414 CHAPTER 9. SOLITON SYSTEMS
The dependence of soliton separation onq 0 can be studied analytically by using the
inverse scattering method [29]. A perturbative approach can be used forq 0 1. In the
specific case ofr=1 andθ=0, the soliton separation 2qsat any distanceξis given
by [30]
2exp[ 2 (qs−q 0 )]= 1 +cos[ 4 ξexp(−q 0 )]. (9.2.7)
This relation shows that the spacingqs(ξ)between two neighboring solitons oscillates
periodically with the period
ξp=(π/ 2 )exp(q 0 ). (9.2.8)
A more accurate expression, valid for arbitrary values ofq 0 , is given by [32]
ξp=
πsinh( 2 q 0 )cosh(q 0 )
2 q 0 +sinh( 2 q 0 ). (9.2.9)
Equation (9.2.8) is quite accurate forq 0 >3. Its predictions are in agreement with
the numerical results shown in Fig. 9.6 whereq 0 = 3 .5. It can be used for system
design as follows. IfξpLDis much greater than the total transmission distanceLT,
soliton interaction can be neglected since soliton spacing would deviate little from its
initial value. Forq 0 =6,ξp≈634. UsingLD=100 km for the dispersion length,
LTξpLDcan be realized even forLT= 10 ,000 km. If we useLD=T 02 /|β 2 |and
T 0 =( 2 Bq 0 )−^1 from Eq. (9.2.1), the conditionLTξpLDcan be written in the form
of a simple design criterion
B^2 LTπexp(q 0 )
8 q^20 |β 2 |. (9.2.10)
For the purpose of illustration, let us chooseβ 2 =−1ps^2 /km. Equation (9.2.10) then
implies thatB^2 LT 4 .4 (Tb/s)^2 -km if we useq 0 =6 to minimize soliton interactions.
The pulse width at a given bit rateBis determined from Eq. (9.2.1). For example,
Ts= 14 .7psatB=10 Gb/s whenq 0 =6.
A relatively large soliton spacing, necessary to avoid soliton interaction, limits the
bit rate of soliton communication systems. The spacing can be reduced by up to a factor
of 2 by using unequal amplitudes for the neighboring solitons. As seen in Fig. 9.6, the
separation for two in-phase solitons does not change by more than 10% for an initial
soliton spacing as small asq 0 = 3 .5 if their initial amplitudes differ by 10% (r= 1 .1).
Note that the peak powers or the energies of the two solitons deviate by only 1%.
As discussed earlier, such small changes in the peak power are not detrimental for
maintaining solitons. Thus, this scheme is feasible in practice and can be useful for
increasing the system capacity. The design of such systems would, however, require
attention to many details. Soliton interaction can also be modified by other factors,
such as the initial frequency chirp imposed on input pulses.
9.2.3 Frequency Chirp
To propagate as a fundamental soliton inside the optical fiber, the input pulse should
not only have a “sech” profile but also be chirp-free. Many sources of short optical
pulses have a frequency chirp imposed on them. The initial chirp can be detrimental to