Towards optimal point spread function design for resolving closely spaced emitters in three dimensions

The past decade has brought many innovations in optical design for 3D super-resolution imaging of point-like emitters, but these methods often focus on single-emitter localization precision as a performance metric. Here, we propose a simple heuristic for designing a point spread function (PSF) that allows for precise measurement of the distance between two emitters. We discover that there are two types of PSFs that achieve high performance for resolving emitters in 3D, as quantified by the Cram\'{e}r-Rao bounds for estimating the separation between two closely spaced emitters. One PSF is very similar to the existing Tetrapod PSFs; the other is a rotating single-spot PSF, which we call the crescent PSF. The latter exhibits excellent performance for localizing single emitters throughout a 1-$\mu$m focal volume (localization precisions of 7.3~nm in $x$, 7.7~nm in $y$, and 18.3~nm in $z$ using 1000 detected photons), and it distinguishes between one and two closely spaced emitters with superior accuracy ($25$-$53\%$ lower error rates than the best-performing Tetrapod PSF, averaged throughout a 1-$\mu$m focal volume). Our study provides additional insights into optimal strategies for encoding 3D spatial information into optical PSFs.