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I am currently having some struggle to understand the connection between the following three concepts in optics:

  1. Law of Reversibility (Geometrical Optics): The direction of light rays does not matter because the light travels from point A to point B and from B to A via the same way.

  2. Fermat's principle: Light travels the way of shortest and/or longest optical path length.

  3. Faraday effect: A magnetic field can be used to change the polarisation of light which makes it possible to build optical isolators.

Questions:

  1. The Law of Reversibility is apparently violated by the Faraday effect. What are the exact preconditions for the Law of Reversibility to be valid? Is there any derivation from wave optics? Unfortunately I couldn't find something regarding this via Google.

  2. How is the connection between Fermat's principle and the Faraday effect? How can the optical path length be different for the same path but different directions?

Maybe question 1 and question 2 are the same, I am not quite sure about that... I would be grateful if somebody could help me.

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    $\begingroup$ I think the answer has something to do with the fact that $\vec{B} \to - \vec{B}$ under time reversal in Maxwell's equations, but I'm not familiar enough with the Faraday effect to write that as an answer. $\endgroup$
    – Michael Seifert
    Commented Mar 11, 2021 at 14:50
  • $\begingroup$ From what I understand, any optical isolator build with the Faraday effect must absorb the rays and heat up, this generates sufficient entropy to enable the directionality. $\endgroup$
    – boyfarrell
    Commented Mar 11, 2021 at 18:34
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    $\begingroup$ By Faraday effect are you referring to Faraday rotation? If so, it doesn't change the polarization of electromagnetic waves, it just causes left-handed modes to propagate more slowly than right-handed ones. $\endgroup$
    – honeste_vivere
    Commented Mar 12, 2021 at 15:40
  • $\begingroup$ @honeste_vivere This circular birefringence you describe is the cause of the rotation of the polarization of light passing through the medium. This is described for example here on Wikipedia. $\endgroup$
    – A. P.
    Commented Mar 20, 2021 at 0:34
  • $\begingroup$ The reversibility law is part of a more general reciprocity theorem. The quick answers is that is not valid under magnetic fields. $\endgroup$
    – Mauricio
    Commented Jan 14, 2023 at 13:58

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You've mentioned some light phenomenon and tried to relate them with theories.

The connection between Fermat's principle and the Faraday effect

The Fermat's principle is mathematical treatment of geometrical optics, while the Faraday effect is light phenomenon that is based on the EM wave nature of light.

Fermat's principle is used to explain how light bends as it passes through lenses and mirrors, while the Faraday effect is used to explain how the polarization of light is rotated when it passes through a magnetic field.

Both of these effects have different applications, Fermat's principle is used in the design and analysis of optical systems such as cameras, telescopes, and microscopes, while the Faraday effect is used in applications such as isolator, remote sensing, navigation, and telecommunications. The faraday effect induces birefringence in material check my answer here.

Validity of Law of Reversibility

The Law of Reversibility is reciprocal to geometrical optics (minus dispersion) and is only valid under the assumption of straight-line light propagation and linear optical properties of the medium. In general, Geometric optics is an approximation that ignore several phenomenon and is useful when we take the shortcut in many cases where using the wave theory is just extra work.

The optical path length can be different for the same path but different directions in the presence of a magnetic field due to the Faraday effect. The magnetic field can affect the refractive index of the medium, which in turn affects the optical path length of light traveling through the medium. This is why the Law of Reversibility is violated by the Faraday effect.

PS: The optical path is the product of refractive index of that medium and length of path:

Light theories, Phenomenon and Branches of optics

There are three main branches of optics that differentiate different light phenomenons. Different light theories explains these phenomenon assuming light as photons, particles, energy, waves or rays.

Quantum Mechanics unifies all theories and describes light as a wave-particle duality, It says that Wave nature of light dominates when light interacts with light. The particle nature of light dominates when the light interacts with matter (micro-scopic particles) . Following picture visually maps and explains the the relations of light theories, phenomenon and optic branches.

Validitiy of theories for different light Phenomenon and optic branches

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