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 term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work. The fascinating wave phenomenon of ‘bound states in the continuum’ spans different material and wave systems, including electron, electromagnetic and mechanical waves. In this Review, we focus on the common physical mechanisms underlying these bound states, whilst also discussing recent experimental realizations, current applications and future opportunities for research.

Bound states in the continuum

Recent progress in understanding the topological properties of condensed matter has led to the discovery of time-reversal-invariant topological insulators. A remarkable and useful property of these materials is that they support unidirectional spin-polarized propagation at their surfaces. Unfortunately topological insulators are rare among solid-state materials. Using suitably designed electromagnetic media (metamaterials) we theoretically demonstrate a photonic analogue of a topological insulator. We show that metacrystals-superlattices of metamaterials with judiciously designed properties-provide a platform for designing topologically non-trivial photonic states, similar to those that have been identified for condensed-matter topological insulators. The interfaces of the metacrystals support helical edge states that exhibit spin-polarized one-way propagation of photons, robust against disorder. Our results demonstrate the possibility of attaining one-way photon transport without application of external magnetic fields or breaking of time-reversal symmetry. Such spin-polarized one-way transport enables exotic spin-cloaked photon sources that do not obscure each other.

Photonic topological insulators

This review discusses how low-energy, valence excitations created by swift electrons can render information on the optical response of structured materials with unmatched spatial resolution. Electron microscopes are capable of focusing electron beams on sub-nanometer spots and probing the target response either by analyzing electron energy losses or by detecting emitted radiation. Theoretical frameworks suited to calculate the probability of energy loss and light emission (cathodoluminescence) are revisited and compared with experimental results. More precisely, a quantum-mechanical description of the interaction between the electrons and the sample is discussed, followed by a powerful classical dielectric approach that can be in practice applied to more complex systems. We assess the conditions under which classical and quantum-mechanical formulations are equivalent. The excitation of collective modes such as plasmons is studied in bulk materials, planar surfaces, and nanoparticles. Light emission induced by the electrons is shown to constitute an excellent probe of plasmons, combining sub-nanometer resolution in the position of the electron beam with nanometer resolution in the emitted wavelength. Both electron energy-loss and cathodoluminescence spectroscopies performed in a scanning mode of operation yield snap shots of plasmon modes in nanostructures with fine spatial detail as compared to other existing imaging techniques, thus providing an ideal tool for nanophotonics studies.

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Optical excitations in electron microscopy

We demonstrate the self-assembled formation of concentric quantum double rings with high uniformity and excellent rotational symmetry using the droplet epitaxy technique. Varying the growth process conditions can control each ring's size. Photoluminescence spectra emitted from an individual quantum ring complex show peculiar quantized levels that are specified by the carriers' orbital trajectories.

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Self-assembly of concentric quantum double rings

The dispersion relation and the optical transmittance of a two-dimensional photonic crystal composed of the hexagonal array of cylindrical air holes fabricated in a dielectric slab were analyzed by group theory and the numerical calculation based on the finite-difference time-domain method. The decay rate of the leaky modes that exist above the light line (the dispersion relation in air) in the band diagram was also evaluated, from which the absence of the coupling between certain internal eigenmodes and the external radiation field was shown. This phenomenon was related to symmetry mismatching by the group-theoretical argument. It was also shown that a certain leaky band has a quality factor as large as 3000 over its entire spectral range. These features as well as the opaque frequency regions due to symmetry mismatching were clearly demonstrated by the calculated optical transmission spectra.

Dispersion relation and optical transmittance of a hexagonal photonic crystal slab

This paper investigates the topological phase transition in honeycomb lattice photonic crystals with and without time-reversal and space-inversion symmetries through extensive analysis on bulk and edge states. In the system with both the symmetries, there appear multiple Dirac cones in the photonic band structure, and the mass gaps are controllable via symmetry breaking. The zigzag and armchair edges of the photonic crystals can support novel edge states that reflect the symmetries of the photonic crystals. The dispersion relation and the field configuration of the edge states are analyzed in detail in comparison to electronic edge states. Leakage of the edge states to free space, which is inherent in photonic systems, is fully taken into account in the analysis. A topological relation between bulk and edge states, which has been discussed in the context of electronic quantum Hall effect, is also examined in the photonic system with leaky edge states.

Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states

Making use of a droplet-epitaxial technique, we realize nanometer-sized quantum ring complexes, consisting of a well-defined inner ring and an outer ring. Electronic structure inherent in the unique quantum system is analyzed using a microphotoluminescence technique. One advantage of our growth method is that it presents the possibility of varying the ring geometry. Two samples are prepared and studied: a single-wall ring and a concentric double ring. For both samples, highly efficient photoluminescence emitted from a single quantum structure is detected. The spectra show discrete resonance lines, which reflect the quantized nature of the ring-type electronic states. In the concentric double ring, the carrier confinement in the inner ring and that in the outer ring are identified distinctly as split lines. The observed spectra are interpreted on the basis of single electron effective mass calculations.

https://arxiv.org/pdf/cond-mat/0509625.pdf

Optical transitions in quantum ring complexes

This paper investigates the superprism effect in opal-based three-dimensional photonic crystals which consist of an face-centered cubic (fcc) array of close-packed dielectric spheres embedded in a host. The light propagation angle inside the opals is determined by the group velocity vector of the bulk eigenmodes having the strongest coupling with the incident light at the boundary of the opal. It is found that the superprism effect takes place quite generally in the frequency range where many flat bands exist owing to the zone folding. The numerical simulations of the superprism effect in bare opal systems indicate that the effect can be used as frequency selector devices as well as beam splitters.

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Tetsuyuki Ochiai

Papers

Self-assembly of concentric quantum double rings

Dispersion relation and optical transmittance of a hexagonal photonic crystal slab

Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states

Optical transitions in quantum ring complexes

Superprism effect in opal-based photonic crystals