Figure 1. Single Bracewell configuration.

In the following we describe how planet photons can be isolated from their parent star using a nulling interferometer. Figure 1 shows the simplest nulling interferometer - the single Bracewell configuration - proposed by Bracewell in 1979. This comprises two collecting apertures separated by baseline length Β, phased such that the light from an on-axis source is canceled in the single-mode spatial filter at the beam combiner output. This is the nulled or dark output port; all the on-axis photons exit from the bright port to the left of the figure. To implement this scheme requires that a phase difference of π independent of wavelength, be introduced between the two arms. The corresponding response of the interferometer on the sky is shown in both upper panels. It is a sinusoidal corrugation with a null running through the star at the center, and an angular periodicity of λ / Β. If the array is rotated about the line of sight to the star, then this corrugated pattern rotates with respect to the star and the offset planet. While the star remains on the null, the planet follows the circular locus and the detected planet photon rate (lower right) rises and falls as the peaks and troughs of the response sweep through the location of the planet. The main disadvantage of the single Bracewell configuration is that the response on the sky is symmetric. As a result there is ambiguity in the location of the planet, the exozodiacal dust emission can have a similar signature to the planet, and (most important) it is not possible to implement an effective chopping scheme.

These disadvantages are overcome with the Dual Bracewell configuration, an example of which is illustrated in Figure 2. There are now four collecting apertures. In this case, they are deployed along a line with equal spacing, phased as indicated. This configuration is essentially two single Bracewell baselines, which are then cross-combined with a third beam combiner with a relative phase shift of π / 2. The resulting response on the sky of this four-element phased array is shown in the top panel. The structure is more complex than before, and there is a clear left-right asymmetry. We will refer to this as the 'left' chop state, since there is a large peak in the response immediately to the left of the star.

Figure 2. Dual Bracewell configuration.

By changing the sign of the relative phases of the collectors, we obtain the mirror image response on the sky, as shown by the dashed line in Figure 3. This is the "right" chop state. By switching the phasing of the instrument back and forth between these two states, the response on the sky is chopped from left to right and back. Taking the difference of the photon rates obtained gives the "chopped" response denoted by the heavy line in Figure 3 (upper left panel) and the 3D view shown in the upper right. The chopped response is purely asymmetric, and the chopped photon rate has both positive and negative excursions. It is now possible to distinguish the side of the star on which the planet is located, and to discriminate against any symmetric sources of emission (e.g., star, exozodiacal dust). Any source of noise (e.g., stray light) that contributes equally to the left and right chop states is also removed.

The lower right panel of Figure 3 shows the variation of the chopped planet photon rate with the rotation angle of the array. This characteristic signature depends on the location of the planet relative to the star. As we change the "azimuthal" offset of the planet, the signature pattern is shifted left or right with respect to the array rotation angle. Increasing the radial offset of the planet from the star means that the circular locus in the upper right panel of Figure 3 expands and passes through more peaks and valleys of the response, resulting in a signature pattern with higher "frequency. " In general, the data must be inverted to obtain the fluxes and locations of any planets that are present.

Figure 3. Chopped Dual Bracewell configuration.

The approach that has been used most commonly to do this is correlation mapping, first suggested by Angel and Woolf (1997). The principle is described in Figure 4. The process is closely analogous to the Fourier transform used for standard interferometric image synthesis. The cross-correlation process generates a "dirty map" (a term borrowed from radio synthesis imaging), which must be deconvolved to extract the point-like planets. The example in Figure 4 shows the noise-free dirty map for a single point source, and therefore represents the point-spread function (PSF) for the array. Because we are dealing with a phased array in which more than two collectors are combined in a single output, the PSF is more complex than for a standard imaging array in which each baseline is measured independently. There are satellite peaks in addition to the main peak, each of which has sideobes, and the PSF varies with the position in the map. These properties depend on the array configuration. Several approaches to deconvolution are possible.

Up to this point the analysis has been for a single wavelength. The measurements in practice span a broad range of wavelengths (nominally 6.5-18 µm). Independent of the desire to do spectroscopy, the measurement must be broken out into a number of spectroscopic channels to avoid smearing together the different planet signatures (photon vs. array-rotation angle) obtained at each wavelength. Each of these channels is processed independently to obtain a correlation map. The correlation maps can then be co added (with appropriate weighting) to obtain the net correlation map. The wide range of wavelengths greatly extends the UV coverage of the array, suppressing the sidelobes of the PSF.