Multi-Objective Optimization of Chiral Metasurface for Sensing Based on a Distributed Algorithm

We propose a distributed multi-objective optimization (DMO) method for designing chiral plasmonic metasurface that satisfies multiple objectives simultaneously. We aim to improve the refractive index sensitivity of the archetypical Born-Kuhn type chiral plasmonic metasurface while ensuring that circular dichroism (CD) is as pronounced as possible at a designated resonance wavelength. By leveraging distributed technology, the proposed method significantly improves the time efficiency of the optimization process. The simulation results demonstrate approximately 33% enhancement in sensitivity by DMO, as well as greater than 100% boost in time efficiency compared to stand-alone optimization approaches. These findings highlight the potential of the proposed method to guide the design of chiral plasmonic metasurface sensors, enabling the simultaneous optimization of multiple objectives and facilitating advancements in chiral optics and sensing applications.


Multi-Objective Optimization of Chiral Metasurface
for Sensing Based on a Distributed Algorithm Xianglai Liao , Lili Gui , Shulei Bi, Ang Gao, Zhenming Yu , and Kun Xu Abstract-We propose a distributed multi-objective optimization (DMO) method for designing chiral plasmonic metasurface that satisfies multiple objectives simultaneously.We aim to improve the refractive index sensitivity of the archetypical Born-Kuhn type chiral plasmonic metasurface while ensuring that circular dichroism (CD) is as pronounced as possible at a designated resonance wavelength.By leveraging distributed technology, the proposed method significantly improves the time efficiency of the optimization process.The simulation results demonstrate approximately 33% enhancement in sensitivity by DMO, as well as greater than 100% boost in time efficiency compared to stand-alone optimization approaches.These findings highlight the potential of the proposed method to guide the design of chiral plasmonic metasurface sensors, enabling the simultaneous optimization of multiple objectives and facilitating advancements in chiral optics and sensing applications.

I. INTRODUCTION
C HIRAL metasrufaces are photonic devices with unique chiral characteristics, composed of subwavelength-scale structural units.These units are precisely designed and arranged to achieve highly selective responses and elaborate control over light.Chiral metasurfaces exploit optical chirality, which refers to the asymmetric response of left-handed and right-handed excitation light.Circular dichroism (CD), a crucial chiroptical phenomenon representing the differential absorption when metasurfaces interact with incident light of different handedness, is employed to evaluate the performance of chiral metasurfaces, and it is influenced by the handedness and parameters of the chiral metasurfaces.Chiral metasurfaces have significant importance in various optical applications, including molecular recognition [1], biosensing [2], and optical information processing [3].Additionally, chiral metasurfaces offer several advantages for refractive index sensing applications, such as high sensitivity, a broad operation wavelength range, fast response, and high selectivity [4], [5].These features make them an effective and versatile solution for refractive index sensing.The intensity of CD at the resonance wavelength and the sensitivity to refractive index change are key for outstanding sensing performances, which are influenced by the structural parameters.Therefore, optimizing the structures of the chiral metasurfaces is essential to enhance these characteristics.
Heuristics algorithms (HAs), such as genetic algorithms (GAs) [6], [7], [8], particle swarm algorithms [9], and simulated annealing algorithms [10], have been recognized as valuable solutions for addressing the high computational demands of numerical simulations.HAs and their variations have been successfully employed in the design of optical devices, including plasmon biosensors [8], Fabry-Perot cavities [6], and nanophotonic routers [7], [12], among others.For instance, A. Mirzaei and colleagues applied genetic algorithms to optimize the scattering cross-section of plasmonic nanowires [13], while T. Feichtner and colleagues employed evolutionary algorithms to discover new binary pattern nanoantennas with strong light localization [14].Similarly, Z. Li and colleagues utilized genetic algorithms to design binary-pattern chiral plasmonic metasurfaces to achieve strong circular dichroism effects [15].
Although HAs have undoubtedly improved the efficiency of metasurface design, they are often tailored for specific individual objectives, which can be limited when metasurface devices need to fulfill multiple objectives simultaneously.Very recently, several research groups have delved into the realm of multi-objective optimization for photonic devices.P. R. Wiecha and colleagues applied evolutionary multi-objective optimization algorithms to design dielectric nanoantennas and maximize scattering at different wavelengths [16].S. Es-Saidi and colleagues combined multi-objective genetic algorithms and neural networks to optimize metal-dielectric diffraction gratings for specific color production [17].E. B. Whiting et al. proposed a multi-objective optimization method for freeform optimization of metasurfaces with arbitrary shapes, enabling the achievement of multiple functions [18].Additionally, M. Elsawy et al. presented a statistical learning-based global multi-objective optimization method for metalens design [19].However, research on the multi-objective design of chiral metasurfaces is currently limited.Moreover, the optimization process of chiral metasurfaces by HAs is still time-consuming due to more than thousands of numerical calculations, particularly as the solution space expands, posing challenges in terms of iteration difficulty and time constraints.
In this article, we propose an algorithm called the Distributed Multi-Objective Optimization (DMO) method for efficient design of chiral metasurfaces with multiple objectives for refractive index sensing tasks.Specifically, we utilize the Non-Dominated Sorting Genetic Algorithm (NSGA) [20] to address the multiple target problem.The DMO algorithm operates in two stages.In the first stage, NSGA interacts with a deep neural network (DNN) for forward prediction, generating a set of solutions in a smaller parameter space.These solutions serve as the initial population for the second stage, along with randomly generated individuals.The second stage employs a distributed approach to explore the vast parameter space and rapidly identify superior solutions, thereby searching for a new Pareto front.This method can also be applied to the design of other types of metasurfaces or photonic devices, with optimization goals not limited to the spectrum.Typical applications include imaging as indicated in [21] and [22].

