Lie algebraic theory of geometrical optics and optical aberrations

Abstract

A new method, employing Lie algebraic tools, is presented for characterizing optical systems and computing aberrations. It represents the action of each separate element of a compound optical system, including all departures from Gaussian optics, by a certain operator. These operators can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. New insight into the origin and possible correction of aberrations is provided. With some effort, it should be possible to produce, by manual calculations, explicit formulas for the third-, fourth-, and fifth-order aberrations of a general optical system including systems without axial symmetry. With the aid of symbolic manipulation computer programs, it should be possible to compute routinely explicit formulas for aberrations of seventh, eighth, and ninth order, and probably beyond.