Nonlinear Bessel Vortex Beams

• Nonlinear Bessel vortex beams are propagation-invariant optical beams that combine the conical structure of Bessel beams with nonlinear phase and absorption effects to yield intrinsic orbital angular momentum.
• Their stability is achieved through a self-consistent spiral inward energy flux and balance of Kerr nonlinearity with multiphoton absorption, as modeled by the nonlinear Schrödinger equation.
• They enable practical applications in long-range optical guiding, high-fidelity OAM transfer, and passive beam cleaning, with implementations in nonlinear materials processing and optical manipulation.

Nonlinear Bessel vortex beams are a class of propagation-invariant optical beams that combine the nondiffracting, conical wave structure of Bessel beams with nonlinear phase and absorption effects in optical media, exhibiting intrinsic orbital angular momentum (OAM) through embedded phase singularities. These beams, when propagating in nonlinear media (Kerr-type with multi-photon absorption, or quadratic media for harmonic generation), present a rich phenomenology including the formation of stationary bound states, nonlinear stabilization against modulational instabilities, robust OAM transfer, and novel soliton-like behavior.

1. Mathematical Formulation and Nonlinear Propagation

The canonical description of nonlinear Bessel vortex beams (NBVBs) employs the paraxial nonlinear Schrödinger equation (NLSE) with embedded vorticity and nonlinear absorption: $$\partial_z A = \frac{i}{2k}\nabla_\perp^2 A + i\frac{k n_2}{n} |A|^2 A - \frac{\beta^{(M)}}{2} |A|^{2M-2} A$$ where $A(r, \varphi, z)$ is the slowly varying envelope, $k$ the linear wavenumber, $n_2$ the Kerr nonlinear index, and $\beta^{(M)}$ the M-photon absorption coefficient (Porras et al., 2017, Porras et al., 2016, Riquelme et al., 2018, Porras et al., 2014). The ansatz for a stationary NBVB of topological charge $m$ is: $$A(r, \varphi, z) = a(r) e^{i m\varphi} e^{i \delta z}$$ with an amplitude $a(r)$ and propagation constant shift $\delta < 0$, leading to an ODE for $a(r)$ that encapsulates linear, nonlinear, and absorptive effects.

For quadratic (χ^2) nonlinear media undergoing second harmonic generation (SHG), coupled NLSEs for the fundamental and harmonic fields describe the nonlinear wave mixing and resultant topological configurations (Buldt et al., 2021).

2. Structure, Energy and OAM Conservation, and Reservoir Dynamics

In the presence of nonlinear absorption, NBVBs maintain stationary profiles by supporting a spiral-inward flux of energy and OAM from an “intrinsic reservoir” in their outer rings. The continuity equation connecting the intensity $I = |A|^2$ and the radial current $j_r(r)$ is: $$\partial_z I + \nabla_\perp \cdot \mathbf{j} = -\beta^{(M)} I^M$$ with the negative divergence of the spiral current exactly compensating the multiphoton loss in the nonlinear core (Porras et al., 2014, Porras et al., 2017). The same mechanism sustains OAM, enabling stable propagation and continuous OAM transfer even under strong absorption.

Explicitly, in the far field NBVBs are represented as an “unbalanced” superposition of Hankel functions: $$A(r) = \frac{1}{2} \left[b_{\rm in} H_m^{(2)}(\kappa r) + b_{\rm out} H_m^{(1)}(\kappa r)\right]$$ where $b_{\rm in} > b_{\rm out}$ and the difference in their intensities matches exactly the nonlinear loss over the transverse plane: $$|b_{\rm in}|^2 - |b_{\rm out}|^2 = \int_0^\infty 2\pi r \beta^{(M)} |A|^{2M} dr$$ (Porras et al., 2014, Porras et al., 2016, Porras et al., 2017).

3. Stability and Nonlinear Attractor Behavior

NBVBs, unlike vortex solitons (which demand precise balance of self-focusing/defocusing and diffraction), achieve stability through the dissipative action of nonlinear absorption, which suppresses azimuthal modulational instabilities and beam collapse (Porras et al., 2017, Porras et al., 2016, Riquelme et al., 2018). Linear stability analysis reveals that stable NBVBs exist for parameter regions defined by the ratio of nonlinear to absorptive terms (e.g., Kerr parameter α below a critical threshold for a given M and cone angle). Multi-charged NBVBs ($m > 1$) also possess stability windows, and the onset of instability is characterized by the growth rate of azimuthal perturbation modes computed from the non-Hermitian spectrum of the linearized equations.

