1. INTRODUCTION

In contrast to traditional optical devices with complicated and bulky components [1], metasurfaces have recently attracted much attention due to their remarkable ability to manipulate electromagnetic waves at subwavelength dimensions [2–5]. They have been utilized for a wide range of applications, such as metagratings [6–8], metalenses [9–11], and holography [12–15]. In particular, a large-angle metagrating is one of the most important applications of metasurfaces and has been widely used in high-performance imaging [9,10,16], high-resolution spectroscopy [17], and other applications. Generally, the electromagnetic response of a metagrating is determined both by meta-atom arrangements and material property. In comparison with discrete structures where the meta-atoms are arranged according to spatially discontinuous phase distribution, continuous metasurfaces can suppress the coupling effect between adjacent meta-atoms, offering an ideal candidate for high-efficiency and broadband metasurfaces [18–20]. It also has been shown that dielectric metasurfaces are ultra-high-efficiency devices without intrinsic absorption losses like metal [21,22]. Nevertheless, due to the impedance mismatch of reflections, the efficiency of large-angle metagratings decreases sharply with the increase of deflection angle [23]. Several approaches have been demonstrated to overcome the impedance mismatch of reflections and enable high-efficiency capability, such as homo-metagratings [24] and freeform metalenses [25]. Particularly, the corresponding bandwidth is greatly limited while a metagrating operates in the large-angle condition, thus impeding the practical applications of metagratings. Therefore, it is of vital importance to design a broadband high-efficiency large-angle metagrating.

Recently, an adjoint-based topology optimization approach has been utilized for a wide range of metasurface designs [26–37]. In fact, it needs only two full-wave electromagnetic simulations to efficiently update gradient information of structure variation in every iteration operation [38–41]. Particularly, sufficiently large degrees of freedom can be designed in topology optimization, leading to the non-intuitive functionality design of metasurfaces. Although several different freeform high-efficiency metasurface have been demonstrated by topology optimization [20,25], a broadband metagrating for high-efficiency large-angle deflection remains unexplored.

 

Fig. 1. (a) Schematic depiction of initial continuous catenary metagrating. (b) Schematic depiction of initial discrete geometric phase metagrating. (c) Forward simulation of topology optimization. (d) Adjoint simulation of topology optimization. Here, $r$ denotes the optimization location and ${r^\prime}$ denotes the object location.

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Here, we numerically investigate the broadband topology-optimization problem of a high-efficiency large-angle deflection metagrating. Specifically, three different broadband optimization approaches are comprehensively investigated, including single-wavelength optimization (SO), max-min optimization (MO), and average optimization (AO). Two different structures with different materials (Si and GaAs), namely, a continuous catenary metagrating and discrete geometric phase metagrating, are used as the initial structures of topology optimization. It is shown that, compared to SO and MO, the AO approach is more appropriate to optimize the broadband high-efficiency metagrating. Specifically, a freeform metagrating of up to about a 70° deflection angle at a central wavelength of 10.6 µm with diffraction efficiency exceeding 80% over a broad bandwidth of 4.36 µm is demonstrated based on the AO approach. Moreover, the quasi-catenary metagrating optimized from initial catenary structures has the evident advantage on bandwidth compared to that optimized from initial discrete structures. Our results may find potential applications in broadband achromatic metalenses and other broadband meta-devices.

