Hippopede Curves For Modeling Radial Spin Waves In An Azimuthally Graded Magnonic Landscape

We propose a mathematical model for describing radially propagating spin waves emitted from the core region in a magnetic patch with n vertices in a magnetic vortex state. The azimuthal anisotropic propagation of surface spin waves (SSWs) into the domain, and confined spin waves (or Winter's magnons) in domain walls, increases the complexity of the magnonic landscape. In order to understand the spin wave propagation in these systems, we first use an approach based on geometrical curves called "hippopedes"; however, it provides no insight into the underlying physics. Analytical models rely on generalized expressions from the dispersion relation of SSWs with an arbitrary angle between magnetization M and wave number k. The derived algebraic expression for the azimuthal dispersion is found to be equivalent to that of the hippopede curves. The fitting curves from the model yield a spin wave wavelength for any given azimuthal direction, number of patch vertices, and excitation frequency, showing a connection with fundamental physics of exchange-dominated surface spin waves. Analytical results show good agreement with micromagnetic simulations and can be easily extrapolated to any n-corner patch geometry.