Bloch state + Photonic crystals master equation

  • #1

Main Question or Discussion Point

Hello all!

I've been trying to go from the second to the third equation shown in the image.
Here, [itex]\vec{H}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}\vec{u}_\vec{k}(\vec{r})[/itex] is the Bloch state for some periodic dielectric arrangement.

344wmxl.png


I have tried using the identity that [itex]\vec{\nabla}\times\phi\vec{F}=\nabla\phi\times\vec{F}+\phi\vec{\nabla} \times \vec{F} [/itex], and got to the result [itex]\vec{\nabla}\left(\frac{1}{\epsilon}\left[i\vec{k}\times\vec{u}+\vec{\nabla}\times\vec{u}\right]\right)=\frac{\omega^2}{c^2}\vec{u}[/itex], but can't see where to go from here.

Any help is appreciated.
Thanks in advance.
 

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