Quadrupole Solitons Research Articles

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier

Nonlocal soliton in non-parity-time-symmetric coupler

Parity-time (PT) symmetric is not a necessary condition for achieving a real spectrum and some studies about realizing real spectra in non-PT-symmetric systems with arbitrary gain–loss profiles have been presented recently. By tuning the free parameters in non-PT-symmetric potentials, phase transition could also be induced. Above phase transition point, discrete complex eigenvalues bifurcate out from continuous real eigenvalues in the interior of the continuous spectrum. In this work, we investgate the existence and stability of solitons in nonlocal nonlinear couplers with non-PT-symmetric complex potentials both below and above phase transition. There are several discrete eigenvalues in the linear spectra of the non-PT-symmetric system used here. With the square-operator iteration method, we find that different continuous families of solitions can bifurcate from different discrete linear eigenvalues. Moreover, linear-stability analysis collaborated with direct numerical propagation simulations demonstrates that the nonlocal solitions can be stable in a range of parameter values. we first address the cases below the phase transition. To be specific, when we fix the coupling coefficient and vary the degree of nonlocality, it’s found that fundamental solitons, dipole solitons, tripolar solitons, quadrupole solitons bifurcate from the largest,the second-largest, the third-largest and the fifth-largest discrete eigenvalue, respectively. These nonlocal solitons are all stable in the low power region. With an increase of the degree of nonlocality, the stability region shrinks for the fundamental solitons while it widens for the dipole and multiplole solitons. At the same time, the power of all the stable solitons increases with the increase of the degree of nonlocality. By varying the coupling coefficient, the arrangement of soliton families emerging in the discrete interval of the linear spectrum can be changed. For example, the dipole solitons bifurcate from the third-or fourth-largest discrete eigenvalue while the tripolar solitons bifurcate from the fifth largest discrete eigenvalue. Above phase transition,the fundamental solitons are unstable in the low and high power region but are stable in the moderate power region. The stability region shrinks with the increasing degree of nonlocality. We also find the family of dipole solitons bifurcates from the second-largest discrete eigenvalue, but all the dipole solitons are unstable. In addition, we find that the eigenvalues in linear-stability spectra of solitons emerge as conjugation pairs.

Transformation of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation by means of Lévy-index management

The structure and stability of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation with a gradually decreasing Lévy index, α, are numerically studied. It is found that the solitons adiabatically compress with the decrease of Lévy index, and new species of stable ones are produced by means of this technique. It is known that, under the action of the normal diffraction (α = 2), the nonlocal cubic self-trapping can support, at most, quadrupole solitons and vortex ones with winding number m = 2 as stable modes in the one- and two-dimensional space, respectively. In contrast to that, we find that the application of the Lévy index management (the gradual decrease of α) leads to the formation of stable five-poles and sextupoles in one-dimensional, and vortices with m = 3 in two-dimensional. Weak dissipation does not essentially affect the observed results.

Stable high-dimensional solitons in nonlocal competing cubic-quintic nonlinear media

We find and stabilize high-dimensional dipole and quadrupole solitons in nonlocal competing cubic-quintic nonlinear media. By adjusting the propagation constant, cubic, and quintic nonlinear coefficients, the stable intervals for dipole and quadrupole solitons that are parallel to the x-axis and those after rotating 45° counterclockwise around the origin of coordinate are found. For the dipole solitons and those after rotation, their stability is controlled by the propagation constant, the coefficients of cubic and quintic nonlinearity. The stability of quadrupole solitons is controlled by the propagation constant and the coefficient of cubic nonlinearity, rather than the coefficient of quintic nonlinearity, though there is a small effect of the quintic nonlinear coefficient on the stability. Our proposal may provide a way to generate and stabilize some novel high-dimensional nonlinear modes in a nonlocal system.

Periodic interconversion evolution of a double-charge discrete vortex and out-of-phase quadrupole solitons in Lieb photonic lattices with self-focusing saturable nonlinearity media

We demonstrate the periodic interconversion evolution of the double-charge vortex (DCV) and out-of-phase (OOP) quadruple solitons in two-dimensional (2D) optically induced Lieb photonic lattices by single-site excitation under appropriate self-focusing nonlinearity conditions. The numerical results show that the DCV evolves into a rotating quasi-vortex soliton, utilizing an OOP quadrupole-like beam as a transition state to reverse the topological charge and the direction of rotation periodically; The OOP quadrupole beam whose incident orientation along the crystalline c-axis or the diagonal axes of the lattices self-traps into a stable localized mode nearly invariant during propagation, whose Fourier spectrum reveals that they are localized states close to the high-symmetry points X and M of Brillouin Zone. By imposing an initial orientation slightly twisted relative to the lattices’ crystalline c-axis, the localized quadruple state tends to evolve into a dynamical rotating DCV and undergoes periodic charge-flipping when the direction of rotation is reversed. Additionally, analysis of different twisting angle shows that the oscillation period will become longer when the initial orientation approaches the lattices’ crystalline c-axis. Our findings could give some quantitative insight into the experimental feasibility of observing such phenomena.

