Quantum Theory of Superresolution for Two Incoherent Optical Point Sources

Abstract

Rayleigh’s criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century. In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern far-field superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision. Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measurements with linear optics and photon counting can estimate their separation from the far field almost as precisely as conventional methods do for isolated sources, rendering Rayleigh’s criterion irrelevant to the problem. Our results demonstrate that superresolution can be achieved not only for fluorophores but also for stars.

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Plain Language Summary

In the 19th century, scientists discovered that the wave nature of light causes pointlike sources to appear blurry, where the size of the blurred spot sets the fundamental resolution of an instrument such as a telescope. In a seminal 1879 paper, Lord Rayleigh suggested that a pair of spots corresponding to stars had to be separated by at least the spot size in order to resolve the two stars. Researchers have since found “Rayleigh’s criterion” to be an excellent rule of thumb for both telescopes and microscopes. Here, we show that, contrary to conventional wisdom, Rayleigh’s criterion is not fundamental, and cleverer optical techniques can measure the separation between two light sources much more accurately than previously realized. Our calculations, based on quantum-information theory, reveal that the light actually carries much more information about the source positions than previously thought, and Rayleigh’s criterion is a problem only because conventional imaging techniques are unable to extract all of the information.

By considering two weak sources emitting in the optical regime, we derive the fundamental quantum limit on the precision of separating the sources. We find that Rayleigh’s criterion has no influence on this limit. To reach the limit, our new optical method, called spatial-mode demultiplexing (SPADE), splits the incoming light into specially designed channels. This process is extremely sensitive to the separation between the light sources, allowing the separation to be estimated as accurately as quantum mechanics allows and without regard to Rayleigh’s criterion.

We expect that our findings will pave the way for significant advances in both the study of binary stars and the imaging of single molecules by improving the imaging resolution by orders of magnitude.