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Let the two tangential aperture fields be $F_y(y,z)$ and $F_z(y,z)$, extended by zero off the aperture. The normal component does not enter $\mathbf{n}\times {\mathbf{E}}_{\mathrm{inc}}$, which equals $-F_z{\mathbf{e}}_y+F_y{\mathbf{e}}_z$. Apply the Weyl plane-wave representation and curl to both components. With the same kernel normalization as part (a),This is the electromagnetic aperture angular spectrum. The two electromagnetic polarization vectors are linearly independent and both orthogonal, bilinearly, to $\mathbf{q}$; they need not be orthogonal to each other. Arbitrary tangential electromagnetic polarization is therefore represented, with its outgoing longitudinal component fixed by $\kappa E_x+\nu E_y+\eta E_z=0$ rather than independently prescribed. Away from grazing components one could equally express the spectrum in a transverse-electric and transverse-magnetic basis. The same limiting outgoing branch handles grazing and evanescent components.

For each component the magnetic amplitude follows from Faraday's law as $\mathbf{B}=\mathbf{q}\times \mathbf{E}/\omega$. Together with ${\mathbf{q}}^2=k^2$ and transversality of an electromagnetic plane wave, this verifies the source-free Maxwell equations. If the doubled-kernel aperture convention is used, multiply the entire expression by two, exactly as in part (a).