Metasurfaces, engineered materials with subwavelength structures, have revolutionized the field of optics by offering unprecedented control over light. Their ability to manipulate electromagnetic waves at will has led to a plethora of applications, ranging from advanced imaging and sensing to optical computing and communication. Designing these intricate structures, however, presents significant challenges. Traditional methods often rely on computationally intensive simulations and iterative optimization processes. Recently, machine learning (ML) has emerged as a powerful tool to accelerate and enhance metasurface design, offering new avenues for creating complex and high-performance optical devices. Among these innovative approaches, physics-informed machine learning (PIML) stands out by integrating fundamental physical principles into the learning process. This integration is crucial for ensuring the physical plausibility and performance of the designed metasurfaces, especially when dealing with nonlinear systems. This article explores the application of machine learning techniques in metasurface design, with a particular focus on physics-informed neural networks and topology optimization, highlighting their theoretical underpinnings and practical applications in achieving advanced functionalities.
Machine Learning for Metasurface Design
Basic Principle
Machine learning offers a paradigm shift in metasurface design by moving away from purely simulation-driven approaches to data-driven methodologies. The core principle of applying machine learning to metasurface design is depicted in Fig. 2. This process generally begins with generating a comprehensive dataset of electromagnetic (EM) responses corresponding to various metasurface geometries. For a simple structure like a cylindrical meta-atom, this involves using forward solvers to simulate the EM response for a wide range of parameter combinations.
Fig. 2: Basic principle of machine learning for metasurface design. This flowchart illustrates the process of training both forward and inverse neural networks for efficient metasurface design. It includes steps from problem definition and data collection to model validation and optimization.
These datasets serve as the training ground for deep neural networks (DNNs). Two primary types of networks are typically trained: forward and inverse networks. A forward network learns to predict the EM response of a metasurface given its geometric parameters as input. Conversely, an inverse network is trained to perform the reverse task: to determine the geometric parameters required to achieve a desired EM response. This inverse design capability is particularly valuable, as it directly addresses the core challenge of metasurface engineering – finding the right structure for a specific optical function.
The optimized designs obtained from the inverse network are then evaluated using forward solvers to verify their performance and ensure the predicted response aligns with simulations. This validation step is crucial for the reliability of the machine learning-driven design process. This approach significantly accelerates the design cycle, reducing the reliance on computationally expensive iterative simulations and enabling the exploration of a much broader design space. [Huang, M. et al. Machine–learning-enabled metasurface for direction of arrival estimation. Nanophotonics 11, 2001–2010 (2022).],[Koziel, S. et al. Machine-learning-powered EM-based framework for efficient and reliable design of low scattering metasurfaces. IEEE Trans. Microwave Theory Tech. 69, 2028–2041 (2021).],[Naseri, P. et al. A generative machine learning-based approach for inverse design of multilayer metasurfaces. IEEE Trans. Antennas Propag. 69, 5725–5739 (2021).],[Zhang, Q. et al. Machine‐learning designs of anisotropic digital coding metasurfaces. Adv. Theory Simul. 2, 1800132 (2019).],[Ma, W. et al. Pushing the limits of functionality‐multiplexing capability in metasurface design based on statistical machine learning. Adv. Mater. 34, 2110022 (2022).],[Lin, H. et al. Machine-learning-assisted inverse design of scattering enhanced metasurface. Opt. Express 30, 3076–3088 (2022).]
Cases and Approaches
Early applications of machine learning in metasurface design demonstrated its potential to surpass traditional methods in terms of both accuracy and efficiency. In 2019, An et al. pioneered the use of deep learning to optimize the optical response of metasurfaces, achieving results that outperformed conventional techniques. [An, S. et al. A deep learning approach for objective-driven all-dielectric metasurface design. ACS Photonics 6, 3196–3207 (2019).] Their work was notable for being among the first to successfully model complex 3D metasurface structures using machine learning. However, their approach was limited to relatively simple and fixed structures, as illustrated in Fig. 3a. The complexity of structures and the associated increase in input parameters can lead to a decrease in optimization speed and accuracy, thereby limiting the design freedom. Furthermore, experimental validation of their theoretical and simulation results was absent, necessitating further research to confirm its practical viability.
Fig. 3: Cases and approaches of machine learning for metasurface design. This figure showcases various metasurface designs and machine learning frameworks applied in recent research. It includes (a) a simple metasurface model, (b) a binary coded metasurface, (c) a deep neural network framework for phase manipulation, and (d) a flowchart for multi-functional metasurface design.
