Abstract

We show that the interaction between localized surface plasmons sustained by a metallic nano-antenna and delocalized phonons lying at the surface of an heteropolar semiconductor can generate a new class of hybrid electromagnetic modes. These plasphonic modes are investigated using an analytical model completed by accurate Green dyadic numerical simulations. When surface plasmon and surface phonon frequencies match, the optical resonances exhibit a large Rabi splitting typical of strongly interacting two-level systems. Based on numerical simulations of the electric near-field maps, we investigate the nature of the plaphonic excitations. In particular, we point out a strong local field enhancement boosted by the phononic surface. This effect is interpreted in terms of light harvesting by the plasmonic antenna from the phononic surface. We thus introduce the concept of active phononic surfaces that may be exploited for far-infared optoelectronic devices and sensors.

© 2013 OSA

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  4. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
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  5. R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
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  7. F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
    [CrossRef] [PubMed]
  8. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
    [CrossRef] [PubMed]
  9. G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
    [CrossRef] [PubMed]
  10. N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
    [CrossRef] [PubMed]
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  27. L. Novotny, “Strong coupling, energy splitting, and level crossings: a classical perspective,” Am. J. Phys.78, 1199–1202 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  32. E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
    [CrossRef]

2011 (2)

A. Manjavacas, F. G. de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011).
[CrossRef] [PubMed]

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

2010 (6)

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

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

H. C. Kim and X. Cheng, “Infrared dipole antenna enhanced by surface phonon polaritons,” Opt. Lett.35, 3748–3750 (2010).
[CrossRef] [PubMed]

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010).
[CrossRef] [PubMed]

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

2008 (2)

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

2007 (3)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[CrossRef]

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007).
[CrossRef] [PubMed]

2006 (3)

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

2005 (4)

C. Girard, “Near field in nanostructures,” Rep. Prog. Phys.681883–1933 (2005)
[CrossRef]

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

M. S. Anderson, “Surface enhanced infrared absorption by coupling phonon and plasmon resonance,” Appl. Phys. Lett.87, 144102 (2005).
[CrossRef]

D. Lockwood, G. Yu, and N. L. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun.136, 404–409 (2005).
[CrossRef]

2004 (1)

G. Yu, N. L. Rowell, and D. J. Lockwood, “Anisotropic infrared optical properties of GaN and sapphire,” J. Vac. Sci. Technol. A22, 1110–1114 (2004).
[CrossRef]

2002 (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
[CrossRef] [PubMed]

1998 (1)

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

1996 (1)

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

1995 (1)

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526 (1995).
[CrossRef] [PubMed]

1993 (1)

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci.280, 217–230 (1993).
[CrossRef]

1983 (1)

1966 (1)

A. Mooradian and G. B. Wright, “Observation of the interaction of plasmons with longitudinal optical phonons in GaAs,” Phys. Rev. Lett.16, 999–1001 (1966).
[CrossRef]

Abstreiter, M.

M. Abstreiter, G. Cardona, and A. Pinczuk, Light Scattering by Free Carrier Excitations in Semiconductors (Springer-Verlag, Berlin, 1984).

Adato, R.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Aizpurua, J.

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Aksu, S.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Alexander, J.

Altug, H.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Anderson, M. S.

M. S. Anderson, “Surface enhanced infrared absorption by coupling phonon and plasmon resonance,” Appl. Phys. Lett.87, 144102 (2005).
[CrossRef]

Artar, A.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Atkinson, R.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010).
[CrossRef] [PubMed]

Barchiesi, D.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

Barnes, W. L.

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Bell, R. J.

Bell, R. W.

Bell, S. E.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New YorkWiley-Interscience, 1983).

Borghese, F.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Bower, C.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Bozhevolnyi, S.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci.280, 217–230 (1993).
[CrossRef]

Brandl, D. W.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

Bustos, F.

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Camden, J. P.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Capasso, F.

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

Cardona, G.

M. Abstreiter, G. Cardona, and A. Pinczuk, Light Scattering by Free Carrier Excitations in Semiconductors (Springer-Verlag, Berlin, 1984).

Cheng, X.

Cornelius, T. W.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Courjon, D.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

Crozier, K. B.

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

Cubukcu, E.

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

de Abajo, F. G.

A. Manjavacas, F. G. de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011).
[CrossRef] [PubMed]

Denti, P.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Dereux, A.

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526 (1995).
[CrossRef] [PubMed]

Di Stefano, O.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Dickson, W.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Dintinger, J.

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Ebbesen, T. W.

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Esser, N.

