  In this study, we use the transient thermal grating optical technique
\textemdash a non-contact, laser-based thermal metrology technique with
intrinsically high accuracy \textemdash to investigate room-temperature
phonon-mediated thermal transport in two nanoporous holey silicon membranes
with limiting dimensions of 100 nm and 250 nm respectively. We compare the
experimental results to ab initio calculations of phonon-mediated thermal
transport according to the phonon Boltzmann transport equation (BTE) using two
different computational techniques. We find that the calculations conducted
within the Casimir framework, i.e. based on the BTE with the bulk phonon
dispersion and diffuse scattering from surfaces, are in quantitative agreement
with the experimental data, and thus conclude that this framework is adequate
for describing phonon-mediated thermal transport through holey silicon
membranes with feature sizes on the order of 100 nm.
