Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 7D / b / i / Solution

The phase has wavevector $\mathbf{k}=k{\mathbf{e}}_z$. The electric field has fixed direction $({\mathbf{e}}_x+{\mathbf{e}}_y)/\sqrt{2}$, so it has linear polarization at $45^{\circ }$ to the $x$ and $y$ axes, perpendicular to the wavevector. Substitution into the electromagnetic wave equation givesFor the usual positive-frequency convention, $\omega =c|k|$ and propagation is in the direction $\mathrm{sgn}(k){\mathbf{e}}_z$. If $k>0$, this reduces to $\omega =ck$. The following wave formulas assume the nonconstant case $\omega \ne 0$.