Advancing statistical learning and artificial intelligence in nanophotonics inverse design

2.1 Topology optimization

Gradient-based inverse designs exploit the first-order derivative (gradient) of the cost function to build approximate models that are minimized at each iteration [22–38]. Traditional topology optimization implements the gradient-descent optimization scheme [39, 40], while advanced techniques exploit convex optimization such as the trust-region [41], and moving asymptotes [42, 43]. In topology optimization based on density variables, the design parameters β are typically the permittivity ϵ _r (x) probed at discrete spatial lattice points x. The resulting design space is a 2D binary image, in which the pixel density measures the distribution of different classes of materials [27–34]. Other topology approaches express β following a re-parameterization strategy such as, e.g., the level-set method [36–38]. In two-dimensional material profiles, the level-set method represents design variables via horizontal slices of a 3D surface, defined as the level-set function. During the optimization, the continuous movement of a cross-sectional plane simulates the variation of design variables. In contrast to explicit curve representation such as analytic equations, the level-set method traces topology changes, including merging, splitting, generating, and vanishing different shapes.

In topology optimization, the iterative calculation of the gradient is the most time-consuming operation. Yablonovitch et al. [44] developed a fast numerical implementation of this operation based on the adjoint method [45, 46]. This approach performs a forward simulation and an adjoint simulation, distinguished by the applied incident source. In the simple example of a density-based optimization task, with the FOM represented by the field intensity |E(x ₀)|² measured at a certain diffraction order, the forward simulation calculates the electric field E ^fwd*(x) at one grid point x under normal incident light. The adjoint simulation, on the other hand, calculates the electric field E ^adj(x) under incident light from the target diffraction order. This formulation allows expressing the gradient with respect to the permittivity at the grid point ∂FOM/∂ϵ _r (x) as [44]:

with E f w d * ( x ) representing the complex conjugate of electric field from forward simulation. Equation (1) is rapidly evaluated by using only the local values of the electric field, and significantly speeds up calculations.

Figure 2A reports a group of broad-band photonic devices designed by density-based topology optimization. In the work [33], Hammond et al. also investigated foundry design-rule constraints, including both systematic limitations and fabrication errors. These are introduced as inequalities in the form of g _k ≤ G _k . For each k, G _k is a positive constant indicating the constraint for a certain metric g _k , including line width, line spacing, minimum area, and minimum enclosed area [47, 48]. To keep into account constraints of under and over-etching, which unavoidably occur at each fabrication step, Hammond et al. applied a conic image filter [49] on the design pattern. The modified design β ̄ satisfying all design-rule constraints listed above simulates the real-world fabrication scheme, including dilation and erosion. The topology design utilizes a classical binary representations of a silicon structure on SiO₂ substrate. At each design point, the algorithm interpolates the permittivity as: ϵ r ( β ̄ ) = ϵ min + β ̄ ( ϵ max − ϵ min ) , where ϵ _min is the permittivity of SiO₂ and ϵ _max is the permittivity of silicon. Constraining the parameter β ̄ in [0,1] allows continuously varying the permittivity across the design area and, during the initial stage of the optimization, it facilitates convergence by exploring all possible continuous distributions of design parameters. The optimization then introduces regularization terms that penalize all parameters that are different from zero or one, which does not represent silicon or SiO₂. After convergence, the algorithm produces a discrete material distribution satisfying fabrication’s constraints. The above study contains three independent mirror, bend, and T-splitter designs working at 1500–1600 nm. The devices share a minimum line width of 90 nm, a minimum line spacing of 90 nm, a minimum area (for islands) of 0.08 μm² and a minimum enclosed area (for holes) of 0.2 μm². An increasing threshold coefficient that weights the penalization term ensures clear boundaries at the end of optimization. The algorithm iterates for 210 iterations and incorporates fabrication constraints during the last 70 iterations. Figure 2A shows the resulting devices, which achieve a simulated transmission efficiency above 94% within a processing time of 6–8 h. In 2020, Augenstein et al. [50] introduced mechanical loads into the optimization of nanophotonic devices, enabling additive manufacturing and further enriching the library of design-rule constraints.

