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Abstract
Optical nano-structure designs usually employ computationally expensive and time-intensive electromagnetic (EM) simulations that call for resorting to modern-day data-oriented methods, making design robust and quicker. A unique dataset and hybrid image processing model combining a CNN with gated recurrent units is presented to foresee the EM absorption response of photonic nano-structures. An inverse model is also discussed to predict the optimum geometry and dimensions of meta-absorbers. Mean-squared error of the order of 10−3 and an accuracy of 99% is achieved for trained models, and the average prediction time for the DL models is around 98% faster than that of simulations. This idea strengthens the proposition that efficient DL-based solutions can substitute the traditional methods for designing nano-optical structures.
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1. Introduction
Absorbers have been a part and parcel in various disciplines of electromagnetics (EM) including EM interference, wirelessly connected communication devices such as artificial intelligence (AI), Internet of Things (IoTs), microwave sensors, wearable electronics and stealth technology [1]. Although, EM-based wireless communication has transformed the world into a global village, but there remains serious inevitable challenge of EM pollution which is rendered to be a fourth leading cause of pollution [2] having adversarial effects on human health as well as being a cause of interference of communication channels resulting in poor signal strength. Thus, the origination of EM absorbers became imperative which later on were used in many other applications as well including the energy applications employing sun spectrum. The concern, thereby, of the design engineers was to realize broadband excellent absorption, having compact footprint, light weight and low cost.
In this context, the metasurfaces have provided a platform by offering a solution to the aforementioned challenges, they are artificial thin-film structures exhibiting unique and unusual interesting on-demand applications via controlling the incident EM waves [3–5]. They are planar structures featured by sub-wavelength dimensions [6,7] to realize their functionality and offer integrability with on chip nanophotonics [8–10]. They are periodic/ aperiodic arrangements of unit-elements known as “meta-atoms” [11]. They have drawn an enormous attention in many fields related to physics [12], material science [13,14], optical imaging [15–18], and bio-medical imaging [19]. Among these applications, metasurface based solar thermal absorbers have played a vital role for energy harvesting purposes in augmentation with PV cells [20]. Such designs are meant for achieving higher efficiencies than obtained using PV cells solely as their performance is constrained by Shockely Queisser (SQ) limit [21]. A broadband absorber is a key component in an ideal energy conversion system i.e. solar thermo-PhotoVoltaic (STPV) system [22,23]. The broadband absorber design calls for a cautious material selection for a particular meta-atom geometry. The material should have peculiar characteristics including high thermal, mechanical and chemical stabilities along with high extinction coefficient. Preliminary absorber designs were plasmonic materials-oriented because of their favorable lossy characteristics [24]. However, they stood unfit for real-world applications being rare, unstable against environmental effects, low melting point, inherently soft particularly in nanostructured devices and expensive for large scale production [25–27]. Therefore, there arises a need for some well-suited alternatives devoid of the routine challenges.
In this regard refractory materials provide a perfect solution. The term refractory, refers to a material that withstands high temperatures and chemical exposure without undergoing decomposition, while maintaining its structural integrity and strength. The refractory metals and their derivatives exhibit desired behavior with improved device performance with higher melting points than 2000 °C [28]. They have the tendency to behave like plasmonics in the visible region while simultaneously providing the benefits of environmental sustenance. They have offered near unity absorption over broadband, flexible and tunable low cost, easy to fabricate complementary metal-oxide semiconductor (CMOS) compatible designs [28–31], which are the desired attributes for an STPV system.
