1. Introduction

Color in nature is mainly derived from both pigments and structural color. While traditional pigments have long played a vital role in various aspects of human life, their widespread application is hindered by inherent limitations, such as toxicity, poor stability, and environmental unfriendliness. In contrast, structural color has the advantages of high spatial resolution, long-term stability and environmental friendliness prompting many scholars to investigate [1]. Structural color is generated from light scattering, absorption, diffraction or interference resulting from the interaction between incident light and nanostructures [2–4]. Structural color have been successfully applied to anti-counterfeiting [5] and information encryption [6] in addition to color decoration [7] because of its excellent properties. Jiao Geng et al. [8] proposed a method for high-speed laser-written structural color of full-color inkless printing. This technique addresses the pressing demands of low-cost, large-scale, and flexible manufacturing, the method provides a powerful technique for the broad prospects of structural color in practical applications.

Nanostructures generate color by manipulating resonant light at visible wavelengths through different geometries. Through proper design and arrangement, metasurfaces can arbitrarily control the amplitude, phase and polarization of electromagnetic waves. Metasurface devices consisting of dielectric nanostructures can provide high-resolution and low-loss full-color displays [9]. The Mie resonance wavelength of dielectric nanostructures, closely tied to material properties and structure geometry, significantly influences device performance [10]. However, traditional forward design method for metasurfaces is a trial-and-error process based on numerical simulation. It firstly needs to determine the material and structure based on intuition and physical effects, then experiment with aspects such as composition and geometry of the structure. The process requires complex electromagnetic simulations, leading to time-consuming and inefficient design approach for multi-objective optimization. Therefore, the inverse design method ensures the accuracy and speed of identifying structures that achieve the desired color. It is based on the established mapping relationships and can directly provide geometric parameters, greatly accelerating the design of optical devices and significantly improving the efficiency of design. The inverse design optimization algorithm has been widely employed in the design of various types of structure-colored devices through intelligently finding the structure corresponding to the realization of a specific function [11–14].

The commonly used inverse design optimization algorithms include genetic algorithms (GA) [15–17], particle swarm optimization (PSO) [18,19], and deep learning [20–22]. These algorithms utilize inputting the target color and associated constraints through intelligent optimization in conjunction with electromagnetic simulation tools like finite-difference time-domain (FDTD) to determine the corresponding structural design. C. Liu et al. [15] used genetic algorithms to design a multiplexed metasurface, which each unit structure consists of multiple meta-atoms. This approach differs from traditional forward design methods that typically utilize unit structures with only one meta-atom, thereby limiting design freedom and device performance. Utilizing algorithms improves the device's performance while circumventing the impracticality of traditional design methods when dealing with a large number of parameters. Meanwhile, Q. Guan et al. [23] used machine learning to inversely design a multifunctional structure consisting of a selective emitter on top of a stepwise nanocavity. With the aid of algorithms, this kind of device design with a large design parameter space and multi-realization goals not only speeds up the design process but also makes the device achieve great flexibility among various functions. Therefore, algorithms provide an avenue for fundamental research and application development in optics-related fields.

The typical optimization algorithm design heavily relies on computational power and entails lengthy iterations of complex, variably dimensioned structural simulations. This leads to significant time consumption, especially when high simulation accuracy is required. Deep learning has gained popularity for its rapid and high-accuracy capability in device inverse design. However, the deep learning-assisted design process necessitates prior simulations and large datasets, making it less efficient compared to simpler models when dealing with systems consisting of a small number of features. For systems with a small number of features, utilizing the tensor completion algorithm is relatively efficient. Such algorithms can rapid and precise predictions when operating on datasets that are only partially acquired [24–27]. Tensor is a high-dimensional generalization of matrices and vectors, represented as multidimensional arrays. Tensor excels at capturing spatial structures inherent in practical data and based on its multilinear data analysis, enabling a comprehensive understanding and precise information extraction. The core problem of tensor completion algorithm is to estimate the missing data using known data and has found extensive application in color image and video restoration [28–31], hyperspectral image recovery [32–34], seismic data reconstruction [35], and higher-order network link analysis [36,37]. S. Etter et al. [38] proposed a parallelization strategy for the hierarchical Tucker combined with alternating least squares (ALS), which distribute the vertices of the network and their corresponding computations across an array of distributed memory processors. They obtain a linear system suitable for parallelization and effectively address the time and memory consumption of single computational nodes. Similarly, L. Zhang et al. [39] presented a method for hyperspectral image (HSI) compression and reconstruction leveraging multidimensional or tensor data processing techniques. Specifically, they employed the Tucker decomposition in tandem with alternating optimization, representing the observed HSI cube as a third-order tensor, and reconstruction was achieved via multilinear projection. Experimental results for specific applications of hyperspectral remote sensing images indicate that the reconstructed HSI data utilizing this approach exhibits high quality. The higher-order tensor is promising for modeling data with multidimensional nature, so it is possible to apply it to structural color design that is closely related to structural geometry, providing an effective method for the inverse design of devices. This paper models reflectance spectra and geometric parameters as a tensor and formulates the device inverse design problem as a low-rank tensor completion problem. Specifically, by combining Tucker factorizations and ALS to capture the geometry-color relationship in structural colors, thereby to predict the spectral data of other geometric parameters based on partial spectral data. To the best of our understanding, this work is the inaugural application of this tensor schema to structural color design, which is particularly advantageous for capturing the latent features in high-dimensional data.