Figure 2:

Topology optimization in nanophotonic inverse design (A) design-rule-constrained nanophotonic devices optimized with foundry design rule checks. The panels depict the optimization process and efficiency of top: mirror, middle: bend and bottom: T-splitter. Adapted from [33]. (B) High-NA, high-efficiency metalenses designed with linearized topology optimization. Top left: SEM images of the fabricated metalenses. Top right: experimental and simulated efficiencies of metalenses with NAs of 0.2, 0.5, and 0.8, respectively. Middle row: measured intensity at the focal planes of the designed metalenses. Bottom row: transmission efficiencies in the wavelength range 580–700 nm of the designed metalenses. Adapted from [31]. (C) Two dielectric metalenses with NA = 0.78 and 0.94, respectively (left) and the corresponding focusing efficiencies during the optimization process (right). Adapted from [34].

Topology optimization reported also the implementation of metalenses at ultraviolet [51], visible [52, 53], infrared [54, 55], and millimeter [56] wavelengths with high numerical aperture (NA). Unlike conventional refractive lenses, metalenses focus incident light with sub-wavelength elements, which simulate the required surface curvature [57]. Researchers investigated non-periodic metalenses where the optical path varies among different designs to optimize the FOM, that is, the focusing efficiency at the focal length. Both relative (focused power/transmitted power) and absolute (focused power/incident power) efficiencies are accepted as part of the standard metric for metalenses. The optical properties, including deflection angle and phase response, are optimized at specific spatial points along the lens radius. As a result of this procedure, the numerical complexity of this class of designs is notably higher than the periodic metasurfaces mentioned above. To address this challenge, Phan et al. [31] studied a computationally efficient approach for the design of large-area metalens. The idea is to decompose the desired phase response along the radius into wavelength-scale segments and apply topology optimization to design isolated metasurfaces with scattering properties that linearly fit the corresponding segments. By integrating all discrete parts, the design converges to a functional metalens, with a time complexity that reduces from O(L ^2.4) to O(L) for L being the degree of freedom in one-dimensional design space. Figure 2B depicts a series of 200 μm-wide metalenses designed to focus incident plane wave with NAs ranging from 0.2 to 0.8. The algorithm performs adjoint-based topology optimization on each 2 μm linear segment. Experimental results reveal efficiencies above 89% for all NAs in a wavelength range from 580 nm to 700 nm. The concept of linearization vastly reduces the system complexity compared to monolithic optimization. However, since the parallel tasks use perfectly matched layer (PML) boundary conditions [58, 59], the algorithm does not consider mode couplings between metasurface sections. This factor limits the linearization approach for an increasing number of segments, introducing strong couplings that finally obstruct convergence to an actual structure. Thus, the scheme searches for the best compromise between computational efficiency and linear fitting errors.

Recently, Mansouree et al. [34] reported an efficient design for high-NA metalenses with rectangular parameterization. Rather than directly modeling the permittivity at all design areas [60], the proposed method defines design variables as the widths of rectangular sub-wavelength structures. The design optimizes two metalenses with desired NAs of 0.78 and 0.94, formed by a total of N = 19,200 of 430 nm-thick amorphous Si bars. The nanostructures are symmetric in all four quadrants of the space, reducing the number of design variables to N = 4800. To calculate the gradient of the width variables, the authors implement an adjoint method with a perfect electric conductor (PEC) boundary condition. Figure 2C shows the resulting focal planes, located at 20 μm and 8.33 μm above the device surface for the metalens with NA = 0.78 and NA = 0.94, respectively. The rectangular parameterization inherits both advantages from periodic unit cells and monolithic design. It reduces the redundancy of bulk density variables while keeping the design diversity for complex, non-periodic nanostructures. The optimized metalenses with NAs of 0.78 and 0.94 achieve focusing efficiencies of 78% and 55%, respectively. This work assesses the performances of the proposed parameterization technique by designing two unit-cell metalenses as a control group under identical conditions. By comparing the simulated focusing efficiencies, the work reports a 10% performance improvement.

2.2 Heuristics and meta-heuristics

At variance with classical optimizers, heuristics and meta-heuristics exploit randomness in the searching process, imitating the behavior of different types of natural systems and phenomena. The primary example of heuristic methods is simulated annealing (SA), which originates in the context of non-differentiable, combinatorial problems, such as, e.g., the traveling salesman [61]. Problems of this type are mathematical ‘hard’ in the sense that the straightforward solution increases in time with a non-polynomial function, becoming quickly impossible to solve by any classical hardware.

