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 twodimensional 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
) halfspace (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.185200, 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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 '09
 Electromagnet, Region, Wave mechanics, plane wave incident, PML, Perfectly Matched Layer

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