High-NA Achromatic Metalenses by Inverse Design

TL;DR

This paper employs inverse design to create broadband, achromatic metalenses with high numerical apertures, surpassing traditional unit-cell methods in efficiency and achieving the first high-NA achromatic focusing.

Contribution

It introduces a novel inverse design approach for high-NA achromatic metalenses, overcoming limitations of standard unit-cell techniques and establishing new efficiency benchmarks.

Findings

Achieves high theoretical efficiencies at low NA.

First to demonstrate achromatic high-NA focusing at NA 0.9 and 0.99.

Provides computational bounds on unit-cell approach efficiencies.

Abstract

We use inverse design to discover metalens structures that exhibit broadband, achromatic focusing across low, moderate, and high numerical apertures. We show that standard unit-cell approaches cannot achieve high-efficiency high-NA focusing, even at a single frequency, due to the incompleteness of the unit-cell basis, and we provide computational upper bounds on their maximum efficiencies. At low NA, our devices exhibit the highest theoretical efficiencies to date. At high NA -- of 0.9 with translation-invariant films and of 0.99 with "freeform" structures -- our designs are the first to exhibit achromatic high-NA focusing.

Tables

Table 1. Table 1 : Constant-z geometry parameters of every grid point (25-nm-long) of the nine metasurfaces in Fig. 1(c). The grid number starts from the left side of the metasurface. Value of one indicates TiO 2 and zero indicates air.

Grid Number	NA=0.1	NA=0.2	NA=0.3	NA=0.4	NA=0.5	NA=0.6	NA=0.7	NA=0.8	NA=0.9
1	0	0	0	1	1	1	1	1	0
2	1	0	0	1	1	1	0	0	1
3	0	0	0	1	0	0	0	1	1
4	0	0	1	1	0	0	1	1	1
5	1	1	1	1	1	1	1	0	1
6	0	1	1	1	0	1	1	0	1
7	1	1	1	1	1	1	1	0	1
8	1	0	1	1	1	0	0	0	1
9	1	0	0	1	0	1	0	0	1
10	1	1	1	1	1	0	0	0	0
11	1	1	1	1	1	1	1	0	1
12	1	1	1	1	1	1	1	1	1
13	1	1	1	1	0	1	0	1	1
14	1	1	0	1	0	1	0	1	1
15	1	0	0	1	1	1	0	1	1
16	1	0	0	0	0	1	0	1	0
17	0	1	0	0	0	0	0	1	0
18	0	1	1	1	0	1	1	1	1
19	0	1	0	1	0	0	1	0	1
20	0	1	0	1	0	0	1	1	1
21	0	0	0	1	0	0	1	0	0
22	0	1	1	1	1	1	1	1	0
23	1	1	0	1	1	0	0	1	0
24	1	1	0	1	1	1	1	1	1
25	1	1	0	1	1	1	0	1	0
26	0	0	1	1	1	0	1	1	1
27	0	1	1	0	1	0	0	1	0
28	0	1	0	1	1	0	1	1	0
29	0	1	1	0	1	0	1	1	0
30	0	1	0	1	1	0	0	1	0
31	0	1	0	1	0	0	0	1	0
32	0	0	0	0	1	1	1	1	0
33	0	1	0	0	0	0	1	0	0
34	1	0	0	0	1	1	1	1	0
35	1	1	1	1	0	0	1	0	0
36	1	1	1	1	0	1	1	1	1
37	0	1	1	1	1	0	1	0	1
38	0	1	1	1	1	1	1	0	1
39	0	1	0	0	1	1	0	0	1
40	0	0	0	0	1	0	1	0	0
41	0	1	1	0	1	0	1	0	1
42	1	1	1	0	1	0	1	0	1
43	0	1	0	0	1	0	1	0	1
44	1	1	1	0	1	0	1	1	1
45	1	1	1	0	1	0	1	1	1
46	1	0	1	0	1	0	0	0	0
47	0	1	0	0	1	0	0	1	0
48	0	1	0	0	0	0	0	1	0
49	0	1	1	0	1	0	0	1	1
50	0	1	1	0	1	1	0	0	1
51	0	1	1	0	1	0	0	0	1
52	1	1	0	0	0	1	1	0	1
53	1	1	1	1	1	1	1	1	1
54	0	1	1	0	1	0	0	1	1
55	1	0	1	0	0	0	0	0	1
56	1	1	1	0	0	0	0	1	0
57	1	1	0	0	0	1	1	1	1
58	0	1	0	0	0	0	1	0	1
59	0	1	1	0	0	1	1	1	1
60	0	1	1	0	0	1	0	1	0

