Chiral diffraction from aperiodic monotile structure

Periodic systems have been the central focus of research across various disciplines, including physics and chemistry. However, aperiodic systems have attracted significant attention due to their unique properties and phenomena that cannot be observed in periodic systems. In mathematics, studies on structures that can only tile the plane aperiodically have a long history. Penrose discovered a set of two different tiles, known as Penrose tiles, that can only form aperiodic tilings¹. The Penrose tilings are classified as quasiperiodic structures which do not have periodicity but exhibit long-range order². Mathematical studies on such quasiperiodic systems eventually led to the discovery of quasicrystals by Shechtman³ and he received the Nobel Prize in Chemistry in 2011. This discovery had a groundbreaking impact on crystallography, fundamentally altering the definition of crystals⁴. The concept of quasiperiodic structures and quasicrystals has been applied to various fields. In nanophotonics, the concept of quasiperiodicity has been introduced in photonic crystals, photonic quasicrystals^5,6,7, leading to interesting photonic functionalities, such as the realization of complete photonic bandgaps^8,9,10,11 and laser oscillation^12,13,14,15. More recently, studies on metasurfaces with quasiperiodic arrangements^16,17 and topological photonic systems using quasiperiodicity^18,19 have highlighted photonics as a versatile platform for both fundamental research and practical applications of quasiperiodic structures.

A natural mathematical question following the discovery of Penrose tiles is the existence of an aperiodic monotile, a single shape that can tile the plane only non-periodically. This problem, known as the Einstein problem, remained unsolved for 50 years after the discovery of the Penrose tiles. However, in 2023, Smith et al. discovered this aperiodic monotile, attracting considerable attention²⁰. The first identified aperiodic monotile, resembling the shape of a hat, has been named the Smith hat tile as shown in Fig. 1a. Smith discovered the Smith hat tile from the honeycomb framework (Fig. 1b) and it is possible to generate an aperiodic tiling, hat tiling, using the hat tile and its inverted structure (illustrated by a blue tile), as illustrated in Fig. 1c. Although the hat tiling employs two types of tiles related by space inversion, unlike the Penrose tiling, these tiles share the same shape. Therefore, the hat tile is considered the first discovered aperiodic monotile. Smith’s team subsequently discovered a truly single-type aperiodic monotile²¹, referred to as the specter tile. Whether this aperiodic tiling exhibits properties similar to quasiperiodic systems or possesses entirely new characteristics is an intriguing question. Many theoretical studies have been reported, and shown interesting properties originating from the aperiodic monotiles^{22,23,24,25,26,27,28,29,30,31}. However, an experimental study aimed at uncovering features due to the aperiodic monotile tiling remains unexplored.

In this study, we report an experimental investigation of optical diffraction from the aperiodic hat tiling. By using the hat tiling, we found a quasicrystalline structure which has threefold rotational (C₃) symmetry without mirror symmetry. We define a point set by placing a point at the centroid of each hat tile as shown in Fig. 1b, d. In this work, we use the term quasilattice to denote a non-periodic point set with quasiperiodic order. We therefore refer to the present centroid-based point set as the monotile quasilattice because it exhibits long-range quasiperiodic order. Typical quasicrystals exhibit noncrystallographic rotational symmetries forbidden in conventional crystals and lack the usual crystallographic symmetries (C₂, C₃, C₄, and C₆). In contrast, our monotile quasilattice with C₃ symmetry represents a distinct class of aperiodic structures. Furthermore, unlike quasicrystals, such as the Penrose structure, the monotile quasilattice is chiral, lacking mirror symmetry. In this paper, chirality refers to the absence of mirror symmetry within a two-dimensional plane, which is distinct from the conventional chirality defined in three-dimensional space. We show that these symmetry properties give rise to chiral diffraction patterns. Experimentally, we fabricated the aperiodic pattern on SiN films and observed optical diffraction. The observed diffraction patterns exhibited clear Bragg peaks that were insensitive to the illumination position, providing evidence of long-range order and experimentally confirming its quasiperiodic nature. We further find a pronounced dependence on the helicity of incident circular polarization, consistent with the structural chirality. Our results reveal the physical properties arising from the aperiodic monotile and extend the study of aperiodic systems beyond conventional quasiperiodic structures and quasicrystals.

