## 1. Introduction

[2] Periodic structures have many electromagnetic applications ranging from frequency selective surfaces to composite meta-materials [*Munk*, 2000; *Vardaxoglou*, 1997; *Ouchetto et al.*, 2006]. Rigorous simulations are often used to understand interactions between electromagnetic signals and these structures [*Dechant and Elezzabi*, 2004; *Chae et al.*, 2004]. To take advantage of the nature of periodic structures, periodic boundary conditions have been developed and implemented in the finite difference time domain (FDTD) method [*Taflove*, 2000; *Kao and Atkins*, 1996; *Roden et al.*, 1998; *Ren et al.*, 1994; *Marek and MacGillivray*, 1993; *Ko and Mittra*, 1993]. However, all these implementations assume that periodic structures are illuminated by plane wave incidences. That is, these methods are only valid when the incident electromagnetic signal is a plane wave. For some applications where electromagnetic responses from finite-sized sources are required, “brute force” FDTD simulations were performed [*Luo et al.*, 2002a, 2002b]. In the “brute force” simulations, rather than using a single periodic element, many repetitive cells were used to approximate the structure's infinite extension. Oftentimes, at least 30 unit cells are required in directions where infinite repetitions exist. However, such approximation truncates the actual structures, which could lead to significant reflections from the truncated boundary. In addition, this approach requires significantly more computer memory and CPU time compared to the modeling of a single periodic element.

[3] The purpose of this paper is to present a novel FDTD method to analyze the behaviors of arbitrary finite-sized electromagnetic sources over infinite periodic structures. Only a single periodic element needs to be modeled using this method and the field distribution at any location of the structure can be obtained. This approach not only eliminates the simulation domain truncation errors associated with the “brute-force” approach, it also leads to significant savings in both computer memory and CPU time. This approach is based on a source spectral expansion method combined with a spectral FDTD method [*Cangellaris et al.*, 1993; *Aminian and Rahmat-Samii*, 2005, 2006; *Yang et al.*, 2006; *Qiang et al.*, 2006, 2007a, 2007b; *Kokkinos et al.*, 2006; *Li and Sarris*, 2008]. In this approach, the finite-sized electromagnetic source is first expanded into its spectral components. Multiple spectral-FDTD simulations in the complex-domain are carried out using spectral electromagnetic sources. The final results obtained from multiple spectral-FDTD simulations are superposed to restore the electromagnetic signals from the original source. In addition to describing the basic principles of this technique, the guidelines of how to use this new method, which cover the convergence analysis, the selection of spectral sampling rate, and the acceleration schemes of this algorithm are also presented. Numerical examples are presented to validate this new method.

[4] The remainder of this paper is organized as follows. In section 2, the methodology of this approach is presented in detail, which includes the source expansion and the spectral-FDTD method with complex periodic boundary conditions. In section 3, some properties of this algorithm are discussed. This includes the convergence analysis, selection of appropriate spectral sampling rate, and acceleration scheme for this method. Following several numerical examples in section 4, conclusions are given in section 5.