We report analytical solutions for spatial solitons supported by layers of a quadratically nonlinear material embedded into a linear planar waveguide. A full set of symmetric, asymmetric, and antisymmetric modes pinned to a symmetric pair of the nonlinear layers is obtained. The solutions describe a bifurcation of the subcritical type, which accounts for the transition from the symmetric to asymmetric modes. The antisymmetric states (which do not undergo the bifurcation) are completely stable (the stability of the solitons pinned to the embedded layers is tested by means of numerical simulations). Exact solutions are also found for nonlinear layers embedded into a nonlinear waveguide, including the case when the uniform and localized nonlinearities have opposite signs (competing nonlinearities). For the layers embedded into the nonlinear medium, stability properties are explained by comparison to the respective cascading limit.
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 overview the properties of nonlinear guided waves and (bright and dark) spatial optical solitons in a periodic medium created by a sequence of linear and nonlinear layers. First, we consider a single layer with a cubic nonlinear response (a nonlinear waveguide) embedded into a periodic layered linear medium, and describe nonlinear localized modes (guided waves and Bragg-like localized gap modes) and their stability. Then, we study modulational instability as well as the existence and stability of discrete spatial solitons in a periodic array of identical nonlinear layers, a one-dimensional …
Narrow linewidth lasers and optical frequency combs generated with mode-locked lasers revolutionized optical frequency metrology. The advent of soliton Kerr frequency combs in compact crystalline or integrated ring optical microresonators opens new horizons for applications. These combs, as was naturally assumed, however, require narrow-linewidth single-frequency pump lasers. We demonstrate that a regular multi-frequency Fabry-Perot laser diode self-injection locked to an optical whispering gallery mode (WGM) microresonator can be first efficiently transformed to a single-frequency ultra-narro…
We report on the observation of bound states of gain-guided solitons (GGSs) in a dispersion-managed erbium-doped fiber laser operating in the normal net cavity dispersion regime. Despite of the fact that the GGS is a chirped soliton and there is strong pulse stretching and compression along the cavity in the laser, the bound solitons observed have a fixed pulse separation, which is invariant to the pump strength change. Numerical simulation confirmed the experimental observations.
We present eight types of spatial optical solitons which are possible in a model of a planar waveguide that includes a dual-channel trapping structure and competing (cubic-quintic) nonlinearity. Among the families of trapped beams are symmetric and antisymmetric solitons of "broad" and "narrow" types, composite states, built as combinations of broad and narrow beams with identical or opposite signs ("unipolar" and "bipolar" states, respectively), and "single-sided" broad and narrow beams trapped, essentially, in a single channel. The stability of the families is investigated via eigenvalues of…
We show dissipative spatial solitons in nonlinear optical micro-resonators in which the refractive index is laterally modulated. In addition to "normal" and "staggered" dissipative solitons, similar to those in spatially modulated conservative systems, a narrow "mid-band" soliton is shown, having no counterparts in conservative systems.
Embedded solitons are exceptional modes in nonlinear-wave systems with the propagation constant falling in the system's propagation band. An especially challenging topic is seeking for such modes in nonlinear dynamical lattices (discrete systems). We address this problem for a system of coupled discrete equations modeling the light propagation in an array of tunnel-coupled waveguides with a combination of intrinsic quadratic (second-harmonic-generating) and cubic nonlinearities. Solutions for discrete embedded solitons (DESs) are constructed by means of two analytical approximations, adjusted,…
We develop a method for achieving scalable transmission stabilization and switching of $N$ colliding soliton sequences in optical waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss. We show that dynamics of soliton amplitudes in $N$-sequence transmission is described by a generalized $N$-dimensional predator-prey model. Stability and bifurcation analysis for the predator-prey model are used to obtain simple conditions on the physical parameters for robust transmission stabilization as well as on-off and off-on switching of $M$ out of $N$ soliton sequences. Nume…
For the one-dimensional nonlinear Schroedinger equation with a complex potential, it is shown that if this potential is not parity-time (PT) symmetric, then no continuous families of solitons can bifurcate out from linear guided modes, even if the linear spectrum of this potential is all real. Both localized and periodic non-PT-symmetric potentials are considered, and the analytical conclusion is corroborated by explicit examples. Based on this result, it is argued that PT-symmetry of a one-dimensional complex potential is a necessary condition for the existence of soliton families.
Existence of localized modes supported by the PT-symmetric nonlinear lattices is reported. The system considered reveals unusual properties: unlike other typical dissipative systems it possesses families (branches) of solutions, which can be parametrized by the propagation constant; relatively narrow localized modes appear to be stable, even when the conservative nonlinear lattice potential is absent; finally, the system supports stable multipole solutions.
