# Momentum of light in anisotropic media

This question is related to the Abraham-Minkowski controversy that has already been discussed extensively here and in the research community. But I want to ask about an aspect of this momentum controversy that I could not find in literature: There are two common expressions to calculate momentum of light:

\begin{equation} \vec p_1 = \hbar \vec k \end{equation} with $\vec k = \vec D \times \vec H$ the wave vector; and \begin{equation} \vec p_2 = \frac{1}{c} \vec S \end{equation} with $\vec S = \vec E \times \vec H$ the Poynting vector.

For an anisotropic medium with \begin{equation} \vec D = \epsilon_0 \hat \epsilon_r \vec E \end{equation} with $\hat \epsilon_r$ being a tensor with three different entries on the diagonal, $\vec D$ and $\vec E$ do obviously not point in the same direction, so $\vec k$ and $\vec S$ also not point in the same direction. In this case, which definition of momentum has to be used and why?

"In this case, which definition of momentum has to be used..."

The momentum density (momentum transferred per unit time per unit area) associated with the fields will be described by the Poynting vector. An expression similar to that which you give for $p_2$:

$\vec{g} = \frac{1}{c^2}\vec{S}$

"... and why? ..."

The $\vec{k}$ vector just indicates the spatial dependence of the phase of the travelling wave. On the other hand, the Poynting vector indicates where the energy density of the wave is moving, in other words, the direction in which the "amplitude" of the wave is propagating to.

Thinking of an individual photon, the Poynting vector gives a measure of the instantaneous group velocity of the wave-packet's wavefunction, something which can be directly related to the momentum of the photon.

An example that illustrates this in an anisotropic medium!: Let us think of a photon that travels through a birrefringent medium as shown in (Fig. 6.2. (b)): the $\vec{k}$ vector never changes (ie. always points to the right), as the wave fronts are always parallel to the material's walls. Nonetheless, the photon exits this anisotropic material at a different vertical position: it must have had vertical momentum while it propagated through the medium, as indicated by $\vec S$. Sidenote: there is an additional co-travelling momentum due to the electrons rearranging themselves in the medium in response to the travelling wave. This is named the Minkowski momentum and is described by something similar to the expression you give for $p_1$.

• Thanks for your answer. I will think about it. Would be nice though, if you cite some papers that talk about these things in the case of anisotropic media. – Tornado Jun 14 '18 at 6:45
• Sure thing. I found something that supports my claim, check it out: arxiv.org/ftp/arxiv/papers/1207/1207.6676.pdf -Cheers! – Gyromagnetic Jun 14 '18 at 13:58
• I have been pondering about your answer and the paper that you linked. In the paper, it's explicitely said that they postulate the momentum to be $\vec p = \vec S/c^2$. They don't derive it from first principles, and neither do you in your answer. Considering the large amount of papers that argue in favour of the Minkowski momentum $\hbar \vec k$, I find your argument not satisfying. E.g. this paper states that the wave momentum has to be always parallel to the wave vector: arxiv.org/abs/1409.5807 – Tornado Jun 16 '18 at 20:35
• I stand by my comments. I read the abstract of the paper you posted and it seems to be a different problem than that you proposed. To begin with, they talk of a uniform, isotropic medium. And then they say that the momentum is not necessarily described by the Poynting vector when the medium is moving. I agree with everything they present (in the abstract), I just cannot see how it relates to the this question. – Gyromagnetic Jun 17 '18 at 17:30
• You're right, I missed that they only talk about isotropic media. If no other answers come, I will award you the bounty later. – Tornado Jun 18 '18 at 7:28