Ultrashort optical solitons in transparent nonlinear media with arbitrary dispersion

We consider propagation of ultrashort optical pulses in nonlinear fibers and suggest a new theoretical framework for description of pulse dynamics and exact characterization of solitary solutions. Our approach deals with a proper complex generalization of the nonlinear Maxwell equations and completely avoids the use of the slowly varying envelope approximation. The only essential restriction is that fiber dispersion does not favor both the so-called Cherenkov radiation, as well as the resonant generation of the third harmonics, as these effects destroy ultrashort solitons. Assuming that it is not the case, we derive a continuous family of solitary solutions connecting fundamental solitons to nearly single-cycle ultrashort ones for arbitrary anomalous dispersion and cubic nonlinearity.

We showed recently [1] that a single soliton solution of the highly-nonlinear Schrödinger equation becomes multi-stable for a certain class of nonlinearities. This implies that more than one amplitude profile and propagation constant of a single soliton may exist for the same total power carried by the soliton. The multistable solitons exist only if the nonlinear component of the susceptibility as a function of intensity is either changing its sign or its derivative has a sufficiently sharp peak. This can result in such effects as bistable (or multistable, in general) self-trapping of light in nonlinear media as well as bistable propagation of soliton pulses in nonlinear optical fibers. Both of these processes may be described by the same nonlinear Schrödinger equation with a nonlinear term in the form Ef(|E|2) where f(|E|2) is an arbitrary function of the field intensity |E|2 with f(0) = 0. The so called cubic nonlinear Schrödinger equation with f(|E|2) proportional |E|2, corresponds to Kerr-nonlinearity in optical propagation and does not give rise to bistable solitons). By evaluating the propagation constant of a solitary solution δ as a function of its total power P for each given f(|E|)2), we found a class of nonlinear functions f(I) for which the dependence δ(P) becomes multivalued. We conjectured earlier [1] that stable (unstable) solitary solutions are those with dδ/dP > 0 (dδ/dP < 0). The first computer results by Enns and Rangnekar [2] supported this conjecture for a certain non-Kerr nonlinearity. In our further collaborative research [3], we discovered that depending on the nonlinear function f(I), the stable solitary solutions (dδ/dP > 0) fall into two classes: “weak” solitons (stable only against sufficiently small perturbation), and “robust” solitons (stable against any perturbation, in particular, against collisions with other solitons). It was shown [3] that while the condition dδ/dP > 0 is sufficient for the “weak” stability, some extra conditions imposed on the function f(I) warrant existence of “robust” solitons (one of them e.g. is f(I)/I2 = o(1) as I → ∞; stability against collapsing behavior). It was shown most recently by Enns [4] that at least for some of the nonlinear functions of (|E|2) giving rise to bistable solitons, the nonlinear Schrödinger equation passes the so called Painleve test and is therefore completely integrable. This confirms that the “robust” solitary solutions are indeed solitons. The propagation of single pulses in nonlinear fibers with an appropriate nonlinearity may provide the first known opportunity to obtain a temporal (dynamic) bistability as contrasted to the known kinds of optical bistability which are so far based on the equilibrium regimes.

We propose a simple scheme to generate high-energy ultrashort pulses by combination and compression of multiple input pulses which share the same chirp profile. First, the multiple raised-cosine pulses in the input pulse train are modulated by a phase modulator, in which each modulation cycle covers two, three, four, or five pulses. Then, the modulated pulses are launched into a nonlinear fiber with the exponentially decreasing dispersion. We find that these pulses initially coalesce into a single pulse whose pulse profile is nearly hyperbolic secant, which then undergoes self-similar compression. Thus in the proposed method, first the combination of the multiple optical pulses occurs and then self-similar compression takes over. Besides, we also report the generation of ultrashort pulses by combination and compression of multiple hyperbolic secant pulses with the same chirp. The numerical results reveal that the resulting ultrashort pulse possesses a large portion of the input pulses for both raised-cosine and hyperbolic secant pulses. However, the compression factor and energy ratio are relatively higher for the hyperbolic secant pulses when compared with the raised-cosine pulses.

