420 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 46, NO. 4, APRIL 1998 Time-Domain Analysis of Periodic Structures at Oblique Incidence: Orthogonal and Nonorthogonal FDTD Implementations J. Alan Roden, Stephen D. Gedney, Member, IEEE, Morris P. Kesler, James G. Maloney, Member, IEEE, and Paul H. Harms, Member, IEEE Abstract—A novel implementation of periodic boundary condi- tions incorporated into the finite-difference time-domain (FDTD) technique in both orthogonal and nonorthogonal grids is pre- sented in this paper. The method applied is a field-splitting ap- proach to the discretization of the Floquet-transformed Maxwell equations. As a result, computational burden is reduced and the stability criterion is relaxed. The results of the two methods are compared to experimental data. Index Terms—FDTD, nonorthogonal, PBG, periodic media. I. INTRODUCTION T HE finite-difference time-domain (FDTD) technique is a robust analysis tool applicable to a wide variety of complex problems [1]. It is particularly useful when nonlin- earities exist and transient analysis is required. Frequently, problems are encountered in which a periodicity exists in one or more dimension of the problem geometry, as illustrated in Fig. 1(a). Taking advantage of this periodicity can lead to greater efficiency and accuracy when solving the problem numerically. Typically, each periodic feature is referred to as a cell and the periodicity of these cells is accounted for using Floquet theory. For a normally incident plane wave, accounting for this periodicity in either an orthogonal or nonorthogonal FDTD method is quite straightforward as there is no phase shift between each periodic cell [2]. However, when a plane- wave source is obliquely incident, there is a cell-to-cell phase variation between corresponding points in different unit cells which causes the time-domain implementation to become more difficult. For oblique incidence plane waves, a Floquet field mapping may be applied, which results in a set of mapped fields which possess the same cell-to-cell field relations as exist for the Manuscript received January 14, 1997; revised January 14, 1998. This work was supported in part by the National Science Foundation through the NSF CAREER Award ECS-9624628 and the Army Research Office under Contract DAAHO4-94-G-0243 with the University of Kentucky. J. A. Roden, M. P. Kesler, J. G. Maloney, and P. H. Harms are with Signature Technologies Laboratory, Georgia Tech. Research Institute, Atlanta, GA 30332 USA. S. D. Gedney is with the Department of Electrical Engineering, University of Kentucky, Lexington, KY 40506-0046 USA. Publisher Item Identifier S 0018-9480(98)02753-7. (a) (b) Fig. 1. (a) Periodic geometry in the -direction with a plane-wave incident at an angle . (b) Typical cell of the computational grid. normally incident unmapped fields [3]. The resulting equations may add considerable complexity to the FDTD solution [4] and lead to a more stringent stability relation for higher angles of incidence. At present, the periodic methods which are available have only been applied using the orthogonal FDTD method. In this paper, the Floquet-mapped periodic FDTD equations are solved using an alternative approach referred to as the split- field update method. This technique is shown to be simple to implement and a stability analysis shows the technique to have a less strict stability criterion than previous implementations. The split-field update method is then applied in a general curvilinear space using the nonorthogonal FDTD technique. The use of Floquet-mapped FDTD in nonorthogonal grids may lead to further computational savings due to fewer and 0018–9480/98$10.00  1998 IEEE