Unfolding unstable skyrmionic polarization textures

1. Problem Statement

Optical skyrmions are topologically non-trivial polarization textures in light, analogous to magnetic skyrmions in condensed matter. A widely accepted theoretical framework for "Full Poincaré beams" (Stokes skyrmions) suggests that the skyrmion number ($Q_{sk}$) of a superposition of two vortex beams with topological charges $\ell_1$ and $\ell_2$ is given by the difference of their charges:
$$Q_{sk} = \ell_2 - \ell_1$$
However, this formula relies on a critical, often unspoken assumption: that the phase singularities (zeros) of the two orthogonal polarization components coalesce at the exact same spatial point ($r=0$).

The authors identify a fundamental instability in this assumption. In a 2D transverse plane, the coalescence of two first-order zeros from orthogonal polarization states is a codimension-4 event. This means it requires four specific real conditions to be met simultaneously, making it an unstable configuration. Any arbitrarily small perturbation (e.g., aberrations, noise, or slight misalignment) will cause these singularities to split. The paper addresses the question: What happens to the topological charge (skyrmion number) when these unstable singularities split?

2. Methodology

The authors employ a combination of rigorous topological theory, numerical simulation, and experimental verification.

• Theoretical Framework:

  • They model the vector wave function $\Psi(r)$ as a superposition of two Laguerre-Gaussian (LG) vortex beams with charges $\ell_1$ and $\ell_2$ and polarization states $u_1$ and $u_2$.
  • They introduce an arbitrary weak perturbation $\tilde{\epsilon}$ to simulate the splitting of singularities.
  • They define the skyrmion number via a solid angle integral over the Poincaré sphere (PS).
  • Crucially, they reformulate the skyrmion number calculation using a line integral (Stokes' theorem approach) around the singularities. By treating the zeros (poles) of the numerator and denominator of the complex polarization ratio independently, they derive a general expression for $Q_{sk}$ that does not assume coalescence.

• Numerical Simulation:

  • Simulations were performed using LG vortices with varying radial indices.
  • Perturbations were introduced by adding weak Gaussian beams to the vortex components.
  • The authors analyzed the effect of intensity thresholding (ignoring low-intensity regions) and spatial undersampling (low-resolution meshes) on the calculated skyrmion number.

• Experimental Setup:

  • A Sagnac interferometer equipped with a Spatial Light Modulator (SLM) was used to generate arbitrary polarization textures.
  • An incident linearly polarized beam ($\lambda = 780$ nm) was split into counter-propagating orthogonal components, modulated independently in amplitude and phase, and recombined.
  • Stokes parameters were reconstructed using a quarter-wave plate, linear polarizer, and a 12-bit CMOS camera to measure the polarization texture under varying perturbation strengths.

3. Key Contributions

• Correction of the Skyrmion Number Formula: The paper demonstrates that the standard formula $Q_{sk} = \ell_2 - \ell_1$ is only valid in the idealized, unperturbed case where singularities coalesce.
• Discovery of the "Unfolding" Mechanism: When singularities split due to even infinitesimal perturbations, the skyrmion number changes abruptly. The new, stable skyrmion number is determined solely by the highest-order topological charge:
  $$Q_{sk} = \max(|\ell_1|, |\ell_2|)$$
• Explanation of Discrepancies: The authors explain why the "difference" formula ($\ell_2 - \ell_1$) is frequently observed in literature and experiments. It arises as an artifact of:
  1. Intensity Thresholding: Excluding low-intensity regions (where the new singularities and rapid polarization changes occur) during numerical integration.
  2. Undersampling: Failing to resolve the fast polarization variations near the split singularities with a low-resolution mesh.

4. Results

• Theoretical Derivation: Using Cauchy's argument principle and line integrals, the authors proved that for a superposition of two vortices with $|\ell_2| > |\ell_1|$, the skyrmion number in the presence of splitting is $Q_{sk} = \ell_2$. This holds regardless of the orthogonality of the polarization states.
• Numerical Evidence:
  • In unperturbed simulations ($\epsilon = 0$), integrating over a disk avoiding the origin yields $Q_{sk} = \ell_2 - \ell_1$.
  • When a perturbation splits the singularities, the calculated $Q_{sk}$ jumps to $\ell_2$.
  • The simulation confirmed that if the integration excludes the low-intensity region where the new singularity resides (via thresholding), the result artificially reverts to $\ell_2 - \ell_1$.
• Experimental Verification:
  • Experiments with vortex superpositions (e.g., $\ell_1=1, \ell_2=3$) showed that as perturbation strength increased, the measured skyrmion number jumped from $\approx 2$ ($\ell_2-\ell_1$) to $\approx 3$ ($\ell_2$).
  • The "jump" corresponded to the experimental resolution of an additional polarization singularity (a new zero) that appeared at a low-intensity location.
  • Once the perturbation was strong enough to resolve this split, further increases did not change the skyrmion number, confirming the stability of the new topological state.

5. Significance

• Fundamental Topology: The work clarifies that the "difference" rule is a fragile, idealized case, while the "maximum" rule is the robust, physical reality for perturbed systems. This redefines the understanding of topological stability in optical fields.
• Implications for Complex Media: The findings are critical for applications involving light propagation through complex media (e.g., atmospheric turbulence, biological tissue) where aberrations inevitably split singularities. It suggests that optical skyrmions in such environments may possess higher topological charges than previously thought.
• Beyond 2D: The authors note that this unfolding mechanism is particularly relevant for 3D spin angular momentum textures and wavelength-scale confined structures, where singularity splitting cannot be neglected.
• Methodological Guidance: The paper provides a warning for experimentalists and simulators: accurate topological characterization requires high spatial resolution and careful handling of low-intensity regions to avoid missing the "hidden" singularities that define the true topology.

In conclusion, the paper reveals that unstable high-order polarization singularities unfold into stable configurations with a skyrmion number equal to the highest constituent vortex charge, challenging the long-held belief that skyrmion numbers in superpositions are simply the difference of their constituent charges.