Light is economic. It takes the minimal path. It follows the Principle of Least Time!
When light is reflected, the angles of incidence of original ray and reflected ray are the same. This might seem natural. But you can derive it from the Principle of Least Time – also called Fermat’s Principle.
If we want a ray of light to go from one point to another AND hit a mirror in between, what path would it take? It could go “straight down” to the surface of the mirror and take a long path to the second point. It could also try to hit the mirror as close as possible to the end point. Or the ray might feel like hitting somewhere in the middle.
You can calculate the lengths of these paths, or you can stare at the geometry – like Richard Feynman. Think about the reflected ray “reflected again” – into the depth of the mirror. The minimal path is obtained when there is no kink in this hypothetical extension. This happens exactly when light hits the mirror in the point of equal angles of incidence.
The speed of light is constant along the whole path – in air. But the ray of might enter a different medium, like glass or water. Then its (phase) velocity would be lower. Light still travels the minimal path, and this accounts for refraction: The ray of light seems kinked when entering the water.
Feynman’s metaphor was: How would you optimize your path on a beach, trying to save a swimmer screaming for help? (And as expected from a 1960s textbook, the swimmer was a “beautiful girl”). The path is unlikely to be a straight line: In the water you cannot swim as fast as you can run on the beach. So, you stay on the beach a little longer, make a turn when entering the water, and swim a shorter distance.
The path of least time is a minimum: If your path would deviate a little bit from this optimum, it would only be changed in the second order. This is possible when the effects of beach and water roughly compensate each other: You reduce the time on the beach, and you extend your time in the water a bit – taking into account that your velocity in the water is slower. You end up with the law of refraction (Snell’s Law) – relating the angles of incidence to the ratio of the refractive indices of the two materials.
The result is the same you would get by drawing tiny circular wavefronts: Each point in the straight wave front of a ray (of finite size) becomes the original of a spherical wave. The envelope of all these tiny wave becomes the new wavefront. In air and in water, these circles would have different sizes, finally resulting in the angle between incoming ray and refracted ray.
I feel it is fascinating that you can deduce so much from the principle of least time.
But not being a computer yourself, it is useless trying to analyze multiple reflections and refraction in real life.
You can just look at this miracle of nature, and shapes and lines descending into delightful chaos!
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I have created Celebration of Dispersion last year, but only digitized it properly now.
This crystal has turned out to be my favorite model: It was featured also in First Encounter and Edge of Darkness.
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