Broadband achromatic metalens design based on predictive neural network and particle swarm optimization-genetic algorithm

The main challenge in designing broadband achromatic metalenses is to achieve the desired phase distribution at different wavelengths and positions. Finding the exact relationship between phase modulation and the size or shape of individual nanopillars is a critical but time-consuming step. This paper presents a novel joint design framework predictive neural networks and particle swarm optimization-genetic algorithms, which combines predictive neural network (PNN) and particle swarm optimization-genetic algorithm (PSO-GA). The proposed framework aims to accurately predict the phase response of nanopillars using PNN, increase the number of phase data points to establish a one-to-one correspondence between the phase and nanopillar parameters in the design of broadband achromatic metalenses, and optimize the parameters of an individual nanostructure of the metalens using PSO-GA. To validate the efficacy of the proposed method, a broadband achromatic metalens for line polarization light in the range of 260–350 nm is designed. Numerical simulations demonstrate that the designed metalens exhibits achromatic focusing. The method proposed in this paper may find wider application in the design of more complex metasurface devices.


Introduction
Recently, there has been considerable interest in metasurfaces composed of subwavelength features, which offer the advantage of being simple to integrate into optical devices due to their smaller size and simpler design.Metasurfaces have garnered significant research attention due to their remarkable ability to manipulate the phase, amplitude, and polarization of incident waves in a highly flexible manner [1][2][3].Numerous ultra-compact nanoscale optical devices have been developed, including tiny spectrometers [4], gratings [5], beam deflectors [6], polarizers, and metalens [7][8][9][10].In the field of imaging research, metalenses with small imaging capabilities have drawn a lot of attention among all applications for planar optical elements.
Metalenses are thought to be the most efficient optical components to replace traditional lenses, but they retain the same aberration problem as conventional lenses, rendering achromatic metalenses crucial for imaging and display [11].Achieving achromatic metalenses requires the careful arrangement of nanostructured units to compensate for dispersion and satisfy the wavelength and position-dependent phase distributions of the metalens [12].Traditional methods, such as parametric scanning using numerical simulations including finite-difference time-domain (FDTD) and finite element method, are commonly employed for selecting suitable nanostructures.It has been pioneered to design achromatic metalenses in the ultraviolet (UV) band, as well as multifunctional metalenses [13][14][15][16].However, due to the extensive computational requirements for designing broadband metalenses, traditional methods are less effective and adaptable [17].How to improve the focusing performance of achromatic metalenses while efficiently decreasing the time and computing resources needed is a crucial problem in the design of broadband achromatic metalenses.
Deep learning has become a key technique for solving automatic prediction and decision making of huge amounts of information in artificial intelligence, and is widely used in the study of photonics and optics [18][19][20].Recently, the combination of deep neural networks and genetic algorithms has been used in the design of nanophotonic devices such as metalenses [21], absorbers [22], and orbital angular momentum generators [23].Deep neural networks containing multiple hidden layers can efficiently learn the hidden relationships between the parameters of an individual nanopillar structure and electromagnetic responses, replacing or supplementing traditional simulations to determine the phase modulation capability of different parameters of an individual nanopillar structure at a much faster rate [24].However, previous methods for metalens design based on deep learning require obtaining a substantial number of phase datasets at individual wavelengths across a continuous wavelength range [17,25].The time-consuming process of this data acquiring process may result in extended preparation and training periods for the dataset.
In this paper, we develop a method to design broadband achromatic metalenses by combining a predictive neural network and a particle swarm optimization-genetic algorithm predictive neural networks and particle swarm optimization-genetic algorithms (PNN-PSO-GA).The real and imaginary parts of the phase response of a single wavelength nanopillar structure are used to generate the dataset, which is utilized for the phase response at broadband wavelengths based on the trained predictive neural network.To optimize the structural parameters of the broadband metalens, a particle swarm optimization-genetic algorithm (PSO-GA) is employed to enhance the accuracy of selecting nanopillars with the assistance of the predictive neural network [26,27].To validate the effectiveness of our method, an UV broadband achromatic metalens is designed to operate within the wavelength range of 260-350 nm.Numerical experimental results demonstrate that the designed achromatic metalens using our proposed method exhibits excellent achromatic performance across the whole broadband range.Additionally, as compared to traditional design methods, the proposed method offers the benefits of low average focal error and high design efficiency.

