Dielectric Catenary Metasurface for Broadband and High-Efficiency Anomalous Reflection

Abstract

This paper proposes a broadband and high-efficiency anomalous reflection device based on a dielectric catenary metasurface, addressing the bottleneck problems of low efficiency and narrow bandwidth in traditional discrete metasurfaces. By designing a silicon-based equal-strength catenary structure, the efficient control of circularly polarized light beams within a wide angular range in the infrared band has been achieved. Simulation results show that the designed metasurface exhibits excellent beam steering performance when the deflection angle reaches 65°. Furthermore, to characterize the diffraction efficiency of the metasurface within a large angular range, the results indicate that under oblique incidence (0–60°), the diffraction efficiency of the ±1st order exceeds 80%, and the undesired higher-order diffractions are significantly suppressed. This ultrahigh working efficiency is attributed to the nearly perfect polarization conversion and continuous phase profile of the dielectric catenary structure. By combining catenary optics with the low-loss properties of the dielectric material, this design provides a new solution for the design of efficient, broadband, and wide-angle planar optical devices.

1. Introduction

Metasurfaces, as two-dimensional artificial electromagnetic materials, have opened new avenues for optical device miniaturization and integration through their capability to flexibly manipulate optical wavefronts at subwavelength scale [1,2,3,4,5,6]. Up to now, a variety of meta-devices with diverse functions have emerged, such as vortex light generators [7,8,9,10], metalenses [11,12,13,14], and polarization converters [15,16,17,18]. Among their functionalities, anomalous reflection—serving as the physical basis for beam steering, wavelength division multiplexing, and related applications—has emerged as a key research focus. According to the generalized Snell’s law, the anomalous reflection angle is determined by the discrete phase gradient at the interface. To achieve high efficiency and wide bandwidth, it is required that the phase gradient remains linear and continuous within a wide spectral range, which poses a severe challenge to the design of metasurface units. However, current designs face significant challenges in efficiency and bandwidth. For instance, plasmonic structures suffer from low energy conversion efficiency due to intrinsic ohmic and radiative losses [19,20,21], while their resonant nature inherently limits operational bandwidth [22,23].

Among these reported metasurfaces, geometric phase metasurfaces based on the Pancharatnam–Berry (PB) phase are particularly promising: rotating anisotropic nanostructures enable full 0-to-2π phase coverage while preserving structural symmetry [24,25,26,27,28]. However, conventional discrete units (e.g., nanopillar arrays) struggle to achieve the continuous linear phase gradient required for anomalous reflection due to phase sampling intervals and discretization errors, leading to higher-order diffraction noise and reduced efficiency [29]. The catenary curve, a mathematically intrinsic shape, enables continuous phase control across the 0–2π range and generates wavelength-independent phase shifts derived from the PB phase. Owing to this unique continuous phase modulation capability, catenary structures have been introduced into metasurface design [30,31,32,33]. By leveraging spatially varying geometric phases, catenary-based configurations achieve full 0-to-2π phase coverage within a single unit cell, effectively suppressing quantization errors and mode coupling losses induced by discretization [29]. Recent studies on all-metallic catenary configurations have demonstrated improved efficiency and bandwidth via quasi-continuous phase modulation [34,35]; however, they remain constrained by high metallic reflection losses and near-field coupling effects, hindering broadband high-efficiency performance. To address these limitations, dielectric metasurfaces have shown remarkable potential due to their low-loss characteristics, high refractive index contrast, and multimodal resonances [36,37,38,39,40,41,42].

Herein, we present a dielectric catenary metasurface for broadband high-efficiency anomalous reflection. By maintaining a fixed duty cycle and height while leveraging the low-loss characteristics of silicon-based dielectric metasurfaces, we systematically investigated the relationship between structural parameters and optical responses, achieving high polarization conversion efficiency across the operational wavelength range of 5–11 μm. Furthermore, the broadband anomalous reflection performance is validated, demonstrating that the structure retains exceptional light manipulation capabilities even at a deflection angle of 65°. Additional characterization of diffraction efficiency under wide-angle incidence (0–60°) reveals that ±1st-order diffraction efficiencies exceed 80%, while higher-order diffractions (e.g., ±2nd orders) are suppressed to near-zero levels. This work provides a novel strategy for designing high-efficiency broadband optical devices, offering promising applications in compact LiDAR systems and free-space communication.

2. Theory and Methods

The proposed metasurface comprises an array of catenary-shaped silicon beams patterned on a gold substrate, as illustrated in Figure 1a. Each catenary structure follows a “catenary of equal strength” profile defined by the following equation:

$$y={\displaystyle \frac{\Lambda }{\pi }}\ln (|\sec {\displaystyle \frac{\pi x}{\Lambda }}|)$$

(1)

where Λ is the horizontal length of a single catenary, and the range of x is (−Λ/2, Λ/2), as shown in Figure 1b. According to the PB phase principle, under circularly polarized illumination, the cross-polarized reflected wave will be imprinted with a geometric phase that is twice the inclination angle ξ(x) of the catenary curve. Therefore, the phase modulation along the non-uniform catenary profile can be defined as follows:

$$\Phi (x)=2\sigma \xi (x)=2\sigma \arctan ({\displaystyle \frac{dy}{dx}})={\displaystyle \frac{2\sigma \pi x}{\Lambda }}$$

(2)

where σ = ±1 represents left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) incident light. Compared with discrete structures, the catenary structure varies continuously along the coordinate, and a linear phase shift covering 0–2π is obtained (as x changes from −Λ/2 to Λ/2), which effectively reduces the high-order noise problem caused by discrete unit structures.

