## Abstract

This study presents the design and fabrication of a dielectric metasurface with free-form phase distribution and with a large period unit cell. The dielectric metasurface is fabricated using i-line stepper, dry etching, and nanoimprint technology. The phase distribution of the meta-device is the combination of a blazed grating for deflection and an aspherical lens for eliminating the off-axis aberration. The optical measurement result shows the off-axis focusing spot is with loss aberration and the corresponding Strehl ratio is 0.34. The diffraction efficiency is around 2%. The low efficiency is mainly attributed to the rounding of the rectangular nanostructures during the pattern transfer and relatively thin thickness. Moreover, the polarization-dependency of this large period metasurface is also discussed.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

A lens with lightweight and low aberrations always is the goal for optical engineers. Aberrations cause the image blurred or distorted and thus reducing the optical performance of a lens. Conventionally, people use a multi-piece lens set and aspherical lens to eliminate the aberration. However, this can cause increases in the size, weight, and cost of the lens assembly. To eliminate the off-axis aberrations such as coma, astigmatism, and field curvature, a straightforward method is to achieve an equal optical path for each focusing ray. For example, the Acuña-Romo equation [1] describes the solution for designing spherical aberration-free singlet lens with bi-aspheric surface. For a flat lens with collimated incident light, the corresponding phase distribution is hyperbolic [2,3,4], as shown in Eq. (1):

where*ϕ*is the phase distribution of the metalens;

*x*and

*y*are the coordinates along the metalens plane.

*λ*is the free space wavelength, and

*F*is the focal length of the metalens. Nevertheless, such phase profiles are only valid for collimated incident light with any tilt angle and only for on-axis focusing. In fact, Eq. (1) represent that to be free from spherical aberrations, the optical path length of an axial ray and the optical path length traveled by a marginal ray must be equal. For simple cases, for example, point-to-point focusing and collimated beam focusing, Eq. (1) works well. However, apparently, Eq. (1) fails to produce exact point-to-point correspondence between a multi-point object and its image owing to complicated relations arisen from the off-axis point and tilted optical ray. For more complicated applications, usually, another analytical expression modified from Eq. (1) to solve the required phase distribution is needed [5,6].

However, there is no exact solution of phase distribution for an aplanatic lens for off-axis focusing, multi-field angle, and large field of view. At this time, an iterative method to achieve a merit phase distribution is needed. In this paper, the phase distribution of the meta-device is defined by a series of polynomials as follows:

*x*,

*y*) coordinate. The

*x*-

*y*polynomial results in a non-rotationally symmetric free form surface as the coefficients with (

*m*+

*n*) value is even number. For example, for

*m*= 1 and

*n*= 0, the polynomial is

*A*

_{2}

*x*; for

*m*= 0 and

*n*= 1, the polynomial is

*A*

_{3}

*y*. The phase distribution allows us to design a meta-device with multi-function, for example, focusing, deflecting, and eliminating the aberration simultaneously.

## 2. Device description

Figure 1 shows the schematic of the designed meta-device, which is basically the combination of an aspherical lens and blazed grating. The blazed grating deflects the incident light efficiently while the aspherical lens plays a role to focus and eliminating the off-axis aberrations. The phase distribution of the meta-corrector is:

where parameter*A*

_{2}(2.5 × 10

^{5}) is the phase distribution function of the blazed grating; parameter

*A*

_{4}(-5.537673 × 10

^{6}) and

*A*

_{6}(-5.537673 × 10

^{6}) is the phase distribution of a spherical lens. As a deflection is introduced by the

*A*_{2}coefficient, it accompanies off-axis aberrations, such as coma and astigmatism. These aberrations were eliminated by non-radial-symmetric

*A*

_{11}

*×*

^{4}term where

*A*

_{11}is 3.172364 × 10

^{−4}. Once the coefficients of the polynomial are optimized, the bulk sections of 2π phase shifts of the corresponding curvature are eliminated to be a modulo-2π phase structure. The saw-tooth-like phase distribution is then pixelized to multi-phase-level with a pixel size of 2 µm.

Here, we utilize the Pancharatnam-Berry (PB) phase principle to realize the phase distribution of our meta-device [7,8]. The basic building block of the unit cell of the PB-phase meta-device is shown as Fig. 2(a), which consisting of a Si nano-fin on a Si substrate. In our design, the dimension of meta-device is 4 mm. The meta-device is patterned with periodical unit cells where the period is 2 µm. Each nano-fin has its base rod fixed with dimensions of L=1500 nm, W=600 nm. Generally, in the case of an appropriate thickness nano-fin with the incline angle between the x-axis and the long axis of the nano-fin, the phase modulation for the EM wave through the meta-device is twice of the *θ _{p}*. The phase distribution of nano-fin with a light source with circular polarization was simulated by the finite-difference time domain (FDTD) based commercial software Lumerical. The boundary condition had been setup as follow: periodic boundary condition in x, y-axis direction and the perfect match layer in z-axis. Finally, we have chosen the phase modulation points corresponding to

*θ*= 55°, 119°, 8°, 68°, 137°, 24°, 86°, 151°, 37°, 102°, 168° as shown in Fig. 2(b). The phase distribution is very like a blazed grating owing that the special frequency for focusing is much smaller than that for deflection in our design. In general, the simulated reflectivity is higher than 60%.

