
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39289417
71566
10.1038/s41598-024-71566-1
Article
Skyrmion engineering with origami
Wakayama Toshitaka wakayama@saitama-med.ac.jp

1
Aizawa Kohei 1
Higuchi Yudai 1
Higashiguchi Takeshi 2
1 https://ror.org/04zb31v77 grid.410802.f 0000 0001 2216 2631 Faculty of Health and Medical Care, Saitama Medical University, 1397-1 Yamane, Hidaka, Saitama 350-1241 Japan
2 https://ror.org/05bx1gz93 grid.267687.a 0000 0001 0722 4435 Department of Electrical and Electronic Engineering, Faculty of Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585 Japan
17 9 2024
17 9 2024
2024
14 216737 6 2024
29 8 2024
© The Author(s) 2024
2024
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Skyrmion structures play critical roles in solid-state systems involving electric, magnetic and optical fields. Previous approaches to the study of skyrmions have involved specific structures in magnetic materials, liquid crystals and polymers in addition to two-dimensional arrays used for electrical control. These methods have encountered limitations and constraints on both the microscopic and macroscopic scales related to the physical properties of materials. The present work demonstrates an origami-based skyrmion engineering strategy that suggests a new approach to topological control. This technique utilizes the unique properties of orientational origami, combining polarization techniques with rotationally symmetric, periodically folded designs. This strategy enables the transformation of flat sheets into three-dimensional structures with associated changes in optical topology, similar to the characteristics of proteins. Topological defects such as misalignments and dislocations in folded molecularly oriented sheets lead to the creation of skyrmion clusters at boundaries having different orientational orders. The strategy reported herein involves the construction of unique metamaterial platforms that could provide new applications for twistronics in graphene and photonic crystals.

Subject terms

Mechanical engineering
Mechanical properties
Optical manipulation and tweezers
http://dx.doi.org/10.13039/501100001691 Japan Society for the Promotion of Science 21H03842 http://dx.doi.org/10.13039/501100005308 Amada Foundation AF-2023226-B3 Wakayama Toshitaka http://dx.doi.org/10.13039/100008732 Uehara Memorial Foundation 2022 Higashiguchi Takeshi issue-copyright-statement© Springer Nature Limited 2024
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pmcThe concept of topology, which interconnects the natural sciences1–3 and the arts4,5, is associated with exploration of the universality that underlies shape and structure. Topological strategies have improved our understanding of the natural sciences1–3 and led to transformative innovations involving novel materials6–11 and advanced technologies12. Folds, connections, boundaries and holes all play crucial roles in the manipulation of mechanical or electromagnetic topologies and periodic structures13–17. Such manipulations can affect topological properties and may also provide information concerning the manner in which continuous deformation changes topological properties. Origami and kirigami engineering18 in particular suggest the possibility of exotic topologies in the fields of graphene twistronics19,20 as an approach to tailoring the behavior of electrons and light21,22 and producing metamaterials to control various forces23. Skyrmions are vortex models having topological features that are important in specific fields of study and have potential applications in materials science, information engineering and optics. Previous research has enabled the manipulation of skyrmions based on the implementation of two-dimensional (2D) arrays24 having micro-periodic structures serving as fixed singularities25–27. Recent studies have induced Pancharatnam-Berry phases and spin–orbit interactions by using topological vortices within the momentum space of polarized light surrounding bound states across two-dimensional periodic structures28. Additionally, active control of optical skyrmions through thermally driven surface topography of polymerizable liquid crystals has been achieved29. However, these approaches are constrained by the physical properties of the microstructures and limited by spatial scale constraints of several 10 μm, because of defects and discontinuities in liquid crystals. Meanwhile, mechanical metamaterials13–18, 23 have been geometrically designed on isotropic material substrates. Recently, polarization conversion has been performed using a fusion of electromagnetic metasurfaces and mechanical metamaterials such as kirigami30 and origami31–33. However, this is limited to spatially uniform mode conversion and does not yet support the conversion of modes within spatial optical topologies. The present work demonstrates a strategy for skyrmion engineering based on a design platform that fuses origami and polarization, using anisotropic materials to produce mechanical metamaterials. Specifically, exotic polarization topologies were formed via periodic structural deformations, including higher order skyrmions and skyrmion clusters. This work involved topological structure transformations that have not yet been achieved in work using metasurfaces or other methods. The size of the topology generated by our method primarily depends on the thickness of the molecularly oriented sheets. However, the topology created through origami-based skyrmion engineering can overcome spatial scale limitations by utilizing two-dimensional materials, such as graphenes and carbon nano-tube sheets. This research has the potential to allow the application of skyrmions in condensed matter physics. Our origami-based skyrmion engineering strategy enables active control over the spatial topology of light, which is anticipated to lead to developments in high-density optical memory, vortex-multiplexed optical communications, and new optical devices.