In the same year, Zhang et al. introduced a binary coded metasurface structure optimized using machine learning, shown in Fig. 3b. [Zhang, Q. et al. Machine‐learning designs of anisotropic digital coding metasurfaces. Adv. Theory Simul. 2, 1800132 (2019).] This design offered a higher degree of freedom compared to An et al.’s work, enabling the use of larger datasets for network training. This resulted in a network with high phase response accuracy and rapid optimization capabilities. Importantly, Zhang et al. provided experimental validation, demonstrating the practical feasibility of their method. However, their work primarily focused on coding optimization for unit cell structures and did not extend to array optimization strategies. Additionally, while the microwave band application facilitated fabrication due to relaxed feature sizes, translating this approach to the optical band poses manufacturing challenges due to the required precision.
Jiang et al. (2021) expanded the capabilities of machine learning for metasurface design by demonstrating that DNNs can predict not only phase but also group delay for meta-atoms operating in the visible light spectrum, as shown in Fig. 3c. [Jiang, L. et al. Neural network enabled metasurface design for phase manipulation. Opt. Express 29, 2521–2528 (2021).] However, they noted that significant deviations or complexity in the desired phase spectrum could lead to inherent errors between the network-predicted spectrum and the spectrum obtained from forward solver simulations.
Ma et al. (2022) introduced an innovative approach for designing multi-functional metasurfaces in the near-infrared band by integrating optimization algorithms with machine learning, as depicted in Fig. 3d. [Ma, W. et al. Pushing the limits of functionality‐multiplexing capability in metasurface design based on statistical machine learning. Adv. Mater. 34, 2110022 (2022).] Their method pushes the boundaries of metasurface functionality, approaching physical limitations for multiplexing capabilities. The number of functions a metasurface can perform is constrained by factors such as physical size, fabrication technology, meta-atom coupling, and material losses. Their design process, validated experimentally, demonstrated the successful implementation of multi-functional metasurfaces.
Lin et al. (2022) also combined machine learning and optimization algorithms to optimize all meta-atoms within a metasurface array in the microwave band. [Lin, H. et al. Machine-learning-assisted inverse design of scattering enhanced metasurface. Opt. Express 30, 3076–3088 (2022).] This approach was used to design and experimentally validate a retroreflector, further showcasing the power of machine learning in designing complex metasurface devices.
Analysis and Conclusion
While machine learning offers significant advantages in metasurface design, traditional methods often rely on creating a large library of potential designs and simulating their responses. This is followed by classical search methods to identify optimal parameters, which can be inefficient and time-consuming. The machine learning techniques discussed here still require numerical solvers to generate training data, representing an upfront computational cost. However, once trained, the networks can predict metasurface parameters without repeatedly using numerical solvers, leading to significant time savings in the design process.
A critical challenge in applying general neural network frameworks to metasurface design is ensuring the physical validity of the solutions and the manufacturability of the resulting parameters. As highlighted by Jiang et al., [Jiang, L. et al. Neural network enabled metasurface design for phase manipulation. Opt. Express 29, 2521–2528 (2021).] these frameworks can sometimes produce physically meaningless negative parameters. To address this, constrained optimization frameworks and validation using numerical solvers are crucial to eliminate technically infeasible designs. The following sections explore methods specifically designed to address these limitations by incorporating physics into the machine learning process.
Physics-Informed Neural Networks for Metasurface Design
Basic Principle
Physics-informed neural networks (PINNs) represent a significant advancement in machine learning for metasurface design by integrating known physical laws, such as Maxwell’s equations, directly into the neural network framework. This is achieved by incorporating the governing partial differential equations (PDEs) into the loss function of the network. Detailed theoretical background and implementation specifics of PINNs are available in Chen et al. [Chen, Y. Y. et al. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Express 28, 11618–11633 (2020).]. Fig. 4 illustrates the basic principle of using PINNs for metasurface design.
Fig. 4: Basic principle of Physics-Informed Neural Networks for metasurface design. This flowchart outlines the PINN design process, emphasizing the integration of physics principles and loss functions into the neural network architecture for more efficient and physically accurate metasurface designs.
