The application of topology, the mathematics studying conserved properties through continuous deformations, is creating new opportunities within photonics, bringing with it theoretical discoveries and a wealth of potential applications. This field was inspired by the discovery of topological insulators, in which interfacial electrons transport without dissipation even in the presence of impurities. Similarly, the use of carefully-designed wave-vector space topologies allows the creation of interfaces that support new states of light with useful and interesting properties. In particular, it suggests the realization of unidirectional waveguides that allow light to flow around large imperfections without back-reflection. The present review explains the underlying principles and highlights how topological effects can be realized in photonic crystals, coupled resonators, metamaterials and quasicrystals.Frequency, wavevector, polarization and phase are degrees of freedom that are often used to describe a photonic system. In the last few years, topology -a property of a photonic material that characterizes the quantized global behavior of the wavefunctions on its entire dispersion band-has been emerging as another indispensable ingredient, opening a path forward to the discovery of fundamentally new states of light and possibly revolutionary applications. Possible practical applications of topological photonics include photonic circuitry less dependent on isolators and slow light insensitive to disorder.Topological ideas in photonics branch from exciting developments in solid-state materials, along with the discovery of new phases of matter called topological insulators [1, 2]. Topological insulators, being insulating in their bulk, conduct electricity on their surfaces without dissipation or backscattering, even in the presence of large impurities. The first example was the integer quantum Hall effect, discovered in 1980. In quantum Hall states, two-dimensional (2D) electrons in a uniform magnetic field form quantized cyclotron orbits of discrete eigenvalues called Landau levels. When the electron energy sits within the energy gap between the Landau levels, the measured edge conductance remains constant within the accuracy of about one part in a billion, regardless of sample details like size, composition and impurity levels. In 1988, Haldane proposed a theoretical model to achieve the same phenomenon but in a periodic system without Landau levels [3], the so-called quantum anomalous Hall effect.Posted on arXiv in 2005, Haldane and Raghu transcribed the key feature of this electronic model into photonics [4,5]. They theoretically proposed the photonic analogue of the quantum (anomalous) Hall effect in photonic crystals [6], the periodic variation of optical materials, molding photons the same way as solids modulating electrons. Three years later, the idea was confirmed by Wang et al., who provided realistic material designs [7] and experimental observations [8]. Those studies spurred numerous subsequent theoretical [9...

No abstract

In 1929, Hermann Weyl derived [1] the massless solutions from the Dirac equation -the relativistic wave equation for electrons. Neutrinos were thought, for decades, to be Weyl fermions until the discovery of the neutrino mass. Moreover, it has been suggested that low energy excitations in condensed matter [2][3][4][5][6][7][8] [16,17]. Nevertheless, Weyl points are yet to be experimentally observed in nature. In this work, we report on precisely such an observation in an inversion-breaking 3D double-gyroid photonic crystal without breaking time-reversal symmetry.Weyl points are sources of quantized Berry flux of ±2π in the momentum space. Their charges can be defined by the corresponding Chern numbers of ±1, as shown in Fig. 1a. So, Weyl points robustly appear in pairs and can only be removed through pair annihilation. Since the Berry curvature is strictly zero under PT symmetry, -the product of parity (P, inversion) and time-reversal symmetry (T ), isolated Weyl points only exist when at least one of P or T is broken. In Ref.[9], frequency-isolated Weyl points were predicted in PT -breaking DG photonic crystals. We chose to break P instead of T , in the experiment, to avoid using magnetic materials and applying static magnetic fields. This also allows our approach to be directly extended to photonic crystals at optical wavelengths. This P-breaking DG is shown in its bodycentered-cubic (bcc) unit cell in Fig. 1b. At the presence of T , there must exist even pairs of Weyl points. The two pairs of Weyl points illustrated in the Brillouin zone (BZ), in Fig. 1c, are thus the minimum number of Weyl points possible. The bandstructure plotted in Fig. 1d shows two linear bandcrossings along Γ − N and Γ − H. The other two Weyl points have identical dispersions due to T .We work at the microwave frequencies around 10GHz for the accessible fabrication of 3D photonic crystal. The current * linglu@mit.edu . The P-breaking defects are introduced in each layer of red gyroid in Fig. 1b.We approximate each gyroid network by three sets of holedrilling, as illustrated in Fig. 2a in a unit cell of the bodycentered-cubic bcc lattice. Similar methods of drilling and angled etching has been used in fabrication of 3D photonic crystals at microwave [18] and near infrared wavelengths [19]. The three cylindrical air holes, of the blue gyroid, alongx,ŷ andẑ go through (0, , 0, 0)a respectively. All air holes have a diameter of 0.54a, where a is the cubic lattice constant. Gyroids approximated by this drilling approach have almost identical bandstructures as those defined by the level-set iso-surfaces in Ref. [9].The second (red) gyroid is the inversion counterpart of the arXiv:1502.03438v1 [cond-mat.mtrl-sci]

