1. Introduction

Metasurface, a flat ultrathin optical element, consists of an array of artificially fabricated subwavelength antennas to manipulate electromagnetic wave. Dielectric metasurfaces are in the spotlight due to its high efficiency and compact integrated platform that can replace conventional optical elements. Also, multi-functional metasurfaces for simultaneous amplitude and phase control [1–3], polarization multiplexing [3–7], full space control [8–10], or multi-spectral control [3,4,11–29], allow high potential in diverse display and imaging applications. In the conventional optics, lenses are important optical devices since they are essential in various optical systems such as imaging and optical information processing. As in conventional optics, one of the most desired applications is metalens, which has advantages of large numerical aperture (NA), ultrathin form factor, and multi-functionality. Based on these advantages, various applications of metalens such as imaging [6,30,31], spectroscopy [14–16], full-color routing [13], microscopy [27], light-field imaging [29], and augmented reality devices [32] have been studied. However, like conventional refractive lenses, chromatic and monochromatic aberrations are the main problems of metalenses as depicted in Fig. 1(a). Therefore, recent studies are mainly focused on the correction of aberrations since improvement of focusing quality is significantly required for the most practical applications of metalens [17–29,33,34]. Moreover, compared to the aberration correction of conventional lens, the metasurface lens is more advantageous since it has a higher design degree of freedom given by phase and dispersion control of meta-atoms.

 

Fig. 1. Schematics of the doublet metalens for simultaneous correction of chromatic and monochromatic aberrations at the three primary colors. (a) A metalens with chromatic and monochromatic aberrations. (b) Correction of both chromatic and monochromatic aberrations with the doublet metalens system. In this study, the diameter of the front-side metalens is 300 µm, and the diameter of the back-side metalens is 720 µm. The thickness of fused silica substrate is fixed at 500 µm.

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Recently, several research groups demonstrated the achromatic metalenses working in the broadband visible or near infrared range [21–25,28]. Chen et al. and Wang et al. utilized geometric phase for basic focusing and controlled group velocity dispersion for compensation of focal lengths by adjusting the size of nano-pillars [23,24]. Shrestha et al. utilized diverse geometry to cover both basic phase and group velocity range [25]. However, the limitations in propagation phase dispersion due to limited refractive indices of high-index dielectrics and heights of meta-atoms make metalens have small diameter under 100 µm with numerical aperture under 0.2 or larger diameter with low numerical aperture under 0.1. To relieve such drawbacks, achromatic metalenses working at several discrete wavelengths have been reported that are targeted to several specific wavelengths since many display devices operate at three primary colors of RGB pixels. Spatial multiplexing or stacking of metalenses operating at different wavelengths were reported to realize such multi-wavelength achromatic metalens [18–20]. However, these metalenses show large noise and low efficiency due to interference with other structures. Because of this problem, Shi et al. suggest independent phase modulations at the three target wavelengths with single structure [26]. By combining guided-mode resonances, high height of nano-structure, and reflecting platform, large phase dispersion was obtained so that full phase modulations were achieved independently at the three target wavelengths. Then, they match the ideal phase profiles at the target wavelengths and the phase modulation of the nanostructures. In this case, however, low focusing quality at non-targeted wavelengths results in large noise.

Meanwhile, the correction of monochromatic aberrations has also been widely studied. Spherical aberration-free phase profile gives diffraction limited focusing only for normally incident light. But this kind of phase profile shows large monochromatic aberrations for obliquely incident light. Adopting the doublet lens idea from the conventional refractive optics, monochromatic aberrations were corrected in doublet metalenses at one target wavelength by combining basic focusing metalens and correction metalens into each side of a transparent substrate [33,34]. However, chromatic aberrations still remain significantly problematic for full color focusing and imaging in these previous doublet studies. Wiener deconvolution is a method to sharpen image from the chromatic aberration, but still has problems in noise amplification [33]. Thus, even if simultaneous correction of both chromatic and monochromatic aberrations is a significant issue, it has not been solved using ultrathin metalenses with high NA to our best knowledge.

