Terahertz (THz) is broadly defined in 0.1 THz to 10 THz, lying between the infrared and microwave range of the spectral region [1]. The increasing efforts in THz science and technology stem from promising applications in spectroscopy, imaging, and communications [2]. In recent years, breakthroughs in THz sources and detectors have greatly promoted the development of this research area. However, the lack of active THz components to realize wave manipulation and modulation has hindered the substantive progression of THz applications. One of the key THz components mostly desired for this field, such as THz imaging and communications [3, 4], is high-performance THz modulators, used to maintain the amplitude, phase, polarization, and propagation of the THz wave. The THz region has a broad range of applications including biotechnology, telecommunication, military, and environmental applications [5–9]. The advances of THz technology and the growing interest in its applications have accelerated the development of THz sources, THz detectors, THz fibers, and other devices [10]. Among them, remarkable progress has been attained in designing effective THz fibers. The THz fiber has been used in many applications, such as filtering, sensing, interferometry, quantum key distribution, and coherent communication [11].

A phase or an amplitude THz modulator requires prohibitively thick crystals or the non-trivial implementation of efficient THz waveguides and their operation at high voltages. To overcome these hurdles, several THz modulators have been reported based on active materials with tunable optical properties, such as the two-dimensional electron gas (2DEG) in InGaAs/GaAs systems, conventional semiconductors, metamaterials, superconductors, and phase-transition materials [12–16].

Graphene, as a two-dimensional semiconductor material, has been proven useful for THz wave modulation due to its unique electrical, optical, and mechanical properties and has great potentialities in the field of thermology, chemistry, electricity, and optics [17–22]. Compared to other materials, graphene with its linear energy dispersion with zero bandgap energy exhibits an ultrahigh carrier mobility to modulate THz wave flexibly, along with a broad spectral response and plasmonics in infrared and THz frequencies [23–25]. Moreover, optical properties of graphene can be easily tuned by altering its Fermi level via electrical means, an optical stimulus, or chemical doping, such as the gate voltage [26] and optical pump [27]. A kind of square porous-core photonic crystal fiber (PCF) for polarization-maintaining terahertz wave guidance proposed in [28] offers a high birefringence of 0.063 and a low EML of 0.081 cm⁻¹ at 1 THz. Moreover, a very low flattened dispersion profile is observed over a wide frequency domain of 0.85–1.9 THz. The zero flattened dispersion can be controlled. Furthermore, a highly birefringent single-mode-suspended core fiber in the THz regime was proposed in [29], which can provide a high birefringence of the order of 10–2 over a wide frequency region in a single-mode operation. [30] proposed an ultrasensitive tunable terahertz (THz) sensor, which consists of single-layer graphene-based gratings integrated with a Fabry–Perot cavity through numerical simulations.

Therefore, the status of the modulators can be controlled based on a graphene switch between the ON and OFF states [31]. Graphene THz modulators have been widely used in the THz region [32]. Meanwhile, THz graphene waveguide modulators have been designed and demonstrated [33].

THz fibers have attracted much attention recently, owing to their performance advantages, including low loss and dispersion, good flexibility of structure, and good confinement ability [34]. The development of 3D printing technology has greatly improved the flexibility of designs and the fabrication of THz fibers, and reduced the material loss and manufacturing cost [35]. In this paper, a graphene-coated D-shaped THz fiber modulator is proposed. The proposed modulator with a single layer of graphene on the surface of the D-shaped fiber can share the advantage of both graphene-based and fiber-structured modulators. The performance of the D-shaped THz graphene fiber-based modulator is investigated by the finite element method. A tunable confinement loss is observed by adjusting the Fermi level of graphene via chemical doping. The confinement loss of Y polarization is higher than that of X polarization at the operating frequency of 0.64 THz.

