= Bessel-Gaussian beam
{c}
A \b[<Bessel-Gaussian beam>] is a structured optical beam whose Bessel-like transverse oscillations are limited by a Gaussian envelope. For a circularly symmetric zero-order waist-plane model,
$$
U(r,0)=U_0J_0(k_t r)e^{-r^2/w_0^2},
$$
where $J_0$ is a <Bessel function of the first kind>, $k_t$ sets the radial oscillation scale and $w_0$ sets the Gaussian envelope. This specifies the initial field; propagating it requires a wave model rather than keeping the same product unchanged at every plane.
An ideal <Bessel beam> can be constructed from a ring of <plane waves> with the same longitudinal <wavenumber> and different transverse directions. Their coherent angular sum gives $J_0(k_t r)$. The ideal infinitely extended beam has unbounded transverse <power>. Multiplication by the Gaussian envelope makes the <power> integral finite, since $|J_0|\le1$ and
$$
\int_0^\infty |U(r,0)|^2\,2\pi r\,dr
\le |U_0|^2\frac{\pi w_0^2}{2}.
$$
Finite transverse extent also means that indefinitely unchanged propagation is no longer guaranteed.
For an original paraxial scale example, let $\lambda=532$ nm, $w_0=0.50$ mm and $k_t=k\sin\theta$ with $\theta=1\,{}^\circ$. The first radial zero of $J_0$ is at $k_t r\simeq2.4048$, giving $r\simeq11.67\,\mathrm{\mu m}$. The geometric overlap estimate $w_0/\tan\theta\simeq28.6$ mm supplies a finite propagation-distance scale, not an exact invariant-range theorem or a measured beam length. The envelope and central-lobe scales answer different questions.
A primary 2023 experiment compared Bessel-Gaussian <Gouy phase> theory with interferometric measurements. Its apparatus used a 532 nm laser, beam expansion, spatial light modulators, focusing lenses and a camera translated along a rail; Figure 3 documents the interferometer geometry.<Gouy phase of Bessel-Gaussian beams: theory vs. experiment by Lyubomir Stoyanov and others>[Figure 3, p. 13691, apparatus and propagation-plane measurement]{ref} Its beam parameters should not be identified with the invented example above.
A ringed <light intensity> image alone does not determine the whole propagated complex field. Measuring <wave phase>, aperture effects and longitudinal evolution distinguishes a finite structured beam from the mathematical ideal. In particular, the <Gouy phase> of such a mode should not automatically be replaced by the fundamental <Gaussian beam> formula.
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