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The paradigmatic example of a topological phase of matter, the two-dimensional Chern insulator1-5, is characterized by a topological invariant consisting of a single integer, the scalar Chern number. Extending the Chern insulator phase from two to three dimensions requires generalization of the Chern number to a three-vector6,7, similar to the three-dimensional (3D) quantum Hall effect8-13. Such Chern vectors for 3D Chern insulators have never been explored experimentally. Here we use magnetically tunable 3D photonic crystals to achieve the experimental demonstration of Chern vectors and their topological surface states. We demonstrate Chern vector magnitudes of up to six, higher than all scalar Chern numbers previously realized in topological materials. The isofrequency contours formed by the topological surface states in the surface Brillouin zone form torus knots or links, whose characteristic integers are determined by the Chern vectors. We demonstrate a sample with surface states forming a (2, 2) torus link or Hopf link in the surface Brillouin zone, which is topologically distinct from the surface states of other 3D topological phases. These results establish the Chern vector as an intrinsic bulk topological invariant in 3D topological materials, with surface states possessing unique topological characteristics. Ministry of Education (MOE) National Research Foundation (NRF) Submitted/Accepted version We acknowledge funding from the Singapore National Research Foundation Competitive Research Program (grant no. NRF-CRP23-2019-0007) and Singapore Ministry of Education Academic Research Fund Tier 3 (grant no. MOE2016-T3-1-006). P.Z., Z.G. and Y.Y. acknowledge funding from the National Natural Science Foundation of China (grant nos. 52022018, 52021001, 12104211, 6101020101 and 62175215) and Chinese Academy of Engineering (grant no. 2022-XY-127).

Terahertz (THz) radiation has broad applications ranging from medical imaging to spectroscopy due to its useful properties, such as strong absorption by organic materials. One viable source of high-intensity THz radiation is the Smith-Purcell (SP) effect, which 1 involves charge carriers moving over a periodic surface. Conventional SP emitters use electron beams to generate charge carriers, necessitating bulky electron acceleration stages. Here, we propose a compact design for generating THz SP radiation using mobile charge carriers within 2D materials. This circumvents the beam alignment and beam divergence challenge, allowing for a reduction in the electron-grating separation from tens of nm to 5 nm or less, leading to more efficient near-field excitation and a potentially chip-level THz source. In such a configuration, we show that the optimal electron velocity and the corresponding maximum radiation intensity can be predicted from the electron-grating separation. For graphene on a silicon grating, we numerically demonstrate how SP radiation is excited by hot electrons, including how the radiation intensity can be enhanced by graphene surface plasmons and modified by tuning the Fermi level in the graphene sheet. Due to the high carrier concentration in graphene, the radiation intensity may be further enhanced through coherent interference. This study can be extended to a broad variety of charge carriers in 2D materials, including electrons, holes, and trions, mobilized through various means such as photoexcitation and external electric fields. Utilizing intrinsic mobile carriers in 2D materials may thus allow for compact, tunable, and low-cost THz sources. Agency for Science, Technology and Research (A*STAR) National Research Foundation (NRF) Submitted/Accepted version This work was supported by the National Research Foundation Singapore via Grant No. NRF-CRP26-2021-0004. L.W. gratefully acknowledges the Singapore University of Technology and Design for the Start-Up Research Grant SRG SMT 2021169 and Kickstarter Initiative (SKI) SKI 2021-02-14, and the National Re-search Foundation Singapore via Grant No. NRF2021-QEP2-02-P03 and NRF2021-QEP2-03-P09. S.L. acknowledges the A∗STAR National Science Scholarship. W.J.D acknowledges the support of the National Research Foundation Singapore via Grant No. NRF-CRP17-2017-08. L.J.W. acknowledges the support of the National Research Foundation Singapore Grant No. NRF2020-NRF-ISF004-3525. J.H.T. acknowledges the support of the Agency for Science, Technology and Research (A*STAR) AME IRG GrantA20E5c0084.

This is an introduction to the feature issue of Optical Materials Express on Non-Hermitian Optics and Photonics.