Methods
The main method for designing broadband achromatic metalens involves coordinating the phase of the metalens at each spatial location to address both chromatic and spherical aberration.In order to achieve the desired focusing effect, the phase distribution of the metalens can be represented under the normal incidence of incident light as [28]: In equation (1), the variables f and λ represent the designed focal length and operating wavelength in free space, respectively, and r = √ x 2 + y 2 indicates the distance between the unit cell and the center of the achromatic metalens.To achieve both broadband achromatic properties and efficient focusing of a multi-wavelength beam to a single position, it is necessary for the achromatic metalens to maintain the desired phase across all wavelengths simultaneously.However, due to the constant nanopillar structure, compensatory phase adjustments need to be introduced to account for variations in the phase response at different operating wavelengths.This adjusted phase profile can be represented as follows: ( In equation ( 2), the compensatory phase C (λ) is determined by the operating wavelength.Figure 1 illustrates a schematic diagram of the achromatic metalens.
To design an efficient achromatic metalens for the UV region, this paper utilizes aluminum nitride (ALN) as the nanopillar material.ALN is a commonly used UV optical material known for its high transmittance.The nanopillars are fabricated on an aluminum trioxide (AL 2 O 3 ) substrate.The H-shaped and I-shaped nanopillars are specifically chosen to achieve a 0 ∼ 2π phase delay in the achromatic metalenses.The nanopillar design in this paper consists of three rectangular nanopillars, forming an H shape and an I shape.This predetermined shape allows for the adjustment of four geometric parameters: y max , y min , x max , and x min , providing greater freedom in phase control.The combination of the four parameters determines the propagation phase, and the structure of the nanopillar is schematically illustrated in figure 2(a).The period (P) of the nanopillar must be less than λ/(2NA) in order to satisfy the Nyquist sampling requirement.Therefore, the period (P) in this study remains constant at 200 nm.In this study, the height (H) of the nanopillar is fixed at 600 nm taking into account the requirement for phase coverage 0 ∼ 2π [16].It is crucial to accurately determine the nanopillar configuration at each location during the design process.Previous research on UV achromatic metalenses has mainly relied on the FDTD method to create a database and identify the optimal nanopillars.However, this method is time-consuming and computationally demanding [13,16].Figure 2 provides an overview of the used prediction method to efficiently forecast the phase response at different wavelengths.
To incorporate different-shaped nanopillars, an additional parameter called 'shape' is introduced, with a value of 1 for H-shaped nanopillars and 2 for I-shaped nanopillars.The phase response of the nanopillars is predicted by constructing a database using five parameters: y max , y min , x max , and shape.For H-shaped nanopillars, the ranges of y 1max −y 1min and y 1min are set to 0.0015-0.0415µm and 0.015-0.055µm, while the ranges of x 1max −x 1min and x 1min are set to 0.02-0.06µm and 0.02-0.03µm.The y 2max −y 2min and y 2min ranges for I-shaped nanopillars are adjusted to 0.02-0.06µm and 0.02-0.03µm, and the x 2max −x 2min and x 2min ranges are set to 0.01-0.04µm and 0.015-0.055µm.The wavelength range is set to 260-350 nm.
The phase and transmittance of the transmitted light passing through the nanopillar are determined using the FDTD commercial software package.The real and imaginary portions of each wavelength transmission coefficient, rather than the amplitude and phase, are chosen as the PNN output in this study due to the issue with phase hopping on 0 and 2π .A dataset consisting of 1950 data pairs of structural parameters and the corresponding real/imaginary phase components is constructed.The process to generate the dataset can be found in appendix A. For the neural network model, the Tensorflow and Keras frameworks are utilized.The PNN consists of 1 input layer (y max , y min , x max , x min , and shape), 7 hidden layers, and 1 output layer (Re, Im).After each hidden layer, a batch normalization layer is added to prevent overfitting.The complete architecture of the PNN is depicted in figure 3.
A random sample of 20% of the simulated samples is chosen to serve as the test set, and the remaining samples are used as the training set.The rectified linear unit (ReLU) activation function is used during training, and the Adam optimizer is used to update the weights and bias values.The mean square error (MSE) serves as the loss function to evaluate the performance of the PNN.Before training, the four structural parameters are normalized to ensure consistent scaling across all parameter values.This normalization process scales the parameters to a range of (0, 1), removing any discrepancies in their original values.During training, the weights of each hidden layer are constantly updated over 5000 epochs until the loss function reaches a desirable minimum.The optimal PNN hyperparameters and ablation study are displayed in appendix B. Once the PNN is trained, it can be used to predict the real and imaginary components of the phase for any given nanopillar parameter at a specific wavelength.Furthermore, the initial dataset can be rapidly expanded using both the FDTD method and the trained PNN.This allows for the efficient generation of additional data for further analysis and improvement of the model's performance.
The design of the achromatic metalens involves selecting the nanopillar with the correct phase and group delay from an extended library of the phase.To achieve this, a combination of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) is employed in this study [29].In order to maximize the benefits of both optimization algorithms, the primary objective of PSO-GA is to expand the capabilities of the algorithms by combining genetic and particle swarm algorithms.By incorporating genetic operators into the regular PSO, the hybrid algorithm combines the high convergence efficiency and accuracy capability of PSO with the local search capacity of GA.The method is implemented step-by-step to find the complete resolution using a hybrid PSO-GA method that complements one another and yields improved efficiency.The total phase error between the desired phase and the real phase is the objective function in this study.Figure 4 depicts the flowchart of the PSO-GA optimization algorithm.
Five steps make up the full process of designing an achromatic metalens.First, the initial design parameters are set, such as the material, period and height of the unit nanopillar, as well as the diameter and focal length of the metalens.The creating of the dataset, which consists of two different nanopillar shapes, is explained in appendix A. Second, the nanopillar dataset is used to train the PNN.The detailed training and tuning process is described in section 2 and appendix B of this paper.Third, the optimal compensation phase is calculated using a hybrid optimization algorithm.In the fourth step, equation ( 2) is used to calculate the target phase at each position on the metalens, and the nanopillar with the smallest phase difference from the target is selected as the best nanopillar from the expanded database.Finally, by placing the selected nanopillars in various positions on the metalens, the design of an achromatic metalens can be finished.
The design process of the PNN-PSO-GA framework can be described as follows.
(1) Set the initial design parameters and generate a dataset based on the designed wavelength range.
(2) Use data from numerical simulation to train PNN.
(3) Calculate the compensating phase C(λ) term in the formula for a metalens' phase distribution using a hybrid PSO and GA optimization algorithm.(4) Select the nanopillar from the extended database with the minimum phase error.

