Page 465 - Fiber Optic Communications Fund

446 Fiber Optic Communications

where {a } is the data sequence and f(t) is the pulse shape function. The pump power at the fiber input is
n
2
| ∑ |
2 | ( ) |
p p p0 n b
P (T, 0)= |q (T, 0)| = P | a f t − nT | , (10.190)
| |
| n |
2
|q (T − d Z, 0)| = P (T − d Z, 0). (10.191)
p p p p
Taking the Fourier transform of Eq. (10.191), we find
2
̃
{|q (T − d Z, 0)| }= P ()e id p Z , (10.192)
p
p
p
̃
P ()= [P (T, 0)]. (10.193)
p
p
The phase shift of the signal due to XPM over a fiber length dZ can be found by differentiating Eq. (10.165)
with respect to Z,
2 −Z
d (T, Z)= 2|q (T − d Z, 0)| e dZ. (10.194)
XPM p p
Taking the Fourier transform of Eq. (10.194) and using Eq. (10.192), we obtain
̃
d ̃ XPM (, Z)= 2P ()e −(−id p )Z dZ. (10.195)
p
Let us deviate from XPM and consider a different problem. Suppose we have a linear dispersive fiber of
length Z and let the fiber input in the frequency domain be
̃ q ()= A e i ̃ in (). (10.196)
in in
̃
The input phase () is assumed to be small, and A is a constant. After passing through the dispersive
in
in
̃
̃
fiber, the phase fluctuations () lead to amplitude fluctuations A() at the fiber output given by [22]
in
̃
̃
A ()= A + A(), (10.197)
out
in
( 2 )
Z
2
̃
̃
A()=−A () sin . (10.198)
in in
2
Now let us return to the phase shift due to XPM. Let d ̃ (, Z ) be the phase shift of the signal due to
XPM 0
XPM at Z . After passing through the dispersive fiber of length L − Z where L is the fiber length, this phase
0 0
shift leads to an amplitude shift, as shown in Fig. 10.14, [16, 18],
[ ( )]
2 L − Z
√ 2 0
̃
dA ()=− P s0 d ̃ XPM (, Z ) sin . (10.199)
0
s
2
The nonlinear phase shift due to XPM is distributed over the fiber length, with each infinitesimal phase shift
leading to an infinitesimal amplitude shift at the fiber output. Substituting Eq. (10.195) into Eq. (10.199) and
integrating the XPM contributions originating from 0 to L, we obtain [16–18]
( ( ))
L 2
2
√ L − Z 0
̃
ΔA ()= −2 P s0 ∫ P ()e −(−id p )Z 0 sin dZ 0
s
p
0 2
√
P P () L
s0
p
=− [e −[−id p +ix]Z 0 +ixL − e −ixL−[−id p −ix]Z 0 ]dZ 0
i ∫ 0
√ ixL −ixL
= i P P (){ e L [( − id + ix) , L]− e L [( − id − ix) , L]},
s0 p EFF p EFF p
(10.200)