1. Introduction

Metamaterials, made of well-ordered artificial metal or dielectric subwavelength unit cells, have widespread applications in artificially manipulating the polarization state of EM waves [1,2], Fano resonances [3], plasmonic resonances [4], plasmon-induced transparency [5]. The manipulation of the polarization state of electromagnetic (EM) waves has become highly important for many optical systems, as many optical phenomena are polarization sensitive. Based on these advantages, many metamaterial-based polarization converters have been proposed for linear-to-linear [6], linear-to-circular [7–9] and circular-to-circular [10] polarization. In 2013, Chen et al. [11] first proposed metasurface reflection and transmission polarization converters in while the efficiency of the reflection polarizer was over 80% and that of the transmission polarizer reached 50%. Chiang et al. [12] studied transmission type wideband polarization converters composed of s-type and asymmetric split-ring resonators (SRRs), achieving 60% average conversion efficiency. Zhang et al. [13] used double-ring chain structures to achieve both linear and elliptical polarization conversions, and in 2015, Zhang’s group implemented a polyimide-isolated three-layer metal grating perfect polarization conversion device with high-performance broadband properties [14]. However, because all of the abovementioned structures are composed of metallic materials, the operating frequency and polarization conversion characteristics cannot be changed after the structures are constructed [15–17], restricting the practical applications for these devices.

Graphene has been widely used as an active material in polarization conversion devices [18–20], because of its gate-voltage-dependent properties [21]. Liu et al. [22] etched the square holes on a graphene surface to achieve a tunable properties over a wide range from about 30 to 50 THz. Guo et al. [23] studied a graphene-based L-shaped super-surface structure and achieved frequency tunable cross-polarization, which demonstrated that the resonant frequencies blue shifted as the Fermi energy was increased. Although graphene exhibited controllable optical properties enabling dynamic light manipulation, the coupling of graphene to incident EM waves was not sufficient [24].

Three-dimensional (3D) Dirac semimetals (DS), which represent a novel state of quantum matter considered to be “3D graphene,” have recently aroused great interest in physics and material science as a type of topological semimetals. DS have a similar band structure to graphene, and the energy and momentum satisfy the linear dispersion relation in 3D K space [25,26]. Similar to graphene, the permittivity function of DS can be dynamically controlled by changing the Fermi levels through gate voltage [27–29]. Furthermore, the dielectric functions can also be adjusted by changing the Fermi energy of DS through alkaline surface doping [31,32]. However, DS are more robust against environmental defects or excess conductive bulk states than graphene. Recently, works about tunable properties of Dirac semimetals have also begun to generate widespread interest. Su et al. [33] investigated the optical coupling of terahertz surface plasmons in DS sheets and dynamically tuned the transmission spectra for a small change in Fermi energy in DS. Liu et al. [34] proposed a Dirac semimetal based tunable narrowband absorber at terahertz frequencies, and the absorption frequency could be tuned from 1.381 to 1.395 THz by varying the Fermi energy of DS from 50 to 80 meV.

In this study, a reflective linear-to-circular (LTC) polarization converter is proposed based on a metasurface structure, which consists of a center-cut cross-shaped metallic patterned structure and a sandwiched DS ribbon. The proposed structure operates in the 1.5–2.8 THz ranges with an axis ratio (AR) ≤ 3_dB and efficiency ≥ 90%. Moreover, the LTC polarization converter is excited by x-polarized incident waves, where the reflected waves comprise left-hand circular polarization (LHCP) waves or right-hand circular polarization (RHCP) waves. Finally, to reveal the underlying physical mechanisms, current distributions are used to analyze the influence of the electric and magnetic moments on the polarization conversion of the incident waves. This converter has potential applications in antenna design, EM measurement and stealth technology. Such an ultrathin dual-band superposition-induced broadband LTC converter will have great potential applications in the area of wireless communication, imaging, and detection, and open a new way to design novel functional THz devices.

