Abstract

The position, width, extinction, and electric field of localized plasmon modes in closely-coupled linear chains of small spheres are investigated. A dipole-like model is presented that separates the universal geometric factors from the specific metal permittivity. An electrostatic surface integral method is used to deduce universal parameters that are confirmed against results for different metals (bulk experimental Ag, Au, Al, K) calculated using retarded vector spherical harmonics and finite elements. The mode permittivity change decays to an asymptote with the number of particles in the chain, and changes dramatically from 1/f 3 to 1/f 1/2 as the gap fraction (ratio of gap between spheres to their diameter), f, gets smaller. Scattering increases significantly with closer coupling. The mode sharpness, strength and electric field for weakly retarded calculations are consistent with electrostatic predictions once the effect of radiative damping is accounted for.

©2010 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Plasmonic waves on a chain of metallic nanoparticles: effects of a liquid-crystalline host

Nicholas A. Pike and David Stroud
J. Opt. Soc. Am. B 30(5) 1127-1134 (2013)

Universal scaling of plasmonic refractive index sensors

Yen-Kai Chang, Zong-Xing Lou, Kao-Der Chang, and Chih-Wei Chang
Opt. Express 21(2) 1804-1811 (2013)

Dark spots along slowly scaling chains of plasmonic nanoparticles

Gianluigi Zito, Giulia Rusciano, and Antonio Sasso
Opt. Express 24(12) 13584-13589 (2016)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. U. Kreibig, and M. Vollmer, Optical properties of metal clusters (Springer-Verlag, 1995).
  2. P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
    [Crossref]
  3. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
    [Crossref]
  4. J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
    [Crossref] [PubMed]
  5. C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
    [Crossref]
  6. A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
    [Crossref] [PubMed]
  7. D. Langbein, “Theory of Van der Waals attraction,” in Springer Tracts in Modern Physics (Springer, 1974), pp. 1–139.
  8. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
    [Crossref]
  9. S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
    [Crossref]
  10. S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
    [Crossref]
  11. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. García De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
    [Crossref] [PubMed]
  12. N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
    [Crossref]
  13. S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004).
    [Crossref]
  14. S. L. Zou and G. C. Schatz, “Theoretical studies of plasmon resonances in one-dimensional nanoparticle chains: narrow lineshapes with tunable widths,” Nanotechnology 17(11), 2813–2820 (2006).
    [Crossref]
  15. F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008).
    [Crossref]
  16. M. D. Arnold and M. G. Blaber, “Optical performance and metallic absorption in nanoplasmonic systems,” Opt. Express 17(5), 3835–3847 (2009).
    [Crossref] [PubMed]
  17. T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).
  18. J. H. Weaver, and H. P. R. Frederikse, Optical properties of selected elements, 82 ed. (CRC Press, 2001).
  19. D. W. Mackowski, “Calculation of Total Cross-Sections of Multiple-Sphere Clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994).
    [Crossref]
  20. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996).
    [Crossref]
  21. M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
    [Crossref] [PubMed]
  22. R. Fuchs, “Theory of optical properties of ionic-crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975).
    [Crossref]
  23. C. F. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 2004).
  24. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics (Springer, 1988).
  25. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
    [Crossref]
  26. V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40(11), 2281–2291 (1993).
    [Crossref]
  27. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
    [Crossref]
  28. T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
    [Crossref]
  29. R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
    [Crossref] [PubMed]
  30. W. Y. Chien and T. Szkopek, “Multiple-multipole simulation of optical nearfields in discrete metal nanosphere assemblies,” Opt. Express 16(3), 1820–1835 (2008).
    [Crossref] [PubMed]
  31. N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
    [Crossref] [PubMed]
  32. M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

2009 (6)

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

M. D. Arnold and M. G. Blaber, “Optical performance and metallic absorption in nanoplasmonic systems,” Opt. Express 17(5), 3835–3847 (2009).
[Crossref] [PubMed]

T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
[Crossref]

2008 (3)