In this study, we design and optimize the doublet metalens for simultaneous correction of longitudinal chromatic aberration and four monochromatic aberrations including spherical aberration, coma, astigmatism, and field curvature and high numerical aperture (NA) focusing for the three visible wavelengths. Our target design of the aberration corrected doublet metalens is depicted in Fig. 1(b). To solve optimization problem, we first choose phase modulation method and role of the two metalenses. In the front of the doublet system, multi-wavelength targeted metalens, which has independent phase profiles for red, green, and blue light, is located. This front-side metalens has functions correcting chromatic aberrations of the back-side metalens and monochromatic aberrations of each targeted wavelength by optimization of its phase profile. In the back side of the doublet system, we utilized the metalens with meta-atoms which only permit the transmission at the targeted primary colors and encode local phase profile by geometric phase. This back-side metalens filters the non-targeted wavelengths of front-side metalens, which would harm imaging quality as noises. Also, by phase profile optimization, monochromatic aberrations are also corrected by back-side metalens. Despite inevitable constraint in the optimization process, the color filtering of back-side metalens enables large suppression of noise from un-targeted wavelengths. The target wavelengths are the three primary colors in the visible, 445 nm for blue, 532 nm for green, and 660 nm for red. Also, target correction range of monochromatic aberrations of incident angle is fixed as 0 to 30 degrees, which affects the sizes of both metalenses.

2. The design of doublet metalens

To correct both chromatic and monochromatic aberrations, the doublet system is basically utilized due to its large design degree of freedom. In the doublet metalens system, each side of the substrate can be configured by combining a broadband targeted metalens or a multi-wavelength targeted metalens. If we utilize multi-wavelength targeted metalens at the both sides of a substrate, the noise at non-controlled wavelengths would be problematic. Large numerical aperture can be achieved in this case, but noise at other wavelengths need to be reduced by additional color filtering elements. Otherwise, if we utilize broadband controlled metalens in both sides, we can think some combinatoric cases. When only using broadband geometric phase metalens, which has all the same phase profiles for all wavelength, the numerical aperture of the metalens is only limited by the period of a meta-atom. If the both sides of the doublet metalens consist of this type of metalens, large numerical aperture can be achieved. However, achromatization of the metalens system is impossible in the broadband range, since both sides have the same dispersion power [35]. Meanwhile, this type of metalens has an advantage that the phase modulation is possible without changing the spectrum. If only one side of doublet metalens consists of this type, it is advantageous to filter undesired color range. In other case using metalens with combination of propagation phase and geometric phase which has different dispersive power, the numerical aperture is also limited by the range of the phase dispersion. Therefore, due to this limitation, the doublet system utilizing metalens with the combination of propagation phase and geometric phase suffers low numerical aperture.

Due to the abovementioned advantages and problems, we utilize multi-wavelength targeted metalens in one side and broadband geometric phase metalens in the other side of the substrate. Therefore, the former metalens corrects chromatic aberration of the latter metalens, and the latter filters transmission of undesired color range. In this configuration, with numerical phase profile optimization of both sides, we can correct both chromatic and monochromatic aberrations.

