Page 496 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 477

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that there exists an optimum energy at which the total phase variance is minimum. By setting d ∕dE to
zero, the optimum energy is calculated as
√
T b 3
E = . (10.399)
opt
L 2(N − 1)(2N − 1)
eff a a
2
When N is large, (N − 1)(2N − 1)≈ 2N and using Eq. (10.394), we find that the phase variance is
a a a a
minimum when the deterministic nonlinear phase shift ≈ 0.87 rad. Eqs. (10.397) and (10.398) are derived
d
under the assumption that dispersion is zero. In the presence of dispersion, Eq. (10.397) is modified as [60]
2
2
= + 2E[g (mL ) ], (10.400)
m fr a
2E
where { }
T 0 L tot
g (x)= √ Re ∫ G(s)ds , (10.401)
fr
x
2
a (s)
, (10.402)
G(s)= √
2
4
2
2
(1 + T Δ(s))(T + 3S (s)+ 2iT S(s))
0 0 0
2
T − iS(s)
0
Δ(s)= 2 2 . (10.403)
T [T + i3S(s)]
0 0
T and S(z) are defined in Section 10.9. If a dispersion-managed fiber with zero mean dispersion per span is
0
used, the total variance can be expressed in a form similar to Eq. (10.398) [60],
N a (N − 1)N (2N − 1)E(h ) 2
a
fr
a
a
2
= + , (10.404)
2E 3
{ }
T 0 L a
h = √ Re G(s)ds , (10.405)
fr ∫
0
Comparing Eqs. (10.398) and (10.404), we see that these two expressions are the same except that L ∕T 0
eff
is replaced by h . For a highly dispersive system, h is much smaller than L ∕T and, hence, the variance
fr
eff
fr
0
of nonlinear phase noise due to SPM is much smaller in a highly dispersive system. Eq. (10.404) does not
include contributions due to IXPM. Even if IXPM contributions are included, numerical simulations have
shown that for highly dispersive systems, the variance of nonlinear phase noise (signal–noise interaction)
is much smaller than that due to IFWM and IXPM (signal–signal interactions). In a WDM system, interac-
tion between ASE and XPM leads to nonlinear phase noise as well [76]. Using the digital back propagation
technique discussed in Chapter 11, it is possible to compensate for deterministic (symbol pattern-dependent
signal–signal interactions) nonlinear effects, but not for nonlinear phase noise (signal–ASE interactions).
So, when the DBP is used to compensate for intra- and interchannel nonlinear impairments, nonlinear phase
noise is likely to be one of the dominant impairments.
Example 10.7
A rectangular pulse of peak power 2 mW and pulse width 25 ps is transmitted over a periodically amplified
dispersion-free fiber-optic transmission system operating at 1550 nm. The fiber-optic link consists of 20
amplifiers with an amplifier spacing of 80 km. The parameters of the link are as follows: nonlinear coefficient