Ultra deep nulling interferometry using fractal interferometers

The difficult goal of directly detecting a planet around a star requires the cancellation of, as far as possible, the stellar light and nulling interferometry is one way to do so: the star is put on a central dark fringe while the planet is supposed to be on a bright fringe. One problem is, however, leaks due to the finite angular dimension of the stellar disk, resolved by the interferometer. The solution is to increase the exponent of the term theta(n) which describes the cancellation efficiency with respect to the angular distance to the axis of the central dark fringe. Efficient configurations have been found, using basically guess and check methods until recently. I present here one method to define configurations of telescopes that achieve any given power of theta. The principle is based on a peculiar property of a partition into two sets of the first 2(N) integers; the partition is built using the Prouhet-Thue-Morse sequence which presents some fractal properties. A phase shift (0 or pi) between 2N telescopes is applied according to this partition. I first examine 1-D pattern of identical telescopes, then extend the method to 2-D configurations of identical telescopes, to 1-D arrays and 2-D arrays of non-identical telescopes and finally to arrays where the phase shift between n groups of telescopes is 2k pi/n. I examine then how a non-perfect fractal interferometer behaves and show that its robustness with respect to nulling stability is an important advantage.