# Broadband Achromatic Metasurfaces for Longwave Infrared Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}metalens in the visible region; Wang et al. [32] reported a broadband achromatic Au metalens and Au metasurface grating in the near-infrared range; Ou et al. [33] reported broadband achromatic focusing vortex generators based on all-silicon platforms in the mid-infrared regime.

_{3}, offer new opportunities for infrared metaphotonics [41,42,43,44,45]. In this study, we propose a general method of implementing LWIR achromatic metasurfaces based on all-germanium platforms. To demonstrate the validity of the proposed method, a broadband achromatic metalens (BAML) and a broadband achromatic metasurface grating (BAMG) operating in the LWIR range are presented.

## 2. Principles and Design

#### 2.1. Chromatic Dispersion of a Metasurface

#### 2.1.1. Chromatic Dispersion of a Metasurface Grating

#### 2.1.2. Dispersion of a Metalens

#### 2.2. Achromatic Metasurface Design

_{max}) to an arbitrary point (r = R) on the metalens, such that the phase at r = R is of the form

_{max}). In the general case, $\phi \left({R}_{\mathrm{max}},\omega \right)$ can be Taylor expanded around the central frequency, as

## 3. Results and Discussion

#### 3.1. Meta-Atom

#### 3.2. Broadband Achromatic Metalens

#### 3.3. Broadband Achromatic Metasurface Grating

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Dispersion properties of chromatic metasurfaces and schematic of the achromatic metalens. (

**a**) Angular dispersion of the metasurface grating; (

**b**) dispersion of polychromatic light incident at the edge of the chromatic metalens; (

**c**,

**d**) ray tracing of light at different coordinates; (

**e**) schematic of the achromatic metalens in the LWIR range. In (

**a**) through (

**d**), the red ray represents light at ${\lambda}_{\mathrm{max}}$, the yellow ray represents light at ${\lambda}_{0}$, and the blue ray represents light at ${\lambda}_{\mathrm{min}}$. On the right is a schematic of meta-atoms, where H is the height of the nanopillar, and p is the period of the nanopillar. All meta-atoms have the same height and period.

**Figure 2.**Simulation results for three selected meta-atoms. (

**a**) PCR and phase spectrum for one nanofin, geometry parameters in the inset picture: L1 = 4.4 μm, W1 = 2.2 μm; (

**b**) PCR and phase spectrum for two nanofins, geometry parameters in the inset picture: L1 = 4.3 μm, W1 = 1.3 μm, g1 = 0.5 μm, L2 = 1.9 μm, W2 = 1.9 μm; (

**c**) PCR and phase spectrum for three nanofins, geometry parameters in the inset picture: L1 = 1.3 μm, W1 = 1 μm, g1 = 0.5 μm, L2 = 2.1 μm, W2 = 1 μm, L3 = 1.1 μm, W3 = 1 μm; (

**d**) top and side views of normalized H field intensities for the three meta-atoms.

**Figure 3.**Layout and simulation results of intensity distributions of the metalenses: (

**a**) layout of the broadband achromatic metalens, scale bar is 20 μm; (

**b**) normalized intensity distribution in the x–z plane for the broadband achromatic metalens (left) and PB metalens (right) at a selected wavelength; (

**c**) focal length as a function of the wavelength for the achromatic and PB metalenses, with the theoretically predicted focal length calculated as $f=\frac{{\lambda}_{0}{f}_{0}}{\lambda}$ for the PB metalens; (

**d**,

**f**) intensity distributions at the focal plane for the broadband achromatic metalens and PB metalens, respectively; (

**e**,

**g**) horizontal cuts (red dashed curves) across the focal spots in (

**d**,

**f**) compared with an ideal Airy spot (black curves).

**Figure 4.**Simulation results of performance characterization of the broadband achromatic metalens. (

**a**) FWHM for the broadband achromatic metalens, where the blue dashed line represents the theoretical FWHM of the ideal Airy disk; (

**b**) Strehl ratio for the achromatic metalens, where the blue dashed line represents the diffraction-limited criterion; (

**c**) intensity efficiency of the broadband achromatic metalens.

**Figure 5.**Schematic and deflection angles of the metasurface gratings: (

**a**,

**b**) schematic of the broadband achromatic and PB metasurface gratings, respectively; (

**c**,

**d**) far-field intensities of scattered light vs. angle of refraction for the broadband achromatic and PB metasurface gratings, respectively, where the stars in (

**c**,

**d**) represent the theoretically predicted deflection angles; (

**e**) deflection angles of the broadband achromatic and PB metasurface gratings, where the deflection angle is defined as the angle corresponding to the maximum scattering intensities at the different wavelengths.

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**MDPI and ACS Style**

Song, N.; Xu, N.; Shan, D.; Zhao, Y.; Gao, J.; Tang, Y.; Sun, Q.; Chen, X.; Wang, Y.; Feng, X. Broadband Achromatic Metasurfaces for Longwave Infrared Applications. *Nanomaterials* **2021**, *11*, 2760.
https://doi.org/10.3390/nano11102760

**AMA Style**

Song N, Xu N, Shan D, Zhao Y, Gao J, Tang Y, Sun Q, Chen X, Wang Y, Feng X. Broadband Achromatic Metasurfaces for Longwave Infrared Applications. *Nanomaterials*. 2021; 11(10):2760.
https://doi.org/10.3390/nano11102760

**Chicago/Turabian Style**

Song, Naitao, Nianxi Xu, Dongzhi Shan, Yuanhang Zhao, Jinsong Gao, Yang Tang, Qiao Sun, Xin Chen, Yansong Wang, and Xiaoguo Feng. 2021. "Broadband Achromatic Metasurfaces for Longwave Infrared Applications" *Nanomaterials* 11, no. 10: 2760.
https://doi.org/10.3390/nano11102760