## 4 Design Methodology for SOA-Enabled HONoC
### Worst-Case Crosstalk Noise and SNR Analysis  
For the analysis of the worst-case SNR in the proposed architecture, we use several definitions as follows:  
* \(n_{r}\): the total number of routers in the optical signal path  
Figure 5: Example of cumulated crosstalk noise according to the SOA placement. * \(n_{s}\): the total number of SOAs in the optical signal path
* \(t_{i}\): the number of the routers between the \(i\)th SOA and the \((i-1)\)th SOA in the optical signal path (\(t_{0}=0,\ t_{n_{s}+1}=(\#\) of routers between the last SOA and the destination))
* \(P_{N_{i}}\): the crosstalk noise introduced into the \(i\)th router in the optical signal path
* \(L\): router loss, \(G\): SOA gain, \(K\): crosstalk noise coefficient  
To simplify the worst-case SNR equations, we make the following assumptions
* The insertion loss and the crosstalk noise coefficient are the same regardless of ports. (\(L_{i,j}=L,K_{i,j,n}=K\))
* The mesh size is \(m\times m\). As the crosstalk noise introduced into each router is also amplified by the corresponding SOAs in the optical signal path, these effects are reflected in the SNR as in (8). \[SNR=10\log\left(\frac{P_{5}G^{n_{s}}L^{n_{s}}}{\sum_{i=0}^{n_{s}}G^{n_{s}-i} \sum_{j=T_{i}+1}^{T_{i+1}}\left(P_{N_{j}}L^{n_{s}-j}\right)}\right)\text{ where }T_{i}=\sum_{k=0}^{i}t_{k} \tag{8}\]  
As shown in Equation (8), it is inevitable that the crosstalk noise introduced into the optical signal passes through the SOAs located in the signal path. Therefore, to maximize the SNR, the worst crosstalk candidate must be suppressed before it leaks into the desired optical signal path. The dominant crosstalk power is mainly caused by the crosstalk noise coming from the nearest router in conventional HONoCs. However, in the SOA-integrated HONoC, the worst crosstalk candidates vary depending on the gain of the associated SOAs. If the SOA gain is larger than the insertion loss between the SOAs, the crosstalk noise introduced from a farther router has a greater impact on the SNR than that from a nearer router. For example, when there are the three crosstalk noise sources \(P_{1}\), \(P_{2}\) and \(P_{3}\) as shown in Figure 6, each crosstalk noise power can be expressed as in (9). If the total SOA gain is smaller than the total power loss (\(\prod G_{i}L_{i}<1\)), the magnitude of each crosstalk noise power is ordered as \(N_{1}<N_{2}<N_{3}\); otherwise, \(N_{1}>N_{2}>N_{3}\). \[\text{N}_{1}=P_{1}G_{1}L_{1}G_{2}L_{2}G_{3}L_{3}K \tag{9a}\] \[\text{N}_{2}=P_{2}G_{2}L_{2}G_{3}L_{3}K\] (9b) \[\text{N}_{3}=P_{3}G_{3}L_{3}K\] (9c) \[\text{where }P_{1}=P_{2}=P_{3},G_{1}=G_{2}=G_{3}\]  
Therefore, the worst crosstalk noise differs depending on the SOA gain even with the same placement of SOAs. To quantify the effect of SOAs when the current signal path acts as a crosstalk noise to other adjacent signal paths, we introduce the concept of average insertion loss (\(L_{avg}\)) per SOA as (10). Here, definitions of terms are as follows:  
* \(n_{r,c}\): the total number of routers in the crosstalk path
* \(n_{s,c}\): the total number of SOAs in the crosstalk path  
Figure 6: Example of crosstalk noise paths whose magnitude order varies depending on SOA gains. \(t_{i,c}\): the number of the routers between the \(i\)th SOA and the \((i-1)\)th SOA in the crosstalk path  
\[L_{avg}=\left\{ \begin{array}{c}\
icefrac{{n_{s}}}{{\sqrt{i\in\{2,\cdots,n_{r,c} \}}L_{i}}},\sum\limits_{i=1}^{n_{s,c}}t_{i,c}\
eq n_{r,c}\\ \\ \
icefrac{{n_{s,c}-1}}{{\sqrt{i\in\{2,\cdots,n_{r,c}\}}L_{i}}},\sum\limits_{i=1 }^{n_{s,c}}t_{i,c}=n_{r,c}\end{array} \right. \tag{10}\]  
To analyse the worst crosstalk candidates, we denote the router to be analysed as \(R_{DUT}\) and the SOA farthest from the \(R_{DUT}\) in the crosstalk-inducing path as \(SOA_{DUT}\). In Equation (10), \(L_{avg}\) is determined by the power loss of an optical signal traveling from \(SOA_{DUT}\) to \(R_{DUT}\) and the number of SOAs in that path.