Rapid scaling of semiconductor devices has led to an increase in the number of processor cores and integrated functionalities onto a single chip to support the growing demands of high-speed and large-volume consumer electronics. To meet this burgeoning demand, an improved interconnect capacity in terms of bandwidth density and active tunability is required for enhanced throughput and energy efficiency. Low-loss terahertz silicon interconnects with larger bandwidth offer a solution for the existing inter-/intrachip bandwidth density and energy-efficiency bottleneck. Here, a low-loss terahertz topological interconnect-cavity system is presented that can actively route signals through sharp bends, by critically coupling to a topological cavity with an ultrahigh-quality (Q) factor of 0.2 × 10

Topological photonics seeks to control the behaviour of the light through the design of protected topological modes in photonic structures. While this approach originated from studying the behaviour of electrons in solid-state materials, it has since blossomed into a field that is at the very forefront of the search for new topological types of matter. This can have real implications for future technologies by harnessing the robustness of topological photonics for applications in photonics devices. This roadmap surveys some of the main emerging areas of research within topological photonics, with a special attention to questions in fundamental science, which photonics is in an ideal position to address. Each section provides an overview of the current and future challenges within a part of the field, highlighting the most exciting opportunities for future research and developments., JPhys Photonics, 4 (3), ISSN:2515-7647

Exponential growth in data rate demands has driven efforts to develop novel beamforming techniques for realizing massive multiple-input and multiple-output (MIMO) systems in sixth-generation (6G) terabits per second wireless communications. Existing beamforming techniques rely on conventional optimization algorithms that are too computationally expensive for real-time applications and require complex digital processing yet to be achieved for phased array antennas at terahertz frequencies. Here, we develop an intelligent and self-adaptive beamforming scheme enabled by deep reinforcement learning, which can predict the spatial phase profiles required to produce arbitrary desired radiation patterns in real-time. Our deep learning model adaptively trains an artificial neural network in real-time by comparing the input and predicted intensity patterns via automatic differentiation of the phase-to-intensity function. As a proof of concept, we experimentally demonstrate two-dimensional beamforming by spatially modulating broadband terahertz waves using silicon metasurfaces designed with the aid of the deep learning model. Our work offers an efficient and robust deep learning model for real-time self-adaptive beamforming to enable multi-user massive MIMO systems for 6G terahertz wireless communications, as well as intelligent metasurfaces for other terahertz applications in imaging and sensing. National Research Foundation (NRF) Published version YJ Tan acknowledges the A*STAR Graduate Scholarship funding from the Agency for Science, Technology and Research, Singapore. The authors also acknowledge the funding support from the National Research Foundation, Singapore (Grant No. NRF-CRP23-2019-0005).

The topological characterization of nonequilibrium topological matter is highly nontrivial because familiar approaches designed for equilibrium topological phases may not apply. In the presence of crystal symmetry, Floquet topological insulator states cannot be easily distinguished from normal insulators by a set of symmetry eigenvalues at high symmetry points in the Brillouin zone. This work advocates a physically motivated, easy-to-implement approach to enhance the symmetry analysis to distinguish between a variety of Floquet topological phases. Using a two-dimensional inversion-symmetric periodically-driven system as an example, we show that the symmetry eigenvalues for anomalous Floquet topological states, of both first-order and second-order, are the same as for normal atomic insulators. However, the topological states can be distinguished from one another and from normal insulators by inspecting the occurrence of stable symmetry inversion points in their microscopic dynamics. The analysis points to a simple picture for understanding how topological boundary states can coexist with localized bulk states in anomalous Floquet topological phases., Comment: 4+ pages of main text, plus 3 pages of supplementary material, 7 figures in total, comments welcome

