Abstract

The spectral region around 2μm wavelengths is of special significance for applications in optical communications, which is facing a capacity crunch at the telecommunication band. Development of CMOS-compatible on-chip photonic and optoelectronic devices operating in this spectral region is highly demanded but still at its infancy. In this paper, we present a design of graphene-on-germanium slot waveguides at the wavelength of 2μm. Due to the optical-field enhancement and poor mode confinement, light-matter interactions are enhanced in the graphene-on-germanium slot waveguide. Moreover, the influence of the refractive index contrast and the number of integrated graphene layers on the optical absorption enhancement of the slot waveguide is studied. Based on the graphene-on-slot waveguide configuration, energy-efficient and compact Mach-Zehnder interferometer and microring resonator phase modulators are designed. Our study paves the way for the development of on-chip electro-optic modulators at 2μm wavelengths.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

High-speed and large-capacity data transmission at 2μm wavelengths is a potential candidate for solving the capacity limit at the telecommunication band. The light transmission can be realized by using hollow-core photonic crystal fibers with a theoretical minimum loss below 0.1 dB/km [1,2]. And coincidentally, the thulium-doped fiber amplifiers (TDFA) are capable of providing a gain bandwidth over 100 nm [3] in this spectral region. However, compared with discrete elements at 2μm wavelengths, chip-scale integration has the advantages of lower cost, higher performance, smaller footprint, and higher reliability [4,5]. Moreover, photonic integrated circuits (PICs) can provide subwavelength optical confinement, leading to enhanced light-matter interactions and the reinforced signal-to-noise ratio. In recent years, PICs have been demonstrated for applications in nonlinear frequency generation [6], lasing [7] and photodetection [8] at 2μm wavelengths. Besides, electro-optic modulators are of special significance for achieving transmission systems with large data rates. However, traditional electro-optic modulators are only developed at the wavelength of 1.55μm based on lithium niobate [9], III–V materials [10] and silicon (Si) [11,12] during the past years, while the design and fabrication of waveguide integrated electro-optic modulators at the wavelength of 2μm are still in its infancy, with the demonstrated modulation data rate up to 3Gbit/s by exploiting the free carrier effects in silicon [13] and a 3 dB bandwidth of 20kHz in a thermal-optic silicon modulator [14].

Compared with traditional integration platforms, germanium (Ge) PICs have a host of excellent properties. First, Ge has a high refractive index (RI) around 4.0, enabling high-density photonic and optoelectronic integration. Second, Ge has strong thermal-optic and free-carrier effects, which is applicable for high-efficiency electro-optic modulation. Moreover, the fabrication process of Ge PICs is compatible with Si CMOS processing, ensuring low-cost and high-yield fabrication. Finally, Ge has a wide transparent window which fully covers the wavelength region of 2μm. Therefore, Ge PICs have attracted great attention in recent years. Several Ge platforms, namely Ge-on-Si [1517], Ge-on-insulator [18,19], and Ge-on-Si-on-insulator (Ge-on-SOI) waveguides [20,21], have been widely developed.

On the other hand, graphene is a two-dimensional material which has many unique properties and could be conveniently integrated on PICs. The integration of graphene with PICs has allowed for the realization of high-efficiency optoelectronic devices [2224], by virtue of enhanced light-matter interactions between graphene and propagating light in waveguides through the evanescent field coupling [25]. Based on this configuration, the modulation of the complex effective RI of the waveguide mode can be achieved through electrically tuning the Fermi level of graphene, achieving electro-refractive modulation [26,27] or electro-absorptive modulation [28,29]. However, electro-refractive modulators are challenging compared with electro-absorptive modulators. Since the optical mode is mostly confined in the waveguide and graphene only acts as a little perturbation on the surface of the waveguide, large footprint devices are normally used for phase-shift devices in previous studies. To overcome this limitation, novel structures with larger mode overlap with graphene, such as graphene sandwiched by insulator layers, have been theoretically proposed [30]. However, the growth of insulator and silicon on graphene is challenging in practice.

In this paper, we study graphene-on-Ge waveguide modulators based on planar fabrication techniques at the wavelength of 2μm. Due to the optical field enhancement in the subwavelength low RI slot and less mode confinement of the waveguide mode, the light-graphene interaction is stronger in the TE-mode graphene-on-slot waveguide than that in the TE-mode graphene-on-channel waveguide. Besides, germanium has a higher RI than silicon, thus capable of providing higher light intensity in the slot region, according to the principle of operation of the slot waveguide [31]. Since the TM mode is less stable than the TE mode, in the following discussion we only study optical characteristics of TE-mode graphene-on-waveguide devices. The influence of the RI contrast and the number of graphene layers on the optical interaction enhancement in the graphene-on-Ge slot waveguide is also discussed. Based on this configuration, two types of phase modulators based on Mach-Zehnder interferometer (MZI) and microring resonator are theoretically designed. Due to the merits of the CMOS-compatible fabrication process, our devices are expected to open a way to develop compact graphene-based waveguide-integrated modulators at the wavelength of 2μm.

