Abstract

Nonlocal optical response is one of the emerging effects on the nanoscale for particles made of metals or doped semiconductors. Here we classify and compare both scalar and tensorial nonlocal response models. In the latter case the nonlocality can stem from either the longitudinal response, the transverse response, or both. In phenomenological scalar models the nonlocal response is described as a smearing out of the commonly assumed infinitely localized response, as characterized by a distribution with a finite width. Here we calculate explicitly whether and how tensorial models, such as the hydrodynamic Drude model and generalized nonlocal optical response theory, follow this phenomenological description. We find considerable differences, for example that nonlocal response functions, in contrast to simple distributions, assume negative and complex values. Moreover, nonlocal response regularizes some but not all diverging optical near fields. We identify the scalar model that comes closest to the hydrodynamic model. Interestingly, for the hydrodynamic Drude model we find that actually only one third (1/3) of the free-electron response is smeared out nonlocally. In that sense, nonlocal response is stronger for transverse and scalar nonlocal response models, where the smeared-out fractions are 2/3 and 3/3, respectively. The latter two models seem to predict novel plasmonic resonances also below the plasma frequency, in contrast to the hydrodynamic model that predicts standing pressure waves only above the plasma frequency.

© 2015 Optical Society of America

Full Article  |  PDF Article
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    [Crossref] [PubMed]
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  9. A. Moradi, “Infrared absorption spectra of a spatially dispersive polar semiconductor nanowire,” Solid State Commun. 212, 10–13 (2015).
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  10. T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
    [Crossref]
  11. D. Correas-Serrano, J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Nonlocal response of hyperbolic metasurfaces,” Opt. Express 23(23), 29434 (2015).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  27. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
    [Crossref]
  28. W. Yan, N. A. Mortensen, and M. Wubs, “Hyperbolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21(12), 15026 (2013).
    [Crossref] [PubMed]
  29. A. Moradi, Maxwell-Garnett effective medium theory: quantum nonlocal effects,” Phys. of Plasmas 22(4), 042105 (2015).
    [Crossref]
  30. A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
    [Crossref]
  31. F. Intravaia and K. Busch, “Fluorescence in nonlocal dissipative periodic structures,” Phys. Rev. A 91(5), 053836 (2015).
    [Crossref]
  32. P. Ginzburg and A. Zayats, “Localized surface plasmon resonances in spatially dispersive nano-objects: phenomenological treatise,” ACS Nano 7(5), 43344342 (2013).
    [Crossref] [PubMed]
  33. N. A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Phot. Nanostr. 11(4), 303 (2013).
    [Crossref]
  34. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447 (1998).
    [Crossref]
  35. D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics. An Introduction to Radiation Molecule Interactions (Academic, 1984); see Sec. 3.1: Transverse and longitudinal δ-dyadics.
  36. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms. Introduction to Quantum Electrodynamics (Wiley, 1989); see pp. 36–42.
  37. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103(9), 097403 (2009).
    [Crossref] [PubMed]
  38. J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Dansk Videnskab. Selskab, Mat. Fys. Medd. 28(8), 1–57 (1954).
  39. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
    [Crossref]

2015 (12)

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

X. Chen, J. E. Moore, M. Zekarias, and L. Jensen, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime,” Nature Commun. 6, 8921 (2015).
[Crossref]

S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

R. Carmina Monreal, T. J. Antosiewicz, and S. P. Apell, “Diffuse surface scattering in the plasmonic resonances of ultralow electron density nanospheres,” J. Phys. Chem. Lett. 6, 1847–1853 (2015).
[Crossref] [PubMed]

A. Moradi, “Infrared absorption spectra of a spatially dispersive polar semiconductor nanowire,” Solid State Commun. 212, 10–13 (2015).
[Crossref]

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

F. Intravaia and K. Busch, “Fluorescence in nonlocal dissipative periodic structures,” Phys. Rev. A 91(5), 053836 (2015).
[Crossref]

