Modular optical switch architectures combining wavelength routing based on arrayed waveguide grating (AWG) devices and multicasting based on star couplers hold promise for flexibly addressing the exponentially growing traffic demands in a cost- and power-efficient fashion. In a default switching scenario, an input port of the AWG is connected to an output port via a single wavelength. This can severely limit the capacity between broadcast domains, resulting in interdomain traffic switching bottlenecks. An unexplored solution to this issue is to exploit multiple AWG free spectral ranges (FSRs), i.e., to set up multiple parallel connections between each pair of broadcast domains. In this paper, we study, for the first time, the influence of the FSR count on the throughput of a multistage switching architecture and propose a generic and novel analytical framework to estimate the blocking probability. We assess the accuracy of our analytical results via Monte Carlo simulations. Our study points to significant improvements with a moderate increase in the number of FSRs. We show that an FSR count beyond four results in diminishing returns. Furthermore, to investigate the tradeoffs between the network- and physical-layer effects, we conduct a cross-layer analysis, taking into account pulse amplitude modulation and rate-adaptive forward error correction. We illustrate how the effective bit rate per port increases with an increase in the number of FSRs.