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Abstract
Multiple-wavelength, frequency-adjustable planar metalenses are of great concern in optical imaging and microscopy systems. Here we propose a novel approach to design a high efficiency dual-wavelength achromatic metalens (DAM). This is different from the mostly reported multiple-wavelength achromatic devices which realize their functions at the expense of sacrificing the efficiency by partitioning the structure plane to several sections. In our designed DAM, the doublet approach is adopted and phase shifts on both sides are superimposed to modulate the focal length at a certain wavelength. With elaborate design, focal lengths at two discrete wavelengths can be regulated to own the same value. With such a strategy, DAM for different focal lengths (f = 20μm and 30μm) with focusing efficiency as high as 70% are demonstrated. Our work provides a general design scheme for multiple-wavelength, high efficient wavefront engineering and holds potential applications in designing optical achromatic devices.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical lenses, indispensable elements in optical systems, hold potential applications in various fields related to optical microscopy, spectroscopy and imaging [1]. In traditional lens, reshaping the wavefront depends on phase accumulation along the optical paths, which is achieved by either varying the spatial distribution of refractive index or the surface topography [2]. In such systems, the light bending ability is therefore restricted by the refractive index of the constructed dielectric material. Therefore, the physical dimension of traditional lens inevitably suffers from specific thickness limitation. To solve this problem, Fresnel lens and plasmonic lens are designed, with which the volume of material can be reduced [3]. However, the thickness of the lenses is still on wavelength scale, hindering their applications in miniaturization and integration.
Metasurfaces, two-dimensional equivalents of metamaterials, have emerged as revolutionary metarials to take the place of bulky conventional components. Composed of subwavelength elements, metasurfaces are capable of providing unprecedent superiority for wavefront engineering by the introduction of abrupt phase shifts [4–6]. In virtue of their superiority in wavefront engineering, metasurfaces have opened up a new door for constructing a wide range of ultra-thin optical devices such as wave-plates [7–11], vortex beam generators [12–14], high resolution holograms [15–17], and metalenses [18–31]. However, for a material with normal dispersion, the wavelength dependence of the focal length leads to chromatic aberrations which degrades the quality of imaging systems [19]. Recently, a variety of metasurfaces structures have been theoretically and experimentally demonstrated to eliminate the chromatic aberrations at a broad bandwidth or discrete wavelengths [19,32–38]. Achromatic metalenses in the visible and near infrared wavelengths with certain bandwidths have been experimentally demonstrated [32–34]. Francesco Aieta et.al demonstrated a new metasurface design capable of realizing achromatic wavefront control at three wavelengths, however, with low focusing efficiencies (less than 30%) [19,35]. Another type of metasurface designs physically partition the structure into several sections, each of which works at an individual wavelength [37,38]. In this way, achromatic wavefront control can be realized, however, at the expense of sacrificing the efficiency. Recently, the metasurface doublet lenses are proposed as a new approach for correcting monochromatic aberrations [39,40]. Such an approach also promises to achieve achromatic focusing at discrete wavelengths and address the efficiency challenge simultaneously, however, has not been demonstrated.
Here, by creatively adopting the metalens doublet approach, a dual-wavelength achromatic metalens with high efficiency is demonstrated. Firstly, an operation metalens with the ability of achieving high efficiency dual-band focusing is designed. Then, the correction metalens that aims at eliminating the chromatic aberrations is designed as well. Integrating the two metalenses and utilizing the phase difference superposition principle, an achromatic metalens is achieved. With such a strategy, achromatic focusing with different focal lengths (f = 20μm and f = 30μm) at two discrete wavelengths (λ = 1.0μm and λ = 1.55μm) are demonstrated in detail. Our work will provide a general design scheme for multiband, high efficient wavefront engineering such as beam deflecting, focusing, vortex beam generation and so on.
