Edge Solitons in Nonlinear-Photonic Topological Insulators

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 intersite 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.

Figure 1

(a) Schematic of a photonic lattice of helical waveguides forming a square lattice in the $x-y$ plane. On each sublattice (red and blue; right and left in inset respectively), the helix phase is staggered by $\pi$. (b)–(e) Beam propagation simulation results showing solitonic self-focusing in a topologically nontrivial lattice. The lattice consists of a strip four lattice constants wide, with single-site input on the edge at $z=0$. (b) Peak intensity $I_{\max }$ and inverse participation number ${\mathcal{P}}^{-1}$ at $z=5Z$ (i.e., after five helix cycles), versus input power $P$. Corresponding in-plane intensity profiles at various propagation distances, for (c) $P\to 0$, (d) $P=0.8P_c$, and (e) $P=P_c$, where $P_c$ is the power at which the soliton is most localized (i.e., ${\mathcal{P}}^{-1}$ is the largest).

Figure 2

Self-induced nonlinear edge modes in a topologically trivial lattice. (a),(b) Intensity profiles after propagation through $z=5Z$, starting from a single-site excitation (the dotted circle), for (a) the zero-power (linear) limit $P\to 0$, and (b) input power $P_{\mathrm{es}}$ corresponding to the embedded soliton. (c) Peak intensity $I_{\max }$ and inverse participation ratio ${\mathcal{P}}^{-1}$ at $z=5Z$, versus input power $P$. (d),(e) Local density of states along the edge, for the nontrivial and trivial lattices in the linear limit, plotted using the moving-frame quasienergies defined by ${\beta }_v(k_x)\equiv \beta (k_x)-v{k}_x$.

Figure 3

Stability of nonlinear edge modes in trivial and nontrivial photonic lattices. (a) Relative edge intensity (the ratio of the edge intensity to the initial edge intensity at $z=0$) versus propagation distance $z$. (b) Relative edge intensity at $z=5Z$, after scattering off a defect with the normalized strength $\mathrm{\Delta }\equiv \delta n{k}_{0}Z/(2\pi n_0)$, where $\delta n$ is the detuning of the defect waveguide’s refractive index.

Figure 4

Nonlinear filtering and switching using edge solitons. (a)–(c) Power-dependent filtering by a narrow channel. The intensity profile at $z=8Z$ is plotted for (a) the linear limit $P\to 0$ and (b) the nonlinear regime at $P=0.55\mathrm{MW}$. The input light is injected uniformly on four waveguides marked by green circles, and the lattice boundary is marked by blue dashes. (c) Transmittance through the channel (defined as the total intensity on the edge sites to the right of the channel) versus input power $P$. (d),(e) Optical switching by bulk-edge soliton collisions. (d) Intensity profiles at $z=0$ before the collision, and at $z=5Z$ after the collision. (e) Postcollision peak intensities in the bulk and edge waveguides versus the relative phase $\mathrm{\Delta }\phi$ of the inputs.