Magnetic spheres in microwave cavities

We apply Mie scattering theory to study the interaction of magnetic spheres with microwaves in cavities beyond the magnetostatic and rotating wave approximations. We demonstrate that both strong and ultrastrong coupling can be realized for stand alone magnetic spheres made from yttrium iron garnet (YIG), acting as an efficient microwave antenna. The eigenmodes of YIG spheres with radii of the order mm display distinct higher angular momentum character that has been observed in experiments.

Figure 1

Plane wave with wave vector ${\mathbf{k}}_0$ coming in at an arbitrary angle hits a large spherical cavity modeled by a dielectric spherical shell of radius $R$, thickness $\delta$, and permittivity ${\epsilon }_c$. The spherical cavity is loaded with a magnetic sphere of radius $a$ centered at the origin of the coordinate system.

Figure 2

Scattering intensity $|S_1{|}^2$ as function of scattering angle $\theta$ and frequency $\omega /2\pi$ is shown for (a) a dielectric sphere of radius $a=1.25\mathrm{mm}$ and relative permittivity $\epsilon /{\epsilon }_0=15$ and for (b) the same sphere in a cavity of radius $R=1.6\mathrm{mm}$. In (c) the scattering intensity is plotted for the same cavity as function of frequency and loading rate $a/R$. The dashed lines are guides for the eye.

Figure 3

Panel (a) shows the scattering efficiency factor $Q_{\mathrm{sca}}$ as function of normalized magnetic field $H_0/M_s$ and frequency $\omega /2\pi$ for a YIG sphere of radius $a=2\mathrm{mm}$ and relative permittivity $\epsilon /{\epsilon }_0=15$. Panel (b) shows results for a nonmagnetic dielectric sphere. The character of the microwave modes sufficiently far from the anticrossing with the spin waves is labeled by the spherical harmonic indices $(n,m)$.

Figure 4

Scattering efficiency factor $Q_{\mathrm{sca}}$ plotted as function of normalized magnetic field $H_0/M_s$ and frequency $\omega /2\pi$ for a YIG sphere of radius $a=1.25\mathrm{mm}$ and relative permittivity $\epsilon /{\epsilon }_0=15$ (a) in the center of a spherical cavity of radius $R=1.6\mathrm{mm}$ and (b) without cavity.

Figure 5

Scattering efficiency factor $Q_{\mathrm{sca}}$ as function of normalized magnetic field $H_0/M_s$ and frequency $\omega /2\pi$ for a YIG sphere of radius $a=1.25\mathrm{mm}$ and relative permittivity $\epsilon /{\epsilon }_0=15$ (a) in the center of a spherical cavity of radius $R=1.6\mathrm{mm}$ and (b) in the absence of the cavity. Dashed lines indicate the frequency range in Fig. 4.