Reinforced design method for Moiré metalens with large spacing

Although metalens has attracted many research interests for its advantages of light weight, ultrathin size, and high design freedom in realizing achromatic and aberration-free optical devices, it still lacks adjustability in zoomable optical systems. Moiré metalens, which consists of two cascaded metasurface layers, can realize large focus tuning range by the mutual rotation of the two layers, and becomes a possible solution to realize real application of reconfigurable metalenses. However, due to the spacing between the two metasurface layers, it suffers from aberration caused by diffraction, leading to a dramatically decreased efficiency with the spacing. In this paper, we propose a reinforced design method for moiré metalenses with large spacing based on diffraction optics. Simulation results demonstrate that at the wavelength of 810 nm, when the spacing of the two metasurfaces is 10λ, the focusing efficiency of the reinforced moiré metalens is 3.4 times larger than the traditional moiré metalens. Furthermore, in order to consider the situation that the spacing between the two metasurfaces cannot be controlled precisely, we also propose a reinforced design method for multiplex spacings, which can make the device maintain a high focusing efficiency (3 times larger than the traditional moiré metalens) for the spacing in a range of 6λ∼10λ. The new design method is anticipated to be applied in realizing tunable metalenses in integrated continuously zoomable optical systems.


Supplementary Material
Section S1.Calculation method of moiré metalenses (1) Parameter setting of the sampling points in theoretical calculations For ease of calculation, in the theoretical calculations, we assume that the moiré metalens is composed of two layers of square phase plates with the same size.We set the wavelength of the incident light as 810nm, the size of each metasurface is 11.6μm *11.6μm, the distance p between two adjacent sampling points is 400nm, the number of the sampling points of each plate is 30*30.The lattice is shown in Fig. S1.When there is a rotation between the two metasurfaces, the structures of the two layers are not rotated, but the phase distribution of metasurface 1 (MS1) is changed.To calculate the electric field of the focus, we first obtain the complex amplitude of MS1: U1.By Eq. ( 4), we then obtain the complex amplitude of the input plane of MS2: U2, thus the far field can be calculated by Eq. ( 4), based on the complex amplitude exp{j[φ2+arg(U2)]}, in which φ2 is the phase modulation of MS2.(2) Parameter setting of the nano-rods in simulations For simulation study, we use a commercial simulation software--FDTD Solutions to simulate the structure of the moiré metalens.Unlike using square lattices and metasurfaces in the theoretical calculation, we use circular metasurface structures in the simulation in order to realize a good alignment when there is a rotation between the two metasurfaces.The schematic diagram of each metasurface is shown in Fig. S2(a).The diameter D of each metasurface is 12.8μm and the nano-rods are arranged in concentric circles on the substrate.The ring spacing is 400nm, and the height of each nano-rod is 600nm.The top view of one metasurface and the side view of the moiré metalens are illustrated in Fig. S2 (a) and Fig. S2 (b), respectively.There are 17 rings of nanorods 30 sampling points from the center to the outermost circle.The radius of each ring is proportional to n, in accordance with the law R=400nm *(n-1), and the number of the nanorods on each circle is also proportional to the circle number n, which conforms to the law 6*(n-1).Therefore, the lattice constant p between two adjacent nano-rods is close to 400nm.We chose amorphous silicon as the material of the nanorods, which has a refractive index of 3.52459339.The thickness of the substrate is 1μm and the material is SiO2, with the refractive index of 1.45315.By sweeping radius of a nanorod in the unit cell in Fig. S2 (c), we found that when the radius increases from 20nm to 115nm, the range of its phase modulation covers 0 to 2π with high transmittance at 810 nm (mostly more than 90%) (the result is shown in Ref [25] of the manuscript).For the simulation of the whole moiré metalens, the incident light with a wavelength of 810nm is incident along the positive direction of the z-axis, and the illumination range covers the whole aperture of the moiré metalens.We set the boundary conditions of the x, y, and z directions as perfect matched layers (PML).In the subsequent calculations of the moiré metalenses with spacings, we divide the farfield electric field intensity on the focal plane by the maximum intensity Emax of the focus corresponding to the ideal moiré metalens with no spacing.The relative intensity curve with respect to the ideal moiré metalens can be obtained. (

2) Simulations
In the simulations, since it is impossible to realize the numerical computation of doublelayer moiré metalens with no spacing, we assume that an ideal moiré metalens is only a one-layered metasurface, whose phase modulation is calculated by the well know equation The single-layered metalens used in the simulation software is shown in Fig. S4 (a).
Then we calculate the far-field electric field intensity distribution on the x-z plane and the focal plane according to the phase distribution on the monitor.Find the z coordinate fs of the point with the highest intensity on the x-z plane as the focal length, and plot the electric field intensity distribution of the focal plane of z=fs and record the maximum intensity value Emax', which is 1.922526666549408e+02 in our simulation.
As for the calculation of the moiré metalens with a spacing, which is shown in Fig. S4(b), we divide the electric field intensity on the focal plane by Emax'.Finally, the relative intensity curve with respect to the ideal single-layered metalens can be obtained.

