Analytic conic constants to reduce the spherical aberration of a single lens used in collimated light

We study the formation of caustic surfaces produced by conic lenses, considering a plane wavefront propagating parallel to the optical axis. The shape of the caustic can be modified by changing the parameters of the lens in such a way that if we are able to vanish the caustic, the optical system produces the sharpest diffraction-limited images. Alternatively, caustic surfaces with a large area can be applied to the design of non-imaging optical systems, with potential applications such as diffusers of light for illumination or solar concentrators. Here, we provide analytic equations for the conic constants, principal surfaces, and caustic surfaces, and also approximations at the third and fifth orders formed by conic lenses, in order to reduce the spherical aberration at these orders.