Moreover, a metasurface’s response is governed by the geometry of their basic elements as there is a lack of universal theory for determination of a particular application. Therefore, the design of such structures is crucial in terms of material as well as geometry selection of a meta-atom simultaneously. Under the light of a desired optical output augmented with the fabrication constraints, the conventional design route warrants for lengthy simulations of the geometrical features. This approach overly burdens the CPU becoming computationally costly as the EM solvers compute the solution of intriguing Maxwell Equations by applying boundary conditions on a case-by-case basis. The use of state-of-the-art EM simulation schemes such as finite-difference time domain is cumbersome and the design time is proportional to simulation time of as many as trials on iterative basis are done and to the number of degrees of freedom present in a structure. Various gradient and non-gradient based optimization techniques have been explored in this regard to expedite the inverse design of meta-devices, among these the adjoint variable method has grown to be one of the most popular gradient-based inverse design technique, owing to its low computational cost [32]. This method acquires the objective function's gradient with respect to every design degree of freedom (DoF) by means of two full-field simulations [33,34]. This feature makes it appropriate for nano-structures that need to simultaneously accomplish many design goals and have a vast parameter space [35,36]. Despite of all the advantages of adjoint methods and their capability to handle free-from geometries, they sometimes struggle with low efficiency because, when optimizing, they must inevitably call an EM solver, which drives up their computational cost and memory usage as the number of design parameters that need to be optimized rises. Furthermore, it can suffer from poor convergence rate due to their high chances of falling into local optimal solutions. This hinders the inverse design approach, necessitating the development of other algorithms to deal with these issues. As a solution, the data-driven deep-learning modelling tools have been put forward to expedite or replace the conventional inverse optimization methods not only in optics [37,38] but also in other domains of device designs [39]. A number of metasurface-based designs have enjoyed the benefits offered by such emerging techniques as discussed in subsequent section.
2. Related work
Recent surveys [40,41] have discussed that deep learning techniques have been applied in forward as well as inverse topologies to predict the EM response and the optimized design parameters respectively [42–44] owing to their computational friendly nature [45,46]. Several approaches using fully connected neural networks (FCNN) [47–49] or the combinations of FCNN and convolutional neural network CNN/Transposed CNN [38,50] were proposed to provide accurate forward EM spectrum predictions. But these approaches take either the planar shape information of the input nanostructures only (while completely ignoring its one dimensional measurements), or they take the detailed geometrical parameters of the nano-structure as input but work for a specific shape only. In order to cater multiple shapes as well as the complete geometrical parameters of each shape simultaneously, a combination of CNN and RNN (recurrent neural network) was proposed [51]. It predicted the absorption spectrum accurately while considering all the necessary information, but needed to be retrained via transfer learning to work of each new material. For the inverse design of metasurfaces, generative adversarial networks (GANs) [52,53] have been proposed to predict the shapes of nano-resonators for a corresponding design target. Using GANs do perform well, however it also suffers from certain drawbacks including its extremely data hungry nature, complexity, mode collapse and training instability [54]. Since the discriminator and generator networks of a GAN are always in competition with one another, training becomes really hard, irregular and sluggish [55]. Also, their usage for diverse materials inclusion and robust design targets is constrained by the fact that they are only able to predict 2D planar images of 3D nano-structures, unable to provide information about its 1D geometrical parameters (like the height, width, length etc. of the nano-structures) or the constructive material. In contrast, our approach trains a single inverse model while simultaneously taking into account the planar shape as well as the 1D dimensional measurements. Training a single model results in easy implementation, faster training and easy convergence. Despite of being a single model, it is capable of incorporating material diversity and handle broadband optical spectrums at the input. Some other approaches such as fast forward dictionary search (FFDS) [38], tandem neural networks [56] and loop based cascaded neural networks [57] were also proposed to perform inverse design of meta-atoms but they predict only structural parameters and are specific to one shape only. Also the FFDS method is still time and memory consuming requiring to create a whole database for the optimum inverse design search. A comprehensive comparison for both forward and inverse models from the existing literature is presented in Table 1 and Table 2, respectively.
Table 1. Tabular comparison of the proposed APCNN with the previously reported models for Metasurface’s forward modeling networks.
Table 2. Tabular comparison of the proposed MDPNN with the previously reported models for Metasurface’s inverse modeling networks.
