On the complexity of aperiodic Fourier modal methods for finite periodic structures

The Fourier modal method (FMM) is based on Fourier expansions of the electromagnetic field and is inherently built for infinitely periodic structures. When the infinite periodicity assumption is not realistic, the finiteness of the structure has to be incorporated into the model. In this paper we discuss the recent extensions of the FMM for finite periodic structures and analyze their complexity both with respect to the main discretization parameter N̂ as well as with respect to the number of periods R. We show that among the three FMM-based approaches able to represent finiteness, the aperiodic Fourier modal method with alternative discretization has the lowest computational cost given by either O(N̂ log2R) or O(N̂R) depending on the values of N̂ and R. This result demonstrates that the method is highly suited for rigorous modeling of scattering from large periodic structures. For instance, for N̂ = 100 and R < 1000 the complexity of the aperiodic Fourier modal method with alternative discretization is comparable to the complexity of the standard FMM.