1. Introduction

Graphene-on-silicon photonic integrated circuits (PICs) have attracted tremendous attention in the past few years [1–4]. Due to the ultrahigh carrier mobility and electrically tunable Fermi level [5,6], the optical absorption of graphene can be rapidly modulated through the band-filling effect, which may be used for developing optoelectronic devices with bandwidth as large as 500 GHz [7]. Moreover, by integrating graphene on a silicon waveguide, the propagating light in the waveguide can strongly interact with the graphene through the in-plane evanescent-field coupling [8,9]. Therefore, this platform can take advantages of graphene’s supreme optoelectronic properties without suffering from the weak light-matter interaction in the atomic-layer-thick graphene [10]. So far, a variety of optoelectronic devices have been demonstrated on this platform, including high-speed electro-optical modulators [11–16], ultrafast photodetectors [17], all-optical switching [18], pressure sensors [19] and tunable delay-line devices [20].

On the other hand, mode-division multiplexing (MDM) techniques have been widely studied for high-speed optical interconnects [21–24]. Compared with wavelength-division multiplexing techniques [25], the MDM can scale the data communication bandwidth through multiplexing spatial modes by using a single-wavelength light source, greatly reducing device costs and on-chip heat management complexity [26,27]. To date, many efforts have been made to demonstrate the MDM applications [28–32]. As shown in Fig. 1(a), a typical method is to employ high-order-mode-to-fundamental-mode converters to demultiplex all spatial modes to a fundamental spatial mode, then use a couple of parallel electro-optical modulators to modulate different channels with the fundamental spatial mode separately, finally, multiplex these channels to different carrier spatial modes again via fundamental-mode-to-high-order-mode converters. However, this method inevitably suffers from a high cost, large device footprint, and high insertion loss.

 

Fig. 1. Comparison between the typical MDM method and proposed method. (a) Typical MDM method. (b) Proposed MDM method. In the proposed method, the TE₀ and TE₁ modes could be directly modulated by using the dual-mode modulators.

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Here, we designed graphene-based dual-mode modulators that support individual modulations for two modes by sharing a single graphene-on-silicon waveguide. By tailoring graphene nanoribbons (GNs) on the surface of a silicon waveguide, the zeroth transverse electric (TE₀) and first transverse electric (TE₁) modes could selectively interact with different structured GNs, such that they could be modulated separately and simultaneously in an identical modulator. Based on the proposed method, we designed on-off keying (OOK) and binary phase-shift keying (BPSK) modulators by using dual-mode micro-racetrack resonators (MRRs). Compared with the typical MDM method, as shown in Fig. 1(a), the proposed method in Fig. 1(b) could decrease the number of the electro-optical modulators and eliminate mode converters, which is expected to reduce costs, system footprint, and insertion optical losses induced by fan-in and fan-out devices. By virtue of graphene’s ultrafast optoelectronic properties and our delicate design, this study is expected to enable unprecedented high-speed electro-optical modulators for optical interconnects, which may bring us intriguing applications in tera-scale optical interconnects.

2. Waveguide parametrical study for mode-selective modulation

The proposed graphene-on-silicon waveguide for developing the dual-mode modulators is schematically shown in Fig. 2(a). A dual-mode silicon waveguide is designed based on a commercial silicon-on-insulator (SOI) wafer consisting of a 250-nm-thick top silicon layer and 3-µm-thick buried oxide (BOX). With a waveguide width of 1 µm, the dual-mode silicon waveguide can simultaneously support TE₀ and TE₁ modes, as shown in the inset of Fig. 2(b). There is a 50-nm-thick waveguide slab in our design, which could be used for applying a back-gate voltage to the GNs. Then a silica cladding could be deposited on the chip by using the chemical vapor deposition (CVD) method to insulate the electrical contact between the silicon waveguide and GNs. Finally, two graphene layers, namely, a bottom GN and two top GNs, could be fabricated on the surface of the chip, which could be insulated by using a silica cladding, as shown in Fig. 2(b). For the bottom GN, it was designed above the middle of the silicon waveguide, such that it can be largely overlapped with the electric field of the TE₀ mode, while the top GNs were designed above two edges of the silicon waveguide, thus they can be significantly overlapped with the electric field of the TE₁ mode. Based on this principle, the TE₀ and TE₁ modes could be modulated separately and simultaneously by changing the Fermi levels of the bottom GN and top GNs individually. The possible fabrication processes of the proposed devices are discussed in Appendix 1.

