= Fourier localization of a Gaussian beam
{c}
{title2=$\sigma_x\sigma_{k_x}=1/2$}
\b[<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, take
$$
U(x,y)=U_0e^{-(x^2+y^2)/w_0^2},\qquad
\widetilde U(k_x,k_y)=\int_{\mathbb R^2}U(x,y)e^{-i(k_xx+k_yy)}\,dx\,dy.
$$
Two <Gaussian integrals> give
$$
\widetilde U=\pi w_0^2U_0\exp\!\left[-\frac{w_0^2(k_x^2+k_y^2)}4\right].
$$
Reducing $w_0$ therefore broadens the <angular spectrum method> representation rather than merely shrinking a picture in the same set of propagation directions.<Physical origin of the Gouy phase shift by Simin Feng and Herbert G. Winful>[Equations 4–12, transverse confinement, spectral description and Gaussian phase]{ref}
Widths must state their weighting. Normalize $|U|^2$ and $|\widetilde U|^2$ to define transverse position and transverse-wavenumber distributions. Their one-coordinate standard deviations are
$$
\sigma_x=\frac{w_0}{2},\qquad \sigma_{k_x}=\frac1{w_0},\qquad
\boxed{\sigma_x\sigma_{k_x}=\frac12}.
$$
These 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 $\sigma_{\theta_x}=1/(kw_0)$. 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 $\sigma_{\theta_x}\simeq0.2014$ 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>: $kw_0\gg1$ keeps most amplitude at transverse wavenumbers much smaller than $k$. Strong focusing can require vector polarization constraints and a nonparaxial propagation model.
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