## Abstract

We theoretically present a high-efficiency switchable reflective terahertz polarization converter composed of a periodic array of rectangular-shaped metal-dielectric-graphene sandwich structure on a dielectric substrate supported by a thick metallic film. Graphene sheet together with the rectangular-shaped metal patch provides tunable anisotropic hybrid magnetic plasmon resonance to obtain tunable phase delay of 90° and 180°, corresponding to a quarter-wave plate (QWP) and half-wave plate (HWP), respectively. Results of numerical simulations indicate that the proposed structure can switch functions between a QWP and HWP at a certain frequency simply by adjusting the Fermi energy of graphene. Both the QWP and HWP have high energy conversion efficiency, respectively 83% and 90% at 15.96THz, and high polarization conversion ratio closed to 1.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As one of the basic characteristics of electromagnetic waves, polarization plays a central role in a wide range of fields. Conventional approaches to realize polarization manipulation include using optical gratings and birefringence materials [1]. However, these methods usually rely on relatively long distance to gain phase accumulation, resulting in large weight, bulky volume, and complex synthesis procedures. Meanwhile, it is worth mentioning that in recent years, terahertz (THz) wave has been widely used in sensing [2], detecting [3], imaging, and spectroscopy [4] with the rapid development of THz technology. Among common THz functional devices such as THz switchers, filters, polarizers, phase shifters, and wave plates [5,6], the THz polarization converters [7] plays an important role. Under this background, the advent of metamaterials opens up a direction towards polarization manipulation at THz regime, which can effectively overcome the above-mentioned shortcomings of conventional polarizers. Different kinds of polarization converters based on metamaterial structures have been achieved, such as metallic nanoparticles with different shapes [7–9] and metallic nanoslot [10]. However, these reported polarization converters are not easily to be tuned via simple methods such as electrical control, which limits their applications in practice.

Fortunately, graphene [11], a novel semi-metal material, has attracted tremendous attention due to its unique optical and electrical properties, such as low loss, ability to support plasmonic resonance in terahertz and mid-far infrared bands [12–14], strong field localization [15] and nonlinear optical effects [16]. More importantly, with electrical gating [17,18] and chemical doping [19], graphene shows the ability to tune plasmonic resonance. Thus, it appears to be a good candidate for designing THz polarization devices [20–27]. For example, some researchers have used graphene nanostructures to realize QWP [23] or HWP [24–27] to manipulate the polarization state of light both theoretically and experimentally, whose operating frequencies can be tuned by changing the Fermi energy of graphene. Other tunable materials have also been researched addressing the topic of polarization conversion such as phase change material vanadium dioxide (VO_{2}) [28] and liquid crystals (LCs) [29]**.** However, till now, most of the proposed polarization converters are typically designed to achieve single functionality and cannot switch between QWP and HWP at a certain frequency without refabricating the structure. Only rare papers have been reported for obtaining simultaneous and switchable functions. For example, liquid crystals (LCs)-infiltrated MIM-based metamaterials have been applied into the design of a single-frequency operation electrically tunable THz polarization converter [30]. Meanwhile, switch between HWP and QWP can also be achieved by inserting phase transition material vanadium dioxide (VO_{2}) film into metamaterials [31]. However, it relies on thermal heating to obtain the crucial phase change of VO_{2}. So, Yu *et al.* presented an electrically controllable terahertz polarization converter based on a structure consisting of two-layer graphene wires [32]. Meanwhile, a recent published paper reported a bi-functional switchable polarization converter based on a hybrid graphene-metal metasurface [33]. Nevertheless, complex structures of their unit cells bring great challenges to their fabrication and application. For more convenient manufacture, Zhang *et al.* successfully adopted graphene sheets as the tunable component to realize the function of switchable polarization conversions [34]. However, its relatively low reflection efficiency due to high absorption needs to be further improved.

In this paper, we demonstrate theoretically an alternative and efficient graphene-based electrically controlled terahertz polarization conversion without refabricating the structure between QWP and HWP. In the structure, two magnetic plasmon resonance modes are constructed to obtain adjustable phase delay differences of 90° and 180° at the same frequency and two different fermi levels, corresponding to quarter-wave plate (QWP) and half-wave plate (HWP), respectively. In contrast to previous related works, this proposed structure exhibits better comprehensive performance, including simple structure, convenient fabrication, high conversion rate, and high reflectivity. With the new design idea and simple structure, this work provides a new approach towards the implementation of two different polarization functions (QWP and HWP) in one fixed structure, and may be further applied into other potential applications such as modulators, filters, sensors, and detecting and imaging systems.

