The general equation of the stigmatic lens

Abstract

I present a solution to a problem proposed by Diocles, in ancient Greece. A problem with more than two millennia without an analytical close form solution, the design of the stigmatic lens. Stigmatism refers to the image-formation property of an optical system which focuses a single point source in object space into a single point in image space. Two such points are called a stigmatic pair of the optical system. A stigmatic lens is a lens such that the point object and the point image are stigmatic. In this doctoral thesis, the general equation of the stigmatic lenses was found. The most important implications of this equation are the uniqueness of stigmatism and a new methodology for designing stigmatic lenses in a totally analytical way, free of iterations and numerical approximations. This methodology is the basis for designing more complex stigmatic systems with a freeform shape, an arbitrary number of refractive surfaces, telescopes etc.