Ultrawideband Solid-State Terahertz Phase Shifter Electrically Modulated by Tunable Conductive Interface in Total Internal Reflection Geometry

photonics1
May. 16, 2024

Abstract

Phase modulation plays a crucial role in various terahertz applications, including radar detection, biomedical imaging, and data communication. Existing terahertz phase shifters typically rely on tuning the resonant effect of metamaterial structures to achieve a narrow bandwidth phase shift. However, the terahertz band offers a wide bandwidth resource, which has great advantages in high longitudinal resolution detection, high-capacity communication, spectral imaging, and so on. Here, we propose and demonstrate an ultrawideband terahertz phase shifting mechanism that utilizes an optical conductivity tunable interface combined with a nonresonant metasurface operating in the total internal reflection geometry. This approach effectively modulates the phase of the reflected terahertz signal in an ultrawideband. To implement this mechanism, we designed a structure consisting of graphene-loaded nonresonant periodic metal microslits arranged in the total internal reflection geometry. By controlling the gate voltage of the graphene within a range of ±5 V, an averaged ∼120° continuous phase shift in the frequency range of 0.4 to 1.2 THz was achieved, with a group delay less than 50 ps. Notably, in the frequency range of 1 to 1.2 THz, the phase modulation exhibited a linear relationship with the driving voltage. Our device demonstrated minimal fluctuations in the reflected amplitude, with a deviation of less than 1 dB and an insertion loss of less than 10 dB. Additionally, the modulation speed of this solid-state device reached the kHz level. Remarkably, the phase modulation bandwidth (Δf/f) achieved approximately 100% of the arithmetic center frequency at 0.8 THz, surpassing the definition of ultrawideband, which typically encompasses 20% of the center frequency. To the best of our knowledge, this is the first and most wideband phase shifter developed for the terahertz regime with the lowest recorded group delay to date.

Introduction

Terahertz (THz) technology has immense potential across a range of fields, including radar detection, (1−3) high-rate data communications, (4−8) real-time and high-resolution imaging, (9−11) material characterization, (12−14) and other fields. (15,16) However, due to the unique characteristics and constraints of terahertz waves, the vast majority of these applications are entirely dependent upon the fundamental THz devices, such as sources, detectors, and waveguides, as reviewed in 2017 (17) and 2023. (18) Developing terahertz phase modulators, (19) particularly those with wide bandwidths, remains a challenging task in this field. In 2000, Kersting et al. proposed the first terahertz phase modulator, a semiconductor device that enabled electronic control to modulate the phase of terahertz waves, thus opening the door to research on terahertz phase modulation. Existing techniques primarily rely on metamaterial structures that manipulate their resonant behavior between inductive and capacitive modes to induce a phase change. For example, Chen reported the first solid-state terahertz phase modulator using a metasurface integrated GaAs Schottky diode, achieving a phase shift of approximately π/6 radians. (20) Subsequent designs have explored a photoinduced vanadium-dioxide-coupled nanostructure, (21) a gallium-nitride-based high electron mobility transistor (HEMT), (22) and gate-controlled graphene metasurfaces. (23) However, these metamaterial-based designs are limited to narrow bandwidth operation. One possible theoretical method to broaden the operational bandwidth is to pattern the graphene layer into a metasurface structure. (24−28) Although a graphene-based Brewster angle device demonstrated wideband phase modulation capability, it only achieved a discontinuous phase shift. (29) The utilization of liquid crystal material in the total internal reflection (TIR) geometry has been explored to alter the phase of reflected terahertz light. (30) However, the birefringence of liquid crystals is typically low in the terahertz region, (31) resulting in limited phase shift. Another approach involved a terahertz polarization converter that employed ion-gel gated two graphene layers in the TIR geometry to induce phase changes for both s- and p-polarizations, but the ion-gel device operates at a slow speed and achieves a phase shift of less than 80°. (32) In this study, we propose an ultrawideband phase shifting mechanism utilizing an optically thin conductive layer in the TIR geometry at the interface (referred to as CI-TIR). This approach enables efficient phase modulation of the reflected light while preserving the wideband characteristics for both s- and p-polarizations. For this purpose, we employ a single layer of graphene as the conductive interface. Graphene is chosen for its atomic thickness, tunable sheet conductivity, and nearly frequency-independent conductivity within the terahertz band. (33,34) While previous work has explored the use of graphene for amplitude modulation of terahertz radiation, (35−37) its potential for phase modulation has not been extensively investigated. In this paper, we demonstrate an effective phase modulation technique by utilizing a graphene-loaded nonresonant metasurface composed of a periodic arrangement of metal microslits. This structure achieves a continuous phase shift of approximately 120° in the frequency range of 0.4 to 1.2 THz. Notably, the phase modulation bandwidth (Δf/f) reaches approximately 100% at f = 0.8 THz, surpassing the definition of ultrawideband (UWB), which typically covers 20% of the center frequency. Particularly, in the frequency range between 1 and 1.2 THz, the phase shift exhibits a linear dependence on the driving voltage. The amplitude fluctuation during phase modulation remains below 1 dB across the entire operational bandwidth while incurring an insertion loss of approximately 10 dB. Moreover, the phase modulation speed of our device operates at the kHz level. Overall, our device showcases ultrawideband, linear, low-loss, and near-ideal phase-only modulation capabilities.

