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Review Paper

Applied Science and Convergence Technology 2023; 32(6): 141-150

Published online November 30, 2023

https://doi.org/10.5757/ASCT.2023.32.6.141

Copyright © The Korean Vacuum Society.

Review of Generative Models for the Inverse Design of Nanophotonic Metasurfaces

Seunghwan Moona , b , Jihun Kanga , b , and Jong-Souk Yeoa , *

aSchool of Integrated Technology, College of Computing, Yonsei University, Seoul 03722, Republic of Korea
bBK21 Graduate Program in Intelligent Semiconductor Technology, Yonsei University, Seoul 03722, Republic of Korea

Correspondence to:jongsoukyeo@yonsei.ac.kr

Received: November 5, 2023; Revised: November 30, 2023; Accepted: November 30, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc-nd/4.0/) which permits non-commercial use, distribution and reproduction in any medium without alteration, provided that the original work is properly cited.

Evolving nanotechnologies and further understanding of nanophotonics have recently enabled the control of electromagnetic waves using metasurfaces. Since metasurfaces can provide diverse optical characteristics depending on their geometries, the forward design of metasurfaces conventionally has been employed through an understanding of the physical effects of each geometrical parameter. In contrast, the inverse design approach optimizes the metasurface geometry using computational algorithms. This review discusses recent studies on constructing generative models for the inverse design of nanophotonic metasurfaces. The generative model for inverse design is constructed mainly with three components: an evaluator, a generator, and a criterion. The evaluator, which can be implemented by physical simulators or deep neural networks, determines whether the input metasurface geometry satisfies the target optical characteristics. The generator suggests new possible design candidates that may have optical properties close to the target. The criterion, which includes algorithms based on mathematical optimization and artificial intelligence, manages the operation of a generative model while satisfying the convergence of optimal solutions. Inverse design takes advantage of larger design space for customized applications along with the possibility of investigating new physics, and hence it is expected to improve metasurfaces further with the emerging computational algorithms.

Keywords: Metasurface, Nanophotonics, Inverse design, Generative model, Mathematical optimization, Artificial intelligence

From ancient civilizations to modern society, humans have attempted to control light and have also sought to understand light-matter interactions. The earliest lenses identified in ancient Egyptian and Mesopotamian civilizations lenses were made of polished rock crystal or glass-like materials and were used for magnification and focusing [13]. Around the 3rd century before the common era (CE), the Greek mathematician Euclid wrote extensively on the properties of light and suggested the rules of reflection [4,5]. In the second century CE, another Greek scientist, Ptolemy, examined the behavior of light as it traveled through various media and produced important observations on refraction [57]. Substantial advances in optics were made in the period from the 8th to 14th century CE. Scholars such as Al-Kindi, Alhazen, and others conducted extensive research and created ideas on how light behaves [5]. Alhazen is also referred to as the father of modern optics because he provided theories about light, colors, vision, and concepts of reflection and refraction in his book ‘Book of Optics’ [5,8]. In the late 17th century, Isaac Newton found that white light can be separated into rainbow colors when it passes through a prism [9,10]. By observing the continuous spectrum coming out through a prism, he concluded that white light is constructed with diverse colors, and this led Newton to invent his reflecting telescope, known as the Newtonian telescope [10]. A curved primary mirror was utilized to gather and focus light, reducing chromatic aberration and producing sharper images [10,11]. Theoretical research on light was undertaken from the 17th to the 19th centuries. Scientists such as Christiaan Huygens and Robert Hooke explained light as waves propagating through matter, and this is called wave theory [12]. Thomas Young, an English scientist in the early 19th century, demonstrated the double-slit experiment and showed interference and diffraction of light, which are also wave characteristics [13,14]. Albert Einstein and Max Planck’s research in the late 19th century, however, led to the understanding of light as particles called photons, as described by particle theory [1518]. These observations prompted scientists to infer that light shows both waveand particle-properties depending on the experimental settings.

With this theoretical understanding of light, the concept of optical antennas that manipulate light at the nanoscale was presented by researchers such as Edward Hutchinson Synge, John Wessel, and Lukas Novotny [1922]. Optical antennas can be fabricated using various materials and structures, such as nanostructures, optical fibers, and photonic crystals, and these antennas enable optical signal concentration, directional control, and wavelength adjustment [21,2325]. A metamaterial is a subwavelength structure that can modulate a light signal, and thus its lattice constant to operating wavelength ratio is comparably smaller than that of the photonic crystals [2628]. Furthermore, a metamaterial is an array of metaatoms, and this repeating structure brings specialized optical characteristics according to the geometrical parameters [29,30]. A metasurface is a two-dimensional metamaterial that can be used for diverse applications such as optical communication, nanoscale light sources, and optical/plasmonic biosensors [29,31,32]. By changing the geometry of metasurfaces, it is possible to control the properties of reflected or transmitted light, such as frequency, complex wavenumber, and directionality.

