How to approach the design of a bilateral symmetric optical system

symmetry has been for a century and continues to be a subject of interest in lens design. The study of these systems is relevant for understanding the effects of fabrication and assembly errors in lenses nominally designed as axially symmetric,' for designing reflective systems without a central obstruction,2 or for designing exotic systems that provide a practical solution to a difficult problem.3 The literature on the theory of nonaxially symmetrical optical systems is extensive. For example, anamorphotic systems have been discussed,48 the effects of decenters in lenses have been addressed,"9_'3 the transverse ray aberrations of plane symmetric systems have been analyzed,'4 and many other relevant works on nonrotationally symmetric systems have also been published.'527 The class ofoptical systems that possess a plane of bilateral symmetry is of particular interest because such systems are simple enough to be easily described, practical, and because they exhibit most of the significant differences between axially symmetric systems and those that lack this symmetry. The purpose of this paper is to present a theoretical development for understanding and designing plane symmetric systems with special emphasis on systems made with surfaces that are spherical or slightly aspheric. This development can be considered an extension of the wave theory of rotationally symmetric systems. Although there are other methods for designing plane symmetric systems, the theory presented has value because of the insight it gives2833 and because the concepts used are familiar to the lens designer. Other methods that rely on the use of off-axis or eccentric sections of axially symmetric systems are very powerful because of the inherent symmetry, but they do not provide many new insights. The theory presented in this paper can give a different point of view in the analysis of a plane symmetric system. In the next section, we define several primordial entities of a bilateral symmetric system. The third section establishes an aberration function and consequently the size and position of the image, its defects, and the concept of aberration fields. The fourth section presents an informal derivation of aberration coefficients. In the fifth section, simplifications for reflective confocal systems are addressed, and finally, the last section discusses and summarizes the paper contents.