Theory and Simulation

When light propagates from a dense medium to a less dense medium at an angle greater than the critical angle, total internal reflection occurs, causing the incident light to be reflected back into the dense medium. However, the incident light actually penetrates into the less dense medium as an evanescent wave and travels along the interface for a short distance. This phenomenon is accompanied by a nonzero imaginary part in the reflection coefficient, resulting in a phase shift relative to the incident light. According to Fresnel’s equations, this phase shift is constant and wideband when the materials involved are nondispersive. To manipulate the phase shift, one approach is to modify the refractive index of either the dense or less dense medium, (30) but the tunable range is limited (see Supporting Information). In this Article, we propose a new mechanism, which uses a conductive interface in the TIR geometry (Figure 1) to effectively tune the phase of the reflected terahertz light in a wideband. We study the phase shift in the CI-TIR geometry from the modified Fresnel equations in ref (36) (eqs 1 and 2),

r_{s} = \frac{n_{1} \cos ? θ_{i} - i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} - Z_{0} σ_{s}}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} + Z_{0} σ_{s}}

$r_{s} = \frac{n_{1} \cos ? θ_{i} ? i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} ? Z_{0} σ_{s}}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} + Z_{0} σ_{s}}$

(1)

r_{p} = \frac{i \cdot n_{1} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}} - {n_{2}}^{2} \cos ? θ_{i} - i \cdot Z_{0} σ_{s} \cos ? θ_{i} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}}}{i \cdot n_{1} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}} + {n_{2}}^{2} \cos ? θ_{i} + i \cdot Z_{0} σ_{s} \cos ? θ_{i} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}}}

$r_{p} = \frac{i \cdot n_{1} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}} ? {n_{2}}^{2} \cos ? θ_{i} ? i \cdot Z_{0} σ_{s} \cos ? θ_{i} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}}}{i \cdot n_{1} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}} + {n_{2}}^{2} \cos ? θ_{i} + i \cdot Z_{0} σ_{s} \cos ? θ_{i} \cdot \sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}}}$

(2)

where n₁ is the refractive index of dense media, n₂ is the refractive index of less dense media, θ_i is the supercritical incident angle, Z₀ is the vacuum impedance (377 Ω), and σ_s is the optical sheet conductivity. In both s- and p-polarizations, considering linear, homogeneous, isotropic, and nonmagnetic media, the reflection coefficient exhibits an imaginary part due to the supercritical incident angle. This imaginary part affects the phase of the reflection coefficient and can be adjusted by changing σ_s. To comprehensively analyze the impact of various parameters on the phase described by eqs 1 and 2, a series of calculations were performed. The incident angle was varied from the critical angle to the grazing angle, while the sheet conductivity was adjusted from 0 to 12 mS. Additionally, six combinations of different materials were selected. The theoretical results for both s- and p-polarizations of the air and high-resistivity silicon combination are illustrated in Figure 2, and the results of other combinations can be found in the Supporting Information.