A. Moradi, Maxwell-Garnett effective medium theory: quantum nonlocal effects,” Phys. of Plasmas 22(4), 042105 (2015).
[Crossref]

W. Yan, M. Wubs, and N. A. Mortensen, “Projected-dipole model for quantum plasmonics,” Phys. Rev. Lett. 115(13), 137403 (2015).
[Crossref] [PubMed]

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

S. Raza, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal study of ultimate plasmon hybridization”, Opt. Lett. 40(5), 839–842 (2015).
[Crossref] [PubMed]

D. Correas-Serrano, J. S. Gomez-Diaz, M. Tymchenko, and A. Alù, “Nonlocal response of hyperbolic metasurfaces,” Opt. Express 23(23), 29434 (2015).
[Crossref]

2014 (4)

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
[Crossref]

H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
[Crossref]

2013 (6)

W. Yan, N. A. Mortensen, and M. Wubs, “Green’s function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments,” Phys. Rev. B 88(15), 155414 (2013).
[Crossref]

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
[Crossref]

P. Ginzburg and A. Zayats, “Localized surface plasmon resonances in spatially dispersive nano-objects: phenomenological treatise,” ACS Nano 7(5), 43344342 (2013).
[Crossref] [PubMed]

N. A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Phot. Nanostr. 11(4), 303 (2013).
[Crossref]

W. Yan, N. A. Mortensen, and M. Wubs, “Hyperbolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21(12), 15026 (2013).
[Crossref] [PubMed]

2012 (5)

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
[Crossref] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
[Crossref]

2011 (1)

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
[Crossref]

2009 (1)

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103(9), 097403 (2009).
[Crossref] [PubMed]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447 (1998).
[Crossref]

1989 (1)

1986 (1)

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34(6), 3883–3899 (1986).
[Crossref]

1971 (1)

I. Lindau and P. O. Nilsson, “Experimental verification of optically excited longitudinal plasmons,” Phys. Scr. 3, 87–92 (1971).
[Crossref]

1954 (1)

J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Dansk Videnskab. Selskab, Mat. Fys. Medd. 28(8), 1–57 (1954).

1933 (1)

F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Physik A 81, 363–376 (1933).
[Crossref]

Aizpurua, J.

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
[Crossref]

Alù, A.

Antosiewicz, T. J.

R. Carmina Monreal, T. J. Antosiewicz, and S. P. Apell, “Diffuse surface scattering in the plasmonic resonances of ultralow electron density nanospheres,” J. Phys. Chem. Lett. 6, 1847–1853 (2015).
[Crossref] [PubMed]

Apell, S. P.

R. Carmina Monreal, T. J. Antosiewicz, and S. P. Apell, “Diffuse surface scattering in the plasmonic resonances of ultralow electron density nanospheres,” J. Phys. Chem. Lett. 6, 1847–1853 (2015).
[Crossref] [PubMed]

Barbry, M.

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

Baumberg, J. J.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

Bloch, F.

F. Bloch, “Bremsvermögen von Atomen mit mehreren Elektronen,” Z. Physik A 81, 363–376 (1933).
[Crossref]

Borisov, A. G.

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
[Crossref]

Bozhevolnyi, S. I.

S. Raza, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal study of ultimate plasmon hybridization”, Opt. Lett. 40(5), 839–842 (2015).
[Crossref] [PubMed]

S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

Burrows, A.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

Busch, K.

F. Intravaia and K. Busch, “Fluorescence in nonlocal dissipative periodic structures,” Phys. Rev. A 91(5), 053836 (2015).
[Crossref]

Carmina Monreal, R.

R. Carmina Monreal, T. J. Antosiewicz, and S. P. Apell, “Diffuse surface scattering in the plasmonic resonances of ultralow electron density nanospheres,” J. Phys. Chem. Lett. 6, 1847–1853 (2015).
[Crossref] [PubMed]

Cha, H.

H. Jung, H. Cha, D. Lee, and S. Yoon, “Bridging the nanogap with light: continuous tuning of plasmon coupling between gold nanoparticles,” ACS Nano, Article ASAP (2015).
[Crossref]

Chen, X.

