  Diamond microdisk cavities play a key role in optomechanical and
spin-optomechanical technologies. Previous optomechanical studies of these
devices have focused exclusively on their fundamental radial breathing mode.
Accessing other mechanical modes of these structures is desirable for
identifying routes towards improving their optomechanical properties,
implementing multimode optomechanical systems, and broadening the accessible
range of resonant spin--phonon coupling processes. Here we perform broadband
optomechanical spectroscopy on diamond microdisks, and observe high quality
factor mechanical modes with frequencies up to 10 GHz. Through Fano
interference of their optomechanical response with diamond's Kerr nonlinear
optical response, we estimate that optomechanical coupling of these high
frequency modes can exceed 10 kHz, making them attractive for high-frequency
multimode optomechanics. In combination with their per-phonon stress of a few
kPa, these properties makes them excellent candidates for spin-optomechanical
coupling.