II. STRUCTURE OF CHIRAL METASURFACE
We focus on optimizing a 3D chiral metasurface, the unit cell of which is depicted in Fig. 1.The archetypical Born-Kuhn type chiral plasmonic metasurface has been studied by us, as described in references [23], [24] and by others [25], since it provides clear physical insights and great potential in sensing applications.Reference [25] has already optimized the Born-Kuhn type chiral metasurface for chiral sensing.Our study here shares a commonality with reference [25] in that we both focus on Borh-Kuhn type chiral plasmonic metasurfaces.The objective in both cases is to optimize the design of structural parameters for this chiral metasurface to fulfill specified tasks.In contrast to the research outlined in reference [25], our work here differs in two key aspects.Firstly, the chiral metasurface we design possesses C 4 symmetry, and we do not directly employ a neural network for the inverse design task.Secondly, while reference [25] focuses on a single-target chiral sensing task, our work addresses a dual-target task intended for refractive index sensing.
In detail, the metasurface consists of periodic bilayer gold nanorods, which involves five variable geometrical parameters, namely the length (L) and width (W) of the nanorods, the distance (D) between the bottom of the top layer rods and the top of the bottom layer rods, and the gap (G) between two rods along the X or Y directions.Additionally, the period (P) of the structure is taken into consideration.In the beginning, the metasurface is enclosed by a dielectric spacer layer with a refractive index of 1.30.The CD response is here defined as differential transmission of the metasurface when excited by left-and right-handed circular polarization (LCP and RCP), as illustrated in Fig. 1.The valley and peak of the CD optical spectrum indicate two eigenmodes existing in this nanosystem, one of which can be preferentially excited depending on the handedness of the incident light.It is worth noting that the position of the two resonances (peak and valley) varies with the refractive index of the ambient medium, acting as the basic principle of plasmonic sensing.We should also mention that the great interest of this chiral plasmonic metasurface for sensing originates from the fact that the two resonances may exhibit distinct sensing capabilities, which need to be engineered and optimized by intelligent algorithms.

