We present details of a novel imaging algorithm based on the extended Nijboer-Zernike (ENZ) theory of diffraction. We derive integral expressions relating the electric field distribution in the entrance pupil of an optical system to the electric field in its focal region. The evaluation of these integrals is made possible by means of a highly accurate and efficient series expansion similar to those occurring in standard ENZ theory. Based on these results an ENZ imaging scheme is constructed and evaluated in detail with attention to the convergence properties and computational complexity of the new method.

We present a derivation of the analytic result for on-axis field values of the Rayleigh diffraction integral, a result that was originally presented in a paper by Osterberg and Smith (1961). The method on which our derivation is based is then applied to other diffraction integrals used in acoustics and optics, e.g., the far-field Rayleigh integral, the Debye integral and the separate near-field part of the Rayleigh integral. Having available our on-axis analytic or semi-analytic solutions for these various cases, we compare the various integrals for wave numbers k pertaining to low-frequency acoustic applications all the way up to high-frequency optical applications. Our analytic results are compared to numerical results presented in the literature.