# Nonstandard Concave Mirror Optics, beyond the parabola.

1. May 20, 2012

### 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!

2. May 21, 2012

### 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.

3. May 21, 2012

Thanks.