# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 3 WSS-Based Approach  
Another approach to the design of RNB wavelength/space switches is based on WSSes [20; 21; 22]. As shown in Figure 8a, each input and output \(i\) of an \(r\times r\) WSS carries \(n\) wavelengths, denoted by \(\lambda_{0}^{i}\), \(\lambda_{1}^{i}\),..., \(\lambda_{n-1}^{i}\), for \(0\leq i\leq r-1\). A WSS can switch one wavelength channel from an input to any output, as long as no other signals of the same wavelength are switched to the same output (see Figure 8a). There are many technologies and architectures for implementing the device [28; 29; 30]. A possible implementation is shown in Figure 8b, where each block in the middle represents a nonblocking space switch. Note that a WSS requires space switching, but contains no WCs. Thus it is nonblocking in the space domain, but not in the wavelength domain. Figure 8: **(a)** A \(4\times 4\) WSS with four wavelengths. **(b)** One implementation of wavelength selective switches (WSS). Each block in the middle is a nonblocking space switch. Figure 7: An RNB network with \(n=8\), \(N=4\), and \(r=24\). We can just demux the eight wavelengths into two fibers and use the same technique shown in Figure 6 to design an RNB switch with such parameters. Note that these eight wavelengths are converted to wavelengths \(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) in each middle switch. A typical architecture for making a WSS-based wavelength/space RNB switch is shown in Figure 9a, where there are three stages of \(r\times r\) WSSes and two stages of \(n\)-WC modules that perform _fixed_ wavelength conversions and need not be tunable. Switching is done through dynamically changing the states of the space switches in the architecture. In contrast, the architecture of Section 2 contains no space switches and switching is done through dynamically changing the states of WCs. To understand how to make a WSS-based nonblocking wavelength/space switch, we again expand the topology of the switch in the space domain [31]. Note that in the wavelength dilated topology (as such Figure 9b), node stages are numbered from 1 and links from 0. Each \(r\times r\) WSS is represented by \(n\)\(r\times r\) space switches in the expanded space topology, where \(n\) is the number of wavelengths carried on each input and output fiber. When WSS is expanded in the wavelength domain, all \(r\) input or output links will be repeated \(n\) times in the space domain. Let \(\zeta_{2}\) represent the link number in the original topology, where \(0\leq\zeta_{2}\leq r-1\). Then, we can use a two-tuple (\(\zeta_{1}\), \(\zeta_{2}\)), \(0\leq\zeta_{1}\leq n-1\), to represent a link in the expanded space topology, where \(\zeta_{1}\) represents the group number of each link and each group corresponds to one wavelength in the expanded space topology. The interconnection pattern between two stages of space switches in the expanded space topology must follow the following properties. **Property 4**. : _Each link in the expanded space topology can be represented by a two tuple (\(\zeta_{1}\), \(\zeta_{2}\)), as defined above. Consequently, we let (\(t_{1}\), \(t_{2}\)) denote an output link of stage-\(i\) switches and (\(t_{1}\)\({}^{\prime}\), \(t_{2}\)\({}^{\prime}\)) the input link of stage-\((i+1)\)  
Figure 9: (**a**) The general architecture of a WSS-based wavelength/space switch with \(r=4\), \(n=4\). (**b**) An RNB network dilated in space domain of Figure 9a. (**c**) From the mapping in Figure 9b, we can derive the final implementation with \(r=4\), \(n=4\). switches in the expanded space topology, where \(t_{1}\) and \(t_{1}{}^{\prime}\) represent the corresponding group numbers. If there is a wire connecting (\(t_{1}\), \(t_{2}\)) and (\(t_{1}{}^{\prime}\), \(t_{2}{}^{\prime}\)), then \(t_{2}=t_{2}{}^{\prime}\)._  
In the original topology, the \(l\)th output link of the stage-\(i\) WSS is always connected to the \(l\)th input link of the stage-\((i+1)\) WSS. Therefore, in the expanded space topology, the same property must hold. Thus, we prove the property above.