2. Structural design and simulation

The schematic diagram of the proposed LTC polarization converter is shown in Fig. 1 with the structural parameters, The top layer was composed of a center-cut cross-shaped resonator with a DS ribbon filling in the gap. The dielectric material was the typical lossy polyimide with a dielectric constant of $\epsilon \text{=}3\left(1+\text{i}0.06\right)$. The thicknesses of the DS ribbon and polyimide layer were set to 20 $nm$ and 16$\mu m$, respectively. The thickness of the center-cut cross-shaped resonator was 20$nm$, which was the same as the thickness of the DS ribbon. A fully reflective gold mirror served as the bottomed layer. The ground metallic continuous film was selected to be lossy gold with a conductivity of $\sigma =4.56\times {10}^{7}S/m$ and a thickness of $h_2=0.4\mu m$. The other structural parameters were defined as follows: $p=70\mu m$, $L_1=25\mu m$, $L_2=9\mu m$, $w_1=4\mu m$, $w_2=4\mu m$, $w_3=4\mu m$, $h_1=16\mu m$, and $h_2=0.4\mu m$.

 

Fig. 1 A 3 × 3 unit structure diagram of the proposed polarization converter.

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The proposed converter was simulated using the frequency-domain solver of CST Microwave Studio, periodic boundary conditions were employed in the x- and y- directions, and a perfectly matched layer boundary condition was used for the z-plane. The polarized plane waves along x-direction were normally incident to the structure surface. The reflected wave could be expressed as$E_r=E_{xr}{e}_x+E_{yr}{e}_x=r_{xx}\exp \left(j{\varphi }_{xx}\right)E_{xi}{e}_x+r_{yx}\exp \left(j{\varphi }_{yx}\right)E_{xi}{e}_y$, where $r_{xx}=\left|E_{xr}/E_{xi}\right|$ and $r_{yx}=\left|E_{yr}/E_{xi}\right|$ define the reflection ratio of the x-to-x and x-to-y polarization conversions, respectively, and ${\varphi }_{xx}$and ${\varphi }_{yx}$ are the corresponding phases. If $r_{xx}=r_{yx}$ and $\Delta \varphi ={\varphi }_{xx}-{\varphi }_{yx}=2n\pi \pm \pi /2$ (n is an integer), the linearly polarized waves could be converted to circular polarized wave, where $-$ and $+$ represent LHCP and RHCP, respectively, successfully realizing a circular polarization converter.

In the simulation, the dynamic conductivity [30] of the Dirac semimetals were written as

where G(E) = n(-E)-n(E) with n(E) being the Fermi distribution function, $E_F$is the Fermi level, is the Fermi momentum, $v_F={10}^{6}m/s$is the Fermi velocity, $\epsilon =E/E_F$, $\Omega =\hslash \omega /E_F$, ${\epsilon }_c=E_c/E_F$, $E_c=3$ is the cutoff energy, and g is the degeneracy factor. The permittivity of the 3D Dirac semimetals can be expressed as $\epsilon ={\epsilon }_b+i\sigma /\omega {\epsilon }_0$, where ${\epsilon }_b=1$ and ${\epsilon }_0$ is the permittivity of vacuum. Here, we give the dynamic conductivity of the DS, as shown in Fig. 2. The dotted line stands for the real part of the dynamic conductivity, while the solid line stands for the imaginary part.

 

Fig. 2 (a) The real and (b) imaginary parts of the dynamic conductivity for DS under different Fermi levels in units $e^2/\hslash$ as a function of the normalized frequency $\hslash \omega /E_F$. The parameters are set as g = 40, and $u=3\times {10}^{4}c{m}^2{V}^{-1}{S}^{-1}$ (the intrinsic time$\tau =4.5\times {10}^{-13}s$).