2. INITIAL METAGRATING STRUCTURES AND BROADBAND TOPOLOGY OPTIMIZATION METHODS

A. Initial Metagrating Structures

Figures 1(a) and 1(b) illustrate the initial metagrating employed in topology optimization, which comprises continuous catenary structures made of silicon (Si) with a refractive index of 3.37, and discrete geometric phase structures composed of gallium arsenide (GaAs) with a refractive index of 3.3. Here, the metagratings work in the transmissive mode, which deflects a left circularly polarized (LCP) incident plane wave to a right circularly polarized (RCP) light wave in ${+}{1}$ diffraction order along the $x$-direction. The two initial structures have the same periods in the $x$-direction (${P_x} = {11.28}\;\unicode{x00B5}{\rm m}$) and different periods in the $y$-direction (${P_{y1}} = {3.2}\;\unicode{x00B5}{\rm m}$, ${P_{y2}} = {1.41}\;\unicode{x00B5}{\rm m}$). In the initial continuous catenary structures, the height ${h_1}$ and width ${w_1}$ are 4.7 µm and 1 µm, respectively. In the initial discrete structures, the length $l$ and width ${w_2}$ of GaAs nanoposts are 1.2 µm and 0.65 µm, respectively. The corresponding thickness is 6.8 µm and the rotation angle of each GaAs nanopost increases in turn by a specifical angle of $\theta = 22.5^\circ$. In our simulation, the refractive index of Si and GaAs is 3.37 and 3.3, respectively. These structure parameters of both initial catenary and discrete geometric phase metagratings are selected from the Ref. [20]. During the optimization process, the heights and periods keep constant while other structure parameters update according to the variation of the gradient. Notably, the central wavelength of all the optimized metagratings is 10.6 µm.

B. Broadband Topology Optimization Methods for Metagrating

As shown in Figs. 1(c) and 1(d), the topology optimization includes the forward and adjoint simulation, where $r$ denotes the pixels in the optimization region $\Gamma$ and ${r^\prime}$ indicates the pixels in the object region $\Omega$. During the forward simulation, the incident electric fields ${{\boldsymbol E}_{{\rm in}}}$ illuminate the metagrating from bottom to top and the forward electric field distribution ${{\boldsymbol E}_{{\rm for}}}$ in the metagrating surface can be obtained. During the adjoint simulation, an adjoint source ${\boldsymbol E}_{{\rm obj}}^*$ reversely illuminates the metagrating to calculate the adjoint electric field distribution ${{\boldsymbol E}_{{\rm adj}}}({{\lambda _i}})$ in the metagrating surface. Here, the absolute efficiency is defined as the energy ratio of the first-order diffracted RCP light to the total incident light, and the relative efficiency is defined as the energy ratio of the first-order diffracted RCP light to the energy transmitted through the metagrating. It is worth noting that such definition of absolute efficiency ensures the polarization conversion functionality of metagratings during the optimization process.

For the SO topology optimization approach, the figure of merit (FOM) is defined as a function of electrical field ${{\boldsymbol E}_{{\rm obj}}}({{\lambda _i}})$ at a single wavelength, which can be expressed as

Based on the Lorentz reciprocity, the gradient in the SO approach can be obtained as [20,38]

For the MO approach, the FOM is defined as

 

Fig. 2. (a) Absolute and relative diffraction efficiencies for initial catenary metagrating. Insets: real part of distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (b) Absolute and relative diffraction efficiencies for SO optimized metagrating. (c) Absolute and relative diffraction efficiencies for MO optimized metagrating. Insets: real part distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (d) MO optimization processes. Insets: optimized structures in the first and 100th iterations. (e) Efficient bandwidth performances for SO and MO methods. The incident wavelengths are 6 µm (SO I), 7 µm (SO II), 8 µm (SO III), 9 µm (SO IV), 10 µm (SO V), and [7 µm, 9 µm] (MO). (f) Deflection angle versus wavelengths.

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Another strategy for broadband optimization is the AO method, which calculates the arithmetical average of absolute efficiency at different wavelengths as the optimization object. In contrast to the SO and MO methods, sufficient sample wavelengths are needed in the AO method for efficient bandwidth optimization. The corresponding FOM and gradient of the AO method thus can be defined as

In the topology optimization procedure, the electromagnetic simulations are performed with finite-difference time-domain (FDTD) simulations (Ansys Lumerical FDTD Solutions) for 100 iterations to balance the convergence speed and time cost of optimization. In order to eliminate small features [39], the blurring radius is one of the most crucial parameters to determine its optimization performance. Here, a normalized blurring radius of 0.38 µm is utilized in the blurring process for the two different initial structures. Most importantly, to evaluate both the efficiency and bandwidth performance of topology-optimized metagratings, an efficient bandwidth defined as the width of the wavelength range with relative diffraction efficiency above 80% is used in the following analysis.