 

Fig. 2. Schematic of the graphene-on-silicon waveguide used for developing the dual-mode modulators. (a) Three-dimensional view. (b) Cross-section view. Inset: intensity distributions of the TE₀ and TE₁ modes in the waveguide.

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Before discussing the dual-mode modulators, we designed the graphene-on-silicon waveguide by using commercial finite element method software (COMSOL Multiphysics). The graphene’s parameters we used in the simulation can be found in Appendix 2. As shown in Fig. 3(a), the optical losses of the TE₀ and TE₁ modes in the graphene-on-silicon waveguide as a function of the width of each top GN (highlighted in red) were simulated. In this simulation, the bottom silica cladding thickness was fixed as 5 nm, while the width of the bottom GN was fixed as 300 nm. The Fermi levels of the top GNs and bottom GN were fixed as 0.0 eV and 1.0 eV, respectively, therefore the top GNs dominate optical absorption to the propagating light in the waveguide. As we expected, the optical absorption to the TE₁ mode is larger than that of the TE₀ mode, since the top GNs largely overlap with the electric field of the TE₁ mode. Based on the simulation results, we chose a GN width of 300 nm with which the graphene-on-silicon waveguide exhibits the largest TE₁-to-TE₀ modal extinction ratio (ER), as indicated by the red arrow in Fig. 3(a). Also, we simulated the optical losses of the TE₀ and TE₁ modes in the graphene-on-silicon waveguide as a function of the width of the bottom GN (highlighted in red color), as shown in Fig. 3(b). The bottom silica cladding thickness was fixed as 5 nm, while the width of each top GN was fixed as 300 nm. The Fermi levels of the top GNs and bottom GN were fixed as 1.0 eV and 0.0 eV, respectively. In this simulation, the bottom GN dominates optical absorption to the propagating light in the waveguide, so the optical absorption to the TE₀ mode is larger than that of the TE₁ mode. As shown in Fig. 3(b), we chose a GN width of 350 nm with which the graphene-on-silicon waveguide exhibits the largest TE₀-to-TE₁ modal ER.

 

Fig. 3. Simulation results of the optical losses of the TE₀ and TE₁ modes as a function of the GN width. (a) Optical loss of the graphene-on-silicon waveguide as a function of each top GN (highlighted in red color) width. (b) Optical loss of the graphene-on-silicon waveguide as a function of the bottom GN (highlighted in red color) width. Red arrows indicate the GN widths with which the graphene-on-silicon waveguides exhibit the largest modal ERs.

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After choosing the suitable GNs’ widths, we simulated the wave vector and optical loss of the graphene-on-silicon waveguide as a function of the graphene’s Fermi level, as shown in Fig. 4. We first simulated complex refractive index (RI) (neff) of the graphene-on-silicon waveguide. Then, the wave vector (κ) and optical loss (α) can be calculated as follows [33]:

 

Fig. 4. Simulation results of the optical loss and wave vector of the graphene-on-silicon waveguide as a function of the graphene’s Fermi level. (a) Optical losses of the TE₀ and TE₁ modes with respect to the Fermi level of the top GNs (highlighted in red color). (b) Wave vectors of the TE₀ and TE₁ modes with respect to the Fermi level of the top GNs. (c) Optical losses of the TE₀ and TE₁ modes with respect to the Fermi level of the bottom GN (highlighted in red color). (d) Wave vectors of the TE₀ and TE₁ modes with respect to the Fermi level of the bottom GN. The simulation results indicate that, by tuning the Femi level, the top GNs could introduce a larger phase shift to the TE₁ mode than to the TE₀ mode, while the bottom GN could introduce a larger phase shift to the TE₀ mode than to the TE₁ mode.

