Electrically switchable and tunable infrared light modulator based on functional graphene metasurface

2.1 Structure design and working status of the absorber

The schematic diagram of the modulator proposed in this work is shown in Figure 1. The structure cell comprises a full monolayer graphene sheet sandwiched between a spacer layer and an antenna block patterned with an elliptical nanohole. The symmetry axis of the ellipse pattern is rotated 45° counterclockwise from the x-axis direction, indicating anisotropy in the x–y plane. Here, a and b represent the semi-length of the minor and major axis of the elliptical nanohole pattern. h₁ and h₂ denote the thicknesses of the antenna nanohole and spacer layer. The bottom silver layer serves as a reflector with a thickness of 200 nm, bigger enough than the skin depth to suppress the wave transmission.

Figure 1:

Schematic of the infrared light modulator consisting of a full monolayer graphene sheet with electrically tunable chemical potential and a FP-like ellipse-patterned metasurface. a = 200 nm, b = 600 nm. The bias gate voltage and the polarization status of the incident and reflected light are also shown.

The insert shows the electromagnetic wave’s coordinates. The light is a normal incidence along the z-axis, vertically irradiating from the top, and the nanostructure supports several GMRs. When these two wavevectors match, the incident wave will couple with the inherent guided mode, meeting the resonance condition. As a result of photon localization from the resonance, the absorption of the graphene would be significantly enhanced. Thus, the reflection is generated with graphene’s electrically switchable and tunable optical dielectric. Furthermore, a converter can be realized with the implementation of modulation by the anisotropy antenna layer. Since the absorption of graphene is mainly due to the bias-gate-voltage dependent direct interband transition, a gate voltage is added to realize the dynamic switching between different statuses, i.e., absorber and polarization converter.

The finite-element method is applied to investigate the property of the proposed metasurface using the COMSOL software. In the theoretical simulation, graphene is treated as an effective surface conductor for its ultra-thin thickness, and the Kubo formalisms can describe the surface conductivity of graphene (σ_gra) with the summary of the intra-band term (σ_intra) and the inter-band term (σ_inter) as the following [41, 44]:

Here, e, ω, k_B, T, ℏ, and E_f represent the electron charge, the frequency of incident light, the Boltzmann constant, the temperature in kelvin, the reduced Planck constant, and Fermi energy (or chemical potential), respectively. τ denotes the momentum relaxation time for the charge carrier scattering. It relates with the carrier mobility μ as τ = μE_f/(ev_f ²), where v_f is the Fermi velocity of about 10⁶ m/s [45]. Previous reports showed that the carrier mobility of graphene on SiO₂ substrate could reach 4 × 10⁴ cm²/(V.s) at room temperature. Note that the carrier concentration will not increase infinitely with the increase of τ. It has a critical saturation value when obtaining the highest absorption efficiency. Therefore, either at the low or high frequency resonance, the relaxation time variation causes little wave shift, which indicates that the dynamical tunability is only related to the Fermi level of graphene. Hence, the τ is set with the value of 0.5 ps in the simulation to ensure the credibility of results [41, 46, 47]. As for an electromagnetic wave propagating inside the graphene layer with a finite thickness Δ, it has a relation of Δ × H = −iωε₀E + J = −iωε₀ε_gE. Here J = E_||σ/Δ. E_|| is the electric field along the graphene surface. Therefore, the anisotropic relative permittivity tensor of graphene can be described as:

where ε₀ is the graphene permittivity, and t_g stands for the thickness of the monolayer graphene sheet (typically 0.34 nm). A silver layer is used as the reflector, and its dielectrics permittivity is described by the Drude model εm(ω) = ε∞ − (ω_p)²/(ω² + jωγ). ε∞ is the dielectric constant at the infinite frequency, and the corresponding electron collision and bulk plasma frequencies are 0.018 eV and 9.1 eV, respectively. The proportional ratio of the radiation absorbed by the monolayer graphene is dependent on the electrical energy incident upon its surface and can be obtained as [40]:

Here ω(x, z) is the power dissipation density that depends on a combination of electric field and material loss. In the simulation, periodical boundary conditions are applied to a single calculation cell to approximate an infinite array. The antenna layer and spacer thicknesses are designed to be 280 nm and 520 nm with dielectric constants of 1.48 and 1.45. The background is assumed to be air. A periodicity (P) of 1258 nm is used to suppress the high-order diffraction at the wavelength of 1550 nm. E_f is set as 0.3 eV at the beginning, where the graphene exhibits as a loss material. Only x-polarized incident light is considered in this study to prove the design concept.

