An approach to color correction is described in which the ratio of the group velocity to the speed of light in vacuum (the group index of refraction) in glass is used, in conjunction with the more familiar phase index of refraction, to control longitudinal color in a system of thin lenses in contract. It is shown that at the wavelength of a turning point where the phase power of a lens is locally independent of wavelength, the phase power is equal to the group power. It is further shown that in a lens consisting of three or more elements, if the phase and group powers are equal and the group power has a turning point at the same wavelength, the second derivative of phase power with respect to wavelength is zero at that wavelength (the lens is confocal and achromatic in both phase and group power and the secondary spectrum is locally zero). The group index of refraction and the concept of group aberrations can equally by applied to exact raytracing for thick lens systems. In using easily calculated derivatives of the phase index of refraction the approach affords a computational simplicity that is well suited to computer-aided lens design.
Arthur H. Vaughan,
"Application of group indices of refraction to achromatic lens design", Proc. SPIE 2774, Design and Engineering of Optical Systems, (23 August 1996); doi: 10.1117/12.246720; https://doi.org/10.1117/12.246720