Simulated annealing attempts to replicate the behavior of metals whose temperature decreases slowly, allowing the medium to converge towards a solid material, representing the global minimum of its potential energy. SA optimization imitates this process by setting the inverse design cost function C( β ) as the potential energy of an equivalent molecular dynamics system, with each molecular particle described by a position β _i representing a design parameter that we intend to find. The annealing then proceeds to the global minimization of the particles’ positions β = [β ₁, …, β _n ] by progressively reducing the temperature with a suitably defined cooling scheme. If the temperature decreases sufficiently slowly, the probability distribution of the sequences of β found at each iteration follows the Boltzmann probability density ∼ e − C ( β ) / T of a classical thermodynamic system [62]. At T → 0, the molecular system converges to the set positions β representing the global minima of the cost function, solving the inverse design problem.

Zhao et al. [63] implemented this strategy to design diffusion metasurfaces that scatter the incident light uniformly in all directions. By exploring phase changing materials, this work designs a 2.4 μm × 2.4 μm reconfigurable metasurface [64] composed of 36 rectangular blocks as basic elements. The experimental results show a minimized directional reflection, with a 10 dB radar-cross-section (RCS) [65] reduction under normal incidence for both TE and TM polarizations. Another recent study by Xie et al. [66] proposed a magnetic resonator assembled by disordered nanostructures. The applied SA algorithm optimizes a five-by-five array of all-dielectric sub-wavelength scale resonators. The converged configuration reaches an experimentally measured peak magnetic field enhancement factor of 16.51 at 5.8675 GHz.

The main challenge of SA is that the convergence towards the optimal minimum lies in the heuristic idea of recreating a Boltzmann-type thermodynamic machine, which is not mathematically guaranteed to occur in every case. Ideally, addressing this problem requires sampling the probability space with different SA runs launched with diverse (random) input conditions. This operation, which could be time-consuming, has a drawback: it does not employ the knowledge acquired during the past since different SA runs proceed independently. Meta-heuristics schemes such as particle swarm optimization (PSO) provide a possible approach to address this issue. These optimizations take inspiration from the social behaviors of crowded biological systems, including ant colonies, fish, and birds flocks. Despite the apparent simplicity of their social interactions, these systems are highly efficient in exploring a given terrain when looking for food, flowers, and other equivalents ‘solutions’ to their search problem. In computational PSO, the system is initialized with sufficiently large swarms of randomly chosen particles β ₁, …, β _n , with each β _i representing a candidate solution to the problem. During each iteration, particles generate new configurations of β with the information from the best position found by each particle and neighboring particles. As in SA, the equivalent temperature of the system progressively decreases; unlike SA, however, temperature rescaling does not follow a pre-imposed annealing schedule but relies on the information acquired during the search [67].

Chen et al. [68] reported recent progress on PSO designed polarization beam splitters. The devices strengthen its compatibility to complementary metal-oxide-semiconductor (CMOS) by optimizing the silicon-on-insulator (SOI) structures. As shown in Figure 3A, the dashed area constrains the design space, which completes the functionality as a counter-tapered coupler. To reduce computational complexity, the authors split the coupler into ten silicon stripes within a total coupling length of 5 μm. The algorithm includes particles with positional vectors ps _n represented by widths of the segmented stripes and velocity vectors ve _n employed to update the former. The FOM is defined based on the polarization extinction ratio between output channels. As TE and TM waves are separated into the Bar and Cross port, respectively, the FOM is calculated by P _{TE_Bar}/P _{TE_Cross} + P _{TM_Cross}/P _{TM_Bar}, where P represents the transmitted power. After initialization and EM simulation, each particle records its best position bp _n and the global best position gp _n among the swarm. The update formula for ps _n and ve _n is as follows:

where w _I is the inertial weight, describing the momentum of previous movements. 0 ≤ r ₁ ≤ 1 and 0 ≤ r ₂ ≤ 1 are the cognitive and social rate, representing the weights of individual bp _n and social memory gp _n of past best configurations. Figure 3B shows experimental results revealing high polarization extinction ratios over 16 dB for both TE-type and TM-type grating couplers obtained with PSO. Recent PSO-based researches also provide solutions across a wide range of nanophotonic devices including power splitters [69], solar cells [70], achromatic metalenses [71] and phase changing antennas [72].

