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Abstract
A standardized hybrid deep-learning model based on a combination of a deep convolutional network and a recurrent neural network is proposed to predict the optical response of metasurfaces considering their shape and all the important dimensional parameters (such as periodicity, height, width, and aspect ratio) simultaneously. It is further used to aid the design procedure of the key components of solar thermophotovoltaics (STPVs), i.e., metasurface based perfect solar absorbers and selective emitters. Although these planar meta-absorbers and meta-emitters offer an ideal platform to realize compact and efficient STPV systems, a conventional procedure to design these is time taking, laborious, and computationally exhaustive. The optimization of such planar devices needs hundreds of EM simulations, where each simulation requires multiple iterations to solve Maxwell's equations on a case-by-case basis. To overcome these challenges, we propose a unique deep learning-based model that generates the most likely optical response by taking images of the unit cells as input. The proposed model uses a deep residual convolutional network to extract the features from the images followed by a gated recurrent unit to infer the desired optical response. Two datasets having considerable variance are collected to train the proposed network by simulating randomly shaped nanostructures in CST microwave studio with periodic boundary conditions over the desired wavelength ranges. These simulations yield the optical absorption/emission response as the target labels. The proposed hybrid configuration and transfer learning provide a generalized model to infer the absorption/emission spectrum of solar absorbers/emitters within a fraction of seconds with high accuracy, regardless of its shape and dimensions. This accuracy is defined by the regression metric mean square error (MSE), where the minimum MSE achieved for absorbers and emitters test datasets are 7.3 × 10−04 and 6.2 × 10−04 respectively. The trained model can also be fine-tuned to predict the absorption response of different thin film refractory materials. To enhance the diversity of the model. Thus it aids metasurface design procedure by replacing the conventional time-consuming and computationally exhaustive numerical simulations and electromagnetic (EM) software. The comparison of the average simulation time (for 10 samples) and the average DL model prediction time shows that the DL model works about 98% faster than the conventional simulations. We believe that the proposed methodology will open new research directions towards more challenging optimization problems in the field of electromagnetic metasurfaces.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As the contemporary world is focused on replacing conventional non-renewable energy sources (such as coal, gas, and oil) with renewable energy sources, the sun has turned out to be an immeasurable, constantly replenishable source of green energy. Solar energy can be harvested throughout the year without any turbomachines, providing additional benefits of negligible harmful emissions and environmental pollution. Despite all these advantages, the effective use of solar energy for large-scale applications is obstructed by its poor conversion efficiency dictated by the Shockley-Queisser limit. Recently Solar thermophotovoltaics (STPVs) have emerged as an unprecedented route to boost the efficiency of solar cells [1]. The performance of an ideal STPV system significantly depends upon two of its key components, which are; (i) perfect solar absorbers: capable of absorbing sunlight over the entire solar spectrum covering a wide range of incident angles, (ii) selective emitter: capable of emitting the absorbed light above the bandgap of the solar cell present underneath it. Together the absorber/emitter module harvests all the electromagnetic waves reaching the earth as sunlight by converting their elementary particles into thermal energy, and then effectively emits these photons to excite the underneath solar cell resulting in power generation [2,3]. This results in compactness, improved conversion efficiency, and high power generation, thus overcoming the drawbacks of a typical solar cell like Shockley-Queisser limit and bulkiness which makes them unfit for on-chip realizations. Different techniques have been proposed to design compact perfect solar absorbers and selective emitters for STPV’s, among which, the use of planar metasurfaces has turned out to be the best solution in terms of high conversion efficiency and compactness [4–6]. Metasurfaces are composed of periodic arrays of electrically thin subwavelength-featured resonators that can efficiently manipulate the wavefronts of the incident electromagnetic waves. The geometric shape and the dimensions of these subwavelength resonators can be engineered to achieve a unique optical response that can help to realize interesting applications such as metalenses [7], optical absorbers [8,9], meta mirrors [10,11], holography [12,13], OAM multiplexing [14–17] and non-diffracting beams generation [18] and many other non-linear optical interfaces [19]. The biggest hurdle in the designing of such metasurfaces-based devices is their hit and trial design procedure that lacks generalizability and entirely depends upon the researcher’s knowledge and intuition [20]. The conventional course of action for designing and optimizing metasurfaces involves multiple simulations and parameter sweeping of their unit cells via expensive commercial software that consumes a large amount of time as well as computational resources [21,20]. Hence, there’s a clear need to develop a generalized and time-efficient model to retrieve, analyze and map the optical response of a metasurface on its structural parameters aiding their design procedure.