N. Esser and W. Richter, Raman Scattering from Surface Phonons (Springer Berlin / Heidelberg, 96–168, 2000).

Evans, P. R.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Fofang, N. T.

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

Garcia-Etxarri, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Girard, C.

C. Girard, “Near field in nanostructures,” Rep. Prog. Phys.681883–1933 (2005)
[CrossRef]

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526 (1995).
[CrossRef] [PubMed]

Grabowski, S.

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

Guiton, B. S.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Halas, N. J.

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[CrossRef]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

Harrison, W.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. PressNew York, 2006).
[CrossRef]

Hendren, W.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Hillenbrand, R.

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
[CrossRef] [PubMed]

Huang, M.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Huber, A.

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New YorkWiley-Interscience, 1983).

Iberi, V.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Kampen, T.

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

Karim, S.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Keilmann, F.

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
[CrossRef] [PubMed]

Keller, O.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci.280, 217–230 (1993).
[CrossRef]

Kim, H. C.

Klein, S.

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Kort, E. A.

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

Kotula, P. G.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[CrossRef]

Le, F.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

Leonard, D. N.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Li, S.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[CrossRef]

Lockwood, D.

D. Lockwood, G. Yu, and N. L. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun.136, 404–409 (2005).
[CrossRef]

Lockwood, D. J.

G. Yu, N. L. Rowell, and D. J. Lockwood, “Anisotropic infrared optical properties of GaN and sapphire,” J. Vac. Sci. Technol. A22, 1110–1114 (2004).
[CrossRef]

Long, L. L.

Manjavacas, A.

A. Manjavacas, F. G. de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011).
[CrossRef] [PubMed]

Martin, O. J. F.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526 (1995).
[CrossRef] [PubMed]

Mirin, N. A.

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

Monch, W.

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

Mooradian, A.

A. Mooradian and G. B. Wright, “Observation of the interaction of plasmons with longitudinal optical phonons in GaAs,” Phys. Rev. Lett.16, 999–1001 (1966).
[CrossRef]

Neubrech, F.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Neumann, O.

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

Nienhaus, H.

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

Nordlander, P.

A. Manjavacas, F. G. de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011).
[CrossRef] [PubMed]

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

Novotny, L.

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

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007).
[CrossRef] [PubMed]

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. PressNew York, 2006).
[CrossRef]

Ocelic, N.

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

Ordal, M. A.

Parish, C. M.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Park, T. H.

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

Pennycook, S. J.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Pinczuk, A.

M. Abstreiter, G. Cardona, and A. Pinczuk, Light Scattering by Free Carrier Excitations in Semiconductors (Springer-Verlag, Berlin, 1984).

Pollard, R. J.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010).
[CrossRef] [PubMed]

Pucci, A.

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

Reyes-Coronado, A.

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

Richter, W.

N. Esser and W. Richter, Raman Scattering from Surface Phonons (Springer Berlin / Heidelberg, 96–168, 2000).

Ridolfo, A.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Rowell, N. L.

D. Lockwood, G. Yu, and N. L. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun.136, 404–409 (2005).
[CrossRef]

G. Yu, N. L. Rowell, and D. J. Lockwood, “Anisotropic infrared optical properties of GaN and sapphire,” J. Vac. Sci. Technol. A22, 1110–1114 (2004).
[CrossRef]

Saija, R.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Savasta, S.

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

Schatz, G. C.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Taubner, T.

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
[CrossRef] [PubMed]

Van Labeke, D.

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

Varela, M.

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Wang, H.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

Ward, C. A.

Wei, H.

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

Wright, G. B.

A. Mooradian and G. B. Wright, “Observation of the interaction of plasmons with longitudinal optical phonons in GaAs,” Phys. Rev. Lett.16, 999–1001 (1966).
[CrossRef]

Wurtz, G. A.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

Xiao, M.

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci.280, 217–230 (1993).
[CrossRef]

Xu, H.

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

Yanik, A. A.

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

Yu, G.

D. Lockwood, G. Yu, and N. L. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun.136, 404–409 (2005).
[CrossRef]

G. Yu, N. L. Rowell, and D. J. Lockwood, “Anisotropic infrared optical properties of GaN and sapphire,” J. Vac. Sci. Technol. A22, 1110–1114 (2004).
[CrossRef]

Zayats, A. V.

