Introduction 

Diffraction theory has been greatly advanced in the 1930-1940's by the joint work of Frits Zernike, the later 1953 Nobel Prize winner because of his invention of the phase-contrast microscope (see photograph), and his PhD-student Bernard Nijboer (thesis submitted in 1942). Using the famous circle polynomials developed by Zernike, the work of Nijboer has focused on using these polynomials for an efficient representation of the complex field in the exit pupil and in the focal plane. A very elegant result provided by the circle polynomials is the automatic balancing of aberrations of various orders. This balancing problem, for a long time a holy grail in optical design and the subject of mysterious rules of thumb among the optical designers, was solved in one single stroke by the values of the coefficients of various powers of the radial coordinate in the polynomials. The circle polynomials have also provided an efficient tool to propagate the optical field originated by a point source from the circular exit pupil of an optical system to the image plane. The representation of the wave-front aberration of the focusing beam  in terms of Zernike polynomials leads to a direct representation of the complex amplitude distribution in the image plane. A basic result derived by Zernike in 1934 and extensively used by Nijboer in his thesis is the expression for the complex amplitude distribution of a point source image in the focal plane of an aberrated optical system:


This expression establishes a direct link between the Zernike (cosine- and sine-)coefficients (αnm) in the exit pupil of the aberrated imaging system and the complex amplitude in the image plane that is constructed from Bessel functions of order n+1.

Since Nijboer's thesis in 1942, the analytic progress in the field of scalar diffraction theory related to focused fields has been limited. It was only in 2002 that an extension of the Nijboer-Zernike diffraction theory to the optical field in the focal volume has become available; its application in optics is described below in a global way. The advanced reader can also choose to directly consult the ENZ-document that can be found in the downloads section of this website. This document provides a general mathematical introduction to the subject and summarizes the contents of our various publications on the subject. 

The work described in this document has been carried out by workers at Philips Research Laboratories and at Delft University of Technology, both in the Netherlands. The collaboration is between a mathematician (Augustus J.E.M. Janssen, now at Eindhoven University of Technology) and an optics experimentor Peter Dirksen (Bio & Pic), from Philips Research, and Joseph Braat, a former professor at Delft University and one of his PhD-Students, Sven van Haver.

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