  We have investigated theoretically band structure of spin waves in magnonic
crystals with periodicity in one-(1D), two- (2D) and three-dimensions (3D). We
have solved Landau-Lifshitz equation with the use of plane wave method, finite
element method in frequency domain and micromagnetic simulations in time domain
to find the dynamics of spin waves and spectrum of their eigenmodes. The spin
wave spectra were calculated in linear approximation. In this paper we show
usefulness of these methods in calculations of various types of spin waves. We
demonstrate the surface character of the Damon-Eshbach spin wave in 1D magnonic
crystals and change of its surface localization with the band number and
wavenumber in the first Brillouin zone. The surface property of the spin wave
excitation is further exploited by covering plate of the magnonic crystal with
conductor. The band structure in 2D magnonic crystals is complex due to
additional spatial inhomogeneity introduced by the demagnetizing field. This
modifies spin wave dispersion, makes the band structure of magnonic crystals
strongly dependent on shape of the inclusions and type of the lattice. The
inhomogeneity of the internal magnetic field becomes unimportant for magnonic
crystals with small lattice constant, where exchange interactions dominate. For
3D magnonic crystals, characterized by small lattice constant, wide magnonic
band gap is found. We show that the spatial distribution of different materials
in magnonic crystals can be explored for tailored effective damping of spin
waves.