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3. Results and discussion

Figure 3 shows the simulated magnitude and phase difference of the LTC polarization converter in the wide band state (the Fermi energy of Dirac semimetals is 90 meV). In the frequency range from 1.50 to 2.80 THz, the relative value of the amplitude difference between the two curves was less than 0.2, corresponding to an axis ratio (described later) less than 3dB that meet the circular polarization condition [1,7]. Thus the reflection coefficient magnitudes were approximately equal, and the corresponding phase difference was close to 90° or −270°. Thus, the reflected waves were RHCP waves. Moreover, in the narrow frequency ranges from 1.20 to 1.25 THz and 3.04–3.07 THz, the phase difference was close to −90°and the reflection coefficient magnitudes were also approximately equal (the relative value of the amplitude difference here was less than 0.2, we can approximate ignored the difference), indicating LHCP reflected waves. To describe the performance of the polarization converter, we used the following Stokes parameters

 

Fig. 3 Reflection coefficients for the x-and y-components. (a) Magnitude and (b) phase with the phase difference.

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To better understand the polarization states of the reflected waves, the polarization ellipticity ($V/I$) of the reflected waves and the axis ratio AR for the x-polarized incident waves are shown in Fig. 4. Here, $V/I=-1$ and $V/I=+1$ indicate that the reflected wave is an LHCP wave and RHCP wave, respectively. As shown in Fig. 4(a), the ellipticity was close to unity in the frequency range of 1.50–2.80 THz, further indicating that the reflected waves were RHCP waves. An ellipticity approaching –1 was analogously obtained from the frequency ranges of 1.20–1.25 THz and 3.04–3.07 THz, indicating the reflected waves were LHCP waves. Figure 4(b) shows the dynamic electric field distributions on the metasurface for different time phases at 1.8 THz, where the direction of the arrow represents the polarization direction of the electric field. The waves clearly showed right-hand rotation with the change in the time phases. We defined the axis ratio $AR=10\log \left(\tan \beta \right)$ ($\sin 2\beta =V/I$, where $\beta$ is the ellipticity angle) [1,7] to quantify the circular polarization performance. The energy conversion efficiency was calculated by $\eta ={\left|r_{xx}\right|}^2+{\left|r_{yx}\right|}^2$. These two parameters are shown in Fig. 4(c). The AR was less than 3 dB in the frequency ranges from 1.20 to 1.25 THz, 1.50–2.80 THz, and 3.04–3.07 THz, as indicated by the red solid line. The efficiency $\eta$ was greater than 80%. These results demonstrate the high-efficiency conversion property of our proposed design.

 

Fig. 4 (a) Ellipticity of the proposed design when excited by an x-polarized plane wave. (b) The electric field distributions at 1.8 THz for different time phases from 0° to 160° with an increment of 20°. (c) The axis ratio and efficiency of the proposed design with respect to the frequency.

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Figure 5(a) shows the ellipticity of the LTC polarization conversion with different Fermi energies. When the Fermi energy was 80meV, the LTC-polarization converter converted linear waves to RHCP waves with a lower mode ($f_l$) of 1.4 THz and a higher mode ($f_h$) of 2.9 THz. As the Fermi energy changed from 80 meV to 130 meV, the lower resonant mode shifted to the higher frequency bands from 1.4 to 1.6 THz. On the contrary, the higher resonant mode shifted to the lower frequency bands from 2.9 to 2.3 THz. The two resonance modes were coupled together gradually separate apart, leading to a broadband to dual-band tunable cross-polarization conversion. To explain this phenomenon, the electric field distributions at the top surface of the converter at the lower resonant mode (1.4 THz) and higher resonant mode (2.9 THz) are given in the insert of Fig. 5(a). At the lower resonant mode, the electric field was concentrated at the upper left and lower right parts of the structure, indicating that the lower resonant mode originated from the localized surface plasma (LSP) modes of the long axis of the cross-shaped resonator; At the higher resonant mode, the electric field was concentrated at the shorter axis of the cross-shaped resonator, indicating that the higher resonant mode originated from the LSP modes of the shorter axis of the resonator. To further demonstrate this. Figure 5(b) and (c) present the ellipticity of the converter with different values of the long axis length ($L_1$) and short axis length ($L_2$), respectively. In Fig. 5(b), the fixed value of short axis length ($L_2$) was $9\mu m$. The lower resonant mode shifted from 1.1 to 1.40 THz when $L_1$ varied from 23$\mu m$ to 29$\mu m$, whereas the higher resonant mode remained unchanged. In Fig. 5(c), the fixed value of long axis length ($L_1$) was $25\mu m$. Similarly, as $L_2$ increased from 9$\mu m$ to 12$\mu m$, the higher resonant mode shifted from 3.2 to 2.6 THz. This feature proved that the proposed polarization converter possesses the advantage of high flexibility as two resonant modes can be tunable independently.

