Abstract

The phase of graphene plasmon upon edge-reflection plays a crucial role on determining the spectral properties of graphene structures. In this article, by using the full-wave simulation, we demonstrate that the mid-infrared graphene plasmons are nearly totally reflected at the boundary together with a phase jump of approximately 0.27π, regardless of the environments surrounding it. Appling this phase pickup, a Fabry-Perot model is formulated that can predict accurately the resonant wavelengths of graphene nano-ribbons. Furthermore, we find that the magnitude of the phase jump will either increase or reduce when two neighboring coplanar graphene sheets couple with each other. This could be used to explain the red-shift of resonant wavelength of periodic ribbon arrays with respect to an isolated ribbon. We provide a straightforward way to uncover the phase jump of graphene plasmons that would be helpful for designing and engineering graphene resonators and waveguides as well as their associated applications.

© 2014 Optical Society of America

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References

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2014 (2)

2013 (8)

M. Jablan, M. Soljacic, and H. Buljan, “Plasmons in graphene: fundamental properties and potential applications,” Proc. IEEE 101(7), 1689–1704 (2013).
[Crossref]

J. H. Strait, P. Nene, W. M. Chan, C. Manolatou, S. Tiwari, F. Rana, J. W. Kevek, and P. L. McEuen, “Confined plasmons in graphene microstructures: experiments and theory,” Phys. Rev. B 87(24), 241410 (2013).
[Crossref]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

L. Wang, W. Cai, X. Z. Zhang, P. H. Liu, Y. X. Xiang, and J. J. Xu, “Mid-infrared optical near-field switching in heterogeneous graphene ribbon pairs,” Appl. Phys. Lett. 103(4), 041604 (2013).
[Crossref]

X. L. Zhu, W. Yan, N. A. Mortensen, and S. S. Xiao, “Bends and splitters in graphene nanoribbon waveguides,” Opt. Express 21(3), 3486–3491 (2013).
[Crossref] [PubMed]

H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, M. Freitag, X. S. Li, F. Guinea, P. Avouris, and F. N. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

R. Wang, L. Du, C. Zhang, Z. Man, Y. Wang, S. Wei, C. Min, S. Zhu, and X. C. Yuan, “Plasmonic petal-shaped beam for microscopic phase-sensitive SPR biosensor with ultrahigh sensitivity,” Opt. Lett. 38(22), 4770–4773 (2013).
[Crossref] [PubMed]

H. X. Da, Q. L. Bao, R. Sanaei, J. H. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

2012 (12)

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

V. J. Sorger, R. F. Oulton, R. M. Ma, and X. Zhang, “Toward integrated plasmonic circuits,” MRS Bull. 37(08), 728–738 (2012).
[Crossref]

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490(7419), 192–200 (2012).
[Crossref] [PubMed]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

Q. L. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, F. J. García-Vidal, X. C. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. 109(7), 073901 (2012).
[Crossref] [PubMed]

B. Wang, X. Zhang, X. C. Yuan, and J. H. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

H. Ju Xu, W. Bing Lu, W. Zhu, Z. Gao Dong, and T. Jun Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).
[Crossref]

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85(8), 085416 (2012).
[Crossref]

2011 (5)

J. T. Kim and S. Y. Choi, “Graphene-based plasmonic waveguides for photonic integrated circuits,” Opt. Express 19(24), 24557–24562 (2011).
[Crossref] [PubMed]

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[Crossref]

F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

L. P. Du, G. H. Yuan, D. Y. Tang, and X. C. Yuan, “Tightly focused radially polarized beam for propagating surface plasmon-assisted gap-mode Raman spectroscopy,” Plasmonics 6(4), 651–657 (2011).
[Crossref]

2009 (3)

M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

J. Dorfmüller, R. Vogelgesang, R. T. Weitz, C. Rockstuhl, C. Etrich, T. Pertsch, F. Lederer, and K. Kern, “Fabry-Pérot resonances in one-dimensional plasmonic nanostructures,” Nano Lett. 9(6), 2372–2377 (2009).
[Crossref] [PubMed]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. H. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79(3), 035120 (2009).
[Crossref]

2008 (3)

2007 (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref] [PubMed]

2006 (1)

R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73(15), 153405 (2006).
[Crossref]

2005 (1)

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

2004 (1)

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120(1), 357–366 (2004).
[Crossref] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B Chem. 54(1-2), 3–15 (1999).
[Crossref]

1997 (2)

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[Crossref] [PubMed]

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997).
[Crossref]

1995 (1)

Alonso-González, P.

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Atwater, H. A.

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

Avouris, P.

H. G. Yan, T. Low, W. J. Zhu, Y. Q. Wu, M. Freitag, X. S. Li, F. Guinea, P. Avouris, and F. N. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012).
[Crossref] [PubMed]

Badioli, M.

