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Abstract
Wide-angle, high-efficiency, wide-band, and ultra-compact focusing blocks are crucial for implementation and future evolution of passive millimeter-wave focal plane array imaging systems. The spherical or doublet metalens can attain high-efficiency, wide-angle field-of-view (FOV) but suffer from fabrication difficulties, complex assembly, and low compactness. Here we present an efficient single-metalens design capable of performing high-efficiency diffraction-limited wideband focusing over a wide-angle FOV. This single-metalens design can greatly mitigate the Seidel aberrations by a rational allocation of amplitude-phase of the electromagnetic waves. A proof-of-concept metalens at millimeter-wave band (33 GHz-37 GHz) confirms the validity of our design.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over the past few decades, passive millimeter-wave imaging technology has been rapidly evolving towards high spatial resolution, large field-of-view (FOV), quasi-video frame rate, and miniaturization [1–10]. Especially in recent years, the passive millimeter-wave focal plane array (PMMW-FPA) imaging, superior in safety due to zero radiation, privacy protection, and high frame rate, has been applied in many important scenarios, such as security screening, non-destructive evaluation, non-invasive medical diagnosis, and remote sensing applications [11–19]. The main challenge in these multi-sensor staring imaging systems is to achieve a large FOV with satisfactory spatial resolution and fast image acquisition [20–28]. Thus, wide-angle, high-efficiency, and diffraction-limited wideband focusing lenses or reflectors are crucial for attaining the high-resolution and high-quality image over a large FOV. To further enlarge the FOV and increase the imaging speed, the various one-dimensional or two-dimensional mechanical scanning modes, such as raster scanning [29,30], line scanning [31], zigzag scanning [32], conical scanning [33,34], helical scanning [35,36], and rotary scanning [37], are required. On this basis, there is a great demand for the ultra-compact and low-cost imaging architecture to increase the utility of these imaging techniques. However, angular-dependent Seidel aberrations (e.g., coma, astigmatism, and field curvature) are among the major challenges to realize imaging systems with enhanced performances and functionalities. Conventional refractive curved lenses or compound lenses have been implemented to correct the Seidel aberrations and attain diffraction-limited wide-angle operation [21,37,38]. Such curved or multi-lens architectures, however, suffer from fabrication difficulties, complex assembly, and bulky body. As a superior substitution, the metasurface, featuring subwavelength thickness, advanced beam controlling capability, high degree of function integration, and ease of fabrication and integration, has been applied to synthesize various focusing lenses to correct or mitigate angular-dependent Seidel aberrations [39–47]. Spherical and coma aberrations can be corrected by patterning meta-atoms on a spherical surface, which unfortunately faces with a huge fabrication challenge [39–41]. Besides, similar conformal metalens would increase the difficulty of system integration [42]. An alternative approach is to correct the monochromatic aberrations by cascading and constructing metalens doublets, where the significant portion of focusing is performed by one of the metasurfaces while the other operates as a Seidel aberration corrector [43–46]. Nearly diffraction-limited focusing over an angular FOV up to 60° in the near-infrared (wavelength λ=850 nm) has been demonstrated by a metalens doublet with its longitudinal thickness of 1001200 nm (≈1178λ) [44]. In the visible (λ=532 nm), another metalens doublet enabling the diffraction-limited imaging along the focal plane over an angular FOV of 50° has also been demonstrated, where its longitudinal thickness is 501200 nm (≈942λ) [43]. Compared to the single-metalens, the increased profile of the metalens doublets can be acceptable in the infrared and visible, however, that of their millimeter-wave counterparts may be unreasonable for a practical implementation. The planar single-metalens, nevertheless, has a limited angular FOV of 30° with reduced diffraction-limited focusing performance and low efficiencies of 6-20% [47–49].
