Semiclassical Theory of Superresolution for Two Incoherent Optical Point Sources

Lord Rayleigh suggested in 1879 that two incoherent optical point sources should be separated by a diffraction-limited spot size for them to be resolved [1]. This criterion has since become the most influential measure of imaging resolution. Under the modern advent of rigorous statistics and image processing, Rayleigh’s criterion remains a curse. When the image is noisy, necessarily so owing to the quantum nature of light [2], and Rayleigh’s criterion is violated, it becomes much more difficult to estimate the separation accurately by conventional imaging methods [3–5]. Modern superresolution techniques in microscopy [6–8] can circumvent Rayleigh’s criterion by making sources radiate in isolation, but such techniques require careful control of the fluorescent emissions, making them difficult to use for microscopy and irrelevant to astronomy. Here we show that, contrary to conventional wisdom, the separation between two incoherent optical sources can be estimated accurately via linear optics and photon counting (LOPC) even if Rayleigh’s criterion is severely violated. Our theoretical model here is based on the semiclassical theory of photodetection with Poissonian noise, which is a widely accepted statistical model for lasers [2] as well as weak thermal [9, 10] or fluorescent [5, 11] light in astronomy and microscopy. The semiclassical model is consistent with the quantum model proposed in Ref. [12] for weak incoherent sources and the mathematical formalisms are similar, but the semiclassical model has the advantage of being applicable also to lasers, which are important sources for remote-sensing, testing, and proof-of-concept experiments. The semiclassical theory also avoids a quantum description of light and offers a more pedagogical perspective. Compared with the full semiclassical theory in Ref. [13], the Poissonian model is invalid for strong thermal sources but more analytically tractable. Consider J optical modes and a column vector of complex field amplitudes α = (α1, . . . ,αJ) within one coherence time interval. The amplitudes are normalized such that |α j| is equal to the energy in each mode in units of quanta. The central quantity in statistical optics is the mutual coherence matrix [2, 9]