Understanding the diffraction limit in the context of being provided perfect information on an intensity distribution

Question

As per http://scienceworld.wolfram.com/physics/AiryDisk.html, let the intensity distribution given by diffraction around a circular aperture be proportional to:

$I(r) \propto [\frac{J_1(r)}{r}]^2$

Where $J_1(r)$ is a Bessel function of the first kind. We can rotate this curve to generate a two-dimensional Airy disk: http://en.wikipedia.org/wiki/File:Airy-3d.svg.

My question is as follows: the "diffraction limit" tells us that if we shine a light through two circular apertures that are separated by less than approximately half the wavelength of the light being emitted from the apertures ($\approx \frac{\lambda}{2}$), it becomes impossible for an observer from afar to tell how many circular point sources of light there are, and/or where they are.

My intuition is this rule is more a statement that, due to noise and detector statistics, one simply can't determine intensity profiles to the requisite resolution to allow for determination of the number, position, and respective intensities of the point sources being summed together.

So, with only a finite collection of point sources, is it possible to engineer a "conspiracy" where, provided perfect information about the intensity profile of one or more point sources of light being summed together, one cannot determine the intensity and location of each point source? In other words, can we find a way to overlay and sum the intensities of a finite collection of Airy disks (or more simply Gaussian curves) s.t. information is lost about the curves being summed together?

Edit - I don't mean above that we simply "sum" the intensities of the Airy disks. There will be diffraction between the point sources in the eyes of the observer. Still, it is not at all clear to me that information can be lost due to a conspiracy of point source arrangements and intensities.

Edit 2 - Carlo Beenakker provided a very nice answer to this question in the case of coherent point sources of light, but can information be lost in the manner described if the point sources are incoherent?

Answer 1

Consider the transmission function $T(x,y)$, equal to unity when $(x,y)$ is inside an aperture and equal to zero outside. (The apertures lie on a screen in the $x,y$ plane, and I am assuming monochromatic plane wave illumination with wave number $k$, so that the wave has the same phase at each aperture and $T$ is real. More generally, the transmission function contains both amplitude and phase information.)

The far-field ( Fraunhofer ) diffraction pattern $D(x,y)$ (on a screen at some large distance $L$ from the source) is given by the two-dimensional Fourier transform

$$D(x,y)=\int_{-\infty}^{\infty}dx'\int_{-\infty}^{\infty}dy' T(x',y')\exp\left(-i\frac{k}{L}(x'x+y'y)\right)$$

An inverse Fourier transform of $D$ then uniquely determines the transmission function $T$, and hence the location and size of the apertures.

Note that for this inversion it is essential that both the amplitude and the phase of the diffracted wave are measured.

The classic application of this inversion is the identification of a crystal structure from electron diffraction: the atoms of the crystal can be considered as a collection of point sources, and their location is reconstructed from the diffraction pattern.