In this work, time-efficient and computationally inexpensive deep-learning approach has been employed for broadband realization of absorption behavior for an efficient STPV system. The application of this strategy would circumvent the challenges of the optimization via bypassing the long iterative simulations, irrespective of complex design requirements. The complete shape of a meta-atom is governed by its material and geometry acting therefore as training features for our deep-learning model. The material is incorporated by storing the two-dimensional images distinguished based on its colors for different materials, while the geometrical features define the meta-atom. Image information along with the geometrical features are two main forms of input that thoroughly define a meta-atom and, therefore, both of them are employed for establishing a relationship between the optical response and their design. We have collected a unique dataset having both 2D images as well as 1D geometric information vectors of meta-atoms and their corresponding absorption spectra over the wavelength range 300-1200 nm spanning the VIS-NIR regimes of the EM spectrum. The dataset is made diversified with reference to material nature e.g. it has both metals and refractory materials forming the nano-resonators, qualifying for their absorptive behavior. Two deep learning models i.e., absorption predicting convolutional neural network (APCNN) and meta-atom design predicting neural network (MDPNN) are proposed to predict the absorption spectrum of meta-atoms and to perform their inverse design optimization for the target response respectively. APCNN comprises of a combination of convolutional neural network (CNN) having skipped connections and gated recurrent units (GRUs). The CNN extracts all the useful information from images and pass it onto the GRU layer whose output is then flattened and combined with the normalized 1D structural parameters. The combined array is then passed through FC layers to set-up a relationship between spectral response and meta-atom’s complete structural information. Once trained, the APCNN is successful in realizing a test MSE of 1.8 × 10−4 and generates the absorption spectrum very accurately across a broad range of wavelengths. The inverse model i.e. MDPNN comprises of a regression based deep neural network which takes the desired absorption spectrum at its input. To train this model, ground-truth meta-atom images are first reduced in dimensionality using principal component analysis (PCA) and then flattened and combined with the 1D structural parameters. The combined array of the PCA extracted features and other parameters is then used as the training output of the MDPNN. After training, a test MSE of 8.9 × 10−1 is attained and the output is processed through inverse PCA operation to reconstruct the predicted images. The accuracy of MDPNN is validated with the help of multiple experiments, showing the concurrency between predicted image and the 1D parameters with the original image and parameters. Thus, it is capable of replacing the conventional time-consuming, computationally-expensive and laborious design procedure of metasurface based perfect solar absorbers.
3. Dataset collection
To validate the proposed deep-learning method for design of nanoscale absorber, we have considered a variety of shapes of meta-atoms. Their designs are made up of eight (8) different materials ranging from plasmonics to refractory materials. The example materials include gold (Au), Silver (Ag), Chromium (Cr), Tantalum (Ta), Molybdenum (Mo), Nickel (Ni), Titanium Nitride (TiN) and Zirconium Nitride (ZrN) [58–60]. Each of the materials have been used in four different shapes in stacked MIM array, where M is a metal and I is the insulator which in our case is SiO2. The design of these absorbers have procured strength in energy harvesting STPV applications. As a cautious design is demanding in terms of time and computational power, so their burden is shared with the aid of deep learning models to design the metasurfaces in temporally and computationally ameliorated manner. Other than STPV’s, such Metasurface-based, subwavelength, near-perfect MIM absorbers (employing metallic top and bottom layers with an insulator in between) are widely used for many other application in different wavelength regimes, including radio frequency (RF), terahertz (THz) [61], infrared (MIR and NIR) [62,63] and optical domain [64,65].Therefore, it is quiet useful to collect a versatile dataset and devise a model for rapid optimization of such structures. Although this dataset collection is time-consuming, but it is a one-time process. Using commercial solvers demands iterative optimization procedure with time-consuming parametric sweeps for any slight change in the design target, wavelength or material. Keeping in mind the vast application of MIM absorbers, this process needs to be repeated again and again for every new design target/change of desired application. But, once trained our DL-enabled inverse optimization model, will be able to predict optimum geometry within fraction of seconds for any application lying in its ultra-wide absorption spectrum (300 nm to 1200 nm) with a variety of materials and shapes to choose form. Thus, this one-time data collection saves us from the redundant efforts of solving the Maxwell equations on a case-by-case basis in conventional iterative optimization procedure.