2. Results and discussions

Optical properties of graphene can be described by the optical conductivity σ, which includes contributions from the interband transition and intraband transition in graphene and can be expressed by [32]

$$\begin{array}{l} \sigma = - \frac{{2i{e^2}(\Omega + 2i\Gamma )}}{h}[\frac{1}{{{{(\Omega + 2i\Gamma )}^2}}}\int_\Delta ^\infty {\frac{{{\omega ^2} + {\Delta ^2}}}{\omega }} (\frac{{\delta {n_F}(\omega )}}{{\delta \omega }} - \frac{{\delta {n_F}( - \omega )}}{{\delta \omega }})d\omega \\ - \int_\Delta ^\infty {\frac{{{\omega ^2} + {\Delta ^2}}}{\omega }} (\frac{{{n_F}( - \omega ) - {n_F}(\omega )}}{{{{(\Omega + 2i\Gamma )}^2} - 4{\omega ^2}}})d\omega ] \end{array}, $$
where Ω is the optical frequency, nF(ω) is the Fermi distribution function, ω is the energy of the relativistic Landau levels and Δ is the exciton gap of Landau level energies. Γ(ω) is the scattering rate, which is estimated to be 7.6×1012s−1 [30]. With the calculated optical conductivity, the relative permittivity of graphene can be described by [33]
$${\varepsilon _{eff}} = 1 + i\frac{\sigma }{{\omega {\varepsilon _0}d}}, $$
where d is the thickness of the graphene layer, here we choose 0.7 nm for the calculation. The calculated graphene permittivity as a function of Fermi level at the wavelength of 2μm is shown in Fig. 1. The relative permittivity of graphene is mainly determined by the interband transition when the Fermi level is less than half of the photon energy (0.31 eV). Graphene has relatively large optical absorption in this region. When the Fermi level is larger than 0.31 eV, the relative permittivity is dominated by the intraband transition. The imaginary part drops dramatically and the absolute value of relative permittivity is determined by the real part.

 figure: Fig. 1.

Fig. 1. Calculated graphene permittivity as a function of Fermi level at the wavelength of 2μm.

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Then, we design a graphene-on-Ge slot waveguide device operating at the wavelength of 2μm, as shown in Fig. 2. The slot waveguide consists of two ridge waveguides of 300 nm width (wr) and 300 nm height (h1), which are fully etched to the 2μm-thick buried oxide (BOX) layer, and separated to each other by a narrow slot (s). Due to the discontinuity of the electrical field at the interface with high RI contrast, the optical field confined in the subwavelength low RI slot region is enhanced [31]. When graphene is integrated on the surface of the slot waveguide, the light-graphene interaction can be enhanced. As shown in the schematic of the cross section of the proposed waveguide configuration in Fig. 2(b), after the Ge waveguide planarization, the bottom layer graphene can be integrated on the photonic chip through the wet transferring process. The commercial chemical vapor deposited (CVD) graphene on copper foil is first spinning coated by polymethyl methacrylate (PMMA) photoresist. Then the graphene copper foil is immersed in the ammonium persulfate solution to remove the copper beneath. And then the CVD graphene covered by PMMA is rinsed by the deionized water and transferred to the photonic chip. After drying in the ambient condition, the photonic chip is baked at 150°C for 15minutes to melt the PMMA photoresist, leading to improved contact between graphene and the photonic chip. Next, the photonic chip is baked at 80°C for 2 hours on a hot plate so that the steam could improve the adhesion of graphene. Finally, the photonic chip is immersed in acetone to remove the covered PMMA. Then a thin layer of silicon oxide with thickness (h2) of 10 nm can be deposited on the bottom layer graphene through the atomic layer depositing process. Finally, the upper layer graphene can be integrated on the surface of the thin insulator layer. The two electrodes are set on the two different graphene layers, respectively. For comparison, we design a dual-layer graphene-on-channel waveguide structure with 300 nm height and 600 nm width on the same wafer. It is worth to mention that our proposed dual-layer graphene-on-Ge slot waveguide configuration is much easier to be realized by the standard nanofabrication technology, compared with the configuration of multi-layer graphene embedded in a horizontal slot waveguide [34].

 figure: Fig. 2.

Fig. 2. Schematics of the proposed dual-layer graphene-on-Ge slot waveguide modulator. (a) 3-dimensional schematic of the dual-layer graphene-on-slot waveguide modulator. (b) Cross-section of the dual-layer graphene-on-slot waveguide modulator. The electrodes are set on the surfaces of the two graphene layers, respectively.

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With the above waveguide parameters and the calculated relative permittivity of graphene, the complex RIs of different dual-layer graphene-on-Ge waveguide modes are calculated by using a finite element method simulator (COMSOL Multiphysics). With the calculated imaginary part of the effective RI (I1), the loss factor α (dB/μm) of graphene-on-waveguide is obtained from α = 40πlog(e)I1λ, where λ (μm) is the optical wavelength in vacuum. The result is shown in Fig. 3. For the TE mode slot waveguide, the loss factor increases to 0.231 dB/μm with a slot width of 80 nm, which is 2.7 times that of the TE mode channel waveguide. Besides, the real parts of the effective RIs (neff) of the 80 nm slot waveguide and channel waveguide are 2.19144 and 3.02120, respectively. Therefore the less mode confinement of the slot waveguide may also contribute to the larger optical absorption. The optical loss of the slot waveguide gradually saturates when the slot width is larger than 80 nm. In the following calculation, the slot width is set as 80 nm.

 figure: Fig. 3.

Fig. 3. Calculation of graphene-on-waveguide loss factors with different waveguide configurations. Inset is the schematic of mode distributions of TE mode dual-layer graphene-on-slot waveguide and dual-layer graphene-on-channel waveguide, respectively.

