Pupil Mapping in 2-D for High-Contrast Imaging
  Pupil-mapping is a technique whereby a uniformly-illuminated input pupil, such as from starlight, can be mapped into a non-uniformly illuminated exit pupil, such that the image formed from this pupil will have suppressed sidelobes, many orders of magnitude weaker than classical Airy ring intensities. Pupil mapping is therefore a candidate technique for coronagraphic imaging of extrasolar planets around nearby stars. The pupil mapping technique is lossless, and preserves the full angular resolution of the collecting telescope, so it could possibly give the highest signal-to-noise ratio of any proposed single-telescope system for detecting extrasolar planets. A planet fainter than $10^{-10}$ times its parent star, and as close as about $2 \lambda/D$ should be detectable. We derive the 2-dimensional equations of pupil mapping for both 2-mirror and 2-lens systems. We give examples for both cases. We derive analytical estimates of aberration in a 2-mirror system, and show that the aberrations are essentially corrected with an added reversed set of mirrors. 
R.J. Vanderbei and W.A. Traub