Figure 1. Schematic of the conductive interface total internal reflection geometry. The lighter blue and darker blue rectangles represent less dense and dense material, respectively. The yellow line represents the conductive interface. The green arrow curves represent the incident and reflected light, which has a supercritical incident angle (θ). Due to the evanescent wave in the total internal reflection, there is a phase delay in the reflected light (dashed green curve). The red arrows represent the electric field in p-polarization (in-plane) and s-polarization (out-of-plane).

Figure 2. The theoretical results of phase change along with incident angle and sheet conductivity considering air (n_air ≈ 1) and high-resistivity silicon (n_Si ≈ 3.42) as the less dense and dense material in s-polarization (a) and p-polarization (b). The incident angle is from the critical angle to grazing incident angle, and the sheet conductivity is from 0 to 12 mS. The color bar represents the phase shift of the reflected light in the unit of degrees.

The theoretical results demonstrate that the phase of the reflected light can be adjusted by manipulating the sheet conductivity in both s- and p-polarizations. Since the conductive interface is assumed to be isotropic, the phase shift in the CI-TIR model is wideband. Additionally, the theoretical analysis reveals that the phase shift in s-polarization is more responsive to changes in the sheet conductivity compared to p-polarization. This discrepancy can be attributed to the fact that s-polarization has a greater in-plane electric field component interacting with the conductive interface than p-polarization.

In this Article, high-resistivity silicon is chosen as the dense material due to its low-loss properties in the terahertz band. To achieve an approximately 120° phase shift at an incident angle of 30° in an air/silicon combination, the sheet conductivity of the interface needs to be adjusted within the range of 0 to approximately 8 mS. However, achieving such a high sheet conductivity electrically through a solid-state device in experimental setups can be challenging. To address this issue and reduce the required sheet conductivity while maintaining the operational bandwidth, we utilize a nonresonant electric field enhancement metasurface known as subwavelength periodic metal microslits. (38−40) This metasurface is employed to enhance the electric field intensity of the evanescent wave in the TIR model. The enhancement factor, denoted as η, is defined as the ratio of the period of the microslits (P) to the slit width (w), η = P/w. The mechanism of nonresonant subwavelength metal microslits is distinct from that of metal stripe antenna designs, (41,42) which rely on resonant effects. The influences from metal microslits with different slit widths on amplitude and phase of the reflected terahertz signal were simulated and measured, revealing no resonant frequencies in the range of 0.1 to 3 THz (see Supporting Information). By optimizing this enhancement factor, we can effectively enhance the electric field intensity of the evanescent wave, enabling lower sheet conductivities to achieve the desired phase shift. The reflection coefficient with the enhancement factor can be written as

r_{s} = \frac{n_{1} \cos ? θ_{i} - i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} - η \cdot Z_{0} σ_{s}}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} + η \cdot Z_{0} σ_{s}}

$r_{s} = \frac{n_{1} \cos ? θ_{i} ? i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} ? η \cdot Z_{0} σ_{s}}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} + η \cdot Z_{0} σ_{s}}$

(3)

The phase information presented in eq 3 has not been previously investigated and is derived in this study, along with eq 4. For detailed derivation and phase results, please refer to the second part of the Supporting Information.