2.1 Front-side metalens design

For the front-side metalens, the isotropic square and the square with central hole structures consisting of polycrystalline silicon on glass substrate are utilized as the meta-atoms as shown in Fig. 2(a). Low loss materials such as TiO₂ and GaN show high transmission efficiency in the visible spectrum, but their refractive indices are not sufficient for this design. Such materials require high height or reflection platform for sufficient propagation phase coverage, which induces difficulty in large scale cost-effective fabrication. Moreover, they are inadequate for our transmissive type doublet design. Instead, we utilize higher index material of polycrystalline silicon for larger coverage of the propagation phase. By adjusting geometric parameters, w_f and a_f, phase modulations for the three target wavelengths can be numerically calculated via full-field electromagnetic simulations using commercial software, COMSOL 5.3a (Fig. 2(b)). Refractive index spectra of materials are quoted from experimental results [36,37]. The geometric parameter w_f is in range of 50 nm to 250 nm, and a_f is in range of 40 nm to 170 nm with satisfying w_f – a_f > 80 nm. Also, the phase modulation spectra for each wavelength and a_f = 0, 50 nm cases are shown in Fig. 2(c) and (d). For independent phase control for the target wavelengths, the large change of the phase modulation is required. The most rapid change in phase modulation in this structure occurs around the vertical asymmetric Fabry-Pérot resonance. The examples around the resonance are indicated by the dashed circle in Fig. 2(c), and the magnetic energy density profiles at the resonances implying Fabry-Pérot resonances are depicted in Fig. 2(e). As shown in the figure, second order Fabry-Pérot resonances are observed [24,33]. This structure set can cover the nearly full phase modulation map, which can give independent phase modulation for three target colors. For high transmission efficiency, we select the structures with the diffraction efficiencies above certain values 9%, 25%, and 49% for 445 nm, 532 nm, and 660 nm, respectively. In high-contrast dielectric material, electromagnetic energy is funneled into a local meta-atom, which can be regarded as isolated waveguide [38]. The meta-atoms support a single fundamental mode except some structures with large fill factor. Also, even in the structures with multiple modes, the electromagnetic waves are mostly coupled to the lowest-order fundamental mode. Since the waveguide mode supported by single mode meta-atoms is nearly the same for both normal and oblique incidence, the phase modulation performance is not affected as shown in the simulation results Fig. 9(a) in appendix 7.1. Also, in the multi-mode structures, the ratio of coupling to each mode varies depending on the angle of incidence, but the phase modulation does not change much as shown in the simulation results Fig. 9(a) [39]. However, diffraction efficiency decreases as the incident angle increases. Therefore, in the light propagation simulation, the complex transmission coefficients in normal incidence case would not be problematic to be used in obliquely incident case to check focusing quality and aberrations except for working efficiency calculation.

 

Fig. 2. Unit cell structures of the front-side metalens. (a) Schematics of unit cell structures. The period P and height h_f are fixed at 300 nm and 600 nm, respectively. (b) Phase modulation map of all found structures and the selected structures at the target wavelengths. (c) Phase modulation spectra for each color with respect to w_f in case of the isotropic square. (d) Phase modulation spectra for each color with respect to w_f in case that hole size is a_f = 50 nm. (e) Normalized magnetic intensity profiles for the selected structures and wavelengths. The selected ones are indicated by dashed circles in (c).

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Since we utilize the multi-wavelength targeted metalens for the front-side, its phase profiles can be independently constructed for the three target wavelengths. The phase profile formula with five even order correction terms can be constructed as follows.

2.2 Back-side metalens design

As mentioned above, the meta-atoms of the back-side metalens is designed to have color filtering functionality. In here, we also utilize polycrystalline silicon for filtering of the three target colors, which has large dispersion in the visible spectrum. An anisotropic rectangular nanofin is utilized for meta-atom as depicted in Fig. 3(a). This kind of nanostructure can provide wavelength-independent phase modulation for circularly polarized light by its rotation angle as can be seen by Jones calculus:

 

Fig. 3. Unit cell structure of the back-side metalens. (a) A schematic of unit cell structure. The period P, height h_b, width w_b, and d_b are fixed at 300 nm, 330 nm, 80 nm, and 160 nm, respectively. (b) Cross-polarization conversion ratio with respect to wavelength. (c) The real part of the effective mode index of the lowest order mode for x-polarization and y-polarization with respect to wavelength. (d) Calculated phase modulation difference between x-polarization and y-polarization with respect to wavelength. The black dashed line indicates the π phase difference.

