# A Fermat's principle can be derived from Maxwell equations?

1. Dec 18, 2015

### PFfan01

As we know, the Fermat's principle states: Light takes the path of least time. I wonder whether Fermat's principle can be derived from Maxwell equations. If it can, then Fermat's principle is included in Maxwell equations, or Fermat's principle is not an independent postulate.

2. Dec 18, 2015

### Andy Resnick

Only indirectly: geometrical optics can be derived from Maxwell's equations via the eikonal equation, and Fermat's principle may then be derived by application of variational principles to the path (the eikonal function), so-called 'Hamiltonian optics'.

3. Dec 18, 2015

### PFfan01

But I don't think so. Fermat’s principle is an independent postulate and it cannot be derived from Maxwell EM theory, although there is a description of Fermat’s principle in geometric optics based on EM theory: Light takes the path of least time.

The original form of Fermat’s principle is “Nature always acts by the shortest course”. When applied to geometric optics, this principle requires that light take the path of least time. In optics of light rays (geometrical optics) set up from Maxwell equations, this principle is specifically expressed as: An actual light ray makes $\int{n}ds$ the minimum. Light rays, refractive index $n$, light speed … are all defined in the Maxwell-EM-theory frame, but the conclusion “an actual light ray makes the $\int{n}ds$ minimum” comes from the Fermat’s principle. In other words, Fermat’s principle itself is not included in Maxwell equations. A correct statement probably would be: Maxwell’s equations can be used to formulate the Fermat’s principle in geometric optics.

4. Dec 18, 2015

Staff Emeritus
Then why did you ask the question?

5. Dec 18, 2015

### Staff: Mentor

Do you have a reference for that claim? It doesn't seem likely to me.

6. Dec 18, 2015

### PFfan01

In my manuscript, I claim that Fermat’s principle is a separate postulate independently of the Maxwell EM theory. The reviewer does not agree, and rejected my manuscript, criticizing: “Fermat’s principle cannot be put on the same level as Maxwell’s equations. Maxwell’s equations can be used to derive Fermat’s principle.”

Last edited: Dec 18, 2015
7. Dec 18, 2015

### PFfan01

"Fermat’s principle is an additional physical condition imposed on the direction of EM energy transport." Optik 126 (2015) 2703–2705 http://dx.doi.org/10.1016/j.ijleo.2015.06.053 [Broken]

Last edited by a moderator: May 7, 2017
8. Dec 18, 2015

### Staff: Mentor

Then you should support your claim with that reference.

One job of a reviewer is to represent the readership of a journal. The reviewer's criticism is likely one that others in the community would would share.

Do you believe that this reference would be convincing to the community? I don't have access to it, so I cannot judge it directly. But I have not been too impressed by other solo-authored papers from the same author.

In any case, until your paper is actually published in the professional literature, it cannot be discussed here. You will need to work the rest of this out with the reviewers and editors.