# Relation of dispersion for a plasma

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 ?