Abstract

Expressions for the enhancement of the far-field scattering cross section of a luminescent or Raman- active compound contained within a multilayered nanosphere are derived, where the active compound resides between an outer metallic shell and a metallic core. The quasi-static approximation is assumed for silver and gold particles using a Lorentz–Drude model of the dielectric function. An attempt has also been made to account for the effect of electron scattering from the boundaries of the shell on the enhancement calculation.

© 2009 Optical Society of America

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  1. T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
    [CrossRef]
  2. T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Anal. Chem. 17, 557-570(1998).
    [CrossRef]
  3. H. N. Wang and T. Vo-Dinh, “Multiplex detection of breast cancer biomarkers using plasmonic molecular sentinel nanoprobes,” Nanotechnology 20, 065101 (2009).
    [CrossRef] [PubMed]
  4. S. J. Norton and T. Vo-Dinh, “Plasmonic resonances of nanoshells of spheroidal shape,” IEEE Trans. Nanotechnol. 6, 627-638 (2007).
    [CrossRef]
  5. S. J. Norton and T. Vo-Dinh, “Spectral bounds on plasmon resonances for Ag and Au prolate and oblate nanospheroids,” J. Nanophoton. 2, 029501 (2008).
    [CrossRef]
  6. S. J. Norton and T. Vo-Dinh, “Optical response of linear chains of metal nanospheres and nanospheroids,” J. Opt. Soc. Am. A 25, 2767-2775 (2008).
    [CrossRef]
  7. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
    [CrossRef]
  8. S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
    [CrossRef]
  9. R. D. Averitt, S. L. Westcott, and N. J. Nalas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824-1832 (1999).
    [CrossRef]
  10. J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
    [CrossRef]
  11. C. Radloff and N. J. Halas, “Plasmonic properties of concentric nanoshells,” Nano Lett. 4, 1323-1327 (2004).
    [CrossRef]
  12. U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys. 224, 307-323 (1969).
    [CrossRef]
  13. M. Quinten, “Optical constants of gold and silver clusters in the spectral range between 1.5 eV and 4.5 eV,” Z. Phys. B 101, 211-217 (1996).
    [CrossRef]
  14. B. Khlebtsov and N. Khlebtsov, “Ultrasharp light-scattering resonances of structured nanospheres: effects of size-dependent dielectric functions,” J. Biomed. Opt. 11, 044002(2006).
    [CrossRef] [PubMed]
  15. A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
    [CrossRef]
  16. D.-S. Wang and M. Kerker, “Absorption and luminescence of dye-coated silver and gold particles,” Phys. Rev. B 25, 2433-2449 (1982).
    [CrossRef]
  17. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
    [CrossRef]
  18. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
    [CrossRef]
  19. Even if the polarizabilities of the individual molecules are not isotropic, an effective isotropic polarizability will result if the molecules are randomly oriented.
  20. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 102.
  21. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 99.
  22. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 136-137.
  23. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991), pp. 280-291.
  24. The constants for silver are Δϵk=[1759.47,135.34,258.19,22.90,1749.06,11756.18], ak=[1,1,1,1,1,1], bk=[0.2431,19.681,2.289,0.3292,4.6391,12.25], and ck=[0,17.079,515.022,1718.357,2116.092,10559.42], and for gold are Δϵk=[1589.52,50.20,20.92,148.49,1256.97,9169], ak=[1,1,1,1,1,1], bk=[0.268,1.221,1.747,4.406,12.63,11.21], and ck=[0,4.42,17.67,226.10,475.14,4550.77].
  25. E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
  26. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).
  27. A. E. Neeves and M. H. Birnboim, “Composite structures for the enhancement of nonlinear-optical susceptibility,” J. Opt. Soc. Am. B 6, 787-796 (1989).
    [CrossRef]
  28. R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
    [CrossRef]

2009

H. N. Wang and T. Vo-Dinh, “Multiplex detection of breast cancer biomarkers using plasmonic molecular sentinel nanoprobes,” Nanotechnology 20, 065101 (2009).
[CrossRef] [PubMed]

2008

S. J. Norton and T. Vo-Dinh, “Spectral bounds on plasmon resonances for Ag and Au prolate and oblate nanospheroids,” J. Nanophoton. 2, 029501 (2008).
[CrossRef]

A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

S. J. Norton and T. Vo-Dinh, “Optical response of linear chains of metal nanospheres and nanospheroids,” J. Opt. Soc. Am. A 25, 2767-2775 (2008).
[CrossRef]

2007

S. J. Norton and T. Vo-Dinh, “Plasmonic resonances of nanoshells of spheroidal shape,” IEEE Trans. Nanotechnol. 6, 627-638 (2007).
[CrossRef]