2. Designs and structure
Figure 1 shows the schematic of the designed dual-wavelength achromatic metalens (DAM). The side view of the unit cell is shown in Fig. 1(a). The unit cell consists of an amorphous silicon nanoblock placed on a glass substrate (n = 1.46). The refractive index of the amorphous silicon is extracted from our own experimental measured data as shown in Fig. 1(d). At the two working wavelengths, the refractive index are n = 3.68 at wavelength λ = 1.0μm and n = 3.56 at wavelength λ = 1.55μm, respectively. Figure 1(b) shows the top view of the unit cell, which shows the length a, width b, rotation angle θ of the nanoblock and the lattice constant P. Figure 1(c) shows the focusing principle schematic of the designed DAM, which consists of two cascaded metalenses. The two metalenses along the incident direction are defined as operation metalens and correction metalens, respectively. The designed DAM capable of focusing the incident lights at discrete wavelengths λ = 1.0μm and λ = 1.55μm into a spot (with the same focal length), achieving achromatic focusing.
Fig. 1 (a) Side view of the unit cell that composed of an amorphous silicon nanoblock (with height h) on a glass substrate. (b) Top view of the unit cell, in which the detail parameters of the unit cell are width a, length b and lattice constant P. The nanoblock with an angle θ to impart the required phase. (c) Schematic of the DAM illustrating the achromatic focusing principle, where incident lights with wavelengths λ = 1.0μm and λ = 1.55μm are focused into a spot with the same focal length. The two cascaded metalenses along the z-axis are defined as operation metalens and correction metalens, respectively. The thickness of the glass spacer between them is set as d. (d) The experimental measured refractive index of amorphous silicon as a function of wavelength.
In order to focus the incident circular plane wave, the required parabolic shaped phase profile is imparted by introducing the Pancharatnam-Berry (P-B) phase via rotating the nanoblocks [23,41]. As we know, the P-B phase is exhibited in the circularly polarized light with converted helicity, indicating that each nanoblock acts as a half-wave plate [23,25]. To achieve such a goal, the three-dimensional FDTD method is utilized to optimize parameters of the nanoblock. For our designed DAM composed of two cascaded metalenses, the nanoblocks in each metalens are optimized to function as half-wave plates at certain wavelengths. The detail simulation parameters in our simulations are as follow. In our simulations, the spatial mesh grids are set as Δx = Δy = 10nm and Δz = 20nm. RCP plane wave is adopted to normally illuminate the metasurface. To simulate the unit cells, periodic boundary conditions are applied along the x and y axes and perfectly matched layers (PML) is applied along the z axis. A monitor is set at the transmission area (x-y plane at z = 2μm) to obtain the detail information of electromagnetic field, in which the phase shift can be extracted as angle(Ex). To simulate the metalenses, PML boundary conditions are applied along all the three axes. Similarly, a monitors is set at x-z plane (y = 0) to obtain the detail information of electromagnetic field including the intensity (|E|2) profile can be obtained.
3. Results and discussion
3.1 The optimizing of the two unit cells
To achieve high efficient metalens, the unit cell should own both high polarization conversion efficiency and transmission. The former is to introduce the P-B phase (φ = 2θ) and the latter is to guarantee high focusing efficiency. The unit cells of the operation and correction metalenses are defined as U1 and U2, respectively. The transmission and polarization conversion efficiency of the unit cell U1 as a function of the incident wavelength are shown in Fig. 2(a). It can be observed that the unit cell U1 is optimized to function as half-wave plates (with polarization conversion efficiencies both over 97%.) at both the two discrete wavelengths λ = 1.0μm and λ = 1.55μm. Moreover, the transmissions at both wavelengths are above 96%, indicating its high efficiency performance for light focusing. The optimized parameters of the unit cell U1 are a1 = 500nm, b1 = 190nm, lattice constant P1 = 600nm and height h1 = 700nm. Similarly, the transmission and polarization conversion efficiency of the unit cell U2 for the correction matalens are shown in Fig. 2(c). It can be observed that the polarization conversion efficiency at wavelength λ = 0.72μm is 95.3%, and the transmission is 95.8%. The optimized parameters are a2 = 165nm, b2 = 60nm, lattice constant P2 = 200nm and height h2 = 600nm. Note that, the unit cell U2 is elaborately designed not only to eliminate achromatic aberration but also maintain high efficiency at the same time. As shown in Fig. 2(c), the transmissions for the optimized unit cell U2 at the two working wavelengths (λ = 1.0μm and λ = 1.55μm) are above 95%, owing to the fairly weak light-matter interaction. The simulated phase shifts for the two unit cells U1 and U2 with various rotation angle θ are plotted in Figs. 2(b) and 2(d), respectively. It can be observed that the phase shifts and the rotation angle at the three wavelengths (λ = 0.72μm,1.0μm,1.55μm) all satisfy P-B phase condition (φ = 2θ), with which the phase shift coverage of 0 to 2π can be obtained.
