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Abstract
A transmissive metasurface lens thinner than a wavelength promises a potential way to replace conventional bulky components for wavefront and polarization control of incident light. Here we propose a novel approach for an ultrathin metasurface lens at the working wavelength of 1550 nm composed of silicon cuboids on the silica substrate. Taking advantage of the ‘slow-light’ effect of the electromagnetically induced transparency (EIT) phenomena, the thickness of the proposed transmissive metalens has been reduced to 130 nm (∼1/12λ0), and it can focus incident light to a near diffraction-limited spot. The focusing efficiency of the ultrathin metalens is about 2 times larger than its peer without EIT effect. One key advantage of this metalens design is the reduction of the aspect ratio down to about 1, making this approach significantly degrade the difficulty of metalens manufacturing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Conventional optical components are generally heavy, bulky, costly and time-consuming to manufacture with high precision, considering their phase accumulation mechanism by varying the macroscopic lens thickness at different radial positions [1,2]. These are significant challenges for applications such as miniaturized, integrated, and portable devices. Metasurface optical elements with the features of small size, light weight, and easy to integration offer a potential way to replace the conventional bulky lens considering their excellent properties in manipulation of phase, amplitude and polarization of light waves [3–6]. In recent years, many compact optical devices based on metasurfaces have been demonstrated for the applications of polarization conversion [7,8], polarimeters [9,10], and holography [11–13]. In 2011, Yu et al. [14] proposed a metasurface composed of a V-shaped metallic nanoantenna array, which can be used to provide 2π phase coverage. By designing the shape, size, and orientation of antenna, the phase shift of the radiated light can be manipulated, thus introducing an abrupt phase change at a much smaller distance than the wavelength. Later, Aieta et al. [15] showed an aberration-free flat lens at telecom wavelength using only 60 nm thick gold metasurface. Furthermore, plasmonic lenses based on the Pancharatnam-Berry (PB) phase modulation were theoretically and experimentally presented using rectangular antennas [16] and U-apertures [17], respectively. Although plasmonic metalenses have attracted considerable attentions, the large ohmic losses of the metallic materials have limited its efficiencies for practical applications. Later, the all-dielectric metasurfaces have been put forward to improve the efficiency of metalens [18]. Due to their low absorption loss and relative mature fabrication technology, silicon-based metalenses provide a potential solution for achieving large efficiency and near diffraction-limited focusing in the near-infrared region [19,20]. However, to achieve complete 2$\pi $ phase coverage, one needs to compensate for the smaller refractive index with higher nanostructure height based on waveguiding effects. As a result, high aspect ratio nanostructures are inevitably demanded. Table 1 summarizes the corresponding parameters of previously reported silicon-based metalenses [20–26], where the aspect ratio is defined as the ratio of the height to the minimum width/diameter/gap of the meta-atoms. It can be seen that the aspect ratios of meta-atoms in the previous works are all larger than 4, up to 27.3. Such high aspect ratio may cause the nanostructures to collapse or break easily during fabrication, which poses great challenges for fabrication techniques such as liftoff and dry-etching processing [27,28].
Table 1. Summary of previously reported silicon-based meta-atoms
On the other hand, electromagnetically induced transparency (EIT) in metamaterials has attracted considerable interest due to the EIT behaviors which are similar to those in atomic system [29–33]. Compared with the EIT resonance in atomic system, the EIT resonance in metasurfaces is easier to implement experimentally because of allowing operation at room temperature. Nevertheless, most of these reported works are focusing on the modulations of EIT in metasurfaces, few works have applied the EIT effect to the study of metalenses. In this paper, we propose a near-infrared dielectric ultrathin PB phase metalens based on EIT effect. The unit cell of the metalens consists of two Si cuboids. When the height of the two Si cuboids is 130 nm, a clear EIT phenomenon can be obtained due to the Mie-like resonance interference. Taking advantage of the ‘slow-light’ effect associated with EIT phenomena, the effective refractive index of the meta-atoms can be greatly increased, and the height of the meta-atoms can be reduced to ${\sim} {1 / {12}}$ of the free-space wavelength or below. In this way, it can significantly reduce the aspect ratio of metalens to about 1, which is beneficial for the fabrication of compact optical devices. Our approach put forward a novel way to reduce the aspect ratio of metalens fabrication, which can also be extended to other electromagnetic wave ranges.