2006

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
[CrossRef]

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

B. Khlebtsov and N. Khlebtsov, “Ultrasharp light-scattering resonances of structured nanospheres: effects of size-dependent dielectric functions,” J. Biomed. Opt. 11, 044002(2006).
[CrossRef] [PubMed]

2005

R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
[CrossRef]

2004

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

C. Radloff and N. J. Halas, “Plasmonic properties of concentric nanoshells,” Nano Lett. 4, 1323-1327 (2004).
[CrossRef]

1999

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

R. D. Averitt, S. L. Westcott, and N. J. Nalas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16, 1824-1832 (1999).
[CrossRef]

1998

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Anal. Chem. 17, 557-570(1998).
[CrossRef]

1996

M. Quinten, “Optical constants of gold and silver clusters in the spectral range between 1.5 eV and 4.5 eV,” Z. Phys. B 101, 211-217 (1996).
[CrossRef]

1989

1984

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

1982

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye-coated silver and gold particles,” Phys. Rev. B 25, 2433-2449 (1982).
[CrossRef]

1969

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys. 224, 307-323 (1969).
[CrossRef]

1951

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
[CrossRef]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Averitt, R. D.

Begun, G. M.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
[CrossRef]

Birnboim, M. H.

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

Fragstein, C. V.

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys. 224, 307-323 (1969).
[CrossRef]

Greffet, J.-J.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

Halas, H. J.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Halas, N. J.

C. Radloff and N. J. Halas, “Plasmonic properties of concentric nanoshells,” Nano Lett. 4, 1323-1327 (2004).
[CrossRef]

Hale, G. D.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Henkel, C.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

Hiromoto, M. Y. K.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Huang, L.

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991), pp. 280-291.

Jackson, J. B.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 136-137.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 102.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 99.

Jhu, R. J.

R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
[CrossRef]

Jin, G. F.

R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
[CrossRef]

Kerker, M.

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye-coated silver and gold particles,” Phys. Rev. B 25, 2433-2449 (1982).
[CrossRef]

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
[CrossRef]

Khlebtsov, B.

B. Khlebtsov and N. Khlebtsov, “Ultrasharp light-scattering resonances of structured nanospheres: effects of size-dependent dielectric functions,” J. Biomed. Opt. 11, 044002(2006).
[CrossRef] [PubMed]

Khlebtsov, N.

B. Khlebtsov and N. Khlebtsov, “Ultrasharp light-scattering resonances of structured nanospheres: effects of size-dependent dielectric functions,” J. Biomed. Opt. 11, 044002(2006).
[CrossRef] [PubMed]

Kreibig, U.

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys. 224, 307-323 (1969).
[CrossRef]

Lu, Y.

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

Moody, R. L.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Moroz, A.

A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

Nalas, N. J.

Neeves, A. E.

Norton, S. J.

S. J. Norton and T. Vo-Dinh, “Optical response of linear chains of metal nanospheres and nanospheroids,” J. Opt. Soc. Am. A 25, 2767-2775 (2008).
[CrossRef]

S. J. Norton and T. Vo-Dinh, “Spectral bounds on plasmon resonances for Ag and Au prolate and oblate nanospheroids,” J. Nanophoton. 2, 029501 (2008).
[CrossRef]

S. J. Norton and T. Vo-Dinh, “Plasmonic resonances of nanoshells of spheroidal shape,” IEEE Trans. Nanotechnol. 6, 627-638 (2007).
[CrossRef]

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
[CrossRef]

Oldenburg, S. J.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

Quinten, M.

M. Quinten, “Optical constants of gold and silver clusters in the spectral range between 1.5 eV and 4.5 eV,” Z. Phys. B 101, 211-217 (1996).
[CrossRef]

Radloff, C.

C. Radloff and N. J. Halas, “Plasmonic properties of concentric nanoshells,” Nano Lett. 4, 1323-1327 (2004).
[CrossRef]

Vigoureux, J. M.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

Vo-Dinh, T.

H. N. Wang and T. Vo-Dinh, “Multiplex detection of breast cancer biomarkers using plasmonic molecular sentinel nanoprobes,” Nanotechnology 20, 065101 (2009).
[CrossRef] [PubMed]

S. J. Norton and T. Vo-Dinh, “Optical response of linear chains of metal nanospheres and nanospheroids,” J. Opt. Soc. Am. A 25, 2767-2775 (2008).
[CrossRef]

S. J. Norton and T. Vo-Dinh, “Spectral bounds on plasmon resonances for Ag and Au prolate and oblate nanospheroids,” J. Nanophoton. 2, 029501 (2008).
[CrossRef]

S. J. Norton and T. Vo-Dinh, “Plasmonic resonances of nanoshells of spheroidal shape,” IEEE Trans. Nanotechnol. 6, 627-638 (2007).
[CrossRef]

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Anal. Chem. 17, 557-570(1998).
[CrossRef]

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Wang, D.-S.