Figure 3:

Heuristic optimization in nanophotonic inverse design (A) a silicon polarization beam splitter placed on SiO₂ substrate. Bar and Cross are two polarization-related output channels. (B) The measured spectra of the fabricated beam splitters in (A). (A) and (B) are adapted from [68]. (C) Multi-resonant dielectric nano-antennas with fixed resonance at 550 nm for one polarization X, and ranging resonance peak from 400 nm to 650 nm for the other polarization Y. Adapted from [76]. (D) Plasmonic gold antennas with maximum scattering on the desired direction. Each panel shows the fittest solution at different iterations, starting from initialization to convergence. (E) Optimized examples of directional antennas for various target angels and scattering environments. Left: backscattering in vacuum. Right: scattering towards glass substrate. (D) and (E) are adapted from [78].

The main issue in PSO is the speed of the algorithm’s convergence, which follows the set of worst-performing particles. Evolutionary schemes, such as, e.g., genetic algorithms, try to address this issue by selectively mixing individual information arising from the design population [73]. Spuhler et al. [74] first proposed a genetic algorithm for the design of fiber-waveguide coupler. By splitting the SiO₂/SiON functional area into segments with identical lengths and flexible widths, a population of design candidates containing a distinct width sequence evolves with random mutations, cross-overs, and die-out. A non-intuitive structure differing from any existing design emerges after 1132 optimization steps, with a 2 dB reduction on coupling loss compared to direct butt-coupling devices.

Following this initial genetic approach, a series of photonic devices have been proposed with increasing design complexity during the past decade [75]. Figure 3C reports a cluster of dielectric nano-antennas for polarization-encoded color display obtained by genetic algorithms. The work exploits sub-wavelength nanostructures that alter their optical properties for different light polarizations. To address this design task, Wiecha et al. [76] proposed an evolutionary multi-objective optimization approach, which maximizes simultaneously the scattering efficiency at demanded wavelength, as well as both TE and TM polarizations. The periodic unit cell is a set of four silicon blocks, with horizontal sizes ranging from 60 nm to 160 nm. The entire design space is a 600 nm × 600 nm area, with available parameter combinations larger than 1 × 10¹⁵. The algorithm defines a fitness function that integrates polarization-dependent metrics and evaluates all candidate designs in each iteration. The selected high-performance designs reproduce a new generation by cross-over and random mutation and finally lead to a convergence indicated by the Pareto front, a standard metric for multi-object programming [77]. In this specific task, the Pareto front is the set of design parameters whose modifications enhancing the scattering at one polarization cause the efficiency to drop for the other polarization. The Pareto front keeps expanding as the optimization processes until the device reaches saturation, indicating the best performance. As depicted in the bottom panel of Figure 3C, the X-polarization (TE or TM) scattering peak is at 550 nm. In contrast, the Y-polarization (TM or TE, respectively) scattering peak is evolved gradually from 400 nm to 650 nm, all with consistent efficiency (within ±20%) in both polarizations. Compared to direct search, the evolutionary algorithm rapidly explored the design space within 100 iterations from a population of 20 candidate designs.

The work in [78] exemplifies a light-directional antenna, which maximizes the ratio of direct scattering at a specific direction to the residual scattering directions. Inheriting similar optimization strategies from [76], the authors leveraged a binary-shaped design space containing 40 nm × 40 nm × 40 nm gold blocks (represented by “1”) and air (represented by “0”). The total amount of all possible arrangements extends to 10¹¹¹. As shown in Figure 3D, the algorithm starts with an initial population size N _p = 500 and reaches convergence after 50 generations for fixed azimuthal angle φ = 45°. The scattering antenna evolves spontaneously to form a structure resembling a Yagi–Uda RF antenna [79] with one driving feed, one reflector, and one director. Figure 3E reports examples of scattering antennas optimized under different conditions (e.g., propagation media, diffraction windows) from this work.

