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Abstract
Data-driven deep learning frameworks have significantly advanced the development of modern machine learning, and after achieving great success in the field of image, speech, and video recognition and processing, they have also begun to permeate other disciplines such as physics, chemistry, and the discovery of new drugs and new materials. Our work proposes a deep learning-based model consisting of two parts: a forward simulation network that contains a transposed convolutional network, up and down sampling blocks and dense layers can rapidly predict optical responses from metasurface structures, and an inverse design network that contains convolutional neural networks and dense layers can automatically construct metasurface based on the input optical responses. Our model assists in discovering the complex and non-intuitive relationship between the moth-eye metasurface and optical responses, and designs a metasurface with excellent optical properties (ultra-broadband anti-reflection or nonlinear function of reflectivity), while avoiding traditional time-consuming case-by-case numerical simulations in the metasurface design. This work provides a fast, practical, and robust method to study complex light-matter interactions and to accelerate the demand-based design of nanophotonic devices, opening a new avenue for the development of real nanophotonic applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurface refers a class of artificial materials with special properties. In the field of micro-nano optics, sub-wavelength structures interact with light to produce versatile optical properties, so these micro-nano structures can be designed to provide useful functions. Inspired by naturally occurring nanostructures, such as the anti-reflection propertie observed in moth eyes [1,2], and by studying and creating such micro-nano metasurfaces with reduced reflection properties [3], researchers find applications in fields of renewable energy, optoelectronics and anti-reflection in screens [4].
At present, prediction tasks rely heavily on the iterative, time-consuming and computationally expensive numerical simulations to solve Maxwell’s equations case by case, and there is no universal solution fulfilling inverse design tasks so far. Common inverse design methods include genetic algorithms [5] and topology optimization [6], in addition , particle swarm based optimization methods are also proposed [7]. However, due to the mechanism limitation of the pattern-driven algorithm, its generalization ability can also be regarded as the ability to search unknown space, which is stretched when facing complex problems. Unlike numerical optimization methods, data-driven methods based on machine learning (ML) can represent and generalize complex functions or data to discover relationship within unknown large numbers of variables.
Deep learning (DL) is a type of machine learning that can efficiently learn the inherent laws of data, allowing computational models to learn multi-level abstract representations [8]. By using the superior advantages brought by the back-propagation algorithm in training, it can discover complex and subtle structures in large data sets. Also, it is advanced in some areas, such as speech recognition [9], object recognition [10], and decision making [11]. In recent years, due to the success of deep learning in computer vision and natural language processing, researchers have applied deep learning into other fields, such as materials science [12], chemistry [13], particle physics [14], quantum mechanics [15,16], and microscopy [17]. Neural networks have been applied to solve design and prediction problems in electromagnetism [18–20], but due to the shallow network layers, the generalization ability of these models is not strong enough. Recent works using deep neural network to model nanophotonic structures [21–24], to inversely design micro-nano photonic structures [25], and to optimize micro-nano photonic structures [26–28] are proposed. However, these neural networks are constructed by stacking fully connected layers without in-depth network architecture design regarding specific research problem, leading to prediction accuracy poor or a small number of optical parameters predicted. The issue of predicting high-dimensional complex optical response curves from limited metasurface parameters has always been difficult to solve. There is a precedent by using ensemble learning method to solve such problem in previous work [29], however, any machine learning algorithm can benefit by increasing the amount of computation and training data. A practical design scheme for optical metasurfaces and other metasurfaces should include two main functions: the forward prediction, which is to give the required optical response based on the structure and material of the metasurface, and the inverse design, which can output the geometric parameters and material type from the requested optical responses. The scheme should be in-depth designed regarding specific research problem, with high accuracy and robustness.
In this paper, we propose a deep learning architecture for automatic design of metasurface parameters and optical simulation of metasurfaces. The overall architecture of the network is shown in Fig. 2, which consists of two parts. The first part is used for forward simulation, including two iterative up and down sampling blocks(shown in Fig. 3), composing of convolutional neural network(CNN) and transposed convolutional neural networks , and the dense layers. The second part is used to inverse design, including several CNN layers and dense layers. Once the model is successfully trained, for the part of the forward simulation, just given the parameters of the metasurface, the model can calculate the reflection spectral characteristics and absorption spectral characteristics. For the part of the inverse design, the metasurface parameters, including structural and material parameters, can be calculated by spectral curve based on certain requirements. Once the model has been trained, the mapping of structural and material parameters to optical properties will be included, so the optical properties can be quickly generated according to the structural and material parameters, or vice versa. Our study demonstrate neural networks can learn highly complex mapping between optical metasurface parameters and optical responses and this mapping can be applied to the design of optical metasurfaces.
