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Abstract
Benefiting from the excellent capabilities of arbitrarily controlling the phase, amplitude and polarization of the electromagnetic wave, metasurfaces have attracted much attention and brought forward the revolution of fields ranging from device fabrications to optical applications. Cascaded metasurfaces have been demonstrated to correct the monochromatic aberration and enable a near-diffraction-limited focusing spot over a wide incident angle. However, they can only work under the design wavelength and suffer from the axial chromatic aberration at another wavelength. Here, an achromatic metasurface doublet is proposed to eliminate the axial achromatic aberration and enable high-quality focusing with a wide incident angle at distinct wavelengths. It consists of square nanopillar arrays with spatially varying width to simultaneously realize wavelength-dependent phase controls. The constructed metasurface doublet is further verified numerically and near-diffraction-limited foci are exactly formed at the same plane with an incident angle up to 20° for design wavelengths. We expect that our proposed approach can find optical applications in the fields of holograms, photograms, microscopy and machine vision.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A singlet lens, as a fundamental optical component in imaging systems, is usually composed of two convex/concave/flat surfaces for collimating or focusing light, and also functions as a Fourier transform element. In many optical applications, such as the microscopy and camera, the assembly of multiple singlet lenses through an elaborate optical design is adopted to correct different kinds of aberrations, such as chromatic aberration, spherical aberration, coma aberration, astigmatism, etc. Therefore, the whole optical systems are always bulky even without filters and sensors.
In recent years, the ultrathin and lightweight metasurfaces have attracted widespread attention and been successfully applied to the demonstrations of the beam generators [1–9], optical focusing metalenses [10–18] and holograms [19–25]. These efforts push the engineering optics into a new era termed as Engineering Optics 2.0 [26,27]. Normally, they consist of an arrangement of sub-wavelength structures patterned on flat surfaces, with the capabilities of arbitrarily controlling the phase, amplitude and polarization of light at subwavelength spatial resolution. Benefiting from the novel designs and excitation methods, the dielectric metasurfaces for the generation and detection of focusing vortex beam have been extensively investigated with ultrahigh efficiency, and the multichannel metasurfaces for measuring the topological charges have been meticulously demonstrated [28–30]. In addition, based on holographic principle, the holographic free-electron light source has been theoretically and experimentally realized via point-excitation of holographic plasmonic metasurfaces, which is illustrated using medium-energy free-electron injection to generate highly-directional visible to near-infrared light beams and allows generation of light with prescribed wavelength, direction, divergence and topological charge number [31]. In 2016, a metasurface doublet working at the near-infrared wavelength was experimentally demonstrated to correct the monochromatic aberration and form a near-diffraction-limited focus for an incidence angle up to 30° [32]. Following the same approach, a visible metasurface doublet at 532 nm based on the geometric phase was also demonstrated to have a field of view of 50° [33]. Some wide-angle singlet metalenses have been demonstrated by exploring symmetry transformation from rotational symmetry to transversal symmetry [34,35]. However, when their metalenses were illuminated at different wavelengths, the focal length shifted with the incident wavelength due to the chromatic aberration. Even though some deconvolution algorithms can be used to correct distortions and achieve the improvement of the imaging quality [32,36,37], one would suffer from the complex post-processing operation in some scenarios, such as real-time observations and detections.
Here, we propose an achromatic metasurface doublet (AMD) to realize multiwavelength dispersion controls and eliminate the influence of the chromatic aberration existed in the previously reported metasurface doublets. Our AMD consists of square nanopillar arrays with spatially varying widths. Each nanopillar can be regarded as a cross-sectional truncated waveguide supporting the Fabry-Perot resonance and effectively promise wavelength-dependent phase controls. Final achieved phase profiles of the cascaded metasurfaces match well with the required phase profiles through an advanced optimization algorithm. The constructed AMD is further verified numerically and near-diffraction-limited foci are precisely formed at the same plane with an incident angle of 20° for design wavelengths. Furthermore, an apochromatic metasurface doublet, which brings three wavelengths into focus at the same plane, is also demonstrated for the possible generation of any perceived color. We believe the strategy here can offer a new path to correct the chromatic aberration with a wide incident angle and enable the further optical applications with the metasurfaces, such as the computer display and digital imaging.
2. Design and optimization of the AMD
As it is well known, for a single wavelength, the combination of two cascaded structures with specially optimized phase profiles could be used to reduce the monochromatic aberration over a wide incident angle [32,33,38]. Here and below, two cascaded metasurfaces are also adopted. They consist of a large number of the sub-wavelength nano-structures patterned on both sides of a glass substrate and can separately generate the wavelength-controlled phase profiles, which are simultaneously satisfied at each wavelength to correct the monochromatic aberration. Therefore, the AMD can be achieved with achromatic foci over a wide incident angle, as schematically shown in Fig. 1. The required phase profiles and designs of unit cells are analyzed in the following paragraphs.
