In wavelength-division-multiplexing (WDM) switches, such as arrayed-waveguide-grating (AWG)-based Clos networks, the supporting of multicast traffic must rise to the challenge of route and wavelength assignment (RWA) problem. In this paper, we study the non-blocking multicast RWA problem in two phases with respect to the cascaded combination of an AWG-based broadcast Clos network, called copy network, and a point-to-point AWG-based Clos network. In phase one, input requests generate broadcast trees in the copy network, and then point-to-point connections are established in the AWG-based Clos network in the second phase. The Clos-type AWG-based multicast networks can be constructed from modular AWGs of smaller sizes with the purpose of minimizing the number of wavelengths required and reducing the tuning range of the wavelength selective converters (WSCs). For solving the multicast RWA problem, we extend the rank-based routing algorithm for traditional space-division broadcast Clos networks such that broadcast trees can also be generated in the WDM copy network in a contention-free manner. However, due to wavelength routing properties of AWGs, the subset of requests input to each subnetwork in the middle stage may not satisfy the precondition of the rank-based RWA algorithm. Nevertheless, we prove that this problem can be solved by cyclically shifting the indices of wavelengths in each subnetwork, which provides the key to recursively route the multicast requests in a non-blocking and contention-free manner in the decomposed AWG-based broadcast Clos network. The time complexity of the proposed multicast RWA algorithm is comparable to that of an AWG-based unicast Clos network.