Interference traps waves in an open system: bound states in the continuum

I review the four mechanisms of bound states in the continuum (BICs) in application to microwave and acoustic cavities open to directional waveguides. The most simple are the 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 result of full destructive interference of outgoing resonant modes. The third type of the BICs, the Fabry-Perot BICs, occur in a double resonator system when each resonator can serve as an ideal mirror. At last, 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 by a smooth deformation of the eigenmode. We also review the one-dimensional systems in which the BICs occur owing to full destructive interference of two waves separated by spin or polarization or by paths in the Aharonov-Bohm rings. We widely use the method of effective non-Hermitian Hamiltonian equivalent to the coupled mode theory which detects bound states in the continuum (BICs) by finding zero widths resonances.