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With the above slot waveguide structure, we simulate RIs of graphene-on-waveguide modes with the Fermi level tuned from 0 eV to 2 eV. As shown in Fig. 4(a), by tuning the Fermi level from 0.31 eV to 2 eV, the variation of the neff is 0.0414 for the TE mode slot waveguide and 0.0148 for the TE mode channel waveguide. So the phase shift of the optical mode propagating in the graphene-on-slot waveguide is 2.8 times of that in the graphene-on-channel waveguide. The optical intensity enhancement and less mode confinement of the slot waveguide contribute to the larger phase shift, which agrees with the previous discussion. As shown in Fig. 4(b), when the Fermi level is tuned larger than half of the photon energy, the interband transition is blocked due to Pauli-blocking effect and the graphene-on-waveguide optical absorption is determined by the intraband transition. The optical loss in the graphene-on-slot waveguide is relatively low when the Fermi level is larger than 0.31 eV and is similar to that in the graphene-on-channel waveguide. Thus, the dual-layer graphene-on-slot waveguide structure is suitable for the development of phase modulators with high efficiency and low insertion loss. In the experiment, the two layers of graphene can be symmetrically doped with electrons and holes at high drive voltages, respectively, shifting Fermi levels beyond half of the photon energy, resulting in optical transparency of graphene and the change of effective RI of the graphene-on-waveguide mode.

 figure: Fig. 4.

Fig. 4. Calculated mode parameters for different dual-layer graphene-on-waveguide structures. (a) Real parts of effective RIs as a function of Fermi level for TE mode dual-layer graphene-on-slot waveguide and dual-layer graphene-on-channel waveguide. (b) Waveguide loss factors as a function of Fermi level for different waveguide configurations.

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Then, we study the influence of the RI contrast of the slot waveguide configuration and the number of integrated graphene layers on the optical absorption enhancement in the graphene-on-slot waveguide. We fix the slot width to 80 nm and perform mode calculation of graphene-on-slot/channel waveguide structures with different ridge waveguide materials (silicon nitride (SiN), aluminum nitride (AlN), Si and Ge) and SiO2 as the slot region. All the slot waveguides consist of two ridge waveguides with 300nnm width and 300 nm height. All the channel waveguides are 300 nm high and 600 nm wide. The BOX layer is 2μm thick. The real parts of effective RIs and waveguide loss factors are shown in Table 1. According to the simulation result, only Si and Ge slot waveguides have enhanced optical absorption, compared with channel waveguides. For SiN and AlN waveguides, although there is the electrical field discontinuity at the RI contrast interface that is proportional to the square of the ratio of the RI contrast [26] and less mode confinement in the slot waveguides, channel waveguides may have larger optical interaction area and therefore larger optical absorption.

Tables Icon

Table 1. Real parts of effective RIs and waveguide loss factors of dual-layer graphene-on-slot waveguides and graphene-on-channel waveguides of different materials.

By increasing the number of integrated graphene layers on the slot waveguide, the light graphene interaction can be enhanced, compared with the dual-layer graphene-on-slot waveguide structure. Besides, the Fermi level of each graphene monolayer can be easily tuned by applying the gate voltage. With previously optimized dual-layer graphene-on-Ge slot waveguide structure, we study the influence of the number of integrated graphene layers on the optical absorption enhancement in the graphene-on-Ge slot waveguide. As shown in the inset of Fig. 5(a), multiple layers of graphene are stacked together on the surface of the slot waveguide. There is 10 nm SiO2 between the adjacent graphene layers. The calculated waveguide loss factor as a function of the number of graphene layers is shown in Fig. 5(a). The graphene-on-waveguide loss increases with the graphene layer number and reaches 0.763 dB/μm when the number of graphene layers is 28. The calculated normalized derivative of the waveguide absorption as a function of the number of graphene layers is shown in Fig. 5(b). The increasing rate of the waveguide loss factor gradually decreases with graphene layers and reduces to less than 1/e when the number of layers is larger than 9. The upper layers of graphene have less overlap with the slot waveguide mode and therefore less optical absorption, and the optical absorption gradually saturates when the number of layers is larger than 9. Moreover, the neff increases with the increasing number of graphene layers, as shown in Fig. 5(c), indicating that the mode confinement increases with the number of integrated graphene layers. So the increasing mode confinement with graphene layers may also contribute to the gradually saturated optical absorption of the graphene-on-slot waveguide.

 figure: Fig. 5.

Fig. 5. Influence of the integrated graphene layers on the waveguide loss factor. (a) Calculated waveguide loss factor as a function of the number of graphene layers. Inset is the schematic of the multiple layers of graphene sandwiched by insulator layers on the surface of the slot waveguide. (b) The normalized derivative of the waveguide loss factor as a function of the number of graphene layers. (c) The effective RI as a function of the number of graphene layers.

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Next, we design phase modulators based on the dual-layer graphene-on-slot waveguide configuration. The length of each graphene layer can be tailored to desired lengths by using electron beam lithography to fabricate the photoresist mask and followed by O2 plasma etching for a few seconds. As shown in Fig. 5(a), the two arms of the MZI are connected by two Y-junctions which provide equal power splitting at any wavelength. The lengths of both arms are 100μm. The transmission (T) of the MZI modulator can be obtained from [35]

$$T(\lambda ) = \frac{1}{4} \times [\exp ( - {\alpha _1}L) + \exp ( - {\alpha _2}L) + 2\exp ( - \frac{{{\alpha _1}L + {\alpha _2}L}}{2})\cos )(\Delta \varphi )], $$
where Δφ = 2π/λΔneffL, Δneff is the effective RI difference between the two arms, α1 and α2 are waveguide loss factors of the two arms, respectively. The Fermi level of arm A is fixed at 0.31 eV and the Fermi level of arm B is tuned from 0 eV to 2 eV. Transmissions of MZI modulators as a function of Fermi level are shown in Fig. 6(b) and Fig. 6(c). For the slot waveguide MZI modulator, 3 dB modulation depth can be obtained by tuning arm B to 0.35 eV. When arm B is tuned to 0.4 eV, the transmission reaches the minimum, leading to a 180° phase shift. For the channel waveguide MZI modulator, arm B should be tuned to 0.9 eV and 1.6 eV to achieve 3 dB modulation depth and 180° phase shift, which can hardly be realized in practice due to the breakdown of the silicon oxide spacer layer with Fermi level beyond 1.0 eV. Therefore, a more efficient MZI phase modulator can be obtained with the slot waveguide configuration.

 figure: Fig. 6.

