An international team of researchers in Singapore and Japan has developed a theoretical framework for creating hopfion crystals, which are complex, knot-like patterns of light that repeat across both space and time.
By connecting knot theory with photonics, a new study detailing their work explores new opportunities for using light to encode and process information. The joint study was done in collaboration between scientists from Nanyang Technological University, the Institute of Science, Tokyo, Keio University, and the University of Tokyo.
The idea still remains theoretical but represents potential for creating light-based structures that exhibit properties similar to those of solid materials. This framework could help move photonic technologies beyond transient light beams, opening possibilities for future applications.
The History of Hopfions
Hopfions are three-dimensional topological structures defined by their unique geometry, instead of the material from which they are made. Internally, hopfions consist of “spin” patterns that form tightly wound, interlocking loops. The name “hopfion” is derived from the Hopf fibration, a mathematical concept describing how circles can be linked and knotted in higher-dimensional spaces. Unlike most light fields, which tend to spread out and fade away, hopfions keep themselves tied together and can stick around much longer.
Previously, these complex structures have only been observed in isolated instances, such as brief appearances in magnetic or optical experiments. This new study proposes assembling hopfions into crystal-like structures that could extend across both space and time.
Developing hopfion crystals could lead to the creation of photonic devices that are less susceptible to interference. This could be practical for building secure communications and advanced data processing systems that require robust, stable information carriers based on light.
From Knots to Crystals
The researchers used bichromatic light, which is created by combining two light beams of slightly different colors and wavelengths, each with opposite circular polarizations. When these beams overlap, they create a property called pseudospin, which describes the combined characteristics of the light beams, forming their own spin pattern from the two combined colors. By setting the wavelength difference to a simple ratio, a repeating pattern forms. During each cycle, hopfions with specific twisted configurations appear and organize into chains.
This process is similar to braiding two ropes of different colors. When twisted together in sync, the pattern begins to appear; adding more strands allows for even more complex knots. In this case, the patterns are made from light instead of physical fibers.
The models demonstrated that the complexity of the knots can be controlled by adjusting an integer parameter, which determines the number of times the loops wind.
Scaling into Three Dimensions
The team suggested building three-dimensional hopfion crystals by arranging small emitters, such as microwave and dipole antennas, or grating couplers, with a set phase and polarization. When each emitter is fed with bichromatic light, the setup generates repeating units with different topological patterns. These units come together to form a stable lattice, or crystal-like structure, that repeats across both space and time.
Unlike previous optical hopfions, which rely on carefully engineered diffraction patterns, these crystals are formed through the beating of light waves. This allows them, in theory, to move, or ‘fly’ through space while maintaining their structure, until natural diffraction eventually causes them to disperse.
Potential Applications
If space-time hopfion crystals can be made in practice, they could impact fields such as optical communications, quantum computing, neuromorphic computing, and aerospace. Their stability could help make data transfer more secure and computing systems more robust, which are crucial goals across all of these industries.
Similar topological textures, such as skyrmions, have already shown potential for data storage. If hopfion crystals can be produced, they might enable denser, error-resistant information encoding, support secure communication channels, or offer new ways to manipulate particles with light for technological use.
The authors refer to their work as “the birth of space-time hopfion crystals”, highlighting its significance as a first step toward creating stable, repeating topological structures at optical, terahertz, and microwave frequencies. This advance opens new directions for research in topological photonics.
For now, the results are theoretical, but the authors see a path toward practical applications. There are still technical and engineering obstacles to overcome for space-time hopfion crystals to be created in practice, but their application could lead to significant advances in data processing and photonic technology.
Austin Burgess is a writer and researcher with a background in sales, marketing, and data analytics. He holds a Master of Business Administration and a Bachelor of Science in Business Administration, along with a certification in Data Analytics. His work combines analytical training with a focus on emerging science, aerospace, and astronomical research.