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Fourier localization of a Gaussian beam expresses the tradeoff between a narrow transverse field and a broad distribution of transverse spatial frequencies. At a waist, takeTwo Gaussian integrals giveReducing $w_0$ therefore broadens the angular spectrum method representation rather than merely shrinking a picture in the same set of propagation directions.^[ref]

Widths must state their weighting. Normalize $|U{|}^2$ and $|\tilde{U}{|}^2$ to define transverse position and transverse-wavenumber distributions. Their one-coordinate standard deviations areThese are optical field-width statements under the stated Fourier convention, not an inference that each ray has simultaneously measured position and angle. Using amplitude widths instead would give different numerical factors.

For a strongly paraxial spectrum, ${\theta }_x\simeq k_x/k$ and ${\sigma }_{{\theta }_x}=1/(k{w}_0)$. The usual far-field Gaussian $1/e^2$ light intensity half-angle is $2{\sigma }_{{\theta }_x}=\lambda /(\pi w_0)$. For original $\lambda =632.8$ nm and $w_0=0.50$ mm, ${\sigma }_x=0.25$ mm and ${\sigma }_{{\theta }_x}\simeq 0.2014$ mrad; the conventional divergence half-angle is about 0.4029 mrad. Halving the waist doubles both angular measures.

An illustrative apparatus places a camera in a plane conjugate to the waist and another measurement at a lens's back focal plane, where transverse spatial frequency maps to position in the paraxial model. Comparing widths tests localization and angular spread, provided magnification, focal length and detector response are accounted for. Finite aperture clipping changes the field and produces extra spectral structure, so the exact Gaussian width product cannot simply be assigned to every clipped beam.

The same spectrum also bounds the paraxial approximation: $k{w}_0\gg 1$ keeps most amplitude at transverse wavenumbers much smaller than $k$. Strong focusing can require vector polarization constraints and a nonparaxial propagation model.

The 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: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.^[ref]

With time dependence $e^{-i\omega t}$ and $\mathbf{k}=(k_x,k_y,k_z)$, the relation becomesWhen $k_z\ne 0$, the longitudinal field is therefore ${\tilde{E}}_z=-(k_x{\tilde{E}}_x+k_y{\tilde{E}}_y)/k_z$. 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 ${\tilde{E}}_x=1$ and ${\tilde{E}}_y=0$ forces ${\tilde{E}}_z=-0.75$. 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 ${\tilde{E}}_x=1$ gives ${\tilde{E}}_z=(5/3)i$. 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.