[Paper Review] 4D topological textures in light

David Marco, Miguel A. Alonso|arXiv (Cornell University)|Dec 2, 2022

Liquid Crystal Research Advancements4 citations

TL;DR

This paper proposes experimentally realizable 4D optical fields—termed full 3D polarization (F3DP) fields—that span all possible nonparaxial polarization states through a superposition of five non-coplanar, time-modulated plane waves. These fields form periodic polarization lattices in 3D space and time, exhibiting 4D skyrmionic textures with a quantized Skyrme number of ±1 per spatiotemporal cell, demonstrating topologically protected polarization structures in nonparaxial light.

ABSTRACT

We present 4D topological textures in (quasi)monochromatic nonparaxial optical lattices that contain all possible polarization ellipses with every combination of ellipticity and orientation in 3D space. These fields span the nonparaxial polarization space (a complex projective plane) and a 4-sphere within specific spatiotemporal regions, forming 4D skyrmionic structures. Constructed from five plane waves with adiabatically varying relative amplitudes, they are experimentally realizable in free space by focusing a temporally variant beam with a high numerical aperture lens.

Motivation & Objective

• To develop a theoretical framework for optical fields that fully span the 4D space of nonparaxial polarization states.
• To address the challenge of mapping the 4D polarization space (ellipticity and three Euler angles) into a physically realizable 3D spatial and temporal configuration.
• To demonstrate that such fields can host 4D skyrmionic structures with quantized topological invariants.
• To propose a simple, experimentally feasible configuration using five non-coplanar plane waves with adiabatic amplitude modulation.
• To identify and characterize F3DP cells with unit Skyrme number, distinguishing them from nonskyrmionic counterparts.

Proposed method

• Constructs quasi-monochromatic optical fields as a superposition of five linearly polarized plane waves with non-coplanar propagation directions.
• Implements adiabatic temporal modulation by slowly varying the relative amplitudes between two groups of plane waves, enabling dynamic coverage of all nonparaxial polarization ellipses.
• Defines a Skyrme density as the Jacobian of the mapping between the 4D polarization parameter space (two unit vectors on the Poincaré sphere) and the 3D space-time coordinates.
• Uses a parametric representation of the electric field vector to describe polarization ellipses in terms of ellipticity and orientation angles via Euler angles.
• Analyzes the Skyrme number by integrating the Skyrme density over spatiotemporal cells, yielding quantized values of ±1 for skyrmionic cells.
• Validates results analytically and numerically for specific configurations, including the case α=β=π/4, where the Skyrme density simplifies and the Skyrme number is exactly ±1 in cells A and B.

Figure 1: (a,c) Wavevectors and their polarization for five plane waves that compose a nonparaxial optical field containing all possible polarization states, in (a) the construction frame and (c) the laboratory frame. The field is constructed by adiabatically harmonically varying in time the relativ

Experimental results

Research questions

• RQ1 Can a single optical field dynamically cover all possible nonparaxial polarization states in a 4D spatiotemporal volume?
• RQ2 What is the topological nature of the polarization texture formed when all nonparaxial polarization states are uniformly distributed in space and time?
• RQ3 How can the Skyrme number be computed and quantized in a 4D polarization lattice formed by nonparaxial fields?
• RQ4 Are there configurations where the Skyrme density maintains a consistent sign, leading to more robust topological structures?
• RQ5 Can experimentally realizable fields be constructed that support 4D skyrmionic textures with unit topological charge?

Key findings

• The proposed F3DP fields are constructed from five non-coplanar plane waves with time-varying amplitudes, enabling full coverage of the 4D nonparaxial polarization space within a spatiotemporal cell.
• For the case α=β=π/4, the Skyrme density simplifies to an analytically tractable form that depends on y and t, with a periodic dependence on time.
• The Skyrme number is exactly −1 for cell A and +1 for cell B within the interval Ωt∈[0,π/2], confirming quantized topological charge.
• The Skyrme number accumulates linearly with z, implying that extended regions can host non-zero Skyrme numbers even outside standard F3DP cells.
• Nonskyrmionic F3DP cells exist with the same volume and full polarization coverage but zero Skyrme number due to cancellation of the Skyrme density contributions.
• The field is experimentally feasible, particularly in the focal region of a high numerical aperture lens, and the authors are developing a 4D measurement scheme for such fields.

Figure 2: (a) The dimensions of the F3DP cell can be controlled by changing the angles $\alpha$ and $\beta$ for the wavevectors and the corresponding polarization vectors. The angle between two wavevectors with the same polarization state (blue-blue or red-red pairs) is $2\alpha$ , and the angle bet