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye-coated silver and gold particles,” Phys. Rev. B 25, 2433-2449 (1982).
[CrossRef]

Wang, H. N.

H. N. Wang and T. Vo-Dinh, “Multiplex detection of breast cancer biomarkers using plasmonic molecular sentinel nanoprobes,” Nanotechnology 20, 065101 (2009).
[CrossRef] [PubMed]

Wang, J.

R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
[CrossRef]

Wang, Y.

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

Westcott, S. L.

Zhu, J.

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

Anal. Chem.

T. Vo-Dinh, M. Y. K. Hiromoto, G. M. Begun, and R. L. Moody, “Surface-enhanced Raman spectrometry for trace organic-analysis,” Anal. Chem. 56, 1667-1670 (1984).
[CrossRef]

Appl. Phys. Lett.

S. J. Oldenburg, G. D. Hale, J. B. Jackson, and H. J. Halas, “Light scattering from dipole and quadrupole nanoshell antennas,” Appl. Phys. Lett. 75, 1063-1065 (1999).
[CrossRef]

IEEE Trans. Nanotechnol.

S. J. Norton and T. Vo-Dinh, “Plasmonic resonances of nanoshells of spheroidal shape,” IEEE Trans. Nanotechnol. 6, 627-638 (2007).
[CrossRef]

J. Appl. Phys.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242-1246 (1951).
[CrossRef]

J. Biomed. Opt.

B. Khlebtsov and N. Khlebtsov, “Ultrasharp light-scattering resonances of structured nanospheres: effects of size-dependent dielectric functions,” J. Biomed. Opt. 11, 044002(2006).
[CrossRef] [PubMed]

J. Nanophoton.

S. J. Norton and T. Vo-Dinh, “Spectral bounds on plasmon resonances for Ag and Au prolate and oblate nanospheroids,” J. Nanophoton. 2, 029501 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. C

A. Moroz, “Electron mean free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

Nano Lett.

C. Radloff and N. J. Halas, “Plasmonic properties of concentric nanoshells,” Nano Lett. 4, 1323-1327 (2004).
[CrossRef]

Nanotechnology

H. N. Wang and T. Vo-Dinh, “Multiplex detection of breast cancer biomarkers using plasmonic molecular sentinel nanoprobes,” Nanotechnology 20, 065101 (2009).
[CrossRef] [PubMed]

Opt. Commun.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368-375(2006).
[CrossRef]

Optik (Jena)

R. J. Jhu, J. Wang, and G. F. Jin, “Mie scattering calculation by FDTD employing a modified Debye model for gold material,” Optik (Jena) 116, 419-422 (2005).
[CrossRef]

Phy. Rev. Lett.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phy. Rev. Lett. 96, 113002 (2006).
[CrossRef]

Phys. Lett. A

J. Zhu, Y. Wang, L. Huang, and Y. Lu, “Resonance light scattering characters of core-shell structure of Au-Ag nanoparticles,” Phys. Lett. A 323, 455-459 (2004).
[CrossRef]

Phys. Rev. B

D.-S. Wang and M. Kerker, “Absorption and luminescence of dye-coated silver and gold particles,” Phys. Rev. B 25, 2433-2449 (1982).
[CrossRef]

Trends Anal. Chem.

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” Trends Anal. Chem. 17, 557-570(1998).
[CrossRef]

Z. Phys.

U. Kreibig and C. V. Fragstein, “The limitation of electron mean free path in small silver particles,” Z. Phys. 224, 307-323 (1969).
[CrossRef]

Z. Phys. B

M. Quinten, “Optical constants of gold and silver clusters in the spectral range between 1.5 eV and 4.5 eV,” Z. Phys. B 101, 211-217 (1996).
[CrossRef]

Other

Even if the polarizabilities of the individual molecules are not isotropic, an effective isotropic polarizability will result if the molecules are randomly oriented.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 102.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), p. 99.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), pp. 136-137.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991), pp. 280-291.

The constants for silver are Δϵk=[1759.47,135.34,258.19,22.90,1749.06,11756.18], ak=[1,1,1,1,1,1], bk=[0.2431,19.681,2.289,0.3292,4.6391,12.25], and ck=[0,17.079,515.022,1718.357,2116.092,10559.42], and for gold are Δϵk=[1589.52,50.20,20.92,148.49,1256.97,9169], ak=[1,1,1,1,1,1], bk=[0.268,1.221,1.747,4.406,12.63,11.21], and ck=[0,4.42,17.67,226.10,475.14,4550.77].