3.1 Discriminative models

Discriminative models bypass the repeated EM input–output simulations, often the bottlenecks of the optimization schemes reviewed in the previous section. Discriminative models are essentially unique configurations of deep learning architectures that learn the relationship between designs in the parameter space and their optical responses (reflection, transmission, amplitude or phase) [85–91]. Based on the study of plasmonic nanostructures, Malkiel et al. [92] introduced a bidirectional network composed of geometry-predicting-network (GPN) and spectra-predicting-network (SPN). The authors train the GPN in a supervised manner with a dataset composed of 15,000 randomly generated device layouts containing eight geometrical parameters, including width, height, rotation angle, and the presence of positional elements in the H-shaped framework. Additional to the device data, finite elements simulations (FEM) calculate the corresponding transmission spectra. The GPN predicts the geometrical design parameters by processing 86 sample points from both TE and TM spectra and dispersive material coefficients. After training, the SPN, constructed in a cascade with the GPN output layer, retrieves the spectra from the eight geometrical parameters launched at the input. The network efficiently leverages the limited dataset by using unseen shapes generated by GPN as the training set of SPN. The loss function and gradient propagate to each layer in both networks. The end-to-end training of the network converges within 2 h, after which the time required to complete queries for potential nanostructures lies in the millisecond range. As exemplified in this work, the GPN retrieves classic nanophotonic structures such as nanobars, L-shaped, and split-ring resonators. To efficiently extract the hidden features from nanostructures, the training of a forward model (device-to-response) and an inverse model (response-to-device) is beneficial. Similar to the GPN and SPN introduced above, the two models can be either constructed and trained independently [93] or with inner connections, such as the generative models that will be introduced in the next section.

Other discriminative models leverage the generalization ability of NN in two main directions [94, 95]. On the one hand, they exploit the fact that a trained system predicts novel results for an arbitrarily large number of queries; on the other, they use the information acquired by the NN during training and transplant it via transfer learning [96] in a different task. Work proposed by Qu et al. [89] embedded and demonstrated transfer learning in a nanophotonic inverse design scheme. The approach includes the precise prediction of transmission spectra for an 8-layer thin film and a 10-layer thin film. Figure 4B shows an overall schematic of the transfer learning process. A 7-layer fully connected network (FCN), acting as a base network, is trained on a complete dataset containing 50,000 simulation examples. After the training, a second FCN acting as the transferred network replaces the top n layers of the base network, leaving the rest 7-n layers sharing weights from the trained one. The second network demonstrates the advantage of transfer learning by training 500 examples sliced from simulation results in a different design task, which are insufficient for training a model from scratch. As a quantitative example, Figure 4B illustrates the transfer learning schemes where the source task and target task switch between 8-layer and 10-layer films. By increasing the number of shared layers, the mean square error (MSE) reduces by 50.5% and by 23.7% in the 8-to-10-layer and 10-to-8-layer transfer model, respectively. The improved performance reveals that the knowledge from the complete dataset is preserved and leveraged in training the sliced dataset. Figure 4C shows a test example, comparing the performance of direct learning strategy and transfer learning in a working bandwidth between 400 nm and 800 nm. This work demonstrates that while comprehensive modeling of physical effects could be difficult to achieve, the implicit knowledge carried out by pre-trained NN is a practical approach to mitigate this issue, especially when simulations are time-expensive.

Figure 4:

Discriminative deep learning model for spectra predicting and design parameter retrieval (A) bidirectional network for plasmonic nanostructure design. Left: geometry predicting network with TE&TM spectra and material properties as input. Right: spectra predicting network, concatenated to the first network in a cascaded structure. Adapted from [92]. (B) Knowledge transfer between the design of 8-layered and 10-layered transmissive films. As the shared network layers keep increasing, both transfer schemes (8-layered to 10-layered and the inverse) show better prediction precision compared with direct training. (C) Example of predicted spectra compared to real spectra from the 8-to-10-layer transfer learning tasks. Left: direct learning. Right: transfer learning. (B) and (C) are adapted from [89]. (D) Results of the discriminative design on the plasmonic silver absorber. The device structure is encoded as 2D images, with absorption spectra predicted by the RNN and FCN layers. Adapted from [88].

The examples discussed above include different FCNs to grasp the physical relation between characteristic parameters of a design and the output spectral response furnished by the system. A recent trend of nanophotonic inverse design [97] introduces a high-level hierarchical representation of data for fast image processing and sequential analysis. Sajedian et al. [88] exemplified a design of periodic silver nanoparticles based on a combination of convolutional neural networks (CNN) and recurrent neural networks (RNN). Two-dimensional images of 100 × 100 pixels 2D encode the top view of three-dimensional structures with a definite constant thickness along the Z-axis. The authors use a training dataset with random images comprising a large variety of geometries, width, length, position, and rotation. Finite-difference time-domain (FDTD) simulations [98] computes the absorption spectra of 100,000 structures in the dataset. The applied CNN is ResNet [99], which is a powerful architecture in computer vision for pattern recognition problems. ResNet contains shortcut connections that link non-adjacent network layers, providing a straight path for identity mapping [100]. This feature prevents the gradient vanishing and gradient exploding issue in training deep CNNs and grants the model a more robust capability to extract information from the 2D structure.

