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Abstract
Multilayer metagratings have strong wavefront manipulation capabilities and find important applications in beam splitters. Traditional methods rely on the phase gradient design of generalized Snell’s law, which can achieve highly efficient beam splitters with uniform energy distribution. However, designing arbitrary energy distributions in different channels under two orthogonal polarizations remains a challenge because it requires more complex structures to modulate the energy flow. In this work, we employed a hybrid evolutionary particle swarm optimization (HEPSO) from the combination of particle swarm optimization (PSO) and genetic algorithm (GA) which has a strong ability to find the optimal structures that satisfy the specific energy flow distributions. We used the crossover and mutation operators of GA to improve the global search capabilities, and the velocity updating formula of PSO to replace the selection operator of GA to avoid local optimization. Using this approach, we successfully designed a uniform beam splitter with an efficiency of over $90{\%}$ and two beam splitters with arbitrary energy distributions, achieving an average error of about $0.5{\%}$. The optimal and average efficiencies obtained from running 10 optimizations are $2.2{\%}$ and $4{\%}$ higher than those obtained using PSO alone with 30 populations and 75 iterations. We envision that the proposed method can also provide an idea for other photonics design problems.
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1. Introduction
Beam splitters can split light into multiple beams in different directions and are widely used in many optical and photonic systems such as spectrometers, interferometers, optical communications, and quantum optics [1–5]. Traditional beam splitters include prisms [6,7], photonic crystals [8,9], and waveguides [10,11]. Prisms and photonic crystals are large and bulky optical systems that are not suitable for on-chip integration, while waveguides often have complex curved structures, making them more difficult to fabricate. In contrast, metagratings offer the advantages of small size, easy on-chip integration and fabrication, as they are capable of free wavefront modulation [12–16]. Metagrating is a periodic array whose structure is variable in the grating plane. As a result, metagratings have attracted considerable attention in the field of beam splitters, serving as an important area of research and application [17–22].
The development of metagrating beam splitters has focused on achieving polarization insensitivity, high efficiency, and high uniformity [17,19,22–24]. Zhou et al. [17] designed an Ag metagrating on a ${\rm SiO}_{2}$ substrate to achieve a four-channel beam splitter with an efficiency of more than $24{\%}$. Huang et al. [18] designed a TiO2 metagrating with a ${\rm SiO}_{2}$ spacer layer on an Ag substrate, getting a four-channel beam splitter with an efficiency greater than $24{\%}$ and a five-channel beam splitter with an efficiency greater than $19{\%}$. These are designed to deflect the beam based on the phase gradient calculated from the generalized Snell’s law [25]. While designing beam splitters with arbitrary energy distribution in different channels remains a challenge. The main problem lies in the need for complex structures to modulate the energy flow distribution, and this flexible regulation capability is of great importance in photonic chip integration applications [26,27]. Inverse design, which is commonly used in photonics, allows the design of structures based on complex and specific targets [28–33], thus enabling the modulation of the energy flow distribution. Zu et al. utilised the particle swarm optimization (PSO) to achieve beam splitters that manipulate arbitrary polarization and energy distribution in the one-dimensional case [22]. PSO is an optimization algorithm based on swarm intelligence, which simulates the information sharing and cooperative searching behavior of bird flocking to find the optimal solution [34]. If a more complex structure is needed, PSO requires a larger number of populations and iterations and the optimization results are often limited by the initial values and prone to falling into local optima.