This paper presents an approach to assist in structural color design through a tensor completion algorithm for the prediction of missing data from known data. By combining Tucker factorizations and ALS to capture the geometry-color relationship in structural colors to efficiently identify the structural geometry corresponding to each color generated, while employing a low-pass Gaussian filter to smooth the data. We demonstrate the strategy by designing cylinder, cuboid, and cruciform resonators using Si, cruciform resonators using diamond, and datasets of different sizes, experimentally demonstrating remarkable reliability and accuracy. Numerically and experimentally demonstrated that the algorithm performs exceptionally even with only 10% of the available data. With its capacity to handle complex behavior and large datasets, the algorithm offers a rapid and highly accurate method for investigating the optical properties of nanostructures. Consequently, it holds significant potential for the design of functional devices within the field of optics. Furthermore, this paper proposes designing structural colors of cruciform resonators with diamond in dielectric nanostructures. This approach can break the high loss problem of traditional dielectric materials in the blue-light wavelength region and enhance the corrosion resistance of the structure. Numerical experiments demonstrated that the proper design of diamond nanostructures yields a wide color gamut and a high-narrow reflection spectrum nearing 1. The theoretical analysis substantiates diamond is a promising material in the field of optics.

2. Theory and methodology

2.1 Tensor completion algorithm

We use ordinary lower or upper case for scalars (e.g., x, X), bold lower case for vectors (e.g., x), bold upper case for matrices (e.g., X), and Euclid script letters for tensors (e.g., $\mathrm{{\cal X}}$).The tensor order refers to its dimension or mode, we denote the vector space of an N_th-order tensor of size ${I_1} \times \cdots \times {I_N}$ as ${\mathrm{\mathbb{R}}^{{I_1} \times \cdots \times {I_N}}}$, and the N_th-order tensor $\mathrm{{\cal X}}$ of elements is denoted as ${x_{{i_1} \cdots {i_N}}}$. The tensor completion problem can be formulated as follows:

Specifically, tensor completion is a specific task aimed at filling in missing or unobserved entries within a given tensor. While tensor factorization serves as an intermediate step in the process of refining the tensor, it transforms the tensor into interpretable soft representations that can benefit subsequent tasks. By employing tensor factorizations [40,41], a lower-dimensional approximation can be obtained, finding meaningful potential non-negative components within the original data. In multilinear algebra, tensor factorization can be viewed as an extension of matrix singular value factorizations to tensors [42]. The Tucker model [43,44] provides significant flexibility and is widely employed as a valuable tool in tensor completion, and the Tucker factorization is also known as the higher-order SVD (HOSVD) [45]. Tucker rank in the Tucker factorization can be approximated by the sum of kernel parametrizations, defined as $\mathop \sum \limits_K || {X_{(K )}||}$, where $||{X_{(K )}}||$ represents the sum of singular values of the expansion matrix ${X_{(K )}}$. The diagram of the Tucker factorization model is presented in Fig. 1(a). In this model, three factor matrices correspond to orthogonal bases associated with three distinct spaces, and the rank magnitude of these matrices determines the Tucker rank of the tensor. To perform a low-rank approximation on the observed incomplete tensor, we employ the Tucker factorization. For an N_th-order tensor ${\in} {\mathrm{\mathbb{R}}^{{I_1} \times {I_2} \times \cdots {I_N}}}$, the Tucker factorization with a rank $J = ({{J_1},{J_2}, \cdots ,{J_N}} )$ is expressed as:

 

Fig. 1. (a)The diagram of the Tucker factorization model; (b) The specific flow of the ALS algorithm.

