Abstract

A graphene electro-absorption optical modulator based on double-stripe silicon nitride waveguide is proposed and analyzed. By embedding four graphene layers in the double-stripe silicon nitride waveguide and the graphene layers co-electrode design, the total metal-graphene contact resistance can be reduced 50% and as high as 30.6GHz modulation bandwidth can be achieved theoretically. The calculated extinction ratio and figure of merit are 0.1658dB/um and 9.7, respectively. And the required switching voltage from its minimum to maximum absorption state is 3.8180V and 780.50fJ/bit power consuming can be achieved. The proposed modulator can remedy the lack of high speed modulator on the passive silicon nitride waveguide.

© 2017 Optical Society of America

1. Introduction

Graphene, a single sheet of carbon atoms arranged in a two-dimensional honeycomb lattice, was isolated experimentally by mechanical exfoliation in 2004 [1]. It has many extraordinary properties, such as the extremely high electron mobility of more than 200000cm2/(V·s) [2]. In an un-doped layer of graphene, it has a universal linear absorption coefficient, which values 2.3% over a broad spectrum range from visible to mid-infrared(IR) [3]. Since the Fermi energy of graphene is quite sensitive to the occupation of electronic states, it can be significantly shifted by slightly variations of carrier density, which can be tuned conveniently by applied voltage [4], leading to the gate-controlled optical absorption in graphene.

Utilizing this effect, various geometries of graphene electro-absorption modulators based on silicon-on-insulator (SOI) have been proposed [2,3,5,6]. In 2011, a graphene electro-absorption modulator based on SOI was first reported by Liu et al. [2]. The graphene modulator experimentally demonstrated a mode power attenuation (MPA) of 0.10dB/μm and modulation speed of 1.2GHz. In 2012, Liu et al. experimentally proposed a double-layer graphene optical modulator, which demonstrated a high MPA of 0.16dB/μm, modulation speed of 1.2GHz, insertion loss of 4dB, and power consumption of 1pJ/bit [5]. In 2014, a graphene electro-absorption modulator based on SOI waveguide with low insertion loss was presented experimentally by Muhammad Mohsin et al. [6]. The graphene modulator showed a relatively low modulation speed of 670MHz, insertion loss of 3.3dB and a figure of merit (FOM) Δα/α = 4.8(ratio of extinction to the insertion loss). So far, graphene modulators based on SOI waveguide have been extensively studied. As we known, the main limitation of SOI waveguide is it’s relatively large propagation loss introduced by the surface roughness (about 2~3dB/cm), which is much larger than the typical 0.1dB/cm propagation loss of silicon nitride (Si3N4) waveguide. However, it should be noted that graphene losses will overshadow most of the propagation loss reduction offered by Si3N4 because graphene loss is in dB/um, which is several orders larger than the SOI loss in dB/cm. Therefore the linear propagation losses of the silicon graphene and silicon nitride graphene modulators should be comparable. Besides, Si3N4 waveguide can be fabricated using CMOS-compatible technologies at the length of tens of centimeters with high yield, and comparing with the SOI waveguide, it has relatively lower refractive index contrast, which brings the larger fabrication tolerances [7] and better performance in the fiber-coupling [8]. And Si3N4 stripe waveguide has demonstrated a propagation loss below 0.1dB/m with a fiber-coupling loss less than 1dB [9]. So the silicon nitride graphene modulator may has the potential of easier fabrication and lower insertion loss. In addition, it is well known that Si3N4 has a large bandgap of ~5eV compared to Si (~1.12eV), and due to this large bandgap, Si3N4 has a negligible two-photon absorption (TPA) loss compare with the rather large TPA loss in silicon [10, 11]. So coupled to the fact that graphene has saturable absorption property, it is very possible that Si3N4 devices can offer very low nonlinear losses when compared to silicon [12, 13]. It can be concluded that Si3N4 waveguide shows better performances than the SOI waveguide when it is used to build passive devices. But because the Si3N4 has no electro-optic effect, no high speed electro-optical modulators can be achieved on the Si3N4 waveguide platform. This intrinsic shortcoming can be overcome by embedding graphene in the Si3N4 waveguide. In recent years, graphene-on-silicon nitride - waveguides have also been investigated [12–15]. In 2013, a hybrid structure with graphene layer on top of a partially etched Si3N4 rib waveguide with MPA of 0.066dB/μm was presented theoretically and experimentally by N.Gruhler et al. [12]. In 2015, a graphene-on-Si3N4 electro-optic modulator based on ring resonator was demonstrated experimentally by Christopher T. Phare et al [14]. This modulator operates with a 30GHz bandwidth and a state-of-the-art modulation efficiency of 15dB per 10V. In 2017, Leili Abdollahi Shiramin et al. [15] theoretically proposed a double layer graphene on silicon nitride (DLG-Si3N4) modulator and a double layer graphene embedded in silicon nitride (DLG-E-Si3N4) modulator, which embed the double layer graphene stacks within the waveguide core. In the DLG-Si3N4 configuration, a high MPA of ∼0.12dB/μm of the TE mode can be achieved, and a 200-µm long configuration allows for 23 dB extinction ratio and 2.5GHz modulation bandwidth. In the DLG-E-Si3N4 configuration, an even higher MPA of ∼0.26dB/μm of the TE mode can be achieved, and a MPA of ∼0.01dB/μm of the TM mode was also obtained. As far as we know, almost all the presented Si3N4/graphene modulators are based on the single stripe Si3N4 waveguide. Comparing to the single stripe Si3N4 waveguide, double-stripe Si3N4 waveguide will have much more symmetrical mode field distribution and lower polarization dependence. While the Si3N4/graphene modulator using the double-stripe Si3N4 waveguides has not been proposed.

