Is Fermat’s principle appicable to a plane wave?

[PFfan01]

Fermat’s principle states that light follows the path of least time. In textbooks, a specific formulation of Fermat’s principle is about the optical path between two points, A and B: How can a ray of light, emitted from point A, reach point B? Suppose that there is a plane wave in free space, with the line AB not parallel to the wave vector. In such a case, the actual light ray never goes from A to B because the actual light ray must be perpendicular to the equiphase planes. From this, a question comes to me: Is the Fermat’s principle appicable to a plane wave?

 

[Khashishi]

Not really. It's hard to interpret what "path" means for a wave. Huygens's principle is probably better here.

 

[olivermsun]

How do the ray paths relate to the wave crests in a pure plane wave?

 

[Khashishi]

They are perpendicular

 

[olivermsun]

Ok. So maybe this trivializes the example, but can you think of any ways that a ray path other than a straight one could be identified with a (strictly) plane wave?

 

[Khashishi]

A plane wave refers to something of the form $A e^{i(\mathbf{k} \cdot \mathbf{x}-\omega t)}$ and propagates in a straight line. Anything else, and it wouldn't be a plane wave.

 

[PFfan01]

The (plane wave) phase function defines equiphase planes of motion (wavefronts), with the wave vector as the normal vector. From one equiphase plane to another equiphase plane, the path parallel to the normal vector is the shortest and has the minimum optical length. According to Fermat’s principle, light follows the path of least time, and this path is an actual light ray. ---- Thus the Fermat’s principle for a plane wave can be formulated as: from one equiphase plane to the next, the optical length of an actual ray is the shortest.

It seems to me that the traditional formulation of Fermat’s principle about the optical path between two points has a loophole.

 

[PFfan01]

Ok. So maybe this trivializes the example, but can you think of any ways that a ray path other than a straight one could be identified with a (strictly) plane wave?

I don't think the plane wave example is trivial. A minor fixing can provide a more general challenge as follows.

Suppose that there is a point light source at S in free space, with SA not parallel to AB. In such a case, the actual light ray never goes from A to B because the actual light ray must be perpendicular to the equiphase surface. Thus the challenge persists: Is the traditional formulation of Fermat’s principle valid for general cases?

 

[olivermsun]

I'm still not clear on what you're asking. If the source is at S where S doesn't lie on the ray path between A and B, then what does it have to do with the path from A to B?

 

[PFfan01]

I'm still not clear on what you're asking. If the source is at S where S doesn't lie on the ray path between A and B, then what does it have to do with the path from A to B?

S is not between A and B. The source is at S, and all equiphase surfaces are spherical with the common center S. SA is not parallel to SB or AB.

 

[olivermsun]

So then what is the (attempted) application of Fermat's theorem?
Are you saying that "light travels from A to B along the path of least time" is invalid because there might be light that does not travel from A to B?

 

[PFfan01]

Not really. It's hard to interpret what "path" means for a wave. Huygens's principle is probably better here.

It seems to me that the book by Born and Wolf (M. Born, E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1986, p. 114) interprets what "path" means for a wave.

 

[PFfan01]

...
Are you saying that "light travels from A to B along the path of least time" is invalid because there might be light that does not travel from A to B?

What I mean is: For arbitrarily given two terminal points A and B, the real light ray does not necessarily go from A to B in general. In other words, for given two points A and B, generally the path of least time is not a real light ray.