Page 47 - Fiber Optic Communications Fund

28 Fiber Optic Communications

where is called the group velocity. Even if the number of monochromatic waves traveling together is more
g
than two, an equation similar to Eq. (1.165) can be derived. In general, the speed of the envelope (group
velocity) could be different from that of the carrier. However, in free space,

= = c.
g ph
The above result can be proved as follows. In free space, the velocity of light is independent of frequency,

1 2
= = c = . (1.167)
ph
k 1 k 2
Let
Δ Δk
= − , k = k − , (1.168)
0
1
1
0
2 2
Δ Δk
= + , k = k + . (1.169)
2 0 2 0
2 2
From Eqs. (1.168) and (1.169), we obtain
− 1 Δ
2
= = . (1.170)
g
k − k Δk
2 1
From Eq. (1.167), we have
= ck ,
1 1
= ck ,
2 2
− = c(k − k ). (1.171)
1 2 1 2
Using Eqs. (1.170) and (1.171), we obtain
−
1 2
= c = . (1.172)
g
k − k 2
1
In a dielectric medium, the velocity of light could be different at different frequencies. In general,
ph
1 ≠ 1 . (1.173)
k k
1 2
In other words, the phase velocity ph is a function of frequency,
= (), (1.174)
ph ph

k = = k(). (1.175)
()
ph
In the case of two sinusoidal waves, the group speed is given by Eq. (1.166),
Δ
= . (1.176)
g
Δk
In general, for an arbitrary cluster of waves, the group speed is defined as
Δ d
= lim = . (1.177)
g
Δk→0 Δk dk