Experiment A: the results of the proposed PNN
In this section, Experiment A aims to demonstrate the viability of the proposed method by showing the prediction performance of the trained predictive neural network (PNN).The study focuses on predicting the real and imaginary parts of nanopillars with various sizes and shapes at three wavelengths.The specific findings of these predictions are presented in figures 5(a)-(c).Furthermore, figure 5(d  To further demonstrate the accuracy of the predicted phase, the real and predicted phase of the random nanopillars in the test set at three wavelengths are selected in this paper, as shown in figures 5(e)-(g) showing the polar plots of the real and predicted phase.The majority of the predicted values align closely with the target values, with only a few instances showing slight deviations.These deviations, however, are generally consistent with the real values.In order to assess the overall performance of the neural network, the phase errors of the nanopillars from the validation set at five wavelengths are analyzed.Figure 5(d) illustrates that the phase errors for most nanopillars are below 6 • , indicating the high accuracy of the neural network's predictions.
Figures 6(a) and (b) depict the comparison between the original database, obtained through FDTD simulation, and the extended database predicted by the neural network.These figures aim to provide a deeper understanding of the phase database necessary for designing achromatic metalenses.The results demonstrate that the extended database is denser than the original one, indicating its improved ability to fulfill the phase database requirements in achromatic metalens design.To assess the effectiveness of the PNN optimization, the phase errors of the metalenses at different radial positions are analyzed before and after the optimization process.Figures 6(c) and (d) present the phase errors, highlighting the significant improvement achieved through the optimized network compared to the original database.In this paper, different optimal target phases are utilized for the original and extended databases, resulting in still a small percentage of phase errors at different positions of the expanded metalens that are bigger than the original metalens.However, compared to the phase error of the original database, the total phase error on the all position of the expanded achromatic metalens is lower.These findings emphasize that the optimized network yields more precise design results, showcasing its enhanced performance in achieving accurate phase control for achromatic metalenses.