Fig. 2 (a) Simulated transmission and polarization conversion efficiency of the unit cell U1 . (b) Simulated phase shift for the the unit cell U1 with various rotation angles at wavelengths λ = 1.0μm and 1.55μm. The optimized parameters of U1 are a1 = 500nm, b1 = 190nm, lattice constant P1 = 600nm and height h1 = 700nm. (c) Simulated transmission and polarization conversion efficiency of the the unit cell U2. (d) Simulated phase shift for the unit cell U2 with various rotation angles at wavelength λ = 0.72μm. The optimized parameters of the U2 are a2 = 155nm, b2 = 50nm, lattice constant P2 = 200nm and height h2 = 600nm. The rotation angle θ ranges from 0° to 180° with a step angle of 10°. For these simulations, periodic boundary conditions are applied along the x and y axis and perfectly matched layers (PML) is applied along the z axis.
3.2 Performance of the operation metalens
To achieve a high efficiency DAM, the first key issue is to make sure that the operation metalens owns the characteristics of dual-wavelength focusing and high efficiency. Such an operation metalens can be achieved by constructing of unit cell U1. The corresponding phase profile φ(x) of the operation metalens along x-axis can be derived by the equal optical path principle as
where λ is the incidence wavelength, x is the position of the nanoblock along x-axis, and f is the designed focal length. The required phase is imparted based on the P-B phase via rotating the nanoblock with an angle θ(x) = φ(x)/2. Here, a metalens with focal length fo = 20μm and operating wavelength λo = 1.55μm is designed. Figure 3(a) shows the simulated intensity (|E|2) profile of the designed metalens at x-z plane with incident wavelength λ = 1.55μm. The incident light is right-handed circularly polarized (RCP) that normally illuminates along the z-axis. It can be observed from Fig. 3(a) that a focusing spot is exhibited at the expected position (0,0,20μm). As shown in Fig. 3(b), switching the incident wavelength to λ1 = 1.0μm, the focal length shifts to f1 = 32μm. This is fully in expect due to the change of wavelength but similar phase profiles for these two cases (as depicted in Fig. 3(c)). The phase shifts profiles of the operate metalens along x-axis for the two wavelengths can be expressed asFig. 3 (a) Simulated intensity (|E|2) profile of the operation metalens in x-z plane at y = 0 with incident wavelength λ = 1.55μm. (b) Simulated intensity profile of the operation metalens in x-z plane at y = 0 with incident wavelength λ = 1.0μm. (c) The phase shift along x axis for the operation metalens with λo = 1.55μm, fo = 20μm and λ1 = 1.0μm, f1 = 32μm. (d) Corresponding vertical cut of the two focal spots along x-axis.
Therefore, with nearly the same phase profile, the designed operate metalens is able to achieve focusing effect at the two wavelengths(λo = 1.55μm and λ1 = 1.0μm). Vertical cuts of the two focal spots are shown in Fig. 3(d), in which the full width at half maximum (FWHM) of the two focal spots are 1.41μm. Moreover, the simulation result also indicate that the focusing efficiencies for these two cases are 72%. The focusing efficiency is defined as the fraction of the incident light that passes through a radius which is three times of the FWHM spot size to the incident light [20]. Here, to calculate the focusing efficiency, a monitor with a radius equals to three times of the FWHM spot size is set at the focal spot.
