Edge Solitons Research Articles

P T-symmetric photonic lattices with type-II Dirac cones.

The type-II Dirac cone is a special feature of the band structure, whose Fermi level is represented by a pair of crossing lines. It has been demonstrated that such a structure is useful for investigating topological edge solitons and, more specifically, for mimicking the Klein tunneling. However, it is still not clear what the interplay between type-II Dirac cones and the non-Hermiticity mechanism will result in. Here, this question is addressed; in particular, we report the P T-symmetric photonic lattices with type-II Dirac cones for the first time to our knowledge. We identify a slope-exceptional ring and name it the type-II exceptional ring. We display the restoration of the P T symmetry of the lattice by reducing the separation between the sites in the unit cell. Curiously, the amplitude of the beam during propagation in the non-Hermitian lattice with P T symmetry only decays because of diffraction, whereas in the P T symmetry-broken lattice it will be amplified, even though the beam still diffracts. This work establishes the link between the non-Hermiticity mechanism and the violation of Lorentz invariance in these physical systems.

Floquet valley Hall edge solitons

We introduce traveling Floquet valley Hall edge solitons in a genuinely continuous system consisting of a waveguide array with a dynamically varying domain wall between two honeycomb structures exhibiting broken inversion symmetry. Inversion symmetry in our system is broken due to periodic and out-of-phase longitudinal modulation of the refractive index applied to the constituent sublattices of the honeycomb structure. By combining two honeycomb arrays with different initial phases of refractive index modulation we create a dynamically changing domain wall that supports localized linear Floquet edge states despite the fact that on average two sublattices in each honeycomb structure forming the domain wall have the same refractive index. In the presence of focusing nonlinearity, bright or dark Floquet edge solitons may bifurcate from such linear Floquet edge states. We numerically identified family of these solitons and compared them with the results obtained from the analytical approach, which involved averaging over one longitudinal period in the evolution coordinate. These solitons exhibit localization in both spatial directions – along the interface due to nonlinear self-action and across the interface as the edge states – that allows them to travel along the domain wall over long distances without noticeable shape variations.

Topological solitons in coupled Su-Schrieffer-Heeger waveguide arrays.

We investigate the formation of multipole topological solitons at the edges of two and three coupled parallel Su-Schrieffer-Heeger (SSH) waveguide arrays. We show that independent variations of waveguide spacing in the unit cells (dimers) in coupled waveguide arrays result in the emergence at their edges of several topological edge states with different internal symmetries. The number of emerging edge states is determined by how many arrays are in topologically nontrivial phase. In the presence of nonlinearity, such edge states give rise to families of multipole topological edge solitons with distinct stability properties. Our results illustrate that coupling between quasi-one-dimensional topological structures substantially enriches the variety of stable topological edge solitons existing in them.

Bound-in-continuum-like corner states in the type-II Dirac photonic lattice

Dirac points are special points in the energy band structure of various materials, around which the dispersion is linear. If the corresponding Fermi surface is projected as a pair of crossing lines—or touching cones in two dimensions, the Dirac point is known as the type-II; such points violate the Lorentz invariance. Until now, thanks to its unique characteristics, the Klein tunneling is successfully mimicked and the topological edge solitons are obtained in the type-II Dirac photonic lattices that naturally possess type-II Dirac points. However, the interplay between these points and corner states is still not investigated. Here, we report both linear and nonlinear corner states in the type-II Dirac photonic lattice with elaborate boundaries. The states, classified as in-phase and out-of-phase, hide in the extended bands that are similar to the bound states in the continuum (BIC). We find that the nonlinear BIC-like corner states are remarkably stable. In addition, by removing certain sites, we establish the fractal Sierpiński gasket structure in the type-II Dirac photonic lattice, in which the BIC-like corner states are also demonstrated. The differences between results in the fractal and nonfractal lattices are rather small. Last but not least, the corner breather states are proposed. Our results provide a novel view on the corner states and may inspire fresh ideas on how to manipulate/control the localized states in different photonic lattices.

Vector valley Hall edge solitons in distorted type-II Dirac photonic lattices.