φ = \tan^{- 1} ? \frac{\sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}}}{n_{1} \cos ? θ_{i} - η \cdot Z_{0} σ_{s}} + \tan^{- 1} ? \frac{\sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} - {n_{2}}^{2}}}{n_{1} \cos ? θ_{i} + η \cdot Z_{0} σ_{s}}

$φ = \tan^{? 1} ? \frac{\sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}}}{n_{1} \cos ? θ_{i} ? η \cdot Z_{0} σ_{s}} + \tan^{? 1} ? \frac{\sqrt{{n_{1}}^{2} \sin^{2} ? θ_{i} ? {n_{2}}^{2}}}{n_{1} \cos ? θ_{i} + η \cdot Z_{0} σ_{s}}$

(4)

The introduction of subwavelength metal microslits allows for a reduction in the required sheet conductivity to achieve an amplitude and phase modulation by a factor of 1/η compared to the initial value while preserving the operational bandwidth. Figure 3 illustrates the phase and intensity changes in a silicon/air model with metal microslits of various enhancement factors at an incident angle of 30° in the CI-TIR geometry. The relative phase shift and intensity are referenced to the results obtained from the TIR model without a conductive interface. Figure 3a shows that as the enhancement factor (η) increases, the phase curve exhibits a steeper gradient. The most significant phase change occurs between 0 and 2 mS sheet conductivity for an enhancement factor range of η = 8 to 20. Figure 3a demonstrates that the phase change in simulation surpasses that in calculation when the sheet conductivity exceeds 1 mS, particularly for high enhancement factors such as η = 8, 10, 20. These differences can be attributed to the concentration of the electric field at the edges of the metal microslits in simulation, leading to a stronger modulation effect. In contrast, the theoretical model did not consider the uneven distribution of the electric field. Instead, for simplicity in equation derivation, the interaction effect was treated as an averaged effect. The enhancement effect of the microslits on the evanescent wave was simulated, revealing that the strongest electric field is concentrated at the edges of the slits. The enhancement factor of the electric field along the interface in the slit is close to, but slightly larger than, the averaged theoretical value η. This discrepancy arises because the evanescent wave decays exponentially away from the interface. Please refer to Figure S4 for details. The goal is to achieve a pure phase shift while maintaining a constant intensity. However, achieving this ideal scenario is challenging in practice. Instead, the focus is on minimizing intensity fluctuations to approach a near-ideal phase shift. Figure 3b demonstrates the corresponding intensity changes with sheet conductivity. Initially, the intensity decreases as the sheet conductivity increases from zero, reaching a minimum at a specific sheet conductivity (σ_m), after which it gradually recovers. The phase exhibits the most rapid change when the sheet conductivity is approximately σ_m. The phenomenon can be explained as follows: when the sheet conductivity is zero in the TIR model, the interface behaves as a dielectric, and the incident light is totally reflected. This reflection results in a constant phase change without any intensity attenuation. On the other hand, when the sheet conductivity approaches infinity, the interface acts as a “perfect electric conductor,” leading to total reflection with a 180° phase change. In this case, the reflected intensity remains at unity as well. As the sheet conductivity varies between zero and infinity, the reflected intensity becomes less than unity and reaches its minimum at σ_m. Prior to reaching σ_m, the conductive layer exhibits a more absorptive effect, causing the reflected intensity to continuously decrease. Beyond σ_m, the conductive layer demonstrates a more reflective effect, leading to a recovery of the reflected intensity. The σ_m point serves as a transition point between absorption and reflection, often referred to as the “absorption-to-reflection” (A-R) point. Around this transition point, the phase exhibits the most significant and drastic changes. This behavior is analogous to the inductance-capacitance resonant theory observed in metamaterials, where the inductive and capacitive behavior undergoes a transition at the resonant frequency. In the CI-TIR model with metal microslits, the phase shift is not based on resonant effects, allowing for a wideband phase shift. The range of sheet conductivity that can be achieved using a solid-state electrical graphene device is from 0.5 to 1.5 mS, (36,43) as indicated by the gray rectangular region in Figure 3. Within this range, the theoretical phase changes for designs without slits, η = 2, and η = 3 are relatively small. However, for designs with higher enhancement factors, such as η = 8, 10, and 20, larger phase shifts are obtained. Specifically, the phase changes for η = 8, 10, and 20 are 102, 103, and 46°, respectively. The phase change increases as η increases from 2 to 10. However, as η further increases to 20, the phase change starts to decrease. This behavior is attributed to the “A-R” point, which occurs outside the range highlighted by the gray rectangular region. For designs with η = 8 and 10, the reflected intensity initially decreases, then gradually increases, and remains relatively stable during the phase shift period. It is important to note that for designs with η = 2 and 3, as the sheet conductivity increases from 0.5 to 2 mS, the reflected intensity decreases. On the other hand, for designs with η = 15 and 20, the reflected intensity increases, indicating the presence of an “A-R” transition in these cases.