J. N. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Bao, Q. L.

H. X. Da, Q. L. Bao, R. Sanaei, J. H. Teng, K. P. Loh, F. J. Garcia-Vidal, and C. W. Qiu, “Monolayer graphene photonic metastructures: giant Faraday rotation and nearly perfect transmission,” Phys. Rev. B 88(20), 205405 (2013).
[Crossref]

Q. L. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS Nano 6(5), 3677–3694 (2012).
[Crossref] [PubMed]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Barnard, E. S.

A. Chandran, E. S. Barnard, J. S. White, and M. L. Brongersma, “Metal-dielectric-metal surface plasmon-polariton resonators,” Phys. Rev. B 85(8), 085416 (2012).
[Crossref]

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectral properties of plasmonic resonator antennas,” Opt. Express 16(21), 16529–16537 (2008).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[PubMed]

Bechtel, H. A.

L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[Crossref] [PubMed]

Beermann, J.

T. Sondergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant-nanostrip antennas: analysis and demonstration,” Phys. Rev. B 77(11), 153405 (2008).
[Crossref]

Bing Lu, W.

H. Ju Xu, W. Bing Lu, W. Zhu, Z. Gao Dong, and T. Jun Cui, “Efficient manipulation of surface plasmon polariton waves in graphene,” Appl. Phys. Lett. 100(24), 243110 (2012).
[Crossref]

Boltasseva, A.

T. Sondergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant-nanostrip antennas: analysis and demonstration,” Phys. Rev. B 77(11), 153405 (2008).
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R. Wang, L. Du, C. Zhang, Z. Man, Y. Wang, S. Wei, C. Min, S. Zhu, and X. C. Yuan, “Plasmonic petal-shaped beam for microscopic phase-sensitive SPR biosensor with ultrahigh sensitivity,” Opt. Lett. 38(22), 4770–4773 (2013).
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ACS Nano (2)

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic diagram illustrating the edge reflection of graphene surface plasmon, assuming a reflection co-efficient of |r|·exp(iΔφ). (b) and (c) |Ez| distributions in the vicinity of graphene without and with a boundary. The presence of a boundary will give rise to a self-interference pattern, with the positions of the fringes solely determined by the reflection phase of graphene plasmon. In the simulation, the refractive indices above and below graphene are set to be 1 and 1.5, respectively.
Fig. 2
Fig. 2 (a) The effective mode indices of graphene plasmon plotting against the incident wavenumber at various substrate permittivities ε2. The open scatterers are the values obtained from the self-interference patterns and the solid lines are the theoretical curves. ε1 is set to be 1 in all of the simulations and calculations. (b) The reflection amplitudes and phases at the mid-infrared region extracted from the self-interference patterns (open scatterers), along with the spectral curves from an analytical estimation (dashed lines). The amplitude is shown only for the case of ε1 = ε2 = 1.
Fig. 3
Fig. 3 Plasmonic resonance of an isolated graphene nano-ribbon. (a) Schematic diagram illustrating the excitation of plasmon modes on an isolated graphene ribbon with normal incidence. (b) Resonance wavelengths of the fundamental modes of isolated graphene ribbons from FDTD (open scatterers) as compared with the results from F-P model, assuming a reflection phase of 0.27π (solid lines). (c) Resonance wavelengths of the higher order plasmon modes and the comparison.
Fig. 4
Fig. 4 (a) The absorption contour pattern of periodic ribbon structures at the mid-infrared region, with ribbon width ranging from 100 nm to 500 nm. The width-to-period ratio is fixed at 1/2 in the simulation. (b) Resonance curves as calculated with the F-P formula at the same spectrum region and width range as in (a), assuming a constant reflection phase of 0.27π.
Fig. 5
Fig. 5 (a) Schematic diagram illustrating the modeling of edge-reflection phase under a pair of neighboring coplanar graphene sheets with separation of d. A pair of line dipole sources located symmetrically with respect to the slit center is employed to launch the SPs separately at each of graphene sheets. (b) The splitting of phase results from the coupling between the two graphene sheets.

Equations (10)

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β = ε 0 ( ε 1 + ε 2 ) 2 2 i ω 0 σ g
σ g = e 2 E f π 2 i ω 0 + i τ 1
E z = E z , i + E z , r = E 0 e - i β ( x - x 0 ) e k z z + | r | e i Δ φ E 0 e i β ( x + x 0 ) e k z z
n G S P = β r e a l k 0 = ε 0 ( ε 1 + ε 2 ) 2 2 π 2 c 2 e 2 E f k 0
r = n ' 1 n ' + 1
n ' = π 2 [ 1 1 u 2 ( u n G S P u 2 + n G S P 2 ) 2 d u ] 1
E z = E z l e f t + E z r i g h t = 2 i E 0 e i β w / 2 sin β x 1 + | r | e i ( Δ φ + β w )
β r w + Δ φ = m π m = 1 , 2 , 3 , ...
ω 0 = e 2 E f ε 0 2 w m Δ φ ' ( ε 1 + ε 2 )
E z = E z l e f t + E z r i g h t = 2 E 0 e i β w / 2 cos β x 1 r e i ( Δ φ + β w )

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