Decoupling of topology and texture in optical skyrmions under turbulence

1. Problem Statement

Topological invariants in photonics are often touted as robust mechanisms for information encoding in disordered environments (e.g., atmospheric turbulence). The prevailing assumption is that if a global topological invariant (like the skyrmion number, $N_{sk}$) is conserved, the underlying field structure remains intact and usable.

However, this paper addresses a critical gap: Does the conservation of a global topological index guarantee the preservation of the local, fine-grained polarization texture required for practical information encoding? The authors investigate whether optical skyrmions—vectorial light fields characterized by a global skyrmion number—can maintain their local structural fidelity (texture) when propagating through realistic turbulent channels, or if the global topology and local texture become decoupled.

2. Methodology

The study employs a multi-faceted approach combining controlled experiments, numerical simulations, and analytical modeling to independently track global and local observables.

• Experimental Setup:

  • Generation: Optical skyrmions were generated by superposing a fundamental Gaussian mode and a Laguerre-Gaussian (LG) mode with orthogonal circular polarizations (Right- and Left-Circular Polarization). The relative weights and phase controlled the skyrmion type (Néel, Bloch, or anti-skyrmion) and order.
  • Turbulence Emulation: A hot-air turbulence chamber was used to create a dynamically evolving, distributed refractive-index fluctuation environment. The turbulence strength was calibrated against the Kolmogorov parameter ($C_n^2$) and the Rytov variance ($\sigma_R^2$).
  • Propagation: The beam traversed a 270-meter equivalent channel (achieved via a 1.5m chamber with multiple reflections) at a wavelength of 532 nm.
  • Detection: A polarization-resolved camera system captured single-shot full Stokes field measurements ($S_0, S_1, S_2, S_3$) to reconstruct the vector field.

• Numerical Simulations:

  • Split-step propagation simulations were performed using statistically independent phase screens derived from the Kolmogorov spectrum.
  • The simulations matched experimental parameters (beam waist, wavelength, turbulence strength) to validate the analytical models.

• Analytical Modeling:

  • The authors developed a scaling model treating cumulative distortions as a multiplicative modulation of the topological density field.
  • They utilized the Rytov-Tatarski description of turbulence-induced field fluctuations to derive exponential decay laws for skyrmion number fidelity as a function of topological order.

• Metrics Defined:

  • Global Metric: The Skyrmion Number ($N_{sk}$), calculated via the integral of the topological density over the beam cross-section.
  • Local Metrics:
    • Polarity ($P_{sk}$): Out-of-plane orientation near the core vs. boundary.
    • Vorticity ($V_{sk}$): Integer winding of the in-plane polarization angle.
    • Helicity ($H_{sk}$): Orientation of in-plane rotation relative to the radial direction.

3. Key Contributions

1. Demonstration of Decoupling: The primary contribution is the experimental and theoretical proof that global topological invariance and local texture fidelity are decoupled under turbulence. The global skyrmion number ($N_{sk}$) can remain conserved while the local polarization texture (polarity, vorticity, helicity) rapidly degrades.
2. Hierarchy of Robustness: The study establishes a hierarchy where global invariants are more robust than local texture-resolved information. This hierarchy is exacerbated for higher-order skyrmions.
3. Analytical Scaling Law: The authors derived an analytical model predicting that the fidelity of the skyrmion number decays exponentially with the square of the topological order ($N_0^2$) under turbulence, formalizing the limits of topological protection.
4. Statistical Framework for Helicity: A novel statistical approach using the von Mises distribution was introduced to quantify the degradation of helicity (in-plane orientation), distinguishing between a preserved mean direction and a loss of circular coherence.

4. Key Results

• Global vs. Local Stability:

  • For first-order skyrmions ($N_{sk}=1$), the global skyrmion number remained stable across a broad range of turbulence strengths.
  • Conversely, local metrics ($P_{sk}, V_{sk}, H_{sk}$) degraded rapidly. The polarity became ambiguous due to core displacement; vorticity suffered from mode mixing (crosstalk between integer windings); and helicity lost its defined orientation as the angular distribution broadened.
  • Conclusion: A field can be topologically nontrivial (preserving $N_{sk}$) while carrying almost no usable texture information.

• Order-Dependent Degradation:

  • As the topological order ($N_{sk}$) increased (up to $N_{sk}=5$), the robustness of the global invariant itself diminished.
  • Higher-order textures exhibited faster loss of a well-defined skyrmion number, characterized by broadened distributions and increased variance.
  • The ensemble average $\langle N_{sk} \rangle$ decayed and the variance $\text{Var}(N_{sk})$ grew exponentially with $N_0^2$, following the model:
    $$\langle N_{sk} \rangle = N_0 \exp(\Lambda_{eff} N_0^2)$$
    $$\text{Var}(N_{sk}) = N_0^2 \exp(\Gamma_{eff} N_0^2)$$
    (where $\Lambda_{eff} < 0$ and $\Gamma_{eff} > 0$).

• Experimental Validation: The experimental data matched the numerical simulations and the analytical scaling model, confirming that the decoupling is an intrinsic property of wave propagation in stochastic media, not an artifact of measurement.

5. Significance

• Redefining Topological Protection: The paper challenges the notion that topology alone guarantees disorder resilience. It clarifies that while topological invariants may survive, the information encoded in the local texture (which is necessary for classification, tracking, and high-dimensional encoding) is fragile.
• Practical Implications for Optical Communications: For applications relying on skyrmions for data transmission (e.g., free-space optical communication), relying solely on the conservation of the skyrmion number is insufficient. Systems must account for the rapid degradation of local texture metrics, especially for higher-order modes.
• Fundamental Physics: The work draws parallels between optical skyrmions and other topological systems (like Bose-Einstein condensates and liquid crystals), suggesting a universal limitation where increasing topological order does not linearly increase protection against stochastic perturbations.
• Design Guidelines: The findings provide a quantitative framework for determining the maximum topological order and turbulence strength under which skyrmion-based systems can operate reliably, defining the intrinsic limits of topological protection in disordered wave systems.