Physical model

The physical model is demonstrated in Fig. 1. Figure 1a shows the operation principle of the device. Graphene with conductivity σ g is placed on the dielectric silica substrate. Light with p-polarization is incident from air onto the graphene/silica medium. According to Maxwell’s equations, together with the boundary conditions (Supplementary Note 1 and Supplementary Figure 1), the reflection coefficient for p-polarization light can be expressed by Eq. (1).

$$r_{\mathrm{p}} = \frac{{\sqrt {\varepsilon _{\mathrm{s}}\mu _{\mathrm{s}}} {\mathrm{cos}}\theta _{\mathrm{i}} - \sqrt {\varepsilon _0\mu _0} {\mathrm{cos}}\Phi + Z_0\sigma _{\mathrm{g}}{\mathrm{cos}}\theta _{\mathrm{i}}{\mathrm{cos}}\Phi }}{{\sqrt {\varepsilon _{\mathrm{s}}\mu _{\mathrm{s}}} {\mathrm{cos}}\theta _{\mathrm{i}} + \sqrt {\varepsilon _0\mu _0} {\mathrm{cos}}\Phi + Z_0\sigma _{\mathrm{g}}{\mathrm{cos}}\theta _{\mathrm{i}}{\mathrm{cos}}\Phi }},$$ (1)

$$\sigma _{\mathrm{g}} = i\frac{{e^2E_{\mathrm{F}}}}{{{\mathrm{\pi }}\hbar }}\frac{{i}}{{\omega + i\tau ^{ - 1}}},$$ (2)

where θ i and Φ are the incident angle and refraction angle, respectively; Z 0 is the impedance of air; ε s , μ s are the permittivity and permeability of the silica substrate, and ε 0 , μ 0 are the permittivity and permeability of air. We used the Drude model to describe the graphene conductivity, which can be expressed by Eq. (2)34,35. \({\mathrm{\tau }}^{ - 1} = \frac{{ev_{\mathrm{F}}^2}}{{E_{\mathrm{F}}\mu _{\mathrm{c}}}}\) is the damping rate of carriers. In this model, we use the ideal value of 10,000 cm2 V−1 s–1 as the carrier mobility μ c of graphene36,37, and assume that the Fermi level of the graphene, E F , ranges from 0.00 to 0.35 eV. v f is the Fermi velocity and ħ is the reduced Planck constant. The parameters of the substrate are ε s = 4 and μ s = 1. The modulation depth in this model is defined as

$${\mathrm{MD}} = \left( {1 - \frac{{\left| r \right|^2}}{{\left| {r_{{\mathrm{max}}}} \right|^2}}} \right) \times 100{\mathrm{\% }},$$ (3)

where r is the reflection coefficient, and r max is the baseline defined as the maximum reflection value in this active system. As an example here, we show the result at 0.8 THz.

Fig. 1 Optical arrangement and device configuration. a Optical path diagram of the incident light from air to graphene/substrate medium. b The reflection amplitude and (c) phase as a function of incident angle, when the graphene conductivity is at 0 and 3.7 mS. The yellow arrows indicate the modulation of intensity and phase. d The three-dimensional (3D) diagram of the device, and (e) the SEM image of the cross section of the device. The scale bar is 5 µm Full size image

We first present the simulation results based on the physical model in Fig. 1a. When the Fermi level of graphene is at the Dirac point, the conductivity of the graphene becomes the minimum. In this case, Eq. (1) returns to the traditional Fresnel equation for reflection. At the Brewster angle, the magnitude of reflection intensity reaches zero, as shown in Fig. 1b. In addition, the phase shift of reflection light is switched from π to 0, when the incident angle changes from θ i > θ B to θ i < θ B , as shown in Fig. 1c. While in the case of non-zero Fermi level, the Brewster angle (defined as the incident angle that gives the smallest reflection amplitude in this case) is shifted and determined by \(\sqrt {\varepsilon _{\mathrm{s}}\mu _{\mathrm{s}}} {\mathrm{cos}}\theta _{\mathrm{i}} - \sqrt {\varepsilon _0\mu _0} {\mathrm{cos}}\,\Phi - Z_0\sigma _{\mathrm{g}}{\mathrm{cos}}\theta _{\mathrm{i}}{\mathrm{cos}}\,\Phi = 0\), if ignoring the imaginary part of graphene’s conductivity. The term introduced by the conductivity, Z 0 σ g cosθ i cosΦ, allows the Brewster angle to be tunable. Because the conductivity of graphene is a complex number, the reflection amplitude at the Brewster angle is not zero (Fig. 1b) and the phase changes gradually with the incident angle close to the Brewster angle (Fig. 1c), which is a little different from the abrupt change for conventional Brewster angle reflection. As shown in Fig. 1, when the conductivity changes from 0 to 3.7 mS, the Brewster angle shifts from 63 to 79°. When the incident angle at 63° is fixed, the reflection amplitude increases from 0 to ~0.52, as indicated by the yellow arrow in Fig. 1b. In this case, the modulation depth is up to 100% due to the perfect “off-state” (|r|=0, at 63°). The insertion loss of this device, \({\mathrm{lg}}_{10}\left( {\left| {r_{{\mathrm{max}}}} \right|^2} \right)\), is −5.6 dB. The insertion loss can be reduced to −3 dB through using higher-quality graphene or composing two-monolayer graphene (Supplementary Figure 2 and Supplementary Note 2). When the incident angle is fixed at 68°, a phase modulation range, as large as about 140°, can also be achieved, as indicated by the yellow arrow in Fig. 1c.

