Rotational, Spatial, and Frequency Robustness of the Modified MML. The traditional MML has the advantages of large varifocal range, simple structure, and ease of tuning. However, in practical applications, it still presents certain limitations. For instance, its focal length cannot be continuously adjusted, and it requires high precision in the alignment of the two metasurface components. To overcome these shortcomings, we have developed a metasurface based on the traditional MML but with a completely new phase distribution, resulting in a smoother phase distribution as illustrated in Fig. 1(a). Traditional MML is composed of two identical metasurfaces placed "face to face." The joint transmission function of the two combined conjugate metasurfaces (MS1 and MS2) can be expressed in polar coordinates as follows32:
$$\:{T}_{1}\left(r,\phi\:\right)=\text{exp}\left(round\left(ia{r}^{2}\phi\:\right)\right)$$
$$\:\begin{array}{c}{T}_{2}\left(r,\phi\:\right)=\text{exp}\left(\text{r}ound(-ia{r}^{2}\phi\:)\right) \left(1\right)\end{array}$$
where a is a constant, and r and \(\:\phi\:\) are polar coordinate parameters measured from the center of the phase plate. When MS2 rotates relative to MS1 by an angle θ around the optical axis, the focal length of the combined metalens can be determined as:
$$\:\begin{array}{c}f\left(\theta\:\right)=\frac{\pi\:}{\theta\:}{F}_{0} \left(2\right)\end{array}$$
where \(\:{F}_{0}=1/a{\lambda\:}_{0}\). The Eq. (2) shows that by simply changing the relative rotation angle θ, the overall phase distribution of the combined metalens can be easily adjusted. However, when the relative angle in Eq. (2) is 0, the combined lens cannot focus, making it impossible to continuously adjust the focal length front and back in the initial state. To address this shortage, we can introduce the modified MML with an offset term in the phase distribution of the lens, allowing the combined metalens to focus around an offset focal length. By adding the same offset phase factor Foffset to both metasurfaces elements, the phase distributions of the two metasurfaces, MS1 and MS2, become:
$$\:{T}_{1\text{m}}\left(r,\theta\:\right)=\text{exp}\left(round\left(ia{r}^{2}\theta\:\right)\right)\text{e}\text{x}\text{p}\left(\frac{i\pi\:}{2{\lambda\:}_{0}{F}_{\text{offset}}}{r}^{2}\right)$$
$$\:\begin{array}{c}{T}_{2\text{m}}\left(r,\theta\:\right)=\text{exp}\left(\text{r}ound(-ia{r}^{2}\theta\:)\right)exp\left(\frac{i\pi\:}{2{\lambda\:}_{0}{F}_{\text{offset}}}{r}^{2}\right) \left(3\right)\end{array}$$
$$\:\begin{array}{c}{f}_{m}\left(\theta\:\right)=\frac{\pi\:}{\theta\:+\pi\:}{F}_{\text{offset}} \left(4\right)\end{array}$$
By comparing Eq. (2) and Eq. (4), we can observe that the most immediate effect of the offset term is the adjustment of the varifocal curve through the offset factor \(\:{\text{F}}_{\text{offset\:}}\) (more details in Supplementary Material S1). As shown in the Eq. (4) and Fig. 1(b-c), this modified MML achieves a continuous focal range throughout the entire rotation period. More importantly, the tuning smoothness (df/dθ) is significantly improved. This characteristic of modified MML imparts strong rotational robustness and enables more precise focal length adjustment ability. Due to the intrinsic sector effect of the Moiré lens, as the rotation angle increases, the focusing efficiency decreases. Generally, the best focusing efficiency is near 0 degrees, allowing the modified MML to be used more effectively in small-angle regions. Furthermore, the traditional MML suffers from greater sector loss, resulting in significantly lower focusing efficiency (as shown in the dotted line of Fig. 1(e)). When minor misalignments occur during use, the focusing ability of the traditional MML cannot guarantee the quality of the beam spot. In contrast, the addition of the offset factor endows the modified MML with better spot maintenance capabilities under increased misalignment, as indicated by the solid line in Fig. 1(e). This improvement is due to the rational design of the offset factor, enhancing the modified MML's tolerance to Z-axis misalignment and allowing for optimization based on common misalignment levels in practical scenarios. By balancing these factors, we ensured that the modified MML maintains a continuously adjustable focal length and stable spot quality within the 0–17λ range (corresponding to 0-2mm, which generally meets practical needs). Additionally, we employed geometric phase modulation for the Moiré metasurface, ensuring that its electromagnetic wave modulation is independent of the incident wave frequency and material properties, thereby maintaining high focusing efficiency across a broad bandwidth (as shown in Fig. 1(f-g)). Consequently, the metalens achieves ultra-broadband focusing (focusing efficiency > 40%, bandwidth ~ 1THz, ~ 40% of the central frequency \(\:{\omega\:}_{\text{c}}\))(see the calculation of focusing efficiency in Supplementary Material S2). In summary, by modifying the traditional MML we designed a highly robust metalens with rotational, spatial, and frequency robustness, ensuring exceptional practical value in various application scenarios.
