Experimental demonstration of rotational varifocal moir\'e metalens

This paper reports experimental demonstration of moir\'e metalens with a wide focal length tunability from negative to positive through mutual angle rotation at a wavelength of 900 nm. The moir\'e metalens was developed using high-index contrast transmit array meta-atoms composed of amorphous silicon octagonal pillars designed to have polarization insensitivity and full 2$\pi$ phase coverage. The designed moir\'e metalens was fabricated on a glass substrate using simple a-Si sputter deposition, electron beam lithography through character projection, metal mask lift-off, and reactive-ion silicon etching. The moir\'e metalens has focal length tunability ranges from $-\infty$ to $-1.73$ mm and from 1.73 mm to$\infty$ corresponding to an optical power in the range -578 $-$ 578 m$^{-1}$ at the mutual rotation between $\pm 90^\circ$. Our results reveal a proof of concept for the focal length tuning with mutual rotation of lens components based on moir\'e lens configuration at optical frequencies.


Introduction
Metasurfaces, a planar branch of metamaterials, have attracted significant attention because they can tailor optical wavefront by arranging subwavelength patterns (meta-atoms) on surfaces. 1-3 They can demonstrate unique optical properties that cannot be achieved with natural materials and have high affinity for micro/nanofabrication methods, including lithography, deposition, and etching. Therefore, metasurfaces have opened many research fields and have several applications, including lenses, 4-6 retarders and waveplates, 7-10 vector beam converters, [11][12][13] color filters, [14][15][16] and holography. [17][18][19][20] It should be emphasized that mechanical deformation or change of mutual geometric position of metasurfaces offer novel functionalities and tunability for their optical properties. Reconfigurable metasurfaces have been studied based on this idea, including tunable transmittance, 21 color tuning, 22 and active phase shifters. 23,24 Metasurface lenses, or metalens, has attracted considerable attention owing to their thinness and lightweight. The recent development of lossless dielectric metalens 5,6,25 have also stimulated the attention in this field. Tunable focal length, or varifocal lens, are a promising and expected functionality of metalens. Similar to conventional refractive lens doublet, the longitudinal motion of lens along with the optical axis have been demonstrated based on micro-electromechanical actuators. 26 However, the effect of the pull-in instability of the electrostatic parallel plates actuator limits the travel range of metalens, 27,28 resulting in a narrow tunable range for the focal length.
Since metalens has a high degree of design freedom, it is not necessary to make the tuning method for the focal length similar to conventional refractive lenses. It is desirable to study how diffractive optical element (DOE) techniques or diffractive lenses can be applied to metalens. In fact, a DOE-inspired tunable metalens, the Alvalez metalens, has been experimentally demonstrated with focal length tuning based on the lateral movement of a pair of lenses. [29][30][31] However, the lateral motion of Alvarez lenses limits the effective area of the lenses. Furthermore, the reported focal length range of mechanically tunable metalens is limited to the positive region.
Moiré lenses are a pair of axially asymmetric lenses that achieve tunable focal length with mutual rotation, as presented in Fig. 1(a). 32,33 They offer a wide range of focal lengths from negative to positive and has been realized using DOE. 34 Moiré metalens has also been studied using numerical simulations, 35,36 and has been demonstrated in the microwave frequency band quite recently. 37 However, the demonstration at optical frequencies has not been reported yet due to the difficulty of fabricating at such frequencies.
In this paper, we experimentally demonstrate tunable focal length using moiré metalens at the near infrared frequency band using polarization-insensitive meta-atoms based on highindex contrast transmit arrays (HCTAs), which are made of amorphous silicon (a-Si) in this case. We designed a-Si octagonal pillars with polarization insensitivity and full 2π phase coverage at a wavelength of 900 nm. The metalens were designed to satisfy the phase distribution of moiré metalens, as detailed below, by hexagonally mapping the corresponding pillars to each lattice points. The metalens was fabricated on a silicon substrate using simple a-Si sputter deposition, electron beam lithography through character projection, metal mask lift-off, and reactive-ion etching (RIE) of silicon. The fabricated metalens exhibited focal length tunability at the ranges from −∞ to -1.73 mm and from +1.73 mm to +∞ at a mutual rotation of ±90 • at 900 nm. The results reported here reveal a proof of concept for focal length tuning with mutual rotation of lens components based on moiré lens configuration at optical frequencies, and thus paving way for applications in optical frequencies, including near infrared, visible, and ultraviolet wavelengths. Figure 1 presents a schematic diagram of moiré metalens. It consists of two metalenses, as shown in Fig. 1(a), and their total focal length can be tuned by mutual rotation from negative to positive. Each of the moiré metalens pair is designed with transmission function distributions in polar coordinates (r, ϕ): 32

