Assuming the electric field :

$\vec{E} = E_{0}\,e^{i(kz-\omega t)}\vec{e_{z}}$

and the complex relation by doing $\vec{rot}\,(\vec{rot}\times\vec{E})$

with $\vec{rot}\times \vec{E}= i \vec{k}\times \vec{E}$ :

I get with Maxwell equation : $k^2 \vec{E} = \mu_{0}\omega \vec{J}-i\dfrac{\omega^2}{c^2}\vec{E}$

How can I find the expression of conductivity $\sigma(\omega)$ such $\vec{J}=\sigma\vec{E}$ and deduce the dispersion relation for these waves ?


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