Our proposal was inspired by the concept of skyrmions in solid-state physics. While band diagrams are valuable for examining the topological properties of materials in solid-state physics, they are primarily limited to the absorption and reflection properties of light and do not directly aid in detecting polarization distribution in the topology of light in real space, such as optical vortices and polarization vortices. Therefore, we proposed an origami polarization calculation algorithm that derives the polarization pattern of light based on the molecularly oriented film in molecularly oriented sheets and the arrangement of their spatial elements, to determine the topological patterns of output polarization in Skyrmion Engineering. This research explored the possibility of using axisymmetric folding34 to induce topological polarization ordering as a means of generating skyrmions. To simplify the associated polarization calculations, we initially designed an origami featuring eight blades. This design allowed an assessment of the manner in which the spatial arrangement of the geometric net of the origami could be layered through folding. The results of this work demonstrated a spatial polarization morphology that arose from the relative molecular orientation order stemming from overlap (Fig. 1, Extended Data Fig. 1). The origami geometric net employed in this work comprised eight blades, each labeled with colors and numbers, connected by a “C” mountain fold and folded along all creases (Fig. 1a–c, Extended Data Movie 1). A pivotal aspect of these designs was the trapezoidal blade, which facilitated control of the topological modes to generate skyrmions through partial fold adjustments. This design represented the fundamental structure of the origami, such that the interplay between the folding of the molecularly oriented sheets and light polarization allowed elucidation of the polarization morphology (Fig. 1d, e).Fig. 1 The topological metamorphosis concept. (a) The origami geometry net. The mountain and valley folds are indicated by red and blue lines. The direction (φ = 0°) of the molecular orientational order of the sheet are represented here by the eight colored arrows. (b) Both ends, as indicated by line C, of the molecular-oriented sheet connected with non-oriented tape. (c) The sheet is folded into a valley fold at point O to form an angle θ and is then mountain-folded at points A and B to create a single blade. A unit is defined by the dimensions t, a and b and the angle θ (see Extended Data Figs. 1b and 1c). (d) Eight blades were designed and individually folded. Depending on the folding method, these structures transitioned to the first mode (d1) or the second mode (d2). (e) An enlarged view of blade number 3 as seen from the front of the flat fold depicted in the right column of (d1). Note that the interior of each blade is divided into eight elements. The origami was constructed via the rotationally symmetric assembly of the four elements enclosed within the red box. (f) The output polarization was calculated based on the folding structure of the four elements and the orientated directions of the overlapped sheets.

Here it is helpful to consider a simple example in which the incident polarization, sin, was calculated with horizontal polarization and a specific molecular orientational order with a retardance of Δ = 90°. Using Mueller matrix calculus, the light polarization output, sout, could be determined for each element in the blade as described in the Methods section, and these calculations provided the periodic polarization morphology (Fig. 1f). The present results confirmed that even a single uniaxial molecular-oriented sheet can produce a unique polarization morphology based on the spatially segmented elements of the blade. Using this strategy allowed an origami design to be generated that allowed the tailoring of skyrmions as topological structures.

