in this physics lecture, on slide 15-16, it is found that the ideal surface for a perfect lens (which maps a plane wavefront into a perfect spherical wavefront, i.e. which makes focus all input parallel rays into one point) can be an hyperbola or an ellipsoid according to the refraction index ratio being higher or lower than 1:
Now, I don't understand this result quite well. My doubts are:
Imagine the rays start from right (being parallel). In case of an hyperbola, they are already in glass and then go into air. In case of an ellipsoid, they are in air and then go into glass. None of them are actually the common "thin lens" we usually study in basic optics (air - lens - air). How could we adapt these results to a thin lens? Should it have a hyperbolic/ellipsoid shape on both sides?
I cannot visualize why a spherical surface is not ideal to map a plane wavefront into a spherical wavefront. A spherical surface slow down the input plane wavefront points in a spherical wave. I find strange that this does not occur, whilst the ideal surface are hyperbola and ellipsoid!
Some books propose a different ideal surface for the perfect lens, precisely a cartesian oval. Other sources say the ideal surface should be parabolic, like for a mirror... which is the truth?