INTRODUCTION

Nonlinear optical pulses can propagate in dispersive fibers in the form of bright and dark solitons under certain conditions, as first described by Zakharov and Shabat in 1972[#!ZA!#] and in 1973,[#!ZB!#] respectively. They are stationary solutions of the initial boundary value problem of the nonlinear Schr $\ddot{o}$ dinger equation (NLSE).[#!SA!#] In the anomalous dispersion regime of the fiber, under the boundary condition u(z, t = ±∞) = 0, there exists a class of particle-like, stationary solutions called bright solitons.[#!HA!#] In the normal dispersion region, under the

boundary condition | u(z, t = ±∞)| =constant, one can obtain another class of stationary solutions, which are called dark solitons, since a dip occurs at the center of the pulse.[#!HB!#] Ever since the pioneering work by Zakharov and Shabat[#!ZA!#,#!ZB!#] and Hasegawa and Tappert,[#!HA!#,#!HB!#] optical solitons have been an active topic of research. This is particularly true since advances in experimental techniques for generating ultrashort pulses in the picosecond regime have made it possible to observe soliton effects in single-mode optical fibers. The bright soliton was first successfully observed in single-mode optical fibers by Mollenauer et al. in 1980,[#!MA!#] and the dark soliton was first observed by Emplit et al. in 1987.[#!EA!#]

The characteristics of bright solitons have been studied extensively during the past decade.[#!AA!#,#!MB!#] It was found[#!SA!#] that bright solitons are periodic and highly stable against small perturbations, such as fiber loss, background noise, and amplitude variations.[#!HA!#,#!HB!#] Ideally, when fiber loss is neglected, the fundamental bright soliton can propagate inside an anomalously dispersive fiber over an infinitely long distance without changing its pulse shape. This can occur because, for a fundamental soliton, the effect of dispersion on the pulse is exactly balanced by that of the nonlinear refractive index of the fiber, i.e., the self-phase modulation. Solitons can also survive collisions between them. The interaction force between two neighboring solitons is periodic and decreases exponentially with their separation.[#!GA!#] Another characteristic of bright solitons is that they can be adiabatically amplified under certain conditions when gain is introduced into the fiber, e.g., through Raman amplification.[#!BA!#] The effect of fiber loss on the pulse can thus be compensated for by injecting a cw laser beam at a shorter wavelength into the fiber, whereby stimulated Raman scattering transfers its energy to the soliton.[#!HC!#] Therefore solitons are candidates for information carriers for future optical communications. Much research has been done in this area.[#!DA!#] The possibility of stable, repeaterless, all-optical soliton transmission at a 10–GHz rate across almost 5000 km has been numerically demonstrated[#!HD!#,#!MC!#] and experimentally realized with a rate-length product of approximately 11,000 GHz km.[#!MD!#] More recently, with erbium-doped fiber amplifiers, soliton transmission of 9,000 km at 4 Gbits/s has been realized.[#!ME!#]

Because a dark pulse (with a dip of pulse intensity under constant background),[#!EA!#,#!KA!#,#!WA!#] especially the so called odd dark pulse (for which the electric field changes sign at the center of the pulse), cannot be easily generated, dark solitons have been studied less than their counterparts, bright solitons. However, as a result of recent developments in techniques for synthesizing short optical pulses with almost arbitrary shapes and phases,[#!WB!#] it is possible to observe soliton like propagation of individual dark pulses in single-mode fibers. Because these fibers exhibit normal dispersion over a large spectral region, extending from UV to IR ( λ < 1.3μm), many cw and pulsed laser sources can be used to generate dark solitons. As a result, dark solitons have attracted increasing attention. ...

In the following discussions, we adopt the normalization convention used in Agrawal's book.[#!AB!#] We normalize the field amplitude A (optical power P₀ = A²) into u by

u = $\displaystyle \left(\vphantom{ { 2 \pi n_2 {\tau_0}^2 }\over { \lambda A_{\rm eff} \vert \beta_2 \vert } }\right.$ 2πn₂τ₀² $\displaystyle \over$ λA_eff| β₂| $\displaystyle \left.\vphantom{ { 2 \pi n_2 {\tau_0}^2 }\over { \lambda A_{\rm eff} \vert \beta_2 \vert } }\right)^{{1/2}}_{}$ A,

where A_eff is the effective area of the propagating mode, n₂ = 3.2×10^-16cm²/W is the nonlinear optical Kerr coefficient of the silica fiber, and β₂ is a parameter describing the group velocity dispersion of fiber, defined as the second-order derivative of the propagation constant with respect to the radiant frequency evaluated at the signal frequency. The time variable t is normalized by a characteristic time constant τ₀ (e.g., τ_FWHM = 1.76τ₀ for hyperbolic secant pulses), and the spatial variable z is normalized by the so-called dispersion length,

L_D

τ₀² $\displaystyle \over$ β₂.

As an example, at wavelength λ = 1.06 μm with A_eff = 40 μm², for a pulse with τ₀ = 1 ps, the normalized distance z = 1 corresponds to a real fiber length of L_D = 60 m, and u = 1 represents an optical power of P₀ = 3.5 W. However, when τ₀ = 0.1 ps, L_D = 60 cm, and P₀ = 350 W.