Controlling the three dimensional propagation of spin waves in continuous ferromagnetic films with an increasing out of plane undulation

The role of three-dimensionality in a ferromagnetic medium in ruling the propagation properties of spin-waves (SW) has been one of the main focuses of the research activity in recent years. In this context, we investigate the evolution of the SW dispersion (frequency vs wave vector) induced by a progressive vertical undulation of a ferromagnetic film. The geometric undulation is taken along a single direction and is periodic with constant period, while the amplitude (differential maximum height with respect to the film thickness) is gradually increased from 0 to 60 nm. We study the characteristic modification of the internal effective field and link it to the resulting SW dispersions and spatial profile. These systems display at once features both of a planar film and a discretized medium, and the dispersion curves change not only when SWs propagate along the undulation direction, but also perpendicular to it. We discuss the geometric and magnetic conditions for having either the invariance of the SW group velocity with respect to even major changes in the undulation, or a large group velocity for some edge modes. We address a potential dual-band activity, namely the simultaneous propagation of two independent SW-signals, with separated frequency bands and disjoint oscillation regions.