As we saw in Section 7.4, once the dispersion and nonlinearity profiles of an amplifier are known the shape of the gain profile required to observe self- similar propagation can be found via Eq. (7.30). However, as the shape of the gain profile is largely determined by saturation effects caused by pump depletion, and thus the pumping geometry, such accurate control over its lon- gitudinal variation is not possible. For this reason here we study the evolution of a pulse in an amplifier with an experimentally realistic profile that differs slightly in shape from the ideal profile, yet yields the same total pulse gain.
Chapter 7 Self-Similar Solutions of the NLSE with Distributed Coefficients
We consider the evolution of a pulse in an amplifier with distributed gain but constant dispersion and nonlinearity so that the situation corresponds to case (i) described in Section 7.4.2. We recall that the shape of the gain profile re- quired for self-similar propagation in such a fibre is given by [Eq. (7.47)]:
g(z) = g0 1−g0z.
An experimentally realistic approximation to this, corresponding to counter- directional pumping, is an exponentially increasing profile of the form:
g(z) =g0exp (z/za), (7.56)
wherezadetermines the rate of increase. With the input pulse and fibre param-
eters the same as those used to generate Fig. 7.3(a), to ensure that the total pulse gain also remains the same (∼10 dB) we requireza = 5.44 m. Fig. 7.6 shows the
intensity profile and the chirp of the output pulse from the simulations with the gain profile given in Eq. (7.56) (solid curves), clearly in agreement with the theoretical predictions based on the exact gain profile (circles). Thus this suggests that the self-similar nature of these solutions is sufficiently robust to withstand amplification in an experimental situation where the longitudinal gain may deviate from the ideal profile.
−20 −1 0 1 2 2 4 6 Time (ps) Power (W) −30 0 30 Chirp (THz)
Figure 7.6: Intensity profile (left axis) and chirp (right axis) generated via amplification of the solitary wave solution with a gain profile given by Eq. (7.56) (solid curves) com- pared with the analytic solitary wave predictions for the exact gain profile (circles).
Chapter 7 Self-Similar Solutions of the NLSE with Distributed Coefficients
7.5.3 Evolution in an Amplifier with Higher Order Effects
Third Order Dispersion
As discussed in Chapter 3, although the basic form of the NLSE employed in this chapter [Eq. (7.1)] is remarkably successful in describing many of the fea- tures of pulse propagation in fibre amplifiers [4], it often needs to be extended to include higher order terms when the bandwidth of a pulse becomes large. Thus in this section we consider the effects of higher order dispersion and non- linearity, independently, to establish whether the neglect of such terms in our theoretical analysis was justified. As in Section 7.5.1, in both cases we investi- gate the system described in Section 7.4.1 corresponding to propagation in an amplifier with distributed dispersion and nonlinearity, but constant gain, so that the solitary wave solution is given by Eqs. (7.43)–(7.46) with ν = 1 and
α < 0. Again we choose the input pulse and fibre parameters to be the same as those used to generate Fig. 7.2(a).
The equation describing pulse propagation in a fibre including the effects of third order dispersion is [Eq. (3.7)]:
i∂Ψ ∂z = β2(z) 2 ∂2Ψ ∂T2 + i β3 6 ∂3Ψ ∂T3 −γ(z)|Ψ| 2Ψ + ig(z) 2 Ψ, (7.57)
where we have assumed that the z-dependence ofβ3 is negligible. With the choice ofβ3 = 0.1×10−4ps3m−1, we obtain an output pulse of the form given in Fig. 7.7(a). Although there is still good agreement between the intensity profile of the simulated pulse (solid curves) and the solitary wave solution (circles), clearly the symmetry of the pulse profile and the linearity of the chirp have been destroyed. These intensity and phase distortions are well known consequences of propagation with higher order dispersion which arise due to the cubic spectral phase introduced by the third order dispersive term [4]. Consequently, we expect that for larger values of β3, or alternatively, longer amplifier lengths, the pulses will become highly distorted so that the theoret- ical predictions will no longer be valid. However, these results show that the solitary pulses can endure a small amount of higher order dispersion and with the availability of dispersion flattened fibres to provideβ3 ≈ 0we expect that these solutions should be valid for a wide range of systems.
Chapter 7 Self-Similar Solutions of the NLSE with Distributed Coefficients −5 0 5 0 0.75 1.50 Time (ps) Power (W) −5 0 5 Time (ps) −30 0 30 Chirp (THz) (a) (b)
Figure 7.7: Intensity profiles (left axis) and chirps (right axis) of the output pulses generated under amplification with: (a) third order dispersion and (b) Raman gain. The numerical simulations (solid curves) are compared with the predicted self-similar solitary wave solutions (circles).
Delayed Nonlinear Response
In Section 3.10.2 it was noted that for pulses with bandwidths in excess of
1 THz it is possible for the Raman gain to amplify the low frequency compo- nents by transferring energy from the high frequency components of the same pulse. Modifying Eq. (3.54) to include the effects of longitudinally varying fibre parameters yields:
i∂Ψ ∂z = β2(z) 2 ∂2Ψ ∂T2+i g (z) 2 Ψ−γ(z) 1 + i ω0 ∂ ∂T Ψ ∞ 0 R(T)|Ψ (z, T −T)|2dT, (7.58) where againR(T) = (1−fR)δ(T) +fRhR(T). The output pulse from the simu-
lations is plotted in Fig. 7.7(b) (solid curves) showing excellent agreement with the solitary wave solution (circles). As expected the simulated pulse has un- dergone a slight frequency shift. However, this was calculated to be0.192 MHz
(in the red direction) which, compared to the terahertz bandwidth of the pulse, is negligible. The slightness of the self-frequency shift seen here, relative to the soliton self-frequency shift [29], can be attributed to the fact that the pulse is chirped (which results in a spreading of the frequency components) so that internal pumping of the high frequency components to the low frequencies is less efficient. Although the effects of Raman gain also accumulate over the length of the fibre, the length required for the effects to become noticeable is
Chapter 7 Self-Similar Solutions of the NLSE with Distributed Coefficients
well beyond practical amplifier lengths. Thus these results illustrate the insen- sitivity of the solitary wave solutions to Raman gain, justifying the neglect of this effect in the theoretical analysis.