E = Eε e e
i knÁxÀωt
ð
Þ ,
ð8:63Þ
H = Hε m e
i knÁxÀωt
ð
Þ ,
ð8:64Þ
where ε e and ε m are unit polarization vector; we assume that both E and H are
positive. Notice again that ε e , ε m , and n constitute a right-handed system in this
order.
The energy transport is characterized by a Poynting vector S that is described by
S = E Â H:
ð8:65Þ
Since E and H have a dimension [
V
m ] and [
A
m ], respectively, S has a dimension [
W
m 2 ]. Hence,
S represents an energy flow per unit time and per unit area with respect to the propagation
direction. For simplicity, let us assume that the electromagnetic wave is propagating
toward the z-direction. Then we have
E = Eε e e
i kzÀωt
ð
Þ ,
ð8:66Þ
H = Hε m e
i kzÀωt
ð
Þ
:
ð8:67Þ
To seek a time-averaged energy flow toward the z-direction, it suffices to multiply
real parts of (8.66) and (8.67) and integrate it during a period T at a point of z ¼ 0.
Thus, a time-averaged Poynting vector S is given by
S ¼ e 3
EH
T
Z T
0
cos
2
ωtdt,
ð8:68Þ
where T ¼ 1/ν ¼ 2π/ω. Using a trigonometric formula
cos
2
ωt ¼
1
2
1 þ cos 2ωt
ð
Þ ,
ð8:69Þ
the integration can easily be performed. Thus, we get
S ¼
1
2
EHe 3 :
ð8:70Þ
Equivalently, we have
S ¼
1
2
E Â H
Ã
:
ð8:71Þ
Meanwhile, an energy density W is given by
8.4 Energy Transport by Electromagnetic Waves
309