Metasurfaces, composed of subwavelength artificial elements with periodic or non-periodic distributions, possess the remarkable ability to flexibly control the polarization, amplitude, and phase of electromagnetic (EM) waves by harnessing the field discontinuities at interfaces [1–4]. They offer the capability to efficiently reshape the wavefront of incident waves through deliberate design of phase distribution in microwaves, enabling various applications such as holograms [5,6], vortex waves [7,8], metalenses [9,10], array antennas [11,12], and anomalous reflection [13,14]. In addition, they are also widely used in visible or near-infrared frequency bands [15,16]. Among them, anomalous reflection is one of the most fundamental applications, as it directs incident waves to the desired direction. It manifests important applications in fields such as communication, stealth, and radar technology.

To date, extensive research has typically focused on exploiting metasurfaces to achieve anomalous reflections, specifically by applying the generalized Snell’s law to obtain an accurate linear gradient phase, known as the local approach [17–19]. This gradient phase is usually realized with periodically arranged supercells. However, phase variations introduced by supercells would cause unexpected high-order modes, making it challenging to effectively steer all incident energy to the desired direction. Such an issue of reduced efficiency caused by the local method becomes more pronounced, especially for larger steering angles [20,21]. To address this issue, numerous studies have been conducted based on employing a coding metasurface (CM) and meta-grating technique to improve efficiency at large angles [22–25]. For example, in Ref. [22], a metasurface with two different sized meta-atoms was employed, which primarily enhanced the efficiency to 80%. A 3-bit CM was designed to realize an efficiency of 90% at an elevation angle of $\theta =70°$ [23]. An efficiency of 88.8% was achieved based on a nonplanar groove meta-grating at $\theta =70°$ [24], and a passive aperiodic grating was used to achieve all-angle scanning perfect reflection within $\theta =75°$ [25]. It is worth noting that the meta-grating technique suffers from the issue of complex design and larger electrically sized arrays. Although nearly all of the above approaches have contributed to overcoming the current challenges, they primarily focused on one-dimensional (1D) steering of $\theta$ at an azimuth angle of $\phi =0°$ and gave little consideration for 2D steering involving both $\theta$ and $\phi$. Achieving 2D beam steering, especially with both high efficiency and a large angle, is an even more challenging task. At present, the exploration of inverse design for metasurfaces has triggered substantial attention, aiming to obtain the desired EM performance and functions by effectively optimizing the architecture within a complex and high-dimensional input parameter [26–30]. Nevertheless, there is currently a lack of inverse design of metasurfaces using the input parameters $\theta$ and $\phi$ to achieve arbitrary 2D anomalous reflection with a large angle and high efficiency.

Here, genetic algorithm (GA) is employed for the first time using an inverse design approach to optimize phase distribution of the CM for achieving arbitrary 2D anomalous reflection with high efficiency and a large angle. A novel four-leaf rose-shaped reflective meta-atom is designed to achieve high reflectance and full 360° phase coverage at 8–14 GHz. The optimized CM enables a high efficiency ($\ge 90\%$) and a large angle of anomalous reflection within the range of $\theta \le 70°$ and $0°\le \phi \le 360°$ under normal incidence. Numerical results are consistent with the experimental ones, showing the effectiveness of GA in designing and optimizing CM. The most significant advantage of this work is 2D beam steering with high efficiency and large angles, which offers a practical solution for phased arrays, stealth, and other scenarios requiring precise beam control within a large range.

According to the generalized Snell’s law, ${\theta }_r$ can be determined as follows: $${\theta }_r=\arcsin ({\lambda }_0/nP).$$(1)

Here, ${\lambda }_0$ is the wavelength at the center frequency, the length of each element is $P$ in $x$- and $y$-directions, and $n$ is the element number in a supercell. Since the number of elements is an integer, there is a discrepancy between the target angle and the actual angle for 1D beam steering. The supercell becomes electrically large when the target angle is small, often leading to limited numbers of supercells within fixed physical dimensions and thus causing significant errors. In contrast, the supercell should be ideally much shorter at large target angles; however, this is often practically prohibitive, which ultimately limits the realization of large angles.

