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ee256-08-lecture13 - 1 EE256 Numerical Electromagnetics...

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1 EE256 Numerical Electromagnetics H. O. #19 Marshall 08 July 2008 Summer 2008 LECTURE 13 13.1 PERFECTLY MATCHED LAYER (PML) BOUNDARY CONDITION The Berenger PML method for absorbing waves incident on the boundaries of an FDTD grid is based on reflection/refraction of uniform plane waves at the interface between a lossless dielectric and a general lossy medium. A traditional analytical treatment of this subject was provided to you in Lecture 12, with the attached copy of Section 3.8 of U. S. Inan and A. S. Inan, Electromagnetic Waves , Prentice Hall, 2000. Note that there are some notation differences between the analytical treatment and that given in the rest of this Lecture which is geared toward FDTD modeling. Specifically, in the analytical analysis, the plane of incidence is typically chosen to be the x z plane and the interface between the two media is typically placed at z = 0 and the reflection coefficient is defined typically in terms of the electric field, i.e., Γ E r0 E i0 where E i0 and E r0 are respectively the magnitudes of the incident and reflected wave electric fields at the z = 0 interface. In FDTD analysis, on the other hand, the reflection coefficient is typically defined in terms of the singular wave component, i.e., H z for TE waves or E z for TM waves. In the same vein, the terms ‘perpendicular’ versus ‘parallel’ polarization in the traditional context usually refers to the orientation of the wave electric field with respect to the plane of incidence. On the other hand, in FDTD analysis of a two-dimensional system, it is more convenient to think in terms of the singular wave component ( H z for TE waves or E z for TM waves), and choose the plane of incidence to be the x y plane. We now proceed with the analysis of the problem of a uniform plane wave incident from a lossless dielectric medium (Region 1) to a lossy (allowing for both electric and magnetic losses, as represented respectively by σ and σ m ) half-space (Region 2), staying close to the notation used in Chapter 7 of the textbook, Taflove and Hagness, Computational Electrodynamics . This analysis shows that by careful choice of the medium parameters σ , , and μ we can achieve perfect matching between Region 1 and Region 2, but only for waves incident normally at the interface between the two regions. We then discuss incidence of uniform plane waves on the Berenger 1 PML medium, in which case the perfect matching condition can be realized for any angle of incidence. 1 Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Computational Physics , Vol. 114, pp.185-200, 1994.
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2 x =0 ψ z x Phase fronts λ 1 λ 2 k i k ψ 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 i H i Figure 13.1: Oblique incidence on a lossy medium: Constant amplitude and phase fronts. (The planes of constant amplitude are parallel to the interface, while the planes of constant phase are perpendicular to k ψ , which is at an angle ψ from the vertical.
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