  The study of resonant dielectric nanostructures with high refractive index is
a new research direction in nanoscale optics and metamaterial-inspired
nanophotonics. Because of the unique optically-induced electric and magnetic
Mie resonances, high-index nanoscale structures are expected to complement or
even replace different plasmonic components in a range of potential
applications. Here we study strong coupling between modes of a single
subwavelength high-index dielectric resonator and analyse the mode
transformation and Fano resonances when resonator's aspect ratio varies. We
demonstrate that strong mode coupling results in resonances with high quality
factors, which are related to the physics of bound states in the continuum when
the radiative losses are almost suppressed due to the Friedrich-Wintgen
scenario of destructive interference. We explain the physics of these states in
terms of multipole decomposition and show that their appearance is accompanied
by drastic change of the far-field radiation pattern. We reveal a fundamental
link between the formation of the high-quality resonances and peculiarities of
the Fano parameter in the scattering cross-section spectra. Our theoretical
findings are confirmed by microwave experiments for the scattering of a
high-index cylindrical resonators with a tunable aspect ratio. The proposed
mechanism of the strong mode coupling in single subwavelength high-index
resonators accompanied by resonances with high quality factor helps to extend
substantially functionalities of all-dielectric nanophotonics that opens new
horizons for active and passive nanoscale metadevices.
