  This chapter describes spin-wave excitations in nanosized dots and rings in
the presence of the vortex state. The special attention is paid to the
manifestation of the competition between exchange and dipolar interactions in
the spin-wave spectrum as well as the correlation between the spectrum and the
stability of the vortex. The calculation method uses the dynamic matrix for an
all-discrete system, the numerical diagonalization of which yields the spectrum
of frequencies and spin-wave profiles of normal modes of the dot. We study
in-plane vortices of two types: a circular magnetization in circular dots and
rings and the Landau state in square rings. We examine the influence of the
dipolar-exchange competition and the geometry of the dot on the stability of
the vortex and on the spectrum of spin waves. We show that the lowest-frequency
mode profile proves to be indicative of the dipolar-to-exchange interaction
ratio and the vortex stability is closely related to the spin-wave profile of
the soft mode. The negative dispersion relation is also shown. Our results
obtained for in-plane vortices are in qualitative agreement with results for
core-vortices obtained from experiments, micromagnetic simulations, and
analytical calculations.
