Revisiting anapoles in a single high-index dielectric structure

High-index dielectric structures support electric and magnetic Mie resonance. Through careful manipulation of geometric parameters, destructive interference can be induced between electric multipole moments and toroidal multipole moments. This leads to the formation of anapoles, which are characterized by quenched scattering in the far field and giant enhancement in the near field. Here, we revisit the formation mechanism of anapole states in a single dielectric structure with a high refractive index from an eigenmode perspective. We find that scattering efficiency is mainly determined by the intrinsic phase governed by the leaky mode of the structure and the extrinsic phase induced by the frequency deviation from resonance. It is also demonstrated that the anapole modes in a two-dimensional cylinder and a three-dimensional sphere can only occur in the following two situations: (1) when only one mode is involved, the combined phase of intrinsic and extrinsic phase should be equal to 2 pi at a certain frequency (anapole frequency), which is very close to the resonance frequency. Generally, these types of anapoles are low-order anapoles since low-order resonant modes (i.e., magnetic (electric) dipole and quadrupole) are well separated. (2) If two or more leaky modes are involved, the combined phase for each mode must be 2 pi at the same frequency located between the two resonances. This corresponds to the high-order anapoles. It is also found that more anapole states will emerge with increasing refractive index. Our results may provide new perspectives for designing high-order anapoles with more freedom.