In this paper, we proposed a four graphene layers electro-absorption optical modulator based on the double-stripe Si3N4 waveguides. And the rest of this paper is organized as follows: In Section 2, the optical and electrical properties of graphene are described. In Section 3, the performances of this modulator were studied by the finite element method (FEM). In the final section, the device’s characteristics were summarized and concluded.

2. Gate-variable properties of graphene

In [16], it is proved that graphene need to be treated as an anisotropic material instead of an isotropic material because it is one atom thick and its π electrons cause electric conduction in its plane. The out-of plane conductivity σ can be different from the in-plane conductivity σ|| and the epsilon-near-zero (ENZ) effect does not exist. For the in-plane optical conductivity σ|| of graphene, an analytic expression derived by the random phase approximation is used, which is

σ=σinter+σintra=i8σ0πEthEph+iEsIn[2cos(EF2Eth)]+σ0[12+1πtan1(Eph2EF2Eth)i2πIn(Eph+2EF)2(Eph2EF)2+4Eth2]
where σ0 = e2/4ћ = 60.85(μS) is the universal conductivity of graphene (e is the charge of an electron and ћ is the reduced Planck constant), Eth = kBT is thermal energy in eV (kB is the Boltzmann constant, and T is the temperature), EF is the Fermi energy (that is chemical potential µc) of graphene in eV, Eph = hc/λ is photon energy in eV (h is the Planck constant), and Es = ћ/τ is scattering energy in eV for the scattering time τ. Thus, the graphene conductivity as a function of chemical potential is shown in Fig. 1(a).

 

Fig. 1 (a) Real and imaginary parts of graphene’s conductivity. (b) Graphene’s dielectric constant as a function of chemical potential at wavelength λ = 1550nm, T = 300 K, τ = 0.1ps.

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From the complex optical conductivity, the complex dielectric constant ε||(µc) of graphene can be calculated as

ε(μc)=2.5+iσ(μc)ωε0dG

where dG = 0.7nm is the thickness of monolayer graphene [16–18], ε0 is the vacuum permittivity, and the out-of plane permittivity is assumed to be ε = 2.5 [16]. Thus the obtained dielectric constant of graphene as a function of chemical potential is shown in Fig. 1(b).

As shown in Fig. 2, the carrier density in graphene can be tuned in an electric-field-gated structure [19]. Subsequently the chemical potential of graphene can be dynamically tuned by the applied voltage. The relationship between the carrier density and applied voltage is N = α(V + V0), where α = ε0εr/(de) is estimated using a parallel-plate capacitor model, V is applied voltage, V0 is the voltage offset caused by natural doping, εr and d is the relative dielectric constant and thickness of the dielectric spacer respectively, e is the elementary charge. Thus the chemical potential of graphene shifts according to:

μc=sgn(n)VFπ|N|=sgn(n)VFαπ|V+V0|
where VF = 106m/s is the Fermi velocity, positive (negative) n means electron (hole) doping in graphene [4]. For simplicity, |V + V0| is defined as the applied voltage below.

 

Fig. 2 Tuning the optical absorption of graphene in an electric-gated structure.

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3. Performances of the graphene modulator

The structure of the double-stripe Si3N4 optical waveguide is shown in Fig. 3(a). The optical waveguide contains two identical stripe Si3N4 waveguides. In order to ensure that only a single mode is supported in the waveguide at λ = 1550nm, the width and thickness of each Si3N4 stripe waveguide was optimized to be 1.2μm and 0.17μm, and the two Si3N4 stripe waveguides are separated by a 500nm SiO2 interlayer [7]. The advantages of using double-stripe Si3N4 waveguides are the balancing between the low propagation loss and compact device size induced by high mode confinement [7, 8, 20]. The chemical potential of graphene can be tuned in an electric-gated structure, but the Si3N4 core and the SiO2 cladding are not conductive, thus it is difficult to realize a capacitor structure to apply voltage for a single layer graphene-on-Si3N4 configuration. Hence, in order to facilitate the addition of electrodes and increasing the optical absorption modulation, only the configurations of double graphene layers and four graphene layers were considered. As shown in Fig. 3(b), two layers of graphene are separated by an insulating dielectric spacer with applied voltage between them, namely the graphene-on-graphene(GOG) structure. Thus, as the voltage is applied, one graphene layer is doped by holes and the other by electrons at the same doping level. The thickness of the dielectric spacer is d, the width and length of graphene are W and L, respectively. In Fig. 3(c), it shows the double graphene layers configuration, where the two graphene layers arranged as the GOG structure are placed on the top surface of bottom Si3N4 stripe waveguide. And Fig. 3(d) shows the four graphene layers configuration, where two GOG structures are placed on the top and bottom surfaces of the two Si3N4 stripe waveguides, respectively

