Polarization Independent Achromatic Meta-Lens Designed for the Terahertz Domain

In recent years, metasurface-based focusing elements have gradually become an indispensable type of terahertz lenses. However, the meta-lens often suffers from chromatic aberration due to the intrinsic dispersion of each element, especially in the broadband application scenarios. In this paper, we design and demonstrate a silicon-based achromatic meta-lens working from 0.6 to 1.0 THz, which is polarization insensitive because of the adopted symmetrical structures. The simulated focal length and the full width at half maximum (FWHM) of the foci at different frequencies prove the achromatic behavior of our meta-lens compared with the chromatic counterpart. We also show that the focus shift incongruence of our design originates from the transmission amplitude distribution of the meta-lens. This article not only provides an achromatic planar lens working at terahertz domain but also reveals the importance of the amplitude distribution in the achromatic metasurface design.


INTRODUCTION
In recent years, the terahertz frequency domain has ushered in extensive and in-depth development. Various terahertz techniques and systems have been proposed for sensing, imaging, and communication scenarios, showing promising application prospects [1]. In almost all terahertz systems, the lens is an important component that can directly determine the performance of the system [2,3]. Traditional terahertz lenses manipulate the output wavefront through phase accumulation along the propagation path via thickness variation, which is bound to a bulky size and curved shape, damaging the compactness of the lens. The metasurface, a 2D version metamaterial composed of an array of artificial subwavelength structures with the ability to engineer the phase, amplitude, and polarization features, has provided a new paradigm for the design of terahertz lenses. From the early metal gradient index terahertz lens to the all-dielectric meta-lens, which has attracted wide attention in recent years, metasurface-based focusing elements have gradually become an indispensable type of terahertz lenses. Compared with traditional terahertz lenses, terahertz meta-lenses composed of graphene [4,5], silicon [6,7], metals, etc. [8][9][10][11][12][13][14] can control the amplitude and/or phase distributions of the terahertz wavefront at quite a high freedom level to achieve novel focusing functions [15,16]. Moreover, most meta-lenses have a planar shape and wavelength order thickness with potential outstanding integration capability.
However, compared with the silicon-based traditional terahertz lens, the meta-lens still faces the challenge of large chromatic aberration introduced by the mode dispersion of each meta-atom. For example, as the most mature terahertz technology, the timedomain spectroscopy uses a broadband terahertz pulse to analyze the spectrum of various materials, which has an natural demand for achromatic lenses. Recently in the visible domain, a huge line of meta-lens research has been dedicated to achieving the dispersion engineering. For instance, Khorasaninejad et al. [17] carried out a series of scientific researches on the achromatic meta-lens with a reflection mode. Chen et al. [18,19] designed several achromatic meta-lenses with anisotropic nanostructures in the visible domain. Shrestha et al. [20] proposed an achromatic meta-lens working from 1,200 to 1,650 nm. Wang et al. [21,22] demonstrated an achromatic meta-lens through integrated resonances for imaging. Despite those inspiring advances in the visible light domain, terahertz metasurfaces are critically different from visible light metasurfaces in terms of raw materials and structure design. Only sporadic study of achromatic meta-lenses has been conducted in the terahertz domain; Cheng et al. [23] proposed an achromatic lens based on silicon metasurface with a high aspect ratio. Though the achromatic behavior was demonstrated, the efficiency of their terahertz meta-lens is somehow limited, and the dispersion engineering capability is polarization sensitive due to the adopted geometrical phase modulation method.
In this paper, we propose and investigate a silicon based metalens with a focal length of 11 mm and a diameter of 6.4 mm. Etched silicon pillars with 190 µm height are adopted as the unit cells to accumulate the required transmissive phase distribution. By changing the cross section of the pillars, alteration of the linear dispersion slop of each unit cell is achieved, which satisfies the requirement of an achromatic lens. The numerical investigation shows that our meta-lens has an achromatic bandwidth ranging from 0.6 to 1.0 THz that is polarization insensitive due to the C 4 symmetry of the cross section geometry. The FWHMs of the foci at different frequencies are smaller than those of the chromatic counterpart. We also analyze the deviation between the simulated focal length and the designed value by the Rayleigh-Sommerfeld integration. This study has provided an effective method for solving the chromatic aberration problem in the terahertz lens design and paved the way for the research on terahertz achromatic devices that require dispersion engineering.

