Image Processing Reference
In-Depth Information
electrons that might exist in the medium, as described by a very classical
model for conduction known as the Drude model.
It is important to stress that these wave equations, which describe the prop-
agation of an electromagnetic field, are valid in an inhomogeneous or scatter-
ing media. If we know how the permittivity and permeability (or equivalently
the refractive index) vary as a function of space and frequency, we can predict
how a wave that is incident in that medium will propagate and scatter. The
converse, which is the focus of this topic, is a much harder problem.
2.2 gReen'S FunCtIon
Green's function is very important and gives the electromagnetic field pro-
duced by a point source for a given medium or object. In free space or a vac-
uum, Green's function from a point source is an outgoing spherical wave.
Here
is made of three vector Green's functions. Thus, in a homogeneous (infinite)
medium, Green's function,
G is the Green dyadic and
G , satisfies the equation.
(2.19)
∇×∇×
Grr
(, ,)
ω
k 2
()
ω ωδ
G rr
(, ,)
=
Ir r
(
)
where I is the unit tensor and δ denotes the delta function describing the point
source located at some position r ′. A solution of this equation can be formally
written as
1
2
Grr
(, ,)
=+∇∇
ω
I
Grr
(, ,)
ω
n i
(2.20)
k
where n i is a unit vector and where
e ik
4π|
||
rr
Grr
(, ,)
′ω
=
(2.21)
rr
|
is the outgoing scalar Green's function that satisfies the wave equation. In
terms of Green's function the solution to
∇×∇×
Er rEr
(, )
ω
k
( ,)(, )
ω ω µµ ωω
=
i
( ,)(, )
r Jr
s
(2.22)
2
0
r
for the current distribution J s , located in some volume V , becomes
Er Er
(, )
ω
=
( ,)
ω
+
i
ωµ µω
() (, ,) ()
Grr
ω
J s
r
d
r
3
(2.23)
0
0
r
V
where r is a point located outside V . The field E 0 is the incident wave, which
is typically assumed to be a plane wave, and which one could imagine as
being created by a point source at infinite distance away from the medium in
volume V .    Search WWH ::

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