Nonlinear Fiber Optics - 4 ed. Agrawal

74 Chapter 3. Group-Velocity Dispersion
(3.2.5) becomes
U(L m ,t)= 1 ∫ ∞
]
i
Ũ(0,ω)exp[
2π −∞
2 ω2 (β 21 L 1 + β 22 L 2 ) − iωt dω, (3.4.7)
where L m = L 1 +L 2 is the dispersion-map period, and β 2 j is the GVD parameter of the
fiber segment of length L j ( j = 1, 2). By using D j = −(2πc/λ 2 )β 2 j , the condition for
dispersion compensation can be written as
D 1 L 1 + D 2 L 2 = 0. (3.4.8)
As A(L m ,t) =A(0,t) when Eq. (3.4.8) is satisfied, the pulse recovers its initial width
after each map period even though pulse width can change significantly within each
period.
Equation (3.4.8) can be satisfied in several different ways. If two segments are of
equal lengths (L 1 = L 2 ), the two fibers should have D 1 = −D 2 . Fibers with equal and
opposite values of GVD can be made by shifting the zero-dispersion wavelength appropriately
during the manufacturing stage. Alternatively, if standard fibers with large
anomalous GVD [D ≈ 16 ps/(km-nm)] are employed, dispersion can be compensated
by using a relatively short segment of dispersion-compensating fiber (DCF) designed
to have “normal” GVD with values of D > −100 ps/(km-nm). Several other devices
(such as fiber gratings) can also be used for dispersion management [29].
3.4.3 Compensation of Third-Order Dispersion
When the bit rate of a single channel exceeds 100 Gb/s, one must use ultrashort pulses
(width ∼1 ps) in each bit slot. For such short optical pulses, the pulse spectrum becomes
broad enough that it is difficult to compensate GVD over the entire bandwidth
of the pulse (because of the frequency dependence of β 2 ). The simplest solution to this
problem is provided by fibers, or other devices, designed such that both β 2 and β 3 are
compensated simultaneously.
The dispersion-management conditions can be obtained from Eq. (3.3.2). For a
fiber link containing two different fibers of lengths L 1 and L 2 , the conditions for broadband
dispersion compensation are given by
β 21 L 1 + β 22 L 2 = 0 and β 31 L 1 + β 32 L 2 = 0, (3.4.9)
where β 2 j and β 3 j are the GVD and TOD parameters for fiber of length L j ( j = 1,2). It
is generally difficult to satisfy both conditions simultaneously over a wide wavelength
range. However, for a 1-ps pulse, it is sufficient to satisfy Eq. (3.4.9) over a 4–5 nm
bandwidth. This requirement is easily met by especially designed DCFs [46] and other
devices such as fiber gratings and liquid-crystal modulators.
Several experiments have demonstrated signal transmission at relatively high bit
rates (>100 Gb/s) over distances ∼100 km with simultaneous compensation of both
GVD and TOD [47]–[53]. In a 1996 experiment, a 100-Gb/s signal was transmitted
over 560 km with 80-km amplifier spacing [47]. In a later experiment, bit rate was
extended to 400 Gb/s by using 0.98-ps optical pulses within the 2.5-ps time slot [48].