2. Results and discussion
2.1 Metasurface structure design
The designed moth-eye structure is shown in Fig. 1. Each unit is composed of a half ellipsoid and an extra layer of material covered on the surface. It is a sub-wavelength structure with the size of hundreds of nanometers and the anti-reflection property is based on this structure. Taking into account the convenience of processing and manufacturing this structure and covering this structure with a film composed of different materials, we can attempt to construct an absorber with superior performance.
Fig. 1. Schematic diagram of the designed moth-eye micro-nano structure array. Inset is a zoomed-in structure of a single moth-eye structure.
The main structure of the moth eye is a half ellipsoid, and its three semi-axes are represented by $R_{1}$ (length range 100 nm to 200 nm), $R_{2}$ (length range 100 nm to 200 nm) and $R_{3}$ (length range 150 nm to 300 nm). We added a layer of covering material layer to the moth-eye structure, FDTD simulation shows this layer can affect the reflectivity and absorption. The thickness of this layer of material is represented by the covering thickness (length range 0 to 100 nm), and the type of material is also a variable denoted by $K$, optional materials are gold (Au [30]), silver (Ag [30]) and aluminum oxide ($\textrm{Al}_{2}\textrm{O}_{3}$ [30]). In addition, each unit of the moth-eye array is a rectangle, $L_{1}$ and $L_{2}$ are the lengths of its two sides which depend on the size of the moth-eye structure, the relationship is as follows: $L_{1}$ = $2R_{1}$, $L_{2}$ = $2R_{2}$ and the moth-eye structure base $T$ = 300 nm, its material is silicon [30]. These three parameters($L_{1}$, $L_{2}$, $T$) are fixed or depend on other design parameters and therefore do not participate in the design. Therefore, the structure of the moth-eye metasurface is determined by five design parameters, that is, ellipsoid semi-axis $R_{1}$, ellipsoid semi-axis $R_{2}$, ellipsoid semi-axis $R_{3}$, covering thickness, and covering material type $K$. Considering that the main use scenario is to reduce reflection in the visible light band, for optical absorbers, the absorptivity is also very important, based on these two indicators, the transmissivity can be calculated. The above three spectral characteristics can characterize the flow of light energy. Therefore we focus on the reflectance spectrum and the absorbance spectrum. Based on the existing detection methods and application scenarios of reducing reflections, the wavelength of interest is set in the band from 380 nm to 780 nm and discretized into 200 data points.
2.2 Deep-learning model construction
This model includes forward simulation and inverse design networks, which is trained in a supervised manner respectively.
Firstly, we introduce the architecture of the forward simulation. The network of forward simulation includes iteratively stacked up and down sampling blocks (Fig. 2), the up and down sampling block shown in Fig. 3 mainly consists of two parts, one is the up-sampling part, which contains a four-layer transposed convolutional neural network, where the input data will be expanded in terms of dimension and size. The other part is the downsampling part, which consists of a four-layer convolutional neural network, where the data will be reduced in both dimension and size. It is clear that for the entire up and down sampling block, the input data and output data are of the same dimension and size, so the network block can be used repeatedly and stacked. The stacking composition of dense layers is shown in the upper half of the dashed box in Fig. 2. Forward simulation is a process in which the dimension of data is continuously increased. In this process, there will be serious mismatches [29], that is, it is difficult for the network to obtain enough information from the forward flow of data in the training process to learn, moreover, it’s difficult to converge during the training process and the prediction accuracy is low, so the expected effect cannot be achieved. To solve this problem, we use a network architecture that mixes a transposed convolutional network, repeatedly stacked up and down sampling blocks and dense layers, as well as a segmented training method to ensure that the network can utilize error backpropagation to obtain enough information.
Fig. 2. The deep network architecture of forward simulation and inverse design. For any network, the parameters on the left side are regarded as the input of the network, and the right side is regarded as the output of the network. The upper dashed box is the forward simulation network, the repeatedly stacked up and down sampling blocks in the figure are exactly the same, the input is the metasurface parameters and the output is the spectral curve. In the dashed box below is the inverse design network, the input of the network is the spectral curve, and the metasurface parameters are the output of the network (FC, fully connected layer; Conv, convolutional layer; Tconv, transposed convolutional layer; Maxpool, max pooling).
Fig. 3. Up and down sampling blocks. The up and down sampling block mainly includes two parts, one is the up sampling part, where the data will increase from two aspects of dimension and size. The other part is the down sampling part, where the data is reduced in both dimension and size.