 

Fig. 5 (a) Ellipiticity at different Fermi energies from 80 to 130 meV, and the ellipticity at a Fermi energy of 90 meV (b) with different $L_1$ values and (c) with different $L_2$ values. The electric distributions at the surface of the polarization converter of the lower resonant mode and the higher resonant mode with a Fermi energy of 90 meV are given in the insets of (a).

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The magnitudes and phase differences of the reflection coefficient and the axis ratio and the efficiency are shown in Fig. 6 (a)‐(d), respectively, for 13$\mu m$, 16$\mu m$ and 19$\mu m$. By comparison, the relative difference of amplitude was the smallest when the substrate thickness was 16$\mu m$, and the bandwidth of AR less than 3db was also the widest. Optimal LTC polarization conversion performance was clearly obtained when $h=16\mu m$. The angular stability of the proposed polarization convertor was also investigated. Figure 6(e) shows the simulated AR with the incident angle increasing from 0° to 50°. One can see that the proposed converter can maintain a good circular polarization conversion performance with one ultra-wideband and two narrow bands while the angle of incidence varies from 0° to about 10°. With an increase in the incident angle above 10°, the ultra-wideband became narrow and split into two bands that could not maintain its continuity and the right narrowband gradually disappeared.

 

Fig. 6 (a) Magnitudes (b) phase differences (c) AR and (d) η of different substrate thicknesses. (e) Angle of incidence and frequency-dependence of AR.

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4. Physical mechanisms

To better understand the physical mechanism of the polarization conversion for the x-polarized incident EM wave, we decomposed the x-polarized incident EM wave into two perpendicular components (u and v components). The u-v coordinate system is rotated about the z-axis ${45}^{\circ }$ from the x-y coordinate system, as show in the inset of Fig. 7(a). The linearly polarized incident plane wave can be expressed as: $E_i=E_i\exp \left(jkz\right)e_y=E_i\exp \left(jkz\right)e_u/+E_i\exp \left(jkz\right)e_v/$, and the reflected electric field is given by $E_r=\left\{r_{uu}{E}_i\exp \left[j\left(-kz+{\varphi }_{uu}\right)\right]+r_{uv}{E}_i\exp \left[j\left(-kz+{\varphi }_{uv}\right)\right]\right\}{e}_u/+$$\left\{r_{vu}{E}_i\exp \left[j\left(-kz+{\varphi }_{vu}\right)\right]+r_{vv}{E}_i\exp \left[j\left(-kz+{\varphi }_{vv}\right)\right]\right\}{e}_v/$. Here, $r_{uu}$, $r_{vu}$, $r_{uv}$, and $r_{vv}$ represent the magnitude of the reflection coefficients for the u-to-u, u-to-v, v-to-u, and v-to-v polarization conversions, respectively, and ${\varphi }_{uu}$, ${\varphi }_{vu}$, ${\varphi }_{uv}$, and ${\varphi }_{vv}$ are the corresponding phases. When $r_{uu}=r_{vv}=r$, $r_{uv}=r_{vv}=0$, and $\Delta \varphi ={\varphi }_{uu}-{\varphi }_{vu}=2n\pi \pm \pi /2$, the reflected electric field can be written as $E_r=r{E}_i\exp \left(-jkz\right)\left\{\exp \left(j{\varphi }_{uu}\right)e_u+\exp \left[j\left({\varphi }_{uu}+2n\pi \pm \frac{\pi }{2}\right)\right]e_v\right\}$, and a circular polarization wave will be obtained. As shown in Fig. 7, $r_{vu}$ and $r_{uv}$ were nearly equal to 0, $r_{uu}$ was approximately equal to $r_{vv}$, and the phase difference was close to −90° or 270° in the frequency ranges of 1.50–2.80 THz. Thus, the transformation of linear polarization waves to LHCP waves was achieved. A similar situation occurred in the ranges of 1.20–1.25 THz and 3.04–3.07 THz with a phase difference close to 90°, indicating RHCP waves were obtained.