Nonlinear transmission lines (NLTLs) are nonlinear electronic circuits commonly used for parametric amplification and pulse generation. It has previously been shown that harmonic generation can be enhanced, and shock waves suppressed, in so-called "left-handed" NLTLs, a manifestation of the unique properties of left-handed media. Here, we demonstrate that harmonic generation in a left-handed NLTL can be greatly increased by the presence of a topological edge state. Our NLTL is a nonlinear analogue of the Su-Schrieffer-Heeger (SSH) lattice. Recent studies of nonlinear SSH circuits have investigated the solitonic and self-focusing behaviors of modes at the fundamental harmonic. We find, however, that frequency-mixing processes in an SSH NLTL have important effects that have previously been neglected. The presence of a topological edge mode at the first harmonic can produce strong higher-harmonic signals that propagate into the lattice, acting as an effectively nonlocal cross-phase nonlinearity. We observe maximum third-harmonic signal intensities that are 5 times that of a comparable left-handed NLTL of a conventional design, and a 250-fold intensity contrast between the topologically nontrivial and trivial lattice configurations. Our work may have applications for compact electronic frequency generators, as well as for advancing the fundamental understanding of the effects of nonlinearities on topological states., 15 pages, 9 figures

Confining photons in a finite volume is highly desirable in modern photonic devices, such as waveguides, lasers and cavities. Decades ago, this motivated the study and application of photonic crystals, which have a photonic bandgap that forbids light propagation in all directions1–3. Recently, inspired by the discoveries of topological insulators4,5, the confinement of photons with topological protection has been demonstrated in two-dimensional (2D) photonic structures known as photonic topological insulators6–8, with promising applications in topological lasers9,10 and robust optical delay lines11. However, a fully three-dimensional (3D) topological photonic bandgap has not been achieved. Here we experimentally demonstrate a 3D photonic topological insulator with an extremely wide (more than 25 per cent bandwidth) 3D topological bandgap. The composite material (metallic patterns on printed circuit boards) consists of split-ring resonators (classical electromagnetic artificial atoms) with strong magneto-electric coupling and behaves like a ‘weak’ topological insulator (that is, with an even number of surface Dirac cones), or a stack of 2D quantum spin Hall insulators. Using direct field measurements, we map out both the gapped bulk band structure and the Dirac-like dispersion of the photonic surface states, and demonstrate robust photonic propagation along a non-planar surface. Our work extends the family of 3D topological insulators from fermions to bosons and paves the way for applications in topological photonic cavities, circuits and lasers in 3D geometries. A three-dimensional photonic topological insulator is presented, made of split-ring resonators with strong magneto-electric coupling, which has an extremely wide topological bandgap, forbidding light propagation.

In disordered two dimensional Chern insulators, a single bulk extended mode is predicted to exist per band, up to a critical disorder strength; all the other bulk modes are localized. This behavior contrasts strongly with topologically trivial two-dimensional phases, whose modes all become localized in the presence of disorder. Using a tight-binding model of a realistic photonic Chern insulator, we show that delocalized bulk eigenstates can be observed in an experimentally realistic setting. This requires the selective use of resonator losses to suppress topological edge states, and acquiring sufficiently large ensemble sizes using variable resonator detunings.

Nonlinearities in lattices with topologically nontrivial band structures can give rise to topological solitons, whose properties differ from both conventional lattice solitons and linear topological boundary states. We show that a Su-Schrieffer-Heeger-type lattice with both nonlinearity and nonreciprocal non-Hermiticity hosts a novel oscillatory soliton, which we call a topological end breather. The end breather is strongly localized to a self-induced topological domain near the end of the lattice, in sharp contrast to the extended topological solitons previously found in one-dimensional lattices. Its stable oscillatory dynamics can be interpreted as a Rabi oscillation between two self-induced topological boundary states, emerging from a combination of chiral lattice symmetry and the non-Hermitian skin effect. This demonstrates that non-Hermitian effects can give rise to a wider variety of topological solitons than was previously known to exist., 6+4 pages, 3+3 figures, version to appear in PRB as a Letter

We construct an electrical circuit to realize a modified Haldane lattice exhibiting the phenomenon of antichiral edge states. The circuit consists of a network of inductors and capacitors with interconnections reproducing the effects of a magnetic vector potential. The next nearest neighbor hoppings are configured differently from the standard Haldane model, and as predicted by earlier theoretical studies, this gives rise to antichiral edge states that propagate in the same direction on opposite edges and coexist with bulk states at the same frequency. Using pickup coils to measure voltage distributions in the circuit, we experimentally verify the key features of the antichiral edge states, including their group velocities and ability to propagate consistently in a Möbius strip configuration. Ministry of Education (MOE) This work was supported by the National Natural Science Foundation of China (Grant Nos. 11874274, and 12004425), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20170058, and BK20200630), and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). YiDong Chong was supported by the Singapore MOE Academic Research Fund Tier 3 (Grant No. MOE2016-T3-1-006).

