Electromagnetic aperture angular spectrum ($\mathbf{q}=(\kappa ,\nu ,\eta ),\kappa =\sqrt{k^2-{\nu }^2-{\eta }^2}$)

For an aperture normal to ${\mathbf{e}}_x$, let $F_y,F_z$ be its tangential electric field traces extended by zero outside the aperture. With the outgoing branch of $\kappa$, the Weyl plane-wave representation and the curl aperture integral give a spectrum proportional to $(-(\nu {\hat{F}}_y+\eta {\hat{F}}_z)/\kappa ,{\hat{F}}_y,{\hat{F}}_z)$. Each wave is transverse to $\mathbf{q}$. The longitudinal electric component is regenerated by oblique diffraction even when it vanishes at the input. For $G=e^{ikr}/(4\pi r)$ and a unit prefactor on the stated curl integral, the spectrum has an overall factor $1/2$; a doubled half-space Green kernel gives factor one. The normalization must follow the chosen Green kernel and equivalent-current convention.