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Abstract
This study proposes two types of metasurfaces to adjust the phase of wavefront based on geometric phases. Specifically, a meta-atom etched by two different elliptic cylinders possesses more adjusted parameters and satisfies some demanding design conditions. Under the illumination of circularly polarized light, one of them, metalens, can focus the light with a focusing efficiency of 60 %. The other can generate vortex beams with an efficiency of ∼55%. Such metasurfaces can be used as planar optical components in biological, medical, and intelligent wearable fields.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The metasurface is a subwavelength-spaced structure consists of unit-cells called meta-atoms. In recent years, it has been used in many fields due to its nearly two-dimensional structure and multi-functional integration properties. Compared with the traditional diffraction devices, the well-constructed meta-atoms are helpful to restrain the high-order diffraction and control the wavefront phase accurately. It is worth mentioning that this planar structure can be effectively associated with mature semiconductor technology, reducing the size and high cost of optical systems. Therefore, the ability to arbitrarily control the wavefront is of great interest and has been demonstrated in Metalens [1–10], vortex generators [11–20], holograms [21], absorbers [22,23], and in special beams [24]. Generally, the design of these metasurfaces is divided into five steps: selection of material, choice of meta-atom geometry, setting up the simulation environment, choosing the appropriate fabrication technique, and characterization of the fabricated sample. On the selection of meta-atom to construct, a single dielectric pillar (rectangular column or elliptical cylinder) typically has two tunable parameters (height and width) [9,25]. Two rectangular dielectric cylinders typically have five tunable parameters (two different heights and widths and the direct distance between them), usually used in achromatic metalens [26]. In this article, we build a meta-atom etched by two different elliptic cylinders. It has more freedom (the long and short axes of two different elliptical cylinders, and the length and width of the etched rectangular meta-atom). It is necessary to achieve the aim of multi-wavelength optimization base on the geometric phase. Furthermore, to test its basic performance, we design two metasurfaces such as the metalens and vortex generators. Compared with the former, the latter introduced orbital angular momentum. That is to say, metasurface plays a vortex phase plate combined with a focusing lens. The multi-function integrated optical metasurface is expected to be used in biomedical, medical, and wearable smart devices.
2. Model design
The meta-atom is optimized by using a particle swarm optimization (PSO) tool, which is built into the Lumerical. The optimized geometric parameters are the long and short axes of two different elliptical cylinders and the length and width of the etched rectangular meta-atom. The top view of the best meta-atom is shown in Fig. 1(a). The length and width of the rectangle are respectively ${\textrm{L}_1} = 0.33\; \textrm{um}$, ${\textrm{L}_2} = 0.3\; \textrm{um}$. The parameters of the middle ellipses are ${\textrm{L}_{\textrm{left}}} = 0.2\; \; \textrm{um}$, ${\textrm{S}_{\textrm{left}}} = 0.05\; \textrm{um}$, ${\textrm{L}_{\textrm{right}}} = 0.14\; \textrm{um}$, ${\textrm{S}_{\textrm{right}}} = 0.05\; \textrm{um}$. Besides, the center points of two ellipses are located at $0.1\; \textrm{u}m$ on both sides of the vertical axis. The height of the meta-atom is fixed ($\textrm{H}\; = \; 0.6\; \textrm{um}$) and the period of unit-cell is $\textrm{P}\; = \; 0.4\; \textrm{um}$. In the simulation, the x and y directions are set as periodic boundaries, and the z directions are set as perfect absorption layers. Min mesh step is 0.25 um. We use different mesh accuracy to simulate and compare the results to judge whether the simulation is convergent. To reduce the energy coupling between structures, the maximum value of the structure parameter is set as not exceed $0.35\; \textrm{um}$ during optimization. Figure 1(b) shows a metasurface designed with this meta-atom. Figures 1(c) and 1(d) reveal TE and TM modes under the eigenmode solver with commercial software Ansys Lumerical. In these two different modes, the energy is concentrated in the structure and the air respectively, which effectively increases the effective refractive index of the structure. Detailed explanations can be found in Ref. [27].
Fig. 1. (a) Top view and perspective of meta-atom (b) Schematic diagram of the metasurface. (c) TE mode. (d) TM mode. (e) The transmittance of right-polarized light, the transmittance of left-polarized light, and total reflectivity. (f) The relationship of the rotation angle and the phase of the meta-atom.
Because the designed structure works based on geometric phase (the relationship between the phase shift of wavefront and the rotation angle of the meta-atoms is: $\varphi = 2{\mathrm{\theta }_\varphi }$), the polarization of right-handed circularly polarized light (RCP) will be reversed. The left-handed circularly polarized light (LCP) will be produced after it is illuminated by right-handed circularly polarized light. Figure 1(e) shows the conversion efficiency of transmitted right, left polarized light and reflection without regard to absorbance. To maximize the value of operable left-right circularly polarized light, the figure of merit (FOM) using PSO is set as FOM = T(LCP)-T(RCP). To make the structure effect closer to the Half-Wave plate, Max (FOM) is set as the target of optimization. Figure 1(f) shows the phase plot against rotation angle. The best FOM in this work is approximately 0.9 (structural parameters are accurate to the order of ${10^{ - 2}}\; \textrm{um}$). Besides, the substrate material was selected as silicon dioxide (SiO2). The meta-atoms material is titanium dioxide (TiO2) with a refractive index of 2.4. The ohmic loss of metal material reduces the working efficiency of the device, and the dielectric material (TiO2) is effective to reduce this loss [7]. In terms of practically realize the proposed metasurfaces, we will adopt the manufacturing process mentioned in Ref. [28]. The fabrication process is based on atomic layer deposition and electron beam lithography.