Over the past few years, deep neural networks have been successfully used in solving complex scientific problems, such as biomedical imaging [21], predicting DNA and RNA sequences [22], high-energy physics data analysis [23], etc. Likewise, the latest research in the field of nanophotonics is also focused on using deep learning to discover the latent relationship between the optical response and the parametric feature space of nanostructures [24–26]. Different architectures of neural networks have been proposed to model complex nanophotonic structures [27–29] and infer their optical response. To the best of the authors’ knowledge, the state-of-the-art methodologies take into account only one of the two structural aspects i.e., either the shape of the nanostructure or its geometrical parameters [30]. Also, most of the proposed models have been designed for a specific thin-film material only, and require retraining to make them applicable for any other material [31]. For the accurate optical response estimation, it's important to take into account both shape and the 3d geometrical parameters of the nanostructure simultaneously as done in some inverse models [37]. These geometric parameters could be height, width, aspect ratio, periodicity, etc.
Here, we propose a novel composite model, integrating deep convolutional neural network with skip layers and recurrent neural networks, to predict and analyze the absorption and emission curves of metasurface-based perfect solar absorbers and selective emitters respectively, using the images of their building blocks (unit cells) as input. Since the electromagnetic response of a nanostructure is owing to its design, shape, and size, each input is a 3 channel image that incorporates the shape as well as other structural parameters (such as height, width, aspect ratio, etc) of a unit cell in a very unique way. The geometrical shape of the nanostructure is incorporated in the first channel, followed by a dedicated height channel and an aspect ratio channel respectively. The optical response in the form of absorption/emission spectrum has been used to define the output labels. The input images are fed to a convolutional neural network (CNN) incorporating the residual structure similar to ResNet, to extract the spatial information. Periodicity, which is another important structural parameter, is taken as a secondary input and its array is concatenated with the extracted visual feature vector. It is then passed to a recurrent neural network (RNN) to find the relationship between the image and its optical response spectrum. We have developed a generalized model architecture that is trained separately for solar absorbers and emitters as their optical response exhibits different wavelength ranges. Once the model is trained for a specific refractory material absorber or emitters, transfer learning can be employed to re-use the trained model for other thin-film materials. This eliminates the need to collect huge data for each new material and trainer-train the model from scratch. Despite the fact that the CNNs are generally employed for the classification tools, owing to the numerical nature of our absorption/emission curves we have used CNNs to solve a linear regression problem by combining them with RNNs. The authenticity of the proposed model has been validated by its error metric and its comparative analysis with the conventional design approach. Table 1 shows the comparison of the proposed methodology with the previously reported methodologies to infer the optical response of metasurfaces.
Table 1. Comparison of Different Deep learning based models for Metasurface’s optical response prediction
2. Methodology
Figure 1 depicts the proposed network architecture for predicting the absorption/emission curve from a 3D nanostructures-based unit cell of the target metasurface. Below we discuss different parts of the proposed model and its reasoning, starting from how to represent the metasurface, then moving on to how to extract relevant features, and finally a discussion about using RNN to output optical characteristics.
Fig. 1. Schematic of the proposed plan of action. Layout of the model, from a metasurface unit cell input Xi to output absorption spectrum (Ai) defined by a function f (Xi, Ai; Θ)
2.1 Representing Metasurface as a 3D image
As illustrated in Fig. 2, a metasurface-based perfect solar absorber is composed of periodically repeated nano resonators made up of tungsten laid on a SiO2 dielectric spacer sheet, followed by a ground plane made up of the similar metal as that of the nano resonators i.e., tungsten. Tungsten is a refractory metal and exhibits a high melting point which makes it a suitable candidate for perfect solar absorbers.
