Quadruplets of exceptional points and bound states in the continuum in dielectric rings

The photonic properties of a narrow dielectric ring with a rectangular cross section are studied analytically and numerically. It is shown that exceptional points in such a resonator exist in pairs, with each point adjacent in parametric space to a bound state in the continuum, resulting in the formation of quadruplets of singular photonic states. The appearance of quadruplets is determined by the interaction of two photonic branches, which can anticross or intersect in parametric space during the transition from the strong to weak coupling regime, which is described by the Friedrich-Wintgen model. A dielectric ring is an ideal object for modeling quadruplets due to the ability to arbitrarily change the shape of a rectangular cross section, that is, to accurately scan the areas of intersection of axial and radial Fabry-P\'erot-like resonances along the height or width of the ring. The key role is played by the internal hole as an additional degree of freedom, which allows one to change the mode coupling coefficient and observe exceptional points. The regimes of electric field concentration inside a narrow dielectric ring at the incidence of a plane electromagnetic wave are also demonstrated. The discovery of quadruplets of singular photonic states is of fundamental importance for the development of the photonics of non-Hermitian structures. The proximity of the exceptional point and the high-$Q$ bound state in the continuum in parametric space allows easy switching between gain and loss regimes and opens up new perspectives for applications.