Prof. Ji Chen
Notes 18
Reflection and Transmission of
Plane Waves
ECE 3317
1
Spring 2019

General Plane Wave
Consider a plane wave propagating at an
arbitrary direction
in space.
Denote
so
2
x
y
z
z
jkz
e
sin
cos
sin
sin
cos
z
x
y
z
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
z
xx
yy
zz
z
x z
x
y z
y
z z
z
x r
x
y r
y
z r
z
ˆ
r
ˆ
ˆ
ˆ
ˆ
sin
cos
sin
sin
cos
r
x
y
z
,
,
x y z

General Plane Wave (cont.)
Hence
x
y
z
z
Note:
(wavenumber equation)
or
3
x
y
z
j k x
k y
k z
e
sin
cos
sin
sin
cos
x
y
z
k
k
k
k
k
k
ˆ
r
2
2
2
2
2
2
2
2
2
sin
cos
sin
cos
x
y
z
k
k
k
k
k
2
2
2
2
x
y
z
k
k
k
k

General Plane Wave (cont.)
We define the
wavevector:
The
k
vector tells us which direction the wave is traveling in.
(This assumes that the
wavevector is real.)
4
x
y
z
z
sin
cos
sin
sin
cos
x
y
z
k
k
k
k
k
k
ˆ
ˆ
ˆ
x
y
z
k
x k
y k
z k
2
2
2
2
2
2
x
y
z
x
y
z
k
k
k
k
k
k
k
k
ˆ
r

TM and TE Plane Waves
The electric and magnetic fields are both
perpendicular to the direction of propagation.
There are two fundamental cases:
Transverse Magnetic (TM
z
)
H
z
= 0
Transverse Electric (TE
z
)
E
z
= 0
x
TM
z
y
z
E
H
S
x
TE
z
y
z
E
H
S
Note: The word “transverse” means “perpendicular to.”
5
ˆ
ˆ
TM :
E
E
H
H
ˆ
ˆ
TE :
E
E
H
H
z
z

Reflection and Transmission
As we will show, each type of plane wave (TE
z
and TM
z
) reflects differently from
a material.
#1
x
z
i
r
t
#2
Incident
Reflected
Transmitted
6

Boundary Conditions
Here we review the
boundary conditions at an interface
(from ECE 2317).
++++
Note: The unit normal points towards region 1.
No sources on interface:
The tangential electric and magnetic fields are
continuous. The normal components of the electric
and magnetic flux densities are continuous.
7
1
2
1
2
1
2
1
2
ˆ
D
D
ˆ
E
E
0
ˆ
B
B
0
ˆ
H
H
J
s
s
n
n
n
n
ˆ
n
1
1
,
2
2
,
s
J
s
1
2
1
2
1
2
1
2
ˆ
ˆ
D
D
ˆ
ˆ
E
E
ˆ
ˆ
B
B
ˆ
ˆ
H
H
n
n
n
n
n
n
n
n

Reflection at Interface
First we consider the
(
x
,
z
)
variation of the fields.
(We will worry about the polarization later.)
Assume that the Poynting vector of the incident plane wave lies
in the
xz
plane
(
=
0
). This is called the
plane of incidence
.

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