W. Y. Chien and T. Szkopek, “Multiple-multipole simulation of optical nearfields in discrete metal nanosphere assemblies,” Opt. Express 16(3), 1820–1835 (2008).
[Crossref] [PubMed]

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008).
[Crossref]

2007 (3)

P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
[Crossref]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
[Crossref] [PubMed]

2006 (3)

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. García De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
[Crossref] [PubMed]

S. L. Zou and G. C. Schatz, “Theoretical studies of plasmon resonances in one-dimensional nanoparticle chains: narrow lineshapes with tunable widths,” Nanotechnology 17(11), 2813–2820 (2006).
[Crossref]

2004 (1)

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004).
[Crossref]

2003 (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

2002 (2)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
[Crossref]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

2000 (1)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
[Crossref]

1997 (1)

T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).

1996 (1)

1995 (1)

N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
[Crossref] [PubMed]

1994 (1)

1993 (1)

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40(11), 2281–2291 (1993).
[Crossref]

1975 (1)

R. Fuchs, “Theory of optical properties of ionic-crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975).
[Crossref]

Aizpurua, J.

Arnold, M. D.

M. D. Arnold and M. G. Blaber, “Optical performance and metallic absorption in nanoplasmonic systems,” Opt. Express 17(5), 3835–3847 (2009).
[Crossref] [PubMed]

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
[Crossref]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
[Crossref]

Blaber, M. G.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

M. D. Arnold and M. G. Blaber, “Optical performance and metallic absorption in nanoplasmonic systems,” Opt. Express 17(5), 3835–3847 (2009).
[Crossref] [PubMed]

Brongersma, M. L.

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
[Crossref]

Bryant, G. W.

Chang, C. C.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
[Crossref] [PubMed]

Chern, R. L.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
[Crossref] [PubMed]

Chien, W. Y.

Davis, T. J.

T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
[Crossref]

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

El-Sayed, M. A.

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
[Crossref]

Feldmann, J.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Ford, M. J.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

Freeman, R. G.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Fuchs, R.

R. Fuchs, “Theory of optical properties of ionic-crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975).
[Crossref]

Funston, A. M.

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

García de Abajo, F. J.

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008).
[Crossref]

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. García De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
[Crossref] [PubMed]

Gomez, D. E.

T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
[Crossref]

Hagihara, T.

T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).

Harris, N.

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
[Crossref]

Hayashiuchi, Y.

T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).

Henry, A.-I.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Huang, W. Y.

P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
[Crossref]

Jain, P. K.

P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
[Crossref]

Ke-Da, B.

N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
[Crossref] [PubMed]

Khlebtsov, B.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

Khlebtsov, N.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
[Crossref]

Klar, T. A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Kürzinger, K.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Liu, X. X.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
[Crossref] [PubMed]

Mackowski, D. W.

Mahmoud, M.

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
[Crossref]

Markel, V. A.

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40(11), 2281–2291 (1993).
[Crossref]

McMahon, J. M.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

McPhedran, R. C.

N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
[Crossref] [PubMed]

Melnikov, A.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

Mishchenko, M. I.

Mulvaney, P.

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

Murali, R.

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

Natan, M. J.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Nichtl, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Nicorovici, N. A.

N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
[Crossref] [PubMed]

Novo, C.

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

Okada, T.

T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).

Park, S. Y.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004).
[Crossref]

Ringler, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Romero, I.

Schatz, G. C.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

S. L. Zou and G. C. Schatz, “Theoretical studies of plasmon resonances in one-dimensional nanoparticle chains: narrow lineshapes with tunable widths,” Nanotechnology 17(11), 2813–2820 (2006).
[Crossref]

Schwemer, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Stroud, D.

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004).
[Crossref]

Szkopek, T.

Tabor, C.

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

Van Duyne, R. P.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Vernon, K. C.

T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
[Crossref]

Wunderlich, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Wustholz, K. L.

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Zharov, V.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

Zou, S. L.

