= Transverse polarization constraint in an angular spectrum
{title2=$\mathbf k\cdot\widetilde{\mathbf E}=0$}
The \b[<transverse polarization constraint in an angular spectrum>] requires each source-free electromagnetic <plane wave> component to have electric field orthogonal to its <wavevector> in the algebraic sense:
$$
\mathbf k\cdot\widetilde{\mathbf E}=0.
$$
For an isotropic homogeneous medium with constant permittivity, this follows by inserting a Fourier component into $\nabla\cdot\mathbf D=0$. It restricts the vector fields that can accompany a scalar <angular spectrum method> calculation.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach>[Scalar Diffraction Optics, Angular Spectrum Method, vector divergence constraint following the propagation equations]{ref}
With time dependence $e^{-i\omega t}$ and $\mathbf k=(k_x,k_y,k_z)$, the relation becomes
$$
k_x\widetilde E_x+k_y\widetilde E_y+k_z\widetilde E_z=0.
$$
When $k_z\ne0$, the longitudinal field is therefore $\widetilde E_z=-(k_x\widetilde E_x+k_y\widetilde E_y)/k_z$. It is not an arbitrary third scalar image that can be propagated independently of the first two.
For an original <propagating angular-spectrum component> with $\mathbf k=k(0.6,0,0.8)$, choosing $\widetilde E_x=1$ and $\widetilde E_y=0$ forces $\widetilde E_z=-0.75$. Polarization along $y$ instead gives a purely transverse-to-axis field for this propagation plane. Thus electromagnetic transversality relative to $\mathbf k$ does not require zero field along the laboratory $z$ axis.
For an <evanescent angular-spectrum component> with $\mathbf k=k(1.25,0,0.75i)$, the same choice $\widetilde E_x=1$ gives $\widetilde E_z=(5/3)i$. The dot product is not a <Hermitian inner product>: there is no complex conjugation of $\mathbf k$ in the Fourier-transformed divergence equation. The imaginary longitudinal amplitude represents a phase relationship, not permission to discard the constraint.
At grazing propagation, $k_z=0$, division by $k_z$ fails and the original equation must be used. Boundary conditions, material response and the corresponding <magnetic field> remain necessary to reconstruct a physical solution. In the strongly <paraxial approximation>, longitudinal components may be small, but that is an approximation derived from the allowed spectrum rather than a replacement for the vector condition.
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