Hat tiles fabricated as optical nanostructures

The spots that appeared on the screen were orderly without repeating the same pattern. Researchers from the University of Tokyo, Institute of Science Tokyo and NTT transferred the arrangement of the “hat tile,” which shook up the mathematical world, into an actual nanostructure and observed its unusual diffraction when illuminated. The research paper was published in Nature Communications.

The story began with the “Einstein problem”: whether a plane can be tiled without gaps using only one shape while the overall pattern does not repeat periodically. In 2023, the hat tile was discovered as a solution, but it was unknown what its mathematical arrangement would do in a real physical system. The research team placed a point at the center of each tile and designed a quaslattice that retains the same shape after a 120-degree rotation but does not overlap with its mirror image.

The team etched that arrangement into a thin film using NTT’s microfabrication technology. Using electron-beam lithography, a technique also employed in semiconductor manufacturing, they arranged circular holes with a radius of 100 nm over a large area in a 350 nm-thick silicon nitride film. When illuminated with a green monochromatic laser, the structure produced sharp Bragg peaks in its diffraction pattern despite having no repetition, and the positions of the peaks barely changed even when the illumination position was moved. The result indicates that the structure has long-range order overall, according to the team.

With both a white laser and a green laser, the diffraction pattern also rotated in one direction like a pinwheel. When the arrangement was changed to its mirror image, the direction of rotation was reversed. When the team separately illuminated the structure with right- and left-handed circular polarization—that is, light whose electric-field direction twists as it travels—the brightness of the spots also differed. The fact that the structure does not overlap with its mirror image was reflected in the light pattern and its response to polarization.

So what specifically changes? The achievement lies in transforming mathematical nonperiodic tiles into a design for an experimentally usable material that manipulates light. What has been demonstrated so far is the confirmation of a phenomenon in a fabricated nanostructure, not a product or practical device. The research team says it will explore applications of new optical responses by incorporating the structure into metasurfaces and photonic crystals.