By embedding physics constraints, PINNs require smaller training datasets compared to conventional machine learning methods, leading to substantial reductions in computational time. [Tang, Y. H. et al. Physics-informed recurrent neural network for time dynamics in optical resonances. Nat. Comput. Sci. 2, 169–178 (2022).],[Chen, Z. et al. Physics-informed learning of governing equations from scarce data. Nat. Commun. 12, 6136 (2021).],[Ma, W. et al. Deep learning for the design of photonic structures. Nat. Photonics 15, 77–90 (2021).],[Jenkins, R. P. et al. Establishing exhaustive metasurface robustness against fabrication uncertainties through deep learning. Nanophotonics 10, 4497–4509 (2021).],[Chen, Y. Y. et al. Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data. APL Photonics 7, 010802 (2022).] This is particularly beneficial when dealing with complex metasurface structures, such as multi-pillar meta-atoms, which have a large number of design parameters and require extensive computational resources for traditional simulation-based optimization. The remaining steps in the design process are similar to those used in standard machine learning methods, but the incorporation of physics constraints ensures greater physical realism and efficiency.
Cases and Approaches
Chen et al. (2020) pioneered the application of PINNs in metasurface design by combining neural networks with the Helmholtz equation, a fundamental PDE in electromagnetics. [Chen, Y. Y. et al. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Express 28, 11618–11633 (2020).] Their work aimed to replace an array of small cylinders with a single cylinder that could replicate the electric field response of the array when illuminated by a plane wave. This concept is illustrated in Fig. 5a.
Fig. 5: Cases and approaches of Physics-informed neural networks for metasurface design. This figure presents various examples of PINN applications in metasurface design, including (a) replacing a cylinder array with a single cylinder, (b) predicting time-domain response of graphene structures, (c) Deep Lorentz Neural Networks, and (d) 3D permittivity retrieval using Maxwell’s equations.
As shown in Fig. 5a, they compared the electric field distribution of a small cylinder array with the predicted distribution of a single optimized cylinder designed using PINNs. The PINN-designed single cylinder achieved an electric field distribution with only a 2.82% error compared to the array, demonstrating the method’s accuracy and efficiency in reducing computational complexity, particularly for applications like invisibility cloaks.
Tang et al. (2022) introduced physics-guided and physics-explainable recurrent neural networks (RNNs) to predict the time response of optical resonances in metasurfaces. [Tang, Y. H. et al. Physics-informed recurrent neural network for time dynamics in optical resonances. Nat. Comput. Sci. 2, 169–178 (2022).] RNNs are well-suited for sequential data, such as time-domain signals. By incorporating physical principles, their approach improved prediction accuracy and interpretability. Fig. 5b shows the application of their method to periodic monolayer graphene stripe structures, where physics-informed RNNs accurately predicted the full time-domain response using only a small fraction of the complete sequence. This significantly reduces the data acquisition time and allows for efficient extraction of resonant frequencies and frequency-domain information through Fourier transform.
Khatib et al. (2022) proposed Deep Lorentz Neural Networks (DLNNs), specifically designed for modeling all-dielectric metamaterials. [Khatib, O. et al. Learning the physics of all‐dielectric metamaterials with deep Lorentz neural networks. Adv. Opt. Mater. 10, 2200097 (2022).] DLNNs are based on the Lorentz model, which describes the interaction of electromagnetic fields with materials. Fig. 5c illustrates the working principle of DLNNs. Trained on datasets of simulated EM wave propagation, DLNNs can accurately predict the behavior of all-dielectric metamaterials under varying conditions such as frequency, polarization, and incidence angle. Compared to conventional DNNs, DLNNs require less training data to achieve similar accuracy, highlighting the benefits of physics-informed architectures.
Chen et al. further advanced PINNs by incorporating Maxwell’s equations directly into the framework for retrieving 3D permittivity distributions from near-field data. [Chen, Y. Y. et al. Physics-informed neural networks for imaging and parameter retrieval of photonic nanostructures from near-field data. APL Photonics 7, 010802 (2022).] Fig. 5d shows the retrieval of 3D permittivity information using PINNs, trained with electric field distributions obtained from finite element simulations. This method enables the extraction of information from complex 3D structures, which is crucial for real-world applications in near-field microscopy and medical imaging.
Analysis and Conclusion
Physics-informed neural networks offer a powerful approach to metasurface design by combining the efficiency of machine learning with the rigor of physics. They demonstrate high speed, accuracy, and design flexibility while mitigating the limitations of data-hungry conventional machine learning methods. PINNs are particularly effective in solving inverse problems, such as determining the geometry of a metasurface for a desired EM response. However, a current limitation is that a trained PINN is typically specific to a particular inverse problem and may require retraining for different design scenarios. Despite this, PINNs represent a significant step forward in intelligent metasurface design, offering enhanced physical realism and reduced computational costs.
