Manipulating the Point Spread Function of Light to Increase Quantum Limited Estimation Accuracy for Bio-Medical Applications

Deterministic and Stochastic Super-Resolution techniques can regularly achieve resolutions well below 20nm. However, this is only achieved via sophisticated control of the emission of fluorophores, and it is both time-consuming and invasive because super-resolution assumes individual emitters that can be resolved independently. There are nonetheless scenarios where one would like to localise simultaneously emitting fluorophores that are too closed to be individually resolved. This is in part due to progress made in the determination of the number of simultaneous emitters in unresolved clusters.
Initially we have restricted ourselves to two identical but incoherent sources. In conventional intensity imaging, estimating the positions of two point-like sources becomes harder the closer these sources are to each other. Work by Tsang et al showed that remarkably, this need not be the case: by capturing phase information in the imaging system, the amount of information on the separation of the sources that can be extracted from the radiation does not depend on the separation itself, provided the emitters are incoherent. Unfortunately, the corresponding implementations of such optimum scheme are cumbersome. They involve interferometric arrangements that are ill-suited to microscopy because the position of the centre of mass of the two sources would need to be known a priori, a rare occurrence in bio-imaging. As a practical compromise, Paúr et al found that shaping the point spread function (PSF) of the imaging system such that it exhibits a zero of intensity, a comparatively easier operation, significantly improves the quantum-limited estimation of the separation. While the resolving power of this technique does still drop to zero at short distances, the scaling is much more favourable than imaging with a typical Gaussian point spread function.
While the original work of Paúr et al focused on a one-dimensional demonstration, we are interested in the two-dimensional case, where the direction of separation between the emitters is unknown a priori. By manipulating the phase of the radiated light such that the point spread function has a dark ring, we have shown analytically and numerically that the separation of two fluorophores can be estimate to an accuracy greater than direct imaging. A proof-of-principle experiment is under construction to confirm these predictions.

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