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In Eqs. (2) and (3), θ is the rotation angle of anisotropic nanostructure and σ is +1 or -1 for left-hand circular polarization (LCP) and right-hand circular polarization (RCP), respectively. Therefore, the phase modulation of cross-polarized light is double of rotating angle of structure with the unchanged and wavelength-independent property, which is called geometric phase. Here, we can find a structure which has high LCP-to-RCP conversion ratio near the targeted wavelengths as shown in Fig. 3(b). The cross-polarization conversion ratio is defined as ${|{{t_{xx}} - {t_{yy}}} |^2}/4$ in here. The phase of complex transmission coefficient can be approximately described by effective mode index and height of structure by Eq. (4). If we suppose amplitude is slowly varying with respect to wavelength, the peak and dip are critically affected by the phase difference between x- and y-polarizations. With waveguided mode analysis in this nanofin, we can find the lowest order mode for x- and y- polarizations. Its real part of effective mode indices is plotted in Fig. 3(c). Since the light propagating through the nanofin waveguide mostly couples to the lowest order mode, we can approximately calculate propagation phase for each polarization and difference of them as plotted in Fig. 3(d). Around the peaks, the difference in phase modulation is nearly π except blue color. At the blue wavelength, multiple modes are excited by light illumination, hence the resonance peak is not explained by the propagation phase difference in Fig. 3(d).

Since we only utilize geometric phase, the phase profile of the back-side metalens is all the same for the broadband wavelength range. The phase profile formula with basic focusing terms and five even order correction terms can be constructed as follows.

2.3 Phase profile optimization

The optimization process of the two phase profiles is based on an iterative method. At the first step of the iterative process, sampled rays with different positions, wavelengths, and incident angles propagate through the metalens doublet system, and get arrival points in the image plane. For ray sampling, azimuthal angle of incidence is neglected due to symmetry where only y- and z-components of k-vectors exist. When rays meet the metasurface, k-vectors of sampled rays are updated with the phase gradient of the metalens [38]. With updated k-vectors, rays propagate to targeted focal plane and give arrival points. The metalens doublet system can be represented as a coefficient vector c where its components are the collection of correction coefficients of both metalenses. The ray sampling and propagation process can be represented as a function F. Its input variable is c and output is the vector x. Components of x are the arrival points at the image plane. Next, the loss function G is set using the collection of the arrival points x so that the larger aberration gives the larger value of the loss function. The gradient of G with respect to c can be numerically calculated, and coefficient vector c is updated to reduce the loss function by using the gradient, iteratively. Adaptive moment estimation method is utilized and all of the optimization processes are performed with MATLAB [40]. In this study, the loss functions are set in two ways. First one is determined considering all aberrations while second one is set in regard to all aberrations excepting distortion and transverse chromatic aberrations.

3. Optimization for all aberrations

The lens aberration can be divided into chromatic and monochromatic aberration according to the dependence of wavelength. Specifically, the chromatic aberration can be divided into longitudinal chromatic aberration where the focal distance from the lens is inconsistent with wavelength, and transverse chromatic aberration where the position of the focal point in the focal plane is inconsistent with wavelength.

On the other hand, the monochromatic aberrations can be classified into spherical aberration, coma, astigmatism, field curvature, and distortion. In the viewpoint of ray optics, spherical aberration, coma, and astigmatism occur when the incident rays at each position with the same incident angle do not converge to single point. Field curvature is an aberration in which the focal plane forms a curved surface rather than a plane according to the incident angle. Distortion is aberration that the magnification varies depending on the position in the focal plane, which causes deformation of an image. When distortion occurs, the position of the focal spot in the focal plane does not match the ideal position to keep the magnification constant. Therefore, if there is no aberration, for the all target wavelengths, the rays have to be gathered at a specific point decided by the angle of incidence. This specific point in the focal plane is at $f\tan \theta $ away from the center of focal plane when the incident angle is θ.

The spherical aberration, coma, and astigmatism can be corrected only with gathering all rays at one point for each incident angle. In addition to this condition, field curvature and longitudinal chromatic aberration can be corrected by ensuring the points that gather rays according to each angle of incidence and each target wavelength are located in single plane. If we add the condition that ensures the points that gather rays are the same points according to each wavelength from the very first condition, transverse chromatic aberration can be eliminated. Using the strongest condition that the ray gathering points are the same specific point in single plane, distortion can be corrected.

