Resolving topological obstructions to vectorial structured field control

TL;DR

The paper tackles the challenge of achieving arbitrary vectorial transformations with structured matter while preserving continuity across cascaded devices, highlighting a topological mismatch: $\pi_2(S^2) \approx Z$ versus $\pi_2((S^1)^n) = 0$. It introduces a lifting framework based on Čech cohomology to diagnose when continuous parameter decompositions exist and to guide cascade depth. For skyrmion-beam generation, it shows that three SLMs suffice when aconstructed cohomology class vanishes, while four SLMs generally remove obstructions, with extensions to matter-field generation. Experimental results corroborate the theory, demonstrating that discontinuous decompositions induce phase singularities and errors, whereas continuous decompositions stabilize topological features like skyrmion numbers, offering a topology-guided route to robust vectorial-field control and topological field engineering.

Abstract

The use of structured matter, such as optical retarders, for vectorial control is a well-established and widely employed technique in modern optics, and has driven continued advances in the manipulation of complex, spatially varying vectorial fields. However, achieving arbitrary field conversion typically requires the use of cascaded elements, as intrinsic physical and fabrication constraints fundamentally limit individual devices to a restricted subset of transformations. This results in an overall continuous transformation potentially failing to be continuous at the level of the parameters of the cascade, leading to detrimental engineering consequences such as the introduction of complex, discontinuous aberrations that disrupt important topological properties of the underlying matter field. In this work, we establish a novel mathematical framework for analyzing the topological difficulties that emerge in the decomposition of an overall transformation into individual layers, and for determining the minimal depth required to overcome them. The strategy introduced provides a general pathway for optimizing designs for vectorial field control and matter field generation, with particular significance for the manipulation of topological phases in optical polarization fields, such as Stokes skyrmions, where continuity is of vital importance.

Figures (7)

• Figure 1: Concept. A pictorial representation illustrating the vanishing of $\pi_2(T^2)$, where $T^2 = S^1 \times S^1$ denotes the standard torus. Consider the image of a map $f \colon S^2 \to T^2$, depicted as a blob above. For any such $f$, there exists a homotopy that continuously collapses the image to a point, implying that any two maps from $S^2$ into $T^2$ are homotopic. The figure also illustrates, in red and blue, the trajectories of two representative points throughout this process.
• Figure 2: Experimental results (2 SLM cascade). Target and experimentally measured Stokes fields of four different degree-1 skyrmions, with the corresponding experimental configuration shown above. Throughout this paper, Stokes fields are depicted using color to represent the azimuthal angle on the Poincaré sphere and saturation to represent height shen_optical_2023. The experimental set-up is drawn with I representing the incident field, which is linearly polarized at $45^\circ$, and O representing the output field. Note that the second SLM is sandwiched between half-wave plates (HWPs) whose fast axes are aligned at $22.5^\circ$, such that the composite system acts as a linear retarder with its fast axis aligned at $45^\circ$. Insets highlighting the output Stokes field near $\pm 45^\circ$ linearly polarized light are shown, in which a line artefact is clearly visible. The corresponding phase patterns (given in levels) applied to each SLM are also shown. A clear phase singularity is observed on the second SLM at points where the target field is $\pm 45^\circ$ linearly polarized, in agreement with the established theory. Lastly, the $\ell^2$-error distribution between the generated and target fields is presented, where the errors arising from phase discontinuities are clearly visible.
• Figure 3: Experimental results (3 SLM cascade). Target and experimentally measured Stokes fields of four different degree-1 skyrmions, with the corresponding experimental configuration shown above. The distinguished set $\Sigma$, as defined in Eq. \ref{['eq: sigma']}, is highlighted by dashed gray lines on the target field. The corresponding phase patterns on each SLM, along with the $\ell^2$-error distributions between the generated and target fields, are also shown. For experiment D, insets highlighting the Stokes field near phase discontinuities are included, where a line artefact is clearly visible.
• Figure 4: Experimental results (4 SLM cascade). Target and experimentally measured Stokes fields of four different degree-1 skyrmions, with the corresponding experimental configuration shown above. The corresponding phase patterns on each SLM, along with the $\ell^2$-error distributions between the generated and target fields, are also shown.
• Figure 5: Experimental results (skyrmion). Experimentally measured Stokes fields for different input beams passing through a beam generator designed to produce a standard Néel-type skyrmion from uniformly $45^\circ$ linearly polarized light, using cascades of two and three SLMs. Relevant experimental set-ups and SLM phase distributions are provided for completeness. The corresponding numerically computed skyrmion numbers are also presented, demonstrating instability of the skyrmion number when continuous parameter decomposition is not achieved, and stable skyrmion numbers when it is.