DESIGN AND UNIT CELL CONSTRUCTION
The designed meta-lens is illustrated by Figure 1A. The desired phase distribution of a meta-lens working for a monochromatic wave can be expressed as where ω is the wave frequency, c is the speed of light in vacuum, f is the focal length, and r and r 0 represent the positions of the unit cell and the reference cell on the meta-lens, respectively. L(r) represents the required optical path difference to realize a focusing effect. However, for an achromatic meta-lens working in the frequency range [ω 1 , ω 2 ], L(r) should also be equivalent at different frequencies because the focal point is fixed at different frequencies. [24]. With ω 1 as the minimal frequency in the design process, the required phase distribution for an achromatic metalens is The unit cells for the achromatic meta-lens are thus expected to have a phase distribution ϕ(r, ω 1 ) similar to the chromatic lens designed for a monochromatic wave and a linear dispersed phase whose dispersion slope is determined by L(r). To ensure the achromatic effect, the phase generated by the unit cells should be proportional to the frequency in the working frequency range.
The first step in the construction of the unit cells is to select the right raw material by using the commercial software CST Microwave Studio TM and applying the periodic boundary condition with normal incidence wave. In the visible light domain, the substrate and the resonate cavity are often made of different materials in order to utilize the dispersion feature of the Mie scattering of a closed cavity. However, in the terahertz band, due to the ease of the fabrication and the consideration of absorption, dielectric metasurfaces prefer to etch structures directly onto the semi-insulating silicon substrates. As shown in Figure 2A, when both the substrate and the structure are made of silicon, a larger variation range of the phase and dispersion can be obtained by subwavelength pillar structure, compared to the fused silica structure with silicon substrate. Conversely, when the unit cell is made of a silicon pillar with fused silica substrate, its phase and dispersion have nearly the same ranges as the all silicon unit cells', while the larger absorption and material dispersion of silica may cause problems in the design process.
Considering the aspect ratio in the dry etching process, we set the height of the silicon pillars as 190 µm, and the fabrication accessibility of our selection is supported by our previous work [25]. The periodicity of the unit cell was set as 130 µm, and our meta-lens has 49 unit cells in total with a diameter of 6.4 mm. A focal length of 13.5 mm and the target frequency ranging from 0.6 to 1.0 THz were determined as well for the followed unit cell construction process. Inspired by the previous work of Shrestha et al [20], we appropriately changed the cross-section geometry of the silicon pillars while retaining the C 4 symmetry to hold the polarization-independent characteristics of the structures to engineer the phase and dispersion simultaneously. The specific cross section shapes are presented in Figure 1B, which are square ring, crisscross and hollow regular octagon, respectively. As depicted in Figures 2B,C, the three unit cells have different phases and amplitudes, while the phase has a quasilinear dispersion and the transmission is relatively high in the target frequency range, indicating the silicon pillar structure proposed here has the capability to realize the desired phase range and liner dispersion. Besides, we found that adding a thin layer of pure gold on the dielectric silicon pillars could further endow the freedom level of the phase control. When the structure size was the same, the phase range of the metal-attached structure was significantly increased ( Figure 3A). Hence we adopt this method in the construction of the reference unit cell. Next, by changing the cross section shape and scales in the simulation, we constructed a unit cell base and drew its phase-dispersion library in Figure 3B. We found that for all the investigated unit cells, the phase decreased linearly as the frequency increased from 0.6 to 1.0 THz, and the dispersion slope differs according to the specific cross section shape and geometric parameters. Finally, taking the lens edge as the reference position r 0 in the design, we calculated the phase distribution ϕ(r, ω 1 ) at ω 1 0.6 THz and the dispersion slop L(r) for each position r. Then, different shapes of unit cells whose phase and dispersion slop satisfy the required ϕ(r, ω 1 ) at ω 1 0.6 THz as well as the dispersion slop L(r) were chosen to form the metalens. The whole sketch of our metalens is depicted in Figure 4D. The adopted cross section shape and the geometric parameters are listed in Table 1. It can be seen in Figure 3C that the phase and dispersion of the selected unit cells fit the theoretical design quite well.

