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Abstract
Dynamic and independent amplitude and phase manipulation are the paramount demand for many advanced wavefronts engineering applications. Currently, the coupling issue between the amplitude and phase hinders the efficient modulation wavefront’s further implementation. This paper proposes and numerically demonstrates the bi-layer stacked graphene Pancharatnam–Berry (P–B) phase metasurface and mono-layer graphene P-B phase metasurface to address the above problem. The simulation results show that the proposed models can achieve the independent control amplitude and phase and significantly reduce their coupling strength. Our findings offer a flexible and straightforward method for precise wave reconstruction applications such as holography, optical tweezers, and high-resolution imaging.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) technology for wavefront manipulation has become a current scientific spotlight owing to its promises of myriad applications, including imaging, spectroscopy detection, food quality control, and communications [1–4]. Recently, metasurface, a class of artificially engineered new materials composed of an electromagnetic (EM) atom with length scales much smaller than the wavelength of the incident light, has been intensively studied as the unparalleled platform to manipulate the EM wave from a multi-dimensions perspective, such as phase, amplitude, frequency or polarization. Owing to its desirable performance, e.g., low profile, lightweight, high-flexibility, and integrated multi-functional characteristics, many functional devices for manipulating the wavefront of the EM wave using the metasurface have been proposed until now [5–11]. Moreover, most conventional metasurface’s EM responses are fixed after the structure is fabricated or manufactured. And it can only be modulated by the limited degree of freedom, such as frequency, polarization, and incident direction. As a matter of fact, dynamic high-resolution wavefront manipulation of EM waves has always been underpinning progress in relevant areas [12–15]. So far, many methods have been put forward to develop the active or tunable metasurface, including phase change materials (PCMs), liquid crystals (LCs), semiconductors, 2D materials, elastic materials, etc [16–25]. Among all of the above tunable artificial materials, graphene-based metasurfaces have more advantages like easy modulation, extensive modulation ranges, and high modulation rates. Simultaneously owing to the increasing maturity of the graphene fabrication process, metasurface integration of the graphene has attracted much attention. Recently, many graphene-based patterned metasurfaces have been presented in the terahertz domain, such as superlens, phase modulators, polarization rotators, etc [26–30].
Amplitude, phase, and polarization are the three most fundamental characteristics describing terahertz waves. It’s well known that the equi-phase plane determines the essential radiation properties of electromagnetic waves. Without the cooperation of phase modulation, other dimensions like polarization and amplitude can not be effectively manipulated [31,32]. So phase modulation plays a pivotal role in associating with the polarization states and amplitudes of electromagnetic waves. Additionally, independent control of the amplitude and phase, i.e., complex field modulation, is critically essential for many advanced applications needing precise reconstruction of the desired wavefronts, such as holography, optical tweezers, high-resolution imaging, etc [11,12,33–38]. However, the intrinsic direct coupling of the phase and amplitude based on the multi-resonance mechanism hinders the independent control process implementation [32,39,40]. Although many devices simultaneously exhibit amplitude and phase control schemes proposed in the past few years, those studies essentially have significant coupling effects [41–43]. Further investigation shows that once amplitude changes at resonance, the electromagnetic performance corresponding to the phase does not maintain good stability but rapidly deteriorates significantly. Due to this challenge, most previous studies can only primarily establish the relationship between polarization and phase profile and their inability to control the scattered amplitude independently [44–48]. Recently Pancharatnam-Berry (P-B) or geometric phase has been applied to perform the amplitude-phase independent decouple modulation [49–53]. But generally, the amplitude response of those metasurfaces is controlled by modifying the geometric dimension. Correspondingly, it is difficult to tune the amplitude profile after complete fabrication [54]. In contrast to the above works, the research on freely dynamic modulation of the amplitude and phase mechanism is still relatively scarce and merits further study. To address the above problems, we introduce another freedom to control the amplitude profile of the reflected wave when the incident beam illuminates the metasurface so that this new design achieves not only the independent control of the phase and amplitude but also the dynamic regulation of the amplitude.
