🔬 optics

Fractional optical skyrmions

This paper reports the first experimental creation of fractional optical skyrmions using vectorial superpositions of non-integer orbital angular momentum, revealing abrupt transition jumps in skyrmion number that reinforce the robust integer nature of optical topologies and open new avenues for communication and sensing applications.

Original authors: Yuancong Cao, Ram Nandan Kumar, Hadrian Bezuidenhout, Mingjian Cheng, Lixin Guo, Andrew Forbes

Published 2026-02-18

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CC BY 4.0

Original authors: Yuancong Cao, Ram Nandan Kumar, Hadrian Bezuidenhout, Mingjian Cheng, Lixin Guo, Andrew Forbes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine light not just as a beam that turns things on, but as a dancer spinning with a specific rhythm. For a long time, scientists have known how to make light spin in perfect, whole-number steps. If you think of the light's spin as a clock hand, it could point to 12, then 1, then 2. These are "integer" spins.

Recently, scientists discovered that these spinning light beams can form tiny, invisible knots in their structure called Skyrmions. Think of a Skyrmion like a perfect, swirling tornado of light where the direction of the light's "spin" (polarization) wraps around a sphere exactly once, twice, or three times. These knots are incredibly tough; they don't easily unravel when they hit dust, fog, or other messy environments. This makes them great for sending secret messages or sensing tiny changes in the world.

The Big Discovery: Breaking the Rules

Until now, these light knots could only be made with whole-number spins (1, 2, 3). You couldn't have a "half-knot" or a "1.5-knot." It was like a staircase where you could only stand on the steps, never in between.

In this paper, the researchers did something revolutionary: They built the first "fractional" light knots.

They managed to create light that spins in non-whole numbers, like 1.5 or 2.3. Imagine a staircase where you can now stand perfectly on the landing between two steps.

How They Did It (The Recipe)

To make these fractional knots, the scientists mixed two different types of light beams together:

One beam spinning normally (like a standard integer).
One beam spinning in a "fractional" way (a weird, in-between spin).

They combined these two beams using a special mirror setup (an interferometer) that acts like a high-tech mixing bowl. The result is a new kind of light texture that has never been seen before.

The Surprise: The "Jump" in the Knot

Here is the most fascinating part. The researchers expected that if they slowly changed the spin from 1 to 2, the knot would smoothly morph from a "1-knot" to a "2-knot," passing through a smooth "1.5-knot" stage.

But that's not what happened.

Instead, they found that the knot behaves like a rubber band that suddenly snaps.

As they slowly turned the dial to increase the spin, the knot looked stable and smooth for a while.
Then, right at the halfway point (like 1.5), the knot suddenly jumps. It doesn't slowly change; it abruptly flips its structure.
After the jump, it settles into the next integer shape (like 2).

The Analogy: The Map and the Cliff

Think of the light's structure as a map of a landscape.

Integer Skyrmions are like smooth, rolling hills. You can walk from the bottom to the top without ever falling off.
Fractional Skyrmions are like a cliff edge. You can walk up the hill, but right at the halfway point, the ground just drops off. The "map" of the light's direction has a sudden tear or discontinuity.

Because of this sudden tear, the light doesn't wrap around its sphere perfectly. It only covers part of the sphere before the jump happens. This proves that while you can have fractional spins, the knots (Skyrmions) themselves still want to be whole numbers. The fractional state is just a temporary, unstable bridge between two solid integer states.

Why This Matters

This discovery is a big deal for two main reasons:

New Tools for Communication: Because these fractional knots have these sudden, sharp jumps, they are extremely sensitive to changes. You could use them to build super-sensitive detectors. If the light hits a tiny particle or a change in the air, the "jump" happens at a different spot, alerting you instantly.
More Data: Just like how we used to only use whole numbers for computer bits (0 and 1), but now we are exploring more complex states, this opens up a new "spectrum" of light. We might be able to pack more information into a single beam of light by using these fractional steps.

In a Nutshell

The scientists took light, gave it a weird, half-step spin, and found that it creates a new type of knot. This knot doesn't change smoothly; it jumps. This "jump" reveals that nature prefers whole numbers for these knots, but by understanding the jump, we can use it to build better sensors and faster communication systems. It's like discovering a new musical note that isn't quite a whole step or a half-step, but a "micro-step" that changes how the whole song sounds.

1. Problem Statement

Optical skyrmions are topological states characterized by spatially structured vectorial polarization textures. Historically, these have been generated exclusively using integer Orbital Angular Momentum (OAM) modes, resulting in integer-valued Skyrmion numbers ( $N$ ). These integer topologies are robust, covering the Poincaré sphere an integer number of times via smooth, adiabatic evolution.

The central problem addressed in this work is the limitation of the discrete nature of optical topology. While fractional OAM (non-integer OAM) has been studied as a superposition of integer modes, it was previously unknown if fractional OAM could generate fractional optical skyrmions. A key theoretical challenge is that while fractional OAM states can be decomposed into an orthogonal basis of integer OAM modes, fractional skyrmions cannot be similarly decomposed because integer skyrmionic states do not form an orthogonal basis. Consequently, the topological evolution between integer orders was unexplored, and the existence of a continuous, tunable topological landscape remained an open question.

