# An analytic expression for coronagraphic imaging through turbulence. Application to on-sky coronagraphic phase diversity

## TL;DR

This paper derives an analytic expression for coronagraphic imaging through turbulence, enabling efficient wave-front reconstruction and quasi-static aberration calibration in high contrast exoplanet imaging systems.

## Contribution

It extends Roddier's imaging expression to coronagraphic systems, providing a validated, computationally efficient model for turbulence-affected imaging.

## Key findings

- Validated the model with numerical simulations.
- Achieved sub-nanometric wave-front reconstruction precision.
- Demonstrated potential for real-time aberration calibration during observations.

## Abstract

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.