General formula to design a freeform singlet free of spherical aberration and astigmatism.

In this paper, an analytical closed-form formula for the design of freeform lenses free of spherical aberration and astigmatism is presented. Given the equation of the freeform input surface, the formula gives the equation of the second surface to correct the spherical aberration. The derivation is based on the formal application of the variational Fermat principle under the standard geometrical optics approximation.