The real device in this work is schematically shown in Fig. 1d, e. A graphene/Al 2 O 3 /TiO x sandwich structure is constructed for a solid-state electrical gate to tune the conductivity of graphene (details of the fabrication can be seen in the Methods section). A 50-nm Al 2 O 3 layer is used as a dielectric gate, which has a negligible effect on the light reflection. The back gate electrode is made of 10-nm thickness of a TiO x (sheet resistance ~2000 Ω) film. This thin and high-resistance film ensures the transparency for the THz light. The transparency of TiO x is verified by THz transmission spectrum and the loss of TiO x is also discussed in Supplementary Figure 3 and Supplementary Note 3. The properties of graphene are first characterized by atomic force microscopy (AFM) for surface morphology with measurement of the thickness and also by Raman spectrum (Supplementary Figure 4 and Supplementary Note 4). The tunability of graphene’s conductivity through the electrical gate is tested by the device’s THz transmission spectra (Supplementary Figure 5 and Supplementary Note 5), as well as verified by a graphene field-effect transistor (G-FET) (Supplementary Figure 6 and Supplementary Note 6).

Actively controlled Brewster angle

In order to obtain the Brewster angle of the device, we measured the reflected THz pulse with an incident angle of 55–75°, as well as gate voltage from −12 to 12 V. Three typically reflected THz pulses in the time domain are shown in Fig. 2a–c. When the incident angle is 55°, the pulse keeps an asymmetric “M” shape from −12 to 12 V. The pulse amplitude decreases monotonically with the gate voltage. When the incident angle is 68°, the pulse shape changed from “M” to “W”, indicating a significant phase change during this process. When the incident angle is 75°, the THz pulse keeps an asymmetric “W” shape from −12 to 12 V and the amplitude decreases monotonically. The pulse shape changing from “M” to “W” indicates that the phase of THz wave is changed by 180°. We plot the curve of the amplitude as a function of incident angle under different gate voltages in Fig. 2d (here the result of 0.8 THz is taken as an example). The results for other frequencies are shown in Supplementary Figure 7 and Supplementary Note 7. The “V” curves show similar trends to the theoretical results (Fig. 1b). To verify our physical model, the conductivity of graphene is extracted by fitting the experimental data to Eq. (1), as shown in Fig. 2d, and then the collected data are compared with the conductivity extracted from transmission spectra (Supplementary Figure S5). The fitted conductivity increases from 0.3 to 2.2 mS, and is consistent with the conductivity extracted from the device’s THz transmission spectra (red curve of Fig. 2e), as well as that from G-FET (Supplementary Figure 6c). The fitting results for other frequencies are shown in Supplementary Figure 7f, which is also consistent with that extracted from transmission spectra. With the parameters extracted from fitting, the theoretical reflection amplitude as a function of incident angle for the frequency is achieved, as shown in Supplementary Figure 8 and Supplementary Note 8. Figure 2f is the Brewster angle as a function of gate voltage at different frequencies, and shows that the Brewster angle can be tuned from about 64.5 to 71.5°. The minute change in different frequencies is due to the weak dispersion relation of conductivity in this frequency region. The passive tuning of Brewster angle by metamaterials and metasurface has been reported in previous works34,35. However, these methods are hard to active tuning of Brewster angle. To change the Brewster angle, the unit structure or periodic parameter for the metamaterial or metasurface should be changed. Besides, Brewster angle of metamaterials/metasurface is always frequency-dependent, due to their resonance nature. As far as we know, this work is the first time to realize an active tunable Brewster angle and achieve broadband response.

Fig. 2 Controlling the Brewster angle by adjusting the conductivity of graphene. The reflection THz time-domain signal under different gate voltages with incident angles of (a) 55, (b) 68, and (c) 75°. d Reflection amplitude as a function of incident angle for the frequency of 0.8 THz. Open symbols represent the experimental amplitude and solid lines show the curve fitting according to Eq. (1). e The fitted conductivity for different gate voltages at the frequency of 0.8 THz (open circles) and the conductivity extracted from the transmission spectrum (red line). f The Brewster angle as a function of gate voltage at different frequencies Full size image

