Discrete vector solitons in Kerr nonlinear waveguide arrays

Discrete optics in optomechanical waveguide arrays

The propagation properties of light in optomechanical waveguide arrays (OMWAs) are studied. Due to the strong mechanical Kerr effect, the optical self-focusing and self-defocusing phenomena can be realized in the arrays of subwavelength dielectric optomechanical waveguides with the milliwatt-level incident powers and micrometer-level lengths. Compared with the conventional nonlinear waveguide arrays, the required incident powers and lengths of the waveguides are decreased by five orders of magnitude and one order of magnitude, respectively. Furthermore, by adjusting the deformation of the nanowaveguides through a control light, the propagation path of the signal light in the OMWA can be engineered, which could be used as a splitting-ratio-tunable beam splitter. This Letter provides a new platform for discrete optics and broadens the application of integrated optomechanics.

Vector plasmonic lattice solitons in nonlinear graphene-pair arrays

We investigate the vector plasmonic lattice solitons (PLSs) in nonlinear graphene-pair arrays (GPAs) consisting of periodically arranged double graphene sheets, which are spatially separated. There are two dispersion bands for the Bloch modes in the array due to the coupling of surface plasmon polaritons (SPPs) between the graphene pairs. The vector PLSs composed of two components originate from the nonlinear interaction of Bloch modes in different bands. Both components undergo mutual self-trapping through the balance between diffraction and self-focusing nonlinearity of graphene. Thanks to the strong confinement of SPPs, the vector PLSs can be squeezed into a lateral width of ∼λ/100. The study provides a promising approach to all-optical control on a deep-subwavelength scale.

Subwavelength binary plasmonic solitons

We study the formation of subwavelength solitons in binary metal-dielectric lattices. We show that the transverse modulation of the lattice constant breaks the fundamental plasmonic band and suppresses the discrete diffraction of surface plasmon waves. New types of plasmonic solitons are found, and their characteristics are analyzed. We also demonstrate the existence of photonic-plasmonic vector solitons and elucidate their propagation properties.

Vector solitons in parity-time-symmetric lattices

I study vector solitons involving two incoherently coupled field components in periodic parity-time (PT)-symmetric optical lattices. The specific symmetry of the lattice imposes restrictions on the symmetry of available vector soliton states. While all configurations with asymmetric intensity distributions are prohibited, such lattices support multihump solitons with an equal number of "in-phase" or "out-of-phase" spots in two components, residing on neighboring lattice channels. In the focusing medium, only the solitons containing out-of-phase spots in at least one component can be stable, while in the defocusing medium stability is achieved for structures consisting of in-phase spots. Mixed-gap vector solitons with components emerging from different gaps in the lattice spectrum also exist and can be stable in the PT-symmetric lattice.

Multiband vector plasmonic lattice solitons

We predict multiband vector plasmonic lattice solitons (PLSs) in metal-dielectric waveguide arrays, in both focusing and defocusing nonlinearities. Such vector solitons consist of two components originating from different transmission bands. By simulating the full nonlinear Maxwell's equations, we demonstrate the diffractionless propagation of vector PLSs and their discrete diffraction when only one component is present. Their subwavelength size characteristics and the influences of metallic losses are also studied.

Optical spatial solitons: historical overview and recent advances

Solitons, nonlinear self-trapped wavepackets, have been extensively studied in many and diverse branches of physics such as optics, plasmas, condensed matter physics, fluid mechanics, particle physics and even astrophysics. Interestingly, over the past two decades, the field of solitons and related nonlinear phenomena has been substantially advanced and enriched by research and discoveries in nonlinear optics. While optical solitons have been vigorously investigated in both spatial and temporal domains, it is now fair to say that much soliton research has been mainly driven by the work on optical spatial solitons. This is partly due to the fact that although temporal solitons as realized in fiber optic systems are fundamentally one-dimensional entities, the high dimensionality associated with their spatial counterparts has opened up altogether new scientific possibilities in soliton research. Another reason is related to the response time of the nonlinearity. Unlike temporal optical solitons, spatial solitons have been realized by employing a variety of noninstantaneous nonlinearities, ranging from the nonlinearities in photorefractive materials and liquid crystals to the nonlinearities mediated by the thermal effect, thermophoresis and the gradient force in colloidal suspensions. Such a diversity of nonlinear effects has given rise to numerous soliton phenomena that could otherwise not be envisioned, because for decades scientists were of the mindset that solitons must strictly be the exact solutions of the cubic nonlinear Schrödinger equation as established for ideal Kerr nonlinear media. As such, the discoveries of optical spatial solitons in different systems and associated new phenomena have stimulated broad interest in soliton research. In particular, the study of incoherent solitons and discrete spatial solitons in optical periodic media not only led to advances in our understanding of fundamental processes in nonlinear optics and photonics, but also had a very important impact on a variety of other disciplines in nonlinear science. In this paper, we provide a brief overview of optical spatial solitons. This review will cover a variety of issues pertaining to self-trapped waves supported by different types of nonlinearities, as well as various families of spatial solitons such as optical lattice solitons and surface solitons. Recent developments in the area of optical spatial solitons, such as 3D light bullets, subwavelength solitons, self-trapping in soft condensed matter and spatial solitons in systems with parity-time symmetry will also be discussed briefly.

