# An \(N\)-Port Universal Multimode Optical Router Supporting Mode-Division Multiplexing
## 2 Materials and Methods
Figure 6d displays that the two modes are input from different ports of the multimode switching element, but output from the same port. Since the single-mode optical waveguide has a suppressive effect on higher-order modes, this paper ignores the crosstalk noise generated by the mode demultiplexing process. Combined with above analysis on the source of crosstalk noise, in the three cases of Figure 6a-c, after the two modes are restored to the TE\({}_{0}\) mode, they have no intersection in the single-mode waveguide. In other words, the crosstalk noise between them only comes from the mode multiplexing process, and the corresponding crosstalk noise \(C_{1,1}\), \(C_{1,2}\), and \(C_{1,3}\) can be illuminated as Equations (22) and (23). When the two modes are input from different ports of the multimode switching element and output from the same port, the crosstalk noise \(C_{2,1}\) generated by the TE\({}_{m}\) mode to the TE\({}_{n}\) mode can be illuminated as Equation (24). Correspondingly, the crosstalk noise \(C_{2,2}\) generated by TE\({}_{n}\) mode to TE\({}_{m}\) mode can be obtained by Equation (25). \[P_{mb,mn} = P_{in}\cdot Cmb_{mn} \tag{2}\]9 The four communication situations where two modes are output from the same port of the multimode switching element. (**a–c**) show situations that the two modes are input from the same port. (**d**) shows situation that the two modes are input from different ports. Figure 3.0 The two communication situations where the two modes are output from different ports of the multimode switching element. (**a**) The two modes are input from different ports. (**b**) The two modes are input from same ports. \[\left[ \begin{array}{c}C_{1,1}\\ C_{1,2}\\ C_{1,3}\end{array} \right]=P_{in}\cdot\left[ \begin{array}{cc}L_{dmx0}L_{sc}^{N}C_ {0 mmx}&0\\ L_{dmx0}L_{sc}^{N}C_{0 mmx}&0\\ 0&L_{dmx0}C_{0 mmx}\end{array} \right]\cdot\left[ \begin{array}{c}L_{sp1}\\ L_{sc2}\end{array} \right]\quad m=0 \tag{22}\]  
\[\left[ \begin{array}{c}C_{1,1}\\ C_{1,2}\\ C_{1,3}\end{array} \right]=P_{in}\cdot\left[ \begin{array}{cc}L_{dmxm}L_{sc}^ {N}L_{sb}^{2}C_{mmmx}&0\\ L_{dmxm}L_{sc}^{N}L_{sb}^{2}C_{mmmx}&0\\ 0&L_{dmxm}L_{sc}^{2m}L_{sb}^{2}C_{mmmx}\end{array} \right]\cdot\left[ \begin{array} []{c}L_{sp1}\\ L_{sc2}\end{array} \right]\quad m\
eq 0 \tag{23}\]  
\[\left[ \begin{array}{c}C_{2,1}^{a}\\ C_{2,1}^{b}\\ C_{2,1}^{c}\\ C_{2,1}^{d}\end{array} \right]=P_{in}\cdot\left[ \begin{array}{cc}L_{dmxm}L_{sc }^{N}L_{sb}^{2}C_{m0 mx}&0\\ L_{dmxm}L_{sc}^{N}C_{0 mmx}&L_{dmxm}L_{sb}L_{sc}^{N+n-2}K_{c}L_{mxn}\\ L_{dmxm}L_{sc}^{N}L_{sb}^{2}C_{mmmx}&L_{dmxm}L_{sc}^{N+n-m}-2L_{sb}^{2}K_{c}L_{ mxn}\\ L_{dmxm}L_{sc}^{N}L_{sb}^{2}C_{mmmx}&0\end{array} \right]\cdot\left[ \begin{array}{c}L_{sp1}\\ L_{sc2}\end{array} \right] \tag{24}\]  
\[a:n=0;b:m=0,n>m;c:n>m,m\
eq 0;d:n<m\]  
\[\left[ \begin{array}{c}C_{2,2}^{a}\\ C_{2,2}^{b}\\ C_{2,2}^{c}\\ C_{2,2}^{d}\end{array} \right]=P_{in}\cdot\left[ \begin{array}{cc}0&L_{dmx0} C_{0 mmx}\\ 0&L_{dmxm}L_{sc}^{2n}L_{sb}^{2}C_{mmmx}\\ L_{dmxm}L_{sc}^{m+n}L_{sb}^{2}K_{c}L_{mxm}&L_{dmxm}L_{sc}^{2n}L_{sb}^{2}C_{ mmmx}\end{array} \right]\cdot\left[ \begin{array}{c}L_{sp1}\\ L_{sc2}\end{array} \right] \tag{25}\]  
\[a:n=0;b:n\
eq 0,m>n;c:n\
eq 0,m=0;d:m\
eq 0,n>m\]  
Figure 7a,b describe the situations where the two modes are output from different ports of the multimode switching element. In Figure 7a, the two modes will inevitably intersect in the single-mode waveguide inside the multimode switching element, the corresponding crosstalk noise C\({}_{3,1}\) and C\({}_{3,2}\) generated by the TE\({}_{m}\) mode to the TE\({}_{m}\) mode can be expressed as Equations (26) and (27). Equation (26) represents the crosstalk noise generated from the TE\({}_{m}\) mode to the TE\({}_{m}\) mode, and Equation (27) is the crosstalk noise generated from the TE\({}_{n}\) mode to the TE\({}_{m}\) mode. \[\left[ \begin{array}{c}C_{3,1}^{a}\\ C_{3,1}^{b}\\ C_{3,1}^{c}\\ C_{3,1}^{d}\\ C_{3,1}^{e}\end{array} \right]=P_{in}\cdot\left[