Nonlocal Solitons Research Articles

Nanopteron solution of the Korteweg-de Vries equation

The nanopteron, which is a permanent but weakly nonlocal soliton, has been an interesting topic in numerical studies for many decades. However, the analytical solution of such a special soliton is rarely considered. In this letter, we study the explicit nanopteron solution of the Korteweg-de Vries (KdV) equation. Starting from the soliton-cnoidal wave solution of the KdV equation, the nanopteron structure is shown to exist. It is found that for the suitable choice of the wave parameters, the soliton core of the soliton-cnoidal wave trends to be a classical soliton of the KdV equation and the surrounded cnoidal periodic wave appears as small amplitude sinusoidal variations on both sides of the main core. Some interesting features of the wave propagation are revealed. In addition to the elastic interaction, it is surprising that the phase shift of the cnoidal periodic wave after the interaction with the soliton core is always half its wavelength, and this conclusion is universal to soliton-cnoidal wave interactions.

Large phase shift of spatial soliton in lead glass by cross-phase modulation in pump-signal geometry

We investigate the co-propagation of a strong pump beam and a weak signal beam in lead glass, and find that the large phase shift of the strongly nonlocal spatial optical soliton (SNSOS) can be realized via cross-phase modulation. The theoretical study suggests a synchronous propagation of the pump SNSOS and the signal SNSOS under the required initial condition. A π-phase shift of the signal SNSOS is experimentally obtained by changing the power of the pump SNSOS by about 13 mW around the soliton critical power, which agrees qualitatively with our theoretical prediction. The ratio of the phase shift rate of the signal SNSOS to that of the pump SNSOS shows a close match to the reciprocal of the ratio between their wavelengths.

Transmission of nonparaxial nonlocal lattice solitons at nonlinear interfaces

We report on transmission of nonparaxial nonlocal lattice solitons at an arbitrary angle of incidence to the interface separating two nonlinear media with transverse periodic modulation of the refractive index. The energy distribution of solitons in two media can be varied by the degree of nonlocality, the nonlinear refractive index mismatch parameter at the interface, the nonparaxial parameter, the angle of incidence and the lattice parameters. Importantly, the modulation period and depth can introduce different regimes of soliton propagation in the media including refraction and trapping.

Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities

We investigate analytically and numerically the interactions of dark solitons under competing nonlocal cubic and local quintic nonlinearities. It is shown that the self-defocusing quintic nonlinearity will strengthen the attractive interaction and decrease the relative distance between solitons, whereas the self-focusing quintic nonlinearity will enhance the repulsive interaction and increase soliton separation. We demonstrate these results by approximate variational approach and direct numerical simulation.

Vortex confinement and bending with nonlocal solitons

We investigate the routing of vortex beams in nonlocal media by means of coaxial, co-propagating spatial optical solitons. By introducing a refractive index perturbation in the form of a localized defect or a dielectric interface, the soliton waveguide can be curved and, therefore, can deviate the collinear vortex, effectively routing it, while preventing its destabilization and breakup.

Stabilization of nonlocal solitons by boundary conditions

We discovered that boundary conditions can stabilize nonlocal solitons with an oscillatory periodic response function. These solitons are the equivalent of quadratic solitons consisting of fundamental waves and oscillatory second harmonics, which are unstable unless subject to boundary confinement.

Nonlocal surface dark solitons and their stability analysis

Due to their future applications in optical communication, nonlocal dark solitons in bulk medium and surface bright solitons have received much attention recently. However, nonlocal surface dark solitons have not been investigated till now. In this paper, 1+1 dimensional nonlocal fundamental and second-order surface dark solitons have been found numerically at the interface between thermal nonlinear medium and linear medium. The relation between the wave shape of nonlocal surface dark soliton and propagation constant and nonlocality degree is investigated. Moreover, the stability of them is analyzed theoretically. The numerical simulation results show that 1+1 dimensional nonlocal fundamental surface dark Solitons are always stable in the domain of their existence, while second-order surface dark solitons are oscillatorily unstable and the width of unstable domain depends more greatly on propagation constant than nonlocality degree of nonlocal nonlinear medium. The figure showing the propagation, with the initial input of noise added, confirms the correctness of stability analysis results.

Theoretical investigation of spatial optical solitons in nematic liquid crystals with negative dielectric anisotropy

In this paper, the nonlocal nonlinear medium, nematic liquid crystals (NLCs) with negative dielectric anisotropy, are studied. Theoretical research shows that NLCs with negative dielectric anisotropic have a negative nonlinear coefficient. The analytical expressions for characteristic length of the nonlocal response function and for nonlinear coefficient are given. Secondly, the solutions for spatial optical solitons in NLCs with negative dielectric anisotropic are obtained by numerical computation. Finally, the influences of beam power and bias voltage on the propagation of the beam in NLCs with negative dielectric anisotropic are investigated. The result shows that the critical power of the nonlocal spatial solitons in NLCs with negative dielectric anisotropic varies with bias voltage.

