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Abstract
Optimizing the shape of metasurface unit cells can lead to tremendous performance gains in several critically important areas. This paper presents a method of generating and optimizing freeform shapes to improve efficiency and achieve multiple metasurface functionalities (e.g., different polarization responses). The designs are generated using a three-dimensional surface contour method, which can produce an extensive range of nearly arbitrary shapes using only a few variables. Unlike gradient-based topology optimization, the proposed method is compatible with existing global optimization techniques that have been shown to significantly outperform local optimization algorithms, especially in complex and multimodal design spaces.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces have the potential to replace bulky conventional optical system components like lenses due to their ability to offer unprecedented control of light at subwavelength scales [1,2]. In particular, phase-gradient metasurfaces empower designers to exploit the generalized form of Snell’s law [3], which enables tremendous control over refraction and reflection in optical devices. Already, lenses [4–8], vortex beam generators [3,9], and hologram devices [10,11] have been demonstrated using planar meta-atoms. These designs show that metasurfaces have the potential to revolutionize optical system design. However, to compete with conventional optical systems, metasurface devices need to achieve performances (e.g., bandwidth, field-of-view, transmission efficiency) that approach or even surpass their conventional counterparts. Some of the first optical plasmonic metasurfaces had low efficiencies from high material losses and low coupling efficiency between polarization-sensitive modes [3,12]. Low-loss dielectric meta-atoms that are usually based on canonical shapes such as ellipses or rectangles [13–16] have been used to achieve improved efficiency. Nevertheless, design methodologies that can simultaneously generate arbitrarily shaped meta-atoms and perform nanophotonic optimizations are desired to thoroughly explore the design space. This will realize metasurfaces that leverage the photonic resonance to the greatest extent.
Many optimization techniques have been utilized in nanophotonic meta-device design [17–26]. Recently, it has been shown that using inverse-design (i.e., optimization) techniques to synthesize freeform metasurface topologies has led to significant efficiency improvements compared to previous devices based on canonical shapes demonstrated in the literature [27–29]. However, finding these more complicated and higher performing shapes can require many full-wave function evaluations (e.g., thousands or, potentially, even millions). Fortunately, advanced optimization and machine learning techniques have drastically reduced the time to generate high-performance meta-atoms [30–40]. In most cases, machine learning is used to predict the performance of a specified design, which is many orders of magnitude faster than a similar process based on full-wave simulation. One of the most intriguing recent inverse-design methods is topology optimization [41–43], which exploits the adjoint method to generate gradients for metasurface configurations with an arbitrary number of variables using only two simulations. It should be noted that gradient-based topology optimization suffers from the limitations of local methods such as getting trapped in local minima. Furthermore, topology optimization performs tradeoff studies using the weighted sum of competing objectives, which may not find all Pareto optimal points for nonconvex problems such as metasurface design. Additionally, the commonly used implementation of topology optimization has only been demonstrated for all-dielectric supercell (i.e., wavelength scale) based meta-devices. However, meta-atom based devices have several advantages overs those based on supercells. First, meta-atoms have a smaller domain and can be evaluated quickly, a feature that is in sharp contrast to topology optimization which requires hundreds of evaluations to come up with a single design. Moreover, global optimization techniques are better suited for multi-core computing as entire generations of designs can be evaluated in parallel. Second, many interesting devices can be built from just a few meta-atom designs with well-studied responses. While intercell coupling can be a cause for concern with meta-atom based devices, this can be mitigated with the proper choice of neighboring elements and fine tuning of the synthesized metasurface.
In this paper, a versatile methodology for metasurface inverse-design is presented. This method has the important advantage that it works with plasmonic and dielectric materials for both reflective and/or transmissive configurations as well as enables direct meta-atom and supercell inverse-design. The method exploits global optimization to avoid local minima and allows for trade-off studies to be performed using multi-objective algorithms. Furthermore, both conventional and freeform meta-atom shapes can be readily generated with a relatively few number of variables (as few as one). Additionally, a technique is introduced to transform transmission (reflection) and phase objectives in such a way as to enable one to generate a library of topologically diverse optimized meta-atom designs in a single optimization run. Meta-atoms can be chosen from this library to make a highly efficient phase gradient metasurface. Finally, two beam-steering devices are demonstrated to showcase the power of the proposed design methodology.