2. Independent control of the phase
To focus an incident beam in a diffraction limited spot, the phase profile should be in the following formula [34,35],
where $\lambda$, r, and f are the wavelength, the lens radial coordinate, and the designed focal length, respectively. The input plane beam is modulated into a converging wavefront after passing through the metalens element. In this paper, the hyperbolic phase profile described by Eq. (1) is realized by rotating the meta-atom to a designed angle using the concept of PB phase. When a left-circular polarized (LCP) beam passes through the meta-atom, the transmitted electric field can be described by the Jones vector [36,37],3. Dielectric double-cuboid meta-atoms
To design a metalens for focusing, an all-dielectric metasurface structure is optimized for manipulation of wavefronts with a phase-modulation range of $2\pi$. The structure of the proposed meta-atoms is depicted in Fig. 1(a). The meta-atom composes of two Si cuboids on a SiO2 substrate, which can be regarded as a coupled waveguide. The refractive indices of Si and SiO2 are 3.7 and 1.48, respectively [29]. The size of a unit cell is $P \times P$, the sizes of the two Si cuboids are ${L_1} \times {W_1} \times h$ and ${L_2} \times {W_2} \times h$, respectively, and the gap between the two cuboids is g. The rotation angle $\theta$ is shown in Fig. 1(b), where the two cuboids are rotated with respect to the centre of the square cell. For the case of 1550 nm incident light, the height h of the Si cuboid is fixed at 130 nm or ${\sim} {1 / {12}}$ of the free-space wavelength, the smallest among dielectric metasurfaces reported to date. The other geometrical parameters of the metasurface structure are set as $P = 1040$nm, ${L_1} = 690$nm, ${W_1} = 330$nm, ${L_2} = 330$nm, ${W_2} = 150$nm and $g = 120$nm, respectively.
Fig. 1. (a) The proposed meta-atom structure consisted of dielectric double cuboids. (b) The two cuboids are rotated with respect to the centre of the square cell.
The simulations are performed using the full electromagnetic field simulation software with a plane wave normally illuminated from the substrate and periodic boundary conditions set at the boundaries of the simulation domains for these meta-atoms. To illustrate the influence of EIT effect on the designed metalens, the cases with heights of 600 nm and 130 nm are taken for comparison in Fig. 2. For the case of $h = 600$nm, the EIT phenomenon is not observed in the transmitted linear polarization wave (as seen in Fig. 2(a)). With the same parameters, when the circular polarized waves incident normally upon the structure, there is a transmission peak at the wavelength of 1550 nm, as shown in Fig. 2(c). Fixed the wavelength at ${\lambda _0} = 1550$nm, the corresponding amplitude transmittance and phase of the transmitted cross-polarization wave are plotted with respect to the rotation angle $\theta$ in Fig. 2(e) for LCP incident waves, where $\theta$ is the angle between the long side of cuboid and y-axis. As shown in Fig. 2(e), the transmission efficiency of the double-cuboid metasurface exhibits a very unstable behavior, varying its value between 0.16 and 0.45 as the rotation angle varied. Also, although the phase-modulation range of $2\pi$can be covered, the nonlinearity is very pronounced. These features make the performance of metalens hard to maintain.
Fig. 2. (a, b) Simulated amplitude transmittance of linear polarization wave in the x direction (black dashed curve) and the y direction (red solid curve). (c, d) Simulated amplitude transmittance of cross-circularly polarized wave. (e, f) Amplitude transmittance (black) and phase (red) of transmitted cross-polarization wave (RCP) with respect to rotation angle θ for the LCP incident waves at the wavelength of 1550 nm. Where the corresponding meta-atom height $\textrm{h} = 600$ nm in (a, c, e), and $\textrm{h} = 130$ nm in (b, d, f).
