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Does [itex]\vec{H}=\frac{1}{\eta}\hat{k}\times \vec{E}[/itex] always suit an electromagnetic wave ?

Because I'm getting some inconsistency for a circular wave

For instance:

[itex]\vec{E}=E_0(\hat{e}_x+e^{j\frac{\pi}{2}}\hat{e}_y)e^{-jkz}[/itex] (propagating [itex]\hat{e}_z[/itex])

[itex]\Rightarrow\vec{H}=\frac{E_0}{\eta}(\hat{e}_y-e^{j\frac{\pi}{2}}\hat{e}_x)e^{-jkz}[/itex]

The poynting vector results:

[itex]\vec{S}=\vec{E}\times\vec{H}=\frac{E^2_0}{\eta}e^{-2jkz}(1+e^{j\pi})\hat{e}_z=\vec{0}[/itex] which is quite disconcerting :/

Thx a lot for your help !