III. DISTRIBUTED MULTI-OBJECTIVE ALGORITHM
In this work, we only care about the left-handed Born-Kuhn metasurface, for the convenience of study.Our target is to maximize both CD response (for the best signal-to-noise ratio when recording in experiment) and sensitivity, important for practical sensing applications.Common heuristic optimization algorithms include genetic algorithms, particle swarm algorithms, and simulated annealing algorithms, among others.Reference [26] has compared these three algorithms, demonstrating the superiority of the genetic algorithm over the other two.In the design of chiral metasurfaces, since the time consumption is primarily attributed to electromagnetic calculations, there is minimal discrepancy in the efficiency of the three algorithms for optimal design.Moreover, genetic algorithms excel in parallel optimization and multi-objective optimization compared to other algorithms, requiring fewer adjustments of empirical parameters [26].Consequently, we employ the multi-objective genetic algorithm NSGA to explore the desired structure within the specified parameter space.To expedite the optimization process, we utilize a distributed method.It is important to note that among the multiple objective functions, there may exist certain trade-offs.In other words, improving one objective function may potentially compromise the performance of other objective functions.We refer to a solution that cannot be surpassed by any other solution in all objectives as a Pareto optimal solution [27].The goal of the multi-objective algorithm is to identify all these Pareto optimal solutions, acting as the Pareto optimal front.We define a dominance relationship, or a situation where solution A outperforms solution B in all objectives, as A dominating B. Within the obtained Pareto front, no single solution can completely dominate the others.Fig. 2 illustrates The distributed system consists of two components, namely, the Client and the Server group, as shown in Fig. 2. Client is responsible for obtaining the initial population and executing fundamental operations of NSGA, including fast non-dominated sorting, crowding degree calculation, selection, mutation, and crossover, as depicted in Fig. 3.The Client divides the population into multiple smaller populations and transmits population information to each Server within the Server group.The Servers then execute the FDTD algorithm and calculate the fitness function.Subsequently, the Servers transmit the fitness information back to the Client.The communication protocol between the Client and Servers is the Transport Control Protocol/Internet Protocol (TCP/IP).To prevent premature convergence of the population in the multi-objective genetic algorithm, we incorporate a restart mechanism.This mechanism entails retaining excellent solutions from the stable population while re-randomizing the remaining solutions.
Unlike single-objective optimization algorithms, which yield a unique solution, multi-objective optimization methods provide a set of solutions located on the Pareto front.In addition to the selection, crossover, and mutation operations, NSGA incorporates two crucial operations: a fast non-dominated sorting method and crowding degree calculation.The fast non-dominated sorting divides all solutions in the population into Pareto ranks based on the dominant relationship of each solution.The higher the rank, the greater the likelihood of being selected.The crowding degree calculation plays a crucial role in maintaining population diversity by evaluating the density relationship between each solution in the population and its neighboring solutions.This is accomplished by calculating the crowding distance, as demonstrated in (1) where o represents the o-th target, f i o is the i-th sorted solution of target o, and f o,max and f o,min denote the maximum and minimum values of the non-dominated boundary solutions, respectively.When computing the crowding distance, the average distance between a solution and its neighboring solutions on the left and right sides within the same non-dominated front is calculated.This process is performed for each target, and the distances are summed to determine the crowding distance of the given solution, as illustrated in ( 2) where n is the number of targets.
In the design process of a chiral metasurface for a single target, we have optimized the chiral metasurface using a genetic algorithm to maximize the absolute value of CD at 1035 nm [24].Differently, here we formulate the design of the chiral metasurface as a multi-objective optimization problem, as depicted in (3).
There are five parameters to be optimized, as illustrated in Fig. 1.C1 represents the range constraint for these five parameters, and C2 denotes the method used for calculating the period of the structure.The wavelength sensitivity is evaluated according to the following equation [29] S λ (nm/RIU ) = Δλ peak/valley /Δn, ( where Δλ peak/valley represents the spectral shift of the resonance peak or valley when there is a change of the refractive index of the ambient medium Δn.Additionally, the figure of merit (FOM) takes into account the sharpness of the spectrum and is defined as follows [30]: where FWHM is the full width at half maximum of the CD peak or valley.Both S and FOM are key characteristics for evaluating sensing performance of a plasmonic sensor.The DMO algorithm is shown in Algorithm 1.
The utilization of a multi-objective genetic algorithm is motivated by the need to address the optimization task of the chiral metasurface with multiple objectives.To enhance the efficiency of the optimization design, we employ two approaches.Firstly, we incorporate a pre-trained DNN forward prediction model from prior studies [28], replacing electromagnetic simulations in predicting the spectral response during the objective function calculation in the multi-objective genetic algorithm.However, owing to the non-zero error in the error function during neural network training, there is an inherent error in the calculated objective function.Consequently, the optimized outcome may deviate from the optimal solution.Therefore, it is essential to employ this optimized solution as one of the initializations, followed by another round of optimization using the distributed multi-objective algorithm.In this subsequent optimization, the fitness function is directly computed using electromagnetic simulation to obtain the spectrum.Due to the utilization of the solution optimized in the first stage as one of the initial subpopulations in the second optimization process, the convergence speeds up.

