Resolution in Diffraction-limited Imaging, a Singular Value Analysis

Abstract

In a previous paper, methods of singular function expansions have been applied to the analysis of coherent imaging when the object and image domains are allowed to differ. In this paper the method is extended to incoherent illumination, restricting the analysis to the aberration-free case. While singular functions and singular values for coherent imaging are related in a simple way to the prolate spheroidal functions and their eigenvalues, such relations do not exist for the incoherent imaging case. In spite of this difficulty many properties of singular functions and singular values are derived in this paper and asymptotic estimates are obtained in the limit of large space-bandwidth product. For small values of the space-bandwidth product, the singular values are computed numerically and by means of these results it is shown that super-resolution, in the sense of improving on previous criteria in the presence of noise, can be achieved.