Vector solitons in nonlinear lattices

We consider two-component solitons in a medium with a periodic modulation of the nonlinear coefficient. The modulation enables the existence of complex multihump vector states. In particular, vector solitons composed of dipole and fundamental or dipole and even double-hump components exist and may be stable. Families of unstable scalar solitons can be stabilized in the vectorial form, due to the coupling to a stable second component.

Discrete reduced-symmetry solitons and second-band vortices in two-dimensional nonlinear waveguide arrays

Considering a two-dimensional lattice of weakly coupled waveguides, where each waveguide may carry two orthogonal modes of dipolar character, we present a nonlinear discrete vector model for the study of Kerr optical solitons with profiles having a reduced symmetry relative to the underlying lattice. We describe analytically and numerically existence and stability properties of such states in square and triangular lattices and also reveal directional mobility properties of two-dimensional gap solitons which were recently observed in experiment. The model also describes one-site peaked discrete vortices corresponding to experimentally observed "second-band" vortex lattice solitons, for which oscillatory instabilities are predicted. We also introduce a concept of "rotational Peierls-Nabarro barrier" characterizing the minimum energy needed for rotation of stable dipole modes and compare numerically translational and rotational energy barriers in regimes of good mobility.

Observation of two-dimensional coherent surface vector lattice solitons

We report, to the best of our knowledge, the first experimental observation of vector surface solitons, which form at the edge and in the corner of two-dimensional laser-written waveguide arrays. These elliptically polarized vector states are composed of two orthogonally polarized components. They exist only above a power threshold and bifurcate from scalar surface solitons. The components of a vector soliton may have substantially different degrees of localization in certain parameter ranges.

Walking-vector-soliton caging and releasing

We address the formation and propagation of vector solitons in optical lattices in the presence of anisotropy-induced walk-off between ordinary and extraordinary polarized field components. Stable vector solitons trapped by the lattice form above a threshold power, while decreasing the lattice depth below a critical value results in the abrupt release of the caged solitons that then move across the lattice and may get trapped in a desired lattice channel.

Generalized neighbor-interaction models induced by nonlinear lattices

It is shown that the tight-binding approximation of the nonlinear Schrödinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrödinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.

Symmetry breaking in linearly coupled dynamical lattices

We examine one- and two-dimensional models of linearly coupled lattices of the discrete-nonlinear-Schrödinger type. Analyzing ground states of the system with equal powers (norms) in the two components, we find a symmetry-breaking phenomenon beyond a critical value of the total power. Asymmetric states, with unequal powers in their components, emerge through a subcritical pitchfork bifurcation, which, for very weakly coupled lattices, changes into a supercritical one. We identify the stability of various solution branches. Dynamical manifestations of the symmetry breaking are studied by simulating the evolution of the unstable branches. The results present the first example of spontaneous symmetry breaking in two-dimensional lattice solitons. This feature has no counterpart in the continuum limit because of the collapse instability in the latter case.

Discrete surface solitons in two dimensions

We investigate fundamental localized modes in two-dimensional lattices with an edge (surface). The interaction with the edge expands the stability area for fundamental solitons, and induces a difference between dipoles oriented perpendicular and parallel to the surface. On the contrary, lattice vortex solitons cannot exist too close to the border. We also show, analytically and numerically, that the edge supports a species of localized patterns, which exists too but is unstable in the uniform lattice, namely, a horseshoe-shaped soliton, whose "skeleton" consists of three lattice sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.