The output arising from ResNet follows a time-distributed RNN layer connected to a final predictive FCN stage. Due to the increased complexity of this network, the reported training requires approximately one week to converge. The trained network predicts the absorption spectra of silver structures. As a proof of concept, the authors test the model on a validation dataset containing 1000 sample points on the spectra between 800 nm and 100 nm, which reaches an MSE = 4.259 1 × 10⁻⁵. As shown in Figure 4D, vital characteristics such as peaks and valleys are fitted well with the simulation results. This work extends the design space from geometrical parameters to arbitrary 2D shapes by applying image processing networks.

An empirical principle in machine learning dictates that the higher the NN model complexity, the larger the dataset needed to obtain good prediction accuracy on untrained data [101]. This condition also applies to and challenges NN-based inverse nanophotonic design. To train a more sophisticated NN with linearly expanding layers, the size of datasets can increase drastically, requiring vast amounts of EM simulations. From this point of view, the computational cost of training deep NN scales up by requiring more resources than optimization-based methods. However, the main strength of the deep learning-based inverse design is transfer learning: transferring the knowledge acquired by the progressive solution of many numerical simulations into different tasks. In the future of NN-based inverse design with increased model complexity, we expect a more decisive role of transfer learning techniques in formulating the network model.

3.2 Generative models

An issue with discriminative models is that one trained network can only provide a single, one-to-one mapping between the design variables and the spectral response at the output. However, in optics, various distinct structures can achieve a specific response. Generative models address this issue by computing distributions of possible structures that achieve analogous target responses. The models start the training by sampling a ‘latent space’ constructed from known distributions of random vectors. In most cases, these probabilities are multi-dimensional Gaussian distributions, with enough randomness provided for the model to learn a complex projection from the latent space to the distribution of functional nanophotonic devices. In general, the generative models decode the sampled vector to produce design parameters. Different generative models implement diverse strategies for learning these complex projections.

Generative adversarial networks (GANs) [102] represent the most prominent class of deep learning generative methods developed during the past decade in various research fields, including image translation [103, 104], privacy protection [105], natural language processing [106]. We here summarize a generic schematic of conditional GAN (cGAN) [107] architecture implemented in nanophotonic inverse designs. The essential parts of a GAN comprise a generator G network, which provides the design, and a discriminator D, also known as the critic that evaluates the provided designs. In the context of inverse design, an additional simulator network S serves as an external unit for predicting the optical response from the generated design and measuring the pre-defined FOM corresponding to specific tasks. The models train the generator G and discriminator D, simultaneously and adversarially. As depicted in Figure 5A, the pioneering work of Liu et al. [108] defines z as the random vector sampled from latent space, and T as the conditional variables that indicate the target response, which is the transmission spectra measured across TE and TM polarizations in this work. Under the goal of modeling and designing photonic devices, the generator G produces structural designs G(z, T) resembling actual structures X in the provided dataset. In contrast, the discriminator D is trained to distinguish G(z, T) from X. To carry out this task, the discriminator D predicts a [0,1] ranged value l, which represents the probability of each structure being real (existing in the dataset) or fake (generated by G). The authors train the discriminator D to minimize the distance between l and the actual categories, while the objective of the generator G is to maximize this distance. These adversarial training goals finally converge to a Nash equilibrium [109], representing a balance between the productivity of G and the discriminative ability of D. At this stage, the generated design images G(z, T) share similar geometrical features with X in the training set, while D fails to identify them from the actual structures X. According to various design tasks, the simulator S is pre-trained on labeled datasets, including device structures and the corresponding simulation results (transmission/reflection/absorption spectra, scattering coefficient) in different polarizations and wavelength ranges. It calculates the FOM by predicting the optical response on the generated structure G(z, T). The algorithm then integrates the optimization of FOM into backpropagation and gradient descent of G layers. The generator can produce optimized designs at convergence with all necessary structural constraints and topologies shared in the dataset.

