I am trying to find the wavenumbers $\kappa(\omega)$ of the time-harmonic maxwell equation (eigenvalue problem)
$curl \ curl E = \omega^2 E$
whereas $E(x,y,z) = u(y,z)exp(i\kappa_0x)$ and $\Omega$ equals a cube with side length $1$.
Can someone please help me? It is super urgent, because i need them for my master thesis and have to hand it in tomorrow.
I would be forever thankful.
EDIT:
So if I apply the curl twice I get
$curl curl E = \\(i\kappa_0\partial_yu_2-\partial_y^2u_1-\partial_z^2u_1+i\kappa_0\partial_zu_3, \\ \partial_z\partial_y u_3 -\partial_z^2u_2+i^2\kappa_0u_2 - i\kappa_0\partial_yu_1, \\ i\kappa_0\partial_zu_1 - i^2\kappa_0^2u_3 - \partial_y^2u_3 +\partial_y\partial_zu_2) \ exp(i\kappa_0x)$
which then yields into a system of differential equations.