Cramér-Rao lower bound analysis of phase diversity for sparse aperture optical systems

Aberration detection of subapertures is critical to the resolution of a sparse aperture optical system. In this work, the Cramér-Rao lower bound (CRLB) expression of aberrations was first derived for the subapertures of a sparse aperture optical system. The aberrations were estimated by phase diversity, assuming that the photons follow the Poisson distribution. The derived formula has been applied to the Golay3 sparse aperture system to calculate the CRLB of the subapertures' aberrations with different defocus distances, fill factors, and ratios of photon numbers between the focal and defocal images. According to the CRLB criterion, the results indicate that the accuracy of aberration estimation increases as the fill factor of the sparse aperture system increases. The results also demonstrate that the optimal defocus distance of phase diversity is achieved when the CRLB reaches its minimum. The optimal ratio of photon numbers between the focal and defocal images is around 3∶7. However, for the piston error, more photons in the defocal image yield more accurate estimation results. This method can also be applied to the sparse aperture optical systems with other different configurations.