Abstract

A parameterization of the dispersive conductivity of highly-doped graphene has been developed and is presented for use in finite-difference time-domain simulation of near infrared graphene-based photonic and plasmonic devices. The parameterization is based on fitting a Padé approximant to the conductivity arising from interband electronic transitions. The resulting parameterization provides an accurate spectral representation of the conductivity in the wavelength range 1.3 – 2.3μm which is important for near infrared graphene plasmonics. Finite-difference time-domain simulations of straight graphene plasmonic waveguides of infinite and finite width are presented.

© 2012 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  42. Y. Chen and R. Mittra, “A highly efficient finite-difference time domain algorithm for analyzing axisymmetric waveguides,” Microwave Opt. Technol. Lett.15, 201–203 (1997).
    [CrossRef]
  43. M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett.30, 327–330 (2001).
    [CrossRef]
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  46. W. Kuang, W. J. Kim, A. Mock, and J. D. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron.12, 1183–1195 (2006).
    [CrossRef]

2012 (1)

G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of innite graphene sheets via a class of generalized FDTD schemes,” IEEE Trans. Magn.48, 379–382 (2012).
[CrossRef]

2011 (3)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

A. Mock, “Modal analysis of nanoplasmonic multilayer spherical resonators,” IEEE Photonics J.3, 765–776 (2011).
[CrossRef]

I. Ahmed, E. H. Khoo, O. Kurniawan, and E. P. Li, “Modeling and simulation of active plasmonics with the FDTD method by using solid state and Lorentz–Drude dispersive model,” J. Opt. Soc. Am. B28, 352–359 (2011).
[CrossRef]

2010 (9)

D. R. Andersen, “Graphene-based long-wave infrared tm surface plasmon modulator,” J. Opt. Soc. Am. B27, 818–823 (2010).
[CrossRef]

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics4, 297–301 (2010).
[CrossRef]

E. D. Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express18, 8359–8353 (2010).

W. Zhao, P. Tan, J. Zhang, and J. Liu, “Charge transfer and optical phonon mixing in few-layer graphene chemically doped with sulfuric acid,” Phys. Rev. B82, 245423 (2010).
[CrossRef]

A. Mock and P. Trader, “Photonic crystal fiber analysis using cylindrical FDTD with Bloch boundary conditions,” PIERS Online6, 783–787 (2010).
[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, 803–810 (2010).
[CrossRef] [PubMed]

Y.-W. Song, S.-Y. Jang, W.-S. Han, and M.-K. Bae, “Graphene mode-lockers for fiber lasers functioned with evanescent field interaction,” Appl. Phys. Lett.96, 051122 (2010).
[CrossRef]

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
[CrossRef]

I. T. Rekanos and T. G. Papadopoulos, “FDTD modeling of wave propagation in Cole–Cole media with multiple relaxation times,” IEEE Antennas Wireless Propag. Lett.9, 67–69 (2010).
[CrossRef]

2009 (4)

A. K. Geim, “Graphene: status and propects,” Science324, 1530–1534 (2009).
[CrossRef] [PubMed]

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

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009).
[CrossRef]

H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express17, 17630–17635 (2009).
[CrossRef] [PubMed]

2008 (8)

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys.104, 084314 (2008).
[CrossRef]

P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponomorenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Geim, and K. S. Novoselov, “Graphene-based liquid crystal device,” Nano Lett.8, 1704–1708 (2008).
[CrossRef] [PubMed]

T. Stauber, N. M. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78, 085432 (2008).
[CrossRef]

G. W. Hanson, “Dyadic greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103, 064302 (2008).
[CrossRef]

F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett.101, 196405 (2008).
[CrossRef] [PubMed]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science320, 1308 (2008).
[CrossRef] [PubMed]

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

A. Mock and J. D. O’Brien, “Direct extraction of large quality factors and resonant frequencies from Padé interpolated resonance spectra,” Opt. Quantum Electron.40, 1187–1192 (2008).
[CrossRef]

2007 (5)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007).
[CrossRef] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett.446, 115–118 (2007).
[CrossRef]