In the forward simulation architecture, we adopt a segmented pre-training method, which is similar to the training strategy of the greedy layer-wise training method of deep networks [31]. The network architecture used in each segment is the same, but the dimensions are different. This forward simulation is divided into two segments. The first segment is a 5-dimensional to 25-dimensional mapping. The 5-dimensional data is both structural and material parameters, and the 25-dimensional data is uniformly sampled from a 200-dimensional spectrum. Here, the output tensor layer is given by:
where $f$ is the Rectified Linear Unit (RELU) activation function, $X$ is the row vector of the five designed parameters, $k$ is the output vector dimension, ${W}_{k}$ is seen as a $k\times 5$ matrix and $B$ is a $k\times 1$ bias vector. The output dimension $k$ is chosen to be 25 in the first segment. The second segment is a 25-dimensional to 200-dimensional mapping. During the second segment of training, the network of the first segment will be frozen, and the last two trained networks will be combined to complete the forward simulation. In this step, same as Eq. (1), $k$ is set to 200 and the dimension of $X$ becomes 25.The data stream enters a repeatedly stacked up and down sampling bocks. The downsampling architecture uses a one-dimensional convolutional neural network, while the upsampling architecture uses one-dimensional transposed convolutional neural networks. Iterative up and down sampling bocks (shown in Fig. 3) are used to complete the training of the network and to realize the mapping from low-dimensional data to high-dimensional data. Similar model frameworks have been used to increase the dimension of images [32–34] successfully. We use this architecture to process one-dimensional sequence this time, and the architecture is used as part of the forward simulation to successfully build our model. Finally, we use stacked dense layers to map the data to the dimension we need, and the data is better represented in the dense layer.
The next step is to process the inverse design as shown in the lower half of the dashed box in Fig. 2, and we try to establish the mapping relationship between spectral properties and metasurface parameters. We use an one-dimensional convolutional neural network to capture the features in the spectral characteristic curve, and then pass through the stacked dense layers, where the data is more abstract and better represented. After the supervised iterative training, inverse design network of spectral properties to metasurface parameters, i.e., 200-dimensional to 5-dimensional mapping, is completed.
Each network uses the mean squared error(MSE) as the loss function, the formula is as follows:
for the forward simulation: ${Y}_{i}$ is the data generated by the FDTD simulation for spectral point of each specific moth-eye structure, and $\stackrel {-}{{Y}_{i}}$ is the data generated by the forward simulation networks. For the inverse design: ${Y}_{i}$ is the parameters of metasurface, $\stackrel {-}{{Y}_{i}}$ is the parameter generated by the inverse design networks and the input of the neural network is a specific spectral curve.2.3 Model evaluation
A trained neural network is evaluated on test data set that has never been used in previous training or validation steps.
For the forward simulation network, Fig. 4(a,b,c,d) draws four typical examples in the test set, and use four examples to show the effect of forward prediction of reflection and absorption spectra. The simulated spectral characteristic curves are consistent with our results of proposed forward simulations, and the mapping between the moth-eye metasurface parameters and optical responses is established.
Fig. 4. Evaluation of forward simulation deep learning models. (a,b,c,d) Four forward simulation examples, which show the results predicted by the deep learning forward simulation model compared with numerical simulation (A, absorptivity; R, reflectivity).
In order to truly solve the actual forward simulation problem, we adopt a single forward simulation model shown as Fig. 2, in which the repeatedly stacked up and down sampling blocks shown as Fig. 3 can well capture the inverse flow of errors, and the data is well represented in repeatedly stacked convolution and transpose convolutional network. Four randomly selected forward simulation instances shown in Fig. 4, however, it can be observed in Fig. 4(b), in the 550 nm to 600 nm band, the spectral curve of the numerical simulation has dense fluctuations in a very narrow band, and the neural network cannot predict the curve change well here. The reason for this phenomenon is related to the loss function we used. Here we use MSE as the loss function for training the neural network, this loss function averages the error of the entire output, so that those data points with dense fluctuations and slight variation diluted by the changing trend of the entire curve, making the network fall into a local optimum. To quantitatively measure the error we introduce the mean absolute error (MAE) criterion to illustrate our results, the formula is as follows:
The average MAE error of 3840 test samples is 0.0072. For our data in this experiment, the average time of the traditional FDTD simulation is 91 seconds and the computation time of the forward simulated network is 0.15 seconds.For the inverse design network, the randomly selected test results are shown in Fig. 5. Our proposed inverse design model performs well in the inverse design moth-eye structure parameters. In particular, for the inverse design of material species, which is a classification problem, our design has an accuracy of $100\%$ , while the inverse design of the remaining four numerical parameters has an accuracy of over $95.2\%$.