IV. RESULTS AND DISCUSSION
First of all, we aim to maximize the absolute CD value at both resonance wavelengths simultaneously.Then we could analyze and compare the sensitivity when detecting at these two different wavelengths.In detail, in addition to minimizing the CD at 1035 nm, we also target at maximizing the CD at 1200 nm.We implemented our distributed algorithm on four computers, all equipped with identical configurations, including Intel(R) Xeon(R) CPU E5-2678 v3 processors and 64 GB of RAM.Fig. 4 illustrates the results obtained through the DMO algorithm.In Fig. 4(a), a set of solutions (red dots) are obtained after the multi-objective optimization, where objective 1 represents the CD value at 1035 nm and objective 2 represents the CD value at 1200 nm.The initial population includes the random solutions (blue dots show part of them) and Pareto front obtained by NSGA and DNN (green dots).It is important to note that not all solutions on the Pareto boundary can yield excellent results for both targets.The multi-objective algorithm ultimately provides a set of solutions that do not dominate each other, including the desired solutions.We can choose the desired solution based on We then compare the sensitivity of the two resonances among the three structures obtained in Fig. 4, as presented in Table I.The table clearly shows that the resonance at longer wavelengths in general exhibits higher sensitivity to refractive index changes compared to the resonance at shorter wavelengths (typically,   4 25%-34% improvement with regard to S).In terms of the S and FOM comprehensive properties, one could select the structure C for the optimum sensing performances, considering that the CD signal at the peak wavelength of 1200 nm is intense enough (0.44).
With the assistance of DMO, we are able to further optimize the sensitivity while maintaining a strong CD at resonance.The valley and peak of the CD spectrum are separately chosen as the working wavelength to calculate the refractive index sensing.In this way several structures with high sensitivity are retrieved, as illustrated in Fig. 5.In detail, Fig. 5(a) and (b) are the CD spectra of the metasurfaces that achieve a high sensitivity and strong CD value at valley, while Fig. 5(c) and (d) take the peak of the CD spectra as the research object.The S maximum improves about 33% in Fig. 5(a) compared with the S without optimization [Fig.4(c), 649.00 nm/RIU].The maximum wavelength sensitivity in terms of S reaches up to 889.33 nm/RIU, as depicted in Fig. 5(d).And the maximum FOM can go up to 7.59 RIU-1, as demonstrated in Fig. 5(c).Both metasurfaces in Fig. 5(c) and (d) reach excellent S and FOM in addition to remarkable CD peaks, employing the longer resonance wavelength as the sensing band.The sensitivities of the optimized structures with regard to S are several times larger than those reported in fiber optical plasmonic sensors in [31] (S = 420.00nm/RIU), and [32] (S = 137.28nm/RIU).Besides, the S values of our designed chiral metasurfaces outperform that of the chiral metasurface in [4] and many achiral metasurfaces listed as Table I in [5], and the FOM is also comparable with [4].
We also conducted a test to assess the improvement in time efficiency using the distributed method, as outlined in Table II.