## 2. Structure and theoretical analysis

Figures 1(a) and 1(b) schematically illustrate the structure of the proposed functionally switchable reflective terahertz polarization converter based on metal-dielectric-graphene sandwich structure. It consists of a periodic array of rectangular-shaped metal-dielectric-graphene on a dielectric substrate supported by a thick metallic film, with periodic interval *P* = 500 nm along the *x*-axis and *y*-axis, height *H _{1}* = 60 nm, width

*W*= 200 nm and length

*L*= 500 nm of the rectangular-shaped metal patch, thickness

*D*= 20 nm of the dielectric spacer, thickness

_{1}*D*= 2770 nm of the dielectric substrate, thickness

_{2}*H*= 100 nm of the thick metallic film, thickness

_{2}*H*= 1 nm and Fermi energy

_{G}*E*of the single-layered graphene. A bias voltage

_{F}*V*is imposed, as shown in Fig. 1(c), between the graphene and the thick metal layer, to electrically dope and tune the graphene chemical potential for adjusting

_{g}*E*. As analyzed in these work [35–37], the rectangular-shaped metal-dielectric-graphene array in Fig. 1(a) can be equivalent to an inductance-capacitance (LC) circuit to support strong anisotropic hybrid magnetic plasmon resonance mode labelled as MP

_{F}*[37], where*

_{ij}*i*and

*j*denote the number of loops formed by the electric vector and induced current on the cutting plane perpendicular to

**x**-

**y**(

**v**-direction) and

**x**+

**y**(

**u**-direction) directions, respectively. The resonant frequency of the mode MP

*is equal to the oscillation frequency of the equivalent LC circuit, which is proportional to 1/(*

_{ij}*L*)

_{e}C_{e}^{1/2}, where

*L*is the equivalent inductance and

_{e}*C*is the equivalent capacitance. In this work, we choose two fundamental magnetic plasmon resonance modes (magnetic dipoles) MP

_{e}_{01}and MP

_{10}to realize the electrically controlled polarization switch. The resonant frequencies of MP

_{01}and MP

_{10}are determined by the Fermi energy

*E*of graphene, and also depends on the width

_{F}*W*and length

*L*of the rectangular metal patch respectively. In our model system, the width

*W*and length

*L*have different values, resulting in different equivalent inductances and different resonant frequencies of MP

_{01}and MP

_{10}. Thus, by applying two different appropriate bias voltages to the graphene sheet to obtain two appropriate Fermi energy

*E*and

_{F1}*E*, we can make the two resonant modes MP

_{F2}_{01}and MP

_{10}resonate at the same frequency

*ω*. For simplicity, suppose that a monochromatic

_{0}**x**-polarized plane wave with

**y**magnetic field direction is normally incident into the structure, described by ${\textbf E} = {\textbf x} \cdot \textrm{exp} ( - i\omega t)$ at the surface of the rectangular-shaped metal-dielectric-graphene arrays, where

*ω*is the angular frequency of the incident wave and is close to the resonant frequency

*ω*of MP

_{0}_{01}and MP

_{10}. The incident polarization state can be decomposed into two orthogonal components, ${\textbf E}_{\textbf u}^{{\textbf in}} = ({\textbf x} + {\textbf y}) \cdot \textrm{exp} ( - i\omega t)/\sqrt 2 $ and ${\textbf {E}}_{\textbf v}^{{\textbf in}} = ({\textbf x} - {\textbf y}) \cdot \textrm{exp} ( - i\omega t)/\sqrt 2 $.

When the graphene is electrically doped and possesses a Fermi energy of *E _{F1}*, the incident polarization component ${\textbf E}_{\textbf u}^{{\textbf in}}$ effectively excites MP

_{01}and is resonantly reflected, which introduces a reflection phase delay

*Φ*. The other incident polarization component ${\textbf E}_{\textbf v}^{{\textbf in}}$ cannot effectively excite MP

_{10}_{01}or MP

_{10}and is ordinarily reflected by the bottom metal layer with a phase delay

*Φ*. Thus, the total reflected wave is given by the superposition of the resonant reflection and the ordinary reflection with a phase delay difference (PDD)

_{01}*Φ*=

*Φ*-

_{10}*Φ*. Because the electric vector of MP

_{01}_{01}and MP

_{10}is mainly perpendicular to the graphene sheet and the absorption loss of the graphene sheet is highly suppressed, the resonant reflection coefficient

*r*

_{10}is large and can be easily designed to be approximately equal to ordinary reflection coefficient

*r*

_{01}. Then, if

*Φ*is equal to ${\pm} \pi \textrm{/2} \pm 2n\pi (n = 0,1,2\ldots )$, the total reflected wave will be circularly polarized, which means the structure can be considered as a QWP.

