What shape should the lens be so that it collects all the rays at one point?

Question

I want to understand what shape a biconvex lens should be so that it collects all the rays at one point (without spherical aberration). So I want to get the equation of the lens shape depending on the focal length and refractive index of the lens

Answer 1

You cannot have rigorous stigmatism for more than two conjugate points in general. As you’ve mentioned in the comment, I’ll just construct a rigorous stigmatism between the point at infinity (incoming parallel beams) and the focal point.

A first simplification would be to to have the side with the incoming collimated rays to be flat. This reduces the problem to just a rigorous stigmatism in a dioptre. Reformulating the stigmatism problem in terms of optical path length, you’ll recognize the monofocal definition of a conic section. Explicitly, given a line $L$ (directrix) and a point $F$ (focus) the locus of the points $P$ such that: $$ PF=ed(P,L) $$ with $e$ a fixed ratio (eccentricity) is precisely a conic section. In our case, the directrix is incoming wave front of the collimated beam, the focus is the focus of the lense and the eccentricity is the relative refractive index. In your case, $e>1$ so the shape is a hyperbola.

In general, you could have a curved “upstream” surface. You can use the previous reasoning to send the collimated beam to a point after the first interface. You just need to find the proper second interface which gives you stigmatism between finite points. This is given by an Appollonian circle. For two points $O,O’$ it is defined as the locus of points $P$ such that: $$ OP=r O’P $$ with $r$ a fixed ratio. Note that the previous construction is a special case where the intermediate image point is still at infinity.

Thus, to focalize a collimated beam you’ll need a conic section interface and then a circular one.

Hope this helps.