 

Fig. 3 (a) The cross section of the double-stripe Si3N4 waveguide. (b) The graphene-on-graphene (GOG) structure. (c) The double graphene layers configuration. (d) The four graphene layers configuration.

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In the two and four graphene layers configurations, the electric field magnitude |E| distributions of the TE and TM modes are shown in Figs. 4(a)-4(d). As we can see, the modes are well confined around the two Si3N4 stripes. And electric field magnitude |E| distributions for both configurations are close to circular shape, which could be well coupled to the fiber mode with a spot size converter. As shown in Figs. 4(e)-4(h), the real part and imaginary part of effective mode index (EMI) can be tuned by varying the chemical potential of the graphene for both configurations. For the TE modes, the real part of EMI first increases then decreases with the chemical potential μc, and achieves maxima at μc = 0.4eV. Note that in the vicinity of μc≥0.4eV a small change in the chemical potential yields a substantial change in the real value of EMI. This effect can be made into the electro-refraction modulator [21]. On the other side, a small change in the chemical potential in both directions around μc = 0.4eV (0.3eV<μc<0.5eV) yields a substantial change in the value of MPA. This effect can be made into the electro-absorption modulator. And according to the interband transition condition |μc|<ћω/2≈0.4eV, the large absorption when μc<0.4eV is caused by the interband absorption of the graphene. Due to the symmetrical structure of the four graphene layers configuration, a maximum MPA = 0.196dB/μm and minimum MPA = 0.0072dB/μm were obtained in the four graphene layers configuration, which are about two times of the maximum MPA = 0.1dB/μm and minimum MPA = 0.0036dB/μm obtained in the double graphene layers configuration, respectively. For the TM modes, the real part of EMI also achieves minimum value at μc = 0.4eV. Again the maximum MPA = 0.0040dB/μm in the four graphene layers configuration is doubled comparing with the maximum MPA = 0.0022dB/μm in the two graphene layers configuration. From the results above, by increasing the graphene layer numbers, a higher MPA, which will lead to a smaller footprint and lower power consumption can be achieved. It is clear that the MPA of the TE modes are much larger than the TM modes due to the anisotropic absorption. Therefore, for the rest of our work, we only consider the TE modes.

 

Fig. 4 (a)-(d) The electric field magnitude |E| distributions of the TE and TM modes at μc = 0 eV in the double and four graphene layers configurations. (e)-(h) The Real(Neff) and MPA of the TE and TM modes with different chemical potentials in both configurations.

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As shown in Fig. 4(f), the MPA of the TE mode reduces rapidly when the μc is in the region 0.3eV<μc<0.5eV, namely the operating region in our modulator, but it changes slowly in other regions. These results can be explained through the evaluation of the tangential electric field, E||, at the graphene layer. Because only this field component has an effect on the in-plane absorption of the graphene. It is known that the power absorbed in a unit area can be expressed as Pd = Re[σ||(μc)]|E|||2/2 [22]. Combining with Eq. (2), it can be expressed as

pd=12kIm[ε||(μc)]|E|||2
with |E|||2 = |Ex|2 + |Ez|2, and k = ωε0dG is a constant. We can see that the power absorption is roughly proportional to imaginary part of dielectric constant and the square of the electric field magnitude in the graphene. The Im[ε||(µc)] of graphene is ~7.9 at μc = 0eV and decreases to ~0.29 when μc = 0.7eV as shown in Fig. 1(b). Thus, according to Eq. (4), by substituting the Im[ε||(µc)] and |E|||2 values in the graphene of the TE mode in the four graphene layers configuration, a same normalized MPA decreasing trend can be obtained from μc = 0eV to μc = 0.7eV as shown in Fig. 5.

 

Fig. 5 (a) The plots of the normalized MPA of the TE mode in the four graphene layers configuration by substituting the Im[ε||(µc)] and |E|||2 values in the graphene to Eq. (4).

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As shown in Fig. 4, the four graphene layers configuration has a larger EMI and MPA modulation compared to the double graphene layers configuration for both the TE and TM modes, thus giving advantages of smaller footprint and lower power consumptions. Therefore, we only consider the four graphene layers configuration in the following section.

