9.2. SOLITON-BASED COMMUNICATIONS 415
Figure 9.7: Evolution of a chirped optical pulse for the caseN=1 andC= 0 .5. ForC=0 the
pulse shape does not change, since the pulse propagates as a fundamental soliton.
soliton propagation simply because it disturbs the exact balance between the GVD and
SPM [34]–[37].
The effect of an initial frequency chirp can be studied by solving Eq. (9.1.5) nu-
merically with the input amplitude
u( 0 ,τ)=sech(τ)exp(−iCτ^2 / 2 ), (9.2.11)whereCis the chirp parameter introduced in Section 2.4.2. The quadratic form of
phase variations corresponds to a linear frequency chirp such that the optical frequency
increases with time (up-chirp) for positive values ofC. Figure 9.7 shows the pulse
evolution in the caseN=1 andC= 0 .5. The pulse shape changes considerably even
forC= 0 .5. The pulse is initially compressed mainly because of the positive chirp;
initial compression occurs even in the absence of nonlinear effects (see Section 2.4.2).
The pulse then broadens but is eventually compressed a second time with the tails
gradually separating from the main peak. The main peak evolves into a soliton over
a propagation distanceξ>15. A similar behavior occurs for negative values ofC,
although the initial compression does not occur in that case. The formation of a soliton
is expected for small values of|C|because solitons are stable under weak perturbations.
But the input pulse does not evolve toward a soliton when|C|exceeds a critical valve
Ccrit. The soliton seen in Fig. 9.7 does not form ifCis increased from 0.5 to 2.
The critical valueCcritof the chirp parameter can be obtained by using the inverse
scattering method [34]–[36]. It depends onNand is found to beCcrit= 1 .64 forN=1.
It also depends on the form of the phase factor in Eq. (9.2.11). From the standpoint
of system design, the initial chirp should be minimized as much as possible. This is
necessary because even if the chirp is not detrimental for|C|<Ccrit, a part of the pulse
energy is shed as dispersive waves during the process of soliton formation [34]. For
instance, only 83% of the input energy is converted into a soliton for the caseC= 0. 5
shown in Fig. 9.7, and this fraction reduces to 62% whenC= 0 .8.