Concept and design of Modified Varifocal Terahertz MML. In an integrated terahertz system, such as the broadband THz-FEL utilized in our experiments, maintaining tunable terahertz light output while achieving high integration and miniaturization requires a device capable of modulating the amplitude and phase of terahertz light across a wide bandwidth. As shown in Fig. 2, to meet this demand, we designed and fabricated a broadband varifocal lens operating at terahertz frequencies (center wavelength 2.52 THz) based on the aforementioned modified MML.
The phase distribution and focal length of the lens are adjusted by the relative rotation of two metasurfaces with geometric phases. As illustrated in Fig. 2(a), within one rotation cycle, the varifocal lens achieves an approximate 800-times wavelength (f0 = 118.8 µm, Δf = 95 mm) varifocal range. Although the theoretical varifocal range of a MML is infinite, practical adjustments at large focal lengths are challenging and the focusing efficiency is difficult to maintain, so we focused on the easily adjustable and practically useful range. The inherent broadband robustness of the geometric phase enables stable phase modulation over a certain wavelength range, allowing the metalens to achieve broadband focusing.
As shown in Fig. 2(b), the optical paths on both sides show that our metalens can effectively focus incident light of different frequencies (\(\:{\omega\:}_{1}\), \(\:{\omega\:}_{2}\)), with a bandwidth approximately 40% of the central wavelength \(\:{\omega\:}_{c}\). The optical paths on the left means that by adjusting the rotation angles appropriately, the metasurface lens can focus incident light of different frequencies at the same position (i.e., 𝑓0(\(\:{\omega\:}_{1}\))=𝑓1(\(\:{\omega\:}_{2}\))). The combined characteristics of broadband and variable focus enable the lens to maintain a fixed focal length for incident light at different wavelengths. This simplifies the coupling of a tunable wavelength light source with the lens, allowing for the maintenance of collimated light output through simple lens rotation, significantly enhancing the practicality of frequency-tunable terahertz light sources. Variable focusing, broadband focusing, and fixed focusing are the three main functionalities of our modified MML that will be analyzed and verified below.
Four modified MML metasurfaces fabricated with the SOI process are shown in Fig. 3(a) (details of the fabrication process can be found in the Methods section). The single metasurface is composed of a \(\:{\text{S}\text{i}\text{O}}_{2}\) layer on a silicon substrate, with a geometric phase layer on top, as depicted in the top view of Fig. 3(b). Each microstructural unit, shown in Fig. 3(b), consists of a \(\:{\text{T}}_{{\text{S}\text{i}\text{O}}_{2}}\)=2µm thick \(\:{\text{S}\text{i}\text{O}}_{2}\:\)isolation layer embedded between the high-resistivity silicon substrate and the rectangular silicon pillar. The array of sub-wavelength microstructural units that form the metasurface lens is identical in size, and the modulation of the incident wavefront is achieved solely by altering the orientation angle (α) of the rectangular silicon blocks. For our design with a central operating wavelength of λ = 2.52 THz, we employed COMSOL Multiphysics software to simulate and optimize the geometric parameters of the sub-wavelength microstructural units. To achieve high and stable cross-polarized wave amplitude transmittance and 2π phase modulation while minimizing the coupling effects between adjacent structural units, we determined the geometric parameters of the unit cell as follows: sub-wavelength microstructure period (P = 0.394λ), length (L = 0.326λ), and width (W = 0.09λ). Considering the fabrication difficulty due to the aspect ratio of the rectangular silicon blocks and aiming for the highest possible amplitude transmittance, we selected a thickness (TSi=0.552λ) (details in Supplementary Material S3).
