## Switching energy limits of waveguide-coupled graphene-on-graphene optical modulators |

Optics Express, Vol. 20, Issue 18, pp. 20330-20341 (2012)

http://dx.doi.org/10.1364/OE.20.020330

Acrobat PDF (891 KB)

### Abstract

The fundamental switching energy limitations for waveguide coupled graphene-on-graphene optical modulators are described. The minimum energy is calculated under the constraints of fixed insertion loss and extinction ratio. Analytical relations for the switching energy both for realistic structures and in the quantum capacitance limit are derived and compared with numerical simulations. The results show that sub-femtojoule per bit switching energies and peak-to-peak voltages less than 0.1 V are achievable in graphene-on-graphene optical modulators using the constraint of 3 dB extinction ratio and 3 dB insertion loss. The quantum-capacitance limited switching energy for a single TE-mode modulator geometry is found to be < 0.5 fJ/bit at *λ* = 1.55 μm, and the dependences of the minimum energy on the waveguide geometry, wavelength, and graphene location are investigated. The low switching energy is a result of the very strong optical absorption in graphene, and the extremely-small operating voltages needed as the device approaches the quantum capacitance regime.

© 2012 OSA

## 1. Introduction

1. F. Xia, T. Mueller, Y. M. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. **4**(12), 839–843 (2009). [CrossRef] [PubMed]

2. T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics **4**(5), 297–301 (2010). [CrossRef]

3. 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]

4. B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. **3**, 780 (2012). [CrossRef] [PubMed]

5. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics **5**(7), 411–415 (2011). [CrossRef]

6. Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano **4**(2), 803–810 (2010). [CrossRef] [PubMed]

7. Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. **19**(19), 3077–3083 (2009). [CrossRef]

8. X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science **324**(5932), 1312–1314 (2009). [CrossRef] [PubMed]

9. B. Jalali and S. Fathpour, “Silicon photonics,” IEEE J. Lightwave Tech. **24**(12), 4600–4615 (2006). [CrossRef]

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

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

12. D. A. B. Miller, “Energy consumption in optical modulators for interconnects,” Opt. Express **20**(S2Suppl 2), A293–A308 (2012). [CrossRef] [PubMed]

## 2. Modulator structure

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

_{2}cladding layers. The graphene dual-layer structure is placed either in the center or on top of the waveguide core. For the calculations below, we have made the simplifying assumption that the graphene does not perturb the optical mode, and that the distance between the two graphene layers is sufficiently small compared to the size of the optical mode so that the two layers have the same absorption properties. Using numerical simulations, we have found this to be a reasonable assumption for graphene interlayer spacing of less than ~10 nm. We have further assumed that the graphene width is simply the width of the waveguide core, and that the connecting leads to the graphene (not shown in the figure) do not contribute to optical absorption. This assumption is justified since absorption in the leads could be suppressed by heavily doping the graphene in these regions, and/or by limiting the leads to specific locations along the length of the modulator. As shown in Fig. 1, the calculations further assume that the device has been tuned to operate at the ideal operating conditions for maximum extinction ratio, where both graphene layers are utilized as gate-tunable absorbers [10

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

*E*

_{F}_{(top)}= –

*hc*/2

*λ*and

*E*

_{F}_{(bottom)}= +

*hc*/2

*λ*, where

*E*

_{F}_{(top)}and

*E*

_{F}_{(bottom)}are the Fermi energies relative to the Dirac point in the top and bottom layers respectively,

*h*is Planck’s constant and

*λ*is the wavelength. Such a situation could be achieved either by applying a DC bias between the layers, or by precisely doping the top and bottom graphene to be p- and n-type respectively. While the former approach might be less challenging to fabricate, the latter would allow the modulator to operate at zero DC bias, thus ensuring that no DC power is consumed in the biasing electronics. We also assume that the Si waveguide is undoped and that the bias voltages applied to the graphene layers do not induce charges in the Si so that no additional free carrier absorption is caused. Finally, we assume that no leakage current flows between the graphene sheets.

## 3. Energy calculation

13. F. Y. Gardes, D. J. Thomson, N. G. Emerson, and G. T. Reed, “40 Gb/s silicon photonics modulator for TE and TM polarisations,” Opt. Express **19**(12), 11804–11814 (2011). [CrossRef] [PubMed]

*E*, of the modulator can be expressed as:where

*C*and

_{m}*V*are the modulator capacitance and peak-to-peak voltage, respectively. The ¼ term in (1) comes from the fact that in an NRZ signaling scheme, and for a random bit sequence, one complete charge/discharge cycle occurs on average once every four bits.