The monotile quasilattice with C ₃ symmetry but without mirror symmetry

A hat tiling can be generated by using several inflation rules²⁰. In this study, we use the inflation rule using four initial metatiles, H, P, T, and F²⁰ as shown in Fig. 1e. Each metatile indicated by black lines is composed of one, two, or three hat tiles. By combining the initial metatiles, the first generation of metatiles H₁, P₁, T₁, and F₁ are generated recursively. Among them, the arrangement of metatiles forming H_n possesses exact C₃ symmetry. The early discovered quasicrystals were characterized by rotational symmetries, such as C₅ and C₁₀, which are forbidden in ordinary crystals. These unusual symmetries were long regarded as one of the most distinctive features of quasicrystals. However, in recent years, quasicrystals possessing rotational symmetries allowed in conventional crystals, such as C₃ and C₆, have also been reported ^32,33.

Another important feature of the hat tiling is that the metatiles are arranged with a rotation relative to the underlying honeycomb framework, as shown in Fig. 1f. When the inflation rule is applied to generate the H₁ metatile, the H₁ metatile is found to be rotated clockwise with respect to the underlying honeycomb framework (see Section S1.2 for details). This rotation breaks the mirror symmetry of the structure. Unlike conventional quasicrystals, such as the Penrose structure, the hat tiling is chiral and lacks mirror planes, which gives rise to the chiral diffraction patterns shown later.

To construct the monotile quasilattice, we define the point at the centroid of each T metatile as shown in Fig. 1b. These points form the aperiodic pattern shown in Fig. 1d. Unlike the original hat tiling, the resulting point pattern acquires exact C₃ symmetry at its center because the detailed shape information of the individual hat tiles is removed. In Fig. 1d, the C₃ center of the monotile quasilattice is colored by red, and the colored points surrounding it coincide exactly when the whole structure is rotated by 2π/3. We note that although the hat tiling does not possess perfect C₃ symmetry, it exhibits weak C₃ rotational symmetry (see section S1.1 for further details). Here, “weak” means that, when the aperiodic tiling is extended infinitely, the set of points that break the rotational symmetry has a scale smaller than the spatial dimension of the tiling. Importantly, the pattern preserves the lack of mirror symmetry inherited from the hat tiling, which leads to the chiral diffraction behavior shown later.

Experimental observation of chiral diffraction

For experiments, we fabricated aperiodic structures based on the monotile quasilattice and carried out optical diffraction measurements. A 350-nm-thick SiN film deposited on a Si substrate was patterned using electron beam lithography and etching to create circular holes on the quasilattice of H₆ metatile structure. We define a pseudo-period a by the period of the underlying honeycomb frame (Fig. 1b), and choose a ranging from 600 to 750 nm. To investigate the chiral nature of the monotile quasilattice, mirrored structures of H₆, \(\bar{{H}_{6}}\), were also fabricated as shown in Fig. 2b. In the mirrored hat tiling, the entire structure is reversed, resulting in a hat tiling predominantly composed of blue hat tiles. Typical optical and scanning microscope images of the fabricated H₆ and \({\bar{H}}_{6}\) metatile structures are shown in Fig. 2b. In the case of H₆, the number of holes is 372,100 and its area is approximately 500 μm × 500 μm. In diffraction experiments, a laser was normally incident to the sample surface, and the diffraction pattern in the reflected direction was projected onto a white screen and captured with a camera (see Method for details). Because of the sample holder, a portion of the diffraction pattern around the image center is obscured. To clarify the diffraction patterns in wavenumber space, the real-space axes of the captured images were transformed into wavenumber axes through a coordinate transformation (see Method for details). The wavenumber axis is normalized by the pseudo-period a and the wavelength of the incident laser λ (k₀ = 2π/λ) for direct comparison between experiments and calculations. Figure 2c shows the diffraction pattern under illumination with a white-light source (a supercontinuum laser). A distinctive pinwheel-like diffraction pattern is observed. The observed diffraction pattern clearly lacks mirror symmetry, demonstrating the inherent chirality of the monotile quasilattice. Although the diffraction angles depend on the incident wavelength and the white light is spectrally dispersed, the chiral pattern itself remains unchanged across all wavelengths. Figure 2d, e show the diffraction patterns of H₆ and \(\bar{{H}_{6}}\) metatile structures when a green laser (λ = 532 nm) is injected. Numerous sharp diffraction peaks were clearly observed, indicating the long-range order of our quasilattice. The diffraction pattern exhibits C₆ symmetry reflecting the C₃ symmetry of the structure due to Friedel’s law³⁴. The diffraction patterns obtained from mirrored structures exhibited a reversal of chirality, as shown in Fig. 2e. A related theoretical study reported chiral diffraction in a hat-tiling system ³¹. However, that work considered diffraction from a key tiling based on a nearly mirror-symmetric hexagonal framework, in which the chirality is manifested mainly in the intensity distribution of the diffraction peaks. By contrast, our quasilattice is characterized by C₃ symmetry without mirror symmetry, and the resulting diffraction exhibits a pinwheel-like arrangement.

