Page 48 - Fiber Optic Communications Fund

Electromagnetics and Optics 29

Sometimes it is useful to define the inverse group speed as
1
1 dk
= = . (1.178)
1
g d
could depend on frequency. If changes with frequency in a medium, it is called a dispersive medium.
1
1
Optical fiber is an example of a dispersive medium, which will be discussed in detail in Chapter 2. If the
refractive index changes with frequency, becomes frequency dependent. Since
1
n()
k()= , (1.179)
c
from Eq. (1.178) it follows that
n() dn()
()= + . (1.180)
1
c c d
Another example of a dispersive medium is a prism, in which the refractive index is different for different
frequency components. Consider a white light incident on the prism, as shown in Fig. 1.26. Using Snell’s law
for the air–glass interface on the left, we find
( )
sin 1
−1
()= sin (1.181)
2
n ()
2
where n () is the refractive index of the prism. Thus, different frequency components of a white light travel
2
at different angles, as shown in Fig. 1.26. Because of the material dispersion of the prism, a white light is
spread into a rainbow of colors.
Next, let us consider the co-propagation of electromagnetic waves of different angular frequencies in a
range [ , ] with the mean angular frequency as shown in Fig. 1.27. The frequency components near
1 2 0
the left edge travel at an inverse speed of ( ). If the length of the medium is L, the frequency components
1 1
corresponding to the left edge would arrive at L after a delay of
L
T = = ( )L.
1
1
1
( )
1
g
Similarly, the frequency components corresponding to the right edge would arrive at L after a delay of
T = ( )L.
2
1
2
The delay between the left-edge and the right-edge frequency components is
ΔT = |T − T | = L| ( )− ( )|. (1.182)
1
2
1
2
1
1
φ 1
Rainbow
colors
White light
n 2

Figure 1.26 Decomposition of white light into its constituent colors.