I am investigating three wave interactions for magnetostatic surface spin-waves in a spin-wave active ring oscillator [1]. I would appreciate it if the community could let me know how to represent momentum conservation with a vector diagram for three magnon scattering.
1. Wu, M., 2010. Nonlinear spin waves in magnetic film feedback rings. Solid State Physics, 62, pp.163-224.
Hi Anirban,
in order to answer your question precisely, you need to clarify in which equilibrium state your ring resonator is magnetized. Naively, I assume it is magnetized in the vortex (flux-closure) state. In this case, however, waves running around the circumference are not surface waves but backward-volume waves. In order to achieve propagation in surface-wave geometry, the ring would need to be magnetized in the radial direction (which is hard without any crystalline anisotropy).
Regarding your question, simply writing down conservation of momentum for your system is not immediately possible using a nice vector diagram since (1) your system is not flat and (2) not all wave vector components are conserved in your case. In general, to determine the laws of wave-vector conservation, you need to look for the continuous symmetries of your system. In the case of a ring, the symmetry is broken in radial and in out-of-plane (z) directions. Therefore, there is no wave-vector conservation in these directions during scattering processes. Assuming that your ring is magnetized in a state that is cylindrically symmetric (rotationally symmetric around the z axis), such as the vortex state, the radially magnetized state, or even the fully out-of-plane state, then the canonical momentum in the azimuthal (φ) direction, i.e., the angular momentum, is conserved. In terms of wave scattering, this translates to a selection rule for the azimuthal indices. Consider the case that your system is magnetized in a cylindrically symmetric state, then the modes can be written as snm(r)exp(imφ) with n being the number of nodal lines of the wave in radial direction and m being the azimuthal index, which counts the number of periods in the azimuthal direction and denotes the wave vector in the azimuthal direction. Due to the cylindrical symmetry, during any scattering process, this index needs to be conserved. That is, for three-wave scattering of modes 0 -> (1,2) you would have m0 = m1 + m2. However, if your ring is magnetized in the vortex state, you will have an additional selection rule which does not directly come from wave vector conservation but from the special symmetry of the system. Namely, if the system is in the vortex state and your initial modes is m = 0, then the two secondary modes need to have m1 = -m2, but they also need to have different radial profiles sn1m(r) =/= sn2m(r) which translates to n1 =/= 0. Therefore, if your ring is narrow enough such that modes with n > 0 are at high frequencies, the modes (n, m) = (0, 0) cannot resonantly via three-splitting. You can find more about three-magnon splitting in systems cylindrical symmetry, for example, here: Article Theory of three-magnon interaction in a vortex-state magnetic nanodot
Keep in mind that if your have a waveguide adjacent to the resonator, then the modes in the ring can only approximately be expressed as proportional to exp(imφ) because the coupling to the waveguide breaks the cylindrical symmetry.
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