Reflection coefficients of the co- (R _xx ) and cross- (R _yx ) polarization together with graphene absorption are shown in Figure 2(a). The graphene absorption dramatically increases with declined R _xx , indicating the reduction attributed to the enhanced graphene absorption. In contrast, the cross reflectivity (blue line) is suppressed. The detailed electric field distribution at the horizontal surface of the graphene sheet and the vertical planes (x–z and y–z planes) are shown in 2(b). The incident light forms a well-confined resonant mode, and there is no interference between the incidence and reflection, forming a typical standing wave profile. It also indicates that the modulation structure meets the critical coupling condition, where the leakage rate of the guided mode resonance out of the structure is equal to the absorption rate of that resonance in the absorptive materials. Moreover, the confined electric field localizes compactly at the spacer-antenna boundary where the graphene is deposited, leading to a greatly enhanced graphene-light interaction and further modulated status of the reflected light.

Figure 2:

Calculation results: (a) Simulation intensity of R _xx , R _yx , and absorption of the monolayer graphene under normal x-polarized incidence (τ = 0.5 ps). The insert indicates direct interband transition. (b) Normalized electric field distributions of the structure with an x-polarized incidence of a plane wave. Note that the subscripts i and j in coefficients R _ij refer to the reflection and illumination polarization states.

2.2 Dependence of graphene dielectric on bias-gate controlled Fermi energy

The enhanced graphene absorption mainly attributes to the strengthened induced electric current on its surface caused by the interband transition of electrons. It can be readily understood that, for the frequency region where ℏω_o > 2E_f, the absorbed light is transferred into kinetic energy for electrons to realize the direct interband transition. While for ℏω_o < 2E_f, the transition is suspended due to the declined optical energy (ℏω_o), and then the graphene performs as a lossless material. Therefore, graphene’s Fermi energy (E_f) can be adjusted by controlling the bias-gate voltage. It offers a practical approach to realizing the switchability and turnability of our modulator [48, 49]. Since the interband transition is reflected in the spectral properties of its dielectric, the real and imaginary parts of graphene dielectric permittivity under different Fermi energy are shown in Figure 3.

Figure 3:

The real (a) and imaginary (b) parts of dielectric coefficient of graphene under different Fermi energy.

The dielectric dispersion of graphene is quite different from conventional 2D materials due to its conduction and valence band meeting at the Dirac point where the Fermi level locates. Taking the E_f = 0.37 eV (solid red lines) as a typical analysis, the real and imaginary appear entirely different at two wavelength regions divided by an apparent boundary wavelength of about 1.68 μm. With the wavelength increasing across the boundary, the real part shows a sudden rise and rapid drop. At the same time, the imaginary performs a dramatic decline from a certain big value to almost zero, leading to a sharp peak in the real part and a sudden fall in the imaginary. Since the imaginary component indicates the absorption property of a material, graphene acts as a loss or lossless dielectric on the left or right side of the boundary. It also indicates that the wave region smaller or higher than 1.68 μm corresponds to the wavelength domain of ℏω_o > 2E_f or ℏω_o < 2E_f, individually. Thus, the boundary wave has a corresponding energy, i.e., interband transition energy of 0.74 eV. Further, with varying Fermi energy from 0.36 eV to 0.48 eV, the diving boundary shifts from 1.72 μm to 1.29 μm. As such, the lossy and lossless wavelength domains change reversely. Therefore, by adjusting the Fermi energy using the gate-bias voltage, the graphene can be switched between lossy and lossless material at a specific wavelength.