To tackle the aforementioned issues, this paper will start from the propagation of electromagnetic (EM) wave to find a highly-efficient metalens design method to suppress the third-order Seidel aberrations and further expand the angular FOV while maintaining high focusing efficiency and an ultra-compact profile. The metasurface-enabling arbitrary wave-front manipulation can contribute to realize the planning and the management of the incoming and the outgoing EM wave travelling path. This strategy carries out the energy allocation and the phase arrangement of the EM waves by simply pairing the incident angle of the illuminating beam and the exit angle of the diffracted beam. An incident-exit angle pairing method (IEAPM) is consequently proposed to perform the metalens design, featuring in its extreme simplicity yet surprising effectiveness without redundant iterative optimization and time-consuming meta-atom implementation processes. To enable the PMMW-FPA imaging camera to perform real-time image acquisition with a diffraction-limited imaging quality over a large FOV, an efficient single-metalens design incorporated into the rotary scanner system is presented. A proof-of-concept single-metalens with its longitudinal thickness of 0.19λ0 (@35 GHz) at the millimeter-wave band (33 GHz-37 GHz) is designed and experimentally verified.
The following text of this paper is arranged as follows. In section 2, we provide the IEAPM and make a detailed interpretation. In section 3, a wide-angle, wideband, and polarization-insensitive unit element, namely the regular hexagon slots (RHSs), is constructed. The metalens system configuration is then determined, and under offset illumination of a linearly polarized conical corrugated horn antenna (LP-CCHA), an off-axis focusing metalens is designed by the IEAPM and assembled in a honeycomb lattice by the RHSs. Its diffraction-limited wide-angle focusing performance is numerically analyzed by the efficient generalized Rayleigh-Sommerfeld diffraction theory (GRS) [50], and compared to that of the conventional on-axis focusing metalens. A proof-of-concept prototype of the proposed metalens is fabricated and measured to verify the above numerical results and the proposed design. In Section 4, we conclude our findings.
2. Metalens design method
As shown in Fig. 1, the y = 0 plane is determined by the feed phase center, the geometric center of the metalens, and the focal point. For convenience, we assume that the metalens is located on the z = 0 plane, and its geometric center is the origin. The feed points to the origin at an incident angle θin, and its phase center is placed on an arc with the center at the origin and a radius of fd′. Its amplitude and phase at any point r′=(x′, y′, 0) on the metasurface are denoted by Afeed(r′; θin) and φfeed(r′; θin), respectively. The focal spot with its exit angle of θout is set at (fd×tanθout, 0, fd) on the z = fd plane. Then, the final phase profile to be compensated by the metalens is composed of the focusing phase and the additional phase for compensating the phase disparity generated by the feed, and reads,
Fig. 1. (a) The three-dimensional schematic diagram and (b) its side-view of the wide-angle metalens and the IEAPM.
From the Fig. 1 and Eq. (1), when the locations of the feed and the focal spot are configured by the incident angle θin and the exit angle θout, respectively, the metalens can be uniquely determined and further channels the incoming and outgoing EM waves along the preset travelling path. It means that each pair of incident-exit angles corresponds to a definite metalens design, and also determines the specific travelling path of the EM waves through the metalens. The planning and the management of the EM wave travelling path, that is, the energy allocation and the phase layout of the EM waves, can be thus performed by simply pairing the incident angle of the feed and the exit angle of the focal spot. We call this metalens design method the incident-exit angle pairing method (IEAPM).
For a determined metalens with the incident-exit angle pair, θin, θout${\in} $[0, π/2), when the metalens is rotated along the arc to the incident angle θin′, we assume that the exit angle of the resulting focal spot is θout′, as shown by the green lines with arrows in Fig. 1. According to the theory of wave propagation, the effective amplitude Afocus(θin′, θout′) and the effective phase Фfocus(θin′, θout′) at the focus, accumulated by the EM wave from the feed phase center through any point r′ on the metalens to the focus, can be represented as,
To attain a uniform, well-focused beam array over a wide angular FOV, the effective amplitudes and phases of all focal spots along the focal plane are required to maintain consistent. For a well-determined metalens, according to Eqs. (1) and (2), it means that each pair of incident-exit angles, θin′-θout′, should satisfy the inverse change relationship. That is, if the incident angle increases, the exit angle should decrease correspondingly, as schematically indicated by the blue and green line in Figs. 1(a) and 1(b). The metalens can be thus determined by a befitting incident-exit angle to perform wide-angle diffraction-limited focusing.
Let us review the conventional on-axis focusing metalens design, where the exit angle of focal spot, θout, increases with the growing incident angle of feed, θin. According to Eq. (2), it will lead to a sharp change in amplitude and phase of the focal spots, in turn result in reduced diffraction-limited focusing, reduced efficiency, and limited angular FOV.