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In the case of the top GNs modulation, we fixed the Fermi level of the bottom GN as 1.0 eV and changed the Fermi level of the top GNs from 0.0 eV to 1.0 eV. Since the Fermi level of the bottom GN is more than the half-photon energy (∼0.4 eV), it introduces low optical losses to the TE₀ and TE₁ modes. As shown in Fig. 4(a), when the Fermi level of the top GNs is beyond 0.4 eV, the optical losses of the TE₀ and TE₁ modes quickly reduce to near zero, while the wave vectors continually decrease as shown in Fig. 4(b). The maximum variations of the wave vector were calculated as 0.0036 µm^-1 for the TE₀ mode and 0.0077 µm^-1 for the TE₁ mode, which indicates the phase shift of the TE₁ mode is ∼2 times larger than that of the TE₀ mode when tuning the Femi level of the top GNs. On the other hand, in the case of the bottom GN modulation, we fixed the Fermi level of the top GNs as 1.0 eV and changed the Fermi level of the bottom GN from 0.0 eV to 1.0 eV. Similarly, the graphene introduces low optical losses to both the TE₀ and TE₁ modes when the Fermi level is beyond the half-photon energy (∼0.4 eV), as shown in Fig. 4(c). The maximum variations of the wave vector were calculated as 0.0041 µm^-1 for the TE₁ mode and 0.0077 µm^-1 for the TE₀ mode as shown in Fig. 4(d), indicating the phase shift of the TE₀ mode is ∼1.9 times larger than that of the TE₁ mode when tuning the Fermi level of the bottom GN. From the above results, we can conclude that by tuning the Femi levels of the top GNs and bottom GN, we can separately control the phase shifts of the TE₀ and TE₁ modes in an identical graphene-on-silicon waveguide without introducing high optical losses to the propagating light.

3. Design of the dual-mode modulators

Based on the above study, we designed the dual-mode modulators based on an MRR resonator configuration which could achieve comparable cross-coupling coefficients and simultaneous resonance for two modes [34]. As shown in Fig. 5, the dual-mode MMR with a coupling waveguide length of 228.6 µm and a bending radius of 30 µm, which can simultaneously achieve TE₀ and TE₁ modes coupling, was employed in the dual-mode modulators’ design. After simulating effective RIs and power attenuation coefficients of the TE₀ and TE₁ modes, the optical transmission through the MMR can be calculated by using the transfer matrix method [35,36]. The detailed design of the dual-mode MMR can be found in Appendix 3.

 

Fig. 5. Schematic of the dual-mode modulators based on the MRR. The TE₀ and TE₁ modes could be individually modulated by driving the electrodes contact with the top GNs and bottom GN.

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3.1 Design of the OOK dual-mode modulator