Here, we utilize the strongest condition to correct all the aberrations mentioned above. If there are no aberrations in imaging system, the rays propagated through the system with the incident angle of θ arrive at the single ideal point which is $f\tan \theta $ away from center of target focal plane. If we consider all two chromatic aberrations and all five monochromatic aberrations, the loss function can be set as below considering the above conditions:

The optimized phase profiles of the front-side and the back-side metalenses are shown in Figs. 4(a) and 4(b), respectively. The details of optimized correction coefficients are suggested in the appendix 7.2. For the achromatization, in the front-side metalens, the phase profile of the red color is formed as concave lens, otherwise the phase profile of the blue color is formed as convex lens. For the green color, it shows small deviation around the origin point but shows convex lens form at the edge. Light propagating simulation is performed by angular spectrum method with MATLAB [41].

 

Fig. 4. Optimization results and longitudinal focusing quality analysis in case of considering all aberrations. (a) Initial and optimized phase profiles of the front-side metalens. (b) Initial and optimized phase profiles of the back-side metalens. (c) Longitudinal intensity profiles for the target wavelengths and selected incident angles. Each intensity profile is normalized to its maximum intensity. White bar indicates 6 µm. The center of z-axis is 360 µm and the center of y-axis is matched to focal spot location. (d) Calculated back focal length for each wavelength with respect to incident angles. (e) Calculated focal spot location with respect to the incident angles. The black dashed line indicates ideal focal spot location.

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Longitudinal intensity profiles are shown in Fig. 4(c). Also, back focal length and focal spot location are plotted in Figs. 4(d) and 4(e), respectively. Here, back focal length is defined as the distance between the back-side metalens and the maximum intensity point. Also, focal spot location is defined as the distance between the optical axis of the metalens system and the maximum intensity point of the focal spot in the focal plane. For the normal incidence case, back focal lengths are nearly matched with the deviation of nearly 5 µm, which means the weak correction of longitudinal chromatic aberration. However, the focal spot location for three colors are not matched except normal incidence case, which means the transverse chromatic aberration still exist. In case of blue color, the back focal length is nearly matched but focal spot locations are farther from the origin point than ideal location. In case of green color, the back focal lengths are nearly the same from 0 to 20 degrees of incidence but a large deviation occurs in the large degree of incidence. Also, in case of red color, the back focal lengths increased as the incident angle increased. For blue color, field curvature is corrected, but not for green and red colors. The focal spot location is nearly matched to ideal location for green color, which means distortion is well corrected. As we can see in the intensity profiles, three monochromatic aberrations including spherical aberration, coma, and astigmatism still exist. We can still observe spherical aberrations for normal incidence case in blue and green color. Also, the observed multiply generated focal points come from astigmatism and spherical aberration for oblique incidence case, and still large coma can be detected for all colors.

As discussed above, when we consider all aberrations for optimization, there are still some problems. First, the optimized doublet system still has weak longitudinal chromatic aberrations. Second, it is also not free from some monochromatic aberrations. Spherical aberration, coma, and astigmatism still happen at the all colors, while field curvature is still observed for the green and red colors. Third, distortions are well corrected for the green color, but have transverse chromatic aberrations. Back focal lengths are matched in this system, but this is caused by effective focal length mismatch due to the chromatic aberrations of purely geometric phase metalens. The design degree of freedom is not sufficiently high to correct all aberrations, which is lowered due to the constraint of the back-side metalens for color filtering.