Description, Negative refraction induced by synthetic gauge fields is experimentally verified in twisted multilayer acoustic metamaterials., Negative refraction is a counterintuitive wave phenomenon that has inspired the development of metamaterials and metasurfaces with negative refractive indices and surface phase discontinuities, respectively. Recent theories have proposed an alternative mechanism for negative refraction: Synthetic gauge fields, induced by either dynamical modulation or motion, can shift a material’s dispersion in momentum space, forcing a positive refractive index medium to exhibit negative refraction above a certain threshold. However, this phenomenon has not previously been observed. Here, we report on the experimental demonstration of gauge field–induced negative refraction in a twisted bilayer acoustic metamaterial. The synthetic gauge fields arise in a projected two-dimensional geometry and can be continuously tuned by varying the wave number along the third dimension. Gauge field–induced waveguiding with backward-propagating modes is also demonstrated in a trilayer configuration. These results introduce a mechanism for performing wave manipulation in artificially engineered materials.

Floquet higher order topological insulators (FHOTIs) are a novel topological phase that can occur in periodically driven lattices. An appropriate experimental platform to realize FHOTIs has not yet been identified. We introduce a periodically-driven bipartite (two-band) system that hosts FHOTI phases, and predict that this lattice can be realized in experimentally-realistic optical waveguide arrays, similar to those previously used to study anomalous Floquet insulators. The model exhibits interesting phase transitions from first-order to second-order topological matter by tuning a coupling strength parameter, without breaking lattice symmetry. In the FHOTI phase, the lattice hosts corner modes at eigenphase $0$ or $\pi$, which are robust against disorder in the individual couplings., Comment: 5+ pages for the main text, plus 7 pages for the Supplementary Materials, to be submitted to Phys. Rev. B

Efficient nanoscale light sources are sought after for applications such as sensing, imaging, and the development of photonic circuits. In particular, free electron light sources have gained much attention due to their ability to tune and direct light emission. Here, we show that radiation from free electrons passing through a 100 nm wide nanohole can reach as high as 90% of the theoretical limit. This is accomplished through the introduction of a circular nanoridge around the hole to form a structure we call the nanowell. The power radiated from the nanowell exceeds that of a regular nanohole by over 100 times and that of nanoholes surrounded by other features, such as bullseyes, by similar enhancement factors. Upon varying the structural parameters of the nanowell, the peak output wavelength can be tuned over a broad frequency range from the visible to the near-infrared. This reveals a route to extracting power from free electrons via material nanopatterning. Agency for Science, Technology and Research (A*STAR) Ministry of Education (MOE) Nanyang Technological University Published version This research was supported by the Agency for Science, Technology and Research (A∗ STAR) Science and Engineering Research Council (Grant No. A1984c0043) and Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2016-T3-1-006, Tier 1 Grant Nos. RG187/18 and RG148/20, and Tier 2 Grant No. MOE2019- T2-2-085. L.J.W. acknowledges the support of the Nanyang Assistant Professorship Startup Grant. A.N. acknowledges the A∗ STAR Singapore International Graduate Award (SINGA) scholarship

Applying surfactants on submerged oil droplets permits resonators 10 orders of magnitude softer than what solids permit, with accordingly larger Brownian fluctuations. We measure a 150,000 quality factor, 6nm Brownian-fluctuation amplitude, and 155 Hz eigenfrequency.

We show that free electron radiation from metal films can be enhanced by over two orders of magnitude with the introduction of a cylindrical nanoridge, compared to previously studied structures such as bullseye gratings.