The cross section of the four graphene layers modulator based on double-stripe Si3N4 waveguide is shown in Fig. 6. One graphene layer in the lower GOG and one graphene layer in the upper GOG are connected to the signal electrode, and the remaining two graphene layers in the two GOG structures are connected to the ground electrode. Due to this graphene layers co-electrode design, the series resistance of the modulator is reduced by 50% [18]. In the four graphene layers modulator structure, the RC bandwidth is limited by the graphene sheet resistance, contact resistance and the capacitance of the parallel-plate capacitor. And the frequency response of the devices can be modeled using the equivalent electrical circuit model shown in Fig. 7.

 

Fig. 6 The cross section of the four graphene layers modulator structure.

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Fig. 7 The equivalent electrical circuit of the four graphene layers modulator.

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The series resistance, RS, is the sum of the metal-graphene contact resistance (RcG) and sheet resistance (RshG) of each graphene layer. These parameters are objects of many studies and investigations and we referred to the typical results reported in literatures. In particular, the graphene sheet resistance RshG is varied between 100 and 300Ω/sq, equivalent with a mobility of 6900 and 2300cm2/V·s, respectively, when the carrier density on the graphene layer is in the order of 1012-1013cm−2 [15,23–25]. As for the contact resistance RcG, this still represents one of the major challenges in graphene technology. Its value depends on the specific metal and process used to form the contact. Here the graphene contact resistance is assumed to be RcG = 150Ω·μm, a value achievable e.g. through edge contacting of graphene [26–28]. And Cox is the parallel-plate capacitor formed by the two graphene layers with an insulating dielectric spacer, and it can be calculated as below:

Cox=ε0εrSd
where d is the thickness of the insulating dielectric spacer, S = W*L is the overlap area of two graphene layers, which are shown in Fig. 3(b). Then, the series resistance (RS) and total capacitance (Ctotal) of the graphene modulator are calculated as follows [18]:
1RS=12(RcG+RshG)+12(RcG+RshG)=1RcG+RshG
Ctotal=2Cox
Thus, the 3dB bandwidth of the graphene modulator can be calculated by [3,15,18]:

f3dB=12πRSCtotal

In order to find the minimum distance between the metal electrode and waveguide, so that the optical modes of the waveguide remained undisturbed by the metal contact. We plot the electric field magnitude |E|||2 across the waveguide center of the TE mode in the X direction and the MPA of the metal electrodes in the double-stripe Si3N4 waveguide as shown in Fig. 8. It is obvious that |E|||2 of the TE mode is close to zero, when the distance d0 between the electrode and the waveguide core edges is d0 >1.4μm as shown in Fig. 8(a). Furthermore, to get a more accurate value d0, we considered the absorption of the metal electrodes without graphene embedding in the waveguide. As shown in Fig. 8(b), the mode absorption of the TE mode is quite small when d0 >1.4μm. Specifically, the MPA of the metal electrodes is as small as 0.00077dB/μm with d0 = 1.7μm. At this point, the absorption of the metal electrodes can be ignored and the optical modes of the waveguide remained undisturbed by the metal contacts. Thus, in this work, we considered a 1μm width for each metal electrode and 1.7μm distance between the electrodes and the waveguide core edges on both sides. And we assumed RshG = 100Ω/sq and RcG = 150Ω·μm, respectively.

 

Fig. 8 (a) The plots of the electric field magnitude |E|||2 across the waveguide center of the TE mode in the X direction. (b) The MPA of the metal electrodes in the double-stripe Si3N4 waveguide without graphene.

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From the above formulas, one can get that the insulating dielectric spacer plays an important role in affecting the modulation bandwidth. The higher relative permittivity of the insulating dielectric spacer, the smaller modulation bandwidth can be obtained. Here a 7 nm-thick SiO2 layer was used as the insulating dielectric spacer in the four graphene layers modulator.

As shown in Fig. 9, for the TE mode, we defined the modulator is in “OFF” state with maximum MPA of 0.1830dB/μm at μc = 0.3eV, which corresponds to the applying voltage of about 2.1476V. Then, the modulator is in “ON” state at μc = 0.5eV and the minimum MPA of 0.0172dB/μm is achieved with applying voltage of 5.9656V. This corresponds to an extinction ratio of Δα = 0.1658dB/μm and FOM of Δα/α = 9.7 with applying voltage (ΔV) of 3.8180V. A modulator length of 18.09μm is needed to achieve 3dB modulation and the corresponding power consumption (Ebit = CΔV2/4) is 780.50fJ/bit. And from Eq. (8) a modulation bandwidth as large as f3dB = 30.6GHz can be achieved.

 

Fig. 9 The modulation performances of the four graphene layers modulator with a 7 nm-thick SiO2 insulating dielectric spacer.

