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Abstract
Optical metalens has been attracting more and more attention in recent years. To date, it is still difficult to simultaneously achieve wide field and broadband imaging in the visible region, which is very important in many applications, such as cameras, microscopy, and other imaging devices. In this paper, we design a double-layer metalens to achieve achromatic imaging over a field of view (FOV) of 60° in the visible light range of 470 nm to 650 nm, and its performance is verified by numerical simulations. The numerical aperture (NA) of the metalens is 0.61 and the average focusing efficiency is > 50% at normal incidence. The metalens has an additional advantage of polarization insensitivity.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As an important optical element, the refractive lens has been widely used in numerous devices [1–3]. Traditionally, an aberrations correction refractive lens consists of two or more pieces [4,5], which leads to bulky and heavy optical devices. One solution to overcome the problem is using flat optics [6,7]. Metalens, consisting of optical nanoantennas as thin as tens of nanometers on a surface, has been proposed as a candidate to replace the traditional lens owing to its ability to manipulate the properties of incoming light wavefront [8–10]. A number of applications based on metalens have been demonstrated, such as cameras [11], telescopes [12], augmented reality devices [13], optical imaging encoding [14–16] and optical aligners [17–19].
Early metalenses were designed for imaging applications at normal incidence. For example, Yu et al. designed a single-layer gradient metasurface to achieve around 45% transmission at 710 nm in 2015 [20]. However, large FOV imaging is much more attractive and highly desirable for real applications. For this purpose, an aplanatic metasurface on a curved substrate was proposed by Capasso et al. [21]. The technique allows the curved metasurface placed on substrates to eliminate the coma aberration at large FOV, but the metasurface is complex and bulky, which is not good for integration. To solve this problem, Arbabi et al. designed a miniature flat camera integrating a monolithic metasurface lens doublet based on high-contrast metasurfaces [22], in which a FOV of 60° was obtained at 850 nm. Similarly, Groever et al. demonstrated a compact metalens doublet based on Pancharatnam-Berry (P-B) phase principle [23]. This design obtained a FOV of up to 50°, but the focal length varied with the working wavelength in a visible region [24]. In 2019, Fu et al. reported a highly integrated step-zoom metalens with a FOV of 40° and dual focal lengths based on double-sided metasurfaces on a transparent substrate [25], allowing focusing of both x-polarized and y-polarized beam at 658 nm. Moreover, Guo et al. presented a methodology to extend the FOV by exploring the local catenary optical fields and symmetry transformation from rotational symmetry to transversal symmetry. This design can achieve a FOV of 120° at around 19 GHz [26]. Xu et al. proposed a beam-steering method based on wave-front shaping through a disorder-engineered metasurface, in which a FOV of 160° at 532 nm was achieved [27].
On the other hand, broadband color imaging with metalenses is another challenge for researchers due to chromatic aberration. To reduce chromatic aberration, Li et al. designed a series of flat dielectric metalens that are able to steer the dispersion arbitrarily for three wavelengths (473, 532 and 632.8 nm) in visible region and the light with different wavelengths can be focused on any desired spatial positions [28]. However, the spatial multiplexing results in low focusing efficiency. Avayu et al. introduced dense vertical stacking of independent metasurfaces, where each layer was made from a different material [29]. Through spatial multiplexing, the metalenses can focus different wavelengths (450, 550 and 650 nm) in the same focal plane, but with 5.8-8.7% focusing efficiency. Another way of realizing achromatic metalenses is incorporating an integrated-resonant unit element with P-B phase principle. In 2018, Wang et al. demonstrated that the focal length remains unchanged as the incident wavelength varies from 400 to 660 nm [30]. But the metalens only worked with particular circularly polarized light due to the effect of P-B phase. To make the metalens polarization-insensitive, Chen et al. designed a metalens consisting of anisotropic nanostructures to eliminate the effect of P-B phase [31]. This design allows the metalens working across nearly the entire visible spectrum from 460 nm to 700 nm. Unfortunately, in both designs the NAs are small (0.106 and 0.2, respectively).
To our knowledge, there is few design to simultaneously achieve achromatic and large FOV metalens. Until this year, we notice that Kim et al. designed a doublet metalens for this purpose. [32]. In their design, the meta-atom in the first layer is a square structure with a hole at the center. This layer works with propagation phase principle. The meta-atom in the second layer is an anisotropic rectangular nanofin. This layer works with P-B phase principle. The design can achieve a FOV of 60° at three wavelengths (445, 532 and 660 nm) but with different NAs for different wavelengths. Also, the NAs are very small, 0.33, 0.38, and 0.47, respectively.