Equations

$$\underset{c_i}{\min }F(c_i)=\underset{c_i}{\min }{\Vert {\mathbf{E}}_{\mathrm{tar}}-\mathbf{E}(\mathbf{x})\Vert }_2^2=\underset{c_i}{\min }{\Vert {\mathbf{E}}_{\mathrm{tar}}-\sum _i{c}_i{\mathbf{E}}_i(\mathbf{x})\Vert }_2^2.$$

$$c_i=\int {\mathbf{E}}_i^{\ast }\cdot {\mathbf{E}}_{\mathrm{tar}}.$$

$$\frac{\partial F}{\partial {c_i}^{\ast }}$$

$$=-\int {\mathbf{E}}_i^{\ast }\cdot {\mathbf{E}}_{\mathrm{tar}}+c_i,$$

Full text

1. I Material-dispersion simulation verification
2. II Achromatic metalenses with a glass substrate
3. III Least-squares optimal coefficients
4. IV Summary of optimized metalenses for various numerical apertures using translation invariant geometry
5. V Summary of optimized metalenses for various numerical apertures using freeform geometry
6. VI Analysis of larger device size
7. VII Frequency sensitivity study
8. VIII Metalens design in 3D

I Material-dispersion simulation verification

In this work, Finite Difference Time Domain (FDTD) method is used to solve a full Maxwell equation Oskooi et al. (2010); Taflove et al. (2013). FDTD can cover a wide range of frequencies with a single time domain simulation. However, it requires time-domain modeling of material dispersion. We model the dispersion of TiO2 material with Drude model and validate the material modeling with a theoretical calculation of Fabry-Perot oscillations in a 1D dielectric slab Orfanidis (2002) as shown in Fig. 1.

II Achromatic metalenses with a glass substrate

Metalenses are generally fabricated on transparent substrates. Here, we demonstrate achromatic metalenses with SiO2 substrate (n = 1.5). The incidence power is calculated within the glass substrate. As shown in Fig. 2 (a), the optimized device has 250 nm thickness, 12.5 $\mu$m width, and 8.2 $\mu$m focal length, which corresponds to NA = 0.6. As shown in Fig. 2 (b), a calculated average focusing efficiency is 32 % over visible wavelengths (450 – 700 nm) while the highest efficiency is 39 %. The focusing efficiency is defined as the ratio of the E-field intensity within the first minimum point and the incidence power Chen et al. (2018). As shown in Fig. 2(c), the calculated full-width half-maximum (FWHM) over visible wavelength is very close to the diffraction limited FWHM except for the longer wavelength. Figure. 2(d) shows the optimized structure which is feasible to existing lithography techniques Owa and Nagasaka (2004); Khorasaninejad et al. (2016). Figure. 2(e) shows E-field intensity profiles at the focal plane. Except for the 700nm wavelength, they are very close to diffraction limited focal spots.

III Least-squares optimal coefficients

In the main text, we solve for optimal unit-cell coefficients to achieve a desired field distribution. In the setup of the problem, there is a given field $\bm{E}(\bm{x})$ that can be decomposed into an orthonormal basis of functions $\bm{E}_{i}(\bm{x})$ and coefficients $c_{i}$, and a target field $\bm{E}_{\rm tar}(\bm{x})$ that one would like $\bm{E}(\bm{x})$ to most closely approximately, i.e., there is a functional $F$ of the coefficients $c_{i}$ to be minimized:

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In the main text we stated that the optimal coefficients are given by

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This statement is a well-known result for linear least squares problems. Here, we provide a short proof in this context.