K. Ziegler, “Minimal conductivity of graphene: Nonuniversal values from the kubo formula,” Phys. Rev. B75, 233407 (2007).
[CrossRef]

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B76, 153410 (2007).
[CrossRef]

2006 (2)

W. Kuang, W. J. Kim, A. Mock, and J. D. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron.12, 1183–1195 (2006).
[CrossRef]

H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and H. J. Halas, “Symmetry breaking in individual plasmonic nanoparticles,” Proc. Natl. Acad. Sci. USA103, 10856–10860 (2006).
[CrossRef] [PubMed]

2005 (1)

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

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,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

2001 (1)

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett.30, 327–330 (2001).
[CrossRef]

1998 (1)

S. Dey and R. Mittra, “Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation,” IEEE Microw. Guid. Wave Lett.8, 415–417 (1998).
[CrossRef]

1997 (2)

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in fdtd,” IEEE Microw. Guid. Wave Lett.7, 121–123 (1997).
[CrossRef]

Y. Chen and R. Mittra, “A highly efficient finite-difference time domain algorithm for analyzing axisymmetric waveguides,” Microwave Opt. Technol. Lett.15, 201–203 (1997).
[CrossRef]

1993 (1)

S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables [guided wavestructure analysis],” IEEE Microw. Guid. Wave Lett.3, 127–129 (1993).
[CrossRef]

1992 (2)

A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron. Lett.28, 1451–1452 (1992).
[CrossRef]

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett.2, 165–167 (1992).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

Ahmed, I.

Andersen, D. R.

Antonopoulos, C. S.

G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of innite graphene sheets via a class of generalized FDTD schemes,” IEEE Trans. Magn.48, 379–382 (2012).
[CrossRef]

Asi, A.

A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron. Lett.28, 1451–1452 (1992).
[CrossRef]

Avouris, P.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics4, 297–301 (2010).
[CrossRef]

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

Bae, M.-K.

Y.-W. Song, S.-Y. Jang, W.-S. Han, and M.-K. Bae, “Graphene mode-lockers for fiber lasers functioned with evanescent field interaction,” Appl. Phys. Lett.96, 051122 (2010).
[CrossRef]

Baker, G. A.

G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge University Press, New York, 1996).
[CrossRef] [PubMed]

Bao, Q. L.

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Basko, D. M.

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, 803–810 (2010).
[CrossRef] [PubMed]

Blake, P.

P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponomorenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Geim, and K. S. Novoselov, “Graphene-based liquid crystal device,” Nano Lett.8, 1704–1708 (2008).
[CrossRef] [PubMed]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science320, 1308 (2008).
[CrossRef] [PubMed]

Bonaccorso, F.

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, 803–810 (2010).
[CrossRef] [PubMed]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science320, 1308 (2008).
[CrossRef] [PubMed]

P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponomorenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Geim, and K. S. Novoselov, “Graphene-based liquid crystal device,” Nano Lett.8, 1704–1708 (2008).
[CrossRef] [PubMed]

Bouzianas, G. D.

G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of innite graphene sheets via a class of generalized FDTD schemes,” IEEE Trans. Magn.48, 379–382 (2012).
[CrossRef]

Brimicombe, P. D.

P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponomorenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Geim, and K. S. Novoselov, “Graphene-based liquid crystal device,” Nano Lett.8, 1704–1708 (2008).
[CrossRef] [PubMed]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009).
[CrossRef]

Chandrashenkar, M.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Chen, Y.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Y. Chen and R. Mittra, “A highly efficient finite-difference time domain algorithm for analyzing axisymmetric waveguides,” Microwave Opt. Technol. Lett.15, 201–203 (1997).
[CrossRef]

Cheng, D. K.

D. K. Cheng, Field and Wave Electromagnetics (Addison Wesley, New York, 1992).

Dawlaty, J. M.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

de la Chapelle, M. L.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

de Vries, T.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

de Waardt, H.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
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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,” Science306, 666–669 (2004).
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F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett.101, 196405 (2008).
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Smalbrugge, B.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Smit, M. K.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Soljacic, M.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009).
[CrossRef]

Song, Y.-W.

