# Exploring a New Adaptive Routing Based on the Dijkstra Algorithm in Optical Networks-on-Chip
## 2 Internal Insertion Loss Analysis of ONoCs
### Basic Optical Switching Elements (BOSEs)  
An optical router consists of waveguide crossings, waveguides, optical terminators, and BOSEs. BOSEs are composed of waveguides and microresonators [10]. There are two basic \(1\times 2\) optical switching elements which are the crossing switching element (CSE) and the parallel switching element (PSE). Besides, the CSE consists of two crossing waveguides and a microring resonator adjacently located between two crossing waveguides shown in Figure 2a,b. The PSE is composed of two parallel waveguides and a microring resonator locating between two parallel waveguides shown in Figure 2c,d.  
As shown in Figure 2, PSE and CSE have two states, namely on state and off state. If the wavelength of the optical signal is different from the resonant wavelength, the optical signal will be output to the through port rather than being coupled into the ring. This state is called off state, which is shown in Figure 2a,c. The on states of PSE and CSE are shown in Figure 2b,d. When the wavelength of the optical signal is equal to the resonant wavelength, the optical signal is transmitted from the input port to be coupled with the ring and directed to the drop port. We ignore the crosstalk noise on the add port of the PSE in this analysis according to a previous study [10]. In order to understand the notations in the formula, Table 1 lists the meanings of the relevant notation. The parameters in Table 1 are used for performance evaluation. In addition, the insertion loss is the output power coefficient of optical switching elements, while the transmission loss is the loss coefficient of the optical switching elements in this paper.  
Figure 1: Hybrid optical mesh network-on-chip (NoC) construction.  
The output powers of the through port and drop port as functions of input optical power could be calculated based on (1) and (2) for the off state and (3) and (4) for the on state.  
\[P_{T,off} =P_{in}L_{p,off} \tag{1}\] \[P_{D,off} =P_{in}K_{p,off}\] (2) \[P_{T,on} =P_{in}L_{p,on}\] (3) \[P_{D,on} =P_{in}K_{p,on} \tag{4}\]  
When two optical signals pass through the crossing at the same time in CSE, the optical signals interfere with each other and crosstalk occurs. At the same time, a portion of the light will be reflected back to the input port. When CSE is in the on state, the output power values of different ports are calculated as (5)-(8) according to previous studies [10]. When the CSE is in the off state, the optical signal enters from the input port and is coupled to the ring, and then is directed to the drop port. The formulae for calculating the output power at different ports are shown in (9)-(12).  
\[P_{D,on} =P_{in}L_{c,on} \tag{5}\] \[P_{T,on} =P_{in}K_{p,on}(L_{c}(1+K_{c}L_{p,on})+K_{r}L_{p,on}K_{c})\] (6) \[P_{A,on} =P_{in}K_{p,on}(K_{c}(1+K_{c}L_{p,on})+K_{r}L_{p,on}L_{c}) \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Notation** & **Value** & **Reference** \\ \hline _Crossing loss_ & \(L_{c}\) & \(-0.04\) dB & [22] \\ _Power loss per PSE in ON state_ & \(L_{p,on}\) & \(-0.5\) dB & [23] \\ _Power loss per PSE in ON state_ & \(L_{p,off}\) & \(-0.005\) dB & [23] \\ _Crosstalk coefficient per PSE in OFF state_ & \(k_{p,off}\) & \(-45\) dB & [10] \\ _Crosstalk coefficient per PSE in ON state_ & \(k_{p,off}\) & \(-25\) dB & [10] \\ _Power loss per CSE in OFF state_ & \(L_{c,off}\) & \(-0.04\) dB/\(90^{\circ}\) & [10] \\ _Power loss per CSE in ON state_ & \(L_{c,on}\) & \(-0.5\) dB/\(90^{\circ}\) & [10] \\ _Crossing crosstalk coefficient_ & \(k_{c}\) & \(-38.5\) dB/\(90^{\circ}\) & [10] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Power loss factors of a basic optical device.  
Figure 2: Basic \(1\times 2\) switching elements using microresonators.  
\[P_{R,on}=P_{in}K_{p,on}K_{c} \tag{8}\]  
\[P_{T,off}=P_{in}L_{c,off} \tag{9}\]  
\[P_{D,off}=P_{in}(K_{p,off}+L_{p,off}^{2}K_{c}) \tag{10}\]  
\[P_{A,off}=P_{in}K_{p}L_{p,off}^{2} \tag{11}\]  
\[P_{R,off}=P_{in}K_{r}L_{p,off}^{2} \tag{12}\]