Radiation losses and dark mode for spin-wave propagation through a discrete magnetic micro-waveguide

This paper presents the quantum mechanical type $T$-scattering operator approach to studying the forward volume magnetostatic spin-wave multiple scattering by a finite ensemble of cylindrical magnetic inclusions in a ferromagnetic thin film. The approach is applied to the problem of spin-wave excitation transfer along a linear chain of inclusions. The substantial results are deriving the optical theorem for the $T$-scattering operator and, as a consequence, deriving a formula for collective extinction cross section of inclusion ensemble, where only the first inclusion of the chain is irradiated by an incident narrow spin-wave beam. From this formula it can be shown that only irradiated inclusion makes a direct contribution in the collective extinction cross section of the total number of inclusions. In this case the direct summarized contribution of all the other inclusions from the chain into the spin-wave scattering is invisible; we call such phenomenon the dark mode. Applying a one-multipole and closest neighbor coupling approximation, we reveal a regime of distant resonant transfer for spin-wave excitation along the linear chain of an essentially big but finite number of particles with the dark mode. Because we also found a resonant mechanism of filtering this mode from radiation losses, the revealed regime shows that at resonant conditions the linear chain of magnetic inclusions can play the role of a spin-wave micro-waveguide, which transfers a signal over a big distance in a form of the dark mode, where the controllable level of radiation losses can tend to reach nearly zero values.

Figure 1

The ferromagnetic film with linear chain of cylindrical ferromagnetic inclusions.

Figure 2

The sketch showing two ferromagnetic inclusions with their local coordinate systems relative to the laboratory's coordinate system.

Figure 3

Curves presenting the dependence of the resonant frequency on the inclusion radius $R$ at fixed ratio $R_{12}/R$, when $a_{12}^{\prime \prime }=0,a_{12}^{\prime }\le -0.5$.

Figure 4

Linear decay of scattering amplitudes along the linear chain of inclusions. ${\mathrm{\Omega }}_{\mathrm{res}}=1.007,{\theta }^{\prime \prime }=4.4{10}^{-6},R=6.9\mu m,R_{12}=4R,k_r^{0}R=0.38$.

Figure 5

Exponential decay of scattering amplitudes along the linear chain. ${\mathrm{\Omega }}_{\mathrm{res}}=1.004,{\theta }^{\prime \prime }=0.43,R=7\mu m,R_{12}=3R,k_r^{0}R=0.3$.

Figure 6

Illustrations of the behavior of scattering amplitudes ${\widehat{\widehat{B}}}_j$ of inclusions depending on the inclusion number $j$ for the case $N=24$ (a), $N=23$ (b), and collective extinction factor $F_N$ (c) of a linear chain under condition of dark mode filtering from radiation losses.