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Abstract
We investigate the topology optimization of geometric phase metasurfaces for wide-angle and high-efficiency deflection, where adjoint-based multi-object optimization approach is adopted to improve the absolute efficiency while maintaining the polarization conversion characteristic of geometric phase metasurfaces. We show that, for the initially discrete geometric phase metasurfaces with different materials and working wavelengths, the topology shapes gradually evolve from discrete structures to quasi-continuous arrangements with the increment of optimization iteration operations. More importantly, the finally optimized metasurfaces manifest as catenary-like structure, providing significant improvements of absolute efficiency. Furthermore, for the initial structure with catenary distribution, the corresponding optimized metasurface also has a catenary-like topology shape. Our results on the topology-optimized geometric phase metasurfaces reveal that, from the perspective of numerical optimization, the continuous catenary metasurfaces is superior to the discrete geometric phase metasurfaces.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Subwavelength metasurfaces, composed of two-dimensional arrays of artificial atoms, have attracted considerable interests due to their compact nature and great abilities of manipulating electromagnetic wave at subwavelength [1–3]. Based on the elaborate design of each meta-atom, optical metasurfaces are capable of independent or simultaneous control of amplitude, phase, and polarization of incident light wave. Nowadays, optical metasurfaces have been widely employed as compact and flat optical elements for a wide range of applications in, for instance, broadband achromatic metalens [4–7], holography metadevices [8–10], broadband spin Hall effect [11,12], and polarization optics [13–15]. Specially, geometric (or Pancharatnam-Berry) phase [16,17] enabled by spatial anisotropic distribution of local phase delay plays an important role in polarization-dependent wavefront regulation.
According to distinct arrangement patterns of meta-atoms, geometric phase in optical metasurfaces can be intuitively divided into two categories, i.e., discrete geometric phase and continuous geometric phase. In general, the discrete geometric phase is acquired from discrete arrangement of subcells [18,19], while the spatially continuous distribution patterns generate continuous geometric phase shifts. In contrast with discrete structures, continuous structures can avoid the undesired resonances between adjacent subcells. Especially, one of the most important examples of continuous geometric phase is catenary optics [20,21], which is constructed with catenary function and has significant advantages in both bandwidth and efficiency compared with its discrete counterparts. Recently, catenary metasurface has found tremendous applications in photonic spin-orbit interaction [11] and high-efficiency flat optical devices [22–24]. Although the polarization-dependent phase regulation mechanisms for both discrete-shaped structures and continuous-shaped structures (i.e. catenary metasurfaces) are well-studied and well-explained, it remains a great challenge to unambiguously characterize the distinct influences of discrete geometric phase and continuous geometric phase on a specifical performance of subwavelength metasurfaces.
Recently, topology optimization has been extensively exploited to design freeform metasurfaces with non-intuitively quasi-continuous topology shapes [25–28]. It has attracted increasing attentions in the variety of applications, such as broadband wavelength demultiplexer [29], large-angle metagratings [30], angle-tunable polarization conversion metasurface [31], double wavelength focussing metastructure [32], non-reciprocal pulse router [33] and high-NA achromatic metalens [34]. Actually, adjoint-based topology optimization only needs to calculate Maxwell’s equations twice in every iteration operation [35], offering a potential inverse design approach with high computational efficiency.
Here, from the perspective of inverse design, we investigate the performance differences between discrete geometric phase metasurfaces and continuous geometric phase metasurfaces both for wide-angle ($70^\circ$) and high-efficiency deflection. Specifically, we adopt adjoint-based multi-object topology optimization approach to improve the absolute efficiency of geometric phase metasurfaces while maintaining the polarization conversion functionality of geometric phase. First, two kinds of discrete geometric phase metasurfaces with different materials (Si and GeAs) and working wavelengths ($1.55~\mu$m and $10.6~\mu$m) are used as initial structures, as shown in Fig. 1(a). We find that, in comparison with regular and discrete original structures, the finally optimized freeform metasurfaces have catenary-like quasi-continuous structures, leading to significant improvements of absolute efficiency. Specifically, the absolute efficiency of Si metasurface is optimized from $21.23\%$ to $65.07\%$, while that of GaAs metasurface is improved from $22.19\%$ to $62.96\%$. Then, a catenary metasurface with an equal width shown in Fig. 1(b) is topology optimized using the same strategy. The finally optimized metasurface still manifests as catenary-like topology shape with the improvement of absolute efficiency from $56.90\%$ to $66.46\%$. The optimization-induced topological transformation process from discrete geometric phase structure to catenary-like quasi-continuous structure indicate the great advantages of catenary metasurfaces from the view point of numerical optimization.