In this paper, we proposed two types of metasurface structure design, i.e., bi-layer stacked graphene-based metasurface and mono-layer graphene-based metasurface, to achieve independent control of the amplitude and phase profile. The P-B phase operation mechanism that maintains the spin state of the incident wave is utilized in the design. The proposed metasurface structures can achieve dynamic independent control of the reflected wave’s amplitude and phase in the working frequency band. Among them, the underlying mechanism of both models are the same, the reflected wave’s phase can be controlled by changing the top layer structure’s rotation angle, and the reflected wave’s amplitude can be dynamically tuned by changing the chemical potential of the graphene. The analysis and discussion of the amplitude and phase coupling relations show that the proposed method can reduce their coupling strength effectively. The 1-bit coding metasurface based on the mono-layer graphene has been designed to demonstrate excellent dynamic modulation performance in practical applications. The proposed structure designs have obvious technical advantages over other terahertz meta devices.
2. Working principle of graphene controllability
The graphene can be modeled as an ultra-thin conductive layer. The corresponding surface conductivity can be modeled using the Kubo formula. It involves two processes careering intraband transitions and interband transitions, which can be expressed as $\sigma (\omega )=\sigma _{{\rm intra}}(\omega )+\sigma _{\rm inter}(\omega )$, where $\omega$ is the angular frequency of the wave. For the low terahertz regime, the graphene surface conductivity is dominantly determined by intraband transitions, and interband contributions of graphene conductivity are negligible [55,56]. So the graphene frequency-dependent conductivity obtained from Kubo’s formula can be described as follows [57,58].
According to Eq. (1), the conductivity of graphene is a function of relaxation time, temperature, and chemical potential. It is assumed in simulation that the relaxation time $\tau =0.1$ fs, and the temperature $T=300$ K. Figure 1 is the variation of graphene surface conductivity with frequency and chemical potential. We can see that the conductivity of graphene decreases with increasing frequency while keeping the chemical potential fixed. The conductivity rises as the chemical potential of graphene continually increases while controlling the frequency fixed. Therefore, it is clear that the conductivity of graphene can be actively tuned by the chemical potential $\mu _{\rm c}$ and frequency. The chemical potential can be controlled by using the biased gate-voltage source [60,61]. However, in practical design, the structured graphene pattern layer’s surface conductivity is challenging to obtain analytically. So we need to use the numerical method calculating the transmission spectrum to retrieve it. Figure 2 is the schematic principle to retrieve the effective surface conductivity of the structured graphene pattern based on the transmission line theory. The graphene pattern layer between two dielectrics is equivalent to a load attached to the junction between two transmission lines [62]. The relationship can be described in Eq. (3). The $Z_{0}$ is the air impedance, and $t$ is the transmission coefficient.
Fig. 1. Graphene surface conductivity is the function of the frequency and chemical potential. (a) Real and (b) imaginary parts of the surface conductivity.
Fig. 2. Schematic principle to retrieve the effective surface conductivity of the structured graphene pattern.
3. Bi-layer stacked graphene P-B phase metasurface design
A schematic diagram of the P-B phase metasurface unit cell based on the bi-layer stacked graphene structure is shown in Fig. 3(a). To take full advantage of the ultra-wide terahertz spectrum and overcome the problem of narrow operating bandwidth based on multiple resonance mechanisms, the metasurface design in this section still utilizes the P-B phase mechanism with the advantage of broadband achromatic properties [63,64]. The geometric relationship and dimensions of the proposed metasurface are shown in Fig. 3(b),(c) and (d), where $p=48\,\mu$m, $w_1=6.33\,\mu$m, $w_2=2.16\,\mu$m, $w_3= 6.17\,\mu$m, $d=27.36\,\mu$m, $a_1=24\,\mu$m, $a_2=12\,\mu$m, $b_1=6\,\mu$m, $b_2=6\,\mu$m $\alpha = 40^{\circ }$. The other parameters in Fig. 4(a) are ${\rm d_1}= 3\,\mu$m, ${\rm d_2}= 5\,\mu$m, ${\rm d_3}=3\,\mu$m.
