Information Technology Reference
In-Depth Information
In
the
presence
of
lateral
inhomogeneities,
the
anomalous
field
E
z
z
=
0
=
E
A
(
E
x
,
E
y
,
E
z
H
A
(
H
x
,
H
y
,
H
z
)
,
),
0
appears.
Subtracting
(I.5)
of (I.4), we arrive at equations for the anomalous field:
curl
H
A
N
E
A
=
+
j
,
(1
.
6)
curl
E
A
o
H
A
=
i
.
From these equations we deduce the integral representations for the anomalous
field:
G
E
(
r
r
v
)
j
(
r
v
)
dV
E
A
(
r
)
=
|
,
V
(1
.
7)
G
H
(
r
r
v
)
j
(
r
v
)
dV
H
A
(
r
)
=
|
,
V
where [
G
E
] and [
G
H
] are the electric and magnetic
Green tensors
for horizontally
layered medium:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
G
xx
G
xy
G
xz
G
xx
G
xy
G
xz
G
yx
G
yy
G
yz
G
yx
G
yy
G
yz
[
G
E
]
[
G
H
]
=
,
=
.
(1
.
8)
G
zx
G
zy
G
zz
G
zx
G
zy
G
zz
The Green tensors satisfy the equations
curl
[
G
H
(
r
|
N
[
G
E
(
r
|
r
v
)]
=
r
v
)]
+
[
(
r
−
r
v
)]
,
(1
.
9)
curl
[
G
E
(
r
|
r
v
)]
=
ω
o
[
G
H
(
r
|
r
v
)]
,
i
where [
] is the diagonal matrix consisting of the scalar Dirac
-functions:
⎡
⎣
⎤
⎦
.
(
r
−
r
v
)
0
0
0
(
r
−
r
v
)
0
[
(
r
−
r
v
)]
=
0
0
(
r
−
r
v
)
Here we should clarify how the
curl
of Green's tensor is calculated. Let us write
[G] in the form
=
,
[
G
]
[
G
x
G
y
G
z
]