Fig. 6. Design of dual-layer graphene-on-waveguide MZI phase modulators. (a) Schematic of the dual-layer graphene-on-MZI modulator structure based on the slot waveguide configuration. (b), (c) Calculated transmissions of MZI phase modulators as a function of Fermi levels of arm B for graphene-on-slot waveguide and graphene-on-channel waveguide configurations, respectively.

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Then we study transmission characteristics of dual-layer graphene-on-microring resonator phase modulators. Resonant-structure based modulators like ring resonators have significant savings in footprint despite that these structures may have a limited spectral operation band. The Fermi level of graphene is tuned beyond half of the photon energy of the input light (0.31 eV) so that the microring resonator has a low optical loss. The radius (r) of the microring resonator is 25μm, as shown in Fig. 7(a). The transmission of the microring resonator is described by [35]

$$B(\lambda ) = \frac{{{a^2} + {t^2} - 2at\cos \theta }}{{1 + {a^2}{t^2} - 2at\cos \theta }},$$
where a is the roundtrip loss of the microring resonator, t is the transmission coefficient of the all-pass microring resonator, θ = 4π2rneff. The critical coupling condition is assumed for resonant wavelengths. Transmissions of microring resonators with slot waveguide and channel waveguide configurations are shown in Fig. 7. As shown in Fig. 7(b), resonant wavelengths for the slot waveguide microring resonator are 2004.58 nm, 2002.53 nm, 2000.65 nm, respectively, when the Fermi level is tuned to 0.33 eV, 0.34 eV and 0.35 eV, respectively. While for the channel waveguide microring resonator, resonant wavelengths are 2003.32 nm, 2002.78 nm, and 2002.12 nm, respectively, as shown in Fig. 7(c). Therefore, the slot waveguide microring resonator modulator has a larger resonant wavelength shift with the same graphene doping level, indicating a larger spectral operation range.

 figure: Fig. 7.

Fig. 7. Design of dual-layer graphene-on-microring resonator phase modulators. (a) Schematic of the microring resonator phase modulator based on the dual-layer graphene-on-slot waveguide configuration. (b), (c) Calculated transmissions of microring resonator phase modulators as a function of Fermi level with the slot waveguide and channel waveguide configurations, respectively.

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3. Conclusions

In summary, we present the design of dual-layer graphene-on-Ge slot waveguide phase modulators operating at the wavelength of 2μm. Owing to the field enhancement and less mode confinement of the slot waveguide, graphene has enhanced interaction with light propagating in the waveguide, leading to more efficient effective RI modulation when the Fermi level is tuned from 0 eV to 2 eV, compared with the channel waveguide configuration. The dependence of the optical absorption enhancement on the RI contrast of the slot waveguide and the number of integrated graphene layers is discussed. The dual-layer graphene-on-Ge slot waveguide structure is easy to be fabricated by mature nanofabrication technology and is promising for realizing compact and efficient phase modulators in the near future.

Funding

National Natural Science Foundation of China (NSFC) (61775149, 61805164); Shenzhen Research Foundation for Talented Scholars (000309); Natural Science Foundation of SZU (85302-000170); Shenzhen Science and Technology Innovation Commission (JCYJ20160226192754225, JCYJ20160307111047701, JCYJ20160307145209361); Natural Science Foundation of Guangdong Province (2016A030313059, 2017A010101018).

