We introduce a novel class of spatial complex variable function Gaussian (CVF-Gaussian) beam, which is the product of an arbitrary analytic complex variable function and a Gaussian function, in strongly nonlocal nonlinear media (SNNM). The CVF-Gaussian beam rotates generally during propagation. By choosing the input power of the beam, we can obtain the CVF-Gaussian breather or the CVF-Gaussian soliton. We reveal that stable CVF-Gaussian beam can exist in SNNM with different forms, including rotating dipole, and rotating elliptic donut. A distribution factor which is the parameter for the description of the transverse distribution of the CVF-Gaussian beam is discussed.
We analyze the transverse instabilities of spatial bright solitons in nonlocal nonlinear media, both analytically and numerically. We demonstrate that the nonlocal nonlinear response leads to a dramatic suppression of the transverse instability of the soliton stripes, and we derive the asymptotic expressions for the instability growth rate in both short- and long-wave approximations.
It is studied the bright spatial solitons in nonlocal defocusing Kerr media with parity-time (PT) symmetric potentials. We find that these solitons can exist and be stable over a different range of potential parameters. The influence of the degree of nonlocality on the solitons and the transverse energy flow within the stable solitons are also examined.
We study the formation of dark-bright vector soliton pairs in nonlocal Kerr-type nonlinear medium. We show, by analytical analysis and direct numerical calculation, that in addition to stabilize vector soliton pairs nonlocal nonlinearity also helps to reduce the threshold power for forming a guided bright soliton. With help of the nonlocality, it is expected that the observation of dark-bright vector soliton pairs in experiments becomes more workable.
We are able to detect the details of spatial optical collisionless wave-breaking through the high aperture imaging of a beam suffering shock in a fluorescent nonlinear nonlocal thermal medium. This allows us to directly measure how nonlocality and nonlinearity affect the point of shock formation and compare results with numerical simulations.
We find that a surface soliton in nonlocal nonlinear media can be regarded as a half of a bulk soliton with an antisymmetric amplitude distribution. The analytical solutions for the surface solitons and breathers in strongly nonlocal media are obtained, and the critical power and breather period are gotten analytically and confirmed by numerical simulations. In addition, the oscillating propagation of nonlocal surface solitons launched away from the stationary position is considered as the interaction between the soliton and its out-of-phase image beam. Its trajectory and oscillating period ob…
It is shown that three or more dark/gray solitons can form bound states in nonlocal media. More over dark/gray solitons can form bound states in several balance distances. Numerical simulations indicate that some of such bound states are unstable and will decay into a group of fundamental solitons, while others may be stable. There exist degenerate bound states with the same velocity, Hamiltonian, particle numbers and momentum but decaying in different ways and having different lifetimes.
The fractional Fourier transform (FRFT) naturally exists in the strongly nonlocal nonlinear (SNN) media and the propagation of optical beams in SNN media can be simply regarded as a self-induced FRFT. Through FRFT technic the evolution of fields in SNN media can be conveniently dealt with and an arbitrary square-integrable input field presents generally as a revivable higher order spatial soliton which reconstructs its profile periodically after every 4 times of Fourier transforms. The self-induced FRFT would illuminate the prospect of the SNN media in new applications such as continuously tun…
Dispersive shock waves are fascinating phenomena occurring when nonlinearity overwhelms linear effects, such as dispersion and diffraction. Many features of shock waves are still under investigation, as the interplay with noninstantaneity in temporal pulses transmission and nonlocality in spatial beams propagation. Despite the rich and vast literature on nonlinear waves in optical Kerr media, spatial dispersive shock waves in nonlocal materials deserve further attention for their unconventional properties. Indeed, they have been investigated in colloidal matter, chemical physics and biophotoni…
We investigate the propagation of the spatial soliton in cylindrical strongly nonlocal media by a novel method of image beam of light. The effect of the boundary on the soliton acting as the dynamic force for the soliton steering is equivalent to the force between the soliton beam and the image beam. The trajectory of the soliton is analytically studied which is in good agreement with the experimental results.
