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 θ to the star is as (1-cosθ)
∝ θ2. One goal is to increase the exponent of the term θn 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 θ. The principle is based on a peculiar property found
of a partition into two sets of the integers, done according to the Thué-Morse sequence.
2L 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 π 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 θ2L.
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 2kπ/n is