2.3 Working status of the reflector as polarization converter

In the analysis of Figure 2, E_f is set as 0.3 eV with corresponding interband transition located at 2070 nm, and the graphene layer behaves as an absorption dielectric. In contrast, when setting E_f as 0.45 eV, the interband transition wavelength moves to 1380 nm, and graphene performs as a lossless material in the calculation range of 1500–1600 nm. A dramatically increased R _yx at 1550 nm is found in Figure 4(a) with a suppressed graphene absorption, indicating an efficiently converted cross-polarization component in the reflection. Mathematically, the polarization conversion ratio (PCR) is defined as:

which is used to describe the conversion efficiency. The phase difference (PD) between R _yx and R _xx is deemed as Δφ = φ _yx − φ _xx implying all the possible polarization states (linear, circular, and elliptic) of reflected light. When cos(Δφ) = ±1, the polarization direction would be transformed to y-axis. If the magnitudes and phase shift satisfy the relationship R _yx = R _xx and Δφ = 2nπ ± π/2 (n is an integer), the reflected light turns to be a left- or right-handed circularly polarized. In other cases, it turns out to be elliptic polarization.

Figure 4:

Conversion properties: (a) Co- and cross-polarization reflection intensity coefficients (R _xx and R _yx ) of the proposed light modulator. The insert implies the suspended interband transition. (b) The PCR and PD curves.

The PCR and PD spectra are depicted in Figure 4(b). The PD can only be either −90 or 90 (degrees), indicating the elliptic status of reflection. For the wavelength range with PCR = 0, the reflection is a linear polarization with a phase change of ±π/2, while for 1550 nm, the modulator turns out to be an elliptic polarization converter. Thus, the proposed graphene metasurface can be manually switched from the operational status of a perfect absorber to a polarization converter by adjusting the Fermi energy.

2.4 Mechanism of the metasurface for polarization conversion

To clarify the conversion mechanism, two mutually perpendicular symmetry axis, along the short and long-axis ellipse patterns, are defined as the u- and v-axis shown in Figure 5(a). Then, the eigenmode response of the modulator, whose polarization is along the u- and v-axis, is studied. Since the incident EM (e^ik_zz) is decomposed into two orthogonal components, i.e., E _u ⁱ and E _v ⁱ in u–v coordinates expressed in equation (5), the reflection can be calculated as equation (6).

Figure 5:

Eigenmode response: (a) Reflection coefficients of the u- and v-polarization illustration, and (b) its corresponding phase response (left axis) and differences (right axis). The first and second subscripts in R _ab represent the polarization status of the reflection and incident.

Here, the û and v ̂ denote the unit vectors along the u–v coordinates. Equations reveal that both the amplitude and phase are modulated. Then, with the eigenmode incident, the detailed reflectivity and PD are calculated and shown in Figure 5. It is critical to find that, for each eigenmode, the polarization status is sustained while giving different responses in both amplitude and phase. For the u-polarization wave, it has a relatively small magnitude reflection but a faster phase transition. Moreover, the PD (Δφ) reaches a peak value at 1550 nm, shown in Figure 5(b) as the green line, which indicates the dramatical phase accumulation at the wavelength point [33]. Therefore, the electric field of x-polarized incidence can be decomposed into u and v components to analyze the eigenmode response to each element. Then the reflection is the re-coupling of these two modulated components.

Besides, for an x-polarization incidence, z-component electric field distributions on the graphene surface, at 1520, 1550, and 1570 (nm), are extracted and shown in Figure 6, where the localized surface plasmon (LSP) is efficiently excited around the ellipse pattern forming electric dipoles [50, 51]. For 1520 nm, the LSP is along the horizontal direction parallel to the incident, and no conversion occurs. While for 1550 nm, the induced dipole is mainly along the u-axis and has a response to the u-component of the incident dramatically. Moreover, for the situation of 1570 nm, electric dipoles are efficiently generated in both the u and v axis and have different responses to both components. Further, with recoupling, the reflection’s amplitude and polarization status can be modulated, realizing a converter.

Figure 6:

Distribution of the z-component of the electric field (E _z ) on the graphene layer for normal x-polarization incidence at (a) 1520 nm, (b) 1550 nm, and (c) 1570 nm. In Figures 5 and 6, a and b in E ^a _b denote the incident (i) or reflection (r) and the polarization status.