This work also investigated the use of the present origami strategy to establish relative molecular orientation order within spatially coded areas to create skyrmions. Considering the significant impact of the origami sheet's orientation order magnitude on the polarization morphology, a detailed analysis was performed to examine the extent of specificity. In this process, single blade configurations (from which a repetitive structure forming an assembly of origami sheets could be derived) were used to provide from 1 to 28 spatially organized elements (Fig. 2a, b, Extended Data Fig. 2). In particular, it should be noted that the gray highlighted areas of elements 1, 2, 14, 19, 23, 26 and 28 in Extended Data Fig. 2 exhibit rotational symmetry. Each element had 24 blades stacked on top of one another, resulting in a different number of layers in each element (Fig. 2c). The orientational direction of each layer, presented as a color distribution in Fig. 2c, could be used to ascertain the polarization state. The ellipticity and azimuth distributions were mapped according to the orientation directions of the stacked molecularly oriented sheets (Fig. 2d1, d2) as linearly polarized light with incident polarization at 0°. In the case that the magnitude of the orientational order of the molecular-oriented sheet was Δ = 180°, the ellipticity and azimuth of each element showed an interesting distribution (Fig. 2d3). An optical beam passing through the origami also exhibited a characteristic topological polarization morphology, meaning that a higher order skyrmion structure was obtained (Fig. 2e1, e2, Extended Data Fig. 3). The orientation of the outgoing polarization rotated regularly while maintaining linear polarization, forming a topology of the order of n = 2.Fig. 2 Generation of a topological skyrmion using origami (a) The origami geometry net having 24 blades (mountain and valley folds are indicated by red and blue lines). When both ends of the structure shown in (a1) are glued together, a ring-shaped primary structure (a2) is formed. Repeated mountain and valley folds along the creases yield the flat fold shown in (a4) via the structure shown in (a3). (b). (a4) presents a front view of the red area, representing a single blade, which can be divided into as many as 28 elements depending on the folding process (Extended Data Fig. 2b). Consisting of eight basic elements, numbered 1, 2, 14, 19, 23, 26–28 (all of which are shown in grey in Extended Data Fig. 2b), this assembly can undergo rotational transformation to achieve axisymmetric optical properties, allowing these properties to be spatially determined. This figure indicates the number of sheets stacked in elements from 1 to 28 based on the folding structure. (c) Elements 1 to 28 can have up to 13 molecularly oriented sheets superimposed on one another. The direction of the orientational order of the molecularly oriented sheets resulting from this superposition is mapped in two dimensions. (d) The distributions of the ellipticity (d1) and the azimuth (d2) of the output polarization are mapped with an incident linear polarization of 0°. Here, the magnitude of the orientation order possessed by the molecular-oriented sheet was set to a retardance of Δ. The result of the azimuth at Δ = 180° is shown in (d3). (e) Interestingly, the ellipticity became zero (e1) and the azimuth varied linearly (e2). Two-dimensional mappings demonstrated a polarization topology having a polarization order of 2 with respect to the angle. (f) An origami produced by a retardation film. (g1–g6) Light intensity distributions imaged while rotating the achromatic quarter-wave plate in 30° intervals. (h) The analysis results: ellipticity (h1), orientation (h2), ellipticity and azimuth distributions along the white dashed lines in (h2). The ellipticity was almost zero whereas the azimuth changed in a staircase-like manner, resulting in a higher order skyrmion (h3).

Fig. 3 Topological metamorphosis through origami transformation (a) The topological mode transformation of origami with 24 blades. When this geometry net is modified with mountain and valley folds, the topological mode changes. If two mountain fold in the geometry net in (a1) are set to flat, a topology with mode m = 2 is generated (a2). If three mountain fold positions are changed from mountain fold to valley fold, a mode m = 3 topology shape is formed (a3). An investigation of the polarization morphology associated with the topological mode of the origami produced the results shown in (b1–c2). The 2D maps of ellipticity and azimuth at mode m = 2 (b1, b2) and the 2D maps of ellipticity and azimuth at mode m = 3 (c1, c2) are shown. A more detailed inspection of the azimuth distribution at mode m = 2 reveals that a ± 1/2 topological vortex associated with a second-order polarization was generated, thus producing skyrmion clusters. Scar bar is 20 mm.

