Magnetostatic spin-wave modes of a ferromagnetic multilayer

The spin-wave spectrum of a ferromagnetic multilayer is derived in the dipolar-magnetostatic limit. Restrictions are imposed with respect to in-plane magnetization $M_s$ and propagation of the spin waves in plane and perpendicular to $M_s$. A formula is derived from which the spin-wave branches of a multilayer with a finite number $N$ of magnetic layers can easily be calculated. In the limit of infinitely large $N$ the solutions form a band whose edges and density of states are given by simple expressions. The existence of an additional singular solution depends on the ratio of the thicknesses of magnetic and nonmagnetic layers. We show that these modes can be classified by an integer number $j$, where $j=1,\dots$, $N-1\mathrm{}$, and derive a relation between $j$ and the amplitude distribution. As an experimental test of these theoretical results we have also observed the multilayer modes by means of light scattering, and we find good agreement between theory and experiment.