2. Methodology

Theoretical Framework

Definition: The authors define a fractional optical skyrmion as a coherent superposition of non-orthogonal fractional OAM modes encoded in orthogonal polarization bases (e.g., Horizontal $|H\rangle$ $∣ H ⟩$ and Vertical $|V\rangle$ $∣ V ⟩$ ).
- The field is described as: $|\psi(r, \phi)\rangle = \alpha |m_1\rangle|H\rangle + \beta |m_2\rangle|V\rangle$ , where $m_1$ and $m_2$ are OAM indices (one or both being non-integer).
Modal Decomposition: The fractional OAM mode $|m\rangle$ is expressed as a superposition of integer OAM modes using Fourier coefficients.
Stokes Vector Analysis: The local polarization state is mapped onto the Poincaré sphere using the normalized Stokes vector $\mathbf{S}(r, \phi)$ .
Skyrmion Number Calculation: The topological charge (Skyrmion number) $N(m)$ is calculated numerically via the integral of the spin texture over the transverse plane:
$N(m) = \frac{1}{4\pi} \int_A \mathbf{S} \cdot \left( \frac{\partial \mathbf{S}}{\partial r} \times \frac{\partial \mathbf{S}}{\partial \phi} \right) dr d\phi$
Analytical integration is impossible due to the multivalued phase structure of fractional OAM, necessitating numerical evaluation.

Experimental Setup

Generation: A common-path interferometric architecture was used to ensure high fidelity.
- A He-Ne laser ( $\lambda = 633$ nm) was expanded and incident on a Spatial Light Modulator (SLM) encoding a multiplexed hologram to generate two spatially separated scalar vortex beams with specific OAM indices ( $m_1, m_2$ ).
- A 4f imaging system filtered the first diffraction orders.
- A half-wave plate (HWP) rotated the polarization to a diagonal state.
- A Sagnac interferometer (using a Polarizing Beam Splitter and mirrors) recombined the orthogonal polarization components ( $|H\rangle$ and $|V\rangle$ ) to synthesize the vector field.
Characterization: Full Stokes polarimetry was performed using a rotating quarter-wave plate (QWP) or HWP and a linear polarizer (LP) before a CCD camera. Six intensity projections ( $I_H, I_V, I_D, I_A, I_R, I_L$ ) were measured to reconstruct the Stokes parameters ( $S_0, S_1, S_2, S_3$ ).

3. Key Contributions

First Demonstration of Fractional Skyrmions: The authors successfully created the first optical skyrmions with non-integer topological numbers by structuring light with fractional OAM.
Discovery of Abrupt Transition Jumps: They revealed a new phenomenon where the Skyrmion number does not change linearly with the OAM difference. Instead, the number remains relatively stable near integer values and undergoes abrupt jumps near half-integer values.
Theoretical Distinction: They established that fractional skyrmions are fundamentally distinct from fractional OAM. Unlike fractional OAM (which is a linear superposition of integer modes), fractional skyrmions cannot be decomposed into an orthogonal basis of integer skyrmions due to the non-orthogonality of the underlying states.
Polarization Texture Evolution: They demonstrated that fractional skyrmions exhibit incomplete wrapping of the Poincaré sphere and possess abrupt azimuthal polarization transitions (discontinuities) at specific branch cuts, contrasting with the smooth evolution of integer skyrmions.

4. Results

Stokes Field Discontinuities: For fractional topological states (e.g., $\Delta m = 1.5, 2.5$ ), the Stokes parameters ( $S_1, S_2$ ) exhibit a discontinuity at a specific azimuthal angle ( $\phi_0$ ). This leads to a sudden rotation of the polarization ellipse in the State of Polarization (SOP) distributions, breaking the azimuthal symmetry observed in integer skyrmions.
Nonlinear Evolution of $N(m)$ :
- The Skyrmion number $N(m)$ varies nonlinearly with the OAM difference $\Delta m$ .
- Near Integers: The number changes gradually.
- Near Half-Integers: The number changes rapidly (abrupt transition).
- Experimental data (green points) matched numerical simulations (purple lines) with high precision, confirming the nonlinear evolutionary characteristics.
Mode Dependence: The study investigated different OAM base combinations $[m_1, m_2]$ yielding the same $\Delta m$ . It was found that higher OAM configurations shift the transition curves toward larger $\Delta m$ values and result in a systematic reduction of the fractional Skyrmion number. This is attributed to imperfect spatial matching between modes, which weakens the coupling between spatial and polarization degrees of freedom.
Incomplete Wrapping: The fractional skyrmion number corresponds to a polarization texture that covers only a portion of the Poincaré sphere, unlike integer skyrmions which cover it completely.

5. Significance

New Class of Optical Topologies: This work expands the landscape of optical topology beyond integer quantization, introducing a continuous spectrum of topological states.
Reinforcement of Integer Nature: Paradoxically, the study of fractional skyrmions reinforces the robustness of integer topologies. The "abrupt jumps" observed in the fractional regime highlight that integer values are the stable, preferred states of the system.
Applications:
- Optical Communication: The continuous tunability of the topological number offers new degrees of freedom for high-density information encoding.
- Sensing: The rapid rate of change in the continuous slopes of topological transitions (near half-integers) could be exploited to create highly sensitive topological detectors.
- Fundamental Physics: It provides a bridge between discrete integer topologies and continuous fractional states, offering insights into the dynamics of topological phase transitions in structured light.

In summary, Cao et al. have successfully bridged the gap between integer and fractional optical topologies, demonstrating that fractional skyrmions are a distinct, experimentally realizable class of states with unique discontinuous polarization textures and nonlinear topological evolution.