Nonlinear Fiber Optics - 4 ed. Agrawal

128 Chapter 5. Optical Solitons
depends on their bandwidth, this effect degrades system performance. Experimental
results on a lightwave system operating at 10 Gb/s showed considerable degradation
for a transmission distance of only 455 km [58]. As expected, system performance improved
when GVD was compensated partially using a dispersion-compensating fiber.
The use of optical amplifiers can induce modulation instability through another
mechanism and generate additional sidebands in which noise can be amplified in both
the normal and anomalous GVD regime of optical fibers [53]. The new mechanism has
its origin in the periodic sawtoothlike variation of the average power P 0 occurring along
the link length. To understand the physics more clearly, note that a periodic variation of
P 0 in z is equivalent to the creation of a nonlinear index grating because the term γP 0 in
Eq. (5.1.4) becomes a periodic function of z. The period of this grating is equal to the
amplifier spacing and is typically in the range 50–80 km. Such a long-period grating
provides a new coupling mechanism between the spectral sidebands located at ω 0 + Ω
and ω 0 − Ω and allows them to grow when the perturbation frequency Ω satisfies the
Bragg condition.
The analysis of Section 5.1.1 can be extended to include periodic variations of P 0 .
If we replace P 0 in Eq. (5.1.4) by P 0 f (z), where f (z) is a periodic function, expand
f (z) in a Fourier series as f (z) =∑c m exp(2πimz/L A ), the frequencies at which the
gain peaks are found to be [53]
( 2πm
Ω m = ± − 2γP ) 1/2
0c 0
, (5.1.12)
β 2 L A β 2
where the integer m represents the order of Bragg diffraction, L A is the spacing between
amplifiers (grating period), and the Fourier coefficient c m is related to the fiber loss α
as
c m = 1 − exp(−αL A)
αL A + 2imπ . (5.1.13)
In the absence of grating, or when m = 0, Ω 0 exists only for anomalous dispersion, in
agreement with Eq. (5.1.9). However, when m ≠ 0, modulation-instability sidebands
can occur even for normal dispersion (β 2 > 0). Physically, this behavior can be understood
by noting that the nonlinear index grating helps to satisfy the phase-matching
condition necessary for four-wave mixing to occur when m ≠ 0.
With the advent of wavelength-division multiplexing (WDM), it has become common
to employ the technique of dispersion management to reduce the GVD globally,
while keeping it high locally by using a periodic dispersion map. The periodic variation
of β 2 creates another grating that affects modulation instability considerably. Mathematically,
the situation is similar to the preceding case except that β 2 rather than P 0
in Eq. (5.1.4) is a periodic function of z. The gain spectrum of modulation instability
is obtained following a similar technique [57]. The β 2 grating not only generates new
sidebands but also affects the gain spectrum seen in Figure 5.1. In the case of strong
dispersion management (relatively large GVD variations), both the peak value and the
bandwidth of the modulation-instability gain are reduced, indicating that such systems
should not suffer much from amplification of noise induced by modulation instability.
Figure 5.5 shows the impact of dispersion compensation on the gain spectrum of
modulation instability for a 1000-km lightwave system consisting of 100-km spans of