Abstract

Channeling surface plasmon–polaritons to control their propagation direction is of the utmost importance for future optoelectronic devices. Here, we develop an effective-index method to describe and characterize the properties of a 2D material’s channel plasmon–polaritons (CPPs) guided along a V-shaped channel. Focusing on the case of graphene, we derive a universal Schrödinger-like equation from which one can determine the dispersion relation of graphene CPPs and corresponding field distributions at any given frequency, since they depend on the geometry of the structure alone. The results are then compared against more rigorous theories, having obtained very good agreement. Our calculations show that CPPs in graphene and other 2D materials are attractive candidates to achieve deep subwavelength waveguiding of light, holding potential as active components for the next generation of tunable photonic devices.

© 2017 Optical Society of America

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References

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  41. In fact, more rigorously, for w(z)/2[Q2(z)−ϵk02]1/2≪1.
  42. We remark that if the dielectric material within the V-shaped region differs from the cladding dielectric media, it is not possible to derive a simple, closed-form expression for Q2(z). For this reason, here we consider only the case of a homogeneous dielectric environment.
  43. In writing this equation we have explicitly approximated the width (for small angles) as w(z)≈θz, for the sake of clarity. The expression using the exact form is recoverable upon the replacement θ→2 tan(θ/2) (we have used the exact expression in all our results).
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  47. E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110, 153602 (2013).
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  48. S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B 88, 115401 (2013).
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    [Crossref]

2016 (6)

O. Lotan, C. L. C. Smith, J. Bar-David, N. A. Mortensen, A. Kristensen, and U. Levy, “Propagation of channel plasmons at the visible regime in aluminum V-groove waveguides,” ACS Photon. 3, 2150–2157 (2016).
[Crossref]

S. Xiao, X. Zhu, B.-H. Li, and N. A. Mortensen, “Graphene-plasmon polaritons: from fundamental properties to potential applications,” Front. Phys. 11, 117801 (2016).
[Crossref]

W. Wang, B.-H. Li, E. Stassen, N. A. Mortensen, and J. Christensen, “Localized surface plasmons in vibrating graphene nanodisks,” Nanoscale 8, 3809–3815 (2016).
[Crossref]

P. A. D. Gonçalves, E. J. C. Dias, S. Xiao, M. I. Vasilevskiy, N. A. Mortensen, and N. M. R. Peres, “Graphene plasmons in triangular wedges and grooves,” ACS Photon. 3, 2176–2183 (2016).
[Crossref]

D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, “Trapping and guiding surface plasmons in curved graphene landscapes,” ACS Photon. 3, 875–880 (2016).
[Crossref]

X. Chen, N. C. Lindquist, D. J. Klemme, P. Nagpal, D. J. Norris, and S.-H. Oh, “Split-wedge antennas with sub-5 nm gaps for plasmonic nanofocusing,” Nano Lett. 16, 7849–7856 (2016).
[Crossref]

2015 (5)

T. Christensen, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Localized plasmons in graphene-coated nanospheres,” Phys. Rev. B 91, 125414 (2015).
[Crossref]

M. Riso, M. Cuevas, and R. A. Depine, “Tunable plasmonic enhancement of light scattering and absorption in graphene-coated subwavelength wires,” J. Opt. 17, 075001 (2015).
[Crossref]

E. Bermúdez-Ureña, C. Gonzalez-Ballestero, M. Geiselmann, R. Marty, I. P. Radko, T. Holmgaard, Y. Alaverdyan, E. Moreno, F. J. García-Vidal, S. I. Bozhevolnyi, and R. Quidant, “Coupling of individual quantum emitters to channel plasmons,” Nat. Commun. 6, 7883 (2015).
[Crossref]

C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, L. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, “All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9, 525–528 (2015).
[Crossref]

C. L. C. Smith, N. Stenger, A. Kristensen, N. A. Mortensen, and S. I. Bozhevolnyi, “Gap and channeled plasmons in tapered grooves: a review,” Nanoscale 7, 9355–9386 (2015).
[Crossref]

2014 (5)

S. Raza, N. Stenger, A. Pors, T. Holmgaard, S. Kadkhodazadeh, J. B. Wagner, K. Pedersen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Extremely confined gap surface-plasmon modes excited by electrons,” Nat. Commun. 5, 4125 (2014).
[Crossref]

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon-phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

A. Kumar, K. H. Fung, M. T. H. Reid, and N. X. Fang, “Transformation optics scheme for two-dimensional materials,” Opt. Lett. 39, 2113–2116 (2014).
[Crossref]