3. Metalens system design, analysis, and verification
3.1 The regular hexagon slots (RHSs)
The three-layer metal-dielectric-metal regular hexagon slots (RHSs) in the honeycomb lattice is presented and optimized to achieve the required EM properties for the PMMW-FPA imaging, such as high transmission efficiency, complete 2π phase coverage, wideband, wide-angle, polarization-independent. As shown in Fig. 2(a), the side length of the RHSs is P=$\sqrt 3 $ mm, and its lattice periods in the x-direction and the y-direction are 3P and $\sqrt 3 $P, respectively. The Copper pattern of 0.035 mm thickness is printed on the substrate (Rogers RT5880) with its thickness of d = 0.787 mm. The inscribed circle radius of the outer hexagon of the slot, Rout = 1.45 mm, keeps constant. When only increasing Rin (the inscribed circle radius of the inner hexagon of the slot) from 0.3 mm to 1.35 mm at intervals of 0.01 mm, within the frequency range from 33 GHz to 37 GHz the RHSs can achieve high enhanced transmission efficiency and approximately linearly adjustable 2π phase coverage, as exhibited in Figs. 2(b) and 2(c). Furthermore, the amplitude-phase response of the optimized RHSs at different oblique incident angles is further investigated, and that of three representative RHSs with their Rin = 0.45 mm, 0.85 mm, and 1.25 mm are extracted and listed in Figs. 2(d), 2(e), and 2(f), respectively. The simulation results show that within the oblique incident angle of at least 45°, the complex transmission coefficients of the RHSs keep almost identical to that at the normal incidence. The amplitude attenuation (at most 0.05) is so slight as to be negligible, and the small phase change is less than 22.5°. Especially, for oblique incidence at the incident angles of not exceeding 30°, almost constant transmission amplitude and slight phase deviations (less than 15°) exactly cater for the upper and lower bounds of the phase intervals when discretizing 360° into 12 specific phases at the intervals of 30° [51,52]. Besides, the geometrically symmetrical hexagon-slot structure on the cross section inherently implies that the RHSs are nearly insensitive to the polarization of the incident EM wave as demonstrated by the Finite-Difference Time-Domain method.
Fig. 2. The construction of the regular hexagon slots (RHSs) and the complex transmission coefficient characteristics varying with the Rin, frequency, and angle of incidence. (a) Geometrical details of the RHSs. (b) and (c) Transmission amplitude and phase of RHSs varying with Rin from 0.3 mm to 1.35 mm and frequency from 32 GHz to 38 GHz at the normal incidence. The transmission amplitude and phase of three representative RHSs with (d) Rin = 0.45 mm, (e) Rin = 0.85 mm, and (f) Rin = 1.25 mm versus gradually varied angles of incidence from 0 to 90°. Note that the 12 white dots in (b) and (c) represent the selected specific RHSs constructing the proposed metalens.
It should be emphasized that compared to the three-layer coaxial annular apertures (CAAs) embed into the square lattice presented in the literature [5], the RHSs with the cross-sectional area of 3$\sqrt 3 $P2/2≈7.79 mm2 are more compact, which is beneficial to achieve denser spatial EM information collection at a finer sampling pixel. For example, when assembling the same metasurface disc with its aperture of 150 mm, it can contain 2269 RHSs, but only holds 1964 CAAs.
In general, from Fig. 2, it is shown that the proposed RHSs can offer high transmission efficiency, complete 2π phase coverage, and fairly stable transmission amplitude and phase properties within a wide solid angle range of at least 90° in the frequency range from 33 GHz to 37 GHz, which are capable of assembling the focusing metalens required by the near-field PMMW-FPA imaging systems.
3.2 Metalens system configuration and design
Here, a linearly polarized conical corrugated horn antenna (LP-CCHA) is optimally designed to feed the metalens. Simulation and measurement results show that the LP-CCHA with compact cross section, has low return loss, high radiation efficiency, high and stable power gain, low side-lobe and cross-polarization levels, and highly consistent and rotationally symmetric radiation pattern within the frequency range from 33 GHz to 37 GHz. To reduce the coupling effect between the LP-CCHA and the metalens while maintaining the compactness of the metalens system, the metalens is placed near the lower boundary of the far-field radiation area of the LP-CCHA. As shown in Fig. 3(a), according to the aperture D′=21 mm of the LP-CCHA and the far-field radiation criterion 2D′2/λ0, the metalens is placed fd′=105 mm away from the LP-CCHA at the normal incidence while keeping their geometric centers aligned. The aperture of the metalens is further determined to be D = 150 mm so that it can cover at least −10 dB edge exposure level (more than 93% illumination efficiency) of the LP-CCHA. To keep the prototypical metalens system compact and low-cost, the focal plane is set on the z = fd = 130 mm plane.
