  Extinction spectroscopy is a powerful tool for demonstrating the coupling of
a single quantum emitter to a photonic structure. However, it can be
challenging in all but the simplest of geometries to deduce an accurate value
of the coupling efficiency from the measured spectrum. Here we develop a
theoretical framework to deduce the coupling efficiency from the measured
transmission and reflection spectra without precise knowledge of the photonic
environment. We then consider the case of a waveguide interrupted by a
transverse cut in which an emitter is placed. We apply that theory to a silicon
nitride waveguide interrupted by a gap filled with anthracene that is doped
with dibenzoterrylene molecules. We describe the fabrication of these devices,
and experimentally characterise the waveguide coupling of a single molecule in
the gap.