Fig. 5. Inverse design of deep learning models for evaluation. (a,b,c,d) Four inverse design examples, in comparison with ground truth (T, thickness).
2.4 Demand-based inverse design
A practical inverse design model should output a series of micro-nano structural and material parameters based on the demand curve given by humans. A key issue here is that the demand curve given by humans is quite different from the real spectral curve, and the model we propose solves this problem well.
As shown in Fig. 6(b), we give the spectral characteristic curve of the demand, which is described as follows:
Fig. 6. Demand-based inverse design. (a,d) Metasurface parameters retrieved by inverse design model. (b, d) Demanded, forward simulation network predicted and numerical simulation reflectance spectral curves. (c,f) Absorptivity curves predicted by networks compared with numerical simulation.
The metasurface parameters generate by the network are shown in Fig. 6(a), and the network captures the spectral characteristics we expected precisely. What we require is low reflectivity in a wide frequency band (Fig. 6(b)). The network also generate the reflectivity curve of the forward simulation shown in Fig. 6(b) and the absorptivity curve shown in Fig. 6(c), as a reference.
For the research on the application of moth-eye structure to reduce reflectivity, a key issue is that the parameters of the moth-eye structure cannot be quickly and efficiently calculated within a range and based on human needs, which plagues the practical application of the moth-eye structure. Although previous work has pointed out that as the moth-eye structure gets closer to the pyramid and the height of the pyramid is higher, the reflectivity decreases [35], but such a structure is only at the theoretical level due to its difficulty in fabrication and poor availability. However, to limit the size of the moth-eye micro-nano structure, it has become a practical problem to inverse the design while ensuring that it can be actually manufactured. The traditional method cannot avoid interaction with the simulation environment. Therefore, it is inefficient and consumes a lot of computing resources, especially when material type parameters are added to the scope of the design, increased computation makes design unsustainable. Our proposed method can perform inverse design in a very short time (hundreds of milliseconds), and the simulation results also show that our structure has strong anti-reflection ability in the ultra-broadband.
Another example, as shown in Fig. 6(e), we give the demand spectrum curve described as follows:
To build a metasurface with nonlinear reflection characteristics based on human’s needs, such a requirement is almost impossible for traditional methods, but our inverse design model achieves this function very well. In this example our demand reflectance spectral is depicted by a LeakyRELU function, which is a nonlinear function also is an activation function commonly used in artificial neural networks, which is a nonlinear variant of the RELU function. Compared with the traditional neural network activation function, such as the Sigmoid function, LeakyRELU function is less computationally expensive and easier to avoid the gradient explosion and gradient disappearance problems during the neural network training process. Using the reflection properties of metasurfaces to build optical artificial neural networks may be an idea for future optical computing. The strong generalization ability (search ability) of our inverse design model can serve the design of optical computing components.
Here we give a human demand, but not so physically plausible spectral properties. In this case, the neural network’s implementation of the inverse design reflects the inherent robustness of the neural network model, which can generate the correct response even the input is not so reasonable, it can also be interpreted that the neural network reflects the characteristics of adaptive filtering in this regard. The full-wavelength simulation results show that the inverse design model can indeed solve the on-demand inverse design problem accurately and efficiently.
2.5 Forward simulation aids the exploration of physical properties
Using our proposed deep learning model, researchers can quickly derive related optical responses, thereby assisting their exploration and discovery of optical laws.
When four parameters of the moth eye are fixed (three semi-axis lengths are 150 nm ($R_{1}$), 150 nm ($R_{2}$), 250 nm ($R_{3}$), respectively, and the cover material is selected as silver (Ag)), and the thickness of the covering layer is varied from 0 to 100 nm (Fig. 7(a)). Moreover, data generated by the deep learning model are reflectivity (Fig. 7(b)) and absorptivity (Fig. 7(c)), by comparison, data generated by the numerical simulation are reflectivity (Fig. 7(d)) and absorptivity (Fig. 7(e)). It is obvious that there is no difference in extremum between the two contrasting heatmaps (Fig. 7), which means that the deep learning model has learned a superiorly nonlinear relationship between the wavelength and reflectivity /absorptivity.
Fig. 7. Evaluation of forward simulation deep learning models. (a) A moth-eye matesurface with specific parameters. (b) The results of the forward simulation network (reflectivity), (c) the numerical simulation results (reflectivity) when the cover thickness varies from 0 to 100 nm. (d) The results forward simulation (absorptivity), (e) the numerical simulation results (absorptivity) when the cover thickness varies from 0 to 100 nm.