Fig. 2. Illustration of the 3 channel input images incorporated with the structural parameters of the corresponding unit cells.
As the 3D nanostructures are uniform in the z-direction, the preprocessing of an input Xi starts by taking a cross-sectional image of the frontal view of a specific unit cell. Since the ground plane is a constant layer for perfect solar absorbers, having the same composition and height in all the unit cells so its effect in varying the output response is neglected and only the front view of the unit cell is considered. This image is then binarized and converted into a 100 × 100-pixel binary version where ‘1’ refers to the presence of tungsten nano resonators and ‘0’ represents the underlying SiO2 spacer layer. This pixel resolution is suitable enough to cover all the details of the structure without raising any memory issues during the model compilation. Finally, this binary image is then stacked with two additional layers or channels to incorporate the height and the aspect ratio (ratio of the height to the smallest feature size) of the individual nanostructures in each unit cell to get the ultimate Xi as depicted in Fig. 2. Incorporating height and aspect ratio in the image’s channels will allow us to cater to unit cells having multiple nanostructures where each structure exhibits a different height and aspect ratio.
2.2 Optical response prediction
The optical response of a metasurface adequately relies upon the material, shape, and dimensional variables (such as height, width, periodicity and aspect ratio, etc.) of its unit cell i.e., unit nano-resonant structure that is periodically placed in a specific arrangement with particular orientations to achieve the desired phenomenon. Therefore, we demonstrate a hybrid model combining CNNs and RNN that takes into account all these factors, maps their effect on the output, and predicts the absorption/emission accordingly as shown in Fig. 1. The input of the model will be the images of unit cells along with their geometric parameters and the output will be the corresponding optical absorption and emission spectrums collected by simulating the unit cells in the commercial software CST microwave studio. A CNN consists of multiple layers of trainable convolutional filters. These filters extract the spatial information from the input images by analyzing their subsections and passing on the results to the subsequent layers and so on. Collectively these layers work together to extract task-relevant features from the images, in our case, these might be capturing relationships between number and type of shapes, their location, and orientations, etc. Feature volume extracted from the last layer of CNN is fed to RNN, which is trained to output optical characteristics. Whole network is trained end to end. The final architecture of the proposed model is shown in Fig. 3. Below we discuss details of the architecture.
2.2.1 Feature extraction
Here we have chosen a deep convolutional Neural Network having residual structures inspired by ResNet for feature extraction. Skip connections in our model help to mitigate vanishing gradients [34] and provide stable backpropagation [35], allowing one to train deeper networks thus resulting in faster training and accurate predictions. While structuring our CNN, a large number of factors needed to be optimized. Considering the complexity of our problem, different architectures of the model were tried and tested to find the optimum number and layout of convolutional layers, size of filters in each convolutional layer, batch normalization layers (to avoid overfitting and adding regularization), pooling layers, activation functions for each layer, type of optimizer, hyper parameters and loss functions. To further avoid the problem of the vanishing gradients, the non-linear activation function employed, at some layers, is Leaky ReLU [36] which is mathematically described as:
Instead of the function being zero when x < 0, a leaky ReLU will instead have a small negative slope (of 0.01, or so) thus solving the problem of conventional dying ReLU.
Since the periodicity of a unit cell is a fixed value irrespective to the number of nanostructures in it, therefore after extracting the spatial information from the images via convolutional layers, the periodicity parameter will be added as a secondary input to the network by concatenating its array with the first fully connected layer as shown in Fig. 3.
