  Phase masks coronagraphs can be seen as linear systems that spatially
redistribute, in the pupil plane, the energy collected by the telescope. Most
of the on-axis light must ideally be rejected outside the aperture to be
blocked with a Lyot stop, while almost all off-axis light must go through it.
The unobstructed circular apertures of off-axis telescopes make this possible
but all major telescopes are however on-axis and the performance of these
coronagraphs is dramatically reduced by the central obstruction. Their
performance can be restored by using an additional optimally designed apodizer
that changes the amplitude in the first pupil plane so that the on-axis light
is rejected outside the obstructed aperture of the telescope. The numerical
optimization model is built by maximizing the apodizer's transmission while
setting constraints on the extremum values of the electric field that the Lyot
stop does not block. The coronagraphic image is compared to what a non-apodized
phase mask coronagraph provides and an analysis is made of the trade-offs that
exist between the apodizer transmission and the Lyot stop properties. The
existence of a solution and the mask transmission depend on the aperture and
the Lyot stop geometries, and on the constraints that are set on the on-axis
attenuation. The system throughput is a concave function of the Lyot stop
transmission. In the case of a VLT-like aperture, apodizers with a transmission
of 0.16 to 0.92 associated with a four-quadrant phase mask provide contrast as
low as a few 1e-10 at 1 lambda/D from the star. The system's maximum throughput
is 0.64, for an apodizer with an 0.88 transmission and a Lyot stop with a 0.69
transmission. Optimizing apodizers for a vortex phase mask requires computation
times much longer than in the previous case, and no result is presented for
this mask.
