## 1. Introduction

[2] This paper presents a marching-on-in-time (MOT) and Floquet wave–based scheme for solving an electric field time domain integral equation (TDIE) pertinent to the analysis of transient plane wave scattering from doubly periodic, perfect electrically conducting (PEC), discretely planar, and freestanding structures. The proposed approach uses blocked fast Fourier transform (FFT) based accelerators [*Harrier et al.*, 1985; *Bleszynski et al.*, 2001; *Yilmaz et al.*, 2002] to efficiently evaluate time domain Floquet wave (TDFW) decomposed electromagnetic fields [*Capolino and Felsen*, 2002, 2003; *Felsen and Capolino*, 2000; *Marrocco and Capolino*, 2002] generated by doubly periodic, discretely planar, and temporally band-limited source distributions.

[3] In the past, transient scattering from doubly periodic structures has been analyzed predominantly using finite difference time domain methods [*Veysoglu et al.*, 1993; *Tsay and Pozar*, 1993; *Harms and Mittra*, 1994; *Roden et al.*, 1998; *Holter and Steyskal*, 2002]. These solvers update fields inside a periodic structure's so-called mothercell using the classical Yee scheme [*Yee*, 1996] and impose periodic/absorbing boundary conditions on mothercell walls with normal vectors residing in/perpendicular to the plane of periodicity. Unfortunately, for obliquely excited periodic structures, these periodic boundary conditions call for future fields values to update current ones, and therefore cannot be applied directly. Several avenues for tackling this noncausality problem have been suggested [see *Maloney and Kesler*, 2002, and references therein]. It appears, however, that most fixes proposed to date are either hard to implement or somewhat limited in scope. Transient scattering from periodic structures also can be analyzed using TDIE-based schemes. Indeed, TDIE solvers for analyzing scattering from doubly periodic freestanding or substrate imprinted PEC elements were proposed by *Chen et al.* [2002, 2003]. Just like in their finite difference counterparts, noncausal terms arise when discretizing periodic structure TDIEs for obliquely incident fields using marching-on-in-time (MOT) procedures. *Chen et al.* [2002, 2003], removed these noncausal terms through the introduction of time-shifted temporal current basis functions in conjunction with a prolate-based extrapolation scheme. Unfortunately, even though these periodic structure TDIE solvers now efficiently cope with noncausal artifacts, their high computational complexity precludes them from being applied to the analysis of real-world structures. Generally speaking, the computational cost of MOT-based TDIE solvers can be attributed to their need to evaluate, at each and every time step, fields produced by past currents supported by the structure under analysis. The TDIE solvers of *Chen et al.* [2002, 2003] carry out this operation classically, by direct space-time convolution of the free space Green's function with all currents on the periodic structure. To be more specific, to evaluate the fields due to the past current, there is a double summation over the periodic cells. When the fields are observed on the mothercell, with the marching of time, the region around the mothercell in which the sources have to be take into account becomes larger and larger. This renders the solvers of *Chen et al.* [2002, 2003] computationally expensive.

[4] Here, an improved MOT-based TDIE solver for periodic structures is proposed. Whereas spectral methods for computing frequency domain Green's functions (Ewald representations [*Jordan et al.*, 1986], off-plane plane wave sums [*Jorgenson and Mittra*, 1991], etc.) are commonly used in periodic structure frequency domain integral equation solvers, the proposed solver is the first to do so within periodic structure TDIE simulators. The solver relies on a time domain Floquet wave (TDFW) representation of fields produced by periodic transient current constellations [*Capolino and Felsen*, 2002, 2003; *Felsen and Capolino*, 2000]. Specifically, the proposed solver exploits the fact that TDFW representations of fields produced by quiescent and band-limited sources only involve “propagating modes” (this fact, to the authors' knowledge demonstrated here for the first time for time domain signals, constitutes another important contribution of this paper). Hence TDFWs provide a natural, compact, and computationally efficient means of representing fields produced by band-limited sources residing on practical periodic structures that only support a finite and small number of propagating waves within their operating band, that is, structures with unit cells of linear dimensions on the order of the wavelength at the highest frequency in the incident field. Because the TDFW propagator is not time-local, costly time domain convolutions are carried out using a blocked-FFT scheme (first introduced in [*Harrier et al.*, 1985] for the purpose of solving one dimensional Volterra integral equations, and interpreted/tuned here within the proposed TDFW-TDIE framework). It will be shown that this decomposition and subsequent TDFW representation of the fields provides a means for computing fields produced by “past” currents in a manner consistent with the classical MOT-TDIE framework that is especially effective when the structure under study is discretely planar, viz. comprising a finite set of metallized layers. The computational cost of the new solver is only a fraction of that of periodic structure TDIE solvers not using TDFW concepts.

[5] This paper is organized as follows. Section 2 outlines the proposed TDFW/FFT-based scheme for rapidly computing transient fields produced by periodic current arrangements and its incorporation into an MOT-based TDIE solver for analyzing scattering from discretely planar structures. Section 3 presents numerical results that demonstrate the capability and accuracy of the proposed method. Section 4 relates our conclusions and avenues for future research.