2.2.2 Response prediction
The features extracted by the CNN component are flattened and concatenated with a periodicity input array. Then the concatenated array is passed through some dense layers and fed to the RNN part which maps them temporally onto the output for response prediction. Basic reason behind using RNN here is that, we are dealing with long sequential outputs, each output is a 1000 points vector where each of these points is a unique discrete sample of the same absorption spectrum and they are temporally dependent on previous prediction. Also the sequence of the thousand output points must be maintained to get the accurate curve. Therefore, we have used the RNN component in combination with time distributed layers to apply GRU onto the CNN’s output. A repeat vector layer is used to prepare the flattened output of CNN to be fed to the GRU layers. Here we have three GRU layers (as shown in Fig. 3), to generate a temporal output sequence followed by some time-distributed layers. Time distributed layer is basically a wrapper that allows us to apply a layer to every temporal slice of an input [38]. Time Distributed layers are used in our model to apply the same batch normalization, leaky relu and dense operations on every temporal slice of GRU’s output. Thus the RNN part takes into account the temporal relation between the output points while mapping the input features onto the output. Simple Convolutional network will not allow us to have this kind of temporal dependence.
To further validate this point, we can compare the results predicted by the model after applying a dense layer directly on top of the CNN features without any RNN component to the results predicted by the model using the combination of both CNN and RNN as proposed. Figure 4. shows the comparison of the predicted outputs by the model with GRU and without GRU.
It is evident from Fig. 4, the CNN alone results in outputs with large fluctuations in it. The output is not smooth and it's difficult to trace the original pattern in many cases. Each point of the output behaves as an independent output point with no temporal relation with the previous output point. Whereas as explained in the previous paragraph, adding an RNN component i.e., GRU layers, acts as a temporal regularizor and gives a smooth and accurate output. The temporal output of the final time distributed dense layer represents the predicted absorption spectrum, as shown in Fig. 3.
2.3 Dataset (MetaNet) collection
Two separate datasets are collected to train the proposed model independently for solar absorbers and emitters respectively. Collectively we have named the dataset as “MetaNet”. To meet the challenges of an ideal STPV system, its absorber/emitter metasurfaces need to be designed with special care and require arduous efforts. Below we present a brief overview of the requirements of absorber and emitter, and strategy to collect data for them.
2.3.1 Perfect solar absorber
An ideal Solar thermophotovoltaics (STPV) system must exhibit significantly high solar-to-electric power conversion efficiency, to remarkably exceed the theoretical Shockley-Queisser limit. This is possible only if its intermediate absorber/emitter structure can successfully harvest the entire solar spectrum and thermally emit the absorbed energy in correspondence with the band gap of the PV cell underneath it. Therefore a metasurface based perfect solar absorber for STPVs must be able to absorb all the sunlight within the wavelength range of 300 nm to 1300 nm [32] as dictated by AM1.5 solar spectrum [33]. This absorbed light is then transmitted as heat energy to the underlying selective emitter which then emits a narrow-banded spectrum just above the band gap of the underneath solar cell. The operating temperature of the intermediate absorber/emitter assembly exceeds 1000 K, therefore we need metasurfaces with nano resonators made up of refractory metals (such as tungsten, chromium, tantalum etc.) which exhibit high melting points to withstand the operating temperature. Thus we have opted for tungsten absorbers here.
Let the first part of the MetaNet i.e., the absorbers dataset be defined as DA={ (Xi, Pi, Ai), i = 0,1,3…..N}, where X represents the preprocessed 100 × 100, 3 channel images of 2000 randomly shaped nanostructures (to ensure enough variance) which are simulated in CST microwave studio over a wavelength range corresponding to the solar irradiance spectrum i.e., 300 nm to 1300 nm. All the simulations are carried out with periodic boundary conditions and the resulting 1000 nodes solar absorption spectrum of each simulation is stored as the datasets output Ai. Ai is a 1000 points 1D vector Ai = {a1,a2, a3,…..,a1000}, where each point ai is the absorption value at a single wavelength thus covering the wavelength range 300 nm -1300 nm. Along with the Xi and Ai, periodicity of each unit cell is also stored in the secondary input column vector P. This secondary input is a 1D column vector (2000 × 1) where each value Pi is the periodicity of the corresponding unit cell. This input will be simply concatenated with the first FC layer of the model to map the effect of periodicity on the output response. The composite neural networks model is trained using the optimization function f (Xi, Ai ; Θ) parameterized by Θ, where Θ is a matrix of weights controlling the function’s mapping and is optimized to precisely fit the training data while avoiding overfitting and maintaining generalizability for the test data.
