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Abstract
Although metalens has made breakthroughs in various imaging applications due to its ultrathin, lightweight, and multi-functionality, simultaneously achieving wide field of view (WFOV) and achromatic imaging remains a challenge. Here, we demonstrate a harmonic metalens with a quadratic phase profile that enables WFOV imaging and achromatic imaging at certain discrete wavelengths. First, we quantitatively explain why the quadratic phase enables WFOV imaging using its Fourier Transform (FT). Second, we derive the complex-amplitude transmittance formula of a harmonic metalens. The derived formula integrating with the Angular Spectrum theory can calculate the transmitted field distributions of a harmonic metalens. Third, we propose an achromatic WFOV metalens based on the harmonic diffraction and quadratic phase at a large numerical aperture (NA=0.76), which enables achromatic imaging at wavelengths λ = 600 nm, 1200 nm with a wide field of view (FOV) of 100°. The proposed scheme will undoubtedly expand the applications of metalens in the imaging area.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metalenses are planar metamaterials consisting of many tailored nanostructures with various excellent properties, such as abnormal dispersion and the ability to locally manipulate the amplitude, phase, and polarization of the light field [1–3], which are not available in conventional refractive optical systems. More importantly, metalenses have made breakthroughs in various areas by virtue of planarization and lightweight. However, WFOV and achromatic focusing remain challenges for metalenses in imaging applications.
Various solutions for extending the FOV have been intensely investigated, including the quadratic phase metalens [4–6], the quadratic phase metalens combined with an aperture stop [7,8], cascading doublet metalenses [9,10], etc. The quadratic phase metalens is promising for integrated lightweight applications as it enables WFOV imaging with monolithic metalens. Reference [5] qualitatively explains why the quadratic phase metalens enables WFOV imaging using FT, and calculates the theoretical FOV at different NAs. This theoretical FOV is determined by calculating the errors between the point spread functions (PSFs) of the off-axis field and the on-axis field. Based on this, we quantitatively explain the ability of WFOV imaging using the cross-correlation (COR) between the FTs of the quadratic phase and the ideal imaging phase for each field. We also obtain a curve similar to that in reference [5], showing the theoretical FOV as a function of NA.
For achromatic imaging, several solutions have been proposed at discrete wavelengths or over continuous broadband, such as modulating nanostructures’ phase group delay (PGD) and phase group delay dispersion (PGDD) [11–16], segmenting the aperture to respond to specific wavelengths [17,18], cascading multiple metalenses [19–21] , and optimization algorithm [22,23]. Resonator can also be a good choice for multi-band modulation [24–28]. For the first method, a significant advantage of managing nanostructures’ PGD and PGDD is achieving achromatic imaging over continuous broadband. However, the laborious scanning work to find suitable nanostructures and the limited aperture due to material’s intrinsic characteristics hinder its applications. For the second method, most of these devices are polarization-sensitive because the nanostructures provide phase delay based on geometric phase modulation. The polarization dependence is not conducive to the lightweight of the system. For the third method, cascading multiple metalenses substantially increases the manufacturing costs.
In this paper, we propose a harmonic metalens with a quadratic phase that simultaneously enables WFOV and achromatic imaging at certain discrete wavelengths. The nanostructures of the proposed metalens have simple cross-sections so that suitable ones are easily obtained without laborious scanning work. The aperture can be arbitrarily large in theory because only the phase and transmittance of the nanostructures need to be considered to achieve harmonic diffraction. Moreover, the proposed metalens is polarization-insensitive due to the symmetric nanostructures, which is conducive to a lightweight system. To begin with, we quantitatively explain why the quadratic phase enables WFOV imaging, and calculate the theoretical FOV as a function of NA. In the next part, we modify the complex-amplitude transmittance formula of a harmonic diffractive optical elements (DOE) described as a sum of the Fourier series so that the new formula could calculate the complex-amplitude transmittance of a harmonic metalens. Then the theoretical transmitted field distribution of a harmonic metalens can be calculated using the complex-amplitude transmittance integrating with the Angular Spectrum theory. Finally, we propose two harmonic metalenses with a quadratic phase at a small NA (0.35) and a large NA (0.76), respectively, and investigate their transmitted field distributions and imaging characters in depth. In terms of FOV, the quadratic phase at a large NA has a distinct advantage over the widely used hyperbolic phase, achieving a wide FOV of $100^\circ$. In terms of achromatic imaging, the good agreement between theoretical analysis and simulation results demonstrates that the harmonic metalens can focus the light of harmonic wavelengths on the same image plane.
