Graduate Poynting Vector in Non-Magnetic Dielectric Media

[Twigg]

Hello all,

When light travels in a medium with negligible absorbance, it looks exactly like light in free space but with a different speed relative to that medium given by the refractive index. In free space, the Poynting vector is given by $\vec{S} = \frac{1}{\mu _{0} } (\vec{E} \times \vec{B})$. For a non-magnetic medium ($\mu = 1$), I would naively expect that the Poynting vector expression would be unchanged. Is this the case? If the absorbance is not negligible, then the electric and magnetic fields are no longer in-phase. What does the Poynting vector look like then? If the base expression is unchanged, why does it still apply? If the wave is attenuated, is its Poytning vector complex (since its wavevector is complex)? Lastly, I would think that the intensity (average of the poynting vector over 1 wavelength) of an evanescent wave is 0. Is this true? Sorry for the boatload of questions. If anyone has a reference that goes over this, it may save a lot of time. My understanding is based off Ch 9 of Griffiths E&M textbook and Ch's 3 and 4 of Hecht's book.

 

[Dale]

There has been quite some debate on this topic over the years. Here is my favorite reference on the topic.

http://arxiv.org/abs/0710.0461

 

[Twigg]

@Dale Sorry for the late reply. It took me a while to process that article. You definitely weren't kidding around when you said there had been some debate.

Since my original question is significantly larger in scope that I had anticipated, let me try to identify where I want to go with this. First, I would like to consider a solid, crystalline non-magnetic dielectric medium. I would like to look at reflected waves for which $k_{z}$ is complex (evanescent) at frequencies near a lossy polariton resonance, and determine how much energy they transport in the z direction averaged over a wavelength as a function of their index of refraction and absorbance. If I get that far, I would also like to do something similar for transmitted waves in an Ohmic conductor, looking at energy transport in the z-direction near surface plasmon resonances.

So, based on the kind of material considered, I'm ignoring pressure variations, fluid flow, and magnetization. Based on that, I think I can use the stress tensor given in (33) of the article Dale linked, by removing any term with a p, $\vec{v}$, or $\vec{M}$ in it. That leaves $\vec{g} = \epsilon_{0} \vec{E} \times \vec{B}$, which is the same as in free space. I'm a little surprised that there is no momentum associated with the optical phonons. I thought that ignoring the pressure p and fluid velocity $\vec{v}$ in equation (33) of the article would only neglect acoustic phonons and that any momentum or energy associated with optical phonons would be reflected in terms that depend only on $\vec{P}$. Was I mistaken about that premise? Or is an additional term required for optical phonons? Or is there really no momentum associated with them?

Thanks in advance for any input.