Fig. 3. (a) The system configuration and the design schematic diagram of the optimal OIOO (in red) and the conventional NINO (in black). (b) The OIOO model. (c) The NINO model. All unitless numbers inserted in the figure are in millimeters.
After configuring the metalens system, let’s search a befitting incident-exit angle pair to further determine the metalens according to the IEAPM. In this process, the incident angle θin and the exit angle θout are assigned at intervals of 5° within the angle range of 40°, and then combined into various incident-exit angle pairs. The metalenses corresponding to these angle pairs are further synthesized according to Eq. (1). Their focusing performance in the position, size, shape, intensity, Strehl ratio, and focusing efficiency of the focal spot within an angular FOV of 40° is evaluated and compared with each other by the GRS. The whole process is simulated and completed in MATLAB. Besides, it should be noted that when the incident angle of the LP-CCHA is beyond 40° (the boundary value of the above angle range), the effective illumination efficiency exposed on the metalens is less than 80%. A befitting incident-exit angle pair of 15°-15° is finally determined by the above search process. Under offset illumination of LP-CCHA, an off-axis focusing metalens with its incident-exit angle of 15°-15° is synthesized and assembled by the RHSs, and denoted by OIOO (oblique input, oblique output) for convenience, as shown in Figs. 3(a) (in red) and 3(b). Moreover, as a contrast, a conventional on-axis focusing metalens with its incident-exit angle of 0°-0° is also provided, and denoted by NINO (normal input, normal output), as shown in Figs. 3(a) (in black) and 3(c).
3.3 Numerical analysis and experimental verification of the metalens
To further verify the above results, the prototypical OIOO and its feed, i.e., the LP-CCHA, are machined. As shown in Fig. 4(a), the OIOO and the LP-CCHA are fixed together into the test fixture, and further integrated into the near-field scan test system in the anechoic chamber. Under the illumination of the LP-CCHA at different incident angles, the focusing field patterns generated by the OIOO within the frequency range from 33 GHz to 37 GHz are measured. Figure 4(b) shows the measured focal spots at 35 GHz on several cross sections. As a contrast, the numerical filed patterns at 35 GHz generated by the OIOO and the NINO are also provided by the GRS, as shown in Fig. 4(c) and 4(d). Here, it should be noted that for comparison, when calculating the diffraction field patterns of the NINO, the LP-CCHA is set to rotate along the arc in the positive x-axis region, so that its focal spots are also distributed in the negative x-axis region and keep consistent with these of the OIOO. The focusing parameters characterizing the focal spot location, size, shape, intensity, aberrations, and focusing efficiency, varying with the increasing incident angle from 0 to 40°, are further extracted and drawn in Figs. 4(e), 4(f), and 4(g). These focusing parameters include the full width at half maximum in the x-direction and the y-direction (FWHMx and FWHMy), the out-of-roundness (OOR), the maximum intensity gain (MIG), the encircled energy (EE), the Strehl ratio (SR), and the focusing efficiency (FE). Furthermore, the post-processing of these focusing parameters are also performed and listed in Table 1.
Fig. 4. The near-field scan test system and the calculated and measured diffraction fields of the NINO and the OIOO under the illumination of the LP-CCHA at different incident angles. (a) The near-field scan test system. (b), (c) The measured, calculated focusing field patterns generated by the OIOO. (d) The calculated focusing field patterns generated by the NINO. (e)The FWHMx, the FWHMy, and the OOR of the focal spot. (f) The MIG and the EE of the focal spot. (g) The SR and the FE of the focal spot. “Cal.” and “Mea.” inserted in (e)-(g) are the abbreviation of the calculated and the measured, respectively.
Table 1. The further post-processing of the focusing parameters.
