Linear and nonlinear traveling edge waves in optical honeycomb lattices

Abstract

Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudo-magnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudo-fields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schrödinger equation along the edge. Novel nonlinear states termed edge solitons are predicted analytically and found numerically.

Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

“…(22)- (23) Details for deriving asymptotic solution (36) to the honeycomb lattice system (17)- (18) are presented here. This analysis generalizes the calculation performed in [23] to cover the more general non-synchronized rotation patterns discussed in this paper. The periodic functions ∆h 21 , ϕ and A are all assumed to depend only on the fast variable ζ = z/ǫ, where |ǫ| ≪ 1 and weak nonlinearity of σ = ǫσ is assumed.…”

“…(17)- (18), and the staggered square lattice (22)- (23). The subscripts x and y denote theî and vector components, respectively.…”

“…(17)- (18) are (18) by Z = c 0 L 0 z gives the system considered in [23], namely for the geometric parameter…”

“…The nonlinear edge mode envelope can be seen as a balance between the lattice induced linear dispersion effects and sufficiently strong beam focusing or defocusing. Indeed, many properties associated with the NLS equation can be found in photonic topological insulators, such as modulational instability [21], band gap solitons [22] and soliton propagation in helically driven photonic graphene [23]. Moreover, we construct nonlinear modes that, to leading order, satisfy the focusing NLS equation and whose corresponding linear dispersion is topologically protected.…”

Edge solitons in Lieb topological Floquet insulator

“…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.…”