Abbe's Sine Condition

Abstract

Ernst Abbe (1840–1905), professor of physics and mathematics at the University of Jena, Germany, and major partner in the Carl Zeiss company, made important contributions to the theory and practice of optical microscopy. His compound microscope was a superb optical design based on a theoretical understanding of diffraction and minimization of the effects of aberrations. Abbe enunciated his famous sine condition regarding the axial point in the object plane of a centered image-forming system such as a microscope or a telescope. When this condition is satisfied, “aberration-free” imaging of the object points located in the vicinity of the optical axis is assured. This chapter provides an heuristic description of the sine condition, which, in the words of Conrady, is “one of the most remarkable and labor-saving theorems in the whole realm of applied optics”. As the chapter follows a rather unconventional approach towards explaining the sine condition, it is worthwhile to highlight its main features at the outset. An introduction of the necessary geometric-optical concepts provides the basis for defining the sine condition. This is followed by establishing, for an axial object point, a one-to-one mapping between the principal planes of the imaging system. The wavefront entering the system at the first principal plane (p.p.) is thus related to that emerging from the second p.p. To describe the imaging of near-axis regions, we switch to a wave-optical viewpoint.