Super-localisation of a point-like emitter in a resonant environment: Correction of the mirage effect

In this paper, we show that it is possible to overcome one of the fundamental limitations of super-resolution microscopy: the necessity to be in an optically homogeneous environment. Using recent modal approximation results from [10, 7], we show, as a proof of concept, that it is possible to recover the position of a single point-like emitter in a known resonant environment from far-field measurements, with a precision two orders of magnitude below the classical Rayleigh limit. The procedure does not involve solving any partial differential equation, is computationally light (optimisation in with of the order of ) and is therefore suited for the recovery of a very large number of single emitters.