Page 493 - Fiber Optic Communications Fund

474 Fiber Optic Communications

10.10.2 Gordon–Mollenauer Phase Noise
The optical field envelope in a fiber-optic transmission system can be described by the NLSE in the lossless
form (Eq. (10.246)),
2
u (Z) u 2 2
2
i − =−a (Z)|u| u. (10.372)
Z 2 T 2
Amplifier noise effects can be introduced in Eq. (10.372) by adding a source term on the right-hand side,
which leads to
2
u (Z) u 2 2
2
i − =−a (Z)|u| u + iR(Z, T), (10.373)
Z 2 T 2
where
N a
∑
R(Z, T)= (Z − mL )n(T). (10.374)
a
m=1
Here, N is the number of amplifiers and n(T) is the noise field due to ASE, with statistical properties defined
a
in Eqs. (10.348)–(10.350).
In this section, we first consider the case in which the fiber dispersion is zero. Let us consider the solution
of Eq. (10.373) in the absence of noise. Let
u(Z, T)= A(Z, T) exp [i(Z, T)], (10.375)

and
√
u(0, T)= Ep(T). (10.376)
Substituting Eq. (10.375) in Eq. (10.372), we find
dA √
= 0 → A(Z, T)= A(0, T)= E|p(T)|, (10.377)
dZ
d 2 2
= a (Z)|u(0, T)|
dZ
2 2
= a (Z)E|p(T)| . (10.378)
Solving Eq. (10.378), we find
Z
2 2
(Z, T)= E|p(T)| a (s)ds, (10.379)
∫
0
[ ]
Z
2 2
u(Z, T)= u(0, T) exp i|u(0, T)| a (s)ds . (10.380)
∫
0
We assume that the signal pulse shape is rectangular, with pulse width T . From Eq. (10.346), it follows that
b
2
2
|p(T)| = 1∕T . Since a (Z)= exp (− Z) between amplifiers, we have
b 0
mL a −
2
a (Z)dZ = mL , (10.381)
∫ eff
0
where
1 − exp (− L )
0 a
L = . (10.382)
eff

0