Page 460 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 441

8
Lin
7 Lin + SPM
Lin + SPM + XPM
6

5
Power (mW) 4 3

2
1

0
–60 –40 –20 0 20 40 60
Time (ps)

−1
Figure 10.12 Channel 1 output pulse when (1) Lin: = 0W −1 km ,(2) Lin+SPM: no pulse presents in channel 2, and
(3) Lin+SPM+XPM: both channels are present. Parameters are the same as those of Fig. 10.10.

10.7.1.2 XPM Efficiency
To see the impact of XPM, let us first ignore the pulse broadening due to dispersion and consider only two
channels–pump and signal. Let the central frequency of the signal channel be the same as the reference
frequency , so that d for the signal is zero. We further assume that the pump is much stronger than the
0 n
2
2
signal, i.e., |q | >> |q | . With these assumptions and approximations, Eq. (10.141) becomes
p s
q s ( )
i = −i − 2|q | 2 q , (10.145)
p
s
Z 2
q p q p ( )
( )
i + d p = −i − |q | 2 q , (10.146)
p
p
Z T 2
where the subscripts p and s denote pump and signal, respectively, and the walk-off parameter is
3
2
d = Ω + Ω ; (10.147)
2
p
p
p
2
Ω is the frequency separation between the pump and the signal. To solve Eq. (10.146), we use the following
p
transformation:
′
T = T − d Z, (10.148)
p
′
Z = Z, (10.149)
to obtain
q p ( )
i = −i − |q | 2 q . (10.150)
p
p
Z ′ 2
Let
′
′
′
′
′
′
q (T, Z) ≡ q (T , Z )= A (T , Z )e i p (T ,Z ) . (10.151)
p
p
p