Fig. 1. The schematic of the achromatic/apochromatic metasurface doublet, which brings two/three wavelengths into focus at the same plane over a wide incident angle.
2.1 Optimizations of cascaded phase profiles at a single wavelength
Firstly, the phase profiles of the two cascaded surfaces are required to be determined at each wavelength, and they can be defined as even order polynomials of the radial coordinate r with the following relationship:
where R is the radius of the device, n is the number of the polynomial coefficients, and the coefficients Ai are optimization parameters. In theory, the increase of n would be helpful to a better focusing performance. However, when a near-diffraction-limited focusing quality is achieved, a larger n not only hardly improves the performance, but also leads to the optimization difficulty. Normally, n is less than 5 with a balance between the optimization efficiency and focusing performance.We design the cascaded devices with the focal plane being located at the distance of 40 µm from the back surface. The thickness of the glass substrate, sandwiched by the front and back surfaces, and the radius R are assumed to be 20 µm and 15 µm, respectively. Therefore, the numerical aperture (NA) is 0.35 at the focal plane. The optimization procedure for Ai is illustrated in Fig. 2(b) with the following steps: first, the required parameters are configured in the Zemax optical design software, such as the entrance pupil diameter, incident angles, working wavelength, BK7 glass thickness, focal plane and so on; second, different merit functions can be selected, such as effective focal length (EFFL) and working F number (WFNO); after that, through minimizing the focal spot size with the incident angle up to 20°, the coefficients Ai can be optimized at the design wavelength. The coefficients Ai at three discrete wavelengths of 473 nm, 532 nm and 632.8 nm are presented in Table 1. The required phase profiles are plotted in Figs. 2(c)–2(e), where the phase profile of the front surface operates as a negative lens and that of the back surface as a positive lens. There are different valleys of 6, 5 and 4 in the dashed lines. Therefore, if an achromatic doublet lens is designed, there should be some differences of the phase profiles at distinct wavelengths. For this reason, previously reported doublet metalenses could only achieve the dispersion correction at the design wavelength and observation plane.
Fig. 2. (a) A schematic of focusing with distinct incident angles. (b) The optimization procedure for Ai in Zemax software. (c-e) Required phase profiles of the front and back surface at 473 nm, 532 nm and 632.8 nm, respectively.
Table 1. Optimized coefficients Ai at 473 nm, 532 nm and 632.8 nm.
2.2 Designs of the unit cell and AMD
To simultaneously achieve the required phase profiles at distinct wavelengths in Table 1 or Fig. 2, the unit cell in Fig. 3(a) based on the propagation phase is used in our designs. Compared to the wide-used geometric phase which exhibits the dispersionless phase profile at different wavelengths, the propagation phase could provide anomalous dispersions. It means that we have an access to realize the wavelength-dependent phase profiles through engineering desired propagation phase. The unit cell here contains a TiO2 nanopillar with the height H, period P and varying square width W deposited on a glass substrate, and it can be considered as a truncated waveguide supporting the Fabry−Perot resonance, where the high refractive-index contrast between TiO2 and air confines the light into the nanopillar, thereby provides a high transmission efficiency. The change of the square width could bring the change of the effective refractive index and provide different propagation phase values.
Fig. 3. (a) Schematic of the unit cell. The TiO2 nanopillar with a height of 600 nm and a varying width is placed on a glass substrate. The period of the unit cell is 200 nm. (b, c) Transmitted phase and amplitude with the change of width at 473 nm, 532 nm and 633 nm. (d) Comparisons of the required phase profiles (solid lines) and achieved phase profiles (dot lines) of the front and back metasurface for AMD.
Light response of unit cell is quantitatively calculated in CST microwave studio software, where the square width is swept from 20 nm to 180 nm in steps of 2 nm, and the height and period are set as 600 nm and 200 nm, respectively. Refractive indexes of the BK7 glass and TiO2 are chosen from the Zemax software and Palik’s handbook [39], and added into the CST software. At wavelengths of 473 nm, 532 nm and 632.8 nm, the TiO2 has refractive indexes of 3.097, 2.982 and 2.874 with near-zero imaginary parts, while the BK7 glass has these of 1.523, 1.519 and 1.515, respectively. The simulated phase and amplitude profiles of the co-polarized transmitted light are presented in Figs. 3(b) and 3(c) at 473 nm, 532 nm and 632.8 nm, when the incident light is assumed to be x-polarized light. The folding of the propagation phase between −180° and 180° is very important because there are at least two choices of square widths when a fixed phase is required at each wavelength. At the same time, the unit cell presents different and unique dispersive responses for distinct wavelengths. In other words, the one-to-one relation of the phase is broken at different wavelengths, i.e. there are several widths for a desired phase at the design wavelength, and these widths provide different phases at another wavelength. Therefore, there is an extra degree of freedom to design the wavelength-controlled devices through changing the square width. In additional, the transmitted amplitudes change slightly with varying square widths and are almost above than 0.9, except for some positions caused by the guided mode resonances, as reported in previous Refs. [32].
