# The ideal shape of a plano-convex lens

What's the type of the ideal shape of a surface of a plano-convex lens to focus a parallel beam at the point? Considering the lens turned its flat side to a beam.

Surely it must be aspheric: probably hyperbolic or parabolic - but i have no idea how to figure it out.

The formula of a refraction at spherical surfaces is not that easy but possible to get. But how to deal with hyperbolic or parabolic surfaces?

Or is there a simpler way to answer my question?

• if you are focusing a collimated beam, it's best to have the curved side face the beam, not the flat side. The flat side does nothing to the collimated beam, so you lose some opportunity of correcting spherical aberration. – wcc Mar 18 at 14:09
• Hyperbolic lens. Use equivalence of the optical paths for two rays – user591849 Mar 19 at 9:21
• @user591849 , Oh right! By Fermat's principle 2 different paths are passed for the same time. So if we consider the curve in the lense's section as f(x), we could get an equation - and it turned out to be hyperbola. – SilverLight Mar 19 at 11:19