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Abstract
We designed a high-NA (High numerical aperture) broadband achromatic metalens consisting of elliptical TiO2-based integrated-resonant unit-cells. The phase profile for the designed metalens is constituted by incorporating the geometric phase with the transmission phase, where the geometric phase contributes to the basic phase profile of the metalens and the transmission phase is used to compensate phase profile at different incident wavelengths. With the method of finite element simulation, the results demonstrate that the metalens has an average focusing efficiency of 36.4% in the visible range from 530 nm to 850 nm. Compared with chromatic metalens, the axial chromatic aberration of achromatic metalens is reduced by 2.66 times. Furthermore, the designed metalens has a high-NA of 0.564. This work can be potentially applied in various fields, such as optical communications, microscopy imaging, and optical instruments.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chromatic dispersion is one of the most fundamental characteristics of optical materials, which leads to broadband light converging at different focal points. Thus correcting chromatic aberration is of great significance especially in the fields of imaging systems, communication systems, detection systems. However, conventional achromatic methods are generally bulky, costly, and difficult to manufacture. The above limitations make it difficult to meet the rapid development of lightweight and integrated optics devices. In recent years, metasurface has an excellent ability to precisely control the amplitude, phase, and polarization of light [1–3] because of it is comprised of artificially arranged periodic or aperiodic subwavelength unit-cells. Ultra-compact planar construction, ease of fabrication, and great information capacity make metasurface the prime candidate for different optical applications such as metalenses [4,5], holographic imaging [6], beam shapers [7]. Due to the resonant phase dispersion of the unit-cells and the intrinsic material dispersion, these metasurface-based elements have severe chromatic aberrations. To date, various achromatic metalenses have been proposed in the ultraviolet [8], visible [9–11], infrared [12,13], and terahertz [14,15] spectral ranges. However, achromatic metalenses usually suffer from many limitations such as narrow bandwidth operation, low-NA, and low focusing efficiency.
To solve the chromatic aberration problem, previous works have proposed several investigations. Multiwavelength achromatic metalenses [16–18] and narrowband achromatic metalenses [19] have been used in fluorescence microscopy and LED lighting. But these metalenses are not competent enough for broadband applications such as full-color imaging. In addition, the reflective broadband achromatic metalenses [20,21] consisting of plasmonic nano-scatterers were proposed. However, these metalenses usually have low efficiency owing to the absorption loss of metal. To solve this problem, most achromatic metalenses are designed based on all-dielectric materials. Recently, achromatic metalens has made great progress in terms of broadband and high efficiency by proposing different design principles [11,22], optimization algorithms [17,19,20], and designing multilayer dielectric metasurfaces [23–25]. For instance, F. Balli et al. [26] designed a Hybrid achromatic metalens with an average focusing efficiency greater than 60% for a bandwidth of 800 nm. Y. Wang et al. [27] proposed an achromatic metalens comprises of two stacked nanopillar metasurfaces with an average focusing efficiency of over 64% for a bandwidth of 700 nm. P. Sun et al. [28] designed an achromatic polarization-independent metalens with a maximum focusing efficiency higher than 70% for a bandwidth of 950 nm. However, the most achromatic metalenses have a small NA, which limit the application of metalenses in imaging, displays, sensing systems, and other related fields. Although several studies [12,13,29–31] have been devoted to high-NA broadband achromatic metalenses, the same design has not been found in the wavelength range we studied.
In this paper, we design a broadband achromatic metalens with high-NA of 0.564 that is capable of focusing right-handed circularly polarized (RCP) light from 530 nm to 850 nm. However, RCP light is not the conventional light used in all imaging applications. Different imaging techniques and devices may selectively use RCP light in particular situations. For example, in fluorescence microscopy, using RCP light can reduce sample scattering and polarization effects, thereby enhancing image [32]. The building blocks of the designed metalens are TiO$_2$ nanopillars with the advantages of high refractive index and low loss. By adjusting the rotation angle and geometric parameters of the TiO$_2$ nanopillars, the geometric phase and transmission phase at different wavelengths can be obtained. We have simulated that the metalens can effectively eliminate chromatic aberrations at about 46.4% bandwidth of the central working wavelength. The maximum focusing efficiency of the metalens reaches 58.7% at 570 nm and the average focusing efficiency is nearly 36.4%. This work can promote the development of metasurfaces in miniaturized and integrated optical devices, and has potential application in microscopes, lithography machines, and color display imaging.
