$abbe diffraction limit derivation$

Parallel efforts aim to improve the imaging speed 20. These techniques have specific requirements for fluorescent labels. Stimulated emission depletion (STED) microscopy increases the resolution through shrinking the point-spread function (PSF) by depleting the fluorescence emission in the periphery of the diffraction-limited spot 19. Stochastic optical reconstruction microscopy and photoactivation localization microscopy (PALM) are based on the stochastic switching on of individual molecules at different times 17, 18. However, it yields a resolution improved by only a factor of two. Structured illumination microscopy utilizes spatial modulation of the fluorescence emission with patterned illumination 16. Recent years have witnessed the development of super-resolution far-field fluorescence microscopy that allows the diffraction-limited resolution to be surpassed, unveiling processes at the nano-scale level 15. The ongoing challenge towards high-quality minimally invasive deep-tissue imaging calls for a new solution that combines enhancement in the spatial and temporal resolutions with a footprint reduction. Currently, minimally invasive endoscopes are widely used in neuroscience for in vivo deep brain imaging 10, 11, 12, as well as in clinical studies to assist in detecting cancers, to prescribe the right drugs and to monitor treatment response 13, 14. The numerical aperture (NA) of MM fibres approaches 0.9 8, 9, paving the way towards high-resolution but still diffraction-limited imaging. The recent emergence of spatial wavefront shaping 2, 3 has allowed a conventional step-index multimode (MM) fibre to be utilized as an ultra-thin aberration-free imaging probe 4, 5, 6, 7. Miniaturized endo-microscopy provides large depth penetration and is not limited by the interior of a hollow organ or cavity of the body. Modern microscopy demonstrates a drive towards miniaturization caused by the need to access deep tissues in vivo 1. Optical techniques have long been recognized as indispensable tools for bioimaging. The proposed approach can significantly expand the realm of the application of nanoscopy for bioimaging. 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 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 use the random nature of mode coupling in a multimode fibre, the sparsity constraint and compressive sensing reconstruction. 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. However, these methods typically require complicated setups and long acquisition times and are still not applicable to deep-tissue bioimaging.

$abbe diffraction limit derivation$ $abbe diffraction limit derivation$

The recent development of super-resolution techniques has pushed the limits of spatial resolution. 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. Only more recently have scientists revisited Abbe's limit and successfully increased the resolution of light microscopy through a variety of innovative strategies (see Milestone 21).For several centuries, far-field optical microscopy has remained a key instrument in many scientific disciplines, including physical, chemical, and biomedical research. Rather, other techniques were developed to provide these data, such as replacing photons with electrons in the development of electron microscopy. In fact, Abbe's study influenced the field so greatly that very few attempts were made to overcome the diffraction limit, despite the increasing necessity to enhance resolution and to improve the visualization of cellular structure. But besides these tangible legacies, another long-lasting legacy brought about by Abbe's work included the establishment of physical boundaries in imaging for quite some time. Of course, the success of Abbe, Zeiss and Schott in designing lenses also had an enormous impact on the eventual success of the microscopy manufacturer Carl Zeiss itself. Abbe's quantitative insights greatly enhanced the quality of microscope optics, contributing enormously to improved data collection and an enhanced user experience for the microscopist. Abbe's mathematical foundations of image formation and lens aberrations provided for the proper design of microscope lenses, accomplished in collaboration with Carl Zeiss and Otto Schott.

$abbe diffraction limit derivation$

Abbe diffraction limit derivation