A schematic representation of the proposed graphene-coated D-shaped THz fiber modulator is illustrated in Figure 1. It consists of a hollow core surrounded by eleven cladding layers with the same thickness and a graphene layer coated on a micro-bridge. The diameter of the core is noted as D. The cladding layers are a periodic structure of alternating photosensitive resin layers with a high refractive index and air layers with a low refractive index. The thickness of each layer is presented by t. The micro-bridge is introduced into the cladding layers in order to fabricate the stable structure and coat the graphene layer. The thickness of the micro-bridge is t/2. To implement the alternating high-refractive index photosensitive resin and low-refractive index air layers, we first fabricated a thick layer of high-refractive index photosensitive resin via spin coating.

The micro-bridge was created on top of this thick layer by photolithography and oxygen plasma etching. The micro-bridge acted as a support structure for the remaining fabrication process, which included the deposition of the graphene layer on top of a thin buffer layer and coating this with a relatively thick cladding layer. Once the graphene layer was deposited and coated with the cladding layer, we used another photolithography and oxygen plasma etching process to create air holes in the cladding layer around the micro-bridge. Finally, we removed the micro-bridge to form a suspended photonic crystal membrane structure consisting of the alternating high-refractive index photosensitive resin and low-refractive index air layers, with a graphene layer incorporated within the cladding layer.

High-quality large-area monolayer graphene was grown using the chemical vapor deposition (CVD) method, and stacked multilayers were transferred onto the side-polished region of the fiber for an evanescent field interaction with the guided mode in the fiber [36].

The fiber can be obtained by 3D stereolithography (SLA). In this paper, the refractive index of the air layer and the hollow core are both 1. The absorption coefficient and the refractive index of the photosensitive resin used in this paper can be expressed as in [37].

The relative permittivity of graphene is defined as follows [38, 39]: ε g = 1 + i σ g ε 0 ω t g , (1) where ε 0 is the permittivity in vacuum, ω is the angular frequency of the incident wave, t g is the thickness of the graphene layer, and σ g is the surface conductivity of graphene, which can be calculated from the Kubo formula [40]: σ g = i e 2 k B T π ℏ ω + i τ − 1 E F k B T + 2 ⁡ ln exp E F k B T + 1 + i e 2 4 π ℏ ln 2 E F − ℏ ω + i τ − 1 2 E F + ℏ ω + i τ − 1 , (2) where k B is the Boltzmann constant, T is the Kelvin temperature, E F is the Fermi energy level, ℏ = h / 2 π is the reduced Planck’s constant, and τ is the momentum relaxation time, which can be expressed as follows: τ = μ E F / e v F 2 , (3) where μ is the carrier mobility of graphene and v F is the Fermi velocity. The carrier mobility ranges from 8,000 cm²/(V·s) in the mechanical cleavage of bulk graphite [41] to 230,000 cm²/(V·s) in high-quality suspended graphene [42]. In this paper, T is the room temperature of 300 K. The Fermi velocity v F and the moderate carrier mobility μ are considered to be 106 m/s and 10,000 cm²/(V·s), respectively [43].

The complex refractive index of graphene can be calculated by the relative permittivity of graphene [44]: ε g = n w + i k ω , (4) where n ω and k ω are the real and imaginary parts of the complex refractive index, as shown in Figures 2A, B, respectively. The real part of the complex refractive index first increases and then decreases after 0.3 eV, and it decreases with the increase of the THz wave frequency. The imaginary part of the complex refractive index is proportional to the Fermi energy level, while it shows an opposite trend to the frequency.

The confinement loss of the fiber transmission mode in the form of an evanescent wave is an important parameter in terms of the modulator; the effective material loss should be considered in addition to the confinement loss. The effective material loss can be considered in the context of modulator design, but it is usually not a key parameter of concern. Although the effective material loss is an important consideration in optical systems, confinement loss parameters often take priority in modulator design, and the confinement loss expression is as follows [45]: C L = 8.686 × 2 π λ I m n e f f , (5) where I m n e f f is the imaginary part of the effective mode refractive index and λ is the wavelength of the incident THz wave in vacuum.

