Photonic topological insulator in synthetic dimensions.

Topological phases enable protected transport along the edges of materials, offering immunity against scattering from disorder and imperfections. These phases have been demonstrated for electronic systems, electromagnetic waves1-5, cold atoms6,7, acoustics8 and even mechanics9, and their potential applications include spintronics, quantum computing and highly efficient lasers10-12. Typically, the model describing topological insulators is a spatial lattice in two or three dimensions. However, topological edge states have also been observed in a lattice with one spatial dimension and one synthetic dimension (corresponding to the spin modes of an ultracold atom13-15), and atomic modes have been used as synthetic dimensions to demonstrate lattice models and physical phenomena that are not accessible to experiments in spatial lattices13,16,17. In photonics, topological lattices with synthetic dimensions have been proposed for the study of physical phenomena in high dimensions and interacting photons18-22, but so far photonic topological insulators in synthetic dimensions have not been observed. Here we demonstrate experimentally a photonic topological insulator in synthetic dimensions. We fabricate a photonic lattice in which photons are subjected to an effective magnetic field in a space with one spatial dimension and one synthetic modal dimension. Our scheme supports topological edge states in this spatial-modal lattice, resulting in a robust topological state that extends over the bulk of a two-dimensional real-space lattice. Our system can be used to increase the dimensionality of a photonic lattice and induce long-range coupling by design, leading to lattice models that can be used to study unexplored physical phenomena.

Light emission by free electrons in photonic time-crystals.

Photonic time-crystals (PTCs) are spatially homogeneous media whose electromagnetic susceptibility varies periodically in time, causing temporal reflections and refractions for any wave propagating within the medium. The time-reflected and time-refracted waves interfere, giving rise to Floquet modes with momentum bands separated by momentum gaps (rather than energy bands and energy gaps, as in photonic crystals). Here, we present a study on the emission of radiation by free electrons in PTCs. We show that a free electron moving in a PTC spontaneously emits radiation, and when associated with momentum-gap modes, the electron emission process is exponentially amplified by the modulation of the refractive index. Moreover, under strong electron-photon coupling, the quantum formulation reveals that the spontaneous emission into the PTC bandgap experiences destructive quantum interference with the emission of the electron into the PTC band modes, leading to suppression of the interdependent emission. Free-electron physics in PTCs offers a platform for studying a plethora of exciting phenomena, such as radiating dipoles moving at relativistic speeds and highly efficient quantum interactions with free electrons.

Photonic topological Anderson insulators.

The hallmark property of two-dimensional topological insulators is robustness of quantized electronic transport of charge and energy against disorder in the underlying lattice1. That robustness arises from the fact that, in the topological bandgap, such transport can occur only along the edge states, which are immune to backscattering owing to topological protection. However, for sufficiently strong disorder, this bandgap closes and the system as a whole becomes topologically trivial: all states are localized and all transport vanishes in accordance with Anderson localization2,3. The recent suggestion4 that the reverse transition can occur was therefore surprising. In so-called topological Anderson insulators, it has been predicted4 that the emergence of protected edge states and quantized transport can be induced, rather than inhibited, by the addition of sufficient disorder to a topologically trivial insulator. Here we report the experimental demonstration of a photonic topological Anderson insulator. Our experiments are carried out in an array of helical evanescently coupled waveguides in a honeycomb geometry with detuned sublattices. Adding on-site disorder in the form of random variations in the refractive index of the waveguides drives the system from a trivial phase into a topological one. This manifestation of topological Anderson insulator physics shows experimentally that disorder can enhance transport rather than arrest it.

Synthetic-Space Photonic Topological Insulators Utilizing Dynamically Invariant Structure.

Synthetic-space topological insulators are topological systems with at least one spatial dimension replaced by a periodic arrangement of modes, in the form of a ladder of energy levels, cavity modes, or some other sequence of modes. Such systems can significantly enrich the physics of topological insulators, in facilitating higher dimensions, nonlocal coupling, and more. Thus far, all synthetic-space topological insulators relied on active modulation to facilitate transport in the synthetic dimensions. Here, we propose dynamically invariant synthetic-space photonic topological insulators: a two-dimensional evolution-invariant photonic structure exhibiting topological properties in synthetic dimensions. This nonmagnetic structure is static, lacking any kind of modulation in the evolution coordinate, yet it displays an effective magnetic field in synthetic space, characterized by a Chern number of one. We study the evolution of topological states along the edge, and on the interface between two such structures with opposite synthetic-space chirality, and demonstrate their robust unidirectional propagation in the presence of defects and disorder. Such topological structures can be realized in photonics and cold atoms and provide a fundamentally new mechanism for topological insulators.

