Bessel-Gauss Beams: Properties & Applications

Updated 10 September 2025

Bessel-Gauss beams are paraxial optical fields defined by modulating a Bessel beam with a Gaussian envelope, providing finite-energy and approximate non-diffracting behavior.
They are generated using methods such as axicon illumination, annular apertures, and refractive index mapping, offering tunable beam profiles and efficient energy concentration in the central core.
Their high beam quality, self-healing nature, and robustness in various optical, quantum, and nanofabrication applications make them a pivotal tool in modern photonics.

Bessel-Gauss beams are paraxial optical fields formed by modulating an ideal Bessel beam with a Gaussian envelope. This construction yields a physically realizable, finite-energy beam exhibiting approximate non-diffracting and propagation-invariant properties over a limited distance. Bessel-Gauss beams feature a central lobe surrounded by concentric rings of decreasing intensity, and the Gaussian apodization ensures rapid intensity decay at large radii, circumventing the infinite power requirement of an ideal Bessel beam. Their unique combination of high beam quality, extended depth of focus, and tunable parameters underpins their widespread utility across optical, quantum, and nonlinear science.

1. Mathematical Formulation and Physical Properties

The canonical form of a Bessel-Gauss (BG) beam, for zeroth order ( $n=0$ ) and in paraxial approximation, is given in cylindrical coordinates as: $E_\mathrm{BG}(r,z) = \frac{2}{\alpha(z)} \exp\left(\frac{\chi^2}{\alpha(z)}\right) J_0\left(\frac{2\chi r}{\alpha(z)}\right) \exp\left(-\frac{r^2}{\alpha(z)}\right)$ with

$J_0$ — zeroth-order Bessel function,
$r$ — radial coordinate,
$\chi$ — parameter setting the transverse scale/dilation, related to the semi-aperture or cone angle,
$\alpha(z)=w_0^2+2i z/k$ — complex beam parameter, combining Gaussian waist $w_0$ , propagation distance $z$ , and wave number $k$ .

The BG beam interpolates between a Gaussian ( $\chi=0$ ) and an ideal Bessel beam ( $w_0\rightarrow\infty$ ). The beam's propagation-invariant core endures over the so-called “Bessel zone,” beyond which Gaussian diffraction dominates. The construction generalizes naturally to arbitrary order $n$ (azimuthal index), yielding beams with helical phase (optical angular momentum) and on-axis intensity nulls.

2. Generation Mechanisms and Implementation

Physical realization of BG beams exploits the modulation of a Bessel beam by a Gaussian profile—typically achieved in one of the following architectures:

Axicon illuminated by a Gaussian beam: A conical lens (axicon) imparts the necessary conical phase for Bessel function formation. The input Gaussian truncates the infinite side lobes, yielding a finite Bessel zone. The axicon base angle $\alpha$ and initial beam waist $w_0$ dictate the central core diameter and propagation length (Sheppard, 8 Sep 2025).
Annular apertures and resonators: Placement of an annular pupil in a lens's focal plane, or use of annular-toroidal resonators, provides a spectral filtering approach particularly suited to mode generation in scanning microscopy (Sheppard, 8 Sep 2025).
Heterogeneous refractive index mapping: Integrated planar lightwave circuits can transform a Gaussian input into a BG mode by implementing a numerically optimized 2D graded index region, achieving up to 95% energy in the main lobe and retaining self-healing and non-diffracting properties (Alerigi et al., 2012).

For computational modeling, BG beams can be constructed analytically via Hankel transform methods (Radożycki, 2021), or simulated in truncated systems with efficient series representations based on BG mode superpositions, facilitating high-precision calculations for finite apertures in optical or acoustic domains (Zamboni-Rached et al., 2012).

3. Beam Quality and Coherence Properties

Bessel-Gauss beams are characterized by low divergence and a near-Gaussian main lobe in the far field, conferring high spatial beam quality. The divergence angle can be as small as 0.1 $^\circ$ in optimized high-harmonic generation setups, enabling tightly focused beams for precision spectroscopy (Wang et al., 2011).

Algebraic analysis (SU(1,1) group) situates BG modes as minimum-uncertainty coherent states within the Laguerre-Gauss basis, with their quality factor $M^2$ linked to the order $\ell$ and the “coherent state” parameter $\tau$ . Only the fundamental BG mode ( $\ell=0$ , $\tau=0$ ) reaches the diffraction limit, while OAM-carrying modes intrinsically incur higher divergences (CruzyCruz et al., 2023). The periodic self-focusing and structure revivals observed in parabolic gradient-index media further distinguish their coherence and propagation dynamics.