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Meanwhile, we use ALS to perform Tucker factorization. The fundamental idea of ALS is to solve the least-squares solution of the factorial matrix $\textrm{U}_{k + 1}^{(i )}$ by utilizing the factorial matrix $\textrm{U}_{k + 1}^{({i + 1} )} \cdots \textrm{U}_k^{(N )}$ derived at the kth iteration and the updated factor matrix $\textrm{U}_{k + 1}^{(1 )} \cdots \textrm{U}_k^{({i - 1} )}$ at the (k + 1)th iteration:

ALS is applied to each iteration and each factor matrix until the convergence criterion of the algorithm is met. We set the initial number of iterations k to 0 and establish a maximum iteration threshold k_max of 1000. The specific flow of the ALS algorithm is depicted in Fig. 1(b).

To enhance the quality of the completed data, we employ a Q-order low-pass Gaussian filter for direct filtering during the initialization process and at each iteration step. This recursive low-order approximation is employed to effectively smooth the complemented data. Consequently, the optimization problem in this paper can be expressed as follows:

We employ three indicators to assess the performance of the algorithm: relative square error (RSE), signal-to-interference ratio (SIR), and running time (time/s). RSE quantifies the dissimilarity between the spectral data predicted by the tensor completion algorithm and the original spectral data, defined as:

The SIR qualities verification of predicted spectral data by an algorithm. It serves as an indicator of the algorithm's accuracy, with a higher SIR value suggesting a lower error rate of prediction, defined as:

To ensure the algorithm converges to a stable point, we employ tol parameter as the stopping criterion of the algorithm. The tol is defined as the relative change between two successive estimation tensors:

In this study, we set the value of tol to 1e-12, ensuring a high level of precision in the convergence of the algorithms. All computations were performed on a Windows 10 PC with a 2.5 GHz twelve-core processor and 16GB of RAM. The process of tensor completion algorithm-assisted structural color design is illustrated in Fig. 2(a). The process includes three steps: establishing dataset, processing dataset, and predicting dataset. The first step is to acquire a dataset consisting of spectral data and structural parameters, generated based on FDTD calculations. In the subsequent step, the spectral datasets are arranged into a 3D tensor. The tensor consists of three modes, with mode 1 being the spectral data corresponding to distinct dimensions, while modes 2 and 3 are the geometric parameters of the structure. we form tensors with datasets having 90%, 70%, 50%, 30%, and 10% of known data, respectively, to obtain tensors with missing data, and all missing data values were set to zero. The tensorized dataset is further simplified through the Tucker factorization, which approximates the tensor as a low-rank tensor multiplied by three matrices. The factor matrix is updated using ALS so that the prediction tensor gradually approximates the original tensor and the prediction of missing data is realized. The specific procedure is depicted in Fig. 2(b).

 

Fig. 2. (a) The process of tensor completion algorithm-assisted structural color design; (b) The specific procedure for processing the dataset using the tensor completion algorithm.

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2.2 Color calculation

To assess the various structural color designs, we characterized the colors in the CIE 1931 XYZ color space. The tri-stimulus values were obtained using the following equations:

The chromaticity coordinates x and y obtained from the aforementioned equation can be represented within the CIE 1931 XYZ color space.

3. Results and discussion

3.1 Aid design of Si dielectric arrays

Dielectric arrays of a high refractive index based on strong Mie resonance offer several advantages, such as low optical loss and compatibility with manufacturing processes, hold great promise for a wide range of applications in structural color. We designed a cylinder resonator using Si, the design parameters are the period P of the SiO₂ substrate and the radius R of the Si cylinder, as illustrated in Fig. 3(a). To realize this design, the plasma-enhanced chemical vapor deposition (PECVD) method can be employed to grow a silicon layer on a SiO₂ substrate. Subsequently, high-resolution electron-beam lithography and reactive etching can be utilized to fabricate silicon nanostructures. The dielectric cylinder array design makes for a gradual red shift of the resonance wavelength by varying P and R. Five structures with different resonance wavelengths are designed to cover the visible range and their reflection spectra are depicted in Fig. 3(b). The typical peak observed in Fig. 3(b) arises from the Mie resonance within the resonator. Notably, as the structural parameters change, the distance between the electric dipoles (ED) resonance wavelength and the magnetic dipoles (MD) resonance wavelength increases, leading to a gradual reduction in color saturation, as shown in Fig. 3(c).

