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.
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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.
|
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% |