High-finesse Fabry-Perot cavities with bidimensional Si$_3$N$_4$ photonic-crystal slabs

Light scattering by a two-dimensional photonic crystal slab (PCS) can result in dramatic interference effects associated with Fano resonances. Such devices offer appealing alternatives to distributed Bragg reflectors or filters for various applications such as optical wavelength and polarization filters, reflectors, semiconductor lasers, photodetectors, bio-sensors, or non-linear optical components. Suspended PCSs also find natural applications in the field of optomechanics, where the mechanical modes of a suspended slab interact via radiation pressure with the optical field of a high finesse cavity. The reflectivity and transmission properties of a defect-free suspended PCS around normal incidence can be used to couple out-of-plane mechanical modes to an optical field by integrating it in a free space cavity. Here, we demonstrate the successful implementation of a PCS reflector on a high-tensile stress Si$_3$N$_4$ nanomembrane. We illustrate the physical process underlying the high reflectivity by measuring the photonic crystal band diagram. Moreover, we introduce a clear theoretical description of the membrane scattering properties in the presence of optical losses. By embedding the PCS inside a high-finesse cavity, we fully characterize its optical properties. The spectrally, angular, and polarization resolved measurements demonstrate the wide tunability of the membranes reflectivity, from nearly 0 to 99.9470~$pm$ 0.0025 %, and show that material absorption is not the main source of optical loss. Moreover, the cavity storage time demonstrated in this work exceeds the mechanical period of low-order mechanical drum modes. This so-called resolved sideband condition is a prerequisite to achieve quantum control of the mechanical resonator with light.