4. Optimization excepting distortion and transverse chromatic aberrations

The current design degree of freedom due to the constraint of the back-side metalens does not allow sufficient optimization under the strongest condition for correcting all aberrations. If the focal distances are all the same according to the incident angle, and rays are gathered at one point, which means spherical aberration, coma, astigmatism, and field curvature are eliminated, distortion can be corrected by digital image processing without changing the doublet system [42]. Also, if the focal distances are all the same according to the target wavelengths, transverse chromatic aberration can be corrected by independent post processing of image with independent color channels [43]. Therefore, in this section, we only consider the four monochromatic aberrations which except distortion and the only longitudinal chromatic aberration excepting transverse chromatic aberration. Then, the loss function should reflect first two conditions and can be set as below:

Longitudinal intensity profiles are shown in Fig. 5(c) with singlet system at 532 nm light for comparison. Also, back focal lengths and focal spot locations are plotted in Figs. 5(d) and 5(e), respectively. The back focal lengths are nearly matched with the deviation of nearly 1 µm for the all targeted colors where the longitudinal chromatic aberration is corrected more strongly compared to previous optimization case. Also, it is verified that the field curvature is corrected for the all three colors. As shown in the intensity profiles, spherical aberration, coma, and astigmatism are obviously corrected compared to the previous case and the singlet system. However, the focal spot locations are not matched to ideal location except nearly normal incidence case. Distortion and transverse chromatic aberration still exist in this case, but these can be corrected by pre- or post-processing of image without changing system as previously mentioned.

 

Fig. 5. Optimization results and longitudinal focusing quality analysis without considering distortion and transverse chromatic aberrations. (a) Initial and optimized phase profiles of the front-side metalens. (b) Initial and optimized phase profiles of the back-side metalens. (c) Longitudinal intensity profiles for the target wavelengths and selected incident angles. Each intensity profile is normalized to its maximum intensity. White bar indicates 6 µm. The center of z-axis is 360 µm and the center of y-axis is matched to focal spot location. The profiles for 532 nm with singlet metalens are also plotted for comparison. (d) Calculated back focal length for each wavelength with respect to incident angles. (e) Calculated focal spot location with respect to the incident angles. The black dashed line indicates ideal focal spot location.

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To investigate in-plane focusing performance, focal spot profiles in the target focal plane are suggested in Fig. 6(a) with comparison to the case of singlet system at 532 nm. Also, the cut intensity profiles are plotted for normally incident case at 532 nm in Fig. 6(b). Compared to the singlet system, the doublet system can correct mostly the coma aberration for the target wavelength up to 30 degrees of incidence. The coma aberration for blue and red light are larger than green light in case of 20 and 30 degrees of incidence but well corrected when compared to singlet system. For quantitative analysis, modulation transfer function (MTF) is calculated as shown in Fig. 7. The singlet system at 532 nm light shows nearly diffraction limit performance in normal incidence case but shows dramatical degradations of performance in oblique incidence case. With the doublet system, MTFs are improved for all targeted wavelengths and incidence angles up to 30 degrees. However, for normally incident case, the degradation of MTF can be observed for green light. These kinds of degradation are due to the three reasons. The first is slight defocus within 0.5 µm. The second reason is spherical aberration which can be observed at small peak at the right side of the focus in Fig. 5(c) at 532 nm with normal incidence case. Last one is the noise occurred by the phase modulation mismatch between the meta-atom and the optimized phase profile of the front-side metalens. Therefore, as observed in Fig. 6(b) the intensity profile of the doublet cases, the noise at the sidelobe is slightly larger than the singlet cases, which leads to MTF degradation for 532 nm with normal incidence.

 

Fig. 6. (a) Focal spot profiles without considering distortion and transverse chromatic aberrations. Each intensity profile is normalized to its maximum intensity. White bar indicates 4 µm. The center of x-axis in profiles is x = 0 µm and the center of y-axis is matched to focal spot location. (b) The intensity profiles of normally incidence case at 532 nm for the singlet and doublet metalens in log scale. The white dashed lines in (a) indicate where the intensity was cut.

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Fig. 7. Modulation transfer function (MTF) analysis without considering distortion and transverse chromatic aberrations. (a) MTF of 445 nm light with doublet metalens. (b) MTF of 532 nm light with doublet metalens. (c) MTF of 660 nm light with doublet metalens. (d) MTF of 532 nm with singlet metalens. The solid line indicates MTF of x-direction and dashed line indicates y-direction in Fig. 6.