Skyrmion-like states in two- and three-dimensional dynamical lattices

We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the two-dimensional case, an analog of the baby Skyrmion is built on the square lattice as a discrete vortex soliton of a complex field [whose vorticity plays the role of the Skyrmion's winding number (WN)], coupled to a radial "bubble" in a real lattice field. The most compact quasi-Skyrmion on the cubic lattice is composed of a nearly planar complex-field discrete vortex and a three-dimensional real-field bubble; unlike its continuum counterpart which must have WN=2, this stable discrete state exists with WN=1. Analogs of Skyrmions in the one-dimensional lattice are also constructed. Stability regions for all these states are found in an analytical approximation and verified numerically. The dynamics of unstable discrete Skyrmions (which leads to the onset of lattice turbulence) and their partial stabilization by external potentials are explored too.

Beam interactions in one-dimensional saturable waveguide arrays

The interaction between two parallel beams in one-dimensional discrete saturable systems has been investigated using lithium niobate nonlinear waveguide arrays. When the beams are separated by one channel and in phase it is possible to observe soliton fusion at low power levels. This result is confirmed numerically. By increasing the power, solitonlike propagation of weakly coupled beams occurs. When the beams are out-of-phase the most interesting numerically obtained result is the existence of oscillations which resemble the recently discovered Tamm oscillations.

Discrete vector solitons in one-dimensional lattices in photorefractive media

We construct families of two-component spatial solitons in a one-dimensional lattice with saturable on-site nonlinearity (focusing or defocusing) in a photorefractive crystal. We identify 14 species of vector solitons, depending on their type (bright/dark), phase (in-phase/staggered), and location on the lattice (on/off-site). Two species of the bright/bright type form entirely stable soliton families, four species are partially stable (depending on the value of the propagation constant), while the remaining eight species are completely unstable. "Symbiotic" soliton pairs (of the bright/dark type), which contain components that cannot exist in isolation in the same model, are found as well.

Solitary waves in discrete media with four-wave mixing

In this paper, we examine in detail the principal branches of solutions that arise in vector discrete models with nonlinear intercomponent coupling and four wave mixing. The relevant four branches of solutions consist of two single mode branches (transverse electric and transverse magnetic) and two mixed mode branches, involving both components (linearly polarized and elliptically polarized). These solutions are obtained explicitly and their stability is analyzed completely in the anticontinuum limit (where the nodes of the lattice are uncoupled), illustrating the supercritical pitchfork nature of the bifurcations that give rise to the latter two, respectively, from the former two. Then the branches are continued for finite coupling constructing a full two-parameter numerical bifurcation diagram of their existence. Relevant stability ranges and instability regimes are highlighted and, whenever unstable, the solutions are dynamically evolved through direct computations to monitor the development of the corresponding instabilities. Direct connections to the earlier experimental work of Meier [Phys. Rev. Lett., 91, 143907 (2003)] that motivated the present work are given.

Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double- system

We investigate the simultaneous formation and propagation of coupled ultraslow optical soliton pairs in a cold, lifetime-broadened three-state double-Lambda atomic system. Starting from the equations of motion of atomic response and two-mode probe-control electromagnetic fields, we derive coupled nonlinear Schrödinger equations that govern the nonlinear evolution of the envelopes of the probe fields in this four-wave mixing scheme by means of the standard method of multiple scales. We demonstrate that for weak probe fields and with suitable operation conditions, a pair of coupled optical solitons moving with remarkably slow propagating velocity can be established in such a highly resonant atomic medium. The key elements to such a shape preserving, well matched yet interacting soliton pair is the balance between dispersion effect and self- and cross-phase modulation effects of the system.

Discrete vector on-site vortices

We study discrete vortices in coupled discrete nonlinear Schrödinger equations. We focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals. Stability of the single-component vortex cross in the anti-continuum limit of small coupling between lattice nodes is proved. In the vector case, we consider two coupled configurations of vortex crosses, namely the charge-one vortex in one component coupled in the other component to either the charge-one vortex (forming a double-charge vortex) or the charge-negative-one vortex (forming a, so-called, hidden-charge vortex). We show that both vortex configurations are stable in the anti-continuum limit, if the parameter for the inter-component coupling is small and both of them are unstable when the coupling parameter is large. In the marginal case of the discrete two-dimensional Manakov system, the double-charge vortex is stable while the hidden-charge vortex is linearly unstable. Analytical predictions are corroborated with numerical observations that show good agreement near the anti-continuum limit, but gradually deviate for larger couplings between the lattice nodes.