To minimize the difference between the required phase profile and the achieved phase profile at different wavelengths, a reference phase C(λ) is introduced as follows [40,41]:
As a proof-of-concept example, our AMD is working at the wavelengths of 473 nm and 532 nm with the parameters mentioned in the above sections, i.e. radius of the metasurface, distance from the back surface to the focal plane, period of the unit cell and the thickness of the glass are 15 µm, 40 µm, 0.2 µm and 20 µm, respectively. Following the optimization procedure, the comparisons between the required and achieved phase profiles with wavelength-dependent tuning reference phases are shown in Figs. 3(d) and 3(e). Root-mean-square error (RMSE) is employed to calculate the absolute difference between the required phase profile and the achieved phase profile. The RMSEs between the required phase profile and achieved phase profile are 54.52° and 63.73° at 473 nm, and 84.33° and 83.87° at 532 nm for the front and back metasurface, respectively. It can be predicted that the focusing performance at 473 nm would be better than that at 532 nm because a smaller RMSE means a better approximation. A bigger RMSE means that it is difficult to choose perfect unit cells in the database to simultaneously satisfy the required phase for distinct wavelengths. Therefore, if a reduced RMSE is needed, a straightforward way is to enlarge the database of the phase modulated by the unit cell, such as changing the width and length of a rectangle nanopillar [38], adding two or more nanopillars with distinct geometric parameters in the same unit cell [15], etc.
3. Numerical simulations
To verify the focusing performance of our proposed design, CST is used to analysis the light propagations through the metasurfaces. The light is assumed to be x-polarized at design wavelengths with distinct incident angles of 0°, 10° and 20°. In order to effectively reduce the computation time, a feasible method proposed in previous references is used [16,18,38]. Interaction between light and metasurface is calculated in CST, and the field distribution at a distance of 0.2 µm from the back metasurface is extracted and then used to calculate the follow-up light distributions by the scalar angular spectrum theory.
The light distributions along the propagating direction and field distributions at the focal plane are presented in Fig. 4. The foci are precisely formed if setting the observation plane at a distance of 40.2 µm from the back metasurface, as presented in Fig. 4(a). The full-widths at half-maximum (FWHMs) of the focusing spot along the horizontal direction, the transverse focus shifts (X position) along the focal plane, and the focusing efficiencies (defined as the ratio of the energy concentrated into the spot and the incident energy) are summarized in Table 2 with different incident angles and wavelengths. We can find that the sizes of focusing spots are near-diffraction-limited and there are some energy losses for the oblique incidence, which may be caused by the light leak between these two metasurfaces and the lower transmission of the unit cells under the off-axis illumination. Definitely, the simulated results agree well with our designs and slight deviations may come from the following reasons: (a) there are some differences between the desired phase profiles and the achieved phase profiles, as shown in Figs. 3(d) and 3(e). These differences would cause the formations of multiple foci, where some energy are contributed, thereby our focusing efficiencies under the normal incidence are much lower than that of an ideal Airy spot, even though the average transmitted amplitude of our unit cells is above about 0.9; (b) in our design, the amplitude profile is assumed to be same and only the phase profile, which plays a more important factor for light focusing, is considered in our optimizations, but the achieved amplitude profile is not flat.
Fig. 4. Focusing performances of AMD at 473 nm and 532 nm with distinct incident angles of 0°, 10° and 20°, respectively. (a) Light distributions along the propagating direction. (b) Light distributions at the focal plane. Central positions are the cross points in (a). (c, d) Normalized field distributions along the horizontal direction in (b).
Table 2. Focusing performances of AMD. N., numerical result; Sim., simulated result.