 

Fig. 3. The dielectric cylinder array using Si (a), the corresponding reflectance spectra (b), and the colors in the CIE 1931 XYZ color space (c); (d) The RSE, SIR, and time for ten prediction experiments with known 10% data. (The structural parameters are in order: R = 45 nm, P = 210 nm; R = 55 nm, P = 210 nm; R = 65 nm, P = 200 nm; R = 75 nm, P = 300 nm; R = 90 nm, P = 320 nm.)

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Among the structural geometric parameters of the dielectric cylinder array, we selected approximately 1000 sets of geometric data. These sets encompass a radius range of 45-90 nm with a step size of 1 and a period range of 200-300 nm with a step size of 5. These datasets were arranged in the form of a 3D tensor (mode 1: spectral data, mode 2: radius, mode 3: period), which was constructed as a 301 × 46 × 25 3D tensor with Tucker rank [50,46,25], and employed a tensor completion algorithm to expedite the simulation process. We assume that 90%, 70%, 50%, 30%, and 10% of the known data, respectively, and use the algorithm to make predictions for the remaining data. For each case, we performed ten random experiments, the average values of RSE, SIR and running time were calculated and presented in Table 1. In Table 1, as we expected, the tensor completion algorithm proved to significantly reduce the required simulation time across most cases, with the time reduced to units of seconds. And it maintains high prediction accuracy even with a 90% missing data rate. Figure 3(d) shows the RSE, SIR, and time for ten prediction experiments with known 10% data, from which we can know that the algorithm maintains consistently high accuracy and robustness. It is important to note that the running time of the algorithm is influenced by the hardware configuration of the computer, such as the CPU and available memory. We believe that the running time would be shorter on a higher-performing server or computational infrastructure.

Table 1. Results of the algorithm on 1000 datasets of dielectric cylinder arrays

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To better visualize the predictive capability of the algorithm, we randomly selected five sets of predicted spectral data in two cases where only 50% and 10% of the data were known. We compared the predicted spectral data obtained through the algorithm with the original spectral data acquired from FDTD simulations while characterized into the CIE 1931 color space, as shown in Fig. 4. From the spectral data, it is evident that the predicted curves exhibit an almost perfect match with the original curves when only 50% of known data. With only 10% of known data, the predicted data demonstrate high prediction accuracy, albeit with slight deviations from the original data at certain resonance peaks and valleys. In structural colors, the spectral data is strongly correlated with the structure geometry, and the spectral data exhibits regular variations in response to changes in the geometric parameters. Leveraging multilinear data analysis, the tensor completion algorithm offers swift and remarkably accurate predictions by effectively capturing the complex relationships inherent in the available known data.

 

Fig. 4. The comparison between the predicted spectral data (red dotted line) and the original spectral data (black solid line) in two cases where only 50% (a) and 10% (b) of the known data.

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To assess the predictive capability of the algorithm across structures with a broader color gamut and more intricate geometries, we designed cuboid resonators and cruciform resonators with Si, respectively, these structures are shown in Figs. 5(a) and 5(d). We design the structure by manipulating the width W of the cuboid resonator and the substrate period P to achieve resonance at specific wavelengths, thereby enabling the dielectric cuboid array to cover the visible wavelength band. As illustrated in Fig. 5(b), the resonance wavelength is gradually red-shifted as the width W and the substrate period P are modified, and the reflectivity increases with it. The color gamut achieved by this metasurface structure surpasses that of the dielectric cylinder array. Meanwhile, distinct resonant wavelength can be achieved by adjusting the widths W₁ and W₂ of the cruciform resonator and the substrate period P, to cover the visible wavelength band. The reflectance spectra of the structure with the corresponding chromaticity map coordinates are shown in Figs. 5(e) and (f). The Mie resonance of the cruciform resonator gives rise to two typical peaks at blue-green wavelengths. The Mie resonance provides robust light-matter interactions that enhance the local field distribution, leading to remarkable brightness across a broad spectrum of colors.

