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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.

Simin Feng is the first-listed author, with Herbert G. Winful, of the 2001 primary paper Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful. The full published byline supplies the name; the paper lists a University of Michigan affiliation and a Lucent Technologies present-address note. These identify the contribution at publication without inferring a present appointment or attaching an unverified namesake's portrait.^[ref]

The paper connects the Gouy phase with transverse confinement and the angular spectrum representation of a wave. Its analysis treats a finite beam as a superposition of plane waves with different transverse wave vector components and obtains the Gaussian phase contribution.^[ref]

An original one-coordinate example makes the localization issue quantitative. At a waist, let the field amplitude be $U(x)\propto e^{-x^2/w_0^2}$. Its Fourier transform has amplitude $\tilde{U}(k_x)\propto e^{-w_0^2{k}_x^2/4}$. Normalizing the squared magnitudes as distributions givesFor stipulated $w_0=100\mu \mathrm{m}$, the position spread is $50\mu \mathrm{m}$ and the transverse-wavenumber spread is $10^4{\mathrm{m}}^{-1}$. Halving the waist halves the first and doubles the second. The distributions here are constructed from field amplitudes; no single-photon counting experiment is assumed.

This Fourier localization of a Gaussian beam is part of the mathematics underlying the paper's focusing analysis. It also explains why a perfectly transverse-localized beam cannot retain the single propagation direction of an infinite plane wave. A field model, its phase convention and its transverse geometry must accompany any numerical phase claim.