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

ACS Nano (2)

S. Savasta, R. Saija, A. Ridolfo, O. Di Stefano, P. Denti, and F. Borghese, “Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna,” ACS Nano4, 6369–6376 (2010).
[CrossRef] [PubMed]

H. Wei, A. Reyes-Coronado, P. Nordlander, J. Aizpurua, and H. Xu, “Multipolar plasmon resonances in individual Ag nanorice,” ACS Nano4, 2649–2654 (2010).
[CrossRef] [PubMed]

Am. J. Phys. (1)

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

Appl. Opt. (1)

Appl. Phys. Lett. (2)

E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett.89, 093120 (2006).
[CrossRef]

M. S. Anderson, “Surface enhanced infrared absorption by coupling phonon and plasmon resonance,” Appl. Phys. Lett.87, 144102 (2005).
[CrossRef]

Appl. Surf. Sci. (1)

S. Grabowski, T. Kampen, H. Nienhaus, and W. Monch, “Vibrational properties of GaN(0001) surfaces,” Appl. Surf. Sci.123, 33–37 (1998).
[CrossRef]

J. Vac. Sci. Technol. A (1)

G. Yu, N. L. Rowell, and D. J. Lockwood, “Anisotropic infrared optical properties of GaN and sapphire,” J. Vac. Sci. Technol. A22, 1110–1114 (2004).
[CrossRef]

Nano Lett. (7)

A. Manjavacas, F. G. de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett.11, 2318–2323 (2011).
[CrossRef] [PubMed]

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: a hybrid plasmonic nanostructure,” Nano Lett.6, 827–832 (2006).
[CrossRef] [PubMed]

G. A. Wurtz, P. R. Evans, W. Hendren, R. Atkinson, W. Dickson, R. J. Pollard, A. V. Zayats, W. Harrison, and C. Bower, “Molecular plasmonics with tunable exciton-plasmon coupling strength in J-aggregate hybridized Au nanorod assemblies,” Nano Lett.7, 1297–1303 (2007).
[CrossRef] [PubMed]

N. T. Fofang, T. H. Park, O. Neumann, N. A. Mirin, P. Nordlander, and N. J. Halas, “Plexcitonic nanoparticles: plasmon-exciton coupling in nanoshell-J-aggregate complexes,” Nano Lett.8, 3481–3487 (2008).
[CrossRef] [PubMed]

S. Aksu, A. A. Yanik, R. Adato, A. Artar, M. Huang, and H. Altug, “High-throughput nanofabrication of infrared plasmonic nanoantenna arrays for vibrational nanospectroscopy,” Nano Lett.10, 2511–2518 (2010).
[CrossRef] [PubMed]

A. Huber, N. Ocelic, T. Taubner, and R. Hillenbrand, “Nanoscale resolved infrared probing of crystal structure and of plasmon-phonon coupling,” Nano Lett.6, 774–778 (2006).
[CrossRef] [PubMed]

B. S. Guiton, V. Iberi, S. Li, D. N. Leonard, C. M. Parish, P. G. Kotula, M. Varela, G. C. Schatz, S. J. Pennycook, and J. P. Camden, “Correlated optical measurements and plasmon mapping of silver nanorods,” Nano Lett.11, 3482–3488 (2011).
[CrossRef] [PubMed]

Nat. Mater. (1)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010).
[CrossRef] [PubMed]

Nat. Photonics (1)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007).
[CrossRef]

Nature (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature418, 159–162 (2002).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (1)

J. Dintinger, S. Klein, F. Bustos, W. L. Barnes, and T. W. Ebbesen, “Strong coupling between surface plasmon-polaritons and organic molecules in subwavelength hole arrays,” Phys. Rev. B71, 035424 (2005).
[CrossRef]

Phys. Rev. E (1)

D. Barchiesi, C. Girard, O. J. F. Martin, D. Van Labeke, and D. Courjon, “Computing the optical near-field distributions around complex subwavelength surface structures: a comparative study of different methods,” Phys. Rev. E544285–4292 (1996).
[CrossRef]

Phys. Rev. Lett. (4)

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett.98, 266802 (2007).
[CrossRef] [PubMed]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526 (1995).
[CrossRef] [PubMed]

F. Neubrech, A. Pucci, T. W. Cornelius, S. Karim, A. Garcia-Etxarri, and J. Aizpurua, “Resonant plasmonic and vibrational coupling in a tailored nanoantenna for infrared detection,” Phys. Rev. Lett.101, 157403 (2008).
[CrossRef] [PubMed]

A. Mooradian and G. B. Wright, “Observation of the interaction of plasmons with longitudinal optical phonons in GaAs,” Phys. Rev. Lett.16, 999–1001 (1966).
[CrossRef]

Rep. Prog. Phys. (1)

