Zhao
Huang
^{ab},
Xin
Liu
^{ab} and
Jianfeng
Zang
*^{ab}
^{a}School of Optical and Electronic Information and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, China 430074. E-mail: jfzang@hust.edu.cn
^{b}Innovation Institute, Huazhong University of Science and Technology, Wuhan, China 430074

Received
19th July 2019
, Accepted 25th August 2019

First published on 27th August 2019

Efficiently identifying optical structures with desired functionalities, referred to as inverse design, can dramatically accelerate the invention of new photonic devices, and this is especially useful in the design of large scale integrated photonic chips. Structural color with high-resolution, high-saturation, and low-loss holds great promise in image display, data storage and information security. However, the inverse design of structural color remains an open challenge, and this impedes practical application. Here, we propose an inverse design strategy for structural color using machine learning (ML) technologies. The supervised learning (SL) models are trained with the geometries and colors of dielectric arrays to capture accurate geometry-color relationships, and these are then applied to a reinforcement learning (RL) algorithm in order to find the optical structural geometries for the desired color. Our work succeeds in finding simple and accurate models to describe geometry-color relationships, which significantly improves the efficiency of the design. This strategy provides a systematic method to directly encode generic functionality into a set of structures and geometries, paving the way for the inverse design of functional photonic devices.

Fortunately, advances in artificial intelligence (AI) technology have prompted a settlement of this dilemma. Machine learning (ML) is a statistics technology that trains a machine by telling it what to do. ML plays crucial roles in computer vision, natural language processing, robot control, and other AI applications.^{9,10} The capability of dealing with complex behaviors and big data makes ML a prominent technique in science and engineering.^{11–17} In materials science, ML has succeeded in guiding the chemical synthesis and discovery of suitable compounds with target properties.^{18–21} In biogenetics, analyzing large genomic data through ML has helped to annotate various genomic elements, and this is useful in gene recognition and editing.^{22,23} Moreover, significant progress has been made in photonics using ML technology, such as in the pattern recognition of photonic modes, the analysis of modulation instability in optical fibers and in the rational structure design for waveguides.^{24–29} Most recently, neural networks in a tandem architecture have been proposed to solve the inverse design problems of nanophotonic structures, which provides an opportunity to revolutionize the optical device design. Dianjing Liu et al. proposed a tandem architecture composed of an inverse design network connected to a forward modeling network, and this allowed the deep neural networks to effectively solve the non-unique inverse design problem. Such structures have been applied to the inverse design of topological photonics, core–shell nanoparticles and plasmonic waveguides.^{30–33} Deep learning networks have made great success in solving the inverse design problems. However, such a strategy requires large amounts of data, which makes it less efficient compared with simple models when dealing with a system that consists of few features.

Structural color holds great promise in display and security technology.^{34–41} Devices that consist of metallic or dielectric nanostructures can provide high resolution, and low-loss full color displays.^{42–46} However, the existing design approach is inefficient due to the complexity of the geometry-color relationship, which prevents structural color from reaching scalable fabrication. It is crucial to achieve a new structural color design strategy through the direct identification of structure geometries for each desired color property. Here, we propose an inverse design strategy for structural color using supervised learning (SL) and reinforcement learning (RL). We train the SL models to capture the geometry-color relationships in structural color and introduce a RL algorithm together with the SL models to efficiently identify the structure geometries that generate the desired colors. We demonstrate our strategy by designing both ring and pyramid resonator arrays. Our strategy achieves inverse design with high reliability and accuracy and can be applicable to the design of a wide range of functional materials.

2.2.1. Establishing datasets.
Data acquisition is the key and fundamental part in the inverse design method. High-quality data that contains sufficient information on the dielectric arrays for structural color can provide a guarantee for the ML models with excellent learning performance. To establish the datasets that consisted of accurate and adequate data, we carefully selected and processed the geometry features that dominate the color properties and used the FEM simulation to obtain the accurate color properties. Here, we demonstrate our inverse design strategy of structural color using dielectric ring arrays. Dielectrics with a high refractive index are promising materials for structural color due to their low optical loss, strong Mie resonance, and compatibility with the fabrication process. We selected three geometry parameters: outer diameter D, inner diameter d, and the gap g between ring resonators, as marked in Fig. 2a, and three color properties: (x, y) coordinates in the CIE 1931 color space and reflection peak intensity I that are obtained from the FEM simulation, as key features to establish the dataset in the inverse design of the ring arrays. These features describe completely the structural color system and provide sufficient information for the training of the ML models. We simulated 3876 ring arrays and the color distribution in the CIE 1931 color space is presented in Fig. 2b. As a result, we established the dataset with six features consisting of 3876 ring arrays for the next training process. The reflection spectra of the ring arrays with D = 160 nm, g = 100 nm, and d = 40, 60, 80, and 100 nm, respectively, are shown in Fig. 2c. The typical peaks are attributed to the Mie resonances of the ring resonators. Fig. 2d illustrates the electric field distributions of the cross-section at the two selected resonances, which are denoted by black arrows in Fig. 2c. The Mie resonances offer a strong light matter interaction that enhances the local field distribution, and this results in the high brightness of the colors in a wide range.