Fig. 3. Schematic diagram of the P-B phase metasurface unit cell based on the bi-layer stacked graphene structure. (b), (c) and (d) Geometric relationship and dimensions of the bi-layer stacked graphene P-B metasurface unit cell.
Fig. 4. Theoretical model of the bi-layer stacked graphene metasurface. (a) Schematic of the unit cell model. (b) Transmission line model.
Figure 4 shows the theoretical model of independent control amplitude and phase metasurface unit cell based on the bi-layer stacked graphene proposed in this section. As shown in Fig. 4(a), the theoretical model using the two graphene ribbons, metal ground, metal top layer, and three dielectric media represents the proposed bi-layer stacked graphene metasurface. The two layers of graphene ribbons are represented by their effective surface conductivities, respectively, according to the retrieve method in section 2. The metal used in the model is treated as the Perfect Electric Conductor (PEC). To have a clear physical insight into the proposed stacked graphene metasurface, we have established a transmission line model, as shown in Fig. 4(b). The bottom PEC layer corresponds to a short circuit at one end. The dielectric media layer is represented by a cascaded transmission line with specific characteristic impedance $Z_{\rm c1}$, $Z_{\rm c2}$, and $Z_{\rm c3}$, respectively, which are determined by their permittivity. Here the role of spacer thickness is to maintain impedance matching. The two-dimensional graphene ribbons are represented as corresponding lumped impedances ($Z_{\rm g1}$ and $Z_{\rm g2}$) at the connecting points between different transmission lines. Therefore, the entire metasurface can be modeled as an equivalent transmission line with parallel lumped elements. Here $Z_1$, $Z_2$, $Z_3$, $Z_4$, and $Z_5$ are the input impedance looking from the arrow positions, respectively. $Z_{\rm in}$ is obviously the total input impedance looking from the air. The dielectric material selected in this paper is SiC, which has very little loss in the terahertz band. So the transmission line in Fig. 4 can be treated as a lossless transmission line. The propagation constant of EM waves in dielectric media is $\beta$. Therefore, Eqs. (4–10) are easily obtained according to the transmission line theory.
Furthermore, the S-parameters describing the port and the input impedance have a relationship described as shown in Eq. (10).
Without loss generality, a bi-layer stacked graphene structure studied in this section simplifies the analysis of the problem. However, according to the transmission line theory, it is easy to know that this analysis method is identical to the more than two layers cascade approach. Therefore, the design principle proposed in this section is somewhat general and can be directly extended to the case of more layers of stacked graphene.
The prototype of the designed reflective P-B metasurface is numerically analyzed using the CST Microwave Studio based on the Finite Difference Time Domain (FDTD) method in the terahertz frequency regime. Figure 5 shows the simulation settings of the boundary conditions and excitation. The periodic boundary conditions are used in the $x$ and $y$ directions, the PEC boundary condition is used in the -$z$ direction to extract only the reflection coefficient of the metasurface, and plane wave incidence is selected as the excitation type.
Figure 6 shows the EM response of the metasurface in the frequency domain under the excitation of the linear polarized plane wave incidence when both two graphene strips’ chemical potentials satisfy $\mu _{\rm c1}=\mu _{\rm c2}=0\,{\rm eV}$. In Fig. 6(a), both the amplitude of the co-polarized reflection coefficients are domain over 0.82 in the entire frequency band range from 1.5 THz to 2.3 THz, indicating that the structure has very high efficiency. Figure 6(b) shows the phase angles of the two co-polarized reflection coefficients and the phase difference between them, from which it can be seen that the phase difference is around 180$^\circ$. Thus the designed metasurface unit cell structure satisfies all the conditions required for the reflective P-B phase procedure [63].
Fig. 6. Simulated results of the proposed bi-layer stacked graphene metasurface at the $\mu _{\rm c1}=\mu _{\rm c2}=0\,{\rm eV}$ under the linear polarized wave incidence. (a) The magnitude of the co-polarized reflection coefficients ${\rm r}_{\rm xx}$, and ${\rm r}_{\rm yy}$. (b) The phase angle of the two co-polarized reflection coefficients $\mathrm{\phi} _{\rm xx}$, $\mathrm{\phi} _{\rm yy}$, and the phase difference between them $\mathrm{\phi} _{\rm xx} -\mathrm{\phi} _{\rm yy}$.