Based on the present numerical calculations, origami specimens were fabricated using retardation films comprising molecularly oriented sheets (Fig. 2f). The origami specimens were subsequently placed in a polarization imaging system (Extended Data Figure 4a) and images were captured for each angle of rotation of the achromatic 1/4 waveplate (Fig. 2g1–g6). The origami produced a range of topological variations in polarization, akin to a kaleidoscope, depending on the magnitudes of orientational order exhibited by the molecular-oriented sheet. Polarization imaging showed that the ellipticity was almost zero but the azimuth underwent stair-step-like variations (Fig. 2h1–h3). These results demonstrate the generation of polarization topologies with n = 2, meaning the formation of a higher order skyrmion. This strategy therefore provides new perspectives on the design and application of optical materials.Fig. 4 Skyrmion structures with a polarization vortex of ± 1/2 order. (a) Enlarged views of the image in Fig. 2h2, showing CW (clockwise) (a1) and CCW (counter-clockwise) (a2) images. (b) The angle variations of the azimuth along the white dashed lines in (a1) and (a2) are shown: a 180° change in the azimuth of the polarization was obtained for a single revolution. The number of such changes was investigated with the results shown in Figs. 2 and 3. (c) Distributions of the number of skyrmion clusters with ± 1/2-order vortices caused by metamorphosis of the origami. The folding structure exhibits the fewest defects when in the first-order topological mode, as this mode was the most stable. Conversely, defects were most prevalent when the topological mode was of the second order during the metamorphosis of the origami metamaterials because this structure was the least stable. Pixel size is 0.35 mm in (a1) and (a2).

Softness is a key characteristic of origami and allows for protein-like metamorphoses. As an example, a slight adjustment to an origami fold can induce significant variation in the polarization morphology. The shape of the present origami changed from a flat fold to a three-dimensional (3D) structure with such variations (Fig. 3a) and folding also changed the structure from the 1st through 3rd modes. This work additionally investigated variations in polarization morphology with changes in mode (Fig. 3b, c). The polarization morphologies used in this work corresponded to structures having 8 and 16 blades (Extended Data Fig. 3) and the molecular orientational order was found to exceed λ/2 because of the effect of the tilt angle (Extended Data Figs. 5 and 6).

The work reported herein investigated origami with 24 blades. However, this origami had the potential to adopt more sophisticated polarization morphologies based on increasing the number of blades. Such dynamic topological skyrmion changes in polarization morphology suggest a new level of active control that could not be achieved with previous metasurfaces. These experiments also suggest another interesting possibility for origami. Specifically, the forces acting on the origami were somewhat unstable because a number of the initially designed folding directions were changed. These changes produced torsion that resulted in a transition from a plain weave to a 3D structure (Fig. 3, Extended Data Fig. 5). This twisting, in turn, caused slight misalignment, slippage and dislocation in the sheet overlap. Defects in the origami thus gave rise to exotic polarization topologies to produce a skyrmion cluster.

The topologies associated with defects in the present origami occurred not only in 3D structures (Fig. 3) but also in flat folds (Figs. 2 and 4a). The topological symmetry appeared as a polarization vortex of ± 1/2 order35,36,37 (Fig. 4b). Because it is challenging to identify localized polarization topologies, a quantitative search was conducted by performing 2D matching, meaning a vortex correlation analysis, while varying the initial phase of the ± 1/2-order polarization vortex (see Methods for details and Extended Data Fig. 7). The number of defects was highest for the topological mode m = 2 (Fig. 4c) and the folding became significantly twisted and unstable in conjunction with this mode. This led to a displacement in the folding structure, resulting in the spatial region being coded in a more finely and complex manner and generating microscopic polarization anisotropy. These variations were particularly evident at the boundaries of at least two different molecular-ordered regions. As light propagated through the origami, it underwent a vector beam self-healing effect that contributed to the formation of a smooth polarization topology.