The microstructural unit array can convert circularly polarized (CP) incident waves into the opposite chirality polarization state, accompanied by a phase change of (ϕ = 2α). Typically, when the microstructural units of a metalens provide eight discrete phase values—0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°—the lens can achieve excellent focusing and imaging performance. As shown in Figs. 2(c) and 2(d), under the specified geometric parameters, the microstructural units can provide linear 2π phase modulation and at least 20% transmittance over a bandwidth exceeding 3 THz. The transmittance decreases as the incident light deviates from the central wavelength, which is the primary limiting factor for the working bandwidth of the metalens.
Experiments on Varifocal length, Broadband, and Fixed Focusing Based on THz-FEL. The performance of our modified MML was demonstrated with the high-quality, high-power tunable THz-FEL apparatus (0.1–4.2 THz, > 10 W). As depicted in Fig. 3(a), the system comprises the FEL light source, a beam collimation system, an analyzer, a polarizer, a modified MML, and a terahertz camera (LET, TL-TC160S2, resolution 160×120, pixel size 25 µm). The terahertz camera can be translated back and forth to locate the precise focal point at different relative angles of the two metasurface elements (usually, the metasurface lens exhibits a bifocal phenomenon, but we only focus on one of them due to significant intensity differences). We performed multiple measurements around the estimated focal point, where the square of the Gaussian beam's FWHM (full width at half maximum) varies quadratically with the propagation distance: \(\:{w}^{2}\left(z\right)/{w}_{0}^{2}-{z}^{2}/{z}_{0}^{2}=1\).
As shown in Fig. 4(b), at θ=-90°, we first conducted a two-dimensional Gaussian fit on each measured spot to obtain the FWHM at each position, then performed a quadratic fit of the square of the FWHM against z to determine the focal length and FWHM at this angle θ (details in Supplementary Material S4). According to Eq. (4), our MML theoretically possesses an infinite varifocal range, which is a significant advantage over other varifocal lenses. However, within the range approaching − 180 °, even slight changes in angle can result in substantial focal length adjustments. To obtain accurate validation results, we conducted experiments at various angles within the − 90° to 270° range. The results are illustrated in Fig. 4(c) (notably, the focal lengths for angles exceeding 180 degrees could not be measured due to the size constraints of the camera housing). Simulation results were calculated for the light intensity distribution in the x-z propagation plane and the x-y focal plane at different rotation angles of the metasurface. Given the terahertz camera's pixel size of 25 µm, the FWHM of the spot might have pixel-level errors. Figure 4(c) shows that as the rotation angle increases, the focal length decreases, which can be derived using Eq. (4). As the focal length decreases, the FWHM of the spot becomes smaller. The experimental results agree well with the simulations, demonstrating a varifocal range of 988 λ − 188 λ (117 mm-22 mm), with a zoom ratio exceeding 5 times.
Figure 4(d) presents the bandwidth test results of the metasurface lens, where experiments selected five frequency points: 1.98 THz, 2.3 THz, 2.52 THz, 2.8 THz, and 3 THz. Using the same data processing method as in Fig. 4(c), we obtained the focal lengths and spot FWHM for different incident frequencies. Within approximately 1 THz bandwidth, the focal length of the metasurface lens increases with the incident frequency while the FWHM remains constant. Similarly, in the fixed-focus function test shown in Fig. 4(e), for these five frequency points, we set a fixed focal length as the reference focal length F0 for w = 2.52 THz and θ = 0°. The higher frequency of the incident light increases the lens's focal length, which can be compensated by increasing θ. Thus, by increasing the angle θ, a constant focal length can be maintained as the incident light frequency increases. For different wavelengths, the corresponding angles are θ=-40.7°, -14°, 0°, 17.5°, and 33.9°. This enables focusing of a broadband light source at the same position through approximately 75° rotation of the metasurface lens. Given that most light sources output divergent light, often with varying divergence angles, and require initial collimation, which can degrade source quality and occupy significant space, an ultrathin metasurface lens with broadband fixed-focus capabilities can be seamlessly integrated at the source's emission port. This allows for the synchronized rotation of the lens while adjusting the source frequency, achieving a consistent high-quality collimated light output.