3.3 Performance of the correction metalens
The second key issue is the designing of the correction metalens, which should not only maintain high transmission at the working wavelengths (1.0μm and 1.55μm) but also eliminate the chromatic aberrations. To solve this issue, the unique characteristic of unit cell U2 is utilized. A correction metalens constructed of unit cell U2 with focal length fc = 20μm and operating wavelength λc = 0.72μm is demonstrated. Figures 4(a) and 4(b) show the simulated intensity profiles of the correction metalens at x-z plane with incident wavelengths λc = 0.72μm and λ1 = 1.0μm, respectively. It can be observed that the focal length shifts to f2 = 14μm while switching the incident wavelength to λ1 = 1.0μm. The shift on the focal length can be attributed to the almost uniform phase shift profiles for these two cases (as depicted in Fig. 4(c)). The phase shifts profiles of the correction metalens along x-axis for the two wavelengths can be expressed as
Fig. 4 (a) and (b) Simulated intensity profile of the correction metalens in x-z plane at y = 0 with incident wavelength λ = 0.72μm and λ = 1.0μm. (c) The required phase shift along x axis for the metalens with λc = 0.72μm, fc = 20μm and λ1 = 1.0μm, f2 = 14μm. (d) Simulated phase profile (angle(Ex)) of the correction metalens in x-z plane at y = 0 with incident wavelength λ = 1.55μm.
From Eq. (2) and Eq. (3), we can conclude that focusing effect can be achieved at wavelength λ1 = 1.0μm for both the operation and correction metalenses. Figure 4(d) shows the simulated phase profile (angle(Ex)) of the correction metalens in x-z plane at y = 0 with incident wavelength λ = 1.55μm. It can be observed that the correction metalens exhibits no focusing effect and the phase profile is almost aclinic. Therefore, the correction matelens exhibits focusing effect at wavelength λ = 1.0μm, however exhibits no focusing effect at wavelength λ = 1.55μm, which provides a new way to eliminate the chromatic aberrations (discussed in the next part).
3.4 Performance of the achromatic metalenses
For the designed DAM composed of two cascaded metalenses, at working wavelength λ1, the superposed phase shift profile along x-axis can be derived from Eq. (2) and Eq. (3) as
where φo(x)and φc(x) are the phase shifts introduced by the operation and correction metalenses. f1 and f2 represent focal lengths of the operation and correction metalenses at operate wavelength λ1. From Eq. (4), we plot the phase shift profiles of the operation metalens, correction metalens and the sum of the two phase shifts in Fig. 5(a). It can be observed that the superposed phase shift also exhibits typical parabolic shape, satisfying the phase compensation mechanism to function as a metalens. Hence, the two cascaded metalenses can be treated as a new integrated metalens with certain focal length and operating wavelength. The phase profile along x-axis for the integrated metalens φa(x) can be expressed aswhere fa represents the focal length of the integrated metalens. From Eq. (5), the curve fitting process is implemented and the optimized phase shift profile of the integrated metalens is depicted in Fig. 5(a). It can be observed that the phase shift profile φa(x) shows well agreement with φ(x) when the focal length with certain value (fa = 17μm). Therefore, it is verified that the two cascaded metalenses can be treated as an integrated metalens. Moreover, based on the phase superposition principle, the focal length of the integrated metalens can be modulated by adjusting the focal lengths of the two metalenses.Fig. 5 (a) The phase shift relationship among the operation, correction and integrated metalenses. The detail wavelengths and focal lengths of the two metalenses are set as λo = 1.55μm, fo = 20μm, λc = 0.72μm and fc = 60μm. (b) and (c) Simulated intensity profiles of the DAM in x-z plane at y = 0 with incident wavelength λ = 1.0μm and λ = 1.55μm, respectively. (d) Corresponding vertical cut of the two focal spots along z-axis.
