Pulsed Bessel-Like Needle Beams

• Pulsed Bessel-like needle beams are structured light pulses that maintain quasi-nondiffracting propagation with sub-micron cores and extended focal zones.
• They are generated by superposing Bessel modes using axicons, diffractive elements, or SLMs, enabling precise ultrafast material processing and nonlinear photonics.
• Their engineered spatiotemporal design ensures high homogeneity and efficient filamentation, critical for advanced plasma applications and photonic device fabrication.

Pulsed Bessel-like needle beams are spatially and temporally structured light pulses exhibiting quasi-nondiffracting propagation, sub-micron-scale cores, and extended invariant focal regions. They are generated by the superposition of monochromatic or polychromatic Bessel modes, typically using axicons or related phase-changing elements, and play a central role in structured space–time wavepackets (STWPs). Their propagation invariance and robust localization make them foundational in ultrafast optics, precision material processing, and advanced nonlinear photonics.

1. Mathematical Formulation and Physical Principles

Pulsed Bessel-like needle beams are constructed as solutions to the scalar homogeneous wave equation in free space: $$\nabla^2 E(r, \phi, z, t) - \frac{1}{c^2} \frac{\partial^2 E}{\partial t^2} = 0,$$ assumed in cylindrical coordinates with time-harmonic dependence $e^{-i\omega t}$. The radial part yields

$$\frac{d^2\psi}{dr^2} + \frac{1}{r}\frac{d\psi}{dr} + \left[k_r^2 - \frac{m^2}{r^2}\right]\psi = 0,$$

with regular solutions given by the Bessel function of the first kind: $\psi_m(r) = A_m J_m(k_r r).$

A polychromatic (pulsed) Bessel-like beam is formed via spectral superposition: $$E(r,z,t) = \int d\omega\, A(\omega) J_0[k_r(\omega) r] e^{i[k_z(\omega) z - \omega t]},$$ where $k_r(\omega) = (\omega/c)\sin\theta$, $k_z(\omega) = (\omega/c)\cos\theta$, with cone angle θ defined by the axicon or other phase mask. Needle-like spatial localization arises from apodization, typically via a Bessel–Gauss profile: $$E(r,z,t) = \int d\omega\, A(\omega) e^{-(r/w_0)^2} J_0[k_r(\omega) r] e^{i[k_z(\omega)z - \omega t]}.$$

In the STWP formalism, spatiotemporal localization is achieved by enforcing a one-to-one correlation, $\omega(k_r) = \omega_0 + \alpha k_r$, so that each spatial frequency is linked to a temporal frequency, preventing diffractive or dispersive broadening (Grunwald et al., 3 Dec 2025, Sheppard, 8 Sep 2025, Efremidis, 2017).

2. Generation Techniques and Beam Engineering

Pulsed Bessel-like beams are generated using several phase-front engineering methods:

• Refractive axicons: Conical glass elements impose a linear radial phase, converting a Gaussian (or flat-top) pulse into a ring-shaped angular spectrum. The resulting cone angle θ is set by $\theta = (n-1) \alpha/2$.
• Diffractive axicons / phase plates: Lithographically patterned concentric structures provide robust, achromatic phase control, suitable for broadband or ultrafast pulses.
• Spatial Light Modulators (SLMs): Digitally programmed phase profiles allow dynamic, programmable production of Bessel or higher-order/vortex modes and can also apply pre-compensation for optical aberrations or interface tilts.

Optical configurations may also exploit relay lenses and telescopic imaging to control beam diameter and demagnification, and three-axicon arrangements have been demonstrated for producing ultra-high aspect ratio light needles with Joule-level pulse energy tolerance (Meyer et al., 2019). Finite aperture effects truncate the ideal Bessel beam, leading to propagation-invariant zones with length $L \approx R_0/\tan\theta$, where $R_0$ is the incident beam radius (Stoian et al., 2018, Demirci et al., 2019).

3. Spatial, Spectral, and Temporal Homogeneity

The central lobe of a Bessel-like needle beam exhibits a width set by the first zero of $J_0$, $r_0 \approx 2.405 / (k \sin\theta)$, achieving sub-micron FWHM diameters with high cone angles (Meyer et al., 2019, Stoian et al., 2018). The longitudinal (nondiffracting) zone can exceed centimeters, with aspect ratios $> 10^4:1$ demonstrated (Meyer et al., 2019). The on-axis intensity remains nearly constant (variation < 10%) across this region, with minimal broadening of femtosecond pulse durations.

Spectral and temporal homogeneity metrics for space–time wavepackets are defined as

$$H_s = 1 - \frac{\max_r |\lambda_{peak}(r) - \lambda_0|}{\Delta\lambda}, \quad H_t = 1 - \frac{\max_r |\tau(r) - \tau_0|}{\tau_0},$$

where λ_peak(r) and τ(r) are local spectral centroid and pulse width, respectively. For Bessel–Gauss STWPs, simulations indicate $H_s \approx 0.98$ over 50 nm bandwidth, outperforming spatio-spectrally shaped Gaussian foci, which display radial chirp and degraded $H_s$ (Grunwald et al., 3 Dec 2025).

