Abstract

The effect of inducing a one-dimensional periodic modulation in the conductivity of both a single and double layer of graphene is investigated using analytical modeling. By employing a modal matching approach, we find deep transmission minimums associated with hybridized resonances of the modes supported by low- and high-conductivity regions. By carefully tuning the conductivity profile, we show that an increase, approaching 50%, can be achieved in the resonant absorption when both regions are made dipole resonant. Such plasmonic cavities may be a promising route to eliminating plasmonic losses typically introduced when etching graphene.

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References

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2015 (5)

P. Simonet, D. Bischoff, A. Moser, T. Ihn, and K. Ensslin, “Graphene nanoribbons: relevance of etching process,” J. Appl. Phys. 117, 184303 (2015).
[Crossref]

C. Ye, Z. Zhu, W. Xu, X. Yuan, and S. Qin, “Electrically tunable absorber based on nonstructured graphene,” J. Opt. 17, 125009 (2015).
[Crossref]

A. P. M. Michel, P. Q. Liu, J. K. Yeung, P. Corrigan, M. L. Baeck, Z. Wang, T. Day, F. Moshary, C. F. Gmachl, and J. A. Smith, “Quantum cascade laser open-path system for remote sensing of trace gases in Beijing, China,” Opt. Eng. 49, 111125 (2015).
[Crossref]

D. B. Farmer, D. Rodrigo, T. Low, and P. Avouris, “Plasmon–plasmon hybridization and bandwidth enhancement in nanostructured graphene,” Nano Lett. 15, 2582–2587 (2015).
[Crossref]

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene-based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

2014 (2)

I.-T. Lin and J.-M. Liu, “Enhanced graphene plasmon waveguiding in a layered graphene-metal structure,” Appl. Phys. Lett. 105, 011604 (2014).
[Crossref]

I. Luxmoore, C. Gan, P. Liu, and F. Valmorra, “Strong coupling in the far-infrared between graphene plasmons and the surface optical phonons of silicon dioxide,” ACS Photon. 1, 1151–1155 (2014).
[Crossref]

2013 (6)

W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13, 3698–3702 (2013).
[Crossref]

Z. Li and N. Yu, “Modulation of mid-infrared light using graphene-metal plasmonic antennas,” Appl. Phys. Lett. 102, 131108 (2013).
[Crossref]

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14, 299–304 (2013).
[Crossref]

N. M. R. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).

T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

2012 (5)

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

N. M. R. Peres, A. Ferreira, Y. V. Bludov, and M. I. Vasilevskiy, “Light scattering by a medium with a spatially modulated optical conductivity: the case of graphene,” J. Phys. 4, 245303 (2012).
[Crossref]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
[Crossref]

H. S. Song, S. L. Li, H. Miyazaki, S. Sato, K. Hayashi, A. Yamada, N. Yokoyama, and K. Tsukagoshi, “Origin of the relatively low transport mobility of graphene grown through chemical vapor deposition,” Sci. Rep. 2, 337 (2012).
[Crossref]

R. Alaee, M. Farhat, C. Rockstuhl, and F. Lederer, “A perfect absorber made of a graphene micro-ribbon metamaterial,” Opt. Express 20, 28017–28024 (2012).
[Crossref]

2011 (2)

P.-Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
[Crossref]

2010 (3)

T. Buffeteau, J. Grondin, Y. Danten, J.-C. Lasse, and J. C. Lassègues, “Imidazolium-based ionic liquids: quantitative aspects in the far-infrared region,” J. Phys. Chem. B 114, 7587–7592 (2010).
[Crossref]

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys. 78, 1199–1202 (2010).
[Crossref]

2009 (2)

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

P. Sessi, J. R. Guest, M. Bode, and N. P. Guisinger, “Patterning graphene at the nanometer scale via hydrogen desorption,” Nano Lett. 9, 4343–4347 (2009).
[Crossref]

2008 (1)

L. A. Falkovsky, “Optical properties of graphene,” J. Phys. 129, 012004 (2008).

2007 (4)

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

J. Lee, M. J. Panzer, Y. He, T. P. Lodge, and C. D. Frisbie, “Ion gel gated polymer thin-film transistors,” J. Am. Chem. Soc. 129, 4532–4533 (2007).
[Crossref]

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56, 281–284 (2007).
[Crossref]

2006 (1)