The mobility of nonlocal solitons in fading optical lattices

We study the soliton mobility in nonlocal nonlinear media with an imprinted fading optical lattice. The results show that the transverse mobility of solitons varies with the lattice decay rate and the nonlocality degree, which provides an opportunity for all-optical control of light.

Self-Induced Mode Transformation in Nonlocal Nonlinear Media

We report on the first experimental observation of self-induced optical mode transformations in nonlocal nonlinear media. We show that the quadrupole Hermite-Gaussian mode experiences complex nonlinear dynamics in a nematic liquid crystal, including power-dependent conversion into a radially symmetric Laguerre-Gaussian mode. The physical mechanism responsible for self-induced transformation is the excitation of internal modes of a metastable quadrupole nonlocal soliton and its subsequent transmutation into a robust soliton with a bright peak surrounded by a bright ring. We also observe the onset of transformations of higher-order modes, proving the generic character of this nonlinear phenomenon.

Nonlocal multihump solitons in parity-time symmetric periodic potentials

We report on the existence and stability of nonlocal multihump gap solitons in one-dimensional parity-time symmetric periodic potentials. They can exist in the first gap in defocusing nonlocal nonlinearity and in the semi-infinite gap in focusing nonlocal nonlinearity. These solitons can be stable in the defocusing nonlinearity but are unstable in the focusing nonlinearity. For the multihump solitons, the shapes of the nonlinear contribution to refractive index are also multihump. The stability and shapes of the intensity distribution of these solitons will be changed by the degree of nonlocality. We also study the transverse power flow of these solitons.

Nonlocal defect solitons in parity–time-symmetric photonic lattices with spatially modulated nonlinearity

We report the existence and stability of nonlocal defect solitons (NDSs) in parity–time-symmetric photonic lattices with spatially modulated nonlocal nonlinearity. We reveal that solitons exist in the semi-infinite gap and a defect in the real part of nonlinear lattices plays a significant role in controlling the extent of the stability domains. We show that the existence and stability of the NDSs can be profoundly affected by the parameters of the nonlinear lattice structures.

Interactions of accessible solitons with interfaces in anisotropic media: the case of uniaxial nematic liquid crystals

We investigate, both theoretically and experimentally, spatial soliton interaction with dielectric interfaces in a strongly anisotropic medium with non-locality, such as nematic liquid crystals. We throw light on the role of refractive index gradients as well as optic axis variations in both voltage- and self-driven angular steering of non-local solitons. We specifically address and then employ in experiments a suitably designed electrode geometry in a liquid crystalline cell in order to define and tune a graded dielectric interface by exploiting the electro-optic response of the material through the in-plane reorientation of the optic axis in two distinct regions. We study both refraction and total internal reflection as well as voltage controlled steering of spatial solitons.

Nonlocal dark soliton and its linear stability analysis

Due to its future use in communication area, nonlocal spatial optical soliton has been a hot research topic recently. However, because of its special border condition, little research has been done on spatial dark solitons especially on its linear stability. In this paper, a method to analyze linear stability of nonlocal spatial dark soliton is put forward, moreover a numerical simulation and analysis is done on the linear stability of (1+1)-dimensional fundamental and second-order dark soliton in thermal nonlinear medium. Numerical results show that (1+1)-dimensional fundamental nonlocal dark solitons are always stable in their entire existence domain, while second-order dark solitons are oscillatorily unstable and the width of unstable domain depends on propagation constant and nonlocality degree of thermal nonlinear medium. The propagation graphs of initial input with noise addition confirms the correctness of linear stability analysis results.

Nonlocal dark solitons under competing cubic–quintic nonlinearities

We investigate properties of dark solitons under competing nonlocal cubic-local quintic nonlinearities. Analytical results, based on a variational approach and confirmed by direct numerical simulations, reveal the existence of a unique dark soliton solutions with their width being independent of the degree of nonlocality, due to the competing cubic-quintic nonlinearities.

Anomalous interaction of nonlocal solitons in media with competing nonlinearities

We theoretically investigate properties of individual bright spatial solitons and their interaction in nonlocal media with competing focusing and defocusing nonlinearities. We consider the general case with both nonlinear responses characterized by different strengths and degrees of nonlocality. We employ a variational approach to analytically describe soliton properties. In particular, we prove analytically that the interplay of focusing and defocusing nonlocal nonlinearities leads to attraction or repulsion of solitons depending on their separation distance. We then study the propagation and interaction of solitons using numerical simulations of the full model of beam propagation. The numerical simulations fully confirm our analytical results.

Quadratic solitons for negative effective second-harmonic diffraction as nonlocal solitons with periodic nonlocal response function

We employ the formal analogy between quadratic and nonlocal solitons to investigate analytically the properties of solitons and soliton bound states in second-harmonic generation in the regime of negative diffraction or dispersion of the second harmonic. We show that in the nonlocal description this regime corresponds to a periodic nonlocal response function. We then use the strongly nonlocal approximation to find analytical solutions of the families of single bright solitons and their bound states in terms of Mathieu functions.