We reveal the relevance between the nonlocality and the focusing/defocusing states in nonlocally nonlinear media, and predict a novel phenomenon that the self-focusing/self-defocusing property of the optical beam in the nonlocally nonlinear medium with a sine-oscillation response function depends on its degree of nonlocality. The transition from the focusing nonlinearity to the defocusing nonlinearity of the nonlinear refractive index will happen when the degree of nonlocality of the system goes cross a critical value, and vise verse. Bright and dark soliton solutions are obtained, respectivel…
In a recent paper [A. Alberucci, C. Jisha, N. Smyth, and G. Assanto, Phys. Rev. A 91, 013841 (2015)], Alberucci et al. have studied the propagation of bright spatial solitary waves in highly nonlocal media. We find that the main results in that and related papers, concerning soliton shape and dynamics, based on the accessible soliton (AS) approximation, are incorrect; the correct results have already been published by others. These and other inconsistencies in the paper follow from the problems in applying the AS approximation in earlier papers by the group that propagated to the later papers.…
An perturbation-iteration method is developed for the computation of the Hermite-Gaussian-like solitons with arbitrary peak numbers in nonlocal nonlinear media. This method is based on the perturbed model of the Schrödinger equation for the harmonic oscillator, in which the minimum perturbation is obtained by the iteration. This method takes a few tens of iteration loops to achieve enough high accuracy, and the initial condition is fixed to the Hermite-Gaussian function. The method we developed might also be extended to the numerical integration of the Schrödinger equations in any type of pote…
We report the experimental observation of the interaction and attraction of many localized modes in a two dimensional (2D) system realized by a disordered optical fiber supporting transverse Anderson localization. We show that a nonlocal optically nonlinear response of thermal origin alters the localization length by an amount determined by the optical power and also induces an action at a distance between the localized modes and their spatial migration. Evidence of a collective and strongly interacting regime is given.
The relation between optical beams propagation in strongly nonlocal nonlinear (SNN) media and {propagation} in free space is {demonstrated using} the technique of variable transformation. The governing equation, integral and analytical solutions, and propagation properties in free space can be directly transferred to their counterparts in SNN media through a one-to-one correspondence. The one-to-one correspondence together with the Huygens-Fresnel integral yields an efficient numerical method to describe SNN propagation. The existence conditions and possible structures of solitons and breather…
The effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media is theoretically and numerically investigated. A perturbative approach is developed to solve the nonlocal nonlinear Schroedinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states.
We study gray solitons in nonlocal nonlinear media and show that they are stable and can form bound states. We reveal that gray soliton velocity depends on the nonlocality degree, and that it can be drastically reduced in highly nonlocal media. This is in contrast to the case of local media, where the maximal velocity is dictated solely by the asymptotic soliton amplitude.
Nonlocal communication between two laser light beams is experimented in a photochromic polymer thin films. Information exchange between the beams is mediated by the self-induction of a surface relief pattern. The exchanged information is related to the pitch and orientation of the grating. Both are determined by the incident beam. The process can be applied to experiment on a new kind of logic gates.
We predict the existence of surface dark solitons at the interface between a self-defocusing nonlocal nonlinear medium and a linear medium. The fundamental and higher-order surface dark solitons can exist when the linear refractive index of the self-defocusing media is much larger than that of the linear media. The fundamental solitons are stable and the stabilities of higher-order solitons depend on both nonlocality degree and propagation constant.
We show that ballistic transport of optically excited atoms in an atomic vapor provides a nonlocal nonlinearity which stabilizes the propagation of vortex beams and higher order modes in the presence of a self-focusing nonlinearity. Numerical experiments demonstrate stable propagation of lowest and higher order vortices over a hundred diffraction lengths, before dissipation leads to decay of these structures.
We show that multipole solitons can be made stable via vectorial coupling in bulk nonlocal nonlinear media. Such vector solitons are composed of mutually incoherent nodeless and multipole components jointly inducing a nonlinear refractive index profile. We found that stabilization of the otherwise highly unstable multipoles occurs below a maximum energy flow. Such threshold is determined by the nonlocality degree.
We reveal that lattice interfaces imprinted in nonlocal nonlinear media support surface solitons that do not exist in other similar settings, including interfaces of local and nonlocal uniform materials. We show the impact of nonlocality on the domains of existence and stability of the surface solitons, focusing on new types of dipole solitons residing partially inside the optical lattice. We find that such solitons feature strongly asymmetric shapes and that they are stable in large parts of their existence domain.
