We can write an approximate analytic expression for the Airy disk, i.e. the intensity of a Fraunhofer diffraction pattern of a circular aperture, as: \(I(\theta) = I_0 (\frac{2 J_1}{x})^2\), where \(I_0\) is the peak intensity of the Airy disk, \(x = ka\times sin(\theta) = (\frac{2\pi}{\lambda})a \times sin(\theta)\), where \(\lambda\)is the wavelength, and \(a\)is the aperture radius (src: http://en.wikipedia.org/wiki/Airy_disk, "Mathematical details" section).

However, is it possible to write down an analytic expression for smallest possible diffraction limited donut? I am interested in this in part to better understand the shape of the emission profile in the context of STimulated Emission Depletion (STED: http://en.wikipedia.org/wiki/STED_microscopy) microscopy, where a "donut" shaped laser is superimposed over a "stimulating" laser (exciting e.g. fluorophores) focused to diffraction limit. If this donut shaped laser is sufficiently intense, it can induce stimulated emission of a red-shifted photon for fluorophores some distance from its circumference. If you then filter out photons beyond a cutoff red-shift from the emission peak of the fluorophores, you'll only see emission near the center of the donut letting you sort of "cheat" the diffraction limit.

However, I have no idea how the efficiency for stimulated emission falls off as one moves along a line from the circumference of the depletion donut to the donut's center, and I'd like to be able to at least write down an expression for this stimulated emission efficiency vs. displacement expression that has roughly the right form.