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Abstract
In recent years, metasurface has received significant attention because of its outstanding ability to manipulate electromagnetic waves at a subwavelength scale. In this paper, combined with an adjoint method, the multi-objective level-set optimization of geometric phase metasurface for high-efficiency beam deflection is exploited. It is shown that the geometric shape of the discrete metasurface is optimized towards a continuous shape. The final quasi-continuous metasurface achieves the absolute efficiency improvement from 22.63% to 76.08%; meanwhile, the polarization conversion efficiency maintains above 95.00% during the whole optimization process. An important consideration during the structural design is the tolerance of the structural parameters’ fluctuation in the manufacturing process. We investigate the robust metasurface optimization by geometrically considering contractive and expansile devices directly into the algorithm. The results suggest that when absolute efficiency exceeds 60.00%, compared with non-robust design, the acceptable edge deviation range of robust optimized metasurface is enhanced from 13 nm to 17 nm. Furthermore, this method can also be extended to the optimization design of other mico-/nano- optical devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurfaces consisting of a series of subwavelength units arranged periodically or aperiodically on a two-dimensional plane are the focus of current research due to their advantages of thin thickness and easy processing. By designing the size, shape, and arrangement of each meta-atom, the metasurface can freely modulate the amplitude, phase, and polarization of electromagnetic waves at the subwavelength scale, which opens up a new way for electromagnetic wave manipulation [1,2]. Nowadays, subwavelength metasurfaces have been employed for a myriad of applications in broadband achromatic metalens [3–5], polarization optics [6–8], holography [9–13], quantum photonics [14], and broadband spin Hall effect [15,16], etc.
The geometric phase [17–19] also known as Pancharatnam-Berry phase plays an important role in wavefront control [20–23] because of its simple control mode and broadband dispersion-free characteristics. The geometric phase can be divided into discrete geometric phase and continuous geometric phase according to the arrangement of meta-atoms. In general, the former realizes phase coverage by discretely arranging several meta-atoms with fixed rotation angle difference [24], while the latter is spatially continuous and the generated geometric phase shifts are also continuous [25]. Comparatively, continuous structures can suppress interference between discrete adjacent meta-atoms and result in the advantages of broadband and high-efficiency according to the catenary optics [26,27]. Recently, metasurface based on catenary and its deformation have been widely used in various optical devices, including beam deflectors [28,29], metalens [30,31], and Bessel beam generators [32], etc. However, the parameters of catenary structure are achieved depending on experience or parameter scanning. The design degree of freedom is limited and the obtained catenary structure is not always close to the global optimal solution. Thus, there is an urgent need for an automatic metasurface design method with global optimization capability.
Fortunately, the inverse design procedure can solve the above problems, which is less time-consuming, more convenient, and has larger design space. It is usually guided by optimization algorithms [33], either gradient-based approaches (for example, topology optimization and level-set method) or evolutionary approaches (such as genetic algorithms and particle swarm algorithms). Gradient-based approaches have been extensively exploited in multiple areas because of their fast optimization speed, such as optical proximity correction [34], wide-angle deflector [35,36], and depth-of-focus metalens [37]. Among them, topology optimization need to blur and binarize, which is laborious. By contrast, as a boundary optimization method, the level-set optimization can operate directly on the binary structure, which is more advantageous and can be directly used for fabrication.
In this paper, we elaborate on the adjoint-based multi-objectives level-set optimization for high-efficiency beam deflection. Specially, only two simulations are required to compute the derivative of an objective function with respect to the entire infinite-dimensional space. When a discrete geometric phase metasurface is used as the initial structure, the finally optimized metasurface has a catenary-like quasi-continuous distribution which can improve the absolute efficiency remarkably. Indeed, the absolute efficiency of Si metasurface is optimized from 22.63% to 76.08%. Then, the robust metasurface design is investigated using the same strategy on this basis. These three patterns considering edge deviations by truncating blurred optimized metasurface still manifest as a catenary-like shape with the absolute efficiency from 16.00%, 22.68%, and 34.01% to 67.15%, 73.40%, and 70.22%, respectively. The acceptable edge deviation range of robust optimized metasurface is enhanced from 13 nm to 17 nm compared with a non-robust design. All results verify the optimization capacity and convergence efficiency of the proposed method.
