Dipolar localization of quantized spin-wave modes in thin rectangular magnetic elements

Dipole-exchange spectrum of quantized spin wave modes of a tangentially magnetized rectangular thin-film magnetic element is calculated using the method of tensorial Green’s functions. The strong inhomogeneity of the internal bias magnetic field along the magnetization direction leads to the localization of spin wave modes either at the edges (exchange localization) or at the center (dipolar localization) of the element. The mode intensity distributions along the other in-plane direction are determined by the dipolar boundary conditions and have the usual cosinusoidal form. The approximate theory developed in the paper gives a quantitative description of the resonance fields and qualitative description of the spatial distributions of quantized spin wave modes in a thin square permalloy element (in plane sizes 50×50 μm2, thickness 0.1 μm) recently observed by space-resolved Kerr spectroscopy. The theory shows also that the mode localization in this case (in the element center) is of the dipolar nature.