_{pp}*C*for the modulator can then be calculated aswhere

_{m}*c*is the oxide capacitance per unit area associated with the dielectric separating the two graphene layers,

_{ox}*c*is the quantum capacitance (per unit area) in each graphene sheet,

_{Q}*W*is the modulator width and

_{m}*L*is the modulator length. Once again, in (2), it is assumed that the modulator is operated in the regime that provides maximum extinction ratio, which occurs when the Fermi levels in the top and bottom layers are ±

_{m}*hc*/2

*λ*from the Dirac point energy. In this condition, and in the limit of small spacing between the graphene sheets, both graphene layers contribute equally to optical modulation. The potential movement, Δ

*V*, in each graphene layer required to achieve a specified degree of modulation will be proportional to the thermal energy and can be expressed aswhere

*k*is Boltzmann’s constant,

*T*is temperature,

*e*is the electronic charge, and

*m*is a term we call the “modulation coefficient.” This potential change will need to appear across both graphene layers, and therefore, the minimum peak-to-peak applied voltage needed to achieve a specific degree of modulation can be expressed aswhere

*V*is the voltage dropped across the oxide separating the graphene layers. The quantum capacitance per unit area,

_{ox}*c*, in single-layer graphene [14

_{Q}14. T. Fang, A. Konar, H. Xing, and D. Jena, “Carrier statistics and quantum capacitance of graphene sheets and ribbons,” Appl. Phys. Lett. **91**(9), 092109 (2007). [CrossRef]

*e*is the electronic charge,

*h*is Planck’s constant,

*v*is the Fermi velocity, and

_{F}*λ*is the wavelength. The quantum capacitance relation can be greatly simplified for photon energies much larger than the thermal energy. In this case, the quantum capacitance reduces toThe oxide capacitance per unit area can be expressed using the standard parallel-plate formulawhere 3.9 is the relative dielectric constant of SiO

_{2},

*ε*

_{0}is the permittivity of free space and

*EOT*is the equivalent oxide thickness of the dielectric separating the graphene layers. This is an excellent assumption when

*EOT*<<

*W*,

_{m}*L*, which is likely to be valid for most practical designs.

_{m}*L*in terms of the absorption coefficient,

_{m}*α*

_{0}, so we can write:where

*γ*is a dimensionless proportionality coefficient. Now, in order to determine the performance parameters of the device, we first define an expression for the modulator loss,

*L*. Here, we define the loss (in dB) as:where

*T*

_{max}, is the maximum transmission coefficient within a modulator cycle.

*T*

_{max}, can then be expressed as:In (10), the main exponential accounts for the decaying optical intensity along the length of the modulator. The factor of two inside the exponential represents the fact that both top and bottom graphene layers are contributing to absorption. Finally, the 1/(1 +

*e*) term is simply the Fermi-Dirac occupation probability when

^{m}*E*is at its furthest distance from the Dirac point (i.e., when

_{F}*E*= ± (

_{F}*hc*/2

*λ*+

*mkT*)). It should be pointed out that (10) only accounts for optical modulation associated with Pauli blocking [15

15. Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. **4**(7), 532–535 (2008). [CrossRef]

16. J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B **83**(16), 165113 (2011). [CrossRef]

*E*= ±

_{F}*hc*/2

*λ*by a few

*kT*, and optical absorption due to intra-band transitions is small compared to the residual inter-band absorption. Such effects may have to be taken into account if much larger extinction ratios, very low insertion loss or very long wavelength operation are desired.

## 4. Calculation and optimization results

_{2}cladding have been considered, and the switching energy of various device configurations have been calculated to determine the optimal modulator structure that provides the lowest value of

*w*/

*a*. For simplicity, we have focused on waveguide structures supporting only the fundamental TE and TM modes, since multimode operation is generally undesirable in optical modulators. Three particular waveguide geometries with the graphene-on-graphene layers residing either in the middle or on the top of the Si waveguide core have been simulated.