Propagation of ultra-short optical pulses in cubic nonlinear media

The dynamics of an ultrashort optical pulse in order—disorder ferroelectrics in the presence of defects without limitations on the pulse power is considered in the framework of the microscopic pseudospin formalism within the unidirectional propagation approximation. The effective equations for the electric field amplitude and the ferroelectric polarization are derived and numerically solved. Quasi-soliton modes of propagation of ultrashort optical pulses are revealed, and the dependences on the microscopic Hamiltonian parameters are analyzed. The influence of defects of different types on the propagation of ultrashort pulses is established.

In this paper, the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated, which describes the propagation of ultrashort optical pulse in optical fibers. The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method. The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background. The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions. The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.

The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schrödinger equations

In this paper, we utilize the first integral method with symbolic computation to constructing periodic and solitary wave solutions of complex nonlinear evolution equations. With the aid of Maple, new exact solutions of two nonlinear evolution equations in physics, namely, generalized Zakharov equations (GZ) and higher-order nonlinear Schrodinger equation (HONLS) describing propagation of ultrashort pulses in nonlinear optical fibers are obtained.

In this paper, we propose a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. Furthermore, we construct the multi-dark-soliton solution for the defocusing complex short-pulse equation through the Darboux transformation and reciprocal (hodograph) transformation. One- and two-dark-soliton solutions are given explicitly, whose properties and dynamics are analyzed and illustrated.

The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation

Propagation of ultrashort optical pulses

The guided, single-mode propagation of ultrashort optical pulses is commonly described by a well studied and understood generalized nonlinear Schrödinger equation. Here we present and discuss an extended version for multimode optical fibers and waveguides including polarization effects, high-order dispersion, Kerr and Raman nonlinearities, self-steepening effects, as well as wavelength-dependent mode coupling and nonlinear coefficients. We then investigate the symmetry properties of the nonlinear coupling coefficients for the cases of step-index and circularly symmetric conventional fibers and for microstructured fibers with hexagonal symmetry. Finally, we study the computational complexity of the proposed algorithm.

We treat two physical problems--the propagation of ultra-short optical pulses in a resonant 5-fold degenerate medium, and the creation--and propagation of spin waves in the anisotropic magnetic liquids 3HeA and 3HeB at temperatures below the transition to the A phase at 2.6 mK. The optical pulse problem is governed by a ‘double’ sine-Gordon equation, the 3HeA problem by the sine-Gordon and the 3HeB by a double sine-Gordon with changed sign. The double sine-Gordons are not integrable: we study them analytically and numerically and predict a “wobbling” 4π pulse in the optical case, a result supported by experiments. We solve certain initial value problems and show how they could be used for the creation of spin excitations in 3HeA and 3HeB. Finally we develop singular perturbation theory in terms of sine-Gordon scattering data in order to describe the 4π wobbler and a similar 0π pulse.

We examine the propagation of ultra-short optical light pulses in dispersion-managed birefringent fiber transmission systems, in which the pulse dynamics is governed by the coupled higher-order nonlinear Schrodinger equations with higher-order linear and nonlinear optical effects. We derive the equations of motion in terms of pulse parameters such as amplitude, temporal position, width, chirp, frequency and phase, using a projection operator method, and we obtain the spatial dynamical behavior of picosecond and femtosecond pulse parameters. From our detailed analysis, we show that the stimulated Raman scattering has a strong impact on the pulse dynamics.

This paper investigates the generation of ultra-short optical pulses from directly modulated multiple-quantum-well (MQW) laser diodes. The ultra-short pulse generation is simulated using the rate equations model. The effects of the input signal parameters (bias current value and RF current amplitude and frequency) on the duration and magnitude of the pulses are examined from simulations. Particular interest is paid to the influence of spontaneous emission factor on the chaotic laser behavior. The study is focused on improving the understanding of how the phenomenological laser parameters influence the nonlinear dynamics and on establishing the connections between the modulation signal parameters and the nonlinear dynamic behavior. The numerical simulations carried out for different laser structures show good agreement with previously reported results and demonstrate period bifurcation and chaos for some ranges of the bias current and modulation index.