This research demonstrates systematic metamorphoses of polarization topology created by folding structures and the orientational order of molecularly oriented sheets and shows new possibilities for the topological control of skyrmions10,11, 21, 22. This strategy is not limited to birefringence in transparent materials with molecular orientational order but could also be applied to reflectivity38, absorption39, dichroism40, optical activity21,41 and even non-linear optical effects42. The concept of origami metamaterials can be realized on different scales and with various morphologies and so could also be applied to graphene twistronics19,20. In addition to suggesting new topological physical phenomena such as skyrmion clusters35-37, origami metamaterials are also expected to contribute to the design of novel optical elements1,10–13, 18, 24–27, 35- 37, 41–43, 47, 48, including next-generation high-efficiency solar cells39,49 and high-density optical memory50,51. This concept could also be employed to improve the performance of vortex-multiplexing optical communication systems52.

Methods

Creation of origami geometry nets

The structure of the origami presented herein allowed the rotation of molecularly oriented sheets in an axisymmetric manner. The smallest unit of the origami geometry net consisted of a rectangle having minor and major axes of t and a + b, respectively, as shown in Fig. 1c. A crease with angle θ was made at point O. Each side of the rectangle involved a mountain fold and the crease at angle θ was a valley fold. Several of these folds could be combined to design an origami geometry net. To determine the origami structure, the blade sizes a and b were normalized by w = L/N = a + b, varying the length t, where L and N are the total length of the origami geometry net and the number of blades, respectively. The angle θ was determined by N according to the equation1 θ=N-2N·90∘.

In Extended Data Fig. 1b, θ is shown for N values of 3 to 32.

The lengths a (shown in green) and b (in red) were affected by varying the length of the short axis t. The values of a and b after normalization by the length of the unit, w, were respectively defined as2 a=w2+ttanθ

and3 b=w2-ttanθ.

The shape of the origami varied according to the value of t (Extended Data Fig. 1c and 1d) and the green and red lines in Extended Data Fig. 1c indicate a and b, respectively. The present work developed five designs. From these, we selected a shape that could be divided into three radial regions each with a normalized thickness of t = 0.3. The 3D illustrations were created using the ORI-REVO software package.

The direction of orientational order

The overlap of molecularly oriented sheets in origami will affect the polarization morphology. Therefore, this work investigated the manner in which the overlap of the molecularly oriented sheets varied. Based on an enlarged illustration of one of the 24 blades, as shown in Extended Data Fig. 2b, the folding of 28 elements was examined while varying the number of sheets. Elements, Ω, from 1 to 13, 14 to 18, 19 to 22, 23 to 24 and 26 to 27, along with element 28, exhibited regular changes in their direction of orientational order, φ(Ω, k), that were dependent on the number of sheets and the sheet positions, as observed in the variations described below.

The orientation directions, φ(Ω, k), for various Ω ranges were defined as4 φΩ,k=π12⌊Ω2⌋+⌊k2⌋-π2

for Ω from 1 to 13,5 φΩ,k=π12⌊Ω-14⌋+⌊k2⌋-π3

for Ω from 14 to 18,6 φΩ,k=π12⌊Ω-19⌋+⌊k2⌋-π4

for Ω from 19 to 22,7 φΩ,k=π12⌊Ω-23⌋+⌊k2⌋-π6

for Ω from 23 to 25,8 φΩ,k=π12⌊Ω-26⌋+⌊k-12⌋-π12

9 φΩ,k=π12⌊Ω-28⌋+⌊k-12⌋-0=0

for Ω = 28.

Here, ⌊ ⌋ denotes the floor function. Note that Eqs. (1)–(3) are for an odd number of folds whereas Eqs. (4) and (5) are for an even number.