*E*. The complex 2D conductivity was calculated from the Kubo formula [17

_{t}17. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. **103**(6), 064302 (2008). [CrossRef]

*E*=

_{F}*hc*/2

*λ*+

*kT*ln(2), and assuming a scattering energy of

*T*= 300 K, and

*λ*= 1.55 μm, the calculated conductivities are:

*σ*and

_{intra}*σ*are the complex optical conductivities associated with intra-band and inter-band absorption, respectively, and

_{inter}*σ*=

_{tot}*σ*+

_{intra}*σ*is the total optical conductivity. As stated previously, because the real part of the intra-band conductivity is much smaller than the inter-band portion, intra-band absorption was ignored in the energy calculations in section 3.

_{inter}**n**is the unit vector normal to the graphene layer,

**H**

_{1}and

**H**

_{2}are the magnetic fields on both sides of the graphene layer,

*σ*is the complex conductivity and

**E**is the electric field at the graphene layer.

*E*|

_{t}^{2}, is plotted vs. position in the waveguide. Each structure was analyzed to obtain the values of

*w*/

*a*that are needed to determine the minimum switching energy of the modulator. The simulation suggests that in all of the cases considered here, the presence of the graphene layers has a negligible perturbation to the optical mode, so that the optical absorption is proportional to the real portion of the graphene conductivity. We have also neglected material dispersion, which is a valid approximation for Si and SiO

_{2}in the considered wavelength range (

*λ*= 1.55 – 3.5 μm).

*E*is plotted vs. the normalized waveguide width

*w*for several values of the normalized waveguide thicknesses,

*t*=

*T*/

_{Si}*λ*, under the requirement that the waveguide supports the fundamental TE and TM modes. For each configuration, and for a given value of

*t*, the energy has a minimum at a specific value of

*w*, and for a particular value of

*t*, a global minimum energy value,

*E*

_{min}, is reached.

*λ*= 1.55 μm and

*λ*= 3.5 μm. The wavelength dependence of

*E*

_{min}for the three different modulator designs is also shown in Fig. 4(a) , confirming the wavelength scaling property. A minimum switching energy of 0.48 fJ/bit occurs for Case 1 (the TE-mode / embedded graphene design) at

*λ*= 1.55 μm, a value that increases to 1.09 fJ/bit at

*λ*= 3.5 μm. Comparing the two TE configurations, placing the graphene layers in the middle (Case 1) instead of on the top surface (Case 2) of the waveguide core reduces the optimal switching energy by 40%. This improvement is due to the much higher in-plane electric field in the middle of the Si core compared to on the top surface. The minimum energy associated with the TM mode when the graphene layers are on the top surface (Case 3) is comparable to that for the TE mode (Case 2). Although the linear absorption coefficient for the TE mode is higher than for the TM mode, the TE mode design requires a wider waveguide, and thus leads to an increased device capacitance. Conversely, the TM-mode design allows the waveguide to be narrower, making up for the lower absorption. In addition, we point out that the absorption coefficients calculated from the simulation results in Fig. 3 for the TE-mode / graphene-on-top structure are in excellent agreement with our experimental results for graphene-on-waveguide structures [19].

*E*

_{min}is also plotted vs.

*ER*at

*λ*= 1.55 μm for the three waveguide configurations. It can be seen that as

*ER*increases,

*E*also increases due to the much higher AC voltage needed to shift the Fermi level over a greater energy range in the graphene. As a reminder, free carrier absorption was not included in these calculations, and while this is reasonable assumption at

_{min}*λ*= 1.55 μm, at longer wavelengths, free carrier absorption could become a factor and limit the achievable extinction ratios.

*V*plotted as a function of the

_{pp}*EOT*value of the dielectric separating the graphene layers for different wavelengths. The figure shows that

*V*decreases with increasing wavelength due to the lower quantum capacitance as the Fermi-level is positioned closer to the Dirac point. The reduced quantum capacitance, however, does not lead to lower energy, since this reduction is exactly compensated by the wider waveguide required at longer wavelengths. The figure shows that even in realistic geometries with

_{pp}*EOT*~1-2 nm, peak-to-peak voltages less than 0.25 V are still possible. The minimum energy vs.

*EOT*for the TE-mode / graphene-on-top modulator geometry is shown in Fig. 5(b). Here, it can be seen that the energy increases at the same rate for all wavelengths, and that in realistic geometries with

*EOT*~1-2 nm, the switching energy is on the order of 2-3 fJ/bit.