C. Girard, “Near field in nanostructures,” Rep. Prog. Phys.681883–1933 (2005)
[CrossRef]

Solid State Commun. (1)

D. Lockwood, G. Yu, and N. L. Rowell, “Optical phonon frequencies and damping in AlAs, GaP, GaAs, InP, InAs and InSb studied by oblique incidence infrared spectroscopy,” Solid State Commun.136, 404–409 (2005).
[CrossRef]

Surf. Sci. (1)

O. Keller, M. Xiao, and S. Bozhevolnyi, “Configurational resonances in optical near-field microscopy: a rigorous point-dipole approach,” Surf. Sci.280, 217–230 (1993).
[CrossRef]

Other (4)

M. Abstreiter, G. Cardona, and A. Pinczuk, Light Scattering by Free Carrier Excitations in Semiconductors (Springer-Verlag, Berlin, 1984).

N. Esser and W. Richter, Raman Scattering from Surface Phonons (Springer Berlin / Heidelberg, 96–168, 2000).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (New YorkWiley-Interscience, 1983).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. PressNew York, 2006).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the system studied in this work. A gold antenna (length L, width w=200 nm, height h=100 nm) is located in the vicinity of a polar semiconductor GaN surface. The system is illuminated at normal incidence by an incident electric field E0 oscillating at frequency ω and parallel to the long axis of the metallic antenna.

Fig. 2
Fig. 2

a. Resonance frequencies ω+ (upper branches) and ω (lower branches) obtained from the analytical model as a function of the dipole antenna frequency ω1 and for different dipole–to–surface separations Z. The dots are the results of the Green Dyadic Model (GDM) simulations performed for a gold antenna in contact with the GaN surface (Z=0). The antenna length has been changed from 3 to 6 μm (see color code). b. Analytical (full line) and GDM (dots) Rabi splitting Δ as a function of the antenna-to-surface separation. The dashed line is a fit of the GDM results (green dots) with the analytical model (Eq. (8)) using α0 as an adjustable parameter. c. Extinction spectrum simulated using the GDM for a 4.5 μm antenna located on a GaN substrate.

Fig. 3
Fig. 3

GDM electric near-field enhancement |E(r)/E0| maps simulated for a 4.5 μm gold antenna close to either a GaN surface (Figs. 3(a)-3(b)-3(d)) or a non dispersive (ε(ω) = ε) surface (Fig. 3(c)). In Figs. 3(a)-3(c)-3(d) the antenna is in contact with the surface whereas in Fig. 3(b) it is separated from the surface by a 20 nm air-gap. The maps are shown in planes parallel to the surface at 50 nm above the antenna (Figs. 3(a)-3(b)-3(c)) and 50 nm below the GaN surface (Fig. 3(d)). In Figs. 3(a)-3(b)-3(d) the frequency of the incident electric field is resonant with the Rabi-splitted frequencies ω+ = 800 cm−1, ω+ = 740 cm−1 and ω = 540 cm−1. In Fig. 3(c) |E(r)/E0| is mapped at the surface plasmon frequency ωSP(Z = 0) = 560 cm−1 in the case of a non polar substrate (δ =0). The sign, in Figs. 3(a)–3(d), of the surface polarisation charge at the field intensity maxima was determined from the GDM simulations.

Equations (9)

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

ε ( ω ) = ε ω L 2 ω 2 + i ω γ L ω T 2 ω 2 + i ω γ T
α ( ω ) = α 0 ω 1 2 ω 1 2 ω 2 + i ω γ 1
E ( R , ω ) = [ I α ( ω ) S ( R , ω ) ] 1 E 0 ( R , ω )
S ( R , ω ) = ε ( ω ) 1 ε ( ω ) + 1 1 R 3 ( 1 0 0 0 1 0 0 0 2 )
p ( R , ω ) = α ( ω ) 1 1 ( 2 R ) 3 ε ( ω ) 1 ε ( ω ) + 1 α ( ω ) E 0 ( R , ω )
{ 1 1 ( 2 R ) 3 ε ( ω ) 1 ε ( ω ) + 1 α ( ω ) } = 0
0 = ( ω 2 ω S P 2 ( Z ) ) ( ω 2 ω s p h 2 ) δ 2 ω 1 2 ε 2 ( ε + 1 ) 2 α 0 4 ( Z + h / 2 ) 3
Δ = ω + ω = δ ω 1 ω s p h ε ε + 1 α 0 1 / 2 2 ( Z + h / 2 ) 3 / 2
λ n eff ( l + 1 / 2 ) = L

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