Assuming ray optics model is valid, what lens profile focuses light to a point? It must not be spherical, otherwise there would not be the term 'spherical abberation'.
The problem with your reasoning is that that ray optics model has a limited application. There is a "best form" to a biconvex lens, but the spherical aberration is not zero. Use of aspherics can reduce aberrations, but a singlet will only have zero spherical aberration for an extremely limited set of illumination conditions. For example, parabolic reflectors have zero aberration, but only for on-axis points.
In fact, simply reversing the orientation of a planoconvex lens will result in radically different amounts of spherical aberration.
I was wondering if it was a conic section. I tried to work this out when I was in high school but gave up after a while.lzkelley said:segment of an ellipse
Yes that's a good point. In practice the ray model is not good enough, and you can't realize the zero aberration ideal. I guess I'm asking more of a mathematical question then, but it is optics-related so I posted it here. I was thinking that the specific term 'spherical aberration' was included in the theory of ray optics, so that's why I reasoned that the ideal lens shape in the ray model must not be spherical. I am imagining a plano-convex lens with light coming from infinity, being focused to a point on the axis of the lens.
The way aberrations are discussed in ray optics is very artifical, IMO. Ray tracing involves linear and higher-order approximations to the sine function- linear optics has no aberrations, but there are 5 aberrations in 3rd order optics (7 actually, but 2 of them- piston and tilt- do not affect the PSF) and more for 5th order optics with strange names you have not heard of, etc. etc.
So, you can see how aberrations form in optics- as the linear approximation to a sine function breaks down (say the numerical aperture of a lens increases), higher order terms are required for accuracy, and aberrations come along for the ride as a result.
That sounds like cool stuff you do. Thank you for the telescope links. I've only read a little so far, but I can tell I will find them interesting.