2. Principle and design of the broadband achromatic metalens
As the RCP plane waves with different wavelengths incident from the substrate, the focal point remains focused in a single position, accompanied by polarization conversion to left-handed circularly polarized (LCP) light. The schematic of the broadband achromatic metalens is shown in Fig. 1(a). The required phase profile for each unit-cell at the position $(x_i, y_i)$ in a focusing metalens satisfies the equation as follows:
Where $(x_i, y_i)$ are the in-plane coordinates of a unit-cell center. $f$ and $\lambda$ represent the designed focal length and the light wavelength in free space, respectively. Figure 1(b) shows the required phase distribution of the unit-cell in the central column of the metalens when different wavelengths are incident. Different wavelengths require different phases at the same location, resulting in the focal length changes with the working wavelengt, which is the main cause of chromatic dispersion. Therefore, we need to modify the phase profile $\varphi (x_i, y_i )$ to focus light with different wavelengths at the same position. For the working wavelength $\lambda \in \{\lambda _{\rm min}, \lambda _{\rm max}\}$ ($\lambda _{\rm min}$ and $\lambda _{\rm max}$ are the boundaries of the designed wavelength), Eq. (1) can be converted into the following formula by using the phase division approach:
Fig. 1. Design schematic of the broadband achromatic metalens. (a) The focal length of the metalens will always remain constant when different wavelengths of light incident normally. (b) The required phase distribution at $\lambda _{\rm min}$ and $\lambda _{\rm max}$ wavelengths for the unit-cell in the center column of the metalens, where $\chi$ is the maximum phase difference between $\varphi _{\rm min}$ and $\varphi _{\rm max}$. (c) Phase distribution of the metalens with additional phase factor $\varphi _{\rm shift}(\lambda )$ at the wavelength range of $\lambda \in \{\lambda _{\rm min}, \lambda _{\rm max}\}$. The X-axis in Fig. 1(b) and (c) represents the horizontal coordinates of the center position of the unit-cell.
As shown in Eq. (3), $\varphi (x_i,y_i,\lambda _{\rm max})$ is independent of the working wavelength $\lambda$ and solely related to $\lambda _{\rm max}$. Therefore, it is dispersion-free, and the corresponding phase distribution can be obtained by utilizing the geometric phase. According to the geometric phase principle, phase regulation can be realized only by rotating the angle of the unit-cell. In the case of RCP incident light, after giving each nanopillar a rotation angle $\theta$ [Fig. 2(d)], these rotations yield a 2$\theta$ phase shift. It is worth noting that when the rotation angle $\theta$=0, nanopillars with different geometric parameters will produce different phases at a given wavelength, that is, the transmission phase of the unit-cell, which is denoting $\varphi _{\rm trans}(\lambda )$ in the paper. After giving each nanopillar a rotation angle, the total phase generated by the unit-cell theoretically becomes: $\varphi _{\rm total}(\lambda )=\varphi _{trans}(\lambda )+2\theta$. The geometric parameters of the unit-cell used in conventional metalenses are the same when using geometric phase for phase modulation, and $\varphi _{\rm trans}(\lambda )$ will not affect the performance of the metalens. However, our metalens needs to use the unit-cell with different geometric parameters to obtain the compensation phase, so it is necessary to consider the influence of $\varphi _{\rm trans}(\lambda )$ when using the geometric phase to obtain the basic phase profile. The unit-cell with different geometric parameters has different transmission phase at $\lambda _{\rm max}$. Therefore, the angle of each nanopillar at a given coordinate $(x_i, y_i)$ is equal to half of the phase calculated from Eq. (3) minus the value of $\varphi _{\rm trans}(\lambda )$ at $\lambda _{\rm max}$, that is, $\theta =[\varphi (x_i,y_i,\lambda _{\rm max})-\varphi _{\rm trans}(\lambda _{\rm max})]/2$.
