  The ultimate performance of coronagraphic high contrast exoplanet imaging
systems such as SPHERE or GPI is limited by quasi-static aberrations. These
aberrations produce speckles that can be mistaken for planets in the image. In
order to design instruments, correct quasi-static aberrations or analyse data,
the expression of the point spread function of a coronagraphic instrument in
the presence of residual turbulence is most useful. Here we derive an analytic
expression for this point spread function that is an extension to coronagraphic
imaging of Roddier's expression for imaging through turbulence. We give a
physical interpretation of its structure, we validate it by numerical
simulations and we show that it is computationally efficient. Finally, we
incorporate this imaging model into a coronagraphic phase diversity method
(COFFEE) and validate by simulations that it allows wave-front reconstruction
in the presence of residual turbulence. The preliminary results, which give a
sub-nanometric precision in the case of a SPHERE-like system, strongly suggest
that quasi-static aberrations could be calibrated during observations by this
method.