2.3.2 Selective emitter
The proposed model having the same architecture as optimized for the solar absorbers will be used for emission spectrum prediction. This is justified by the fact that the thermal emitters call for the same design composition as that of the absorbers for an STPV system, but they need to exhibit narrow-band emissions, right above the wavelength analogous to the bandgap of the solar cell for which the STPV system is being designed. Here we have targeted the highly efficient InGaAsSb PV cells having a band gap of 0.54 eV ≈ 2.3 µm. So, the proposed model will be separately trained for metasurface based selective emitters as the wavelength range of their optical response is from 1500 nm to 2500 nm, which differs from that of the absorbers. Training dataset for selective emitters having sufficient variance was collected by simulating 2000 unit cells in CST microwave studio with periodic boundary conditions over a wavelength range of 1500 nm – 2500 nm yielding a 1000 nodes output for the model. The design configuration of the unit cells is the same as that for the absorbers i.e., tungsten nano resonators and ground plane separated by SiO2 dielectric layer. The 3 channel input images for the model are designed in the same way as described earlier and the output also exhibits the same size vectors (i.e., 1000 × 1). These input images will be passed through the composite model to accurately predict the emissions spectrum output.
Similar to the absorber dataset, the emitter’s dataset is defined as DE={(Xi, Pi, Ei), i = 0,1,3…..N}, and it is collected in the same way as the absorbers dataset, but the simulations are now carried out over a different wavelength range corresponding to the bandgap of InGaAsSb PV cells i.e., 1500 nm – 2500 nm. The 1000 nodes emission spectra of each simulation is stored as Ei . A secondary input 1D column vector P (2000 × 1) is also collected for emitters (similar to the way it was calculated for absorbers). Each value Pi represents the periodicity of the corresponding unit cell. This periodicity array will be concatenated with the first FC layer of the model to map its effect on the output. The composite neural networks model is trained using the optimization function f (Xi, Ei ; Θ) parameterized by Θ.
2.3.3 MetaNet properties
Both the parts of the MetaNet exhibit full freedom of shape and structural dimensions to incorporate enough variance and deviation in the input dataset. This is important to boost the model’s capacity to predict the response for unseen geometries and structures. The periodic boundary conditions, linear incident light source and the mesh size are the fixed input parameters for all the simulations, whereas, the following parameters are kept variable and chosen randomly for each unit cell
- 1. Lattice constant (periodicity) of each unit cell
- 2. Number of shapes in each unit cell (with an upper bound of maximum four shapes in each unit cell)
- 3. Types of shapes: circle, triangle, rectangle, ring, or polygon. (as these are the most commonly used shapes in metasurfaces)
- 4. Structural dimensions: width, height, radius, and length.
- 5. Location of each shape in a unit cell: any value of x and y within the lattice constant of the unit cell.
- 6. Orientation angle of each shape: 0–360 degrees.
These variable parameters allow enough flexibility to cover all possible outcomes. In both the datasets DA and DE, the total size of the input is 100 x100 × 3×2000 (where 100 × 100 is the pixel size, 3 is the number of channels and 2000 is the number of samples). Each of these datasets is divided into 70% training data and 30% test data. The size of the output dataset is 1000 × 1×2000 (where 1000 × 1 is the size of each output label, and 2000 is the total number of labels.
3. Results and discussion
3.1 Results obtained for solar absorbers
To train the model for absorbers dataset, Nesterov Adam optimizer is used with a learning rate of 10−3, batch size 64, and 500 training epochs. Since we are dealing with a regression problem so the loss function being employed is the mean squared error i.e., MSE given as follows:
where N is the total number of samples in the dataset, Aactual shows the actual output obtained from simulations and Apredicted is the output predicted by the model. Our trained model for the solar absorbers achieves 7.3 x10−04 mean square error over the test-set. The simulated and the DL model predicted results for some random test samples for the solar absorber’s dataset are shown in Fig. 5. It can be seen that the optical absorption response predicted by the model is in good agreement with the simulated response.Fig. 5. Comparison of the Actual and the Model’s Predicted Outputs for some random tungsten based solar absorber (from test dataset). The perfect agreement between the DL model’s predicted curves and the simulation curves validate the accuracy of the model.