In this work, we designed a five-channel beam splitter with arbitrary energy distribution under different polarizations based on a hybrid evolutionary particle swarm optimization(HEPSO), which combines PSO and genetic algorithm (GA). The device has 23 degrees of freedom and it consists of an infinite array of fully dielectric quasi-3D subwavelength structure (Q3D-SWS) with unitary structure. Q3D-SWS composed of a purposely designed one-dimensional multilayer film and a two-dimensional dielectric metagrating separated by a dielectric spacer [15,16]. It is much easier to manufacture than the three-dimensional (3D) structure. Meanwhile, due to the restrictions imposed by the structure, it cannot be used to modulate the electromagnetic wave in the x,y,z direction as a 3D structure. Therefore we call it a quasi-3D structure [35,36]. Q3D-SWS not only exhibits negligible absorption loss, but also eliminates transmission loss through the use of multilayer films. Our team has achieved anomalous deflection over $99{\%}$ efficiency at both single wavelength [35] and broadband [36], demonstrating that Q3D-SWS has a strong ability to modulate energy flow in three channels. Here, we use a more complex structure to achieve arbitrary energy distribution in four channels under TE polarization and in five channels under TM polarization. The use of PSO or GA alone presents challenges such as susceptibility to local optima, long iteration times, and suboptimal optimization results. To address these issues, we integrated GA operations, including crossover and mutation. The crossover operator was used to increase population diversity, while the adaptive mutation operator provided the ability to escape local optima. We adjusted the mutation coefficient throughout the optimization process to ensure the convergence of the algorithm. In addition, the natural selection process of the GA was replaced by the speed-up operator of the PSO to prevent the algorithm from converging quickly to local optima. We first designed a beam splitter with uniform energy distribution that splits into four beams under TE polarization and five beams under TM polarization, achieving an overall efficiency of more than $90{\%}$. Then, we further designed two beamsplitters with arbitrarily different energy distributions for TE polarization and TM polarization, and the total efficiencies of these beamsplitters were $77.22{\%}$ and $91.91{\%}$, respectively, with an average efficiency error of about $0.5{\%}$.
2. Methods and materials
2.1 Structure and simulation methods
We employ the Q3D-SWS to achieve arbitrary energy distribution beam splitting, as shown in Fig. 1(a). This structure consists of a specially designed multilayer film and dielectric pillar structure, which has been successfully used to achieve $100{\%}$ perfect anomalous reflection at optical frequencies [35]. The grating structure can be divided into three main parts, with a single period shown in Fig. 1(b) and a sectional view shown in Fig. 1(c). The first part consists of an array of Si material pillars, while the second part is a phase layer consisting of ${\rm Si},{\rm SiO}_2$ film layers, labeled in Fig. 1(c) as $ds, A, B, C, D$. The third part is a reflection enhancing layer consisting of a ${\rm Si},{\rm SiO}_2$ multilayer film, labeled in Fig. 1(c) as $H$ and $L$. Thus the total multilayers can be written as $S(HL)^3HDCBAdS$, where $S$ represents the substrate. This all-dielectric structure not only exhibits no absorption loss at optical frequencies, but also eliminates transmission loss by using a highly reflective multilayer film. In addition, by adjusting the thickness of each film layer in the phase layer, the reflection phase and propagation phase of light can be manipulated, providing control over the multiple scattering of light within the Q3D-SWS and realizing the modulation of the energy flow distribution. In the first part of the column array, each periodic structure consists of $2\times 2$ small structures with dimensions of $p_x=p_y=p$, and the one quarter of the structure is shown in Fig. 1(d). We define each substructure as a superposition of two orthogonal rectangles, which allows a higher degree of design freedom, since the substructure can be either a complete rectangle or a cross. The two rectangles are determined by the duty cycle $f_1^i, f_2^i, f_3^i, f_4^i$ respectively, where $i=1,2,3,4$ denotes the $i$th substructure of the structure. $f_1^i,f_2^i$ determine one rectangle, and $f_3^i,f_4^i$ determine the other. When $f_3^i<f_1^i$ and $f_4^i<f_2^i$ or $f_3^i>f_1^i$ and $f_4^i>f_2^i$, one rectangle completely covers the other, causing the substructure to become a rectangle instead of a cross which greatly increases the freedom of design. Due to the symmetry of the designed structure, the $(\pm 1,0)$ or $(0,\pm 1)$ order diffraction efficiency is identical.
