Criteria for the experimental observation of multidimensional optical solitons in saturable media

Analytic methods to find beating transitions of asymmetric Gaussian beams in GNLS equations

In a simple model of propagation of asymmetric Gaussian beams in nonlinear waveguides, described by a reduction to ordinary differential equations of generalized nonlinear Schrödinger equations with cubic-quintic (CQ) and saturable (SAT) nonlinearities and a graded-index profile, the beam widths exhibit two different types of beating behavior, with transitions between them. We present an analytic model to explain these phenomena, which originate in a 1:1 resonance in a 2 degree-of-freedom Hamiltonian system. We show how small oscillations near a fixed point close to 1:1 resonance in such a system can be approximated using an integrable Hamiltonian and, ultimately, a single first order differential equation. In particular, the beating transitions can be located from coincidences of roots of a pair of quadratic equations, with coefficients determined (in a highly complex manner) by the internal parameters and initial conditions of the original system. The results of the analytic model agree with the numerics of the original system over large parameter ranges, and allow new predictions that can be verified directly. In the CQ case, we identify a band of beam energies for which there is only a single beating transition (as opposed to 0 or 2) as the eccentricity is increased. In the SAT case, we explain the sudden (dis)appearance of beating transitions for certain values of the other parameters as the grade-index is changed.

Light bullets in coupled nonlinear Schrödinger equations with variable coefficients and a trapping potential

We analyze three-dimensional (3D) vector solitary waves in a system of coupled nonlinear Schrödinger equations with spatially modulated diffraction and nonlinearity, under action of a composite self-consistent trapping potential. Exact vector solitary waves, or light bullets (LBs), are found using the self-similarity method. The stability of vortex 3D LB pairs is examined by direct numerical simulations; the results show that only low-order vortex soliton pairs with the mode parameter values n ≤ 1, l ≤ 1 and m = 0 can be supported by the spatially modulated interaction in the composite trap. Higher-order LBs are found unstable over prolonged distances.

Staggered and moving localized modes in dynamical lattices with the cubic-quintic nonlinearity

Results of a comprehensive dynamical analysis are reported for several fundamental species of bright solitons in the one-dimensional lattice modeled by the discrete nonlinear Schrödinger equation with the cubic-quintic nonlinearity. Staggered solitons, which were not previously considered in this model, are studied numerically, through the computation of the eigenvalue spectrum for modes of small perturbations, and analytically, by means of the variational approximation. The numerical results confirm the analytical predictions. The mobility of discrete solitons is studied by means of direct simulations, and semianalytically, in the framework of the Peierls-Nabarro barrier, which is introduced in terms of two different concepts, free energy and mapping analysis. It is found that persistently moving localized modes may only be of the unstaggered type.

Cubic-quintic solitons in the checkerboard potential

We introduce a two-dimensional (2D) model which combines a checkerboard potential, alias the Kronig-Penney (KP) lattice, with the self-focusing cubic and self-defocusing quintic nonlinear terms. The beam-splitting mechanism and soliton multistability are explored in this setting, following the recently considered 1D version of the model. Families of single- and multi-peak solitons (in particular, five- and nine-peak species naturally emerge in the 2D setting) are found in the semi-infinite gap, with both branches of bistable families being robust against perturbations. For single-peak solitons, the variational approximation (VA) is developed, providing for a qualitatively correct description of the transition from monostability to the bistability. 2D solitons found in finite band gaps are unstable. Also constructed are two different species of stable vortex solitons, arranged as four-peak patterns ("oblique" and "straight" ones). Unlike them, compact "crater-shaped" vortices are unstable, transforming themselves into randomly walking fundamental beams.

Spatiotemporal optical solitons in nonlinear dissipative media: from stationary light bullets to pulsating complexes

Nonlinear dissipative systems display the full (3+1)D spatiotemporal dynamics of stable optical solitons. We review recent results that were obtained within the complex cubic-quintic Ginzburg-Landau equation model. Numerical simulations reveal the existence of stationary bell-shaped (3+1)D solitons for both anomalous and normal chromatic dispersion regimes, as well as the formation of double soliton complexes. We provide additional insight concerning the possible dynamics of these soliton complexes, consider collision cases between two solitons, and discuss the ways nonstationary evolution can lead to optical pattern formation.

Near-field imaging of nonlinear pulse propagation in planar silica waveguides

A simplified near-field scanning optical microscope is employed to image the propagation of short laser pulses in planar silica waveguides, in the anomalous dispersion regime, under varying conditions of input beam power and width. Our results show a complex evolution of the transverse intensity profiles of the beam when there is a pronounced difference between the input diffraction and dispersion lengths. Numerical simulations confirm that these complex spatial dynamics are intimately related to the temporal and spectral evolution of the pulse.

