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I have been studying how light behaves in dielectric and conductive media, and now I was about to study it in anisotropic media. The introduction to the subject is: "From Maxwell's equations it is shown that the direction of propagation of the energy (rays) and the wave do not coincide in anisotropic mediums, and in addition there are in general two waves propagating in the middle".

I can understand perfectly well that there are two waves, since the refractive index depends on the direction of propagation, but how is it possible that the energy propagates in an other direction than the wave? I think it is not as "fanciful" as it really is, perhaps it is a lack of clarity in my concepts.

I think the answer is related to how we understand a light wave. If we understand the oscillatory character of light as a "rigid" entity that physically moves in one direction and with velocity v is where the confusion arises, but if we understand that it is actually the wave pattern that propagates with phase velocity v, without transporting energy or information, I can understand that they have different directions, but that the one that really "matters" in the physical sense is the one of energy, which is the light we will see with our eyes.

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I recently asked myself the same question. The direction of the wave vector $\vec{k}$ does not coincide with the direction of the Poynting vector $\vec{S}=\vec{E}\times\vec{H} $ in an anisotropic media since $\vec{D}$ and $\vec{E}$ aren't colinear anymore due to the fact that $[\epsilon]$ is now a tensor for an anisotropic media.

As I understood it so far, the direction of $\vec{k}$ is the same as the one of the phase velocity $\vec{v_{\phi}}=\frac{\omega}{k}\vec{u}$, where $\vec{u}$ is the direction of $\vec{k}$. However, the direction of the group velocity $\vec{v}_g=\nabla_k \omega(\vec{k})$ is colinear to the Poyting vector. So, we can say that the phase and the energy don't propagate in the same direction.

Let me know if you've found some insights about this topic ;)

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  • $\begingroup$ I came up with an answer but I don't know if it is correct. What we defined earlier as a "wave", when it enters a material with "certain" properties, will have an electric displacement vector $\vec{D}$ as you said, and since $\vec{E}$ and $\vec{D}$ are not parallel, and $\vec{H}$, $\vec{D}$ and $\hat{s}$ are perpendicular, the direction of propagation of the wave is different from the direction of propagation of the energy. $\endgroup$
    – JL14
    Commented Feb 20 at 18:50
  • $\begingroup$ Now, I ask you the same question I asked myself, does this have any important meaning other than that the mathematical entity that we defined before as the direction of wave propagation has a different direction? In my opinion, the energy now has another direction of propagation that is not the one expected by the formulas, it differs from $\hat{s}$, but the energy still continues to propagate as a wave, in its new direction of propagation $\hat{\rho}$, so in its direction it will again have a phase and group velocity. $\endgroup$
    – JL14
    Commented Feb 20 at 18:51
  • $\begingroup$ Let's suppose our wave of vector $\vec{k}$ is normally incident on an anisotropic cristal . Since the wave is normally incident, optical geometry and refraction law tells us that the ray isn't deflected. The wave vector remains perpendicular to the wavefronts. The ordinary ray goes through the cristal carriyng a Poyting vector $S_o$ colinear to $\vec{k}$ (as if it was an isotropic media). But the direction of propagation of the energy of the extraordinary ray, $S_e$ differs from $\vec{k}$ by a certain angle. The fact that the energy propagates in two different direction gives birefringence. $\endgroup$
    – MauvaiseFoi
    Commented Feb 20 at 19:25
  • $\begingroup$ The two wave vectors $\vec{k}_o$ and $\vec{k}_e$ are colinear i.e. perpendicular to the wavefronts, they follow the refraction law. But, in certain circumstances the energy propagation of the extraordinary ray differs from this direction. $\endgroup$
    – MauvaiseFoi
    Commented Feb 20 at 19:31

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