Based on the aforementioned analysis, we first designed the OOK dual-mode modulator. In our calculation, we considered four different signal modulation patterns, namely, TE₀ → 1 TE₁ → 1, TE₀ → 1 TE₁ → 0, TE₀ → 0 TE₁ → 1, and TE₀ → 0 TE₁ → 0. Then, we optimized the Fermi levels of the top GNs and bottom GN to achieve the above patterns, as shown in Fig. 6. We first chose the Fermi level of 0.45 eV and 0.46 eV for the top GNs and the bottom GN as an initial status, so that both the TE₀ and TE₁ mode can almost reach critical coupling condition simultaneously. Then, the normalized transmission through the MRR resonator was calculated, as shown in Fig. 6(a). Two dips in the normalized transmission spectrum, namely, 1.55303 µm and 1.55306 µm, correspond to the TE₀ mode and TE₁ mode resonance, respectively. The quality (Q) factors corresponding to the two resonance are ∼15,100 and ∼13,600, respectively. At a wavelength of 1.55237 µm, both the TE₀ and TE₁ modes are off-resonance and provide high transmission (∼0.03 dB insertion loss), corresponding to the signal modulation pattern of TE₀ → 1 TE₁ → 1. Then, we separately tuned the Fermi levels of the top GNs and bottom GN to achieve other patterns at the same wavelength. According to the above analysis, with increasing the Fermi levels, the resonant wavelengths display blue-shifts due to the decrease of the effective RI, as shown in Figs. 4(b) and 4(d). The Q factors of the TE₀ and TE₁ resonant modes do not greatly reduce after changing the Fermi levels since the variations in the waveguide optical loss are small when the Fermi levels are more than the half-photon energy (∼0.4 eV), as shown in Figs. 4(a) and 4(c). Moreover, we can get larger resonant wavelength shift for the specific modes when we tuned the Fermi levels of the specific GNs. Based on these properties, we could obtain the patterns TE₀ → 0 TE₁ → 1 and TE₀ → 1 TE₁ → 0, respectively. As shown in Fig. 6(b), by tuning the Fermi level of the top GNs from 0.45 eV to 0.95 eV, the TE₁ mode has 9.25 dB modulation depth while the TE₀ mode only obtains 0.06 dB modulation depth at the wavelength of 1.55237 µm. Similarly, by tuning the Fermi level of the bottom GN from 0.46 eV to 1.00 eV, the TE₀ mode gets 14.77 dB modulation depth, while the TE₁ mode only achieves 0.06 dB modulation depth at the wavelength of 1.55237 µm, as shown in Fig. 6(c). Finally, by simultaneously shifting the Fermi level of the top GNs to 0.72 eV and the Fermi level of the bottom GN to 0.75 eV, both the TE₀ and TE₁ modes are on-resonance at the wavelength of 1.55237 µm, as shown in Fig. 6(d). Therefore, the TE₀ mode has 14.62 dB modulation depth, while the TE₁ mode obtains 15.29 dB modulation depth responding to the pattern of TE₀ → 0, TE₁ → 0. Overall, by adjusting the Fermi levels of the top GNs and bottom GN, we can achieve arbitrary modulation patterns over two spatial modes. All the results are summarized in Table 1.

 

Fig. 6. Normalized transmission through the OOK dual-mode modulator. The red stars indicate the resonant wavelengths of the TE₀ and TE₁ modes before tuning the Fermi levels. (a) Normalized transmission as a function of the wavelength by fixing the Fermi level of 0.45 eV for the top GNs and the Fermi level of 0.46 eV for the bottom GN. (b) Normalized transmission as a function of the wavelength by tuning the Fermi level of the top GNs. (c) Normalized transmission as a function of the wavelength by tuning the Fermi level of the bottom GN. (d) Normalized transmission as a function of the wavelength by tuning the Fermi levels of the top and bottom GNs.

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Table 1. Signal modulation over two spatial modes and its corresponding Fermi levels of the OOK dual-mode modulator.

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3.2 Design of the BPSK dual-mode modulator

Before designing the BPSK dual-mode modulator, we have to optimize the bottom cladding thickness. Different from OOK modulators, BPSK modulators based on MMR resonators usually require a precious control of the resonant wavelength to achieve a π phase delay change [36]. However, in the above OOK dual-mode modulator simulation, the bottom silica cladding thickness was fixed as 5 nm, resulting in a 2π phase delay change even tuning the Fermi level as small as 0.01 eV. Due to this limitation, we have to increase the bottom cladding thickness, which results in weaker light-graphene interactions to finely control the resonant wavelength shift by tuning the Fermi levels of GNs. According to our simulation, after increasing the bottom cladding thickness from 5 nm to 60 nm, the changes in the effective RI with respect to the Fermi level could be reduced by a factor of 2. As a result, we chose 60 nm as the bottom silica cladding thickness for developing the BPSK modulator.