2. Methods
2.1 Quadratic phase to realize WFOV imaging
In this section, we quantitatively explain why the quadratic phase enables WFOV imaging using FT, and calculate the theoretical FOV as a function of NA. The quadratic phase and widely used hyperbolic phase is described as $\varphi _{\text {Quadr}}(r)=-\frac {\pi r^{2}}{\lambda _0 f_{0}}$ and $\varphi _{\text {Hyper}}(r)=-\frac {2 \pi }{\lambda _0}\left (\sqrt {r^{2}+f_{0}^{2}}-f_{0}\right )$, where $r$, $\lambda _0$, $f_{0}$ is the radial coordinate, design wavelength, and focal length of the metalens, respectively. When a plane wave is incident on a metalens with a quadratic phase (or hyperbolic phase), the image position will be $x_{i} = f_{0} * sin (\theta )$ (or $x_{i} = f_{0} * tan(\theta )$) as a function of angle of incidence (AOI, $\theta$). The ideal imaging phases without aberrations for these two positional relationships can be easily derived from Fermat’s principle, namely Eq. (1) and Eq. (2), denoted as $PhaseSin$ and $PhaseTan$, which are functions of spatial coordinate and AOI [29,30]. It is almost impossible for the existing devices to realize the ideal imaging phase, either PhaseSin or PhaseTan.
Figure 1 shows the FT distribution of $PhaseTan$, hyperbolic phase (denoted by $Hyper$), $PhaseSin$, quadratic phase (denoted by $Quadr$) as a function of normalized $k$-vector. Only the $k$-vector components inside the unit circle (white dashed circle) could propagate in free space. The $k$-vector components outside the unit circle represent the evanescent wave, which do not contribute to the far-field focusing. For the quadratic and hyperbolic phase, the FT at oblique incidence can be obtained by translating the FT at normal incidence. However, for the ideal imaging phase, the FT at oblique incidence is not a simple translation of that at normal incidence because its relative distribution also changes. More explicit FT comparisons are provided in Supplement 1. Therefore, more spectrum components within the unit circle do not represent a better imaging performance.
Fig. 1. FTs of PhaseTan, quadratic phase (denoted by Quadr), PhaseSin, hyperbolic phase (denoted by Hyper). column 1 and 2: for the cases of NA = 0.3. column 3 and 4: for the cases of NA = 0.85. The white dashed circle represents the unit circle. The AOI is indicated at the upper-right corner of each subfigure.
We calculate the COR between FTs of the quadratic phase and PhaseSin (the corresponding ideal imaging phase) and between FTs of the hyperbolic phase and PhaseTan for comparison. COR is defined as Eq. (3), a measure of similarity between two series, where $g$ and $h$ indicate two signals. The larger the COR, the more similar the spectral information to the ideal imaging phase, that is, the better the imaging performance. FOV is defined as twice the AOI when the normalized COR is greater than 0.88. This means that high-quality image spots can be formed within the field of view, similar to the definition of FOV in Ref. [5].
Figure 2 shows the FOV of the quadratic phase and hyperbolic phase as a function of NA, where the curves agree well with those determined by errors between PSFs in Ref. [5]. The FOV of the hyperbolic phase decreases approximately monotonically with the increase of NA, while the FOV of the quadratic phase decreases first and then increases. The FOVs of these two phases are very closed at small NAs. When NA is larger than about 0.5, the FOV of the quadratic phase rapidly increases to the theoretical upper limit of 180$^{\circ }$. Although this upper limit is practically impossible to achieve due to many factors, such as the size of nanostructures, the pixel size of the imaging sensor, etc. [6], above analysis manifests that the FOV of the quadratic phases is comparable to that of the hyperbolic phase at small NAs, while the FOV of the quadratic phase is significantly larger than that of the hyperbolic phase at large NAs.
Fig. 2. FOV as a function of NA for the quadratic phase (denoted by Quadr) and hyperbolic phase (denoted by Hyper).