2. Inverse design strategy of unit cells
2.1 Generation of topologically diverse unit cells
Many methods have been used to design meta-atoms. The most prevalent method is to define the structure by a simple geometric shape like a rectangle or ellipse and perform a parametric sweep of the different dimensions (e.g., periodicity, height, width/radius). This approach allows one to efficiently explore the design space and achieve structures that are easy to fabricate. However, changing only the size of a meta-atom often leads to poor efficiency at certain phase options. Another design approach frequently employed is to pixelate the design space into boxes that are then assigned to either one of two materials (e.g., air or Au). This approach is extremely powerful at generating high performing meta-atoms at radio frequencies [26] and for thin, metallic layers at optical frequencies [44–46]. However, the pixelated approach can lead to designs with many small holes or lone pixels, which are difficult to fabricate for some material layouts (e.g., thick dielectrics). Furthermore, unlike performing parametric sweeps of canonical shapes, pixelated designs only come in discrete dimensions, which is limiting because (1) the optimal design dimension might be somewhere in between the two pixels and (2) the discrete nature makes it difficult to represent curved geometries (like rings). These issues can be partially resolved by using smaller boxes, which, however, quadratically increases the number of variables and makes the corresponding optimization more difficult. A third approach is to define the outside boundary of the meta-atom topology by an equation [19] or set of points [32]. While this approach has shown remarkable results, it is restricted to single shapes, excludes designs with holes, and can have very sharp corners. Nevertheless, a universal method capable of generating both conventional and freeform shapes is exceptionally desirable for high performance metasurface design. A comparison of different metasurface design methods is shown in Table 1.
Table 1. Comparison of different meta-atom topology design strategies
To overcome the limitations of previous methods, we propose a new method of meta-atom design based on finding the contour line on a surface. This method can create freeform shapes and find meta-atom designs that achieve higher efficiencies at all desired phase options. It is easy to use, and can quickly create a collection of functional designs, especially when paired with multi-objective optimization. Furthermore, with the multitude of various shapes possible, one can select designs based on ease of fabrication or robustness to fabrication deformities. To this end, the surface contour method is a simple but powerful technique that uses control points to create a unique surface that can then be adjusted to improve design performance. As Fig. 1 shows, an optimizer intelligently chooses the height for each control point along a specified grid. While it may appear in this example that many variables are needed to control the shape, 4-fold symmetry is used so that only 16 (4 × 4) variables are required. Then a three-dimensional continuous surface is created by employing a cubic interpolation between the points. The intersection of the surface with a plane defines the contour for the design. An easy way to find this intersection is to perform a thresholding operation on the surface to create a binary image, and then find the outside boundary of that binary image. Often it is desirable to complete some further functions on this image to enforce fabrication constraints (e.g., prevent sharp corners and enforce minimum feature widths and gaps between features). After this post-processing step, the design can be imported into an electromagnetics solver in either one of two ways. The first method is to break up the design into boxes which can then be imported to the solver. The second method is to use the boundary around the shape(s) and import the curve. Once the design is imported into the desired software, the meta-atom configuration can be analyzed, and the objective function(s) evaluated.
The surface contour method can accurately generate a large range of conventional as well as freeform topologies. Importantly, this method is immediately compatible with global optimizers, which can efficiently and robustly explore the meta-atom design space to maximize performance. To better illustrate the corresponding working principle, Fig. 2 shows a small sampling of the possible designs able to be created with this approach. The control points were optimized to minimize the error between the desired pattern (see Figs. 2(a)–(d) and (f)–(i)) and the shape produced using the surface contour method. The top row of Fig. 2 shows 4-fold symmetrical designs created with 25 (5 × 5) control points, while the bottom row shows designs without 4-fold symmetry that require 81 (9 × 9) control points. Note that this method is completely capable of producing canonical shapes including rectangles and ellipses of various sizes, while creating other shapes from the literature like H- [47] or C-shapes [48]. We emphasize that this is an essential feature that shows that conventional structures are included in the design space and can be similarly found through the optimization process. Figure 2 shows that the shapes are accurate and reasonable approximations to the desired shapes with only minor rounding of the corners. These freeform geometries are highly desirable because they can achieve greater efficiencies than canonical shapes while opening the door to multi-functional meta-atoms. With this wide search space of possible designs available, the next challenge is to efficiently optimize the shapes to simultaneously achieve high efficiency and diverse phase.