The specific calculations of the above parameters are given as follows. The out-of-roundness (OOR) is expressed as 2|FWHMx-FWHMy|/(FWHMx + FWHMy)×100%, and used to measure the elliptical distortion of the focal spot cross-sectional shape. The energy ratio (EE) calculated in this paper refers to the ratio between the electric-field intensity integral sum encircled by the first dark ring of the focal spot on the focal plane and that covered by the entire focal plane. The Strehl ratio (SR) quantifying the aberrations, is defined as the ratio between the peak electric-field intensity of the focal spot generated by the designed metalens and that of an aberration-free lens under the illumination of the LP-CCHA. The focusing efficiency (FE) is defined as the ratio of the electric-field intensity integral sum covered by the first dark ring of the focal spot on the focal plane to that radiated by the LP-CCHA. The total electric-field intensity radiated by the LP-CCHA is approximated by the total electric-field intensity exposed on the metalens divided by the radiation efficiency (RE) and the illumination efficiency (IE) of the LP-CCHA. Here the specific definitions and calculations of the RE and the IE are given. The RE refers to the ratio of the radiation power of the feed antenna (i.e., the LP-CCHA in our manuscript) to its input power. The IE, also known as the overflow efficiency, is defined as the ratio of the power exposed on the metalens to the radiation power of the feed antenna. The RE and the IE of the LP-CCHA are determined by the Finite-Difference Time-Domain method, as listed in Table 2. For comparison with the ideal aberration-free lens, the radius of the first dark ring is uniformly set to 13 mm, which is determined by the first dark ring of the focal spot generated by the ideal aberration-free lens under the normal illumination of the LP-CCHA. In addition, it should be noted that for both the numerical results and the measured ones, the calculation and sampling area element of the electric-field intensity or power density is fixedly set as 1 mm2 (0.12λ0×0.12λ0), that is, the calculation and sampling step in the x-/y-direction is 1 mm (0.12λ0). The diffraction fields of the OIOO within the frequency band from 33 GHz to 37 GHz at intervals of 1 GHz are recorded in a raster scan at a speed of 5 mm/s by the Ka-band waveguide probe.
Table 2. Weight coefficients at several discrete frequencies
Figures 4(b) and 4(c) show that the measured and calculated focal spots on several cross-sectional slices generated by the OIOO agree well. Compared to the calculated focal spots of the NINO (as shown in Fig. 4(d)), well-proportioned shapes, vanishing comet tails, and well-balanced intensities of the focal spots over a larger angular FOV manifest that the angular-dependent Seidel aberrations of the OIOO are visibly mitigated. From Fig. 4(e) and the maximum/average OOR of the focal spots in Table 1, it is shown that more consistent and symmetrical focal spot shapes are attained by the OIOO, implying that the coma aberration is well suppressed. It is further confirmed by the more well-balanced, concentrated intensity distributions, as indicated by MIG and EE in Fig. 4(f) and their post-processing in Table 1. Further, the Strehl ratio quantifying the above aberrations in Fig. 4(g) and Table 1 firmly manifests that the OIOO achieves nearly diffraction-limited focusing at all incident angles. Furthermore, the measured focusing efficiencies (>55%) presented in Fig. 4(g), indicate a relatively weak dependence on the incident angle, which benefits from a rational energy and phase allocation of the EM waves and excellent-performance of RHSs. The above calculated and measured results demonstrate that, compared to the NINO, the simply-designed OIOO attains high-efficiency, almost aberration-free, and approaching diffraction-limited focusing over an at least 40° angular FOV. Such a nearly uniform focusing performance is appealing to the PMMW-FPA imaging for obtaining the consistent image across the entire projection area of the metalens. It is noted that the little differences between the measurement and the numerical calculation are mainly caused by the alignment deviation of test platform. In addition, the FWHMx and the OOR of the focal spot generated by the NINO appear a peak at the incident angle of 30°. This phenomenon could result from a combined effect of the inconsistent transmission amplitude-phase response variations of thousands of RHSs, and the inconsistently spatially-varying incident angle and amplitude-phase profiles of the arriving EM wave from the LP-CCHA at different locations of the metalens, in the oblique incidence cases.