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The diffraction limit of the doublet system is nearly same for all target wavelengths. Due to purely geometric phase metalens, effective focal lengths are inversely proportional to wavelength. Since the cut off spatial frequency of diffraction limit system is proportional to wavelength but inversely proportional to effective focal length, the diffraction limit is nearly the same for three targeted wavelengths. The effective focal lengths are calculated with $y = {f_{eff}}\tan \theta $ in small incident angle regime, where y is the focal spot location, f_eff is effective focal length, and θ is incidence angle. The calculated effective focal lengths are 425.2 µm, 370.3 µm, and 284.6 µm for 445 nm, 532 nm, and 660 nm wavelengths, respectively. The numerical aperture value is determined by effective focal length, so we can calculate NAs of 0.33, 0.38, and 0.47 for 445 nm, 532 nm, and 660 nm wavelengths, respectively. Since the effective focal lengths are not matched despite matching in the back focal lengths, the image plane mismatch for target wavelengths occurs when the distance between object plane and the doublet metalens is finite. However, this mismatch is not problematic when the object is far away from the doublet metalens as described in appendix 7.3. The focusing efficiencies for normally incident light are 9.27%, 24.8%, and 52.6% for 445 nm, 532 nm, and 660 nm, respectively. The focusing efficiencies are decreased as according to the increase of the incident angle. Without considering the diffraction efficiency degradation for oblique incidence, the focusing efficiencies are 7.47%, 20%, and 41.26% for 445 nm, 532 nm, and 660 nm in incident angle of 30 degree, respectively. Here, the focusing efficiency is defined as the ratio of the power in the circular region with three times the full width at half maximum (FWHM) of the focal spot to the power of incident light. This reduction of focusing efficiency is caused by only the effect of the optimized phase profile. If we consider the diffraction efficiency degradation as shown in Fig. 9(b), we expect the focusing efficiencies would be much lowered.

5. Relative intensities and focal lengths in broadband visible range

The designed doublet metalens filters the target wavelength, which can degrade the noise came from non-targeted wavelength. To investigate the effects of local color filtering in nanopillars on noise reduction in image plane, broadband focusing intensities is calculated. The intensity profiles of 440 nm and 450 nm wavelengths are weaker than that of 445 nm wavelength as shown in Fig. 8(a). Second, the intensity profiles of 520 nm and 550 nm are weaker than that of 532 nm as shown in Fig. 8(b). Lastly, the intensity profiles of 640 nm and 680 nm are weaker than that of 660 nm as shown in Fig. 8(c). The back focal lengths are shifted to positive and negative directions for smaller and larger wavelength, respectively. The calculated back focal lengths for broad wavelength range are shown in Fig. 8(d). The relative maximum intensity is also plotted in Fig. 8(e). The peak intensities are formed around the target wavelength. Non-targeted wavelength light is not focused in the target focal plane and has low maximum focusing intensity.

 

Fig. 8. Broadband operation analysis. (a) Longitudinal intensity profiles at 440 nm, 445 nm, and 450 nm wavelengths. Intensities are normalized to the maximum intensity at 445 nm. (b) Longitudinal intensity profiles for 520 nm, 532 nm, and 550 nm wavelengths. Intensities are normalized to the maximum intensity at 532 nm. (c) Longitudinal intensity profiles for 640 nm, 660 nm, and 680 nm wavelengths. Intensities are normalized to the maximum intensity at 660 nm. White bar indicates 8 µm. The center of z-axis is 360 µm and the center of y-axis is matched to focal spot location. (d) Calculated back focal length in normal incidence case. Black dashed line indicates target focal lengths. (e) Relative maximum intensity with respect to wavelength. Intensity plot is normalized to maximum intensity at 670 nm wavelength.

