Environmental Engineering Reference
In-Depth Information
By using the similar scheme, the scattering matrix
S
(
II
,
III
)
through the interface (
x
=
b
) between the region II and III can
be obtained easily. Therefore, the relationships of these expansion
coe
cients denoting the incoming and outing waves can be
determined by the total scattering matrix
S
T
A
III
B
I
S
T
A
I
B
III
S
11
S
12
S
21
S
22
A
I
B
III
.
=
=
(4.52)
In terms of
S
(
I
,
II
),
S
(
II
,
II
), and
S
(
II
,
III
),
S
T
in the quantum
structure shown in Fig. 4.1 can beconstructed by
S
T
=
S
(
I
,
II
)
⊗
S
(
II
,
II
)
⊗
S
(
II
,
III
).
(4.53)
Thus, here it is imperative to demonstrate how to calculate the
elements of the total scattering matrix. Now one considers the two
successive scattering matrices defined by
r
1
t
1
t
1
r
1
,
r
2
t
2
t
2
r
2
.
S
1
=
S
2
=
(4.54)
The resultant scattering matrix of the two matrices is
r
12
t
12
t
12
r
12
S
12
=
S
1
⊗
S
2
=
(4.55)
with
t
1
r
2
(1
r
1
r
2
)
t
1
,
r
12
=
r
1
+
−
t
12
=
t
1
[
r
2
(1
r
1
r
2
)
−
1
r
1
t
2
+
t
2
],
−
r
1
r
2
)
−
1
t
1
,
t
12
=
−
t
2
(1
r
12
=
r
1
r
2
)
−
1
r
1
t
2
+
t
2
.
−
t
2
(1
(4.56)
By using these relations iteratively, expressed in Eq. 4.56, we
can easily get the total scattering matrix
S
T
. In the calculation
of the transmission coe
cient, we can regard that only the
transmitted components are presented in the outgoing lead, i.e.,
B
III
=
0. According to Eq. 4.52, thisleads to
A
III
S
11
A
I
=
(4.57)
with
A
I
being an
N
I
1 vector with elements 1. Then, the
transmission coe
cient from the mode
m
in the incoming lead to
×
