Stable Spin-Wave Trajectories

The propagation of spin waves in nonuniform magnetic fields has been theoretically investigated using a ``geometrical optics'' approximation and a simplified dispersion relation. In this approximation, the beam trajectory is identical with the path of a particle whose potential energy is proportional to the local magnetic field. The latter satisfies Laplace's equation and hence does not assume a minimum inside the sample. A general criterion has been derived for the existence of stable trajectories (a beam which starts out close to the axis of symmetry of the sample will never depart very far from it, even after arbitrarily many reflections). The stability criterion has been evaluated in detail for a number of special cases. Stable trajectories occur only when the potential V(x) along the axis of symmetry has a region in which it is ``concave down'' ${\mathrm{[(\partial }}^2{\mathrm{V/\partial x}}^2\text{)<0]}$ , and when the total energy of the particle lies within one of several bands of stability. The relevance of the theory to magnetoelastic delay lines is discussed.