1 Introduction

The electro-optic modulator is a key device in optoelectronic integrated circuits [1, 2]. It converts electronic signals into high bit-rate photonic data. Many researchers are investigating the development of the electro-optic modulators [3,4,5,6], but implantation of ultra-high-speed compact electro-optics modulator is a critical technical bottleneck. Regular materials have poor electro-optic properties and therefore a conventional electro-optic modulator has a very large footprint [7, 8]. Xu et al. [9] experimentally demonstrated a high-speed electro-optical modulator in compact silicon structures based on a resonant light-confining structure, but it simultaneously decreases the operation bandwidth and thermal stability. Thin quantum-well structures are proposed to improve the optical modulator using quantum-confined Stark effect mechanism, but the involved waveguides were relatively long [10,11,12].

Graphene, a single layer of the carbon atoms arranged in a honeycomb structure, has attracted a great deal of interest because of its exceptional electric, mechanical, and thermal properties [13,14,15,16]. Graphene has the highest intrinsic mobility [17], the largest current density at room temperature [18], the strong coupling with light over a wide frequency range from terahertz (THz) to infrared frequencies [19] and controllable conductivity by shifting the Fermi energy level [20]. Modulation of THz waves is still a challenge in photonics. Recently, a broadband device was proposed for modulation of the THz waves that it achieved a modulation depth greater than 90% between 0.15 and 0.4 THz, and reaching a maximum of 99.3% at 0.24 THz [21]. This modulation depth was reported by applying a gate voltage between − 0.1 and 2 V to a graphene layer in total internal reflection geometry. Also, in a recent study of graphene properties in the THz regime, a tunable THz meta-surface was proposed by patterning a graphene sheet in the cut-wire array and optimizing the geometry of the structure for enhancement of THz absorption [22]. A tunable Goos–Hänchen (G–H) shift reported by a monolayer graphene in the THz regime that the G–H shift changed by electrically controlling the Fermi energy of the monolayer graphene [23]. A monolayer graphene has been realized as the Coherent Perfect Absorber (CPA) and the existence of the quasi-CPA has been reported in the THz regime for suspending monolayer graphene [24].

Many researchers reported the graphene-based modulator in different configurations and characteristics. Theoretically, a new long-wave infrared optical modulator based on graphene is designed [25]. A broadband, high-speed, waveguide-integrated electro-absorption modulator based on monolayer graphene has been shown experimentally [26]. An optical modulator consists of two layers of graphene with a high performance has been reported that two layers of graphene and an oxide layer in between to form a p-oxide-n like junction and it is operated at 1 GHz with modulation depth 0.16 dB/µm [27]. An electro-absorption optical modulator based on a dual graphene layer is demonstrated where two graphene layers are separated by a thin insulating region [28]. An electro-optic modulator based on graphene sandwiched in dielectric or plasmonic waveguides is presented that the dielectric constant of graphene can be tuned with a suitable gate voltage, resulting in “graphene-slot waveguides [29]. An electro-optic modulator is reported on the single-layer graphene in a sub-wavelength thick that it provided uniform modulation with flat frequency response from 1 Hz to 100 MHz [30]. Hao et al. demonstrated an eight-layer graphene embedded Mach–Zehnder modulator with compact footprint 4  ×  30 μm2 and extinction ratio 35 dB [31]. A compact wideband THz modulator with the simple structure, one graphene ribbon embedded in the polythene substrate is investigated that it had a bandwidth about 1 THz and the modulation depth about 28% [32]. A graphene hybrid plasmonic waveguide modulator in which graphene–hBN–graphene layer has been embedded to enhance the absorption of the graphene is studied and this modulator could provide the extinction rate about 40 dB and the modulation bandwidth about 190 GHz [33]. In another study, a graphene-based long-range hybrid plasmonic slot waveguide modulator is presented that it consists of a silica substrate, two high-index silicon strips, two low-index slots, two graphene layers, and a metal [34]. Recently, a plasmonic Mach–Zehnder modulator was studied with a flat frequency response of about 170 GHz consisting of two phase modulators exploiting the Pockels effect of an organic electro-optic material in plasmonic slot waveguides [35]. Zhang et al. studied a graphene-wrapped plasmonic optical modulator which consists of a silver (Ag) nanowire as a single-mode plasmonic waveguide being wrapped with a graphene monolayer as an electrically controllable absorbing material [36]. An electro-absorption optical modulator based on four layers of graphene embedded in a silicon-on-insulator waveguide is investigated and it has been shown that an electro-absorption optical modulator can achieve 34 dB extinction ratio and 100 GHz modulation bandwidth [37]. A few-layered graphene-dielectric multilayer electro-optic modulator is studied in the mid- and far-infrared range that works on electro-absorption mechanism [38]. The unique optical and electronic properties of graphene allow realizing novel active optical devices. Recently, researchers have focused on the study and fabrication of hybrid graphene plasmonic waveguide modulators by combining the versatility of graphene with the sub-wavelength field confinement of plasmonic waveguides [33, 39,40,41,42,43,44].

