R
k
H ¼
Z 1 cos θ À Z 2 cos ϕ
Z 1 cos θ þ Z 2 cos ϕ
¼ ÀR
k
E ,
ð8:61Þ
T
k
H ¼
2Z 1 cos θ
Z 1 cos θ þ Z 2 cos ϕ
:
ð8:62Þ
In Table 8.1, we list the important coefficients in relation to the reflection and
transmission of electromagnetic waves along with their relationship.
In Examples 8.1 and 8.2, we have examined how the reflection and transmission
coefficients vary as a function of characteristic impedance as well as incidence and
refraction angles. Meanwhile, in a nonmagnetic substance a refractive index n can be
approximated as
n %
ffiffiffiffi
ε r
p
,
ð7:57Þ
assuming that μ r % 1. In this case, we have
Z ¼
ffiffiffiffiffiffiffi ffi
μ=ε
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
μ r μ 0 =ε r ε 0
p
% Z 0 =
ffiffiffiffi
ε r
p % Z 0 =n:
Using this relation, we can readily rewrite the reflection and transmission coefficients as a function of refractive indices of the dielectrics. The derivation is left for
the readers.
8.4 Energy Transport by Electromagnetic Waves
Returning to (7.58) and (7.59), let us consider energy transport in a dielectric
medium by electromagnetic waves. Let us describe their electric (E) and magnetic
(H) fields of the electromagnetic waves in a uniform and infinite dielectric medium
such that
Table 8.1 Reflection and transmission coefficients of TE and TM waves
⊥ Incidence (TE)
k Incidence (TM)
R
⊥
E ¼
Z2 cos θÀZ1 cos ϕ
Z2 cos θþZ1 cos ϕ
R
k
E ¼
Z2 cos ϕÀZ1 cos θ
Z1 cos θþZ2 cos ϕ
R
⊥
H ¼
Z1 cos ϕÀZ2 cos θ
Z2 cos θþZ1 cos ϕ ¼ ÀR
⊥
E
R
k
H ¼
Z1 cos θÀZ2 cos ϕ
Z1 cos θþZ2 cos ϕ ¼ ÀR
k
E
T
⊥
E ¼
2Z2 cos θ
Z2 cos θþZ1 cos ϕ
T
k
E ¼
2Z2 cos θ
Z1 cos θþZ2 cos ϕ
T
⊥
H ¼
2Z1 cos θ
Z2 cos θþZ1 cos ϕ
T
k
H ¼
2Z1 cos θ
Z1 cos θþZ2 cos ϕ
ÀR
⊥
E R
⊥
H þ T
⊥
E T
⊥
H
cos ϕ
cos θ ¼ 1 R
⊥ þ T
⊥ ¼ 1
ð
Þ
ÀR
k
E R
k
H þ T
k
E T
k
H
cos ϕ
cos θ ¼ 1 R
k þ T
k ¼ 1
À
Á
308
8 Reflection and Transmission of Electromagnetic Waves in Dielectric Media