In this work, we design a doublet metalens with a FOV of 60° and only a NA of 0.61 for three wavelengths of 473 nm (blue), 532 nm (green) and 632.8 nm (red). We introduce the method and design details in Section 2. The simulated results and analysis are presented in Section 3.
2. Methods and design
Previous works have demonstrated that a doublet metalens has the ability to eliminate off-axis aberrations, such as spherical and coma aberration, but can’t reduce chromatic aberration [24]. The reason is use of P-B phase results in different focal lengths at different wavelengths. To avoid the effect, only propagation phase principle is used in our design.
Figure 1(a) schematically shows our double-layer metalens in which the two layers are supported by a thick SiO2 substrate. Lossless semiconductor TiO2 nanorods are used in both layers to access waveguide-like resonant modes in the visible spectrum [33–35]. The nanorods behave as truncated waveguides with circular cross sections supporting Fabry-Perot resonances and thus generate different phase outputs at different wavelengths. The high refractive index difference between the nanorods and their surroundings leads to weak optical coupling among the nanorods and allows for the implementation of any phase profile with subwavelength resolution by spatially varying the structure parameters of the nanorods. Since the cylinder is isotropic with respect to light in all polarization directions, we use cylindrical nanorods to achieve polarization-insensitive focusing.
Fig. 1. Schematic view of the doublet metalens. The metalens contains two TiO2 layers supported by a SiO2 substrate. The TiO2 layer consists of an array of nanorods. (a) 3-D view of the metalens. (b) Side-view of the metalens with the trace of the beam. (c) Top-view of a unit. (d) Side-view of a unit. $r$ and $h$ are the radius and height of a nanorod, respectively, and $D$ is the lattice constant.
In principle, the first TiO2 layer serves as a modulation layer (ML) to correct off-axis aberrations while the second TiO2 layer is used as a focusing layer (FL), as illustrated in Fig. 1(b). By using cylindrical TiO2 nanorods, the metalens can be used for focusing different polarized beams [36,37]. Thanks to low dispersion of TiO2, the phase dependence of the light frequency is linear [38], which provides potential to achieve achromatic and large FOV focusing in wide waveband.
The ideal phase profile produced by the ML can be defined as odd-order polynomials of the radial coordinate $\rho \,\,\left( {\rho = \sqrt {{x^2} + {y^2}} } \right)$ as the following equation:
The ideal phase profile produced by the FL is obtained by the following equation [24]:
In Eq. (2), the first term is a function of $\rho $ when $f$ and ${\lambda _d}$ are fixed, and can be replaced by the odd-order polynomials of the radial coordinate $\rho $. Therefore, the phase profile of the FL can be transformed into the same form as Eq. (1):
where ${b_n}$ is the phase profile coefficient.The phase profile coefficients ${a_n}$ and ${b_n}$ can be obtained by ray tracing method in which the objective function is based on gradient descent with the largest focal spot along the focal plane. In our calculation, ${\lambda _d}$ is set to 473 nm (blue), 532 nm (green) and 632.8 nm (red), respectively. Considering the computation ability, we set the ${R_{ML}}$ size to 2.5 µm, the thickness of the SiO2 substrate to 5 µm and the focal length to 3.5 µm. It is noted that the initial ${a_n}$ and ${b_n}$ are zero in our design. The optimal parameters (${a_n}$, ${b_n}$ and ${R_{FL}}$) are shown in Table 1. The corresponding phase profiles are shown in Fig. 2. Besides, the optimal ${R_{FL}}$ is 4.3 µm, and thus the NA is 0.61. As can be seen in Fig. 2(a), the ideal phase profiles for the ML at three wavelengths are similar to a Schmidt plate that is widely used to correct spherical aberration [39]. This results in converging chief rays and diverging marginal rays. The needed phase coverage for each wavelength is less than 0.2$\pi $ and the needed phase compensation range for three center wavelengths is less than 0.04$\pi $. For the FL, the ideal phase profiles are similar to a focusing lens (see Fig. 2(b)). The needed phase coverage for each wavelength is about 2$\pi $ if all the phase values are folded into [-$\pi $,$\pi $) and the needed phase compensation range for three center wavelengths is within 0-2$\pi $.
Fig. 2. Ideal phase profile of the metalens. (a) Phase profile of the modulation layer to correct spherical aberrations. (b) Phase profile of the focusing layer.