X. Chen, J. E. Moore, M. Zekarias, and L. Jensen, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime,” Nature Commun. 6, 8921 (2015).
[Crossref]

Christensen, T.

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms. Introduction to Quantum Electrodynamics (Wiley, 1989); see pp. 36–42.

Correas-Serrano, D.

Craig, D. P.

D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics. An Introduction to Radiation Molecule Interactions (Academic, 1984); see Sec. 3.1: Transverse and longitudinal δ-dyadics.

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447 (1998).
[Crossref]

Di Vece, M.

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms. Introduction to Quantum Electrodynamics (Wiley, 1989); see pp. 36–42.

Esteban, R.

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
[Crossref]

Evers, F.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

Fernández-Domínguez, A. I.

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
[Crossref] [PubMed]

Fischer, S. V.

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

Forstmann, E.

E. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer, 1986).
[Crossref]

García-Vidal, F. J.

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
[Crossref] [PubMed]

Gerhardts, R. R.

E. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer, 1986).
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K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
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T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
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S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
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G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
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S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
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S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
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S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
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H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
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H. Jung, H. Cha, D. Lee, and S. Yoon, “Bridging the nanogap with light: continuous tuning of plasmon coupling between gold nanoparticles,” ACS Nano, Article ASAP (2015).
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A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
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A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
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McMahon, J. M.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103(9), 097403 (2009).
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X. Chen, J. E. Moore, M. Zekarias, and L. Jensen, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime,” Nature Commun. 6, 8921 (2015).
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S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

W. Yan, M. Wubs, and N. A. Mortensen, “Projected-dipole model for quantum plasmonics,” Phys. Rev. Lett. 115(13), 137403 (2015).
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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
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S. Raza, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal study of ultimate plasmon hybridization”, Opt. Lett. 40(5), 839–842 (2015).
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S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
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N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

W. Yan, N. A. Mortensen, and M. Wubs, “Hyperbolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21(12), 15026 (2013).
[Crossref] [PubMed]

A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
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N. A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Phot. Nanostr. 11(4), 303 (2013).
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S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

W. Yan, N. A. Mortensen, and M. Wubs, “Green’s function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments,” Phys. Rev. B 88(15), 155414 (2013).
[Crossref]

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
[Crossref]

Nilsson, P. O.

I. Lindau and P. O. Nilsson, “Experimental verification of optically excited longitudinal plasmons,” Phys. Scr. 3, 87–92 (1971).
[Crossref]

Nordlander, P.

H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
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R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
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A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
[Crossref] [PubMed]

Prodan, E.

H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
[Crossref]

Raza, S.

S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

S. Raza, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal study of ultimate plasmon hybridization”, Opt. Lett. 40(5), 839–842 (2015).
[Crossref] [PubMed]

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
[Crossref] [PubMed]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
[Crossref]

Rockstuhl, C.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

Ruppin, R.

Sánchez-Portal, D.

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

Savage, K. J.

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
[Crossref] [PubMed]

Schatz, G. C.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103(9), 097403 (2009).
[Crossref] [PubMed]

Søndergaard, T.

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

Stenger, N.

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
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D. P. Craig and T. Thirunamachandran, Molecular Quantum Electrodynamics. An Introduction to Radiation Molecule Interactions (Academic, 1984); see Sec. 3.1: Transverse and longitudinal δ-dyadics.

Toscano, G.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
[Crossref] [PubMed]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
[Crossref]

Tymchenko, M.

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447 (1998).
[Crossref]

Wang, W.

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
[Crossref]

Wiener, A.

A. I. Fernández-Domínguez, A. Wiener, F. J. García-Vidal, S. A. Maier, and J. B. Pendry, “Transformation-optics description of nonlocal effects in plasmonic nanostructures,” Phys. Rev. Lett. 108(10), 106802 (2012).
[Crossref] [PubMed]

Wubs, M.