2.2 Harmonic metalens to realize achromatic imaging
Here we propose a harmonic metalens scheme to realize achromatic imaging at certain discrete wavelengths. In contrast with the achromatic metalens realized by modulating nanostructures’ PDG and PDGG, the aperture of the harmonic metalens is not limited by material’s intrinsic characteristics. The harmonic metalens, also called multi-order metalens, is inspired by the harmonic DOE since the metalens still belongs to DOE to some extent [31–33]. For the harmonic metalens, the wrapping of the phase profile is done with a modulo-$2\pi p_{\text {design}}$ operation, where $p_{\text {design}} \geq 2$, an integer value, is the harmonic order. Thus the maximum phase that needs to be modulated by the nanostructures is $2\pi p_{\text {design}}$, which is $p_{\text {design}}$ times larger than that of the conventional metalens.
We derive the theoretical complex-amplitude transmittance of a harmonic metalens described as a sum of the Fourier series. Eq. (4) describes the complex-amplitude transmittance of a continuous quadratic phase profile,
Eq. (5) is obtained after a modulo-$2\pi p_{\text {design}}$ phase-wrapping operation to Eq. (4),3. Simulations
In this paper, the metalenses are implemented using TiO2 nanostructures with a square cross-section standing on SiO2 substrate to provide phase modulation based on the waveguide principle (a sketch map can be seen in Fig. S4 of Supplement 1). By varying the width of the square cross-section at an appropriate height, suitable nanostructures can be found without laborious scanning work. To realize the harmonic metalens, nanostructures will be taller than that of the conventional metalens to provide larger phase modulation. The phase and transmittance of nanostructures at harmonic wavelengths for each AOI can be simulated using a commercial tool, namely, FDTD Solutions. We consider both the phase and transmittance of nanostructures to ensure that the diffraction efficiency is as high as possible.
We design two achromatic metalenses with a quadratic phase profile at a small NA (0.35) and a large NA (0.76) and analyze their imaging performance in detail. The simulation results of the metalens at NA$= 0.35$ are described in Supplement 1, section 4. The achromatic WFOV metalens at NA$= 0.76$ is discussed here in depth. The parameters taken for the simulations are aperture diameter $D = 42 \mu m$, focal length $f_0 = 18 \mu m$, design wavelength $\lambda _{0} = 600 nm$, design harmonic order $p_{\text {design}}=2$. For the nanostructures composing this metalens, they have a height of 1.4 $\mu m$, a period of 250 $nm$, and a width ranging from 104 $nm$ to 233 $nm$. We summarize the design parameters in Table 1. See Supplement 1, section 3, for the phase and transmittance provided by each nanostructure and the width distribution of selected nanostructures on the metalens. The aperture proposed here is mainly limited by the computational resources. The largest NA can be achieved to meet the Nyquist sampling criterion ( NA $< \lambda _{0} / 2U$) is $0.7682$ [6]. Theoretically, focal lengths are the same at wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$ with a harmonic order of 2 and 1, respectively. We obtain the simulated electromagnetic near-field distribution of the metalens using FDTD Solutions. A full 3D simulation is done with the boundary condition of perfectly matched layers for all three directions conformal mesh with a grid size about 28$nm$, the source is a plane wave linearly polarized in the $x$-direction, the required memory is about 21 $Gigabyte$, the simulation time for this achromatic WFOV metalens (NA$= 0.76$) is about 8 hours 15 minutes with four processors, our available software version is Lumerical $2021 R1.3$. Then the simulated far-field light distribution is calculated using the near-field distribution integrating with the Angular Spectrum theory. Likewise, we can calculate the theoretical far-field light distribution using Eq. (6) integrating with the Angular Spectrum theory.
Table 1. Design parameters for the achromatic WFOV metalens at NA = 0.76.
As shown in Fig. 3, Eq. (5) calculates the theoretical diffraction efficiencies of the achromatic WFOV metalens at NA $= 0.76$ and wavelengths $\lambda _2 = 600 nm$ and $\lambda _1= 1200 nm$ by substituting the work phase and transmittance of each nanostructure into the formula. The target harmonic order is 2 and 1 for wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$, respectively. The theoretical diffraction efficiencies of the target harmonic orders are 63 % and 82%. As expected, they are significantly higher than other orders, ensuring a high focusing efficiency.
