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

THE subject of quantum electrodynamics is extremely difficult, even for the case of a single electron. The usual method of solving the corresponding wave equation leads to divergent integrals. To avoid these, Prof. P. A. M. Dirac* uses the method of redundant variables. This does not abolish the difficulty, but presents it in a new form, which may be dealt with in two ways. The first of these needs only comparatively simple mathematics and is directly connected with an elegant general scheme, but unfortunately its wave functions apply only to a hypothetical world and so its physical interpretation is indirect. The second way has the advantage of a direct physical interpretation, but the mathematics is so complicated that it has not yet been solved even for what appears to be the simplest possible case. Both methods seem worth further study, failing the discovery of a third which would combine the advantages of both.

http://ieeexplore.ieee.org/iel5/3/22993/01069961.pdf

Quantum electrodynamics

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.

https://link.aps.org/accepted/10.1103/PhysRevLett.113.257401

Topological nature of optical bound states in the continuum.

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Graph models for waves in thin structures

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.

/pdf/existence-theorems-for-trapped-modes-32kghnxnqm.pdf

Existence theorems for trapped modes

We consider a two-dimensional infinitely long acoustic waveguide formed by two parallel lines containing an arbitrarily shaped obstacle The existence of trapped modes that are the eigenfunctions of the Laplace operator in the corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin and Vassiliev (J Fluid Mech 261 1994) for obstacles symmetric about the centreline of the waveguide In our paper we deal with the situation when the obstacle is shifted with respect to the centreline and study the resulting complex resonances, We are particularly interested in those resonances which are perturbations of (real) eigenvalues We study how an eigenvalue becomes a complex resonance moving from the real axis into the upper half-plane as the obstacle is shifted from its original position The shift of the eigenvalue along the imaginary axis is predicted theoretically and the result is compared with numerical computations The total number of resonances lying inside a sequence of expanding circles is also calculated numerically

Complex resonances in acoustic waveguides

Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

A classical pp-wave is a four-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. We generalize this definition to metric compatible spacetimes with torsion and describe basic properties of such spacetimes. We use our generalized pp-waves for constructing new explicit vacuum solutions of quadratic metric-affine gravity.

pp-waves with torsion and metric-affine gravity

The approach of metric-affine field theory is to define spacetime as a real oriented 4-manifold equipped with a metric and an affine connection. The 10 independent components of the metric tensor and the 64 connection coefficients are the unknowns of the theory. We write the Yang-Mills action for the affine connection and vary it both with respect to the metric and the connection. We find a family of spacetimes which are stationary points. These spacetimes are waves of torsion in Minkowski space. We then find a special subfamily of spacetimes with zero Ricci curvature; the latter condition is the Einstein equation describing the absence of sources of gravitation. A detailed examination of this special subfamily suggests the possibility of using it to model the neutrino. Our model naturally contains only two distinct types of particles which may be identified with left-handed neutrinos and right-handed antineutrinos.

https://iopscience.iop.org/article/10.1088/0264-9381/18/12/307/pdf

Dmitri Vassiliev

Papers

Existence theorems for trapped modes

Complex resonances in acoustic waveguides

Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

pp-waves with torsion and metric-affine gravity

Torsion waves in metric-affine field theory