3. Results
The metalens could converge light to a point, according to the focusing formula of eliminating spherical aberration:
where $\varphi (r )$ is radial phase distribution, $\mathrm{\lambda }$ is wavelength in vacuum, f is focal length, and r is radial coordinate. In this work, the numerical aperture of the metalens is set to $\textrm{NA}\; = \; 0.45$, the radius is $r\; = \; 4\; \textrm{um}$ and the focal length is $f\; = \; 8\; \textrm{um}$ at the wavelength of 450 nm.Figures 2(a)–2(c) shows the performance of focusing on the wavelength of 450 nm. Figure2 2(d)-2(f) and 2(g)-2(i) are performances illuminated by 532 nm and 633 nm light. From these pictures, it can be seen clearly that the simulation results are roughly consistent with the design objectives. The focusing efficiency of the metalens is about 60%, which is calculated by dividing the light intensity at the focal point (three times the range of full width at half maximum) by the light intensity through the metalens. It should be noted that the longer the wavelength, the shorter the focal length. In the situation of illuminated by a plane wave of 633 nm, focusing efficiency decrease ∼8%. Besides, we find that about 5% of the energy is lost in the structure calculated from the incident energy minus the energy passing through the metalens and the energy reflected.
Fig. 2. (a)-(c) The far-field intensity distribution, the profile of the focus, and the normalized intensity of the focus at the wavelength of 450 nm. (d)-(f), (g)-(e) The performance of the same metalens at the wavelength of 532 nm and 633 nm.
Also, we use the designed meta-atoms to construct the metasurface that can produce vortex beams. The traditional vortex beam is generated by combining the lens and vortex phase. The metasurface can integrate the functions of these two devices. After the array is constructed in one direction according to the phase distribution of the metalens, each meta-atom rotates and copies around the origin, and finally, a complete surface structure is formed. Due to the special nature of the geometric phase, when doing this operation, it will naturally introduce the phase with the topological charge of 2. The structure of the first array completes a 360-degree rotation, according to $\varphi = 2{\mathrm{\theta }_\varphi }$. As a result, the magnitude of the phase shift is equal to twice the magnitude of the rotation, where the vortex phase is ${\varphi _v} = l{\mathrm{\theta }_\varphi }$, and $l\; = \; 2$. Using a similar method, we construct metasurfaces with topological charge $l = \; 3$ and $l\; = \; 4$.
The results are shown in Fig. 3. Figures 3(a)–3(c) shows the phase distribution of the orbital angular momentum mode. Figures 3(d)–3(f) shows the near-field phase distribution of the simulated metasurface. The phase singularity at the center of the vortex creates the distribution of the hole of the center, which is shown in Figs. 3(g)–3(l). The efficiency of vortex beams generating is respectively 54%, 55%, and 50%. Table 1 shows the efficiency comparisons between recently reported metasurfaces. After forming the metasurfaces, some meta-atoms are too close together causing an inaccurate wavefront phase because of energy coupling. It eventually leads to the decline of efficiency. Besides, further optimization of the meta-atom can also improve the device performance.
Fig. 3. Topological charge $\; = \; 2$, $l\; = \; 3$ and $l\; = \; 4$ . (a)-(c) The phase distribution of the orbital angular momentum mode. (d)-(f) Phase distribution of the near field. (g)-(l) The intensity distribution of the vortex beam.
Table 1. Summary of our result and other references.
4. Discussion
In PSO, the maximum number of generations is twenty and each generation size is ten. After the last iteration, it finds ten different meta-atoms. Each of them possesses high conversion efficiency, and the best one was adopted. Beyond this, other excellent optimization algorithms can be used to find structures with high performance [29–31]. However, all of them require a lot of time. Using geometric phases can effectively save time for the reason of only one meta-atom needs to be found and rotate. Of course, geometric phases have their limitations, such as the need for circularly polarized light to illuminate. In the process of simulation, periodic boundary conditions are used in the cell structure. This is not strict in the construction of the entire metalens. The meta-atoms in the array are not periodically repeated. Also, from the perspective of the entire array, meta-atoms are too close to each other. This will result in an inaccurate wavefront phase distribution, which eventually leads to partial light scattering and a decrease in focusing efficiency. In this paper, the focus is not so accurate. For instance, the focus seems to be ${\sim} \; 8.15\; \textrm{um}$ at the wavelength of 450 nm. But in theory, it is $8.6\; \textrm{um}$. This is because of the extra thickness of the structure. When the light passes through the entire metalens, the phase profile begins to meet the design requirements. This error can be mitigated by enlarging the size of the metalens and increase the distance between meta-atoms. In conclusion, the metasurface can be used as a planar optical element and its multi-functional integration characteristic is very attractive. In this paper, the metalens and the vortex generator are designed with special meta-atoms. Maybe they would be applied to biomedical, artificial intelligence fields.
Funding
Natural Science Foundation of Zhejiang Province (LY20F050001, LY20F050002); National Natural Science Foundation of China (11847112, 61975185).
Disclosures
The authors declare no conflicts of interest.
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