The Strong Spatial Robustness of the Terahertz modified MML. The designed varifocal THz modified MML consists of two identical metasurfaces, MS1 and MS2. The phase distribution in Eq. (3) represents the ideal condition when MS1 and MS2 are perfectly aligned, overlapped at the center with zero axial distance. In practice, to prevent microstructure damage during the relative rotation of two metasurface, a certain distance must be maintained between MS1 and MS2. However, due to diffraction effects, the gap between the two metasurfaces can cause phase distortion in the metalens, affecting its focusing performance (see the calculation of focusing performance in Supplementary Material S5). As Wei et al. have shown34, increasing the distance between the metasurfaces reduces the focusing efficiency of the varifocal lens, and beyond a certain distance, the focusing performance degrades rapidly and becomes ineffective. Our proposed modified MML solves this problem and largely increases the axial displacement tolerance threshold.
In the following analysis, we use numerical simulations and experiments to investigate the impact of the axial distance z on the focusing performance of the varifocal THz modified MML, using a simple yet general case where the relative rotation angle of MS1 and MS2 is θ = 0° (equivalent to θ = 180° in traditional MML). As shown in the spot distribution in Fig. 5(a), the focusing efficiency of the MML significantly decreases with increasing axial distance. Additionally, the focusing efficiency loss due to sector effects exacerbates the focusing challenges for traditional MML under this axial displacement. Compared to the modified MML, the spot size of traditional MML shows more pronounced elongation with the increasing axial distance, becoming spindle-shaped and degrading spot quality around 20λ.
Considering the equivalent phase addition of the two lenses, diffraction effects cause the phase distribution of MS1 to alter as it propagates a distance z before reaching the input plane of MS2. Figure 5(a) demonstrates that as z increases, the phase distribution ϕ1 reaching the input of MS2 becomes increasingly blurred, making the concentric ring phase of the two metasurfaces more indistinct. Simulated phase distributions predict that increasing z impairs the metalens's focusing ability. Experimental adjustments of the axial distance z confirm that the metalens maintains good focusing performance even beyond 42λ (~ 5 mm), though with some degradation, aligning with simulation results.
Furthermore, given the metasurface unit size (P = 46.8 µm) (1 Unit = 46.8 µm), misalignments in the x or y directions during actual use can also affect the lens's focusing performance. Similar simulations and experimental records were used to examine the focal spot when the two metasurfaces are misaligned transversely. Figure 6(a) shows that as the displacement increases, the focal spot gradually shifts off-axis along the x direction and begins to deform. Additionally, the poor robustness of the traditional MML causes it to lose focus more quickly. This asymmetry in the phase distributions ϕ1 and ϕ2 of MS1 and MS2 becomes more pronounced with greater misalignment. Both simulation and experimental results indicate that when the displacement is 4 pixels (187 µm), the focusing effect remains relatively good. However, at an 8-pixel displacement (374 µm), the focal spot becomes distorted and shifts to one side, reducing the focusing effect but still achieving high efficiency. To ensure optimal focusing performance, misalignment between the two metasurfaces should be minimized.
In conclusion, the modified design significantly enhances the robustness of the combined lens, particularly increasing tolerance to axial distance z. The robust MML can maintain excellent focusing performance even with axial gaps greater than 42λ and lateral deviations exceeding 8 pixels.
Imaging Experiments with Modified MML. To enhance the performance characterization of our metasurface lens, we conducted additional experiments following the initial measurements of the lens's fundamental parameters. We placed a simple resolution target (with a stripe spacing and stripe thickness of 2 mm each) in front of the lens. Much like a traditional varifocal lens in visible light imaging systems that can magnify images by adjusting the angle of rotation, we devised a similar imaging setup, as shown in Fig. 7(a). A terahertz detector array with a larger field of view (384 × 288 pixels, 35 µm pixel pitch) was positioned at the focal distance 𝐹0 when 𝜃=0°, and the resolution target was placed in the far field. At 𝜃=0°, where the focal length equals the image distance, the resolution target was not discernible. However, as the relative rotation angle of the metasurface was increased, the image on the detector plane enlarged, and the stripes of the resolution target became increasingly distinct (as inset in Fig. 7(b)). Additionally, Fig. 7(b) demonstrates that the theoretically calculated magnification based on the focusing formula from Eq. (4) closely matches the experimental results. These imaging outcomes confirm that our metalens possesses terahertz-wavelength-scale imaging capabilities, making it well-suited for integrated and miniaturized imaging systems.