Topology Optimization for Metasurface Design
Basic Principle
Topology optimization is a purely physics-based method that has emerged as a highly effective strategy for designing high-performance metasurfaces. Fig. 6 illustrates the topology optimization process. It starts with an initial guess for the metasurface structure and its parameters. The EM response of this structure is then calculated using rigorous electromagnetic solvers like the rigorous coupled-wave analysis (RCWA) method. [Colburn, S. et al. A. Inverse design and flexible parameterization of meta-optics using algorithmic differentiation. Commun. Phys. 4, 65 (2021).],[Rumpf, R. C. Improved formulation of scattering matrices for semi-analytical methods that is consistent with convention. Progr. Electromagn. Res. B 35, 241–261 (2011).]
Fig. 6: Basic principle of topology optimization for metasurface design. This flowchart details the iterative topology optimization process, from defining the problem and initial parameters to calculating loss functions and gradients, and finally achieving the desired metasurface design.
A loss function is defined to quantify the difference between the calculated EM response and the desired target response. Gradient-based optimization algorithms, such as automatic differentiation, are then used to compute the gradient of the loss function with respect to the design parameters. [Colburn, S. et al. A. Inverse design and flexible parameterization of meta-optics using algorithmic differentiation. Commun. Phys. 4, 65 (2021).] This gradient information guides the update of the structural parameters in a direction that minimizes the loss function. This iterative process continues until the loss function reaches a minimum, resulting in an optimized metasurface design that closely achieves the desired EM response.
Cases and Approaches
Lin et al. (2019) demonstrated the effectiveness of topology optimization using the RCWA method for multi-layer metasurface structures. [Lin, Z. et al. Topology optimization of freeform large-area metasurfaces. Opt. Express 27, 15765–15775 (2019).] RCWA is particularly well-suited for metasurface analysis due to its computational efficiency in handling periodic multi-layer structures. They combined RCWA with the adjoint method for optimization, achieving designs with thousands of degrees of freedom, significantly more than previously possible with other methods. They successfully designed a metalens, shown in Fig. 7a, although its complex structure posed fabrication challenges at the nanoscale.
Fig. 7: Cases and approaches of topology optimization for metasurface design. This figure illustrates various topology optimization applications, including (a) a topology-optimized metalens, (b) large-area lenses designed with Aperiodic Fourier Modal Method, (c) lens design with elliptical resonator meta-atoms and automatic differentiation, and (d) topology-optimized catenary metasurface for wide-angle deflection.
Phan et al. (2019) designed large-area lenses using the Aperiodic Fourier Modal Method (AFMM) for finite-sized, isolated devices. [Phan, T. et al. High-efficiency, large-area, topology-optimized metasurfaces. Light Sci. Appl. 8, 48 (2019).] To simplify computation, they divided the lens into smaller sections and linearized the phase profiles for each section, as shown in Fig. 7b. They developed AFMM, which combines a solver for periodic systems with perfectly matched layers, to accurately model aperiodic pillar structures. By considering coupling between adjacent sections and adding gaps to minimize interference, they achieved highly efficient lenses with high numerical aperture (NA).
Colburn et al. (2021) utilized RCWA with automatic differentiation (AD) to optimize metasurface parameters. [Colburn, S. et al. A. Inverse design and flexible parameterization of meta-optics using algorithmic differentiation. Commun. Phys. 4, 65 (2021).] AD efficiently calculates gradients for complex sequential computations by applying the chain rule. Combined with parallel computation on GPUs, AD offers significant speed improvements over the adjoint method. Fig. 7c shows an example of lens design using elliptical resonator meta-atoms and AD, demonstrating improved focus efficiency with increasing iterations.
Xu et al. (2021) employed RCWA with a multi-objective adjoint-based approach to optimize discrete geometric phase metasurfaces. [Xu, M. F. et al. Topology-optimized catenary-like metasurface for wide-angle and high-efficiency deflection: from a discrete to continuous geometric phase. Opt. Express 29, 10181–10191 (2021).] Their optimization led to continuous structures with significantly improved efficiency. Fig. 7d shows the optimized refractor array and the electric field intensity distribution, highlighting the effectiveness of topology optimization in achieving high-performance metasurface designs.
Analysis and Conclusion
Topology optimization has emerged as a leading strategy for metasurface design, particularly with the integration of advanced AD formulations and high-performance computing. These methods provide high flexibility, performance, accuracy, and speed, enabling the design of complex metasurfaces with a large degree of freedom. The choice of optimization strategy often depends on the specific application requirements. For highly complex metasurfaces, combining physics-informed neural networks with elements of topology optimization, such as RCWA or high-performance parallel architectures, could be a promising direction for future research.