After optimizing the thickness of the bottom cladding thickness, we designed the BPSK dual-mode modulator. Compared to the OOK dual-mode modulator, the BPSK dual-mode modulator can separately and simultaneously modulate the TE₀ and TE₁ mode with a π phase delay change in an identical device. Here, we also considered four different signal modulation patterns, namely, TE₀ → 0 phase delay change TE₁ → 0 phase delay change, TE₀ → 0 phase delay change TE₁ → π phase delay change, TE₀ → π phase delay change TE₁ → 0 phase delay change, TE₀→ π phase delay change TE₁ → π phase delay change. To achieve the above modulation patterns in the spatial mode domain at a single wavelength, the Fermi levels of the top GNs and bottom GN were optimized, as shown in Fig. 7.

 

Fig. 7. Normalized transmission and phase delay through the BPSK dual-mode modulator. The black and blue curves represent the simulated results of TE₀ and TE₁ modes, while the solid and dash curves correspond to the simulated results before and after modulation. (a) Normalized transmission and phase delay as a function of the wavelength by fixing the Fermi level of 0.42 eV for the top GNs and the Fermi level of 0.46 eV for the bottom GN. (b) Normalized transmission and phase delay as a function of the wavelength by fixing the Fermi level of 0.43 eV for the top GNs and the Fermi level of 0.45 eV for the bottom GN. (c) Normalized transmission and phase delay as a function of the wavelength by fixing the Fermi level of 0.41 eV for the top GNs and the Fermi level of 0.52 eV for the bottom GN. (d) Normalized transmission and phase delay as a function of the wavelength by fixing the Fermi level of 0.43 eV for the top GNs and the Fermi level of 0.46 eV for the bottom GN.

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We first chose the Fermi level of 0.42 eV and 0.46 eV for the top GNs and the bottom GN as an initial status, so that both the TE₀ mode and TE₁ mode can reach critical coupling condition simultaneously. Then, the normalized transmission and the phase delay through the dual-mode MRR resonator was calculated, as shown in Fig. 7(a). At a wavelength of 1.560117 µm, the TE₀ mode TE₁ mode gets initial phase delays of 0.51π and 0.53π, respectively, corresponding to the signal modulation pattern of TE₀ → 0 phase delay change TE₁ → 0 phase delay change. Then we separately tuned the Fermi levels of the top GNs and bottom GN. We can get specific phase delay change by properly tuning the Fermi levels of the GNs. Based on the above properties, we can obtain the BPSK modulation for the pattern described before. As shown in Fig. 7(b), by tuning the Fermi level of the top GNs from 0.42 eV to 0.43 eV and the bottom GN from 0.46 eV to 0.45 eV, we realize a 1.03π phase delay change for TE₁ mode and a 0.07π phase delay change for TE₀ mode. Similarly, by tuning the Fermi level of the top GNs from 0.42 eV to 0.41 eV and the Fermi level of the bottom GN from 0.46 eV to 0.52 eV, we obtain a 1.04π phase delay change for TE₀ mode and a 0.13π phase delay change for TE₁ mode, as shown in Fig. 7(c). Finally, by simultaneously shifting the Fermi level of the top GNs from 0.42 eV to 0.46 eV and the Fermi level of the bottom GN remain constant, we get phase delay change of 0.96π and 1.15π for TE₀ mode and TE₁ mode. In summary, through tuning the Fermi levels of the top GNs and bottom GN, four BPSK modulation patterns over two spatial modes could be achieved, and all the results are summarized in Table 2.

Table 2. Signal modulation over two spatial modes and its corresponding Fermi levels of the BPSK dual-mode modulator.

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4. Discussion of the duobinary dual-mode modulator

To explore applications of the proposed device in higher-order modulation formats, we here also discussed phase-intensity hybrid modulation based on the graphene-based dual-mode modulators. In Section 3, we have theoretically demonstrated that both the intensity modulator and phase modulator can be designed by using the proposed method. By combining the different intensity and phase states of the TE₀ and TE₁ modes, we considered nine different signal modulation patterns, as shown in Table 3. By using the same principle to optimize the device structure and adjust the Fermi level of graphene, it can be expected that the duobinary modulation could be achieved by in the proposed device. Herein, we won’t go to the design details in this section. Moreover, it is worthwhile to note that the idea of the dual-mode modulators in this work may be transferred to other semiconductor-based (namely, Ge) modulators based on quantum-confined Stark effect or Franz-Keldysh effect [37,38]. Compared to them, graphene-based modulators could have advantages in terms of working speed [7], spectral bandwidth [5], and power consumption [12] due to graphene’s 2D nature.