F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1, 135–152 (2014).
[Crossref]

F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photonics 8, 899–907 (2014).
[Crossref]

2013 (7)

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

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013).
[Crossref]

A. K. Geim and I. V. Grigorieva, “Van der Waals heterostructures,” Nature 499, 419–425 (2013).
[Crossref]

P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express 21, 32432–32440 (2013).
[Crossref]

X. Li, T. Jiang, L. Shen, and X. Deng, “Subwavelength guiding of channel plasmon polaritons by textured metallic grooves at telecom wavelengths,” Appl. Phys. Lett. 102, 031606 (2013).
[Crossref]

E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110, 153602 (2013).
[Crossref]

S. Raza, T. Christensen, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal response in thin-film waveguides: loss versus nonlocality and breaking of complementarity,” Phys. Rev. B 88, 115401 (2013).
[Crossref]

2012 (2)

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Commun. 3, 969 (2012).
[Crossref]

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[Crossref]

2011 (2)

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]

S. Lee and S. Kim, “Long-range channel plasmon polaritons in thin metal film V-grooves,” Opt. Express 19, 9836–9847 (2011).
[Crossref]

2010 (2)

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[Crossref]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[Crossref]

2009 (3)

2008 (2)

2006 (4)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref]

S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14, 9467–9476 (2006).
[Crossref]

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31, 3447–3449 (2006).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

2005 (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref]

2004 (2)

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85, 6323–6325 (2004).
[Crossref]

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069–1071 (2004).
[Crossref]

2003 (1)

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

Ajayan, P. M.

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013).
[Crossref]

Alaverdyan, Y.

E. Bermúdez-Ureña, C. Gonzalez-Ballestero, M. Geiselmann, R. Marty, I. P. Radko, T. Holmgaard, Y. Alaverdyan, E. Moreno, F. J. García-Vidal, S. I. Bozhevolnyi, and R. Quidant, “Coupling of individual quantum emitters to channel plasmons,” Nat. Commun. 6, 7883 (2015).
[Crossref]

Amooghorban, E.

E. Amooghorban, N. A. Mortensen, and M. Wubs, “Quantum optical effective-medium theory for loss-compensated metamaterials,” Phys. Rev. Lett. 110, 153602 (2013).
[Crossref]

Avouris, P.

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

H. Yan, F. Xia, Z. Li, and P. Avouris, “Plasmonics of coupled graphene micro-structures,” New J. Phys. 14, 125001 (2012).
[Crossref]

Baeuerle, B.

C. Haffner, W. Heni, Y. Fedoryshyn, J. Niegemann, A. Melikyan, D. L. Elder, B. Baeuerle, Y. Salamin, A. Josten, U. Koch, C. Hoessbacher, F. Ducry, L. Juchli, A. Emboras, D. Hillerkuss, M. Kohl, L. R. Dalton, C. Hafner, and J. Leuthold, “All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9, 525–528 (2015).
[Crossref]

Bar-David, J.

O. Lotan, C. L. C. Smith, J. Bar-David, N. A. Mortensen, A. Kristensen, and U. Levy, “Propagation of channel plasmons at the visible regime in aluminum V-groove waveguides,” ACS Photon. 3, 2150–2157 (2016).
[Crossref]

Barnes, W. L.

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

A. A. Maradudin, W. L. Barnes, and J. R. Sambles, Modern Plasmonics (Elsevier, 2014).

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]

Beermann, J.

T. Søndergaard, S. M. Novikov, T. Holmgaard, R. L. Eriksen, J. Beermann, Z. Han, K. Pedersen, and S. I. Bozhevolnyi, “Plasmonic black gold by adiabatic nanofocusing and absorption of light in ultra-sharp convex grooves,” Nat. Commun. 3, 969 (2012).
[Crossref]

Bermúdez-Ureña, E.

E. Bermúdez-Ureña, C. Gonzalez-Ballestero, M. Geiselmann, R. Marty, I. P. Radko, T. Holmgaard, Y. Alaverdyan, E. Moreno, F. J. García-Vidal, S. I. Bozhevolnyi, and R. Quidant, “Coupling of individual quantum emitters to channel plasmons,” Nat. Commun. 6, 7883 (2015).
[Crossref]

Bøggild, P.

X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon-phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
[Crossref]

Boltasseva, A.

Bozhevolnyi, S. I.