We have investigated mixed-gap vector solitons involving incoherently coupled fundamental and dipole components in a parity-time (PT) symmetric lattice with saturable nonlinearity. For the focusing case, vector solitons emerge from the semi-infinite and the first finite gaps, while for the defocusing case, vector solitons emerge from the first finite and the second finite gaps. For both cases, we find that stronger saturable nonlinearity is relative to sharper increase/decrease of soliton power with propagation constant and to narrower existence domain of vector solitons. This finding is helpf…
The stabilization of one-dimensional solitons by a nonlinear lattice against the critical collapse in the focusing quintic medium is a challenging issue. We demonstrate that this purpose can be achieved by combining a nonlinearlatticeandsaturationofthequinticnonlinearity. Thesystemsupportsthreespeciesofsolitons, namely, fundamental (even-parity) ones and dipole (odd-parity) modes of on- and off-site-centered types. Very narrow fundamental solitons are found in an approximate analytical form, and systematic results for very broad unstable and moderately broad partly stable solitons, including t…
Dark solitons and localized defect modes against periodic backgrounds are considered in arrays of waveguides with defocusing Kerr nonlinearity constituting a nonlinear lattice. Bright defect modes are supported by local increase of the nonlinearity, while dark defect modes are supported by a local decrease of the nonlinearity. Dark solitons exist for both types of the defect, although in the case of weak nonlinearity they feature side bright humps making the total energy propagating through the system larger than the energy transferred by the constant background. All considered defect modes ar…
We report on the first experimental observation of higher-order modes guided by soliton-induced waveguides in unbiased nematic liquid crystals. We find that the nematicon waveguides operate in a bounded power region specific to each guided mode. Below this region the guided beams diffract, above it the mode mixing and coupling give rise to an unstable output.
We propose two models for the creation of stable dissipative solitons in optical media with the $χ^{(2)}$ (quadratic) nonlinearity. To compensate spatially uniform loss in both the fundamental-frequency (FF) and second-harmonic (SH) components of the system, a strongly localized "hot spot", carrying the linear gain, is added, acting either on the FF component, or on the SH one. In both systems, we use numerical methods to find families of dissipative $χ^{(2)}$ solitons pinned to the "hot spot". The shape of the existence and stability domains may be rather complex. An existence boundary for th…
The evidence that double negative media, with an effective negative permittivity, and an effective negative permeability, can be manufactured to operate at frequencies ranging from microwave to optical is ushering in a new era of metamaterials. They are referred to here as 'left-handed', even though a variety of names is evident from the literature. In anticipation of a demand for highly structured integrated practical waveguides, this paper addresses the impact of this type of medium upon waveguides that can be also nonlinear. After an interesting historical overview and an exposure of some s…
We construct families of fundamental, dipole, and tripole solitons in the fractional Schrödinger equation (FSE)\ incorporating self-focusing cubic and defocusing quintic terms modulated by factors $\cos ^{2}x$ and $\sin^{2}x$, respectively. While the fundamental solitons are similar to those in the model with the uniform nonlinearity, the multipole complexes exist only in the presence of the nonlinear lattice. The shapes and stability of all the solitons strongly depend on the Lévy index (LI)\ that determines the FSE fractionality. Stability areas are identified in the plane of LI and propagat…
We suggest an effective method for controlling nonlinear switching in arrays of weakly coupled optical waveguides. We demonstrate the digitized switching of a narrow input beam for up to eleven waveguides in the engineered waveguide arrays.
We study the influences to the discrete soliton (DS) by introducing linearly long-range nonlocal interactions, which give rise to the off-diagonal elements of the linearly coupled matrix in the discrete nonlinear schrodinger equation to be filled by non-zero terms. Theoretical analysis and numerical simulations find that the DS under this circumstance can exhibit strong digital effects: the fundamental DS is a narrow one, which occupies nearly only one waveguide, the dipole and double-monopole solitons, which occupy two waveguides, can be found in self-focusing and -defocusing nonlinearities, …
We propose a novel method based on the so-called shortcut to adiabatic passage techniques to achieve fast compression of temporal solitons in a nonlinear waveguide. We demonstrate that soliton compression could be achieved, in principle, at an arbitrarily small distance by inverse engineering the pulse width and the nonlinearity of the medium. The proposed scheme could possibly be exploited for various short-distance communication protocols and may be even in nonlinear guided wave-optics devices and generation of ultrashort soliton pulses.