 

Fig. 5. The dielectric cuboid array using Si (a), the corresponding reflectance spectra (b), and the colors in the CIE 1931 XYZ color space (c); The dielectric cruciform array using Si (d), the corresponding reflectance spectra (e), and the colors in the CIE 1931 XYZ color space (f). (The cuboid resonators’ structural parameters are in order: W = 50 nm, P = 250 nm; W = 80 nm, P = 260 nm; W = 90 nm, P = 200 nm; W = 110 nm, P = 220 nm; W = 140 nm, P = 250 nm. The cruciform resonators’ structural parameters are in order: W₁= 100 nm, W₂= 50 nm, P = 270 nm; W₁= 60 nm, W₂= 150 nm, P = 300 nm; W₁= 60 nm, W₂= 180 nm, P = 350 nm; W₁= 70 nm, W₂= 210 nm, P = 350 nm; W₁= 100 nm, W₂= 200 nm, P = 390 nm.)

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Similarly, approximately 1000 sets of geometric data are chosen from the two dielectric arrays and the tensor completion algorithm is applied to forecast the missing data when 90%, 70%, 50%, 30%, and 10% of the data are available respectively. In the dielectric cuboid array, we selected a range of widths of 50-140 nm with a step size of 1 and a period range of 200-260 nm with a step size of 5. The dataset is arranged as a 3D tensor of spectral data × width × period as 301 × 91 × 13 with a Tucker rank of [50,50,13]. In the dielectric cruciform array, we selected a range of widths W₁ of 50-180 nm with a step size of 1 and a period range of 270-370 nm with a step size of 10, with a fixed width W₂ of 100 nm. The dataset is arranged as a 3D tensor of spectral data × width W₁ × period as 301 × 131 × 11 with a Tucker rank of [50,50,11]. In each of the above two tensors, ten experiments were conducted randomly in each case utilizing the algorithm, and the average of the experimental results was documented in Table 2. As depicted in Table 2, the algorithm consistently exhibits rapid speed and exceptional accuracy in both structures and the algorithm attains superior values of RSE and SIR for both structures, surpassing those obtained for the cylinder resonator. In four cases where 90%, 70%, 50%, and 30% of the data are known, the running times are comparable to those observed in the cylinder resonator. However, in the case where only 10% of the data is known, the algorithm requires slightly more time for completion. Nevertheless, there has been a notable reduction in time compared to other optimization algorithms, which require time-consuming repetitions of structural simulations. In the case of only 10% of the data known, we randomly selected five sets of spectral data predicted in each of the two structures for comparison with the original data, as presented in Fig. 6. It is manifest that the algorithm's predicted spectral data for both structures closely fit with the original data, achieving a high level of agreement even at resonance peaks and troughs, indicating the equally robust predictive capability for structures with a larger color gamut and more complex of the tensor completion algorithm proposed in this study.

 

Fig. 6. The comparison between the predicted spectral data (red dotted line) and the original spectral data (black solid line) in the dielectric cuboid array (a) and the dielectric cruciform array (b) in the cases where only 10% of the known data.

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Table 2. Experimental results of algorithm for dielectric cuboid arrays and dielectric cruciform arrays

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To assess the capability of the algorithm across datasets of varying sizes. We selected approximately 1800 and 2800 sets of geometrical parameters in three metasurface structures: dielectric cylinder array, dielectric cuboid array, and dielectric cruciform array, respectively, for experiments. Only 10% of the available data, we performed ten random experiments for each structure, recording the range of parameter in the dataset and the average experimental outcomes of the algorithm in Table 3. The smaller RSE and the larger SIR indicate better prediction accuracy. As depicted in the table, when only 10% of known data, the other two resonators outperform the cylinder resonator regarding RSE and SIR in the same case. While the cylinder resonator exhibits an RSE of approximately 4.836900e-04 across approximately 2800 datasets, the cuboid resonator achieves a significantly lower RSE of only 8.680230e-05, similarly, the RSE of the cruciform resonator is only 1.470650e-04. It is suggested that the spectral data observed in cuboid and cruciform resonators exhibit a notably stronger correlation with geometry. To provide a visualization of the algorithm's predictive capabilities across different datasets, we randomly selected five sets of predicted spectral data from two datasets of three structures respectively, and plotted the original coordinates with the predicted coordinates in Fig. 7.

