Pseudovorticity of 2+1D optical solitons

Technical Summary: Pseudovorticity of 2+1D Optical Solitons

Problem Statement
In the hydrodynamic representation of optical fields, vorticity typically vanishes where the phase is well-defined and is localized only at phase singularities (quantum vortices). However, a distinct quantity known as pseudovorticity characterizes local rotational structures even in the absence of singularities or net orbital angular momentum (OAM). While pseudovorticity has been established as a diagnostic tool for quantum vortices in superfluids and for optical torque in linear wave superpositions, its role in vortex-free nonlinear wavefields remains largely unexplored. Specifically, it is unclear whether conventional bright optical solitons—self-trapped wave packets where phase and amplitude are intrinsically coupled—naturally exhibit pseudovorticity, and if so, what structural hierarchy emerges from their phase-amplitude dynamics.

Methodology
The authors employ a combined experimental and numerical approach to investigate pseudovorticity in bright 2+1D optical solitons propagating in photorefractive crystals.

• Theoretical Framework: The study utilizes the Madelung transform, defining the optical field as $\psi = \sqrt{\rho}e^{i\phi}$. Pseudovorticity ($\omega_{ps}$) is defined as the curl of the current density $\mathbf{j} = \rho \nabla \phi$, yielding $\omega_{ps} = \nabla \rho \times \nabla \phi$. The authors analyze perturbations of the phase field $\phi$ around a stable, radially symmetric soliton solution, considering linear (dipolar) and quadratic (quadrupolar) phase terms.
• Experimental Setup: Experiments are conducted in a KTN:Li photorefractive crystal ($K_{0.964}Li_{0.036}Ta_{0.60}Nb_{0.40}O_3$) maintained in the paraelectric phase (9K above the Curie point). A 532 nm x-polarized Gaussian beam propagates along the z-axis under a static bias field. The system operates in both saturated and unsaturated (Kerr-like) regimes.
• Measurement and Reconstruction: The complex transverse field distribution at the output facet is retrieved using off-axis interferometry and a spatial-frequency filtering algorithm. This allows for the simultaneous reconstruction of amplitude ($\rho$) and phase ($\phi$), from which the pseudovorticity distribution is calculated.
• Numerical Simulation: The authors simulate the propagation using the 2+1D generalized nonlinear Schrödinger equation (NLSE) with a saturable nonlinearity, modeling both single soliton propagation and soliton-soliton fusion.

Key Contributions and Results
The study demonstrates that stable 2+1D optical solitons naturally carry a hierarchy of pseudovorticity multipoles, encoded in the local perturbation of the phase and amplitude.

1. Dipolar Pseudovorticity in Stable Solitons:
   For input conditions lying on the soliton existence curve (both in saturated and Kerr-like regimes), the resulting stable solitons exhibit a dipolar pseudovorticity structure. This dipole arises from a linear phase gradient (transverse velocity) associated with the self-bending of the beam due to the nonlinear Raman effect. The pseudovorticity is positive on one side of the soliton center and negative on the other, with a nodal line aligned with the direction of motion.

2. Quadrupolar Pseudovorticity in Perturbed States:
   When the input beam deviates from the stationary soliton state (e.g., conditions above the existence curve where self-focusing dominates diffraction), the phase-amplitude mismatch becomes more complex. This results in a quadrupolar pseudovorticity pattern. Theoretical analysis shows that second-order phase perturbations (elliptical or hyperbolic phase contours) naturally generate this four-lobed structure with alternating signs in each quadrant.

3. Soliton Fusion Dynamics:
   The authors investigate the collision and fusion of two solitons. In the linear regime, no defined pseudovorticity is observed. As nonlinearity increases, the two individual dipolar structures (associated with the separate solitons) merge. Upon fusion, the system transitions into a well-defined quadrupolar pseudovorticity pattern. This is interpreted as the superposition of two oppositely oriented dipoles, consistent with the geometric symmetry of the fused state.

4. Geometric Interpretation:
   The paper provides a geometric explanation for these findings: pseudovorticity arises wherever the intensity gradient ($\nabla \rho$) and phase gradient ($\nabla \phi$) are non-collinear. In stable solitons, the intrinsic coupling between amplitude and phase creates a specific misalignment that manifests as a dipole. Deviations from the ideal soliton profile introduce higher-order curvature in the phase, leading to quadrupoles.

Significance and Claims
The paper claims to provide the first direct experimental and numerical evidence of pseudovorticity in bright 2+1D optical solitons. Its primary significance lies in revealing a "hidden dynamical structure" in high-dimensional solitons that is not captured by conventional intensity or phase descriptions alone.

• Unified Framework: Pseudovorticity offers a unified framework to characterize the phase-amplitude coupling in nonlinear waves, suggesting that stable high-dimensional solitons naturally carry a hierarchy of multipoles.
• Optical Torque Implications: The authors note that these structures imply that solitons can exert optical torque on absorbing particles even in the absence of topological charge (OAM). Unlike OAM, which induces global rotation, pseudovorticity creates a local torque field capable of inducing rotations in different directions depending on the particle's position relative to the soliton's multipolar structure.
• Limitations: The analysis is currently restricted to the z-component of pseudovorticity (rotations in the transverse plane) under paraxial conditions. The authors acknowledge that full vectorial effects remain to be explored.

The work establishes pseudovorticity as a fundamental property of the self-trapped state in nonlinear media, arising naturally from the interplay between nonlinearity and perturbations, rather than requiring externally imposed phase singularities.