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  1. P. Roberts, F. Couny, H. Sabert, B. Mangan, D. Williams, L. Farr, M. Mason, A. Tomlinson, T. Birks, and J. Knight, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
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  2. E. Desurvire, C. Kazmierski, F. Lelarge, X. Marcadet, A. Scavennec, F. A. Kish, D. F. Welch, R. Nagarajan, C. H. Joyner, R. P. Schneider, and et al., “Science and technology challenges in XXIst century optical communications,” C. R. Phys. 12(4), 387–416 (2011).
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  3. A. M. Heidt, Z. Li, and D. J. Richardson, “High Power Diode-Seeded Fiber Amplifiers at 2 μm—From Architectures to Applications,” IEEE J. Sel. Top. Quantum Electron. 20(5), 525–536 (2014).
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  4. R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
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  5. Z. Cheng, H. K. Tsang, X. Wang, J.-B. Xu, and K. Xu, “Mid-infrared Suspended Membrane Waveguide and Ring Resonator on Silicon-on-Insulator,” IEEE J. Sel. Top. Quantum Electron. 20(1), 43–48 (2014).
    [Crossref]
  6. X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
    [Crossref]
  7. B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon mid-IR Raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
    [Crossref]
  8. X. Wang, Z. Cheng, K. Xu, H. K. Tsang, and J.-B. Xu, “High-responsivity graphene/silicon-heterostructure waveguide photodetectors,” Nat. Photonics 7(11), 888–891 (2013).
    [Crossref]
  9. E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
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  10. P. C. Schindler, D. Korn, C. Stamatiadis, M. F. O’Keefe, L. Stampoulidis, R. Schmogrow, P. Zakynthinos, R. Palmer, N. Cameron, Y. Zhou, R. G. Walker, E. Kehayas, S. Ben-Ezra, I. Tomkos, L. Zimmermann, K. Petermann, W. Freude, C. Koos, and J. Leuthold, “Monolithic GaAs Electro-Optic IQ Modulator Demonstrated at 150 Gbit/s With 64QAM,” J. Lightwave Technol. 32(4), 760–765 (2014).
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  11. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
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  12. K. Xu, L. G. Yang, J. Y. Sung, Y. M. Chen, Z. Z. Cheng, C. W. Chow, C. H. Yeh, and H. K. Tsang, “Compatibility of Silicon Mach-Zehnder Modulators for Advanced Modulation Formats,” J. Lightwave Technol. 31(15), 2550–2554 (2013).
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  13. W. Cao, D. Hagan, D. J. Thomson, M. Nedeljkovic, C. G. Littlejohns, A. Knights, S.-U. Alam, J. Wang, F. Gardes, W. Zhang, S. Liu, K. Li, M. S. Rouifed, G. Xin, W. Wang, H. Wang, G. T. Reed, and G. Z. Mashanovich, “High-speed silicon modulators for the 2  μm wavelength band,” Optica 5(9), 1055 (2018).
    [Crossref]
  14. V. C. Joris, W. M. J. Green, A. Solomon, and Y. A. Vlasov, “Integrated NiSi waveguide heaters for CMOS-compatible silicon thermo-optic devices,” Opt. Lett. 35(7), 1013–1015 (2010).
    [Crossref]
  15. M. Nedeljkovic, J. S. Penadés, C. J. Mitchell, A. Z. Khokhar, S. Stanković, T. D. Bucio, C. G. Littlejohns, F. Y. Gardes, and G. Z. Mashanovich, “Surface-Grating-Coupled Low-Loss Ge-on-Si Rib Waveguides and Multimode Interferometers,” IEEE Photon. Technol. Lett. 27(10), 1040–1043 (2015).
    [Crossref]
  16. Y. C. Chang, V. Paeder, L. Hvozdara, J. M. Hartmann, and H. P. Herzig, “Low-loss germanium strip waveguides on silicon for the mid-infrared,” Opt. Lett. 37(14), 2883–2885 (2012).
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  17. A. Malik, M. Muneeb, Y. Shimura, J. Van Campenhout, R. Loo, and G. Roelkens, “Germanium-on-silicon planar concave grating wavelength (de) multiplexers in the mid-infrared,” Appl. Phys. Lett. 103(16), 161119 (2013).
    [Crossref]
  18. W. Li, P. Anantha, S. Bao, K. H. Lee, X. Guo, T. Hu, L. Zhang, H. Wang, R. Soref, and C. S. Tan, “Germanium-on-silicon nitride waveguides for mid-infrared integrated photonics,” Appl. Phys. Lett. 109(24), 241101 (2016).
    [Crossref]
  19. T.-H. Xiao, Z. Zhao, W. Zhou, C.-Y. Chang, S. Y. Set, M. Takenaka, H. K. Tsang, Z. Cheng, and K. Goda, “Mid-infrared high-Q germanium microring resonator,” Opt. Lett. 43(12), 2885–2888 (2018).
    [Crossref]
  20. A. Malik, S. Dwivedi, L. L. Van, M. Muneeb, Y. Shimura, G. Lepage, C. J. Van, W. Vanherle, O. T. Van, and R. Loo, “Ge-on-Si and Ge-on-SOI thermo-optic phase shifters for the mid-infrared,” Opt. Express 22(23), 28479–28488 (2014).
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  21. U. Younis, S. K. Vanga, A. E. Lim, P. G. Lo, A. A. Bettiol, and K. W. Ang, “Germanium-on-SOI waveguides for mid-infrared wavelengths,” Opt. Express 24(11), 11987–11993 (2016).
    [Crossref]
  22. J. Wang, Z. Cheng, Z. Chen, J. Xu, H. K. Tsang, and C. Shu, “Graphene photodetector integrated on silicon nitride waveguide,” J. Appl. Phys. 117(14), 144504 (2015).
    [Crossref]
  23. J. Wang, Z. Cheng, Z. Chen, X. Wan, B. Zhu, H. K. Tsang, C. Shu, and J. Xu, “High-responsivity graphene-on-silicon slot waveguide photodetectors,” Nanoscale 8(27), 13206–13211 (2016).
    [Crossref]
  24. J. Wang, Z. Cheng, and X. Li, “Progress on Waveguide-Integrated Graphene Optoelectronics,” Adv. Condens. Matter Phys. 2018, 1–9 (2018).
    [Crossref]
  25. Z. Wu, Y. Chen, T. Zhang, Z. Shao, Y. Wen, P. Xu, Y. Zhang, and S. Yu, “Design and optimization of optical modulators based on graphene-on-silicon nitride microring resonators,” J. Opt. 19(4), 045801 (2017).
    [Crossref]
  26. 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 (2012).
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  27. T. Pan, C. Qiu, J. Wu, X. Jiang, B. Liu, Y. Yang, H. Zhou, R. Soref, and Y. Su, “Analysis of an electro-optic modulator based on a graphene-silicon hybrid 1D photonic crystal nanobeam cavity,” Opt. Express 23(18), 23357 (2015).
    [Crossref]
  28. C. T. Phare, Y. H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 GHz bandwidth,” Nat. Photonics 9(8), 511–514 (2015).
    [Crossref]
  29. X. Gan, R. J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013).
    [Crossref]
  30. Z. Lu and W. Zhao, “Nanoscale electro-optic modulators based on graphene-slot waveguides,” J. Opt. Soc. Am. B 29(6), 1490 (2012).
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  31. C. A. Barrios, B. Sánchez, K. B. Gylfason, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Demonstration of slot-waveguide structures on silicon nitride / silicon oxide platform,” Opt. Express 15(11), 6846–6856 (2007).
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  32. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19(2), 026222 (2007).
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  33. Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Direct observation of a widely tunable bandgap in bilayer graphene,” Nature 459(7248), 820–823 (2009).
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  34. X. Yin, T. Zhang, L. Chen, and X. Li, “Ultra-compact TE-pass polarizer with graphene multilayer embedded in a silicon slot waveguide,” Opt. Lett. 40(8), 1733 (2015).
    [Crossref]
  35. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
    [Crossref]