S. L. Zou and G. C. Schatz, “Theoretical studies of plasmon resonances in one-dimensional nanoparticle chains: narrow lineshapes with tunable widths,” Nanotechnology 17(11), 2813–2820 (2006).
[Crossref]

Anal. Bioanal. Chem. (1)

J. M. McMahon, A.-I. Henry, K. L. Wustholz, M. J. Natan, R. G. Freeman, R. P. Van Duyne, and G. C. Schatz, “Gold nanoparticle dimer plasmonics: finite element method calculations of the electromagnetic enhancement to surface-enhanced Raman spectroscopy,” Anal. Bioanal. Chem. 394(7), 1819–1825 (2009).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

S. A. Maier, P. G. Kik, and H. A. Atwater, “Observation of coupled plasmon-polariton modes in Au nanoparticle chain waveguides of different lengths: estimation of waveguide loss,” Appl. Phys. Lett. 81(9), 1714–1716 (2002).
[Crossref]

J. Mod. Opt. (1)

V. A. Markel, “Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure,” J. Mod. Opt. 40(11), 2281–2291 (1993).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Phys. Chem. A (1)

C. Tabor, R. Murali, M. Mahmoud, and M. A. El-Sayed, “On the use of plasmonic nanoparticle pairs as a plasmon ruler: the dependence of the near-field dipole plasmon coupling on nanoparticle size and shape,” J. Phys. Chem. A 113(10), 1946–1953 (2009).
[Crossref]

J. Phys. Chem. C (2)

N. Harris, M. D. Arnold, M. G. Blaber, and M. J. Ford, “Plasmonic resonances of closely coupled gold nanosphere chains,” J. Phys. Chem. C 113(7), 2784–2791 (2009).
[Crossref]

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112(46), 17983–17987 (2008).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 558–589 (2007).
[Crossref]

Nano Lett. (2)

A. M. Funston, C. Novo, T. J. Davis, and P. Mulvaney, “Plasmon coupling of gold nanorods at short distances and in different geometries,” Nano Lett. 9(4), 1651–1658 (2009).
[Crossref] [PubMed]

P. K. Jain, W. Y. Huang, and M. A. El-Sayed, “On the universal scaling behavior of the distance decay of plasmon coupling in metal nanoparticle pairs: a plasmon ruler equation,” Nano Lett. 7(7), 2080–2088 (2007).
[Crossref]

Nanotechnology (2)

S. L. Zou and G. C. Schatz, “Theoretical studies of plasmon resonances in one-dimensional nanoparticle chains: narrow lineshapes with tunable widths,” Nanotechnology 17(11), 2813–2820 (2006).
[Crossref]

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17(5), 1437–1445 (2006).
[Crossref]

Opt. Express (3)

Phys. Rev. B (6)

T. J. Davis, K. C. Vernon, and D. E. Gomez, “Designing plasmonic systems using optical coupling between nanoparticles,” Phys. Rev. B 79(15), 155423 (2009).
[Crossref]

R. Fuchs, “Theory of optical properties of ionic-crystal cubes,” Phys. Rev. B 11(4), 1732–1740 (1975).
[Crossref]

S. Y. Park and D. Stroud, “Surface-plasmon dispersion relations in chains of metallic nanoparticles: an exact quasistatic calculation,” Phys. Rev. B 69(12), 125418 (2004).
[Crossref]

S. A. Maier, M. L. Brongersma, P. G. Kik, and H. A. Atwater, “Observation of near-field coupling in metal nanoparticle chains using far-field polarization spectroscopy,” Phys. Rev. B 65(19), 193408 (2002).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(20), 205402 (2003).
[Crossref]

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62(24), R 16356–R16359, 16359 (2000).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(1), 016609 (2007).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

N. A. Nicorovici, R. C. McPhedran, and B. Ke-Da, “Propagation of electromagnetic waves in periodic lattices of spheres: Green’s function and lattice sums,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(1), 690–702 (1995).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100(20), 203002 (2008).
[Crossref] [PubMed]

Other (7)

T. Hagihara, Y. Hayashiuchi, and T. Okada, “Photoplastic effects in colored KCl single crystals containing potassium metal colloids. I. Preparation of specimens enriched with potassium metal colloids,” Memoirs of Osaka Kyoiku University. Ser. 3. Natural Science and Applied Science  46, 49–56 (1997).