We address the evolution of higher-order excited states, such as vortex and multi-hump solitons, in nonlocal media with synthetic, competing focusing and defocusing nonlinearities with different nonlocal transverse scales. We reveal that introduction of suitable competing effects makes possible the stabilization of vortex solitons with topological charge m>2, as well as one-dimensional multi-hump solitons with number of humps p>4, all of which are highly unstable in natural nonlocal materials with focusing nonlinearities.
We present the experimental observation of scalar multi-pole solitons in highly nonlocal nonlinear media, including dipole-, tri-pole, quadru-pole, and necklace-type solitons, organized as arrays of out-of-phase bright spots. These complex solitons are meta-stable, but with a large parameters range where the instability is weak, enabling their experimental observation.
We analyze the existence, stability, and mobility of gap solitons (GSs) in a periodic photonic structure built into a nonlocal self-defocusing medium. Counter-intuitively, the GSs are supported even by a highly nonlocal nonlinearity, which makes the system quasi-linear. Unlike local models, the variational approximation (VA) predicts the GSs in a good agreement with numerical findings, due to the suppression of undulating tails of the solitons.
We address the properties of optical solitons in thermal nonlinear media with a local refractive index defect that is capable to trap solitons launched even close to the sample boundary despite the boundary-mediated forces that tend to deflect all beams toward the center of the sample. We show that while such forces become more pronounced with increasing the input beam power the defect can trap only light below a critical power above which solitons are ejected. The dynamics of soliton ejection and the subsequent propagation may be controlled.
The effect of finite boundaries in the propagation of spatial nonlocal solitons in media with cylindrical symmetry is analyzed. Using Ehrenfest's theorem together with the Green's function of the nonlinear refractive index equation, we derive an analytical expression for the force exerted on the soliton by the boundaries, verifying its validity by full numerical propagation. We show that the dynamics of the soliton are determined not only by the degree of nonlocality, but also by the boundary conditions for the refractive index. In particular, we report that a supercritical pitchfork bifurcati…
We investigate the breathing of optical spatial solitons in highly nonlocal media. Generalizing the Ehrenfest theorem, we demonstrate that oscillations in beam width obey a fourth-order ordinary differential equation. Moreover, in actual highly nonlocal materials, the original accessible soliton model by Snyder and Mitchell [Science \textbf{276}, 1538 (1997)] cannot accurately describe the dynamics of self-confined beams as the transverse size oscillations have a period which not only depends on power but also on the initial width. Modeling the nonlinear response by a Poisson equation driven b…
An integrated super resolution THz 3D imaging system based on a linear nonlocal achromatic freeform Bessel beam lens and high power oscillator radiator array
City University of Hong Kong（香港城市大学）；Southeast University（东南大学）
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.
We investigate the formation of collisionless shocks along the spatial profile of a gaussian laser beam propagating in nonlocal nonlinear media. For defocusing nonlinearity the shock survives the smoothing effect of the nonlocal response, though its dynamics is qualitatively affected by the latter, whereas for focusing nonlinearity it dominates over filamentation. The patterns observed in a thermal defocusing medium are interpreted in the framework of our theory.
Shallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in $(2+1)$-dimensions. We identify an analogue of surface tension in optics, namely a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a Kadomtsev-Petviashvilli (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality.
We address the stabilization of dipole solitons in nonlocal nonlinear materials by two different approaches. First, we study the properties of such solitons in thermal nonlinear media, where the refractive index landscapes induced by laser beams strongly depend on the boundary conditions and on the sample geometry. We show how the sample geometry impacts the stability of higher-order solitons in thermal nonlinear media and reveal that dipole solitons can be made dynami-cally stable in rectangular geometries in contrast to their counterparts in thermal samples with square cross-section. Second,…
Ring dark and anti-dark solitons in nonlocal media are found. These structures have, respectively, the form of annular dips or humps on top of a stable continuous-wave background, and exist in a weak or strong nonlocality regime, defined by the sign of a characteristic parameter. It is demonstrated analytically that these solitons satisfy an effective cylindrical Kadomtsev-Petviashvilli (aka Johnson's) equation and, as such, can be written explicitly in closed form. Numerical simulations show that they propagate undistorted and undergo quasi-elastic collisions, attesting to their stability pro…
We introduce vector soliton complexes in nonlocal Kerr-type nonlinear media. We discover that under proper conditions the combination of nonlocality and vectorial coupling has a remarkable stabilizing action on multi-humped solitons. In particular, we find that stable bound states featuring several field oscillations in each soliton component do exist. This affords stabilization of vector soliton trains incorporating large number of humps, a class of structures known to self-destroy via strong instabilities in scalar settings.