Light bullets and dynamic pattern formation in nonlinear dissipative systems

In the search for suitable new media for the propagation of (3+1) D optical light bullets, we show that nonlinear dissipation provides interesting possibilities. Using the complex cubic-quintic Ginzburg-Landau equation model with localized initial conditions, we are able to observe stable light bullet propagation or higher-order transverse pattern formation. The type of evolution depends on the model parameters.

Near- and far-field evolution of laser pulse filaments in Kerr media

Measurements of the spatio-temporal and far-field profiles of ultrashort laser pulses experiencing conical emission, continuum generation, and beam filamentation in a Kerr medium outline the spontaneous formation of wave packets with X -like features, thus supporting recent numerical results [M. Kołesik, E. Wright, and J. Moloney, Phys. Rev. Lett. 92, 253901 (2004)]. Numerical simulations show good agreement with experimental data.

Quasistable two-dimensional solitons with hidden and explicit vorticity in a medium with competing nonlinearities

We consider basic types of two-dimensional (2D) vortex solitons in a three-wave model combining quadratic chi((2)) and self-defocusing cubic chi((3))(-) nonlinearities. The system involves two fundamental-frequency (FF) waves with orthogonal polarizations and a single second-harmonic (SH) one. The model makes it possible to introduce a 2D soliton, with hidden vorticity (HV). Its vorticities in the two FF components are S(1,2) = +/-1 , whereas the SH carries no vorticity, S(3) = 0 . We also consider an ordinary compound vortex, with 2S(1) = 2S(2) = S(3) = 2 . Without the chi((3))(-) terms, the HV soliton and the ordinary vortex are moderately unstable. Within the propagation distance z approximately 15 diffraction lengths, Z(diffr), the former one turns itself into a usual zero-vorticity (ZV) soliton, while the latter splits into three ZV solitons (the splinters form a necklace pattern, with its own intrinsic dynamics). To gain analytical insight into the azimuthal instability of the HV solitons, we also consider its one-dimensional counterpart, viz., the modulational instability (MI) of a one-dimensional CW (continuous-wave) state with "hidden momentum," i.e., opposite wave numbers in its two components, concluding that such wave numbers may partly suppress the MI. As concerns analytical results, we also find exact solutions for spreading localized vortices in the 2D linear model; in terms of quantum mechanics, these are coherent states with angular momentum (we need these solutions to accurately define the diffraction length of the true solitons). The addition of the chi((3))(-) interaction strongly stabilizes both the HV solitons and the ordinary vortices, helping them to persist over z up to 50 Z(diffr). In terms of the possible experiment, they are completely stable objects. After very long propagation, the HV soliton splits into two ZV solitons, while the vortex with S(3) = 2S(1,2) = 2 splits into a set of three or four ZV solitons.

Finite-band solitons in the Kronig-Penney model with the cubic-quintic nonlinearity

We present a model combining a periodic array of rectangular potential wells [the Kronig-Penney (KP) potential] and the cubic-quintic (CQ) nonlinearity. A plethora of soliton states is found in the system: fundamental single-humped solitons, symmetric and antisymmetric double-humped ones, three-peak solitons with and without the phase shift pi between the peaks, etc. If the potential profile is shallow, the solitons belong to the semi-infinite gap beneath the band structure of the linear KP model, while finite gaps between the Bloch bands remain empty. However, in contrast with the situation known in the model combining a periodic potential and the self-focusing Kerr nonlinearity, the solitons fill only a finite zone near the top of the semi-infinite gap, which is a consequence of the saturable character of the CQ nonlinearity. If the potential structure is much deeper, then fundamental and double (both symmetric and antisymmetric) solitons with a flat-top shape are found in the finite gaps. Computation of stability eigenvalues for small perturbations and direct simulations show that all the solitons are stable. In the shallow KP potential, the soliton characteristics, in the form of the integral power Q (or width w) versus the propagation constant k, reveal strong bistability, with two and, sometimes, four different solutions found for a given k (the bistability disappears with the increase of the depth of the potential). Disobeying the Vakhitov-Kolokolov criterion, the solution branches with both dQ/dk > 0 and dQ/dk < 0 are stable. The curve Q(k) corresponding to each particular type of the solution (with a given number of local peaks and definite symmetry) ends at a finite maximum value of Q (breathers are found past the end points). The increase of the integral power gives rise to additional peaks in the soliton's shape, each corresponding to a subpulse trapped in a local channel of the KP structure (a beam-splitting property). It is plausible that these features are shared by other models combining saturable nonlinearity and a periodic substrate.