 Bessel-Gaussian beam

A 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,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 $|J_0|\le 1$ andFinite 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 $k_{t}r\simeq 2.4048$, giving $r\simeq 11.67\mu \mathrm{m}$. The geometric overlap estimate $w_0/\tan \theta \simeq 28.6$ 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.^[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.