Design and Fabrication of Moiré Metalens
where a is a constant. The phase distribution of the first lens T 1 is presented in Fig. 1 With mutual rotation angle of θ, the joint transmission function can be expressed as: This equation is similar to that for spherical lenses under paraxial approximation, T = exp(iπr 2 /f λ), where f and λ represent the focal length and the wavelength, respectively.
Therefore, by adjusting the constant a to satisfy equation f −1 = aθλ/π, the optical power f −1 becomes proportional to θ. Note that the round function is used to avoid the sectoring effect. 32 Figures 1(c-e) presents the phase distributions of the second metalens with mutual rotation angles of +30 • , +60 • , and -60 • , respectively, together with the total phase distributions superimposed with the first metalens ( Fig. 1(b)). Positive rotation angles create total phase distribution similar to convex Fresnel lenses, whereas and larger rotation angles give rise to higher phase gradient, which corresponds to short focal length and higher optical power, as presented in Figs. 1(c) and (d). In contrast, as presented in Fig. 1(e), a negative rotation angle creates concave-like phase distribution. We show that adopting moiré metalens leads to focal lengths with a wide tuning range from negative to positive. We adopted a-Si octagonal pillar HCTAs as polarization-insensitive meta-atoms. 25 Electromagnetic simulation was conducted using the commercially available finite element software, COMSOL Multiphysics ver. 5.1 (COMSOL Inc., USA), as presented in Fig. 2. A schematic diagram of the simulation setup is presented in Figure 2(a). An octagonal pillar with a period of 400 nm was situated on the silica glass substrate, with air as the surrounding medium (not shown). X-polarized light at a wavelength of 900 nm was incident from the glass bottom side, whereas the transmittance and phase delay of the pillar was evaluated at the output port (air side). The perfect electric conductor (PEC) boundary condition was applied to the ±x boundaries, whereas the Froquet periodic condition was applied to other boundaries. An a-Si material parameter was used as an Si pillar. 38 Figure 2(b) presents the results of parameter sweep for transmittance and phase/2π, which is obtained by changing the pillar width and height in 10-nm steps. We found that the region around height> 380 nm achieves both high transmittance and full 2π phase coverage. Figure 2(c) presents the x−component of electric field, E x distribution at a pillar height of 400 nm, and widths of 90, 240, and 300 nm, respectively. It is clearly shown that the phase difference between the 240, 300, and 90 nm pillars at the output port (top of the image) correspond to π and 2π, respectively. Therefore, considering fabrication process limitations for high-aspect-ratio structures, we adopted pillars with 400-nm height as a meta atoms.
We used these octagonal pillars as meta-atoms to design moiré metalens with a diameter We fabricated a metalens on a quartz glass substrate, sputter-depositing an amorphous silicon film with a thickness of 400 nm. A lens pattern with a diameter of 2 mm was drawn by  electron beam lithography using character projection (CP) (Advantest F7000-S, Advantest, Japan), employing dedicated octagonal stencil masks to achieve high-throughput drawing.
The GDSII CAD file descried above was converted into a CP shot data with proximity effect compensation. After the resist development, an aluminum mask was patterned on the a-Si film through lift-off process. Si pillars were formed by RIE. Finally, the aluminum mask was removed using a solution of phosphoric, nitric, and acetic acids mixture.   respectively. The data points in Fig. 3(b) are based on the measured widths of pillars.

Results and Discussion
The phase delays of the pillars were measured by the interference method, 39 using a laser diode and a microspectroscopic system (Techno Synergy, DF-1037). Figure 3(c) presents the SEM image of the lens, and 3(d) shows its close-up to the center, which is observed at 45 • angles. We fabricated width-distributed pillars. However, the pillars are cylindrical instead of octagonal due to fabrication limitations.
The phase delay rapidly increases for both the simulation and experiment as the width increases, as presented in Fig. 3(b). Though the experimental transmittance is lower than that of the simulation, the position of the dips is similar. The disparity between the experiment and simulation can be attributed to fabrication error, including the rounded shape of the pillar top presented in 3(d).  almost equivalent to the opening of the iris. Figure 4(d) presents the magnified image of (c).
The mutual rotation angle of θ = 30 deg. are clearly seen.
Using the metalens position x from the image plane and target position l as shown in 4(a) under the certain mutual rotation angle θ, the focal length f (θ) of the moiré metalens was calculated using the following equation, Our results reveal a wide range of tunability, and this is the first report on the focal length tuning from negative to positive. In fact, metalens can be tuned both as a convex and concave lens. Though we adopted alignment and adhesion processes in this paper, MEMSactuated rotational stages have been reported, whereas microrotational varifocal metalens can be realized by proper integration with actuators. In this work, we employed a 2-mmdiameter lens, which can be extended to a cm-scale in principle due to the high throughput of CP-based EB lithography.
In conclusion, we experimentally demonstrated a rotational varifocal metalens or moiré metalens. The metalens was designed using polarization-insensitive a-Si HCTAs as meta- atoms. We investigated the fabricated metalens at a wavelength of 900 nm and demonstrated focal length tuning at the ranges from −∞ to −1.73 mm and from +1.73 mm to +∞ at a mutual rotation between ±90 • , which corresponds to an optical power tuning between ±578 m −1 .

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The authors declare no competing financial interest.