We consider bound states in the continuum (BSCs) or embedded trapped modes in two- and three-dimensional acoustic axisymmetric duct–cavity structures. We demonstrate numerically that, under variation of the length of the cavity, multiple BSCs occur due to the Friedrich–Wintgen two-mode full destructive interference mechanism. The BSCs are detected by tracing the resonant widths to the points of the collapse of Fano resonances where one of the two resonant modes acquires infinite life-time. It is shown that the approach of the acoustic coupled mode theory cast in the truncated form of a two-mode approximation allows us to analytically predict the BSC frequencies and shape functions to a good accuracy in both two and three dimensions.

We consider refractive index sensing with optical bounds states in the continuum (BICs) in dielectric gratings. Applying a perturbative approach we derived the differential sensitivity and the figure of merit of a sensor operating in the spectral vicinity of a BIC. Optimisation design approach for engineering an effective sensor is proposed. An analytic formula for the maximal sensitivity with an optical BIC is derived. The results are supplied with straightforward numerical simulations.

We consider propagating bound states in the continuum in dielectric gratings. The gratings consist of a slab with ridges periodically arranged either on top or on both sides of the slab. Based on the Fourier modal approach, we recover the leaky zones above the line of light to identify the geometries of the gratings supporting Bloch bound states propagating in the direction perpendicular to the ridges. Most importantly, it is demonstrated that if a two-sided grating possesses either mirror or glide symmetry, the Bloch bound states are stable to variation of parameters as long as the above symmetries are preserved.

The article reports on light enhancement by structural resonances in linear periodic arrays of identical dielectric elements. As the basic elements both subwavelength spheres and rods with circular cross section have been considered. In either case it has been demonstrated numerically that high-Q structural resonant modes originated from bound states in the continuum enable near-field amplitude enhancement by factor of 10–25 in the red-to-near infrared range in lossy silicon. The asymptotic behavior of the Q-factor with the number of elements in the array is explained theoretically by analyzing quasi-bound states propagation bands.

We consider light scattering by dielectric gratings supporting optical bound states in the continuum. Due to the presence of instantaneous Kerr nonlinearity critical field enhancement in the spectral vicinity of the bound state triggers the effect of optical bistability. The onset of bistability is explained theoretically in the framework of the temporal coupled-mode theory. As the central result we cast the problem in the form of a single field-driven nonlinear oscillator. The theoretical results are verified in comparison with numerical simulations.

We consider an optical response induced by bound states in the continuum (BICs) in arrays of dielectric spheres. By combining the quasi-mode expansion technique with coupled mode theory (CMT), we put forward a theory of the optical response by high-$Q$ resonance surrounding BICs in momentum space. The central results are analytical expressions for the CMT parameters, which can be easily calculated from the eigenfrequencies and eigenvectors of the interaction matrix of the scattering systems. The results obtained are verified in comparison against exact numerical solutions to demonstrate that the CMT approximation is capable of reproducing Fano features in the spectral vicinity of the BIC. Based on the quasi-mode expansion technique, we derived the asymptotic scaling law for the CMT parameters in the vicinity of the $\mathrm{\Gamma }$-point. It is rigorously demonstrated that the linewidth in the CMT approximation exhibits different asymptotic behavior depending on the symmetry of the BIC.