Why does the spotlight reflected off of a rectangular mirror tend to become circular?

Background and setup

When I was 12 I used to like a girl, we were almost neighbors and it was essential that our parents don't find out. So whenever one of us wanted to call the other they'd signal the other person with a mirror so they get ready near the phone to answer before anyone else in the house.

Here's the setup:

• I lived in A, she lived in B.
• I used a rectangular-shaped mirror.

The observation

• Case A:

When I reflected the light to a close surface, like the small wall in front of me on building A, I saw a spot light that looked like this:

Which was exactly what I'd expect.

• Case B:

When I reflected the light to a further target, building B, the spotlight was also rectangular but with much smoother corners:

• Case C:

This is when I thought it's interesting. When I reflected the light to an even further target, building C, the spotlight looked almost like a circle:

Question

Why did that happen? shouldn't it stay a rectangle? and even if it had to change, why a circle?

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In similar situations, one could observe the wave properties of light except that this is not the case here. The mirror could be imperfect except that it's usually very close to flat so this is not the reason, either. Your observation has another simple reason: the Sun isn't a point. It's round.

That's why the light rays coming from the Sun are not exactly parallel to each other (I mean to other rays from the Sun). In fact, the photons from the Sun come from different angles and the maximum angle between two photons arriving from the Sun is nothing else the apparent size of the Sun

http://www.wolframalpha.com/input/?i=apparent+size+of+sun

which is about 0.54 degrees or 0.0094 radians. It's the angular diameter. Note that it may be calculated as the absolute diameter of the Sun divided by the Sun-Earth distance.

http://www.wolframalpha.com/input/?i=diameter+of+the+Sun+%2F+distance+between+Earth+and+Sun

Now, imagine that you have an infinitely small mirror. The photons from the Sun get reflected in slightly different directions because they are coming from slightly different directions. In fact, the directions from which the photons arrive form a small disk on $S^2$, so the same must hold for the reflected photons: the beam isn't a sharp line, it's a cone diverging from the mirror.

If you multiply the angle 0.0094 by e.g. 20 meters – distance between your sweetheart and the mirror, you get 19 centimeters. So already a tiny mirror will produce an illuminated disk of diameter 19 centimeters (almost 1% of the distance between the mirror and the "screen"). The disk-shaped illuminated spot you see in case C – more precisely, in cases when the mirror-screen distance is much more than 100 times the mirror radius – is nothing else than the shape of the Sun itself.

If your mirror is a 30 cm times 30 cm square, the illuminated area will be a "convolution" (some kind of combination or compromise, if you don't know what a convolution is) of the same square and a disk of diameter 1% of the mirror-screen distance. So if the distance is 20 meters, the radius of curvature of the smooth corners will be about 9.5 centimeters. Each of the four corners will be replaced by a disk of radius 9.5 centimeters and the total illuminated area will be the convex envelope of these four circles, so it will be a square of side 30+9.5+9.5=49 centimeters with round corners whose curvature radius is 9.5 centimeters. The places closer to the center of this "fuzzy reflection of your mirror" will be illuminated more strongly; the intensity of light will decrease towards the boundaries of the 49-centimeter rounded square.

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Such an excellent answer. Very well-put and convincing. – Adnan Nov 21 '12 at 21:15
It's really a great thing. I used to wonder the same about light coming through square holes in tin roofs until I looked more closely and saw the clouds going by. Then it all began to make sense. – krs013 Jun 24 '14 at 22:07