Nonstandard Concave Mirror Optics, beyond the parabola.

[xlearsi31]

Why is the parabola the most bent, concave, effective mirror?
What are some properties of a third order mirror (absolute value of x^3)? There is no uniform focal point. Shouldn't shining a light beam along a normal to the x-axis reflect off of the function multiple times?
Then take this to the nth order. Or make it exponential.
Does the result of the light beam end up converging to a "focal point" after all of its consecutive reflections?
Just some things I am curious about.
Thanks!

 

[haruspex]

If I understand the question, you want to know if some other shape could produce light focussed at a point, albeit with some of the rays taking multiple reflections.
No.
Suppose there were such a point, F. Consider a light ray arriving parallel to the axis and eventually going through F. Now consider a neighbouring axis-parallel ray, just displaced a fraction from the first one. If the mirror is not locally a parabola, this will, after the same number of reflections, just miss F. OK, maybe you get lucky and it passes through F after some more reflections. So now look at a ray halfway between, etc. It's pretty clear that you cannot continuously displace rays and have them all coming through the same point, no two using the same number of reflections.

 

[xlearsi31]

Thanks.