Optical bound states in the continuum (BICs) have recently been realized in photonic crystal slabs, where the disappearance of out-of-plane radiation turns leaky resonances into guided modes with infinite lifetimes. We show that such BICs are vortex centers in the polarization directions of far-field radiation. They carry conserved and quantized topological charges, defined by the winding number of the polarization vectors, which ensure their robust existence and govern their generation, evolution, and annihilation. Our findings connect robust BICs in photonics to a wide range of topological physical phenomena.

Weyl points and line nodes are three-dimensional linear point-and line-degeneracies between two bands. In contrast to Dirac points, which are their two-dimensional analogues, Weyl points are stable in the momentum space and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic crystals, based on the double-gyroid structures, exhibiting frequency-isolated Weyl points with intricate phase diagrams. The surface states associated with the non-zero Chern numbers are demonstrated. Line nodes are also found in similar geometries; the associated surface states are shown to be flat bands. Our results are readily experimentally realizable at both microwave and optical frequencies.Two-dimensional (2D) electrons and photons at the energies and frequencies of Dirac points exhibit extraordinary features [1][2][3][4][5]. As the best example, almost all the remarkable properties of graphene are tied to the massless Dirac fermions at its Fermi level [6,7]. Topologically [8], Dirac cones are not only the critical points for 2D phase transitions but also the unique surface manifestation of a topologically gapped 3D bulk. In a similar way, it is expected that if a material could be found that exhibits a 3D linear dispersion relation, it would also display a wide range of interesting physics phenomena. The associated 3D linear pointdegeneracies are called "Weyl points". In the past year, there have been a few studies of Weyl fermions in electronics [9][10][11][12][13][14]. The associated Fermi-arc surface states, quantumHall-effect [15], novel transport properties [16] and the realization of the Adler-Bell-Jackiw anomaly [17] are also expected. However, no observation of Weyl points has been reported. Here, we present a theoretical discovery and detailed numerical investigation of frequency-isolated Weyl points in perturbed double-gyroid(DG) photonic crystals(PhCs) along with their complete phase diagrams and their topologicallyprotected surface states. PhCs containing frequency-isolated linear line-degeneracies, known as "line nodes", and their flatband surface states are also presented. Unlike the proposed Weyl points in electronic system thus far, our predictions in photonics are readily realizable in experiments.Before proceeding, we first point out one intriguing distinction between the 2D Dirac points and the 3D Weyl points. 2D Dirac cones are not robust; they are only protected by the product of time-reversal-symmetry(T) and parity(P, inversion). In 2D, Dirac cone effective Hamiltonian takes the form of H(k) = v x k x σ x + v z k z σ z ; this form is protected by PT (product of P and T) which requires H(k) to be real. Thus, one can open a gap in this dispersion relation upon introducing a perturbation proportional to σ y = 0 −i i 0 that is imaginary;for example, even an infinitesimal perturbation that breaks just P or just T will open a gap. In contrast, 3D Weyl points are topologically protected gapless dispersions robust...

