If I am not mistaken, a parabola is the shape that a mirror has to be to focus ideal, parallel light rays to a single point. Real light sources are usually not actually parallel though, but are more similar to a point light source. Assuming an ideal point light source, and treating light as ideal rays, is there a name for the shape a mirror need to be to focus light from this source, and if so what is that name? Note that the trivial answer of a spherical mirror centered on the source is not what I am looking for.
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$\begingroup$ Hint: What shape has two foci? $\endgroup$– Qmechanic ♦Commented Feb 19, 2016 at 23:48
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2 Answers
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As you said, trivially a spherical mirror will focus the light from a point in its focal point back into its focal point. If you want two focal points, then you need an elliptical mirror. I don't believe that a simple surface can do more than that. The image of any light source that is not at one of the focal points will have optical errors.
A few years ago there were claims that if you want to do better than that, you will need a material with negative refractive index, see e.g. "Negative Refraction Makes a Perfect Lens" by J. B. Pendry. Other researchers have, in the meantime, revised that claim by showing the the analysis was incomplete and not self-consistent. While I didn't look into the details of this, I would tend to opine (and that's all it is) that having negative refractive index materials will make imaging systems much better and probably close to perfect.
So this still leaves one class of optical imaging systems as a solution to the problem, which are systems with varying index of refraction. That idea seems to go all the way back to Maxwell and is explained e.g. in "Perfect imaging without negative refraction" by Ulf Leonhardt. This, of course, goes beyond your question, since you would need a volume element, rather than just a surface mirror.
answered Feb 19, 2016 at 23:57
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$\begingroup$ Well, that seems really obvious now. It is interesting that the 2d analog of each of these surfaces (parabaloid, sphere, elipsoid) is a conic section. Though, it does seem somewhat intuitive now that I think of it; when the focus is at the source, you have a circle, when they are separate, you get an ellipse, and as the distance between the focus and the source approaches infinity, the ellipse approaches a parabola. $\endgroup$– VaelusCommented Feb 20, 2016 at 0:10
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$\begingroup$ @Vaelus: That's how I usually think if it, too... by moving the second focal point to infinity we degenerate the ellipsoid to a paraboloid. As every owner of a Newtonian telescope can tell you, the parabolic mirror has strong coma, only paraxial beams are actually focused in the focal point and better designs use two (non-flat) mirrors and a correction plate to reduce those errors very significantly. $\endgroup$ Commented Feb 20, 2016 at 0:14
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Agree with the above. What determines the shape of the surface that fits your criteria is determined strictly from geometry and knowledge of the fact that the angle of incidence equals the angle of reflection.
The fact that the image won't be a perfect point, assuming you're using an elliptical shaped reflector, is more related to the issue that the source is an extended body.