Fig. 1. (a) Discrete geometric phase structure for topology optimization. (b) Continuous catenary structure for topology optimization.
2. Initial structures and multi-object topology optimization
2.1 Initial structures
The initial geometric arrangement of regular and discrete geometric phase metasurfaces is shown in Fig. 1(a). Such typical geometric phase structure consists of eight pillars, each of which is respectively rotated by a specifical angle (here $\varphi = 22.5^\circ$). The geometric phase metasurface transforms an normal incident circularly polarized (LCP) light to right circularly polarized (RCP) light in the -1st diffraction order of $x$-direction ($\simeq -70^\circ$). To verify the availability of adjoint-based topology optimization, two different kinds of geometric phase metasurfaces with different materials (Si and GaAs) and different working wavelengths ($1.55~\mu$m and $10.6~\mu$m) are used as initial metasurfaces. Moreover, the initial structure of catenary metasurface for topology optimization is shown in Fig. 1(b). The typical parameters of cell unit of those three kinds of metasurfaces are presented in Table 1. To ensure the relatively-high initial efficiency, the parameters of two discrete geometric phase metasurfaces are pre-optimized with genetic algorithm. Specially, the cell period $P_x$ of Si metasurface is modified to $1.68~\mu$m considering practical fabrication issue, which slightly adjusts the -1st diffraction angle to $-67^\circ$. Notably, silicon dioxide with refractive index of 1.45 is used as substrate material for Si metasurface. However, the substrate material is the same as structure material both for GaAs metasurface and catenary metasurface.
Table 1. Typical parameters of initial metasurfaces for topology optimization.
As a gradient-based iteration optimization approach, the adjoint-based topology optimization needs the non-binarized parameters space of refractive index distribution $n (x,y)$ to update required gradient and modify topology shape during the binarization process. Therefore, the initial structure presented in Fig. 1 is blurred to generate the non-binarized refractive index distribution pattern for local topology optimization [36]. Specially, identically normalized blurring radiuses are utilized in the blurring process for the two different discrete geometric phase structures. However, a relatively smaller normalized blurring radius is used for the catenary structure to realize local topology optimization. We note that the initial regular and discrete geometric phase metasurfaces have low absolute efficiency with comparison to the initial continuous catenary metasurface.
2.2 Multi-object topology optimization for geometric phase metasurfaces
To inversely design the metasurfaces elements that possess the desired performances of both wide-angle and high-efficiency deflection, the adjoint-based topology optimization is utilized. In essence, adjoint-based topology optimization is a kind of gradient optimization algorithm, wherein the gradient is updated in every iteration to maximize a figure of merit (FoM). During the topology optimization process, refractive index distribution distribution $n (x,y)$ of metasurfaces elements is modified based on the updated gradient.
To maintain the geometric phase characteristic of optimized metasurfaces, we consider two different FoMs during topology optimization process, 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 specific diffraction order (-1st order) and the incident light illuminating on the metasurface elements. The FoM of absolute diffraction efficiency could be expressed as
where ${\boldsymbol {E}}$ represents the electric field complex amplitude of transmitted light at -1st diffraction order. On the other hand, the polarization conversion efficiency is defined as the complex amplitude inner product of the transmitted light and the RCP state. The FoM of polarization conversion efficiency is expressed as where $| {\boldsymbol {\hat {E}}}_{\textrm {RCP}} \rangle = \frac {\sqrt {2}}{2}[1,i]^{\textrm {T}}$ indicates RCP state and $\textrm {T}$ denotes transpose operation. Therefore, the final FoM of our multi-object topology optimization is given byThe partial derivative of total FoM with respect to the electric filed ${\boldsymbol {E}}$ is expressed as
Consequently, the optimized gradient could be updated by calculating only two times of field simulations, i.e., forward electric field amplitude ${\boldsymbol {E}}_{\textrm {fwd}}$ and adjoint electric field amplitude ${\boldsymbol {E}}_{\textrm {adj}}$. It is worth mentioning that, in our topology optimization process, both the in-plane symmetries along $x$- and $y$-direction are broken to acquire a more larger optimization space. Moreover, the electromagnetic simulations of optimization process are performed with rigorous coupled-wave analysis (RCWA) solver RETICOLO [37].