In terms of designing metasurfaces with the conventional approach, the effects of the geometrical parameters of the metasurface on its optical properties should be investigated based on a physical understanding. However, there may be physics in nature that is not yet understood that could provide an analytical solution for metasurface design. In addition, when there are multiple required functionalities, various physical phenomena are entangled, and the specifications can be in a trade-off relation with each other, resulting in the high complexity of finding an optimal solution with rather tedious and laborious experimental verifications. In response to this, inverse design approaches that provide optimized solutions through computational algorithms have been suggested [3337]. The computational algorithms for the inverse design of metasurfaces include mathematical optimization and artificial intelligence (AI) techniques, and they can be applied to 1) suggest a new metasurface design, 2) estimate the optical characteristics of the suggested metasurface efficiently, and 3) determine and optimize the metasurface design to satisfy the required specifications. The inverse design can be conducted by human resources in principle; however, for better efficiency and more iterations through automation, inverse design is generally performed by designing a computational system called a generative model. In this review, the basic concepts of computational algorithms that can be used for metasurface inverse design are introduced in Section 2, and recent inverse design studies are discussed in Section 3 by categorizing their role in a generative model.

Mathematical optimization, which is also called mathematical programming in the applied mathematics field, not only has solved economic, industrial, and technical optimization problems through analytical approaches but also has provided fundamental logic for traditional computational algorithms. Among them, topological optimization (TO) and the evolutionary algorithm (EA) are the two most widely used approaches when inversely designing metasurfaces [37, 38]. TO is a geometry optimization method employed to achieve the best performance for a specific constraint within a given design space [39,40]. The main principle of TO is to repeatedly change the material distribution of the design while maintaining structural integrity. The procedure starts with an initial design, normally a solid block or volume, and gradually strips away extraneous material from locations where it does not enhance overall performance. The material is redistributed by the optimization algorithm in a way that improves desirable attributes. EAs are a group of optimization algorithms inspired by the biological processes of natural selection and evolution [41]. EAs are particularly effective at solving optimization problems when multiple solutions or non-linear relationships make it challenging for standard mathematical or gradient-based approaches to succeed. Genetic algorithm (GA), particle swarm optimization (PSO), and ant-colony optimization (ACO) are the representative EAs that are used for metasurface inverse design [37]. The fundamental concept underlying the GA is to build a population of candidate solutions, generally depicted as individuals or chromosomes, and then let them go through evolutionary processes such as selection, reproduction, and mutation to improve oversubsequent generations [42]. Every component of the population represents a potential answer to the optimization issue, and the quality of each answer is assessed using a fitness function that measures how effectively it answers the question. PSO mimics the movement of a flock of birds or fish that updates their velocity according to each individual’s position to maintain the form of the flock [42,43]. The key point in PSO is that the surrounding individuals affect the velocity vector of one individual, affecting its next position. Therefore, in PSO, the position of the individual corresponds to the specific metasurface design, and the velocity function corresponds to how the metasurface will be changed. ACO is an algorithm that mimics ant colonies that have evolved to use pheromones to locate food [44,45]. An ant probabilistically determines its next position as a function of the remaining concentration of volatile pheromone and the distance to the point where the pheromone is located. The metasurface is optimized when the ant colony finds the optimal route to food.