_{p}PB-phase metasurfaces working with circularly polarized light. It is known that PB-phase can only pass an equivalent phase delay with an opposite sign to right-handed circular polarization (RCP) and left-handed circular polarization (LCP) light [9]. In theory, the PB-phase modulation is independent of the period of unit cell. As mentioned, the period of the unit cell is 2 µm which is much larger than the working wavelength λ = 633 nm. The PB-phase modulation under a unit cell much larger than the working wavelength will be discussed.

On the other hand, here, we consider a nano-fin dielectric PB-phase metasurface. A nano-fin is an optical birefringent structure [10]. This artificial form-birefringent structure induces a polarization-dependent phase and play a role of polarization conversion [11]. Therefore, the thickness of the nano-fin accumulates the polarization-dependent phase and affects the polarization conversion efficiency (PCE) [12]. As the geometry parameters of the nano-fin provides a phase delay as a half-waveplate, the PCE is 100%. Nevertheless, regarding a non-perfect polarization conversion, the output LCP carries an ideal PB-phase modulation for an input RCP light [13]. Therefore, in this paper, we didn’t pay too much attention to control the thickness of the nano-fin. It mostly affects the PCE but not phase distribution of the meta-device.

## 3. Fabrication and measurement result

#### 3.1 Fabrication process

Most metasurfaces are currently fabricated using electron-beam (e-beam) lithography [14] or photolithography [15]. E-beam lithography is expensive and time-consuming and is difficult to achieve mass production. In addition, the resolution of the photolithography suffers from light diffraction. Here, we utilize nanoimprint lithography, which have greatly benefited from the rapid development of nanofabrication of large-scale, high pattern resolution, and low-loss dielectric metasurfaces. We first prepared a Si-based meta-device by stepper lithography and ICP-RIE dry etching. The unit cell of this meta-device is Si nano-fin on a Si substrate. In order to decrease the costs, this Si structure is utilized as a master template for creating negative replicas. Then a Polydimethylsiloxane (PDMS) layer was spin-coated on the Si-based meta-device and thermally annealed for curing. After demolding, a negative replicas PDMS stamp of nano-fins can be made. Finally, the meta-device structure is transferred onto a polyimide (PI) layer on a glass substrate by using nanoimprint technique. Figure 3(a) shows the scanning electron microscope (SEM) image of Si-based meta-device. Figure 3(a) shows the SEM image of Si-based meta-device. The orange dash line indicates the nano-fin arranged along the *x*-axis. It can be obviously seen that the *θ _{p}* of nano-fin changes, which corresponds to the modulation of the phase distribution. The SEM image of PI meta-device is shows as Fig. 3(b). Although it can be seen the rectangular structure becomes rounded, we still can observe the nano-fin structure of meta-device after twice pattern transfer.

#### 3.2 Far-field diffraction pattern

Figure 4(a) shows the far-field diffraction pattern of the PI film meta-device with a collimated incident light with a wavelength of 633 nm (He-Ne laser). Due to the period of the unit cell is 2µm, the diffraction pattern shows multiple diffraction orders, such as (1, 0), (-1, 0), (1, 1), (2, 0) diffraction orders, and so on. The diffraction angle of (1, 0) order is 18.45°. In between the zero-order and the (1, 0) order diffraction beam, it shows a focal spot (marked by a white circle). On the contrary, we can barely observe a focal spot between the zero-order and the (-1, 0) order diffraction beam. This is because the phase distribution of the blazed grating benefits the diffraction efficiency of (1, 0) diffraction order as expected. Figure 4(b) shows the far-field diffraction pattern of the dielectric meta-device illuminated by a collimated white light source. Similar to the diffraction pattern shown in Fig. 4(a), we can still observe a focal spot between the zero-order and the 1st order diffraction beam. The focused beam between the zero-order and the (-1, 0) order diffraction beam shows a relatively low intensity. Moreover, owing to the diffraction nature of the meta-device, one can observe the diffraction pattern is a focused dispersive line. The diffraction pattern shows a smaller spot than that of (1, 0) and (-1, 0) diffraction orders. This reveals that the free-form phase distribution simultaneously provides a deflection and focusing ability. Nevertheless, a conventional metasurface is usually with a unit cell smaller than the working wavelength. This criterion is for the 1^{st} diffraction order is absent for normal incident light. For oblique incident angle, the unit cell of the metasurface should be smaller than half of the working wavelength. In this paper, the period of the unit cell is 2 µm which is larger than the working wavelength 633 nm. As a consequence, the device suffers from low diffraction efficiency and multi-diffraction orders. However, it still capable for demonstrating the function of a meta-device for low-spatial frequency applications [16].