Topological edge states have recently garnered a lot of attention across various fields of physics. The topological edge soliton is a hybrid edge state that is both topologically protected and immune to defects or disorders, and a localized bound state that is diffraction-free, owing to the self-balance of diffraction by nonlinearity. Topological edge solitons hold great potential for on-chip optical functional device fabrication. In this report, we present the discovery of vector valley Hall edge (VHE) solitons in type-II Dirac photonic lattices, formed by breaking lattice inversion symmetry with distortion operations. The distorted lattice features a two-layer domain wall that supports both in-phase and out-of-phase VHE states, appearing in two different band gaps. Superposing soliton envelopes onto VHE states generates bright-bright and bright-dipole vector VHE solitons. The propagation dynamics of such vector solitons reveal a periodic change in their profiles, accompanied by the energy periodically transferring between the layers of the domain wall. The reported vector VHE solitons are found to be metastable.

Topological edge solitons in the non-Hermitian nonlinear Su-Schrieffer-Heeger model

The Su-Schrieffer-Heeger model is the simplest one-dimensional model showing the characteristic features of topological insulators. Its most interesting property is the appearance of a solution of the edge state or edge soliton in a topologically nontrivial phase determined by the system parameters. Recently, the authors have been investigating generalizations of such a system in two different aspects, both through the inclusion of nonlinearity in the model, and considering the effects of gain and loss. This paper provides an example of accounting for both of these mechanisms. It is shown that for a given gain parameter, there is a region of loss parameters where protected edge solitons are also implemented. Passing the critical value of the loss parameter, the stationary edge soliton becomes oscillating. It is interesting that there are regimes in which the edge soliton, while maintaining spatial localization, demonstrates chaotic temporal dynamics. Analytical estimates characterizing the properties of solutions are given.

Zero-energy edge states and solitons in strained photonic graphene

Zero-energy edge states and solitons in strained photonic graphene

Rotating topological edge solitons.

We address the formation of topological edge solitons in rotating Su-Schrieffer-Heeger waveguide arrays. The linear spectrum of the non-rotating topological array is characterized by the presence of a topological gap with two edge states residing in it. Rotation of the array significantly modifies the spectrum and may move these edge states out of the topological gap. Defocusing nonlinearity counteracts this tendency and shifts such modes back into the topological gap, where they acquire the structure of tails typical of topological edge states. We present rich bifurcation structure for rotating topological solitons and show that they can be stable. Rotation of the topologically trivial array, without edge states in its spectrum, also leads to the appearance of localized edge states, but in a trivial semi-infinite gap. Families of rotating edge solitons bifurcating from the trivial linear edge states exist too, and sufficiently strong defocusing nonlinearity can also drive them into the topological gap, qualitatively modifying the structure of their tails.

Topological edge breathers in a nonlinear Su-Schrieffer-Heeger lattice

We show the existence of breathing edge modes in the Su-Schrieffer-Heeger model with cubic (Kerr) on-site nonlinearity, bifurcating from stationary edge solitons with propagation constant inside the topological gap of the linear model. These edge breathers are exact solutions to the nonlinear equations of motion, with time-periodic intensity oscillations and tails exponentially decaying from the edge. They bifurcate from two localized internal eigenmodes of the stationary edge soliton, having eigenfrequencies inside the topological gap and all higher harmonics above the linear spectrum. Numerical Floquet analysis for solutions obtained from a Newton scheme shows that edge breathers may be linearly stable even in regimes of large-amplitude oscillations, mainly manifested as time-periodic power exchange between the edge site and its next-nearest neighbor.

Topological edge solitons in χ(2) waveguide arrays.

We address the formation of χ(2) topological edge solitons emerging in a topologically nontrivial phase in Su-Schrieffer-Heeger (SSH) waveguide arrays. We consider edge solitons, whose fundamental frequency (FF) component belongs to the topological gap, while the phase mismatch determines whether the second harmonic (SH) component falls into topological or trivial forbidden gaps of the spectrum for the SH wave. Two representative types of edge solitons are found, one of which is thresholdless and bifurcates from the topological edge state in the FF component, while the other exists above a power threshold and emanates from the topological edge state in the SH wave. Both types of soliton can be stable. Their stability, localization degree, and internal structure strongly depend on the phase mismatch between the FF and SH waves. Our results open up new prospects for the control of topologically nontrivial states by parametric wave interactions.