Fig. 3. Diffraction efficiency for the achromatic WFOV metalens at NA $= 0.76$ as a function of diffraction order at wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$.
Fig. 4, Fig. S7 and Fig. S8 show light distributions of the achromatic WFOV metalens at normal and oblique incidence on the $x-z$ plane, respectively. The simulated light distribution and brightest position are in good agreement with those obtained by the theoretical calculation. For the wavelength $\lambda _2 = 600 nm$, the brightest focus in both subfigures (Fig. 4(a) and Fig. 4(b)) locates around $z = 18 \mu m$, indicating the second harmonic order of interest. Also, a faint focus around $z = 36 \mu m$ in both subfigures indicates the first harmonic order, which is not of interest for the wavelength $\lambda _2 = 600 nm$. For the wavelength $\lambda _1 = 1200 nm$, the focus of the first harmonic order is around $z = 18 \mu m$ as well (Fig. 4(c) and Fig. 4(d)). The brightest positions of wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$ are focused around $z = 18 \mu m$ for both the theoretical calculations and simulations.
Fig. 4. Light distributions on the $x-z$ plane for the achromatic WFOV metalens at NA $= 0.76$. Left column: theoretical calculation results. Right column: the simulation results. The white dashed lines pass through the center of foci of the target harmonic orders at wavelength $\lambda = 600 nm$.
The centered PSFs of the achromatic WFOV metalens for different AOIs are shown in Fig. 5 (for the harmonic wavelength $\lambda _2 = 600 nm$) and Fig. 6 (for the harmonic wavelength $\lambda _1 = 1200 nm$). The theoretical PSFs calculated using Eq. (6) (Fig. 5(a) and Fig. 6(a)) and simulated PSFs (Fig. 5(b) and Fig. 6(b)) keep virtually unchanged throughout the AOI of $0^{\circ }$ to $50^{\circ }$. For comparison, we design two hyperbolic phases at the same NA (0.76) and two harmonic wavelengths $\lambda = 600 nm$ and $\lambda = 1200 nm$, respectively. The theoretical PSFs for these two hyperbolic phases (Fig. 5(c) and Fig. 6(c)) diverge severely and rapidly as the AOI increases despite converging quite tightly at the normal incidence. The FOV of the hyperbolic phase at NA $= 0.76$ is relatively small, which is consistent with the quantitative explanation shown in Fig. 2.
Fig. 5. Centered PSFs. a) The theoretical calculation results and b) the simulation results for the achromatic WFOV metalens at NA $= 0.76$ and wavelength $\lambda _2 = 600 nm$. c) The theoretical calculation results and d) the simulation results for the metalens with a hyperbolic phase at NA $= 0.76$ and design wavelength $\lambda _d = 600 nm$ for comparison. The AOI is indicated at the upper-right corner of each subfigure. Scale bars represent $1 \mu m$.
Fig. 6. Centered PSFs. a) The theoretical calculation results and b) the simulation results for the achromatic WFOV metalens at NA $= 0.76$ and wavelength $\lambda _1 = 1200 nm$. c) The theoretical calculation results and d) the simulation results for the metalens with a hyperbolic phase at NA $= 0.76$ and design wavelength $\lambda _d = 1200 nm$ for comparison. The AOI is indicated at the upper-right corner of each subfigure. Scale bars represent $1 \mu m$.
Fig. 7 shows the full width at half maximum (FWHM) results of the achromatic WFOV metalens obtained by theoretical calculations using Eq. (6) (solid lines), simulations (markers), and the theoretical FWHM [34,35] of an ideal lens calculated using Eq. (8) for comparison (dashed lines). The theoretical and simulated FWHMs of the achromatic WFOV metalens are consistent throughout the AOI from $0^{\circ }$ to $50^{\circ }$ and very close to those of an ideal lens, which ensure a good imaging performance.
To investigate the imaging performance of the achromatic WFOV metalens, we also calculate the image position, focusing efficiency, and PSF as a function of AOI.