Fig. 1. Schematic diagram of quasi-3D subwavelength structure. (a) 3D structure, (b) unit view, (c) sectional view and (d) top view of the one quarter of the structure, where $i=1,2,3,4$ denotes the $i$th substructure of the structure.
Among the theoretical algorithms for solving periodic electromagnetic fields, Rigorous Coupled Wave Analysis (RCWA) is an effective method [37,38]. Since the designed structure exhibits periodicity, we use the RCWA solver toolkit in our self-developed software called WGallop [39] as the electromagnetic field solver. The structure is an infinite periodic array in both $x$ and $y$ directions, and to ensure convergence and feasibility of design optimization, a total of 12 harmonics are considered in the $x,y$ directions. The refractive indices of ${\rm Si},{\rm SiO}_2$ and substrate materials are $3.5,1.48,$ and $1.51$, respectively. The incident electromagnetic wave has a wavelength of $1550nm$, and the diffraction efficiencies are calculated for ${\rm TE},TM$ polarization at each order. See Supplement 1 for more details.
The metagrating we designed consists of a significant number of 23 optimization parameters. This large number of parameters results in longer computation times for each calculation. In addition, the increased number of optimization parameters expands the solution space, making it difficult to obtain the ideal structure in a single optimization round. To address these challenges and improve efficiency, we propose a strategy that uses the optimal solution from the previous round as the initial value for the subsequent optimization round.
2.2 Hybrid evolutionary particle swarm optimization
PSO and GA are widely used metaheuristics that use stochastic global search algorithms to simulate the evolution of organisms in nature or the behavior of flocks of birds foraging for food. These algorithms have gained increasing popularity in the design of super-configurable metagrids for various functions, such as efficient absorption of solar energy spectra [40], perfect absorption of broadband spectra [41], highly efficient large-angle beam steering [42] and guided-mode resonance grating filter [43]. However, these optimization approaches are typically used with fewer variables [18,40] or are limited to one-dimensional metagratings [22,42], resulting in faster computation. In contrast, the structure we are currently designing is more complex, with a greater number of optimization variables, and involves two-dimensional metagratings that require more computationally intensive calculations. As a result, traditional algorithms struggle to solve such problems and often experience long stagnation in the initial stages, leading to unsatisfactory results. Therefore, using an optimization algorithm with superior performance can significantly improve design efficiency. In general, hybrid intelligent algorithms outperform single algorithms in solving complex problems [44]. To address this issue, we propose a hybrid evolutionary particle swarm optimization (HEPSO), as shown in Fig. 2.
Lin et al. [34] proposed an improvement to the parameters in the PSO by assigning different weights to the inertia factor $w$ and learning factors $c_1 ,c_2$ for individual particles based on their ranking in the population after sorting by fitness. This modification allows better exploration of a larger solution space to find the global best solution, while allowing rapid convergence of bad solutions to the vicinity of good solutions, thereby accelerating the convergence speed. Inspired by this idea of sorting, we introduce a dynamically changing factor into the weight factor of each particle to improve the local search capability in the later stages of the algorithm. In our proposed algorithm, all particles at time t are sorted based on their fitness values, where the smaller values represent worse solutions and the larger values indicate better solutions. The inertia factor and the learning factor for the kth particle (sorted) are given by:
