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:

iu_z - 1/2u_tt + | 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: $\begin{mathletters} \begin{eqnarray} t' &=& t e^{ \Gamma z }, \\ z' &=& { e^{2 ... ...1 \over 2 \Gamma }, \\ u &=& v e^{ \Gamma z } . \end{eqnarray}\end{mathletters}$ Under this transformation, the NLSE has the new form

iv_z' - $\displaystyle \slantfrac$ 12v_t't' - | v|²v

- $\displaystyle {\Gamma t' \over 2 \Gamma z' + 1}$ v_t'.

(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 $\displaystyle \left(\vphantom{ i{e^{2\Gamma z}-1 \over 2\Gamma}}\right.$ i $\displaystyle {e^{2\Gamma z}-1 \over 2\Gamma}$ $\displaystyle \left.\vphantom{ i{e^{2\Gamma z}-1 \over 2\Gamma}}\right)$ e^Γz tanh(te^Γz),	(4)
Γ	=	g(e^-2Γ_pz + e^-2Γ_p(L-z)) - Γ_s,	(5)
g	=	$\displaystyle {\Gamma_p(\Gamma_s + \beta)L \over {\rm sinh}(\Gamma_pL)}$ e^Γ_pL,	(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 Γ_pL 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.