First, Fig. 8(a) shows the image position as a function of AOI. We can first determine six image positions by finding the coordinates of the brightest point on the aforementioned PSFs (Fig. 5 and Fig. 6). All the image positions of each AOI are obtained by interpolating the six determined image positions. It can be seen from Fig. 8(a) that at wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$, the theoretical image positions calculated using the theoretical PSFs (Fig. 5(a) and Fig. 6(a)) agree with each other (denoted by solid lines with triangle markers in Fig. 8(a)). The same is true for the simulated image positions calculated using the simulated PSFs (Fig. 5(b) and Fig. 6(b)) at wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$ (denoted by lines with square markers in Fig. 8(a)). Therefore, the theoretically calculated and simulated final images have no obvious lateral chromatic aberrations. For reference, the image position as a function of AOI for undistorted imaging is $f_0 * tan(\theta )$, where $f_0$ indicates the theoretical or simulated focal length at normal incidence (denoted by black dashed lines in Fig. 8(a)). The theoretical and simulated image positions deviate slightly from the reference image positions, resulting in image distortions (Fig. 9(b) and Fig. 9(d)).
Fig. 8. a) Image position and b) focusing efficiency for the achromatic WFOV metalens at NA $= 0.76$ as a function of AOI.
Fig. 9. Image simulation results for the achromatic WFOV metalens at NA $= 0.76$. a) Optical schematic layout of the imaging system. b) The original object. c) The un-deteriorated distorted image calculated using the theoretical image position relationship (solid lines with triangle markers in Fig. 8(a)). d) The deteriorated distorted image calculated by the convolution between theoretical PSFs and the theoretical un-deteriorated distorted image (Fig. 9(c)). e) The un-deteriorated distorted image calculated using the simulated image position relationship (solid lines with square markers in Fig. 8(a)). f) The deteriorated distorted image calculated by the convolution between simulated PSFs and the simulated un-deteriorated distorted image (Fig. 9(e)). g) and h) The theoretical calculation results of a chromatic metalens with design wavelength = 600nm. i) and j) The simulated results of a chromatic metalens with design wavelength = 600nm.
Second, Fig. 8(b) shows the theoretical and simulated focusing efficiency as a function of AOI. Focusing efficiency here is defined as the ratio of the optical power within four times the FWHMs around the focus to that of the incident beam. We can determine six focusing efficiencies using the theoretical and simulated PSFs mentioned above (Fig. 5 and Fig. 6). The focusing efficiencies of all AOIs can also be obtained by interpolating the determined ones. Notice that the focusing efficiency remains relatively stable with the increase of AOI. It will be incorporated in the final imaging performance as an intensity weight of each FOV. Compared with the theoretical calculation, the significant reduction in focusing efficiency obtained by simulation may be attributed to the following two aspects. One is the low transmittance of the selected nanostructures. To achieve 2nd-order harmonic diffraction, the selected nanostructures are taller than the conventional ones, and some nanostructures are close to $200 nm$ in width with very low transmittance as shown in Supplement 1, Fig. S5. The other is the rapid change of phase profile in the boundary region of the metalens in the case of large NA. This violates the local periodic approximation, so the transmittance of nanostructures obtained under periodic boundary conditions may be incorrect.
Finally, Fig. 9 shows the imaging performance of the achromatic WFOV metalens at NA $= 0.76$. The imaging system (Fig. 9(a)) can not be treated as a linear space-invariant system because we are investigating the WFOV imaging performance. We must take the PSF of each image point into consideration. The PSF of each image point is also calculated by interpolating the determined ones. The original object is a checkerboard, as shown in Fig. 9(b). The image data is divided into three channels of $R$, $G$, and $B$. The $R$ channel corresponds to the wavelength $\lambda _2 = 600 nm$, the $B$ channel corresponds to the wavelength $\lambda _1 = 1200 nm$, and the $G$ channel is the average effect of the wavelengths $\lambda _2 = 600 nm$ and $\lambda _1 = 1200 nm$ as there are only two harmonic wavelengths for this achromatic WFOV metalens. The $R$, $G$, and $B$ channel corresponds to the wavelength $\lambda = 600 nm$, $800 nm$, and $1200 nm$, respectively, for the achromatic metalens at $NA = 0.35$ discussed in Supplement 1, section 3, as there are three harmonic wavelengths used into the calculation. Fig. 9(c) and Fig. 9(e) are the un-deteriorated distorted images calculated by distorting the original checkerboard according to the theoretical and simulated image positional relationships (Fig. 8(a)). The final deteriorated image of the achromatic WFOV metalens is obtained by the convolution of the un-deteriorated distorted image and PSF weighted by the focusing efficiency of each image point. Fig. 9(d) shows the theoretical image of the achromatic WFOV metalens at a NA $= 0.76$ calculated using the theoretical PSFs and image postional relationship. Fig. 9(f) shows the simulated image of the achromatic WFOV metalens at a NA $= 0.76$ calculated using the simulated PSFs and image postional relationship.
