Dynamic origin of azimuthal modes splitting in vortex-state magnetic   dots
  A spin wave theory explaining experimentally observed frequency splitting of dynamical excitations with azimuthal symmetry of a magnetic dot in a vortex ground state is developed. It is shown that this splitting is a result of the dipolar hybridization of three spin wave modes of a dot having azimuthal indices |m|=1: two high-frequency azimuthal excitation modes of the in-plane part of the vortex with indices m = +/-1 and a low-frequency m= +1 gyrotropic mode describing the translational motion of the vortex core. The analytically calculated magnitude of the frequency splitting is proportional to the ratio of the dot thickness to its radius and quantitatively agrees with the results of time resolved Kerr experiments. 
Konstantin Y. Guslienko, Andrei N. Slavin, Vasyl Tiberkevich and   Sang-Koog Kim