Table 1. Ideal phase profile coefficients at three wavelengths.
To determine the phase dependence of the radius and height of the nanorods, we performed parameter scanning using finite difference time domain (FDTD) method at three wavelengths. The nanorod radius varies from 30 nm to 150 nm, and the height from 0.6 µm to 1.2 µm (see Figs. 3(a)-(f)). According to the Nyquist sampling criterion ($D < {\lambda _d}/\textrm{2NA}$), D is set to 300 nm. We used ${\varphi _R}$, ${\varphi _G}$ and ${\varphi _B}$ to represent the phase shift produced by a nanorod at 632.8 nm, 532 nm and 473 nm wavelengths. Similarly, we used ${\varphi _{R\textrm{0}}}$,${\varphi _{G\textrm{0}}}$ and ${\varphi _{B\textrm{0}}}$ to represent the ideal phase required at the three wavelengths. Therefore, for one nanorod, the phase error is ${\varphi _0} = |{\varphi _R} - {\varphi _{R0}}|+ |{\varphi _G} - {\varphi _{G0}}|+ |{\varphi _B} - {\varphi _{B0}}|$. Due to its periodic nature, each absolute term should less than π. For a metalens, the phase error is ${\varphi _i} = \sum {{\varphi _0}} $. At RGB wavelengths, the perfect achromatic and wide field imaging can be achieved when ${\varphi _i}\textrm{ = 0}$. Figures 3(a)-(f) shows the phases and transmittances produced by different nanorods at RGB wavelengths. Figures 3(g)-(h) shows minimum ${\varphi _i}$ at different heights for the ML and FL. As shown in Fig. 2(a), the required phase compensation range for the ML is relatively narrow and thus small ${\varphi _i}$ is needed. In Fig. 3(g), the lowest ${\varphi _i}$ occurs at the rod height of around 740 nm, where the transmittance at B wavelength is relatively low (see blue areas in Fig. 3(d)). Considering large transmittance for all wavelengths and easy fabrication, we choose $h$ = 660 nm as the rod height of the ML (see white dash lines in Fig. 3). For the FL, the required phase compensation range is large, so low-height nanorods cannot generate sufficient phase coverage, especially at long wavelength. It is worth noting that the values of minimum ${\varphi _i}$ are very close at different heights (see Fig. 3(h)). Therefore, we choose $h$ = 940 nm as the rod height of the FL (see black dash lines in Fig. 3). Figure 3(i) gives out the phase values generated by the nanorods with the heights of 0.66 µm and 0.94 µm and the radii of 30 nm and 150 nm. In 0.66 µm case, 30 nm and 150 nm radius produce a phase compensation range of 0-1$\pi $ at all wavelengths, which can meet ideal dispersion requirement of ML. Similarly, in 0.94 µm case, both radii produce a phase compensation range of 0-2$\pi $ at all wavelengths, which can meet ideal dispersion requirements of FL.
Fig. 3. (a, b, c) Simulated phase shift and (d, e, f) transmittance as a function of nanorod radius and height at RGB wavelengths. The white dashed line indicates the nanorod height of the ML (660 nm) and the black dashed line indicates the nanorod height of the FL (940 nm). (g) Minimum ${\varphi _i}$ for ML with respect to different Height $h$. (h) Minimum ${\varphi _i}$ for FL with respect to different Height $h$. (i) Phases generated by the nanorods with the heights of 0.66 µm and 0.94 µm and the radii of 30 nm and 150 nm.
3. Results and analysis
In order to verify our design, a FDTD method is used to analyze the focusing performance of the metalens at different incident angles and RGB wavelengths, as shown in Figs. 4–6. The incident light is a linearly p-polarized plane wave. As seen from all the figures (a), the focal spots are almost in the same focal plane at the three wavelengths when the incidence angles are <30°. In all the figures (b), it can be observed that all the focal spots have similar shapes. It is mentioned that the focal spots with large FOVs (>20°) have slight deformation at 532 nm wavelength due to the influence of curvature of field (see Fig. 5(a)). To make a comparison with diffraction-limited focusing performance, we calculated the full-width half-maximum (FWHM) values, as denoted by the dash lines in all the figures (c). For 473 and 632.8 nm wavelengths, it can be seen that the FWHM values are almost same at all incident angles and close to the diffract limit (${\lambda _d}\textrm{/2NA}$). As predicted at 532 nm wavelength, the FWHM values increase at the incidence angles >20° and slightly deviates from the diffract limit.