S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

S. Raza, M. Wubs, S. I. Bozhevolnyi, and N. A. Mortensen, “Nonlocal study of ultimate plasmon hybridization”, Opt. Lett. 40(5), 839–842 (2015).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Projected-dipole model for quantum plasmonics,” Phys. Rev. Lett. 115(13), 137403 (2015).
[Crossref] [PubMed]

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
[Crossref]

W. Yan, N. A. Mortensen, and M. Wubs, “Green’s function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments,” Phys. Rev. B 88(15), 155414 (2013).
[Crossref]

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

W. Yan, N. A. Mortensen, and M. Wubs, “Hyperbolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21(12), 15026 (2013).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

G. Toscano, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20(4), 4176–4188 (2012).
[Crossref] [PubMed]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
[Crossref]

Xu, H.

G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

Yan, W.

W. Yan, M. Wubs, and N. A. Mortensen, “Projected-dipole model for quantum plasmonics,” Phys. Rev. Lett. 115(13), 137403 (2015).
[Crossref] [PubMed]

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

W. Yan, N. A. Mortensen, and M. Wubs, “Green’s function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments,” Phys. Rev. B 88(15), 155414 (2013).
[Crossref]

W. Yan, N. A. Mortensen, and M. Wubs, “Hyperbolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21(12), 15026 (2013).
[Crossref] [PubMed]

W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
[Crossref]

Yanai, A.

A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
[Crossref]

Yoon, S.

H. Jung, H. Cha, D. Lee, and S. Yoon, “Bridging the nanogap with light: continuous tuning of plasmon coupling between gold nanoparticles,” ACS Nano, Article ASAP (2015).
[Crossref]

Zayats, A.

P. Ginzburg and A. Zayats, “Localized surface plasmon resonances in spatially dispersive nano-objects: phenomenological treatise,” ACS Nano 7(5), 43344342 (2013).
[Crossref] [PubMed]

Zekarias, M.

X. Chen, J. E. Moore, M. Zekarias, and L. Jensen, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime,” Nature Commun. 6, 8921 (2015).
[Crossref]

Zhang, H.

H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
[Crossref]

ACS Nano (2)

T. Christensen, W. Yan, S. Raza, A.-P. Jauho, N. A. Mortensen, and M. Wubs, “Nonlocal response of metallic nanospheres probed by light, electrons, and atoms,” ACS Nano 8(2), 1745–1758 (2014).
[Crossref] [PubMed]

P. Ginzburg and A. Zayats, “Localized surface plasmon resonances in spatially dispersive nano-objects: phenomenological treatise,” ACS Nano 7(5), 43344342 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. C (1)

H. Zhang, V. Kulkarni, E. Prodan, P. Nordlander, and A. O. Govorov, “Theory of quantum plasmon resonances in doped semiconductor nanocrystals,” J. Phys. Chem. C 118, 16035–16042 (2014).
[Crossref]

J. Phys. Chem. Lett. (1)

R. Carmina Monreal, T. J. Antosiewicz, and S. P. Apell, “Diffuse surface scattering in the plasmonic resonances of ultralow electron density nanospheres,” J. Phys. Chem. Lett. 6, 1847–1853 (2015).
[Crossref] [PubMed]

J. Phys.: Condens. Matter (1)

S. Raza, S. I. Bozhevolnyi, M. Wubs, and N. A. Mortensen, “Nonlocal optical response in metallic nanostructures,” J. Phys.: Condens. Matter 27, 183204 (2015).

Kgl. Dansk Videnskab. Selskab, Mat. Fys. Medd. (1)

J. Lindhard, “On the properties of a gas of charged particles,” Kgl. Dansk Videnskab. Selskab, Mat. Fys. Medd. 28(8), 1–57 (1954).