While there have been efforts to consider inverse design as a mathematical optimization issue, designing metasurfaces based on AI has also been researched. Recent AI techniques have been developed focusing on machine learning (ML), which provides models and algorithms that let computers learn from experience, without having to be explicitly programmed, through supervised, unsupervised, or reinforcement learning [46,47]. Deep learning is an ML technique that utilizes deep neural networks (DNN), and it is effective in training machines from massive volumes of data [48,49]. DNNs mimicked the structure of the human nervous system by constructing layers of interconnected nodes for learning data representations in hierarchies [49]. A fully connected layer (FC), also called a multi-layer perceptron, is a fundamental DNN model in which each node is connected to all neurons in the previous layer, so every input affects the output node [50,51]. A design parameter is the number of neurons in an FC, which affects the output’s dimensionality. The weights and biases of the neurons are modified during training using optimization techniques such as gradient descent and backpropagation to reduce the discrepancy between the projected outputs and the ground-truth labels [51]. The rectified linear unit (ReLU) or sigmoid function is usually used after the output layer of the FC as an activation function for incorporating non-linearity into the neural networks [51,52]. The convolutional neural network (CNN) is another type of DNN that is widely applied to treat images as input or output data sets [53,54]. A CNN is implemented using kernels, which are small matrices (usually square matrices with a dimension of 3 × 3) used for image processing [55]. Convoluting a 3 × 3 kernel [−1, −1, −1; −1, 9, −1; −1, −1, −1] with an image consisting of a 1,024 × 1,024 data array can result in the edge sharpening effect of the input image. By using different kernels, it is possible to extract post-processed images with different strengthened characteristics: these are called feature maps [56,57]. Normally, an activation function such as ReLU is applied to the feature map for assigning non-linearity so that the neural network can learn more complex patterns [52]. The feature maps are compressed by passing through a pooling layer to reduce the computational costs and to extract spatially invariant features. The convolution set composed of a kernel, an activation function, and a pooling layer may be repeatedly applied several times [50,53,54]. Usually, the FC is connected to the last part of the CNN and used for image classification, but if an image generated in the middle of the CNN is output, the CNN can be used as an image-generating DNN [58]. Various research is being conducted by introducing the geometry of the metasurface as a pixelated image and learning it through a CNN [59]. As long as the search space is relatively small (pixelized images), the FC can be the best choice because it deals with every possible case; however, when the search space is large, such as high-definition photographs, the CNN is normally used for image processing.

To train these neural networks well, a large number of datasets is required, but it is not easy to acquire numerous datasets for the design of a metasurface and its optical properties. Therefore, the transfer learning (TL) technique, which uses models pre-learned by other datasets, is also being applied [6063]. In TL, the range of fine-tuning and freezing is determined by the number of data to be learned and the correlation between the data to be learned and the previously learned data [64,65]. For instance, if the number of data to be learned is sufficiently large and not related to the previously learned data, it is best to train all weights in the DNN using a new data set. Otherwise, it is necessary to decide whether to fine-tune the weights of only the activation layer, or whether to fine-tune the weights of several layers along with the activation layer. In addition to these various AI techniques, quantum machine learning (QML), which utilizes the uncertainty principle in artificial neurons, has also been demonstrated recently [6670]. The DNN for QML is called a quantum neural network (QNN), and its performance has been improved with the evolving quantum computers with the increasing number of qubits.

By employing these mathematical optimization algorithms and DNNs, it is possible to construct a generative model that converges a metasurface design to the optimal solution. Furthermore, it is not required to select only one type of algorithm, and it is possible to utilize multiple algorithms simultaneously or sequentially for a generative model. With the development of new mathematical optimization algorithms, DNNs, and computer architectures, the generative models for metasurface design and their applications can show endless possibilities.

The term ‘Generative Model’ is often considered a classification of a DNN, as opposed to the ‘Discriminative Model’; however, this should not be considered a strict classification with apparent definitions [36]. The reason is that some generative models, such as generative adversarial networks (GAN), contain the structure of a discriminative model within them [71]. Additionally, because it is possible to construct a generative model even without using a DNN, the generative model cannot be considered a subcategory of DNNs. Therefore, in this review, the generative model for metasurface inverse design is divided into an evaluator, a generator, and a criterion, as shown in Fig. 1, to discuss both DNN- and non-DNN-based algorithms. Through this, it is possible to classify which studies among various inverse design research have made novel contributions to the generative model among the evaluator, generator, and criterion. An evaluator calculates or estimates optical output according to the specific metasurface design through physical simulations or DNNs. A generator inversely suggests possible solutions that may satisfy input constraints. A criterion is an overall algorithm for providing appropriate metasurface design by managing the feedback loop between the evaluator and the generator. For metasurface design, a simulation program based on Maxwell’s equations can be an example of an evaluator, but it is also possible to design an evaluator using a DNN to reduce the computation time compared to conventional methods. In addition, an evaluator is applied to determine how well the inverse design presented by the generator satisfies the required optical properties. A DNN can also be used when designing a generator, providing a metasurface design according to the required optical properties. The criterion is an algorithm or feedback system to improve the performance of the generative model so it can be applied to design a metasurface that satisfies the required optical properties.

