## 2 Optical Interconnection Architecture
### Backplane Transfer Function  
Once each RR is designed (_i.e._, setting its physical parameters on the basis of the target filter-function physic [8]), we have used the Advanced Simulator for Photonic Integrated Circuits (ASPIC) [9; 10] to calculate the transfer function of the backplane. Figure 4 shows the transfer function of a certain TX-RX pair from the architecture shown in Figure 2 with \(S=4\) planes and \(N\)/\(S=8\) TX/RX per plane. The grey vertical bars represent optical channels at 10 Gb/s with 0.4 nm (50 GHz) spacing. Typical parameters used to assess the performance are: in-band insertion losses (\(IL\)), bandwidth at \(-3\) dB from the resonance peak (\(B_{3\ dB}\)), adjacent channel cross-talk (XT) (\(X_{\lambda}\)) and non-adjacent channel XT (\(X_{\lambda\lambda}\)). When considering the transfer function of all TX-RX pairs, other parameters are defined: \(<\)\(IL\)\(>\) is the average insertion-loss value across filter resonance peaks; \(IL_{m}\) and \(IL_{M}\) are the best and worst insertion loss, respectively; \(B_{m}\) and \(B_{M}\) are the smallest and largest \(-3\) dB bandwidth, respectively; and \(X_{FL}\) represents the worst-case floor XT (_i.e._, the gap between \(IL_{M}\) and the ground XT). Besides the amplitude response of the architecture, in this study we also take into account the phase response \(\Phi(\omega)\). Actually, the most important parameter which is used to be investigated is the _group delay_, which is defined as: \(\tau_{g}=-d\Phi(\omega)/d\omega=\lambda^{2}d\Phi(2\pi cd\cdot\lambda)\). The group delay represents the time that a signal takes to pass through a filter. Usually, frequency components of the signal spectrum emerge from the filter with a changed time relationship as they propagate with different velocities and it can cause signal distortion and pulse broadening [11]. It is also worth mentioning that the derivative of the group delay with \(\omega\) is the second order dispersion which is expressed in (ps\({}^{2}\)) and it can be calculated as \(D=d\tau_{g}/d\omega\) or, equivalently, by \(D=\lambda^{2}d\cdot d\tau_{g}/(2\pi cd\lambda)\) expressed in (ps\({}^{2}\)/nm). Figure 3: (**a**) Three stages BTR; (**b**) Drop port transfer function of a three stages BTR. Figure 5a shows a typical \"_drop_\" port transfer function of a ring resonator and its group delay. Optical \(-3\) dB bandwidth is set to 20 GHz. Figure 5b shows the \"_through_\" port transfer function and its group delay. It is evident that for the latter the dispersion is negligible as the group delay is constant and all the frequency components near to the resonant frequency roughly experience the same \"pass-through\" time. On the other hand, drop port exhibits a group delay distortion due to a non-linear phase response of the filter. At the resonance, the group delay increase with the _finesse_, which is the ratio between the _FSR_ and the bandwidth, whereas, for the through, port it becomes no longer significant. Second order dispersion is zero at the resonance peak, leading to a small contribution of the third order dispersion on the pulse shape. In our architectures, on the basis of the number of stages of the PSC, signals pass through a given number of ring resonators. Thus, transfer functions show different values of the performance parameters since they are related to the particular path through the architecture a certain signal is routed. Figure 4: Backplane transfer function of a certain TX-RX pair. Figure 5: Transfer function “-” and group delay “- -” at the (**a**) drop port and (**b**) through port of an RR. In Figure 6 we report the transfer functions just before the RSC of the architecture in Figure 2 with \(S=4\) planes and \(N/S=4\) TX/RX per plane. The first picture (architecture receiver plane 1) is given by two drop port pass-through. In fact, it has the highest peak of the group delay and the smallest bandwidth. It is worth noting that compared to Figure 5a, the group delay is almost doubled while retaining its shape and the insertion loss is increased of about 1 dB.