Y.-W. Song, S.-Y. Jang, W.-S. Han, and M.-K. Bae, “Graphene mode-lockers for fiber lasers functioned with evanescent field interaction,” Appl. Phys. Lett.96, 051122 (2010).
[CrossRef]

Spencer, M. G.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Stauber, T.

T. Stauber, N. M. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78, 085432 (2008).
[CrossRef]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science320, 1308 (2008).
[CrossRef] [PubMed]

Strait, J.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Stuchly, M. A.

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in fdtd,” IEEE Microw. Guid. Wave Lett.7, 121–123 (1997).
[CrossRef]

Su, C. Y.

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
[CrossRef]

Sun, Z.

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, 803–810 (2010).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, Massachusetts, 2000).

Tan, P.

W. Zhao, P. Tan, J. Zhang, and J. Liu, “Charge transfer and optical phonon mixing in few-layer graphene chemically doped with sulfuric acid,” Phys. Rev. B82, 245423 (2010).
[CrossRef]

Tan, W. D.

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
[CrossRef]

Tang, D. Y.

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
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H. Zhang, D. Y. Tang, L. M. Zhao, Q. L. Bao, and K. P. Loh, “Large energy mode locking of an erbium-doped fiber laser with atomic layer graphene,” Opt. Express17, 17630–17635 (2009).
[CrossRef] [PubMed]

Torrisi, F.

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, 803–810 (2010).
[CrossRef] [PubMed]

Trader, P.

A. Mock and P. Trader, “Photonic crystal fiber analysis using cylindrical FDTD with Bloch boundary conditions,” PIERS Online6, 783–787 (2010).
[CrossRef]

Tsiboukis, T. D.

G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of innite graphene sheets via a class of generalized FDTD schemes,” IEEE Trans. Magn.48, 379–382 (2012).
[CrossRef]

Turkiewicz, J. P.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Vahldieck, R.

S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables [guided wavestructure analysis],” IEEE Microw. Guid. Wave Lett.3, 127–129 (1993).
[CrossRef]

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett.2, 165–167 (1992).
[CrossRef]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

Valdes-Garcia, A.

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

van Otten, F. W. M.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

van Veldhoven, P. J.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Veksler, D.

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Vial, A.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

Wang, F.

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, 803–810 (2010).
[CrossRef] [PubMed]

Wang, H.

H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and H. J. Halas, “Symmetry breaking in individual plasmonic nanoparticles,” Proc. Natl. Acad. Sci. USA103, 10856–10860 (2006).
[CrossRef] [PubMed]

Wu, Y.

F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett.101, 196405 (2008).
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H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and H. J. Halas, “Symmetry breaking in individual plasmonic nanoparticles,” Proc. Natl. Acad. Sci. USA103, 10856–10860 (2006).
[CrossRef] [PubMed]

Xia, F.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics4, 297–301 (2010).
[CrossRef]

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

Xiao, S.

S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables [guided wavestructure analysis],” IEEE Microw. Guid. Wave Lett.3, 127–129 (1993).
[CrossRef]

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett.2, 165–167 (1992).
[CrossRef]

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K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

Zhang, H.

Zhang, J.

W. Zhao, P. Tan, J. Zhang, and J. Liu, “Charge transfer and optical phonon mixing in few-layer graphene chemically doped with sulfuric acid,” Phys. Rev. B82, 245423 (2010).
[CrossRef]

Zhang, Y.

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,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

Zhao, L. M.

Zhao, W.

W. Zhao, P. Tan, J. Zhang, and J. Liu, “Charge transfer and optical phonon mixing in few-layer graphene chemically doped with sulfuric acid,” Phys. Rev. B82, 245423 (2010).
[CrossRef]

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E. D. Fabrizio, A. E. Nikolaenko, and N. I. Zheludev, “Graphene in a photonic metamaterial,” Opt. Express18, 8359–8353 (2010).

Zhu, Y.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Zie, G. Q.

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
[CrossRef]

Ziegler, K.