Due to the wideband reception property of the PMMW-FPA imaging, the wideband focusing intensities on the z = 112 mm plane at the incident angle of 0°, 15°, and 30°, which are obtained by computing a weighted sum of the intensities at several discrete frequencies in the working frequency range from 33 GHz to 37 GHz, are also provided and shown in Fig. 5. As listed in Table 2, weight coefficients at different frequencies are determined by the illumination efficiency (IE) of the metalens and the radiation efficiency (RE) of the LP-CCHA, which are extracted by the Finite-Difference Time-Domain method.
Fig. 5. The calculated and measured normalized wideband focal spots on the z = 112 mm plane generated by the NINO and the OIOO, and their MTFs in the x-direction and the y-direction as well as focusing efficiencies. (a) The calculated wideband focal spots of the NINO and their locations, FWHMx, FWHMy, and MIG. (b), (c) The calculated, measured wideband focal spots of the OIOO and their locations, FWHMx, FWHMy, and MIG. (d), (e) The MTF in the x-direction and the y-direction of the wideband focal spots generated by the OIOO. (f) The calculated and measured wideband focusing efficiencies of the OIOO varying with increasing incident angle from 0° to 40°. DL (diffraction-llimited) is the MTF of focal spot generated by the aberration-free lens under the illumination of the LP-CCHA.
Highly-consistent normalized wideband focal spots on the z = 112 mm plane between the measurement ones and the calculation ones can be attained by the OIOO, as shown in Figs. 5(b) and 5(c). Compared to the NINO (as shown in Figs. 5(a)), the OIOO at different incident angles generates more symmetrical wideband focal spots in a larger transverse space, implying that the angular-dependent aberrations are greatly improved. It is further confirmed by the wideband focusing parameters including the location, FWHMx, FWHMy, and MIG of the wideband focal spots, as inserted in Figs. 5(a), 5(b), and 5(c). It is noted that the area covered by the yellow circle in Figs. 5(a)–5(c) is the orthographic projection area of the OIOO. Figures 5(d) and 5(e) show that under the illumination of the LP-CCHA at 0°, 15°, and 30° incident angles, the measured modulation transfer functions (MTF) in the x-direction and y-direction indicate that the approaching diffraction-limited image resolution in the x-direction and y-direction can reach 4.5 mm and 4.8 mm. The contrast in the x-direction is weakly reduced due to the oblique incidence of the LP-CCHA on the y = 0 plane. Furthermore, from Fig. 5(f), the measured wideband focusing efficiencies at different incident angles show a relatively weak dependence on the incident angle, varying from 43.12% to 52.78% as the incident angle changes from 0° to 40°. Nearly uniform and superior wideband focusing performance of the OIOO presents a critical benefit to implementation and future evolution of the PMMW-FPA imaging. As illustrated in Figs. 5(b) and 5(c), when combining with the rotary scanning, this metalens can enable the PMMW-FPA imaging device to perform high-consistent, high-resolution, and high-contrast image over at least 80° angular FOV. It is noted that, affected by the chromatic aberrations and non-ideal wideband transmission characteristics of RHSs, the weighted wideband focusing performance of the metalens is slightly reduced compared to its single frequency focusing performance at 35 GHz, such as larger focal spots and lower focusing efficiencies. In addition, there are slight differences between the measurement and the numerical calculation, which is caused by the alignment of the test.
4. Conclusions
In this paper, an efficient single-metalens design is presented for performing high-efficiency, approaching diffraction-limited focusing over a wide-angle, large-angle FOV and a wide frequency band. This single-metalens can effectively suppress the angular-dependent aberrations including coma, astigmatism, and field curvature by a rational allocation of energy and phase of the EM waves, which is governed by a simple yet very effective incident-exit angle pairing method. The experiment results firmly validate this efficiently-designed wideband metalens. Within an over 40° angular FOV, the prototypical metalens can achieve approaching diffraction-limited wideband focusing performance with high-efficiency more than 43%. When cooperating with the rapid rotary scanning, this metalens can enable the PMMW-FPA imaging device to perform high-consistent, high-resolution, and high-contrast image over a more than 80° angular FOV. It is experimentally demonstrated that our design provides a straightforward solution to the single-metalens for performing high-efficiency, wide-angle, and approaching diffraction-limited focusing, in turn offers a compelling boost to the ultra-compact PMMW-FPA imaging equipment.
Funding
National Natural Science Foundation of China (no. 61301013, no. 61671178, no. 61731007).
Disclosures
The authors declare no conflicts of interest.
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