Aberration reduction by multiple relays of an incoherent image

Consider a generally aberrated one-dimensional (1D) optical pupil P illuminated by quasi-monochromatic light of mean wavelength lambda. In past work it was found that, if the pupil's intensity point-spread function (psf) is multiply convolved with itself, as in an imaging relay system, and then ideally (stigmatically) demagnified, the resulting psf s(x) approaches a fixed Cauchy form s(x) = deltax( pi2x2 + deltax2)(-1), which is independent of the aberrations of the pupil. Here deltax is the Nyquist sampling interval given by deltax = lambdaf/2 with f the f/number of the pupil. This Cauchy form for this intensity psf s(x) also manifestly lacks sidelobes. The overall questions that we examine are how far do these effects carry over to the case of a circular, two-dimensional (2D) pupil, and to what extent do practical imaging considerations compromise the theoretical results? It is found that, in the presence of spherical aberration of all orders, the resulting theoretical psf of a large number of self-convolutions approaches a "circular" Cauchy form, S(r) = 2deltar[pi2r2 + (4deltar/pi)2](-3/2), where deltar is the Nyquist sampling interval lambdaf/2 with f the f/number of the (now) circular pupil. Thus, for these aberrations the 1D effect does carry over to the 2D case: The output psf does not depend on the aberrations and completely lacks sidelobes. However, when all aberrations are generally present, the output psf s(r, theta) does depend on the aberrations, although its azimuthal average over theta still preserves the circular Cauchy form, as a superposition of Cauchy functions. Imaging requirements for achieving these ideal effects are briefly discussed as well as probability laws for photons that are implied by the above-mentioned PSF's s(x) and S(r). Real-time super resolution is not attained, since the stigmatic imaging demanded of the demagnification step requires the use of a larger-apertured lens. Rather, the approach achieves significant aberration suppression.