Fig. 2. A demonstration of traditional and uniquely shaped meta-atom designs generated by the surface contour method. The control points (pink spheres) shape the surface topology and, thus, the meta-atom geometry.
2.2 Multi-objective optimization of meta-atom based metasurfaces
While a large search space of unique shapes is desirable, it creates a challenge to efficiently find optimal meta-atom designs. Therefore, to realize a rapid inverse-design strategy one must pair the shape generation engine with a highly efficient optimization algorithm. To this end, state-of-the-art global optimizers based on evolutionary algorithms have demonstrated tremendous success in complex radio frequency and optical meta-device design [18,20–22,24,49–52] and are well-suited for the proposed method. Global optimizers can escape local minima in pursuit of the true global minimum within the hyperdimensional design space. Furthermore, many multi-objective variants of these global algorithms are available that can readily optimize multiple competing design goals [53–55]. This enables a quick and efficient method for designing optimal meta-atom geometries for a wide range of objectives including multi-functional performances. As an added benefit, it is simple to incorporate secondary objectives in addition to efficiency using multi-objective optimization (e.g., minimize maximum field in the structure, which could be beneficial for high power applications). A marked difference between single- and multi-objective optimization is in the number of final solutions they generate. Multi-objective algorithms, unsurprisingly, yield a set of solutions at the end of the optimization cycle called the Pareto Set which lie on a virtual surface called the Pareto Front. Interestingly, through analysis of the Pareto Front one can infer the inherent trade-offs between the various design goals (objectives) in the chosen problem enabling designers to better understand the underlying physics that governs the meta-device’s performance.
With the shape generation method and optimization algorithm chosen, there are two main strategies one can employ for designing phase-gradient metasurfaces. The first strategy is to design a supercell geometry. This can easily be done with the proposed method using efficiency in the desired diffraction order(s) as the desired goal(s). The second strategy is to design a collection of meta-atom geometries that have high efficiency and diverse phase. This second approach is slightly trickier because while efficiency can be maximized, maximized (or minimized) phase is not desirable. Rather, phase diversity is truly a distinct objective. One could run a single objective optimization for each specific phase goal. However, for optimal performance, this strategy requires a priori knowledge of the best phase goals, and it can be incredibly inefficient since multiple optimizations must be performed. Fortunately, multi-objective optimization enables one to generate the entire library of meta-atoms in a single optimization run. To do this, one can perform a coordinate transformation on the transmittance, T, and phase, ϕ, objectives. Two new objectives are obtained by simply rotating the transmittance versus phase plot by -135°:
Fig. 3. Depiction of the relationship between the transformed objectives and (a) transmittance and (b) transmission phase.
3. Example designs
In the following sections, two example designs are presented to demonstrate the versatility and effectiveness of using the proposed multi-objective optimization enabled meta-atom design method. The first example is a plasmonic gold nanodisk based reflective beam-steering metasurface [56]. The second example showcases how shape-optimized silicon meta-atoms vastly outperform designs based on elliptical pillars. However, unlike the gold nanodisk design, the silicon unit cells are affected by nearest neighbor coupling due to their relatively weak field confinement. Nevertheless, for beam-steering designs, all undesired coupling effects can be mitigated by fine-tuning the supercell selection in a secondary optimization procedure. In both cases, the multi-objective optimization algorithms created a library of meta-atoms that outperform their canonical counterparts. Our purpose is not to showcase any specific device or functionality, but rather to demonstrate a design methodology compatible with conventional simulation tools that leads to significantly better performing metasurfaces than traditional approaches.
