Trapped modes with extremely high quality factor in the subwavelength ring resonator composed of dielectric nanorods

In this paper, we investigate numerically the trapped modes with near zero group velocities supported in the ring array composed of dielectric nanorods, based on a two-dimensional model. Two sorts of trapped modes in the ring array have been identified: the BCR trapped modes which correspond to the bound modes below the light line at the edge of the first Brillouin zone in the corresponding planar structure (namely the infinite linear chain); the quasi-BIC trapped modes corresponding to the bound states in the continuum supported in the infinite linear chain. According to the whispering gallery condition, the BCR trapped modes can be supported in the ring array only when the number of dielectric elements N is even, while the quasi-BIC ones always exist no matter whether N is odd or even. For both two kind of trapped modes, the lowest one of each kind possesses the highest Q factor, which are ~105 for BCR kind and ~1011 for quasi-BIC kind with N=16 respectively, and the radiation loss increases dramatically as the structural resonance increases. Finally, the behavior of the Q factor with N is explained numerically for the lowest one of each kind of trapped modes. The Q factor scales as Q~exp(0.662N) for the quasi-BIC trapped mode and Q~exp(0.325N) for the BCR one. Intriguingly, the Q factor of the quasi-BIC trapped mode can be as large as ~105 even at N=8. Compared to the finite linear chain, the structure of ring array exhibits overwhelming advantage in Q factor with the same N because there is no array-edge radiation loss in the ring array. We note that the principles can certainly be extended to particles of other shapes (such as nanospheres, nanodisks, and many other experimentally feasible geometries).