## 5. Discussion

20. M. R. Watts, W. A. Zortman, D. C. Trotter, R. W. Young, and A. L. Lentine, “Vertical junction silicon microdisk modulators and switches,” Opt. Express **19**(22), 21989–22003 (2011). [CrossRef] [PubMed]

24. S. Ren, Y. Rong, S. A. Claussen, R. K. Schaevitz, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Ge/SiGe quantum well waveguide modulator monolithically integrated with SOI waveguides,” IEEE Photon. Technol. Lett. **24**(6), 461–463 (2012). [CrossRef]

21. J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics **2**(7), 433–437 (2008). [CrossRef]

22. N.-N. Feng, D. Feng, S. Liao, X. Wang, P. Dong, H. Liang, C.-C. Kung, W. Qian, J. Fong, R. Shafiiha, Y. Luo, J. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “30GHz Ge electro-absorption modulator integrated with 3 μm silicon-on-insulator waveguide,” Opt. Express **19**(8), 7062–7067 (2011). [CrossRef] [PubMed]

23. R. K. Schaevitz, E. H. Edwards, J. E. Roth, E. T. Fei, Y. Rong, P. Wahl, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Simple electroabsorption calculator for designing 1310 nm and 1550 nm modulators using germanium quantum wells,” IEEE J. Quantum Electron. **48**(2), 187–197 (2012). [CrossRef]

24. S. Ren, Y. Rong, S. A. Claussen, R. K. Schaevitz, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Ge/SiGe quantum well waveguide modulator monolithically integrated with SOI waveguides,” IEEE Photon. Technol. Lett. **24**(6), 461–463 (2012). [CrossRef]

^{2}/Vs, and contact resistance = 400 Ω-μm) and RC-limited bandwidth definition reported previously [10

**100**(17), 171107 (2012). [CrossRef]

*EOT*at different wavelengths was calculated and the results are shown in Fig. 6(a) . It can be seen that indeed the bandwidth decreases significantly as the graphene layer separation is decreased. The bandwidth is also reduced at longer wavelengths [10

**100**(17), 171107 (2012). [CrossRef]

*λ*

^{3}, which is a result of the fact that energy-delay scales as

*C*

_{m}^{3}, and

*C*is proportional to

_{m}*λ*.

## 6. Summary

## References and links

1. | F. Xia, T. Mueller, Y. M. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol. |

2. | T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics |

3. | 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 |

4. | B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun. |

5. | Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics |

6. | Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano |

7. | Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. |

8. | X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science |

9. | B. Jalali and S. Fathpour, “Silicon photonics,” IEEE J. Lightwave Tech. |

10. | S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett. |

11. | M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. |

12. | D. A. B. Miller, “Energy consumption in optical modulators for interconnects,” Opt. Express |

13. | F. Y. Gardes, D. J. Thomson, N. G. Emerson, and G. T. Reed, “40 Gb/s silicon photonics modulator for TE and TM polarisations,” Opt. Express |

14. | T. Fang, A. Konar, H. Xing, and D. Jena, “Carrier statistics and quantum capacitance of graphene sheets and ribbons,” Appl. Phys. Lett. |

15. | Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys. |

16. | J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B |

17. | G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. |

18. | J. D. Joannopoulos, |

19. | H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” arXiv:1205.4050v1, 2012. |

20. | M. R. Watts, W. A. Zortman, D. C. Trotter, R. W. Young, and A. L. Lentine, “Vertical junction silicon microdisk modulators and switches,” Opt. Express |

21. | J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics |

22. | N.-N. Feng, D. Feng, S. Liao, X. Wang, P. Dong, H. Liang, C.-C. Kung, W. Qian, J. Fong, R. Shafiiha, Y. Luo, J. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “30GHz Ge electro-absorption modulator integrated with 3 μm silicon-on-insulator waveguide,” Opt. Express |

23. | R. K. Schaevitz, E. H. Edwards, J. E. Roth, E. T. Fei, Y. Rong, P. Wahl, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Simple electroabsorption calculator for designing 1310 nm and 1550 nm modulators using germanium quantum wells,” IEEE J. Quantum Electron. |

24. | S. Ren, Y. Rong, S. A. Claussen, R. K. Schaevitz, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Ge/SiGe quantum well waveguide modulator monolithically integrated with SOI waveguides,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(250.7360) Optoelectronics : Waveguide modulators

(250.4110) Optoelectronics : Modulators

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: June 13, 2012

Revised Manuscript: August 11, 2012

Manuscript Accepted: August 11, 2012

Published: August 20, 2012

**Citation**

Steven J. Koester, Huan Li, and Mo Li, "Switching energy limits of waveguide-coupled graphene-on-graphene optical modulators," Opt. Express **20**, 20330-20341 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20330

