Advanced Shaping of Quasi-Bessel Beams for High-Intensity Applications

Quasi-Bessel beams produced by axiparabolas are increasingly used in high-intensity laser applications, yet their longitudinal profiles exhibit unwanted oscillations that limit their effectiveness. Here we identify the physical origin of these distortions and develop a general strategy to control the on-axis intensity of extended focal lines. By combining analytical insight with numerical and experimental validation, we show how both smooth and sharply structured longitudinal profiles can be reliably produced. This establishes a robust framework for tailoring quasi-Bessel beams in regimes relevant to laser-plasma acceleration, advanced photon sources, and other high-field applications.

💡 Research Summary

The paper addresses a critical limitation of quasi‑Bessel beams generated by axiparabolas: unwanted oscillations in the on‑axis intensity along the extended focal line. Such modulations, while sometimes useful, generally degrade performance in high‑intensity applications such as laser‑plasma accelerators, plasma waveguides, and advanced photon sources.

Analytical foundation – The authors begin by modeling a radially symmetric, monochromatic top‑hat beam incident on an axiparabola that may include a central aperture. Using Fresnel diffraction and the stationary‑phase approximation, they derive an expression for the on‑axis field E(z) in terms of the radial phase Ψ(r)=k r²/(2z)−ϕ(r). The stationary point rₛ(z) maps each axial position to a specific mirror radius. Near the aperture edge (r→r_h) and the outer beam edge (r→R), the simple stationary‑phase term is insufficient; additional boundary contributions generate a complex exponential factor that produces the observed intensity ripples. The condition 2π|Ψ″|≫1/rₛ² must hold for the ripples to be negligible, a criterion rarely satisfied near the beam centre or at abrupt truncations.

Numerical validation – Using the Axiprop Python library (which implements discrete Hankel and Fourier transforms), the authors simulate a 50 mm‑diameter beam focused by an axiparabola designed for a 50 mm‑long constant‑intensity segment starting at z=200 mm. The simulated axial intensity matches the analytical prediction (including the boundary terms) at the far end of the focal line, but deviates near the start, especially when the central hole radius r_h is reduced. This confirms that the oscillations stem from the sharp truncation of the Fresnel integral rather than from the hole itself.

Amplitude‑mask strategy – To suppress the ripples, the authors replace the top‑hat illumination with a super‑Gaussian profile of very high order (n=100) and substitute the hard central hole with a smooth Gaussian aperture. The modified incident field is inserted into the stationary‑phase integral, yielding a new expression for E(z) that predicts dramatically reduced oscillations. Axiprop simulations and a He–Ne laser experiment (λ=633 nm) employing an SLM to generate the smooth amplitude mask confirm the suppression. The measured intensity, however, shows a modest decline along the plateau, which the authors attribute to a trefoil aberration intrinsic to the axiparabola. By incorporating a trefoil phase term Φ_trefoil into the model, the simulated curve reproduces the experimental data, and the authors suggest that a deformable mirror could correct this aberration.

Phase‑only optimization – Recognizing that amplitude shaping can waste energy or damage optics, the authors explore a phase‑only approach. They note that the ripples arise when Ψ″(r) is too small near the beam centre, allowing interference between contributions from different radial zones. By artificially increasing the second derivative of the phase for radii r<r₀, they flatten Ψ″ in that region. Practically, they replace the original phase Φ(r) by a quadratic continuation (Eq. 10) that holds a constant curvature for r<r₀. Axiprop simulations with r₀=10 mm and r₀=15 mm show progressive damping of the oscillations; the larger r₀ case nearly eliminates them, and the on‑axis intensity follows the simple stationary‑phase formula (Eq. 4). Analytically, the resulting intensity scales as I(z)∝z(k₀−zΦ″(r₀))³, producing a long, smooth leading edge.

Phase‑only experiment – To verify this concept experimentally, the authors use a 405 nm diode laser and two cascaded SLMs (required because a single SLM cannot provide the full phase range). Implementing the quadratic phase for r₀=10 mm, they record the axial intensity with a CMOS camera on a translation stage. The measured profile exhibits strongly damped ripples and a gradual rise, matching the simulations. As before, the overall intensity is lower than the ideal constant‑plateau case, likely due to residual aberrations.

Implications and outlook – The work establishes a robust, physics‑based framework for tailoring the longitudinal intensity of quasi‑Bessel beams in high‑field regimes. By either smoothing the incident amplitude or engineering the phase curvature, one can design arbitrary on‑axis profiles—from flat plateaus to sharply structured patterns—without sacrificing the high‑damage‑threshold advantage of reflective axiparabolas. This capability directly benefits laser‑plasma acceleration (by stabilizing the accelerating cavity length and preventing unwanted injection), advanced X‑ray sources (by controlling betatron emission), and any application requiring long, non‑diffracting high‑intensity beams. Future directions include extending the method to broadband, ultrashort pulses, incorporating nonlinear propagation effects, and developing adaptive feedback loops for real‑time aberration correction.