C. Bauer, P. Geiser, J. Burgmeier, G. Holl, and W. Schade, “Pulsed laser surface fragmentation and mid-infrared laser spectroscopy for remote detection of explosives,” Appl. Phys. B 85, 251–256 (2006).
[Crossref]

2005 (1)

D. C. Fernandez, R. Bhargava, S. M. Hewitt, and I. W. Levin, “Infrared spectroscopic imaging for histopathologic recognition,” Nature Biotech. 23, 469–474 (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,” Sci. 306, 666–669 (2004).
[Crossref]

2003 (1)

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003).
[Crossref]

2002 (1)

R. D. Shannon, “Refractive index and dispersion of fluorides and oxides,” Phys. Chem. Ref. Data 31, 931 (2002).
[Crossref]

AbdollahRamezani, S.

Ajayan, P. M.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14, 299–304 (2013).
[Crossref]

W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13, 3698–3702 (2013).
[Crossref]

Alaee, R.

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Alu, A.

P.-Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[Crossref]

Arik, K.

Aussenegg, F. R.

W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003).
[Crossref]

Avouris, P.

D. B. Farmer, D. Rodrigo, T. Low, and P. Avouris, “Plasmon–plasmon hybridization and bandwidth enhancement in nanostructured graphene,” Nano Lett. 15, 2582–2587 (2015).
[Crossref]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Baeck, M. L.

A. P. M. Michel, P. Q. Liu, J. K. Yeung, P. Corrigan, M. L. Baeck, Z. Wang, T. Day, F. Moshary, C. F. Gmachl, and J. A. Smith, “Quantum cascade laser open-path system for remote sensing of trace gases in Beijing, China,” Opt. Eng. 49, 111125 (2015).
[Crossref]

Bauer, C.

C. Bauer, P. Geiser, J. Burgmeier, G. Holl, and W. Schade, “Pulsed laser surface fragmentation and mid-infrared laser spectroscopy for remote detection of explosives,” Appl. Phys. B 85, 251–256 (2006).
[Crossref]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6, 630–634 (2011).
[Crossref]

Bhargava, R.

D. C. Fernandez, R. Bhargava, S. M. Hewitt, and I. W. Levin, “Infrared spectroscopic imaging for histopathologic recognition,” Nature Biotech. 23, 469–474 (2005).
[Crossref]

Bischoff, D.

P. Simonet, D. Bischoff, A. Moser, T. Ihn, and K. Ensslin, “Graphene nanoribbons: relevance of etching process,” J. Appl. Phys. 117, 184303 (2015).
[Crossref]

Bludov, Y. V.

N. M. R. Peres, Y. V. Bludov, A. Ferreira, and M. I. Vasilevskiy, “Exact solution for square-wave grating covered with graphene: surface plasmon-polaritons in the terahertz range,” J. Phys. 25, 125303 (2013).

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

N. M. R. Peres, A. Ferreira, Y. V. Bludov, and M. I. Vasilevskiy, “Light scattering by a medium with a spatially modulated optical conductivity: the case of graphene,” J. Phys. 4, 245303 (2012).
[Crossref]

Bode, M.

P. Sessi, J. R. Guest, M. Bode, and N. P. Guisinger, “Patterning graphene at the nanometer scale via hydrogen desorption,” Nano Lett. 9, 4343–4347 (2009).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
[Crossref]

Buffeteau, T.

T. Buffeteau, J. Grondin, Y. Danten, J.-C. Lasse, and J. C. Lassègues, “Imidazolium-based ionic liquids: quantitative aspects in the far-infrared region,” J. Phys. Chem. B 114, 7587–7592 (2010).
[Crossref]

Buljan, H.

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

Burgmeier, J.

C. Bauer, P. Geiser, J. Burgmeier, G. Holl, and W. Schade, “Pulsed laser surface fragmentation and mid-infrared laser spectroscopy for remote detection of explosives,” Appl. Phys. B 85, 251–256 (2006).
[Crossref]

Camara, N.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Centeno, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Chen, J.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Chen, P.-Y.

P.-Y. Chen and A. Alu, “Atomically thin surface cloak using graphene monolayers,” ACS Nano 5, 5855–5863 (2011).
[Crossref]

Corrigan, P.

A. P. M. Michel, P. Q. Liu, J. K. Yeung, P. Corrigan, M. L. Baeck, Z. Wang, T. Day, F. Moshary, C. F. Gmachl, and J. A. Smith, “Quantum cascade laser open-path system for remote sensing of trace gases in Beijing, China,” Opt. Eng. 49, 111125 (2015).
[Crossref]

Danten, Y.

