Comparison of Solutions of the General Nonlinear Amplitude Equation and a Modified Schrӧdinger Equation

We review the dynamics of narrow and broad-band optical pulses in nonlinear dispersive media. A major problem that arises during the development of theoretical models, which describe accurately and correctly the behavior of these pulses, is the limited application of the nonlinear Schr?dinger equation. It describes very well the evolution of nanosecond and picosecond laser pulses. However, when we investigate the propagation of femtosecond and attosecond light pulses, it is necessary to use the more general nonlinear amplitude equation. We show that in this equation two additional terms are included and they have a significant impact on the phase of the pulse. We perform numerical simulations and show the temporal shift of the position of fundamental solitons. This effect depends on the initial duration of the laser pulses. To clarify the influence of the additional terms on the parameters of the optical pulses, we consider the nonlinear amplitude equation, which is a modified nonlinear Schr?dinger equation.     

Optical pulse compression of ultrashort laser pulses in an argon-filled planar waveguide

We investigate the possibility of optical pulse compression of high energy ultrashort laser pulses in an argon-filled planar waveguide, based on two level coupled mode theory and the full 3D nonlinear Schr?dinger equation. We derive general expressions for controlling the spatial beam profile and the extent of the spectral broadening. The analysis and simulations suggest that the proposed method should be appropriate for optical pulse compression of ultrashort laser pulses with energies as high as 600 mJ.     

Effect of a pulsed magnetic field on the nonlinear dynamics of vortexlike domain walls in magnetic films

The Landau-Lifshitz equation is numerically solved to study the nonlinear dynamic behavior of domain walls with the 2D vortexlike magnetization distribution in magnetically uniaxial films that have in-plane anisotropy and are exposed to a pulsed magnetic field. It is shown that a pulsed magnetic field H _p may induce transitions between various steady wall motions that differ in magnetization distribution. Solitary rectangular pulses, as well as a regular train of rectangular pulses, may be used to control the period of nonlinear dynamic transformations of the wall internal structure and the related period of variation of the wall velocity.     

Generation and weak beam control of two-dimensional multicolored arrays in a quadratic nonlinear medium

We report on the first experimental observation of 2D multicolored transverse arrays in a quadratic nonlinear medium under the pump of two crossly overlapped femtosecond beams. The 2D reproducible patterns are caused by cascaded noncollinear quadratic nonlinear couplings between the input pulses and quadratic spatial solitons originated from spatial breakup of one of the input beams with spatial ellipticity. A seed supercontinuum pulse is then diffracted and amplified with phase preservation, resulting in the formation of up-converted multicolor 2D transverse arrays. By seeding with weak second harmonic pulses, the 2D multicolored transverse patterns are suppressed due to weak beam control of the induced quadratic spatial solitons.     

Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory

We analyze interaction of nonlinear pulses in active–dispersive–dissipative nonlinear media. A particular example of such media is a viscous thin film coating a vertical fibre. Experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place. In such events, larger pulses catch up with smaller ones and annihilate them. However, for certain flow conditions and after a certain distance from the inlet, no more coalescence is observed and the flow is described by quasi-equilibrium solitary pulses interacting continuously with each other through attractions and repulsions, and, eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole. This experimental study represents the first evidence of formation of bound states in low-Reynolds-number interfacial hydrodynamics. To gain theoretical insight into the interaction of the pulses and formation of bound states, we derive a weakly nonlinear model for the flow, the generalized Kuramoto–Sivashinsky (gKS) equation, that retains the fundamental mechanisms of the wave evolution, namely, dominant nonlinearity, instability, stability and dispersion. Much like in the experiments, the spatio-temporal evolution of the gKS equation is dominated by quasi-stationary solitary pulses which continuously interact with each other through coalescence events or attractions/repulsions. To understand the latter case, we utilize a weak-interaction theory for the solitary pulses of the gKS equation. The theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses, or, alternatively, the evolution of the separation distances. By analyzing the fixed points of this system, we obtain bound states of interacting pulses. For two pulses, we provide a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The interaction theory and resulting bound states are corroborated by computations of the full equation. We also find qualitative agreement between the theory and the experiments.     

Ultrahigh-bandwidth, on-chip all-optical pulse erasure using the χ(3) process in a nonlinear chalcogenide waveguide

We demonstrate on-chip all-optical pulse erasure based on four-wave mixing and cross-phase modulation in a dispersion engineered chalcogenide (As(2)S(3)) rib waveguide. We achieve an erasure efficiency of ~15 dB for picosecond pulses in good agreement with numerical simulations using the nonlinear Schr?dinger equation. The combined effect of the high instantaneous optical nonlinearity (γ = 9900 (W km)(-1)) and small group-velocity dispersion (D = 29 ps/nm km), which reduces pulse walk-off, will enable all-optical pulse erasure for ultrafast signal processing.     

Comparisons between sine-Gordon and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse

The sine-Gordon (SG) equation and perturbed nonlinear Schrödinger (NLS) equations are studied numerically for modeling the propagation of two space dimensional (2D) localized pulses (the so-called light bullets) in nonlinear dispersive optical media. We begin with the (2 + 1) SG equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed NLS equation in 2D for SG pulse envelopes, which is globally well posed and has all the relevant higher order terms to regularize the collapse of standard critical (cubic focusing) NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles. Efficient semi-implicit sine pseudospectral discretizations for SG and perturbed NLS are proposed with rigorous error estimates. Numerical comparison results between light bullet solutions of SG and perturbed NLS as well as critical NLS are reported, which validate that the solution of the perturbed NLS as well as its finite-term truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of cubic focusing NLS. In contrast, standard critical NLS is in qualitative agreement with SG only before its collapse. As a benefit of such observations, pulse propagations are studied via solving the perturbed NLS truncated by reasonably many nonlinear terms, which is a much cheaper task than solving SG equation directly.     

