Image Processing Reference
InDepth 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
I
is the unit dyad; a dyadic Green's function
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
.
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