i,j}=P_{in,i}\cdot L_{i,j,(x,y)}\quad i,j\in\{0,1,2,\ldots,N-1\} \tag{30}\]  
\[C_{ij}(x,y)=\sum_{k=0}^{N-1}P_{k}(x,y)\cdot K_{i,j,k}(x,y)\quad i,j,k\in\{0,1,2,\ldots,N-1\} \tag{31}\]OSNR represents the ratio of signal power to noise power. More importantly, it is a crucial parameter to measure the performance of an optical router. Equation (32) can be used to calculate OSNR. Based on Equation (32), we can establish the BER, which is presented in Equation (33). BER is the ratio of received error messages to the total amount of information transmitted, which is an important index to judge the accuracy of information transmission.