J. H. Weaver, and H. P. R. Frederikse, Optical properties of selected elements, 82 ed. (CRC Press, 2001).

U. Kreibig, and M. Vollmer, Optical properties of metal clusters (Springer-Verlag, 1995).

D. Langbein, “Theory of Van der Waals attraction,” in Springer Tracts in Modern Physics (Springer, 1974), pp. 1–139.

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1964).

C. F. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, 2004).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, Springer Tracts in Modern Physics (Springer, 1988).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 (Color online) An example geometry showing a chain of n = 3 spherical metal particles (a), and the optimization process (b). Only longitudinally polarized excitation is considered. Extinction as a function of wavelength (blue line) is shown for a chain with 3 Ag spheres of radius a = 7.5nm and d = 16.5nm center-spacing, giving a gap fraction f = d/(2a)-1 = 0.1 corresponding to the diagram on the left. In this example a dipole near 400nm dominates – in general we ignore higher order modes such as the quadrupole seen here emerging near 350nm. Varying the gap fraction moves the peak, as indicated by dots. In the remainder of this paper, we only plot the dipole peaks.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2 (Color online). Summary of peak behavior as a function of wavelength due to changes in particle separation. Peak extinction (a), field enhancement (b), particle separation (c) and the associated mode permittivity shift (d) compared to that of an isolated sphere. Electrostatic limits (lines) are shown for the infinite chain (dashed) and bisphere (solid). The lines are limited in extent by mode behavior at short wavelengths and calculation limits at long wavelengths. Also included are VSH-Mackowski extinctions for bispheres (dots), VSH-GMM field enhancements for bispheres (crosses), and FEM results for infinite chains (circles), all damped by radiation. Details of methods can be seen in later sections.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3 (Color online) Shift of the longitudinal dipole mode permittivity, (a) as a function of the gap fraction and (b) as a function of the chain length. The VSH results for finite spheres are indicated with markers, where different colors indicate different metals. Electrostatic limits are shown as lines. The dashed line in (a) is the infinite chain result, and the solid line is the bisphere. In (b) the lower line has a gap fraction d/2a-1 = 1, and the upper is 1/30.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4 (Color online) Peak quality factor compared to that predicted by Eq. (9), as a function of the measured contribution of metal absorption to energy extraction. The line shows the expected relationship, and all VSH (Mackowski) results have been included as dots, covering all separations and chain lengths. This partially confirms the veracity of the damping correction used for later results.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5 (Color online) Dipole peak strength for different chain lengths n, as a function of gap fraction (a), and as a function of the corresponding peak shift (b), which shows the universal scaling of peak extinction. Electrostatic limits are plotted as lines, for both the bisphere (solid) and infinite chain (dashed). Retarded calculations, corrected for scattering by using Ce 2/Ca on the left hand side of Eq. (2), are overlaid using markers. This includes VSH results for all chain lengths (dots), and finite elements for infinite chains only (circles).

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6 (Color online) Comparison of peak scattering predicted by Eq. (3) as a function of phase length of the chain, where Eq. (3) used cluster α based on (1) and (2) with uncorrected Ce . The curved dotted line is inversely proportional to the x axis. All VSH (Mackowski) results, covering all separations and chain lengths, are shown as dots. The conclusion is that only optically short chains scatter like the small cluster modeled by Eq. (3).

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7 (Color online) Maximum electric field enhancement coefficient, for different lengths of chains, as a function of gap fraction (a), and as a function of the corresponding peak shift (b). Electrostatic surface integral limits are shown as lines with infinite chains (dashed) and bisphere (solid). Retarded results corrected for radiative damping are shown as markers. This includes VSH-GMM for all chain lengths (crosses), and finite elements for infinite chains only (circles).

