Cophasing a wide field multi-aperture array by phase-diversity: influence of aperture redundancy and dilution

Future high-resolution space telescoeps will use discontinuous apertures, either primary segments or sub-telescopes. One of the most critical points in the operation of such instruments will be the cophasing of the sub-apertures. Cophasing sensors are available on ground interferometers, allowing sub-aperture piston/tip/tilt measurement on unresolved or slightly resolved starts. But when observing a very extended object, such as the Earth as seen from space, no reference star can be found in the field. In this case, the cophasing measurement must be derived from the observed object itself, which is a major issue for extended objects. Phase diversity is one of the very few solutions to this problem. Phase diversity consists in the joint estimation of the object and the instrument aberrations from the analysis of several images obtained with different but perfectly known aberrations, for example the focal image and a slightly defocused image. Theoretical analysis and numerical simulations were carried out to investigate how our phase diversity algorihtm behaves when estimating sub-aperture piston and tip/tilt in various conditions. Our study shows that the aperture configuration has a major impact on performance. For diluted apertures, when the optical transfer function has zeros within the frequency domain of interest, it can be shown that at least two possible solutions can be derived by phase diversity. For redundant apertures, when several pairs of sub-aperture contribute to the same spatial frequency, the piston or tip/tilt estimation is degraded for sub-apertures contributing to redundant frequencies.