The well-known (1+1D) nonlinear Schrödinger equation (NSE) governs the propagation of narrow-band pulses in optical fibers and others one-dimensional structures. For exploration the evolution of broad-band optical pulses (femtosecond and attosecond) it is necessary to use the more general nonlinear amplitude equation (GNAE) which differs from NSE with two additional non-paraxial terms. That is way, it is important to make clear the difference between the solutions of these two equations. We found a new analytical soliton solution of GNAE and compare it with the well-known NSE one. It is shown …
We study theoretically nonlinear propagation of light in a graphene monolayer. We show that the large intrinsic nonlinearity of graphene at optical frequencies enables the formation of quasi one-dimensional self-guided beams (spatial solitons) featuring subwavelength widths at moderate electric-field peak intensities. We also demonstrate a novel class of nonlinear self-confined modes resulting from the hybridization of surface plasmon polaritons with graphene optical solitons.
Vector solitons in nonlinear lattices with competing SPM and XPM terms
Guangzhou Maritime University（广州航海学院）；Tel Aviv University（特拉维夫大学）；Universidad de Tarapacá（塔拉帕卡大学）；Horia Hulubei National Institute of Physics and Nuclear Engineering（霍里亚·胡利贝伊国家物理与核工程研究所）
We address the properties of fully three-dimensional solitons in complex parity-time (PT)-symmetric periodic lattices with focusing Kerr nonlinearity, and uncover that such lattices can stabilize both, fundamental and vortex-carrying soliton states. The imaginary part of the lattice induces internal currents in the solitons that strongly affect their domains of existence and stability. The domain of stability for fundamental solitons can extend nearly up to the PT-symmetry breaking point, where the linear lattice spectrum becomes complex. Vortex solitons feature spatially asymmetric profiles i…
An approximate description of surface waves propagating along periodically grooved surfaces is intuitively developed in the limit where the grooves are narrow relative to the period. Considering acoustic and electromagnetic waves guided by rigid and perfectly conducting gratings, respectively, the wave field is obtained by interrelating elementary approximations obtained in three overlapping spatial domains. Specifically, above the grating and on the scale of the period the grooves are effectively reduced to point resonators characterised by their dimensions as well as the geometry of their ap…
We address the dynamics of two-dimensional (2D) truncated Airy waves and three-component solitons in the system of two fundamental-frequency and second-harmonic fields, coupled by quadratic (chi^(2)) terms. The system models second-harmonic-generating optical media and atomic-molecular mixtures in Bose-Einstein condensates. In addition to stable solitons, the system maintains truncated-Airy-waves states in either one of the fundamental-frequency components, represented by exact solutions, which are stable, unlike the Airy waves in the degenerate (two-component) chi^(2) system. It is also possi…
We demonstrate that spatial solitons undergo two consecutive spontaneous symmetry breakings (SSBs), with the increase of the total power, in nonlinear photonic crystals (PhCs) built as arrays of alternating linear and nonlinear stripes, in the case when maxima of the effective refractive index coincide with minima of the self-focusing coefficient, and vice versa, i.e.,the corresponding linear and nonlinear periodic potentials are in competition. This setting may be induced, as a virtual PhC, by means of the EIT (electromagnetically-induced-transparency) technique, in a uniform optical medium. …
It has been recently discovered that stabilization of two-dimensional (2D) solitons against the critical collapse in media with the cubic nonlinearity by means of nonlinear lattices (NLs) is a challenging problem. We address the 1D version of the problem, i.e., the nonlinear-Schrödinger equation (NLSE) with the quintic or cubic-quintic (CQ) terms, the coefficient in front of which is periodically modulated in space. The models may be realized in optics and Bose-Einstein condensates (BECs). Stability diagrams for the solitons are produced by means of numerical methods and analytical approximati…
Soliton crystals are periodic patterns of multi-spot optical fields formed from either time or space entanglements of equally separated identical high-intensity pulses. These specific nonlinear optical structures have gained interest in recent years with the advent and progress in nonlinear optical fibers and fiber lasers, photonic crystals, wave-guided wave systems and most recently optical ring microresonator devices. In this work an extensive analysis of characteristic features of soliton crystals is carried out, with emphasis on their one-to-one correspondance with Elliptic solitons. In th…
The interaction of moving discrete solitons with a linear Gaussian defect is investigated. Solitons with profiles varying from hyperbolic secant to exponentially localized are considered such that the mobility of soliton is maintained; the condition for which is obtained. Studies on scattering of the soliton by an attractive defect potential reveal the existence of total reflection and transmission windows which become very narrow with increasing initial soliton amplitude. Transmission regions disappear beyond the small-amplitude limit. The regions of complete reflection and partial capture co…