Simulation of origami polarization calculus

The magnitude of the molecular orientational order (that is, the retardance Δ) for each origami was given the symbol Δ while the direction of orientational order (the slow axis) was denoted as φ. In the case of the Muller matrix calculations, the polarization calculations were then defined as5310 SoutN,Ω=MΔ,ϕn⋯MΔ,ϕ5·MΔ,ϕ4·MΔ,ϕ3·MΔ,ϕ2·MΔ,ϕ1·Sin=∏k=1nMΔ,ϕk·Sin

11 MΔ,ϕk=100cos22ϕk+sin22ϕkcosΔλ00cos2ϕksin2ϕk1-cosΔλsin2ϕksinΔλ0cos2ϕksin2ϕk1-cosΔλ0-sin2ϕksinΔλsin22ϕk+cos22ϕkcosΔλ-cos2ϕksinΔλcos2ϕksinΔλcosΔλ

and12 Sin=1,1,0,0,

where N, Ω, R, n, k, Δ, M and φ are the blade number, element number, rotation matrix, maximum number of stacks, number of stacks, magnitude of the orientational order of the molecular-oriented sheet, Muller matrix and orientation direction of the molecular-oriented sheet, respectively. The calculations between Muller matrix and Stokes vectors M can be expressed as Eq. (10). All polarization calculations were performed using the Mathematica 13 software package (Wolfram).

Fabrication of origami for skyrmions

A retarder sheet (product #14-359, Edmund Optics) with a retardance level of 280 nm at 560 nm was employed as the molecular-oriented material to be folded in this work. The origami was assembled by folding this sheet according to a predefined geometry net in which red and blue lines designated mountain and valley folds, respectively (Fig. 2a1). The dimensions of each fold (that is, each blade) were set to a normalized thickness of t = 0.3, as shown in Fig. 1 and Extended Data Fig. 3. Optical adhesive sheets (MCS62, MeCan Imaging Inc.) were employed to join the components of the origami and the folding sequence alternated between mountain and valley folds along the designated creases. To visually demonstrate the folding sequence, the process was captured on video using photographic paper (Extended Data Movies 1, 2 and 3).

Polarization imaging

The rotating quarter-wave plate technique was employed to acquire images showing changes in polarization morphology. This involved assessing the spatial distributions of the Stokes vector sin = (s0, s1, s2, s3), where s0, s1, s2 and s3 are the Stokes parameters. The spatial distributions of the four Stokes parameters were obtained by capturing intensity images while rotating the achromatic quarter-wave plate in 10° increments. The polarization information, including the Stokes parameters, was encoded in the intensity profiles as a function of the rotation angle. Using Mueller matrix calculus54, the intensity distribution was determined as13 s0x,y,λ,θ=Ix,y,λ,θ=∑k=0∞αkicoskθ+βkisinkθ=142s0x,y,λ+s1x,y,λ-2s3x,y,λsin2θ+s1x,y,λcos4θ+s2x,y,λsin4θ

The Stokes parameters were obtained using a discrete Fourier transform method55. The four Stokes parameters were calculated on the basis of the associated amplitudes by Fourier analysis as14 s0x,y,λ=4α0x,y,λ-s1x,y,λ2,

15 s1x,y,λ=-2β4x,y,λ,

16 s2x,y,λ=4α4x,y,λ

and17 s3x,y,λ=4β2x,y,λ.

These parameters were subsequently employed to ascertain the 2D distributions of the ellipticity, ε, and the associated azimuth, ϕ, via the equations5418 εx,y,λ=s3x,y,λs0x,y,λ+s1x,y,λ2+s2x,y,λ2

and19 ϕx,y,λ=12tan-1s2x,y,λs1x,y,λ.

The application for polarization imaging was created using the Python 3.12 software package. A light board (MLT-A3N, Mutoh Industries) was employed as the light source, as shown in Extended Data Fig. 4a, along with an achromatic quarter-wave plate (AQWP10M-580, Thorlabs), a polarizer and a CMOS camera (UI-3590CP-C-HQ, iDS). Intensity distributions were assessed as functions of the rotation angle of the achromatic quarter-wave plate (Extended Data Movies 4, 5 and 6).