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As mentioned above, the material and thickness of insulating dielectric spacer play a role in determining the modulation speed and consumption. This is because the insulating dielectric spacer will affect the capacitance and then the modulation performances such as the switching voltage ΔV, 3dB bandwidth and the energy per bit. Here the SiO2 and Al2O3 were chosen as the insulating dielectric spacer and the corresponding modulation performances were simulated and shown in the Table 1.

Tables Icon

Table 1. The performance summary of modulators.

For the modulator with 7nm SiO2 insulating dielectric spacer, a maximum 3dB modulation bandwidth of 30.6GHz can be achieved because relative low dielectric constant of the SiO2 and large insulating dielectric spacer thickness will reduce the parallel-plate capacitor between two graphene layers. But the required applying voltage ΔV is relative larger comparing to the Al2O3 insulating dielectric spacer due to the lower dielectric constant of the SiO2, which will bring the larger power consumption comparing to the modulator with Al2O3 insulating dielectric spacer. For the modulator with 7nm Al2O3 insulating dielectric spacer, a 3dB modulation bandwidth of 10.5GHz can be achieved. Since the power consumption of modulator is proportional to the applied voltage ΔV in graphene [17], a relative low power consumption of 262.90fJ/bit were obtained. It is demonstrated that the higher dielectric constant of the insulating dielectric spacer, the lower power consumption and smaller 3dB modulation bandwidth can be obtained. It can be concluded that the smaller the dielectric constant and the larger thickness of the insulating dielectric spacer, the larger the modulation bandwidth can be achieved. But the cost is the increased power consumption. So there is a trade-off between the modulation bandwidth and the power consumption.

So far, the graphene-assisted electro-absorption modulators are mainly based on Si and Si3N4 waveguides. In general, the extinction ratio(Δα) and FOM of graphene modulator based on Si waveguide is larger than that of Si3N4 waveguide based graphene modulator because the larger refractive index contrast of the SOI waveguide will bring much higher mode confinement, which will increase the interactions between the graphene and the optical waveguide mode. For example, an extinction ratio of Δα = 0.31dB/μm and FOM of 12.2 have been obtained for silicon graphene modulator with double graphene layers [17]. And an extinction ratio of about 0.26dB/μm and FOM of about 21 have been achieved for silicon graphene modulator with double graphene layers [15]. Although the silicon graphene modulators have better performances on the extinction ratio and FOM than our silicon nitride graphene modulator, but they may suffer from the intrinsic two-photon absorption which will plague silicon at communications wavelengths [10]. Besides, the nano-scale silicon waveguide width needs much more expensive fabrication costs and complex fiber coupling method than the micro-scale silicon nitride waveguide (1.2μm waveguide width). In the case of graphene-assisted modulators based on Si3N4 waveguides, the reported double layer graphene on silicon nitride (DLG-Si3N4) modulator in [15] can achieve a higher FOM of 23, which is more than two times of 9.7 obtained in our modulator because of the lower insertion loss (α) of the two graphene layers than the four graphene layers in our structure. While, due to the four graphene layers configuration and the graphene layers co-electrode design, our modulator has a larger extinction ratio of Δα = 0.1658dB/μm and a higher modulation bandwidth about 30.6GHz, compared to the extinction ratio of 0.115dB/μm and maximum bandwidth about 23GHz of the DLG-Si3N4 configuration.

4. Conclusion

To summarize, a four graphene layers electro-absorption optical modulator based on double-stripe Si3N4 waveguide was presented. The proposed graphene modulator can achieve as high as 30.6GHz modulation bandwidth with 18.09μm long active region to meet the 3dB modulation. And a switching voltage of ΔV = 3.8180V and 780.50fJ/bit power consuming can be achieved. By combing the advantages of the low insertion loss of the double-stripe Si3N4 waveguide and the high carrier mobility in graphene, a compact and high speed optical modulator base on the Si3N4/graphene hybrid waveguide can be achieved, which will remedy the lacking of high speed modulator on the Si3N4 waveguide platform.

Funding

This work was supported by the National Science Foundation of Jiangsu Province Grant No. BK 20161429.