For any given $i$, we can compute the value of $c_{i}$ for which $\partial F/\partial c_{i}$ equals 0. In fact, we must carefully consider both the real and imaginary parts of $c_{i}$; instead, we can use the complex CR calculus KreutzDelgado2009, whereby $c_{i}$ and its conjugate ${c_{i}}^{*}$ are formally considered independent variables, and the derivatives of $F$ with respect to each must be zero. Since $F$ is real-valued the two derivatives contain redundant information, and only one must be set to zero; for simplicity in the ultimately derivation, we choose the derivative with respect to ${c_{i}}^{*}$:

[TABLE]

where in the latter expression we have used the orthonormality of the basis functions, $\int{\bm{E}}^{*}_{j}\cdot\bm{E}_{i}=\delta_{ij}$. Thus the condition for zero first derivative requires Eq. 2 to hold; as the only extremum it must also be the minimum, thus proving that it is the solution to the minimization problem.

IV Summary of optimized metalenses for various numerical apertures using translation invariant geometry

We demonstrated low to high NA metalenses in the main text. Here, we attach detailed optimized structures and efficiencies at the visible wavelength. As shown in Fig. 3, the optimized structures, in the macroscopic view, have a gradually varying TiO2 filling ratio over device radial direction. The speed of this variation increases over increasing NA. Figure 4 shows achieved efficiencies for various NA (0.1 – 0.9) using translation invariant geometry. The detailed geometric parameters are attached at the end of this supplementary material.

V Summary of optimized metalenses for various numerical apertures using freeform geometry

The numerical aperture (NA) versus efficiencies plot is shown in the main text. The optimized geometries and efficiencies data are shown in Figs. 5, 6. It might be hard to figure out any physical insight from the optimized geometries. Again, in the macroscopic view, they seem to have a gradually varying TiO2 filling ratio over device radial direction. The detailed geometric parameters are attached in the separate supplementary text files.

VI Analysis of larger device size

Large area design is a particular problem in the metalens community Kamali et al. (2018). To address this issue, we optimize metalenses with different device widths from 10 $\lambda$ to 60 $\lambda$. NA is fixed to 0.3 and then the device width is adjusted to see the focusing efficiency variation. The optimized structure is translational invariant geometry with the 250-nm-thick TiO2 material. As shown in Fig. 7, the average efficiencies are close to 40 % for different device sizes. There is no sign of significant efficiency drop for larger devices. Note that it was tested up to the 60 $\lambda$ long device where our computational capacity can handle.

VII Frequency sensitivity study

Early metalens works have demonstrated discrete operating wavelengths Aieta et al. (2015); Khorasaninejad et al. (2015, 2016). Then, in order to achieve a pure achromatic operation, all of phase profile, group delay, and group delay dispersion are modeled by the unit cell approach Chen et al. (2018). Also, densely selected wavelengths Wang et al. (2018); Avayu et al. (2017); Mohammad et al. (2018) could also realize a continuous achromatic metalenses. The former should work well for a certain bandwidth centered at the selected wavelength. However, many studies using the latter method do not check the performance over continuous bandwidth. Here, we optimize an achromatic metalens with 10 discrete wavelengths ranging from 450 nm to 700 nm for NA = 0.99, which corresponds to $\Delta\lambda=$ 35 nm. As shown in Fig. 8, the selected ten wavelengths show 31 % average efficiency while the average efficiency calculated with finer $\Delta\lambda$ is only about 18 %. This implies that many existing achromatic metalenses which were evaluated at well separated wavelengths may only work at those wavelengths.

VIII Metalens design in 3D

In the main text, we demonstrated 2D achromatic metalenses due to the limit of our computational resource. Here, we present a relatively small 3D achromatic Metalens to validate our methodology in 3D. The diameter of the lens is 7.5 $\mu$m and thickness is 250 nm. The focal length is 15.0 $\mu$m which corresponds to NA = 0.25. Figure. 9 (a) shows the optimized 3D achromatic metalens in top view. The black region indicates TiO2 and the white region indicates air. The focal length remains nearly constant at three target frequencies (450, 650, 850 nm). The electric-field intensity profiles at the focal plane (z = 15.0 $\mu$m) is shown in Fig. 9(b). Note that this is a preliminary result of designing 3D achromatic metalens. To further improve this result, it may require (1) smaller feature sizes (2) a better angular resolution in order decomposition (3) a good guess on initial geometry parameters. The two formers can be resolved with a greater computing resource while the latter may need theoretical study Angeris et al. (2019).

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