Fig. 4. (a) Normalized longitudinal intensity profiles for different incident angles at $\lambda $ = 632.8 nm. The white dashed line indicates the position of the focal plane. (b) Normalized focal spot profiles for different incidence angles at.. = 632.8 nm. Scale bars: 500 nm. (c) FWHMs along x-direction with respect to incident angle at $\lambda $ = 632.8 nm.
Fig. 5. (a) Normalized longitudinal intensity profiles for different incident angles at $\lambda $ = 532 nm. The white dashed line indicates the position of the focal plane. (b) Normalized focal spot profiles for different incidence angles at $\lambda $ = 532 nm. Scale bars: 500 nm. (c) FWHMs along x-direction with respect to incident angle at $\lambda $ = 532 nm.
Fig. 6. (a) Normalized longitudinal intensity profiles for different incident angles at $\lambda $ = 473 nm. The white dashed line indicates the position of the focal plane. (b) Normalized focal spot profiles for different incidence angles at $\lambda $ = 473 nm. Scale bars: 500 nm. (c) FWHM along x-direction with respect to incident angle at $\lambda $ = 473 nm.
To measure the focusing quality at RGB wavelengths, we calculated its modulation transfer function (MTF) and focusing efficiency. The MTF is obtained by taking the modulus of the discrete Fourier transformation of the intensity distribution at the intersection between the focal plane and the plane of incidence (x-direction) [40]. As can be seen in Figs. 7(a)-(c), the MTF of our metalens is very close to the diffraction limit at all FOVs, making the light energy almost concentrated in the focal spot and thus leading to high focusing efficiency (see Fig. 7(d)). The focusing efficiency is calculated as focal spot energy in an area of diameter 3× FWHM divided by the incident light energy [7]. At normal incidence, the focusing efficiency of the red light is high up to 70%. The focusing efficiencies of green and blue lights are also higher than 59.2% and 38.7%, respectively.
Fig. 7. MTF curve with different incident angles at (a)$\lambda $ = 473 nm, (b)$\lambda $ = 532 nm, (c) $\lambda $ = 632.8 nm. (d) Focusing efficiency with respect to incident angle at RGB wavelengths.
In order to demonstrate that our design can achieve wideband achromatic, we calculated the focusing performance of multiple wavelengths from 440 to 680 nm, as shown in Fig. 8. As seen from Figs. 8(a) and 8(b), the metalens can achieve good focusing performance in the wavelength range of 470 to 650 nm with FOV of 60°. All the focal spots are in the same plane within this range. Outside the operative range, the focal points deviate from the same plane due to the effect of field curvature. If one wants to improve the operative range, more center design wavelengths are needed, which inevitably leads to more complicate nanostructures. Figure 8(c) gives out the FWHM comparison with the diffract limit. It is obvious that the FWHM values are close to the diffract limit. All focusing efficiencies are over 38% at normal incidence and the average efficiency is over 50% (see Fig. 8(d)). With the increase of incidence angle, the focusing efficiency decreases. At 30° incident angle, the average focusing efficiency is 27%.
Fig. 8. Normalized longitudinal intensity profiles for different incident wavelengths at (a) 0° FOV, (b) 30° FOV. The white dashed line indicates the position of the focal plane. (c) FWHM along x-direction with respect to wavelength at 0° and 30° FOV. (d) Focusing efficiency with respect to wavelength at 0° and 30° FOV.
4. Conclusion
In this study, we have proposed a doublet metalens to achieve achromatic with FOV of 60° in the visible light range of 470 nm to 650 nm. The NA of the metalens is 0.61 and the average focusing efficiency is > 50% at normal incidence. It is noticed that for practical use the size of our metalens can be increased by increasing the number of designed central wavelengths and developing more resonant cells [41]. This provides a possibility for the metalens to replace the traditional large and bulky achromatic and aberration correction lenses, and provides a convenience for the development of ultra-small achromatic large FOV lenses, which have the characteristics of ultra-thin, ultra-flexible and simple. It is expected to be integrated into cameras, microscopes, telescopes and VR/AR devices, reducing the volume while ensuring the quality of the images. In the future, we expect that the proposed work can be applied in a variety of fields, such as information technology, integrated optics, optical imaging, biomedicine, tablet display and wearable electronic devices.
Funding
Natural Science Foundation of Sichuan Province (2019YJ0172).
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.
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