I review the four mechanisms of bound states in the continuum (BICs) in the application of microwave and acoustic cavities open to directional waveguides. The most simple are symmetry-protected BICs, which are localized inside the cavity because of the orthogonality of the eigenmodes to the propagating modes of waveguides. However, the most general and interesting is the Friedrich–Wintgen mechanism, when the BICs are the result of the fully destructive interference of outgoing resonant modes. The third type of BICs, Fabry–Perot BICs, occurs in a double resonator system when each resonator can serve as an ideal mirror. Finally, the accidental BICs can be realized in the open cavities with no symmetry like the open Sinai billiard in which the eigenmode of the resonator can become orthogonal to the continuum of the waveguide accidentally due to a smooth deformation of the eigenmode. We also review the one-dimensional systems in which the BICs occur owing to the fully destructive interference of two waves separated by spin or polarization or by paths in the Aharonov–Bohm rings. We make broad use of the method of effective non-Hermitian Hamiltonian equivalent to the coupled mode theory, which detects BICs by finding zero-width resonances.

In 1985 Fridriech and Wintgen proposed a mechanism for bound states in the continuum based on full destructive interference of two resonances which can be easily applied to the two- and three-dimensional wave systems. Here we explicitly show that this mechanism can be realized in one-dimensional quantum potential well, owing to destructive interference of electron paths with different spin in tilted magnetic field. Due to one-by-one correspondence between the spin of the electron and the polarization state of light, we have found numerous bound states in the continuum in the one-dimensional photonic system and experimentally confirmed them. The experimental set-up consists of the one-dimensional photonic crystal conjugated with a liquid-crystalline anisotropic defect layer and covered by metal film.

We present a two-dimensional photonic crystal design of four defect dielectric rods, which form a microcavity with eigenfrequencies residing in the propagating band of a directional waveguide. In this system, a nonrobust bound state in the continuum (BSC) occurs as a result of full destructive interference of the monopole and quadrupole modes, with the same parity at certain values of the material parameters of the defect rods. Due to the Kerr effect, a robust BSC arises in a self-adaptive way without necessity to tune the material parameters. The absence of the superposition principle in that nonlinear system gives rise to coupling of the BSC with injected light, resulting in a novel transmission resonance.

Bound states in the continuum (BICs) provide a viable way of achieving high-Q resonances in both photonics and acoustics. In this work, we propose a general method of constructing Friedrich-Wintgen (FW) BICs and accidental BICs in a coupled acoustic waveguide-resonator system. We demonstrate that FW BICs can be achieved with arbitrary two degenerate resonances in a closed resonator, regardless of whether they have the same or opposite parity. Moreover, their eigenmode profiles can be arbitrarily engineered by adjusting the position of the attached waveguide. This suggests an effective way of continuously switching the nature of the BICs from FW BICs to symmetry-protected BICs or accidental BICs. Also, such BICs are sustained in the coupled waveguide-resonator system with shapes such as rectangles, ellipses, and rhomboids. These interesting phenomena are well explained by the two-level effective non-Hermitian Hamiltonian, where two strongly coupled degenerate modes play a major role in forming such FW BICs. Additionally, we find that such an open system also supports accidental BICs in geometry space instead of momentum space via tuning the position of the attached waveguide, which is attributed to the quenched coupling between the waveguide and eigenmodes of the closed cavity. Finally, we fabricate a series of three-dimensional coupled resonator waveguides and experimentally verify the existence of FW BICs and accidental BICs by measuring the transmission spectra. Our results complement the current BIC library in acoustics and provide nice routes for designing acoustic devices, such as acoustic absorbers, filters, and sensors.

We study spin-dependent electron transmission through one- and two-dimensional curved waveguides and quantum dots with account of spin-orbit interaction. We prove that for a transmission through an arbitrary structure there is no spin polarization provided the electron transmits in an isolated energy subband and only two leads are attached to the structure. In particular there is no spin polarization in the one-dimensional wire, for which a spin-dependent solution is found analytically. The solution demonstrates the spin evolution as dependent on a length of wire. The numerical solution for transmission of electrons through the two-dimensional curved waveguides coincides with the solution for the one-dimensional wire if the energy of electron is within the first energy subband. In the vicinity of edges of the energy subbands there are sharp anomalies of spin flipping.