The prototype of the proposed meta-atom using one of the shapes under study is shown in Fig. 1, comprising of two different information, one is the 2D cross section image and the other are its corresponding 1D geometrical parameters. The image data are processed via convolutional neural network to extract features, which are subsequently fed to the recurrent neural network (RNN) along with the 1D geometrical parameters vector. This 1D vector comprises of the lattice size i.e. the period (P), heights of insulator layer which acts as a spacer between to metals i.e. (hs) and nano-resonator pattern height i.e. (hp). The CST Microwave Studio is used for dataset collection. Meta-atoms with eight different materials in their four different shapes and three varying geometrical parameters have been used. Each material is distinguished from the other based on its color in the RGB images. The geometrical parameters are varied from P : 200-500 nm (in steps of 25 nm), hs : 40-80 nm (in steps of 10 nm) and hp : 20-60 nm (in steps of 10 nm), thus the training instances for a given material in case of single shape are 325 in number. The total number of training examples for all materials (8) and all shapes (4) are 32 × 325 i.e. 10400. The output which in this case is the electromagnetic (EM) absorption has 500 data points in the output vector for a given example. The total output vector size is 10400 × 500. The number of images is same as that of number of times the simulations are carried out to get a total of 10400 training samples, where each of the image in its original form is of 64 × 64 pixels as shown in Fig. 1. The dataset has a considerable variance due of different shapes. The absorption spectrum spans the broadband wavelength range to cover the entire optical domain of the EM spectrum. Once the model is trained, it predicts the EM response in matter of seconds thereby bypassing the need for a strong computing system. This technique is extendable such that it may cover any of the different structures and evaluate any other optical properties as well.
Fig. 1. The schematic diagram of forward model with CNN for image processing and RNN incorporating the overall input information for predicting the absorption spectrum over 300-1200 nm. The design materials with their melting points are included. 3D RGB image of meta-atom structures with their 2D cross sectional view and geometrical features which in combination of 64 × 64 pixels image and 1D 3-valued vector is used for training the deep-learning model. The comparison of ground truth obtained in the light of Maxwell Equations using CST-simulations and the model predicted absorption is included in the right section of the figure.
4. Methodology
Image processing based hybrid sequence model is presented to solve the forward problem i.e. to predict the electromagnetic absorption curve (over the complete solar spectrum) for a given meta-atom. This model is robust enough to predict the response with the same accuracy irrespective of the nano-resonator’s material, shape and dimensions. An inverse model based on the combination of principle component analysis (PCA) and deep neural network is also presented to solve the inverse design problem of meta-atoms to meet the desired target response. These two models are explained separately in the following sections.
4.1 Absorption predicting convolutional neural network (APCNN)
The underlying material, geometry, and structural dimensions of a meta-atom collectively affect the output EM response. Therefore, it is intensely important to integrate all these factors into the proposed APCNN and map the output response onto these factor. To address this, we have presented a combination of deep convolutional neural network with residual layers and gated recurrent units followed by fully connected layers at the output. The layout and details of this model are presented in Figs. 2 and 3. The intent of this model is the estimation of EM absorption spectrum of the metasurface.
Fig. 2. Illustration of feature extraction from meta-atom’s images via convolutional neural network.
4.1.1 Image features derivation:
The first part of the proposed APCNN comprises of a deep convolutional Neural Network having skip connections with a ResNet like architecture. These skip connections stabilize the back-propagation by minimizing the vanishing gradients, enabling us to train deeper networks rapidly while providing accurate predictions. This CNN takes 64 × 64 pixel RGB images of the 3D meta-atoms as input 1, and extracts all the important features from these images.
These images are first multiplied to the input 2 (the 1D parameters input) such that the first channel of each image is multiplied to the first 1D parameters i.e., hs (Spacer Height), the second channel is multiplied to hp (dielectric resonator’s height) and the third channel is multiplied with the period of the meta-atom. This multiplication differentiates two similar images having different resonator or spacer height. The multiplied version of the images is then fed to the CNN as the first input. Numerous parameters had to be optimized when structuring the proposed CNN. In the light of the complicated nature of the problem under considerations, a variety of model layouts were explored and evaluated to find the best architecture. The architecture of the final model is detailed in Fig. 3 and it exhibits a number of convolutional layers, batch normalisation layers, pooling layers, activation layers etc. Leaky-ReLU is used as the non-linear activation function. The features extracted by CNN are then passed on to the gated recurrent units (GRU) followed by FC layers.