where $w_1=0.4,w_2=0.5,w_3=0.4$, $c_a=0.5,c_b=2$. $N=30$ is the number of populations, and $max\_iter=75$ is the maximum number of iterations,The position vector of the particle at t times $x_{i}=\{x_{i1}^t,x_{i2}^t,x_{i3}^t, \ldots x_{id}^t\}$, $(1\leq i\leq N)$, speed vector $v_{i}=\{v_{i1}^t,v_{i2}^t,v_{i3}^t, \ldots v_{id}^t\}$, $(1\leq i\leq N)$,where $d$ is the number of optimization variables.The best position of the ith particle at time t is $Pbest$, and the best fitting particle at time t is $Gbest$, $r_1,r_2$ is a random number from 0 to 1.Then the updated equation for the velocity and position of the kth particle is:Crossover and mutation are fundamental operators in GA. To improve the global search capability of the algorithm, we introduce crossover and mutation operations at specific iteration steps within the framework of PSO. Building on the previous concept of sorting individuals based on their fitness, we divide the population into two groups: the top $30{\%}$ and the bottom $30{\%}$. The crossover operation is performed only on individuals within the latter group, with these individuals serving as Parent 1. Parent 2 is selected from the top $30{\%}$ and provides superior genes for crossover while remaining unchanged. The resulting offspring from the crossover process replace Parent 1 for subsequent optimization. Individuals with relatively lower fitness have a higher probability of undergoing crossover, allowing them to escape unfavorable solution regions more efficiently. As the population converges in later stages, the effectiveness of the crossover operator in improving the global search capability diminishes, leading to a decrease in the crossover probability with each iteration. Crossover is performed when t is divisible by 5. The crossover procedure for the kth particle and the corresponding crossover probability are as follows:
Mutation manipulation is a way to increase population diversity and prevent the algorithm from converging to local optima. To achieve this, we apply a mutation operation to individuals in the population after the top $30{\%}$ based on fitness. The mutation is performed by generating a new random particle in both position and velocity space. Individuals with lower fitness have a higher probability of being mutated, and this probability increases with each iteration. The mutation operation is performed when t is divisible by 3. The probability of mutation for the kth particle is given by
We eliminate the natural selection process of the GA, as it can cause the algorithm to converge to a local optimum prematurely. Instead, we implement the velocity update formula from PSO, which assigns importance to the optimal individuals in the population rather than enforcing their retention.
The figure of merit (FoM) function we used during the optimization is:
3. Results and discussion
3.1 Optimization results
In this work, we developed three different multifunctional beam splitters, each with different efficiencies, as shown in Fig. 3. These beam splitters have polarization-dependent characteristics. Specifically, when the incident light is TM polarized, it is split into five beams. Conversely, when the light is TE polarized, the incident beam is split into four beams. The use of this beam splitter enables parallel processing of laser operations and scanning laser projection, thereby increasing the speed of laser processing [22].
Fig. 3. Beam splitter optimization results at 1550 nm. (a) Spectra of a uniform energy beam splitter. (b) Spectra of a beam splitter with an energy ratio of 1:2:1 at TM polarization and 3:1 at TE polarization. (c) Spectra of a beam splitter with an energy ratio of 3:5:2 at TM polarization and 2:1 at TE polarization. (d) Efficiency of the three designs.
The primary design goal of the first structure is to achieve high efficiency uniformity in the individual beam splitting channels, ultimately resulting in a significant overall diffraction efficiency, as shown in Fig. 3(a). Since the bandwidth of the simulation is only 10 nm, the chosen structure exhibits minimal dispersion, and therefore does not significantly impact the final results. As a result, we used a constant refractive index. This design goal is consistent with the focus of most articles in the field [19,23,24]. Under TM polarization, the diffraction efficiencies for the $(\pm 1,0)$ and $(0,\pm 1)$ orders are $18.52{\%}$ and $18.12{\%}$, respectively, while the diffraction efficiency for the $(0,0)$ order is $19.11{\%}$, resulting in a total diffraction efficiency of $92.39{\%}$. Similarly, under TE polarization, the diffraction efficiencies for the $(\pm 1,0)$ and $(0,\pm 1)$ orders are $23.38{\%}$ and $24.01{\%}$, respectively, with a diffraction efficiency for the $(0,0)$ order of $0.02{\%}$, resulting in a total diffraction efficiency of $94.78{\%}$. These outstanding design results for the high-efficiency uniform energy distribution beam splitter effectively demonstrate the capabilities of our Q3D-SWS and HEPSO in rapidly searching for the desired structure.