We show another example, in this case, three parameters (lengths of two semi-axes are 150 nm ($R_{1}$) and 150 nm ($R_{2}$), and the cover material is selected as silver (Ag)) are all fixed, while $R_{3}$ (changes from 150 nm to 300 nm) and the thickness (10 nm, 25 nm, 40 nm, 55 nm, 70 nm, 85 nm) are varied. The heatmap (Fig. 8) shows how reflectivity and absorptivity vary with different parameters.
Fig. 8. Evolution of absorptivity (a) and reflectivity (b) at different wavelengths (380 nm to 780 nm) by varying the $R_{3}$ length (150 nm to 300 nm) with the specific cover thickness (10 nm, 25 nm, 40 nm, 55 nm, 70 nm, 85 nm).
According to Fig. 8(b), we can see that the reflectivity increases and then decreases in some areas in the figure with increasing cover thickness, which indicates that the deep learning model learns superiorly nonlinear relationships and complex evolutions between optical properties and metasurface parameters. Deep learning forward simulation can further discover new features, for example, when researchers find points of interest, they can further improve the subdivision accuracy of parameters to quickly retrieve and assist prototyping. Figure 8(a) shows the evolution of the absorptivity, exhibiting an almost opposite trend to reflectivity evolution, which is consistent with physical principles and validates the reliability of our model.
We further process the obtained absorptivity and reflectivity data to obtain the evolution of transmissivity under the same conditions. As shown in Fig. 9, on the overall trend, with the increase of the cover thickness, the transmissivity further decreases. However, from a local point of view, the transmissivity also shows a trend of nonlinear change. When the coverage thickness is 10 nm, the wavelength limit is 600 nm, and the transmissivity of long-wave light is higher. With the increase of the cover thickness, this limit shows a trend of gradually moving to the long-wave band until it disappears. Under the premise of designing an anti-reflective moth-eye structure, we do not want the material to have a high transmittance, and these heat maps will help us filter parameters and determine the range of parameters.
Fig. 9. Evolution of transmissivity at different wavelengths (380 nm to 780 nm) by varying the $R_{3}$ length (150 nm to 300 nm) with the specific cover thickness (10 nm, 25 nm, 40 nm, 55 nm, 70 nm, 85 nm).
For the numerical simulation part, in order to observe the overall evolution trend, a total of 1200 sets of simulations were carried out, using AMD Thread ripper 3960X personal computer for numerical simulation, the time required for numerical simulation was 10 hours, compared to about 1 second for deep learning models. Although it seems more cost-effective to use simulation software for a single simulation, as the workload increases, especially when a prototype with certain functions is required to be designed, the advantages of deep learning models are undoubtedly revealed.
3. Conclusion
In conclusion, we demonstrate the use of deep neural networks for inverse design and fast optical simulations. For the forward simulation, the network successfully learns the rules of the optical simulation and can generate an optical response curve almost indistinguishable from the FDTD simulation (MAE=0.0072) based on the given metasurface parameters. Once the network is successfully trained, this simulation process based on neural network computing avoids the traditional rule-based computing process and has an unparalleled speed advantage. For the inverse design, we successfully constructed a model based on the input spectral characteristic curve to realize the parametric design of moth-eye structure. This approach enables the design of metasurface parameters to greatly reduce the time and computational cost of manually designing them. It is worth mentioning that this work has solved the problems of structural parameter design and material selection of micro-nano structures. The prediction accuracy of this model for materials is $100\%$, and the prediction accuracy for other parameters exceeds $95.2\%$. The model allows one to quickly retrieve the desired structure, and the model still responds correctly even when the input does not quite conform to the rules of physics. In the future, this data-driven approach is expected to solve problems in the field of optics together with a pattern-driven approach. We believe that artificial intelligence technology represented by deep learning will be more quickly applied to accelerate optical simulation and solve micro-nano optical problems.
4. Methods
We use FDTD for numerical simulation, collected 55,400 samples and used 44,320 for training, 5,540 for validation, and 5,540 for test. The deep learning model is built under the open-source machine learning framework of TensorFlow.
Funding
Education and Scientific 313 Research Foundation for Young Teachers in Fujian Province, China (grant numbers JAT200034); Natural Science Foundation of Fujian Province (202J011303121).
Acknowledgments
The authors are grateful for the financial support from the Education and Scientific Research Foundation for young teachers in Fujian Province, China (grant numbers JAT200034) and Natural Science Foundation of Fujian Province (Grant No. 2021J01130121).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Data can be obtained by contacting the authors by email.
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