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6. Conclusion

In this study, we propose the doublet metalens design for simultaneous correction of longitudinal chromatic aberration and four monochromatic aberrations for three wavelengths by introducing novel metalens combination and numerical optimization. Moreover, our designed doublet metalens has high numerical apertures of 0.33, 0.38, and 0.47 for 445 nm, 532 nm, and 660 nm wavelengths, respectively. By correcting various aberrations excepting distortion and transverse chromatic aberrations, which can be corrected by pre- or post-image processing without changing doublet metalens system, longitudinal chromatic aberration and four monochromatic aberrations including spherical aberration, coma, astigmatism, and field curvature are well corrected simultaneously with precedented quality.

We also expect further progress of the proposed design methods. Increasing the design degree of freedom by cascading multiple multi-wavelength metalens or high index materials with lower loss could further improve focusing quality and efficiency, respectively. We expect that the proposed work would be fruitful for high quality engineering of practical metasurface optic system which should meet multiple performance indicators.

7. Appendix

7.1. Optical responses for obliquely incident light

For each selected structure in the section 2.1, the phase differences for obliquely incident light are calculated as the equations shown below. The averages of the phase differences and standard deviations are plotted in Fig. 9(a). Also, the average diffraction efficiencies of the utilized structures for each wavelength are plotted in Fig. 9(b).

(8)$$\Delta \phi = \phi (\theta )- \phi (0 ).$$

 

Fig. 9. (a) Phase modulation difference of oblique incidence for selected structures. (b) Diffraction efficiencies with respect to the incident angle for each wavelength.

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7.2. Optimized correction coefficients

Tables 1 and 2 show the optimized correction coefficients for each optimization case in sections 3 and 4, respectively.

Table 1. Optimized correction coefficients for section 3

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Table 2. Optimized correction coefficients for section 4

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7.3. Image plane mismatch in case of finite distance between object plane and doublet metalens

If the object plane is at infinity, the image planes are matched and the light with targeted wavelengths shows achromatic focusing. However, if the object plane has finite distance with doublet metalens, the image plane is different for different colors due to mismatch of effective focal lengths. Fig. 10(a) shows this kind of image plane mismatch. By calculating the formula shown below, the image plane mismatch distance can be analytically calculated in the viewpoint of ray optics. Eqs. (9) and (10) are the image equations for two different colors. Then, by Eqs. (11) and (12), the distance between two image planes can be derived as Eq. (13). The image plane mismatch is no problematic if the difference is less than the depth of focus determined by ${\lambda \mathord{\left/ {\vphantom {\lambda {\textrm{N}{\textrm{A}^2}}}} \right.} {\textrm{N}{\textrm{A}^2}}}$. As shown in Fig. 10(b), object plane farther than 10 cm makes the mismatch distance less than the depth of focus.

(9)$$\frac{1}{{{z_0}}} + \frac{1}{{{z_1}}} = \frac{1}{{{f_{eff,1}}}} = \frac{1}{{{f_{eff,2}} + d}},$$

(10)$$\,\frac{1}{{{z_0} + d}} + \frac{1}{{{z_2}}} = \frac{1}{{{f_{eff,2}}}},$$

(11)$${z_1} = \frac{{{z_0}({{f_{eff,2}} + d} )}}{{{z_0} - {f_{eff,2}} - d}},$$

(12)$${z_2} = \frac{{{f_{eff,2}}({{z_0} + d} )}}{{{z_0} - {f_{eff,2}} + d}},$$

(13)$$\delta = ({{z_1} - d} )- {z_2} = \frac{d}{{{{({{z_0} - {f_{eff,2}}} )}^2} - {d^2}}}[{{d^2} + d({{z_0} + {f_{eff,2}}} )+ 2{z_0}{f_{eff,2}}} ].$$

 

Fig. 10. Image plane mismatch analysis for finite distance between the object plane and the doublet metalens. (a) Schematics of image plane mismatch. (b) The distance between two image planes with respect to the distance between the object plane and the doublet metalens. The differences are calculated with f_eff,2 = 370 µm, and d = 50, 100 µm. The black dashed line indicates δ = 500 nm.

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Funding

National Research Foundation of Korea (2020R1A2B5B02002730).

Disclosures

The authors declare no conflicts of interest.

References

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