On the other way, when $h = 130$nm as depicted in Fig. 2(b), the EIT phenomenon emerges when the incident E-field is along the y-axis (i.e., called y-linear polarization). And a remarkable EIT resonance peak located between two dips at the wavelength of 1570 nm with transmission amplitude exceeds 97%. With the same parameters, when the circular polarized waves incident normally upon the structure, there is also a transmission peak located at the wavelength of 1550 nm, as shown in Fig. 2(d). Fixed the wavelength at ${\lambda _0} = 1550$nm, the corresponding amplitude transmittance and phase of the transmitted cross-polarization wave are plotted with respect to the rotation angle $\theta$ in Fig. 2(f) for LCP incident waves. As expected, for the wavelength of ${\lambda _0} = 1550$nm, the double-cuboid metasurface is optimized for stable transmission efficiency and linear phase-modulation in the range of $2\pi$. Obviously, compared to $h = 600$nm, the meta-atoms with $h = 130$nm can achieve a relatively high transmission efficiency and a better phase control due to the EIT effect. Furthermore, the resulting maximum aspect ratio of the meta-atom is only 1, which is significantly lower than the previous values listed in Table 1.
Due to the EIT effect, a steep phase change is accompanied near the wavelengths where the EIT phenomenon occurs, as shown in Fig. 3(a), where a phase abrupt shift about $2\pi$ when the wavelength varies from 1550 nm to 1580 nm. Note that the height of the Si cuboid is $h = 130$nm, which is less than ${{{\lambda _0}} / {10}}$. The effect of larger effective index compensates for the smaller height due to the fact that the phase of the transmitted light passing through the meta-atom is,
where h and ${n_{eff}}$ represent the height and the effective index of the Si cuboid, respectively. In order to show more visually the variation of ${n_{eff}}$ around the EIT when $h = 130$nm, Fig. 3(b) illustrates $\Delta {n_{eff}}$ for different wavelengths. At the EIT peak of 1570 nm, it can be seen that resonant phase cancellation leads to $\Delta {n_{eff}} = 0$. Since the phase gradient around EIT varies greatly, the effective index difference of the Si cuboid for x- and y-polarized light increases significantly ($\Delta {n_{eff}} > 3.5$), so that phase modulation with stable transmission efficiency can be achieved at a very small cuboid height.Fig. 3. (a) Simulated phase spectra of linear polarization wave in the x direction (black dashed curve) and the y direction (red solid curve) when the meta-atom height $\textrm{h} = 130$ nm. (b) The effective index difference of meta-atom for x- and y-polarized light. (c) Simulated electric field distribution at the EIT resonance peak and dips in the y-linear polarization.
The electric field distributions of the EIT peak and dips in Fig. 2(b) are shown in Fig. 3(c). It can be found that the electric field intensity around the big and the small Si cuboid is the strongest at the wavelength of 1553 nm and 1576 nm, respectively. And the electric field resonance mainly occurs in the gap between the two Si cuboids at the EIT peak of 1570 nm. Due to the different sizes of the two Si cuboids, the coupling strengths with the incident light are different, resulting in their bright-mode resonance at different positions [38,39]. At 1570 nm, the two bright-mode resonances are excited simultaneously by the incident light, which leads to the appearance of an EIT window. In this double cuboid meta-atom, the EIT effect is based on the bright-bright mode coupling, which is different from the interference between the bright and dark modes in the EIT generated by the bar and ring structure [29].
Figure 4 gives the variation of the amplitude transmittance and phase with respect to wavelength for different rotation angles θ, i.e., 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, and 157.5°. According to the amplitude transmittance curves in Fig. 4(a), one of the transmission peaks located at the 1550 nm is slight affected by the rotation angle, indicating the realization of a stable transmission conversion efficiency with a value greater than 0.45, which is consistent with the explanation in Fig. 2(f). The other transmission peak in Fig. 4(a) is located near 1576 nm and becomes very unstable at different rotation angles, which is the reason why it is not considered. From Fig. 4(b), it can be also seen that the phase shows an almost linear variation with respect to the rotation angle in the concerned wavelength ranges from 1530 nm to 1560 nm. The variation of incident wavelength only introduces an additional constant phase, implying that the lens might work for a certain frequency range. As shown in Fig. 4, in the concerned wavelength ranges of 1530 nm-1560 nm, the transmittance varies between 0.35 and 0.48, and the corresponding phase shift $\Delta \varphi$ due to the dispersion being $0.45\pi$, such that the corresponding group delay (slope of the phase versus angular frequency plot) being 50 fs. Such large group delay is attributed to the high effective index of the Si cuboids. As mentioned above, the high effective index is due to the EIT effect that substantially reduces the meta-atom height.
