Nonlinear Fiber Optics - 4 ed. Agrawal

166 Chapter 5. Optical Solitons
Normalized Intensity
2
1.5
1
0.5
2
1
τ R
= 0.5
0
−4 −3 −2 −1 0 1 2 3 4
Normalized Time, τ
Figure 5.22: Temporal intensity profiles of kink solitons in the form of an optical shock for
several values of τ R . (After Ref. [206]; c○1992 APS.)
shifts at a much smaller rate. Its spectral and temporal shift would become apparent
over distances longer than z = 8L D .
A question one may ask is whether Eq. (5.5.20) has soliton-like solutions. It turns
out that pulselike solutions do not exist when the Raman term is included, mainly because
the resulting perturbation is of non-Hamiltonian type [142]. This feature of the
Raman term can be understood by noting that the Raman-induced spectral red shift
does not preserve pulse energy because a part of the energy is dissipated through the
excitation of molecular vibrations. However, a kink-type topological soliton (with infinite
energy) has been found and is given by [206]
u(ξ ,τ)=[e −bτ sech(bτ)] 1/2 exp(ib 2 ξ /2). (5.5.21)
where b = 3/(2τ R ).
Kink solitons appear in many physical systems whose dynamics are governed by
the sine–Gordon equation [70]. In the context of optical fibers, the kink soliton represents
an optical shock front that preserves its shape when propagating through the
fiber. Figure 5.22 shows the shock profiles by plotting |u(ξ ,τ)| 2 for several values
of τ R . Steepness of the shock depends on τ R such that the shock front becomes increasingly
steeper as τ R is reduced. Even though the parameter N increases as τ R is
reduced, the power level P 0 (defined as the power at τ = 0) remains the same. This can
be seen by expressing P 0 in terms of the parameter T R using Eqs. (5.2.3) and (5.5.9)
so that P 0 = 9|β 2 |/(16γT 2 R ). Using typical values for fiber parameters, P 0 ∼ 10 kW.
It is difficult to observe such optical shocks experimentally because of large power
requirements.
The kink soliton given in Eq. (5.5.21) is obtained assuming u(ξ ,τ)=V (τ)exp(iKξ ),
and solving the resulting ordinary differential equation for V (τ). The solution shows