Nano Lett. (1)

M. Barbry, P. Koval, F. Marchesin, R. Esteban, A. G. Borisov, J. Aizpurua, and D. Sánchez-Portal, “Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics,” Nano Lett. 15(5), 3410–3419 (2015).
[Crossref] [PubMed]

Nanophotonics (1)

S. Raza, N. Stenger, S. Kadkhodazadeh, S. V. Fischer, N. Kostesha, A.-P. Jauho, A. Burrows, M. Wubs, and N. A. Mortensen, “Blueshift of the surface plasmon resonance in silver nanoparticles studied with EELS,” Nanophotonics 2(3), 161–166 (2013).
[Crossref]

Nature (1)

K. J. Savage, M. M. Hawkeye, R. Esteban, A. G. Borisov, J. Aizpurua, and J. J. Baumberg, “Revealing the quantum regime in tunnelling plasmonics,” Nature 491, 574–577 (2012).
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Nature Commun. (5)

S. Raza, S. Kadkhodazadeh, T. Christensen, M. Di Vece, M. Wubs, N. A. Mortensen, and N. Stenger, “Multipole plasmons and their disappearance in few-nanometer silver nanoparticles,” Nature Commun. 6, 8788 (2015).
[Crossref]

R. Esteban, A. G. Borisov, P. Nordlander, and J. Aizpurua, “Bridging quantum and classical plasmonics with a quantum-corrected model,” Nature Commun. 3, 825 (2012).
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G. Toscano, J. Straubel, A. Kwiatkowski, C. Rockstuhl, F. Evers, H. Xu, N. A. Mortensen, and M. Wubs, “Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics,” Nature Commun. 6, 7132 (2015).
[Crossref]

N. A. Mortensen, S. Raza, M. Wubs, T. Søndergaard, and S. I. Bozhevolnyi, “A generalized non-local optical response theory for plasmonic nanostructures,” Nature Commun. 5, 3809 (2014).
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X. Chen, J. E. Moore, M. Zekarias, and L. Jensen, “Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime,” Nature Commun. 6, 8921 (2015).
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N. A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Phot. Nanostr. 11(4), 303 (2013).
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A. Moradi, Maxwell-Garnett effective medium theory: quantum nonlocal effects,” Phys. of Plasmas 22(4), 042105 (2015).
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F. Intravaia and K. Busch, “Fluorescence in nonlocal dissipative periodic structures,” Phys. Rev. A 91(5), 053836 (2015).
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A. Yanai, N. A. Mortensen, and U. Levy, “Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation,” Phys. Rev. B 88(20), 205120 (2013).
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W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B 86(20), 205429 (2012).
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W. Yan, N. A. Mortensen, and M. Wubs, “Green’s function surface-integral method for nonlocal response of plasmonic nanowires in arbitrary dielectric environments,” Phys. Rev. B 88(15), 155414 (2013).
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T. Christensen, W. Wang, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Classical and quantum plasmonics in graphene nanodisks: role of edge states,” Phys. Rev. B 90(24), 241414(R) (2014).
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S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84(12), 121412(R) (2011).
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W. Yan, M. Wubs, and N. A. Mortensen, “Projected-dipole model for quantum plasmonics,” Phys. Rev. Lett. 115(13), 137403 (2015).
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A. Moradi, “Infrared absorption spectra of a spatially dispersive polar semiconductor nanowire,” Solid State Commun. 212, 10–13 (2015).
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[Crossref]

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Tables (1)

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Table 1 Overview of Response Models, and Where in the Text they are Discussed.

Equations (48)

