PROPAGATION AND AMPLIFICATION
As discussed in Section
, when smaller values of δ are
used, pulses of better characteristics are obtained. This can be seen in
Fig. 1(d), where
a = 1.33 and a pure fundamental dark soliton is
generated. .... ...In what follows, we will examine the
possibility of amplification and compression of dark solitons with a
constant gain and show that the stimulated Raman scattering can be used to
amplify dark solitons as well as to compensate for the fiber loss.
We first examine the solution of a modified NLSE with a constant gain:
iuz - 1/2utt + | u|2u = iΓu, |
|
|
(2) |
where Γ is assumed to be a constant, appropriate for the Raman
amplification under strong pumping without depletion. The solution of a
similar equation to Eq. (
), but in the anomalous dispersion regime,
in which bright solitons are amplified and compressed by the gain, has been
analyzed by Blow et al.[#!BA!#] To solve the equation, we make the
following variable transformation:
Under this transformation, the NLSE has the new form
ivz' - 12vt't' - | v|2v |
= |
- vt'. |
(3) |
The solution of Eq. (
) when Γ = 0 is well known and has the
form
exp[iσ(z, t)]κtanhκt, where κ
is the form factor and the phase variable satisfies
∂σ/∂z = κ2.[#!ZA!#] Therefore, when the right-hand-side of
Eq.(
) is zero, an exact solution for v(z', t) can be obtained from
Eq. (
). On the other hand, when
z→∞ and hence
z'→∞ or
Γ→ 0, the right-hand side
of Eq. (
) becomes infinitely small. Under these conditions, we can
treat the right-hand side of Eq. (
) as a perturbation to the NLSE.
...
u(z, t) |
= |
exp i![$\displaystyle {e^{2\Gamma z}-1 \over 2\Gamma}$](img9.png) eΓz tanh(teΓz), |
(4) |
Γ |
= |
g(e-2Γpz + e-2Γp(L-z)) - Γs, |
(5) |
g |
= |
eΓpL, |
(6) |
κ(z) |
= |
κ0 exp(βz). |
(7) |
...
In summary, we have studied the propagation properties of dark solitons
under the influence of gain. The dark soliton can be amplified and
compressed adiabatically when the gain coefficient remains small, e.g.,
Γ < 0.1. As the gain increases above this value, secondary gray
solitons will be generated. Stimulated Raman scattering can be utilized to
provide the gain. When the product
ΓpL is kept small, dark
solitons can be amplified adiabatically with high quality. Such a property
of dark solitons enable us to obtain dark solitons with short durations for
the ease of observation and transmission.