The positions of the observed diffraction peaks do not depend on the position of the incident laser spot, providing important evidence that the monotile quasilattice possesses quasicrystalline order (see section S2 for details). While a quasicrystal shows perfect rotational symmetry only around specific points and not elsewhere, its characteristic diffraction pattern reflecting that symmetry can be observed irrespective of the illumination position. This invariance originates from the weak rotational symmetry that extends over the entire structure, representing a fundamental aspect of quasicrystalline order. Indeed, our quasilattice exhibits weak C₃ symmetries around multiple points (see section S1.1 for details). The observed distinct Bragg peaks, whose positions are independent of the illumination point, serve as clear experimental evidence of the quasicrystalline order of the monotile quasilattice.

To analyze the observed diffraction patterns, theoretical calculations were performed. By placing delta functions on the monotile quasilattice and performing a Fourier transform, diffraction patterns can be calculated²³. The Fourier amplitude of the diffraction patterns is written as

Here, k_x and k_y are the wavevector components, and x_i and y_i are the coordinates of each point. The origin of the coordinate system is set to the center of the H-metatile. Diffraction intensity is obtained by I = ∣F∣². We computed I at each k-point for H₆ metatile structure, as shown in Fig. 2f. In the calculated results, the intensity I was normalized using the value at the origin of wavevector space. Figure 2g, h are magnified views of 2d, f, respectively. It is evident that the positions of the measured diffraction peaks are in good agreement with the calculations (for direct cut line comparison, see S3 in Supplementary Information). In the calculations, delta functions are used, which result in strong diffraction peaks even at large wave vectors. In contrast, in the experiment, the holes have a finite size, leading to a reduction of the diffraction peak intensity as the wave vector increases. Logarithmic plot and wide wavenumber view of Fig. 2f are presented in section S4.1. In Fig. 2f, asterisk-like features can be observed at positions with C₆ symmetry (see Fig. S6d). The centers of the asterisk-like features coincide with the positions of diffraction peaks of the honeycomb lattice since the monotile quasilattice can be considered as the honeycomb lattice with lattice defects (see section S4.1). The essential properties of the hat-tiling structure do not depend on the specific choice of metatiles. Therefore, the chiral diffraction patterns and circular-polarization dependence observed in this study appear in quasilattices constructed from metatiles other than the H metatile.

Origin of chiral diffraction patterns

The origin of the chiral pinwheel structure in the diffraction patterns is rotation of the asterisk-shaped structure. We found that this rotating angle θ_chiral can be analytically calculated from the twisting angle inherent in the hat tiling. As shown in Fig. 3a–c, the metatiles of the hat tiling gradually tilt away from the underlying honeycomb frame as the inflation rule is applied. To quantitatively evaluate this twisting, we consider the vector \({\vec{A}}_{n}\) drawn by red arrows in Fig. 3a–c. The \({\vec{A}}_{n}\) points from the center of H_n to the center of H_n−1 metatile indicated by the red line. It was found that when vector \({\vec{A}}_{n}\) is expressed in terms of \({\vec{a}}_{1}\) and \({\vec{a}}_{2}\), the coefficients A_n1 and A_n2 follow the Fibonacci sequence F_n+2 = F_n+1 + F_n (see section S1.2 for details). Therefore, we can calculate θ_chiral by the triangle as shown in Fig. 3d. This triangle is composed of sides whose lengths correspond to the 2n + 1-th and 2n − 1-th terms of the Fibonacci sequence. In the limit of \(n \rightarrow \infty\), their ratio F_2n+1/F_2n−1 converges to ϕ². The θ_chiral calculated using the triangle is analytically given by

Figure 3e is a magnified view of the asterisk-shape region in the calculated diffraction patterns in a logarithmic scale. The white dashed lines are eye guides to indicate the line with an angle of θ_chiral. It is evident that the diffraction peaks forming the asterisk align with the direction defined by θ_chiral. This fact means that the geometric features of the hat tiling in real space are directly reflected in the structure of diffraction in wavenumber space. Closer inspection of the diffraction pattern shown in the inset of Fig. 3e (the light-green rectangle) reveals that the relatively weak peaks do not align strictly along straight lines but instead exhibit a meandering arrangement. This feature is observed in the diffraction patterns for other generations of H metatiles (see section S4.2). Moreover, we found self-similarity characterized by the golden mean ϕ, which is observed in Fibonacci lattice and other quasiperiodic structures³⁵ (see section S4.3).