Observation of discrete surface solitons

We report the first observation of discrete optical surface solitons at the interface between a nonlinear self-focusing waveguide lattice and a continuous medium. The effect of power on the localization process of these optical self-trapped states at the edge of an AlGaAs waveguide array is investigated in detail. Our experimental results are in good agreement with theoretical predictions.

Incoherent blocker soliton interactions in Kerr waveguide arrays

We have observed the incoherent interaction between a highly confined (blocker) soliton and wide, moving signal beams of a different wavelength in a one-dimensional discrete Kerr medium. Digital switching of the blocker solitons to successive adjacent channels was measured with increasing signal power via both one and two cascaded interactions in an AlGaAs waveguide array, operations equivalent to a reconfigurable three-output router.

Observation of second-band vortex solitons in 2D photonic lattices

We demonstrate second-band bright vortex-array solitons in photonic lattices. This constitutes the first experimental observation of higher-band solitons in any 2D periodic system. These solitons possess complex intensity and phase structures, yet they can be excited by a simple highly localized vortex-ring beam. Finally, we show that the linear diffraction of such beams exhibits preferential transport along the lattice axes.

Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes

We construct a variety of novel localized topological structures in the 3D discrete nonlinear Schrödinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from multipole patterns and diagonal vortices to vortex "cubes" (stack of two quasiplanar vortices) and "diamonds" (formed by two orthogonal vortices).

Polarization instability, steering, and switching of discrete vector solitons

We study the dynamics of discrete vector solitons in arrays of weakly coupled birefringent optical waveguides with cubic nonlinear response. We start with a modulational instability analysis, followed by approximate analytical solutions in the form of strongly localized modes. Next, we compute the effective Peierls-Nabarro potential for these modes and obtain the spatial average of the power transfer between both polarizations modes as a function of their relative phase. Finally, we combine the concepts of polarization mode instability with discreteness-induced beam trapping by the array, and demonstrate numerically the amplification of a weak signal by a strong pump of the other polarization, combined with simultaneous discretized all-optical switching.

Polarization dependent properties of waveguide arrays: band-structure anomaly and high-band localizations

We experimentally study polarization dependent linear and nonlinear dynamics in waveguide arrays. We found that in certain arrays, the band structure and the modal shapes of the array modes are markedly different for the two polarizations, in a manner that cannot be simply explained using the effective index approximation. Specifically, one of the gaps was found to be missing for the TM polarization. In the nonlinear regime, we observe mixed-polarization nonlinear localizations in high bands, such as Band-2 Floquet-Bloch vector solitons. The band structure anomaly enabled the excitation of a multiband moving breather.

Nonlinear beam interactions in 1D discrete Kerr systems

The interaction between parallel beams in one-dimensional discrete Kerr systems has been investigated using arrays of coupled channel waveguides. The experiments were performed in AlGaAs waveguides at 1550 nm which corresponds to photon energies just below one half the semiconductor's bandgap. The input intensity and relative input phase between the input beams was varied and the output intensity patterns were recorded. Observed was behavior ranging from a linear response, to soliton interactions between moderately and then strongly localized spatial solitons. Finally the influence of multiphoton absorption and asymmetric beam inputs on these interactions was investigated at very high intensities.

Formation of discrete solitons in light-induced photonic lattices

We present both experimental and theoretical results on discrete solitons in two-dimensional optically-induced photonic lattices in a variety of settings, including fundamental discrete solitons, vector-like discrete solitons, discrete dipole solitons, and discrete soliton trains. In each case, a clear transition from two-dimensional discrete diffraction to discrete trapping is demonstrated with a waveguide lattice induced by partially coherent light in a bulk photorefractive crystal. Our experimental results are in good agreement with the theoretical analysis of these effects.

X - waves in nonlinear normally dispersive waveguide arrays

We theoretically demonstrate that optical discrete X-waves are possible in normally dispersive nonlinear waveguide arrays. We show that such X-waves can be effectively excited for a wide range of initial conditions and in certain occasions can be generated in cascade. The possibility of observing this family of waves in AlGaAs array systems is investigated in terms of pertinent examples.