 

Fig. 7 (a) Magnitudes, (b) phases, and phase differences of the reflection coefficients in the u–v coordinate system

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The physical mechanisms of the polarization conversion can be further elucidated using the current distributions on the metasurface and on metal ground in the u–v coordinate system. Electric resonance should be generated when the currents on the metasurface are parallel to those induced on the metal ground. In contrast, magnetic resonance should be formed by current loops in the dielectric substrate if the surface currents on the metasurface are antiparallel to the induced currents. Here, we choose four frequencies, namely 1.22 THz for LHCP and 1.49, 2.24 and 2.64 THz for RHCP, to analyze the current distributions on the metasurface and metal ground, as shown in Fig. 8.

 

Fig. 8 The current distributions (columns 1 and 2 show the u-polarized incident wave, and columns 3 and 4 show the v-polarized incident wave) with the equivalent electric moments (red double-headed arrows) and equivalent magnetic moments (blue circles) at (a)–(e) 1.22 THz, (f)–(j) 1.49 THz, (k)–(o) 2.24 THz, and (p)–(t) 2.64 THz.

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At 1.22 and 1.49 THz, the currents on the metasurface and those on the ground gave rise to magnetic resonance, generating equivalent magnetic moments $m_1$ and $m_3$, respectively, under the u-polarized incident waves. Similarly, $m_2$and $m_4$ were generated for 1.22 and 1.49 THz, respectively, when the incident waves were polarized along v-direction. In Fig. 8(e), $m_1$and $m_2$ manipulate the magnitudes and phases of the reflected electric fields along the u- and v-axis, respectively. If the u- and v-components of the reflected electric fields have equal magnitudes and a phase difference of 90°, ideal LTC polarization conversion will be achieved for left-hand polarized waves. In Fig. 8(j), the principle is identical for right-hand LTC polarization conversion. At 2.24 and 2.64 THz, the magnetic resonances were formed and magnetic moments $m_6$ and $m_8$, respectively, were obtained under the v-polarized incident waves. However, the electric resonances were generated under the u-polarized incident waves; thus electric moments $p_5$ and $p_7$ were obtained at 2.24 and 2.64 THz, respectively, because the currents on the metasurface were parallel to those induced on the metal ground. In Fig. 8(o), $p_5$ and $m_6$ respectively manipulate the magnitudes and phases of the reflected electric fields along the u- and v-axis, respectively. An LTC polarization conversion can be achieved if the approximate magnitudes of the u- and v-components of the reflected electric fields are equal and the phase difference is 90°. The same principle is used as presented in Fig. 8(t). Because of its multiple resonances, the proposed design can be used to perform linear-to-circular polarization in an ultra-wideband by optimizing the structural parameters.

5. Conclusions

In conclusion, a metasurface of DS was proposed for realizing dynamically tunable linear-to-circular polarization conversion. The metasurface successfully converted the x-polarized normally incident EM wave to a circularly polarized wave. Because of the multiple characteristic resonances in the proposed design, ultra-wideband operation was obtained. The conversion band demonstrated a blue shift from 1.2 to 1.3 THz when the Fermi energy increased from 80 to 130 meV in the lower resonance mode. The design operated in one wideband and two narrowbands, the limitations of this device were that the bandwidth is narrow. Thus, the proposed converter may have various practical applications in the area of wireless communication, imaging, and detection, and open a new way to design novel functional THz devices.