## 1. Introduction

[2] Periodic electromagnetic structures have received increased attentions in recent years for their applications in the design of frequency selective surfaces (FSS), electromagnetic band gap (EBG) structures, and negative index materials. The finite-difference time-domain (FDTD) method has been used to analyze these structures. To take advantage of the periodic replication of these structures, periodic boundary conditions (PBC) was developed and implemented in various forms so that only a single unit needs to be simulated [*Maloney and Kesler*, 2000]. Some representative PBCs include the Sin-Cosine method [*Harms et al.*, 1994], the unit cell shifting method [*Holter and Steyskal*, 1999], the split-field method [*Roden et al.*, 1998], and the spectral FDTD method [*Aminian and Rahmat-Samii*, 2004].

[3] In this paper, a simple and efficient FDTD/PBC algorithm is presented to analyze the reflection and transmission properties of periodic structures. Compared to many previous PBC techniques, a distinguish feature of the proposed algorithm is that periodic structures are simulated under a constant horizontal wavenumber instead of at a given incident angle. The idea of FDTD computation with constant wavenumbers was originated in the analysis of guided wave structures and eigenvalue problems [*Xiao et al.*, 1992; *Cangellaris et al.*, 1993], and then expanded to the plane wave scattering problems in [*Aminian and Rahmat-Samii*, 2004]. The novelty of this paper is the direct computation of E and H fields rather than the indirect calculation using auxiliary fields [*Aminian et al.*, 2005]. Thus, the conventional Yee's scheme is used to update the E and H fields. As a consequence, the proposed method possesses several advantages, such as the simplicity in the algorithm implementation, the same stability condition and numerical errors as those of the conventional FDTD method, and good computational efficiency near the grazing incident angles.

[4] The implementation procedure of the proposed FDTD/PBC algorithm is discussed and two key issues are addressed. The first one is the plane wave excitation in this algorithm. Since the new algorithm uses a constant horizontal wavenumber instead of an incident angle, the conventional plane wave incident approach is no longer applicable. Alternatively, a one-field plane wave excitation approach is proposed in this paper. For example, only horizontal E incident field is incorporated for the TE^{z} plane wave case and only horizontal H incident field for the TM^{z} case. The second implementation issue is the horizontal resonance associated with the numerical scheme. Due to the implementation of periodic boundary conditions (PBC), the energy exits from one side PBC will re-enter the computational domain from the opposite side PBC. Thus, if a wave propagates horizontally, its energy will not be absorbed by the perfectly matched layers (PML). As a consequence, a horizontal resonance occurs and the fields never decay to zero in the time domain. It is revealed in this paper that the horizontal resonance can be suppressed by properly designing the excitation waveforms.

[5] The developed algorithm is used to analyze several periodic structures such as a dielectric slab and a frequency selective surface (FSS) consisting of dipole elements. The simulation results are compared with analytical results, results from a split field FDTD method, and results from commercial software such as Ansoft HFSS and Designer. Good agreement between these results is observed, which demonstrates the validity and accuracy of the new algorithm.