Figure 1. (a) Categorized diagram of fundamental computational algorithms for inverse design of metasurfaces. The algorithms are classified based on their operating basics; however, it is also possible to use two or more algorithms at once in an appropriate combination. For example, when implementing the GA for inverse design, a well-trained CNN based on CAD image and their optical spectrum dataset can be used to estimate a given nanostructure’s optical spectrum. (b) Schematic of a generative model for inverse design. Once a metasurface design is put in the generator, it generates several design variations based on the input design and its algorithm. When the generated designs pass through the evaluator, the only designs that sufficiently satisfy the target optical properties become the input design of the next routine. This routine is repeated several times until it generates an appropriate design based on the generative model’s criterion.

3.1. Evaluator

Since evaluators should reflect physical phenomena that occur in nature, it may be reasonable to fabricate a metasurface according to a given design and measure its optical properties to perform the role of the evaluator. However, evaluation based on practical fabrication or characterizations demands a significant amount of time, effort, and cost to carry out a single evaluation and to complete enough iterations for learning, and an unpredictable error range is inevitable. Therefore, physical simulations governing Maxwell’s equations are usually used as evaluators for metasurface design. These physical simulations start by segmenting a computer-aided design (CAD) image of a given structure into spatial mesh structures. Therefore, both finite difference method (FDM) and finite element method (FEM) simulations calculate physical equations based on a shape function. The difference between these approaches is that while FDM approximates the geometry using cubic or square meshes, FEM approximates the physical differential equations as polynomials [72]. For optical simulations whose governing physics is Maxwell’s equations, FDMs and FEMs can be categorized according to the domain of differential equations: time- and frequency-domains. FDMs with time-domain and frequency-domain differential equations are called finite-difference time-domain (FDTD) and finite-difference frequency-domain (FDFD) methods, respectively [72]. FEM normally employs frequency-domain differential equations, while the time-domain finite element method (TDFEM) refers to a FEM based on time-domain differential equations [72]. Since these methods calculate the optical properties of metasurfaces based on physics principles, they can provide logical understanding based on causal relationships if the calculation converges sufficiently to produce correct results. Most of the studies that have used physical simulation as an evaluator employed FDTD or FEM because there are several more commercialized simulation programs compared to FDFD and TDFEM [73]. Many inverse design problems can be solved without necessarily using a specific simulation method, and thus commercialized software is sufficient, but there are special problems that essentially require the use of FDFD or TDFEM [73]. As an example of the FDM serving as an evaluator, Cai et al. [74] employed an FDTD analysis to assess the characteristics of randomly generated metasurface structures. Also, Feichtner et al. [75] conducted FDTD simulations to optimize plasmonic antennas exhibiting plasmon resonances in the visible wavelength range. Wei et al. [76] employed a physical propagation module based on the FDTD method to validate the precision of holographic image reconstructions generated by physics-driven neural networks. Rodriguez et al. [77] used FDFD simulations to inversely design a localized surface plasmon resonance (LSPR) waveguide, depending on the polarization. In addition, Li et al. [78] provided an accelerated FDFD method for inverse design by using a reduced solution matrix size. In the area of FEM, Liu et al. [79] performed a FEM simulation to evaluate the transmittance spectra of generated metasurface structures. However, these physical simulation methods also require a lot of time because they integrate all calculation results from each mesh cell by considering the boundary conditions. Furthermore, it is necessary to confirm convergence by considering the mesh size and computing power according to the material’s structure to guarantee reliability of the results.

Unlike physical simulations, which must solve Maxwell’s equations for all unit cells in each structure, a DNN-based evaluator predicts optical characteristics without solving numerous equations. After training a DNN with more than thousands of data sets of metasurface design and its optical characteristics achieved by a physical simulation, it can output an estimated optical property according to the input metasurface structure in a few seconds [36,37]. As shown in Fig. 2, Liu et al. [80] trained an FC to estimate the transmittance spectrum of multilayer films, and they sought to overcome the property of nonuniqueness of the optimal solution by providing every possible solution using the FC. Lin et al. [81] optimized a CNN for estimating the optical response of a plasmonic metasurface using 25000 sets of training data. Mall et al. [82] utilized two CNNs, one acting as a generator and the other as a discriminator, to create a GAN, and in this setup, the evaluator efficiently classified 1,500 generated datasets of optical structures, enabling optimization of optical structures within a significantly reduced timeframe. Expanding on the utilization of CNNs, several studies have explored the incorporation of combined layers in conjunction with other DNNs. Jiang et al. [83], for example, integrated one layer of CNN and two layers of FC to create an evaluator aimed at optimizing the wavelength and deflection angle of metasurfaces, and the addition of the FC helped in generalizing from the features extracted by the CNN and having higher evaluation power. Also, Sajedian et al. [84] combined a CNN with a recurrent neural network (RNN) to create a model that predicts the absorbance spectrum of a plasmonic nanostructure using 100,000 simulation data sets.