3. Topology optimized results
3.1 Topology optimized Si metasurface
The topology optimization process of Si metasurface is presented in Fig. 2. As shown in Fig. 2(a), the absolute efficiency gradually improves from $1.62\%$ to $66.04\%$ with the increment of iteration operation. However, the evolution of polarization conversion efficiency has a completely different trend during the iteration operations, i.e., it first has a notable decrease and then slowly increases to $97\%$. In fact, the evolution difference between the two efficiency curves can be explained from the perspective of multi-object optimization. At the beginning of topology optimization process, the blurred structure has remains a discrete geometric phase distribution [as shown in Fig. 2(c)] and it thus possesses a very high polarization conversion efficiency. In contrast, the corresponding absolute efficiency is extremely low due to the complex scattering and coupling in such unusual structure. Therefore, during the earlier stage of optimization process, the optimization of absolute efficiency dominates to rapidly improve the total FoM, as shown in Fig. 2(b). Meanwhile, the topology shape of Si metasurface changes rapidly and the corresponding geometric phase property has not been persevered, resulting in an inhibition of polarization conversion efficiency. Subsequently, the topology shape is basically formed with the increment of optimization iteration. For instance, as shown in Fig. 2(d), the topology shape of Si metasurface after 100 iteration oprations is similar to the final shape, although there still has unpractical index distributions. Therefore, during the later stage of optimization process, the corresponding FoM has no apparent improvements. In fact, the main purpose of the final stage of optimization process is to completely binarize the refraction index distribution of metasurface.
Fig. 2. Topology optimization process of Si metasurface. (a) Evolutions of absolute efficiency and polarization conversion efficiency. (b) Evolution of FoM over the topology optimization process. (c-e) Optimized topology shapes of Si metasurface in different iterations, corresponding to three different colored circles in (b).
As shown in Figs. 2(c)–2(e), the topology optimization approach enables us to intuitively observe the transition process of metasurface structure, which clearly reveals that, with the increment of optimization iterations, the corresponding topology shape gradually evolves from discrete structure to quasi-continuous one. Figure 2(e) presents the finally optimized freeform metasurface, which manifests as a kind of quasi-continuous catenary-like distribution. More importantly, the topology optimized Si metasurface has an isolated rectangle substructure on the right part of cell structure, which shows an excellent agreement with catenary metasurface [24]. Actually, the topology optimization process demonstrates that, from the view point of inverse design, the continuous catenary structure outperforms than the initial discrete geometric phase structure presented in Fig. 1(a).
In the following, we analyse the overall performance of topology optimized Si metasurafce by using finite-difference time-domain (FDTD) simulations (Ansys Lumerical FDTD Solutions). Figure 3(a) shows the top view of optimized catenary-like geometric phase metasurface with wide-angle and high-efficiency deflection. Figure 3(b) presents the FDTD simulation result of Re$(E)$ profile in the xoz plane, which is composed of six-period structure illuminated with LCP light. One can clearly see that the vast majority of transmitted light deflects from the -1st diffraction order ($\theta \approx -67^{\circ }$). We further compare the performance difference before and after topology optimization. As shown in Fig. 3(c), the original Si metasurface has only $21.23\%$ absolute efficiency for RCP light at central wavelength of 1550 nm. We note that the LCP component of transmitted light has a resonance peak around 1575 nm, which originates from the adjoint resonance in the discrete structure. In contrast, the topology optimized Si metasurface has about $65.07 \%$ absolute efficiency in 1550 nm for RCP light, as shown in 3(d). More importantly, the LCP component is well suppressed (about $2.11 \%$ absolute efficiency) and the efficiency ratio between RCP and LCP component is as high as 15 dB, which evidently indicates that the geometric phase characteristic is well maintained during topology optimization process. We note that the total absolute efficiency of RCP component and LCP component is not completely consistent with the absolute efficiency presented in the Fig. 2(a), owing to the slight difference of electromagnetic simulations between RETICOLO and Lumerical FDTD Solutions.
