Bound states in the continuum are waves that, defying conventional wisdom, remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their existence was first proposed in quantum mechanics and, being a general wave phenomenon, later identified in electromagnetic, acoustic, and water waves. They have been studied in a wide variety of material systems such as photonic crystals, optical waveguides and fibers, piezoelectric materials, quantum dots, graphene, and topological insulators. This Review describes recent developments in this field with an emphasis on the physical mechanisms that lead to these unusual states across the seemingly very different platforms. We discuss recent experimental realizations, existing applications, and directions for future work.

⇤ These authors contributed equally to this work.The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. Such systems can be implemented by metallic mirrors, by photonic band-gap materials 1 , by highly disordered media (Anderson localization 2 ) and, for a subset of outgoing waves, by translational symmetry (total internal reflection 1 ) or rotation/reflection symmetry 3, 4 . Exceptions to these examples exist only in theoretical proposals [5][6][7][8] . Here we predict and experimentally demonstrate that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an "embedded eigenvalue" 9 -namely a bound state in a continuum of radiation modes-that is not due to symmetry incompatibility [5][6][7][8][10][11][12][13][14][15][16] . Such a bound 1 state can exist stably in a general class of geometries where all of its radiation amplitudes vanish simultaneously due to destructive interference. This method to trap electromagnetic waves is also applicable to electronic 12 and mechanical waves 14,15 .The propagation of waves can be easily understood from the wave equation, but the localization of waves (creation of bound states) is more complex. Typically, wave localization can only be achieved when suitable outgoing waves either do not exist or are forbidden due to symmetry incompatibility. For electromagnetic waves, this is commonly implemented with metals, photonic bandgaps, or total internal reflections; for electron waves, this is commonly achieved with potential barriers. In 1929, von Neumann and Wigner proposed the first counterexample 10 , in which they designed a quantum potential to trap an electron whose energy would normally allow coupling to outgoing waves. However, such artificially designed potential does not exist in reality. Furthermore, the trapping is destroyed by any generic perturbation to the potential. More recently, other counterexamples have been proposed theoretically in quantum systems 11-13 , photonics 5-8 , acoustic and water waves 14,15 , and mathematics 16 ; the proposed systems in refs. 6 and 14 are most closely related to what is demonstrated here. While no general explanation exists, some cases have been interpreted as two interfering resonances that leaves one resonance with zero width 6,11,12 . Among these many proposals, most cannot be readily realized due to their inherent fragility. A different form of embedded eigenvalue has been realized in symmetry-protected systems 3, 4 , where no outgoing wave exists for modes of a particular symmetry.To show that an optical bound state is feasible even when it is surrounded by symmetrycompatible radiation modes, we consider a practical structure: a dielectric slab with a square array 2 of cylindrical holes (Fig. 1a), an example of photonic crystal (PhC) slab 1 . The periodic geomet...

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.

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...