Generalized formulas for ray-tracing and longitudinal spherical aberration.

In this work, we generalize the paraxial ray-tracing formulas to include nonparaxial rays. For a refracting (reflecting) spherical surface, a new single meridional formula is derived. This formula can be easily reduced to a paraxial formula. It can also be applied to any aspheric (or general) surface with a known equation. Also, a new exact ray-tracing procedure for a centered system of spherical surfaces is derived. In this procedure, we apply just two simple equations for each surface of the system, which, to the best of our knowledge, makes it the shortest analytical ray-tracing technique ever. This procedure can be applied in some other applications. For example, it can be reduced to a new single paraxial formula that can be easily used to trace a paraxial ray propagating through a system of spherical surfaces. Also, it is applied to derive an exact meridional formula for both thick and thin lenses that can also be reduced to a new paraxial formula different from the Gaussian one. These results led us to easily derive an exact formula for the longitudinal spherical aberration for both thick and thin lenses and also for a single refracting (reflecting) spherical surface. Numerical examples are provided and discussed.