  The purpose of this paper is to present the results of a theoretical study
pertaining to the feasibility of PIAA units using Deformable Mirrors. We begin
by reviewing the general derivation of the design equations driving PIAA. We
then show how to solve these equations for square apertures and show the
performance of pure PIAA systems in the ray optics regime. We tie these design
equations into the study of edge diffraction effects, and provide a general
expression for the field after a full propagation through a PIAA coronagraph.
Third, we illustrate how a combination of pre and post apodisers yields to a
contrast of 10^10 even in the presence of diffractive effects, for
configuration with neither wavefront errors or wavefront control. Finally we
present novel PIAA configurations over square apertures which circumvent the
constraints on the manufacturing of PIAA optics by inducing the apodisation
with two square Deformable Mirrors (DM). Such solutions rely on pupil size
smaller than currently envisioned static PIAA solutions and thus require
aggressive pre and post-apodizing screens in order to mitigate for diffractive
effect between the two mirrors. As a result they are associated to significant
loss in performance, throughput in particular.