Intensity modulator

From the result in Fig. 2c, we observe that a nearly zero reflection can be achieved at an incident angle of 65° when the gate voltage is 12 V. This indicates great potential to use our device as an intensity modulator with deep modulation depth at this incident angle. Figure 3a shows the evolution of the time-domain waveform reflected from the device as the voltage changed from −12 to 14 V. The waveforms maintain a similar shape, while the peak-to-peak value decreases from 1.37 to 0.067, indicating a nearly unchanged phase and a deeply modulated intensity. The maximum reflected pulse is achieved when the gate voltage is −12 V, and we define it as the baseline for calculating the modulation depth. The reflection amplitude can be achieved through Fourier transform (Supplementary Figure 9 and Supplementary Note 9). The modulation depth of the peak to peak is up to 99.7%, as shown in the inset of Fig. 3a. By Fourier transformation of each signal and comparing them to the baseline, a modulation depth of higher than 99.3% is achieved over the frequency range of 0.5−1.6 THz and the maximum modulation depth is 99.9%, indicating the broadband and spectrally flat modulation. The high modulation depth is due to the nearly zero reflection of THz light near the Brewster angle. Experimental insertion loss of this intensity modulator is about ~−12 dB (detailed calculation is shown in Supplementary Note 2 and Supplementary Figure 2), which is comparable with or better than that of the high-performance narrow bandwidth metadevices38,39,40. This experimental insertion loss is mainly due to the poor quality of commercial graphene and the fabrication process.

Fig. 3 The device operates as a THz intensity modulator, when the incident angle is 65°. a Reflected THz signal in a time domain. b The modulation depth as a function of frequency. The simulated E-field with a conductivity of (c) 0.5 mS, and (d) 2.5 mS under the incident angle of 65°. The dashed line presents the graphene on quartz Full size image

In order to further clarify this phenomenon, we simulated the electric field distribution at the frequency of 0.8 THz, when graphene’s conductivity is 0.5 and 3.5 mS, corresponding to the gate voltage of 14 and −12 V in the experiment. As shown in Fig. 3c, when the conductivity is low (0.5 mS), the incident angle is very close to the Brewster angle of the device, which results in a very weak reflection being observed. While graphene’s conductivity is 3.5 mS, the Brewster angle is moved far away from the current incident angle, as shown in Fig. 2c. This results in a considerably strong reflection field, as illustrated in Fig. 3d. The fitted reflection amplitude as a function of gate voltage shows a high degree of matching with our theoretical results, as well as experimental results (as shown in Supplementary Figure 10 and Supplementary Note 10).

The performance of the device, both of the modulation depth and operation band, is much higher than that of previous results6,7,8,9,10,11,12,–13,23–30,41–43. More importantly, owing to the solid-state-based modulations, our device provides a high modulation speed, as shown in Fig. 4a. The rising time of modulation is about 0.1 ms, as shown in Fig. 4b, corresponding to the modulation speed of about 10 kHz. When the modulation speed is up to 5 and 10 kHz, the modulation depth decreases to 80% and 60%, respectively, as shown in Fig. 4c, d. Theoretically, the modulation speed of a solid-state dielectric-gating device is limited by the resistance–capacitance (RC) time constant. In our device, the effective series resistance (R) comes from the graphene and TiO x . Here, we take a resistance of 250 Ω for the graphene and 1000 Ω for the TiO x , which are calculated by the half of averaged square resistance over the gate voltage range. A capacitance (C) of ~140 nF is calculated by adopting a 50-nm Al 2 O 3 and an active device (graphene) area of 1.0 × 1.0 cm. Therefore, the calculated RC time constant is 1.75 × 10–4 S, corresponding to a modulation speed of ~6 kHz, which agrees well with the experimental result. Higher modulation speed (shorter RC time constant) can be realized through replacing TiO x by a higher-conductivity material (e.g., graphene) and reducing the device size. For instance, by utilizing graphene to replace TiO x and reducing the device size to ~1.0 × 1.0 mm (about two times the wavelength of 0.8 THz), the modulation speed will be ~2.4 MHz, which is comparable with other solid-state THz modulators in previous reports4,7,24,27.

Fig. 4 Modulation speed of the intensity modulator. a The modulated terahertz beam signal under the driving signal of 1-kHz square wave with ±10 V; (b) the zoom-in of the rising edge in (a); the normalized modulation depth under±10-V square wave with (c) 5 kHz and (d) 10 kHz Full size image

Phase modulator

According to the physical model, a broadband phase modulation with a deep modulation range can be achieved by utilizing the phase-jump property across the Brewster angle. Here, we chose the incident angle of 68° to discuss this issue. The reflected THz time-domain signal changes from an asymmetry “W” shape at the gate voltages of 16 V to an asymmetry “M” shape at the gate voltages of –12 V, which indicates that there should be a large phase shift from V g = 16 V to V g = −12 V, as shown in Fig. 5a. The phase shift in the frequency domain is extracted from the time-domain pulse by Fourier transform and referenced to the phase at a gate voltage of 16 V, as shown in Fig. 5b. Over the broad frequency range from 0.5 to 1.6 THz, as the gate voltage decreased from −8 to 16 V, the relative phase shift decreases monotonically with a modulation range over 140°, showing that an ultra-broadband and deep phase modulation is achieved, which is highly consistent with our simulation results. We also noticed that there is a relatively larger modulation range at low frequencies (e.g., –180° at 0.4 THz) and a relatively smaller modulation range at high frequencies (e.g., 140° at 1.6 THz). This frequency dependency is expected to contribute to the frequency-dependent imaginary conductivity of graphene described by the Drude model (Supplementary Note 11 and Supplementary Figure 11).