Page 473 - Fiber Optic Communications Fund

454 Fiber Optic Communications

10.8 Intrachannel Nonlinear Impairments
In quasi-linear systems, dispersive effects are much stronger than nonlinear effects and the fiber nonlinearity
can be considered as a small perturbation in the linear system. Since the dispersive effects are dominant in
quasi-linear systems, neighboring pulses overlap and this system is also known as a strongly pulse-overlapped
system [35] or pseudo-linear system [36]. In contrast, in classical soliton systems, dispersion is balanced by
nonlinearity, the pulses are well confined within the bit period. In quasi-linear systems, the pulses that are
separated by several bit periods could interact nonlinearly because of the strong pulse overlap among the
pulses. In this section, we consider single-channel nonlinear impairments such as intrachannel four-wave
mixing (IFWM) [35–46] and intrachannel cross-phase modulation (IXPM) [47–49]. The variance in signal
distortion due to nonlinear effects is used as a measure to compare different fiber-optic systems. IXPM and
IFWM can be considered as deterministic signal–signal nonlinear impairments because if we know the bit
pattern, these effects can be undone using digital back propagation (DBP) at the transmitter or receiver (see
Chapter 11). In contrast, the nonlinear signal–ASE interaction such as Gordon–Mollenauer phase noise is
stochastic and the DBP can not compensate for it. In single-channel systems, the nonlinear interaction can be
divided into three types: (i) intrapulse SPM; (ii) IXPM; and (iii) IFWM. SPM has already been discussed in
Section 10.5. Here we discuss IXPM and IFWM.

10.8.1 Intrachannel Cross-Phase Modulation
IXPM is the phase modulation of a pulse by another pulse of the same channel. Consider the interaction
between two pulses q (T, Z) and q (T, Z) separated by T at the fiber input. Let the total field envelope be
1 2 b
q(T, Z)= q (T, Z)+ q (T, Z). (10.239)
1 2
Substituting Eq. (10.239) in Eq. (10.81), we find
2
(q + q ) (q + q )
1
2
2
2
1
2
i − + |q + q | (q + q )=−i(q + q )∕2. (10.240)
1
2
1
2
1
2
Z 2 T 2
The last term on the left-hand side can be written as
2
2 ∗
2
2
2
2
2 ∗
|q + q | (q + q )=(|q | + 2|q | )q +(|q | + 2|q | )q + q q + q q . (10.241)
2
1
1
2
2
1
1
1
2
2
1 2
2 1
The last two terms in Eq. (10.241) represent the intrachannel four-wave mixing, and this will be considered
2
in the next section. The term 2|q | q represents the phase modulation of q due to q . If the IXPM terms
1
2
1
2
2 ∗
2
2
2 ∗
2|q | q and 2|q | q and the IFWM term q q + q q were to be absent, the pulses would experience intra-
2
1
2
1
2 1
1 2
pulse SPM only and there would be no change in the temporal position of the pulses as a function of the
propagation distance. However, due to IXPM, pulses could attract or repel each other. Figs. 10.20 and 10.21
show the nonlinear interaction between adjacent pulses. In this example, pulses repel each other, leading to
performance degradation. The timing jitter due to IXPM can be calculated using a variation approach [42, 49]
or a perturbation technique [48]. The repulsion between pulses can be explained as follows. The phase modu-
lation caused by the IXPM leads to instantaneous frequency change of a pulse. In a dispersive fiber, different
frequency components travel at different speeds and, therefore, the frequency change due to IXPM translates
into group speed changes. Therefore, the first pulse moves faster than the second pulse and it arrives at the
fiber output earlier than the second one, leading to a temporal separation longer than the bit interval. In the
absence of IXPM, pulses would have the same group speed and the separation between pulses would be equal
to the bit period. For systems based on OOK, ‘1’ and ‘0’ occur randomly and the timing shift caused by IXPM
is random, leading to time jitters and performance degradation.