The image size is close to $40 \mu m * 40 \mu m$, less than 100 PSFs, which is very small because of the short focal length. The diffractive effect becomes apparent, so the image looks a little blurry. Actually, the image performance is close to that of a diffractive limited system since the FWHMs are essentially equal to the FWHMs of an ideal lens, as analyzed in Fig. 7. Except for the distortion, the images shown in Fig. 9(d) and Fig. 9(f) are virtually free of monochromatic aberrations. The sharpness in the middle and edge area of the images are similar. While the image positions for these two operating wavelengths match well, a slight color blur effect can be seen in the images. This may be caused by the significant change in FWHM at wavelengths from $600nm$ to $1200nm$. The color displayed on the final image depends on the size of the imaging spot (i.e., FWHM) and the positional relationship at each wavelength. For comparison, the images of a hyperbolic metalens without achromatic engineering (named chromatic metalens) are obtained by theoretical calculation and simulation, shown in Fig. 9(g)–Fig. 9(j). The chromatic metalens has the same focal length and aperture diameter as the achromatic metalens at NA $= 0.76$, while its design wavelength is $600nm$. Significant color separation and blurred images are obtained.
We design another achromatic metalens with a quadratic phase at NA $= 0.35$ and harmonic order $p_{\text {design}} = 4$. All detailed results of theoretical calculations and simulations can be found in Supplement 1, section 3. This metalens enables achromatic imaging at three harmonic wavelengths $\lambda = 600 nm$, $800 nm$, and $1200 nm$ by virtue of the harmonic diffraction. But its FOV is about $24^\circ$, only comparable to that of the hyperbolic phase, which is consistent with the FOV limit analysis shown in Fig. 2.
In summary, we demonstrate that the harmonic metalens with a quadratic phase can simultaneously achieve achromatic and WFOV imaging at a large NA.
4. Discussion and conclusion
In this paper, we propose an achromatic WFOV metalens to simultaneously achieve achromatic imaging at certain discrete wavelengths and WFOV imaging by virtue of harmonic diffraction and quadratic phase profile. Firstly, we investigate the COR between FTs of the quadratic phase and ideal imaging phase to quantitatively explain why the quadratic phase enables WFOV imaging. Secondly, we derive the complex-amplitude transmittance formula of a harmonic metalens described as a sum of Fourier series. Integrating with Angular Spectrum theory, the derived formula can calculate the theoretical field distribution transmitted from a harmonic metalens. Thirdly, we design two achromatic metalenses with a quadratic phase at a small NA (0.35) and a large NA (0.76), respectively, and deeply analyze their imaging characteristics, including image positions, focusing efficiencies, PSFs, and final images. The metalens at a small NA (0.35) can realize achromatic imaging thanks to harmonic diffraction, but its FOV is at the same level as the hyperbolic phase. The other one at a large NA (0.76) enables simultaneously achromatic and WFOV imaging, of which FOV ($100^{\circ }$) is significantly higher than that of the hyperbolic phase.
The nanostructures of metalenses in this paper are taller than the conventional ones because they need to provide larger phases to realize high-order harmonic diffraction. A wider wavelength range for achromatic imaging can be realized by higher-order harmonic diffraction, which requires taller nanostructures to provide greater phase modulation. Considering fabrication, the height of nanostructures is limited. Only $4th$-order and $2nd$-order harmonic diffraction are achieved in this paper by the proposed achromatic metalenses at NA $= 0.35$ and $0.76$, respectively.
