🔬 optics

Optical hopfions with arbitrary two winding numbers

This paper presents a systematic method for generating tunable optical hopfions with arbitrary poloidal and toroidal winding numbers using tailored superpositions of Laguerre-Gaussian modes, experimentally demonstrating record-high orders up to 5 and 3 to enable robust topological information carriers for applications in photonics and communications.

Original authors: Xinji Zeng, Jinwen Wang, Yun Chen, Guang Liu, Zhenyu Guo, Yongkun Zhou, Xin Yang, Chengyuan Wang, Dong Wei, Haixia Chen, Yijie Shen, Andrew Forbes, Hong Gao

Published 2026-04-24

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Xinji Zeng, Jinwen Wang, Yun Chen, Guang Liu, Zhenyu Guo, Yongkun Zhou, Xin Yang, Chengyuan Wang, Dong Wei, Haixia Chen, Yijie Shen, Andrew Forbes, Hong Gao

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 you are holding a piece of string. If you tie it into a simple loop, that's easy. But what if you could tie that string into a knot that exists not just on a flat table, but floating in 3D space, twisting and turning in a way that seems impossible to untie without cutting the string?

That is essentially what this paper is about, but instead of string, the scientists are using light.

Here is a simple breakdown of their discovery:

1. The "Magic Knot" of Light (Hopfions)

In the world of physics, there are special shapes called Hopfions. Think of them as 3D knots made of invisible magnetic or light fields.

The Problem: For a long time, scientists could only make very simple, low-level knots (like a basic loop). They were stuck with a "one-size-fits-all" approach.
The Goal: The researchers wanted to build any kind of knot they wanted, with as many twists and turns as they desired. They wanted to be able to dial up the complexity at will.

2. The Recipe: Mixing Colors and Twists

To make these light knots, the team used a special type of laser beam called a vector beam. Imagine this beam as having two "hands": a Left-Handed hand and a Right-Handed hand (representing two different polarizations of light).

To create the knot, they didn't just turn a dial; they acted like a master chef mixing ingredients:

The Ingredients: They used "Laguerre-Gaussian" modes. Think of these as different flavors of light swirls. Some swirl once, some twice, some three times.
The Secret Sauce (Gouy Phase): Usually, to make a knot, you need to twist the light sideways. But in this experiment, they couldn't do that easily. Instead, they played with something called the Gouy phase.
- Analogy: Imagine two runners on a track. They start together. If one runner speeds up slightly at the start and slows down later, they will end up in a different position relative to each other, even if they run the same distance. The scientists used this "timing trick" (phase difference) to make the two hands of the light twist around each other in 3D space, creating the knot without needing to physically twist the beam sideways.

3. The Result: A New World of Knots

By carefully mixing these "flavors" of light, they successfully created Hopfions with arbitrary winding numbers.

The "p" and "q" numbers: Imagine a knot has two ways it can twist:
- p (Poloidal): Twisting around the thickness of the tube (like wrapping a ribbon around a finger).
- q (Toroidal): Twisting around the hole in the middle of the tube (like a pretzel).
The Achievement: Previous experiments could only make simple knots like (1,1) or maybe (3,1). This team successfully built knots like (5,2) and (3,2). They proved they could build a knot with 5 twists one way and 2 the other, or any combination they wanted.

4. Why Does This Matter? (The "So What?")

You might ask, "Why do we need fancy light knots?" Here are three reasons:

Super-Secure Data: Think of a knot as a code. A simple loop is easy to copy or break. A complex, high-order knot (like a (5,3) Hopfion) is incredibly hard to replicate or mess up. This could lead to super-secure optical communication where information is carried in the shape of the light itself. If someone tries to intercept it, the knot might unravel, alerting you immediately.
Tiny Data Storage: These knots are stable. In the future, we might use them to store massive amounts of data in tiny 3D spaces, much like how we use magnetic "skyrmions" (2D cousins of Hopfions) in hard drives today.
A Blueprint for Everything: The cool part is that while they did this with light, the math works the same for magnets and other materials. They have provided a "recipe book" that other scientists can use to build these complex structures in magnets, liquid crystals, or even quantum computers.

The Bottom Line

This paper is like the difference between learning to tie a simple shoelace and learning to tie a complex, artistic macramé pattern. The researchers have figured out how to tie any knot they want using light. They've turned the abstract math of "knot theory" into a real, controllable tool that could revolutionize how we send information and store data in the future.

1. Problem Statement

Hopfions are three-dimensional topologically nontrivial structures characterized by interlinked field lines, described mathematically by a Hopf fibration mapping from a 3-sphere ( $S^3$ ) to a 2-sphere ( $S^2$ ). Their topology is defined by two winding numbers: the poloidal winding number ( $p$ ) and the toroidal winding number ( $q$ ).

Current Limitations: While hopfions have been realized in various physical systems (magnets, liquid crystals, Bose-Einstein condensates), experimental realizations have been restricted to low-order topologies (typically $p=1, q=1$ ).
Tunability Gap: Existing optical methods for generating hopfions rely on the longitudinal orbital angular momentum (OAM) of beams to control $p$ , but the highest reported poloidal order is limited ( $p=3$ ). Crucially, the toroidal winding number $q$ has not been independently controlled in free space because transverse OAM has not been effectively utilized in the spatial dimension under paraxial conditions.
Goal: The authors aim to develop a general, systematic method to generate optical hopfions with arbitrary and tunable winding numbers $(p, q)$ in free space, overcoming previous order limitations and enabling independent control of both topological indices.

2. Methodology

The authors propose a versatile approach using tailored superpositions of Laguerre-Gaussian (LG) modes in free space to construct vector beams with specific polarization textures.