Figure 5:

Generative adversiral network (GAN) implemented in nanopotonic inverse design (A) GAN’s generic schematic applied in nanophotonic design, which includes a generator, discriminator(also known as critic), and simulator. The generated device pattern is further optimized for fabrication by a smoothing procedure. Adapted from [108]. (B) A GAN-based silver structure design under arbitrary reflection spectra. The panel shows nine examples of device binary images generated by GAN and the corresponding reflection spectra as condition (solid line) and predicted response (dash line). Adapted from [110]. (C) Basic training strategy of GAN in the design of optical cloaks. The panel shows a successive data augmentation technique, where FEM simulations evaluate the newly generated geometries to complement the initial dataset. (D) Left: average and minimal scattering coefficient among the top-1000 configurations generated by the network. Right: the scattered field in an optimized 2-dimensional optical cloak. (C) and (D) are adapted from [112].

Rho et al. [110] proposed an alternative deep cGAN model for predicting reflection spectra of silver nano-antennas. The cross-sectional design space of the surface reflectors is limited in a square of 500 nm × 500 nm, represented by 64 × 64 binary images. FDTD simulations provide reflection spectra with 200 frequency points sampled from 250–500 THz. This work further optimizes the generator network architecture by constructing two separate convolutional blocks for the feature extraction of the random vector z and conditional variables, specifically, user-defined reflection spectra. Then the model fuses the output of two convolutional blocks into one 1024-channel tensor and eventually produces the design image. Instead of building an independent simulator model shown in Figure 5A, the FOM is measured directly from FDTD simulation on generator-suggested designs. In the early stage of training, the generator produces chaotic distributions of pixel values. As the training proceeds, the output images gradually show precise geometric shapes without additional regularization. Figure 5B shows examples of generated designs. The solid black spectra are the design objectives corresponding to the dataset’s structural images (black). Moreover, the structures drawn in red are the GAN-optimized designs, which share a similar topology yet comprise deformations such as erosion, dilation, and mirroring. The mean absolute error (MAE) between the reconstructed reflection spectra (red dash line) and simulated spectra is 0.0322. This result reveals GAN’s strength in modeling one-to-many mappings. Furthermore, the authors test the model on user-defined reflection spectra generated by curve functions. Despite the non-existing nature of such reflection patterns, the model achieves a <5% MAE between the reconstructed and the user-defined spectra, revealing a good approximation. An advantage of GANs is to explore the latent space implicitly, with seed vectors sampled from pre-defined probability distributions [111]. The generated randomness helps the networks find equivalent structures that differ from the training set, further enriching the existing dataset. Recently, Blanchard et al. [112] proposed a successive training strategy employing GAN-based data augmentation for the design of dielectric optical cloaks. The design space is binary images of a split ring resonator (SRR) with a central PEC circular boundary. When the incident light propagates through the cloak, the wavefront remains identical. This work defines a relative scattering coefficient to measure the difference between background field and scattered field. Similar to the architecture described in Figure 5A, the proposed approach comprises a generator, a discriminator, and a forward simulator that is pre-trained to predict scalar scattering coefficients. The authors freeze the weights of the forward network during GAN training while its output participates in the backpropagation. Minimizing the relative scattering coefficient demands the forward loss to be calculated simply by the distance between the predicted scalar value and constant 0. A total number of 13,000 FEM simulation results initialized the dataset, where the cloaking performance varies according to the shell geometries.

Figure 5C illustrates the unsupervised data augmentation technique applied every 60 iterations, where the forward simulator first evaluates newly generated configurations. The 1000 best geometric structures with the lowest scattering coefficients are simulated using FEM and added to the dataset. The model then utilizes the updated dataset to refine the prediction accuracy of the forward simulator and subsequently train the GAN. As a consequence of such a joint training strategy, GAN-generated geometries converge to a stage with minimum scattering coefficients by backpropagation, with high-quality samples increasingly dominating the dataset. Figure 5D exemplifies the learning curve of the average and the best scattering suppression at every 60 epochs. The right panel visualizes the normalized H_z field of the optimal configuration, on which the ratio of the scattered field to background field reaches 0.0089, implying almost uniform propagation.

