A new technique for computing the field amplitude in the Fresnel region of a lens

An important problem in optics involves the computation of the field amplitude in the Fresnel region of a circular lens. When the observation point lines in the image plane, it is well known that the field can be written in closed form in terms of Bessel functions, i.e. the Airy pattern. In this paper, it is shown that away from the image plane the field can be expressed in terms of incomplete Weber integrals. This closed-form representation is much more accurate than the commonly used technique involving fast Fourier transforms (FFTs). This is demonstrated by comparing the closed-form results with those produced by an FFT. The closed-form solution is used to study the resolution of two point sources under the assumption that the observation plane is misaligned from the image plane.