Implementation of a PEMC Boundary Condition in the 2-D FDTD Technique Vahid Nayyeri and Mohammad Soleimani Antenna Research Laboratory Iran University of Science and Technology (IUST) Tehran, Iran nayyeri@iust.ac.ir , soleimani@iust.ac.ir Mojtaba Dehmollaian School of Electrical and Computer Engineering University of Tehran Tehran, Iran m.dehmollaian@ece.ut.ac.ir Abstract— In this paper, a two-dimensional FDTD technique is presented to solve scattering problems involving PEMC boundaries. Basically, new FDTD updating equations are introduced to realize the PEMC boundary condition. To achieve the formulation, a modified Yee's Cell at the PEMC boundary is proposed. Next, the method is validated by simulating scattered fields from a PEMC square cylinder and comparing results for two special cases of PEC and PMC cylinder. Keywords- Electromagnetic scattering, finite-difference time- domain (FDTD) method, perfect electromagnetic conductor (PEMC) I. INTRODUCTION Perfect electromagnetic conductor (PEMC) was recently introduced by Lindell and Sihvola [1]. Since PEMC is a perfect conductor, field cannot convey energy into a PEMC medium [2]. As a result PEMC is considered as an ideal medium which has the following boundary condition [3]: ( ) 1 ˆ 0 i.e., n H ME E H t t M - × + = = . (1) where M is a scalar real parameter denoting the PEMC admittance. PEMC boundary is a nonreciprocal generalization of both the perfect electric conductor boundary (M→±∞) and the perfect magnetic conductor boundary (M = 0). While wave interaction with PEMC medium has been focused in many studies analytically, there are only a few works on the topic of numerical modeling of problems involving PEMC objects and boundaries. In [4] scattering by PEMC spheres is studied using surface integral equation approach. However this method is valid only for the spheres whose radius is smaller than half wavelength. Furthermore, in [5] FDTD method is implemented in 1-D problem involving PEMC plates and reflection of plane wave from stratified media backed by a PEMC plate is calculated. In problems involving boundary conditions such as PEC, PMC, SIBC, etc., an updating equation of the electric / magnetic fields at the boundary is required. On the other hand, for a PEMC, the boundary condition (2) is a nonreciprocal mixed one, involving both the electric and magnetic fields. This paper describes a 2- D extension of the FDTD scheme in order to model EM wave interaction with PEMC objects. Indeed, once the PEMC boundary is implemented, the method can be easily applied to any problem involving such a boundary condition. First in Section II, briefly by implementing the PEMC boundary condition in the 2-D FDTD algorithm, updating equations on a PEMC boundary are derived. To drive FDTD updating equations on the PEMC boundary, both central difference and backward difference schemes are used. Then in Section III, the interaction of plane wave with a PEMC square cylinder is simulated using the proposed FDTD method. Finally the method is validated by comparison of results for two special cases of PEC (M→±∞) and PMC (M = 0) cylinders. II. 2-D IMPLEMENTATION OF THE PEMC BOUNDARY CONDITION IN THE FDTD TECHNIQUE The 2-D Yee's mesh including both TM z and TE z fields' components is illustrated in Fig. 1. A PEMC boundary is assumed at grids with index i=I+1 and denoted by bold line in that figure. It is noted that due to the rotation of wave polarization by a PEMC boundary (PEMC is a non-reciprocal medium), both TE z and TM z fields' components should be considered in the algorithm. Based on Yee's algorithm, the electric and magnetic fields are not co-located in the FDTD mesh. However, PEMC boundary condition requires the electric and magnetic fields to be co-located in space and time. So the Yee's cell is modified and both tangential components of electric field at PEMC boundary are considered. Now let us first start with Maxwell's curl equation for the magnetic field, H D t ∂ ∇× = ∂ G G . By discretizing and decomposing the equation, at grid (I+1, j+½) we obtain 1 1 1 2 2 1 1 1, 1, 2 2 2 1 1 1 1, , 2 2 2 n n y y n n z z x E I j E I j t H I j H I j εδ + + + + + = + + Δ - + + - + + ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎣ ⎝ ⎠ ⎝ ⎠⎦ , (2.a) ( ) ( ) 1 1 1 2 2 1 1 2 2 1 1 1, 1, 2 2 2 1 1 1 1, , 2 2 2 1, 1 1, n n z z n n y y x n n x x y E I j E I j t H I j H I j t H I j H I j εδ εδ + + + + + + + = + + Δ + + + - + + Δ - + + - + ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎣ ⎦ . (2.b) 978-1-4673-0462-7/12/$31.00 ©2012 IEEE