Download Full Size | PPT Slide | PDF

Equations (29)

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

α = A V ε − ε b ε b + L ( ε − ε b ) ,
C e = k ℑ { α } ,
C s ≈ k 4 6π | α | 2 .
| E E i | = | B A ε ε − ε b α V | = | B ε ε b + L ( ε − ε b ) | .
ε p = ε b ( 1 − 1 / L ) ,
ε p = ε ′ − ε ″ d ε ″ d λ − ε ″ λ [ d ε ′ d λ ± ( d ε ′ d λ ) 2 + ( d ε ″ d λ ) 2 − ( ε ″ λ ) 2 ] .
ε p ε s − 1 .
α max ≈ i A V ( ε ′ − ε b ) 2 ε ″ ε b .
Q ≡ ω Δ ω ≈ ω 2 ε ″ d ε ′ d ω
α R R ← α 1 − i k 3 α / 6π .
C e ≈ C e R R C e R R C a R R .
C s ~ C s R R ( C e R R C a R R ) 2 .
α ≈ [ α 1 − 1 − S n ] − 1 ,
S n = p s n / ( 4π d 3 ) ,
E → ( r → ) = ∫ P → ( r → ' ) ⋅ n ^ ( r → ' ) ( r → − r → ' ) | r → − r → ' | 3 d S ' + E → 0 ,
p ( i ) Λ = G ( i j ) p ( j ) ,
G ( i j ) = ( r → ( i ) − r → ( j ) ) ⋅ n ^ ( i ) | r → ( i ) − r → ( j ) | 3 Δ S ( j ) ,
ε p = Λ + 2π Λ − 2π .
A m = 1 V ∑ ( n z ( j ) [ p − 1 ] ( m j ) ) ( p ( i m ) z ( i ) Δ S ( i ) ) ,
A ≈ lim χ ″ → 0   α ″ χ ″ V ( χ ′ ) 2 ,
B m ( i ) = ∑ n z ( j ) [ p − 1 ] ( m j ) p ( i m ) Δ S ( i ) .
G 0 = Δ θ ' 2 sin θ ' K ( cos ( θ ' − θ ) − cos ( θ ' + θ ) 1 − cos ( θ ' + θ ) ) / ( a 1 − cos ( θ ' + θ ) ) ,
G q = Δ θ ' 2 sin θ ' K [ ξ ] − E [ ξ ] r q ( cos θ ' + r q / 2 ) 1 − cos [ θ − θ ' ] + r q ( cos θ ' − cos θ + r q / 2 ) a 1 − cos [ θ + θ ' ] + r q ( cos θ ' − cos θ + r q / 2 ) ,
ξ = cos [ θ − θ ' ] − cos [ θ + θ ' ] 1 − cos [ θ − θ ' ] + r q ( cos θ ' − cos θ + r q / 2 ) .
G 0 ( θ = θ ' ) ≈ Δ θ ' a ( ln [ 8 sin θ ] + 1 − ln [ Δ θ / 2 ] ) .
G Σ 2 = G 0 − G 1 ( − θ ' ) ,
G Σ ∞ = ∑ q = − ∞ ∞ G q = G 0 + ∑ q = 1 ∞ [ G q + G q ( − θ , − θ ' ) ] .
∑ q = 10 ∞ G q → 2 π Δ θ ' sin θ ′ ∑ p = 0 3 g p ( a / d ) p + 2 [ ζ ( p + 2 ) − ∑ q = 1 9 1 q p + 2 ] ,
g 0 = cos ( θ ) g 1 = 3 cos 2 θ − 2cos θ cos θ ' − 1 g 2 = 15 2 cos 3 θ − 9 cos 2 θ cos θ ' + 9 2 cos θ cos 2 θ ' − 6cos θ + 3cos θ ′ g 3 = 9 2 − 24 cos 2 θ + 24 cos θ cos θ ' + 27 cos 2 θ cos 2 θ ' − 30 cos 3 θ cos θ ' − 10 cos θ cos 3 θ ' + 35 2 cos 4 θ − 9 cos θ '

Metrics