3.1 Gold nanodisk-based reflective metasurface
In Sun et al. [56], a nanodisk geometry was explored for the synthesis of a highly reflective metasurface beam-steerer. The simple geometry enabled rapid exploration of the design space as there was only a single variable; however, significant performance improvements can be realized by implementing the proposed shape optimization framework. Figure 4(a) shows the design which is based on gold nanodisks placed over a MgF2 spacer (n = 1.892) on top of a gold ground plane. The individual unit cells were simulated in a doubly periodic environment using COMSOL Multiphysics v5.3 [57] at a wavelength of 850 nm. A Drude model was used to describe the gold permittivity [58]. To design the optimized shapes, a 3 × 5 control point uniformly-spaced grid oriented along the x- and y-directions, respectively, was implemented and optimized with BORG [55]. The results of the shape optimization study are indicated by the colored markers in Fig. 4(b). The results shown contain about 18,000 solutions, but this far exceeds the number of evaluations required to find optimal designs, because the Pareto Front is very close to convergence after only 2,000 iterations. Since the average time to simulate a design using a full-wave solver was approximately 30 seconds, it is reasonable to simulate the entire library of unit cells in about a day on a normal workstation computer. This study could even be accelerated by using a faster electromagnetics solver like, for example, the rigorous coupled wave analysis (RCWA) method. For comparison purposes, the reflectance and reflection phase for the simple nanodisk designs are indicated by the black markers in Fig. 4(b). A comparison of the performance of the conventional nanodisks to the shape-optimized designs shows that the optimized designs significantly increase in reflectance near the −60° reflection phase angle. Furthermore, the optimized shapes include designs with reflection phases between 90° and 150°, which were unattainable by the rectangular nanodisk. Since the overall performance of a metasurface is usually limited by its worst performing element, this improvement leads to a significant increase in diffraction efficiency of the steered beam.
Fig. 4. (a) The unit cell arrangement. Lx = 120 nm, Ly = 300 nm, tgp = 130 nm, ts = 50 nm, tms = 30 nm, and Lms = 90 nm. (b) A comparison between the shape optimization designs and the rectangular nanodisk designs. Each circle dot represents an individual unit cell design, and each black diamond represents the solution from sweeping the length of a rectangular nanodisk. The white triangles represent the meta-atoms chosen to construct the nanodisk supercell, and the light pink triangles represent the meta-atoms chosen for the shape-optimized supercell design.
It should be noted that this method is more general when compared against topology optimization techniques, due to the fact that conventional topology optimization implementations are only compatible with supercells (not unit cells) and have some convergence difficulties with metallic structures. State-of-the-art topology optimization methods cannot currently create unit cell designs because the derivations for calculating the gradient are based on maximizing power, which does not depend on reflection (transmission) phase. Furthermore, topology optimization methods require an interpolation scheme between the two materials (e.g., air and Au) used to design the nanophotonic structure. For lossless dielectric materials, a linear interpolation is adequate. However, for complex permittivity values, like those found in metals at optical frequencies, the final result depends on the specific interpolation scheme that is implemented. It has been shown that a linear interpolation results in suboptimal performance, while a nonlinear interpolation based on refractive index and the extinction coefficient can achieve improved performance [59]. Nevertheless, with the surface contour method, no material value interpolation scheme is required, and it is as trivial to optimize a metallic design as it is a dielectric one.
