# Light follows the shortest path, but to where?

I'm a high school student and I'm trying to understand why waves bend when they refract. I read a few answers on the site, and the explanation that they follow the shortest path makes sense enough to me, however that leaves an unanswered question, what point is the photon trying to reach?

• Photons don't try. A photon in free space has momentum. That's a vector quantity---magnitude and direction. You know from Conservation of Momentum that as long as the photon does not interact with anything, then its momentum (magnitude and direction) will not change. Apr 30, 2015 at 16:43
• In light of the discussion, maybe someone could try and tackle this from a (basic) path integral point of view... Apr 30, 2015 at 17:31

Actually, the light beam does not follow the shortest path, but rather the faster path. Else the light would not bend but go straight there. This is Fermat’s principle.

what point is the photon trying to reach?

Good question. This point you are talking about, is in fact your eye.

A straw in a glass of water visually bends at the interface.

Look at the end of the straw, and you see it at another location than it actually is at. The photon was emitted from the straw but bended along the way, and now it hits your eye and you can see the end of the straw, but not at the true location because of the bending.

Schematically, this is a good illustration:

The faster way for the light beam (or photon, if you will) to reach your eye is not the straight way (meaning, not the shortest way)! Because it moves slower in water than in air. The optimal route to reach your eye as fast as possible is to move a smaller portion in the water and a larger portion in the air.

Ants moving as light

The way light behaves can be described as the most logical way to reach a destination faster. From this article:

Humans intuitively employ this rule, e.g., when a lifeguard has to infer the fastest way to traverse both beach and water to reach a swimmer in need.

I have heard a story, which I believe this article is investigating, about ants. When put in oil, which slows them down, they tend to follow not the straight, but the faster route, which always is a portion in the slower medium and a larger portion in the faster medium.

Ants explaining light? Fascinating.

Addition - explaining refraction from the wave point-of-view

It seems that the actual question is why the refraction / bending of light actually happens, rather than which point the photon is trying to reach. To address that I will rather go with a wave explanation than a beam/photon explanation. At least, for me this is much more intuitive.

Think of light waves as water waves. The waves travel towards the beach. When they hit shallow water, they are slowed down:

If the waves come in at an angle, then the part of the wave that hits the shallow water first, is also slowed down first. The rest of the wave will gradually also slow down until the whole wave is slowed down. Since one part of the wave is slower, this part will look as if it was bended.

Shallow water corresponds to a medium inwhich light is slow. I hope this has answered your further questions.

• What about a mirror? Surely the fastest path from my eyebrows would be to bend down over my face info my eyes. Apr 30, 2015 at 17:03
• Another question, suppose we have a laser pointing direcly into my left eye, then we put something in front of the laser such that it bends and now hits my right eye. How did it pick which eye? The one it could reach the quickest? Apr 30, 2015 at 17:07
• @KristofferRyhl (2 comments up) good followup question, which I think illustrates the risk of saying that the photon is trying to reach your eye. It's not really trying to reach anywhere, but once you know where it wound up, Fermat's principle tells you how it got there. That being said, someone could build on Steeven's answer to go into a lot of detail about why that works. (Not me, at least not now; I don't have time) Apr 30, 2015 at 17:07
• @DavidZ Well with a mirror, the photon did reach my eye, but didn't follow the shortest path? Apr 30, 2015 at 17:10
• the straight line to your eye, however, is blocked by your brow, so you can't see that bit! Apr 30, 2015 at 17:11