Next, we calculate the response of the phase and magnitude of the reflection coefficient ${\rm R}_{rr}$ when the chemical potentials of two graphene strips satisfy $\mu _{\rm c1}=\mu _{\rm c2}=0.5\,{\rm eV}$. To clearly show the modulation process, here, only select the four rotation angles, i.e., 0$^\circ$, 36$^\circ$, 108$^\circ$ and 144$^\circ$. As shown in Fig. 8(a), the different phase angles remain parallel, indicating their good linearity performance. And the phase angle difference is the two times rotation angle, which demonstrates the function of the P-B phase mechanism for phase tuning working properly. The co-polarized reflection coefficient in Fig. 8(b) is around 0.6, much smaller than the amplitude in Fig. 7(b). So the reflected wave amplitude at the metasurface can be effectively modulated by changing the chemical potentials of the two graphene strips. Although the amplitudes of the reflection coefficients corresponding to different rotation angles do not overlap entirely in this process, the overall deviation is insignificant, so it can be assumed that the phase modulation still has a negligible effect on the related amplitude response.
Fig. 7. Simulated results of the proposed bi-layer stacked graphene metasurface at the $\mu _{\rm c1}=\mu _{\rm c2}=0\,{\rm eV}$. (a) The P-B phase of the reflection coefficient ${\rm R}_{\rm rr}$ and (b) the magnitude of the reflection coefficients ${\rm R}_{\rm rr}$, and ${\rm R}_{\rm lr}$.
Fig. 8. Simulated results of the proposed bi-layer stacked graphene metasurface at the $\mu _{\rm c1}=\mu _{\rm c2}=0.5\,{\rm eV}$. (a) The P-B phase and (b) the magnitude of the reflection coefficient ${\rm R}_{\rm rr}$.
Furthermore, we demonstrate the variation of the phase and amplitude of the reflection coefficient ${\rm R}_{rr}$ in the frequency domain for the same operating conditions when the chemical potentials of two graphene strips satisfy $\mu _{\rm c1}=\mu _{\rm c2}=1\,{\rm eV}$. From Fig. 9(a), it can be seen that the except for a bit of nonlinearity enhancement at higher frequencies, there is a dominant linearity characteristic in the frequency band. Therefore, the effect on the P-B phase during manipulation amplitude by changing the chemical potential is also tiny. As shown in Fig. 9(b), the reflection amplitude decreases further as the chemical potential increases to 1 eV. Although the fluctuation range, in this case, becomes more prominent than in Fig. 8(b), the overall fluctuation is manageable within 0.2. The change is minor at 1.83 THz, which can be controlled within 0.1.
Fig. 9. Simulated results of the proposed bi-layer stacked graphene metasurface at the $\mu _{\rm c1}=\mu _{\rm c2}=1\,{\rm eV}$. (a) The P-B phase and (b) the magnitude of the reflection coefficient ${\rm R}_{\rm rr}$.
From the above discussion, the independent control of the amplitude and phase using the metasurface based on bi-layer stacked graphene can achieve the desired goal and significantly reduce the coupling strength between phase and amplitude responses. Specifically, when the P-B phase mechanism regulates the phase, the degree of influence on the amplitude response is minimal. The effect on the P-B phase response is minimal when the chemical potential is changed to control the amplitude. However, it should also be noted that the magnitude of the coupling based on the bi-layer stacked graphene metasurface designed here is related to the level of the chemical potential of graphene. Specifically, from Fig. 7(b), Fig. 8(b), and Fig. 9(b), it can be concluded that the lower the chemical potential, the lower the corresponding coupling strength. So the following section proposes a new, improved model to address this issue.