Fig. 3. Topology optimized results of Si metasurface. (a) Top view of optimized catenary-like Si geometric phase metasurface. (b) Real part distribution of $E$ in the xoz plane. Black rectangle represents the corresponding metasurface area. Absolute efficiency of transmitted RCP and LCP component for initial (c) and topology optimized (d) Si metasurface. Insets in (c) and (d) show the corresponding structures.
3.2 Topology optimized GaAs metasurface
Figure 4 presents the whole topology optimization process of GaAs metasurface working at $10.6~\mu$m. In comparison with optimization process of Si metasurface shown in Fig. 2(a), both the absolute efficiency and the polarization conversion efficiency of GaAs metasurface have different evolution curves due to the differences of material and working wavelength, as shown in Fig. 4(a). Specifically, the polarization conversion efficiency of GaAs metasurface has a sharper decreased process during the earlier stage of optimization process. Meanwhile, the corresponding absolute efficiency improves more slowly. We note that even after 100 iteration operations the absolute efficiency is still lower than $50\%$. The final is absolute efficiency optimized to $64.59\%$ and the final polarization conversion efficiency is improved to $97.21\%$. As a result, the multi-object FoM is optimized from $1.19\%$ to $62.79\%$ after 200 iteration operations, as shown in Fig. 4(b). The topology shapes of GaAs metasurface at different iteration operations are shown in Figs. 4(c)-4(e). The optimized topology shape evolves from discrete geometric structure to quasi-continuous geometric structure, which is similar to optimized Si metasurface shown in Figs. 2(c)-2(e). In comparison with Si metasurface, however, the finally optimized GaAs metasurface has four separate substructures, leading to a kind of catenary-like arrangement. It is worth mentioning that a rectangle substructure also appears on the right part of cell structure, which is similar to catenary metasurface [24]. Therefore, the topology optimization process of GaAs metasurface demonstrates again the general advantages of continuous catenary structure compared with discrete geometric phase structure.
Fig. 4. Topology optimization process of GaAs metasurface. (a) Evolution of absolute efficiency and polarization conversion efficiency. (b) Evolution of FoM over the topology optimization process. (c-e) Optimized topology shapes of GaAs metasurface in different iterations, corresponding to three different colored circles in (b).
Figure 5(a) shows the top view of optimized catenary-like GaAs metasurface. The simulation result of Re$(E)$ profile in Fig. 5(b) shows that the optimized metasurface has a wide diffraction angle ($\theta = -70^{\circ }$). The absolute efficiency of initial GaAS metasurface is presented in Fig. 5(c), which has only $22.19\%$ for RCP light at $10.6~\mu$m. Meanwhile, the LCP component of transmitted light has negligible absolute efficiency without any resonance peak, which is completely different with that of Si metasurface presented in Fig. 3(c). As shown in Fig. 5(d), the absolute efficiency of RCP component is improved to $62.96\%$ at $10.6~\mu$m after 200 optimization iterations, while that of LCP component is suppressed at a very low level ($3.38\%$), leading to a significant improvement over traditional design of GaAs diffraction grating [38]. As a result, the efficiency ratio between RCP and LCP component is up to 12.7 dB. Obviously, it illustrates that the optimized GaAs metasurface has remarkable geometric phase characteristic.
Fig. 5. Topology optimized results of GaAs metasurface. (a) Top view of optimized catenary-like GaAs metasurface. (b) Real part distribution of $E$ in the xoz plane. Black rectangle represents the corresponding metasurface area. Absolute efficiency of transmitted RCP and LCP component for initial (c) and topology optimized (d) GaAs metasurface. Insets in (c) and (d) show the corresponding structures.
