Diffraction-free natural optical skyrmions and their subwavelength confinement around vortices

Technical Summary: Diffraction-free Natural Optical Skyrmions and Their Subwavelength Confinement around Vortices

Problem Statement
Diffraction is a fundamental property of waves emitted by sources of finite extent, causing wavefronts to spread and features to separate indefinitely as they propagate. Suppressing this spreading has been a century-long goal in optics and physics. Previous approaches, such as Airy and Bessel beams or localized wave packets (e.g., X-waves), rely on idealized wave engineering. While these can exhibit quasi-non-diffracting behavior over finite distances, they either require infinite power content (idealized) or eventually spread in real-world settings. Furthermore, existing propagating optical skyrmions (topological textures of light) have been shown to experience standard diffraction spreading similar to conventional light beams. A critical gap remains in achieving a naturally occurring, non-diffracting light structure that maintains subwavelength lateral confinement without relying on idealized, infinite-energy solutions.

Methodology
The authors combine theoretical analysis with experimental verification to investigate the polarization structure surrounding the phase singularity of optical vortices (e.g., Laguerre-Gauss beams).

• Theoretical Framework: The study models the transverse electric field of a vortex beam near its phase singularity. Using the Fresnel diffraction integral and Gauss's divergence law ($\nabla \cdot \mathbf{E} = 0$), the authors derive the existence of an axial electric field component ($\psi_z$) that arises due to tight transverse localization. They define a "Transverse-Axial" (TA) polarization texture using the quotient $\psi_z/\psi_\perp$. This quotient is shown to map all TA polarization states onto a Transverse-Axial Poincaré Sphere (TA-PS), forming a skyrmionic texture. The theory predicts that while the intensity components ($\psi_\perp$ and $\psi_z$) diffract and attenuate, the polarization texture itself remains invariant during propagation.
• Experimental Setup: Experiments were conducted using a 532 nm laser beam. A vortex beam with topological charge $\ell=1$ was generated using a Spatial Light Modulator (SLM) and focused by a lens ($f=40$ mm). The beam waist was approximately $6.1 \, \mu\text{m}$ ($\sim 11.5\lambda$).
• Characterization: The team employed off-axis interferometry to reconstruct the complex amplitude of the transverse field. The axial field was then reconstructed computationally using Gauss's divergence law. By varying the Spin Angular Momentum (SAM) of the beam (via a quarter-wave plate), they modulated the polarization state from Right-Hand Circular (RCP) to Left-Hand Circular (LCP).
• Analysis: The researchers measured the Transverse-Axial Stokes parameters ($S_1, S_2, S_3$) to visualize the skyrmion texture. They tracked the Full Width at Half Maximum (FWHM) of the polarization texture (specifically the $s_3=0$ contour) over a propagation distance of more than 1100 wavelengths.

Key Contributions and Results

1. Discovery of Natural Diffraction-Free Skyrmions: The paper demonstrates that a specific polarization texture, termed a "natural optical skyrmion," exists inherently around the phase singularity of optical vortices. Unlike previously known propagating skyrmions, this texture does not spread during propagation.
2. Subwavelength Confinement: The skyrmion is confined within a tube of radius $r_{sk}$ that can be tuned from the wavelength scale ($\sim \lambda$) down to deep-subwavelength scales (experimentally observed below $0.1\lambda$). The radius is determined by the interplay between the Orbital Angular Momentum (OAM, $\ell$) and the Spin Angular Momentum (SAM, $\sigma_z$) of the vortex beam.
   • When SAM and OAM are anti-parallel (e.g., RCP with $\ell > 0$), the size is on the order of the wavelength.
   • When SAM and OAM are parallel, the theoretical radius approaches zero; experimentally, confinement below $0.1\lambda$ was achieved.
3. Experimental Validation of Non-Diffraction: The experiments confirmed that the skyrmionic polarization texture remained confined over a propagation distance of 1128 wavelengths ($\sim 1100\lambda$). This distance is three orders of magnitude greater than the characteristic diffraction distance ($z_R \sim \lambda$) expected for a feature of that size. In contrast, standard light fields confined to this scale would spread significantly within one wavelength.
4. Robustness and Control: The study verified that the non-diffracting nature holds regardless of the convergence or divergence of the nesting vortex beam. The size of the skyrmion was shown to be controllable via the polarization state (SAM), matching theoretical predictions with high precision.

Significance and Claims
The authors claim that this work provides a "factual, real-world form of ideal non-diffracting propagation." The significance lies in the fact that this phenomenon:

• Requires no idealization: It occurs naturally in any real-world vortex beam carrying finite power, unlike Bessel or Airy beams which require infinite energy to be truly non-diffracting.
• Breaks previous conceptions: It challenges the understanding that subwavelength confinement inevitably leads to rapid diffraction spreading.
• Offers a new platform: The natural skyrmion combines arbitrary subwavelength confinement with propagation invariance, properties that had not been observed together in a factual situation prior to this work.

The paper suggests that these findings could open new perspectives for applications requiring precise spatial confinement, such as optical super-resolution and metrology. Additionally, the authors note the potential for extending these concepts to other wave systems, including acoustic, water, and electron vortices, and exploring analogous phenomena in transverse polarization textures within anisotropic media. However, the paper maintains a modest tone regarding immediate applications, focusing primarily on the fundamental discovery of this propagation-invariant physical state.