Table 3. Signal modulation over two spatial modes of the expected duobinary dual-mode modulator

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5. Conclusion

In conclusion, we designed the graphene-based dual-mode modulators. Through controlling the Fermi levels of the top GNs and bottom GN above the dual-mode waveguide, the input modes, namely, TE₀ and TE₁ modes, in the MMR resonators could be separately and simultaneously modulated. The study could open an avenue to develop high-speed and high-density chip-integrated optoelectronic MDM devices for tera-scale optical interconnects.

Appendix 1. Proposed fabrication processes of the dual-mode modulators

The fabrication processes of the proposed dual-mode modulators are summarized in Fig. 8. First, a rib silicon waveguide can be fabricated on a commercial SOI wafer via electron beam lithography (EBL) and inductively coupled plasma (ICP) etching processes [39]. Then, a silica cladding can be grown on the silicon chip by using the CVD method. After the silica cladding deposition, a chemical mechanical polishing (CMP) method can be used to planarize the surface of the chip to ensure a good attachment of graphene [40]. Afterward, a commercial CVD-grown graphene sheet can be transferred to the surface of the chip by using a wet transferring method [41]. After that, the chip-integrated graphene sheet can be patterned to the bottom GN via EBL and oxygen ion etching processes [42]. Next, the CVD and CMP methods can be used again to fabricate the second silica cladding layer on which the top GNs can be fabricated by using another wet transferring, EBL, and oxygen plasma etching processes, respectively. Then, via can be fabricated through EBL and wet etching process to make a connection between GNs and electrodes. Finally, the electrodes can be fabricated by using electron beam evaporation and lift-off processes [43]. It is worthwhile to note that all the proposed processes and waveguide-integrated double-layer graphene devices have been demonstrated in previous studies [44,45], which indicates the fabrication feasibility of the proposed devices.

 

Fig. 8. Schematic of the proposed fabrication processes of the dual-mode modulators. All the proposed fabrication processes are compatible with the CMOS technology

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Appendix 2. Graphene’s relative permittivity

To validate GNs’ modulation on the TE₀ and TE₁ modes in the graphene-on-silicon waveguide, herein we calculated graphene’s relative permittivity as a function of its Fermi level. First, the in-plane optical conductivity of graphene can be expressed by Kubo formula, which depends on its intraband and interband transitions, as follows:

(3)$${\sigma _{\textrm{total}}}\textrm{ = }{\sigma _{\textrm{intra}}}\textrm{ + }\sigma _{\textrm{inter}}^{\prime}\textrm{ + }i\sigma _{\textrm{inter}}^{\prime\prime},$$

where,

(4)$${\sigma _{\textrm{intra}}} = {\sigma _0}\frac{{4{\mu _\textrm{c}}}}{\pi }\frac{1}{{\hbar ({\Gamma _1} - i\omega )}},$$

(5)$$\sigma _{\textrm{inter}}^{\prime} = {\sigma _0}(1 + \frac{1}{{\pi }}{\tan ^{ - 1}}\frac{{\hbar \omega - 2}}{{\hbar {\Gamma _2}}} - \frac{1}{{\pi }}{\tan ^{ - 1}}\frac{{\hbar \omega + 2}}{{\hbar {\Gamma _2}}}),$$

(6)$$\sigma _{\textrm{inter}}^{\prime\prime} ={-} {\sigma _0}\frac{1}{{2{\pi }}}\textrm{In}[\frac{{{{(2{\mu _\textrm{c}} + h\omega )}^2} + {h^2}\Gamma _2^2}}{{{{(2{\mu _\textrm{c}} - h\omega )}^2} + {h^2}\Gamma _2^2}}].$$