K. Ziegler, “Minimal conductivity of graphene: Nonuniversal values from the kubo formula,” Phys. Rev. B75, 233407 (2007).
[CrossRef]

ACS Nano (1)

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, 803–810 (2010).
[CrossRef] [PubMed]

Appl. Phys. Lett. (3)

Y.-W. Song, S.-Y. Jang, W.-S. Han, and M.-K. Bae, “Graphene mode-lockers for fiber lasers functioned with evanescent field interaction,” Appl. Phys. Lett.96, 051122 (2010).
[CrossRef]

W. D. Tan, C. Y. Su, R. J. Knize, G. Q. Zie, L. J. Li, and D. Y. Tang, “Mode locking of ceramic Nd:yttrium aluminum garnet with graphene as a saturable absorber,” Appl. Phys. Lett.96, 031106 (2010).
[CrossRef]

J. M. Dawlaty, S. Shivaraman, J. Strait, P. George, M. Chandrashenkar, F. Rana, M. G. Spencer, D. Veksler, and Y. Chen, “Measurement of the optical absorption spectra of epitaxial graphene from terahertz to visible,” Appl. Phys. Lett.93, 131905 (2008).
[CrossRef]

Chem. Phys. Lett. (1)

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett.446, 115–118 (2007).
[CrossRef]

Electron. Lett. (1)

A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” Electron. Lett.28, 1451–1452 (1992).
[CrossRef]

IEEE Antennas Wireless Propag. Lett. (1)

I. T. Rekanos and T. G. Papadopoulos, “FDTD modeling of wave propagation in Cole–Cole media with multiple relaxation times,” IEEE Antennas Wireless Propag. Lett.9, 67–69 (2010).
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IEEE J. Sel. Top. Quantum Electron. (1)

W. Kuang, W. J. Kim, A. Mock, and J. D. O’Brien, “Propagation loss of line-defect photonic crystal slab waveguides,” IEEE J. Sel. Top. Quantum Electron.12, 1183–1195 (2006).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (4)

S. Xiao, R. Vahldieck, and H. Jin, “Full-wave analysis of guided wave structures using a novel 2-D FDTD,” IEEE Microw. Guid. Wave Lett.2, 165–167 (1992).
[CrossRef]

S. Xiao and R. Vahldieck, “An efficient 2-D FDTD algorithm using real variables [guided wavestructure analysis],” IEEE Microw. Guid. Wave Lett.3, 127–129 (1993).
[CrossRef]

S. Dey and R. Mittra, “Efficient computation of resonant frequencies and quality factors of cavities via a combination of the finite-difference time-domain technique and the Padé approximation,” IEEE Microw. Guid. Wave Lett.8, 415–417 (1998).
[CrossRef]

M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in fdtd,” IEEE Microw. Guid. Wave Lett.7, 121–123 (1997).
[CrossRef]

IEEE Photonics J. (1)

A. Mock, “Modal analysis of nanoplasmonic multilayer spherical resonators,” IEEE Photonics J.3, 765–776 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

IEEE Trans. Magn. (1)

G. D. Bouzianas, N. V. Kantartzis, C. S. Antonopoulos, and T. D. Tsiboukis, “Optimal modeling of innite graphene sheets via a class of generalized FDTD schemes,” IEEE Trans. Magn.48, 379–382 (2012).
[CrossRef]

J. Appl. Phys. (2)

G. W. Hanson, “Dyadic greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103, 064302 (2008).
[CrossRef]

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys.104, 084314 (2008).
[CrossRef]

J. Opt. Soc. Am. B (2)

Microwave Opt. Technol. Lett. (2)

Y. Chen and R. Mittra, “A highly efficient finite-difference time domain algorithm for analyzing axisymmetric waveguides,” Microwave Opt. Technol. Lett.15, 201–203 (1997).
[CrossRef]

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett.30, 327–330 (2001).
[CrossRef]

Nano Lett. (1)

P. Blake, P. D. Brimicombe, R. R. Nair, T. J. Booth, D. Jiang, F. Schedin, L. A. Ponomorenko, S. V. Morozov, H. F. Gleeson, E. W. Hill, A. K. Geim, and K. S. Novoselov, “Graphene-based liquid crystal device,” Nano Lett.8, 1704–1708 (2008).
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Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007).
[CrossRef] [PubMed]

Nat. Nanotechnol. (1)

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

Nat. Photonics (2)