The graphene has an ultra-thin layer and it can embed inside a multilayer structure as absorber tunable layer. Therefore, different kinds of electro-optic modulators can be designed by using the graphene layer. The presence of graphene layer in multilayer structure creates a large design freedom such that characteristics of this graphene-based optical modulator can be optimized to achieve the requirements of the application at hand. Also, the graphene-based modulators could be used as hybrid mode-locking devices due to the saturation of interband absorption at high peak power in graphene [45]. In this paper, we design and demonstrate an electro-optic modulator with two graphene layers that it has a sub-wavelength thick, high modulation speeds and low insertion-loss in reflective modulator structure. This structure is a planar and reflective-type structure that can be used in lasers to achieve direct control of intra-cavity dynamics. This modulator can be realized using graphene as a loss-tunable layer embedded inside a multilayer structure. It offers the solution to optical amplitude modulation without phase distortions, active mode-locking and suppression of Q-switching instabilities [30]. Lee et al. designed an electro-optic modulator with a single graphene layer. They reported a modulation depth of 4% for their modulator [30]. We swept the position of the graphene layer in the multilayer structure and found a suitable placing for the graphene layer. We calculated a modulation depth %7.5 for the structure with single layer of graphene and a modulation depth %9 for the structure with two layers of graphene. Therefore, our proposed structures improve the modulation depth. This paper is organized as follows. In “Structure and analysis” section, we introduce the structure and the numerical results demonstrated for our proposed modulator. In “Conclusion” section, the paper is concluded.

1.1 Structure and analysis

The interband absorption of graphene is determined by its optical conductivity [46]:

$$\sigma = \frac{{e^{2} }}{8\hbar }\left[ {\tanh \left( {\frac{{\hbar \omega + 2E_{\text{f}} }}{{4K_{B} T}}} \right) + \tanh \left( {\frac{{\hbar \omega - 2E_{\text{f}} }}{{4K_{B} T}}} \right)} \right],$$
(1)

where e is the electron charge, \(\hbar\) is the reduced Plank constant, T is the effective carrier temperature, and \(\omega\) is optical frequency. Approximately, the absorbed optical power in graphene is proportional to its optical conductivity. Therefore, if graphene’s Fermi level changes it causes change of the optical absorption in graphene. In the parallel-plate capacitor model, the charge density of electric-field-gated graphene is dependent on the applied voltage (V) and the dielectric constant (ε), and thickness (d) of the dielectric. The Fermi energy (\(E_{\text{f}}\)) of graphene is tunable by the applied voltage as:

$$E_{f} \propto \sqrt {n + n_{0} } \propto \sqrt {\frac{\varepsilon }{d}\left| {V + V_{0} } \right|} ,$$
(2)

where \(n_{0}\) is the doping concentration in the graphene and \(V_{0}\) is an auxiliary voltage to create this doping concentration \(n_{0}\). The permittivity of the graphene as a function of the chemical potential at wavelength λ = 1550 nm is shown in Fig. 1.

Fig. 1
figure 1

The permittivity of graphene as a function of the chemical potential at λ = 1550 nm and temperature T = 300 K

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The permittivity ε of graphene is changed by chemical potential μ. Therefore, the effective mode index (EMI) of the waveguide with graphene layer is tunable by the applied voltage. It is clear from Fig. 1 where the absolute value of the permittivity approaches zero at about µ = 0.52 eV. For µ < 0.52 eV, the graphene operates as a dielectric material because both the real and imaginary components of the permittivity are positive. For µ > 0.52 eV, the graphene shows properties of metallic material because the real component of ε is negative and the interband transition dominates. The changes of graphene’s permittivity ε influence EMI in the waveguide [47]. The real part of the EMI affects the phase of light, known as electro-refraction [48], and the imaginary part of the EMI induces light absorption, defined as electro-absorption [37].

For designing of this modulator, a 100-nm silver film deposited on sapphire substrate by thermal evaporation that this silver film provides high reflectivity (~ 98%) for infrared light at normal incidence and also operates as a back-gate electrode. A 20-nm layer of silicon nitride (Si3N4) deposited on the silver film for preventing its oxidization and tantalum pentoxide (Ta2O5) layer is coated the silver mirror. Then, a single layer of graphene can be grown on the Ta2O5 layer. In the structure with a single layer of graphene, the ring-shaped top contact can be deposited on the graphene. For ring-shaped top contact, the excess graphene outside the top contact annulus can be removed by oxygen plasma method to limit the device capacitance. In the structure with two layers of graphene, Ta2O5 has been coated on the graphene layer and then the ring-shaped top contact can be deposited on the graphene. In both structures (with a single layer and two layers of graphene), the thickness of Ta2O5 layer would be an important parameter to tailor the optical properties of the modulator to the required characterization. The position of the graphene layer must be selected at the suitable location in the standing wave. This standing wave is created between the incident light and the light that was reflected from the silver back-gate electrode.

