  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.
