Page 445 - Fiber Optic Communications Fund

426 Fiber Optic Communications

where n is the linear refractive index and the second term of Eq. (10.51) represents the nonlinear contribution
0
to the refractive index. Typically, the nonlinear part of the refractive index is much smaller than the linear part.
From Eq. (10.51), we have
( 2 ) 1∕2
3 E | (3)
|
n = n 0 1 + | 0| xxxx
4n 2
0
2
≅ n + n |E | , (10.52)
2
0
0
where
(3)
3 xxxx
n = (10.53)
2
8n
0
is called the Kerr coefficient. In Eq. (10.52), we have used the following approximation:
(1 + x) 1∕2 ≈ 1 + x∕2, if x ≪ 1, (10.54)
which is valid since the nonlinear part of the refractive index is much smaller than its linear part. From
Eq. (10.52), we see that the change in refractive index (n − n ) is directly proportional to the optical intensity
0
2
|E | . This effect is called the Kerr effect.
0
2
2
For silica, n ≈ 3 × 10 −20 m /W. If a light beam of intensity 1 W/m is incident on silica media of
2
2
cross-sectional area 1 m , the change in refractive index is 3 × 10 −20 , which is very small. However, silica
2
fiber has an effective cross-sectional area of 100 μm or less, and the change in refractive index due to the
Kerr effect is comparable with the variations in refractive index due to dispersion, leading to interesting
nonlinear phenomena such as soliton formation.
10.3 Fiber Dispersion
As mentioned in Chapter 2, a pulse propagating in a fiber broadens due to fiber dispersion. When the fiber
nonlinear effects are ignored, the complex field envelope in a field is given by Eqs. (2.119) and (2.107) as
2
̃ q(, z)= ̃q(, 0)e −z∕2+i 1 z+i 2 z∕2 . (10.55)
Differentiating Eq. (10.55) with respect to z, we find
̃q(, z) ( 2 )
= ̃q(, z) − + i + i . (10.56)
2
1
z 2
Since n
n
 −1 {(−i) ̃q(, z)} = q(t, z) , (10.57)
t n
taking the inverse Fourier transform of Eq. (10.56), we find
2
q(t, z) q(t, z) i q(t, z)
2
=− q(t, z)− 1 − . (10.58)
z 2 t 2 t 2
As mentioned in Chapter 2, the term exp (i z) of Eq. (10.55) introduces a constant time shift due to propa-
1
gation and it could be dropped as we are primarily interested in assessing the quality of the signal at the fiber
output. Let
Z = z, (10.59)
T = t − z. (10.60)
1