The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, which was previously limited to topological states at boundaries of materials, to those at boundaries of boundaries, such as corners . So far, all HOTI realisations have assumed static equilibrium described by time-invariant Hamiltonians, without considering time-variant or nonequilibrium properties. On the other hand, there is growing interest in nonequilibrium systems in which time-periodic driving, known as Floquet engineering, can induce unconventional phenomena including Floquet topological phases and time crystals. Recent theories have attemped to combine Floquet engineering and HOTIs, but there has thus far been no experimental realisation. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along z axis serving as an effective time-dependent drive. Direct acoustic measurements reveal Floquet corner states that have time-periodic evolution, whose period can be even longer than the underlying drive, a feature previously predicted for time crystals. The Floquet corner states can exist alongside chiral edge states under topological protection, unlike previous static HOTIs. These results demonstrate the unique space-time dynamic features of Floquet higher-order topology., 6 pages, 4 figures

Unconventional Weyl points (WPs), carrying topological charge 2 or higher, possess interesting properties different from ordinary charge-1 WPs, including multiple Fermi arcs that stretch over a large portion of the Brillouin zone. Thus far, such WPs have been observed in chiral materials and acoustic metamaterials, but there has been no clean demonstration in photonics in which the unconventional photonic WPs are separated from trivial bands. We experimentally realize an ideal symmetry-protected photonic charge-2 WP in a three-dimensional topological chiral microwave metamaterial. We use field mapping to directly observe the projected bulk dispersion, as well as the two long surface arcs that form a noncontractible loop wrapping around the surface Brillouin zone. The surface states span a record-wide frequency window of around 22.7% relative bandwidth. We demonstrate that the surface states exhibit a novel topological self-collimation property and are robust against disorder. This work provides an ideal photonic platform for exploring fundamental physics and applications of unconventional WPs., 6 pages, 4 figures

Recent studies have revealed the counterintuitive possibility that increasing disorder can turn a topologically trivial insulator into a nontrivial insulator, called a topological Anderson insulator (TAI). Here, we propose and experimentally demonstrate a photonic TAI in a two-dimensional disordered gyromagnetic photonic crystal in the microwave regime. We directly observe the disorder-induced topological phase transition from a trivial insulator to a TAI with robust chiral edge states. We also demonstrate topological heterostructures that host edge states at interfaces between domains with different disorder parameters. Ministry of Education (MOE) Published version The authors thank Professor Terry Loring for helpful discussions. This work was sponsored by the Singapore Ministry of Education under Grants No. MOE2016-T3-1- 006, No. MOE2018-T2-1-022 (S), Tier 1 RG187/18, and by National Key Research and Development Program of China under Grant No. 2016YFB1200100, and by the program of China Scholarships Council under Grant No. 201806075001, and by the National Natural Science Foundation of China under Grant No. 11774137, and by State Key Laboratory of Acoustics, Chinese Academy of Science under Grant No. SKLA202016.

Topological defects (TDs) in crystal lattices are elementary lattice imperfections that cannot be removed by local perturbations, due to their real-space topology. In the emerging field of topological photonics, photonic topological edge states arise from the nontrivial topology of the band structure defined in momentum space and are generally protected against defects. Here we show that adding TDs into a valley photonic crystal generates a lattice disclination that acts like a domain wall and hosts photonic topological edge states. Unlike previous topological waveguides, the disclination forms an open arc and functions as a free-form waveguide connecting a pair of TDs of opposite topological charge. This interplay between the real-space topology of lattice defects and momentum-space band topology provides a novel scheme to implement large-scale photonic structures with complex arrangements of robust topological waveguides and resonators. Ministry of Education (MOE) Published version This work was supported by the Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2016- T3-1-006, Tier 1 Grants No. RG187/18, and Tier 2 Grant No. MOE2018-T2-1-022(S).

