Fractional optical skyrmions

Abstract

Optical topologies in the form of Skyrmions have attracted significant interest of late, where their integer Skyrmion number has been shown to be robust to complex media. Here we create the first fractional Skyrmions by structuring light as a vectorial superposition of non-integer orbital angular momentum. We unravel the map structure to reveal a new phenomenon, the abrupt transition jumps in skyrmion number, which serves to reinforce the integer nature of skyrmion topologies. Our experimental demonstration agrees well with simulation, opening a new spectrum of optical topologies to explore, with exciting possibilities in optical communication and sensing.

Figures (5)

• Figure 1: Evolution and generation of fractional skyrmionic textures. (a) Schematic illustration of fractional skyrmions acting as a bridge between integer states (exemplified by $N=1$ and $N=2$). This demonstrates a continuous evolutionary process, the mechanism of which remains largely underexplored. (b) Realization of skyrmions with arbitrary topology via the superposition of two fractional Laguerre-Gaussian (LG) beams with orthogonal polarization states (e.g., $|H\rangle$ and $|V\rangle$). The case shown corresponds to $m_1=0, m_2=1.5, \text{and } \Delta m=1.5$. (c) Comparison of polarization-texture mappings corresponding to integer and fractional order skyrmions. The integer topology (e.g., $N = 1$ and $2$) exhibits smooth, adiabatic evolution, resulting in complete coverage. However, the fractional topology (e.g., $N = 0.5$ and $1.5$) exhibits abrupt azimuthal polarization transitions, leading to incomplete polarization-texture mapping.
• Figure 2: Experimental setup for generating and detecting fractional skyrmions. A collimated Gaussian beam ($\lambda = 633$ nm) illuminates a spatial light modulator (SLM) encoded with a multiplexed hologram to generate two spatially separated integer or fractional LG modes. A $4f$ imaging system, formed by lenses L2 and L3 with an aperture (AP1) at the Fourier plane, filters the first diffraction orders. A half-wave plate (HWP) sets the polarization to $45^\circ$ before entering a Sagnac interferometer composed of a polarizing beam splitter (PBS) and mirrors (M2, M3). The PBS splits and subsequently recombines the orthogonal polarization components ($|H\rangle$ and $|V\rangle$), synthesizing the vector field $\ket{\psi(r,\phi)} = \ket{m_1}\ket{H} + \ket{m_2}\ket{V}$. The field is analyzed via Stokes polarimetry using a rotating quarter-wave plate (QWP) or HWP and a linear polarizer (LP), and recorded by a CCD camera. BE: beam expander; OL: objective lens; M: mirror; L: lens; CCD: charge coupled device camrea.
• Figure 3: Stokes fields and state of polarization of fractional topologies. The Stokes parameters ($S_0$, $S_1$, $S_2$, and $S_3$) and the corresponding states of polarization (SOP) for fractional topological states described by Eq. \ref{['Eq_1']} are shown for (a) $m_1 = 0$ and $m_2 = 1.5$ ($\Delta m = 1.5$), (b) $m_1 = 0$ and $m_2 = 2.5$ ($\Delta m = 2.5$), and (c) $m_1 = 0$ and $m_2 = 3.5$ ($\Delta m = 3.5$). The left and right columns present the numerically simulated and experimentally measured results, respectively. By precisely controlling the relative phases ($\alpha$ and $\beta$) in Eq. \ref{['Eq_1']}, excellent agreement between simulations and experiments is achieved. Owing to the fractional topological charge $m_2$, a discontinuity appears at the azimuthal angle $\phi = \phi_0$ in the Stokes fields, such that $S_i(\phi_0^-) \neq S_i(\phi_0^+)$, which leads to a sudden rotation of the polarization ellipse in the SOP distributions.
• Figure 4: Fractional topological Skyrmion numbers and evolution of topological textures. (a) Evolution of Skyrmion numbers within the interval $\Delta m \in [0, 5]$. The purple solid lines represent simulation results, while the green data points (with a step size of 0.1) denote the mean values of 15 experimental measurements, with error bars indicating the standard deviation. The topological textures at half-integer OAM differences ($\Delta m = 0.5, 1.5, 2.5, 3.5,$ and $4.5$) are shown, where the purple and green arrows represent the simulation and experimental results, respectively. The fractional skyrmion number exhibits a nonlinear variation between adjacent integer topological charges. Specifically, this trend is characterized by rapid changes in the vicinity of half-integer values, while remaining relatively gradual near integer regions. Furthermore, this evolutionary behavior becomes increasingly pronounced as $\Delta m$ increases. The experimental results are in good agreement with the simulation curves, verifying the nonlinear evolutionary characteristics of fractional Skyrmions. (b)The continuous evolution of the topological texture from$\Delta m$ = 0 to $\Delta m$= 1 (step by 0.2). The top row (simulation) and bottom row (experiment) visualize the bifurcation of the Skyrmion core mediated by the azimuthal discontinuity.
• Figure 5: Evolution of fractional skyrmions under different OAM configurations. Evolutionary trajectories are shown for five OAM combinations within the interval $\Delta m \in [0, 1]$: $(m_1=0, m_2 \in [0,1])$, $(m_1=1, m_2 \in [1,2])$, $(m_1=2, m_2 \in [2,3])$, $(m_1=3, m_2 \in [3,4])$, and $(m_1=4, m_2 \in [4,5])$. Higher OAM configurations lead to a systematic shift of the transition curves toward larger $\Delta m$. The non-linear profile of these curves reveals the fundamental nonlinear dynamics of fractional topological evolution.