Page 440 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 421

We expect that the displacement x(t) should also change harmonically in the steady state and try a trial solution

x(t)= B exp (−it). (10.7)
Substituting Eqs. (10.7) and (10.6) into Eq. (10.5), we obtain

E q
0 e
B = 2 (10.8)
2
m( − )
0
and
q e
x(t)= 2 E . (10.9)
x
2
m( − )
0
The dipole moment of an atom is defined as
q 2 e
p = q x = 2 E . (10.10)
e
x
x
2
m( − )
0
In general,
q 2 e
p = E. (10.11)
2
2
m( − )
0
Our next step is to determine the electromagnetic field generated by the oscillating electron charge cloud.
Each atom acts as a current source since the oscillating electron cloud can be imagined as a tiny current
element. One of Maxwell’s equations in the presence of a current source is (Eq. (1.47))
E
∇× H = J + . (10.12)
0
t
Consider an incremental volume dV = Adx of an atomic system as shown in Fig. 10.2. Using Eq. (10.10), the
current I is given by
dq dq dx 1 dq dp x
I = = = . (10.13)
dt dx dt q dx dt
e
Let N be the number of atoms per unit volume. The charge in volume dV is
dq = q NdV = q NAdx (10.14)
e e
or
dq
= q NA. (10.15)
e
dx
Using Eq. (10.15) in Eq. (10.13), we obtain
dp x
I = NA . (10.16)
dt
Since J = I∕A, we obtain
x
dp x
J = N . (10.17)
x
dt
In general,
dp
J = N . (10.18)
dt