Multimode fibers (MMFs) have the potential to carry complex images for endoscopy and related applications, but decoding the complex speckle patterns produced by mode-mixing and modal dispersion in MMFs is a serious challenge. Several groups have recently shown that convolutional neural networks (CNNs) can be trained to perform high-fidelity MMF image reconstruction. We find that a considerably simpler neural network architecture, the single hidden layer dense neural network, performs at least as well as previously-used CNNs in terms of image reconstruction fidelity, and is superior in terms of training time and computing resources required. The trained networks can accurately reconstruct MMF images collected over a week after the cessation of the training set, with the dense network performing as well as the CNN over the entire period., 17 pages, 10 figures

Berry phase associated with energy bands in crystals can lead to quantised observables like quantised dipole polarizations in one-dimensional topological insulators. Recent theories have generalised the concept of quantised dipoles to multipoles, resulting in the discovery of multipole topological insulators which exhibit a hierarchy of multipole topology: a quantised octupole moment in a three-dimensional bulk induces quantised quadrupole moments on its two-dimensional surfaces, which in turn induce quantised dipole moments on one-dimensional hinges. Here, we report on the realisation of an octupole topological insulator in a three-dimensional acoustic metamaterial. We observe zero-dimensional topological corner states, one-dimensional gapped hinge states, two-dimensional gapped surface states, and three-dimensional gapped bulk states, representing the hierarchy of octupole, quadrupole and dipole moments. Conditions for forming a nontrivial octupole moment are demonstrated by comparisons with two different lattice configurations having trivial octupole moments. Our work establishes the multipole topology and its full hierarchy in three-dimensional geometries., The concept of topological corner states in two dimensional topological insulators can be generalised to higher dimensions. Here, authors present a three dimensional acoustic metamaterial that exhibits the full hierarchy of topological multipole states including corner, hinge, surface and bulk states.

We present a theoretical and experimental study of topological modes running along string-like disclinations in two-dimensional electromagnetic lattices. Two-dimensional honeycomb lattices can host lattice defects of five- and seven-membered rings, which serve as the fixed termination points for string-like disclination. The strings act upon the Dirac cone states in a manner analogous to Dirac strings. When a band gap is opened by breaking sublattice symmetry, gap-spanning interface states appear along the strings, similar to topological kink states in valley Hall systems. But unlike the valley Hall case, the edges can have arbitrary orientations and terminate at fixed points in the bulk. The theoretical and numerical analysis is rigorously verified through microwave experiments.

The classification of topological insulators predicts the existence of high-dimensional topological phases that cannot occur in real materials, as these are limited to three or fewer spatial dimensions. We use electric circuits to experimentally implement a four-dimensional (4D) topological lattice. The lattice dimensionality is established by circuit connections, and not by mapping to a lower-dimensional system. On the lattice’s three-dimensional surface, we observe topological surface states that are associated with a nonzero second Chern number but vanishing first Chern numbers. The 4D lattice belongs to symmetry class AI, which refers to time-reversal-invariant and spinless systems with no special spatial symmetry. Class AI is topologically trivial in one to three spatial dimensions, so 4D is the lowest possible dimension for achieving a topological insulator in this class. This work paves the way to the use of electric circuits for exploring high-dimensional topological models., Higher-dimensional topological phases are predicted but cannot be realised in real materials as they are limited to three or fewer dimensions. Here, Wang et al. realise a four-dimensional topological insulator associated with a nonzero second Chern number using electric circuits.

Crystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a one-dimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a one-dimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in three-dimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space., Here, the authors introduce a 3D Weyl metamaterial hosting modes bound to a 1D topological lattice defect. The modes carry nonzero orbital angular momentum locked to the direction of propagation, and they experimentally demonstrate the ability to emit acoustic vortices into free space.

The current understanding of topological insulators and their classical wave analogs, such as photonic topological insulators, is mainly based on topological band theory. However, standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only short-range order, exhibiting unique phenomena such as the glass-to-liquid transition. Here, we experimentally investigate amorphous variants of a Chern number-based photonic topological insulator. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic bandgap materials., Photonics: finding one-way pathways in unlikely places Optical circuits that provide high-speed data traffic even in the presence of unexpected defects can now be fabricated from materials with less than perfect crystallinity. In photonic topological insulators, interactions between crystal energy bands produce unique ‘edge states’ that enable photons to move in one direction. Yidong Chong and Baile Zhang from Nanyang Technological University in Singapore and colleagues have investigated how disorder impacts the formation of these edge states. The team placed gyromagnetic cylindrical rods into a waveguide and arranged them in patterns ranging from periodic to completely unstructured. Probing the waveguide with microwave radiation revealed that short-range interactions enabled the one-way states to persist in amorphous structures until the lattice transformed into a liquid-like structure. The amorphous edge states proved robust enough to pass photons around rectangular obstacles and across large air gaps.