However, when the graphene is electrically doped and possesses the Fermi energy of *E _{F2}*, the incident polarization component ${\textbf E}_{\textbf u}^{{\textbf in}}$ cannot effectively excite MP

_{01}or MP

_{10}and is ordinarily reflected by the bottom metal layer with a phase delay

*Φ*. The other incident polarization component ${\textbf E}_{\textbf v}^{{\textbf in}}$ effectively excites MP

_{01}_{10}and is resonantly reflected, which introduces a reflection phase delay

*Φ*. For the same reason, the resonant reflection coefficient

_{10}*r*

_{10}can be easily designed to be approximately equal to the ordinary reflection coefficient

*r*

_{01}. Therefore, if

*Φ*is equal to ${\pm} \pi \pm 2n\pi (n = 0,1,2\ldots )$, the polarization of the total reflected wave will be along the

**y**-direction, which shows the structure can convert the polarization direction of incident linearly polarized plane wave from

**x**-direction to

**y**-direction after reflection and implies the structure can be considered as a HWP.

In the design, the phase delay *Φ _{01}* of the ordinarily reflected component correlates to the thickness of the substrate and cannot be tuned by changing Fermi energy. Therefore, the tunable phase difference

*Φ*is determined by the phase delay

*Φ*of the resonantly reflected component. Because the resonant reflection phase delay

_{10}*Φ*is mainly related to the deviation of the working frequency from the resonant frequency (that is, the degree of detuning),

_{10}*Φ*can be tuned by the Fermi energy of graphene and the width

_{10}*W*and length

*L*of the rectangular metal patch, respectively. The detailed design idea is as follows. First, we select the operating frequency according to our needs. Then, we find two different Fermi levels

*E*and

_{F1}*E*and the corresponding width

_{F2}*W*and length

_{0}*L*by simulation calculations so that the resonance frequencies of the two magnetic resonances (MP

_{0}_{01}and MP

_{10}) are the same as the operating frequency. Again, because the resonant reflection phase delay

*Φ*changes with length and width are more sensitive than the resonance frequency changes with length and width, we can get phase delay difference of 90° and 180° by fine-tuning the width

_{10}*W*and length

*L*based on the corresponding width

*W*and length

_{0}*L*, respectively. Therefore, electrically controlled polarization switching between QWP and HWP in one device is theoretically realized through the proposed structure. At the same time, key parameters are determined.

_{0}## 3. Simulations and results

To verify the theoretical prediction, frequency domain solver in CST Microwave Studio is then used to emulate the design. In the simulation, the metal material is treated as a dispersive medium described by the Drude model, with relative permittivity expressed by ${\varepsilon _m}(\omega ) = {\omega _\infty } - {\omega _p}^2/({\omega ^2} + i\omega \gamma )$, where *ω _{∞}*,

*ω*and γ are 1.0, 1.38×10

_{p}^{16}rad·s

^{−1}and 1.23×10

^{13}s

^{−1}[38], respectively. Both the dielectric substrate and the dielectric spacer are considered to be nondispersive whose relative permittivity

*ɛ*= 4. The single-layered graphene is modeled as an anisotropic effective media of thickness

_{r}*H*= 1 nm with the in-plane relative permittivity

_{G}*ɛ*= 2.5 +

_{in}*iσ*(

*ω*)/(

*ωɛ*) [39,40] and the out-of-plane relative permittivity

_{0}H_{G}*ɛ*= 2.5. Here, we focus on the THz range and the relatively high graphene Fermi energy

_{out}*E*(

_{F}*E*> 0.6 eV) where

_{F}*ω*is much lower than the interband threshold 2

*E*thus interband transitions can be ignored. Under the approximation, the conductivity

_{F}*σ*(

*ω*) can be expressed by the Drude model [41–43] as

*k*is the Boltzmann constant,

_{B}*T*= 300 K is the temperature,

*ω*is the frequency of incident wave,

*e*is the elementary charge,

*E*is the Fermi energy of graphene, $\tau = \mu {E_F}/e{v_F}^2$[44] is the Drude relaxation lifetime, where the mobility

_{F}*μ*is 10000 cm

^{2}/(V·s) and the Fermi velocity

*v*is 10

_{F}^{6}m/s. In our simulations, with a linearly polarized plane wave used as the excitation source, the unit cell boundary condition is applied in

**x**and

**y**directions (or in

**u**and

**v**directions), while the open boundary condition is applied in

**z**-direction. It should be noted that the choice of operating frequency of the polarizer can be arbitrary as the structural geometry and Fermi energy can always be tuned to match the desired operating frequency in the THz region.

