Fermat's principle ↩

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shorthands: {}
aliases: [Principle of least time]
status:
created: 2021-10-24 21:50:33
modified: 2022-01-20 18:56:07

Original form: the path taken by a ray between two given points is the path that can be traveled in the least time.
Better, "weaker" form: the path taken by a ray between two given points is the path that can be traveled in time that is stationary with respect to variations in the path. (The deviation in the path is at most a second-order change in traversal time.)

The principle works just as well when instead of time, we search for paths where the optical path length is minimized.

This principle links ray optics with wave optics.

Examples

Deriving Snell's law using Fremat's principle

We have two media with refractive indices of and . They are homogeneous, so light propagates along a straight line in them, so we only look at such paths. Point is where the lightray goes through the boundary surface between the media.

We give the path some variation by moving the point along the axis by some value. Then we look at the OPL of the varied path and find where it is extremal:

If :

Then using the trigonometric relations:

Applying Fermat's principle: parabolic reflector

Parabolic reflectors have a focal point that depends on the parameters of the parabola. Spherical waves coming from this focal point are converted into plane waves by the parabola reflector.

The line segment is parallel with the axis (optical axis) and all paths have the same OPL.

Now we use a strange, but useful simplification: let . Also for simplicity: . Then the OPL is much simpler:

If a ray goes along the axis, we know it's OPL:

But all the other rays have the same OPL, so:

Then we can see that the reflective surface is indeed a parabola: