## Abstract

Formulas for the four gaussian constants *A, B, C*, and *D* characterizing the paraxial properties of a symmetric optical system are derived for various single lenses in which the refractive index varies continuously from point to point. Particular attention is paid to those lenses with either a cylindrical or an axial index distribution and questions pertaining to the depth of an axial distribution are considered; in particular, it is shown that the depth of an axial distribution has little effect on the paraxial properties. More general distributions are treated by expanding the gaussian constants as power series in the lens thickness. These results are applied to the problem of estimating tolerances on the inhomogeneities. It is shown that the choice of index distribution can have important effects on the relationship between the power of the lens and its Petzval field curvature.

© 1971 Optical Society of America

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