Orthonormal Expansion of the Aberration Difference Function and Its Application to Image Evaluation

Method of orthonormal expansion of aberration function is applied to the evaluation of the normalized image contrast M introduced by H. H. Hopkins, the decrease of which is proportional to the variance of the wave aberration difference function V over the domain S common to the two pupils displaced relative to each other. The difference function V is expanded in terms of Zernike's circle polynomials, the domain S being approximated by an ellipse fitted to S. The orthogonal coefficients of V are derived from a linear transformation of the orthogonal coefficients of wave aberration expanded over the pupil domain. A general formula of this transformation matrix is presented. A method of calculating the normalized image contrast M is described in terms of orthonormalized aberrations together with some numerical examples.