2.2.2. Training of the SL models.
The ML models play key roles in data analysis, behavior prediction and in guiding decisions. The well-trained SL models in the structural color system can capture the geometry-color relationships and provide reliable and accurate prediction results. In the training process, three SL models: the forward kernel ridge regression (KRR) model, the support vector classification (SVC) model, and the backward KRR model, are trained with the input dataset. The forward KRR model is trained to predict (x, y) coordinates based on given D, d and g values. The SVC model is trained to classify the color brightness based on given D, d and g values. The backward KRR model is trained to estimate the initial geometries based on desired (x, y) coordinates (the details for the SL models are given in the Methods section). The performances of these trained SL models have been improved significantly using optimization techniques such as grid searching and cross-validation, which provide a guarantee for the successful inverse design of structural color.

To predict (x, y) based on the given D, d and g values, we employed four regression models: KRR, Gaussian process regression (GPR), decision tree regression (DTR) and multilayer perceptron regression (MLPR), due to their abilities to handle nonlinear relationships. We randomly split the input dataset into two parts. A dataset of 800 ring arrays served as the test set and a dataset of the remaining arrays served as the training set (the learning performance of the KRR model with a different training ratio is given in ESI Fig. 1†). The four regression models were trained with the identical training set and were then applied to the identical test set. The coefficient of determination (R^{2}) and the training-test time (T) were used as evaluation standards to evaluate the performance of each regression model. The training-test time is the amount of time the regression models spend to be trained with the training set and then obtain the predicted colors using the test set. The DTR model finds it difficult to handle complex nonlinear relationships. The GPR and MLPR models suffer from time-consuming training processes. The KRR model is good at dealing with nonlinear relationships and has a simple training process, which makes it best suitable for color prediction. Among the four employed regression models, the KRR model is less time-consuming and reproduces the best agreement to the true coordinates. R^{2} and T for the KRR model are 0.989 and 1.6 seconds, respectively. The outstanding performance of the KRR model is attributed to the kernel function that is well suited in predicting the color properties of the dielectric arrays and the low algorithm complexity of the KRR model. Fig. 3 presents the test results of the KRR model. The scatter plots of the true (x, y) coordinates and the predicted (x, y) coordinates are illustrated in Fig. 3a, indicating great consistency between the true and the predicted coordinates. The counts of regular residual according to the distance between the true coordinates and the predicted coordinates are depicted in Fig. 3b, showing obvious convergence behaviors (the performances of the other three regression models are given in ESI Table 1 and ESI Fig. 2 and 3†). We plotted the true colors (solid black stars) and the predicted colors (solid red dots) in the CIE 1931 color space, as shown in Fig. 3c. The predicted colors are located very close to the corresponding true colors. Compared to a few days required by the FEM simulation, the computing time using the pre-trained KRR model can be dramatically decreased to a few seconds to predict the color properties from thousands of geometries with high accuracy (Intel(R) Core(TM) i5-4590 CPU), and this provides a useful tool in the inverse design of structural color. We denoted the trained KRR model as the forward KRR model and applied the forward KRR model in the optimization process of the RL algorithm, which will be discussed in the inverse design process.

To further uncover the relationship between the geometries and color properties, the heat map of the Pearson correlation coefficient matrix is presented in Fig. 4a. The positive values of the correlation coefficient indicate positive correlations, while the negative values indicate negative correlations between pairs of features. The larger the absolute value is, the bigger the relevance between features is. The correlation coefficients between D and the color properties x, y and I is 0.54, 0.43 and 0.48, respectively, which are significantly larger than the coefficient values between d and the color properties of −0.27, −0.1 and −0.33. It turns out that D plays the most significant role in determining the color properties, and d also plays a part. In addition, the correlation coefficient between g and the color properties x, y and I is one order of magnitude smaller than that between D and the color properties, and this shows that g hardly affects the color properties. It can be explained that optical resonances in high-index dielectric resonators mainly rely on the geometries of a single resonator rather than coupling between adjacent resonators, due to the field confinement within a single resonator. The datasets are grouped by g values to obtain 12 groups that consist of 323 ring arrays with the same g value. We labelled the ring arrays according to their corresponding color properties and trained the SVC models to classify the geometries in each dataset.