The second and third structures demonstrate the ability to achieve arbitrary energy ratios for a beam splitter under different incidences of polarized light. The design objective of the second structure is to achieve a ratio of 1:2:1 for the diffraction efficiencies of $(\pm 1,0)$ order, $(0,\pm 1)$ order, and $(0,0)$ order when TM-polarized light is incident, and a ratio of 3:1 for the diffraction efficiencies of $(\pm 1,0)$ order and $(0,\pm 1)$ order when TE-polarized light is incident. The designed structure achieves a diffraction efficiency of $10.73{\%}$ for the $(\pm 1,0)$ order, $20.99{\%}$ for the $(0,\pm 1)$ order, and $10.73{\%}$ for the $(0,0)$ order when TM-polarized light is incident, meeting the requirements of a 5-channel beam splitter with a 1:2:1 efficiency ratio. When TE-polarized light is incident, the diffraction efficiency is $30.05{\%}$ for the $(\pm 1,0)$ order, $9.73{\%}$ for the $(0,\pm 1)$ order, and $0.71{\%}$ for the $(0,0)$ order, meeting the requirements of a 4-channel beam splitter with a 3:1 efficiency ratio. The design results are shown in Fig. 3(b), with an average error of about $0.4{\%}$. Similarly, the third structure is designed to achieve an energy ratio of 2:1 when TE polarized light is incident and 3:5:2 when TM polarized light is incident. The efficiency of the designed structure is $15.28{\%}, 24.66{\%}, and 9.94{\%}$ for the $(\pm 1,0)$, $(0,\pm 1)$, and $(0,0)$ orders, respectively, under TM-polarized light incidence. Under TE polarized light incidence, the efficiencies are $30.11{\%}, 16.08{\%}, and 1.62{\%}$ for the $(\pm 1,0)$, $(0,\pm 1)$, and (0.0) orders, respectively. The design results are shown in Fig. 3(c), with an average error of about $0.5{\%}$. The structural parameters information is in the Supplement 1.
Figure 4 shows the distribution of the electric field strength in the xz plane for the third structure under TM polarization, with the magnetic field perpendicular to the xz plane, and under TE polarization incidence, with the electric field perpendicular to the xz plane. Figures 4(a)-(b) show the field distributions at $y=\frac {p}{2}$ and $y=-\frac {p}{2}$ for TM polarization, while Figures 4(d)-(e) show the field distributions at $y=\frac {p}{2}$ and $y=-\frac {p}{2}$ for TE polarization. These figures show that the Q3D-SWS multilayer film efficiently reflects the electric field, confirming the effectiveness of our highly reflective film design. Furthermore, the light excites a robust electric field upon entering the structure, which mainly couples with the grating layers to achieve in-plane modulation. The light then passes through the multilayer film for interlayer phase modulation, achieving lateral modulation of the energy flow distribution which can be easily observed in Figs. 4(a)-(b) and Figs. 4(d)-(e). Most of the diffraction order energy is emitted from the top of the grating, and different coupling strengths between the electric field and the structure lead to different diffraction order energy distributions. Figure 4(c) and Figure 4(f) show the far-field diffraction orders. Figure 4(c) corresponds to the TM-polarization incidence and confirms that the energy is split into five channels with an efficiency of $30.11{\%}$ for the $(\pm 1,0)$ order, $13.77{\%}$ for the $(0,\pm 1)$ order, and $10.77{\%}$ for the $(0,0)$ order. Figure 4(f) corresponds to TE-polarization incidence and shows the energy segmented into four channels, with efficiencies of $31.67{\%}$ for the $(\pm 1,0)$ order and $18.08{\%}$ for the $(0,\pm 1)$ order, which is very close to our design result. The fields of the first and second structure are distributed in the Supplement 1.
