# Dielectric Metalens for Superoscillatory Focusing Based on High-Order Angular Bessel Function

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Method

_{10}(·) is the 10th order Bessel function of the first kind; sgn(·) denotes the sign function; k

_{r}is the transverse wave number; f is the focal length of the tightly focusing lens; and θ is the discrete opening angle in the focusing model. Different from the traditional half-band principle that has uniform width [12], the width of the half-band here is dependent on the angular function of Equation (1).

_{r}, we numerically calculated the focal field according to the Richards–Wolf vectorial diffraction integral [44,45], whose electric components are expressed as:

**P**(θ,φ) is:

_{max}= NA, A(θ,φ) corresponds to the uniform complex amplitude of input beam under the modulation of phase Φ

_{0}(θ), a = cos(φ) and b = sin(φ) are the x- and y-polarized components corresponding to the radial vector beams. Figure 2a shows the modulation phase of such a diffractive optical element when k

_{r}= 0.43k

_{0}with k

_{0}= 2π/λ the wave number, and the interval of θ is 0.1 radian. Figure 2b and 2c exhibit the numerically calculated intensity distributions of the total field and its longitudinal component by using the Richards–Wolf vector diffraction theory. The wavelength is 633 nm and the NA of the focusing lens is 0.9. Figure 2d shows the intensity distribution of the total light field along the radial direction. Clearly, the full width at half maximum (FWHM) of the central hotspot is about 0.29λ/NA, which is significantly smaller than the criterion of 0.38λ/NA [12], indicating the superoscillation phenomenon. Figure 2e illustrates the intensity distributions of focal fields corresponding to the metalens designed from 20th and 50th order Bessel functions of the first kind with k

_{r}= 0.68k

_{0}and 0.66k

_{0}, respectively. It can be seen that for high-order Bessel functions, superoscillation can be generated by finding a suitable k

_{r}. Here, we use the 10th-order Bessel function with a different k

_{r}to demonstrate our approach.

_{0}($\sqrt{{r}^{2}+{f}^{2}}$− f), and then generate the combined modulation phase of the metalens, whose profile is shown in Figure 2f. In order to realize polarization modulation, we select geometries whose phase retardation of two orthogonal eigenstates is π, namely, half-wave retardant meta-atoms, as the meta-atom, under the premise of the incidence of linear polarization. For the case of horizontally polarized incidence, the metalens thus can be equivalent to a half-order q-plate with a rotation angle of local meta-atom characterized as α = φ/2.

_{2}as high refractive index materials and substrates to fabricate the metalens, whose meta-atom is schematically shown in Figure 3a. As shown, the meta-atom consists of a poly-Si rectangle nanopillar deposited on the glass substrate. The height and period of the nanopillar are H = 570 nm and P = 450 nm, respectively, and the refractive index is n = 3.36329 + 0.01162i. We calculated the response of the meta-atom by using a finite-difference time-domain (Lumerical software, Ansys Canada Ltd., Vancouver, Canada) simulation and selected 16 geometric configurations meeting the polarization and phase modulation conditions, that is, two linearly polarized eigenstates (E

_{x}and E

_{y}) that keep a π phase retardation difference, i.e., δ = |φ

_{x}– φ

_{y}| = π, while the propagation phase increases linearly in an interval of 2π, i.e., ${\phi}_{0}^{n}$ = (${\phi}_{x}^{n}$+ ${\phi}_{y}^{n}$)/2 = nπ/16. Figure 3b depicts the transmission amplitude, propagation phase φ

_{0}, and retardation difference δ of these 16 configurations. In addition, for the central singularity, we picked out a configuration with near zeroth transmission amplitudes, i.e., E

_{x}≈ E

_{y}≈ 0. Its geometric parameters are L = 276 nm and W = 234 nm. We fabricated the metalens with a transmission-type configuration by using standard electron-beam lithography and inductively coupled plasma etching [46,47]. Figure 3c shows the scanning electron microscope images of the metalens and its local structure. The sample is composed of 800 × 800 elements with a lattice constant of 450 nm along the x- and y-axes. The experiment is carried out with the setup shown in Figure 3d. A linearly polarized beam from the He-Ne laser is converted into a horizontal one after passing through the half-wave plate, and then a normal incident into the metalens. The superoscillating focal field is generated at the focal plane of the metalens, we used a microscopic measurement system consisting of a 100× objective lens (Mitutoyo, NA = 0.9), tube-lens, and a CCD camera (DMK, 23U445) to observe the focal field.

