Selection rules for cavity-enhanced Brillouin light scattering from magnetostatic modes

We identify experimentally the magnetostatic modes active for Brillouin light scattering in the optical whispering gallery modes of a yttrium iron garnet sphere. Each mode is identified by magnetic-field dispersion of ferromagnetic-resonance spectroscopy and coupling strength to the known field distribution of the microwave drive antenna. Our optical measurements confirm recent predictions that higher-order magnetostatic modes can also generate optical scattering, according to the selection rules derived from the axial symmetry. From this we summarize the selection rules for Brillouin light scattering. We give experimental evidence that the optomagnonic coupling to nonuniform magnons can be higher than that of the uniform Kittel mode.

Figure 1

Spatial variation of (a) an optical WGM and (b)–(i) dynamic magnetization modes of a YIG sphere. The labels above the spheres are the angular momentum and radial mode numbers $\{l,m,q\}$ (with subscript $m$ for magnetic modes; see text). (a) The in-phase intensity of the electric field for an optical WGM with $l=m=200$ (in the actual experiment, $l,m\sim 1000$). (b)–(i) The color indicates the intensity of the dynamic magnetization, while arrows indicate the in-phase direction. Negative mode indices are indicated by an overbar. The static magnetization ${\mathbf{M}}_0$ is indicated in (b).

Figure 2

Experimental setup. Linearly polarized input light is evanescently coupled into the YIG sphere via a rutile prism. Photodiode I measures the transmitted input beam in order to identify the WGM resonances. Photodiode II measures the polarization rotated output from the YIG sphere, with a scanning Fabry-Pérot etalon in the beam path providing spectral resolution. A permanent NdFeB magnet saturates the magnetization perpendicular to the WGM orbit. The microwave measurements are made with the loop antenna using a vector network analyzer (VNA).

Figure 3

Identification of magnetostatic modes from magnetic-field dispersion. (a) Microwave reflection coefficient $|S_{11}|$ as a function of magnetic field and reduced frequency $(\omega -{\omega }_H)/{\omega }_M+\frac{1}{3}$ measured in a uniform applied magnetic field generated by an electromagnet. (b) Calculated eigenfrequencies of magnetostatic modes with indices $\{l_m,m_m,q_m\}$, overlaid with dips from (a). This is used to identify the magnons excited by the microwave antenna. (c) Same as (a) but in a nonuniform magnetic field of a permanent magnet showing normal mode splittings and additional magnons. (d) Same as (c), but with a rutile coupling prism in place indicating the microwave modes of the prism alongside that of the YIG sphere.

Figure 4

Optical measurements of BLS efficiency. (a) The reflected input optical power at photodiode I as a function of input laser wavelength detects the WGM resonances. The laser detuning ($x$ axis) is measured relative to one of the WGMs. Resonances corresponding to mode families $q=1,2$ are observed. The free spectral range for modes with $q=1$ is indicated. (b) BLS signal as a function of input laser wavelength for several magnons marked by arrows in (c) with matching color. (c) BLS intensity maxima of the $q=1$ WGM resonance (coded by color intensity) for each measured point of microwave frequency ($y$ axis, reduced frequency) and magnetic field ($x$ axis). (d) Same as (c), but with the static magnetic field inverted.

Figure 5

Identification of BLS active magnetostatic modes. (a) Data points corresponding to those in Fig. 4 that are above a threshold set by the noise floor. The dot colors correspond to the assigned magnetostatic mode. Positive and negative magnetic fields are indicated by filled and open points, respectively. Points where mode identification is unclear are marked in gray. (b) Theoretical dispersion of the modes observed in (a).