Superresolution and singularities in phase images

The Rayleigh criterion and the Airy radius $r_0$ are not adequate for characterizing spatial resolution in phase and some other functional images. An essential feature of phase images is a possible formation of wavefront dislocations which depend on the position in space of the so-called singular lines $[I(x,y,z=0]$, in the neighborhood of which the phase gradient $\operatorname{grad}\varphi\approx I^{-1/2}$ increases and the intensity tends to zero. Based on this gradient phase behavior, the minimal length $L$ dependent on the signal-to-noise ratio $(S/N)$ is proposed as the phase resolution criterion, and a formula for the energy-dependent superresolution, $\Xi=r_0/L\cong 2(S/N)^{1/2}$, is devised. Measurements on a 100-nm-diameter latex sphere using the Airyscan coherent phase microscope confirmed that a marked $(\Xi\cong 5)$ superresolution can be achieved.