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D ( r 1 , ω ) = ε 0 d r 2 ε ( r 1 , r 2 , ω ) E ( r 2 , ω ) ,
I = k ^ k ^ + ( I k ^ k ^ ) = k ^ k ^ + e 1 e 1 + e 2 e 2 = ( 1 0 0 0 1 0 0 0 1 ) ,
L k k ^ k ^ and T k I k ^ k ^ .
L k L k = L k , T k T k = T k , L k T k = T k L k = 0 , L k + T k = I .
ε ( k , ω ) = ε L ( k , ω ) + ε T ( k , ω ) = ε L ( k , ω ) L k + ε T ( k , ω ) T k = ( ε L ( k , ω ) 0 0 0 ε T ( k , ω ) 0 0 0 ε T ( k , ω ) ) ,
ε ( r , ω ) = 1 ( 2 π ) 3 d 3 k 1 e i k 1 r ε ( k 1 , ω ) .
ε L ( r , ω ) = 1 ( 2 π ) 3 d 3 k 1 e i k 1 r ε L ( k 1 , ω ) L k = 1 ( 2 π ) 3 0 d k 1 k 1 2 ε L ( k 1 , ω ) d 2 k ^ 1 e i k 1 r k ^ 1 k ^ 1 .
d 2 k ^ 1 e i k 1 r k ^ 1 k ^ 1 = 1 k 1 2 d 2 k ^ 1 e i k 1 r = 4 π k 1 2 sin ( k 1 r ) k 1 r .
ε L ( r , ω ) = 1 4 π 2 d k 1 ε L ( k 1 , ω ) sin ( k 1 r ) k 1 r .
ε T ( r , ω ) = ε T ( r , ω ) I + 1 4 π 2 d k 1 ε T ( k 1 , ω ) sin ( k 1 r ) k 1 r ,
f ( r ) = f ( r ) r ( I r ^ r ^ ) + f ( r ) r ^ r ^ ( for r 0 ) ,
d r ε L , T ( r , ω ) = d 3 k 1 [ 1 ( 2 π ) 3 d r e i k 1 r ] ε L , T ( k 1 , ω ) = ε L , T ( k = 0 , ω ) ,
× × G ( r , ω ) + ( ω / c ) 2 d r 1 ε ( r r 1 , ω ) G ( r 1 , ω ) = I δ ( r ) ,
k 2 T k G ( k , ω ) + ( ω / c ) 2 ε ( k , ω ) G ( k , ω ) = I .
G L ( k , ω ) = 1 ( ω / c ) 2 ε L ( k , ω ) , G T ( k , ω ) = 1 k 2 ( ω / c ) 2 ε T ( k , ω ) .
G L ( r , ω ) = 1 4 π 2 d k 1 G L ( k 1 , ω ) sin ( k 1 r ) k 1 r ,
G T ( r , ω ) = G T ( r , ω ) I + 1 4 π 2 d k 1 G T ( k 1 , ω ) sin ( k 1 r ) k 1 r ,
ε ( k , ω ) = ε ( ω ) I ε ( r , ω ) = ε ( ω ) δ 3 ( r ) I ,
ε D ( ω ) = ε core ( ω ) + χ D ( ω ) , with χ D ( ω ) = ω p 2 ω ( ω + i γ D ) .
ε ( k 1 , ω ) = ε L ( ω ) L k + ε T ( ω ) T k ,
ε L ( r , ω ) = ε L ( ω ) ( 2 π ) 3 d 3 k 1 e i k 1 r L k = ε L ( ω ) δ ( r ) = ε L ( ω ) [ 1 3 δ 3 ( r ) I + 1 4 π r 3 ( I 3 r ^ r ^ ) ] ,
ε T ( r , ω ) = ε T ( ω ) ( 2 π ) 3 d 3 k 1 e i k 1 r T k = ε T ( ω ) δ ( r ) = ε T ( ω ) [ 2 3 δ 3 ( r ) I 1 4 π r 3 ( I 3 r ^ r ^ ) ] ,
ε ( r , ω ) = 2 ε T ( ω ) + ε L ( ω ) 3 δ 3 ( r ) I + ε L ( ω ) ε L ( ω ) 4 π r 3 ( I 3 r ^ r ^ ) .
r ^ ε ( r , ω ) r ^ = 2 ε T ( ω ) + ε L ( ω ) 3 δ 3 ( r ) ε L ( ω ) ε T ( ω ) 2 π r 3 ,
