Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy

Abstract

Rayleigh's criterion is extensively used in optical microscopy for determining the resolution of microscopes. This criterion imposes a resolution limit that has long been held as an impediment for studying nanoscale biological phenomenon through an optical microscope. However, it is well known that Rayleigh's criterion is based on intuitive notions. For example, Rayleigh's criterion is formulated in a deterministic setting that neglects the photon statistics of the acquired data. Hence it does not take into account the number of detected photons, which, in turn, raises concern over the use of Rayleigh's criterion in photon-counting techniques such as single-molecule microscopy. Here, we re-examine the resolution problem by adopting a stochastic framework and present a resolution measure that overcomes the limitations of Rayleigh's criterion. This resolution measure predicts that the resolution of optical microscopes is not limited and that it can be improved by increasing the number of detected photons. Experimental verification of the resolution measure is carried out by imaging single-molecule pairs with different distances of separation. The resolution measure provides a quantitative tool for designing and evaluating single-molecule experiments that probe biomolecular interactions.Cramer-Rao lower bound ͉ photon statistics ͉ Fisher information matrix ͉ fluorescence microscopy A ccording to Rayleigh's criterion, the resolution of an optical microscope is defined as the minimum distance between two point sources such that their presence can be distinguished in the image (1). It is widely believed that the resolution limit imposed by this criterion precludes the single-molecule study of molecular interactions at distances of Ͻ200 nm. Despite the widespread use of Rayleigh's criterion, it is well known that this criterion is based on heuristic notions (2). Formulated within a deterministic framework, Rayleigh's criterion neglects the stochastic nature of the photon-detection process and hence does not consider the total photon count in the acquired data. This formulation is not surprising, because Rayleigh's criterion was developed at a time when the unaided human eye was used as the detector. Therefore, Rayleigh's criterion is not well adapted to current microscope setups, in which highly sensitive photon-counting cameras are used. Recent singlemolecule experiments have shown that Rayleigh's criterion can, in fact, be surpassed in an optical microscope setup (3-5). Thus Rayleigh's resolution limit is an inadequate performance criterion for current quantitative imaging techniques.By adopting a stochastic framework, we propose a resolution measure that overcomes the shortcomings of Rayleigh's criterion and provides a quantitative measure of a microscope's ability to determine the distance between two point sources. Unlike Rayleigh's criterion our resolution measure predicts that the resolution of a microscope can be improved by increasing the number of photons collected from the point sources. The resolution measu...

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