Fig. 4. The electric field distribution of the third design results. TM polarization: (a), Electric field distribution at y=p/2. (b), Electric field distribution at y=-p/2. (c), Diffraction order distribution. TE polarization: (d), Electric field distribution at y=p/2. (e), Electric field distribution at y=-p/2. (f), Diffraction order distribution.
3.2 Algorithms comparison
We performed a comparison between HEPSO and the original algorithm, as shown in Fig. 5. Due to the large number of optimization parameters in this problem and the considerable time required for each computation, it is not feasible to use a large number of populations and extensive iterations simultaneously to achieve the desired results. Therefore, we adopt a strategy of substituting the optimal solution from the previous round as the initial value for subsequent optimizations, which allows us to achieve the desired results more efficiently. In the algorithm comparison test, we selected 30 populations, performed 75 iteration steps, and repeated each algorithm 10 times. After extensive testing, we found that the GA did not produce better results than the initial value. Therefore, we only present the comparison between HEPSO and PSO. Figure 5(a) shows the average optimization results and the $95{\%}$ confidence intervals of the 10 optimizations performed for the three structures. The obtained optimal and average efficiencies were $2.2{\%}$ and $4{\%}$ higher, respectively, than those obtained using PSO alone. Figures 5(b)-(d) show the convergence mean curves and the standard deviation filled regions for the optimized results. The two algorithms have the same number of calculations per iteration. HEPSO performs additional crossover and mutation operations that take significantly less time than the simulation. The total number of 30 particles used by both PSO and HEPSO remains constant, so both methods require 30 simulations per iteration, making their time consumption equivalent. Thus, there is a linear relationship between the number of iterations and the time consumption, and only the correlation between the algorithm and the number of iterations needs to be examined. It is evident that the HEPSO significantly outperforms the PSO when an iterative initial value is incorporated, indicating its superior search capability. Incorporating initial values into the optimization process is a commonly used technique when tackling complex optimization problems, such as the subsequent wideband optimization. This further demonstrates the superiority of our designed algorithm in solving complex problems. See Supplement 1 for more details.
Fig. 5. (a) Comparison of means and $95{\%}$ confidence intervals of HEPSO and PSO. (b-d) Convergence mean curves and standard deviation filled regions for HEPSO and PSO with the number of iterations utmost to 75 when optimizing structures 1, 2, and 3.
3.3 Bandwidth optimization
In practical, the performance of the beam splitter can be affected by wavelength drift. Conventional lasers typically have a bandwidth of $\pm$10nm. However, in the previous design, we only considered a single wavelength beam splitter operating at 1550nm, as shown in Fig. 3. To account for practical applications and demonstrate the optimization capability of HEPSO in a multi-objective broadband scenario, we uniformly sampled 11 points in the range of 1545-1555nm (a bandwidth of 10nm) and simultaneously optimized their efficiencies, as shown in Fig. 6. To improve the efficiency of the optimization process, we used the results obtained at a single wavelength as the initial values for the broadband optimization. Similar to the single wavelength optimization, we used 30 particles and performed 75 iterations. The FoM is
Fig. 6. Results of broadband beam splitter design for 1540-1560nm. (a) Spectra of a uniform energy beam splitter. (b) Spectra of a beam splitter with an energy ratio of 1:2:1 at TM polarization and 3:1 at TE polarization. (c) Spectra of a beam splitter with an energy ratio of 3:5:2 at TM polarization and 2:1 at TE polarization. (d) Efficiency of the three designs.
