The fourth-order index coefficients of GRIN-rod lenses are estimated using the magnitudes of the third-order wave-front spherical aberration reported by Cline and Jander and with the relationship between the third-order wave-front spherical aberration and the fourth-order index coefficient derived for the model of a near one-quarter pitch GRIN-rod lens in a double-pass configuration using the optical path length which is found analytically by algebraic manipulation of the asymptotic solution for the meridional ray equation derived by Streifer and Paxton.
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Toyokazu Sakamoto (1986) studied this question.
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