  Phase-induced amplitude apodization (PIAA) coronagraphs are a promising
technology for imaging exoplanets, with the potential to detect Earth-like
planets around Sun-like stars. A PIAA system nominally consists of a pair of
mirrors which reshape incident light without attenuation, coupled with one or
more apodizers to mitigate diffraction effects or provide additional
beam-shaping to produce a desired output profile. We present a set of equations
that allow apodizers to be chosen for any given pair of mirrors, or conversely
mirror shapes chosen for given apodizers, to produce an arbitrary amplitude
profile at the output of the system. We show how classical PIAA systems may be
designed by this method, and present the design of a novel 4-mirror system with
higher throughput than a standard 2-mirror system. We also discuss the
limitations due to diffraction and the design steps that may be taken to
mitigate them.