We show theoretically that a photonic topological insulator can support edge solitons that are strongly self-localized and propagate unidirectionally along the lattice edge. The photonic topological insulator consists of a Floquet lattice of coupled helical waveguides, in a medium with local Kerr nonlinearity. The soliton behavior is strongly affected by the topological phase of the linear lattice. The topologically nontrivial phase gives a continuous family of solitons, while the topologically trivial phase gives an embedded soliton that occurs at a single power, and arises from a self-induce…
Dynamics of a chain of interacting parity-time invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrodinger (DNLS) equation is demonstrated. Approximate solutions for solitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart of the discrete equations. These solitons are mobile, featuring nearly elastic collisions. Stationary solutions for narrow solitons, which are immobile due…
We demonstrate a simple method for controlling nonlinear switching of discrete solitons in arrays of weakly coupled optical waveguides, for both cubic and uadratic nonlinearities. Based on the effective discrete nonlinear equations describing the waveguide arrays in the tight-binding approximation, we develop the concept of the array engineering by means of a step-like variation of the waveguide coupling. We demonstrate the digitized switching of a narrow input beam for up to eleven neighboring waveguides, in the case of the cubic nonlinearity, and up to ten waveguides, in the case of quadrati…
The capability to store light for extended periods of time enables optical cavities to act as narrow-band optical filters, whose linewidth corresponds to the cavity's inverse energy storage time. Here, we report on nonlinear filtering of an optical pulse train based on temporal dissipative Kerr solitons in microresonators. Our experimental results in combination with analytical and numerical modelling show that soliton dynamics enables storing information about the system's physical state longer than the cavity's energy storage time, thereby giving rise to a filter width that can be more than …
We reveal the existence of continuous families of guided single-mode solitons in planar waveguides with weakly nonlinear active core and absorbing boundaries. Stable propagation of TE and TM-polarized solitons is accompanied by attenuation of all other modes, i.e. the waveguide features properties of conservative and dissipative systems. If the linear spectrum of the waveguide possesses exceptional points, which occurs in the case of TM-polarization, an originally focusing (defocusing) material nonlinearity may become effectively defocusing (focusing). This occurs due to the geometric phase of…
We demonstrate ultrafast soliton-based nonlinear balancing of dual-core asymmetry in highly nonlinear photonic crystal fiber at sub-nanojoule pulse energy level. The effect of fiber asymmetry was studied experimentally by selective excitation and monitoring of individual fiber cores at different wavelengths between 1500 nm and 1800 nm. Higher energy transfer rate to non-excited core was observed in the case of fast core excitation due to nonlinear asymmetry balancing of temporal solitons, which was confirmed by the dedicated numerical simulations based on the coupled generalized nonlinear Schr…
We theoretically study the spatial focusing of surface polaritons (SPs) in a negative index metamaterial (NIMM)-atomic gas interface waveguide system, based on cross phase modulation (XPM) in a tripod type double electromagnetically induced transparency (EIT) scheme. In the linear region, we realize the low loss stable propagation of SPs, and the group velocities of the probe and signal fields are well matched via double EIT. In the nonlinear region, we show that giant enhancement of the XPM can be obtained. Using a narrow optical soliton in free space, we realize spatial focusing of the SPs s…
We propose a scheme for the coupler-free excitation of surface polaritonic solitons and breathers in a planar plasmonic waveguide structure comprising of a transparent layer, a metal layer and a layer of three-level lambda-type atomic medium. In the proposed system, an enhanced Kerr nonlinearity is achieved via electromagnetically induced transparency (EIT) in the bottom atomic medium, which can be controlled through proper modulation of the frequency detunings and Rabi frequencies of the driving laser fields. This nonlinearity balances the dispersion in the system, thus providing the necessar…
The light-cone gauge approach to $T\overline{T}$ deformed models is used to derive the $T\overline{T}$ deformed matrix nonlinear Schrödinger equation, the Landau--Lifshitz equation, and the Gardner equation. Properties of one-soliton solutions of the $T\overline{T}$ deformed nonlinear Schrödinger and Korteweg--de Vries equations are discussed in detail. The NLS soliton exhibits the recently discussed phenomenon of widening/narrowing width of particles under the $T\overline{T}$ deformation. However, whether the soliton's size is increasing or decreasing depends not only on the sign of the defor…
