  Material absorption is a key limitation in nanophotonic systems; however, its
characterization is often obscured by scattering and diffraction loss. Here we
show that nanomechanical frequency spectroscopy can be used to characterize the
absorption of a dielectric thin film at the parts-per-million (ppm) level, and
use it to characterize the absorption of stoichiometric silicon nitride
(Si$_3$N$_4$), a ubiquitous low-loss optomechanical material. Specifically, we
track the frequency shift of a high-$Q$ Si$_3$N$_4$ trampoline resonator in
response to photothermal heating by a $\sim10$ mW laser beam, and infer the
absorption of the thin film from a model including thermal stress relaxation
and both radiative and conductive heat transfer. A key insight is the presence
of two thermalization timescales, a rapid ($\sim0.1$ sec) timescale due to
radiative thermalization of the Si$_3$N$_4$ thin film, and a slow ($\sim100$
sec) timescale due to parasitic heating of the Si device chip. We infer the
extinction coefficient of Si$_3$N$_4$ to be $\sim0.1-1$ ppm in the 532 - 1550
nm wavelength range, comparable to bounds set by waveguide resonators and
notably lower than estimates with membrane-in-the-middle cavity optomechanical
systems. Our approach is applicable to a broad variety of nanophotonic
materials and may offer new insights into their potential.
