2.6. NONLINEAR OPTICAL EFFECTS 65
Figure 2.19: SPM-induced frequency chirp for Gaussian (dashed curve) and super-Gaussian
(solid curve) pulses.
If fiber losses are compensated periodically using optical amplifiers,φNLin Eq.
(2.6.14) should be multiplied by the number of amplifiersNAbecause the SPM-induced
phase accumulates over multiple amplifiers. To reduce the impact of SPM in lightwave
systems, it is necessary thatφNL 1. If we useφNL= 0 .1 as the maximum tolerable
value and replaceLeffby 1/αfor long fibers, this condition can be written as a limit on
the input peak power as
Pin< 0. 1 α/(γNA). (2.6.15)
For example, ifγ=2W−^1 /km,NA=10, andα= 0 .2 dB/km, the input peak power is
limited to below 2.2 mW. Clearly, SPM can be a major limiting factor for long-haul
lightwave systems.
Cross-Phase Modulation
The intensity dependence of the refractive index in Eq. (2.6.12) can also lead to another
nonlinear phenomenon known ascross-phase modulation(XPM). It occurs when two
or more optical channels are transmitted simultaneously inside an optical fiber using
the WDM technique. In such systems, the nonlinear phase shift for a specific channel
depends not only on the power of that channel but also on the power of other chan-
nels [80]. The phase shift for thejth channel becomes
φNLj =γLeff(
Pj+ (^2) ∑
m=j
Pm
)
, (2.6.16)
where the sum extends over the number of channels. The factor of 2 in Eq. (2.6.16)
has its origin in the form of the nonlinear susceptibility [31] and indicates that XPM is
twice as effective as SPM for the same amount of power. The total phase shift depends
on the powers in all channels and would vary from bit to bit depending on the bit pattern
of the neighboring channels. If we assume equal channel powers, the phase shift in the