4.1.2 Absorption spectrum prediction:
The features derived through CNN are fed to a 100 units GRU layer due to its capability to handle long sequential data. Here each output vector is a 500 rows 1D column vector, and all these 500 points are distinct steps separated from a common spectrum. Therefore each distinct step temporally relies on the preceding step. Also the order of these points must be reserved to predict the perfect curve, therefore the application of GRU’s is the best choice here. Output of the GRU layer is then flattened and concatenated with the second input i.e. the 1D input parameters including hs, hp and P. The concatenated array is then passed through the fully connected (FC) layers resulting in the prediction of the absorption curve at the output. A batch size of 32, LR of 10−3 and an Adam optimizer is used to train the proposed APCNN. This problem involves the prediction of a continuous absorption spectrum making it a regression problem. The most commonly used evaluation metric for regression networks is mean squared error, therefore it is used for evaluation here, given as:
Here Ai′ and Ai refers to the predicted and the simulated absorption curves respectively. N is the total number of dataset samples which in our case is 10400. The forward model presented here is meant to be applied in energy applications particularly in STPV systems to design meta-absorbers and predicting their absorption curve without needing any iterative equation solving simulations.
4.2 Meta-atom design predicting neural network (MDPNN)
The inverse model MDPNN comprises of a regression based deep neural network. It is trained to predict the optimum shape and geometric parameters of an absorber meta-atom to achieve the target EM response. The proposed MDPNN takes the target absorption spectra as input, processes it via multiple up sampling and down sampling layers and predicts the best suitable meta-atom structure corresponding the target spectra. Thus each input here is a 500 rows 1D column vector, representing the absorption spectra over the wavelength range 300 to 1200 nm. The hyper-parameters of the proposed MDPNN are as follows: Optimizer - Adam, Training epochs - 5000, number of hidden layers - 9 and number of neurons per layer - 500-600-900-1000-3000-5000-3000-2000-3843 as included in Fig. 4. The features of the RGB meta-atom images, which served as the input of the APCNN model, are now extracted using PCA with number of components equal to 20. The important PCA features of the three channels are combined and flattened into [3840 × 1] vectors. The flattened vector for each image is combined with its corresponding 1D parameters set i.e., hs, hp and P resulting in [3843 × 1] vectors, which are used as the ground truth outputs to train the proposed MDPNN. An Adam optimizer with an optimal batch size 32 and learning-rate of 10−3 is used to train the proposed MDPNN. Since this problem involves the prediction of continuous PCA vectors, therefore mean squared error is used here as the loss function given as:
Here, Si and Si′ represents the ground-truth and predicted set of PCA extracted meta-atom image features combined with the 1D parameters respectively. The image features are separated and then passed through inverse PCA transform to generate the predicted meta-atom shape.
5. Results and discussion
The results of the trained APCNN and MDPNN models are discussed in the following sections.
5.1 Results of the trained APCNN
APCNN is trained with a 10−3 learning rate (LR) and an Adam optimizer to achieve a minimum possible value of loss function stated in Eq. (1). Optimum training is achieved in 500 epochs. Once trained, the model reaches an MSE of 1.8 × 10−4 over the test dataset. The contrast between the predicted and simulated absorption curves for some randomly selected test examples is shown in Fig. 5. The span of dataset is kept large as in order to make the model learn the intricate relationship of the absorption response of a meta-atom with its geometrical layout, the dataset must have enough variance and standard deviation. Therefore, a huge amount of data-samples were collected where some of them had near-perfect absorption while others had it not so perfect. This dataset was divided into test and train sets, and the performance of APCNN was analyzed upon the test dataset on basis of how well the APCNN’s predicted absorption curves matches with the simulated absorption curves. It is evident that the trained APCNN provides accurate predictions with a negligible difference between the predicted and simulated absorption spectra. Figure. 5 also depicts that the model’s accuracy and performance is equally good for all refractory materials as well as the selected metals.
Fig. 5. Predicted versus Simulated Absorption curves for random test samples belonging to different materials and dimensions. The inset of each graph shows the corresponding nano-resonators shape, the color of the nano-resonator depicts the material with their 1D parameters in the order [hs, hp and P].