Shortcuts to adiabaticity for rapid soliton compression in nonlocal media
Shanghai University（上海大学）；University of the Basque Country UPV/EHU（巴斯克大学）；Quantum Advanced Research Center (QuARC), CSIC（CSIC量子先进研究中心）；Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC（CSIC马德里材料科学研究所）
We consider soliton formation in thermal nonlinear media bounded by rectangular cross-sections and uncover a new class of nonlinear stationary topological state. Specifically, we find that stationary higher-order vortex states in standard shapes do not exist, but rather they take the form of multiple, spatially separated single-charge singularities nested in an elliptical beam. Double-charge states are found to be remarkably robust despite their shape asymmetry and the phase-singularity splitting. States with higher topological charges are found to be unstable.
Intense light propagating in a nonlinear medium can generate an ensemble of interacting filaments of light, or spatial solitons. Using nematic liquid crystals, we demonstrate that they undergo a collective behavior typical of complex systems, including the formation of clusters and sound-like vibrations, as well as the reduction of the configurational entropy, controlled by the degree of nonlocality of the medium.
The generation of rogue waves is investigated via a nonlocal nonlinear Schrodinger (NLS) equation. In this system, modulation instability is suppressed and is usually expected that rogue wave formation would also be limited. On the contrary, a parameter regime is identified where the instability is suppressed but nevertheless the number and amplitude of the rogue events increase, as compared to the standard NLS (which is a limit of the nonlocal system). Furthermore, the nature of these waves is investigated; while no analytical solutions are known to model these events, numerically it is shown…
We discuss the modulational instability (MI) of plane waves in nonlocal Kerr media with the sine-oscillation response function, which can model the nematic liquid crystal with negative dielectric anisotropy. The results in the framework of the nonlocal nonlinear Schrödinger equation show that the MI in this system has two extraordinary properties. First, MI exists in both cases of negative Kerr coefficient $(n_{2}<0)$ and positive Kerr coefficient $(n_{2}>0)$. Second, the maximum gain of MI is not affected by the intensity of the plane wave, and the wave number %spatial frequency of the maximu…
Solitary waves in one-dimensional periodic media are discussed employing the nonlinear Schrödinger equation with a spatially periodic potential as a model. This equation admits two families of gap solitons that bifurcate from the edges of Bloch bands in the linear wave spectrum. These fundamental solitons may be positioned only at specific locations relative to the potential; otherwise, they become nonlocal owing to the presence of growing tails of exponentially-small amplitude with respect to the wave peak amplitude. Here, by matching the tails of such nonlocal solitary waves, higher-order lo…
We demonstrate the existence of two species of stable bright solitons, fundamental and dipole ones, in one-dimensional self-defocusing nonlocal media, with the local value of nonlinearity coefficient having one or several minima and growing at any rate faster than |x| at large values of coordinate x. The model can be derived for a slab optical waveguide with thermal nonlinearity. The most essential difference from the local counterpart of this system is the competition between two different spatial scales, the one determining the modulation pattern of the nonlinearity coefficient, and the corr…
Sorbonne Université, CNRS, Institut des NanoSciences de Paris（索邦大学，法国国家科学研究中心，巴黎纳米科学研究所）；University of the Witwatersrand（威特沃特斯兰德大学）；Università degli Studi di Napoli Federico II（那不勒斯费德里科二世大学）；Scuola Superiore Meridionale（南方高等学院）
Dispersive shock waves in thermal optical media belong to the third-order nonlinear phenomena, whose intrinsic irreversibility is described by time asymmetric quantum mechanics. Recent studies demonstrated that nonlocal wave breaking evolves in an exponentially decaying dynamics ruled by the reversed harmonic oscillator, namely, the simplest irreversible quantum system in the rigged Hilbert spaces. The generalization of this theory to more complex scenarios is still an open question. In this work, we use a thermal third-order medium with an unprecedented giant Kerr coefficient, the M-Cresol/Ny…
Zhejiang Key Laboratory of 3D Micro/Nano Fabrication and Characterization, Department of Electronic Science and Engineering, School of Engineering, Westlake University（西湖大学工程学院电子科学与工程系三维微纳加工与表征浙江省重点实验室）；Westlake Center for Micro/Nano Fabrication, Westlake University（西湖大学微纳加工中心）；Eastern Institute of Technology（宁波东方理工大学）
