From Fractals to Exoplanets: building ultra-nulling interferometers

Abstract

The difficult goal of directly detecting a planet around a star requiresto cancel as much as possible the stellar light.Since the first proposal by Bracewell of a nulling interferometer, where thestar is put on a central dark fringe, several interferometricconfigurations have been presented in order to improve the quality ofthe rejection, especially to avoid the leaks due to the finite angular dimensionof the stellar disk, resolved by the interferometer. In the Bracewell interferometer, the behaviour of the nulling efficiency vs the angular distance θ to the star is as (1-cosθ)∝ θ². One goal is to increase the exponent of the term θⁿ whichgives the cancellation efficiency. I present one method to define configurations of telescopes positions, sizes and phase-shift that canachieve any given power of θ. The principle is based on a peculiar property foundby 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 rankin the Thué-Morse sequence and, to the telescopes of one of the groups, is applied a π phase shift. The result is a fractal-like distribution of the telescopes where redundancy is minimum andwhose interferometric combination produces a very efficient nulling in θ^2L.I first examine 1-D patterns of identical telescopes, then extend the method to 2-D configurations, then show that the lattercan be used to define 1-D arrays of non identical telescopes, according to some algebra of interferometers. The generalizationto arrays where the phase shift between n groups of telescopes is 2kπ/n is finally proposed.