Abstract

The examination of the singularities of the spherical aberration function ′(χ) undertaken in a previous paper [ J. Opt. Soc. Am. A 1, 952 ( 1984)] was aimed at coming to grips with the problem of the generic properties of the power series representing ′. The present paper deals with these series. It begins by extending certain previous results to the complex domain, leading to the selection of an appropriate independent variable χ for specifying rays. A lemma is derived concerning the Taylor coefficients of the product of two analytic functions with different radii of convergence. This makes it possible to give the generic asymptotic form of the spherical-aberration coefficients en as n → ∞, namely, en ~ jnW−1χ0−2n The problem of determining the values of j, W, and χ0 from the en alone is discussed in detail. In practice one can calculate the en only to some order 2N + 1, i.e., traditionally one approximates ′(χ) by Σn=1Nenχ2n+1. A closed, rather simple expression is obtained here for the neglected remainder, ΔN(χ). Finally, a fairly detailed numerical study is undertaken of the aberration series of a particular system as a partial check on the research just described.

© 1984 Optical Society of America

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