Optimal transfer functions for bandwidth-limited imaging

Abstract

One of the fundamental limits of classical optical microscopy is the diffraction limit of optical resolution. It results from the finite bandwidth of the optical transfer function (or OTF) of an optical microscope, which restricts the maximum spatial frequencies that are transmitted by a microscope. However, given the frequency support of the OTF, which is fully determined by the used optical hardware, an open and unsolved question is what is the optimal amplitude and phase distribution of spatial frequencies across this support that delivers the ``sharpest'' possible image. In this paper, we will answer this question and present a general rule how to find the optimal OTF for any given imaging system. We discuss our result in the context of optical microscopy, by considering in particular the cases of wide-field microscopy, confocal image scanning microscopy (ISM), 4pi microscopy, and structured illumination microscopy (SIM). Our results are important for finding optimal deconvolution algorithms for microscopy images, and we demonstrate this experimentally on the example of ISM. They can also serve as a guideline for designing optical systems that deliver best possible images, and can be easily generalized to nonoptical imaging such as telescopic imaging, ultrasound imaging, or magnetic resonance imaging.