4. Mono-layer graphene P-B phase metasurface design
Unlike above the bi-layer stacked graphene metasurface, this section proposes a new, improved model based on the mono-layer graphene P-B phase metasurface. Its unit cell structure is schematically shown in Fig. 10. The specific geometric dimensions are shown in Fig. 10(b), (c) and (d), in which $p = 48\, \mu$m, $w_1 = 6.33\, \mu$m, $w_2 = 2.16 \, \mu$m, $w_3 = 6.17 \, \mu$m, $d = 27.36 \, \mu$m, $r_1 = 13 \, \mu$m, $r_2 = 10 \,\mu$m, $d_1= 0.7 \, \mu$m, $d_2= 10.3 \, \mu$m, and $\alpha = 40^{\circ }$. And the simulation settings in this section are identical to those in the previous study. Figure 11 shows the simulated results of the proposed mono-layer graphene P-B phase metasurface at the $\mu _{\rm c}=0\,{\rm eV}$ under the right-handed circular polarized wave incidence. Figure 11(a) is the phase of the reflection coefficient ${\rm R}_{\rm rr}$. The introduced P-B phase angle is $60^{\circ }$ for a $30^{\circ }$ rotation step, which satisfies the P-B phase modulation function. Figure 11(b) shows the amplitude responses corresponding to the reflection coefficients ${\rm R}_{\rm rr}$ and ${\rm R}_{\rm lr}$, where the co-polarization component is significantly higher than the cross-polarization component over the whole frequency range. Briefly, the results show the excellent performance of the P-B phase modulation. And those agree well with the previous study of bi-layer stacked graphene metasurface.
Fig. 10. (a) Schematic of the unit cell structure of the mono-layer graphene P-B phase metasurface. (b), (c) and (d) Geometric relationship and dimensions of the mono-layer graphene P-B metasurface unit cell.
Fig. 11. Simulated results of the proposed mono-layer graphene metasurface at the $\mu _{\rm c}=0\,{\rm eV}$. (a) The P-B phase of the reflection coefficient ${\rm R}_{\rm rr}$ and (b) the magnitude of the reflection coefficients ${\rm R}_{\rm rr}$, and ${\rm R}_{\rm lr}$.
Here the mono-layer graphene metasurface’s EM responses related to the variation rotation angles under the different chemical potentials will be demonstrated to present a better performance of independently controlling the phase and amplitude. Figure 12 and Fig. 13 show the phase and magnitude of the reflection coefficient ${\rm R}_{\rm rr}$ at different chemical potentials when the metasurface rotation angle is $0^{\circ }$ and $90^{\circ }$, respectively. Notably, in Fig. 12(a) and Fig. 13(a), the different phase angles of the reflection coefficient ${\rm R}_{\rm rr}$ exhibit the excellent stable linear property, and complete overlap in the frequency range from 1.5 to 2.3 THz, despite the graphene’s chemical potential changing from 0 to 1 eV. Here it is also noted that the specific phase angle in Fig. 12(a) and Fig. 13(a) is consistent with the result shown in Fig. 11(a), respectively. In addition, from the amplitude modulation in Fig. 12(b) and Fig. 13(b), it can be seen that the reflection amplitude changes as the variation of the chemical potential of graphene. Specifically, at 1.9 THz, when the chemical potential increases from 0 eV to 1 eV, the amplitude decreases from 0.95 to 0.75. And the modulation effect is very significant. Meanwhile, it can be seen from the comparison of Fig. 12(b) with Fig. 13(b) that the amplitude response is almost identical when changing the metasurface unit cell’s rotation angle from $0^{\circ }$ to $90^{\circ }$. Therefore, from the above discussion, it can be concluded that the amplitude and phase independent control of the metasurface based on mono-layer graphene in this section can achieve the desired goal and significantly reduce the degree of coupling between phase and amplitude.
Fig. 12. (a) The P-B phase and (b) magnitude of the reflection coefficient ${\rm R}_{\rm rr}$ at different chemical potentials when the metasurface unit cell’s rotation angle is $0^{\circ }$.
Fig. 13. (a) The P-B phase and (b) magnitude of the reflection coefficient ${\rm R}_{\rm rr}$ at different chemical potentials when the metasurface unit cell’s rotation angle is $90^{\circ }$.
