Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

Light propagating in linear and nonlinear waveguide lattices exhibits behaviour characteristic of that encountered in discrete systems. The diffraction properties of these systems can be engineered, which opens up new possibilities for controlling the flow of light that would have been otherwise impossible in the bulk: these effects can be exploited to achieve diffraction-free propagation and minimize the power requirements for nonlinear processes. In two-dimensional networks of waveguides, self-localized states--or discrete solitons--can travel along 'wire-like' paths and can be routed to any destination port. Such possibilities may be useful for photonic switching architectures.

Discretizing light behaviour in linear and nonlinear waveguide lattices.

One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized. As a result, the material is transformed from a conductor to an insulator. Anderson's work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron-electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics. Indeed, both weak and strong localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being 'frozen' fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.

Transport and Anderson localization in disordered two-dimensional photonic lattices

Nonlinear periodic lattices occur in a large variety of systems, such as biological molecules, nonlinear optical waveguides, solid-state systems and Bose-Einstein condensates. The underlying dynamics in these systems is dominated by the interplay between tunnelling between adjacent potential wells and nonlinearity. A balance between these two effects can result in a self-localized state: a lattice or 'discrete' soliton. Direct observation of lattice solitons has so far been limited to one-dimensional systems, namely in arrays of nonlinear optical waveguides. However, many fundamental features are expected to occur in higher dimensions, such as vortex lattice solitons, bright lattice solitons that carry angular momentum, and three-dimensional collisions between lattice solitons. Here, we report the experimental observation of two-dimensional (2D) lattice solitons. We use optical induction, the interference of two or more plane waves in a photosensitive material, to create a 2D photonic lattice in which the solitons form. Our results pave the way for the realization of a variety of nonlinear localization phenomena in photonic lattices and crystals. Finally, our observation directly relates to the proposed lattice solitons in Bose-Einstein condensates, which can be observed in optically induced periodic potentials.

Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices

Discrete Solitons in Optics

We show that discrete solitons can be navigated in two-dimensional nonlinear waveguide arrays. This can be accomplished by using vector interactions between two classes of solitons - signals and blockers. Disrete solitons in such two dimensional array networks exhibit a rich variety of functional operations, e.g. blocking, routing, logic functions, and time-gating.

Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays

We show that reflection losses suffered by discrete solitons along sharp bends in two-dimensional waveguide-array networks can be almost eliminated. Analysis indicates that this can be accomplished by appropriately engineering the corner site of the bend. Our analytical results are verified using numerical simulations.

Minimizing bending losses in two-dimensional discrete soliton networks

We show that the discrete diffraction properties of a nonlinear optical zigzag waveguide array can be significantly modified, by exploiting the topological arrangement of the lattice itself. This introduces extended interactions (beyond nearest-neighbors), which, in turn, affect the lattice dispersion relation within the Brillouin zone. As a result, we demonstrate that new families of discrete soliton solutions are possible which are stable over a wide range of parameters. Our method opens new opportunities for diffraction management that can be employed to generate low power spatial discrete optical solitons.

Discrete solitons in nonlinear zigzag optical waveguide arrays with tailored diffraction properties

We demonstrate that optical discrete solitons are possible in appropriately oriented biased photorefractive crystals. This can be accomplished in optically-induced periodic waveg- uide lattices that are created via plane wave interference. Our method paves the way towards the observation of entirely new families of discrete solitons.

Discrete solitons in photorefractive optically-induced photonic lattices

We report the first experimental observation of discrete solitons in an array of optically-induced waveguides. The waveguide arrays are induced in photorefractives by interfering pairs of plane waves, and the solitons form when the screening nonlinearity is employed. We demonstrate both in-phase and staggered bright solitons in 1-D and 2-D arrays. This constitutes the first experimental observation of 2D discrete solitons in any medium.

Nikos K. Efremidis

Papers

Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays

Minimizing bending losses in two-dimensional discrete soliton networks

Discrete solitons in nonlinear zigzag optical waveguide arrays with tailored diffraction properties

Discrete solitons in photorefractive optically-induced photonic lattices

Discrete solitons in optically-induced photonic lattices