# Design of Multifunctional Tunable Metasurface Assisted by Elastic Substrate

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}O

_{3}) embedded in PDMS substrate. The width w and length l of Si and Al

_{2}O

_{3}is 80 nm and 370 nm, respectively. The thickness of Si and Al

_{2}O

_{3}are fixed as t

_{Si}= 110 nm and t

_{Al2O3}= 50 nm. The thickness of PDMS substrate is denoted as t

_{PDMS}, the periods of the unit cell along the x and y-directions is Px and Py, respectively. Since the Young’s modulus of Si and Al

_{2}O

_{3}are much larger than that of PDMS, Si and Al

_{2}O

_{3}can be regarded as rigid materials during the stretching process. Thus, when the flexible substrate is mechanically stretched, the deformation of PDMS substrate will cause the physical distance between the nanoblocks, that is, the period of the metasurface, to change, while the optical constants and geometry of the nanoblocks remain unchanged. The numerical simulation is performed by three-dimensional finite-difference time-domain (FDTD) models. An x-polarized plane wave at a wavelength of 532 nm is normally incident from the bottom of PDMS substrate. The optical constants of Si and Al

_{2}O

_{3}are taken from Ref [44] and the refractive index of 1.41 is used for PDMS. Perfectly matched layers in the z-direction and periodic boundary conditions along the x- and y-directions are applied in the simulation. In an unstrained state, the periods of the metasurface along the x and y-directions are set as Px

_{0}= 450 nm and Py

_{0}= 120 nm, the thickness of PDMS substrate is t

_{0}= 3300 nm.

## 3. Results

_{0}. The thickness of PDMS is reduced according to its Poisson’s ratio (ν), considered to be 0.5 here [29]. Hence, the thickness of PDMS is t

_{PDMS}= t

_{0}− vεt

_{0}= (1 − ε/2) t

_{0}, as shown in Figure 2b. In the case of normal incidence of x-polarized light, the operation of the metasurface will alter with the change of Px, and this phenomenon can be briefly numerically analyzed by the diffraction formula [45]:

_{t}(n

_{i}) is the refractive index of the refracted (incident) medium, θ

_{t}(θ

_{i}) is the anomalous refraction (incident) angle, λ

_{0}is the free-space wavelength of the light, m is the diffraction order of the metasurface, Px is the period of the unit cell along the x-direction. In our design, the incident and transmitted medium are both air (n

_{t}= n

_{i}= 1), θ

_{i}= 0. Importing these parameters into Equation (1), we can obtain the relationship between the splitting angle and Px as below:

_{0}/P is greater than 0.5 but less than 1, the metasurface will have three diffraction orders: −1, 0, and +1 [46]. Hence, for the operating wavelength of 532 nm, only −1, 0, and +1 diffraction orders exist when the period of the metasurface is in the range from 532 nm to 1064 nm. Accordingly, we analyze the two cases where Px is less than the wavelength λ and Px is greater than the wavelength λ.

_{L}+ T

_{R})/T × 100%, is one of the key factors to measure the beam splitting performance. We define that when the conversion efficiency is higher than 90%, the metasurface exhibits efficient beam splitting performance. Figure 5b depicts the transmission intensity of the three parts of the outgoing light and the total transmission when Px varies from 532 nm to 765 nm (ε = 70%). The elastic limit of PDMS is around 200% [50], so a period Px of 765 nm is achievable by stretching the PDMS substrate. It can be seen that T

_{R}and T

_{L}are always the same at any wavelength. According to Equation (2), the angle between the emergent light beams propagating on the left and right sides of the normal and the normal direction is the same, which we define as the splitting angle θ. Figure 5c depicts the conversion efficiency CE and splitting angle θ of the metasurface when the period Px increases from 532 nm to 765 nm. The conversion efficiency remains higher than 90% within the period Px region from 532 nm to 723 nm, where the splitting angle varies from 90° to 47.4°. Thus, the device operates as an efficient dynamic equal-power splitter (Behavior-ΙΙΙ) when the period Px varies from 532 nm to 723 nm, the corresponding working mechanism is depicted in Figure 5a.