Three-dimensional surface profile measurements of origami metamaterials

As the origami structure transitioned from a flat fold to a 3D morphology, the shape changes were monitored using a 3D surface profile measurement system (MINI, Revopoint) based on pattern projection. The data cloud was acquired using a rotating stage (Dual-Axis-Turntable, Revopoint) linked to a 3D camera system generated using a 3D construction application (RevoScan 5, Revopoint). The 3D profiles illustrating the origami transformations can be seen in Extended Data Fig. 5. Inclination angles were derived from these point cloud data.

The effect of the inclination angle resulting from the origami transformation on the retardance of the sheet was also investigated and the relationship between retardance and the tilt angle is summarized in Extended Data Fig. 6. The results of the polarization topology metamorphoses induced by the origami transformations are presented in Fig. 3 and can be seen to be significantly more complex than the ellipticity and azimuth distributions presented in Fig. 2d. This complexity is attributed to variations in the retardance with changes in the slow axis angle. Given the complexity of these calculations, it is anticipated that future work will lead to even more interesting and exotic topologies. Despite these challenges, it was possible to observe unique variations of the polarization topology during the present experiments. The relationship between topological changes and wavelength was also examined and the results are presented in Extended Data Figs. 4, 8 and 9.

Vortex correlation analysis

A vortex correlation analysis was performed to quantify the number of skyrmions having ± 1/2-orders vortices. Because the azimuth distribution provided angular information, a conventional 2D correlation algorithm could not be used. Therefore, a vortex correlation analysis was developed to analyze the azimuth distribution (using an image size of 640 × 480 pixels with pixel reduction) obtained by polarization imaging. Numerical skyrmion vortices with 1/2-orders and sizes of 7 × 7 pixels were created and matched with the azimuth distributions having sizes of 640 × 480 pixels. The vortices were matched in two ways, either CW or CCW, by varying the initial phase of the azimuth distribution φth(x, y) with a size of 7 × 7 pixels from 0° to 360° in 1° intervals. The vortex correlation method removed the azimuth distribution data with a size of 7 × 7 pixels from the distribution having a size of 640 × 480 pixels and ascertained the difference, Δφa(x, y), between the local vortices, φEX(x, y), with 1/2-orders and sizes of 7 × 7 pixels using the equation20 Δφax,y=2φExx,y-φthx,y.

This process gave the relative angle considering reverse rotation as21 Δφbx,y=360∘-Δφa.

A minimum function was then used to select the smaller of Δφa(x,y) and Δφb(x,y) as22 Δφx,y=minΔφa,Δφb.

This value was normalized by π to calculate the arc length on the unit circle, resulting in the equation23 Lx,y=Δφx,y180.

These steps were performed over a total of 49 pixels. To increase the degree of change, the reciprocal of the area-divided value was defined as the correlation value. This was written as24 Cx,y=49∬Lx,ydxdy2.

This calculation process was carried out on the azimuth distribution with a size of 640 × 480 pixels to obtain a correlation map with a size of 634 × 474 pixels (Extended Data Fig. 9).

Supplementary Information

Supplementary Movie 1.

Supplementary Movie 2.

Supplementary Movie 3.

Supplementary Movie 4.

Supplementary Movie 5.

Supplementary Movie 6.

Supplementary Figures.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-71566-1.

Acknowledgements

We thank N. Aizawa for assisting with the assembly of the origami metamaterials shown in Fig. 2 and A. Zama for assisting with the polarization imaging shown in Extended Data Fig. 6. T.W. and T.H. acknowledge support from the Japan Society for the Promotion of Science (JSPS) (Grant Nos. 21H03842, 23H03656, 23H01416, 24K03312, and 24H00838), the Uehara Memorial Foundation (2022) and the Amada Foundation (Grant No. AF-2023226-B3).

Author contributions

T.W. and K.A. designed the sample structure. T.W., K.A. and T.H. discussed the experiments. K.A. fabricated the sample and performed the experiments. T.W. and K.A. analyzed the experimental data and performed the origami polarization simulations. T.W. and Y.H. developed the vortex correlation method. T.W., K.A. and Y.H. generated the figures. T.W. and T.H. wrote the manuscript. All authors interpreted the results and reviewed the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request and online.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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