2. Multi-objective level-set optimization
2.1 Initial structure
The initial discrete geometric phase metasurface is shown in Fig. 1. This structure is composed of an array with eight rectangular pillars separated by a SiO2 substrate, each of which is individually rotated multiples of a certain angle (here is 22.5°). A left circularly polarized (LCP) light is utilized as illumination source, which is normally incident to the metasurface along the –z-axis. This geometric phase metasurface transforms LCP to right circularly polarized (RCP) light in the -1st diffraction wave along x-direction. In order to facilitate comparison, all parameters of the initial discrete geometric phase metasurface are consistent with [35]. According to the generalized Snell’s law [38], the resulted beam deflection angle can be calculated by plugging known conditions into Eq. (1),
After getting all the parameters of the initial discrete geometric phase metasurface, the corresponding model is built in finite-difference time-domain (FDTD) software and the absolute efficiency for RCP light is calculated, which is 22.63%.
Fig. 1. Initial discrete geometric phase metasurface for high-efficiency beam deflection based on multi-objectives level-set optimization. The green rectangular pillars are Si with n = 3.47 and the light yellow substrate is SiO2 with n = 1.45. The incident beam is LCP light and the deflected beam is RCP light. The deflection angle is approximately -67.3°.
2.2 Optimization frame
To realize high-efficiency beam deflection, the adjoint-based multi-objectives level-set optimization method is adopted to design the metasurface inversely. Meanwhile, the gradient ascent method is applied to maximize the figure of merit (FOM). Firstly, we determine the cost function according to the purpose of optimization. Subsequently, the adjoint method is introduced to calculate the effects of the structure changes on the cost function, and the evolutionary gradient of boundary is calculated through two simulations of forward field and adjoint field. Finally, the level-set method is employed to update the metasurface until the iteration stop condition is met. During the optimization process, a binary pattern is used for iterating, where 1 represents Si and 0 represents air. The refractive index distribution of the metasurface will be updated continuously as the number of iterations increases.
To ensure the geometric phase properties of the optimized metasurface, the final FOM requires to consist of two items, including absolute diffraction efficiency and polarization conversion efficiency. Here, the absolute diffraction efficiency is defined as the energy ratio between the transmitted light at a target diffraction order (-1st order) and the incident light. The cost function of absolute diffraction efficiency is expressed as
where E represents the electric field complex amplitude of transmitted light of -1st diffraction order. Furthermore, the polarization conversion efficiency is defined as the complex amplitude inner product of the transmitted light (-1st order) and the RCP state. The cost function of polarization conversion efficiency is expressed asThe partial derivative of final FOM for the electric field E is
The introduction of adjoint method makes the optimization gradient can be calculated from the forward electric field amplitude E(x) and the adjoint electric field amplitude ${{\boldsymbol E}^A}(x )$ by running only two simulations. Among them, ${{\boldsymbol E}^A}(x )$ is expressed as [35]
The level-set method is a powerful tool for modeling time-varying objects. In level-set optimization, a directed distance function is used to represent the change of object and the zero level set means the boundary of an object. Now, we reformulate the inverse metasurface optimization problem by finding an optimal level set function that seeks to maximum FOM. To solve this problem, an iterative scheme is applied by deriving a time-dependent evolution equation for the level set function ϕ. Once the gradient is obtained, then the upwind difference calculation method is utilized to update the zero level set and the corresponding target pattern. The updating formula of level set function [40] is:
where k is iteration number and S means evolution velocity.Herein, all of the simulations and calculations are implemented on our workstation with Intel Xeon Gold 6256 CPU, 3.60 GHz, and 512 GB of RAM. Full-field electromagnetic wave calculations are performed using a commercially available simulation software, Lumerical FDTD solutions. The transmission is recorded with an analysis group placed upon the source and the electric field distributions are detected by 3D field monitors. In the simulation models, the boundary condition is perfectly matched layers (PML) at z direction, periodic boundary conditions in forward field simulations and Bloch boundary conditions in adjoint field simulations at x and y directions, respectively. The illumination source (LCP) is composed of two plane waves with the same amplitude, orthotropic polarization and a phase difference of 90°. The wavelength is 1.55 µm. Of course, the whole process can also be performed by the rigorous coupled-wave analysis (RCWA) solver RETICOLO [41].
2.3 Results and analyses of the optimized Si metasurface
The convergence condition of FDTD simulations is set as 1e-4. During the multi-objectives level-set optimization process, there are two simulations will be run each iteration including forward field simulation and adjoint field simulation, which takes about 150 seconds. Therefore, it takes about 8.5 hours to complete 200 iterations. The optimization results of Si metasurface are presented in Fig. 2. From Fig. 2(a), one can see that the absolute efficiency gradually improves from 22.63% to 76.08% after the whole iteration process, meanwhile, the polarization conversion efficiency maintains a high level with above 95.00%. As shown in Fig. 2(b), the FOM also gradually improves with the increment of iteration numbers. Figure 2(c) depicts that the geometric shape gradually evolves from discrete structures to continuous ones. What’s more, besides an isolated rectangle substructure on the left of the unit metasurface, the optimized pattern is similar to a catenary structure.