Photonic Floquet topological insulators.

Topological insulators are a new phase of matter, with the striking property that conduction of electrons occurs only on their surfaces. In two dimensions, electrons on the surface of a topological insulator are not scattered despite defects and disorder, providing robustness akin to that of superconductors. Topological insulators are predicted to have wide-ranging applications in fault-tolerant quantum computing and spintronics. Substantial effort has been directed towards realizing topological insulators for electromagnetic waves. One-dimensional systems with topological edge states have been demonstrated, but these states are zero-dimensional and therefore exhibit no transport properties. Topological protection of microwaves has been observed using a mechanism similar to the quantum Hall effect, by placing a gyromagnetic photonic crystal in an external magnetic field. But because magnetic effects are very weak at optical frequencies, realizing photonic topological insulators with scatter-free edge states requires a fundamentally different mechanism-one that is free of magnetic fields. A number of proposals for photonic topological transport have been put forward recently. One suggested temporal modulation of a photonic crystal, thus breaking time-reversal symmetry and inducing one-way edge states. This is in the spirit of the proposed Floquet topological insulators, in which temporal variations in solid-state systems induce topological edge states. Here we propose and experimentally demonstrate a photonic topological insulator free of external fields and with scatter-free edge transport-a photonic lattice exhibiting topologically protected transport of visible light on the lattice edges. Our system is composed of an array of evanescently coupled helical waveguides arranged in a graphene-like honeycomb lattice. Paraxial diffraction of light is described by a Schrödinger equation where the propagation coordinate (z) acts as 'time'. Thus the helicity of the waveguides breaks z-reversal symmetry as proposed for Floquet topological insulators. This structure results in one-way edge states that are topologically protected from scattering.

Erratum: Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States [Phys. Rev. Lett. 116, 163901 (2016)].

This corrects the article DOI: 10.1103/PhysRevLett.116.163901.

Topological Optical Waveguiding in Silicon and the Transition between Topological and Trivial Defect States.

One-dimensional models with topological band structures represent a simple and versatile platform to demonstrate novel topological concepts. Here we experimentally study topologically protected states in silicon at the interface between two dimer chains with different Zak phases. Furthermore, we propose and demonstrate that, in a system where topological and trivial defect modes coexist, we can probe them independently. Tuning the configuration of the interface, we observe the transition between a single topological defect and a compound trivial defect state. These results provide a new paradigm for topologically protected waveguiding in a complementary metal-oxide-semiconductor compatible platform and highlight the novel concept of isolating topological and trivial defect modes in the same system that can have important implications in topological physics.

Observation of unconventional edge states in 'photonic graphene'.

Graphene, a two-dimensional honeycomb lattice of carbon atoms, has been attracting much interest in recent years. Electrons therein behave as massless relativistic particles, giving rise to strikingly unconventional phenomena. Graphene edge states are essential for understanding the electronic properties of this material. However, the coarse or impure nature of the graphene edges hampers the ability to directly probe the edge states. Perhaps the best example is given by the edge states on the bearded edge that have never been observed-because such an edge is unstable in graphene. Here, we use the optical equivalent of graphene-a photonic honeycomb lattice-to study the edge states and their properties. We directly image the edge states on both the zigzag and bearded edges of this photonic graphene, measure their dispersion properties, and most importantly, find a new type of edge state: one residing on the bearded edge that has never been predicted or observed. This edge state lies near the Van Hove singularity in the edge band structure and can be classified as a Tamm-like state lacking any surface defect. The mechanism underlying its formation may counterintuitively appear in other crystalline systems.

Accelerating Self-Imaging: The Airy-Talbot Effect.