Fig. 2. Unit-cell of broadband achromatic metalens. (a) Three-dimensional schematic of the unit-cell, which consists of TiO$_2$ nanopillars on a SiO$_2$ substrate. (b and c) Front and top views of the unit-cell showing height H, semi-minor axis b, semi-major axis a of the nanopillar, and unit-cell dimensions P $\times$ P. (d) The desired phase is imparted by rotating an angle $\theta$ of the nanopillar based on the geometric phase. (e) RCP-to-LCP conversion efficiency (black curves) and phase profile (blue curves) of the unit-cell with a compensation phase of 725°. Its geometric parameters are a = 110 nm, b = 34 nm, and H = 750 nm. (f) Phase profile as a function of wavelength for different rotation angles of the nanopillar. (g) Simulated magnetic field distribution in the y-z plane at 530 nm (left), 690 nm (middle), and 850 nm (right). (h) Magnetic field distribution at 690nm for a unit-cell with different rotation angles.
Equation (4) indicates that $\Delta \varphi (x_i,y_i,\lambda )$ is a wavelength-related function and has a linear relation with 1/$\lambda$. Such phase can be acquired by optimizing the geometric parameters of each resonance unit-cell. In addition, the compensation phase of the designed metalens can be optimized by introducing an additional phase factor $\varphi _{\rm shift}(\lambda )$ as shown in Fig. 1(c). This phase shift doesn’t affect the focusing characteristics of the metalens [14]. On this basis, the total phase profile of the metalens can be rewritten as
Establishing a database by adjusting unit-cell parameters is critical to design broadband achromatic metalens. For the broadband achromatic metalens to operate efficiently in the visible region, the refractive index contrast between the nanoscatterer and its surroundings needs to be considered. TiO$_2$ is an optical material widely used in the visible range. It has the characteristics of low surface roughness, negligible absorption loss, and sufficiently high refractive index in the designed wavelengths [4]. Therefore, we employ TiO$_2$ nanopillar on a SiO$_2$ substrate as building blocks of the metalens in this article. Such metasurface devices can be fabricated by electronic lithography followed by reactive-ion etching [33]. The three-dimensional diagram of the unit-cell is shown in Fig. 2(a), and the relevant geometric parameters are defined in Fig. 2(b) to (d). The inverse of the lattice constant $\frac {1}{P}$ is equal to the sampling rate of the target phase. According to Nyquist sampling criteria, the target phase profile can be accurately satisfied when the sampling rate is larger than twice the highest spatial frequency of the phase ($\frac {1}{P}\geqslant \frac {2NA}{\lambda _{\rm min}}$) [34]. Therefore, we set the lattice constant for all unit-cells to 260 nm, and implement the phase compensation by varying the semi-major axis a, semi-minor axis b, and height H of the nanopillar. Here, the semi-major axis a varies in the range of 100 –110 nm, semi-minor axis b varies in the range of 20 – 80 nm, and height H varies in the range of 700 – 800 nm. To simulate the transmission spectrum and phase distribution in an array structure, we adopt cell boundary conditions in the x and y directions and open boundary conditions in the z direction when scanning the unit-cell. Based on these settings, we simulate the unit-cell and build a database. These simulations were implemented using CST software combined with Matlab program.
Figure 2(e) shows the simulated circularly polarized conversion efficiency (black curves) and phase response (blue curves) of a unit-cell as a function of wavelength. It can be seen that there is a quadric relationship between phase and wavelength, and the compensation phase between $\lambda _{\rm min}$ and $\lambda _{\rm max}$ within the working wavelength is 725°. The phase modulation mechanism of the unit-cell originates from the waveguide effect. The introduced phase shift can be expressed as $\phi _{\rm WG}=(\frac {2\pi }{\lambda })n_{\rm neff}H$ [14], where $n_{\rm neff}$ and H are the effective refractive index of the waveguide mode and the nanopillar height, respectively. The manipulation of $n_{\rm neff}$ is closely related to the nanopillar’s geometry parameters, which is the key to obtaining a large compensation phase. By adjusting the size and height of the nanopillar, we obtain a compensation phase from 630° to 990° within the designed wavelength. In addition, the conversion efficiency of the unit-cells needs to be taken into account so that the designed metalens has a relatively stable output wavefront. As shown in Fig. 2(e), the largest conversion efficiency is up to 90%, and the average value is about 50% over the whole bandwidth. To ensure that the unit-cell has a high conversion efficiency, we sample the compensation phase when selecting the unit-cell. However, high-NA metalens has a sharp spatial phase variation profile, resulting in a compensation phase difference between the ideal compensation phase and the actual compensation phase when the phase sampling is relatively rough. To comprehensively reduce the influence of the compensation phase difference and conversion efficiency on the focusing performance of the metalens, we use a non-equal interval sampling scheme. One unit-cell is selected every 10 degrees in the range of 660° to 960°, and every 30 degrees at other ranges. Compared to the unoptimized achromatic metalens, using this sampling method can improve the focusing efficiency of the designed metalens at certain wavelengths. Figure 4(b) shows the changes in focusing efficiency before and after optimization. It can be observed that the maximum focusing efficiency of the optimized metalens increases from 51.5% to 58.7%, and the average focusing efficiency increases from 34.5% to 36.4%.