Experiment B: Numerical experiment of achromatic focusing
To validate the proposed achromatic metalens design, we conducted FDTD simulations over a wavelength range of 260 nm to 350 nm.The focal length and diameter of the designed achromatic metalens are set as f = 10 µm and D = 8 µm.The Corresponding numerical aperture is NA = 0.371, which is calculated from the formula sin(arctan D 2f ). Figure 7(a) illustrates the simulated normalized light intensity distribution of the achromatic metalens in the propagation plane.It can be observed that the intensity distribution remains consistent across the wavelength range, indicating the achromatic behavior of the metalens.To compare the achromatic characteristics, a chromatic metalens operating at a single wavelength of 260 nm is also designed and simulated.Figure 7(b) displays the light intensity distribution of the chromatic metalens in the propagation plane, revealing significant chromatic aberration.In contrast, the achromatic metalens maintains a nearly constant focal length within the 260-350 nm wavelength range.The normalized intensity distribution along the x-directional cross-section of the achromatic metalens is depicted in figures 7(c) and (d), along with the distribution of light intensity at the focal point (in the x-y plane).The results demonstrate that the point spread function (PSF) at the focal point maintains good symmetry, and the focus at all wavelengths is nearly diffraction-limited, exhibiting minimal aberrations.
To further evaluate the performance of the achromatic metalens, we conducted numerical simulations using the FDTD method to determine the focal length, full width at half maximum (FWHM), and focusing efficiency of the proposed achromatic metalens in the direction of the x-axis.The focal length shift at various wavelengths is depicted in figure 8(a).For the chromatic metalens, the offset reaches 2.96 µm within the 260-350 nm wavelength range.In contrast, the achromatic metalens exhibits a significantly smaller offset of 0.59 µm.The average focal length of the achromatic metalens designed using the original database is 10.375 µm, and the average focal length of the expanded database is 10.256 µm, which is 1.19% smaller than the original average focal length.The variation of the average focal length is described by the formula ∆f/f, where ∆f is the difference between the mean focal length of the original and extended databases, and f is the focal length.The PSF at various wavelengths can be used to determine the FWHM of the point of focus.According to in figure 8(b), the FWHM for the remaining wavelengths is very close to the diffraction limit λf/D, with the exception of 260 nm, where there is only a slight divergence.The absolute and relative focusing efficiencies of the achromatic metalens are also calculated.The relative focusing efficiency is defined as the ratio of the energy within the focal region to the total transmitted power, while the absolute focusing efficiency is defined as the ratio of the power within a focal region with a width of 3 × FWHM to the total incident power [17].The average relative focusing efficiency of the proposed achromatic metalens reaches 64.59%, and the average absolute efficiency is 26.83%, as demonstrated in figure 8(c).These calculations and   analyses demonstrate the high-performance of the achromatic metalens, as it exhibits a minimal focal length shift, an excellent diffraction-limited focal spot, and a stronger focus effect.