4. Nonlinear Effects, Filamentation, and Structured Wavepackets

At high intensities (≥ 10¹³ W/cm²), nonlinear effects such as Kerr self-focusing, multiphoton ionization, and plasma generation alter propagation. Sufficiently large cone angles stabilize the Bessel core in a quasi-stationary filament regime, preserving localization and enabling the creation of underdense plasma channels (Stoian et al., 2018).

Interfering pulsed Bessel beams (needle beam arrays or superpositions with distinct spot sizes) generate axially modulated intensity profiles. In the nonlinear regime, periodic self-imaging is retained, with the modulation period set by the wavevector mismatch and tunable by spot size selection. The resultant corrugated plasma strings are promising for advanced plasma photonics applications (Mansourimanesh et al., 13 Aug 2025).

In the context of STWPs, interpretation as a coherent superposition of differential needle beams provides a rigorous analytic basis. Each constituent travels with a specific group delay, and the overall wavepacket preserves full spatiotemporal localization upon propagation. Interference in arrays produces Talbot self-imaging effects in both space and time, with the fundamental Talbot distance $z_T = p^2 / \lambda_0$ (Grunwald et al., 3 Dec 2025).

5. Precision Material Processing and Practical Applications

Pulsed Bessel-like needle beams are extensively utilized in ultrafast laser materials processing:

• Ultrafast cleaving and stealth dicing: High-intensity, propagation-invariant beams enable high-aspect-ratio subsurface structuring, with minimal collateral damage and heat-affected zones (HAZ). Beveled cleaving at high tilt angles (up to 20° inside glass) is realized using digital-holography axicon-SLM systems with phase pre-compensation for interface aberrations, verified via pump-probe microscopy (Jenne et al., 2020, Meyer et al., 2019).
• Sub-micron drilling and nanostructuring: Single or multiple pulses produce micron- and sub-micron voids and channels in transparent materials with exceptional aspect ratios and precise diameter control (Stoian et al., 2018, Demirci et al., 2019).
• Ultrafast welding and photonic device fabrication: Robust tolerance to z-position, high focal intensity uniformity, and long depth-of-focus facilitate reliable joining of dissimilar materials and writing of embedded photonic structures (Stoian et al., 2018).
• Plasma filamentation and nonlinear photonics: Extended, axially modulated plasma columns and localized filament-induced material modifications are accessible at high pulse energies (Mansourimanesh et al., 13 Aug 2025, Meyer et al., 2019).

Design criteria include selecting axicon angle (θ), incident beam diameter, and spectral bandwidth for application-specific localization and temporal resolution, balancing the tradeoff between axial extent and pulse duration (Grunwald et al., 3 Dec 2025, Meyer et al., 2019).

6. Advanced Control, Emerging Directions, and Limitations

High-speed, dynamic switching of STWPs is enabled by combining thin-film axicons, dispersive elements, and MEMS-based reflective components, providing μs–ns reconfigurability of propagation parameters (Grunwald et al., 3 Dec 2025). Beams carrying orbital angular momentum (OAM) with self-torque are realized via radially chirped spiral gratings, where the local topological charge varies with z, facilitating ultrafast multiplexing.

Ongoing and prospective research areas include:

• Intracavity Q-switching of needle beams for new laser architectures.
• Filamentation management for controlled plasma and nonlinear phenomena.
• Implementation in optical fibers and metamaterials for integrated photonics.
• Engineering at attosecond and nanojet scales for high-resolution optical delivery.

Key limitations of Bessel-like needle beams stem from finite aperture truncation (limiting propagation-invariant zone), material and angular dispersion in refractive optics (especially for broadband/ultrashort pulses), nonlinear beam distortions at high fluence, and sidelobe energy (addressed by advanced apodization or pupil filtering strategies) (Sheppard, 8 Sep 2025, Stoian et al., 2018).

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References

• (Grunwald et al., 3 Dec 2025) Needle beams and structured space-time wavepackets
• (Sheppard, 8 Sep 2025) Bessel beams, propagationally invariant beams, and axicons: An historical and tutorial review
• (Stoian et al., 2018) Ultrafast Bessel beams; advanced tools for laser materials processing
• (Jenne et al., 2020) High-quality Tailored-edge Cleaving Using Aberration-corrected Bessel-like Beams
• (Meyer et al., 2019) Extremely high-aspect-ratio ultrafast Bessel beam generation and stealth dicing of multi-millimeter thick glass
• (Demirci et al., 2019) Direct micro-structuring of Si(111) surfaces through nanosecond-laser Bessel beams
• (Efremidis, 2017) Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type
• (Mansourimanesh et al., 13 Aug 2025) Formation of axially modulated plasma strings by filamentation of interfering femtosecond Bessel beams