T
⊥
E ¼
1
À1
cosθ
Z 1
cos θ
Z 1
1
À1
cos θ
Z 1
cos ϕ
Z 2
¼
2Z 2 cos θ
Z 2 cos θ þ Z 1 cos ϕ
:
ð8:52Þ
Similarly, defining
R
⊥
H H r =H i and T
⊥
H H t =H i ,
ð8:53Þ
where R
⊥
H and T
⊥
H are said to be a reflection coefficient and transmission coefficient
with the magnetic field, respectively, we get
R
⊥
H ¼
Z 1 cos ϕ À Z 2 cos θ
Z 2 cos θ þ Z 1 cos ϕ
,
ð8:54Þ
T
⊥
H ¼
2Z 1 cos θ
Z 2 cos θ þ Z 1 cos ϕ
:
ð8:55Þ
In this case, rewrite (8.42) as a relation among H i , H r , and H t using (8.45) and (8.46).
Derivation of (8.54) and (8.55) is left for readers. Notice also that
R
⊥
H ¼ ÀR
⊥
E :
ð8:56Þ
This relation can easily be derived by (8.45).
Example 8.2: TM Wave In a manner similar to that described above, we obtain
information about the TM wave. Switching a role of E and H, we assume that H is
polarized along the y-axis with E polarized in the zx-plane. Following the aforementioned procedures, we have
E i cos θ þ E r cos θ ¼ E t cos ϕ,
ð8:57Þ
H i þ H r ¼ H t :
ð8:58Þ
From (8.57) and (8.58), similarly we get
R
k
E ¼
Z 2 cos ϕ À Z 1 cos θ
Z 1 cos θ þ Z 2 cos ϕ
,
ð8:59Þ
T
k
E ¼
2Z 2 cos θ
Z 1 cos θ þ Z 2 cos ϕ
:
ð8:60Þ
Also, we get
8.3 Transverse Electric (TE) Waves and Transverse Magnetic (TM) Waves
307