The components of $E$ waves are, \begin{equation} e_x=k_x k_z C_E^{\mathrm{rect}} \mathrm{cos}\left(k_x x\right)\mathrm{sin}\left(k_y y\right) \end{equation} \begin{equation} e_y=k_y k_z C_E^{\mathrm{rect}} \mathrm{sin}\left(k_x x\right)\mathrm{cos}\left(k_y y\right) \end{equation}
what it the physical meaning of $\hat{z} \cdot \mathrm{rot}_t e=0$ and when $\hat{z} \cdot \mathrm{rot}_t e\neq0$
My first guess was that there is no rotating field of $e$ exist in the direction of $z$?