An Analytical Solution of Vector Diffraction for Focusing Optical Systems

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Abstract
Vector diffraction theory for optical systems has been of interest for a long time. Ignatovsky and Wolf have formulated these problems in terms of diffraction integrals and Wolf has presented very interesting results. Usually, the quadrature of diffraction integrals is numerically intensive, therefore these problems have remained of interest and many authors have worked on the Ignatovsky-Wolf formulation or some variation thereof. This paper presents yet another method of solving diffraction integrals. Since a certain part of the kernel of these integrals is Riemann integrable in the interval [0, π], the Weierstrass theorem says that it can be approximated by a uniformly convergent series of orthogonal functions. Thus it is possible to expand these functions into a series of Gegenbauer polynomials of the first kind. Once these expansions are substituted in the diffraction integrals, the resulting integrals are readily evaluated, over the surface of unit sphere, in terms of the spherical Bessel functions and Gegenbauer polynomials. The results are particularly simple if the image plane is the focal plane. In this paper, we evaluate the diffraction integrals for several optical systems of arbitrary numerical aperture with or without obscuration, and for a parabolic reflector. The results presented here are in agreement with previously published results. The numerical computations are easy since all the functions are evaluated by adding a finite series. The calculations which for the basis of results presented in this paper were performed on a personal computer.

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