Particle-like topologies of light in turbulent complex media

1. Problem Statement

Optical singularities—specifically phase vortices (scalar) and polarization singularities (vector)—serve as fundamental building blocks for structured light, carrying information via Orbital Angular Momentum (OAM) and complex topological textures. A longstanding debate exists regarding the relative stability of these two types of singularities when propagating through complex, random media like atmospheric turbulence.

• The Conflict: One perspective suggests phase singularities are more robust due to the quantized nature of phase winding, which prevents continuous elimination. Conversely, another view argues that polarization textures might persist better if the medium scrambles spatial modes while leaving the polarization sector coherent.
• The Gap: Previous studies often yielded contradictory conclusions because they relied on different metrics for stability (e.g., defect location vs. OAM spectrum) and lacked a controlled comparison where both singularity types experienced identical perturbations. The specific question of whether phase and polarization vortices exhibit equivalent wandering statistics in a one-sided complex channel (affecting only spatial degrees of freedom) remained unresolved.

2. Methodology

The authors employed a rigorous combination of numerical simulations and controlled experiments using atmospheric turbulence as the testbed.

• Beam Generation:
  • Phase Singularity (Scalar): A single Laguerre-Gaussian (LG) mode with azimuthal index $\ell$ and radial order $p=0$, encoded in a fixed polarization state ($\psi_{ph} = u_{LG}^\ell \hat{e}$).
  • Polarization Singularity (Vector): A coherent superposition of a fundamental Gaussian mode ($\ell=0$) in the horizontal polarization and an LG vortex mode ($\ell$) in the vertical polarization ($\psi_{pol} = u_{LG}^0 \hat{e}_H + u_{LG}^\ell \hat{e}_V$). This creates a polarization singularity (C-point) where the local polarization state becomes undefined.
• Turbulence Simulation:
  • Theoretical Model: Atmospheric turbulence was modeled as a one-sided channel ($1_A \otimes T_B$), where $A$ is the polarization degree of freedom (DoF) and $B$ is the spatial DoF. Since air is non-birefringent, the channel operator $T_B$ acts identically on both polarization components, leaving the polarization DoF ($A$) unaltered.
  • Numerical: Kolmogorov turbulence models were used to generate random phase screens.
  • Experimental: A Spatial Light Modulator (SLM) encoded phase screens onto the beams. To ensure identical perturbation, the same phase screen was applied to both components of the vector beam.
• Measurement & Analysis:
  • Scalar Fields: Measured via interferometry with a reference beam to reconstruct complex amplitude and phase.
  • Vector Fields: Measured using Stokes polarimetry (6 projections) to reconstruct the polarization phase.
  • Metric: The primary metric was the positional deviation radius ($\sigma$) of the singularity. The authors tracked the "wandering" of singularities across an ensemble of 500 turbulence realizations, calculating the variance of the singularity position relative to the beam's center of mass.

3. Key Contributions

• Theoretical Proof of Equivalence: The authors analytically demonstrated that in a one-sided channel (non-birefringent), the position of a polarization singularity is strictly locked to the phase singularity of the underlying scalar component carrying the OAM. Since the Gaussian component (reference) remains non-singular, the polarization singularity inherits the exact wandering statistics of the scalar LG vortex.
• Experimental Benchmarking: They established the first direct experimental comparison showing that phase and polarization vortices exhibit identical dynamics under identical turbulent conditions.
• Resolution of Ambiguity: The study clarifies that the perceived difference in stability in previous literature arises from the choice of metric and the specific nature of the perturbation, rather than an intrinsic difference in the robustness of the topological descriptors themselves.

4. Key Results

• Identical Wandering Statistics: For topological charges $\ell = 1, 2, 3$, the positional deviation radius ($\sigma$) for both scalar and vector vortices broadened monotonically with increasing turbulence strength ($D/r_0$). The curves for phase and polarization singularities overlapped almost perfectly.
• Charge Dependence: Higher-order vortices ($\ell > 1$) showed larger deviation radii due to the instability of multi-charge cores, which break into unit-charge singularities. This effect was identical for both scalar and vector cases.
• Beam Size Normalization: When beam waists were corrected to decouple the mode size from the topological charge ($w_0 \to w_0/\sqrt{|\ell|+1}$), the dependence of wandering on $\ell$ vanished. This confirmed that turbulence-induced OAM crosstalk is driven by the ratio of beam size to turbulence scale, not the charge itself.
• Quantitative Agreement: A linear fit comparing the deviation radii of polarization vs. phase vortices yielded a slope of 0.8291 (experiment) vs. 0.9971 (theory). The slight deviation is attributed to experimental noise and finite ensemble size, but the trend confirms statistical equivalence.

5. Significance and Implications

• Design of Robust Optical Links: The findings provide a quantitative basis for selecting structured light encodings for free-space optical communication. Engineers can now choose between phase or polarization encoding based on other system constraints (e.g., detector complexity) without worrying about a fundamental loss of topological stability in non-birefringent turbulence.
• Sensing and Metrology: The results validate the use of polarization singularities as reliable markers for environmental sensing and turbulence-aware metrology. Since they behave identically to phase singularities, they offer an alternative degree of freedom for tracking in noisy environments.
• Theoretical Clarity: The work resolves a half-century-old debate regarding the resilience of different optical topologies, establishing that in non-birefringent media, the "particle-like" nature of singularities is governed by the spatial perturbation of the shared field component, regardless of whether the singularity is defined by phase or polarization.

In conclusion, the paper demonstrates that polarization and phase vortices are dynamically equivalent in one-sided turbulent channels, providing a unified framework for understanding and harnessing optical topologies in complex media.