2018 (3)

2017 (1)

Z. Wu, Y. Chen, T. Zhang, Z. Shao, Y. Wen, P. Xu, Y. Zhang, and S. Yu, “Design and optimization of optical modulators based on graphene-on-silicon nitride microring resonators,” J. Opt. 19(4), 045801 (2017).
[Crossref]

2016 (3)

W. Li, P. Anantha, S. Bao, K. H. Lee, X. Guo, T. Hu, L. Zhang, H. Wang, R. Soref, and C. S. Tan, “Germanium-on-silicon nitride waveguides for mid-infrared integrated photonics,” Appl. Phys. Lett. 109(24), 241101 (2016).
[Crossref]

J. Wang, Z. Cheng, Z. Chen, X. Wan, B. Zhu, H. K. Tsang, C. Shu, and J. Xu, “High-responsivity graphene-on-silicon slot waveguide photodetectors,” Nanoscale 8(27), 13206–13211 (2016).
[Crossref]

U. Younis, S. K. Vanga, A. E. Lim, P. G. Lo, A. A. Bettiol, and K. W. Ang, “Germanium-on-SOI waveguides for mid-infrared wavelengths,” Opt. Express 24(11), 11987–11993 (2016).
[Crossref]

2015 (5)

J. Wang, Z. Cheng, Z. Chen, J. Xu, H. K. Tsang, and C. Shu, “Graphene photodetector integrated on silicon nitride waveguide,” J. Appl. Phys. 117(14), 144504 (2015).
[Crossref]

T. Pan, C. Qiu, J. Wu, X. Jiang, B. Liu, Y. Yang, H. Zhou, R. Soref, and Y. Su, “Analysis of an electro-optic modulator based on a graphene-silicon hybrid 1D photonic crystal nanobeam cavity,” Opt. Express 23(18), 23357 (2015).
[Crossref]

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

X. Yin, T. Zhang, L. Chen, and X. Li, “Ultra-compact TE-pass polarizer with graphene multilayer embedded in a silicon slot waveguide,” Opt. Lett. 40(8), 1733 (2015).
[Crossref]

M. Nedeljkovic, J. S. Penadés, C. J. Mitchell, A. Z. Khokhar, S. Stanković, T. D. Bucio, C. G. Littlejohns, F. Y. Gardes, and G. Z. Mashanovich, “Surface-Grating-Coupled Low-Loss Ge-on-Si Rib Waveguides and Multimode Interferometers,” IEEE Photon. Technol. Lett. 27(10), 1040–1043 (2015).
[Crossref]

2014 (4)

2013 (4)

X. Gan, R. J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013).
[Crossref]

X. Wang, Z. Cheng, K. Xu, H. K. Tsang, and J.-B. Xu, “High-responsivity graphene/silicon-heterostructure waveguide photodetectors,” Nat. Photonics 7(11), 888–891 (2013).
[Crossref]

K. Xu, L. G. Yang, J. Y. Sung, Y. M. Chen, Z. Z. Cheng, C. W. Chow, C. H. Yeh, and H. K. Tsang, “Compatibility of Silicon Mach-Zehnder Modulators for Advanced Modulation Formats,” J. Lightwave Technol. 31(15), 2550–2554 (2013).
[Crossref]

A. Malik, M. Muneeb, Y. Shimura, J. Van Campenhout, R. Loo, and G. Roelkens, “Germanium-on-silicon planar concave grating wavelength (de) multiplexers in the mid-infrared,” Appl. Phys. Lett. 103(16), 161119 (2013).
[Crossref]

2012 (3)

2011 (1)

E. Desurvire, C. Kazmierski, F. Lelarge, X. Marcadet, A. Scavennec, F. A. Kish, D. F. Welch, R. Nagarajan, C. H. Joyner, R. P. Schneider, and et al., “Science and technology challenges in XXIst century optical communications,” C. R. Phys. 12(4), 387–416 (2011).
[Crossref]

2010 (4)

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

V. C. Joris, W. M. J. Green, A. Solomon, and Y. A. Vlasov, “Integrated NiSi waveguide heaters for CMOS-compatible silicon thermo-optic devices,” Opt. Lett. 35(7), 1013–1015 (2010).
[Crossref]

2009 (1)

Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Direct observation of a widely tunable bandgap in bilayer graphene,” Nature 459(7248), 820–823 (2009).
[Crossref]

2007 (2)

2006 (1)

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon mid-IR Raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[Crossref]

2005 (1)

2000 (2)

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[Crossref]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000).
[Crossref]

Alam, S.-U.

Anantha, P.

