Fermat's principle light and time

[pi-r8]

I don't understand this idea. My terxtbook says that Fermat's principle is that light travels by the path that takes the least amount of time. Does that mean that light will go in crazy, curved paths if those are faster? How does it "know" which path will be the fastest?

For example, let's say I'm shining a flashlight towards a block of some medium with a really high index of refraction, so light traveling in this medium goes really slowly. If light is going in a straight line from the flashlight to a point directly behind the medium, it's going to take a long time to get there. On the other hand, if it follows a curved path up and over the medium, then down behind it, the total time would be much less. Is that what happens?

 

[NewScientist]

Your understanding of his principle is flawed somewhat, read on Wikipedia or anywhere else to see why!

 

[Danger]

This isn't one of my areas, so I'll just give a brief response.
Light does not 'travel' through a refractive medium. When a photon enters the medium, it's absorbed by an atom. A new photon is then emitted, which usually matches the original. That new photon then gets absorbed by another atom, and so on and so on...

 

[robphy]

Here is a good starting point:
http://www.eftaylor.com/leastaction.html
http://www.eftaylor.com/quantum.html

 

[pi-r8]

The wikipedia entry on Fermat's principle isn't particularly helpful... just the same old stuff about how you can use it to derive snell's law (which I had to do as a homework problem this year).

 

[pi-r8]

robphy- I know about the principle of least action. What I don't understand is the actual, physical meaning of Fermat's principle. Even if, as Danger said, photons are constantly absorbed and emitted, rather than just traveling straight through, I don't understand why light would take the shortest time path, rather than a straight path.

 

[NewScientist]

The modern, full version of Fermat's Principle states that the optical path length must be extremal, which means that it can be either minimal, maximal or a point of inflection (a saddle point). Minima occur most often, for instance the angle of refraction a wave takes when passing into a different medium or the path light has when reflected off of a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected off of an elliptical mirrored surface.

Hope that helps

 

[pi-r8]

um... not really. How does that explain my example of the flashlight and the block? In real life the light goes straight through the block... but the minimum path would be to go around the block, and the maximum path would be infinitely long. I don't know what the point of inflection path would be though... is that it?

 

[robphy]

robphy- I know about the principle of least action. What I don't understand is the actual, physical meaning of Fermat's principle. Even if, as Danger said, photons are constantly absorbed and emitted, rather than just traveling straight through, I don't understand why light would take the shortest time path, rather than a straight path.

The thing you have to remember about the Fermat Principle or the Principle of Stationary Action is that the extrema/stationary-points are local... meaning you have to nudge the path a little bit and compare it to that original path... then nudge it again etc... I believe that if you try to work out your curve-around-the-medium path, it won't be a locally-stationary path.

Although the E.F. Taylor articles are good, the best thing to read is probably the lecture on Least Action in the Feynman Lectures... or Feynman's [little] QED book.

Why does light take the stationary time path?
Of course, one answer is philosophical... That's just what nature does [in this formulation of the physics].
If you want something more constructive, realize that, in the neighborhood of the classical path, there is not as much cancellation of phases [i.e. destructive interference] as compared with that of the other, non-classical paths.