Question regarding Ray Optics and Probability

[Ayan Ghazi]

Let there be a sphere whose inner surface consists of a perfectly reflecting surface.
It has a hole on it which allows a ray of light to enter.
Give the angle made with the normal of the hole when the ray of light enters such that the probability
that the ray comes out is the least?Assuming the ray can enter through any angle with the hole.
(eg.if the angle with the normal is 0 then probability that the ray comes out is 1 as the ray reflects from the
surface and retraces itself resulting in the ray coming out in one reflection only)

 

[Buzz Bloom]

Give the angle made with the normal of the hole when the ray of light enters such that the probability
that the ray comes out is the least?

Hi Ayan:

I think the problem is simpler with respect to one aspect, and more complicated with respect to other aspects, than it superficially seems.

The simplification is that the problem requires a solution in only two dimensions rather than three.

I am guessing that you assume the hole is circular. One complication is that the ray angle will be affected on entry by diffraction bending. The amount of bending depends on the size of the hole and the wave length distribution of of the photons in the ray. You may want to make an assumption (probably unrealistic) that there is no entry diffraction.

A second complication is that the size of the hole determines how close the center of the ray needs to be to the center of the hole in order for the ray to to escape.

Hope this is helpful.

Regards,
Buzz