Assuming a coding metasurface is composed of $N\times N$ meta-atoms, a design approach based on the far-field scattering theory is applied in this work to achieve arbitrary 2D beam steering with a large angle and high efficiency. When the CM is normally illuminated by a plane wave, as shown in Fig. 1, the far-field scattering can be expressed as [31] $$F(\theta ,\phi )=\sum _{m=1}^N\sum _{n=1}^N\exp \{-i[{\phi }_{m,n}+k_{0}P(m-1/2)\sin \theta \cos \phi +k_{0}P(n-1/2)\sin \theta \sin \phi ]\}.$$(2)

Here, ${\phi }_{m,n}$ represents the abrupt reflection phase of meta-atoms in the $m$-th row and $n$-th column of CM; $k_0$ is the wavenumber in the free space. To obtain the phase distribution corresponding to the specific $\theta$ and $\phi$, Eq. (2) is constructed as the new fitness function to iteratively optimize the target far-field pattern, enabling GA to continuously maximize the desired beam intensity. Therefore, $\theta$ and $\phi$ in Eq. (2) are used as inputs for the GA. The core of this work is to iteratively search for the optimal phase arrangement $\phi (m,n)$ that maximizes the scattered field at a specific angle based on Eq. (2). This optimization problem is thereby translated into finding the maximum value of $F(\theta ,\phi )$, where the fitness function is represented as $$\text{Fitness}=F(\theta ,\phi ).$$(3)

To achieve arbitrary 2D anomalous reflection, a low-profile meta-atom needs to be designed to reduce the coupling between meta-atoms, with a simple structure that allows full phase coverage by adjusting only a few parameters. Here, a four-leaf rose-shaped meta-atom is designed, and meta-atoms with varying petal lengths are arranged in different configurations, as shown in Fig. 1. The meta-atom comprises a metallic pattern, a dielectric spacer, and a metal backplate. Among them, the metal pattern is a four-leaf rose-shaped structure modeled by the equation $f(\phi )=a·\sin (2\phi )$; $0\le \phi \le 2\pi$ and is chosen for isotropic parametric control and thus dual-polarization operation. The dielectric layer is F4B material with a relative permittivity of 2.65 and a tangent loss of 0.001. Both the pattern and backplate layer are made of copper film with a thickness of 0.035 mm and a conductivity of $5.8\times {10}^7\mathrm{S}/\mathrm{m}$. To minimize coupling between the meta-atoms of the coding metasurface optimized by GA, a period of $0.3\lambda$ and a thickness of 0.05 $\lambda$ have been optimized. The thickness $t$ and period $P$ of the dielectric layer are 1.5 and 10 mm, respectively. As a result, a low-profile meta-atom design has been achieved. It is worth noting that the structural parameters are not directly related to the $\theta$ and $\phi$. Instead, according to the equivalent circuit analysis, different parameters induce changes in the phase of the meta-atoms; then by arranging the meta-atoms in different ways, $\theta$ and $\phi$ can be controlled.

GA is a global optimization algorithm that simulates the process of biological evolution, which has a stronger global search capability due to its parallel search over a population and genetic operations such as selection, crossover, and mutation. This allows GA to avoid local optima more effectively. The parallel nature and adaptability of GA generally make it more efficient and robust when dealing with complex, nonlinear, and multi-modal optimization problems. Figure 3 shows the flowchart of GA optimization. Initially, 30 populations are generated, each consisting of $35\times 35$ binary-coded chromosomes. $\theta$ and $\phi$ are input into Eq. (2) to calculate the fitness of the far-field scattering patterns for these populations. The far-field patterns are summed at specific angles, and the absolute values are squared. The fitness values for 30 populations are then calculated and ranked, and the chromosomes with higher fitness values are then selected. If the required fitness is not met and the maximum iterative algebra is not reached, the far-field intensity at specific angles is continuously optimized for maximization, while the total energy at other angles is minimized. Stochastic universal sampling, multi-point crossover, and chromosome mutation are employed to enhance optimization efficiency. The crossover rate for the multi-point crossover operation is set to 0.7, and the mutation rate $P_m$ is defined as 0.7/Lind. Here, $P_m$ is inversely proportional to the length of the chromosome (denoted by Lind), meaning that as the chromosome length increases, the probability of mutation for each gene decreases. Then, the offspring are reintegrated into the parent population to guarantee chromosomal diversity. Finally, the GA optimization process ends when the desired fitness level is achieved and the maximum iterative algebra is reached.