3.3 Topology optimized catenary metasurface
Figure 6 presents the topology optimization process of catenary metasurface with an equal width. As shown Fig. 6(a), the initial absolute efficiency of blurred catenary metasurface is $9.71 \%$, which is relatively high than that of two discrete metasurfaces, due to higher absolute efficiency of initial catenary metasurface shown in Fig. 7(c). Based on such initial optimal condition, the absolute efficiency is rapidly increased in the beginning of iteration operations, which is totally different from the discrete scenarios. Finally, the absolute efficiency is promoted to $67.33\%$ by using topology optimization. Meanwhile, the polarization conversion efficiency possesses a particularly high value in most optimization processes thanks to the continuous and liner geometric phase nature of catenary structure. Therefore, the multi-object FoM curve presented in Fig. 6(b) has almost the same evolution profile with the absolute efficiency evolution curve. The optimized topology shapes of catenary metasurface at different iteration operations are indicated in Figs. 6(c)-6(e). As shown in Fig. 6(c), owing to the periodic boundary, an non-binarized refractive index distribution appears in the right part of initially blurred structure, offering an additional optimization space around the corresponding area. Therefore, an additional structure emerges in the right region in the following optimization process, as shown in Figs. 6(d) and 6(e). Moreover, Fig. 6(d) shows that there exists a nearly overlapped region in the neighbouring catenary structures of optimized topology shape at 100 iteration. However, such overlapped pattern gradually disappears as the topology optimization continues, finally forming a catenary-like structure with two independent pillars located at the left and right boundaries, as shown in Fig. 6(e).
Fig. 6. Topology optimization process of catenary metasurface. (a) Evolutions of absolute efficiency and polarization conversion efficiency. (b) Evolution of FoM over the topology optimization process. (c-e) Optimized topology shapes in different iterations, corresponding to three different colored circles in (b).
Fig. 7. Topology optimized results of catenary metasurface. (a) Top view of freeform catenary metasurface. (b) Real part distribution of $E$ in the xoz plane. Black rectangle represents the corresponding metasurface area. Absolute efficiency of transmitted RCP and LCP component for initial (c) and topology optimized (d) catenary metasurface. Insets in (c) and (d) show the corresponding structures.
The periodic pattern of optimized catenary-like metasurface presented in Fig. 7(a) shows that, compared with initial catenary metasurface, the tail of final catenary structure is optimized to be more smooth while its intermediate section still manifests as catenary-like topology shape. And the Re$(E)$ profile shown in Fig. 7(b) indicates the wide-angle deflection of transmitted light. As shown in Fig. 7(c), the absolute efficiency of initial catenary metasurface for RCP component is as high as $56.90\%$ at $10.6~\mu$m, while the transmitted energy of corresponding LCP component is especially low (about $0.95\%$). Figure 7(d) shows the absolute efficiency of finally optimized catenary-like metasurface. The final absolute efficiency of transmitted RCP component at central wavelength is improved to $66.46\%$, while that of transmitted LCP component is $1.97\%$. The efficiency ratio between RCP and LCP component is about 15.28 dB, indicating the great polarization conversion feature of optimized metasurface. Moreover, we note that the optimized absolute efficiency profile is more flat than the initial profile within 400 nm bandwidth.
4. Discussion and conclusion
In conclusion, the adjoint-based topology optimization of discrete and continuous geometric phase metasurfaces for wide-angle and high-efficiency deflection are investigated. Using two kinds of discrete geometric phase metasurfaces as initial structures, we successfully observed the topological transformation process from discrete geometric phase structure to catenary-like quasi-continuous structure. In comparison with initially discrete structure, the finally optimized catenary-like metasurfaces provide significant improvements of absolute efficiency without compromise of polarization conversion efficiency. We then applied the same optimization strategy to the catenary meatsurface and found that the corresponding optimized metasurface also manifested as a catenary-like quasi-continuous shape. The topology optimization results of discrete and continuous geometric phase metasurfaces demonstrate the inbuilt advantage of continuous catenary structures. As topology optimization is a kind of local optimization algorithm, we conclude that the catenary and catenary-like metasurfaces are locally optimal structures to a certain extent. In addition, the freeform metasurface emerges from the efficiency-oriented topology optimization procedure [30], instead of effective media theory. Therefore, there is no need for unit cell to be completely local as the coupled quasi-continuous structure is considered. It means that the traditional concept of unit cell is lapsed in the topology optimization design and the optimized freeform metasurface is not a traditional metasurface in common sense. Finally, we emphasize that the design robustness to fabrication error and character dimension should be further considered for practical applications in the future.
Funding
International S and T Cooperation Program of Sichuan Province (2020YFH0002); National Natural Science Foundation of China (61675208, 61822511).
Disclosures
The authors declare no conflicts of interest.
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