Nonlinear Topological Photonic Insulator in Synthetic Space

AbstractAs an emerging field, nonlinear topological photonics attracts great attention nowadays, where pioneering works show interesting phenomena including topological edge solitons and nonlinearity‐induced coupling of light into topological edge states. In this work, the nonlinear effects in a photonic topological insulator in the synthetic space including the frequency axis of light, in a ring resonant array under dynamic modulations are studied, where the four‐wave mixing nonlinear process is included. Such nonlinear process not only induces local interactions, but also creates long‐range interactions in the frequency dimension. These results show destruction on topologically protected edge states when the nonlinear effects are increasing. Moreover, it also breaks the symmetry in the synthetic space as well as the band structure, which reflects different propagation behaviors for the different input sources with positive and negative frequency shifts and exhibits short‐time robustness in weak nonlinear cases. This work hence explores the role of nonlinear effects in affecting the edge modes in the synthetic space, which may be useful in quantum many‐body simulations and find possible applications in information processing.

Vector valley Hall edge solitons in superhoneycomb lattices

Topological edge solitons that bifurcate and inherit topological protection from linear edge states and, therefore, demonstrate immunity to disorder and defects upon propagation, attract considerable attention in a rapidly growing field of topological photonics. Valley Hall systems are especially interesting from the point of view of realization of topological edge solitons because they do not require external or artificial magnetic fields or longitudinal modulations of the underlying potential for the emergence of the topological phases. Here we report on the diverse types of vector valley Hall edge solitons forming at the domain walls between superhoneycomb lattices, including bright-dipole, bright-tripole, dark-bright, and dark-dipole solitons. In contrast to conventional scalar topological solitons, such vector states can be constructed as envelope solitons on the edge states from different branches and with different Bloch momenta. Such vector solitons can be remarkably robust, they show stable long-distance propagation and can bypass sharp bends of the domain wall. The existence of the counter-propagating valley Hall edge solitons at the same domain wall allows us to study their structural robustness upon collisions that can be nearly elastic. Our results illustrate richness of soliton families in the valley Hall systems and open new prospects for the light field manipulation and design of the nonlinear topological functional devices.

Observation of Edge Solitons in Topological Trimer Arrays.

We report the experimental observation of nonlinear light localization and edge soliton formation at the edges of fs-laser written trimer waveguide arrays, where transition from nontopological to topological phases is controlled by the spacing between neighboring trimers. We found that, in the former regime, edge solitons occur only above a considerable power threshold, whereas in the latter one they bifurcate from linear states. Edge solitons are observed in a broad power range where their propagation constant falls into one of the topological gaps of the system, while partial delocalization is observed when considerable nonlinearity drives the propagation constant into an allowed band, causing coupling with bulk modes. Our results provide direct experimental evidence of the coexistence and selective excitation in the same or in different topological gaps of two types of topological edge solitons with different internal structures, which can rarely be observed even in nontopological systems. This also constitutes the first experimental evidence of formation of topological solitons in a nonlinear system with more than one topological gap.

Valley Hall edge solitons in honeycomb lattice with an armchair-type domain wall

Valley Hall edge solitons in honeycomb lattice with an armchair-type domain wall

Floquet Edge Multicolor Solitons

AbstractTopological insulators are unique physical structures that are insulators in their bulk, but support currents at their edges which can be unidirectional and topologically protected from scattering on disorder and inhomogeneities. Photonic topological insulators can be crafted in materials that exhibit a strong nonlinear response, thus opening the door to the exploration of the interplay between nonlinearity and topological effects. Among the fascinating new phenomena arising from this interplay is the formation of topological edge solitons—hybrid asymmetric states localized across and along the interface due to different physical mechanisms. Such solitons have so far been studied only in materials with Kerr‐type, or cubic, nonlinearity. Here, the first example of the topological edge soliton supported by parametric interactions in nonlinear media is presented. Such solitons exist in Floquet topological insulators realized in arrays of helical waveguides made of a phase‐matchable material. Floquet edge solitons bifurcate from topological edge states in the spectrum of the fundamental frequency wave and remain localized over propagation distances drastically exceeding the helix period, while travelling along the edge of the structure. A theory of such states is developed. It is shown that multicolor solitons in a Floquet system exists in the vicinity of (formally infinite) set of linear resonances determined by the Floquet phase matching conditions. Away from resonance, soliton envelopes can be described by a period‐averaged single nonlinear Schrödinger equation with an effective cubic nonlinear coefficient whose magnitude and sign depend on the overall phase‐mismatch between the fundamental frequency and second harmonic waves. Such total phase‐mismatch includes the intrinsic mismatch and the geometrically‐induced mismatch introduced by the array, and its value reveals one of the genuine effects exhibited by the Floquet quadratic solitons. The results open fundamental new prospects for the exploration of a range of parametric frequency‐mixing phenomena in photonic Floquet quadratic nonlinear media.