Figure 6(a) shows the result of the uniform energy beam splitter design. A noticeable improvement in the smoothness of the spectral curve is observed compared to the results of the single wavelength design, which ensures improved broadband performance. In addition, under TM polarization incidence, the diffraction efficiencies for the $(\pm 1,0)$ and $(0,\pm 1)$ orders are $19.10{\%}$ and $18.96{\%}$, respectively, while the diffraction efficiency for the $(0,0)$ order is $17.82{\%}$, resulting in a total diffraction efficiency of $93.92{\%}$. Similarly, when TE polarized light is incident, the diffraction efficiencies for the $(\pm 1,0)$ and $(0,\pm 1)$ orders are $23.70{\%}$ and $22.77{\%}$, respectively, with a diffraction efficiency for the $(0,0)$ order of $1.18{\%}$, resulting in a total diffraction efficiency of $94.13{\%}$. These results are comparable to those of the single wavelength design.
Similarly, as shown in Fig. 6(b)-(c), the design results for arbitrary energy beam splitters exhibit superior broadband performance compared to the single-wavelength design results. In the second designed structure, under TM polarized light incidence, the diffraction efficiency is $19.84{\%}$ for the $(\pm 1,0)$ order, $8.9{\%}$ for the $(0,1)$ order, and $10.21{\%}$ for the $(0,0)$ order. For TE polarized light incidence, the diffraction efficiency is $29.99{\%}$ for the $(\pm 1,0)$ order, $9.32{\%}$ for the $(0,1)$ order, and $0.52{\%}$ for the $(0,0)$ order. Similarly, in the third structure, under TM polarization, the diffraction efficiencies are ${\rm 12}.67{\%},23.95{\%} ,9.95{\%}$ for the $(\pm 1,0)$, $(\pm 1,0)$, and $(0,0)$ orders, respectively. Under TE polarization, the diffraction efficiencies are $30.00{\%},14.70{\%} ,0.02{\%}$ for the $(\pm 1,0)$, $(\pm 1,0)$, and $(0,0)$ orders, respectively. All design results significantly improve the overall broadband performance while maintaining the efficiency of a single wavelength at 1550 nm, validating the multi-objective optimization capability of HEPSO. The structural parameters information is in the Supplement 1.
4. Conclusion
In conclusion, we developed a hybrid algorithm called HEPSO and used it to realize a multilayer metagrating beam splitter with arbitrary energy distributions at 1550 nm for different polarization incidences. Specifically, the beam splitter has four channels for TE polarized incidence and five channels for TM polarized incidence. First, we designed a beam splitter with uniform energy distribution, achieving an overall efficiency of more than $90{\%}$. Then, we designed two non-uniform beam splitters with ratios of 3:1 and 2:1 for TE incidence, and 1:2:1 and 3:5:2 for TM incidence, with average design errors of about $0.4{\%}$ and $0.5{\%}$, respectively. Finally, to address practical considerations, we optimized the broadband beam splitters within a bandwidth range of $\pm$10 nm, significantly improving their broadband performance while maintaining high efficiency at 1550 nm. Our results demonstrate the exceptional performance of HEPSO in multi-parameter optimization scenarios and broadband optimization problems, requiring fewer populations and iterations. Compared to a single optimization algorithm, HEPSO achieved a $2.2{\%}$ improvement in optimal values and a $4{\%}$ improvement in average values over ten optimizations using 30 populations and 75 iterations. By calculating the electric field, we found the ability of HEPSO to modulate the energy flow distribution, showing great promise for solving other metagratings optimization problems.
Funding
National Key Research and Development Program of China (Grant No.2022YFF0604802); National Natural Science Foundation of China (61621001, 61925504, 6201101335, 62020106009, 62192770, 62192771, 62192772); Science and Technology Commission of Shanghai Municipality (17JC1400800, 20JC1414600, 21JC1406100); Major Projects of Special Development Funds in Zhangjiang National Independent Innovation Demonstration Zone, Shanghai (Grant No. ZJ2021-ZD-008); Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare no conflict of interest.
Data Availability
All data needed to evaluate the conclusions in the paper are present in the main text and the supplemental document. The datasets generated and analyzed during this study are available from the corresponding author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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