An Iterative Solution for Spin-Wave Dispersion in a Magnonic Ring

The dispersion relation of forward volume dipolar spin waves in a magnonic curved waveguide is investigated by solving Walker´s equation in cylindrical coordinates with appropriate boundary conditions. Dispersion for exchange spin waves is then calculated using an iterative method. We validate our results by comparing the dispersion relation for higher bending radius with that of a straight waveguide. We also observe good agreement with the dispersion plots from micromagnetic simulations. For a ring, we impose periodic boundary conditions along the azimuthal direction to obtain standing wave mode patterns. The frequency-mode number characteristics of standing waves follow the dispersion relation for propagating spin waves in a curved waveguide. The maximum spin wave amplitude is not necessarily at the center of the waveguide. Exchange interactions cause the maximum to occur at a lower radius for higher mode numbers.