Plane waves: sign of Re(ε), Re(μ) in passive media, attenuation angle

[eliotsbowe]

Hello,
I'm having some issues with plane waves propagating through a medium which is:
- linear
- spatially and temporally homogeneous
- spatially non-dispersive
- isotropic
- temporally dispersive
- passive

I know that permittivity, permeability and the k-vector are complex in temporally dispersive media.

There are different notations of the above-mentioned quantities, so I'm going to briefly introduce those which I'm used to:
$$\epsilon (\omega) = \epsilon ' (\omega) - j \epsilon''(\omega)$$$$\mu (\omega) = \mu ' (\omega) - j \mu''(\omega)$$$$\underline{k} = \underline{\beta} - j \underline{\alpha}$$$$\underline{k} \cdot \underline{k} = \omega^2 \epsilon \mu$$Where$$\epsilon'' , \mu'' \geq 0$$ in a passive medium and $$\alpha>0$$.

During class, my professor stated that, in the medium in question, the so-called "attenuation angle" (the angle between the attenuation vector alpha and the phase vector beta) is 90° or lower, because:$$\underline{\alpha} \cdot \underline{\beta} \geq0$$
The statement was derived from the following equation:
$$(\underline{\beta} - j \underline{\alpha}) (\underline{\beta} - j \underline{\alpha}) = \omega^2 \epsilon \mu$$
(Tearing Re[] and Im[] apart we have:)
[PLAIN]http://img834.imageshack.us/img834/706/immagine1iz.png

My professor said $$Im[\epsilon \mu] < 0$$ and my issue is right here.
I carried out the product:
$$\epsilon \mu = (\epsilon ' - j \epsilon'' ) (\mu' - j \mu'') = \epsilon' \mu' - j \epsilon' \mu'' - j \epsilon'' \mu' - \epsilon'' \mu''$$$$Im[\epsilon \mu] = - \epsilon' \mu'' - \epsilon'' \mu'$$

Mu'' and epsilon'' are non-negative, but what about mu' and epsilon' ?
Are they both positive in a passive medium?

Thanks in advance.

 

[DrDu]

A magnetic response of a medium is equivalent to spatial dispersion as i omega E=rot B whence a material with mu different from mu_0 can always be described as a material with mu=mu_0 and a k dependent epsilon (that is, spatial dispersion), at least at non-zero frequency.
However, I don't see that the answer to your question depends on the medium being non-dispersive. So if you write mu=mu_0, this implies that mu''=0.
Epsilon will then be a function of k but still epsilon''>0. Hence epsilon'' mu' <0 which is all you need.

 

[eliotsbowe]

In that case, the inequality would be prooved. But my problem is: no approximation was made about epsilon and mu.

 

[DrDu]

I am making no approximations besides the ones you mention. My argument is more general than yours as I don't presuppose absence of spatial dispersion.
Have a look at chapter 3.2 in
http://mesoscopic.mines.edu/mediawiki/images/a/ab/Spatial_dispersion.pdf

 

[eliotsbowe]

Well, that was some helpful pdf. Thanks!