We report a new type of phononic crystals with topologically nontrivial band gaps for both longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design, gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononic analogue of the electronic quantum (anomalous) Hall effect. We investigate the response of both hexagonal and square gyroscopic lattices and observe bulk Chern numbers of 1 and 2, indicating that these structures support single and multimode edge elastic waves immune to backscattering. These robust one-way phononic waveguides could potentially lead to the design of a novel class of surface wave devices that are widely used in electronics, telecommunication, and acoustic imaging. DOI: 10.1103/PhysRevLett.115.104302 PACS numbers: 46.40.Cd, 46.40.Ff, 73.43.-f Topological states in electronic materials, including the quantum Hall effect [1] and topological insulators [2,3], have inspired a number of recent developments in photonics [4,5], phononics [6][7][8][9], and mechanical metamaterials [10][11][12][13]. In particular, in analogy to the quantum anomalous Hall effect [14], one-way electromagnetic waveguides in two-dimensional systems have been realized by breaking time-reversal symmetry [15][16][17][18].Very recently, unidirectional edge channels have been proposed for elastic waves using Coriolis force in a noninertial reference frame [19], but such a rotating frame is very difficult to implement in solid state devices. Moreover, one-way propagation of scalar acoustic waves has also been proposed by introducing rotating fluids [20][21][22]. However, it is important to recognize that elastic waves in solids have both transverse and longitudinal polarizations, while acoustic waves in fluids are purely longitudinal. As a result, it is challenging to achieve topological protection for elastic waves on an integrated platform.Here, we present a robust strategy to create topologically nontrivial edge modes for both longitudinal and transverse polarizations in a solid medium. In particular, we introduce gyroscopic phononic crystals, where each lattice site is coupled with a spinning gyroscope that breaks timereversal symmetry in a well-controlled manner. In both hexagonal and square lattices, gyroscopic coupling opens band gaps that are characterized by Chern numbers of 1 and 2. As a result, at the edge of these lattices both single-mode and multimode one-way elastic waves are observed to propagate around arbitrary defects without backscattering.To start, we consider a hexagonal phononic crystal with equal masses (m 2 ¼ m 1 ) connected by linear springs [red and black rods in Figs. 1(a) and 1(b)]. The resulting unit cell has 4 degrees of freedom specified by the displacements of m 1 and m 2 (U ¼ ½u for wave vectors μ within the first Brillouin zone. Here, ω denotes the angular frequency of the propagating wave and M ¼ diagfm 1 ; m 1 ; m 2 ; m 2 g is the mass matrix. Moreover, K is the 4 × 4 stiffness matrix and is a functi...

The Dirac cone underlies many unique electronic properties of graphene 1 and topological insulators 2 , and its band structure-two conical bands touching at a single point-has also been realized for photons in waveguide arrays 3 , atoms in optical lattices 4 , and through accidental degeneracy 5,6 . Deformations of the Dirac cone often reveal intriguing properties; an example is the quantum Hall effect, where a constant magnetic field breaks the Dirac cone into isolated Landau levels 7 . A seemingly unrelated phenomenon is the exceptional point [8][9][10][11] , also known as the parity-time symmetry breaking point [12][13][14][15] , where two resonances coincide in both their positions and widths. Exceptional points lead to counter-intuitive phenomena such as loss-induced transparency 16 , unidirectional transmission or reflection [17][18][19][20][21][22][23] , and lasers with reversed pump dependence [24][25][26] or single-mode operation 27, 28 . These two fields of research are in fact connected: here we discover the ability of a Dirac cone to evolve into a ring of exceptional points, which we call an "exceptional ring." We experimentally demonstrate this concept in a photonic crystal slab. Angle-resolved reflection measurements of the photonic crystal slab reveal that the peaks of reflectivity follow the conical band structure of a Dirac cone from accidental degeneracy, whereas the complex eigenvalues of the system are deformed into a two-dimensional flat band enclosed by an exceptional ring. This deformation arises from the dissimilar radiation rates of dipole and quadrupole resonances, which play a role analogous to the loss and gain in parity-time symmetric systems. Our results indicate that the radiation that exists in any open system can fundamentally alter its physical properties in ways previously expected only in the presence of material loss and gain.Closed and lossless physical systems are described by Hermitian operators, which guarantee realness of the eigenvalues and a complete set of eigenfunctions that are orthogonal to each 2 other. On the other hand, systems with open boundaries 10, 29 or with material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] are non-Hermitian 8 and have non-orthogonal eigenfunctions with complex eigenvalues where the imaginary part corresponds to decay or growth. The most drastic difference between Hermitian and non-Hermitian systems is that the latter exhibit exceptional points (EPs) where both the real and the imaginary parts of the eigenvalues coalesce. At an EP, two (or more) eigenfunctions collapse into one so the eigenspace no longer forms a complete basis, and this eigenfunction becomes orthogonal to itself under the unconjugated inner product [8][9][10][11] . To date, most studies of EP and its intriguing consequences concern parity-time symmetric systems that rely on material loss and gain [12][13][14][16][17][18][19][20][21][22][23][24][25][26][27][28] , but EP is a general property that requires only non-Hermiticity. Here, we...

We employed ab initio calculations to identify a class of crystalline materials of MSi (M=Fe, Co, Mn, Re, Ru) having double-Weyl points in both their acoustic and optical phonon spectra. They exhibit novel topological points termed "spin-1 Weyl point" at the Brillouin zone center and "charge-2 Dirac point" at the zone corner. The corresponding gapless surface phonon dispersions are two helicoidal sheets whose isofrequency contours form a single noncontractible loop in the surface Brillouin zone. In addition, the global structure of the surface bands can be analytically expressed as double-periodic Weierstrass elliptic functions.