Abstract

Fresnel–Kirchhoff’s formula for the diffraction effects of a rectangular aperture in the presence of aberrations is expanded into a series involving only one-dimensional integrations. The resulting nonorthogonal polynomials are expanded in terms of Legendre functions. Special functions are defined on the basis of the orthogonality of Legendre’s polynomials. These polynomials involve both linear combinations of Bessel functions of order n+12 (where n is a nonnegative integer) and arguments containing only a single optical coordinate. The general solution is then expressed by products of these Bessel functions. Finally, some particular cases of the general solution are cited.

© 1964 Optical Society of America

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