Ma et al. [113] introduced the concept of conditional variational autoencoder (cVAE) [114, 115] in the inverse design of reflective metasurfaces. Depicted in Figure 6A, the cVAE contains three correlated sub-models, respectively, recognition model, generation model, and prediction model. The authors train the prediction model on simulated data. After convergence, the model can predict reflection spectra that cycle into the other two models as additional information. Acting as the ‘encoder’, the recognition model projects the input geometry to a 20-dimensional space by assigning mean value μ and variance σ to the pre-defined Gaussian distribution. A re-parameterization technique is applied to sample latent vectors from this probability density function, described as follows:

where i follows a standard Gaussian distribution. The latent vector z is then decoded by the generator model to reconstruct the input geometry, given the reflection spectra as conditional variables similarly to the GAN training scheme mentioned above. The training proceeds to show the formation of a clear unit cell pattern with accurate optical response in the range 40–100 Thz. VAE is a highly structured model, where adjacent points in the latent space correspond to continuous change on device structures. Exploring the converged Gaussian distribution reveals three groups of devices in the scatter plot of Figure 6B, respectively, cross, split-ring, and H-shaped nanostructures. The authors further characterize the reflection properties by locating the most robust resonance and dividing the nanostructures into two types with resonance frequency under and above 60 THz. As depicted in the right panel of Figure 6B, different types of spectra distribute among all clusters. This result demonstrates that prior-defined device form factors can be exploited in the design while each form sufficiently covers a wide range of optical responses. Figure 6C shows an example of generated structures under given conditions, which include TE to TE (R _xx ), TM to TM (R _yy ), and TE to TM (R _xy ) reflection spectra. The required response contains two resonance valleys on R _xx and R _yy , respectively. Two generated structures show in the below panels, where their numerical simulations show major agreement with the required spectra at resonant points.

Figure 6:

Variational autoencoder (VAE) assisted nanopotonic design (A) a c-VAE based metamaterial design system, including a prediction model, recognition model, and generation model. (B) Visualization of the reduced latent space, where three types of geometries and two-types of optical responses are depicted on the plot. (C) Examples of on-demand metamaterial reflector design. Two equivalent structures are generated under the given spectra condition. (A), (B) and (C) are adapted from [113]. (D) Examples of broadband power splitters with 2.25 × 2.25 μm² footprint. A 550 nm working bandwidth is achieved for all devices with arbitrary splitting ratios. Adapted from [116].

In the second example of cVAE-assisted inverse design, Tang et al. [116] developed a set of nano-patterned power splitters with user-defined split ratios. The devices are silicon-based square couplers with a footprint of 2.25 μm × 2.25 μm, which distribute 400 independent etching hole positions across the square area. The configurations of etching holes on the coupler is a 20 × 20 vector scaled in the range [0,1], each variable indicating the absence (<0.3) or a rescaled diameter size (>0.3) of one etching hole. The dataset contains 15,000 simulated transmittances of randomly generated hole vectors. Two ideally flat transmission spectra constitute the dependent variables to achieve the desired split ratio of the output ports. Above the original c-VAE architecture with a latent space, the authors introduced the adversarial block [117], an alternative branch of NN that validates the sampled latent vector z. This network estimates a diversity of optical responses from points across the latent space distribution, following the fact that the actual response of a power splitter is commonly irregular and not smooth as the ideal condition. The network obtains the highest performance by forcing the latent space distribution to generate various yet effective samples that correspond to the defined split ratio. Figure 6D exemplified two prototypes generated by the model, with splitting ratios of 6:4 and 8:2, respectively. The optimized design achieved over 87% total transmission efficiency across the working bandwidth from 1250 nm to 1800 nm. The training in [116] also implements an active learning method, where data augmentation based on FDTD simulations operates after each stage of training. Besides VAE, other autoencoder-like models exclude the random sampling process by directly reducing the input dimensionality to a fixed-length hidden vector. This lightweight model also accelerates nanophotonic applications with low computational burden, such as the design of reflective metasurface [118] and photonic topological insulators [119].

Illustrated in the above examples, the main advantage of generative versus discriminative models is producing novel data. Generative models learn the representation of the data structure and can supplement and refine the training set. Among various application fields such as text generation [120], texture filling [121], and image translation [104], the representation-learning of generative models also show significant improvements for inverse design tasks. Initially trained on a dataset containing nanostructures and optical properties, the converged network produces an optimized design distribution given certain conditions. The typical approach of external validation is EM simulation, which labels the generated design with the corresponding spectral response and eventually contributes to enhancing the original dataset. Besides this, direct experimental measurement on the fabricated devices also serves as an effective data generation method, compared to the intense computational demand of numerical simulations.