Firstly, consider the condition where *E _{F}* = 0.7 eV with the electric fields of two incident plane waves parallel to

**u**-direction and

**v**-direction, respectively. Figure 2 shows the reflection coefficients,

*R*and

_{uu}*R*, and PDD for the two orthogonal fundamental components as functions of frequency, where almost equal reflection coefficients (0.91) and a PDD of near 90° at 15.96 THz can be observed. And as shown in the insets of Fig. 2, the electric fields of incident plane waves parallel to

_{vv}**u**-direction and

**v**-direction can excite MP

_{10}mode and MP

_{01}mode, respectively. Yet we can see clearly that the electric field of mode MP

_{01}is much stronger, namely the mode MP

_{01}can be more effectively excited by the incident plane wave along

**v**-direction while the mode MP

_{10}cannot be effectively excited, consistent with the previous theoretical predictions and further verifying the function of QWP at 15.96 THz.

Key parameters used for characterization of the proposed QWP include axial ratio (AR), circular polarization conversion ratio (CPCR), polarization-averaged power reflectivity, and absorption of the structure. AR is defined as the ratio between the two orthogonal electric-field amplitudes. Figure 3 shows AR of 1 and PDD of nearly 90° at 15.96 THz, indicating efficient conversion of linearly-polarized into circularly-polarized. The proportion of circular polarization among total reflected wave (CPCR) is also calculated, written by the ratio between the reflected circular state of polarization and the total field intensity:

*δ*is the PDD between the reflected component along

**u**-direction and

**v**-direction. Here the absorption is not considered. The above formula results in a value of 1 when the component along

**v**-direction is phase-shifted by 90° with the

**u**-direction component and their amplitudes are equal in magnitude, which signifies complete conversion of an incident

**x**-polarized linearly polarized (LP) beam into circularly polarized (CP) wave. As can be seen from Fig. 4, the simulated CPCR at 15.96 THz is closed to 1, suggesting that nearly all of the reflected beam is circularly-polarized.

The polarization-averaged power reflectivity (*R _{c}*), as shown in Fig. 5, is computed as the geometrical average between the reflected intensities of components along

**u**-direction and

**v**-direction, and the absorption (

*A*) of the structure is also calculated, as ${R_c} = ({R_{uu}}^2 + {R_{vv}}^2)/2$ and

_{c}*A*= 1 -

_{c}*R*, respectively. The reflected power up to about 83% at 15.96 THz is adequate for certain imaging applications. Compared with previous graphene-based QWP such as quarter-wave plate based on graphene gratings [23], the proposed QWP exhibits higher energy and polarization conversion ratios, which may originate from the fact that the electric field of the hybrid magnetic plasmon resonance in rectangular-shaped metal-dielectric-graphene is mainly perpendicular to the non-structured graphene sheet and the absorption of graphene is suppressed.

_{c}The numerical results presented reveal that the proposed structure can achieve a high-power efficiency and total circular polarization conversion from **x**-polarized incident wave at a certain frequency.

Then, consider the case where *E _{F}* = 0.93 eV with electric field of incident wave parallel to

**x**-axis. As can be seen from Fig. 6, the proposed structure presents PDD very close to 180° at 15.96 THz, matching the phase requirement of ${\pm} \pi \pm 2n\pi (n = 0,1,2\ldots )$ for HWP. Polarized reflections

*R*and

_{xx}*R*and linear polarization conversion ratio (LPCR), defined as $LPCR = {R_{yx}}^2/({R_{xx}}^2 + {R_{yx}}^2)$, as functions of frequency are shown in Fig. 7. Here the absorption is not considered. The inset shows the distributions of the

_{yx}**z**components of electric field on the cutting plane perpendicular to

**z**-direction at a distance of

*D*/2 above graphene sheet at 15.96 THz. It can be clearly seen that the mode MP

_{1}_{10}can be more effectively excited and is the operation mode. It can be observed that

*R*exhibits a broad peak near 15.96 THz, where

_{yx}*R*has a corresponding dip, indicating that the structure can realize energy transfer from

_{xx}**x**to

**y**polarization after reflection, namely the proposed structure can be regarded as an HWP at 15.96 THz. We also find that LPCR reaches 0.999 at 15.96 THz, indicating that almost all the reflected waves are

**y**-polarized.

Moreover, in order to investigate the absorption loss of the structure, polarization power reflectivity *R _{l}* and absorption

*A*are calculated as ${R_l} = {R_{xx}}^2 + {R_{yx}}^2$ and

_{l}*A*= 1 -

_{l}*R*, respectively. As shown in Fig. 8, the reflected power is approximately 90% and the absorption loss is about 10% at 15.96 THz. In contrast to previous high-efficiency graphene-based HWP such as polarization converters based on graphene metasurface with twisting double L-shaped unit structure array [25], our proposed HWP has higher energy and polarization conversion ratios.

