CONCLUSIONS

We have discussed the possibility of using the waveguide Mach–Zehnder interferometer to generate a variety of dark solitons under constant background. Under optimal operation, 30% less input power and driving voltage are required than for complete modulation. The generated solitons can have good pulse quality and stimulated Raman scattering process can be utilized to compensate for fiber loss and even to amplify and compress the dark solitons. Generally speaking, when a constant gain coefficient is included in the NLSE, adiabatic amplification of the dark soliton is possible, as long as the gain Γ is kept small ...

When a fundamental dark soliton is adiabatically amplified in the presence of ISRS, the spectral shift and thus the temporal shift follow a simple rule, Eq. ([*]), which takes into consideration the exponentially increasing amplitude and linear dependence of the shift on the propagation distance. We find that such a simple model can accurately describe the behavior of fundamental dark solitons subject to adiabatic amplification and ISRS. The propagation properties of even dark pulses are also studied, with special attention to the distribution of energies among secondary gray solitons. Despite their more complicated nature, our results demonstrate that the partition of the energy is similar for quite different input pulse shapes, as long as they have the same background intensity and total energy for the input pulse. One can use the partition rule obtained here to predict the amplitude of secondary solitons produced from an input even dark pulse.

The authors thank the reviewers for their constructive comments. This research was supported by National Science Foundation grant ECS-91960-64.


\begin{references} \bibitem{ZA} V. E. Zakharov and A. B. Shabat, \lq\lq Exact theory ... ...ckground pulses'', J. Opt. Soc. Am. B {\bf 6,} 329--334 (1989). \end{references}

Figure: The dark solitons generated by the waveguide Mach-Zehnder interferometer. The amplitude of the input cw light is chosen to be a = π/2 for (a)-(c). The parameter δ is (a) 0.8, (b) 0.5, and (c) 0.2. Part (d) is the case of optimal operation when a = 1.33, and δ = 0.7. In all cases, the output pulse shapes are plotted as solid curves while the dashed curves are input pulse shapes. The pulses shown here are at a propagation distance of z = 4.
\begin{figure} \end{figure}
Figure: Dark solitons under constant gain. Pulse shapes (solid) when Γ=0.05 (a) and 1(b), after certain propagation distance, Γz=1.6, as compared to input pulse shapes (dashed). (c): The pulse duration, relative to its input, as a function of Γz at various Γ. The solid curve is the adiabatic approximation obtained by perturbation method. Three values of Γ are used: Γ = 0.05 (dotted); 0.2 (dash-dotted); and 1 (dashed). Negative Γz depicts the case of loss.
\begin{figure} \end{figure}
Figure: The pulse shapes of amplified dark solitons. (a) δ = 0.5, β = 2ln1.05, ΓpL = 2, after 8 amplifying cycles (solid); (b) δ = 0.5, β = 2ln1.02, ΓpL = 2, after 16 amplifying cycles (solid); (c) δ = 0.5, β = 2ln1.02, ΓpL = 0.5, after 16 amplifying cycles (solid); (d) The input pulse is the same as in Fig. 1(c), β = 2ln1.05, after 8 amplification periods (solid). The input pulse shapes are plotted as dashed curves.
\begin{figure} \end{figure}
Figure: (a) The shape of a fundamental dark soliton after a propagation distance of 40 (solid). The normalized time delay τd = 0.01. The dashed curve is the input pulse shape. (b) The trace of the soliton (solid) as a function of propagation distance for the situation described by (a). The dotted curve represents the case for a fundamental bright soliton under similar conditions.
\begin{figure} \end{figure}
Figure: The shape of a higher-order dark soliton [2 tanh(t)] after a propagation distance of 12 for τd = 0.01 (solid). The dotted curve is the pulse if τd = 0, i.e., without ISRS.
\begin{figure} \end{figure}
Figure: (a) The shape of an adiabatically amplified fundamental dark soliton (solid). Γ = 0.05, z = 16, and τd = 0.01. The dotted curve corresponds to the pulse shape without ISRS; (b) The trace of the soliton (solid) for the case of (a). The dotted curve is a fit as described by Eq. (11) in the text.
\begin{figure} \end{figure}
Figure: Even dark pulses when the input pulse (dashed curve) is κ0| tanh(t)|. (a) κ0 = 1.56, and z = 8 (solid curve), (b) κ0 = 4 and z = 3.75 (solid curve). In (c), three different input pulses are assumed: 8| tanh(t)| (solid curve), 8[1-exp(-t2/τg2)]1/2 (dotted curve), and 8[1 - sech(t/τs)] (dashed curve). The propagation distance is z = 8.
\begin{figure} \end{figure}
Figure: Even dark pulses generated from MZI. The pulse after MZI is 2 cos (π/2sech2t) (dashed curve) and the shape of secondary dark solitons is shown by the solid curve for z = 4.
\begin{figure} \end{figure}
Figure: The loss compensated even dark pulses. The input pulse is 2 cos (π/2sech2t) (dotted curve), the secondary solitons with fiber losses compensated by stimulated Raman scattering is shown by the solid curve. For comparison, the pulse shape without fiber losses is shown by the dashed curve (same as Fig. 8). The propagation distance is 4.
\begin{figure} \end{figure}


Table: Amplitudes of Secondary Even Dark Pulses
    Input Pulse Shape      
ΔnValues κ0|tanht| κ0[1 - exp(- t2/τg2)]1/2 κ0[1 - sech(t/τs)] Avg. Range
Δ1 0.34 0.30 0.21 0.28 ±25%
Δ2 1.56 1.41 1.26 1.41 ±11%
Δ3 2.47 2.26 2.28 2.34 ±6%
Δ4 3.52 3.25 3.31 3.36 ±6%
Δ5 4.45 4.26 4.42 4.38 ±6%
Δ6 5.52 5.35 5.50 5.50 ±5%