## 3 General 3-Stage ASA Switch Architecture  
Conventionally, C(\(n\), \(m\), \(r\)) is used to represent a 3-stage Clos network, where each \(n\)\(\times\)\(m\) (or \(m\)\(\times\)\(n\)) switch in stage 1 (or stage 3) connects to all \(r\)\(\times\)\(r\) switches in stage 2. Similarly, we use A(\(n\), \(m\), \(r\)) to denote a 3-stage ASA switch architecture, where each \(n\)\(\times\)\(m\) (or \(m\)\(\times\)\(n\)) AWG in stage 1 (or stage 3) connects to all \(r\)\(\times\)\(r\) space switches in stage 2. Thus the 3-stage ASA switch proposed in [24] is also denoted by A(\(N\), \(N\), \(N\)) (see Figure 2b), where \(N\) is the number of wavelengths in the AWG. Considering the scalability of 3-stage ASA switches, without loss of generality, we assume that the size of each space switch is of the power of \(N\), (i.e., \(r\)\(=\)\(N^{t-1}\)), for any positive integer \(t-1\), as \(N\)\(\times\)\(N\) AWGs are used. An A(\(N\), \(N\), \(N^{t-1}\)) switch is a general 3-stage \(N^{t}\)\(\times\)\(N^{t}\) ASA switch architecture for any positive integer \(t\). An example with an A(3, 3, 9) switch is given in Figure 3. The same as in a 3-stage \(N\)\(\times\)\(N\) ASA switch, for each signal [[\(g_{\rm s}\), \(m_{\rm s}\)], \(w\), [\(g_{\rm d}\), \(m_{\rm d}\)]] in a general 3-stage \(N^{t}\)\(\times\)\(N^{t}\) ASA switch, member fields are used for wavelength routing in stage 1 and stage 3, according to the relationship among \(m_{\rm s}\), \(m_{\rm d}\), and \(w\) satisfying Equation (2). Group fields are used for space switching in stage 2, that is, signal [[\(g_{\rm s}\), \(m_{\rm s}\)], \(w\), [\(g_{\rm d}\), \(m_{\rm d}\)]] is switched from the \(g_{\rm s}\)th input to the \(g_{\rm d}\)th output of the (\(m_{\rm s}\)\(+\)\(w\))th space switch in stage 2. From the topology of an A(\(N\), \(N\), \(N^{t-1}\)) switch architecture, we can see that the \(k\)th space switch in the middle stage connects the \(k\)th output (or input) of each AWG in the first (or last) stage for \(0\)\(\leq\)\(k\)\(\leq\)\(N-1\) (Figure 3). **Theorem 1**. : _Assume N is odd and \(t\) is a positive integer. An A(\(N\), \(N\), \(N^{t-1}\)) switch is nonblocking if signals from inputs with the same member field are destined to distinct outputs._  
**Proof. ** We prove this property by showing (i) each input can reach all outputs, and (ii) two different signals using the same wavelength will not pass through the same link. 1. Equation (3) guarantees that each input [\(g_{\rm s}\), \(m_{\rm s}\)] of an AWG in the first stage using \(N\) different wavelengths \(w\) can reach all outputs [\(g_{\rm d}\), \(m_{\rm d}\)] of an AWG in the last stage, if \(N\) is odd and the \(k\)th output of the AWG in the first stage connects the \(k\)th input of the AWG in the last stage [24]. Given an A(\(N\), \(N\), \(N^{t-1}\)) switch, since the \(k\)th space switch in the middle stage connects the \(k\)th output (or input) of each AWG in the first (or last) stage, it is implied that the \(k\)th output of each AWG in the first stage connects the \(k\)th input of all AWGs in the last stage through  
Figure 2: **(a)** The optical data path and the electronic control path of a 3-stage ASA switch. **(b)** The data plane of a 3-stage 9 \(\times\) 9 ASA (AWG, Space switching, AWG) switch. **(c)** A 2-stage switch cascading two 3 \(\times\) 3 AWGs. **(d)** Two signals ([0, 0], 0, [0, 0]) and ([1, 0], 0, [0, 0]) collide at the same link in stage 2.  
the \(k\)th space switch in the middle stage. Thus, each input can reach all outputs if it uses \(N\) different wavelengths. 2. Given an A(\(N\), \(N\), \(N^{t-1}\)) switch, each link in stage 1 carries only signals from the same \(N\times N\) AWG in the first stage. Recall that all the \(N^{2}\) signals of an \(N\times N\) AWG can traverse the device simultaneously without blocking each other (Equation (1)). This implies that two different signals using the same wavelength will not pass through the same link in stage 1.