T. Buffeteau, J. Grondin, Y. Danten, J.-C. Lasse, and J. C. Lassègues, “Imidazolium-based ionic liquids: quantitative aspects in the far-infrared region,” J. Phys. Chem. B 114, 7587–7592 (2010).
[Crossref]

Day, T.

A. P. M. Michel, P. Q. Liu, J. K. Yeung, P. Corrigan, M. L. Baeck, Z. Wang, T. Day, F. Moshary, C. F. Gmachl, and J. A. Smith, “Quantum cascade laser open-path system for remote sensing of trace gases in Beijing, China,” Opt. Eng. 49, 111125 (2015).
[Crossref]

De, J. G.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

de Abajo, F. J. G.

Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14, 299–304 (2013).
[Crossref]

Dubonos, S. V.

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

Ensslin, K.

P. Simonet, D. Bischoff, A. Moser, T. Ihn, and K. Ensslin, “Graphene nanoribbons: relevance of etching process,” J. Appl. Phys. 117, 184303 (2015).
[Crossref]

Falkovsky, L.

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76, 153410 (2007).
[Crossref]

Falkovsky, L. A.

L. A. Falkovsky, “Optical properties of graphene,” J. Phys. 129, 012004 (2008).

L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56, 281–284 (2007).
[Crossref]

Fang, Z.

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A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
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J. Lee, M. J. Panzer, Y. He, T. P. Lodge, and C. D. Frisbie, “Ion gel gated polymer thin-film transistors,” J. Am. Chem. Soc. 129, 4532–4533 (2007).
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Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
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N. M. R. Peres, A. Ferreira, Y. V. Bludov, and M. I. Vasilevskiy, “Light scattering by a medium with a spatially modulated optical conductivity: the case of graphene,” J. Phys. 4, 245303 (2012).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
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W. Rechberger, A. Hohenau, A. Leitner, J. R. Krenn, B. Lamprecht, and F. R. Aussenegg, “Optical properties of two interacting gold nanoparticles,” Opt. Commun. 220, 137–141 (2003).
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Rodrigo, D.

D. B. Farmer, D. Rodrigo, T. Low, and P. Avouris, “Plasmon–plasmon hybridization and bandwidth enhancement in nanostructured graphene,” Nano Lett. 15, 2582–2587 (2015).
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Sato, S.

H. S. Song, S. L. Li, H. Miyazaki, S. Sato, K. Hayashi, A. Yamada, N. Yokoyama, and K. Tsukagoshi, “Origin of the relatively low transport mobility of graphene grown through chemical vapor deposition,” Sci. Rep. 2, 337 (2012).
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C. Bauer, P. Geiser, J. Burgmeier, G. Holl, and W. Schade, “Pulsed laser surface fragmentation and mid-infrared laser spectroscopy for remote detection of explosives,” Appl. Phys. B 85, 251–256 (2006).
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Z. Fang, Y. Wang, A. E. Schlather, Z. Liu, P. M. Ajayan, F. J. G. de Abajo, P. Nordlander, X. Zhu, and N. J. Halas, “Active tunable absorption enhancement with graphene nanodisk arrays,” Nano Lett. 14, 299–304 (2013).
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P. Sessi, J. R. Guest, M. Bode, and N. P. Guisinger, “Patterning graphene at the nanometer scale via hydrogen desorption,” Nano Lett. 9, 4343–4347 (2009).
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W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13, 3698–3702 (2013).
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W. Gao, G. Shi, Z. Jin, J. Shu, Q. Zhang, R. Vajtai, P. M. Ajayan, J. Kono, and Q. Xu, “Excitation and active control of propagating surface plasmon polaritons in graphene,” Nano Lett. 13, 3698–3702 (2013).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6, 7806–7813 (2012).
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P. Simonet, D. Bischoff, A. Moser, T. Ihn, and K. Ensslin, “Graphene nanoribbons: relevance of etching process,” J. Appl. Phys. 117, 184303 (2015).
[Crossref]

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T. M. Slipchenko, M. L. Nesterov, L. Martin-Moreno, and A. Nikitin, “Analytical solution for the diffraction of an electromagnetic wave by a graphene grating,” J. Opt. 15, 114008 (2013).
[Crossref]

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A. P. M. Michel, P. Q. Liu, J. K. Yeung, P. Corrigan, M. L. Baeck, Z. Wang, T. Day, F. Moshary, C. F. Gmachl, and J. A. Smith, “Quantum cascade laser open-path system for remote sensing of trace gases in Beijing, China,” Opt. Eng. 49, 111125 (2015).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