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References

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  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
    [Crossref] [PubMed]
  2. M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
    [Crossref] [PubMed]
  3. S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
    [Crossref]
  4. F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
    [Crossref] [PubMed]
  5. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
    [Crossref] [PubMed]
  6. M. Mohsin, D. Schall, M. Otto, A. Noculak, D. Neumaier, and H. Kurz, “Graphene based low insertion loss electro-absorption modulator on SOI waveguide,” Opt. Express 22(12), 15292–15297 (2014).
    [Crossref] [PubMed]
  7. X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
    [Crossref]
  8. J. J. Feng and R. Akimoto, “A three-dimensional silicon nitride polarizing beam splitter,” IEEE Photonics Technol. Lett. 26(7), 706–709 (2014).
    [Crossref]
  9. J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
    [Crossref] [PubMed]
  10. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
    [Crossref]
  11. K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
    [Crossref] [PubMed]
  12. N. Gruhler, C. Benz, H. Jang, J. H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21(25), 31678–31689 (2013).
    [Crossref] [PubMed]
  13. J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
    [Crossref]
  14. C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
    [Crossref]
  15. L. A. Shiramin and D. V. Thourhout, “Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide,” IEEE J. Sel. Top. Quantum Electron. 23(1), 1–7 (2017).
  16. M.-S. Kwon, “Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide,” IEEE Photonics J. 6(3), 1–9 (2014).
    [Crossref]
  17. J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
    [Crossref]
  18. S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
    [Crossref] [PubMed]
  19. C. C. Lee, S. Suzuki, W. Xie, and T. R. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. Express 20(5), 5264–5269 (2012).
    [Crossref] [PubMed]
  20. L. Zhuang, D. Marpaung, M. Burla, W. Beeker, A. Leinse, and C. Roeloffzen, “Low-loss, high-index-contrast Si3N4/SiO2 optical waveguides for optical delay lines in microwave photonics signal processing,” Opt. Express 19(23), 23162–23170 (2011).
    [Crossref] [PubMed]
  21. C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
    [Crossref] [PubMed]
  22. Z. L. Lu and W. S. Zhao, “Nanoscale electro-optic modulators based on graphene-slot waveguides,” J. Opt. Soc. Am. B 29(6), 1490–1496 (2012).
    [Crossref]
  23. D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in Graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
    [Crossref] [PubMed]
  24. A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
    [Crossref]
  25. V. Sorianello, M. Midrio, and M. Romagnoli, “Design optimization of single and double layer Graphene phase modulators in SOI,” Opt. Express 23(5), 6478–6490 (2015).
    [Crossref] [PubMed]
  26. J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
    [Crossref] [PubMed]
  27. M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
    [Crossref]
  28. Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
    [Crossref] [PubMed]

2017 (2)

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

L. A. Shiramin and D. V. Thourhout, “Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide,” IEEE J. Sel. Top. Quantum Electron. 23(1), 1–7 (2017).

2016 (1)

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

2015 (4)

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
[Crossref]

V. Sorianello, M. Midrio, and M. Romagnoli, “Design optimization of single and double layer Graphene phase modulators in SOI,” Opt. Express 23(5), 6478–6490 (2015).
[Crossref] [PubMed]

2014 (5)

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
[Crossref] [PubMed]

M.-S. Kwon, “Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide,” IEEE Photonics J. 6(3), 1–9 (2014).
[Crossref]

M. Mohsin, D. Schall, M. Otto, A. Noculak, D. Neumaier, and H. Kurz, “Graphene based low insertion loss electro-absorption modulator on SOI waveguide,” Opt. Express 22(12), 15292–15297 (2014).
[Crossref] [PubMed]

J. J. Feng and R. Akimoto, “A three-dimensional silicon nitride polarizing beam splitter,” IEEE Photonics Technol. Lett. 26(7), 706–709 (2014).
[Crossref]

2013 (2)

N. Gruhler, C. Benz, H. Jang, J. H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21(25), 31678–31689 (2013).
[Crossref] [PubMed]

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (4)

2010 (3)

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in Graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
[Crossref] [PubMed]

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

2008 (1)

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Agarwal, A. M.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Ahmed, S.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Ahn, J. H.

Ahn, J. R.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Akimoto, R.

J. J. Feng and R. Akimoto, “A three-dimensional silicon nitride polarizing beam splitter,” IEEE Photonics Technol. Lett. 26(7), 706–709 (2014).
[Crossref]

An, J.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Ang, L. K.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Ang, P. K.

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

Bae, M.-H.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Barton, J. S.

Bauters, J. F.

Beeker, W.

Benz, C.

Blumenthal, D. J.

Bowers, J. E.

Bruinink, C. M.

Burla, M.

Cardenas, J.

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Chee, A. K. L.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Cheng, Z. Z.

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

Chun, S.-H.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Corbett, B.

J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
[Crossref]

Crommie, M.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Daniel Lee, Y.-H.

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Danneau, R.

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Efetov, D. K.

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in Graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
[Crossref] [PubMed]

Eggleton, B. J.

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

Fallahi, A.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Feng, J. J.

J. J. Feng and R. Akimoto, “A three-dimensional silicon nitride polarizing beam splitter,” IEEE Photonics Technol. Lett. 26(7), 706–709 (2014).
[Crossref]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Geim, A. K.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Geng, B.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Girit, C.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Goldberg, B. B.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Gosciniak, J.

J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Gruhler, N.

Hao, Y.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Heck, M. J. R.

Heideman, R. G.

Jaiswal, M.

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

Jang, H.

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Jiang, X.

Jin, Y.

John, D. D.

Ju, L.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Kim, H.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Kim, P.

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in Graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
[Crossref] [PubMed]

Kim, Y. D.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Kim, Y. S.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Kimerling, L. C.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Kitt, A.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Koester, S. J.

