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

The study of classical wave physics has been reinvigorated by incorporating the concept of the geometric phase, which has its roots in optics, and topological notions that were previously explored in condensed matter physics. Recently, sound waves and a variety of mechanical systems have emerged as excellent platforms that exemplify the universality and diversity of topological phases. In this Review, we introduce the essential physical concepts that underpin various classes of topological phenomena realized in acoustic and mechanical systems: Dirac points, the quantum Hall, quantum spin Hall and valley Hall effects, Floquet topological phases, 3D gapless states and Weyl crystals. This Review describes topological phenomena that can be realized in acoustic and mechanical systems. Methods of symmetry breaking are described, along with the consequences and rich phenomena that emerge.

Topological phases in acoustic and mechanical systems

Advanced Functional Materials

Rapidly growing demands for fast information processing have launched a race for creating compact and highly efficient optical devices that can reliably transmit signals without losses. Recently discovered topological phases of light provide novel opportunities for photonic devices robust against scattering losses and disorder. Combining these topological photonic structures with nonlinear effects will unlock advanced functionalities such as magnet-free nonreciprocity and active tunability. Here, we introduce the emerging field of nonlinear topological photonics and highlight the recent developments in bridging the physics of topological phases with nonlinear optics. This includes the design of novel photonic platforms which combine topological phases of light with appreciable nonlinear response, self-interaction effects leading to edge solitons in topological photonic lattices, frequency conversion, active photonic structures exhibiting lasing from topologically protected modes, and many-body quantum topological phases of light. We also chart future research directions discussing device applications such as mode stabilization in lasers, parametric amplifiers protected against feedback, and ultrafast optical switches employing topological waveguides.

Nonlinear topological photonics

Hydrodynamic and hydromagnetic stability

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 pseudomagnetic 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 pseudofields 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 Schr\"odinger equation along the edge. Nonlinear states termed edge solitons are predicted analytically and are found numerically.

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Linear and nonlinear traveling edge waves in optical honeycomb lattices

/pdf/defect-mediated-snaking-a-new-growth-mechanism-for-localized-305rvjr5q2.pdf

Defect-mediated snaking: A new growth mechanism for localized structures

/pdf/dispersive-shock-waves-in-the-kadomtsev-petviashvili-and-two-4x8mvdgsol.pdf

Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

https://northumbria-test.eprints-hosting.org/id/document/269207

Edge Solitons in a Nonlinear Mechanical Topological Insulator

The propagation of linear and nonlinear edge modes in bounded photonic honeycomb lattices formed by an array of rapidly varying helical waveguides is studied. These edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial). An asymptotic theory is developed that establishes the presence (absence) of typical edge states, including, in particular, armchair and zigzag edge states in the topologically nontrivial (trivial) case. In the presence of topological protection, nonlinear edge solitons can persist over very long distances.

/pdf/strong-transmission-and-reflection-of-edge-modes-in-bounded-1nyguidu8j.pdf

Yi-Ping Ma

Papers

Linear and nonlinear traveling edge waves in optical honeycomb lattices

Defect-mediated snaking: A new growth mechanism for localized structures

Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

Edge Solitons in a Nonlinear Mechanical Topological Insulator

Strong transmission and reflection of edge modes in bounded photonic graphene.