Fig. 2. The optimization results of Si metasurface. (a) Convergence curves of absolute diffraction efficiency and polarization conversion efficiency. (b) Evolution of FOM during the optimization process. (c) Optimized Si metasurface shapes in different iterations.
In the following, we analyze the performance of optimized Si metasurface for wide-angle and high-efficiency beam deflection. Figure 3(a) is the top view of the optimized continuous geometric phase metasurface. The catenary-like shapes are formed after the periodic arrangement of multiple optimized unit metasurfaces. Figure 3(b) shows the electromagnetic simulation result of Re(E) in the xoz plane. It is clear that most transmitted light deflects from the -1st diffraction order. The deflection angle is approximately -67.3°. The following is the performance comparison of the metasurfaces before and after optimization. As shown in Fig. 3(c), at the wavelength of 1550 nm, the original Si metasurface has only 22.63% absolute efficiency for RCP light while the efficiency is 76.08% for optimized metasurface. In addition, the LCP component is well suppressed which suggests that the geometric phase characteristic is well maintained during multi-objectives level-set optimization process. As expected, the optimized quasi-continuous metasurface has high absolute efficiency (above 60.00%) in the band of 1.5 µm -1.6 µm.
Fig. 3. Performance analysis of optimized continuous Si metasurface. (a) The catenary-like geometric phase metasurface is composed of multiple optimized unit structures. (b) The distribution of Re(E) in the xoz plane. The black dotted box represents the region of the metasurface. (c) Absolute efficiencies of transmitted LCP and RCP components for original and optimized Si metasurface.
In this section, the multi-objectives level-set inverse optimization method is used to realize the high-efficiency beam deflection. We note that the performance of the quasi-continuous structure is superior to the discrete structure by the above analysis. The whole optimization process achieves good convergence, and compared with previous work [35], the absolute efficiency of our optimized structure is 11.01% higher than that of the literature. Moreover, under the same design requirements, we also obtain a catenary metasurface with equal width through parameter scanning in FDTD solutions. The typical parameters are presented in Table 1. The simulation result shows that the absolute efficiency for RCP light is 60.49% and our optimized catenary-like metasurface is 15.59% higher than it.
Table 1. Parameters of catenary metasurface with equal width
3. Robust design of Si metasurface
3.1 Modeling edge derivations
One of the challenges hindering the metasurface from theoretic design to actual application is the nanostructures fabrication. Some process parameters fluctuation, such as over- and under-dosing, and over- and under-etching, may greatly affect the performance of optical devices. To represent geometric variations intuitively, the process disturbances that are geometrically contractive and expansile relative to the ideal pattern are considered during the robust metasurface design. The perturbations here just represent the edge derivations of a nanostructure.
Firstly, the refractive index distribution of the ideal pattern is defined as P(xi), xi represents position. The basic parameters are the same as in section 2.1. It is truncated with different thresholds after blurring the ideal pattern to obtain the patterns considering the change of process parameters. A blurred initial pattern $\tilde{P}({{x_i}} )$, is produced by convolving P(xi) with the Gaussian distribution [36],
In our optimization, the blur radius σ, is set as 20 nm. To quantify the degree of shrinkage and expansion with a single parameter, the edge deviation Δ, which is relevant to the blurring and thresholding parameters, is calculated by this equation [36],
where η is the threshold to truncate blurred patterns. The edge deviation represents the distance that the boundary of target pattern will shift. Devices with negative and positive edge deviations correspond to contractive and expansile patterns, respectively.The contractive and expansile patterns $\bar{P}(x )$, are generated by thresholding the blurred pattern:
Fig. 4. Schematics of the metasurface considering the change of process parameters. (a-c) The three graphs show contractive, ideal, and expansile patterns with edge deviations of -10 nm, 0 nm, and +10 nm, respectively. The Si and air are shown in green and light yellow colors, respectively. The dashed red line in each graph represents the contour of the ideal device.
3.2 Results of robust metasurface optimization
The initial structure is also a discrete geometric phase metasurface. The goal is to maximize the FOM, which is the same as section 2.2. For every iteration, forward and adjoint simulations are run to calculate the gradient at each pixel of the device. The gradient update results in the change of the refractive index distribution of the device and the FOM is improved gradually.