We consider a semi-infinite spatially dispersive dielectric with unequal transverse and longitudinal susceptibilities. The effect of the boundary is characterized by arbitrary reflection coefficients for polarization waves in the material that propagate to the surface. Specific values of these coefficients correspond to various additional boundary conditions (ABC) for Maxwell's equations. We derive the electromagnetic reflection and transmission coefficients at the boundary and investigate their dependence on material parameters and ABC. We also investigate the electromagnetic zero-point and t…
We consider an integrable model which describes light beams propagating in nonlocal nonlinear media of Cole-Cole type. The model is derived as high frequency limit of both Maxwell equations and the nonlocal nonlinear Schroedinger equation. We demonstrate that for a general form of nonlinearity there exist selfguided light beams. In high frequency limit nonlocal perturbations can be seen as a class of phase deformation along one direction. We study in detail nonlocal perturbations described by the dispersionless Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction met…
We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrödinger models. We show that stability of bright solitons in presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane, by means of both numerical simulations and analytical estimates. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on t…
We present an analytical description of the nonlocal optical response of plasmonic nanowire metamaterials that enable negative refraction, subwavelength light manipulation, and emission lifetime engineering. We show that dispersion of optical waves propagating in nanowire media results from coupling of transverse and longitudinal electromagnetic modes supported by the composite and derive the nonlocal effective medium approximation for this dispersion. We derive the profiles of electric field across the unit cell, and use these expressions to solve the long-standing problem of additional bound…
We investigate beam diffraction and spatial modulation instability of coherent light beams propagating in the non-paraxial regime in a nonlinear Kerr medium. We study the instability of plane wave solutions in terms of the degree of non-paraxiality, beyond the regime of validity of the nonlinear Schroedinger equation. We also numerically analyze the way non-paraxial terms breaks the integrability and affect the periodical evolution of higher order soliton solutions. We discuss non-locality and effective dissipation introduced by the non-linear coupling with evanescent waves.
We predict the existence and address the stability of two-dimensional surface solitons featuring topologically complex shapes, including dipoles, vortices, and bound states of vortex solitons, at the interface of nonlocal thermal media. Unlike their counterparts in bulk media, surface dipoles are found to be stable in the entire existence domain. Surface vortices are found to exhibit strongly asymmetric intensity and phase distributions, and are shown to be stable, too. Bound states of surface vortex solitons belong to a novel class of surface solitons having no counterparts in bulk media. Suc…
We address spectral tunneling of walking spatial solitons in photorefractive media with nonlocal diffusion of the nonlinear response and an imprinted shallow optical lattice. In contrast to materials with local nonlinearities, where solitons travelling across the lattice close to the Bragg angle suffer large radiative losses, in photorefractive media with diffusion nonlinearity resulting in self-bending solitons survive when their propagation angle approaches and even exceeds the Bragg angle. In the spatial frequency domain this effect can be considered as tunneling through the band of spatial…
We consider dispersive optical shock waves in nonlocal nonlinear media. Experiments are performed using spatial beams in a thermal liquid cell, and results agree with a hydrodynamic theory of propagation.
We study the propagation of light beams through optical media with competing nonlocal nonlinearities. We demonstrate that the nonlocality of competing focusing and defocusing nonlinearities gives rise to self-organization and stationary states with stable hexagonal intensity patterns, akin to transverse crystals of light filaments. Signatures of this long-range ordering are shown to be observable in the propagation of light in optical waveguides and even in free space. We consider a specific form of the nonlinear response that arises in atomic vapor upon proper light coupling. Yet, the general…