## 3. Results

_{r}= 0.08k

_{0}, 0.29k

_{0}, and 0.43k

_{0}, respectively. The simulated and measured intensity distributions of the focal fields are shown in Figure 5a–c. Figure 5d gives the line-scan intensity profiles corresponding to the simulated (red) and measured (black) results indicated by the dashed lines. All results are obtained with the same incident condition and normalized by the maximum intensity. From these simulated results, it can be seen that the central hotspots in three cases both present the superoscillation phenomenon, as shown in Figure 5d. Among them, for the case of k

_{r}= 0.29k

_{0}, the FWHM of the superoscillating hotspot can be reduced to 0.25λ/NA. However, the relative intensity of this hotspot with the sideband is the smallest. In this case of k

_{r}= 0.43k

_{0}, although the resulting hotspot has the largest field of view and the lowest relative intensity of the nearest sideband, the size of the central hot spot is the largest, with a magnitude of about 0.29λ/NA. These results illustrate that the adopted angular Bessel modulation method has high applicability, and can generate superoscillating focal fields for various requirements by optimizing its parameters. Comparing the theoretical and experimental results, one can find that the experimental results are basically consistent with the theoretical predictions. In practice, the smallest size of the hotspot is about 0.32λ/NA when k

_{r}= 0.29k

_{0}, which is still smaller than the superoscillatory criterion of 0.38λ/NA, indicating the superoscillatory focusing capability of this type of metalens. In addition, we would like to note that the sizes of the measured focal spots are greater than these theoretical ones. The reason is mainly due to the non-uniform transmission and direct transmission components due to the phase delay errors caused by the fabrication errors and imperfections of the chosen geometry, as shown in Figure 4.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the superoscillatory focusing metalens. The arrows depict the instantaneous directions of electric component of the light field input and output from the metalens.

**Figure 2.**Generation of superoscillatory focal field from the tightly focusing of vector light field. (

**a**) Binary phase of the diffractive optical element designed from angular Bessel function with k

_{r}= 0.43k and NA = 0.9; (

**b**) Simulated total intensity distribution of the focal field; (

**c**) The intensity distribution of the longitudinal component I

_{z}at the focal plane; (

**d**) Intensity distribution along the radial direction. (

**e**) Intensity distributions of focal fields corresponding to the metalens designed from 20th and 50th order Bessel functions of the first kind with k

_{r}= 0.68k

_{0}and 0.66k

_{0}, respectively; (

**f**) Modulation phase of the metalens generated from the combination of tightly focusing phase and binary phase in 2a.

**Figure 3.**Design and characterization of the metalens. (

**a**) Schematic illustration of an element consisting of a poly-Si nanopillar and glass substrate. The geometric parameters of the element are denoted as H (height), L (length), W (width), and P (period), the rotation angle is denoted as α; (

**b**) Transmission amplitude (E

_{x}and E

_{y}) and phase retardation [δ = φ

_{x}− φ

_{y}and φ

_{0}= (φ

_{x}+ φ

_{y})/2] of eigenstates within 16 selected elements; (

**c**) Scanning electron microscope images of the metalens and its local structure. The sample is composed of 800 × 800 elements with a lattice constant of 450 nm along x- and y-axes. The scale bar is 500 nm; (

**d**) Sketch of experimental setup, HWP: half-wave plate.

**Figure 4.**Intensity distribution of light field transmitted from the metalens in the case of horizontally polarized beam incidence. (

**a**) Total intensity; (

**b**) Horizontal component; (

**c**) Vertical component. The arrows depict the orientation of polarization analyzer.

**Figure 5.**Experimental results of three superoscillatory metalenses. (

**a**–

**c**) Intensity distributions of the simulated and measured focal fields generated by metalenses with parameters of k

_{r}= 0.08k

_{0}, 0.29k

_{0}, and 0.43k

_{0}; (

**d**) Normalized line-scan intensity profiles at the focal plane (indicated by dashed lines): simulation (red) and experiment (black).

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Fan, X.; Huang, Y.; Guo, X.; Zhou, L.; Li, P.; Zhao, J.
Dielectric Metalens for Superoscillatory Focusing Based on High-Order Angular Bessel Function. *Nanomaterials* **2022**, *12*, 3485.
https://doi.org/10.3390/nano12193485

**AMA Style**

Li Y, Fan X, Huang Y, Guo X, Zhou L, Li P, Zhao J.
Dielectric Metalens for Superoscillatory Focusing Based on High-Order Angular Bessel Function. *Nanomaterials*. 2022; 12(19):3485.
https://doi.org/10.3390/nano12193485

**Chicago/Turabian Style**

Li, Yu, Xinhao Fan, Yunfeng Huang, Xuyue Guo, Liang Zhou, Peng Li, and Jianlin Zhao.
2022. "Dielectric Metalens for Superoscillatory Focusing Based on High-Order Angular Bessel Function" *Nanomaterials* 12, no. 19: 3485.
https://doi.org/10.3390/nano12193485