## Abstract

The mathematical process of combining the known characteristic functions of a number of optical systems to obtain the characteristic function of the system created by adjoining these component systems is shown to permit a significant simplification: If the intermediate variables are solved as power series to a given order in the external variables, then it is possible to determine the power series for the resulting characteristic function to at least twice this order. This doubling of order is shown to be a direct result of the extremal property of the rays. By way of illustration, the point characteristic of a refracting plane is determined explicitly as a power series accurate to within terms of degree 16.

© 1982 Optical Society of America

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