4. Polarization, Nonparaxiality, and Beam Engineering

Vector extensions of BG beams allow construction of cylindrically polarized modes—radial and azimuthal—by appropriate superpositions of scalar beams with azimuthal numbers $\pm1$ (Madhi et al., 2014). The paraxial approximation remains extremely accurate for BG beams; nonparaxial corrections remain negligible (relative error $\sim10^{-6}$ ) even when the Bessel aperture angle and Gaussian divergence are comparable.

Nonparaxial generalizations and accelerating variants adopt engineered input phase profiles, enabling beams whose Bessel-like cross-section remains invariant even as the main lobe follows curved or nonplanar trajectories. Such “accelerating Bessel-like beams” maintain subwavelength core widths and non-diffracting character along arbitrary paths for advanced optical manipulation (Chremmos et al., 2013, Corato-Zanarella et al., 2017).

Advanced superposition strategies—integrating over dilation $\chi$ or OAM indices $n$ —yield a hierarchy of paraxial beams, including power-law and special hyperbolic Bessel-Gauss (SHBG) modes, providing finely tunable intensity and phase structures useful for tailored beam shaping, trapping, and nonlinear optics (Radożycki, 2023, Radożycki, 2023).

5. Robustness and Nonlinear/Quantum Applications

Experimental, theoretical, and numerical studies have demonstrated that BG beams possess several robust and application-critical features:

Non-diffracting and self-healing: The Bessel component ensures that the central core reconstructs after encountering obstructions or propagation through scattering media. For example, efficient recovery of the main lobe below 0.5% distortion occurs after passing through opaque obstacles (Alerigi et al., 2012).
Diffusion invariance: In atomic media subject to coherent diffusion, the entire complex BG mode (amplitude, phase) endures, with no broadening beyond uniform amplitude decay. This property is unique to propagation-invariant (narrow ring-spectrum) solutions and is absent for standard Gaussian beams (Smartsev et al., 2020).
High-fidelity quantum entanglement: BG modes provide a continuous radial parameter (via scaling variable $k_r$ ) for OAM entanglement, enabling broader, flatter OAM spectra and high-dimensional, maximally entangled photon pairs. Experiments confirmed robust violations of the CHSH inequality (e.g., $S=2.78\pm0.05$ ) and high state fidelities $F=0.96$ in BG-based OAM-entangled photons (McLaren et al., 2012).

BG beams have been applied to the generation of isolated attosecond pulses via high-order harmonic generation, where their low divergence and favorable phase-matching properties in neutral rare gases yield efficient, well-confined supercontinua (attosecond pulse divergence $\sim$ 0.1 $^\circ$ ) (Wang et al., 2011, Davino et al., 2021). By scanning the gas target across the oscillatory focus of the BG beam, one can optimize phase-matching, ionization fraction, and XUV emission volume for improved attosecond yield and spatial filtering.

6. Micro- and Nanofabrication Applications

BG beams confer notable advantages in high-precision materials processing. In laser ablation and nanomachining of Si and black silicon, the central core's tight focus and extended axial reach enable ablation widths at or below the theoretical FWHM spot (as small as 5 μm), reduced heat-affected zones, and high aspect ratio grooves, with thresholds up to 50 $\times$ lower than those of Gaussian beams (Demirci et al., 2019, Zheng et al., 8 May 2025). For black-Si, the combination of anti-reflective, nanotextured surfaces and the high-aspect-ratio, extended focal region of the Gauss-Bessel beam led to precise modification at extremely low fluences and aspect ratios up to 8. This enables fabrication of deep sub-wavelength features and efficient surface nanostructuring.

7. Applications, Limitations, and Future Prospects

Bessel-Gauss beams are central in ultrafast spectroscopy, optical trapping, high-resolution microscopy, and device fabrication, including chip-scale photonic components and noninvasive biomedical imaging. Their extended depth of focus, tunable propagation, and robust self-healing enable manipulation and interrogation of matter at microscopic and attosecond time scales. Quantum information protocols using BG modes promise high-dimensional encoding and enhanced state purity.

Limitations include the finite propagation-invariance (bounded Bessel zone), energy spread into rings (reducing central core efficiency compared to Gaussian beams), and trade-offs between spot size and longitudinal extent. Recent algebraic methods, index-mapping converters, and generalized mode constructions suggest ongoing advances in beam shaping and integration, enhancing applicability in photonics, quantum optics, and laser–matter interaction. Extending analytic and computational frameworks to nonparaxial and vectorial regimes, as well as further exploiting BG beams' diffusion invariance and structured entanglement, represent continuing directions for foundational and applied research.