Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 4 / b

For a fundamental electric field in the $xy$ plane, $E_z=0$. The specified tensor then givesWriting $E_x=E_0\cos \varphi$ and $E_y=E_0\sin \varphi$ yieldsThe source electric polarization is therefore along $z$ and is largest along the in-plane diagonals. A radiated wave also needs a transverse component of this source: propagation along $z$ would not generate a propagating $z$-polarized output. Propagation in the $xy$ plane provides the simple strictly transverse example. For a tilted extraordinary wave, the electric displacement, rather than generally the entire electric field, is perpendicular to the wave vector; the tensor source must be projected onto that wave's allowed polarization. This distinction matters when using the type-I geometry below.