Symmetry properties of aberrated point-spread functions

Abstract

The symmetry properties of point-spread functions of optical imaging systems with circular pupils aberrated by a Zernike-circle polynomial aberration were discussed by Nijboer [ “ The diffraction theory of aberrations,” Ph.D. dissertation ( University of Groningen, Groningen, The Netherlands, 1942)]. Although it was not pointed out by him, his analysis and some of his results are valid only for systems with large Fresnel numbers. The symmetry properties for systems with small and large Fresnel numbers are discussed. The analysis is extended to systems with annular pupils aberrated by a Zernike-annular polynomial aberration. This analysis is further extended to pupils with nonuniform but radially symmetric illumination such as Gaussian. It is shown, in particular, that whereas, for uniform pupils, the axial irradiance of the imaging-forming light cone for a primary spherical aberration is symmetric about the defocused point with respect to which the aberration variance is minimum, it is asymmetric for nonuniform pupils. The discussion is equally valid for focused beams of light. Computer-generated pictures of point-spread functions of systems with circular and annular pupils aberrated by primary aberrations illustrating their symmetry properties are given.