Moreover, this periodic catenary array functions as a blazed diffraction grating, capable of directing light into specific diffraction orders, as shown in Figure 1c, where p denotes the local period, w is the width of silicon beam, w/p represents the duty cycle. Under circularly polarized illumination at normal incidence, the catenary structure enables polarization conversion of the circularly polarized light, which is then reflected at a predetermined deflection angle. The anomalous reflection angle θ can be calculated using the following equation:

$$\theta =arcsin({\displaystyle \frac{\lambda }{2\pi }}{\displaystyle \frac{d\Phi }{dx}})=arcsin({\displaystyle \frac{\sigma \lambda }{\Lambda }})$$

(3)

3. Results and Discussion

As illustrated in Figure 2a, the catenary structure can be discretized into multiple subunits located on the gold substrate. The sub-unit has a period p, and the silicon (refractive index n = 3.4) beam has a height h = 1.3 μm and width w, with a fixed duty cycle w/p = 51.7%. Numerical simulations are performed using the finite element method (FEM) to evaluate the polarization conversion efficiency. Unit cell boundary is employed in the x- and y-directions, and open boundary is applied in the z-direction. The minimal mesh is set to be λ/10, and the permittivity of gold is from Ref. [43]. Propagation direction was aligned with the z-axis during electromagnetic wave excitation. To ensure simulation fidelity, the mesh refinement process maintained a minimum resolution of two grid points per nanostructural feature throughout the parametric sweep. This rigorous meshing protocol eliminated numerical dispersion artifacts while enabling convergence validation for subwavelength meta-atoms, thereby guaranteeing the physical accuracy of the extracted polarization conversion spectra. When the sub-units are illuminated by circularly polarized light, the simulated reflectance across the 4–14 μm spectral range is shown in Figure 2b. Notably, as the period p varies from 0.08 μm to 0.58 μm, the polarization conversion efficiency remains nearly invariant, exhibiting p-independent behavior, and sustains high performance (>0.8) across the 5–14 μm wavelength range. Remarkably, the efficiency approaches unity (≈1) within the 8–11 μm band, demonstrating exceptional operational stability.

Furthermore, numerical simulations were performed to demonstrate the broadband anomalous reflection of the catenary metasurface. Figure 3 displays the simulated electric-field and far-field distributions under LCP illumination at wavelengths of 5 μm and 10 μm. The results reveal that nearly all incident energy is deflected into the predetermined direction, with negligible power in non-targeted diffraction orders. Furthermore, even under a large deflection angle of 65° at the long wavelength, the structure maintains superior beam steering performance, validating its robustness for wide-angle applications. Notably, residual intensity at the zeroth order discernible in Figure 3a indicates minor unmodulated energy, primarily arising from unavoidable phase-matching errors during unit optimization and phase-distortion perturbations caused by near-field coupling between adjacent meta-atoms.

Figure 4 illustrates the diffraction efficiencies of various diffraction orders with wavelength under different incident angles. Here, diffraction efficiency is defined as the intensity ratio of diffracted light in a given order to the incident light intensity. We can observe that within the range of 0–60°, the diffraction efficiency of the +1 order remains remarkably stable, with an average efficiency exceeding 0.9 in the 5–11 μm wavelength range. Meanwhile, the intensities of the ±2 and −1 diffraction orders are suppressed to extremely low levels, being very close to 0. Additionally, as the incident angle increases, the efficiency of the 0-order diffraction increases gradually, and its average efficiency under a 60° incidence does not exceed 0.12, as depicted in Figure 4d. The results demonstrate that the designed dielectric catenary metasurface can maintain a high operating efficiency within the 5–11 μm band and exhibits strong wide-angle manipulation capabilities.

4. Conclusions

In summary, we demonstrate a dielectric catenary metasurface for broadband high-efficiency anomalous reflection. By maintaining a fixed duty cycle and height while harnessing the low-loss properties of silicon-based dielectric metasurfaces, we systematically investigate the relationship between structural parameters and optical responses, achieving high polarization conversion efficiency across the operational wavelength range of 5–11 μm. The broadband anomalous reflection capability is further validated, with results confirming exceptional beam-manipulation performance even at a deflection angle of 65°. Extended characterization of diffraction efficiency under wide-angle incidence (0–60°) reveals ±1st-order efficiencies exceeding 80%, while higher-order diffractions are suppressed to near-zero levels. The designed device can be fabricated using silicon dry etching processes. It is critical to emphasize that manufacturing precision must be maximized during fabrication: deviations in silicon beam width compromise local phase matching, while height inaccuracies degrade polarization conversion efficiency—both factors contribute to diminished device efficiency. We believe the proposed methodology may pave the way for future designs of broadband, high-efficiency, and wide-angle planar optical devices.