Fermat's Principle: Is Light Travel Time Min?

[Zubair Ahmad]

Is time taken by light always minimum between two points. If yes then what is wrong with Fermat's original statement?

 

[jbriggs444]

Is time taken by light always minimum between two points. If yes then what is wrong with Fermat's original statement?

I believe that a more correct statement is that light will take a path that results in a local extremum.

 

[Zubair Ahmad]

I believe that a more correct statement is that light will take a path that results in a local extremum.

Of what. Time or path

 

[Drakkith]

Of what. Time or path

I believe it is a minimum time path. If it was a minimum distance path, the light would simply travel straight through instead of refracting when entering a lens. If the light is traveling through a constant medium then the minimum time path and the minimum distance path are the same.

From wiki:

In optics, Fermat's principle or the principle of least time, named after French mathematician Pierre de Fermat, is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light.[1] However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path.[2] In other words, a ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse.

This is essentially saying that light takes the minimum time path. The last sentence means that if the light deviated an infinitesimal amount from its shortest-time path in any direction, these new paths all take the same amount of time to traverse. Note that these paths would take longer than the least-time path.

 

[Zubair Ahmad]

I believe it is a minimum time path. If it was a minimum distance path, the light would simply travel straight through instead of refracting when entering a lens. If the light is traveling through a constant medium then the minimum time path and the minimum distance path are the same.

From wiki:
This is essentially saying that light takes the minimum time path. The last sentence means that if the light deviated an infinitesimal amount from its shortest-time path in any direction, these new paths all take the same amount of time to traverse. Note that these paths would take longer than the least-time path.

It means there is a path for which time is always minimum and on either side infinitesimally apart there are paths for which travel time is same but different from the actual path (greater than actual path)

 

[Drakkith]

It means there is a path for which time is always minimum and on either side infinitesimally apart there are paths for which travel time is same but different from the actual path (greater than actual path)

I believe that is correct.

 

[jbriggs444]

Of what. Time or path

Time.

The key is that a local extremum in path length will guarantee that the first derivative of path length with respect to [nearly parallel] launch angle will be zero. If one measures path length in terms of time then this guarantees that, on arrival, the phase angles for light arriving over a narrow range of paths will be identical. You get constructive self-interference.

By contrast, if the first derivative of path length with respect to launch angle is non-zero the path lengths vary with launch angle. You get destructive self-interference.

This is the principle of diffraction in action. It's just that we're doing it without double slits.

 

[Zubair Ahmad]

Give an example where light takes longer time

 

[jbriggs444]

Give an example where light takes longer time

Say that I hold a candle in my hand while you stand 10 meters to the north. Consider a range of paths that go at an approximate 30 degree angle east of north to to a line midway between us. The paths then turn left and proceed at an approximate 30 degree angle west of north to your location.

The length of these paths will depend on launch angle. A 31 degree path, for instance, will be longer than a 29 degree path.

 

[Zubair Ahmad]

Tell in a nutshell the Fermat's Principle which is applicable everywhere.