Topological Edge States and Gap Solitons in the Nonlinear Dirac Model

Smirnova, Daria; Smirnov, L. A.; Leykam, Daniel; Kivshar, Yuri S.

doi:10.1002/lpor.201900223

Cited by 86 publications

(69 citation statements)

References 41 publications

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“…Thus far, nonlinear topological effects have been investigated far less than their linear counterparts, although nonlinearity inherently exists in many topological photonic platforms, such as waveguide arrays, coupled resonators, and metamaterials [19][20][21][22][23][24][25][26][27][28][29][30][31] . Seeking unique functionalities and device applications, research in nonlinear topological photonics has been focused mainly on edge solitons in topological structures 21,23,[32][33][34] , nonlinearity-induced topological transitions 24,25 , nonlinear frequency generation [35][36][37] , and topological lasing [38][39][40] . Despite these efforts, the fundamental issue of the nonlinear coupling of eigenmodes in topological systems remains largely unexplored.…”

Section: Introductionmentioning

confidence: 99%

“…SSH models have been implemented in a variety of platforms, including photonics and nanophotonics [43][44][45][46][47][48][49] , plasmonics 50,51 , and quantum optics [52][53][54][55] , and particularly in the context of topological lasing 38,[56][57][58] . Such SSH-type models with driven nonlinearity have also attracted great attention 19,24,30,32,[34][35][36]59 . In particular, nonlinearity has been employed for spectral tuning 30 and time-domain pumping 59 of topological edge states and for the generation of topological gap solitons 32,34 in such systems.…”

Section: Introductionmentioning

confidence: 99%

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Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

et al. 2020

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The flourishing of topological photonics in the last decade was achieved mainly due to developments in linear topological photonic structures. However, when nonlinearity is introduced, many intriguing questions arise. For example, are there universal fingerprints of the underlying topology when modes are coupled by nonlinearity, and what can happen to topological invariants during nonlinear propagation? To explore these questions, we experimentally demonstrate nonlinearity-induced coupling of light into topologically protected edge states using a photonic platform and develop a general theoretical framework for interpreting the mode-coupling dynamics in nonlinear topological systems. Performed on laser-written photonic Su-Schrieffer-Heeger lattices, our experiments show the nonlinear coupling of light into a nontrivial edge or interface defect channel that is otherwise not permissible due to topological protection. Our theory explains all the observations well. Furthermore, we introduce the concepts of inherited and emergent nonlinear topological phenomena as well as a protocol capable of revealing the interplay of nonlinearity and topology. These concepts are applicable to other nonlinear topological systems, both in higher dimensions and beyond our photonic platform.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

et al. 2020

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“…In the literature, this importance of the linear Dirac equation with a varying mass has been noticed [21,22,23]. In the past few years, people began to use similar nonlinear models to describe nonlinear topological modes [24]. The direct reduction of the envelope equation from Maxwell's equations with a modulated honeycomb material weight has not been done before our work.…”

Section: Connections To Previous Studies and Outlinementioning

confidence: 99%

“…The direct reduction of the envelope equation from Maxwell's equations with a modulated honeycomb material weight has not been done before our work. For example, the nonlinear terms in [24] is added artificially without any reasonable explanations. Our current work provides a complete and consistent theory, including the conditions for the existence of Dirac points and the associated linear spectrum of Maxwell's operator, the reduction of the nonlinear envelope equation.…”

