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486 Fiber Optic Communications

Table 10.3 FWM tones on the central channel with the degeneracy factor.
Tone number Number of tones j k l Type

1 2 −2 1 −1 ND
2 2 −2 2 0 ND
3 2 −1 1 0 ND
4 2 −1 2 1 ND
5 1 1 1 2 D
6 1 −1 −1 −2 D

Example 10.11

For a single-span dispersion-free fiber, find the nonlinear distortion up to second order using the perturbation
theory.

Solution:
When = 0, from Eq. (10.252), we have
2
du 1 2 2
i =−a (Z)|u | u . (10.465)
0
0
dZ
2
For a single-span system, a (Z)= exp (−Z). From Eq. (10.255), we have u = k. Integrating Eq. (10.465)
0
and using u (T, 0)= 0, we obtain
1
2
u (T, Z)= iZ |k| k, (10.466)
1
eff
where
1 − exp (−Z)
Z eff = . (10.467)

From Eq. (10.253), we have
du
2 4 4
= i exp (−Z)(2iZ |k| k − iZ |k| k)
eff
eff
dZ
exp (−Z)− exp (−2Z) 4
= (−|k| k). (10.468)

Integrating Eq. (10.468), we obtain
[ ]
1 − exp (−Z) 1 − exp (−2Z) 4
u (T, Z)=− − |k| k
2 2 2
2
4
|k| k 2
=− Z . (10.469)
2 eff
The total solution up to second order is
2
u = u + u + u 2
1
0
( 2 4 2 )
|k| Z eff
2
= k 1 + i|k| Z − . (10.470)
eff
2