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H. S. Song, S. L. Li, H. Miyazaki, S. Sato, K. Hayashi, A. Yamada, N. Yokoyama, and K. Tsukagoshi, “Origin of the relatively low transport mobility of graphene grown through chemical vapor deposition,” Sci. Rep. 2, 337 (2012).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovi, A. Centeno, P. Godignon, A. Zurutuza, N. Camara, J. G. De, R. Hillenbrand, and F. Koppens, “Optical nano-imaging of gate-tuneable graphene plasmons,” Nature 487, 77–81 (2012).

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, 611–622 (2010).
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Lumerical Solutions Inc., http://www.lumerical.com/tcad-products/fdtd/ .

The research materials supporting this publication can be publicly accessed at http://hdl.handle.net/10871/20870 .

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of graphene sheets with a periodically modulated conductivity with a period of L and a width of w . (a) Conductivity profile described with a single Fourier term. (b) Conductivity profile with many Fourier terms. Inset: conductivity as a function of the distance along the graphene surface, given by Eq. (4).

Fig. 2.
Fig. 2.

(a) Modeling results for a single Fourier term (blue dashed line), three Fourier terms (red dotted–dashed line), 10 Fourier terms (black solid line), and the FDTD comparison in cyan circles. (b)–(d) Real part of the in-plane electric field ( E z ) normalized to the incident field at 22, 27, and 34 THz as labeled on (a).

Fig. 3.
Fig. 3.

Transmission as a function of frequency and the difference in Fermi level for the high- and low-conductivity regions. The structure is an array of modulated high- and low-conductivity regions with a period of 200 nm. The high-conductivity region has a width of 100 nm. Dashed line is the position of the dipole resonance of the equivalent high region given by Eq. (8).

Fig. 4.
Fig. 4.

(a) Transmission through a single layer as a function of frequency and width of the high-conductivity level region. The periodicity of the structure is 200 nm and the high-conductivity region has a Fermi level of 0.3 eV. The position of the dipole frequency for the high-conductivity region (solid line) and the low-conductivity region (dashed line) is shown according to Eq. (8). (b) Real part of the in-plane electric field, normalized to the incident field, at the crossover between the high Fermi level region and low Fermi level region, from (a).

Fig. 5.
Fig. 5.

(a) Transmission through a double layer of graphene with modulated conductivity regions as a function of the frequency and the interlayer separation. The width of the high-conductivity region is 100 nm in a period of 200 nm and has a Fermi level of 0.3 eV. The minima (1) and maxima (2) in the modulation in the transmission are indicated. Inset: transmission through the equivalent single layer. (b) Real part of the in-plane electric field, normalized to the incident field, for 100 nm conductivity region of 0.3 eV, separated by 30 nm. The regions of high and low conductivity are indicated.

Fig. 6.
Fig. 6.

Transmission through a double-layer structure as a function of frequency and the width of the high Fermi level region. The interlayer separation is 30 nm. The dipole and quadrupole frequency for the region of high-conductivity region (solid line) and the dipole, quadrupole and hexapole frequency for the low-conductivity region (dashed line) are shown according to Eq. (8).

Equations (8)

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

B ( α ) , y = A exp i x k 0 + m F ( α ) , m exp i m L z exp i q ( α ) , m x ,
E ( α ) , x = C exp i x k 0 + m G ( α ) , m exp i m L z exp i q ( α ) , m x ,
E ( α ) , z = D exp i x k 0 + m H ( α ) , m exp i m L z exp i q ( α ) , m x ,
σ ( z ) = σ d { w L + p 2 p π [ sin 2 p π w L cos 2 p π z L + sin 2 p π z L ( 1 cos 2 p π w L ) ] } ,
σ d ( ω ) = i e 2 | μ | π ( ω + i τ 1 ) .
( ϵ 1 q ( 1 ) , 0 + ϵ 2 q ( 2 ) , 0 ) ω H ( 2 ) , 0 + 1 ϵ 1 p σ p H ( 2 ) , p = 2 ω ϵ 1 k 0 D ,
( ϵ 1 q ( 1 ) , m + ϵ 2 q ( 2 ) , m ) ω H ( 2 ) , m + 1 ϵ 1 p σ p H ( 2 ) , p = 0 ,
ω 0 = η e 2 E f 2 ϵ avg ϵ 0 w ,

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