Theoretical Framework:
- The optical field is defined by the Stokes vector $\mathbf{S} = (S_x, S_y, S_z)$ .
- The target hopfion is constructed by superimposing Right-Hand Circularly Polarized (RCP) and Left-Hand Circularly Polarized (LCP) components.
- Key Innovation: Instead of relying solely on OAM (which is difficult to control independently for $q$ ), the authors manipulate the Gouy phase evolution. By carefully designing the superposition of LG modes with different radial ( $p$ ) and azimuthal ( $l$ ) indices, they engineer the phase difference between the RCP and LCP components. This phase evolution drives the polarization filaments to wind around the propagation axis ( $z$ -axis) $q$ times.
Construction Strategy:
- For $(p, 1)$ -order hopfions: The authors superimpose LG modes with adjacent OAM values for the two orthogonal polarization components to ensure effective spatial overlap. The poloidal winding $p$ is controlled by the OAM difference.
- For $(1, q)$ -order hopfions: The toroidal winding $q$ is achieved by creating a specific Gouy phase difference. This involves summing multiple LG modes in one polarization component (e.g., RCP) with alternating signs to induce a $q$ -fold winding of the polarization around the $z$ -axis during propagation.
- General $(p, q)$ -order: The two strategies are combined. The complex amplitude of the vector beam is defined by specific linear combinations of LG modes (Eq. 6 in the paper), where coefficients $\alpha$ and $\beta$ are tuned to optimize the topological structure.
Experimental Setup:
- A 780 nm laser beam is shaped and split into RCP and LCP components using a beam displacer (BD).
- A Spatial Light Modulator (SLM) is used to modulate the complex amplitude and phase of each component independently.
- Digital Propagation: Instead of using physical lenses to propagate the beam, the authors use the Angular Spectrum Method to calculate the required phase patterns on the SLM for specific propagation distances ( $z$ ). This allows for precise reconstruction of the 3D field without optical alignment errors.
- Characterization: A Stokes polarimetry setup (HWP, QWP, PBS, CCD) is used to measure the full 3D polarization distribution and reconstruct the isopolarization filaments.

3. Key Contributions

Arbitrary Order Generation: The paper reports the first experimental realization of optical hopfions with arbitrary tunable winding numbers $(p, q)$ $(p, q)$ .
- Demonstrated poloidal orders up to $p=5$ .
- Demonstrated toroidal orders up to $q=3$ (and simulated up to $q=4$ ).
Independent Control Mechanism: The authors successfully decoupled the control of $p$ and $q$ . They achieved independent control of the toroidal winding number $q$ by manipulating the Gouy phase evolution, a feat previously unattained in free-space optics.
Systematic Design Route: They provided explicit analytical expressions for the complex amplitude of the vector beams required to generate any $(p, q)$ hopfion, offering a blueprint for future topological photonics.
Topological Verification: The study experimentally verified the topological stability and structure of these high-order states, including the calculation of the Hopf index ( $N_H$ ), which theoretically equals $p \times q$ .

4. Results

Experimental Realization:
- $(p, 1)$ States: Successfully generated hopfions with $p=1, 3, 4, 5$ . The experimental polarization distributions in the $xy$ and $xz$ planes matched simulations, showing $p$ -petal flower structures and skyrmion-anti-skyrmion pairs.
- $(1, q)$ States: Successfully generated hopfions with $q=2, 3$ . The results showed that the polarization filaments encircled the $z$ -axis $q$ times, forming complex knot structures (e.g., a trefoil knot for $q=3$ ).
- Combined $(p, q)$ States: Demonstrated the generation of $(3, 2)$ and $(5, 2)$ hopfions. The experimental isopolar filaments formed the predicted torus knots (e.g., a trefoil knot for $3,2$).
Hopf Index Calculation:
- The experimentally calculated Hopf indices ( $N_H$ ) closely matched the theoretical values ( $p \times q$ ). For example, for $(3,2)$ , the experimental $N_H$ was $5.2751$ (theoretical $6$), and for $(5,2)$ , it was consistent with the trend, confirming the topological nature of the structures.
- Deviations from ideal integers were attributed to finite integration volumes and sampling limitations, but the topological integrity remained intact.
Parametric Sensitivity: The study analyzed the impact of superposition coefficients ( $\alpha, \beta$ ). It identified a "hopfion region" where stable topological structures exist, distinct from regions producing skyrmion tubes or topologically trivial fields.
Non-Coprime Winding Numbers: The supplementary material discusses cases where $p$ and $q$ are not coprime (e.g., $2,2$). In these cases, the single torus knot splits into multiple disjoint linked curves, yet the overall topological charge remains the product $pq$.

5. Significance

Advancement in Topological Photonics: This work significantly expands the family of realizable optical topological solitons, moving beyond basic orders to high-dimensional, arbitrary-order states.
Information Carriers: The ability to encode information in arbitrary $(p, q)$ pairs suggests a new paradigm for high-capacity optical communications. Each distinct pair could represent a unique topological state, potentially increasing channel density and robustness against perturbations.
Cross-Disciplinary Impact: The design principles are transferable to other physical systems hosting vector-field topological excitations, such as chiral magnets, ferroelectrics, and liquid crystals. The optical platform serves as a "conceptual blueprint" for synthesizing complex 3D topological textures in condensed matter physics.
Future Applications: The method opens avenues for exploring 3D spintronics, light-matter interactions, and the development of robust topological devices for memory and logic applications.

In summary, Zeng et al. have established a robust, systematic, and experimentally verified method to generate optical hopfions with arbitrary topological complexity, bridging the gap between theoretical topological mathematics and practical optical engineering.