Figure 3. The function of phase shift and intensity with sheet conductivity and different terahertz field enhancement factors (η) in the CI-TIR geometry. The design with five enhancement factors and no microslits is chosen for calculation and simulation. The gray rectangle represents the tunable range of sheet conductivity from 0.5 to 1.5 mS. (a) The relative phase change of the reflected light as a function of optical sheet conductivity. (b) The reflected intensity as a function of optical sheet conductivity. The solid lines represent theoretical calculation results, and the empty dots represent simulation results.

Experimental Methods

The structure of the graphene-based device and experimental setup used in the experiment are depicted in Figure 4. The device consists of a high-κ material Al₂O₃ layer, which is 10 nm thick and grown by atomic layer deposition (ALD), serving as the insulation layer. The Al₂O₃ layer is deposited on a high-resistivity silicon substrate (R_□ > 6 kΩ·cm, double-side polished). On the Al₂O₃/Si substrate, a golden microslit pattern with dimensions of 5 × 5 mm is fabricated using standard lift-off photolithography and metallization techniques. The metal microslits have a fixed period of 20 μm, while the slit width varies from 2 to 10 μm, denoted as D2 to D10 in the paper. After the metallization process, a piece of CVD-grown graphene (5 × 5 mm) was transferred onto the metal microslit pattern. After the metallization process, single-layer graphene, grown through chemical vapor deposition (CVD), is transferred onto the metal microslit pattern. The quality of the graphene layer is verified using Raman spectroscopy to ensure its single-layer nature (more details can be found in the Supporting Information). To control the conductivity of graphene, a gated voltage is applied across the graphene layer. The voltage is swept from −5 to 5 V in steps of 1 V, taking care not to exceed the breakdown voltage of the Al₂O₃ insulation layer. For the experimental measurement of phase shift, a commercial terahertz time-domain spectroscopy (THz-TDs) system from TERA-K15 (Menlo Systems Inc.) is utilized. A high-resistivity silicon prism with an isosceles triangular shape, where each base angle is set to 30°, is employed to provide the necessary supercritical incident angle. The incident terahertz light used in the experiment is s-polarized, meaning it is perpendicular to the plane of the metal microslits (out-of-paper). The graphene-based devices were measured sequentially by placing them on the prism surface. Before each measurement, both the device bottom and prism surface were cleaned and pressed tightly by two probes. These probes not only applied gate voltage between the graphene and the device substrate but also minimized the air gap between the device and prism, ensuring tight contact. The THz-TDs system is also used to measure the sheet conductivity of graphene in the transmission geometry using a device without metal microslits. The measured results show that the graphene layer is p-doped, and its sheet conductivity can be tuned from approximately 0.3 mS (at −5 V) to 1.5 mS (at 5 V) in the frequency range of 0.4 to 1.2 THz (see Supporting Information).

Figure 4. Diagram of the graphene-loaded metal microslits device in the total internal reflection geometry. A 120° isosceles triangle shaped high-resistivity silicon prism introduces a 30° supercritical incident angle to the terahertz beam. The incident terahertz light is s-polarized, and the orientation of metal microslits is perpendicular to the electric field. The evanescent wave at the interface is compressed into the metal slit. The thickness of the insulation Al₂O₃ layer is 10 nm. The dimension of the Si substrate is 1 × 1 cm in size and 500 μm in thickness. The metal microslit area is 5 × 5 mm in size and 100 nm in thickness, with period and slit width represented by P and w, respectively. P = 20 μm and w ranges from 2 to 10 μm. By applying an external bias voltage on the graphene layer, the reflected terahertz signal can be shifted in the time-domain. The inset shows the top view of the Al₂O₃/Si gated graphene-loaded metal microslit device with dimensions.