_{PDMS}can be expressed as Px = (1 + ε)Px

_{0}, Py = Py

_{0}− νεPy

_{0}= (1 − ε/2)Py

_{0}, and t

_{PDMS}= t

_{0}− νεt

_{0}= (1 − ε/2)t

_{0}, as shown in Figure 7. For this case, Py

_{0}is set as 130 nm in the unstrained state, while other parameters remain unchanged.

## 4. Discussion

_{+1}and T

_{−1}as a function of Px are shown in Figure 10d. Conversion efficiency remains higher than 90% within the period region from 532 nm to 735 nm, and the corresponding splitting angle varies from 90° to 46.4°. As the Px increases from 532 nm to 615 nm, the split ratio is effectively decreased from 2.38 to 1.03. For larger Px, the tunability of the split ratio is reduced, and the split ratio stabilizes around 1. Thus, in this case, the metasurface can exhibit a transmissive window/reflective mirror/split-ratio-variable splitter transition under external mechanical stretching.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Experimental Feasibility

## References

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**Figure 1.**(

**a**) Schematic of the proposed mechanically reconfigurable metasurface. (

**b**) The three-dimensional structure diagram of a unit cell of the metasurface.

**Figure 2.**The schematic of the metasurface under uniaxially stretching with a strain ratio of ε in (

**a**) x–y top view and (

**b**) x–z side view.

**Figure 3.**The optical response of the metasurface when Px is smaller than the wavelength λ. (

**a**) Transmission and reflection of the metasurface versus period. (

**b**) U as a function of Px. The working mechanism of the design operating as (

**c**) transmissive window and (

**d**) reflective mirror.

**Figure 4.**(

**a**) Reflection as a function of the period Px and wavelength. (

**b**) Reflection as a function of wavelength for the metasurface with Px = 470 nm and Px = 510 nm.

**Figure 5.**The optical response of the metasurface when Px is larger than the wavelength λ. (

**a**) The working mechanism of the design operating as equal-power splitter. (

**b**) Total transmission and intensity of the three emergent beams of the metasurface as a function of period. (

**c**) Conversion efficiency and splitting angle as a function of period.

**Figure 6.**(

**a**) The range of period Px of metasurfaces that can be used as efficient dynamic beam splitters for specific wavelengths. (

**b**) The corresponding splitting angle of metasurfaces with different period Px for specific wavelengths.

**Figure 7.**The schematic of the metasurface under uniaxially stretching with a strain ratio of ε in (

**a**) x–y top view and (

**b**) x–z side view. Both ends of PDMS in the y-direction are not fixed.

**Figure 8.**(

**a**) Transmission and reflection of the metasurface and (

**b**) U versus period when Px is smaller than the wavelength λ. (

**c**) Intensity of the three emergent beams as a function of period. (

**d**) Conversion efficiency and total transmission when Px is greater than the wavelength λ.

**Figure 9.**The schematic of (

**a**) the metasurface composed of an array of L-shaped antennas embedded in a PDMS substrate in x–y top view and (

**b**) unit cell in x–y view.

**Figure 10.**(

**a**) Transmission and reflection and (

**b**) U of the metasurface versus period when Px is smaller than the wavelength λ. (

**c**) Intensity of the three emergent beams of the metasurface as a function of period. (

**d**) Conversion efficiency and the split ratio between T

_{R}and T

_{L}as a function of period.

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**MDPI and ACS Style**

Li, J.; Fan, H.; Ye, H.; Wu, T.; Sun, Y.; Wang, X.; Liu, Y.
Design of Multifunctional Tunable Metasurface Assisted by Elastic Substrate. *Nanomaterials* **2022**, *12*, 2387.
https://doi.org/10.3390/nano12142387

**AMA Style**

Li J, Fan H, Ye H, Wu T, Sun Y, Wang X, Liu Y.
Design of Multifunctional Tunable Metasurface Assisted by Elastic Substrate. *Nanomaterials*. 2022; 12(14):2387.
https://doi.org/10.3390/nano12142387

**Chicago/Turabian Style**

Li, Jing, Hongjie Fan, Han Ye, Tiesheng Wu, Yuhang Sun, Xueyu Wang, and Yumin Liu.
2022. "Design of Multifunctional Tunable Metasurface Assisted by Elastic Substrate" *Nanomaterials* 12, no. 14: 2387.
https://doi.org/10.3390/nano12142387