NUMERICAL DEMONSTRATION AND ANALYSIS
The performance of our achromatic meta-lens was also evaluated using the CST Microwave StudioTM. A full meta-lens was constructed by the selected unit cells with the center of the meta-lens set at the origin of the simulation environment, the   Figure 4. To inspect the achromatic feature of our meta-lens, a control meta-lens designed only for 0.6 THz and composed of pure crisscross pillars with other conditions remaining the same was investigated as well. It is quite clear that both the achromatic meta-lens and the control show focusing capabilities. By analyzing the maximal value of the electric field intensity, we verified that the foci of the achromatic meta-lens at 0.6, 0.8 and 1.0 THz lay at z 11.20, 11.80 and 11.85 mm, respectively, which only has an maximal offset of 650 µm ( Figure 4A). In contrast, the foci of the control meta-lens at the corresponding frequencies lay at z 11.20, 11.80 and 13.30 mm, respectively, which has a much larger focal shifts of 2 mm as Figure 4B showed. Figure 4C depicts the evolution of the focal length as frequencies changes for the achromatic metalens and the control. The focal length of the achromatic meta-lens varies less than that of the control. We also found that the maximal electric field intensity point deviated 6 mm from the foci at 0.8 and 1.0 THz. Our achromatic meta-lens design is successful since it demonstrated a clearer focusing behavior and a more stable focal lengths over the whole interested frequency band than the control. Figure 5A depicts the intensity distributions of our achromatic meta-lens at the focal planes (z 11.2 mm) for the monitored frequencies, respectively. It can be seen that the focus spot shows a perfect circle shape and a typical Gaussian intensity distribution. Fringes around the spot are a common phenomenon in diffraction focusing devices caused by   0  130  61  1,690  130  73  130  130  61  1,820  130  78  260  130  62  1,950  130  84  390  130  63  2,080  130  89  520  130  66  2,210  130  93  650  130  62  2,340  130  97  780  130  58  2,470  130  102  910  130  56  2,600  130  106  1,040  130  42  2,730  45  31  1,170  130  49  2,860  40  27  1,300  130  56  2,990  33  20  1,430  130  62  3,120  60  The evident reduction of the FWHM is a typical feature for achromatic lens, indicating a higher spatial resolution in broadband imaging or microscopy applications which is a well-known advantage of achromatic lens. The polarization dependence of the achromatic meta-lens was manifested by y 0 cut of the electric field intensities as shown in Figure 6 It is clear that the distributions of the electric field are almost the same, with only small variances compared with the electric field distribution shown in Figure 5, which confirms the C 4 symmetry of each unit cell.

DISCUSSION
Although the simulation results have verified the feasibility of the designed achromatic meta-lens, the numerically obtained focal length is around 11.2 mm, which deviates 2.3 mm from the designed focal length of 13.5 mm. In addition, the focal lengths at 0.6 and 1.0 THz also differ by 650 µm. We believe this focal shift and the residual chromatic aberration are due to two reasons. The first one is the hexahedral mesh method in the simulation, which divides the space along Cartesian coordinates and will thus cause a calculation error because of the mismatch between the mesh grid and the tilted edge of the octagon pillars.
The second reason is that focal shift is a prevalent phenomenon of the lens design [26], and we analyzed it from the perspective of the reversible light path. By assuming a point terahertz source located at z 13.5 mm, it is easy to obtain the resulted  amplitude and the phase distributions on the meta-lens. Though the phase distribution has been well satisfied in the whole interested bandwidth, the required amplitude variance, which equals f /r, was not considered in our numerical investigation. To investigate the influence of this amplitude distribution on the focal shift, we numerically testified an achromatic meta-lens with and without the amplitude distribution by the Rayleigh-Sommerfeld diffraction formula, respectively [27]. As shown in Figure 7, without the f /r amplitude distribution, the calculated focal lengths of the achromatic meta-lens are in accordance with the simulated results, verifying our calculation program. While by adding the right f /r amplitude distribution to the constructed achromatic meta-lens, the foci of this 'ideal' achromatic metalens at 0.6, 0.8, and 1.0 THz now all move to z 13.5 mm, which is highly consistent with our design and shows even smaller chromatic features. This comparison inspires the researchers to pay more attention to the amplitude distribution in the design of achromatic meta-lenses, especially for meta-lenses with a large focal length.

CONCLUSION
In summary, we designed and numerically investigated an achromatic meta-lens composed of subwavelength silicon pillars working at terahertz frequencies. The achromatic behavior of the meta-lens was verified by the numerical simulation, which showed that its focal length changed by 650 µm when frequency ranged from 0.6 to 1.0 THz. The variation of the focal length is merely 5.80%. The FWHM of the achromatic meta-lens foci ranges from 1.063 to 1.628 mm, indicating a higher spatial resolution compared with that of the chromatic counterpart. The incongruence of the focal lengths with the design was well explained by adding the correct amplitude distribution in the Rayleigh-Sommerfeld diffraction calculation. The achromatic metalens proposed here not only provides an achromatic planar lens working at terahertz domain but also reveals the importance of the amplitude distribution in achromatic metasurface design.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

AUTHOR CONTRIBUTIONS
YG and JG discussed the design principle. YG and RJ completed the unit cell construction. YG performed the CST simulations, the matlab calculations, and the data analysis. YG wrote the paper, and JG modified the writing. All the authors commented on the paper.