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Unformatted text preview: 1 EE256 Numerical Electromagnetics H. O. #21 Marshall 15 July 2008 Summer 2008 LECTURE 14 14.1 PERFECTLY MATCHED UNIAXIAL MEDIUM We mentioned in Lecture 13 that the important property of the Berenger PML medium was that it hosted different conductivities in the x and y directions, i.e., it was inherently an anisotropic medium. In this Lecture, we discuss an alternative formulation of the Perfectly Matched Layer (PML) absorbing boundary condition in terms of an anisotropic medium having tensor permittivity and permeability. Following the notation in Section 7.5 of Taflove and Hagness Computational Electrodynamics , we con- sider a time-harmonic plane wave incident on the x = 0 wall, with its arbitrarily polarized magnetic field given by: H i = H i xy + ˆ z H i z = H e- j ( β 1 x x + β 1 y y + β 1 z z ) [14.1] where we note that H =ˆ x H x + ˆ y H y +ˆ z H z is a constant vector and H i xy represents the component of H i lying in the x- y plane as shown in Figure 14.1. Note that the propagation direction of this incident wave is given by the ˆ k i vector given by: ˆ k i = ˆ x β 1 x + ˆ y β 1 y + ˆ z β 1 z q β 2 1 x + β 2 1 y + β 2 1 z = k xy + ˆ z β 1 z β 1 [14.2] where k xy is the component of the unit vector ˆ k i lying in the x- y plane, as shown in Figure 14.1. Note that the total magnetic field in Region 1 is in general given by [14.1] plus a reflected wave, with an expression similar to [13.4 a ], depending on the particular polarization of the wave. The half-space region x> , referred to as Region 2, is characterized by the tensors-↔ 2 = 2 a 0 0 b 0 0 b and-↔ μ 2 = μ 2 c 0 0 d 0 0 d [14.3] Note that when we consider arrival of plane waves on the x =0 wall, it is appropriate to consider Region 2 to be rotationally symmetric about the x-axis, hence the fact that and μ along the y and z axes are equal. Plane electromagnetic waves in Region 2 satisfy Maxwell’s curl equations, which can be written in tensor form as: 2 x =0 z x Phase fronts λ 1 k 1xy Region 1 ( ε 1 µ 1 , σ =0) Region 2 ( ε 2 µ 2 , σ , σ m ) θ i y Planes of constant phase Planes of constant amplitude E i k xy H iz λ 2 = λ 1 k 2xy H ixy Figure 14.1: Oblique incidence on the uniaxial PML medium. (Note that Region 2 is now anisotropic, since it has different loss rates in the x and y directions. ˆ k 2 β 2 × E 2 = ω-↔ μ 2 H 2 [14.4 a ] ˆ k 2 β 2 × H 2 =- ω-↔ 2 E 2 [14.4 b ] where ˆ k 2 β 2 = ˆ x β 2 x + ˆ y β 2 y + ˆ z β 2 z is the wave vector in Region 2. A wave equation can be derived from [14.4] as: ˆ k 2 β 2 ×-↔ 2- 1 ˆ k 2 β 2 × H 2 + ω 2-↔ μ 2 H 2 = 0 [14.5] which can be written in matrix form as: cω 2 μ 2 2- β 2 2 y b- 1 β 2 x β 2 y b- 1 β 2 x β 2 y b- 1 dω 2 μ 2 2- β 2 2 x b- 1 dω 2 μ 2 2- β 2 2 x b- 1- β 2 2 y a- 1 H x H y H z = 0 [14.6] Note that the dispersion relation (i.e., an equation equivalent to [13.8] giving the relationship betweenNote that the dispersion relation (i....
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- Electromagnet, plane wave, Wave mechanics, BERENGER, uniaxial PML medium