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I came across this problem in the context of spectroscopy today. Because of it's simplicity, I'm sure it's a question that's been posed by some mathematician ages ago, I just can't figure out how to formulate it mathematically:

Suppose you have a mirror that forms a perfect ring and has an aperture in it with width $\delta w$. A ray of light enters the aperture at angle $\theta$ relative to the tangent at the aperture. There are several questions that I have regarding this system, some of which would be of practical use and some of which are mere curiosity:

  1. Is this a chaotic system? I know a little about chaotic systems from my undergrad math major, and this looks like one to me, but I don't know where to start working on it if it is.
  2. How can this problem be formulated mathematically? I would assume that there is some sort of iterated function approach, but I can't figure out what the iterated function would be (probably because as a chemist I'm more comfortable with cartesian coordinates than polar coordinates).
  3. What is the optimal angle for achieving the most passes through the area within some $\epsilon$ of the center of the ring before exiting through the aperture, assuming (a) $\epsilon$ is small compared to $r$, the radius of the ring, or (b) $\epsilon$ is not small compared to the radius of the ring? (This is one of the questions of practical importance)
  4. Bonus: Suppose there is a second aperture placed at an angle $\phi$ away from the first (with $\phi$ relative to the center of the circle and to the first aperture). Can this angle be optimized in conjunction with the variables in $\phi$?

To summarize, the variables are $r$, $\theta$, $\delta w$, $\epsilon$, and (if we're lucky) $\phi$. Assume that $\epsilon$ and $\delta w$ are given (obviously the optimal size for $\delta w$ is infinitesimal, but this isn't practical of course). Also (for the physicists out there), assume $\delta w$ is wide enough that the beam can be treated classically, i.e. no diffraction. Also, there needs to be a reasonable way to calculate the exit angle based on the entry angle (if this is a truly chaotic system, I realize this will be practically impossible).

Thanks in advance for any help, and let me know if I've left out any important details. I'm comfortable with any math that a typical undergraduate math major should know by the time they graduate; anything more and I'll need a reference to read up on.

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Will someone with tag adding rights add the tags chaotic-systems and iterated-functions, or something to that effect? –  David Hollman Dec 14 '10 at 15:15
    
Also, is this the type of question that is better asked on MathOverflow? Maybe I'll give it a few days, and if I don't get a good answer, I'll move it. –  David Hollman Dec 14 '10 at 15:24
    
A sketch would be helpful –  Ross Millikan Dec 14 '10 at 19:31
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2 Answers

up vote 3 down vote accepted

The dynamics are those of a repeated constant rotation, $\theta \to \theta + c$ where $c$ is the angle on the circle (completed to fill in the aperture) cut out by the line on which the light entered. Exit will occur after $n$ reflections when $\theta_1 + nc$ is in the aperture interval, where $\theta_1$ is the point of first impact between light and mirror. This type of dynamics is integrable, not chaotic.

If the ring-shaped mirror is not a perfect circle but a more general convex curve this goes under the name "convex billiards", or elliptical billiards in the case of an ellipse-shaped mirror. The dynamics can be more interesting (e.g., for polygonal mirrors) in those cases and the book by Tabachnikov cited in the other answer discusses all of these more general problems.

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Thanks. I see now why the problem is not a chaotic one and that it has a relatively simple solution in the case of a circular ring. –  David Hollman Dec 14 '10 at 21:29
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I'm not an expert, but circular billiards is a special case of elliptic billiards, and what you want to know can probably be found in Billiards and Geometry (pdf file) by Serge Tabachnikov.

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