On the focal point of a lens: beyond the paraxial approximation

In this paper we study a planar-convex lens where the focal point is calculated numerically and analytically beyond the paraxial approximation within the context of geometrical optics. We consider this problem as an appropriate and useful example to fill the gap found in physics and optics courses between the simplicity of the paraxial approximation and the complexity of the theory of aberrations, and it can be used as an introduction to non-paraxial behaviour even when teaching general physics courses. We show in a simple way how beyond the paraxial approximation the focal distance is not unique, and how it depends on the distance of the incoming ray to the optical axis. We show the importance of the caustic surface, which is calculated analytically, and its effect oil the position of the point with the highest concentration of light, which is defined as the optimal focal distance of the lens. Finally, we also present some simulations showing light distributions in screens placed at different distances from the lens, to illustrate our results.