e 1 , 2 ( r ) ε ( r , ω ) e 1 , 2 ( r ) = 2 ε T ( ω ) + ε L ( ω ) 3 δ 3 ( r ) + ε L ( ω ) ε T ( ω ) 4 π r 3 ,
× × E ( r , ω ) + ( ω / c ) 2 ε core ( ω ) E ( r , ω ) = i μ 0 ω J ( r , ω )
β 2 ω ( ω + i γ D ) J ( r , ω ) + J ( r , ω ) = i ε 0 ω χ D ( ω ) E ( r , ω ) ,
k 2 T k E + ( ω / c ) 2 ε core ( ω ) E = i μ 0 ω J and β 2 k 2 ω ( ω + i γ D ) L k J + J = i ε 0 ω χ D E ,
ε T ( ω ) = ε core ( ω ) + χ D ( ω ) = ε D ( ω ) ,
ε L ( k , ω ) = ε core ( ω ) ω p 2 ω ( ω + i γ ) β 2 k 2 .
ε T ( r , ω ) = ε T ( ω ) δ ( r ) = ε D ( ω ) [ δ ( r ) I + ( 1 4 π r ) ] = ε D ( ω ) [ 2 3 δ 3 ( r ) I I 3 r ^ r ^ 4 π r 3 ] ,
ε core ( ω ) d k 1 sin ( k 1 r ) k 1 r = π ε core ( ω ) r ,
d k 1 [ ω p 2 ω ( ω + i γ ) β 2 k 1 2 ] sin ( k 1 r ) k 1 r = π [ ω p 2 ω ( ω + i γ ) ] 1 e i q r r = π χ D ( ω ) 1 e i q r r ,
ε L ( r , ω ) = ε D ( ω ) ( 1 4 π r ) + χ D ( ω ) ( e i q r 4 π r ) ,
ε ( r , ω ) = ε D ( ω ) δ ( r ) I + χ D ( ω ) ( e i q r 4 π r ) .
ε ( r , ω ) = [ ε core ( ω ) + 2 3 χ D ( ω ) ] δ ( r ) I + χ D ( ω ) e i q r 4 π r 3 [ ( i q r 1 ) ( I 3 r ^ r ^ ) q r r 2 r ^ r ^ ] .
ε av . ( r , ω ) = I 3 Tr [ ε ( r , ω ) ] = [ ε core ( ω ) + 2 3 χ D ( ω ) ] δ ( r ) I I 3 q 2 χ D ( ω ) e i q r 4 π r .
ε L ( ω ) = ε core ( ω ) + χ free ( ω ) , with χ free ( ω ) = ω p 2 ω ( ω + i γ b ) Δ 2 ,
ε T ( k , ω ) = ε core ( ω ) ω p 2 ω ( ω + i γ b ) Δ 2 β T 2 k 2 ,
ε ( r , ω ) = I δ ( r ) [ ε core ( ω ) + 1 3 χ free ( ω ) ] χ free ( ω ) e i Q r 4 π r 3 [ ( i Q r 1 ) ( I 3 r ^ r ^ ) + Q 2 r 2 ( I r ^ r ^ ) ] ,
ε L ( k , ω ) = ε T ( k , ω ) = ε ( k , ω ) , so that ε ( k , ω ) = ε ( k , ω ) I .
[ β 2 ω ( ω + i γ D ) 2 I + I ] J ( r ) = i ε 0 ω χ D ( ω ) E ( r ) [ β 2 k 2 I ω ( ω + i γ D ) + I ] J ( k ) = i ε 0 ω χ D E ( k ) ,
ε ( r , ω ) = I [ ε core ( ω ) δ ( r ) + ω p 2 β 2 e i q r 4 π r ] ,
f ( r ) = f ( r ) r ( I r ^ r ^ ) + f ( r ) r ^ r ^ ,
( 1 4 π r ) = ( I 3 r ^ r ^ 4 π r 3 ) + 1 3 δ ( r ) I = δ ( r ) ,
f ( r ) = I [ ( r ) ( 1 r d d r ) ] f ( r ) + r ^ r ^ [ ( r ) 2 ( 1 r d d r ) 2 ] f ( r ) .
1 k 2 sin ( k 1 r ) k 1 r = j 1 ( k 1 r ) k 1 r I j 2 ( k 1 r ) r ^ r ^ ,
j n ( x ) ( x ) n ( 1 x d d x 1 x d d x ) n sin x x .

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