Random lasers employing multiple scattering and interference processes in highly disordered media have been studied for several decades. However, it remains a challenge to achieve a broadband multimode random laser with high scattering efficiency, particularly at long wavelengths. Here, we develop a new class of strongly multimode random lasers in the terahertz (THz) frequency range in which optical feedback is provided by multiple scattering from metal pillars embedded in a quantum cascade (QC) gain medium. Compared with the dielectric pillars or air hole approaches used in previous random lasers, metal pillars provide high scattering efficiency over a broader range of frequencies and with low ohmic losses. Complex emission spectra are observed with over 25 emission peaks across a 0.4 THz frequency range, limited primarily by the gain bandwidth of the QC wafer employed. The experimental results are corroborated by numerical simulations that show the lasing modes are strongly localized.

Weyl fermions are hypothetical two-component massless relativistic particles in three-dimensional (3D) space, proposed by Hermann Weyl in 1929. Their band-crossing points, called ‘Weyl points’, carry a topological charge and are therefore highly robust. There has been much excitement over recent observations of Weyl points in microwave photonic crystals and the semimetal TaAs. Here, we report on the experimental observation of ‘type-II’ Weyl points of light at optical frequencies, with the photons having a strictly positive group velocity along one spatial direction. We use a 3D structure consisting of laser-written waveguides, and show the presence of type-II Weyl points by observing conical diffraction along one axis when the frequency is tuned to the Weyl point; and observing the associated Fermi arc-like surface states. The realization of Weyl points at optical frequencies allows these novel electromagnetic modes to be further explored in the context of linear, nonlinear, and quantum optics. NRF (Natl Research Foundation, S’pore) MOE (Min. of Education, S’pore) Accepted version

Non-Hermitian systems containing gain or loss commonly host exceptional point degeneracies, not the diabolic points that, in Hermitian systems, play a key role in topological transitions and related phenomena. Non-Hermitian Hamiltonians with parity-time symmetry can have real spectra but generally nonorthogonal eigenstates, impeding the emergence of diabolic points. We introduce a pair of symmetries that induce not only real eigenvalues but also pairwise eigenstate orthogonality. This allows non-Hermitian systems to host Dirac points and other diabolic points. We construct non-Hermitian models exhibiting three exemplary phenomena previously limited to the Hermitian regime: Haldane-type topological phase transition, Landau levels without magnetic fields, and Weyl points. This establishes a new connection between non-Hermitian physics and the rich phenomenology of diabolic points. Ministry of Education (MOE) Published version We are grateful to M. C. Rechtsman for helpful discussions. This work was supported by the Singapore MOE Academic Research Fund Tier 3 Grant No. MOE2016-T3- 1-006, Tier 1 Grant No. RG187/18 and Tier 2 Grant No. MOE 2018-T2-1-022(S).

We experimentally realize a photonic analogue of the anomalous quantum Hall insulator using a two-dimensional (2D) array of coupled ring resonators. Similar to the Haldane model, our 2D array is translation invariant, has a zero net gauge flux threading the lattice, and exploits next-nearest neighbor couplings to achieve a topologically nontrivial band gap. Using direct imaging and on-chip transmission measurements, we show that the band gap hosts topologically robust edge states. We demonstrate a topological phase transition to a conventional insulator by frequency detuning the ring resonators and thereby breaking the inversion symmetry of the lattice. Furthermore, the clockwise or the counterclockwise circulation of photons in the ring resonators constitutes a pseudospin degree of freedom. The two pseudospins acquire opposite hopping phases, and their respective edge states propagate in opposite directions. These results are promising for the development of robust reconfigurable integrated nanophotonic devices for applications in classical and quantum information processing. MOE (Min. of Education, S’pore) Published version

Homogenization of layered media beyond the asymptotic (long-wavelength) limit remains a challenge (see e.g. [1, 2] ad references therein). In [3–5], we developed local and nonlocal homogenization procedures valid for any reasonable size and composition of a periodic lattice cell. On the fine scale, fields are approximated via a basis set of Bloch waves propagating in different directions, while the coarse-scale basis consists of the corresponding generalized plane waves. To satisfy Maxwell's boundary conditions as accurately as possible, the amplitudes of the plane waves are calculated as surface averages of the Bloch waves, whereupon the dispersion relations are approximated by solving a least-squares problem [3, 5]. In general, the effective tensor contains, in addition to the permittivity and permeability entries, magnetoelectric coupling terms.