Figure 2. Examples of evaluators for the generative model. (a) The optical properties of metasurfaces provided by a generator were evaluated using a physical simulator based on FDTD. The meta-lens structure was optimized to achieve maximum intensity at a certain focal length. Reproduced under the terms of the Creative Commons CC-BY license [74]. Copyright 2020, the authors. (b) FC was utilized to estimate the transmission function of SiO2 and Si3N4 multilayers. By employing the FC evaluator, it is shown that there are two or more optimized metasurfaces for one problem (DA and DB). Reprinted with permission from [80]. Copyright 2019, American Chemical Society. (c) For discrimination in the conditional GAN, CNN layers were utilized since they can be trained based on the geometric feature points of the 325 nm thick silicon meta-grating. The discriminator was constructed by one CNN layer and two FC layers. Reprinted with permission from [83]. Copyright 2018, American Chemical Society. (d) To predict the absorption spectrum of the designed metasurfaces in near-infrared wavelength, a combined network of a CNN and an RNN was utilized. After the CNN obtains the geometrical features, the RNN estimates optical characteristics based on the features. Reproduced under the terms of the Creative Commons CC-BY license [84]. Copyright 2018, the authors.

3.2. Generator

For the conventional metasurface design strategy, understanding the physics between the metasurface structure and incident light is important, and therefore the value of a geometrical parameter is changed within a certain range to analyze the effect of that parameter on the optical properties. For example, when optimizing a nanocylinder array design to elucidate the effect of wire width on the peak wavelength of LSPR, measuring and calculating the optical spectrum by varying the cylinder diameters from 80 to 120 nm at 20 nm intervals can provide knowledge that the LSPR peak will be red-shifted to around 700 nm with the increase of diameter [85]. With this physical understanding, for example, if a new metasurface with an LSPR peak wavelength at 750 nm is required, it is logically valid to suggest a nanocylinder array with a diameter larger than 120 nm. This ‘causality’ is the main consideration when a new metasurface is suggested or generated during the forward design process.

However, inverse design strategies do not focus on the causality of design parameters and the output spectrum. For inverse design using the EA, several generation algorithms have been provided by mimicking biological development, differentiation, and reproductions. The GA, which is the most widely used EA, implements a generator based on crossover and mutation characteristics of chromosomes. Lin et al. [86] studied GA-based optimization of metasurface retroreflectors by generating metasurface designs with single-point crossover and uniform mutation operators to keep the number of chromosomes more than 20 for every iteration. Jiang et al. [87] also demonstrated a GA with single-point crossover and binary mutation, which are types of uniform mutation, while applying 70 and 10 % crossover and mutation probabilities, respectively. Rong et al. [88] employed a uniform crossover method for a GA that is operated by exchanging genes according to the uniform probability function. In PSO, the velocity function serves as a generator to suggest a new position of a particle at each iteration, while each particle’s position corresponds to a unique metasurface design. Thompson et al. [89] designed a broadband reflector through PSO by updating the velocity vector and choosing the value of cognitive and social parameters based on a uniform distribution. As schematically explained in Fig. 3, Nugroho et al. [90] demonstrated the velocity vector of PSO by multiplying the cognitive and social parameters with additional random numbers chosen between 0 and 1 for each to design a plasmonic metasurface for hydrogen detection. When utilizing ACO as a metasurface inverse design, the generator is operated by a probability function that determines the direction of movement of ants depending on the concentration of pheromones [91].

Figure 3. Examples of implemented generators for the generative model. (a) During the palladium nanodisk optimization process using PSO, the figure-of-merit of the generated population, along with the 15 iterations, increased and converged into the optimal solution. Reproduced under the terms of the Creative Commons CC-BY license [90]. Copyright 2022, the authors. (b) After training an FC as an evaluator, the FC could provide the scattering spectrum according to the thickness of the shells constructing the nanoparticle. By backpropagating the FC, it was also possible to suggest the thickness values to satisfy target optical characteristics after several iterations. Reproduced under the terms of the Creative Commons CC-BY license [93]. Copyright 2018, the authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. (c) The transpose CNN layers were used as a generator by extracting features from the input image sets followed by training to generate metasurfaces according to the contrast vector which contains the information of the target transmittance spectrum. Reproduced under the terms of the Creative Commons CC-BY license [94]. Copyright 2020, the authors. InfoMat published by UESTC and John Wiley & Sons Australia, Ltd. (d) By training the autoencoder constructed with CNNs, the performance of both the encoder and the decoder was enhanced to extract features and reconstruct images, respectively. This study showed an example of inverse design using an autoencoder, and this neural network structure can be modified to be used as a generator of VAE-GAN to construct a generative model. Reprinted with permission from [102]. Copyright 2023, American Chemical Society.

