Biomedical Engineering Reference
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E
y
or
E
x
Constant
e
x
/ε
5
Constant
e
-x
/ε
3
x
a
ε
5
ε
3
y
Constant
e
-
y
/η
2
n
2
cos (
k
y
Y
+
B
)
b
E
y
or
H
z
n
4
Constant
e
y
/η
4
FIGURE 5.6
Field distribution of the fundamental mode
11
for a guide immersed in several dielectrics.
are
E
y
and
H
x
. Figure 5.6 shows the fundamental mode
E
11
, characterized by
extinction coefficients
n
2
,
e
3
,
n
4
, and
e
5
in the regions where the mode is expo-
nential, and by propagation constants
k
x
and
k
y
in region 1. In cases where
n
3
=
n
5
, the fields will be symmetric with respect to the
x
= 0 plane.
Quantitative expressions for
k
x
,
k
y
,
n
, and
e
are now determined.
Assuming that the plane wavelets that make a mode impinge on the inter-
faces at grazing angles, that is, 1/
n
1
(
n
1
−
n
y
) ≪ 1, then
k
,
e
3
, and
h
2
are given by
−
1
p
a
π
⎛
⎜
A A
a
+
⎞
⎟
3
5
k
x
=
+
(5.8)
1
π
−
1
q
b
π
⎛
⎜
2
2
⎞
⎟
1
+
n A n A
n b
+
2
2
4
4
(5.9)
k
y
=
π
2
1
1
2
⎡
−
2
⎤
−
2
2
2
π
p
a
π
q
b
⎛
⎜
2
2
⎞
⎟
−
⎛
⎞
⎟
⎛
⎜
A A
a
+
⎞
⎟
−
⎛
⎞
⎟
n A n A
+
⎢
⎢
1
3
5
2
3
4
4
⎥
⎥
(5.10)
k
z
=
k
+
+
1
1
⎜
⎜
1
π
π
n b
1
2
⎣
⎦
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