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This question is exactly what I would like to ask. I know this is a duplicate, but I cannot comment and I do not find answers to that question.

So, let me state again the problem. Let us consider Classical Electrodynamics, which is described by Maxwell's Equations. Then we have Geometrical Optics, which is based on Fermat's Principle. Knowing that light is an e.m. field, one would be happy to derive Fermat's Principle from Maxwell's Equations. This is done in many treatises on the subject, but in the limiting case of very short wavelength AND very slow variation of the index of refraction in the medium in which the phenomenon takes place. Namely, one derives an approximate wave equation by neglecting variations of the index of refraction, and from there obtains the Eikonal Equation by a perturbative expansion in terms of the wavelength. Lastly, the Eikonal Equation leads to Fermat's Principle.

Now, the conclusions of Geometrical Optics (limited to the path of light rays) are exact in the case of a sharp flat boundary between two homogeneous media. I mean, Maxwell's Equations (and the corresponding continuity conditions) imply that a plane wave in this scenario follow a path described by Snell's law, and Geometrical Optics describes exactly the same path. So it seems that Fermat's Principle is correct even when there is a discontinuity in the index of refraction. But in this case we cannot neglect the variations of the latter when deriving the Eikonal Equation. So our derivation of Fermat's Principle from Maxwell's Equations is not valid.

In conclusion: Maxwell's Equations are valid from the outset; Fermat's Principle is valid for very short wavelengths, but for any variation of the refractive index; but we do not have a link from Maxwell to Fermat which is valid whatever be the refractive index. Do we miss something?

Qmechanic
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Enrico
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The mentioned law, what governs the refraction of light rays, the Snell's law can be derived directly from the wave equation without the application of the approximations, which lead to the Eikonal Equation. This derivation uses plane waves, so it's in harmony with the light ray picture too, where the rays are orthogonal to the surfaces of the same phase in the plane waves. This derivation is in all electrodynamics books. So if the refraction coefficient is changing suddenly at the surfaces of change, you must apply the Snell's law and you can apply the Eikonal equation inside the material, where these changes are smooth or not present. In the language of PDE, for negligible wavelengths compared to the sizes of the problem, you should apply the Eikonal equation with boundary conditions, which can be determined by using Snell's law. Since both the Eikonal equation and Snell's law can be taken as a consequence of Fermat's principle, you can solve the light ray propagation problem by using only Fermat's principle. In practice it just means that the light rays are straight lines as long as the refraction coefficients is constant and at the boundaries these rays change direction according to Snell's law.

Alex Erdei
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