In Eqs. (4)–(6), σ₀ = e²/4ℏ = 60.8 µs is the universal optical conductance of graphene, µ_c is Fermi level, and ω is the frequency of incident light. Relaxation rates we used in the calculation are Г₁ = 8.3×10¹¹ s⁻¹ and Г₂= 10¹³ s⁻¹ [13], respectively. Although the conductivity tensor model could completely describe graphene’s optical properties [46], the random phase approximation method is precise enough for the graphene-based modulator design [3], which has been verified by the experimental results [47]. Therefore, we used a complex conductivity to model graphene. Then, the complex permittivity ɛ(ω) of graphene could be obtained from its complex optical conductivity σ(ω) as follows [48]:

(7)$$\; \varepsilon (\omega ) = \textrm{1 + }\frac{{i\sigma (\omega )}}{{\omega {\varepsilon _\textrm{0}}{d_\textrm{g}}}},$$

where, d_g= 0.7 nm is the graphene’s thickness in our calculation, and ɛ₀ is vacuum permittivity. The calculated complex permittivity of graphene as a function of the Fermi level is shown in Fig. 9. At a wavelength of 1.55 µm, when the Fermi level is more than the half-photon energy (∼0.4 eV), the absolute value of the graphene permittivity is dominated by its intraband transitions, sharply decreasing the absorption of graphene. Therefore, in this region, we could modulate the graphene-induced phase shifts to the propagating light without introducing high optical losses to the TE₀ and TE₁ modes.

Appendix 3. Design of the micro race-track resonator (MRR)

Figure 10 shows a schematic of the proposed dual-mode MRR. The MRR is composed of a dual-mode waveguide which can simultaneously support TE₀ and TE₁ modes. When the TE₀ mode or TE₁ mode is on resonance, the transmission of the TE₀ mode or TE₁ mode significantly decrease. Otherwise, the light can pass through the dual-mode MRR. Assuming the intensity of the reflected light is negligible, the device transmission T_n can be expressed as follows [34,49]:

(8)$$\; {T_\textrm{n}} = \frac{{{I_{\textrm{pass}}}}}{{{I_{\textrm{input}}}}} = \frac{{{\alpha ^2} - 2r\alpha \cos \varphi + {r^2}}}{{1 - 2\alpha r\cos \varphi + {{(r\alpha )}^2}}},$$

where r is the self-coupling coefficient, α is the single-pass amplitude transmission of the MRR, and φ is the single-pass phase shift. Moreover, κ² and r² satisfy κ² + r² = 1, where κ is the cross-coupling coefficient determining by the coupling waveguide length as well as the gap width between the coupling waveguide and MRR, as shown in Fig. 10. When the dual-mode MRR is under critical coupling (T_n≈0), the coupled power is equal to the power loss in the ring, which is α²= 1 - κ². Generally, due to the different effective RIs between TE₀ mode and TE₁ mode, the cross-coupling coefficients κ are usually different for the TE₀ and TE₁ modes. Therefore, the TE₀ and TE₁ modes cannot simultaneously reach the critical coupling condition in general.

 

Fig. 9. Complex relative permittivity of graphene as a function of its Fermi level. The wavelength and thickness of graphene used in our calculation are 1.55 µm and 0.7 nm. Insets: when the graphene’s Fermi level increases, the interband transition is significantly suppressed.

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Fig. 10. Schematic of the proposed MRR. The dual-mode waveguide is employed in the MRR which can simultaneously support TE₀ and TE₁ modes.

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In this work, to enable the critical coupling condition for both the TE₀ mode and TE₁ mode, we optimized the coupling waveguide length and the gap width, as indicated in Fig. 11, which can maximize the modulation depths of the TE₀ and TE₁ modes. First, we simulated propagating light intensities in the coupling waveguide and MRR with a gap width of 100 nm by using a beam-propagation-method software tool (RSoft). With a special coupling waveguide length of 30.38 µm, the TE₀ and TE₁ modes have the same cross-coupling coefficient. By assuming the TE₀ and TE₁ modes have a similar optical loss, we can almost achieve simultaneously reach the critical coupling condition for the two modes. This assumption is reasonable since graphene does not introduce significant losses to both the TE₀ and TE₁ modes after the Fermi level is beyond half of the photon energy. Therefore, it can be expected that the TE₀ and TE₁ modes can simultaneously reach critical coupling condition.

