From Fractals to Exoplanets: building ultra-nulling interferometers

Abstract
The difficult goal of directly detecting a planet around a star requires to cancel as much as possible the stellar light. Since the first proposal by Bracewell of a nulling interferometer, where the star is put on a central dark fringe, several interferometric configurations have been presented in order to improve the quality of the rejection, especially to avoid the leaks due to the finite angular dimension of the stellar disk, resolved by the interferometer. In the Bracewell interferometer, the behaviour of the nulling efficiency vs the angular distance to the star is as (1- ) . One goal is to increase the exponent of the term which gives the cancellation efficiency. I present one method to define configurations of telescopes positions, sizes and phase-shift that can achieve any given power of . The principle is based on a peculiar property found by Prouhet of a partition into two sets of the integers, done according to the Thué-Morse sequence. 2 ^L telescopes regularly spaced on a line, are distributed into two groups, following their rank in the Thué-Morse sequence and, to the telescopes of one of the groups, is applied a phase shift. The result is a fractal-like distribution of the telescopes where redundancy is minimum and whose interferometric combination produces a very efficient nulling in . I first examine 1-D patterns of identical telescopes, then extend the method to 2-D configurations, then show that the latter can be used to define 1-D arrays of non identical telescopes, according to some algebra of interferometers. The generalization to arrays where the phase shift between n groups of telescopes is 2k /n is finally proposed.