After optimizing the unit cell configurations, the best performing unit cells from the nanodisk and shape-optimized designs are then selected from the library and arranged into supercells. With the library of unit cells, there are many different phase options from which one can choose. This is very advantageous because there is a flexibility in choosing designs which are most suitable for fabrication. While there are perhaps many ways to select the optimal set of unit cells, the method chosen was to first find the best design in every one-degree interval. Then a set of five-unit cells is chosen from this subset that have a separation of 72° between neighboring unit cells and maximum total reflectance. This approach worked remarkably well for this problem and required no additional adjustments. Figure 5(a) shows the optimized unit cells selected from Fig. 4(b). Interestingly, the first optimized meta-atom contains very little gold which could probably be removed. The next three meta-atoms have three gold segments with differing gap widths between their segments. The final meta-atom can be considered as a perturbed nanorod. It would be difficult to arrive at these designs without advanced inverse-design methods due to the large solutions space afforded by the surface contour method. Figure 5(b) shows the rectangular nanodisk design achieves 72% diffraction efficiency into the +1 order under normally incident illumination [56] while the shape-optimized supercell achieves 85%; a significant increase. Furthermore, from Fig. 5(c) one can see that the steered beam closely resembles an ideal plane wave, which indicates rather little energy is coupled into the other diffraction orders. Simulations indicate that there is close to 15% absorption in the structure.
Fig. 5. (a) The shape-optimized supercell design. (b) The reflectance from the simulated nanodisk design and the shape-optimized supercell design. (c) The field plot showing the reflected field from the shape-optimized supercell.
Interestingly, while neither wideband nor wide field-of-view performances were considered during optimization, the shape-optimized supercell outperforms the rectangular nanodisk design. Figure 6(a) shows that the optimized shapes have a broader bandwidth than the nanodisk designs. This is surprising because the design was only optimized at a single wavelength (850 nm). There is a dip in performance at 790 nm, but since this method uses multi-objective optimization, other frequencies and incident angles could easily be incorporated into a future optimization. Additionally, the designs had higher and more constant performance over a wider-field-of-view (see Fig. 6(b)). Interestingly, as predicted by the generalized Snell’s law [3], incidence angles greater than 17.0° lead to the incident wave coupling into a surface wave, hence the poor performance for positive incidence angles.
Fig. 6. The reflectance into the +1 diffraction order as a function of (a) wavelength and (b) incidence angle.
These results show how the proposed multi-objective shape optimization framework can produce a library of high-performance meta-atoms which can be used to synthesize phase-gradient metasurfaces. To further expand upon this method, the next section demonstrates how it can be applied to handle multiple performance objectives as a path forward for realizing multi-functional metasurface devices.
3.2 Polarization dependent meta-atom design
In this second example, the multi-objective shape optimization method is used to design polarization-dependent meta-atoms. Previously, polarization dependent metasurfaces have been demonstrated [16,60,61] using canonical shapes, but here we compare designs based on elliptical nanopillars to the shape optimized unit cells. The unit cell consists of silicon (ɛr = 12.09) pillars on top of silica (ɛr = 2.08) illuminated from the silica side by an incident plane wave at 1.55 µm wavelength. For speed and efficiency, the unit cells are simulated in a doubly periodic environment with a RCWA code [62] that can be run from MATLAB. Using the coordinate transformation-based multi-objective approach, the solution spaces for both the elliptical and shape-optimized pillars were explored for both x- and y-polarized incident light (see Figs. 7(b) and 7(d)). While the optimization was performed using BORG, it is compatible with any multi-objective optimization algorithm. The shape-optimized designs were created using a 3 × 3 control point grid with 4-fold symmetry using the same unit cell periodicity and silicon height as the elliptical pillars. 55,000 and 73,000 designs were evaluated for the elliptical and the shape optimized unit cells, respectively. By using the RCWA code and parallelizing the simulations, it took less than 10 hours to simulate the entire set of 73,000 designs. Interestingly, using the coordinate transformation method with this problem requires four objectives obtained by applying Eq. (1) to the transmittance and phase for each polarization of light. Ideally, the entire space in Figs. 7(b) and 7(d) is covered (each circle represents a different unit cell) with 100% efficiency for both polarizations (corresponding to red in the color bar). One can see that while the elliptical pillars are unable to achieve this, the shape-optimized designs cover nearly the entire phase space with high efficiency options.
Fig. 7. The geometry setup of the (a) elliptical pillar and (c) shape-optimized unit cells. The dimensions are Λx = Λy = 775 nm and h = 700 nm. The transmission behaviors are mapped out in (b) and (d) for the elliptical and shape-optimized designs, respectively. Each circle represents a unique unit cell design and its position indicates the co-polarization transmission phase for x- and y-polarized light. The color indicates the unit cell’s minimum transmittance for x- and y-polarized light.
