  We have investigated azimuthal spin-wave modes in magnetic vortex structures
using time-resolved Kerr microscopy. Spatially resolved phase and amplitude
spectra of ferromagnetic disks with diameters from 5 $\mu$m down to 500 nm
reveal that the lowest order azimuthal spin wave mode splits into a doublet as
the disk size decreases. We demonstrate that the splitting is due to the
coupling between spin waves and the gyrotropic motion of the vortex core.
