# Chasing exoplanets with the help of a chessboard and Pascal's triangle

Direct detection of an exoplanet (a planet outside our solar system) that is orbiting a star requires the stellar light to be dimmed as much as possible. This is due to the huge contrast in brightness (10^{6–10}) between the parent star and its orbiting planet, as well as the small angular separation (10^{−6} rd) of the telescope. In the mid-infrared, one way to image the planet against the background light of the star is by *nulling interferometry,* which uses at least two telescopes coherently recombined in the way proposed by Bracewell^{1}, so that the brighter light from the star is cancelled out by light wave interference, allowing the orbiting planet to be seen.

When a π phase shift is applied to one of the arms of the telescope, a system of fringes with a central dark fringe is projected onto the sky. The star image, put on the dark fringe, is strongly attenuated, while the planet, if on a bright fringe, can be detected. Obtaining a π phase shift that brings different wavelengths to focus together (is achromatic) is mandatory because the wavelength domain where spectroscopic life signatures are to be found is broad^{2} (typically 6-18 μm), and it's a photon-starving experiment.

Various methods^{3} have been presented in order to approach an achromatic π phase shift in a large domain of wavelength-field reversal, including top-roof mirrors, and crossing a focus with stacks of dielectric (non-conducting) plates. Unfortunately, they typically make the two arms of the interferometer asymmetrical and introduce several additional optical components. We propose a new solution^{4,5} using a twin mirror made of cells of different thicknesses transposed as a chessboard pattern. It is the peculiar distribution of the cells' thickness that makes the π phase shift quasi-achromatic on a broad domain.

Figure 1 depicts our two-telescope interferometer. (A multi-axial recombination is shown here, but an uniaxial one is also valid.) The first telescope issues a beam of parallel (or collimated) light rays. Then, we introduce a cellular mirror (or a transparent plate) where cells produce optical path differences (OPDs) that are odd multiples of half the central wavelength (λ_{o}/2). Symmetrically, a cellular mirror where cells produce OPDs which are even multiples of λ_{o}/2 is put into the beam from the second telescope. At λ_{o}, each cell introduces a phase shift of either (2*k* + 1)π (odd mirror) or of 2*k* π (even mirror) on the fraction of the wave it reflects. Thus, a π phase shift between the two arms is applied for λ = λ_{o}. If now one develops as a Taylor's series the amplitude at the recombined output, with respect to *x* = Δ λ / λ_{o} = (λ − λ_{o})/ λ_{o}, it appears that there is a peculiar distribution of the cells’ thickness for which the first terms of the development vanish, up to any given order 2n (provided that there are 2^{n} cells per mirror). This *magic* distribution of the thickness, i.e. the number *m*_{k} of cells of thickness *k* λ_{o}/2 , is given by the binomial coefficients which are recursively found thanks to Pascal's triangle. The main consequence is that, as long as *x*< 1, the nulling can be effective since the first non-vanishing term is *x*^{n}. A quasi-achromatic nulling is thus reached when *n* is large enough (in practice *n* > 5).

The Pascal's triangle solution provides the distribution in *z* of the cells, but it does not say anything on the distribution of cells in *x* and *y* (on the surface of the mirrors). We have found one which is optimum in terms of *darkening* the residual point spread function (PSF) of the star image close to the axis. The pattern is built according to a recursive scheme, in the line of the ideas developed by Rouan^{6}. Figure 2 shows the resulting pattern for the pair of mirrors (odd and even) when *n*=5.

We have developed a simulator based on a fully analytical solution to predict the performance of our configuration. In Figure 3 we compare, for a planet 10^{−6} fainter than its star, the nulling efficiency versus Δλ/λ_{o} for different orders of the best chessboard. It is worth noting the flatness of the star-to-planet ratio versus λ for the highest orders. The bandpass is typically [0.85λ_{o} − 1.7λ_{o}], i.e. one octave. Note that a realistic phase error between the two arms is introduced in the model.

An 8×16 device with a cell pitch of 600 μm was manufactured by SILIOS-France using an electron etching technique. The two masks (odd and even) are put side by side. The first images – in Fizeau recombination – show, as expected, a central dark fringe which is close to the model prediction, but more important which is fixed when λ ≠ λ_{o}, as illustrated in Figure 4. (The dark fringe would shift with classical mirrors.)

Our achromatic phase shifter for nulling interferometry is based on the use of a pair of chessboard mirrors with 2^{n} cells of different thicknesses producing 2kπ and (2k+1)π phase shift at λ=λ_{o}. Thanks to a peculiar diophantine relation between the odd and even integers that describe the cells thickness, it is possible to reach a quasi-achromatic π phase shift on one octave around λ_{o}. In addition, for a proper distribution of the cells on the surface of the mirror, the planet-to-star contrast is largely improved. The predicted performance of the contrast enhancement, even when realistic phase errors are considered, is 10^{6} within Δλ/λ_{o} = 0.9, well within the specifications of a space telescope such as the European Space Agency's DARWIN^{7} mission.

Our approach presents several advantages: a very compact and robust system, a fully symmetric design with respect to the two arms of the interferometer, and finally the concept can be extended to multi-telescope interferometers with a phase shift other than π. The use, in the future, of adaptive segmented mirrors based on micro optical electro mechanical systems (MOEMS) technology, could allow adjustment of the wavelength and a fine correction of error that would be beneficial to future exoplanet detection programs.

*Icarus*123, pp. 249-255, 1996.doi:10.1006/icar.1996.0155