 

Fig. 3. (a), (b) Topology optimization results with AO method. Insets: real part distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (c) AO IV optimization process. Insets: optimized structures in the first and 100th iterations. (d) Efficient bandwidth performances for AO method. The incident wavelengths are [6 µm, 8 µm, 10 µm] (AO I), [7 µm, 9 µm, 11 µm] (AO II), [6 µm, 8.5 µm, 11 µm] (AO III), and [6.5 µm, 8.5 µm, 10.5 µm] (AO IV). (e) Deflection angle versus wavelengths.

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3. TOPOLOGY-OPTIMIZED RESULTS

A. MO-Optimized Quasi-catenary Metagrating from Initial Catenary Structure

Figure 2(a) presents the absolute and relative diffraction efficiencies for the initial catenary metagrating. It has an efficient bandwidth of 2 µm at a central wavelength of 10.6 µm. Nevertheless, a high diffraction efficiency within the entire bandwidth is challenging to be achieved, particularly for wavelengths less than 8 µm. Note that the optimization performance of the SO method depends heavily on the selection of optimization wavelength. As illustrated in Fig. 2(b), we first investigated the influence of different optimized wavelengths for metagrating performance. In the SO method for the optimized wavelength of 6 µm, the maximum relative efficiency is about 95.3%, corresponding to scenario SO I in Fig. 2(b). With the optimized wavelength deviating considerably from the initial 10.6 µm, there is a consequent reduction in the efficient bandwidth falling from 2 µm to 1.4 µm. The maximum efficient bandwidth can reach 3 µm (SO II and SO V) by choosing 7 um and 10 µm as the optimized wavelengths.

To further increase the efficient bandwidth, two sets of parameters with the best performance (SO II and SO V) are chosen as the initial parameters for MO. The same initial structures were employed for two optimization methods. For each iteration, the refractive index profile is updated based on the gradient with minimum absolute efficiency. As shown in Fig. 2(c), the 80% efficient bandwidth of deflection efficiency is 4 µm (from 6.2 µm to 10.2 µm), and the highest deflection efficiency of 97.15% can be achieved at 8.75 µm. It means that the MO algorithm is an efficient design method for broadband optical devices, without the need for extensive parameter scanning. To better comprehend the enhancement of our inverse design relative to the traditional forward design, we compare the diffraction effect of metagratings at the wavelength of 10.6 µm. Compared with the initial structure, the deflected electric field in the freeform metagrating optimized by the MO method is more uniform, leading to a high-efficiency large-angle deflection performance. Figure 2(d) displays the MO evolution curves of the catenary metagrating optimization process, for 7 µm and 9 µm optimized wavelengths. The efficiency for both wavelengths gradually and synchronously increase, while the oscillation occurs due to the changes of the optimization object during the process. As shown in Fig. 2(e), the bandwidths of SO I, SO II, SO III, SO IV, and SO V are evidently smaller than that of MO, confirming that MO is efficient for addressing bandwidth optimization problems. Figure 2(f) displays the deflection angle curves of the catenary metagrating in the given bandwidth range from 6 µm to 12 µm. We note that using the minimum efficiency value within a broadband width as the merit function in the optimization requires setting multiple wavelength monitoring points, leading to the significant increase of computational complexity. In fact, we have tested different optimization wavelengths and found that the optimization performance and the computational demands could be well balanced by the two-wavelength-based max-min optimization method.

 

Fig. 4. (a) Absolute and relative diffraction efficiencies for initial discrete metagrating. Insets: real part distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (b) Absolute and relative diffraction efficiencies for initial metagrating. (c) Absolute and relative diffraction efficiencies for MO optimized metagrating. Insets: real part distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (d) Efficient bandwidth performances for SO and MO methods. Insets: optimized structures in the first and 100th iterations. The incident wavelengths are 6 µm (SO I), 7 µm (SO II), 8 µm (SO III), 9 µm (SO IV), 10 µm (SO V), and [7 µm, 9 µm] (MO). (e) Bandwidth versus the optimized method. The incident wavelengths are the same as those in Fig. 2. (f) Deflection angle versus wavelengths.

