  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).