C. L. C. Smith, N. Stenger, A. Kristensen, N. A. Mortensen, and S. I. Bozhevolnyi, “Gap and channeled plasmons in tapered grooves: a review,” Nanoscale 7, 9355–9386 (2015).
[Crossref]

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[Crossref]

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W. Wang, B.-H. Li, E. Stassen, N. A. Mortensen, and J. Christensen, “Localized surface plasmons in vibrating graphene nanodisks,” Nanoscale 8, 3809–3815 (2016).
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C. L. C. Smith, N. Stenger, A. Kristensen, N. A. Mortensen, and S. I. Bozhevolnyi, “Gap and channeled plasmons in tapered grooves: a review,” Nanoscale 7, 9355–9386 (2015).
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Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388–2395 (2013).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
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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).
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F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photonics 8, 899–907 (2014).
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W. Wang, B.-H. Li, E. Stassen, N. A. Mortensen, and J. Christensen, “Localized surface plasmons in vibrating graphene nanodisks,” Nanoscale 8, 3809–3815 (2016).
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X. Zhu, W. Wang, W. Yan, M. B. Larsen, P. Bøggild, T. G. Pedersen, S. Xiao, J. Zi, and N. A. Mortensen, “Plasmon-phonon coupling in large-area graphene dot and antidot arrays fabricated by nanosphere lithography,” Nano Lett. 14, 2907–2913 (2014).
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In fact, more rigorously, for w(z)/2[Q2(z)−ϵk02]1/2≪1.

We remark that if the dielectric material within the V-shaped region differs from the cladding dielectric media, it is not possible to derive a simple, closed-form expression for Q2(z). For this reason, here we consider only the case of a homogeneous dielectric environment.

In writing this equation we have explicitly approximated the width (for small angles) as w(z)≈θz, for the sake of clarity. The expression using the exact form is recoverable upon the replacement θ→2 tan(θ/2) (we have used the exact expression in all our results).

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Supplementary Material (1)

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» Supplement 1: PDF (1962 KB)      Supplementary material

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Figures (5)

Fig. 1.
Fig. 1.

V-shaped groove carved in a dielectric substrate of relative permittivity ε onto which a graphene sheet was deposited. After the deposition of graphene, the channel is then filled with the same dielectric material as the underlying substrate. The channel width as a function of the z coordinate follows w(z)=2ztan(θ/2) for triangular cross sections.

Fig. 2.
Fig. 2.

Illustration of the field distributions along the height of the channel, Zn(ζ), akin to the first three 2D CPP modes in a triangular structure with θ=15°. We depict the eigenfunctions in an energy diagram along with the potential V(ζ) (black line), as in typical quantum mechanical problems. The vertical axis for each Zn (in a.u.) starts at the position of the corresponding eigenvalue, Eθ(n).

Fig. 3.
Fig. 3.

Two-dimensional distributions of the electric field magnitude, |Ex(n)(x,z)|, for CPPs in several triangular geometries: (a), (c), (d) fundamental and (b) second-order guided plasmonic modes. The resonant frequencies and opening angles are indicated in each panel. The field is depicted only in the inner region. Parameters: EF=0.5  eV and ε=2.1.

Fig. 4.
Fig. 4.

Dispersion relation of guided graphene CPPs sustained in a V-shaped graphene configuration embedded in homogeneous dielectric environments with different dielectric constants ϵ, for two different opening angles; see figure’s labels. As an eye-guide, the solid black lines indicate the dispersion of GPs in a corresponding flat interface. We assume a Fermi energy of EF=0.5  eV in the calculations. For comparison, the dashed lines show the dispersion relation of graphene CPPs obtained using a more rigorous theory [36].

Fig. 5.
Fig. 5.

Universal description of 2D CPPs, namely Eθq/q0 as a function of the angle θ (EIM). The results obtained using a rigorous theory [36] (full) are also plotted for comparison. The dashed lines indicate the point where the EIM model seems to surpass its regime of validity. The EIM curves follow a simple analytical expression of the form an+bnθ1 with an={0.33,0.36} and bn={28.7,14.1} (with n denoting the mode order); see Supplement 1.

Equations (7)

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

Ex(r)=X(x,z)Z(z)eiqy,
2X(x,z)x2+[ϵk02Q2(z)]X(x,z)=0,
2Z(z)z2+[Q2(z)q2]Z(z)=0
1+coth[w(z)2κQ]+iσ2D(ω)ωϵϵ0κQ=0
Q2(z)=ϵk02+ϵ22fσ2[1+4fσϵw(z)+1+8fσϵw(z)],
θ22Z(ζ)ζ2+V(ζ)Z(ζ)=EθZ(ζ)
V(ζ)=8+ζ+ζ2+16ζ8ζ,

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