 # Laser and Mirrors talzamir Posers and Puzzles 02 Jun '12 08:11
1. 02 Jun '12 08:11
Two flat mirrors have been joined together so that there is a two-degree angle between them. A laser beam is fired from some point on the line that divides the angle equally in two angles, one degree each. The beam reflects back and forth between the two mirrors and finally passes through the point it was fired from.

What is the greatest number of times that the laser beam can reflect before returning to its point of origin? In what direction should the beam be fired to accomplish this?
2. 05 Jun '12 21:28
Originally posted by talzamir
Two flat mirrors have been joined together so that there is a two-degree angle between them. A laser beam is fired from some point on the line that divides the angle equally in two angles, one degree each. The beam reflects back and forth between the two mirrors and finally passes through the point it was fired from.

What is the greatest number of ti ...[text shortened]... returning to its point of origin? In what direction should the beam be fired to accomplish this?
That is positively diabolical🙂 Here is one thought: No matter what the angle you launch the laser beam, there will be a smaller wavelength that can shoot at a smaller yet angle until you get to a point where the light is more X ray than visible and penetrates the reflector, so it would seem to depend on the quality of the reflecting material vs the wavelength of the photons. You can see what I mean? Suppose you use light at one micron, IR. There would be a point on a perfect reflector at which at the apex of the cone the IR would not be able to reflect so that would define the maximum number of reflections for that wavelength.

But at 0.1 micron, 100 nanometers in the ultra violet band, that same light can fit into a much tighter space and still reflect and then 10 nanometers can fit into a tighter space than that but at some wavelength most mirrors turn transparent so there will be a limit to the smallest wavelength you can consider for this problem.

There is also the problem that in order for the light beam to strike exactly where it left, there would have to be an almost quantum jump of angle or it will pass right by the origin of light so only certain angles will actually hit the origin dead on.
3. 06 Jun '12 14:36
Gets extremely diabolical when taken in that direction. ^_^ I was thinking of it in the simpler sense, that the angles between the incoming and the reflected beam on either side of the normal of the surface are the same. Basic high school physics, not the advanced stuff with wavelengths, let alone the quantum models where light goes to places where classically it shouldn't be able to reach, and where partially darkened mirrors actually work better than the old-fashioned fully reflecting ones.
4. 07 Jun '12 02:181 edit
Originally posted by talzamir
Two flat mirrors have been joined together so that there is a two-degree angle between them. A laser beam is fired from some point on the line that divides the angle equally in two angles, one degree each. The beam reflects back and forth between the two mirrors and finally passes through the point it was fired from.

What is the greatest number of ti ...[text shortened]... returning to its point of origin? In what direction should the beam be fired to accomplish this?
The maximum number of reflections will occurr when the beam is fired between 0-1 degree below horizontal ( perpedicular to the line that divides the angle) where at 0 it will never return, and at 1 it will immidiately return to the starting point. Intuition is telling me that the maximum number of reflections would be infinite. Am I warm or completely bonkers?
5. 07 Jun '12 09:38
You are on the right track. The beam would be at an almost perpendicular angle at one of the mirrors, from which it bumps into the other mirror, likewise at an almost perpendicular angle, etc etc. As the two mirror are almost parallel there will be heaps of reflections before the beam of light returns to its starting point; but since they are not exactly parallel, it is possible that the light eventually does return, after a finite number of reflections.

There is, I think, a fairly elegant way to calculate both the number of reflections, as well as the the length of the path the beam takes before returns to its point of origin, the latter of course requiring that the distance between the origin of the beam and the the place where the two mirrors are joined is known.
6. 07 Jun '12 22:33
I thought of this by reflecting the whole situation over the mirror OA. The mirror OB is shown in the reflected area as OB', the field of the angle AOB as B'OA, and the origin of the beam of light C, as C'. The light from C hits the mirror OA and bounces off from it, but the mirror image of the reflected beam, in B'OA, continues in a incoming (incident?) beam. Then reflect again over OB', etc etc etc. In each field is a mirror image of the point of origin, and farthest one to reach is 89 reflections away. The length of the beam is 2 / cos (1 degree) times the distance from the origin of the beam to the place where the mirrors join, so very close to the distance times 2.
7. 08 Jun '12 17:401 edit
Originally posted by talzamir
I thought of this by reflecting the whole situation over the mirror OA. The mirror OB is shown in the reflected area as OB', the field of the angle AOB as B'OA, and the origin of the beam of light C, as C'. The light from C hits the mirror OA and bounces off from it, but the mirror image of the reflected beam, in B'OA, continues in a incoming (incident?) be gin of the beam to the place where the mirrors join, so very close to the distance times 2.
What I don't see is the beam returning at all. You can aim the beam so it is perpendicular to the other mirror and it will of course come right back at you exactly then be gone forever.

But aiming it slightly towards the apex, where the mirrors meet, it looks to me like the place where the mirrors touch would not allow a reflection to start up going back.

Like I said, the size of the opening would get to a point where the gap would be a lot smaller than whatever wavelength you use, say one micron, and the gap gets down to say 0.1 micron, there can be no reflection since that gap is smaller than the wavelength being sent, in this case, 1 micron.

It seems to me there would be beam travel only in one direction, towards the apex where the mirrors meet and no reflection back.

Picture the mirrors as being as perfect as you can make it, say a coating of 100% pure aluminum or gold and the thickness of the gold layer is even to within one atomic size, then the gap between the mirrors would eventually reach a point where there was about one gold atom width apart, so at some point along that path, say when the gap was 2 microns apart, reflections could still take place but the gap when it is less than 0.5 microns would mean there was no room for a reflection.