Here, eight $35\times 35$ CMs are optimized in two scenarios (one scenario involves the same $\phi$ with different $\theta$, while the other involves the same $\theta$ with different $\phi$) without loss of generality. In scenario one, four CMs with $\theta =40°$, 50°, 60°, and 70° ($\phi =330°$) for anomalous reflection are optimized and then simulated in CST Studio Suite. The CMs are illuminated by normally incident $x$-polarized plane waves along $z$-axis, where the boundary is set to open add space along $x$- and $y$-axes. Figure 4(a) shows the optimized phase distributions of four CMs when the fitness function evolution curve stabilizes near 4000 iterations; see Fig. 4(d). These phase distributions present a quasi-oblique-line feature with different slopes. Figure 4(b) depicts a 2D far-field scattering pattern at 10.3 GHz. The brightest circular spot represents maximum energy of far-field intensity, and its direction angles all perfectly align with the theoretically predesigned ones. Meanwhile, Fig. 4(c) shows the normalized cross-section far-field scattering pattern at $\phi =330°$; it is evident that all optimized CMs reflect the incident $x$-polarized plane waves to predetermined directions. Here, efficiency is defined as the ratio of the power of the main reflected beam to the total incident energy. Then, the efficiencies in four cases are calculated as 97%, 97%, 96%, and 90%, respectively. Consequently, this demonstrates the effectiveness of GA in optimizing phase distributions for achieving high-efficiency ($\ge 90\%$) and large-angle ($\le 70°$) anomalous reflection.

In scenario two, four CMs with $\phi =60°$, 150°, 240°, and 330° ($\theta =30°$) for anomalous reflection are optimized and then simulated. Figure 5(a) shows the optimized phase distributions of four CMs and this optimization process can be seen in Fig. 5(d). The fitness function evolution curve remains stable near 4000 iterations. Again, quasi-oblique-line phase distributions were observed with different slopes. The 2D far-field scattering patterns of $x$-polarized reflected waves are displayed in Fig. 5(b). The brightest spot indicates maximum far-field energy, and the target angle has a good agreement with the predetermined angle. The normalized cross-section far-field scattering patterns at $\theta =30°$ are presented in Fig. 5(c). The efficiencies in four directions are 93%, 95%, 93%, and 95%, respectively. Thus, the results further indicate that optimizing the phase distributions of CMs using GA can achieve arbitrary 2D anomalous reflection with high efficiency ($\ge 90\%$) and a large angle range ($\theta \le 70°$ and $0°\le \phi \le 360°$). It is worth noting that, in general, it is a more challenging problem to achieve a large $\theta$ of reflection, so that arbitrary two-dimensional direction within the maximum $\theta$ can be achieved in theory in both scenarios. We also assume that the difference between the theoretical calculation angle and the actual simulation angle is reasonable within the range of $\pm 2°$. Therefore, a minimum angular increment of 1° can still achieve a specific two-dimensional reflection, but finer precision is generally not considered, as it is limited by the accuracy of the simulation software and the angle deviation caused by unavoidable errors in experimental processing.

Table 2 compares the key characteristics of our work with that of previous work. Our work shows comprehensive advantages in both efficiency and arbitrary 2D angle. Specifically, even though some work focuses on efficiency or large-angle steering, they are all limited to 1D direction. In contrast, our work simultaneously considers efficiency, angle, and 2D direction by using GA.Table 2.

Comparison of Performances between This Study and Previous Research

For verification, two prototypes corresponding to case IV in Fig. 4 and Fig. 5 for 2D anomalous reflection at $\theta =30°$ and 70° ($\phi =330°$), respectively, are fabricated and experimentally characterized. The measurement setup and photographs of samples are shown in Fig. 6(a). A 2D rotary table is used to measure the far-field scattering patterns of samples, where a received horn antenna was fixed on an automatically rotated arm around the sample within $-90°$ to 90° in steps of 1°. The horn antenna acts as the transmitter, positioned 1.6 m from the metasurface to satisfy the far-field condition. In the experiments, the sample is rotated along $z$-axis by $-30°$ to measure the scattering field pattern of the 2D anomalous reflection. Therefore, we only need to rotate the arm of the turntable to test $\theta$ within the plane of [$-90°$, 90°] once $\phi$ is determined. Simulated and measured normalized far-field scattering patterns are displayed in Figs. 6(b) and 6(c) under $x$-polarized plane wave excitation. As expected, the patterns exhibit a generally good agreement in both cases, which proved that our optimized metadevice is able to achieve 2D reflection with a large angle. Meanwhile, a slight angle deviation of only $\pm 2°$ is observed between numerical and experimental results. In addition, the efficiency is over 95%/92% for simulation/measurement at $\theta =30°$, but is over 90%/86% at $\theta =70°$. Such minor deviation and slightly reduced efficiency are mainly attributed to inevitable errors in fabrications and EM interferences in the experimental setup. To sum up, the above results demonstrate an excellent performance in both efficiency and arbitrary angle for 2D reflection.