 

Fig. 7. The comparison of the predicted coordinates with the original coordinates is presented for the dielectric cylindrical array (a), the dielectric cuboidal array (b), and the dielectric cruciform array (c) across the 1800 and 2800 datasets.

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Table 3. Experimental results of algorithm for different size datasets within three structures

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To demonstrate the high reliability of the tensor completion algorithm proposed in this paper in terms of speed in structural color design, we present a comparative analysis of the optimization time of the GA, deep learning algorithm and tensor completion algorithm in Table 4. In prior work [46], we optimized the inner diameter r, outer diameter R, thickness H, and substrate period P of the ring resonator using GA to find the structural parameters that meet the objective function. The GA utilized a population size of 100 and averaged 40 iterations, necessitating approximately 4,000 simulations for each structural optimization. Meanwhile, L. Gao et al. [47] propose a deep neural network for the inverse design of structural color. The foundational structure is a cylinder resonator using Si and designed the parameters of diameter D, height H, inter-cylinder gap G, and substrate period P. The work employed 3,900 sets of electromagnetic simulation data for training the neural network, 400 sets for validation purposes, and 360 sets for testing the model's performance. They utilized FDTD to simulate a total of 4,660 parameter combinations. Remarkably, in our study, for the 2,800 datasets of dielectric cylinder arrays, it was essential to simulate only 10% of these datasets, and the algorithm predicted 90% of missing data leveraging 10% of the known data. Results show that the algorithm operates both rapidly and reliably even with as little as 10% of the available data, thereby yielding a substantial reduction in computational time. For systems with a small number of features, utilizing the tensor completion algorithm is relatively efficient, leading to a marked enhancement in design efficiency. However, e.g., X. Han et al. [48] propose a systematic method based on neural networks that can complete an inverse design process with a high degree of freedom to generate device structures. The parameters of the device structure involving the choice of materials, shapes, unit cell size, and layered permutations offer various design freedoms. The average error rate for each point between the real spectrum and the generated spectrum is 4-5% when they feed 5200 parameter combinations selected from the validation set. As evidenced above, the benefit of deep learning is realized when engaging with systems characterized by multiple features, showcasing its highly efficient role in aiding the design process. On the contrary, the deep learning-assisted design process's complexity is relatively higher compared to that of a simpler model for systems with a small number of features.

Table 4. Comparison of optimization time of GA, deep learning and this work

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Table 5 provides the comparison between the coordinates obtained by the algorithm within the 2800 dataset and the corresponding original coordinates of each of the five sets of parameters of the three structures. The algorithm yields predicted coordinates that closely align with the original coordinates, even when only 10% of the data is available, as shown in Table 5. Furthermore, we present the SIR for various datasets in the three structures through a box line plot in Fig. 8. Notably, the algorithm exhibits consistent and robust prediction performance within the same structure, irrespective of the dataset's size. This emphasizes the algorithm's stable and reliable nature when it comes to predicting performance across different dataset sizes within a given structure.

 

Fig. 8. SIR for various datasets in the three structures.

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Table 5. Comparison of the original and predicted coordinates of the algorithm for the three structures within the 2800 dataset

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3.2 Aid design of diamond dielectric arrays

To demonstrate the extensive applicability of the proposed tensor completion algorithm, we conducted further demonstration by replacing the Si of the resonator with diamond and designing the cruciform resonator. Dielectric materials, such as Si and titanium dioxide [49], are frequently employed in structural color design due to their remarkable potential for achieving high resolution and a wide color gamut. However, these materials often encounter limitations in practical applications, primarily stemming from their susceptibility to corrosion and their high absorption in the blue wavelength region. Diamond possesses a remarkable set of physical and chemical properties. Its high refractive index within the visible wavelength range, negligible losses, high thermal conductivity, high density, and resistance to acid and base conditions make it a desirable choice for exploiting its potential in this field. In recent years, significant advancements have been made in artificial growth techniques. High-pressure and high-temperature (HPHT) methods and chemical vapor deposition (CVD) can synthesize high-quality diamond, and the CVD method is compatible with the principles of the semiconductor industry [50]. M. P. Hiscocks et al. [51] first showcased the viability of large-area all-diamond integrated optics through a combination of photolithography, reactive ion etching (RIE), and focused ion beam (FIB) techniques. Meanwhile, J. Gu et al. [52] leveraged diamond to design structured colors based on Mie resonance that help generate encrypted patterns and exhibit long-lasting durability and high resolution. Additionally, diamond use in photonic structures, such as optical waveguides [51] and resonators [53], has unveiled the immense potential of diamond for future applications in optical devices.