However, it was observed that nearest-neighbor coupling arose when forming a supercell from the Si designs. Therefore, unlike the Au nanodisk designs, these unit cells need to undergo a secondary optimization in the supercell environment to refine their behaviors and mitigate any negative couplings effects. Nevertheless, these designs can be readily implemented into devices where similar unit cells are grouped together and there are minimal transitions between different structures. A possible application would be a vortex beam generator [3,9], because each octant contains the same unit cell which significantly reduces the total number of coupling environments.
Using BORG, the meta-atom shapes in a four-element supercell are optimized again to maximize transmittance into the -1 and +1 diffraction orders for x- and y-polarized incident light, respectively. This second optimization continued to use 3 × 3 control points to manipulate the shape of each unit cell resulting in a total of 36 variables and 2 objectives. This second optimization ran for about 92,000 evaluations. The different supercell structures were optimized in parallel for approximately 48 hours. Figure 8 shows the results from the optimization study for both the elliptical and shape-optimized supercells. Once again, the shape-optimized designs significantly outperform the elliptical geometries. This is expected from analyzing Figs. 7(b) and 7(d). Moreover, it should be noted that the entire elliptical pillar solution space is a subset of the shape-optimized space. Interestingly, there is a slight trade-off in efficiency when going from the single polarization beam-steering to polarization-selective steering. If maximum y-polarized beam-steering is desired, then the x-polarized light steering efficiency must decrease, and vice versa. The supercell design shown in Fig. 9(a) corresponds to the one closest to the yellow star in Fig. 8, since it maximizes the efficiency for both polarizations. This design was validated in COMSOL Multiphysics v5.3 and achieves steering efficiencies of 83% and 82% into the -1 and +1 diffraction order for x- and y-polarized incident light, respectively. This performance could potentially be further improved by increasing the design space by optimizing the entire structure rather than individual unit cells, or by increasing the height of the nanopillars. Figure 9(b) shows the electric field distribution in the vicinity of the supercell. The light is incident from the bottom (silica side) and is steered by the metasurface into the desired direction. There is very little power transmitted into the other diffraction orders.
Fig. 8. Pareto Fronts from the multi-objective optimization of the 4-element supercell using (blue) elliptical pillars and (red) shape-optimized meta-atoms.
Fig. 9. Silicon metasurface supercell performance summary. (a) The diffraction efficiencies for the shape-optimized supercell polarization-dependent beam splitter. The inset shows the shape-optimized supercell with its individual meta-atom elements. (b) The field plots for the optimized design for (left) x- and (right) y-polarized light.
4. Conclusions
This work demonstrates a simple and, yet, powerful methodology for rapidly generating meta-atom building blocks including those based on canonical shapes (e.g., rectangles, ellipses) as well as nearly arbitrary geometries. While all conventional shapes exist within a subset of the solution space afforded by the proposed method, it was shown that metasurfaces whose meta-atoms are based on unique geometries outperform those comprised of canonical shapes. Furthermore, the method can generate a complete library of meta-atoms with diverse phase while simultaneously maximizing their transmission or reflection efficiencies in a single optimization run. Additionally, a diverse set of performance objectives can be simultaneously optimized and the trade-offs between them analyzed. Another benefit is that the approach can be combined with any optimization algorithm or full-wave solver including both commercial and in-house tools. Finally, the method offers much of the geometrical variety afforded by topology optimization while not suffering from its limitations; namely, relying on gradient-based local optimizers, evaluating in serial fashion, and implementation difficulties when working with plasmonic materials. However, future work could include the hybridization of the surface contour method with an adjoint field analysis to achieve faster convergence while maintaining the benefits of global optimization. Advanced optimization of the freeform geometries afforded by the surface contour method leads to improved meta-device performance.
Funding
Defense Advanced Research Projects Agency (HR00111720032).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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