## Abstract

Based on the convolution and superposition theorem, we propose a
reflective anisotropic metasurface to realize the functions of
deflection and superposition of vortex beams, bifocal focusing, and
focusing vortex beam. At frequency of 1.04THz, two deflection vortex
beams with topological charges of (*l*=-1 and
*l*=+2) and
(*l*=+1 and *l*=-2)
are generated under *x*- and
*y*-polarized terahertz wave incidence, respectively.
At focal plane, 1200µm from the top layer of the proposed
metasurface, one can see that the bifocal focusing along
*y*-axis and *x*-axis are produced under
*x*- and *y*-polarized terahertz wave
incidence, respectively. Similarly, focusing vortex beams with
*l*=+1 and *l*=-2
are realized under *x*- and
*y*-polarized terahertz wave incidence, respectively.
The designed metasurface can flexibly manipulate terahertz wave under
different polarization waves incidence and has potential application
prospects in fields of terahertz communication.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Metasurfaces are two-dimensional artificial composite electromagnetic
materials in which subwavelength metal particles are periodically arranged
to adjust the polarization, amplitude, phase of electromagnetic waves
[1,2] to realize beam deflection [3–5], metalens [6–8], vortex beam [9–11] and holographic imaging [12–14]. In recent years, different function metasurfaces
have been reported. More recently, multifunctional metasurface have
received much attention [15–17]. In 2017, Cai et al. [18] revealed microwave bifunctional meta-device to exhibit
focusing and wave bending functionalities depending on the incident
polarization. In 2018, Zhuang et al. [19] designed a bifunctional metasurface composed of four
cross-shaped patches and three dielectric layers to achieve diffuse
reflection and transmission focusing. In 2019, Kou et al. [20] demonstrated a linear-polarized
terahertz focusing metasurface composed of the upper and lower two
composite cruciform metals and a medium dielectric layer under a normal
incidence. In 2020, Luan et al. [21] presented an asymmetric meta-device composed of the upper
meta-pattern, middle grating and lower metasurface for focusing, vortex
and Bessel beam generation operating in optical region. Rajabalipanah et
al. [22] designed a two-dimensional
reflective array based on vertical graphene-based meta-atoms to realize
polarization-multiplexed meta-hologram. In 2021, Liu et al. [23] proposed an all-dielectric
metasurface to convert orthogonally polarized fundamental Gaussian modes
to HG_{10} and HG_{11} modes. However, these mentioned
metasurfaces less use the convolution and the superposition theorem to
increase the diversity of metasurface functions. Therefore, it’s
necessary and meaningful to design metasurface combined with the
convolution and the superposition theorem.

In this paper, we design a new reflective anisotropic metasurface, which
consists of a metal pattern layer, polyimide layer and metal plate bottom
layer. The metal pattern is composed of a rectangular metal plate with a
hollow cross and a metal cross structure. Using the convolution and the
superposition theorem, the proposed metasurface realizes the functions of
deflection and superposition of vortex beams, bifocal focusing, and
focusing vortex beam. The simulation results are consistent with the
pre-design, and the simulation results show that the proposed metasurface
can flexibly manipulate terahertz wave under *x*-polarized
and *y*-polarized terahertz wave incidence.

## 2. Structure design and theoretical analysis

The functional diagram realized by the reflective anisotropic metasurface
is shown in Fig. 1(a). The
metal particle from top to bottom is the metal pattern, the polyimide
layer and bottom metal plate, as illustrated in Fig. 1(b). The metal pattern consists of a
rectangular metal plate with a hollow cross and a metal cross structure.
The structural parameters of the rectangular metal plate with a hollow
cross are 90µm and 30µm, respectively. The material is gold,
the conductivity
σ_{Au }= 4.561×10^{7}
S/m and the thickness is 1 µm. The thickness of polyimide
layer(ɛ_{r }= 3.5) is 30µm with
period of *P *= 100µm. The
electromagnetic simulation software CST Studio Suite is used to simulate
the designed structure. The phases 0°, 90°, 180° and
270° represent “1”, “2”,
“3” and “4”, respectively. Figure 2 shows that the reflection amplitudes of
the designed anisotropic unit cells are larger than 0.92, and the
reflection phases satisfy the gradient phases of 0°, 90°,
180° and 270° distributed at the operating frequency of
1.04THz, under both *x*-polarized and
*y*-polarized terahertz wave incidence. The optimized size
parameters of the anisotropic metal particles are shown in
Table 1.

## 3. Performance analysis and discussion

#### 3.1 *Deflection and superposition of vortex
beams*

The designed metasurface electric field distribution and the scattering function of the far field are the Fourier transform pair, which can be given by [24]

Among them, *φ*_{1} and
*φ*_{2} are the phase distributions of
the two different modes, and *φ*_{0} is
the phase distribution of the mixed mode obtained after superposition.
The phase distribution required for each element structure to produce
a deflected vortex beam can be calculated by [26]

*λ*is the wavelength of the operating frequency,

*l*is the number of topological charge, Φ

_{1}(

*x*,

*y*) is the phase at the (

*x*,

*y*) on the metasurface,

*θ*and

*k*

_{0}is the deflection angle and the wave number, and

*Γ*is the length of the encoding period. At the operating frequency of 1.04THz, the period

*Γ*

_{1 }= 400µm under

*x*-polarized incidence wave, the calculated deflection angle ${{\theta}_\textrm{1}} \approx \mathrm{46^\circ }$. The period

*Γ*

_{2 }= 800µm under

*y*-polarized incidence wave, the calculated deflection angle ${{\theta}_\textrm{2}} \approx \mathrm{13^\circ }$.