Stability of quadrupole solitons in nonlocal media

We study quadruple-mode (quadrupole) solitons in nonlocal media with different degrees of nonlocality. The existence and stability of such quadrupole solitons is addressed systematically. The solution of quadrupole solitons is obtained analytically with variational approach. The propagation dynamics of quadrupole solitons is found numerically with the split-step method. The repulsion between the out-of-phase petals of quadrupole solitons can not be balanced by the weak nonlocality, leading to the formation of four separated scalar solitons. Completely stable quadrupole solitons can be achieved when the nonlocality is strong.

Multi-pole solitons in nonlocal nonlinear media with fourth-order diffraction

This paper studies the existence and stability of solitons in nonlocal nonlinear media (NNM) with the fourth-order diffraction, and reveals that the coupling constant of the fourth-order diffraction plays a significant role in destabilizing the propagations of the solitons. Such as in the local nonlinear media, the fundamental solitons are stable in the low coupling constant domains, but cannot exist in the high coupling constant regions. For the exponential-decay response, the maximal number of peaks in stable multipole solitons are dipole and quadrupole in NNM with and without the fourth-order diffraction, respectively. For the Gaussian-shaped response, the solitons exhibit novel structures, where the inner humps of the tripole and quadrupole solitons decrease as the β decreases. The evolutions of solitons with perturbation are also investigated to confirm their stability.

Evolution of vortex and quadrupole solitons in the complex potentials with saturable nonlinearity

We investigate the evolution of vortex and quadrupole solitons in the parity-time (PT)-symmetric optical lattices with self-focusing saturable nonlinearity. It is found that the saturable parameter plays a significant role on the existence and stability of vortex solitons. There exists a threshold for saturable parameter, above which the single-charged vortex solitons can be stable. The strength of gain–loss component of the PT-symmetric potential can affect the existence and stability of vortex solitons dramatically. We find that the single-charged vortex solitons can exist even at the phase transition point of the PT-symmetric potential, but diffuse fast during propagation. In addition, it is also found that the direction of the angular momentum has little effect on the properties of vortex solitons. Dramatically, the stable regions of out-of-phase quadrupole solitons are broader than that of the single-charged vortex solitons, while the stable regions of in-phase quadrupole solitons are narrower than that of the single-charged vortex solitons.

Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential.

We report results for solitons in models of waveguides with focusing or defocusing saturable nonlinearity and a parity-time ([Formula: see text])-symmetric complex-valued external potential of the Scarf-II type. The model applies to the nonlinear wave propagation in graded-index optical waveguides with balanced gain and loss. We find both fundamental and multipole solitons for both focusing and defocusing signs of the saturable nonlinearity in such [Formula: see text]-symmetric waveguides. The dependence of the propagation constant on the soliton's power is presented for different strengths of the nonlinearity saturation, S The stability of fundamental, dipole, tripole and quadrupole solitons is investigated by means of the linear-stability analysis and direct numerical simulations of the corresponding (1+1)-dimensional nonlinear Schrödinger-type equation. The results show that the instability of the stationary solutions can be mitigated or completely suppressed, increasing the value of SThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Multi-pole dark solitons in nonlocal and cubic-quintic nonlinear medium

In this paper, we mainly simulate the characteristics of the ground state dark soliton and the multipole dark soliton in the nonlocal and cubic-quintic nonlinear medium. Firstly, the influences of the degree of nonlocality on the amplitude and the width of the dark soliton in the self-defocusing cubic-and self-focusing quantic-nonlinear medium are studied. Secondly, we find the nonlinear parameters affecting the amplitude values of solitons, but the refractive index induced by the light beam is always a fixed value. The numerical results show that the ground state dark soliton can be propagated stably alone the z axis, and the stable states of the dipole soliton and the dark tri-pole and quadru-pole solitons are stable. However, some quadru-pole dark soliton is unstable after propagating the remote distance. Furthermore, we also discuss the characteristics of the ground state dark soliton and the dark dipole soliton in the local cubic-nonlinear and nonlocal quantic nonlinear media. Both the amplitude and the beam width of the dark ground state soliton and dark dipole soliton are also affected by the degree of nonlocality and nonlinearity. Two boundary values of the induced refractive index change with the variations of the three nonlinear parameters. The dark soliton and the dipole dark soliton are more stable in the self-focusing cubic nonlinear and the nonlocal self-defocusing quantic nonlinear medium than those in the self defocusing cubic nonlinear and nonlocal self-focusing quantic nonlinear medium. The powers of single dark soliton and dark tri-pole soliton decrease monotonically with the increase of propagation constant when the cubic-quintic nonlinearities are certain values and these degrees of nonlocalities are taken different values. Furthermore, we also analyze linear stabilities of various nonlocal spatial dark solitons. And we find that the dipole dark soliton is unstable when the propagation constant is in the region[-0.9,-1.0]. These properties of linear stabilities of other multi-pole dark solitons are the same as their propagation properties.