Some features of one-soliton solutions of the Korteweg - de Vries equation

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.     

An improved nonlinear optical pulse propagation equation

A new equation for self-focusing of extremely focused short-duration intense pulses is derived using a method that treats diffraction and dispersion to all orders with nonlinearity present, and self-consistently determines the nonlinear derivative terms present in the propagation equation. It generalizes both the previous formulation of linear optical pulse propagation to the nonlinear regime, and the cw nonlinear regime propagation to the pulsed regime by including temporal characteristics of the pulse. We apply the new equation and propagate a tightly focused picosecond pulse in silica and explicitly show the effects of spatial-derivative nonlinear coupling terms.     

Propagation of femtosecond pulses in large-mode-area, higher-order-mode fiber

We demonstrate propagation of 14 nJ femtosecond pulses through a large-mode-area, higher-order-mode (HOM) fiber with an effective area of 2100 microm2. The pulses propagate stably in the LP07 mode of the fiber through lengths as long as 12 m. The strongly chirped pulses exiting the amplifier fiber are dechirped by the high-order-mode fiber, resulting in pulses with a peak power of 61 kW after propagation in 5 m of the positive-dispersion fiber. A small amount of self-phase modulation is observed in the compressed pulses and is described well by a nonlinear Schr?dinger equation model that takes into account the measured effective area and dispersion of the HOM fiber.     

Stability criterion for dissipative soliton solutions of the one-, two-, and three-dimensional complex cubic-quintic Ginzburg-Landau equations

The generation and nonlinear dynamics of multidimensional optical dissipative solitonic pulses are examined. The variational method is extended to complex dissipative systems, in order to obtain steady state solutions of the (D + 1)-dimensional complex cubic-quintic Ginzburg-Landau equation (D = 1, 2, 3). A stability criterion is established fixing a domain of dissipative parameters for stable steady state solutions. Following numerical simulations, evolution of any input pulse from this domain leads to stable dissipative solitons.     

Circularly polarized few-optical-cycle solitons in Kerr media: A complex modified Korteweg-de Vries model

We consider the propagation of few-cycle pulses (FCPs) in cubic nonlinear media exhibiting a “crystal-like” structure, beyond the slowly varying envelope approximation, taking into account the wave polarization. By using the reductive perturbation method we derive from the Maxwell–Bloch–Heisenberg equations, in the long-wave-approximation regime, a non-integrable complex modified Korteweg-de Vries equation describing the propagation of circularly polarized (CP) FCPs. By direct numerical simulations of the governing nonlinear partial differential equation we get robust CP FCPs and we show that the unstable ones decays into linearly polarized half-cycle pulses, whose polarization direction slowly rotates around the propagation axis.     

Propagation of optical pulses at an absorption line in the presence of a weak nonlinearity

We examine the propagation of short pulses of light in a resonantly absorbing, weakly nonlinear medium within the limits of a model described by the nonlinear Schrödinger equation. The possibility of transforming pulses of various forms into a soliton signal due to the effects of self-interaction is studied. On the basis of the study of spectra for the associated linear problem, we investigate the break-up of an initial pulse into solitons. We have obtained solutions for two particular cases of the initial pulse.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 36–39, February, 1989.     

Kuznetsov–Ma soliton and Akhmediev breather of higher-order nonlinear Schrdinger equation

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrdinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability(MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov–Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrdinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.     

Snake instability of a spatiotemporal bright soliton stripe

We study experimentally the snake instability of the bright soliton stripe of the (2+1)-dimensional hyperbolic nonlinear Schr?dinger equation. The instability is observed, through spectral measurements, on spatially extended femtosecond pulses propagating in a normally dispersive self-defocusing semiconductor planar waveguide.     

Dynamic Alignment of D2 Enhanced by Two Few-cycle Pulses

利用数值方法求解D2分子刚性转子模型在超短飞秒激光脉冲作用下的薛定谔方程,计算了双原子分子D2在两束固定强度的飞秒激光脉冲作用下,延迟时间对于分子取向的影响.结果表明,只要选取合适的延迟时间,就能很好改善分子取向;保持两束激光强度不变,通过调整两束脉冲的宽度,选择合适的延迟时间能够进一步有效提高D2分子的取向程度.     

Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with S=2) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either S=0 and S=1, or S=1 and S=2.     

Optical filaments and optical bullets in dispersive nonlinear media

We have investigated the evolution of picosecond and femtosecond optical pulses governed by the amplitude vector equation in the optical and UV domains. We have written this equation in different coordinate frames, namely, in the laboratory frame, the Galilean frame, and the moving-in-time frame and have normalized it for the cases of different and equal transverse and longitudinal sizes of optical pulses or modulated optical waves. For optical pulses with a small transverse size and a large longitudinal size (optical filaments), we obtain the well-known paraxial approximation in all the coordinate frames, while for optical pulses with relatively equal transverse and longitudinal sizes (so-called light bullets), we obtain new non-paraxial nonlinear amplitude equations. In the case of optical fields with low intensity, we have reduced the nonlinear amplitude vector equations governing the light-bullet evolution to the linear amplitude equations. We have solved the linear equations using the method of Fourier transform. An unexpected new result is the relative stability of light bullets and the significant decrease in the diffraction enlargement of light bullets with respect to the case of long pulses in the linear propagation regime.