FDTD Analysis of Periodic Structures at Arbitrary Incidence Angles: A Simple and Efficient Implementation of the Periodic Boundary Conditions Fan Yang* (1) , Ji Chen (2) , Rui Qiang (2) , and Atef Elsherbeni (1) (1) Department of Electrical Engineering, The University of Mississippi, University, MS 38677 (2) Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204 fyang@olemiss.edu , jchen18@uh.edu , rui.qiang@mail.uh.edu , atef@olemiss.edu Introduction Periodic structures are widely used in electromagnetics, such as frequency selective surfaces (FSS), electromagnetic band gap (EBG) structures, and negative index materials. When the finite-difference time-domain (FDTD) method is used to analyze these structures, an important issue is how to implement the periodic boundary conditions (PBC) in the time domain simulation. Various PBCs have been developed in the last decade [1], and recently a spectral FDTD method was introduced in [2] that is capable to analyze arbitrary incidence angles. Starting from the constant wave number concept proposed in [2], this paper describes a simple procedure to implement the periodic boundary conditions in the FDTD simulation. In stead of using auxiliary fields P and Q, we use directly E and H fields. The principle of the approach is discussed and the plane wave excitation is highlighted. The validity of the approach is demonstrated through numerical examples. FDTD/PBC Algorithm A. Constant wavenumber method. For a periodic structure with a periodicity of a along the x direction, the PBC in the frequency domain is expressed as below: ) exp( ) , , ( E ) , , 0 ( E a jk z y a x z y x x = = = , (1) where k x is the horizontal wave number determined by both the frequency and incident angle: C k k x θ ω θ sin sin 0 = = . (2) When (1) is transformed into the time domain, future time data are needed in the updating equation: ) / sin , , , ( ) , , , 0 ( C a t z y a x E t z y x E θ + = = = , (3) which is the fundamental challenge in formulating PBC in the FDTD method. To circumvent this problem, a constant wavenumber method was proposed in [2]. If k x is a constant, (1) can be transformed into the time domain equation as below: ) exp( ) , , , ( ) , , , 0 ( a jk t z y a x E t z y x E x = = = . (4) Note that exp(jk x a) is a constant and no future time data are needed in (4). 1-4244-0123-2/06/$20.00 ©2006 IEEE 2715