Metasurfaces for Quantum Optics Applications
Metasurfaces are not only transforming classical optics but also holding significant promise for quantum optics. Since their inception, metasurfaces have found applications in diverse areas, including antennas, [Xie, P. et al. Wideband RCS reduction of high gain Fabry-Perot antenna employing a receiver-transmitter metasurface. Prog. Electromagn. Res. 169, 103–115 (2020).],[Li, H. P. et al. Phase- and amplitude-control metasurfaces for antenna main-lobe and sidelobe manipulations. IEEE Trans. Antennas Propag. 66, 5121–5129 (2018).] radar cross-section modification, [Yuan, F. et al. RCS reduction based on concave/convex-chessboard random parabolic-phased metasurface. IEEE Trans. Antennas Propag. 68, 2463–2468 (2020).] specialized beam generation, [Zhang, K. et al. A review of orbital angular momentum vortex beams generation: from traditional methods to metasurfaces. Appl. Sci. 10, 1015 (2020).],[Bao, Y. J. et al. A minimalist single-layer metasurface for arbitrary and full control of vector vortex beams. Adv. Mater. 32, 1905659 (2020).] active metasurfaces, [Shaltout, A. M. et al. Spatiotemporal light control with active metasurfaces. Science 364, eaat3100 (2019).] and now, increasingly, quantum optics. Quantum optics is a rapidly growing field with profound implications for quantum computation, communication, sensing, and fundamental physics research. [Wang, K. et al. Quantum metasurface for multiphoton interference and state reconstruction. Science 361, 1104–1108 (2018).],[Georgi, P. et al. Metasurface interferometry toward quantum sensors. Light Sci. Appl. 8, 70 (2019).],[Zhou, J. X. et al. Metasurface enabled quantum edge detection. Sci. Adv. 6, eabc4385 (2020).],[Gao, Y. J. et al. Multichannel distribution and transformation of entangled photons with dielectric metasurfaces. Phys. Rev. Lett. 129, 023601 (2022).],[Guo, J. et al. Active-feedback quantum control of an integrated, low-frequency mechanical resonator. Preprint at https://arxiv.org/abs/2304.02799 (2023).] Metasurfaces are poised to become a cornerstone technology in future quantum photonics.
In 2018, Wang et al. pioneered the use of flat metasurfaces to replace bulky conventional optical components for achieving non-classical multiphoton interference. [Wang, K. et al. Quantum metasurface for multiphoton interference and state reconstruction. Science 361, 1104–1108 (2018).] As shown in Fig. 8a, they successfully reconstructed single-photon and two-photon states using a polarization-insensitive detector, demonstrating the feasibility of manipulating multi-photon quantum states with metasurfaces.
Fig. 8: Cases of metasurface applications in quantum optics. This figure highlights recent advancements in quantum metasurfaces, including (a) quantum state reconstruction, (b) entanglement and disentanglement of photon states, (c) optical switch for quantum edge detection, and (d) multi-channel quantum entanglement distribution and transformation.
Georgi et al. (2019) presented a quantum system using a metasurface to entangle and disentangle two-photon spin states, shown in Fig. 8b. [Georgi, P. et al. Metasurface interferometry toward quantum sensors. Light Sci. Appl. 8, 70 (2019).] Their system outperformed conventional optical elements in terms of performance and compactness.
Zhou et al. (2020) developed a polarization-entangled photon source based on a metasurface that functions as an optical switch for edge detection mode, illustrated in Fig. 8c. [Zhou, J. X. et al. Metasurface enabled quantum edge detection. Sci. Adv. 6, eabc4385 (2020).] By switching the photon state between “ON” and “OFF,” the imaging could be toggled between a solid and an outlined cat image, showcasing the potential for quantum information processing.
Gao et al. (2022) demonstrated a multi-channel metasurface capable of transforming polarization-entangled photon pairs, shown in Fig. 8d. [Gao, Y. J. et al. Multichannel distribution and transformation of entangled photons with dielectric metasurfaces. Phys. Rev. Lett. 129, 023601 (2022).] They further showed that using two metasurfaces could enable even more channels for entangled photon pair distribution, opening up exciting possibilities for quantum information processing and communication.
The intelligent design methods discussed in previous sections, particularly physics-informed machine learning and topology optimization, hold the key to improving the performance, accuracy, and design speed of metasurfaces for quantum optics applications. These advanced techniques could pave the way for the next generation of quantum photonic devices, enabling complex quantum information processing and communication technologies.