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B. AO-Optimized Quasi-catenary Metagrating from Initial Catenary Structure

For the AO method, we employed four sets of different parameters, each of which is chosen at equally spaced wavelengths within the target range. As shown in Fig. 3(a), for different parameter sets, the finally optimized efficiencies have distinct variations. For example, the optimized relative efficiency of the third parameter set (AO III) was 91% at a wavelength of 6 µm, while that of the second parameter set (AO II) is only 31%. This phenomenon demonstrates that the performance of the AO method also depends on the selection of optimized parameters. As shown in Fig. 3(b), the deflection efficiency of the metagrating using the fourth parameter (AO IV) set can be larger than 80% from 6.1 µm to 10.46 µm, and the highest deflection efficiency is 94.15% at 9.76 µm. It is worth noting that the performance of the AO method (4.36 µm) is better than that of the MO method (4 µm). Figure 3(c) shows the AO evolution of an optimized quasi-catenary metagrating for different wavelengths. During the entire optimization process, the efficiency curves for 8.5 µm and 10.5 µm consistently increase. However, the efficiency curve for 6.5 µm has a clear reduction process in the iterations. It confirmed that, for the AO method, the average gradient can only ensure an overall increase in the average efficiency, but not the individual wavelength. In fact, it is challenging to maintain a consistent upward trend for all three wavelengths simultaneously. It is worth mentioning that, as shown in the insets in Fig. 3(c), the optimized structure shows a quasi-catenary structure. As shown in Fig. 3(d), the fourth parameter set (AO IV) has a highest efficient bandwidth of 4.36 µm and the other three sets have an efficient bandwidth of about 3 µm. The AO method yields significant improvements in the efficient bandwidth. Figure 3(e) displays the deflection angle curves of the catenary metagrating in the given bandwidth range from 6 µm to 12 µm.

 

Fig. 5. (a), (b) Topology optimization results with AO method. Insets: real part distribution of ${{\rm E}_{\rm{RCP}}}$ in the $x - z$ plane for incident light at 10.6 µm. (c) AO IV optimization processes. Optimized structures in the first and 100th iterations. (d) Efficient bandwidth performances for AO method. The incident wavelengths are [6 µm, 8 µm, 10 µm] (AO I), [7 µm, 9 µm, 11 µm] (AO II), [6 µm, 8.5 µm, 11 µm] (AO III), and [6.5 µm, 8.5 µm, 10.5 µm] (AO IV). (e) Deflection angle versus wavelengths.

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C. MO-Optimized Quasi-continuous Metagrating from Initial Discrete Structure

To demonstrate the generality of the above approaches, we further designed the freeform broadband high-efficiency metagratings with different initial structures and materials. Figure 2(a) displays the absolute and relative diffraction efficiencies for the initial discrete structure. As shown in the inset, the real part of the ${{\rm E}_{\rm{RCP}}}$ profile indicates that only a few parts of incident light are deflected in the 70° deflection direction. It can be seen that, compared with the initial catenary structure, the initial discrete structure has a lower efficient bandwidth as the traditional design methods do not have the capability to account for lattice coupling among meta-atoms. Therefore, as the topology optimization approach is a kind of local optimization algorithm, such structure is not a good choice for the initial structure.

