Edge Solitons in Nonlinear-Photonic Topological Insulators

Abstract

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-induced local nonlinear shift in the inter-site coupling. The solitons can be used for nonlinear switching and logical operations, functionalities that have not yet been explored in topological photonics. We demonstrate using solitons to perform selective filtering via propagation through a narrow channel, and using soliton collisions for optical switching.PACS numbers: 42.65. Jx, 42.65.Tg, 42.65.Sf, 42.82.Et Topologically nontrivial photonic bands, analogous to electronic topological insulators, have now been realized and studied in a variety of photonic structures [1][2][3][4][5][6][7][8][9][10]. These "photonic topological insulators" (PTIs) feature edge states that are topologically protected against certain classes of disorder, and have interesting potential applications as robust waveguides and delay lines. Thus far, PTIs have mostly been studied in the linear limit, in which existing concepts of band topology can be directly applied to the electromagnetic wave equations, such as by mapping the propagation equations for a linear photonic lattice to the linear Schrödinger equation [6]. Even recent studies of PTIs arising in optically nonlinear systems, such as exciton-polaritons in quantum wells, have focused on topological edge states that are linear perturbations around a steady-state nonlinear background [11][12][13]. There have been only a handful of investigations into the non-perturbative nonlinear dynamics that could arise in PTIs [14][15][16][17][18]. Notably, Lumer et al. discovered a localized stationary soliton lying in the bulk of a two-dimensional (2D) PTI, which can be interpreted as a point region of a different topological phase that is "selfinduced" by topological edge states circulating around it [14]. Ablowitz et al. have found evidence for moving edge solitons in weakly nonlinear 2D PTIs, though this was done by taking broad envelope superpositions of existing topological edge states, and reducing the system to a 1D nonlinear Schrödinger equation [15]. In 1D lattices, nonlinear dynamics of boundary states and self-induced topological transitions have also been studied [16,17]. This paper describes a class of moving lattice edge solitons that arise in experimentally feasible 2D PTIs with Kerr nonlinearity. Unlike in Ref. [15], the solitons are derived ab initio, without using broad envelope approximations, in a realistic photonic lattice; furthermore, they can arise whether the underlying lattice is top...

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