## 1 Introduction  
Wavelength-division-multiplexing (WDM) optical networks, in which about 100 wavelength channels can be packed into a fiber and each channel can support a data rate of up to 100 Gb/s [1], have been widely deployed. An optical cross-connect (OXC) [2; 3; 4] switch, involving switching in both the wavelength and the space domain [5; 6; 7; 8], is a key component in such a network.  
An arrayed waveguide grating (AWG) is a passive device that can route multiple wavelengths to different destinations simultaneously. Each input of an \(N\times N\) AWG can simultaneously send \(N\) signals of different wavelengths, each destined to a different output, and the \(N\times N\) AWG can forward all \(N^{2}\) signals without blocking [9]. The multi-wavelength capability makes such devices more attractive for constructing an OXC than other switching technologies, such as semiconductor optical amplifiers (SOAs) [10]. This capability has been exploited in many prior works [6; 7; 8; 9; 11; 12; 13; 14; 15].  
However, AWGs have poor scalability due to crosstalk [16; 17] and deviation of the passband center frequencies from the ITU-T (International Telecommunications Union Standardization Sector) grid [18]. Currently the size limit of a realistic AWG device is likely to be less than fifty [16; 19; 20], and 32 \(\times\) 32 AWGs are commercially available [21]. Solving the scalability issue of AWGs has been attempted in many prior works [8; 14; 22; 23; 24; 25; 20]. The approach given in [22] expanded the port count from \(N\) to \(N^{2}\), but used \(2N^{2}\) costly SOA-based tunable wavelength converters (TWCs), which consumed a substantial amount of power [10]. Another approach, which used two stages of smaller AWGs to construct a 90 \(\times\) 90 AWG, was proposed in [20; 23], while still another approach, given in [8; 14], expanded the size of an AWG-based circuit switch from \(N\) to \(N^{2}\) by using TWCs. In [24], an ASA (AWG, Space switching, AWG) switch, using AWGs, space switches, and AWGs in its three stages, was proposed for packet switching. The 3-stage ASA switch expanded the size from \(N\) to \(N^{2}\) by using \(N\times N\) AWGs but did not use any wavelength converters. Combining two techniques proposedin [20; 24], an ultra-scalable AWG-based optical packet switch with a total capacity close to \(4\times 10^{16}\) bps was presented in [25].  
In this paper, we generalize the 3-stage ASA switch proposed in [24], and expand its switch size to \(N^{t}\) for any positive integer \(t\). Note that \(N^{t-1}\times N^{t-1}\) space switches are required for the general 3-stage \(N^{t}\times N^{t}\) ASA switch architecture. Considering the scalability of space switches, we further propose a rearrangeable nonblocking (RNB) multi-stage \(N^{t}\times N^{t}\) ASA switch architecture using only \(N\times N\) AWGs and \(N\times N\) space switches, where the RNB function can accommodate a new connection by rearranging some existing connections [26]. Since packets in a frame are routed in the network simultaneously, rearrangements are no longer required for packet switching. This leads to that rearrangeable nonblocking can usually be treated as strictly nonblocking for packet switching [26].  
The rest of the paper is organized as follows: Section 2 introduces the preliminaries of the 3-stage ASA switch architecture [24]. Section 3 proposes the general 3-stage ASA switch architecture. The RNB general multi-stage ASA switch architecture is proposed in Section 4, and Section 5 concludes the discussion in this paper.