Similarly, at working wavelength λ2, the superposed phase shift profile along x-axis can be expressed as
where λ2 = λo represents the designed operate wavelength of the operation metalens. fo is the designed focal length of the operation metalens and C is the phase shifts introduced by the correction metalens. As shown in Fig. 4(d), the simulated phase profile (angle(Ex)) of the correction metalens in x-z plane at y = 0 with incident wavelength λ = 1.55μm is depicted. The almost aclinic phase profile of the correction metalens indicates that the phase shifts C is approximate to be a constant. Therefore, at wavelength λ = 1.55μm, the effect of the correction metalens on operation metalens is negligible and the focal length of the integrated metalens depends entirely on the operation metalens.That is to say the focal length of the integrated metalens at wavelength λ = 1.55μm is fo.Finally, to eliminate the chromatic aberration at the two working wavelengths(λ1 = 1.0μm and λ2 = 1.55μm), the condition ultimately turns into fo = fa + d1/2, where d1 = d + h1 + h2 is the thickness of the DAM. d, h1 and h2 are the thickness of the glass spacer, operation metalens and correction metalens, respectively. Therefore, by properly modulating the focal lengths of the two metalenses (operation and correction) and the thickness of glass spacer, achromatic focusing can be obtained at the two discrete wavelengths. Utilizing the phase shift profiles in Fig. 5(a), a DAM with achromatic focal length f = 20μm is designed. Figures 5(b) and 5(c) show the simulated intensity profiles of the designed DAM in x-z plane at y = 0 with incident wavelength λ = 1.0μm and λ = 1.55μm, respectively. The corresponding vertical cut of the two focal spots along z-axis is depicted in Fig. 5(d). As shown in Figs. 5(b)-5(d), a focal spot is exhibited with focal length f = 20.5μm for the two incident wavelengths, indicating the achromatic characteristic of our designed DAM. Moreover, the simulation results also indicate that the focusing efficiencies for working wavelength λ1 = 1.0μm and λ2 = 1.55μm are 53% and 70%, which are much higher than the previous values [19,35,37,38].
To further verify our designing strategy, a DAM with designed achromatic focal length f = 30μm is also demonstrated. With similar designing strategy, the detail values for the two metalenses are λo = 1.55μm, fo = 30μm; λc = 0.72μm, fc = 60μm. The corresponding results are shown in Fig. 6, in which achromatic focusing with focal length f = 31.2μm for the two discrete wavelengths are demonstrated. The small deviation can be attributed to the unperfect phase profile introduced by P-B phase (As shown in Figs. 2(b) and 2(d)). Moreover, the focusing efficiencies for the two working wavelengths λ1 = 1.0μm and λ2 = 1.55μm are as high as 54% and 70%, respectively. It worth noting that dual-wavelength achromatic focusing with focal lengths (f = 20μm and f = 30μm) are demonstrated, whereas the design strategy can be scalable to any values. These results will open up a new door for designing the multiple-wavelength optical achromatic devices, holding potential applications in microscopy, imaging and spectroscopy systems.
Fig. 6 (a) The phase shift relationship among the operation, correction and achromatic metalenses. (b) and (c) Simulated intensity profiles of the DAM in x-z plane at y = 0 with incident wavelength λ = 1.0μm and λ = 1.55μm, respectively. (d) Corresponding vertical cut of the two focal spots along z-axis.
4. Conclusions
In conclusion, by creatively adopting the metalens doublet approach, we demonstrated a new strategy to design high efficiency DAM. The unit cells of the two metalenses were elaborately designed to guarantee high efficiency at the two discrete wavelengths (λ = 1.0μm and λ = 1.55μm). The two metalenses were cascaded to form a integrated metalens, in which the phase superimposition principle were utilized to modulate the focal length at a certain wavelength. By properly modulating the focal lengths of the two metalenses, achromatic focusing can be obtained at the two discrete wavelengths. With such a strategy, DAMs that operate at the two discrete wavelengths with focal lengths f = 20μm and f = 30μm were demonstrated in detail. By virtue of the unique configuration, the designed DAMs overcome the disadvantage of mostly previous achromatic metalenses and the focusing efficiency for the two working wavelengths λ1 = 1.0μm and λ2 = 1.55μm are as high as 54% and 70%, respectively. These results provide extra degrees of freedom for designing high efficiency multiple-wavelength achromatic devices, holding potential applications in microscopy, imaging and spectroscopy systems.
Funding
National Key R&D Program of China (2018YFA0306200, 2017YFA0205800); National Natural Science Foundation of China (11334008, 61705249, 61290301 and 61521005); Fund of Shanghai Science and Technology Foundation (16JC1400400, 16ZR1445300, 16JC1400404); Shanghai Sailing Program (16YF1413200); Youth Innovation Promotion Association CAS (2017285); Key research project of Frontier Science of CAS (QYZDJ-SSW-JSC007); Yao Foundation.
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