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

Kurz, H.

Kwon, M.-S.

M.-S. Kwon, “Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide,” IEEE Photonics J. 6(3), 1–9 (2014).
[Crossref]

Lee, C. C.

Lee, J. H.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Lee, S. W.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Leinse, A.

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Li, M.

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

Lipson, M.

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Liu, M.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Liu, Y.

Loh, K. P.

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

Lu, R.

Lu, Z. L.

Magnuson, C. W.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Marpaung, D.

Midrio, M.

Mohsin, M.

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Mosig, J. R.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Neumaier, D.

Ng, D. K. T.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Ng, S. K.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Noculak, A.

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Ooi, K. J. A.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Otto, M.

Ozyilman, B.

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

Pachoud, A.

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

Park, Y. D.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Pernice, W. H. P.

Perruisseau-Carrier, J.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Phare, C. T.

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

Roeloffzen, C.

Romagnoli, M.

Ruoff, R. S.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Schall, D.

Schibli, T. R.

Seo, J.-T.

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Shen, Y. R.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Shiramin, L. A.

L. A. Shiramin and D. V. Thourhout, “Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide,” IEEE J. Sel. Top. Quantum Electron. 23(1), 1–7 (2017).

Shu, C.

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

Sorianello, V.

Suk, J. W.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Suzuki, S.

Swan, A. K.

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Tamagnone, M.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Tan, D. T. H.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
[Crossref]

Tang, L.

Thourhout, D. V.

L. A. Shiramin and D. V. Thourhout, “Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide,” IEEE J. Sel. Top. Quantum Electron. 23(1), 1–7 (2017).

Tian, C.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Tsang, H. K.

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

Ulin-Avila, E.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Wang, J. Q.

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

Wang, Q.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Wang, T.

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Wang, Z.

Xie, W.

Xiong, C.

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

Xu, C.

Yang, J.

Yang, L.

Ye, S.

Yin, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Zettl, A.

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Zhang, Y.

S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

Zhang, Y. B.

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

Zhao, W. S.

Zhuang, L.

ACS Nano (2)

J. W. Suk, A. Kitt, C. W. Magnuson, Y. Hao, S. Ahmed, J. An, A. K. Swan, B. B. Goldberg, and R. S. Ruoff, “Transfer of CVD-grown monolayer graphene onto arbitrary substrates,” ACS Nano 5(9), 6916–6924 (2011).
[Crossref] [PubMed]

Y. D. Kim, M.-H. Bae, J.-T. Seo, Y. S. Kim, H. Kim, J. H. Lee, J. R. Ahn, S. W. Lee, S.-H. Chun, and Y. D. Park, “Focused-laser-enabled p-n junctions in graphene field-effect transistors,” ACS Nano 7(7), 5850–5857 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. 100(17), 171107 (2012).
[Crossref]

EPL (1)

A. Pachoud, M. Jaiswal, P. K. Ang, K. P. Loh, and B. Ozyilman, “Graphene transport at high carrier densities using polymer electrolyte gate,” EPL 92(2), 27001 (2010).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

L. A. Shiramin and D. V. Thourhout, “Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide,” IEEE J. Sel. Top. Quantum Electron. 23(1), 1–7 (2017).

IEEE Photonics J. (1)

M.-S. Kwon, “Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide,” IEEE Photonics J. 6(3), 1–9 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (2)

J. J. Feng and R. Akimoto, “A three-dimensional silicon nitride polarizing beam splitter,” IEEE Photonics Technol. Lett. 26(7), 706–709 (2014).
[Crossref]

J. Q. Wang, Z. Z. Cheng, C. Shu, and H. K. Tsang, “Optical absorption in graphene-on-silicon nitride microring resonators,” IEEE Photonics Technol. Lett. 27(16), 1765–1767 (2015).
[Crossref]

J. Opt. (1)

X. Zhang, Y. B. Zhang, C. Xiong, and B. J. Eggleton, “Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides,” J. Opt. 18(7), 074016 (2016).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. D Appl. Phys. (1)

J. Gosciniak, D. T. H. Tan, and B. Corbett, “Enhanced performance of graphene-based electro-absorption waveguide modulators by engineered optical modes,” J. Phys. D Appl. Phys. 48(23), 235101 (2015).
[Crossref]

Nano Lett. (1)

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

K. J. A. Ooi, D. K. T. Ng, T. Wang, A. K. L. Chee, S. K. Ng, Q. Wang, L. K. Ang, A. M. Agarwal, L. C. Kimerling, and D. T. H. Tan, “Pushing the limits of CMOS optical parametric amplifiers with USRN:Si7N3 above the two-photon absorption edge,” Nat. Commun. 8, 13878 (2017).
[Crossref] [PubMed]

Nat. Photonics (3)

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “CMOS-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4(1), 37–40 (2010).
[Crossref]