A precise adjustment of the thickness of the Ta2O5 layer can change the position of the graphene layer in the standing wave and it causes change in graphene absorption. Therefore, we can manage the insertion loss and accessible modulation strength by the appropriate placing of the graphene layer. The graphene layers must be located at the position of the maximum light intensity for more influence on the propagating mode. In Fig. 2, the electric field distribution of the propagating mode has been shown for proposed structure with single graphene layer at wavelength λ = 1550 nm. The effective optical absorption of the graphene and the insertion loss of the structure can be calculated by using the modified Fresnel transfer matrices [46]. The value of refractive indices has been selected as 2.1 for Ta2O5 layer, 1.8 for Si3N4 layer and n = 0.51 + 10.8i for silver at 1.55 μm wavelength. The profile of refractive index of structure is shown in Fig. 3 for single layer and two layers of graphene in proposed structures.

Fig. 2
figure 2

The electric field distribution of the propagating mode for proposed single graphene layer structure at wavelength λ = 1550 nm

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Fig. 3
figure 3

The profile of refractive index for proposed structures with a single layer (upper picture) and two layers (lower picture) of graphene

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In this proposed modulator, a bias voltage is applied to the graphene layers as the modulating signal and changes its Fermi energy, \(E_{\text{f}}.\) The Fermi energy controls the propagation speed of the surface plasmon polariton (SPP) waves on the gated graphene structure. Therefore, it can increase the interaction of the hybrid plasmonic–photonic mode with the graphene layer and finally, increase the light absorption and the modulation depth. In this structure, the hybrid plasmonic–photonic mode will experience high or low absorption depending on the applied voltage on the structure. The higher Fermi energy causes the high absorption state and the low absorption state is achieved by lower Fermi energy level. Therefore, application of a variable electric gate to graphene layer modulates the signal in the output of structure. One of the key figures of merit used to characterize a modulator is the modulation depth. The modulation depth is calculated using the extinction ratio that it is the ratio of the maximum and minimum transmittance of the structure. Therefore, the modulation depth is defined as [38]:

$$M = \frac{{T_{{\rm max} } - T_{{\rm min} } }}{{T_{{\rm max} } }} \times 100,$$
(3)

where \(T_{ \hbox{max} }\) and \(T_{ \hbox{min} }\) are the maximum and minimum of the intensity, respectively. The modulation depth versus the chemical potential of graphene has been calculated for single and two layers of graphene in the structure and shown in Fig. 4.

Fig. 4
figure 4

Modulation depth as a function of the chemical potential for a single layer (black line) and two layers (red line) of graphene in the structure

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It is shown in Fig. 4 that increasing the applied voltage enhances the modulation depth up to a certain value. Further increase in the applied voltage does not yield remarkable improvements in the modulation depth. Also, it is clear in Fig. 4 that the modulation depth of the structure with two layers of graphene is more than the structure consisting of single layer of graphene. Therefore, the modulation depth of the proposed modulator increases with increasing the applied voltage and the numbers of graphene layers. Our proposed structures with a single layer and two layers of graphene show the modulation depth %7.5 and %9, respectively (Fig. 4). Experimentally, an electro-optic modulator with a single layer of graphene has been reported by Lee et al. [30]. They measured the modulation depth of %4 by a continuous-wave laser at 1.55 μm wavelength. For this measuring of modulation depth, they focused on the device a laser beam with a spot size ~ 5 μm diameter and the reflected beam received by a photodiode. Finally, the modulation depth has been extracted with a lock-in amplifier.

The 3-dB modulation bandwidth is calculated by

$$f_{{ 3 {\text{dB}}}} = \frac{1}{2\pi RC}.$$
(4)

Here, C is the capacitance and R is the resistance. The capacitance is calculated by \(C = \varepsilon \frac{S}{d}\), where S is the area of capacitance and d is the distance between the electrodes. The resistance consists the graphene–electrode contact resistance \(R_{\text{C}}\) and the graphene sheet resistance \(R_{\text{g}}\). However, the graphene sheet resistance is very small compared with the contact resistance. Therefore, we consider the resistance as the contact resistance \(R_{\text{C}} = 400 \varOmega - \mu m\) [24]. The 3-dB modulator bandwidth is calculated to be 1.07 GHz for a structure consisting of the single layer of graphene and to be 1.18 GHz for the structure with two graphene layers. The strength of the mutual interaction is improved by two layers of graphene and it causes to achieve a higher 3-dB bandwidth with lower energy consumption. However, the 3-dB bandwidth is inversely proportional to the capacitance and therefore, the capacitance must be sufficiently small to achieve a high 3-dB bandwidth. The output-light power distribution of structure for single layer and two layers of graphene is shown in Fig. 5. Full width half maximum (FWHM) of the output power distribution is very narrow for two layers of graphene compared with a single layer of graphene.

Fig. 5
figure 5

The output power distribution in the cavity structure containing a single layer (blue line) and two layers (green line) of graphene

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2 Conclusion

We have shown an electro-optics modulator with one and two graphene layers with sub-wavelength thickness. The modulated output signal is obtained from the structure by application of a variable electric gate to graphene layer. Our simulations show that increasing the number of graphene layers improves the modulation depth. The proposed modulator was very compact and it is applicable in optical circuits and mode-locking laser systems.