- #1
eliotsbowe
- 35
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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
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.
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
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.
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