Page 457 - Fiber Optic Communications Fund

438 Fiber Optic Communications

Eq. (10.131). From the last term on the left-hand side of Eq. (10.131), we see that nonlinear effects in the fiber
generate frequency components of the form Ω +Ω −Ω . For a frequency band centered around Ω , the only
n
l
j
k
frequency components that are of importance are Ω +Ω −Ω =Ω . All the other frequency components
j
n
l
k
generated through fiber nonlinearity have no effect on the frequency band centered around Ω . Therefore,
n
collecting all the terms that oscillate at frequency Ω and noticing that
n
2
2
= + Ω + Ω , (10.133)
1
0
n
n
n
2
we obtain
[ ] 2 N∕2−1 N∕2−1 N∕2−1
q n ( ) q n q n ∑ ∑ ∑
2
∗ iΔ jkln z
i + + d n − + q q q e =−i q n (10.134)
1
j k l
z t 2 t 2 2
j=−N∕2 k=−N∕2 l=−N∕2
where
d = Ω , (10.135)
n 2 n
Δ = + − − , (10.136)
jkln j k l n
Ω =Ω +Ω −Ω . (10.137)
n
k
l
j
2
If j = k = l = n, the last term on the left-hand side is |q | q , which represents SPM. If j = n and k = l ≠ j,
n
n
2
the corresponding term in the summation is |q | q , which represents cross-phase modulation (XPM). All
n
k
other terms in the above summation represent four-wave mixing (FWM).
As before, using a reference frame that moves at the group speed of the reference channel at ,
0
T = t − z, (10.138)
1
Z = z, (10.139)
we find
( ) 2
q n q n q n
2
i + d n − +
Z T 2 T 2
⎧ ⎫
⎪ ⎪
N∕2 N∕2−1 N∕2−1 N∕2−1
∑ ∑ ∑ ∑
⎪ 2 2 ∗ iΔ jkln Z ⎪
|q | q + 2 |q | q + q q q e (10.140)
⎨ n n k n j k l ⎬
k=−N∕2 j=−N∕2 k=−N∕2 l=−N∕2
⎪ ⎪
k≠n j+k−l=n
⎪ ⎪
⎩SPM XPM FWM ⎭

=−i q .
n
2
10.7.1 Cross-Phase Modulation
In this section, we focus on XPM by ignoring the FWM terms of Eq. (10.140). The nonlinear interaction due
to SPM and XPM is described by
{ }
( ) 2 N∕2−1
q n q n q n 2 ∑ 2
2
i + d n − + |q | q + 2 |q | q n =−i q . (10.141)
n
n
k
n
Z T 2 T 2 2
k=−N∕2