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photonics4, 297–301 (2010).
[CrossRef]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Notzel, and M. K. Smit, “Lasing in metal-coated nanocavities,” Nat. Photonics1, 589–594 (2007).
[CrossRef]

Opt. Express (2)

Opt. Quantum Electron. (1)

A. Mock and J. D. O’Brien, “Direct extraction of large quality factors and resonant frequencies from Padé interpolated resonance spectra,” Opt. Quantum Electron.40, 1187–1192 (2008).
[CrossRef]

Phys. Rev. B (6)

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B71, 085416 (2005).
[CrossRef]

K. Ziegler, “Minimal conductivity of graphene: Nonuniversal values from the kubo formula,” Phys. Rev. B75, 233407 (2007).
[CrossRef]

L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B76, 153410 (2007).
[CrossRef]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B80, 245435 (2009).
[CrossRef]

T. Stauber, N. M. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B78, 085432 (2008).
[CrossRef]

W. Zhao, P. Tan, J. Zhang, and J. Liu, “Charge transfer and optical phonon mixing in few-layer graphene chemically doped with sulfuric acid,” Phys. Rev. B82, 245423 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett.101, 196405 (2008).
[CrossRef] [PubMed]

PIERS Online (1)

A. Mock and P. Trader, “Photonic crystal fiber analysis using cylindrical FDTD with Bloch boundary conditions,” PIERS Online6, 783–787 (2010).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and H. J. Halas, “Symmetry breaking in individual plasmonic nanoparticles,” Proc. Natl. Acad. Sci. USA103, 10856–10860 (2006).
[CrossRef] [PubMed]

Science (4)

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine structure constant defines visual transparency of graphene,” Science320, 1308 (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,” Science306, 666–669 (2004).
[CrossRef] [PubMed]

A. K. Geim, “Graphene: status and propects,” Science324, 1530–1534 (2009).
[CrossRef] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

Other (4)

A. Taflove and S. C. Hagness, Computational Electrodynamics (Artech House, Massachusetts, 2000).

D. K. Cheng, Field and Wave Electromagnetics (Addison Wesley, New York, 1992).

G. A. Baker and P. Graves-Morris, Padé Approximants (Cambridge University Press, New York, 1996).
[CrossRef] [PubMed]

A. Mock and J. D. O’Brien, “Dependence of silicon-on-insulator waveguide loss on lower oxide cladding thickness,” in Integrated Photonics and Nanophotonics Research and Applications Topical Meeting (Optical Society of America, Boston, MA, USA, 2008),p. IWG4.

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Figures (6)

Fig. 1
Fig. 1

(a) Plot of the real and imaginary part of the graphene conductivity calculated using Eq. (3) for different values of μc. Other physical values are set at T = 300 K and Γ = 11meV/h̄. (b) Plot of the real and imaginary part of the intraband conductivity term for μc = 0.6 eV. (b) Plot of the real and imaginary part of the interband conductivity term for μc = 0.6 eV.

Fig. 2
Fig. 2

Comparison between the interband conductivity calculated using Eq. (5) and the Padé approximant fit. (a) Real part, (b) imaginary part.

Fig. 3
Fig. 3

Comparison between infinite graphene sheet TM SPP dispersion relations calculated using the actual interband dispersion term (Eq. (5)) and the Padé fit. Agreement is very good for wavelengths greater than 1.3 μm. Also shown is the SPP dispersion relation calculated using only the intraband (Drude) term showing that including the interband term is essential for accurate SPP simulation.

Fig. 4
Fig. 4

(a) Diagram of computational domain for FDTD analysis of infinite graphene sheet. A two-dimensional region is shown, but only a one-dimensional grid is used representing the geometry variation in the y direction. (b) Comparison between exact dispersion relation (using both the actual interband conductivity formula Eq. (5) and the Padé fit) and the dispersion calculated using the FDTD method.

Fig. 5
Fig. 5

Cross sectional field profiles of dominant electric field components of TM SPP mode supported by an infinite graphene sheet. On the right a one-dimensional plot of the field along the y direction is shown. For an infinite sheet, only the one-dimensional fields are calculated in the FDTD simulation. The two-dimensional field profiles are created by extruding the one-dimensional plot along the x direction. The wavelength is 1.82 μm.

