5
Reflection and Transmission
5.1
Propagation Matrices
In this chapter, we consider uniform plane waves incident
normally
on material inter
faces. Using the boundary conditions for the fields, we will relate the forwardbackward
fields on one side of the interface to those on the other side, expressing the relationship
in terms of a 2
×
2
matching matrix
.
If there are several interfaces, we will propagate our forwardbackward fields from
one interface to the next with the help of a 2
×
2
propagation matrix
. The combination of
a matching and a propagation matrix relating the fields across different interfaces will
be referred to as a
transfer
or transition matrix.
We begin by discussing propagation matrices. Consider an electric field that is lin
early polarized in the
x
direction and propagating along the
z
direction in a lossless
(homogeneous and isotropic) dielectric.
Setting
E
(z)
=
ˆ
x
E
x
(z)
=
ˆ
x
E(z)
and
H
(z)
=
ˆ
y
H
y
(z)
=
ˆ
y
H(z)
, we have from Eq. (2.2.6):
E(z)
=
E
0
+
e
−
jkz
+
E
0
−
e
jkz
=
E
+
(z)
+
E
−
(z)
H(z)
=
1
η
E
0
+
e
−
jkz
−
E
0
−
e
jkz
=
1
η
E
+
(z)
−
E
−
(z)
(5.1.1)
where the corresponding forward and backward electric fields at position
z
are:
E
+
(z)
=
E
0
+
e
−
jkz
E
−
(z)
=
E
0
−
e
jkz
(5.1.2)
We can also express the fields
E
±
(z)
in terms of
E(z), H(z)
. Adding and subtracting
the two equations (5.1.1), we find:
E
+
(z)
=
1
2
E(z)
+
ηH(z)
E
−
(z)
=
1
2
E(z)
−
ηH(z)
(5.1.3)
Eqs.(5.1.1) and (5.1.3) can also be written in the convenient matrix forms:
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5.1.
Propagation Matrices
151
E
H
=
1
1
η
−
1
−
η
−
1
E
+
E
−
,
E
+
E
−
=
1
2
1
η
1
−
η
E
H
(5.1.4)
Two useful quantities in interface problems are the
wave impedance
at
z
:
Z(z)
=
E(z)
H(z)
(wave impedance)
(5.1.5)
and the
reflection coefficient
at position
z
:
Γ(z)
=
E
−
(z)
E
+
(z)
(reﬂection coefficient)
(5.1.6)
Using Eq. (5.1.3), we have:
Γ
=
E
−
E
+
=
1
2
(E
−
ηH)
1
2
(E
+
ηH)
=
E
H
−
η
E
H
+
η
=
Z
−
η
Z
+
η
Similarly, using Eq. (5.1.1) we find:
Z
=
E
H
=
E
+
+
E
−
1
η
(E
+
−
E
−
)
=
η
1
+
E
−
E
+
1
−
E
−
E
+
=
η
1
+
Γ
1
−
Γ
Thus, we have the relationships:
Z(z)
=
η
1
+
Γ(z)
1
−
Γ(z)
Γ(z)
=
Z(z)
−
η
Z(z)
+
η
(5.1.7)
Using Eq. (5.1.2), we find:
Γ(z)
=
E
−
(z)
E
+
(z)
=
E
0
−
e
jkz
E
0
+
e
−
jkz
=
Γ(
0
)e
2
jkz
where
Γ(
0
)
=
E
0
−
/E
0
+
is the reﬂection coefficient at
z
=
0. Thus,
Γ(z)
=
Γ(
0
)e
2
jkz
(propagation of
Γ
)
(5.1.8)
Applying (5.1.7) at
z
and
z
=
0, we have:
Z(z)
−
η
Z(z)
+
η
=
Γ(z)
=
Γ(
0
)e
2
jkz
=
Z(
0
)
−
η
Z(
0
)
+
η
e
2
jkz
This may be solved for
Z(z)
in terms of
Z(
0
)
, giving after some algebra:
Z(z)
=
η
Z(
0
)
−
jη
tan
kz
η
−
jZ(
0
)
tan
kz
(propagation of
Z
)
(5.1.9)
152
5.
Reflection and Transmission
The reason for introducing so many field quantities is that the three quantities
{
E
+
(z), E
−
(z), Γ(z)
}
have simple propagation properties, whereas
{
E(z), H(z), Z(z)
}
do not. On the other hand,
{
E(z), H(z), Z(z)
}
match simply across interfaces, whereas
{
E
+
(z), E
−
(z), Γ(z)
}
do not.
Eqs. (5.1.1) and (5.1.2) relate the field quantities at location
z
to the quantities at
z
=
0.
In matching problems, it proves more convenient to be able to relate these
quantities at two arbitrary locations.
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 Spring '10
 taflove
 Magnetic Field, Geometrical optics, Reflection coefficient, Fresnel equations, Transmission coefficient

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