The simulated magnetic field distribution in the y-z plane of the unit-cell at the wavelength of 530 nm, 690 nm, and 850 nm are shown in Fig. 2(g). The induced optical field is highly concentrated inside the nanopillars, so the electromagnetic coupling effect between neighboring unit-cells can be ignored. This means that the phase design for each unit-cell is still accurate when they are arranged in a square lattice of the unit array. Figure 2(f) shows the relationship between the rotation angle and the transmission phase of the unit-cell under different incident wavelengths. The slopes are linear for different rotation angles over a given bandwidth. It indicates that the compensation phase implemented by the unit-cell is independent of the rotation angle. This property is crucial for us to design a broadband achromatic metalens. The magnetic field distribution of the unit-cell with different rotation angles at 690 nm wavelength is shown in Fig. 2(h). It can be seen that the resonance patterns are almost identical. This explains the physical origin of phase compensation being constant at different rotation angles. It should be noted that all simulations in Fig. 2 are performed under the same unit-cell.
3. Characterizations of the broadband achromatic metalens
By employing the design principle proposed above, a broadband achromatic metalens in the visible range is numerically designed and demonstrated. Given the limitations in computing power and storage capacity, we simulated an achromatic metalens with a diameter of 5.46 $\mu$m and a focal length of 4 $\mu$m. As a comparison, we also designed a chromatic metalens working at 530 nm using geometric phase. Here, chromatic metalens has the same diameter and focal length as achromatic metalens. It should be emphasized that the designed metalens is capable of continuous aberration correction over the entire working wavelength range.
To compare the simulation results of the two types of metalenses, we select nine wavelengths from 530 nm to 850 nm to study the focal performance respectively. Figure 3(a) and (b) summarize the normalized intensity profiles in the x-z plane of the chromatic and achromatic metalens at each sampled wavelength. The white dotted line represents the position of the focal length at 530 nm. It can be seen from Fig. 3(a) that the focal length of chromatic metalens has a significant downward shift with increasing wavelength, indicating the existence of negative dispersion. In contrast, the focal length of achromatic metalens in Fig. 3(b) has a smaller fluctuation range, meaning that chromatic aberration is significantly reduced. However, since the polarization conversion efficiency of the selected unit-cells at longer wavelengths is lower than that at shorter wavelengths, longer wavelengths show darker focuses. In addition, the spot size along the z direction decreases at long wavelengths, but due to phase mismatch, the energy is dispersed to other places, resulting in lower focusing efficiency and the generation of secondary focus [35]. Figure 3(c) presents the measured intensity profiles in the focal plane at different wavelengths, and Fig. 3(d) represents a normalized cross-sectional view of the corresponding focal spots in Fig. 3(c). All cross-sections are normalized to a single value with reference to the intensity of 530 nm, so that the decrease in focusing efficiency is visually clear. The results indicate that the focal spots of each wavelength have no obvious distortion, and the metalens has a well-behaved focusing effect.
Fig. 3. Simulation results of metalenses in the wavelength range from 530 nm to 850 nm. Normalized intensity distributions in the plane y = 0 (x-z plane) for (a) the chromatic metalens and (b) the broadband achromatic metalens at nine different selected wavelengths. The white dotted line indicates the focal length at 530 nm. (c) Normalized intensity distribution in the x-y plane for nine selected wavelengths at the focal point of the broadband achromatic metalens. (d) Cross-sectional views of the corresponding focal spots in (c).