By adjusting the widths W₁ and W₂ of the resonators and the period P of the SiO₂ substrate, the dielectric cruciform array can arbitrarily manipulate the amplitude of electromagnetic waves across the visible spectrum, and the resonant wavelength is gradually red-shifted, as illustrated in Figs. 9(a) and (b). The dielectric array exhibits a significantly high-narrow reflectivity approaching 1 at various resonant wavelengths within the visible range. This breakthrough overcomes the limitations of typically suffering from high losses in the blue wavelength region. Furthermore, Fig. 9(c) shows the dielectric array's ability to achieve a relatively wide color gamut, further emphasizing its potential for generating vibrant and diverse colors.

 

Fig. 9. The dielectric cruciform array using diamond (a), the corresponding reflectance spectra (b), and the colors in the CIE 1931 XYZ color space (c). (The structural parameters are in order: W₁= 190 nm, W₂= 210 nm, P = 340 nm; W₁= 190 nm, W₂= 250 nm, P = 340 nm; W₁= 220 nm, W₂= 270 nm, P = 360 nm; W₁= 40 nm, W₂= 460 nm, P = 460 nm; W₁= 500 nm, W₂= 510 nm, P = 510 nm.)

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We selected approximately 1000 sets of spectral data, a range of widths W₂ of 190-280 nm with a step size of 1 and a period range of 340-395 nm with a step size of 5, with a fixed width W₁ of 280 nm. These datasets were arranged as a 3D tensor (mode 1: spectral data, mode 2: width W₂, mode 3: period), where the tensor size is 301 × 91 × 12 with a Tucker rank of [50,50,12]. Employing the algorithm to predict the missing data in three cases where 90%, 50%, and 10% of the available data. Ten random prediction experiments were conducted for each case, and the average of the experimental outcomes was documented in Table 6. The algorithm demonstrates remarkable efficiency by averaging a time of 1.4809s to predict the missing data with 90% of the known dataset. Even when only a meager 10% of the available data, the algorithm exhibits impressive predictive capabilities, yielding small tolerance (RSE = 9.401540e-04) and exceptional robustness (SIR = 19.0216 dB) within a time frame of less than one minute. We randomly selected five sets of predicted spectral data, and as evident from Fig. 10, it is apparent that the predicted spectral data fits closely with the original spectral data. Remarkably, the algorithm successfully captures the intricate details of the spectral data, showcasing its ability to accurately predict even the high-narrow resonance peaks at non-target wavelengths for the unknown data, as shown at A in Fig. 10. The experimental findings illustrate the algorithm's robustness and accuracy in making reliable predictions, even when the structure's material is changed. The tensor completion algorithm exhibits remarkable effectiveness and broad applicability, showing its potential in the field of structural color design.

 

Fig. 10. The comparison between the predicted spectral data (red dotted line) and the original spectral data (black solid line) in the dielectric cruciform array using diamond in the cases where only 10% of the known data.

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Table 6. Results of the algorithm for the diamond-based dielectric cruciform array

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4. Conclusion

This paper presents a tensor completion algorithm designed to efficiently and accurately predict the missing dataset using only a partially acquired dataset. We address the device inverse design problem by formulating it as a low-rank tensor completion problem. In contrast to other traditional inverse design methods, this strategy circumvents the need for intricate electromagnetic theory, resulting in expedited design processes for structural colors. Moreover, it avoids the prerequisite of extensive pre-training data, rendering it particularly efficient for devices characterized by a small number of features. Numerical simulations and experimental validations, the effectiveness of our proposed method in addressing the computational burden and time-consuming nature of simulations, this strategy provides a promising candidate for the design of metasurfaces. Meanwhile, we introduce diamond as a viable material for structural color design, leveraging its exceptional properties to overcome the limitations encountered by traditional dielectric materials in the blue wavelength region and enhance the corrosion resistance of the structure. Numerical simulations validate that the utilization of the structural color of cruciform resonators with diamond enables the realization of a broad color gamut and achieves a high-narrow reflection spectrum nearing 1. The theoretical analysis substantiates the immense potential of diamond as a prominent material in the realm of optics, paving the way for innovative applications in various optical devices.