Quantum cascade lasers are compact, electrically pumped light sources in the technologically important mid-infrared and terahertz region of the electromagnetic spectrum1,2. Recently, the concept of topology3 has been expanded from condensed matter physics into photonics4, giving rise to a new type of lasing5,6,7,8 using topologically protected photonic modes that can efficiently bypass corners and defects4. Previous demonstrations of topological lasers have required an external laser source for optical pumping and have operated in the conventional optical frequency regime5,6,7,8. Here we demonstrate an electrically pumped terahertz quantum cascade laser based on topologically protected valley edge states9,10,11. Unlike topological lasers that rely on large-scale features to impart topological protection, our compact design makes use of the valley degree of freedom in photonic crystals10,11, analogous to two-dimensional gapped valleytronic materials12. Lasing with regularly spaced emission peaks occurs in a sharp-cornered triangular cavity, even if perturbations are introduced into the underlying structure, owing to the existence of topologically protected valley edge states that circulate around the cavity without experiencing localization. We probe the properties of the topological lasing modes by adding different outcouplers to the topological cavity. The laser based on valley edge states may open routes to the practical use of topological protection in electrically driven laser sources.

At photonic Dirac points, electromagnetic waves are governed by the same equations as two-component massless relativistic fermions. However, photonic Dirac points are known to occur in pairs in “photonic graphene” and other similar photonic crystals, which necessitates special precautions to excite only one valley state. Systems hosting unpaired photonic Dirac points are significantly harder to realize, as they require broken time-reversal symmetry. Here, we report on the observation of an unpaired Dirac point in a planar two-dimensional photonic crystal. The structure incorporates gyromagnetic materials, which break time-reversal symmetry; the unpaired Dirac point occurs when a parity-breaking parameter is fine-tuned to a topological transition between a photonic Chern insulator and a conventional photonic insulator phase. Evidence for the unpaired Dirac point is provided by transmission and field-mapping experiments, including a demonstration of strongly non-reciprocal reflection. This unpaired Dirac point may have applications in valley filters and angular selective photonic devices., Observing the photonic analogue of an unpaired Dirac point is hard, as it requires breaking the time-reversal symmetry. Here, the authors use gyromagnetic materials to do that, and thus succeed in observing an unpaired Dirac point in a planar photonic crystal operating at microwave frequencies.

A laser consisting of two independently pumped resonators can exhibit mode bifurcations that evolve out of the exceptional points (EPs) of the linear system at threshold. The EPs are non-Hermitian degeneracies occurring at the parity–time-reversal (PT) symmetry-breaking points of the threshold system. Above threshold, the EPs become bifurcations of the nonlinear zero-detuned laser modes, which can be most easily observed by making the gain saturation intensities in the two resonators substantially different. Small pump variations can then switch abruptly between different laser behavior, e.g., between below-threshold and PT-broken single-mode operation. MOE (Min. of Education, S’pore) Published version

The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological “corner states,” as well as three-dimensional (3D) systems having one dimensional (1D) topological “hinge states.” Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamond-lattice acoustic metamaterial. The bulk acoustic band structure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedronlike structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices. Published version

A general homogenization procedure for periodic electromagnetic structures, when applied to layered media with asymmetric lattice cells, yields an effective tensor with magnetoelectric coupling. Accurate results for transmission and reflection are obtained even in cases where classical effective medium theory breaks down. Magnetoelectric coupling accounts for symmetry breaking in reflection and transmission when a non-symmetric structure is illuminated from two opposite sides.