Discrete light propagation and self-trapping in liquid crystals

We investigate light propagation and self-localization in a voltage-controlled array of channel waveguides realized in undoped nematic liquid crystals. We report on discrete diffraction and solitons, as well as all-optical angular steering and the formation of multiband vector breathers. The results and are in excellent agreement with both coupled mode theory and full numerical simulations.

Stability properties of multiwavelength, incoherent, dissipative spatial solitons

We have investigated the interaction between two dissipative spatial solitons of different frequencies in periodically patterned semiconductor optical amplifiers. The experimental results are in good agreement with the theory. Simulations suggest that multiwavelength interactions do not produce stable bound solitons unless the system's modeling equations are completely symmetric.

Optical multiband vector breathers in tunable waveguide arrays

We investigate multiband optical breathers in a voltage-adjustable array of coupled waveguides in nematic liquid crystals. Symmetric breathers resulting from vector composition of modes from two bands are observed over large propagation distances and are described in terms of the Floquet-Bloch theory.

All-optical switching and amplification of discrete vector solitons in nonlinear cubic birefringent waveguide arrays

We study all-optical switching based on the dynamic properties of discrete vector solitons in waveguide arrays. We employ the concept of polarization mode instability and demonstrate simultaneous switching and amplification of a weak signal by a strong pump of the opposite polarization.

Stabilization of vector solitons in optical lattices

We address the properties and dynamical stability of one-dimensional vector lattice solitons in a Kerr-type cubic medium with harmonic transverse modulation of the refractive index. We discovered that unstable families of scalar lattice solitons can be stabilized via cross-phase modulation (XPM) in the vector case. It was found that multihumped vector solitons that are unstable in uniform media where the XPM strength is higher than that of self-phase modulation can also be stabilized by the lattice.

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.

Two-dimensional higher-band vortex lattice solitons

We study self-localized second-band vortex states in two-dimensional photonic lattices and find stable ring solitons whose phase forms an array of counterrotating vortices. We also identify composite solitons in which a second-band vortex is jointly trapped with a mode arising from the first band and study their stability. When such a composite entity is unstable, it disintegrates while exchanging angular momentum between its constituents, eventually stabilizing into another form of composite soliton.

Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger equation, we find discrete vortex solitons with various values of the topological charge S. Stability regions for the vortices with S=0,1,3 are investigated. The S=2 vortex is unstable and may spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of microresonators.

Switching of discrete optical solitons in engineered waveguide arrays

We demonstrate a simple concept for controlling nonlinear switching of discrete solitons in arrays of weakly coupled optical waveguides, for both cubic and quadratic nonlinear response. Based on the effective discrete nonlinear equations describing light propagation in the waveguide arrays in the tight-binding approximation, we demonstrate the key ideas of the array engineering by means of a steplike variation of the waveguide coupling. We demonstrate the digitized switching of a narrow input beam for up to 11 neighboring waveguides, in the case of the cubic nonlinearity, and up to 10 waveguides, in the case of the quadratic nonlinearity. We discuss our predictions in terms of the physics of the engineered Peierls-Nabarro (PN) potential experienced by strongly localized nonlinear modes in a lattice, and calculate the PN potential for the quadratic nonlinear array for the first time.

Observation of two-dimensional lattice vector solitons

We demonstrate the formation of fundamental and dipolelike vector solitons in an optically induced two-dimensional photonic lattice. Such vector solitons are realized by mutual trapping of two beams in the lattice. Our theoretical results are in good agreement with experimental observations.

Dipole solitons in optically induced two-dimensional photonic lattices

Dipole solitons in a two-dimensional optically induced photonic lattice are theoretically predicted and experimentally demonstrated for the first time to our knowledge. It is shown that such dipole solitons are stable and robust under appropriate conditions. Our experimental results are in good agreement with theoretical predictions.

Observation of vortex-ring "discrete" solitons in 2D photonic lattices

We present the experimental observation of both on-site and off-site vortex-ring solitons of unity topological charge in a nonlinear photonic lattice, along with a theoretical study of their propagation dynamics and stability.

Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays

We demonstrate that both the linear (diffraction) and the nonlinear dynamics of two-dimensional waveguide arrays are considerably more complex and versatile than their one-dimensional counterparts. The discrete diffraction properties of these arrays can be effectively altered, depending on the propagation Bloch k-vector within the first Brillouin zone of the lattice. In general, this diffraction behavior is anisotropic and therefore permits the existence of a new class of discrete elliptic solitons in the nonlinear regime.