Section: Connections To Previous Studies and Outlinementioning

confidence: 99%

Linear and nonlinear electromagnetic waves in modulated honeycomb media

Zhu

2019

Stud Appl Math

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Wave dynamics in topological materials has been widely studied recently. A striking feature is the existence of robust and chiral wave propagations that have potential applications in many fields. A common way to realize such wave patterns is to utilize Dirac points which carry topological indices and is supported by the symmetries of the media. In this work, we investigate these phenomena in photonic media. Starting with Maxwell's equations with a honeycomb material weight as well as the nonlinear Kerr effect, we first prove the existence of Dirac points in the dispersion surfaces of transverse electric and magnetic Maxwell operators under very general assumptions of the material weight. Our assumptions on the material weight are almost the minimal requirements to ensure the existence of Dirac points in a general hexagonal photonic crystal. We then derive the associated wave packet dynamics in the scenario where the honeycomb structure is weakly modulated. It turns out the reduced envelope equation is generally a two-dimensional nonlinear Dirac equation with a spatially varying mass. By studying the reduced envelope equation with a domain-wall-like mass term, we realize the subtle wave motions which are chiral and immune to local defects. The underlying mechanism is the existence of topologically protected linear line modes, also referred to as edge states. However, we show that these robust linear modes do not survive with nonlinearity. We demonstrate the existence of nonlinear line modes, which can propagate in the nonlinear media based on high-accuracy numerical computations. Moreover, we also report a new type of nonlinear modes which are localized in both directions.

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“…Topological effects in optical systems of helical waveguides can be combined with nonlinear self-action, enabling a plethora of phenomena including modulational instabilities [14,15], and the existence of topological edge solitons. Edge solitons have been obtained numerically in continuous [15], discrete [16][17][18], and Dirac [19] models. They are essentially two-dimensional objects, propagating along the boundary of the topological insulator with the velocity imposed by the group velocity of the Floquet edge state on which soliton is constructed.…”

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confidence: 99%

Edge solitons in Lieb topological Floquet insulator

et al. 2020

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We describe topological edge solitons in a continuous dislocated Lieb array of helical waveguides. The linear Floquet spectrum of this structure is characterized by the presence of two topological gaps with edge states residing in them. A focusing nonlinearity enables families of topological edge solitons bifurcating from the linear edge states. Such solitons are localized both along and across the edge of the array. Due to the non-monotonic dependence of the propagation constant of the edge states on the Bloch momentum, one can construct topological edge solitons that either propagate in different directions along the same boundary or do not move. This allows us to study collisions of edge solitons moving in the opposite directions. Such solitons always interpenetrate each other without noticeable radiative losses; however, they exhibit a spatial shift that depends on the initial phase difference.Topological insulation [1, 2] is a fundamental phenomenon that spans across several areas of physics. Topological photonics, initiated by the seminal paper [3], is one of such areas attracting nowadays growing attention [4,5]. Floquet insulators [6,7] are a particular form of topological insulators, which are characterized by special topological invariants [8]. In such systems, a periodic modulation in the evolution coordinate breaks time-reversal symmetry, resulting in the appearance of the in-gap unidirectional edge states. Photonic Floquet topological insulators have been realized with helical waveguide arrays [9] and were also explored in more complex modulated structures [10-12], including quasicrystals [13].Topological effects in optical systems of helical waveguides can be combined with nonlinear self-action, enabling a plethora of phenomena including modulational instabilities [14,15], and the existence of topological edge solitons. Edge solitons have been obtained numerically in continuous [15], discrete [16][17][18], and Dirac [19] models. They are essentially two-dimensional objects, propagating along the boundary of the topological insulator with the velocity imposed by the group velocity of the Floquet edge state on which soliton is constructed. In previously considered Floquet insulators such solitons, obtained for a given interface, were always co-propagating and suffering from considerable radiative losses due to strong localization [15]. In such setting it was practically impossible to implement interactions of edge solitons with appreciably different Bloch momenta of the carrier waves on finite distances in finite samples.In this Letter we show that it is possible to obtain edge solitons that move, along the same edge, in the opposite directions or even remain immobile in specially designed Floquet insulators with non-monotonous topological branches of the spectrum. We obtain them in different gaps of real-world continuous system -dislocated Lieb array of helical waveguides. Counter-propagating Floquet edge solitons interact almost elastically, but acquire phase-dependent spatial shift after collision....

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Topological Edge States and Gap Solitons in the Nonlinear Dirac Model

Cited by 86 publications

References 41 publications

Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

Nontrivial coupling of light into a defect: the interplay of nonlinearity and topology

Linear and nonlinear electromagnetic waves in modulated honeycomb media

Edge solitons in Lieb topological Floquet insulator

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