These properties explain the appearance of robust tubular filamentation, rotating necklaces, and speckle-like breakup regimes in high-intensity experiments with axicon-generated Bessel beams (Porras et al., 2016, Porras et al., 2017).

4. Vortex Conversion, Annihilation, and Topological Control

A unique property of NBVBs in nonlinear absorbing media is their ability to trap, combine, and annihilate nested vortices of various topological charges. Vortices initially embedded off-axis within a nonlinear Bessel beam drift inwards due to the negative radial phase gradient, whereupon they merge and the beam undergoes a mode conversion, resulting in a stationary NBVB whose total charge is the algebraic sum of incident and trapped charges: $m_{\rm final} = n_{\rm initial} + s$ where $n_{\rm initial}$ is the NBVB charge and $s$ the sum of nested vortex charges (Riquelme et al., 2018). This process is robust, order-preserving, and statically stable, enabling passive generation of multiply charged vortex beams, fast annihilation of dipoles, and all-optical N-to-1 combiners for OAM-guided information channels.

The phenomenon provides a powerful method for “beam cleaning,” as randomly distributed topological defects in speckled backgrounds are automatically attracted, fused, and annulled in the nonlinear Bessel zone (Riquelme et al., 2018).

5. Nonlinear Generation and Harmonic Conversion of Bessel Vortex Beams

Efficient nonlinear frequency conversion of Bessel-Gauss vortex beams, especially for high-order or “perfect” vortices (annular beams with order-independent radius), leverages quasi-phase-matched (QPM) media (e.g., chirped MgO:PPLN) to realize high single-pass SHG efficiencies with OAM preservation (Chaitanya et al., 2016). In these schemes, a spiral phase plate and axicon produce a Bessel-Gauss vortex, which, after Fourier transformation by a lens, yields a perfect vortex with ℓ-invariant annular geometry. The SHG process in QPM crystals produces second-harmonic vortices whose ring structure and OAM integrity are preserved up to order 12, with conversion efficiencies up to 27%.

Such nonlinear upconversion avoids mode degradation and singularity splitting common in birefringent phase-matched crystals, instead yielding output with single well-defined OAM singularities (Chaitanya et al., 2016).

6. Topological Soliton Lattices and Second-Harmonic Vector Solitons

In quadratic nonlinear media, the interplay of Bessel-like conical input and phase-matched SHG enables the spontaneous emergence of topological vector solitons in the second harmonic field. The resulting patterns can exhibit central solitons, multiple knotted cores, or linked solitons surrounded by ellipsoidal rings, with topology and spatial geometry set by the fundamental beam’s charge (Buldt et al., 2021, Buldt et al., 2022). These soliton cores behave analogously to screw dislocations and display particle-like invariant propagation characteristics, including 90° scattering interactions reminiscent of O(3) σ-model vortex-bion solutions.

Such structures are robust over meter-scale propagation distances, with their topology and phase properties fully controlled by the parameters of the input Bessel beam and the nonlinear medium.

7. Applications and Physical Significance

Nonlinear Bessel vortex beams have direct implications for:

• Long-range optical guiding via stable filaments with OAM (Porras et al., 2017, Porras et al., 2016)
• High-fidelity OAM transfer for optical manipulation, quantum information, and microscopy (Chaitanya et al., 2016, Porras et al., 2017)
• Passive beam cleaning and topological defect management without optical feedback or alignment tolerances (Riquelme et al., 2018)
• Nonlinear materials processing with ring-shaped or helical ablation profiles (Porras et al., 2017)
• Exploration of integrable soliton dynamics, nonlinear collapse arrest, and singular optics in dissipative environments (Buldt et al., 2021, Buldt et al., 2022)

The experimental realization of these states spans from femtosecond-pulse-driven nonlinear crystals to structured wavefront engineering with spatial light modulators, providing a versatile platform for both applied and fundamental studies in nonlinear and singular optics.

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For direct equations, parameter regimes, and detailed dynamical scenarios, refer to the original research articles: (Porras et al., 2017, Porras et al., 2016, Porras et al., 2014, Riquelme et al., 2018, Chaitanya et al., 2016, Buldt et al., 2021, Buldt et al., 2022).