## 2 Optical Interconnection Architecture
### Ring Resonator Based Backplane  
The switching backplane we aim at investigating in this work is shown in Figure 2. A Binary-Tree Ring Router (BTRR), used to select the output plane, acts as PSC whereas an RR array demultiplexer represents the RSC. We assume that only point-to-point connections are established between TXs and RXs, with no output contention, _i.e._, each RX is to be connected to at most one TX using one single wavelength.  
Figure 1: General multiplane switching architecture.  
The principle of the PSC tree-structure is the same as that based on a cascade of Mach-Zehnder filters, the so-called interleaving. An interleaver is a multi-stage demultiplexer with T stages where at each stage an RR spatially discriminates the incoming wavelength-channels partitioning them into two alternate sub-groups, sending half of them towards the \"through\" port and half of them towards the \"drop\" port. Note that, compared to Figure 1, this is a completely passive switching architecture as we do not investigate the use of EDFA before the RSCs. The number of stages T has to be large enough to completely discriminate all the \(S\) switching planes, thus T = (log2(\(S\))). The \(j\)-th stage will be composed of 2(\(j-1\)) RRs, with 1 \(\leq\)\(j\)\(\leq\) T and after \(j\) stages 2\(j\) planes can be selected.  
Ring resonators are infinite-impulse response filters with a periodic transfer function of the input frequency [4]. The family of resonance wavelengths \(\lambda_{res}\) at which the transfer function has its peaks is \(\beta n_{eg}\beta L_{r}=2\pi N\), where \(\beta=2\pi\lambda_{res}\), \(n_{eg}\) is the effective refractive index, \(L_{r}\) is the geometrical ring length and \(N\) is an integer number (\(N\geq 1\)). The period of the transfer function is knows as Free Spectral Range (\(FSR\)) and is given by \(FSR=c/(2\pi n_{g}R)\), where \(c\) is the speed of the light, \(n_{g}\) is the group index and \(R\) is the ring radius. The \(FSR\) is inversely proportional to the ring radius \(R\). The smallest achievable radius is constrained by the fabrication technology and optical losses inside the ring increase as well with bending [8].  
Let us consider a tree-structure composed of three stages as in Figure 3a. If we assume that \(\lambda_{(i,j)}\) is the resonance wavelength of the \(i\)-th ring of the \(j\)-th stage, the first ring of the first stage has to be dimensioned with a \(FSR=2\Delta\lambda\) and \(\lambda^{res}_{11}=\lambda_{0}\), then both rings of the second stage have \(FSR=4\Delta\lambda\) but \(\lambda^{res}_{12}=\lambda_{0}\), \(\lambda^{res}_{22}=\lambda_{1}\). Each ring of the third stage is dimensioned with \(FSR=8\Delta\lambda\) where \(\lambda^{res}_{14}=\lambda_{0}\), \(\lambda^{res}_{24}=\lambda_{2}\), \(\lambda^{res}_{34}=\lambda_{3}\), \(\lambda^{res}_{44}=\lambda_{1}\) are the resonance wavelengths of ring resonators respectively. As an example, Figure 3b shows general ring drop port transfer function of the first, second, and third stage respectively. The \(-3\) dB bandwidth of drop port transfer function can be very narrow, thus, showing higher filter selectivity than the through port [4].  
Figure 2: Ring Resonator-based switching architecture.  
The RR array demultiplexer RSC has to separate at most \(N\)/\(S\) optical channels (see Figure 2) spaced as \(2^{T}\Delta\lambda\); therefore each RR of the considered array is designed with a \(FSR_{RSC}=N\)/\(S\)\(2^{T}\Delta\lambda\) centered at a fixed wavelength. In order to set up a connection between a TX and a given RX on a certain Output plane to which the \(\lambda_{x}\) is assigned, the TX has to be tuned on the \(\lambda_{x}\).