In order to reduce device sensitivity to edge deviations, the geometric errors are taken into account. The flow chart of the entire iteration is shown in Fig. 5. There are three gradients corresponding to contractive, ideal, and expansile patterns in each robust optimization iteration. The three gradients are combined into a synthetical gradient to update the level-set function and optimized pattern. The final gradient is computed as,
The weights wq, can be tuned and there is 1 for every gradient.
Fig. 5. Flow chart of the robustness deflection device optimization implementation. During each iteration, the simulations of forward and adjoint fields for the contractive, ideal, and expansile patterns will be run and produce three gradients. The final gradient for updating the level-set function is the weighted sum of three gradients. At last, the optimized pattern is updated by the upwind finite difference method.
In the multi-objectives level-set optimization that does not consider robustness in section 2.3, the gradient of every iteration is computed for only the ideal device pattern. However, in the optimization process of robust design, the gradients for ideal, contractive, and expansile patterns are all calculated to comprehensively consider the effects of different process conditions.
Figure 6 shows the optimized results of robust Si metasurface. The optimized metasurface also has a catenary-like quasi-continuous distribution, as shown in Fig. 6(a). From top to bottom of Fig. 6(d), the three panels represent optimized Si metasurface for contractive, ideal, and expansile patterns, respectively. The three patterns are very similar to that optimized without considering robustness. This means that Eq.13 is suitable for our optimization and the final gradient is valid for every kind of metasurface optimization with different edge deviations. Figure 6(b) is the evolution of FOM over the optimization process. The red, orange, and blue lines are the results of contractive, ideal, and expansile patterns, respectively. The three lines all show good convergence of proposed optimization algorithm. As shown in Fig. 6(c), the calculated absolute efficiencies of the devices with or without robustness optimization as functions of edge deviation are plotted. For the non-robust optimized device, a peak efficiency at edge deviation of +1 nm (the boundary of the metasurface expands outward by 1 nm relative to the ideal metasurface through theoretical calculation) is 74.24% while the efficiency diminishes to some extent as the edge deviation changes. The final gradient considers comprehensively the gradient variation of different process parameters when given robustness optimization, thus the performance of optimized device is more stable when the process parameters change. The multi-objectives level-set optimized device with robustness maintains an efficiency above 60.00% for edge deviations ranging from -8 nm to +9 nm while the optimized device without robustness is only from -4 nm to +9 nm. Its peak efficiency at edge deviation of +1 nm is 72.90%, which is lower than the maximum efficiency of the non-robust device. This difference is attributed to the incorporation of robustness constraint and the peak efficiency will be further reduced if we consider robust optimization for larger ranges of edge deviation.
Fig. 6. The results of robustness optimization based on multi-objectives level-set method. (a) Optimized metasurface. (b) Convergence diagram of FOM in the optimization process. Red, orange, and blue lines are the results of contractive, ideal, and expansile patterns, respectively. (c) The absolute efficiencies of robust and non-robust optimized metasurfaces for differing edge deviations. (d) From top to bottom, the three panels represent optimized Si metasurface for contractive, ideal, and expansile patterns, respectively.
This section takes robustness optimization into account by geometrically varying devices in every iteration. The optimized results suggest that the device optimized for robustness is indeed less sensitive to edge deviations than that optimized for non-robustness. More importantly, if multiple process parameters would change in actual production, a weighted gradient can be used to comprehensively consider the influence of different process parameters on the device performance.
4. Conclusion
In conclusion, the multi-objectives level-set optimization algorithm combined with adjoint method of geometric phase metasurface for high-efficiency beam deflection is investigated and a robust metasurface optimization is proposed considering the edge deviations in the fabrication process. We have successfully observed the transformation process from discrete geometric phase structure to continuous structure. Optimized catenary-like continuous geometric phase metasurface achieves a large increase in absolute efficiency without affecting the polarization conversion efficiency. We further verify that the performance of the continuous geometry phase metasurface is better than that of the discrete geometry phase metasurface, and also confirm the advantages of catenary structures. Moreover, compared with the catenary structure with an equal width, the metasurface obtained by global automatic optimization is a more satisfying solution for the goal of this paper. Moreover, we apply the same strategy and achieve robust optimization considering the fabrication errors by truncating blurred patterns with different thresholds. The results show the optimized metasurface with robustness is more tolerant of edge deviations compared with that without robustness design. The proposed method can also be applied to other fields of inverse optimization design, especially the introduction of adjoint method makes it easier to solve the systems which are difficult to model mathematically.
Funding
West Light Foundation of the Chinese Academy of Sciences; National Natural Science Foundation of China (62192773); Sichuan Province Science and Technology Support Program (2022YFG0001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon request.
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