W. Li, P. Anantha, S. Bao, K. H. Lee, X. Guo, T. Hu, L. Zhang, H. Wang, R. Soref, and C. S. Tan, “Germanium-on-silicon nitride waveguides for mid-infrared integrated photonics,” Appl. Phys. Lett. 109(24), 241101 (2016).
[Crossref]

Ang, K. W.

Attanasio, D. V.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[Crossref]

Bao, S.

W. Li, P. Anantha, S. Bao, K. H. Lee, X. Guo, T. Hu, L. Zhang, H. Wang, R. Soref, and C. S. Tan, “Germanium-on-silicon nitride waveguides for mid-infrared integrated photonics,” Appl. Phys. Lett. 109(24), 241101 (2016).
[Crossref]

Barrios, C. A.

Ben-Ezra, S.

Bettiol, A. A.

Birks, T.

Bossi, D. E.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[Crossref]

Bucio, T. D.

M. Nedeljkovic, J. S. Penadés, C. J. Mitchell, A. Z. Khokhar, S. Stanković, T. D. Bucio, C. G. Littlejohns, F. Y. Gardes, and G. Z. Mashanovich, “Surface-Grating-Coupled Low-Loss Ge-on-Si Rib Waveguides and Multimode Interferometers,” IEEE Photon. Technol. Lett. 27(10), 1040–1043 (2015).
[Crossref]

Cameron, N.

Cao, W.

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19(2), 026222 (2007).
[Crossref]

Cardenas, J.

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

Casquel, R.

Chang, C.-Y.

Chang, Y. C.

Chen, L.

Chen, Y.

Z. Wu, Y. Chen, T. Zhang, Z. Shao, Y. Wen, P. Xu, Y. Zhang, and S. Yu, “Design and optimization of optical modulators based on graphene-on-silicon nitride microring resonators,” J. Opt. 19(4), 045801 (2017).
[Crossref]

Chen, Y. M.

Chen, Z.

J. Wang, Z. Cheng, Z. Chen, X. Wan, B. Zhu, H. K. Tsang, C. Shu, and J. Xu, “High-responsivity graphene-on-silicon slot waveguide photodetectors,” Nanoscale 8(27), 13206–13211 (2016).
[Crossref]

J. Wang, Z. Cheng, Z. Chen, J. Xu, H. K. Tsang, and C. Shu, “Graphene photodetector integrated on silicon nitride waveguide,” J. Appl. Phys. 117(14), 144504 (2015).
[Crossref]

Cheng, Z.

T.-H. Xiao, Z. Zhao, W. Zhou, C.-Y. Chang, S. Y. Set, M. Takenaka, H. K. Tsang, Z. Cheng, and K. Goda, “Mid-infrared high-Q germanium microring resonator,” Opt. Lett. 43(12), 2885–2888 (2018).
[Crossref]

J. Wang, Z. Cheng, and X. Li, “Progress on Waveguide-Integrated Graphene Optoelectronics,” Adv. Condens. Matter Phys. 2018, 1–9 (2018).
[Crossref]

J. Wang, Z. Cheng, Z. Chen, X. Wan, B. Zhu, H. K. Tsang, C. Shu, and J. Xu, “High-responsivity graphene-on-silicon slot waveguide photodetectors,” Nanoscale 8(27), 13206–13211 (2016).
[Crossref]

J. Wang, Z. Cheng, Z. Chen, J. Xu, H. K. Tsang, and C. Shu, “Graphene photodetector integrated on silicon nitride waveguide,” J. Appl. Phys. 117(14), 144504 (2015).
[Crossref]

Z. Cheng, H. K. Tsang, X. Wang, J.-B. Xu, and K. Xu, “Mid-infrared Suspended Membrane Waveguide and Ring Resonator on Silicon-on-Insulator,” IEEE J. Sel. Top. Quantum Electron. 20(1), 43–48 (2014).
[Crossref]

X. Wang, Z. Cheng, K. Xu, H. K. Tsang, and J.-B. Xu, “High-responsivity graphene/silicon-heterostructure waveguide photodetectors,” Nat. Photonics 7(11), 888–891 (2013).
[Crossref]

Cheng, Z. Z.

Chow, C. W.

Couny, F.

Crommie, M. F.

Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Direct observation of a widely tunable bandgap in bilayer graphene,” Nature 459(7248), 820–823 (2009).
[Crossref]

Desurvire, E.

E. Desurvire, C. Kazmierski, F. Lelarge, X. Marcadet, A. Scavennec, F. A. Kish, D. F. Welch, R. Nagarajan, C. H. Joyner, R. P. Schneider, and et al., “Science and technology challenges in XXIst century optical communications,” C. R. Phys. 12(4), 387–416 (2011).
[Crossref]

Dimitropoulos, D.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon mid-IR Raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[Crossref]

Dwivedi, S.

Englund, D.

X. Gan, R. J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013).
[Crossref]

Farr, L.

Fathpour, S.

B. Jalali, V. Raghunathan, R. Shori, S. Fathpour, D. Dimitropoulos, and O. Stafsudd, “Prospects for silicon mid-IR Raman lasers,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1618–1627 (2006).
[Crossref]

Freude, W.

Fritz, D. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[Crossref]

Gan, X.

X. Gan, R. J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013).
[Crossref]

Gao, Y.

X. Gan, R. J. Shiue, Y. Gao, K. F. Mak, X. Yao, L. Li, A. Szep, D. Walker, J. Hone, T. F. Heinz, and D. Englund, “High-contrast electrooptic modulation of a photonic crystal nanocavity by electrical gating of graphene,” Nano Lett. 13(2), 691–696 (2013).
[Crossref]

Gardes, F.