Thus, the proposed model can replace conventional lengthy and iterative simulations and provide absorption response of meta-atoms having different shapes and materials within a split second. A comprehensive ablation analysis of the proposed APCNN with respect to the input image resolution, different LRs and batch sizes is shown in Table 3.
Table 3. Ablation Study of the proposed APCNN with respect to the input image resolution, different LRs and batch sizes.
5.2 Results of the trained MDPNN
The proposed MDPNN is trained to provide an optimization tool capable of predicting the optimum shape and the structural parameters of a meta-atom corresponding to a specific design target. MDPNN is trained with a 10−3 learning rate and an Adam optimizer to achieve a minimum possible value of loss function stated in Eq. (2). Optimum training is achieved in 5000 epochs performed in batches of 32 samples each. Once trained, the model reaches an MSE of 8.9 × 10−1 over the test dataset. The predicted vectors are of shape [3843 × 1] for each input. First 3840 points represents the PCA extracted features of meta-atom images while last 3 are 1D structural parameters i.e., [hs, hp and P]. PCA extracted features are then processed via inverse PCA transformation to get the predicted shape image of the meta-atom. The performance of the inverse design is depicted in Fig. 6, where we provide a near-perfect absorption as input and achieve meta-atom’s geometrical layout as the output. A comparison of the predicted and the ground-truth meta-atom’s shape and geometric parameters is shown in Fig. 6. Figure. 6(a) and (b) shows the predicted shape and its corresponding 1D parameters in the order of a metal based test example whereas (c) and (d) depicts the predicted shape and the corresponding 1D parameters of a refractory material test example. These results prove that the trained MDPNN perfectly predicts the shape and the geometric parameters simultaneously regardless of the material of the meta-absorbers. Therefore, it serves as a generalized tool to optimize meta-absorbers for an ultra-wide wavelength range (300 - 1200 nm) using any metal or refractory material resonators. A comprehensive ablation analysis of the proposed MDPNN with respect to the no. of components of PCA extracted features of the output image, different LRs and training epochs is shown in Table 4.
Fig. 6. Comparison of the predicted and the ground-truth absorber meta-atom’s shape and structural parameters. (a) and (c) shows the ground truth and predicted shapes whereas (b) and (d) depicts the ground truth and the corresponding 1D parameters.
Table 4. Ablation Study of the proposed MDPNN with respect to the PCA extracted features of the output image, different learning rates and number of training epochs.
6. Conclusion
To conclude, the design methodology of flat optic devices can be made swift over time and less computationally hectic by taking aid from trained deep-learning models which accurately and efficiently predict the optical response of meta-devices in a split second. The presented work illustrates that image processing can be used to extract all useful information from meta-atom images and map them on its EM response accurately serving as a replacement of computationally-intensive numerical simulations. The success of these methods lie in dataset size and type which ideally must have a significant variance. The dataset here is an accumulation of eight materials in four different shapes with all of them behaving well as absorbers. The collected data size has proved to be sufficient enough that the models perfectly predicted the absorption spectrum and the optimum meta-atom geometry. The forward model, Absorption Predicting Composite Neural Network (APCNN) comprises of an integration of deep convolutional neural network with residual layers and a portion of gated recurrent units based recurrent neural network. APCNN is trained with 10−3 learning rate and an Adam optimizer achieves a minimum possible value of a regressive loss function. Once trained, the model reaches an MSE of 1.8 × 10−4 for test dataset. An inverse model which employs a combination of principle component analysis (PCA) and deep neural network is also presented to solve the inverse design problem of meta-atoms to achieve the desired response. Inverse model MDPNN is also trained with a 10−3 LR and an Adam optimizer to achieve a minimum possible value of a regressive loss function for 5000 training epochs. The inverse trained model reaches an MSE of 8.9 × 10−1 for test dataset. Thus, it is safely stated that the trained DL models serve as a generalized tool for meta-absorber design optimization over an ultra-wide wavelength range (300–1200 nm) while using any metal or refractory material resonators.
Disclosures
The authors declare no conflicts of interest.
Data Availability
The datasets used and analyzed during the current study are available from the corresponding authors on reasonable request.
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