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
[Crossref]

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. Photonics 8(7), 556–563 (2014).
[Crossref]

Nature (1)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

Opt. Express (8)

M. Mohsin, D. Schall, M. Otto, A. Noculak, D. Neumaier, and H. Kurz, “Graphene based low insertion loss electro-absorption modulator on SOI waveguide,” Opt. Express 22(12), 15292–15297 (2014).
[Crossref] [PubMed]

J. F. Bauters, M. J. R. Heck, D. D. John, J. S. Barton, C. M. Bruinink, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Planar waveguides with less than 0.1 dB/m propagation loss fabricated with wafer bonding,” Opt. Express 19(24), 24090–24101 (2011).
[Crossref] [PubMed]

N. Gruhler, C. Benz, H. Jang, J. H. Ahn, R. Danneau, and W. H. P. Pernice, “High-quality Si3N4 circuits as a platform for graphene-based nanophotonic devices,” Opt. Express 21(25), 31678–31689 (2013).
[Crossref] [PubMed]

S. Ye, Z. Wang, L. Tang, Y. Zhang, R. Lu, and Y. Liu, “Electro-absorption optical modulator using dual-graphene-on-graphene configuration,” Opt. Express 22(21), 26173–26180 (2014).
[Crossref] [PubMed]

C. C. Lee, S. Suzuki, W. Xie, and T. R. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. Express 20(5), 5264–5269 (2012).
[Crossref] [PubMed]

L. Zhuang, D. Marpaung, M. Burla, W. Beeker, A. Leinse, and C. Roeloffzen, “Low-loss, high-index-contrast Si3N4/SiO2 optical waveguides for optical delay lines in microwave photonics signal processing,” Opt. Express 19(23), 23162–23170 (2011).
[Crossref] [PubMed]

C. Xu, Y. Jin, L. Yang, J. Yang, and X. Jiang, “Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide,” Opt. Express 20(20), 22398–22405 (2012).
[Crossref] [PubMed]

V. Sorianello, M. Midrio, and M. Romagnoli, “Design optimization of single and double layer Graphene phase modulators in SOI,” Opt. Express 23(5), 6478–6490 (2015).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in Graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010).
[Crossref] [PubMed]

Science (2)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004).
[Crossref] [PubMed]

F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-variable optical transitions in graphene,” Science 320(5873), 206–209 (2008).
[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 (a) Real and imaginary parts of graphene’s conductivity. (b) Graphene’s dielectric constant as a function of chemical potential at wavelength λ = 1550nm, T = 300 K, τ = 0.1ps.
Fig. 2
Fig. 2 Tuning the optical absorption of graphene in an electric-gated structure.
Fig. 3
Fig. 3 (a) The cross section of the double-stripe Si3N4 waveguide. (b) The graphene-on-graphene (GOG) structure. (c) The double graphene layers configuration. (d) The four graphene layers configuration.
Fig. 4
Fig. 4 (a)-(d) The electric field magnitude |E| distributions of the TE and TM modes at μc = 0 eV in the double and four graphene layers configurations. (e)-(h) The Real(Neff) and MPA of the TE and TM modes with different chemical potentials in both configurations.
Fig. 5
Fig. 5 (a) The plots of the normalized MPA of the TE mode in the four graphene layers configuration by substituting the Im[ε||(µc)] and |E|||2 values in the graphene to Eq. (4).
Fig. 6
Fig. 6 The cross section of the four graphene layers modulator structure.
Fig. 7
Fig. 7 The equivalent electrical circuit of the four graphene layers modulator.
Fig. 8
Fig. 8 (a) The plots of the electric field magnitude |E|||2 across the waveguide center of the TE mode in the X direction. (b) The MPA of the metal electrodes in the double-stripe Si3N4 waveguide without graphene.
Fig. 9
Fig. 9 The modulation performances of the four graphene layers modulator with a 7 nm-thick SiO2 insulating dielectric spacer.

Tables (1)

Tables Icon

Table 1 The performance summary of modulators.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

σ = σ int e r + σ int ra = i 8 σ 0 π E t h E p h + i E s I n [ 2 cos ( E F 2 E t h ) ] + σ 0 [ 1 2 + 1 π tan 1 ( E p h 2 E F 2 E t h ) i 2 π I n ( E p h + 2 E F ) 2 ( E p h 2 E F ) 2 + 4 E t h 2 ]
ε ( μ c ) = 2 .5 + i σ ( μ c ) ω ε 0 d G
μ c = sgn ( n ) V F π | N | = sgn ( n ) V F α π | V + V 0 |
p d = 1 2 k Im [ ε || ( μ c ) ] | E || | 2
C o x = ε 0 ε r S d
1 R S = 1 2 ( R c G + R s h G ) + 1 2 ( R c G + R s h G ) = 1 R c G + R s h G
C total = 2 C o x
f 3 d B = 1 2 π R S C t o t a l

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