  Optomechanical (OM) cavities enable coupling of near-infrared light and
GHz-frequency acoustic waves in wavelength-scale volumes. When driven in the
phonon lasing regime, an OM cavity can perform simultaneously as a nonlinear
mixer and a local oscillator -- at integer multiples of the mechanical
resonance frequency -- in the optical domain. In this work, we use this
property to demonstrate all-optical frequency down- and up-conversion of
MHz-bandwidth orthogonal frequency division multiplexed signals compliant with
the IEEE 802.16e WiMAX wireless standard at microwave frequencies. To this end,
we employ a silicon OM crystal cavity (OMCC), supporting a breathing-like
mechanical resonance at $f_m\approx$ 3.9 GHz and having a foot-print $\approx$
10 $\mu$m$^{2}$, which yields frequency conversion efficiencies better than -17
dB in both down- and up-conversion processes at mW-scale driving power. This
work paves the way towards the application of OMCCs in low-power all-photonic
processing of digitally-modulated microwave signals in miniaturized silicon
photonics chips.
