Stable vortex solitons supported by competing quadratic and cubic nonlinearities

Semidiscrete optical vortex droplets in quasi-phase-matched photonic crystals

What we believe is a new scheme for producing semidiscrete self-trapped vortices ("swirling photon droplets") in photonic crystals with competing quadratic (χ(2)) and self-defocusing cubic (χ(3)) nonlinearities is proposed. The photonic crystal is designed with a striped structure, in the form of spatially periodic modulation of the χ(2) susceptibility, which is imposed by the quasi-phase-matching technique. Unlike previous realizations of semidiscrete optical modes in composite media, built as combinations of continuous and arrayed discrete waveguides, the semidiscrete vortex "droplets" are produced here in the fully continuous medium. This work reveals that the system supports two types of semidiscrete vortex droplets, viz., onsite- and intersite-centered ones, which feature, respectively, odd and even numbers of stripes, N. Stability areas for the states with different values of N are identified in the system's parameter space. Some stability areas overlap with each other, giving rise to the multistability of states with different N. The coexisting states are mutually degenerate, featuring equal values of the Hamiltonian and propagation constant. An experimental scheme to realize the droplets is outlined, suggesting new possibilities for the long-distance transmission of nontrivial vortex beams in nonlinear media.

Vortex Solitons in Quasi-Phase-Matched Photonic Crystals

We report solutions for stable compound solitons in a three-dimensional quasi-phase-matched photonic crystal with the quadratic (χ^{(2)}) nonlinearity. The photonic crystal is introduced with a checkerboard structure, which can be realized by means of the available technology. The solitons are built as four-peak vortex modes of two types, rhombuses and squares (intersite- and onsite-centered self-trapped states, respectively). Their stability areas are identified in the system's parametric space (rhombuses occupy an essentially broader stability domain), while all bright vortex solitons are subject to strong azimuthal instability in uniform χ^{(2)} media. Possibilities for experimental realization of the solitons are outlined.

Mode-locking in quadratically nonlinear waveguide arrays

A two-dimensional theoretical model is constructed to describe optical mode-locking (ML) in quadratically nonlinear waveguide arrays (QWGAs). Steady-state solutions of the considered model are obtained by a modified pseudo-spectral renormalization algorithm, and the mode-locking dynamics of the model are investigated through direct simulation of the nonlinear evolution and a linear stability analysis of the solutions. It is shown that stable mode-locking of elliptic steady-state solutions in quadratically nonlinear waveguide arrays are possible for a wide range of parameters, suggesting that quadratically nonlinear materials are well suited for producing stable mode-locked states for a wide range of applications.

Spatiotemporal mode locking in quadratic nonlinear media

A theoretical model is developed to characterize spatiotemporal mode locking (ML) in quadratic nonlinear media. The model is based on the two-dimensional nonlinear Schrödinger equation with coupling to a mean term (NLSM) and constructed as an extension of the master mode-locking model. It is numerically demonstrated that there exists steady-state soliton solutions of the ML-NLSM model that are astigmatic in nature. A full stability analysis and bifurcation study is performed for the ML-NLSM model, and it is manifest that spatiotemporal ML of the astigmatic steady-state solutions is possible in quadratic nonlinear media.

Collapse and stability of necklace beams in Kerr media

We investigate the spatial dynamics of optical necklace beams in Kerr media. For powers corresponding to less than the critical power for self-focusing per bead, we experimentally confirm the confinement of these necklace beams as proposed in [Phys. Rev. Lett. 81, 4851 (1998)10.1103/PhysRevLett.81.4851]. At higher powers, we observe a transition from collective necklace behavior to one in which the beads of the necklace collapse independently. We observe that, below the transition power, the perturbed necklace still behaves in a collective manner with coupling between individual beads but that, at higher powers, it undergoes a similar transition to a decoupled state of the necklace.