It is well known that any perceived color can be generated through combining three colors of blue, green and red, such as hologram, photogram and digital imaging technology. Therefore, we extend our strategy to the design of an apochromatic metasurface doublet, which works at three visible wavelengths of 473 nm, 532 nm and 632.8 nm, corresponding to blue, green and red light. The required phase profiles and the achieved phase profiles are plotted in Figs. 5(a)–5(c). The RMSEs between the required phase profile and achieved phase profile are 67.71° and 80.36° at 473 nm, 29.4° and 57.42° at 532 nm, and 67.71° and 101.11° at 632.8 nm for the front and back metasurface, respectively. The focusing performances at the 532 nm should be the best in this apochromatic design from the RMSE values, even better than the AMD design. Although it is difficult to simultaneously satisfy the required phase profiles at three wavelengths, the achromatic focusing behavior still appears if the observation plane is set at a distance of 40.2 µm from the back surface, as presented in Fig. 5(d). It is also found that there is only one focusing spot along the propagating direction at 532 nm because of the lower RMSEs. Notably, if there is a more suitable unit cell which can provide more choices of the width at a required phase, it is possible to design a super-achromatic metasurface doublet in a similar way, which works for more than three wavelengths.
Fig. 5. Design and focusing performance of the apochromatic metasurface double at 473 nm, 532 nm and 632.8 nm. (a, b, c) Comparisons of the required phase profiles (solid lines) and the achieved phase profiles (dot lines) of the front and back metasurfaces. (d) Light distributions along the propagating direction with incident angles of 0°, 10° and 20°, respectively.
4. Discussion
To address the question of the stability of the doublet metasurface, the impacts of the glass thickness and the phase noise are simulated through the scalar angular spectrum theory, which is valid and has been demonstrated in previous works [34,42]. To simply the case, the AMD in Fig. 4 and corresponding achieved phase profiles are used. The thicknesses of the glass are chosen as 10 µm, 20 µm and 30 µm, and the distributions along the propagating direction with the incident angle of 20° are shown in Fig. 6(a). We can find that the focal plane shifts with the change of the thickness and AMD focuses these two wavelengths into the new focal plane. Even though the sandwiched glass is replaced by the air, the focusing spots are still formed. These focusing behaviors can be easily explained by the fact that the front and back phase profiles are acted as two lenses with different focal lengths, and their combination with different geometric parameters has the function of a zoom lens. Next, we add white phase noises (may be caused by the surface shape, imprecise fabrication, etc.) with different levels on the achieved phase profiles and analyze the focusing performances. The slightly degraded focusing spots can be observed with signal-to-noise (SNR) level being 5 dB at 473 nm but being 10 dB at 532 nm, as shown in Fig. 6(b). The different tolerance of noise comes from smaller RMSEs at 473 nm in our design. Therefore, our AMD can work under relatively large phase noises.
Fig. 6. Focusing performances with an incident angle of 20° at 473 nm and 532 nm: (a) different thickness and material as marked in the figure between the front and back surface; (b) noises on achieved phase profiles with different SNR levels. (c) Doublet device at 473 nm with a glass thickness of 500 µm and corresponding field distributions with incident angles of 0°, 10° and 20°. The results are calculated through the scalar angular spectrum theory.
It should be noted that our proof-of-concept AMD is challenging to fabricate at the current stage due to the sandwiched glass with the thickness of 20 µm. However, as demonstrated in Fig. 6(a), we can replace the glass by air and measure the focusing spot at the new focal plane. In this case, the metasurfaces can be fabricated and aligned through the method in [32,33]. Another solution is changing the glass thickness to a standard thickness of the glass substrate. Here, we give the Zemax optimization at 473 nm with a glass thickness being 500 µm in Fig. 6(c). Optimized coefficients Ai are −209.918, 286.144 and 180.173 for the front phase profile, and −1749.949, −10.098 and 4.668 for the back phase profile. The focal plane is located at the distance of 300 µm from the back surface, and the radius R is 300 µm. Based on our proposed achromatic strategy, the AMD with the above parameters can also be constructed (due to the limitation of the computer memory, the electromagnetic distribution is hardly simulated in CST software and not provided here).
5. Conclusion
To summarize, the achromatic metasurface doublet proposed here has been demonstrated to correct the chromatic aberration and realize diffraction-limited focusing with a wide incident angle of 20° at distinct wavelengths. It consists of a large number of the sub-wavelength nano-structures, which can provide different and unique dispersions through the change of the geometric size, thus there is another degree of freedom to engineer the wavelength-dependent wavefronts. One can even achieve a superachromatic metalens doublet with a certain bandwidth through the elaborate dispersion engineering. We believe our compact and ultrathin metasurface doublet, instead of conventional bulky diffractive optical elements, can find many applications in the fields of the hologram, photogram, spectroscopy, machine vision and microscopy.
Funding
National Natural Science Foundation of China (61905031, 61905073); Fundamental Research Funds for the Central Universities (531118010189); Opening Fund from the State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences (SKLOTNM- KFKT-201802, SKLOTNM-KFS2019-1).
Acknowledgments
We would like to thank the State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences for the software sponsorship.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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