The influence of each parameter of the device structure on the confinement loss and the full width at half-maximum (FWHM) are calculated and analyzed. The thickness of the graphene layer is a single graphene layer of 0.335 nm [46]; the Fermi energy level is 1 eV, and the graphene layer is in the first layer of the cladding layer in this part. The first is the diameter of the core. It can be seen from Figures 3A, B that the confinement loss of the two polarization states decreases with the increasing diameter of the core. When the diameter of the core is greater than 4 mm, the confinement loss is lower with a frequency range of 0.5 THz to 1 THz. Meanwhile, owing to the stronger power guided in the core region at 6 mm, the confinement loss is lower, as shown in Figure 3D and Figure 4. This is because the increasing diameter of the core makes the THz wave less likely to leak from the core to the graphene layer directly above the core.

The depth of polarization states refers to the degree to which the polarization state of a light beam deviates from being fully polarized. Furthermore, the full width at half-maximum is commonly measured as the ratio between the intensity of the polarized component and the total intensity of light. Furthermore, the changes of the FWHM of X- and Y-polarization with the diameter of the core are shown in Figure 3C. The FWHM of X-polarization reaches the middle at 4.5 mm. Moreover, the larger the diameter of the core, the larger the number of modes in the core region. In order to achieve an effective single-mode operation, the diameter of the core should be reduced.

The FWHM of the transmission spectrum in our modulator is primarily affected by two factors: the thickness of graphene coating and the dispersion properties of the fiber. The thickness of graphene coating affects the effective refractive index of the fiber, which, in turn, can shift the frequencies at which resonances occur. Specifically, thicker coatings will shift resonant frequencies to lower values, resulting in a wider bandwidth and a correspondingly smaller FWHM. Conversely, thinner coatings will shift the resonant frequencies to higher values and lead to a narrower bandwidth and a larger FWHM. Anomalous dispersions can be used to compensate for the dispersion of graphene coating, leading to a narrower bandwidth and a smaller FWHM. Other factors that can affect the FWHM include the shape and structure of the fiber, and any impurities or defects present in the material. These factors can introduce additional losses or alter the distribution of the mode field energy within the fiber, leading to changes in the FWHM of the transmission spectrum.

The distribution of the mode field energy in our modulator is determined by a combination of factors, including the geometry and dimensions of the fiber structure, the dispersion properties of the fiber, and the presence of any impurities or defects that may alter the distribution of energy within the fiber. One important factor affecting the distribution of mode field energy is the shape and dimensions of the fiber structure. In our D-shaped fiber modulator, specific shapes and dimensions of the fiber can lead to coupling between different modes within the fiber, resulting in the redistribution of energy within the fiber. Additionally, the thickness of graphene coating can also affect the distribution of the mode field energy by altering the effective refractive index of the fiber and leading to shifts in the resonant frequencies of the device.

Regarding the choice of the photosensitive resin, we selected a specific type of photosensitive resin because it offered a high refractive index, excellent photoresistive properties, and was compatible with our fabrication process. Specifically, the photosensitive resin we used had a refractive index, which is higher than that of air and similar to that of silicon dioxide, making it a good candidate for fabricating a graphene-coated D-shaped terahertz fiber modulator consisting of an alternating high-refractive index and low-refractive index layers.

The confinement loss versus frequency for different thicknesses of the cladding layers of X- and Y-polarization are shown in Figures 5A, B, respectively. When the thickness of the cladding layers is 800 μm, the confinement loss of the two polarization states is lower, owing to the stronger power in the core region, as shown in Figure 5D and Figure 6. Figure 5C shows that the FWHM reaches a minimum at 900 μm.