DNNs also can be used for generators. Ghorbani et al. [92] trained an FC consisting of 11 layers by alternating dense and dropout layers to generate dual-polarized metasurfaces based on input vectors containing the target electromagnetic wave reflection properties. Peurifoy et al. [93] provided an FC-based generator by operating a backpropagation algorithm to a well-trained FC-based evaluator for several iterations. Additionally, Han et al. [94] also demonstrated inverse design using an FC-based generator based on backpropagation that guarantees high degrees of freedom. Asano et al. [95] trained a CNN as an evaluator to predict Q-factors of nanocavities and then applied the error backpropagation method, which changes the CNN to operate as a generator. An et al. [96] constructed several layers of a CNN to generate ‘H-shaped’ metaatom patterns by varying four geometrical parameters.

The decoder part of the variational autoencoder (VAE) can also be used for the generator. An autoencoder is a serial connection of an encoder DNN and a decoder DNN, and it self-trains how to compress data efficiently with representative information such as statistical distributions [97,98]. When the input data are compressed by an encoder, the decoder inversely estimates the input data based on the compressed information [97]. By comparing the difference between the input data of the encoder and the output data of the decoder, the encoder is trained to produce core information while reducing the size of the data [97]. Using the autoencoder structure, a different type of data generation model, that is, a VAE, can be trained. The basic structure of the autoencoder and VAE are the same, but the VAE focuses on training the decoder by putting a probabilistic distribution in the encoding process to be trained as a generator [98,99]. The decoder of the VAE should estimate and generate data that are similar to the input data of the encoder. It is also possible to use only the VAE only for inverse design without it being included in the decoder part of the generative model [100102]. From the perspective of the use of VAE as a component of a generator of generative models, the VAE-GAN was demonstrated and employed for metasurface inverse design [103-105]. Liu et al. [104] designed VAE-conditional GAN to optimize metasurfaces that provide holographic images. In addition, We et al. [105] suggested an efficient method of generating a training dataset based on VAE-GAN for further AI-based inverse design. The denoiser of a diffusion model can also be applied to a generator. The diffusion model starts by noising the input data with several continuous steps until it becomes fully white noise, and then, the denoiser continuously recovers the input data [106,107]. Similar to the VAE, the diffusion model itself can be used for inverse designing metasurfaces [108,109]. Meanwhile, diffusion-GAN is an example where the denoiser of the diffusion model serves as a generator when solving image-classifying problems [110]. To the best of our knowledge, it has not yet been demonstrated for inversely designing metasurfaces. However, since the diffusion models have shown excellent performance as image generators, diffusion-GAN should be further studied in terms of efficiently providing proper metasurface geometry.

3.3. Criterion

A criterion that uses a logical relationship between geometrical parameters and optical response is known as a forward design strategy. Inverse design starts by using a black box called a DNN instead of the researcher’s brain neural network. But the important point is that, for a successful inverse design, the criterion must decrease the difference between the target and the sample’s optical properties and converge the metasurface geometry to the optimized solution. In forward design, optimization based on physical understanding can be seen to function as a criterion. The criterion can be categorized into mathematical optimization-based and AI-based criteria. The former is operated by offering certain constraints with mathematical logic, and the latter automatically trains itself by optimizing the weights of nodes through iterations.