_{l}To investigate the influence of different structural parameters, LPCR and CPCR under several different values of *W* and *L* have been calculated. As can be seen from Figs. 9 and 10, both LPCR and CPCR show a downward trend as *W* and *L* shift from their optimal value. This can be explained by the theory of inductance-capacitance circuit. Since the equivalent inductance and equivalent capacitance of the proposed structure directly relate to *W* and *L*, variations of *W* and *L* will unavoidably lead to the change of oscillation frequency of the equivalent LC circuit, namely the magnetic resonant frequency of the MP_{01} or MP_{10} mode. Thus, working frequency will deviates more from the resonant frequency than the original design, resulting in PDD deflecting from 90° or 180° and consequently reducing its waveplate performance.

Meantime, LPCR and CPCR show different sensitivity towards changes of *W* and *L*. When LPCR of HWP significantly declines as *L* deviates from the optimal value of 500 nm, CPCR of QWP remains in place within the working bandwidth. Conversely, as *W* gradually deviates from 200 nm, CPCR shows much dramatic attenuation than LPCR. Such contrast intuitively suggests the design ideas of the proposed structure as mentioned above, that is, to make MP_{01} resonance responsible for QWP, and MP_{10} responsible for HWP.

The polarization conversion performances under different *E _{F}* are also simulated. As shown earlier, the proposed structure acts as efficient QWP when

*E*= 0.7 eV, and efficient HWP when

_{F}*E*= 0.93 eV, respectively, both at 15.96 THz. While as shown in Fig. 11, with other

_{F}*E*, the proposed structure will also work as HWP and QWP respectively, at different frequency. According to the calculation results, peaks of CPCR and LPCR will deviate from 15.96 THz with different degrees of blue-shifting, as

_{F}*E*increases from

_{F}*E*= 0.7 eV and

_{F1}*E*= 0.93 eV, respectively. Therefore, to obtain the best performance of polarization conversion at the chosen specific frequency (15.96 THz), the Fermi levels for switchable functions are set to be

_{F2}*E*= 0.7 eV and

_{F1}*E*= 0.93 eV.

_{F2}In addition, it should be noted that, 15.96 THz is just a specific frequency used to demonstrate the validity of the proposed structure. For practical application, working bandwidth of the proposed structure should also be discussed. As QWP, as can be seen from Figs. 2 and 4, PDD stables at 90°±5° from 15.61 THz to 16.34 THz, where the device presents high CPCR≥0.99. And as HWP, as can be seen from Figs. 6 and 7, the device also maintains stable performance (LPCR≥0.99) from 15.38 THz to 16.28 THz, where PDD stables at 180°±10°. Namely, the device can be considered as conventional QWP or HWP with broad working bandwidth. More importantly, within the overlapping section of the working bandwidth (15.61 THz to 16.28 THz) of the QWP and HWP, the device can realize broadband switchable function by adjusting *E _{F}* properly without refabricating its structure.

Finally, we compare this work with other reported papers about HWP-QWP switching based on graphene or other tunable materials, such as liquid crystals and vanadium dioxide, as shown in Table 1. Comparison result shows that the proposed structure can achieve effective QWP-HWP switching, with simple structure and convenient fabrication, high conversion rate, and high reflectivity. However, the structure should still be further optimized to gain a broader relative working bandwidth to the central working frequency.

## 4. Conclusion

This work illustrates how a high-efficiency functionally switchable reflective terahertz polarization converter with fixed structural parameters but different Fermi energy of graphene can be created based on hybrid magnetic plasmon resonance. Detailed simulations and calculations confirm the functions of QWP at 15.96 THz when *E _{F}* = 0.7 eV and of HWP at the same frequency when

*E*= 0.93 eV with high conversion efficiency. Thus, functions of the proposed structure can be switched between QWP and HWP at same frequency without refabricating the structure. Given these notable advances, the proposed structure can be effectively used as a multifunctional wave-plate, demonstrating a conceptually new pathway for functionally switchable polarization modulation. It can also be further explored to be utilized in other interesting applications such as THz imaging or communication systems.

_{F}## Funding

National Natural Science Foundation of China (11674396).

## Acknowledgments

The author thanks his co-workers for giving him the pertinent comments and constructive suggestions on this work.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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