Endo-microscopy beyond the Abbe and Nyquist limits.

For several centuries, far-field optical microscopy has remained a key instrument in many scientific disciplines, including physical, chemical, and biomedical research. Nonetheless, far-field imaging has many limitations: the spatial resolution is controlled by the diffraction of light, and the imaging speed follows the Nyquist-Shannon sampling theorem. The recent development of super-resolution techniques has pushed the limits of spatial resolution. However, these methods typically require complicated setups and long acquisition times and are still not applicable to deep-tissue bioimaging. Here, we report imaging through an ultra-thin fibre probe with a spatial resolution beyond the Abbe limit and a temporal resolution beyond the Nyquist limit simultaneously in a simple and compact setup. We use the random nature of mode coupling in a multimode fibre, the sparsity constraint and compressive sensing reconstruction. The new approach of super-resolution endo-microscopy does not use any specific properties of the fluorescent label, such as depletion or stochastic activation of the molecular fluorescent state, and therefore can be used for label-free imaging. We demonstrate a spatial resolution more than 2 times better than the diffraction limit and an imaging speed 20 times faster than the Nyquist limit. The proposed approach can significantly expand the realm of the application of nanoscopy for bioimaging.