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 e_n as n → ∞, namely, e_n ~ jn^−W−1χ₀⁻²ⁿ The problem of determining the values of j, W, and χ₀ from the e_n 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 ${\mathrm{\Sigma }}_{n=1}^N\hspace{0.17em}{e}_n{\chi }^{2n+1}$. A closed, rather simple expression is obtained here for the neglected remainder, Δ_N(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

Power series of geometrical optics. I.

H. A. Buchdahl
J. Opt. Soc. Am. A 1(9) 952-957 (1984)

Power series of geometrical optics. IV

H. A. Buchdahl and G. W. Forbes
J. Opt. Soc. Am. A 3(8) 1142-1151 (1986)

Power series of geometrical optics. III.

H. A. Buchdahl
J. Opt. Soc. Am. A 2(6) 847-851 (1985)

Power series of geometrical optics. V. Characteristic coefficients of the refracting plane

H. A. Buchdahl
J. Opt. Soc. Am. A 3(10) 1620-1628 (1986)

Weighted truncation of power series and the computation of chromatic aberration coefficients

G. W. Forbes
J. Opt. Soc. Am. A 1(4) 350-355 (1984)