Stable solitons of even and odd parities supported by competing nonlocal nonlinearities

We introduce a one-dimensional phenomenological model of a nonlocal medium featuring focusing cubic and defocusing quintic nonlocal optical nonlinearities. By means of numerical methods, we find families of solitons of two types, even-parity (fundamental) and dipole-mode (odd-parity) ones. Stability of the solitons is explored by means of computation of eigenvalues associated with modes of small perturbations, and tested in direct simulations. We find that the stability of the fundamental solitons strictly follows the Vakhitov-Kolokolov criterion, whereas the dipole solitons can be destabilized through a Hamiltonian-Hopf bifurcation. The solitons of both types may be stable in the nonlocal model with only quintic self-attractive nonlinearity, in contrast with the instability of all solitons in the local version of the quintic model.

Stable vortex solitons in nonlocal self-focusing nonlinear media

We reveal that spatially localized vortex solitons become stable in self-focusing nonlinear media when the vortex symmetry-breaking azimuthal instability is eliminated by a nonlocal nonlinear response. We study the main properties of different types of vortex beams and discuss the physical mechanism of the vortex stabilization in spatially nonlocal nonlinear media.

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.

Parametric light bullets supported by quasi-phase-matched quadratically nonlinear crystals

We present a comprehensive analysis of the dynamics of three-dimensional spatiotemporal nonspinning and spinning solitons in quasi-phased-matched (QPM) gratings. By employing an averaging approach based on perturbation theory, we show that the soliton's stability is strongly affected by the QPM-induced third-order nonlinearity (which is always of a mixed type, with opposite signs in front of the corresponding self-phase and cross-phase modulation terms). We study the dependence of the stability of the spatiotemporal soliton (STS) on its energy, spin, the wave-vector mismatch between the fundamental and second harmonics, and the relative strength of the intrinsic quadratic and QPM-induced cubic nonlinearities. In particular, all the spinning solitons are unstable against fragmentation, while zero-spin STS's have their stability regions on the system's parameter space.

Two-dimensional solitons with hidden and explicit vorticity in bimodal cubic-quintic media

We demonstrate that two-dimensional two-component bright solitons of an annular shape, carrying vorticities (m,+/-m) in the components, may be stable in media with the cubic-quintic nonlinearity, including the hidden-vorticity (HV) solitons of the type (m,-m) , whose net vorticity is zero. Stability regions for the vortices of both (m,+/-m) types are identified for m=1 , 2, and 3, by dint of the calculation of stability eigenvalues, and in direct simulations. In addition to the well-known symmetry-breaking (external) instability, which splits the ring soliton into a set of fragments flying away in tangential directions, we report two new scenarios of the development of weak instabilities specific to the HV solitons. One features charge flipping, with the two components exchanging angular momentum and periodically reversing the sign of their spins. The composite soliton does not directly split in this case; therefore, we identify such instability as an intrinsic one. Eventually, the soliton splits, as weak radiation loss drives it across the border of the ordinary strong (external) instability. Another scenario proceeds through separation of the vortex cores in the two components, each individual core moving toward the outer edge of the annular soliton. After expulsion of the cores, there remains a zero-vorticity breather with persistent internal vibrations.

Multicolor vortex solitons in two-dimensional photonic lattices

We report on the existence and stability of multicolor lattice vortex solitons constituted by coupled fundamental frequency and second-harmonic waves in optical lattices in quadratic nonlinear media. It is shown that the solitons are stable almost in the entire domain of their existence, and that the instability domain decreases with the increase of the lattice depth. We also show the generation of the solitons, and the feasibility of the concept of lattice soliton algebra.

Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation

Vortex solitons with the topological charge S=3 , and "quasivortex" (multipole) solitons, which exist instead of the vortices with S=2 and 4, are constructed on a square lattice in the discrete nonlinear Schrödinger equation (true vortices with S=2 were known before, but they are unstable). For each type of solitary wave, its stability interval is found, in terms of the intersite coupling constant. The interval shrinks with increase of S . At couplings above a critical value, oscillatory instabilities set in, resulting in breakup of the vortex or quasivortex into lattice solitons with a lower vorticity. Such localized states may be observed in optical guiding structures, and in Bose-Einstein condensates loaded into optical lattices.