How can refraction be explained using Fermat's principle?

[junglebeast]

I understand that refraction is a bending of light that occurs when light passes between two mediums with different optical density. Because the light is represented as a wave-front, one side of the wave-front hits the medium first and slows down (or speeds up) first, which causes the wave to bend.

This is the explanation I've always heard, and I never questioned it until recently...when I realized it's just glossing over the truth of the matter.

The truth is the above explanation does not make sense unless the "rays" composing the wave have a cohesive force. This cohesive force would do the work of actually bending the wave front. Without a cohesive force, it would just change which photons are composing the wave front without any actual bending occurring.

It is intuitive to think of a cohesive force existing which is why the commonly used example of a car that drives from pavement into sand will be turned. The cohesive force there is obviously the electromagnetic force creating molecular bonds between the atoms of the car. But photons have no mass and no charge, so what would give them cohesion?

 

[mgb_phys]

Yes the analogy is wrong. The real answer is that each individual photon takes the path that minimises the total time for the trip (Fermat's law) - how they know what path to take gets a bit quantum.

A good picture is imagine you have to reach a swimmer in the water at the other end of the beach. You can run faster than you can swim so you have to pick the optimal place along the beach to enter the water so that the overall time is least.

 

[junglebeast]

Yes the analogy is wrong. The real answer is that each individual photon takes the path that minimises the total time for the trip (Fermat's law) - how they know what path to take gets a bit quantum.

A good picture is imagine you have to reach a swimmer in the water at the other end of the beach. You can run faster than you can swim so you have to pick the optimal place along the beach to enter the water so that the overall time is least.

Thanks for your reply, but I'm not following. Fermat's principle says that "the path taken between two points by a ray of light is the path that can be traversed in the least time"...

In the case of refraction, we want to use the principle to predict where the light will go (and hence where the light will end up)...but the principle above does not predict where it will go, it only predicts how it will get someplace given that you know where someplace is.

But if light could simply choose the destination point such that the shortest path leads to that point, then the traversal time is minimized by choosing a destination point equal to the starting point...

 

[Born2bwire]

Thanks for your reply, but I'm not following. Fermat's principle says that "the path taken between two points by a ray of light is the path that can be traversed in the least time"...

In the case of refraction, we want to use the principle to predict where the light will go (and hence where the light will end up)...but the principle above does not predict where it will go, it only predicts how it will get someplace given that you know where someplace is.

But if light could simply choose the destination point such that the shortest path leads to that point, then the traversal time is minimized by choosing a destination point equal to the starting point...

You can use Fermat's principle to find the rules for refraction. These are the kind of problems that are solved in Lagrangian mechanics, finding the path of motion that is an extremum of the action. You do not specify a destination, but only the rule that you want need to make the action stationary and in doing so you find the solution for the path.

You do not need to use Lagrangian mechanics for this problem though. It can be done using simple trigonometry and relating the velocities and distances and etc.