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

Abstract

A systematic approach for deriving tight-binding approximations in general longitudinally driven lattices is presented. As prototypes, honeycomb and staggered square lattices are considered. Timereversal symmetry is broken by varying/rotating the waveguides, longitudinally, along the direction of propagation. Different sublattice rotation and structure are allowed. Linear Floquet bands are constructed for intricate sublattice rotation patterns such as counter rotation, phase offset rotation, as well as different lattice sizes and frequencies. An asymptotic analysis of the edge modes, valid in a rapid-spiraling regime, reveals linear and nonlinear envelopes which are governed by linear and nonlinear Schrödinger equations, respectively. Nonlinear states, referred to as topologically protected edge solitons are unidirectional edge modes. Direct numerical simulations for both the linear and nonlinear edge states agree with the asymptotic theory. Topologically protected modes are found; they possess unidirectionality and do not scatter at lattice defect boundaries.

Topological dipole Floquet solitons

“…Nonlinearity allows the formation of edge solitons -unique states that exhibit topological protection and simultaneously feature a rich variety of shapes and interactions. Edge solitons were predicted in photonic Floquet insulators in continuous [26,34,39,40] and discrete [41][42][43][44] models, and in polariton microcavities [28,45,46]. Their counterparts in nontopological photonic graphene were observed in [47].…”

Liquid-crystal-based topological photonics

“…To do so, we need a simplified description of the waveguides and of their couplings that allows to capture the essential features of the system (such as the Rabi oscillations described above) without having to describe the full liquid-crystal configuration. Hence, we use a time-dependent Hückel method (74,75) to develop a TB model for the photonic waveguides (SI Appendix). The TB Hamiltonian H TB obtained by using this method for the evolution of a system of N waveguides reads…”

Radiative decay of edge states in Floquet media

“…As with other perturbations which break time-reversal symmetry, Floquet systems give rise to chirality (uni-directionality) of modes and topological stability against defects. However, Floquet materials have the advantage of a larger design-space, allowing one to tune a system to convert between being topologically trivial and non-trivial [8], and even be more conducive to nonlinearly localized modes [1,2,21,29]. Floquet systems also display a greater variety of topological properties than their static counterparts; these systems are periodic in one additional dimension and hence have a larger family of topological invariants defined on their high dimensional Brillouin zone [34,36].…”

“…Derivation of the pseudo-magnetic potential, A(T ). We briefly explain how a "pseudo-magnetic potential" in (3.12), A(t), arises in the context of nearly monochromatic light-propagation in a planar array of undulating waveguides; see, for example, [1,2,23,31]. Consider a planar array of waveguides, whose index of refraction variations along the transverse (x) direction are described by the timeindependent potential U ε (x); see the left panel of Fig.…”