Figure 4(a) shows the focal lengths functioning versus wavelength for the chromatic and the achromatic metalens corresponding to Fig. 3(a) and (b). Since the phase realized by the chromatic metalens only matches the desired phase at 530 nm, the focal position at 530 nm is close to the set focus position, after which the focal length gradually shifts downwards. It can be calculated that when using 530nm as the reference wavelength $\lambda _{\rm 0}$, the axial chromatic aberration of the chromatic metalens reaches 3.4$\lambda _{\rm 0}$, while the axial chromatic aberration of the achromatic metalens is only 1.28$\lambda _{\rm 0}$. Compared with the chromatic metalens, the axial chromatic aberration of the achromatic metalens is reduced by 2.66 times. Figure 4(b) plots the focusing efficiency and the full-width half-maximum (FWHM) of the focus at different sampling wavelengths of the optimized achromatic metalens. The focusing efficiency is defined as the fraction of the incident light that passes through a circular aperture in the plane of focus with a radius equal to three times the Rayleigh diameter. It can be seen from Fig. 4(b) that the maximum focusing efficiency reaches 58.7% at 570 nm, and then the overall focusing efficiency shows a downward trend. This change is mainly caused by the conversion efficiency fluctuation of the resonant unit-cell. To obtain the uniformly distributed focusing efficiency, we can eliminate the unit-cell with low polarization conversion efficiency or directly optimize the geometric parameters of the unit-cell. However, when the numerical aperture of the metalens increases, the required compensation phase variation range will be larger. It is difficult for general unit-cell to meet the two conditions of large compensation phase range and high conversion efficiency at the same time. Therefore, in order to expand the phase coverage of the unit-cell to meet the compensation phase required for the designed high-NA metalens, sometimes it is inevitable to use some unit-cells with low transmittance when constructing the metalens. This affects the working efficiency of the metalens to a certain extent. According to Fig. 4(b), we calculate that the simulated FWHM values of the focal points are located in the range of 0.72$\lambda$–0.8$\lambda$. These values are close to the theoretical diffraction-limited width $\frac {\lambda }{2NA}=0.89\lambda$, indicating that the focusing effect of the designed metalens is close to the diffractive limit for the entire operational wavelength range. ($\lambda$ corresponds to the selected nine wavelengths, as shown at the top of Fig. 3.) This work is helpful for the exploration and application of metasurface optics in imaging and other fields.
Fig. 4. Performance of broadband achromatic metalens. (a) The focal lengths functioning versus wavelength for the chromatic and achromatic metalens. The red dotted line represents the design focal length of the metalens. (b) The focusing efficiency of optimized and unoptimized metalenses and FWHM of the focal points as a function of incident wavelength.
4. Conclusion
Based on the modulation principle of geometric phase and transmission phase coupling, the dispersion control in the visible range is achieved by design the metalens with a periodically arranged resonant unit-cells. The maximum diameter of a broadband achromatic metalens is related to the compensation phase range of the unit-cell when the NA is fixed. Theoretically, our method is also applicable to broadband achromatic metalens with larger diameters. The coverage of the compensation phase can be further improved by introducing more types of integrated resonant unit-cells, realize higher NA and larger size metalens in the visible range. In addition, the incident light is required to be circularly polarized because of the manipulation of the wavefront information of the metalens depends on the geometric phase, which can be solved by using a symmetrical element structure. The metalens designed with the unit-cell are simpler and easier to manufacture compared to complex nanostructures. The simulation results show that the metalens has a high-NA of 0.564 and stable focusing efficiency in the working wavelength range of 320 nm. The maximum focusing efficiency is up to 58.7%, and the average focusing efficiency is 36.4%. Although the design principles and methods of this optical device in this paper is only applied in the visible range, it can be extended to other wavelengths such as the infrared region.
Funding
National Natural Science Foundation of China (12104413, 12104414); Henan Provincial Science and Technology Research Project (232102211078, 222102210099).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this manuscript is described in figures but the underlying data may be obtained from the authors upon reasonable request.
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