We present a new type of self-imaging phenomenon: self-imaging along curved trajectories. Unlike the Talbot effect, where self-imaging occurs for periodic wave patterns propagating along a straight line, here the field is generally not periodic and is self-imaged along curved trajectories. In the paraxial regime, self-imaging along a parabolic trajectory can ideally go on indefinitely. In the nonparaxial regime the self-imaging is along a circular trajectory and lasts as long as the beam bends. We demonstrate this accelerating self-imaging effect experimentally, and discuss generalizations to higher dimensions.

Observation of a Topological Transition in the Bulk of a Non-Hermitian System.

We present the first experimental observation of a topological transition in a non-Hermitian system. In contrast to standard methods for examining topological properties, which involve probing edge (or surface) states, we monitor the topological transition by employing bulk dynamics only. The system is composed of a lattice of evanescently coupled optical waveguides, and non-Hermitian behavior is engineered by inducing bending loss by spatially "wiggling" every second waveguide.

Self-localized states in photonic topological insulators.

We propose solitons in a photonic topological insulator: self-localized wave packets forming topological edge states residing in the bulk of a nonlinear photonic topological insulator. These self-forming entities exhibit, despite being in the bulk, the property of unidirectional transport, similar to the transport their linear counterparts display on the edge of a topological insulator. In the concrete case of a Floquet topological insulator, such a soliton forms when a wave packet induces, through nonlinearity, a defect region in a honeycomb lattice of helical optical waveguides, and at the same time the wave packet populates a continuously rotating outer (or inner) edge state of that region. The concept is universal and applicable to topological systems with nonlinear response or mean-field interactions.

Nonlinearly induced PT transition in photonic systems.

We study the effect of nonlinearity on systems with periodic parity-time (PT) symmetry, and show that nonlinearity can transform the system from broken to full PT symmetry and vice versa. Furthermore, we show that even when the nonlinearity is insufficient to induce a transition from broken to full PT symmetry, still, the wave functions neither decay nor diverge, despite the fact that the system has a complex eigenvalue spectrum. Rather, the amplitudes of the wavefunctions oscillate around the transition point. Our results apply to a wide variety of systems in optics and beyond.

Amplified emission and lasing in photonic time crystals.

Photonic time crystals (PTCs), materials with a dielectric permittivity that is modulated periodically in time, offer new concepts in light manipulation. We study theoretically the emission of light from a radiation source placed inside a PTC and find that radiation corresponding to the momentum bandgap is exponentially amplified, whether initiated by a macroscopic source, an atom, or vacuum fluctuations, drawing the amplification energy from the modulation. The radiation linewidth becomes narrower with time, eventually becoming monochromatic in the middle of the bandgap, which enables us to propose the concept of nonresonant tunable PTC laser. Finally, we find that the spontaneous decay rate of an atom embedded in a PTC vanishes at the band edge because of the low density of photonic states.

Causality effects on accelerating light pulses.

We study accelerating and decelerating shape-preserving temporal Airy wave-packets propagating in dispersive media. We explore the effects of causality, and find that, whereas decelerating pulses can asymptotically reach zero group velocity, pulses that accelerate towards infinite group velocity inevitably break up, after a specific critical point. The trajectories and the features of causal pulses are analyzed, along with the requirements for the existence of the critical point and experimental schemes for its observation. Finally, we show that causality imposes similar effects on accelerating pulses in the presence of local Kerr-like nonlinearities.

Topological insulator vertical-cavity laser array.

Topological insulator lasers are arrays of semiconductor lasers that exploit fundamental features of topology to force all emitters to act as a single coherent laser. In this study, we demonstrate a topological insulator vertical-cavity surface-emitting laser (VCSEL) array. Each VCSEL emits vertically, but the in-plane coupling between emitters in the topological-crystalline platform facilitates coherent emission of the whole array. Our topological VCSEL array emits at a single frequency and displays interference, highlighting that the emitters are mutually coherent. Our experiments exemplify the power of topological transport of light: The light spends most of its time oscillating vertically, but the small in-plane coupling is sufficient to force the array of individual emitters to act as a single laser.

Use of polycrystalline Nd:YAG rods to achieve pure radially or azimuthally polarized beams from high-average-power lasers.