Different positions of the graphene layer represent the different distances from the core to the graphene layer; the numbers 1 and 5 are used to indicate the different positions of the graphene layer of the D-shaped THz fiber. In particular, region 1 represents that the graphene layer is in the first layer, while region 5 represents that the graphene layer is in the fifth layer, as shown in Figure 7. In Figure 8 and Figure 9, it is clearly observed that when the graphene layer is in the first layer, the confinement loss of the two polarization states is lowest due to the strongest power in the core region and the FWHM of X- and Y-polarization is lowest and greatest, respectively. This is because as the core to the graphene layer decreases, the channel of THz wave leakage from the fiber core to the graphene layer becomes narrower, which makes leakage more difficult. The periodicity of the structure can create photonic band gaps, where certain frequencies are fully reflected, leading to unique transmission behaviors.

Based on the aforementioned investigation, an optimized structure of the proposed modulator is obtained. The structural parameters are = 4.5 mm and = 800 μm, and the graphene layer located in the first layer is the cladding layer. The confinement loss versus the frequency of different graphene Fermi levels of X- and Y-polarization is shown in Figures 10A, B, respectively. The confinement loss of X-polarization first increases before 0.6 THz and then decreases in the frequency range of 0.6 THz to 0.67 THz with the Fermi level gradually reducing from 1 eV to 0.01 eV, while it remains stable at other frequencies. The confinement loss of Y-polarization first takes on a loss peak around 0.64 THz when the Fermi level is less than 0.3 eV and then decreases rapidly, reduced to a loss peak with the decreasing Fermi level. These results mean that with the weakening of graphene electric doping, the leakage of the evanescent wave from the core to the graphene layer changes and the confinement loss of the proposed structure decreases. The loss peak in the transmission spectrum of our modulator is a result of several mechanisms that contribute to the attenuation of the signal as it propagates through the fiber. One factor contributing to the loss peak is the coupling between different modes within the fiber. In our modulator, the periodic structure of the graphene coating creates a series of resonant cavities that can couple the energy between different modes within the fiber. This coupling can lead to the energy being lost from the fundamental mode and being transferred to higher-order modes, resulting in increased attenuation at certain frequencies. Another factor contributing to the loss peak is scattering losses due to defects or impurities within the fiber. These defects can scatter energy out of the guided mode and into the surrounding modes or the cladding, resulting in increased attenuation. In our modulator, the use of graphene as a coating helps in reducing absorption losses by providing a highly conductive surface that can efficiently carry current and avoid energy losses due to its resistance.

In order to demonstrate the THz fiber-based modulator, the working frequency used to observe the various confinement losses of X- and Y-polarization is 0.64 THz. The confinement loss of X polarization only has a small frequency shift, as shown in Figure 11A. The extinction ratio mainly affected by the confinement loss in the ON and OFF states is an important parameter in terms of the performance of the modulator. As shown in Figure 11B, the minimum value of the Y-polarization confinement loss is 0.15932 dB/cm and the maximum value is 0.26779 dB/cm. The extinction ratio is 22.55% and the modulation depth is 40.51%. The X-polarization confinement loss has a minimum value of 0.08794 dB/cm and a maximum value of 0.11654 dB/cm. The extinction ratio is 12.23%, and the modulation depth is 24.54%. The two polarization states are both proportional to the Fermi level. In Figure 11C, the minimum value of the Y-polarization FWHM is 0.19974 THz and the maximum value is 0.22881 THz. As can be seen from Figure 11D, when the modulator is in the ON state, the incident THz wave is strongly confined within the core region. However, with the increasing conductivity of graphene, transverse resonant coupling between the core mode and the graphene layer mode is stronger until the OFF state.

In conclusion, a graphene-coated D-shaped THz fiber modulator is proposed. The performance of the D-shaped THz graphene fiber-based modulator is investigated. The inference of geometric parameters of the structure of the proposed modulator is also analyzed, and the optimal structure is obtained. The results show that the confinement loss and the FWHM can be tuned by adjusting the Fermi level of graphene. The modulation depths of the two polarization states at an operating frequency of 0.64 THz are 40.51% and 24.54%, respectively. The proposed THz fiber modulator shows potential in future THz fiber communications.