Helicity dependence of incident circular polarization

To further investigate the diffraction properties of the monotile quasilattice, we measured the diffraction pattern under circularly polarized incident light (see section S5 for other polarization dependence). In systems, such as the honeycomb lattice and the Penrose tiling, which possess not only rotational symmetry but also mirror symmetry, Bragg peaks appear only at wave vectors that respect these symmetries. Consequently, no dependence on the helicity of the incident circular polarization emerges. On the other hand, the monotile quasilattice possesses rotational symmetry but lacks mirror symmetry, allowing Bragg peaks to appear at various wave vectors and these peaks can depend on the helicity of the incident circular polarization. To evaluate the degree of circular polarization, we calculated

from measured diffraction patterns. Here, I_LCP and I_RCP are the intensities of diffraction for left-handed and right-handed circular polarization incident, respectively. For these measurements, a monochromatic camera with a wide dynamic range was used.

Figure 4a, d present η maps obtained by illuminating H₆ and \({\bar{H}}_{6}\) metatile structures (a = 700 nm) with circularly polarized light, respectively. In Fig. 4, a two-dimensional moving average has been applied using data from the surrounding 5 × 5 pixels to enhance the visibility of the red and blue peaks. As shown in Fig. 4a, d, the diffraction pattern exhibits an intriguing dependence on incident circular polarization. To make the complex distribution of the red and blue peaks more visible, the positive and negative regions of η are plotted separately in Fig. 4b, c, e, f. By comparing Fig. 4b, c, the shapes formed by the red and blue peaks are distinct from each other upon closer inspection. Moreover, these shapes are interchanged for the mirrored structure, as shown in Fig. 4e, f. Figure 4b, c, e, f share the same shape. The same features are also observed in other cases of a (see section S5). However, interestingly, the shapes of the red or blue peaks are different for each a. This result indicates that the observed circular-polarization dependence depends not only on the properties of the monotile quasilattice but also on other parameters, such as the pseudo-period and the diffraction angle.

The observed circular-polarization dependence has never been observed in conventional quasicrystals with inversion symmetry, but is a phenomenon inherent to the chiral structure of the monotile quasi lattice. If the structure has C₆ symmetry, such circular-polarization dependence does not appear unless chiral structures are placed at the lattice points. To induce circular-polarization dependence, it is necessary to break inversion symmetry and lower the symmetry of the unit cell from C₆ to C₃, as in valleytronic systems. In contrast, in our monotile quasilattice, we globally reduce the overall symmetry of the structure to C₃ by introducing quasiperiodic defects into a honeycomb lattice, providing a route to realizing circular-polarization dependence.

Experimental setup and analysis

In the diffraction measurements, a collimated laser beam was normally incident on the sample, and the diffraction pattern was captured by a camera, as shown in Fig. S1a. To obtain Fig. S1c, a supercontinuum laser (SuperK EVO, NKT Photonics) was used. For monochromatic measurements (Figs. 2 and 4), the laser diode with a wavelength of 532 nm (CPS532-C2, Thorlabs) was used. The spot size (diameter) of the green laser is about 1 mm. To record images, we used a color camera (RX100 VII, SONY) and a monochromatic camera (Kiralux CS126MU, Thorlabs) with a camera lens (MVL25M43, Thorlabs). The polarization of the incident light was controlled using a polarizer and a waveplate (Fig. S1b). To convert the vertical and horizontal axes of the captured image into wavevector space, a coordinate transformation was performed using the equations in Fig. S1c. We set L and k₀ to be 75 mm and 2π/(532 nm), respectively. Here, x and y are obtained from recorded images. Figure S1d shows a typical image before and after the coordinate transformation.

The data which support the figures and other findings within this paper are available in Figshare (https://doi.org/10.6084/m9.figshare.29313743).

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Authors and Affiliations

1. Department of Physics, Tokyo Institute of Technology, Meguro, Japan

   Yuto Moritake & Masaya Notomi

2. Institute of Industrial Science, The University of Tokyo, Meguro, Japan

   Yuto Moritake

3. Basic Research Laboratories, NTT, Inc., Atsugi, Japan

   Masato Takiguchi & Masaya Notomi

4. NTT Nanophotonics Center, NTT, Inc., Atsugi, Japan

   Masato Takiguchi & Masaya Notomi

5. Device Technology Laboratories, NTT, Inc., Atsugi, Japan

   Takuma Aihara

Contributions

Y.M. wrote the manuscript, designed the structure, and carried out all measurements, calculations, and analysis. M.T. and T.A. carried out sample fabrication. All authors discussed the results. M.N. supervised the project.

Corresponding authors

Correspondence to Yuto Moritake or Masaya Notomi.

Competing interests

The authors declare no competing interests.

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

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