Results and Discussions

The experimental results of the devices (D2 to D10) in the TIR geometry are presented in Figure 5. In Figure 5a, the peak-to-peak value of the time-domain waveform for D2 increases as the voltage changes from −5 V (0.3 mS) to 5 V (1.5 mS). This indicates that the graphene layer exhibits a more reflective effect, and its A-R point is left outside the conductivity tunable region. Conversely, for devices D4 to D10, the peak-to-peak values of the time-domain waveforms show an opposite trend, suggesting that the graphene layer exhibits a more absorptive effect, and the A-R points of these devices are located right outside the conductivity tunable region. These experimental observations are consistent with the theoretical predictions shown in Figure 3. The time-domain waveform of D3 stands out, as it exhibits a small variation in its peak-to-peak value and a noticeable time delay, as shown in Figure 5b. This indicates a significant phase shift in the frequency domain. To analyze the amplitude and phase of the reflected terahertz waveforms, the fast Fourier transform (FFT) is employed to convert the time-domain waveform data into the frequency domain. The relative phase change under various voltages, denoted as Δφ(ω) = |φ_V(ω) – φ_5 V(ω)|, is calculated with reference to the phase at 5 V, as shown in Figure 6. In the case of D3, this relative phase change is over 100° from 0.4 to 1.2 THz, with an average value of approximately 120° (Figure 6b). The group delay of this modulator is calculated by GD = −dφ/dω, which is less than 50 ps in this frequency range. Around f = 0.5 THz, the experimental phase shift measures approximately 106°. The theoretical phase shift for D3 (with η ≈ 7), within the range of 0.3 to 1.5 mS, is predicted to be 102°, which aligns well with the experimental result. The phase shift gradually decreases from D3 to D10 due to the decline in the enhancement factor η. It is worth noting that the phase shift in all devices exhibits a slight increase with increasing frequency. This can be attributed to the slight frequency-dependent conductivity of graphene. In the terahertz band, the imaginary part of conductivity increases slightly significantly with frequency, while the real part decreases slightly with frequency. (29) According to eq 5, this results in a larger phase shift as the frequency increases. Further detailed analysis can be found in the Supporting Information. The time-domain waveform of D9 appears smaller than that of D8 and D10, which could be attributed to misalignment in the experiment or different loss from the air gap between the device and prism. However, it is important to note that the relative amplitude fluctuation does not influence the phase shift results. The synergistic interaction between graphene and metal microslits enhances the phase shifting effect from graphene. The phase shift of the graphene device without metal microslits is less than 20° (please see Supporting Information), which is lower than that of the η = 2 device.

r_{s} = \frac{n_{1} \cos ? θ_{i} - i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} - η \cdot Z_{0} ({σ_{s}}^{'} + i {σ_{s}}^{″})}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} + η \cdot Z_{0} ({σ_{s}}^{'} + i {σ_{s}}^{″})} = \frac{(n_{1} \cos ? θ_{i} - η \cdot Z_{0} {σ_{s}}^{'}) - i \cdot (\sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} + {σ_{s}}^{″})}{(n_{1} \cos ? θ_{i} + η \cdot Z_{0} {σ_{s}}^{'}) + i \cdot (\sqrt{n_{1}^{2} \sin^{2} ? θ_{i} - n_{2}^{2}} + {σ_{s}}^{″})}

