444 CHAPTER 9. SOLITON SYSTEMS
cally, the action of modulator is to change the soliton amplitude as
B(zm,t)→Tm(t−tm)B(zm,t), (9.5.37)whereTm(τ)is the transmission coefficient of the modulator located atξ=ξm. The
moment method or perturbation theory can be used to show that timing jitter is reduced
considerably by modulators.
The synchronous modulation technique can also be implemented by using a phase
modulator [133]. One can understand the effect of periodic phase modulation by re-
calling that a frequency shift,δω=−dφ(t)/dt, is associated with any phase variation
φ(t). Since a change in soliton frequency is equivalent to a change in the group veloc-
ity, phase modulation induces a temporal displacement. Synchronous phase modula-
tion is implemented in such a way that the soliton experiences a frequency shift only if
it moves away from the center of the bit slot, which confines it to its original position
despite the timing jitter induced by ASE and other sources. Intensity and phase modu-
lations can be combined together to further improve the system performance [134].
Synchronous modulation can be combined with optical filters to control solitons
simultaneously in both the time and frequency domains. In fact, this combination per-
mits arbitrarily long transmission distances [135]. The use of intensity modulators also
permits a relatively large amplifier spacing by reducing the impact of dispersive waves.
This property of modulators was exploited in 1995 to transmit a 20-Gb/s soliton train
over 150,000 km with an amplifier spacing of 105 km [136]. Synchronous modula-
tors also help in reducing the soliton interaction and in clamping the level of amplifier
noise. The main drawback of modulators is that they require a clock signal that is
synchronized with the original bit stream.
A relatively simple approach uses postcompensation of accumulated dispersion for
reducing the timing jitter [137]. The basic idea can be understood from Eq. (9.5.29)
or Eq. (9.5.32) obtained for the timing jitter of DM and standard solitons, respectively.
The cubic term that dominates the jitter at long distances depends on the accumulated
dispersion through the parameterddefined in Eq. (9.5.30). If a fiber is added at the end
of the fiber link such that it reduces the accumulated GVD, it should help in reducing
the jitter. It is easy to include the contribution of the postcompensation fiber to the
timing jitter using the moment method. In the case of DM solitons, the jitter variance
at the end of a postcompensation fiber of lengthLcand GVDβ 2 cis given by [124]
σc^2 =σt^2 +(SspTm^2 /E 0 )[ 2 NAC 0 dc+NA(NA− 1 )ddc+NAdc^2 ], (9.5.38)whereσt^2 is given by Eq. (9.5.29) anddc=β 2 cLc/TM^2. If we definey=−dc/(NAd)as
the fraction by which the accumulated dispersionNAdis compensated and retain only
the dominant cubic terms in Eq. (9.5.38), this equation can be written as
σc^2 =NA^3 d^2 Tm^2Ssp
E 0(
1
3
−y+y^2)
. (9.5.39)
The minimum value occurs fory= 0 .5 for whichσc^2 is reduced by a factor of 4. Thus,
timing jitter of solitons can be reduced by a factor of 2 by postcompensating the accu-
mulated dispersion by 50%. The same conclusion holds for standard solitons [137].