# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 3 WSS-Based Approach
According to the decomposition method proposed in Section 2, we construct a variant of a three-stage Clos network that can satisfy Property 4 in the expanded space topology. One is given in Figure 11a. Note that the topology shown in Figure 11a is equivalent to that in Figure 3, where switches labeled 0 in the first and last stages in Figure 11a correspond to switches 0, 4, and 8 in Figure 3. For the interconnection pattern between stage-1 and stage-2 space switches, let \((t_{1},t_{2})\) and \((t_{1}\)', \(t_{2}\)') be defined as before. We can then describe the interconnection pattern as:  
\[(t_{1},t_{2})\rightarrow(t_{1}\prime=(t_{1}+(t_{2}\bmod r))\bmod r,\,t_{2} \prime=t_{2}) \tag{10}\]  
The interconnection connecting the last and the second last space switches is the mirror image of the mapping described by Equation (10). In addition, the network from node stage-2 to node stage-(\(n-1\)) is divided into \(n\)\(r\times r\) 3-stage Clos networks derived by the decomposition method proposed in Section 2. The topology of this network is given in Figure 11a. From this topology, we can derive from the original wavelength/space switch how many converters are needed on each link and  
Figure 10: (**a**) An RNB network dilated in space domain with \(r=12\), \(n=3\). Note colors used for switches correspond to wavelengths \(\lambda_{0}\), \(\lambda_{1}\), and \(\lambda_{2}\). (**b**) The WSS-based implementation with \(r=12\), \(n=3\). what conversions will be performed by them (Property 5). The result is shown in Figure 11b. Similar to Property 6, the WSS-based wavelength/space switch given in Figure 10b is rearrangeably nonblocking. It is worth noting that a \(k\)-nary Benes network is a special case under this condition, where each WSS switch is with \(r=k\), \(n=k\). A WSS-based 3-nary Benes network is given in Figure 12.  
**Case 3. \(n>r\)**  
Under this condition, the approach can work if \(n=r^{k}\) for any positive integer \(k\). We again use the rule described by Property 4 to construct a variant of a three-stage C(\(r\), \(r\), \(n\)) network in the expanded space topology, as shown in Figure 11a. Let \((t_{1}\), \(t_{2})\) and \((t_{1}\)\({}^{\prime}\), \(t_{2}\)\({}^{\prime}\)) be defined as before. The mapping \((t_{1}\), \(t_{2})\rightarrow(t_{1}\)\({}^{\prime}\), \(t_{2}\)\({}^{\prime}\)) describing the interconnection pattern between stage-1 and stage-2 space switches in Figure 11a is given as:  
\[ \begin{array}{l}t_{1}t=(\lfloor t_{1}/r\rfloor+t_{2})\bmod n,\\ t_{2}t=t_{2}.\end{array}  \tag{11}\]  
Figure 11: A WSS-based implementation with \(r=12\), \(n=4\), where each WSS switch is with \(r=4\) or \(3\), \(n=4\). (**a**) Construct a variant of a three-stage C (4, 4, 12) network in the expanded space topology. (**b**) The final architecture can be derived from (**a**). Figure 12: A WSS-based implementation with \(r=9\), \(n=3\), where each WSS switch is with \(r=3\), \(n=3\). (**a**) Construct a variant of a 3-nary Benes network in the expanded space topology. (**b**) The final architecture can be derived from (**a**). The interconnection connecting stage-2 and stage-3 space switches is the mirror image of the mapping described by Equation (11). Again, we can use the Clos theory to prove that the switch in Figure 13a is an RNB switch. We then decompose each of the middle-stage switches into another three-stage switch. This decomposition process continues until all switches have the size of \(r\times r\). Each decomposition does not change the nonblocking property of the switch. The final result is that there will be \(n\)\(r\times r\) switches in each stage and each corresponds to one wavelength in the expanded space topology. Since the overall switch is RNB, the original network is RNB.