We report maintenance of perfect radial-polarization purity in high-power, cw pump chambers using strengthened, polycrystalline Nd:YAG laser rods. Although the cubic symmetry of single-crystal rods caused threefold symmetric birefringence due to shear stresses at the ends of the pumped zone, polycrystalline rods macroscopically behaved as isotropic material and enabled polarization preservation. Elimination of this source of depolarization prevents the major source of bifocusing aberrations in a chain of amplifiers.

Photonic Floquet topological insulators in a fractal lattice.

We present Floquet fractal topological insulators: photonic topological insulators in a fractal-dimensional lattice consisting of helical waveguides. The helical modulation induces an artificial gauge field and leads to a trivial-to-topological phase transition. The quasi-energy spectrum shows the existence of topological edge states corresponding to real-space Chern number 1. We study the propagation of light along the outer edges of the fractal lattice and find that wavepackets move along the edges without penetrating into the bulk or backscattering even in the presence of disorder. In a similar vein, we find that the inner edges of the fractal lattice also exhibit robust transport when the fractal is of sufficiently high generation. Finally, we find topological edge states that span the circumference of a hybrid half-fractal, half-honeycomb lattice, passing from the edge of the honeycomb lattice to the edge of the fractal structure virtually without scattering, despite the transition from two dimensions to a fractal dimension. Our system offers a realizable experimental platform to study topological fractals and provides new directions for exploring topological physics.

Generalized laws of refraction and reflection at interfaces between different photonic artificial gauge fields.

Artificial gauge fields the control over the dynamics of uncharged particles by engineering the potential landscape such that the particles behave as if effective external fields are acting on them. Recent years have witnessed a growing interest in artificial gauge fields generated either by the geometry or by time-dependent modulation, as they have been enablers of topological phenomena and synthetic dimensions in many physical settings, e.g., photonics, cold atoms, and acoustic waves. Here, we formulate and experimentally demonstrate the generalized laws of refraction and reflection at an interface between two regions with different artificial gauge fields. We use the symmetries in the system to obtain the generalized Snell law for such a gauge interface and solve for reflection and transmission. We identify total internal reflection (TIR) and complete transmission and demonstrate the concept in experiments. In addition, we calculate the artificial magnetic flux at the interface of two regions with different artificial gauge fields and present a method to concatenate several gauge interfaces. As an example, we propose a scheme to make a gauge imaging system-a device that can reconstruct (image) the shape of an arbitrary wavepacket launched from a certain position to a predesigned location.

Thermally induced birefringence in nonsymmetrically pumped laser rods and its implications for attainment of good beam quality in high-power, radially polarized lasers.

The ability to generate and then amplify radially polarized light opens up the possibility of achieving very high-power, near diffraction-limited beams from rod-based solid-state lasers. Residual bifocusing rapidly degrades beam quality. Residual bifocusing results from nonradially symmetric pump distributions. We analyze how a nonradially symmetric pump distribution induces a nonradially symmetric stress map. This manifests itself as nonradially symmetric birefringence, and as depolarization to radially/azimuthally polarized beams (or as deterioration in birefringence compensated linearly polarized lasers). Here we analytically describe the birefringence terms of a nonradially symmetric strain map. The model results are supported by radial-depolarization measurements in our 2 kW Nd:YAG pump chambers. For the current level of depolarization, beam quality degradation per rod is DeltaM(2)=4 because of bifocusing alone. The degradation per rod can be reduced substantially by improving pump uniformity.

Topological insulator laser: Theory.

Topological insulators are phases of matter characterized by topological edge states that propagate in a unidirectional manner that is robust to imperfections and disorder. These attributes make topological insulator systems ideal candidates for enabling applications in quantum computation and spintronics. We propose a concept that exploits topological effects in a unique way: the topological insulator laser. These are lasers whose lasing mode exhibits topologically protected transport without magnetic fields. The underlying topological properties lead to a highly efficient laser, robust to defects and disorder, with single-mode lasing even at very high gain values. The topological insulator laser alters current understanding of the interplay between disorder and lasing, and at the same time opens exciting possibilities in topological physics, such as topologically protected transport in systems with gain. On the technological side, the topological insulator laser provides a route to arrays of semiconductor lasers that operate as one single-mode high-power laser coupled efficiently into an output port.