The achromatic WFOV metalenses proposed in this paper have at least three advantages. First, suitable nanostructures have simple cross-sections which can be found without laborious scanning work by simply changing their widths. Second, the aperture can be arbitrarily large in theory because only the phase and transmittance of the nanostructures need to be considered to achieve harmonic diffraction. Third, the proposed metalens is polarization-insensitive due to the symmetric nanostructures, which is conducive to a lightweight system. The achromatic WFOV metalens can find numerous applications in a wide range, such as AR/VR, phone cameras, and microscopies.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. H. He, H. Yang, Z. Xie, and X. Yuan, “Broadband mid-infrared metalens with polarization-controlled at-will chromatic dispersion,” Nanoscale 14(34), 12476–12482 (2022). [CrossRef]
2. G. Cao, H.-X. Xu, L.-M. Zhou, Y. Deng, Y. Zeng, S. Dong, Q. Zhang, Y. Li, H. Yang, and Q. Song, “Infrared metasurface-enabled compact polarization nanodevices,” Mater. Today 50, 499–515 (2021). [CrossRef]
3. H. Yang, Y. Jiang, Y. Hu, K. Ou, and H. Duan, “Noninterleaved metasurface for full-polarization three-dimensional vectorial holography,” Laser Photonics Rev. 16(11), 2200351 (2022). [CrossRef]
4. M. Pu, X. Li, Y. Guo, X. Ma, and X. Luo, “Nanoapertures with ordered rotations: symmetry transformation and wide-angle flat lensing,” Opt. Express 25(25), 31471–31477 (2017). [CrossRef]
5. A. Martins, K. Li, J. Li, H. Liang, D. Conteduca, B.-H. V. Borges, T. F. Krauss, and E. R. Martins, “On metalenses with arbitrarily wide field of view,” ACS Photonics 7(8), 2073–2079 (2020). [CrossRef]
6. E. Lassalle, T. W. Mass, D. Eschimese, A. V. Baranikov, E. Khaidarov, S. Li, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Imaging properties of large field-of-view quadratic metalenses and their applications to fingerprint detection,” ACS Photonics 8(5), 1457–1468 (2021). [CrossRef]
7. M. Y. Shalaginov, S. An, F. Yang, P. Su, D. Lyzwa, A. M. Agarwal, H. Zhang, J. Hu, and T. Gu, “Single-element diffraction-limited fisheye metalens,” Nano Lett. 20(10), 7429–7437 (2020). [CrossRef]
8. F. Zhang, M. Pu, X. Li, X. Ma, Y. Guo, P. Gao, H. Yu, M. Gu, and X. Luo, “Extreme-angle silicon infrared optics enabled by streamlined surfaces,” Adv. Mater. 33(11), 2008157 (2021). [CrossRef]
9. A. Arbabi, E. Arbabi, S. M. Kamali, Y. Horie, S. Han, and A. Faraon, “Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations,” Nat. Commun. 7(1), 13682 (2016). [CrossRef]
10. B. Groever, W. T. Chen, and F. Capasso, “Meta-lens doublet in the visible region,” Nano Lett. 17(8), 4902–4907 (2017). [CrossRef]
11. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13(3), 227–232 (2018). [CrossRef]
12. Y. Wang, Q. Chen, W. Yang, Z. Ji, L. Jin, X. Ma, Q. Song, A. Boltasseva, J. Han, V. M. Shalaev, and S. X. Xiao, “High-efficiency broadband achromatic metalens for near-ir biological imaging window,” Nat. Commun. 12(1), 1–7 (2021). [CrossRef]
13. S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, C. Hung Chu, J.-W. Chen, S.-H. Lu, J. Chen, B. Xu, C.-H. Kuan, T. Li, S. Zhu, and D. P. Tsai, “Broadband achromatic optical metasurface devices,” Nat. Commun. 8(1), 1–9 (2017). [CrossRef]
14. W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13(3), 220–226 (2018). [CrossRef]
15. W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10(1), 1–7 (2019). [CrossRef]
16. Z. Li, P. Lin, Y.-W. Huang, J.-S. Park, W. T. Chen, Z. Shi, C.-W. Qiu, J.-X. Cheng, and F. Capasso, “Meta-optics achieves rgb-achromatic focusing for virtual reality,” Sci. Adv. 7(5), eabe4458 (2021). [CrossRef]
17. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength metasurfaces through spatial multiplexing,” Sci. Rep. 6(1), 32803 (2016). [CrossRef]
18. D. Lin, A. L. Holsteen, E. Maguid, G. Wetzstein, P. G. Kik, E. Hasman, and M. L. Brongersma, “Photonic multitasking interleaved si nanoantenna phased array,” Nano Lett. 16(12), 7671–7676 (2016). [CrossRef]
19. Y. Zhou, I. I. Kravchenko, H. Wang, J. R. Nolen, G. Gu, and J. Valentine, “Multilayer noninteracting dielectric metasurfaces for multiwavelength metaoptics,” Nano Lett. 18(12), 7529–7537 (2018). [CrossRef]
20. O. Avayu, E. Almeida, Y. Prior, and T. Ellenbogen, “Composite functional metasurfaces for multispectral achromatic optics,” Nat. Commun. 8(1), 14992 (2017). [CrossRef]
21. Z. Huang, M. Qin, X. Guo, C. Yang, and S. Li, “Achromatic and wide-field metalens in the visible region,” Opt. Express 29(9), 13542–13551 (2021). [CrossRef]
22. M. M. Elsawy, A. Gourdin, M. Binois, R. Duvigneau, D. Felbacq, S. Khadir, P. Genevet, and S. Lanteri, “Multiobjective statistical learning optimization of rgb metalens,” ACS Photonics 8(8), 2498–2508 (2021). [CrossRef]
23. H. Li, X. Xiao, B. Fang, S. Gao, Z. Wang, C. Chen, Y. Zhao, S. Zhu, and T. Li, “Bandpass-filter-integrated multiwavelength achromatic metalens,” Photonics Res. 9(7), 1384–1390 (2021). [CrossRef]
24. B.-X. Wang, Y. He, P. Lou, and W. Xing, “Design of a dual-band terahertz metamaterial absorber using two identical square patches for sensing application,” Nanoscale Adv. 2(2), 763–769 (2020). [CrossRef]
25. B.-X. Wang, Y. He, P. Lou, and H. Zhu, “Multi-band terahertz superabsorbers based on perforated square-patch metamaterials,” Nanoscale Adv. 3(2), 455–462 (2021). [CrossRef]
26. B.-X. Wang, C. Xu, G. Duan, J. Jiang, W. Xu, Z. Yang, and Y. Wu, “Miniaturized and actively tunable triple-band terahertz metamaterial absorber using an analogy i-typed resonator,” Nanoscale Res. Lett. 17(1), 35 (2022). [CrossRef]
27. B.-X. Wang, C. Xu, H. Zhou, and G. Duan, “Realization of broadband terahertz metamaterial absorber using an anti-symmetric resonator consisting of two mutually perpendicular metallic strips,” APL Mater. 10(5), 050701 (2022). [CrossRef]
28. B.-X. Wang, W. Xu, Y. Wu, Z. Yang, S. Lai, and L. Lu, “Realization of a multi-band terahertz metamaterial absorber using two identical split rings having opposite opening directions connected by a rectangular patch,” Nanoscale Adv. 4(5), 1359–1367 (2022). [CrossRef]
29. P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017). [CrossRef]
30. A. Kalvach and Z. Szabó, “Aberration-free flat lens design for a wide range of incident angles,” J. Opt. Soc. Am. B 33(2), A66–A71 (2016). [CrossRef]
31. W. Liu, H. Cheng, J. Tian, and S. Chen, “Diffractive metalens: from fundamentals, practical applications to current trends,” Adv. Phys.: X 5(1), 1742584 (2020). [CrossRef]
32. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34(14), 2462–2468 (1995). [CrossRef]
33. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34(14), 2469–2475 (1995). [CrossRef]
34. M. Khorasaninejad, A. Y. Zhu, C. Roques-Carmes, W. T. Chen, J. Oh, I. Mishra, R. C. Devlin, and F. Capasso, “Polarization-insensitive metalenses at visible wavelengths,” Nano Lett. 16(11), 7229–7234 (2016). [CrossRef]
35. M. Khorasaninejad, W. T. Chen, A. Y. Zhu, J. Oh, R. C. Devlin, C. Roques-Carmes, I. Mishra, and F. Capasso, “Visible wavelength planar metalenses based on titanium dioxide,” IEEE J. Sel. Top. Quantum Electron. 23(3), 43–58 (2017). [CrossRef]