 

Fig. 11. Simulation results of the cross-coupling coefficient as a function of the coupling waveguide length when the gap width between the coupling waveguide and MRR is 100 nm. With the coupling waveguide length of 30.38 µm, TE₀ and TE₁ modes can achieve critical coupling conditions simultaneously.

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Then, we swept the gap width from 100 nm to 400 nm and simulated the coupling waveguide length with which both the TE₀ and TE₁ modes could simultaneously reach critical coupling conditions, as shown in Fig. 12(a). With increasing the gap width, the required coupling waveguide length significantly increases. Moreover, the cross-coupling coefficient (κ) as a function of the gap width was simulated as shown in Fig. 12(b). When the gap width increases from 100 nm to 400 nm, the cross-coupling coefficient κ decreases from 0.272 to 0.130. The results indicate that we need to design the coupling waveguide length and gap width based on the optical loss of the TE₀ and TE₁ modes to ensure both the two modes can simultaneously reach critical coupling conditions. Finally, we calculated the Q factors of the MRR as a function of the gap width, as shown in Fig. 12(c). The Q factor of the MRR is defined as follows [34,50]:

(9)$$\; Q = \frac{{\pi {L_{\textrm{ef}f}}{n_{\textrm{eff}}}}}{{{\lambda _0}\arccos (\frac{{1 + {r^4}{\alpha ^2} - 4{r^2}\alpha }}{{ - 2{r^2}\alpha }}) + {{(r\alpha )}^2}}},$$

where, L_eff is the MRR’s effective length, n_eff is the modal effective refractive index, and λ₀ is the resonant wavelength. As shown in Fig. 12(c), under the critical coupling condition, the Q factors of the TE₀ and TE₁ modes increases from ∼3,000 to ∼25,000 and ∼2,600 to ∼22,000 with increasing the gap width from 100 nm to 400 nm. Here, we chose the Fermi level of 0.45 eV for the top GNs and 0.46 eV for the bottom GN, corresponding to the optical losses of 0.0021916622 dB/µm for the TE₀ and 0.0021910637 dB/µm TE₁ modes, which are simulated by using the commercial software (COMSOL Multiphysics). Due to the similar optical loss, the single-pass amplitude transmissions α of the TE₀ and TE₁ modes are almost the same (0.92) in Eq. (9). With the given optical losses, we chose a gap width of 344 nm in our design for achieving critical coupling condition, which corresponds to the coupling waveguide length of 228.5 µm, the cross-coupling coefficient of 0.15, the Q factor of ∼15,100 and ∼13,600 for the TE₀ and TE₁ modes, respectively. It is worth noting that when the Fermi levels are future increased, the resonant wavelengths of MRR display blue shift but the Q factors of the TE₀ and TE₁ modes hardly change. Therefore, we tuned the Fermi levels of the GNs within this range in our modulators, which corresponds to the resonant wavelength shifts for the TE₀ and TE₁ modes without changing their Q factors.

 

Fig. 12. Simulation results of the coupling waveguide length, cross-coupling coefficient, and Q factor as a function of the gap width when the TE₀ and TE₁ modes almost achieve critical coupling conditions simultaneously. (a) Coupling waveguide length with respect to the gap width. (b) Cross-coupling coefficients of the TE₀ and TE₁ modes with respect to the gap width. (c) Q factors of TE₀ and TE₁ modes with respect to the gap width.

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Funding

National Natural Science Foundation of China (61805175); Japan Society for the Promotion of Science (JP18K13798); National Young Thousand Talents Plan.

Disclosures

The authors declare no conflicts of interest.

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