Starting from the mathematical optimization-based criterion, Andkjær et al. [111] demonstrated two types of plasmonic metasurfaces through TO under the design domain containing 680842 and 696172 elements, respectively. Sell et al. [112] provided optimized high-angle deflecting meta-grating structures using adjoint-based TO. In addition, Phan et al. [113] enhanced the performance of TO for designing a large-scale metasurface with high complexity by discretizing the target optical spectrum and optimizing the metasurface geometry for each section, as shown in Fig. 4. Lin et al. [114] showed a larger search space with 105~106 degrees of freedom for TO by pixelating the metasurface geometry [114]. For the case of EA, Zhu et al. [115] used the GA to design a metasurface with the desired phase spectrum in both x- and y-polarization by applying specific probabilities of crossover and mutation to the selection operator. Lin et al. [116] developed a genetic-type tree search algorithm, which is a modification of the Monte-Carlo tree search by applying the evolution strategy in a GA, and then utilized the algorithm to optimize beam-steering metasurfaces for various steering angles. Zagaglia et al. [117] used PSO for optimizing the lattice parameters in the x and y direction, core diameter, the length of the nanowire, and the shell thickness of the nanowire base simultaneously to achieve a sharp peak for optical sensing applications. ACO was employed by Lewis et al. [118] to optimize an antenna structure by maximizing antenna efficiency and minimizing the resonant frequency at once. Zhu et al. [91] demonstrated both conventional ACO and multiobjective lazy ACO (MOLACO) for inversely designing three-dimensional frequency-selective surfaces and presented a case where the MOLACO can provide better performance in terms of the center frequency, rejection bandwidth, and transmission bandwidths. In addition, Zhu et al. [119,120] and Campbell et al. [121] provided several other cases where MOLACO was implemented for nanophotonic inverse design.

Figure 4. Examples of managing criteria for the generative model. (a) Although TO can provide high-performance metasurfaces for diverse applications, it requires extensive computational costs with an increase in the design space. Therefore, some technical solutions to address this problem were developed, for instance, discretizing the desired optical spectrum into wavelength scale can effectively decrease optimization time, which exponentially increases with the size of the metasurface. Reproduced under the terms of the Creative Commons CC-BY license [113]. Copyright 2019, the authors. The panel labels of the original figures were erased to prevent misleading. (b) By employing unsupervised clustering, the genetic-type tree search algorithm can estimate the optimal metasurface in the virtual design space through initialization, virtualization, optimization, and refinement steps. During the optimization, the amount of modification is decided by scores of selected metasurface designs, and the genetic-type seeds management methods were applied to obtain computational efficiency. Reprinted with permission from [116]. Copyright 2021, American Chemical Society. (c) Implementation of generalized MOLACO with a modified Pareto locus search algorithm can provide an optimized metamaterial design in three-dimensional search space. The path along which ants move within the search space generates a metamaterial with contiguous geometries that can ensure effective wet etching in further fabrication procedures. Reprinted with permission from [120]. Copyright 2019, American Chemical Society. (d) Full workflow diagram of conditional GAN system, which trains the discriminator and generator together through adversarial competition and fine-tuning. In the conditional GAN, information about specific boundary conditions is provided to the discriminator and generator to form conditional images rather than random images. Reproduced under the terms of the Creative Commons CC-BY license [125]. Copyright 2022, the authors, Advanced Photonics Research published by Wiley‐VCH GmbH.

If both the evaluator and generator are designed based on the DNN structure, it is possible to improve the performance of one strategy by repeating routines. A representative example is a GAN, which implies an adversarial criterion on a generative model. In the GAN framework, the evaluator tries to distinguish the real image and the generatormade (fake) image while the generator tries to generate an image that looks more real such that it cannot be distinguished by the evaluator [122124]. If the evaluator correctly distinguishes the generatormade image, the generator gets feedback, and if the generator successfully deceives the evaluator, the evaluator gets feedback [122124]. Since both the evaluator and generator are improved by competing, it is called the adversarial criterion. Liu et al. [71] demonstrated a GAN for achieving customer-defined optical spectra with high fidelity using nanophotonic metasurfaces. Kianin et al. [125] employed a conditional GAN, which uses auxiliary information as inputs of the network, for vortex beam generation and wave manipulation of inversedesigned metasurfaces. Mall et al. [82] demonstrated a cyclical DNN framework by applying a pseudo-GA to a conditional GAN and a simulation neural network to train a generative model efficiently.

Figure 5 shows simplified possible combinations of generative models. The generative model operated by TO can use physical simulations and DNNs as an evaluator, but it does not use a generator because the TO only searches for optimal solutions in a pre-defined design space. When using the EA as a criterion, it is possible to use all types of evaluators presented in the figure, but a DNN-based generator is not allowed. For the adversarial criterion, which also means the GANbased system, both the evaluator and generator should have the form of DNNs since it trains the evaluator and generator at once. When training the evaluator, the data set generated by physical simulation is generally used. However, since the amount of data to train an evaluator is too large, a virtual data set generated by modifying and combining several simulation results is occasionally used. The generator is generally trained through iterative backpropagation; however, it is also possible to utilize a pre-trained decoder from a VAE or a denoiser from the diffusion model. This figure does not describe the strict constraints, but provides a supportive reference when constructing a generative model. For example, when utilizing conventional TO, generators should not be included for metasurface inverse design. However, by modifying or combining the algorithms to obtain a certain structure for the criterion, it can be possible to use other types of evaluators or generators to satisfy the convergence condition.

