## 4 Analytical Model: Signal-to-Crosstalk Ratio
### Signal-to-Crosstalk Ratio Model  
We start by calculating the SCR of a simple example: a \(4\times 4\)-Benes. Then, we propose a general equation that can be adapted to any architecture. The \(4\times 4\)-Benes structure is shown in Figure 6. The four inputs \(P_{in1},\ P_{in2},\ P_{in3},\ P_{in4}\) are connected to the four outputs through the appropriate colored paths. In this example, we have chosen the worst combination connection among the \(4!\) possible ones. We are working on the assumption that the first order crosstalk caused by MZI (\(m\)) and waveguide crossing (\(x\)) is not negligible, and so is the second-order crosstalk generated by MZI (\(m^{2}\)). The other contributions are neglected: for instance, optical crosstalk caused by an MZI could be the origin of another optical crosstalk generated at a waveguide crossing, or vice versa. This case is not considered here because it has a small influence on the total SCR. Table 2 details the output power following each stage of the structure in Figure 6. The wanted-signal is written in black, crosstalk generated by MZI and waveguide crossing are respectively mentioned in blue and red. Assuming that all signals are coherent and in-phase, we can write for the input signals \(P=P_{in1}=P_{in2}=\ P_{in3}=P_{in4}\). The SCR at the worst output (out2 or out3) is then: \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Stage Count** & **W. Crossing Count** \\ \hline Benes & \(2 log_{2}(N)-1\) & \(2(N-log_{2}(N)-1)\) \\ Dilated Benes & \(2 log_{2}(N)\) & \(2(2N-log_{2}(N)-3)^{*}\) \\ DLN & \(2 log_{2}(N)-1\) & \(3N-2 log_{2}(N)-4\) \\ S\&S & \(2 log_{2}(N)\) & \((N-1)^{2}\) \\ \hline \hline \end{tabular} \({}^{*}\) Valid for \(N>2\), for \(N=2\): Crossing count = \(1\).  
\end{table} 
Table 1: Topology comparison for the first case in terms of stage and waveguide crossing counts \(N\) is the number of inputs/outputs. \[10 log\bigg{(}\frac{Pl_{MZI}{}^{3}l_{X}{}^{2}}{mPl_{MZI}{}^{3}(l_{X}{}^{2}+l_{X}+1)+ m^{2}Pl_{MZI}{}^{3}(l_{X}+l_{X}+1)+xPl_{MZI}{}^{3}(l_{X}{}^{2}+l_{X}{}^{2})}\bigg{)} \tag{5}\]  
which simplifies to:  
\[10 log\bigg{(}\frac{l_{X}{}^{2}}{m(l_{X}{}^{2}+l_{X}+1)+m^{2}(2l_{X}+1)+2 xl_{X} {}^{2}}\bigg{)} \tag{6}\]  
Note that if we neglect the effect of waveguide crossings, i.e., \(x\approx 0\), \(l_{X}\approx 1\) and crosstalk of second-order \(m^{2}\approx 0\), SCR is equal to \(-10 log(3m)\), which is the equation obtained in [15]. Based on the example of \(4\times 4-\)Benes structure, we notice that the wanted-signal travels through 3 MZIs and 2 waveguide crossings, which represents the worst path. It can also be noted that crosstalk powers at a given output come from different paths that have the same number of stages but a different number of waveguide crossings, so the total insertion loss for each crosstalk power is not always the same. In a large-scale network, it will be difficult to calculate the exact path of each crosstalk power, thus, we assume for simplification purposes that all crosstalk powers pass through an average number of waveguide crossings. Conversely, the number of stages is always the same for all crosstalk powers. All parameters we have taken into consideration in our model are included in Equation (7). \[\hat{SCR}=\ 10 log\bigg{(}\frac{P_{J,MZI}{}^{num_{MZI}}l_{X}{}^{num_{X}}}{K_{MZI} (N,m)Pl_{MZI}{}^{num_{MZI}}l_{X}{}^{M_{X}}+K_{X}(N,x)Pl_{MZI}{}^{num_{MZI}}l_{X }{}^{M_{X}}}\bigg{)} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Input and**} & **After S1** & **After S2** & **After S3** \\ \hline \multirow{4}{*}{\(P_{1}\)} & \multirow{4}{*}{\(P_{1}\)} & \(P_{1}l_{MZI}{}^{2}\) & \(P_{1}l_{MZI}{}^{3}\) \\  & & \(+mP_{2}l_{MZI}{}^{3}(l_{X}{}^{2}+l_{X}+1)+m^{2}P_{MZI}{}^{3}(l  
We note \(num_{MZI}\) and \(num_{X}\) respectively the number of stages and waveguide crossings in the worst path for a given architecture (Table 1). \(K_{MZI}(N,m)P\) and \(K_{X}(N,x)P\) are, respectively, the total cumulative crosstalk powers at the worst-output generated by MZIs and waveguide crossings.