Valley Hall edge solitons in a photonic graphene.

We predict the existence and study properties of the valley Hall edge solitons in a composite photonic graphene with a domain wall between two honeycomb lattices with broken inversion symmetry. Inversion symmetry in our system is broken due to detuning introduced into constituent sublattices of the honeycomb structure. We show that nonlinear valley Hall edge states with sufficiently high amplitude bifurcating from the linear valley Hall edge state supported by the domain wall, can split into sets of bright spots due to development of the modulational instability, and that such an instability is a precursor for the formation of topological bright valley Hall edge solitons localized due to nonlinear self-action and travelling along the domain wall over large distances. Topological protection of the valley Hall edge solitons is demonstrated by modeling their passage through sharp corners of the Ω-shaped domain wall.

Topological edge solitons and their stability in a nonlinear Su-Schrieffer-Heeger model.

We study continuations of topological edge states in the Su-Schrieffer-Heeger model with on-site cubic (Kerr) nonlinearity, which is a 1D nonlinear photonic topological insulator (TI). Based on the topology of the underlying spatial dynamical system, we establish the existence of nonlinear edge states (edge solitons) for all positive energies in the topological band gap. We discover that these edge solitons are stable at any energy when the ratio between the weak and strong couplings is below a critical value. Above the critical coupling ratio, there are energy intervals where the edge solitons experience an oscillatory instability. Though our paper focuses on a photonic system, we also discuss the broader relevance of our methods and results to 1D nonlinear mechanical TIs.

Four-wave mixing Floquet topological solitons.

We consider a topological Floquet insulator realized as a honeycomb array of helical waveguides imprinted in a weakly birefringent medium. The system accounts for four-wave mixing occurring at a series of resonances arising due to Floquet phase matching. Under these resonant conditions, the system sustains stable linearly polarized and metastable elliptically polarized two-component edge solitons. Coupled nonlinear equations describing the evolution of the envelopes of such solitons are derived.

Dark topological valley Hall edge solitons

Abstract Topological edge solitons propagating along the edge of a photonic topological insulator are localized self-sustained hybrid states that are immune to defects/disorders due to the protection of the edge states stemming from the nontrivial topology of the system. Here, we predict that exceptionally robust dark valley Hall edge solitons may form at the domain walls between two honeycomb lattices with broken inversion symmetry. The underlying structure can be created with femtosecond laser inscription, it possesses a large bandgap where well-localized dark edge solitons form, and in contrast to systems with broken time-reversal symmetry, it does not require external magnetic fields or complex longitudinal waveguide modulations for the realization of the topological phase. We present the envelope equation allowing constructing dark valley Hall edge solitons analytically. Such solitons propagate without radiation into the bulk of the lattice and can circumvent sharp corners, which allows observing their persistent circulation along the closed triangular domain wall boundary. They survive over huge distances even in the presence of disorder in the underlying lattice. We also investigate interactions of closely located dark topological valley Hall edge solitons and show that they are repulsive and lead to the formation of two gray edge solitons, moving with different group velocities departing from group velocity of the linear edge state on which initial dark solitons were constructed. Our results illustrate that nonlinear valley Hall systems can support a rich variety of new self-sustained topological states and may inspire their investigation in other nonlinear systems, such as atomic vapors and polariton condensates.

Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse.

The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In the case of localized simple-wave pulses propagating through a quiescent medium, this theory provided a new approach to derivation of an asymptotic formula for the number of solitons eventually produced from such a pulse.