Nonlocal solitons in dual-periodicPT-symmetric optical lattices

We numerically study the nonlocal gap solitons in parity-time (PT) symmetric optical lattices built into a nonlocal self-focusing medium. We state the existence, stability, and propagation dynamics of such PT gap solitons in detail. Simulated results show that there exist stable gap soltions. The influences of the degree of nonlocality on the soliton power, the energy flow density and the stable region of the PT gap solitons are also examined.

Nonlinear optical beams in bounded nematic liquid crystal cells

Stable nonlinear beams, both solitary waves (nematicons) and optical vortices, can form in a nematic liquid crystal due to a balance between the nonlinear, nonlocal response of the nematic and the diffractive spreading of the light beam. The `huge' nonlinearity of a nematic liquid crystal makes it ideal for the experimental development of photonic devices as nonlinear effects occur over millimetre distances. In this work, a simple and fast method to analyse the trajectory of a nonlinear beam within a finite liquid crystal cell, based on a classical method not explored in this context, the method of images, is developed. With the orientation of the nematic molecules modelled using images, the evolution of the beam is obtained by using both asymptotics and modulation theory. The efficiency of this new method is shown by comparisons with a standard Fourier series solution for the nematic response and full numerical solutions of the governing equations. It is found that only a small number of images is required compared with the usual Fourier series technique in order to obtain excellent agreement with full numerical solutions. Finally, the contrasting effect of the cell boundaries on a nematicon and a vortex is explored. References A. Alberucci and A. Assanto, Propagation of optical spatial solitons in finite-size media: interplay between nonlocality and boundary conditions, J. Opt. Soc. Amer. B, 24, 2314--2320 (2007), doi:10.1364/JOSAB.24.002314. A. Alberucci, G. Assanto, D. Buccoliero, A. S. Desyatnikov, T. R. Marchant, and N. F. Smyth, Modulation analysis of boundary-induced motion of optical solitary waves in a nematic liquid crystal, Phys. Rev. A, 79, 043816 (2009), doi:10.1103/PhysRevA.79.043816. A. Alberucci, M. Peccianti and G. Assanto, Nonlinear bouncing of nonlocal spatial solitons at boundaries, Opt. Lett., 32, 2795--2797 (2007), doi:10.1364/OL.32.002795. G. Assanto, A. Fratalocchi and M. Peccianti, Spatial solitons in nematic liquid crystals: from bulk to discrete, Opt. Express,, 15, 5248--5259 (2007), doi:10.1364/OE.15.005248. G. Assanto, M. Peccianti and C. Conti, Nematicons: optical spatial solitons in nematic liquid crystal, Opt. Photon. News, 14, 44--48 (2003), doi:10.1364/OPN.14.2.000044. C. Conti, M. Peccianti, G. Assanto, Route to nonlocality and observation of accessible solitons, Phys. Rev. Lett., 91, 073901 (2003), doi:10.1103/PhysRevLett.91.073901. R. Courant and D. Hilbert, Methods of Mathematical Physics Vol. 1, Interscience Publishers, New York (1965). W. L. Kath and N. F. Smyth, Soliton evolution and radiation loss for the nonlinear Schr{o}dinger equation, Phys. Rev. E, 51, 1484--1492 (1995), doi:10.1103/PhysRevE.51.1484. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, Wiley, New York (1995). I. C. Khoo, Nonlinear optics of liquid crystalline materials, Phys. Rep., 471, 221--267 (2009), doi:10.1016/j.physrep.2009.01.001. Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, (2003). A. A. Minzoni, L. W. Sciberras, N. F. Smyth and A. L. Worthy, Propagation of optical spatial solitary waves in bias-free nematic-liquid-crystal cells, Phys. Rev. A, 84, 043823 (2011), doi:10.1103/PhysRevA.84.043823. A. A. Minzoni, N. F. Smyth and A. L. Worthy, Modulation solutions for nematicon propagation in nonlocal liquid crystals, J. Opt. Soc. Amer. B, 24, 1549--1556 (2007), doi:10.1364/JOSAB.24.001549. A. A. Minzoni, N. F. Smyth, A. L. Worthy and Yu. S. Kivshar, Stabilization of vortex solitons in nonlocal nonlinear media, Phys. Rev. A, 76, 063803 (2007), doi:10.1103/PhysRevA.76.063803. G. B. Whitham, Linear and Nonlinear Waves, J. Wiley and Sons, New York, (1974). A. I. Yakimenko, Yu. A. Zaliznyak and Yu. S. Kivshar Stable vortex solitons in nonlocal self-focusing nonlinear media, Phys. Rev. E, 71, 065603(R) (2005), doi:10.1103/PhysRevE.71.065603.

Nonlocal bright spatial solitons in defocusing Kerr media supported by symmetric potentials

The bright spatial solitons in nonlocal defocusing Kerr media with parity-time () symmetric potentials are studied. The influence of the degree of nonlocality on the solitons and the transverse energy flow within the stable solitons are examined. We also find that these solitons can exist and be stable over a different range of the propagation constant and potential parameters. Interestingly, there exists a threshold value for the degree of nonlocality or potential parameters.