Electromagnetic diffraction in optical systems, II. Structure of the image field in an aplanatic system

TLDR

The study investigates the electromagnetic field structure near the focus of an aplanatic system imaging a point source, focusing on low‑aperture configurations with angular semi‑aperture 0–90°. Expressions for the electric and magnetic vectors were derived for linearly polarized and unpolarized incident fields, symmetry and polarization properties examined, and the spatial distribution of energy density and Poynting vector mapped through extensive computer calculations. The analysis shows that, in the limit of zero aperture, the field reduces to a single scalar function equivalent to classical Airy–Lommel–Struve theory, and that the detailed energy‑flow patterns and scalar‑theory correspondence have direct implications for the resolving power of low‑apert.

Abstract

An investigation is made of the structure of the electromagnetic field near the focus of an aplanatic system which images a point source. First the case of a linearly polarized incident field is examined and expressions are derived for the electric and magnetic vectors in the image space. Some general consequences of the formulae are then discussed. In particular the symmetry properties of the field with respect to the focal plane are noted and the state of polarization of the image region is investigated. The distribution of the time-averaged electric and magnetic energy densities and of the energy flow (Poynting vector) in the focal plane is studied in detail, and the results are illustrated by diagrams and in a tabulated form based on data obtained by extensive calculations on an electronic computor. The case of an unpolarized field is also investigated. The solution is riot restricted to systems of low aperture, and the computational results cover, in fact, selected values of the angular semi-aperture a on the image side, in the whole range 0 ≤ α ≤ 90°. The limiting case α → 0 is examined in detail and it is shown that the field is then completely characterized by a single, generally complex, scalar function, which turns out to be identical with that of the classical scalar theory of Airy, Lommel and Struve. The results have an immediate bearing on the resolving power of image forming systems; they also help our understanding of the significance of the scalar diffraction theory, which is customarily employed, without a proper justification, in the analysis of images in lowaperture systems.