# Construction of Nonblocking Wavelength/Space Switches with AWGs and WSSes
## 2 A New Decomposition Technique for Constructing AWG/WC-Based Nonblocking Switches  
### Wavelength Dilated Network  
A space/wavelength nonblocking switch is shown in Figure 1a [11], which contains only two types of devices: AWGs and WCs. An \(N\times N\) (\(N\)-input-port and \(N\)-output-port) AWG operates on a set of \(N\) wavelengths {\(\lambda_{0}\), \(\lambda_{1}\),..., \(\lambda_{N-1}\)}. Let \(i\), \(o\), and \(\lambda_{w}\) represent the input port, output port, and wavelength of a signal, respectively, wherein \(w\in\{0,1,\,\ldots,N-1\}\). The relationship among the three parameters of \(i\), \(o\), and \(w\) is specified by the equation below:  
\[w=(i+o)\bmod N \tag{1}\]  
Each input can transmit \(N\) signals of different wavelengths through the device without blocking. For an \(N\times M\) AWG with \(M\leq N\), we can take the first \(M\) output ports of an \(N\times N\) AWG and apply the same routing relationship as described in Equation (1). The same applies to an \(M\times N\) AWG. The \(t\)-WC module in Figure 1 refers to a set of \(t\) WCs. Let \(\psi=\{\lambda_{0},\,\lambda_{1},\,\ldots,\lambda_{t-1}\}\) be the set of input wavelengths, and \(\Gamma=\{\lambda_{0}{}^{\prime},\,\lambda_{1}{}^{\prime},\,\ldots,\,\lambda_{t -1}{}^{\prime}\}\) be the set of output wavelengths. Each WC inside the \(t\)-WC module converts an input wavelength \(\lambda_{i}\) to an output wavelength \(\lambda_{j}{}^{\prime}\), where \(\lambda_{i}\in\psi\) and \(\lambda_{j}{}^{\prime}\in\Gamma\), respectively. Note that each WC must be _tunable_ as the wavelength conversion is not fixed. The switch in Figure 1a has \(r\) input links and each link carries \(n\) wavelengths. Therefore, an input (or output) channel in the switch is defined by a two tuple (link, wavelength). The switch uses three stages of WC modules: \(n\)-WC, \(r\)-WC, and \(n\)-WC modules, respectively. These WC modules are  
Figure 1: (**a**) An \(n\)-wavelength \(r\times r\) rearrangeable nonblocking (RNB) switch with \(r>n\). (**b**) Topological transformation in the space domain—a 3-stage C(\(n\), \(n\), \(r\)) network. interconnected with two AWGs with the size \(r\times n\) and \(n\times r\). This switch is rearrangeable nonblocking (RNB), in the sense that whenever an input channel and an output channel is available, a connection between the two can always be set up by the switch, although some existing connections may need to be re-arranged. Given a set of connections, the looping algorithm [23] is the standard routing algorithm to route all the connections through the switch. To see why the network is rearrangeable nonblocking, we can transform the entire switch into the space domain (called the wavelength dilated switch), as shown in Figure 1b. This transformation technique is well established in the literature [11; 24]. In the wavelength-dilated space topology, an \(n\)-WC (or \(r\)-WC) module is represented by an \(n\times n\) (or \(r\times r\)) space switch and AWGs become connecting _wires_ in Figure 1b. The two networks are equivalent to each other. The transformed space network of Figure 1b is a three-stage C(\(n\), \(n\), \(r\)) Clos network [25]. This equivalent network is rearrangeable nonblocking. Consequently, its original wavelength/space network is also rearrangeable nonblocking. However, the problem becomes complicated once we consider the actual values of \(t\), \(n\), and \(r\). Note that the current maximum value of \(n\) is around 90 [26], and the current maximum value of \(N\), the size of an AWG, is around 32 [27]. If \(r>N\) or \(n>N\), these cases cannot be implemented. Neither can the case \(r>n\) due to the number of wavelengths limitation. To avoid the design constraints imposed by the values above, [11] presents a decomposition technique, decomposing the AWGs and WC modules recursively in the original topology. The technique is quite complicated. In the following, we describe a new decomposition technique. The decomposition is carried out in the dilated topology.