# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 2 A New Decomposition Technique for Constructing AWG/WC-Based Nonblocking Switches
### A New Decomposition Scheme
Since each stage of the space switches corresponds to a stage of WC modules in the original network, we need to minimize \(s\). It is easy to see that under the requirement that each factor \(n_{i}\) is the largest factor of \(r/(n_{1}\times n_{2}\times\dots\times n_{i-1})\) with \(1<n_{i}\leq n\), a compact factorization of \(r\) leads to the minimum number of stages. **Property 3. **_The compact factorization of \(r\) is unique._  
**Proof. ** Suppose \(r\) has two distinct compact factorizations: \(n_{1}\times n_{2}\times\dots\times n_{s}\) and \(n_{1}{}^{\prime}\times n_{2}{}^{\prime}\times\dots\times n_{s}{}^{\prime}\). Assume \(i\) is the smallest index, such that \(n_{i}\
eq n_{i}{}^{\prime}\), that is, \(n_{j}=n_{j}{}^{\prime}\) for all \(j\leq i-1\). It implies that either  
Figure 4: An implementation of Figure 3 in the original space and wavelength domains. The arrayed-waveguide grating (AWGs) and wavelength converters (WC) modules marked in color are transformed from the components of the same color in Figure 3.  
\(n_{i}\) or \(n_{i}{}^{\prime}\) is not the largest factor of \(r/(n_{1}\)\(\times\)\(n_{2}\)\(\times\)\(\ldots\)\(\times\)\(n_{i-1})\) (\(=r/(n_{1}{}^{\prime}\)\(\times\)\(n_{2}{}^{\prime}\)\(\times\)\(\ldots\)\(\times\)\(n_{i-1}{}^{\prime})\)), which contradicts the definition of compact factorization. Hence, the proof is complete. \(\Box\)  
Figures 5 and 6 show another example with \(r\) = 24 and \(n\) = 4. This case was not discussed in [11], which only presents the case where \(r\) is a power of \(n\). In the example above, we assume \(n\) = \(N\). Figure 5: Assume \(r\) = 24, \(n\) = 4. The compact factorization of \(r\) is 4 \(\times\) 3 \(\times\) 2. (a) A three-stage C (4, 4, 24) network. (b) Each 24 \(\times\) 24 middle switch in (a) is replaced with a multi-stage network generated by the decomposition scheme. Figure 6: An RNB network with \(r\) = 24, \(n\) = 4. Its transformed space switch is shown in Figure 5. The AWGs and WC modules marked in color are transformed from the components of the same color in Figure 5.  
Let us consider the case \(n>N\) (this case was not discussed in [11] either). Consider the example where \(n=8\), \(N=4\), and \(r=24\) (see Figure 7). All we need to do is to demux the \(n\) wavelengths into \(n/N\) fibers and the remaining systems will be exactly the same as described in Figure 6. Note that these \(n\) wavelengths are converted to wavelengths \(\lambda_{0}\), \(\lambda_{1}\),..., \(\lambda_{\text{N}-1}\) in the remaining systems. Thus, only \(N\) wavelengths are used internally in the final network (i.e., after the first stage of WC modules and before the last stage of WC modules). From the above discussion, we can see that our decomposition technique is carried out in the _space_ domain. As long as the network is nonblocking in the equivalent space network, the original space/wavelength network is nonblocking. This makes our decomposition simpler than the decomposition technique presented in [11], which is completed in both the space and wavelength domains. Since we only need to deal with the space domain, our technique is also more flexible in designing networks of different sizes.