This paper is introductory to a study concerned with replacing the trial-and-error methods of optical designing by analytical procedures. The aim is to give Hamilton’s characteristic function up to any order for a single general lens as well as for systems that have suitable image-forming qualities. This paper lays the groundwork for expressing various characteristic functions in terms of the optical data of the system (radii, thicknesses, indices of refraction). The theory is made practical by a technique that leads to linear equations for the system data. The characteristic functions (or, what is equivalent, their first derivatives) are derived for systems imaging one or two surfaces sharply as well as for concentric systems, to the fifth and in some cases to the seventh order. This development agrees with results previously published in closed form by Herzberger. Moreover, the characteristic function is given to the fifth order for a thick singlet with aspheric surfaces surrounded by media of arbitrary refractive index. The methods can easily be extended to any order desired.
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