Fig. 6
Fig. 6

(a) Waveguide dispersion for an infinite graphene sheet (blue circles) and a graphene sheet 1.0 μm in width (red circles). (b) Cross sectional field profiles of dominant electric field components of 1.0 μm wide graphene SPP mode. The wavelength is 6.1 μm (corresponding to left most red circle data point).

Tables (1)

Tables Icon

Table 1 Padé Expansion Coefficients for Fit Displayed in Fig. 2

Equations (15)

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

ε ( ω ) = ε r ( ω ) i σ ( ω ) / ω
ε ( ω ) = ε r ( ω ) + σ i ( ω ) / ω i σ r ( ω ) / ω .
σ ( ω , μ c , Γ , T ) = j e 2 / π h ¯ 2 ω j 2 Γ 0 ε ( f d ( ε ) ε f d ( ε ) ε ) d ε + + j e 2 / π h ¯ 2 ( ω j 2 Γ ) 0 f d ( ε ) f d ( ε ) ( ω j 2 Γ ) 2 4 ( ε / h ¯ 2 ) d ε
σ intra ( ω , μ c , Γ , T ) = j 8 σ 0 k B T / h ω j 2 Γ [ μ c k B T + 2 ln ( e μ c / k B T + 1 ) ]
σ inter ( ω , μ c , Γ , T ) j σ 0 π ln [ 2 | μ c | ( ω j 2 Γ ) h ¯ 2 | μ c | + ( ω j 2 Γ ) h ¯ ] .
n = 2 π h ¯ 2 v F 2 0 ε [ f d ( ε ) f d ( ε + 2 μ c ) ] d ε
a 0 + a 1 ω + + a M ω M 1 + b 1 ω + + b N ω N = σ inter ( ω )
a 0 + a 1 j ω + a 2 ( j ω ) 2 1 + b 1 j ω + b 2 ( j ω ) 2 = σ inter ( ω )
a 0 + a 1 j ω + a 2 ( j ω ) 2 b 1 j ω σ inter ( ω ) b 2 ( j ω ) 2 σ inter ( ω ) = σ inter ( ω )
a 0 a 2 ω 2 + b 1 ω Im [ σ inter ( ω ) ] + b 2 ( ω ) 2 Re [ σ inter ( ω ) ] = Re [ σ inter ( ω ) ]
a 1 ω b 1 ω Re [ σ inter ( ω ) ] + b 2 ( ω ) 2 Im [ σ inter ( ω ) ] = Im [ σ inter ( ω ) ] .
[ 1 0 ω 1 2 ω 1 η ( ω 1 ) ω 1 2 γ ( ω ) 0 ω 1 0 ω 1 γ ( ω 1 ) ω 1 2 η ( ω 1 ) 1 0 ω 2 2 ω 2 η ( ω 2 ) ω 2 2 γ ( ω 2 ) 0 ω 2 0 ω 2 γ ( ω 2 ) ω 2 2 η ( ω 2 ) 1 0 ω 3 2 ω 3 η ( ω 3 ) ω 3 2 γ ( ω 3 ) ] [ a 0 a 1 a 2 b 1 b 2 ] = [ γ ( ω 1 ) η ( ω 1 ) γ ( ω 2 ) η ( ω 2 ) γ ( ω 3 ) ]
F i β ( x , y , z ) = f i β ( x , y ) exp ( j β z )
E y | i , j + 1 2 n + 1 = E y | i , j + 1 2 n Δ t ε i , j + 1 2 Δ x [ H z | i + 1 2 , j + 1 2 n + 1 2 H z | i 1 2 , j + 1 2 n + 1 2 ] β Δ t ε i , j + 1 2 H x | i , j + 1 2 n + 1 2 Δ t ε i , j + 1 2 J x | i , j + 1 2 n + 1 2
H x | i , j + 1 2 n + 1 2 = H x | i , j + 1 2 n Δ t μ i , j + 1 2 Δ y [ E z | i , j + 1 n E z | i , j n ] + β Δ t μ i , j + 1 2 E y | i , j + 1 2 n

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