Page 439 - Fiber Optic Communications Fund

420 Fiber Optic Communications

Electron charge cloud
E x

Nucleus

x
(a) (b)

Figure 10.1 Classical electron oscillator model: (a) in equilibrium; (b) in the presence of an external field.

When an external electric field intensity E is applied to an atom, the electron charge cloud is displaced
x
from its equilibrium position, as shown in Fig. 10.1(b). The equation of motion for the center of the electron
charge cloud is given by Newton’s law,

2
d x
m = F ext = q E , (10.1)
e x
dt 2
where x(t) is the displacement of the center of the electron charge cloud, m is the electron mass, and q e
is the electron charge. When the center of the electron cloud moves away from the equilibrium position
(Fig. 10.1(b)), there is a force of attraction between the nucleus and the electron charge cloud. If the displace-
ment x(t) is small, the restoration force can be approximated as

F restoration =−Kx, (10.2)

where K is a constant. The negative sign indicates that the restoration force acts in a direction opposite to
the external force. The situation is similar to the case of a simple pendulum pushed away from the equilib-
rium position by an external force; there is a restoration force due to gravitation which pulls it back to the
equilibrium position. The net force acting on the electron is given by

F = F + F = q E − Kx. (10.3)
net ext restoration e x
Combining Eq. (10.3) with Newton’s law, we obtain
2
d x
m = F net = q E − Kx (10.4)
e x
dt 2
or
2
d x 2 ( q e )
+ x = E , (10.5)
x
0
dt 2 m
where =(K∕m) 1∕2 is the natural frequency of oscillation. Suppose the applied field is of the form
0
E = E exp (−it). (10.6)
x
0