We divided the colors into three categories: red, green and blue, according to the distance between the (x, y) coordinates of the ring arrays and the coordinates of the standard Red Green Blue (sRBG) in the CIE 1931 color space, as presented in Fig. 4b. We trained the SVC model with linear kernel using the input dataset consisting of 323 ring arrays where g = 20 nm (the details of other g groups are given in ESI Fig. 4†). We classified the geometries into three categories and drew optimal separating hyperplanes (black lines), as depicted in Fig. 4c. The mean accuracy (MA) of the classification results for the total dataset was 0.939 (MA of other g groups are given in ESI Table 2†). The direction of the separating hyperplanes indicates that ring arrays with a large D value and a small d value tend to produce red colors, while ring arrays with a small D value and a large d value tend to produce blue colors. Geometries of ring arrays that generate a green color lie between the red group and the blue group. It can be explained that an increase in the length of the ring resonator in the polarization direction will decrease the frequency of the Mie resonances, resulting in a redshift in the reflection peak.

In addition to the frequency of the reflection peaks, peak intensity is another key factor in determining the color quality. Dielectric arrays that suffer from low reflectivity are inefficient and energy consuming for structural color. It is necessary to select the qualified colors with high brightness to achieve industry standard. We selected I = 60% as a threshold to distinguish between the qualified and the unqualified colors and to label ring arrays with peak intensities above and below 60%, respectively. Fig. 4d depicts qualified and unqualified colors in the CIE 1931 color space, showing that ring arrays can produce high brightness colors in a wide range. We trained the SVC model with a radial basis function (RBF) kernel using the input dataset consisting of 323 ring arrays where g = 40 nm (the details of other g groups are given in ESI Fig. 5†). We classified the geometries into qualified and unqualified categories and drew a nonlinear separating hyperplane, as depicted in Fig. 4e. The MA of the classification results for the total dataset was 0.998, thus providing reliable and rapid screening for the qualified colors (MA = 1.0 for g = 10 nm to 110 nm and MA = 0.993 for g = 120 nm). We applied the SVC model to select the qualified colors in the optimization process of the RL algorithm.

2.2.3. The RL algorithm.
Usually, it is unlikely to directly be able to obtain the geometry based on the desired color simply by training a SL model, due to the non-uniqueness of a solution. For a desired color, several geometries always exist. We solved this problem by implementing a RL algorithm. The geometry parameters are first slightly changed and the feedback is obtained according to the color variation in every RL round. If the color becomes closer to the desired color in the CIE color map, the geometry parameters will be changed in the same way in the next RL round. After several RL rounds, the geometry parameters that generate the desired color will be obtained. In the RL algorithm, the initial geometry parameters and optimization algorithm are two key factors that determine the efficiency and precision of the inverse design.

To obtain sensible initial geometry parameters, we imposed some restrictions on the geometry features. For a target color, we predefined two geometry features and predicted the remaining geometry feature. Due to the largest relevancy between D and (x, y), we trained a KRR model, referred to as the backward KRR model, to predict D based on the desired (x, y), and predefined d and g values. The backward KRR model was trained with the training set and was then applied to the test set. Fig. 5a presents the scatter plot of the true D values and the predicted D values. R^{2} and T are 0.982 and 0.6 seconds, respectively. The counts of regular residual are depicted in Fig. 5b, showing obvious convergence behaviors. The backward KRR model can predict a relatively reasonable geometry based on the desired color, serving as the initial geometry in the optimization process of the RL algorithm. Since there are several geometry parameters to be optimized, we used the greedy algorithm to improve the optimization efficiency.

Fig. 5c illustrates the RL algorithm. In the first step, the backward KRR model was used to generate the initial geometry for the desired (x, y) coordinates. We chose predefined values of d = 120 nm and g = 60 nm. Different predefined d and g values may lead to different local optimal geometries but similar colors (the details of predefining d and g are discussed in ESI Table 3†). In the second step, the greedy algorithm was implemented to optimize each geometry feature in turn. For each geometry feature, the initial value was increased and decreased, respectively, until the distance between the predicted color and the desired color reached a minimum, and then the two minimums were compared to find the current optimal value for that optimization round. The (x, y) coordinates of the current geometry were predicted using the forward KRR model and the difference between the desired (x, y) coordinates supplied the feedback to guide us in updating the geometry. Within every update, the SVC model was employed to screen for the qualified colors. After several optimization rounds, D, d and g converged on one of the possible geometries with the desired colors. The greedy algorithm eliminated the coupling among the geometry features to efficiently find the optimal solution. The accuracy of this algorithm can be guaranteed by the sensible initial geometry obtained from the backward KRR model and the sufficient restrictions imposed by the forward KRR model and SVC model. We applied the FEM simulation to verify the geometries and demonstrate the inverse design process in the CIE 1931 color space. The distances between the colors generated by the final geometries and the desired colors are shown in the CIE 1931 color space, indicating the high precision of our inverse design strategy. Fig. 5d illustrates the FEM simulated reflection spectra of the final geometries, which show high reflection.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr06127d |

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