Fig. 4. (a) Amplitude transmittance and (b) phase of transmitted cross-polarization wave with respect to wavelength for different rotation angles θ, i.e., 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, and 157.5°.
4. Metalens design and focusing performance
Taking advantage of the EIT effect of the proposed double-cuboid meta-atoms and utilizing the PB phase modulation method described above, a metalens is designed for a working wavelength of ${\lambda _0} = 1550$nm. In the design, the metalens radius is $322.58{\lambda _0}$ (0.5 mm) and the preset focal length is fixed as $645.16{\lambda _0}$ (1 mm); therefore, the numerical aperture (NA) of the metalens is 0.447.
The focusing performance of the proposed metalens has been theoretically investigated by using Fresnel-Kirchhoff integration, neglecting the actual coupling between meta-atoms [40,41]. The optical intensity profile in the x-z propagation plane is shown in Fig. 5(a). The z coordinate corresponding to the peak intensity value gives the focal length for the given wavelength. It can be seen that the light transmitted by the metalens is focused at $z \sim 645.16{\lambda _0}$, which agrees well with the theoretical designed focal length. The corresponding 2D optical intensity profile of the foci is also presented in Fig. 5(b), which shows clearly symmetric intensity distribution on the focal plane for the simulated wavelength. Figure 5(c) gives the normalized intensity distribution curve in the x-direction. The intensity of the maximum sidelobe is less than 3% of the central lobe intensity at the simulated wavelength, and therefore the sidelobes on the focal plane should have no significant influence on the field-of-view of the metalens. It is found that the corresponding full-width-at-half-maximum (FWHM) of the focal spot is $1.12{\lambda _0}$ (1736nm), close to the theoretical diffraction-limited value of ${{0.5{\lambda _0}} / {NA}}$.
Fig. 5. Results of focusing simulations under normal incident waves: (a) Optical intensity profile in the x-z propagation plane. (b) Optical intensity profile on the focal plane at $z = 645.16{\lambda _0}$. (c) Normalized intensity distribution along the x axis.
In addition, the designed metalens has a certain broadband achromatic adjustment ability due to it is constructed based on PB phase principle. Figure 6(a) shows the wide-band simulated focusing characteristics of the metalens. It can be seen that the metalens can focus well in the range of incident wavelength from 1530 nm to 1560 nm, and the focal length remains unchanged. Figure 6(b) displays the simulated FWHM and the focusing efficiency. It is found from Fig. 6(b) that when the incident wavelength varies from1530 nm to 1560 nm, the corresponding FWHM changes from 1712nm to 1751nm, which is always close to the diffraction limit of ${{0.5{\lambda _0}} / {NA}}$. In addition, the focusing efficiency of such a metalens can reach up to 19% in the considered wavelength range, which is defined as the ratio of the optical power in a circular area with a diameter of 3 × FWHM to the incident power [42,43]. As a comparison, the focusing efficiency for the case of $h = 600$nm is also evaluated, and the maximum is only 9%. It is clear that the designed metalens based on EIT effect has a relatively high focusing efficiency with a much lower aspect ratio of 1.
Fig. 6. (a) Optical intensity profile in a wide wavelength range. (b) FWHMs of the focal spots and focusing efficiencies.
5. Conclusion
In conclusion, utilizing the effect of EIT, an ultrathin dielectric metalens with 130 nm height has been proposed at the working wavelength of 1550 nm. The metalens is realized through the PB phase by locally rotating the double silicon cuboid meta-atoms. Comparing to the stuicture without EIT effect, the optimized structure with EIT effect has a stable transmission efficiency and a linear phase-modulation feature in the range of $2\pi$. Numerical simulations show that the designed metalens has a FWHM near the diffraction limit and a focusing efficiency up to 19%. In our specific case, the metalens thickness is approximately 1/12 of the free-space wavelength and the meta-atom aspect ratio is as low as 1. This EIT-based ultrathin metalens with such a low aspect ratio can substantially reduce the fabrication difficulty and benefit the mass production, which can be extended to other electromagnetic wave ranges.
Funding
National Natural Science Foundation of China (12002326, 12204445); National Key Research and Development Program of China (2020YFA0714001).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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