We obtain simple exact formulas for both caustic and refracted wavefronts through smooth surfaces by considering an incident plane wavefront propagating along the optical axis, providing a condition for total internal reflection (TIR). On the other hand, a formula to provide the maximum slopes for refracted rays outside of the lens is directly related to the condition to obtain the inflection points on the refracting surface. Demanding simultaneously both conditions can potentially provide the shape for an optimized surface which reduces the gap produced by TIR and to refract efficiently all the light outside of the lens. This has a wide potential in applications on the field of non-imagining systems and illumination.
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