= Solution
The phase has <wavevector> $\mathbf k=k\mathbf e_z$. The electric field has fixed direction $(\mathbf e_x+\mathbf e_y)/\sqrt2$, so it has <linear polarization> at $45^\circ$ to the $x$ and $y$ axes, perpendicular to the <wavevector>. Substitution into the <electromagnetic wave equation> gives
$$
\boxed{\omega^2=c^2k^2.}
$$
For the usual positive-frequency convention, $\omega=c|k|$ and propagation is in the direction $\operatorname{sgn}(k)\mathbf e_z$. If $k>0$, this reduces to $\omega=ck$. The following wave formulas assume the nonconstant case $\omega\ne0$.
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