Power series of geometrical optics. IV

A previous paper in this series [ J. Opt. Soc. Am. A 1, 958 (1985)] dealt with generic properties of the Taylor series representing the spherical aberration function ?'(?) of a symmetric system of spherical refracting surfaces. The treatment was, however, essentially incomplete owing to an erroneous claim that ?'(?) had only real singularities. It is in fact possible for the generic asymptotic behavior, for sufficiently large values of n, of the aberration coefficients e_n to be dominated by complex singularities of ?'(?). In this paper the earlier work is amended to allow for the effects of the presence of complex singularities. The behavior of the function ?'(?) near its singularities being known, a representation of the aberration coefficients as Fourier integrals makes it possible to invoke a known theorem to establish the required asymptotic relations. The possibility that a set of singularities, not confined to the circumference of a circle, may significantly affect the form of e_n even when n is large (though not impracticably large) leads to the formulation of a conjecture about the asymptotic behavior of Taylor coefficients. As in earlier work, the question of classifying different types of singularities is briefly explored. Procedures are described for assigning numerical values to the constant parameters that occur in asymptotic relations, given sets of computed values of e_n. Some incidental points of principle are disposed of in the course of a lengthy numerical illustration. Since hitherto in practice no manifest, distinctive traces of the presence of dominant complex singularities have been encountered, the design of an appropriate, even if artificial, system first had to be contrived. The results of the various calculations fully support theoretical conclusions.