Section S3. Calculation method of focusing efficiency (1) Theoretical calculation
In the first step, we need to calculate its full width at half maximum (FWHM) based on the relative electric field intensity curve.The specific method is to find the peak intensity point M in the relative intensity curve, and record its electric field intensity value EM and the corresponding x-coordinate xM.On both sides of M, we look for two points with half the intensity of EM on the curve, and record their x-coordinate as x1 and x2, respectively.Then we can obtain the value of FWHM of the curve, which is |x2-x1|.After obtaining the FWHM, we can calculate the focus efficiency.The specific method is to draw a square region on the focal plane, with the point of maximum intensity as the center and 3 times the FWHM as the side length.Then sum the whole electric field intensity in this area as EA, and divide by the sum of the intensity Etotal on the entire focal plane, i.e., Efficiency=EA/Etotal.
(2) Simulations Similar to the theoretical calculation, we first calculate the far-field electric intensity distribution and the FWHM of the electric field intensity curve according to the monitor.The x-z view of the structure is shown in Fig. S6(a).Then we draw a square region on the focal plane with the point of maximum intensity as the center and 3 times the FWHM as the side length, and sum the whole electric field intensity in the region as EB.
When calculating the total incident energy, we record the sum of the electric field intensity on the monitor as the total incident light energy Etotal', without all the structures, as shown in Fig. S6(b).The focusing efficiency is expressed as EB/ Etotal'.(1)

Section S5. Focal tuning for a general MML and a reinforced MML
From the text, we can know that, theoretically, the focusing range of the moiré metalens can be altered from infinity to close to zero as the mutual rotation angle of the doublet changes.However, no one has discussed the relationship between the focal tuning range and the spacing of the moiré metalens.Therefore, we selected the moiré metalens with three different spacings of 0.1λ, 5λ, and 10λ, and changed the mutual rotation angle to investigate its focal tuning range by means of theoretical calculation and simulation.

Fig. S1 .
Fig. S1.Schematic diagram of the sampling points on each metasurface for theoretical calculations.

Fig
Fig. S2.A schematic diagram of the structure of moiré metalens in the simulation.(a) Top view of one-layer metasurface of the moiré metalens.(b) Side view of the moiré metalens.(c) Swept unit cell.

Fig. S3 .
Fig. S3.Schematic diagram of the normalization method for theoretical calculation of electric field intensity of moiré metalens.(a) Electric field intensity distribution of the x-z plane.(b) Electric field intensity distribution of the focal plane.(c) The normalized electric field intensity distribution curve.

Fig. S4 .
Fig. S4.Structures in FDTD Solutions software.(a) The structure of an ideal single-layered

Fig. S5 .
Fig. S5.Schematic diagram of the theoretical calculation method of focusing efficiency.(a) The electric field intensity distribution curve on the focal plane of the moiré metalens.(b) Electric field intensity distribution on the focal plane and the calculation range of focusing efficiency.

Fig. S6 .
Fig. S6.The structures in the FDTD Solutions software.(a) A schematic diagram of a moiré metalens with a certain spacing.(b) Schematic diagram after removing all structures.
Fig. S7.Electric field intensity distributions of the x-z plane of the general moiré metalenses under different mutual rotation angles (d=0.1λ).

Fig. S8 .
Fig. S8.Electric field intensity distributions of the x-z plane of the reinforced moiré metalenses under different mutual rotation angles (d=0.1λ).

Fig. S10 .
Fig. S10.Electric field intensity distributions of the x-z plane of the reinforced moiré metalenses under different mutual rotation angles(d=5λ).

Fig. S11 .
Fig. S11.Electric field intensity distributions of the x-z plane of the general moiré metalenses under different mutual rotation angles(d=10λ).

Fig. S12 .
Fig. S12.Electric field intensity distributions of the x-z plane of the reinforced moiré metalenses under different mutual rotation angles(d=10λ).