#### 3.3 Analysis of off-axis focusing point

We measure the intensity distribution of the focal spot at a series of z-position. The *x*-cross-section and *y*-cross-section as a function of *z*-position is shown in Fig. 5(a) and Fig. 5(b). From Fig. 5(a), we can calculate the diffraction angle of the focal spot is 11°. From Fig. 5(b), we can find that the focal length of the focal spot is 42 mm. The intensity distribution at the focal plane is shown in Fig. 5(c). The corresponding spot size is about 23.2 µm. It can be seen that the shape of the off-axis focal spot is a circle and symmetric. Barely asymmetric aberration, such as coma and astigmatism, can be observed. As shown in Fig. 5(d), the corresponding Strehl ratio of meta-device is S = 0.34 compared to the ideal diffraction-limited distribution with a numerical aperture (NA) of 0.05. The spot size of the corresponding diffraction-limited distribution is 15.2 µm. Generally, the Strehl ratio is considered corresponding to diffraction-limited performance. For S<=0.3, a lens can be regarded as being with large aberrations [17].

#### 3.4 Polarization conversion efficiency of large period PB-phase metasurfaces

It is known that PB-phase can carry geometric phase delay as the input RCP converted to LCP [13]. In this part, we would like to measure the polarization-dependency of a large period PB-phase metasurface. We measure the diffraction efficiency of the focal spot. The incident polarization is linear polarized with a series of azimuth angle. The diffraction efficiency (red line) and the ellipticity angle (blue line) of the focal spot as a function of azimuth angles, the oscillation direction of electric field with respect to the *x* axis, is shown in Fig. 6. The diffraction efficiency of the focal spot oscillates for a varying azimuth angle.

The peak and valley value of the diffraction efficiency is 2.9% and 1.8%, respectively. As a reference, the diffraction efficiency for circularly polarized light is 0.4% which can be regarded as the PB-phase effect. The diffraction owing to the PB-phase modulation is relatively lower than total diffraction efficiency. The low efficiency is because the rounding and taper of the rectangular nanostructures during the pattern transfer and the relative thin thickness. The low efficiency is because the rounding and taper of the rectangular nanostructures during the pattern transfer and the relative thin thickness. Both rounding and taper structure decrease the anisotropy of the nano-fin. A thin thickness leads to a short optical path difference. All these make the phase modulation of the metasurface is lower than 2π and, therefore, lead to a low diffraction efficiency. The rounding and taper nano-fin generated as pattern transferred from Si mother mold to PDMS stamp. During the transferring process, the air bubbles exist between Si mold and PDMS. As the result, PDMS mold did not perfectly inherit the structure from Si mold. The final structure of the PI metasurface is rounding and taper when we transferred the pattern from PDMS stamp. We believe the air bubble quality can be reduced by providing a vacuum environment during nanoimprinting. Moreover, a nano-fin is a 2-fold rotational symmetry. Therefore, one can find that the oscillation in diffraction efficiency repeat as the azimuth angle increases 180°. Theoretically, the optical anisotropy of the nano-fin induces form-birefringence. Consequently, the input linear polarization state will change after passing through the metasurface. However, owing that the thickness of the PI film is thin and the period of the metasruface is large, structure-induced form-birefringence is weak. Therefore, the polarization state barely changes. However, we can still observe that the ellipticity angle is minimum as the diffraction efficiency is maximum and vice versa. On the other hands, it is challenge for nanoimprinting process to fabricate a high aspect ratio structure, which is essential for high efficiency dielectric metasurfaces. For a low aspect ratio, as the refractive index of PI is increased, the efficiency can also be improved. For example, G. Yoon et al. experimental demonstrated a metalens by nanoimprinting process with high refractive index polymer environmental resin (PER) [18]. The refractive index of PER can be increased from 1.5 to 1.9 by adding TiO_{2} nanoparticles in PER, which is an alternative method to enhance the efficiency.

## 4. Conclusions

This study presents the design and fabrication of a dielectric metasurface with free-form phase distribution and with a large period unit cell. The dielectric metasurface is fabricated using i-line stepper, dry etching, and nanoimprint technology. A dielectric metasurface consisting of PI nano-fins is fabricated. The phase distribution of the meta-device is the combination of a blazed grating for deflection and an aspherical lens for eliminating the off-axis aberration. The optical measurement result shows the off-axis focusing spot is with loss aberration. The corresponding Strehl ratio is 0.34 compared to an ideal point spread function with NA = 0.05. The diffraction efficiency is around 2%. The low efficiency is mainly attributed to the rounding of the rectangular nanostructures during the pattern transfer and relatively thin thickness. Moreover, it is found that the diffraction efficiency and the polarization state of the focal spot are polarization-dependent even for such a large period unit cell. The nanoimprinted free-form metasurface holds a high potential for mass production in the future.

## Funding

Ministry of Science and Technology, Taiwan (MOST 109-2124-M-008-002-MY3, MOST107-2628-E-008-004-MY3).

## Acknowledgments

The authors would like to acknowledge financial support from the Ministry of Science and Technology, Taiwan (grant no. MOST107-2628-E-008-004-MY3 and MOST 109-2124-M-008-002-MY3).

## Disclosures

The authors declare no conflicts of interest.

## 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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