Sort: Year | Journal | Reset

### References

- F. Xia, T. Mueller, Y. M. Lin, A. Valdes-Garcia, and P. Avouris, “Ultrafast graphene photodetector,” Nat. Nanotechnol.4(12), 839–843 (2009). [CrossRef] [PubMed]
- T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics4(5), 297–301 (2010). [CrossRef]
- M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature474(7349), 64–67 (2011). [CrossRef] [PubMed]
- B. Sensale-Rodriguez, R. Yan, M. M. Kelly, T. Fang, K. Tahy, W. S. Hwang, D. Jena, L. Liu, and H. G. Xing, “Broadband graphene terahertz modulators enabled by intraband transitions,” Nat. Commun.3, 780 (2012). [CrossRef] [PubMed]
- Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nat. Photonics5(7), 411–415 (2011). [CrossRef]
- Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano4(2), 803–810 (2010). [CrossRef] [PubMed]
- Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater.19(19), 3077–3083 (2009). [CrossRef]
- X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, “Large-area synthesis of high-quality and uniform graphene films on copper foils,” Science324(5932), 1312–1314 (2009). [CrossRef] [PubMed]
- B. Jalali and S. Fathpour, “Silicon photonics,” IEEE J. Lightwave Tech.24(12), 4600–4615 (2006). [CrossRef]
- S. J. Koester and M. Li, “High-speed waveguide-coupled graphene-on-graphene optical modulators,” Appl. Phys. Lett.100(17), 171107 (2012). [CrossRef]
- M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett.12(3), 1482–1485 (2012). [CrossRef] [PubMed]
- D. A. B. Miller, “Energy consumption in optical modulators for interconnects,” Opt. Express20(S2Suppl 2), A293–A308 (2012). [CrossRef] [PubMed]
- F. Y. Gardes, D. J. Thomson, N. G. Emerson, and G. T. Reed, “40 Gb/s silicon photonics modulator for TE and TM polarisations,” Opt. Express19(12), 11804–11814 (2011). [CrossRef] [PubMed]
- T. Fang, A. Konar, H. Xing, and D. Jena, “Carrier statistics and quantum capacitance of graphene sheets and ribbons,” Appl. Phys. Lett.91(9), 092109 (2007). [CrossRef]
- Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P. Kim, H. L. Stormer, and D. N. Basov, “Dirac charge dynamics in graphene by infrared spectroscopy,” Nat. Phys.4(7), 532–535 (2008). [CrossRef]
- J. Horng, C.-F. Chen, B. Geng, C. Girit, Y. Zhang, Z. Hao, H. A. Bechtel, M. Martin, A. Zettl, M. F. Crommie, Y. R. Shen, and F. Wang, “Drude conductivity of Dirac fermions in graphene,” Phys. Rev. B83(16), 165113 (2011). [CrossRef]
- G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103(6), 064302 (2008). [CrossRef]
- J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd edition (Princeton University Press, 2008).
- H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” arXiv:1205.4050v1, 2012.
- M. R. Watts, W. A. Zortman, D. C. Trotter, R. W. Young, and A. L. Lentine, “Vertical junction silicon microdisk modulators and switches,” Opt. Express19(22), 21989–22003 (2011). [CrossRef] [PubMed]
- J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics2(7), 433–437 (2008). [CrossRef]
- N.-N. Feng, D. Feng, S. Liao, X. Wang, P. Dong, H. Liang, C.-C. Kung, W. Qian, J. Fong, R. Shafiiha, Y. Luo, J. Cunningham, A. V. Krishnamoorthy, and M. Asghari, “30GHz Ge electro-absorption modulator integrated with 3 μm silicon-on-insulator waveguide,” Opt. Express19(8), 7062–7067 (2011). [CrossRef] [PubMed]
- R. K. Schaevitz, E. H. Edwards, J. E. Roth, E. T. Fei, Y. Rong, P. Wahl, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Simple electroabsorption calculator for designing 1310 nm and 1550 nm modulators using germanium quantum wells,” IEEE J. Quantum Electron.48(2), 187–197 (2012). [CrossRef]
- S. Ren, Y. Rong, S. A. Claussen, R. K. Schaevitz, T. I. Kamins, J. S. Harris, and D. A. B. Miller, “Ge/SiGe quantum well waveguide modulator monolithically integrated with SOI waveguides,” IEEE Photon. Technol. Lett.24(6), 461–463 (2012). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.