I'm playing with Algodoo optics. I'm trying to focus parallel lasers using refraction. I thought an ellipse or parabola would work, but not precisely. As you can see in this picture, the ellipse focuses the outside beams too near, and the parabola focuses them too far. I guess the perfect lens is in between these two. What is the cross section formula of a perfect lens? Or if this is a bug in Algodoo, I'd like to know that.
Based on the suggestion by mmesser, I created an ellipse and adjusted the eccentricity so that all beams are refracted to the far focal point of the ellipse. Then I cut away the ellipse with a circle centered on that focal point. This technique can be scaled to produce a perfect meniscus lens for any refractive index and desired focal length. I tried the same using a parabola and it does not work, because there is no internal focus point.
I didn't prove this rigorously. maybe I'll try tomorrow.