As shown in the preceding bi-layer stacked graphene P-B phase metasurface, the coupling effect between the phase and amplitude is related to the level of the chemical potential, which will apparently affect the achieving of the efficiently independent modulation. Furthermore, responses of the phase and magnitude of the reflection coefficient ${\rm R}_{\rm rr}$ when the chemical potential satisfies $\mu _{\rm c}=1\, {\rm eV}$ are shown in Fig. 14. By comparing with the same responses of the co-polarized reflection coefficient ${\rm R}_{\rm rr}$ based on the bi-layer stacked graphene metasurface in the previous section, i.e., Fig. 9(b), it can be found that fluctuation range of the amplitude is manageable within 0.01 at 1.9 THz. So the new design proposed in this section can achieve a better desirable decoupling performance between magnitude and phase at the higher chemical potential level than the bi-layer stacked graphene metasurface. Therefore, the proposed new, improved mono-layer graphene metasurface is fully equipped with perfect independent control of amplitude and phase and the ability to modulate the amplitude dynamically.
Fig. 14. Simulated results of the proposed mono-layer graphene metasurface at the $\mu _{\rm c}=1\,{\rm eV}$. (a) The P-B phase of the reflection coefficient ${\rm R}_{\rm rr}$ and (b) the magnitude of the reflection coefficients ${\rm R}_{\rm rr}$.
Recently, coding metasurfaces combining the binary operations of digital systems have attracted much attention [16,42,65,66]. Based on the previous analysis and discussion, the proposed mono-layer graphene model is applied to a 1-bit coding metasurface to demonstrate its functional performance in a practical application, i.e., the dynamic tuning of the radiation characteristics. And the proposed 1-bit coding metasurface in this section consists of two coding states, i.e., "0" and "1", and their related phases are $0^{\circ }$ and $180^{\circ }$, respectively. Its far-field bi-static RCS patterns corresponding to the coding sequences "0101/1010" under the different chemical potentials are shown in Fig. 15. Furthermore, Fig. 15(a) refers to the case when the chemical potential is 0 eV, while Fig. 15(b) refers to the case when the chemical potential is 1 eV. As can be seen from them, when the chemical potential of graphene increases from 0 eV to 1 eV, the magnitude of the four-beam scattering pattern decreases from 2.55e-5$\rm \,m^{2}$ to 1.75e-5$\rm \, m^{2}$. Therefore, the application example of this 1-bit coding metasurface further demonstrates the excellent performance of the dynamic independent amplitude and phase control metasurface, which is critically important for the many precise wavefronts applications, such as holography and high-resolution imaging, etc [43,67].
Fig. 15. Far-field bi-static RCS pattern of the 1-bit coding metasurface with coding sequences "0101/1010" under the different chemical potentials. Chemical potential of the graphene is (a) 0 eV, (b) 1 eV, respectively.
5. Conclusion
This paper proposes two graphene-based metasurfaces, i.e., bi-layer stacked P-B phase metasurface and mono-layer P-B phase metasurface, to address the coupling issues existing in the conventional metasurface design. The decoupling principle of amplitude and phase response is achieved by combining the P-B phase and using the graphene functional material. Specifically, the underlying mechanism of both models are the same, the phase is modulated by in-plane rotation of the unit cell structure, and the amplitude is dynamically tuned by changing the chemical potential of graphene. Their modulation performance and the degree of coupling between phase and amplitude have been studied in detail. Finally, the 1-bit coding P-B phase metasurface based on the mono-layer graphene model is designed to demonstrate excellent dynamic control amplitude capability. The results in this paper will benefit the many advanced wavefronts engineering applications.
Funding
National Natural Science Foundation of China (51877015, 62001068, U1831117); Natural Science Foundation of Chongqing (cstc2021jcyj-bsh0198).
Acknowledgments
X. Wu thanks the Project supported by the graduate research and innovation foundation of Chongqing, China (Grant NO. CYS17039), Project NO. 2018CDYJSY0055, NO. 106112017CDJXSYY0002 Supported by the Fundamental Research Funds for the Central Universities and supports from the China Scholarship Council (NO.CSC201806050092).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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