Figure 5. A simplified schematic of possible combinations of the evaluator, generators, and criterion for constructing a generative model for metasurface inverse design. To implement conventional TO as a criterion, physical simulations and DNNs can be used for an evaluator, and it is not required to have a generator since TO searches within specific design space boundaries. When utilizing the EA, it is also possible to apply physical simulations and DNNs as an evaluator and a generator based on probabilistic modification of the parent population or random number generation is required. By employing both the evaluator and generator with DNNs, it is possible to construct a generative model based on the GAN framework. The DNN-based evaluators can be trained using a dataset achieved by physical simulations, and the DNN-based generators can be trained through the backpropagation method or by applying the decoder network of a VAE. This simplified diagram only shows fundamental combinations of the components or algorithms and does not describe all the possible cases. Therefore, based on the structure of the generative model for metasurface inverse design, by improving or combining the presented algorithms to be utilized for criterion, evaluator, generator, or even training methodologies, it is possible to enhance the performance of the generative models.

In this review, we discussed metasurface inverse design research, including mathematical optimization and deep learning approaches, by focusing on the contribution of the generative model. Since the essential steps for inversely designing metasurface are discrimination and generation, it is necessary to understand the inverse design prior arts systematically according to their role and which criterion they used for the overall generative model. For an evaluator, the conventionally and most widely used evaluator is a physical simulator based on Maxwell’s equation. Otherwise, since evaluation for the output optical spectrum is similar to classification problems using AI, FCs, and CNNs are trained based on the simulation data set and can be used as an evaluator. The generator part can be demonstrated through programming design recombination or random generation. Well-trained FCs or CNNs can be used for the generator through backpropagation, or trained DNNs under VAEs or diffusion models can also be employed for the generator directly. The criteria were discussed by categorizing mathematical optimization and AI approaches; however, the criteria can be modified and combined diversely to achieve better performance depending on the applications.

Several directions can increase the performance of the inverse design model. 1) Because inverse design cannot handle all design spaces within limited computational power, it cannot always find the absolute best value. Therefore, the problem must be accurately defined by applying various constraints and boundary conditions to obtain the optimal solution with the given cost. 2) When dealing with threedimensional metamaterials with complex functionality, developing both an evaluator and generator with a DNN can provide a more appropriate solution for exponentially increasing computation costs. 3) One of the biggest problems when using physical values as a data set for training a DNN is the high cost in terms of human and computational resources to obtain enough data. Therefore, TL and few shot learning algorithms should be considered to efficiently train DNNs. 4) One of the purposes of inverse design is to investigate unexplored physics in nature, as mentioned in the introduction. For this purpose, it is helpful to estimate the decision-making process of the DNN. However, because of the high connectivity inside a DNN, it is difficult to understand how a DNN generates output with the input. Therefore, algorithms to acquire the interpretability of DNNs should be studied using a well-trained DNN evaluator. 5) The new era of AI techniques that uses QNN can process more information in a faster time by reducing the time complexity with qubits. Computer scientists have demonstrated AI algorithms in the form of a combination of classical ML and QML. In addition, they have been replacing more steps of AI algorithms with quantum platforms to increase computing efficiency. It is also known that QNNs can be trained more efficiently with fewer data, fewer iterations, and fewer validation processes. Inverse design is a multidisciplinary research field that applies mathematics and computer science to nanophotonics. This integrative technology has shown several possibilities for overcoming the limitations of conventional forward design approaches. It is expected that inverse design research will provide new insights into how humans can better understand and control light in the future.

This research was supported by the National Research Foundation (NRF) of Korea under the “Korean-Swiss Science and Technology Program” (2019K1A3A1A1406720011) and was also supported by the BK21 FOUR (Fostering Outstanding Universities for Research) funded by the Ministry of Education (MOE) of Korea and National Research Foundation (NRF) of Korea. The authors thank Jongho Jung, Sungmin Lee, and Kichang Lee, Ph.D. students in the School of Integrated Technology, Yonsei University, for providing supportive discussions about deep neural network algorithms.

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