Gardes, F. Y.

M. Nedeljkovic, J. S. Penadés, C. J. Mitchell, A. Z. Khokhar, S. Stanković, T. D. Bucio, C. G. Littlejohns, F. Y. Gardes, and G. Z. Mashanovich, “Surface-Grating-Coupled Low-Loss Ge-on-Si Rib Waveguides and Multimode Interferometers,” IEEE Photon. Technol. Lett. 27(10), 1040–1043 (2015).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Girit, C.

Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Direct observation of a widely tunable bandgap in bilayer graphene,” Nature 459(7248), 820–823 (2009).
[Crossref]

Goda, K.

Green, W. M. J.

V. C. Joris, W. M. J. Green, A. Solomon, and Y. A. Vlasov, “Integrated NiSi waveguide heaters for CMOS-compatible silicon thermo-optic devices,” Opt. Lett. 35(7), 1013–1015 (2010).
[Crossref]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophotonic waveguides,” Nat. Photonics 4(8), 557–560 (2010).
[Crossref]

Griol, A.

Guo, X.

W. Li, P. Anantha, S. Bao, K. H. Lee, X. Guo, T. Hu, L. Zhang, H. Wang, R. Soref, and C. S. Tan, “Germanium-on-silicon nitride waveguides for mid-infrared integrated photonics,” Appl. Phys. Lett. 109(24), 241101 (2016).
[Crossref]

Gusynin, V. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys.: Condens. Matter 19(2), 026222 (2007).
[Crossref]

Gylfason, K. B.

Hagan, D.

Hallemeier, P. F.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications systems,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[Crossref]

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Nanoscale (1)

J. Wang, Z. Cheng, Z. Chen, X. Wan, B. Zhu, H. K. Tsang, C. Shu, and J. Xu, “High-responsivity graphene-on-silicon slot waveguide photodetectors,” Nanoscale 8(27), 13206–13211 (2016).
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Nature (1)

Y. Zhang, T. T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Direct observation of a widely tunable bandgap in bilayer graphene,” Nature 459(7248), 820–823 (2009).
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Opt. Express (6)

Opt. Lett. (4)

Optica (1)

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Figures (7)

Fig. 1.
Fig. 1. Calculated graphene permittivity as a function of Fermi level at the wavelength of 2μm.
Fig. 2.
Fig. 2. Schematics of the proposed dual-layer graphene-on-Ge slot waveguide modulator. (a) 3-dimensional schematic of the dual-layer graphene-on-slot waveguide modulator. (b) Cross-section of the dual-layer graphene-on-slot waveguide modulator. The electrodes are set on the surfaces of the two graphene layers, respectively.
Fig. 3.
Fig. 3. Calculation of graphene-on-waveguide loss factors with different waveguide configurations. Inset is the schematic of mode distributions of TE mode dual-layer graphene-on-slot waveguide and dual-layer graphene-on-channel waveguide, respectively.
Fig. 4.
Fig. 4. Calculated mode parameters for different dual-layer graphene-on-waveguide structures. (a) Real parts of effective RIs as a function of Fermi level for TE mode dual-layer graphene-on-slot waveguide and dual-layer graphene-on-channel waveguide. (b) Waveguide loss factors as a function of Fermi level for different waveguide configurations.
Fig. 5.
Fig. 5. Influence of the integrated graphene layers on the waveguide loss factor. (a) Calculated waveguide loss factor as a function of the number of graphene layers. Inset is the schematic of the multiple layers of graphene sandwiched by insulator layers on the surface of the slot waveguide. (b) The normalized derivative of the waveguide loss factor as a function of the number of graphene layers. (c) The effective RI as a function of the number of graphene layers.
Fig. 6.
Fig. 6. Design of dual-layer graphene-on-waveguide MZI phase modulators. (a) Schematic of the dual-layer graphene-on-MZI modulator structure based on the slot waveguide configuration. (b), (c) Calculated transmissions of MZI phase modulators as a function of Fermi levels of arm B for graphene-on-slot waveguide and graphene-on-channel waveguide configurations, respectively.
Fig. 7.
Fig. 7. Design of dual-layer graphene-on-microring resonator phase modulators. (a) Schematic of the microring resonator phase modulator based on the dual-layer graphene-on-slot waveguide configuration. (b), (c) Calculated transmissions of microring resonator phase modulators as a function of Fermi level with the slot waveguide and channel waveguide configurations, respectively.

Tables (1)

Tables Icon

Table 1. Real parts of effective RIs and waveguide loss factors of dual-layer graphene-on-slot waveguides and graphene-on-channel waveguides of different materials.

Equations (4)

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σ = 2 i e 2 ( Ω + 2 i Γ ) h [ 1 ( Ω + 2 i Γ ) 2 Δ ω 2 + Δ 2 ω ( δ n F ( ω ) δ ω δ n F ( ω ) δ ω ) d ω Δ ω 2 + Δ 2 ω ( n F ( ω ) n F ( ω ) ( Ω + 2 i Γ ) 2 4 ω 2 ) d ω ] ,
ε e f f = 1 + i σ ω ε 0 d ,
T ( λ ) = 1 4 × [ exp ( α 1 L ) + exp ( α 2 L ) + 2 exp ( α 1 L + α 2 L 2 ) cos ) ( Δ φ ) ] ,
B ( λ ) = a 2 + t 2 2 a t cos θ 1 + a 2 t 2 2 a t cos θ ,

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