TABLE II OPTIMIZATION TIME FOR DIFFERENT NUMBER OF NODES
We calculated the time required for the algorithm to iterate ten generations using different population sizes (24,48,96).With more nodes involved in distributed computing, the time efficiency becomes better improved.We observe a roughly 1/N node scaling of the simulation time for the stand-alone case, where N node represents the number of nodes in the distributed system.The time efficiency improves about 100% (N node = 2) compared with the single node situations.If the number of samples is sufficiently large and the running time is extensive, the genetic algorithm theoretically converges to the global optimal point.Nevertheless, practical constraints, such as a high-dimensional parameter space, limited sampling, and restricted running time, make it challenging to ensure global optimality.To enhance the likelihood of reaching the global optimum, it is important to increase the population size and running generations.However, increasing the population size leads to longer search time.In such cases, the distributed method becomes even more important, as it helps to save time significantly.For example, in the case of a single machine, when the population size is 24, the required time is 3.55 hours.When the population size increases to 96, the time increases to 11.59 hours.In the case of distributed machines, when the population size is 24, the optimization time decreases to 0.79 hours, resulting in time saving of only 2.76 hours.However, when the population size is 96, the optimization time decreases to 2.93 hours, resulting in time saving of 8.66 hours.Therefore, for larger population sizes, it is evident that adding distributed nodes significantly improves efficiency.This method enables efficient design of the chiral metasurface to simultaneously optimize the sensitivity of the refractive index sensing and the CD intensity at resonance wavelength.The DMO is not only applicable to the design of the chiral metasurface discussed in this manuscript but is also well suited for other photonic devices that can be optimized using genetic algorithms and involve multiple objectives.Nevertheless, it is essential to acknowledge the limitation of the algorithm when in scenarios involving an extensive number of parameters or objectives.For instance, structures with hundreds of dimensional parameters and more than tens of objectives may pose challenges for this algorithm.

V. CONCLUSION
In conclusion, we have proposed a distributed multi-objective design method for chiral metasurfaces.The NSGA algorithm combines with the DNN and FDTD algorithms to search for the Pareto front of the multiple objectives in two stages.In the first stage, the DNN is utilized to calculate the fitness function; while in the second stage, the FDTD is employed to calculate the fitness function in a more extensive solution space.The distributed strategy is adopted to accelerate the optimization process, resulting in a roughly scaling of the simulation time needed for a single node.We have optimized the refractive index sensitivities regarding S and FOM of the Born-Kuhn type chiral metasurfaces, at valley and peak of the CD spectra respectively, in addition to maintaining prominent CD response.The sensitivity of S = 866.66nm/RIU, and FOM = 7.59 RIU-1, and a significant absolute CD value of 0.43 at the longer resonance wavelength is ultimately realized in a single metasurface.This method ensures high-quality results while minimizing the required time and can be extended to multi-objective design of other metasurfaces and nanophotonic devices.

Fig. 1 .
Fig. 1.Unit cell of the 3D chiral metasurface and the dimensional parameters needed to be optimized.

Fig. 2 .Fig. 3 .
Fig. 2. Architecture of the DMO method for the design of the chiral metasurface.

Algorithm 1 : 1 . 1 : 2 : 3 : 8 : 12 :
Distributed Multi-Objective Optimization Algorithm.First stage: Output: A set of Pareto front solutions R Initialize the population P, and P 1 = P. Initialize the crossover probability p c = 0.95, mutation probability p m = 0.1.While i < total_generation: 4: Calculate the fitness functions of population P 1 using DNN in [26].5: Fast non-dominated sort.6: Crowding degree calculation.7: Operations including selection, crossover, and mutation.Obtain the new population P e , and P 1 = P e .9: End while Second stage: Client: Output: A set of Pareto front solutions R 2 .10: Initialize the population P, and P 2 = P + R 1 .11: Initialize the crossover probability p c = 0.95, mutat0on probability p m = 0.1.While i < total_generation: Diverse the P 2 as N sub-populations, denoted as P 2 = {P s1 , P s2 , … P sN }, and separately transmits the P sN to server N. 13: Calculate the fitness functions of sub-population P S1 using FDTD.14: Receive the fitness from Server N. 15: Fast non-dominated sort.16: Crowding degree calculation.17: Operations including selection, crossover, and mutation.18: Obtain the new population P e , and P 2 = P e .19: End while Service N: 20: Calculate the fitness functions of population P sN using FDTD.21: Communicate with the Client through TCP, and transmits the fitness of the P sN to Client.specificrequirements.Fig. 4(b) shows the spectra of the leftmost solution A on the Pareto boundary in Fig. 4(a), which achieves a CD value of -0.50 for objective 1.However, its effect on objective 2 is the least favorable among the solutions on the Pareto front.Similarly, Fig. 4(c) corresponds to the transmission and CD spectra of solution B, where objective 2 achieves a value of 0.39.Likewise, Fig. 4(d) corresponds to the spectra of solution C. The absolute CD at 1035 nm is -0.30 and the CD value at 1200 nm is 0.44.

TABLE I COMPARISON
OF SENSITIVITY S AND FOM FOR STRUCTURE IN FIG.