Experiment C. Comparison of design efficiency
The necessity for an enormous database will make the design of achromatic metalens more difficult and time-consuming.However, the proposed method offers a solution by utilizing predictive neural networks to quickly learn the relationship between the parameters of nanostructures and phase response.This approach allows for the augmentation of the phase response database, enabling the identification of the most optimal nanopillar structure.The experiments conducted in this section aim to compare the efficiency of traditional design methods and the deep learning design methods [30,31].We counted the amount of time required to design this work, as displayed in table 1.To maintain consistency in the evaluation, several combinations of structural parameters are simulated, and the average simulation times are determined for 900 H-shaped and 1050 I-shaped nanopillars.
To compare the design efficiency between traditional numerical simulation method and the proposed methods, this paper calculates the average time required for generating nanopillars with different H and I shapes using predictive neural networks.Additionally, the duration of the entire design process is assessed.In this study, the FDTD method is utilized to generate approximately 1950 high-precision data points, sampling four wavelength points within the range of 260 nm to 350 nm.This data generating process took 33 h and 20 min to complete.
For each wavelength sample point, we used a predictive neural network to predict 29 282 data points, which took approximately 1 h to complete.In contrast, performing the same calculation using the traditional numerical simulation method would take approximately 503 h.This represents a 93% reduction in time compared to traditional methods, as indicated in the achromatic design column of table 1.To ensure experimental rigor, the same workstation equipped with an NVidia 3060Ti GPU, 128GB of RAM, and a sixteen-core CPU running at 3.2 GHz was used for all validations.The results in table 1 demonstrate that adopting a hybrid deep learning method not only reduces the amount of computational time saved but also minimizes the computational resources required for the task.

Conclusion
A combined design framework for PNN-PSO-GA is proposed in this paper.By leveraging predictive neural networks, the original nanopillar database is extended, and PSO-GA is utilized to optimize the structural parameters of the metalens.The use of deep learning methods provides enhanced design efficiency compared to traditional methods in the development of broadband achromatic metalenses.The paper demonstrates the design and numerical analysis of a broadband achromatic metalens for continuous 260-350 nm line polarization light.The experimental results indicate that the achromatic metalens performs well across the entire broadband range.These designed metalenses offer several advantages over traditional lenses, such as compact size, lightweight, and ease of integration, making them suitable for applications in UV laser, lithography, communication, and image sensor systems.Furthermore, the proposed method in this paper not only improves the design efficiency of metalenses, but also enables the design of components with different phase distributions with great flexibility, thus promising to be extended to any application for the optimal design of more complex metasurface devices.Reference Kanwal et al [32] Liu et al [13] Guo et al [16] Manchen et al [33] This

Figure 2 .
Figure 2. Flow chart of predicted phase response.(a) Dataset consisting of two nanocolumn structures.(b) Structure of the neural network, where the input layer is five parameters and the output layer is the real and imaginary part responses.(c) The real and imaginary part information of the phase are predicted by the PNN.
) displays the loss function curves during the training of the neural network specifically for the 290 nm wavelength.The training of the network has been finished when the losses of the validation and training sets converge to a constant value after 5000 epochs.

Figure 5 .
Figure 5. Prediction results and loss functions of PNN.(a)-(c) The real and imaginary parts of the phase of the random parameters of nanostructures predicted by PNN at 260 nm, 290 nm and 350 nm.Where the solid line represents the prediction and the dotted line represents the real value.(d) Training loss and validation loss of PNN.The loss is measured using the mean squared error (MSE) for the loss function.(e)-(g) Phase prediction of PNN at 260 nm, 290 nm and 350 nm wavelengths.(h) Histogram of the phase error between the real response values and the PNN predicted response values in the test set for the four wavelengths.

Figure 6 .
Figure 6.Phase distribution and radial phase error for both the initial database and the extended database.(a) The original FDTD simulation database.(b) The extended database predicted by the PNN.(c), (d) Phase errors before (blue line) and after (green line) optimized by PNN at different radial positions at 260 nm and 320 nm.

Figure 7 .
Figure 7. Normalized intensity distributions of (a) the proposed achromatic metalens and (b) a chromatic metalens in the axial plane (x-z cross section), where the black dashed line indicates the position of the focal plane; (c) intensity distribution and (d) intensity profiles at the focal point for a wavelength range of 260-350 nm.

Figure 8 .
Figure 8. Performance of the proposed broadband achromatic metalens: (a) focal length variation of the achromatic metalens at different wavelengths; (b) FWHM of the focal points; (c) focusing efficiency of the achromatic metalens at different wavelengths.

Table 1 .
Time comparisons between traditional numerical simulation and PNN-PSO-GA method.

Table C1 .
Summary of different metalens designs.
However, all of the publications in the above table are based on traditional numerical simulation methods for design.The design method presented in this work, which combines deep learning and intelligent optimization methods, is anticipated to increase the effectiveness of designing hypersurfaces for more intricately formed nanopillars.