$r_{s} = \frac{n_{1} \cos ? θ_{i} ? i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} ? η \cdot Z_{0} ({σ_{s}}^{'} + i {σ_{s}}^{″})}{n_{1} \cos ? θ_{i} + i \cdot \sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} + η \cdot Z_{0} ({σ_{s}}^{'} + i {σ_{s}}^{″})} = \frac{(n_{1} \cos ? θ_{i} ? η \cdot Z_{0} {σ_{s}}^{'}) ? i \cdot (\sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} + {σ_{s}}^{″})}{(n_{1} \cos ? θ_{i} + η \cdot Z_{0} {σ_{s}}^{'}) + i \cdot (\sqrt{n_{1}^{2} \sin^{2} ? θ_{i} ? n_{2}^{2}} + {σ_{s}}^{″})}$

(5)

Figure 5. (a–i) The reflected terahertz waveforms in the time-domain of D2 to D10 at different gate voltages in the TIR geometry. The driving voltage in experiment is swept from −5 to 5 V.

Figure 6. (a-i) The calculated corresponding relative phase shift of D2 to D10 in the reflected terahertz signal compared with the phase at 5 V.

To demonstrate the continuous phase shift of device D3, a finer voltage sweeping experiment was conducted with a step size of 0.2 V. The shape of terahertz time-domain waveform exhibits minimal distortion and demonstrates a gradual delay of approximately 400 fs as the driving voltage is swept from 5 to −5 V (refer to Supporting Information). The insertion loss of our design arises from two sources: 1. the entrance and exit surfaces of the Si prism; 2. the conductive interface. The insertion loss caused by the Si prism, in this study, is approximately 5 dB, which can be mitigated by implementing antireflection structures on the entrance and exit surfaces. (44−46) Therefore, the insertion loss of D3 reported in this Article primarily accounts for the loss from the active layer. Figure 7a presents the relative reflected intensity of D3 compared to the bare Si prism (without the device) in the frequency range of 0.4 to 1.2 THz, which is typically less than 10 dB. This is significantly lower than the insertion losses observed in metamaterial-based narrowband phase modulators. For an ideal phase shifter, it is desirable to minimize the intensity fluctuation while shifting the phase. The reflected intensity fluctuation of D3 is less than 2 dB compared to its average value across the frequency range of 0.4 to 1.2 THz. Figure 7b illustrates the voltage dependence of the phase shift calculated from 0.4 to 1.2 THz. From 1 to 1.2 THz, there is a linear relationship between the voltage and phase shift in the range of −5 to 0 V. At other frequencies, the phase shift exhibits a quasilinear relationship with the voltage. The inset in Figure 7b provides an example at f = 1.2 THz, demonstrating the linearity of the phase shift as a function of the driving voltage from −5 to 0 V. During the 130° linear phase shift process, the reflected intensity fluctuation remains within ±2 dB compared to its mean value. This can be compensated by adjusting the source output power or incorporating a wideband amplitude modulator (47) to achieve an ideal phase shifter.

Figure 7. The insertion loss and phase shift of D3 under finer gate voltages. (a) Reflected intensity as a function of gate voltage at different frequencies. The reflected intensity is referred the Si prism without the device. (b) Relative phase shift referred to the phase at 5 V from 0.4 to 1.2 THz. The inset shows the linearity of the phase shift as a function of driving voltage and the intensity fluctuation.

Table 1 represents representative results of terahertz electrical free-space phase modulators. Electrical vanadium dioxide and liquid crystal modulators, due to their slow operation speed, are not listed in Table 1. Modulators based on metasurface with semiconductors or graphene exhibit narrow operation bandwidth. While the graphene-controlled Brewster angle device offers broadband capabilities, its phase shifting is discontinuous, thereby limiting its applications. The device demonstrated in this paper achieves a continuous phase shifting, reaching up to 120° in the frequency range of 0.4 to 1.2 THz, with a moderate level of insertion loss.

