Nulling interferometry : Lommel's integrals applied to a Fresnel's diffraction effect

Abstract

Nulling Interferometry applied to the search and characterization of earth-like exoplanets requires to eliminate the star's contribution at a rejection level (Rej = collected energy/residual energy)larger than $10^{6}$ over a large bandwidth (6 to 18 $\mu$m). Nulling test-benches are in development in several laboratories so as to master such high a rejection. One approach relies on a Mach-Zehnder set-up with Achromatic Phase Shifters (APS). One APS concept is based on the focus-crossing property, providing an intrinsically achromatic phase shift by $\pi$. Using a confocal configuration for the focus-crossing approach, a Fresnel's diffraction effect degrades the rejection. Usual optical engineering softwares fail in assessing rejection performance and an analytical approach is needed. We describe the bench optical configuration and the Fresnel's diffraction effect as well as a possible way for correction. Then we describe the analytical method, based on Lommel's integrals, to evaluate the expectable rejection.