Propagating properties of spatial solitons in the competing nonlocal cubic-quintic nonlinear media

We study the new spatial optical solitons and their propagating properties in the one-dimensional nonlocal cubic-quintic (C-Q) nonlinear model by the numerical method. We obtain multi-bright solitons and multipole soliton solutions in the one-dimensional nonlocal C-Q nonlinear model. The propagation of bright solitons is stable in the competing nonlocal cubic self-defocusing and quintic self-focusing nonlinear media when these nonlocal and nonlinear parameters are in the appropriate value domain. Considering the different nonlinear cubic effects, the interaction between two optical solitons with the same phase in the general nonlocal media displays the attraction or the repulsion for different nonlocal and nonlinear parameters. We find that the interval of two solitons affects the interaction between them. The refractive index is changed with the propagating constant when the nonlocal constant d3 is 10. Moreover, the triplepole, quadrupole and pentapole solitons can propagate steadily when the nonlocal parameters are appropriate, but hexa-pole (or above) solitons propagate unsteadily for any nonlocal parameter. Furthermore, we investigate the multi-pole solitons and their propagation stabilities by the Newton difference method and the Fourier split step method, obtain the stable propagation conditions for dipole, triplepole and quadrupole solitons, and find that the propagation of the pentapole and higher-order pole solitons is unstable. We also discuss the interactions of multi-pole solitons when they propagate along the axis z. The interactions are attraction or repulsion when the nonlocal and the nonlinear parameters are different. Meanwhile, we simulate the evolution of the refractive index along the axis z when the spatial optical solitons are multi-pole solitons. Finally, we study the relation between the power of soliton and the propagation constant under different degree of nonlocality. The power of the single bright soliton does not monotonically increase with the increasing propagation constant when the degree of nonlocality d3 is 10. We also derive the relation between the power of dipole bright solitons with the cubic nonlinearity parameter and the propagation constant under different degree of nonlocality. The power decreases monotonically with the increasing propagation constant when the cubic nonlinearity is a certain value or with the increasing cubic nonlinearity when the propagation constant is a certain value.

四极孤子在自聚焦光诱导四方晶格中的动态演化

四极孤子在自聚焦光诱导四方晶格中的动态演化

Surface multipole solitons on photorefractive media with Bessel optical lattices

We find the existence conditions for new surface crescent, dipole, tripole, and quadrupole solitons formed at the interface of a focusing photorefractive medium and a medium imprinted with a Bessel optical lattice. We demonstrate by using numerical simulations that the crescent and the dipole solitons show oscillatory behaviors in their amplitude and shape while the tripole and the quadrupole solitons maintain a remarkable rigidity during propagation. Based on a linear stability analysis, we classify the stability region of the tripole and the quadrupole surface solitons in terms of the Bessel optical lattice strength and the Bessel index.

Dipole and quadrupole solitons in optically-induced two-dimensional defocusing photonic lattices

Dipole and quadrupole solitons in a two-dimensional optically induced defocusing photonic lattice are theoretically predicted and experimentally observed. It is shown that in-phase nearest-neighbor and out-of-phase next-nearest-neighbor dipoles exist and can be stable in the intermediate intensity regime. There are also different types of dipoles that are always unstable. In-phase nearest-neighbor quadrupoles are also numerically obtained, and may also be linearly stable. Out-of-phase, nearest-neighbor quadrupoles are found to be typically unstable. These numerical results are found to be aligned with the main predictions obtained analytically in the discrete nonlinear Schrödinger model. Finally, experimental results are presented for both dipole and quadrupole structures, indicating that self-trapping of such structures in the defocusing lattice can be realized for the length of the nonlinear crystal (10 mm).

Dipole and Quadrupole Solitons in Optically Induced Two‐Dimensional Photonic Lattices: Theory and Experiment

Dipole and quadrupole solitons in a two‐dimensional photorefractive optical lattice are investigated both theoretically and experimentally. It is shown theoretically that out‐of‐phase dipole solitons and quadrupole solitons exist and are linearly stable in the intermediate‐intensity regime. In‐phase dipole and quadrupole solitons, however, are always linearly unstable, but their instabilities are rather weak in the low‐intensity regime. Experimentally, both types of dipole solitons are observed, and the experimental results agree qualitatively with the theoretical predictions. In addition, we have observed the anisotropic effect of the photorefractive crystal in the dipole‐soliton formation.

Multipole composite spatial solitons: theory and experiment

We discuss the properties of composite (or vector) spatial optical solitons created by the incoherent interaction of two optical beams and associated with higher-order modes guided by a soliton-induced waveguide in a bulk medium. Such stationary (2+1)-dimensional self-trapped localized structures include, in particular, vortex- and dipole-mode vector solitons and also incorporate higher-order multipole spatial solitons in a bulk medium, such as quadrupole solitons and necklace-type composite beams. We overview our theoretical and experimental results for the structure, formation, and stability of these self-trapped composite optical beams and also discuss the effects of anisotropy and of the nonlocality of the photorefractive nonlinearity on their properties. Additionally, we demonstrate, analytically and experimentally, that an array of the dipole-mode vector solitons can be generated as a result of the transverse instability of a quasi-one-dimensional two-hump soliton stripe in a saturable nonlinear optical medium.