Table 1. Reported Electrical THz Free-Space Phase Modulator Types and Performance

Time	Types	Frequency	Phase	Loss	Ref.
2009	Metasurface with doped GaAs	0.88–0.9 THz	32°	6 dB	(20)
2015	Metasurface with HEMT	0.2–0.4 THz	48°	12 dB	(48)
2017	Metasurface with HEMT	0.7–1 THz	57°	13 dB	(49)
2018	Metasurface with HEMT	0.32–0.37 THz	136°	15 dB	(22)
2018	Metasurface with graphene	1.2 THz	180°	50 dB	(50)
2018	Graphene-controlled Brewster angle	0.5–1.6 THz	140° (discontinuous)	10 dB	(24)
2024	Graphene in the TIR geometry	0.4–1.2 THz	120°	10 dB	This paper

Modulation speed is another crucial parameter for terahertz modulators. In this study, the modulation speed was evaluated by measuring the direct current (DC) conductivity change of the graphene layer. The THz time-domain system used in this work was not sufficiently fast for direct speed measurements. However, the optical sheet conductivity of graphene is related to its DC conductivity, making it a reasonable proxy for representing the phase modulation speed. For device D3, the measured DC conductivity change speed was approximately 3 kHz (refer to Supporting Information for more details), indicating that the phase modulation speed is in the kHz range. To improve the modulation speed of graphene-based devices, one approach is to reduce the resistance-capacitance (RC) time constant. This can be achieved by reducing the size of the graphene or by employing alternative insulation materials. In fact, a graphene modulator operating in the MHz range has been reported by reducing the graphene size and using different insulation materials. (41) It is worth noting that the tunable conductivity interface in the TIR geometry, as demonstrated in this work, is not limited to graphene. Other materials and techniques can also be explored for achieving phase modulation. For instance, utilizing other semiconductor systems such as two-dimensional electron gases in heterostructures holds the potential for improving the operation speed to the GHz range (48) in the future.

The current device serves as a proof-of-concept design, and there are several methods to further increase the phase shift. First, by using a different Si prism with a near supercritical incident angle, the phase shift can be enhanced. Second, improving the quality of the insulation layer and graphene can expand the conductivity tunable range. Third, optimizing the parameters of the metal grating, such as using finer metal microslits, can lead to a deep subwavelength scale and enable the device to operate over an even wider bandwidth. (39) This optimization can further enhance the phase modulation capabilities. Using a material with a larger conductivity variation, for example, increasing from 3 to 8 mS, will not result in a wider phase shift in devices with an enhancement factor larger than 8, as shown in Figure 3a. However, as depicted in Figure 3b, the reflected intensity will experience a significant increase, which will help reduce the insertion loss. Additionally, an alternative approach is to utilize a Fresnel rhomb made of high-resistivity silicon. The Fresnel rhomb can introduce multiple reflections to the terahertz wave, thereby increasing the overall phase shift up to 2π with minimal additional loss inside the rhomb. By combining these methods and further optimizing the device design, it is possible to achieve higher phase shifts in future iterations of the terahertz modulator.

Conclusions

In summary, an ultrawideband terahertz phase shifter based on the TIR geometry was proposed and experimentally demonstrated. The key component of the device was a tunable conductive interface realized by a graphene-loaded nonresonant metasurface. By adjusting the gate voltage, the device achieves continuous and tunable phase shifting of the reflected terahertz waveform. The average phase shift over the frequency range of 0.4 to 1.2 THz is approximately 120°, with group delay less 50 ps and minimal amplitude fluctuation of less than 1 dB. The insertion loss of the graphene device, excluding the loss introduced by the Si prism, is below 10 dB, surpassing the performance of metamaterial-based terahertz modulators in terms of bandwidth and insertion loss. The phase modulation speed of the device is in the kHz range, which can be further improved through parameter optimization or by utilizing other semiconductor systems, potentially reaching the GHz range. In the future, there is the possibility of extending the single element to a phase modulator array. This could be valuable for applications such as terahertz ultrawideband phase array radar or imaging systems, where the precise control of phase shifting across multiple elements is required. The remarkable picosecond-level group delay exhibited in our design highlights its significant potential in high-rate terahertz communication systems.

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