Page 38 - Fiber Optic Communications Fund

Electromagnetics and Optics 19

The integral of the cosine function over one period is zero and, therefore, the second term of Eq. (1.118) does
not contribute after the integration. The average power density  z av is proportional to the square of the electric
field amplitude. Using complex notation, Eq. (1.111) can be written as
[ ] [ ]
̃
̃
 = Re E Re H (1.119)
z x y
[ ̃ ̃ ∗ ][ ̃ ̃ ∗ ]
1 [ ] [ ] 1 E + E x E + E x
x
x
̃
= Re E Re E ̃ x = . (1.120)
x
2 2
2
̃ 2
̃ ∗
2
The right-hand side of Eq. (1.120) contains product terms such as E and E . The average of E and E ∗ x 2 over
x
x
x
the period T is zero, since they are sinusoids with no d.c. component. Therefore, the average power density
is given by
T ̃ 2
x
 av = 1 | ̃ | 2 dt = |E | , (1.121)
E
z 2T ∫ | x|
0 2
̃
2
since |E | is a constant for the plane wave. Thus, we see that, in complex notation, the average power density
x
is proportional to the absolute square of the field amplitude.
Example 1.4
Two monochromatic waves are superposed to obtain
̃
E = A exp [i( t − k z)] + A exp [i( t − k z)]. (1.122)
2
2
1
1
2
1
x
Find the average power density of the combined wave.
Solution:
From Eq. (1.121), we have
T
1 2
 av = 2T ∫ | ̃ | dt
E
z
| x|
{ 0 T
1 2 2 ⋆
= T|A | + T|A | + A A exp [i( − )t − i(k − k )z]dt
2
1
2
1
1
2
1 2
2T ∫ 0
T }
+ A A ⋆ exp [−i( − )+ i(k − k )z] dt. (1.123)
2 1 ∫ 1 2 1 2
0
Since integrals of sinusoids over the period T are zero, the last two terms in Eq. (1.123) do not contribute,
which leads to
2 2
1
2
 z av = |A | + |A | . (1.124)
2
Thus, the average power density is the sum of absolute squares of the amplitudes of monochromatic waves.
1.8 3-Dimensional Wave Equation
From Maxwell’s equations, the following wave equation could be derived (see Exercise 1.6):
2
2
2
2
1
+ + − = 0, (1.125)
2
x 2 y 2 z 2 t 2