460 CHAPTER 9. SOLITON SYSTEMS
(a) (b)Figure 9.26: (a) Frequency shift during collision of two 50-ps solitons with 75-GHz channel
spacing. (b) Residual frequency shift after a collision because of lumped amplifiers (LA=20
and 40 km for lower and upper curves, respectively). Numerical results are shown by solid dots.
(After Ref. [206];©c1991 IEEE; reprinted with permission.)
Consider first the ideal case of constant-dispersion lossless fibers so thatb=1in
Eq. (9.7.9). In that case, integration in Eq. (9.7.9) is trivial, and the frequency shift is
given by
∆δ 1 (Z)= 4 (ZcoshZ−sinhZ)/(Ωchsinh^3 Z). (9.7.10)Figure 9.26(a) shows how the frequency of the slow-moving soliton changes during the
collision of two 50-ps solitons when channel spacing is 75 GHz. The frequency shifts
up over one collision length as two solitons approach each other, reaches a peak value
of about 0.6 GHz at the point of maximum overlap, and then decreases back to zero
as the two solitons separate. The maximum frequency shift depends on the channel
spacing. It occurs atZ=0 in Eq. (9.7.10) and is found to be 4/( 3 Ωch). In physical
units, the maximum frequency shift becomes
∆fmax=( 3 π^2 T 02 fch)−^1. (9.7.11)Since the velocity of a soliton changes with its frequency, collisions speed up or
slow down a soliton, depending on whether its frequency increases or decreases. At
the end of the collision, each soliton recovers the frequency and speed it had before
the collision, but its position within the bit slot changes. The temporal shift can be
calculated by integrating Eq. (9.7.9). In physical units, it is given by
∆t=−T 0∫∞−∞∆δ 1 (ξ)dξ=4 T 0
Ω^2 ch=
1
π^2 T 0 fch^2. (9.7.12)
If all bit slots were occupied, such collision-induced temporal shifts would be of no
consequence since all solitons of a channel would be shifted by the same amount.
However, 1 and 0 bits occur randomly in real bit streams. Since different solitons of
a channel shift by different amounts, interchannel collisions induce some timing jitter
even in lossless fibers.