The optimization results using the SO method based on the initial discrete structure are shown in Fig. 4 (b). With the increase of optimized wavelength, the peak intensity shifts to the larger angle and finally close to the pre-designed 70° deflection. By optimizing the deflection efficiencies at wavelengths of 7.5 µm and 9 µm simultaneously, the bandwidth is improved from 3.1 µm and the highest deflection efficiency is 92.9% at 7.7 µm, as shown in Fig. 4(c). Figure 4(d) displays the MO evolution curves of the discrete metagrating optimization process for 7.5 µm and 9 µm optimized wavelength. The inset shows the topology structure of the final optimized metagrating, which exhibits the quasi-continuous structure. As shown in Fig. 4(e), the efficient bandwidth of the optimized quasi-continuous metagratings using SO I, SO II, SO III, SO IV, and, SO V are 0.9 µm, 2 µm, 2.6 µm, 1.8 µm, and 2.4 µm, respectively. Compared with the optimization results from the initial catenary structure as presented in Fig. 2(d), the efficient bandwidth optimized from the initial discrete structure is as low as 3.1 µm. Therefore, it is demonstrated that the quasi-catenary metagrating optimized from initial catenary structures has the evident advantage on bandwidth compared to that optimized from initial discrete structures. Figure 4(f) displays the deflection angle curves of the catenary metagrating in the given bandwidth range from 6 µm to 12 µm.

D. AO-Optimized Quasi-continuous Metagrating from Initial Discrete Structure

Figure 5(a) shows the deflection efficiency of an optimized freeform quasi-continuous metagrating from the initial discrete structure with the AO method under different parameters. Figure 5(b) displays the best optimization performance of the AO method, where the deflection efficiency is significantly improved in the whole bandwidth. Specifically, the efficient bandwidth is primarily confined within the range from 7.68 µm to 10.5 µm. Figure 5(c) shows the efficiencies at three different wavelengths, demonstrating that the efficiencies of discrete geometric phase metagratings all experience a notable increase. Moreover, the optimized structure shows a quasi-continuous structure. As shown in Fig. 5(d), compared with the initial discrete structure, the efficient bandwidths of all four optimization sets are significantly improved. However, it remains lower than the quasi-catenary metagrating as shown in Fig. 3(d), indicating again the advantage of catenary structure for topology optimization. Moreover, Fig. 5(e) displays the deflection angle curves of the catenary metagrating in the given bandwidth range from 6 µm to 12 µm. Notably, the efficiency of the optimized metagrating using the AO approach keeps its efficiency close to 80% at the central wavelength of 10.6 µm, as shown in Fig. 5(b). In contrast, the efficiency of the optimized grating using the MO approach has dropped to about 70% at the central wavelength of 10.6 µm, as shown in Fig. 4(c). Even though the metagrating using the MO method has a slightly wider bandwidth, it is suggested that the AO approach is more effective for optimizing metagratings for broadband high-efficiency applications.

4. DISCUSSION AND CONCLUSION

In conclusion, different topology optimization methods, including SO, MO, and AO, are comprehensively investigated for the inverse design of broadband optical metagratings with high-efficiency large-angle deflection. Specifically, two different types of metagrating structures, namely, continuous catenary metagrating and discrete geometric phase metagrating, are used as the initial structures of topology optimization. As a result, a quasi-catenary mtagrating with a ${\sim}{70}^\circ$ deflection angle at a central wavelength of 10.6 µm and a high efficiency exceeding 80% over a bandwidth of 4.36 µm is inversely designed with the AO method in the infrared range. It is also shown that, compared to SO and MO, the AO approach is more appropriate to optimize the broadband high-efficiency metagrating. Moreover, it is demonstrated that the catenary structure is a better choice for the topology optimization problem of a broadband metagrating than the discrete structure, as the topology optimization approach is a kind of local optimization algorithm.

The freeform metagrating can be fabricated with conventional processing techniques. For example, the freeform Si metagrating can be fabricated by using electron beam lithography and a top-down etching technique. First, a photoresist is spin-coated onto the Si film and patterned using electron beam lithography. The desired structures are etched into the Si layer using the reactive ion etching process. After removing the photoresist, the final freeform Si metagrating is obtained. Our results provide helpful insights into the inverse design of broadband metasurfaces and meta-devices, which are not limited to the infrared range but are also applicable to the visible spectrum and other wavelength ranges.