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

  1. Aug 31, 2011 #1
    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:
    [tex]\epsilon (\omega) = \epsilon ' (\omega) - j \epsilon''(\omega)[/tex][tex]\mu (\omega) = \mu ' (\omega) - j \mu''(\omega)[/tex][tex]\underline{k} = \underline{\beta} - j \underline{\alpha}[/tex][tex]\underline{k} \cdot \underline{k} = \omega^2 \epsilon \mu[/tex]Where[tex]\epsilon'' , \mu'' \geq 0[/tex] in a passive medium and [tex]\alpha>0[/tex].

    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:[tex]\underline{\alpha} \cdot \underline{\beta} \geq0[/tex]
    The statement was derived from the following equation:
    [tex](\underline{\beta} - j \underline{\alpha}) (\underline{\beta} - j \underline{\alpha}) = \omega^2 \epsilon \mu [/tex]
    (Tearing Re[] and Im[] apart we have:)
    [PLAIN]http://img834.imageshack.us/img834/706/immagine1iz.png [Broken]

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

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

    Thanks in advance.
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Sep 1, 2011 #2

    DrDu

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    Science Advisor

    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.
     
  4. Sep 1, 2011 #3
    In that case, the inequality would be prooved. But my problem is: no approximation was made about epsilon and mu.
     
  5. Sep 1, 2011 #4

    DrDu

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    Science Advisor

    Last edited by a moderator: Apr 26, 2017
  6. Sep 2, 2011 #5
    Well, that was some helpful pdf. Thanks!
     
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