9.7. WDM SOLITON SYSTEMS 459
Note that the two solitons are propagating at different speeds because of their different
frequencies. As a result, the XPM term is important only when solitons belonging to
different channels overlap during a collision.
It is useful to define thecollision length Lcollas the distance over which two solitons
interact (overlap) before separating. It is difficult to determine precisely the instant at
which a collision begins or ends. A common convention uses 2Tsfor the duration of
the collision, assuming that a collision begins and ends when the two solitons overlap
at their half-power points [206]. Since the relative speed of two solitons is∆V=
(|β 2 |Ωch/T 0 )−^1 , the collision length is given byLcoll=(∆V)( 2 Ts)or
Lcoll=2 TsT 0
|β 2 |Ωch≈
0. 28
q 0 |β 2 |Bfch, (9.7.4)
where the relationsTs= 1. 763 T 0 andB=( 2 q 0 T 0 )−^1 were used. As an example, for
B=10 Gb/s,q 0 =5, andβ 2 =− 0 .5ps^2 /km,Lcoll∼100 km for a channel spacing of
100 GHz but reduces to below 10 km when channels are separated by more than 1 THz.
Since XPM induces a time-dependent phase shift on each soliton, it leads to a shift
in the soliton frequency during a collision. One can use a perturbation technique or the
moment method to calculate this frequency shift. If we assume that the two solitons
are identical before they collide,u 1 andu 2 are given by
um(ξ,τ)=sech(τ+δmξ)exp[−iδmτ+i( 1 −δm^2 )ξ/ 2 +iφm], (9.7.5)whereδm=^12 Ωchform=1 and−^12 Ωchform=2. Using the moment method, the
frequency shift for the first channel evolves with distance as
dδ 1
dξ=b(ξ)∫∞−∞∂|u 1 |^2
∂τ|u 2 |^2 dτ. (9.7.6)The equation forδ 2 is obtained by interchanging the subscripts 1 and 2. Noting from
Eq. (9.7.5) that
∂|um|^2
∂τ
=
1
δm∂|um|^2
∂ξ, (9.7.7)
and usingδm=±^12 Ωchin Eq. (9.7.6), the collision-induced frequency shift for the
slower moving soliton is governed by [206]
dδ 1
dξ=
b(ξ)
Ωchd
dξ[∫∞
−∞sech^2(
τ−
Ωchξ
2)
sech^2(
τ+
Ωchξ
2)
dτ]
. (9.7.8)
The change inδ 2 occurs by the same amount but in the opposite direction. The integral
overτcan be performed analytically to obtain
dδ 1
dZ=
4 b(Z)
Ωchd
dZ(
ZcoshZ−sinhZ
sinh^3 Z)
, (9.7.9)
whereZ=Ωchξ. This equation provides changes in soliton frequency during inter-
channel collisions under quite general conditions.