Figure 5 (a)-(d), shows some highly competent absorbers unit cells, which exhibit above 80% absorbance throughout the solar spectrum. Figure 5(e) shows a unit cell giving high absorption efficiency in the visible domain only, such absorbers are quite useful in certain applications. Thus, we can use the model to check if the proposed absorber structure covers the entire spectrum or not, and how much efficiency it offers. A detailed analysis of some designs, their dimensions and efficiencies calculated by taking absorption area under the curve (AUC) % over the entire spectrum under consideration i.e., 300–1300 nm is given in Table 2.
Table 2. Design Specifications and Efficiencies (absorption AUC% (300-1300 nm)) of different variations of Tungsten based Solar absorbers calculated from the model’s predicted outputs.
3.2 Results obtained for the selective emitter
Similar to the case of absorbers, the proposed model was trained with the MSE. Nesterov Adam Optimizer with a learning rate of 10−3 is used during training for 500 epochs. The simulated and the DL model predicted results for some selected test samples for the selective emitter’s dataset are shown in Fig. 6, for qualitative evaluation. It can be seen that the emission response lies within the External quantum efficiency (EQE) range of the targeted InGaAsSb PV cells. The emittance peaks are well above the band gap of InGaAsSb PV cell and are situated at the maximum EQE thus providing optimum thermal selectivity. It can also be observed that the emission response predicted by the model (for the selected unit cells) is in good agreement with the simulated response. Thus, we can use the model to analyze whether the emissions band of the selective emitter under consideration matches with the band edge of the underneath solar cell or not and adjust its selectivity range as per the requirements of the application in hand. Similarly, this model can also be used to extract any type of optical response, may it be transmission, reflection, phase response etc., from the structures exhibiting different shapes and geometrical parameters.
Fig. 6. Comparison of Actual and the Model’s Predicted Outputs for some random tungsten based selective emitters and verification of their presence with in the EQE range of the targeted InGaAsSb PV cells
3.3 Transfer learning from tungsten based absorbers to tantalum based absorbers
The optical response of a metasurface based solar absorber/emitter is significantly dependent upon the composition and material of its nano resonators. Changing the material will change the effective refractive index (η) of the overall structure and therefore the absorption/emission response will change. So, we need to make the proposed model adaptable for different materials without needing to retrain it from scratch and collect a huge amount of data set for each new material. Here in our case, changing the material will only affect the output labels, i.e., for two unit cells having square shaped nano resonators with similar dimensions but different materials, the input image will remain the same but the optical response curve will be different for each of them.
Transfer learning (TL) is a technique which offers immense potential to deal with such correlated problems where the input distribution remains same for each problem but the labels distribution change. Basic concept of transfer learning implies that the model stores knowledge gained while solving one problem and applies it to a different but closely related problem. Since our basic problem of analyzing the absorption response of different unit cells remains the same, and only the material of the unit cells is changing, we can successfully use Transfer learning to address this issue. For instance, to validate the proposed claim, we have collected a small dataset of 500 inputs and outputs for solar absorber unit cells having tantalum based nano resonators via CST simulations. Tantalum (Ta) is another refractory material having a high melting point and good stability which makes it a suitable choice for perfect solar absorbers. Using a small dataset of Tantalum, we were able to shift the proposed model (initially trained on tungsten absorbers) via transfer learning to accurately predict the absorption spectrum of chromium based solar absorbers. In this way we can apply the proposed trained model to any other material without training the model from scratch by simply collecting a small dataset of the desired material and apply transfer learning as illustrated in Fig. 7.