The discretized phase distribution shown in Fig. 3 is arranged to obtain the
metasurface M1, which consists of 24×24 metal particles. The 3D
far-field scattering pattern and the two-dimensional(2D) scattering
E-pattern when two different polarizations terahertz wave is incident
on the metasurface M1 are obtained by calculation, as given in
Fig. 4. As one can see,
when the *x*-polarized terahertz wave incidence, two
vortex beams with different deflection directions and different modes
appear, which are (*l*=-1,
*φ*_{1 }= 0°*,
θ*_{1 }= 46°) and
(*l*=+2,
*φ*_{2 }= 270°*,
θ*_{1 }= 46°),
respectively. When the *y*-polarized terahertz wave
incidence, two vortex beams with different deflection directions and
different modes are generated, which are
(*l*=+1,
*φ*_{3 }= 180°*,
θ*_{2 }= 13°) and
(*l*=-2,
*φ*_{4 }= 90°*,
θ*_{2 }= 13°),
respectively. Figure 5
shows the purity of the OAM mode. One can see that the mode purities
of OAM with topological charges *l*=-2,
*l*=-1,
*l *= 1 and
*l *= 2 are 0.471, 0.2316, 0.362,
and 0.3878, respectively.

#### 3.2 *Bifocal focusing*

The compensation phase required to achieve focusing for each unit cell can be calculated by the following equation [27]:

*λ*is the wavelength,

*m*and

*n*are the deviation distance in the positive direction of the

*x*-axis and

*y*-axis, respectively,

*m*=

*n*= 600µm, and the focal length

*z*=1200µm.

_{f}According to the discretized phase distribution of bifocal focusing,
the metasurface M2 is arranged as shown in Fig. 6(c). Figure 7 provides the electric field distributions and
normalized electric field intensity curves in the focal plane at
*z _{f}*=1200µm when

*x*-polarized and

*y*-polarized terahertz wave is incident. In this article, we set the operating frequency of 1.04THz and the focal plane at

*z*=1200µm. Figures 7(a) and 7(d) shows two focal points along

_{f}*y*-axis and

*x*-axis under

*x*-polarized and

*y*-polarized terahertz wave incidence. Corresponding normalized electric field intensity curves are illustrated in Figs. 7(b) and 7(e). From the figure, one can see that the full width at half maximum (FWHM) of normalized electric field intensity are FWHM1 = 241µm≈0.837λ and FWHM2 = 259µm≈0.899λ under

*x*-polarized terahertz wave incidence, FWHM3 = 259µm≈0.899λ and FWHM4 = 241µm≈0.837λ under

*y*-polarized terahertz wave incidence. The focal point position moves from the center to the sides as the focal length increases under different polarization, as depicted in Figs. 7(c) and 7(f). At the focal plane

*z*=1200µm, we can find that the maximum value of electric field intensity of four focal points locates in (

_{f}*x*= -582µm,

*y*= 0µm), (

*x*= 582µm,

*y*= 0µm), (

*x*= 0µm,

*y*=-582µm) and (

*x*= 0µm,

*y*= 582µm). Here, the four pre-designed positions are (

*x*= -600µm,

*y*= 0µm), (

*x*= 600µm,

*y*= 0µm), (

*x*= 0µm,

*y*=-600µm) and (

*x*= 0µm,

*y*= 600µm). One can see that there is a slight deviation between the simulation results and the preset values.

#### 3.3 *Focusing vortex beam*

The helical phase distribution of the focusing vortex beam can be obtained by adding the phase distribution of the focusing lens and the vortex beam [28]:

The metasurface M3 is arranged according to the discretized phase
distribution that realizes the focusing vortex beam, as shown in
Fig. 8. To characterize
the performance of focusing vortex beam, the electric field
distributions and normalized electric field intensity curves in the
focal plane at *z _{f}*=1200µm are
provided in Fig. 8 under

*x*-polarized and

*y*-polarized terahertz wave incidence. At the operating frequency of 1.04THz and in the focal plane of

*z*=1200µm, when the

_{f}*x*-polarized terahertz wave incidence, a focusing vortex beam with

*l*=+1 appears. The electric field distribution and the phase distribution in the focal plane are shown in Fig. 9(a) and 9(b), respectively. When the

*y*-polarized terahertz wave incidence, a focusing vortex beam with

*l*=-2 appears, the electric field distribution and the phase distribution in the focal plane are displayed in Fig. 9(d) and 9(d).

## 4. Conclusion

To sum up, we proposed a reflective anisotropic metasurface, which can
produce different regulation functions for *x*-polarized
and *y*-polarized terahertz wave incidence without changing
the metasurface arrangement. The designed metasurface based on the
convolution and superposition theorem can realize three functions, such as
deflection and superposition of vortex beams, bifocal focusing and
focusing vortex beam. The simulation results are consistent with the
theoretical predesigns. The reflective anisotropic metasurface is simple
and can flexibly control terahertz wave. The results display that the
designed terahertz metasurface device has widely application prospects in
the field of terahertz system in the future.

## Funding

National Natural Science Foundation of China (61831012, 61871355); Talent project of Zhejiang Provincial Department of Science and Technology (2018R52043); Zhejiang Key R & D Project of China (2021C03153, 2022C03166); Research Funds for the Provincial Universities of Zhejiang (2020YW20, 2021YW86).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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