Fig. 7. Illustration of Transfer Learning Application on the proposed model, where a source model trained on tungsten absorber is fine tuned to transfer the learned knowledge to predict outputs for tantalum absorbers
To validate the capacity of the model to adapt multiple target domains, transfer learning is applied on the pre-trained tungsten absorber model. We fine-tuned the trained network for a small dataset (500 samples) of chromium based absorbers via transfer learning for 100 epochs. The mean squared error, averaged on 100 testing samples, was 3.2 × 10−04, indicating that our model is able to predict the response with high accuracy. The simulated and the DL model predicted results for some tantalum-based test samples shows a good agreement validating the accuracy of the model as shown in Fig. 8.
Fig. 8. Comparison of the simulated and predicted outputs of some random tantalum based absorber unit cells.
This transfer learning methodology can also be used to make the model adaptable for different wavelength ranges. Typically the proposed network will give predictions only for the wavelength range for which it is trained. But the use of transfer learning can make it applicable for different wavelength ranges also. Since transfer learning (TL) is a technique which offers immense potential to deal with such correlated problems where the input distribution remains same for each problem but the labels distribution change. If we change the wavelength range, it will only change the label distribution so transfer learning can be definitely used to fine tune the pre-trained network for new wavelength ranges.
3.4 Simulation time versus DL model prediction time
We have trained the model using Google’s provided GPU’s via Google colaboratory, it took approximately 30 minutes to train the DL model whereas 10 minute for transfer learning (because the dataset for transfer learning consisted of only 500 samples). To further analyze the practicality and benefits of the proposed model, we compared the performance of the trained model with the CST Microwave studio simulations in terms of time taken to generate the output results of some random test samples as shown in Fig. 9. It can be clearly seen that once the proposed Deep learning model is trained, its performance is much faster and memory efficient than the conventional commercial simulation software. The average CST simulation time for 10 random samples of input is 136.1 seconds whereas the average DL model prediction time for the same 10 samples is 2.54 seconds. Thus we can conclude that the DL model predicts the optical response 98.13% (136.1-2.54/136.1 × 100) faster than the simulations. By providing adequate amounts of data and training structures, this model can be extended to a large-scale to completely replace the commercial software’s.
Fig. 9. Comparative Analysis of the Computation Time taken by the conventional CST simulations (range shown by the left vertical axis ≅ 125 sec to 150 sec) and the proposed DL based model (range shown by the right vertical axis ≅ 1.5 sec to 4 sec) to predict the absorption/emission spectrum.
3.5 Ablation study of the three dimensional input
To elaborate the importance and benefits of our input representation scheme, we have shown an ablation study of the 3 channel inputs and their features in Table 3.
Table 3. Ablation study of the three dimensional Input.
4. Conclusion
This paper presents a unique and time-efficient deep learning-based composite model that integrates deep convolutional neural network and recurrent neural network to accurately predict the absorption and emission response of metasurface-based perfect solar absorbers and selective emitters. The proposed technique eliminates the need for conventional time consuming and computationally exhaustive numerical simulations for absorber's/emitter's optimization to perfectly fit an ideal STPV system. Images of the 3D nanostructures and other critical structural parameters (such as height, aspect ratio, periodicity, etc.) are provided as an input to the model to efficiently learn the dependence of the output absorbance on all these factors. Once trained, the model achieves a test loss as low as 7.3 × 10−04 and 6.2 × 10−04, and accurately predicts the absorption/emission spectra precisely similar to the one achieved from simulations. To validate the proposed model's generalizability, it is subjected to transfer learning to cater to the effect of diverse materials and accurately predict the absorption response of any metasurface absorber regardless of its underlying material. The comparative analysis of the time consumed by conventional simulation (via CST) and the proposed DL model to generate the absorption/emission spectra shows that the average simulation time for 10 random samples is 136.1 seconds and the average DL model prediction time for the same samples is 2.54 seconds. The proposed model works approximately 98.13% faster than the conventional approach. Furthermore, apart from absorbance, the proposed DL model can be trained to predict any other optical response such as transmission, reflection, phase response, etc., for nanostructures exhibiting different shapes and geometrical parameters only in a fraction of seconds and without any heavy computational resource's requirement.
Disclosures
The authors declare that there are 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.
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