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, 1976).

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Figures (7)

Fig. 1
Fig. 1

Outer layer and core are metallic and the intermediate layer contains a luminescent or Raman-active compound.

Fig. 2
Fig. 2

(a) Enhancement with the silver outer shell and core present with r 1 = 20 mm , r 2 = 15 nm , and r 3 = 10 nm ; (b) enhancement with the silver core only present. Note that the enhancement is significantly greater with the outer shell present and the appearance of the second, lower-energy mode.

Fig. 3
Fig. 3

(a) Enhancement with the gold outer shell and core present with r 1 = 20 nm , r 2 = 15 nm , and r 3 = 10 nm ; (b) enhancement with the gold core only present. Note that for gold, the lower- energy mode dominates, although the higher-energy mode still exceeds that for the core-only system.

Fig. 4
Fig. 4

Repeat of Fig. 2 with all dimensions doubled ( r 1 = 40 nm , r 2 = 30 nm , r 3 = 20 ). (a) Enhancement with the silver outer shell and core present; (b) enhancement with the silver core only present.

Fig. 5
Fig. 5

Repeat of Fig. 3 with all dimensions doubled ( r 1 = 40 nm , r 2 = 30 nm , r 3 = 20 ). (a) Enhancement with the gold outer shell and core present; (b) enhancement with the gold core only present.

Fig. 6
Fig. 6

Far-field enhancement of a single molecule whose radial position varies within the active layer over the interval r 3 r r 2 along the z axis at wavelength 417 nm . The silver shell-core particle is assumed with r 1 = 20 nm , r 2 = 15 nm , and r 3 = 10 nm .

Fig. 7
Fig. 7

(a) Plot of the profile of the electric field along the axis of a silver shell–core particle at the wavelength 417 nm (the peak of the higher energy plasmon mode). The solid and dashed curves are the real and imaginary parts of the axial component of the electric field, respectively. (b) Plot of the profile of the electric field along the axis of the particle at the wavelength 875 nm (the peak of the lower energy plasmon mode). The jump in the field at each boundary is proportional to the surface charge density on that interface.

Equations (69)

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

ψ 0 = ( E 0 r + B 0 / r 2 ) cos θ ,
ψ 1 = ( A 1 r + B 1 / r 2 ) cos θ ,
ψ 2 = ( A 2 r + B 2 / r 2 ) cos θ ,
ψ 3 = A 3 r cos θ ,
E 0 = r ^ ( E 0 + 2 B 0 / r 3 ) cos θ + θ ^ ( E 0 + B 0 / r 3 ) sin θ ,
E 1 = r ^ ( A 1 + 2 B 1 / r 3 ) cos θ + θ ^ ( A 1 + B 1 / r 3 ) sin θ ,
E 2 = r ^ ( A 2 + 2 B 2 / r 3 ) cos θ + θ ^ ( A 2 + B 2 / r 3 ) sin θ ,
E 3 = r ^ A 3 cos θ + θ ^ A 3 sin θ ,
ψ 0 | r = r 1 = ψ 1 | r = r 1 ,
ϵ 0 ψ 0 r | r = r 1 = ϵ 1 ψ 1 r | r = r 1 ,
ψ 1 | r = r 2 = ψ 2 | r = r 2 ,
ϵ 1 ψ 1 r | r = r 2 = ϵ 2 ψ 2 r | r = r 2 ,
ψ 2 | r = r 3 = ψ 3 | r = r 3 ,
ϵ 2 ψ 2 r | r = r 3 = ϵ 3 ψ 3 r | r = r 3 .
E 0 r 1 + B 0 / r 1 2 = A 1 r 1 + B 1 / r 1 2 , ϵ 0 ( E 0 2 B 0 / r 1 3 ) = ϵ 1 ( A 1 2 B 1 / r 1 3 ) , A 1 r 2 + B 1 / r 2 2 = A 2 r 2 + B 2 / r 2 2 , ϵ 1 ( A 1 2 B 1 / r 2 3 ) = ϵ 2 ( A 2 2 B 2 / r 2 3 ) , A 2 r 3 + B 2 / r 3 2 = A 3 r 3 , ϵ 2 ( A 2 2 B 2 / r 3 3 ) = ϵ 3 A 3 .
[ 1 / r 1 2 r 1 1 / r 1 2 0 0 0 2 ϵ 0 / r 1 3 ϵ 1 2 ϵ 1 / r 1 3 0 0 0 0 r 2 1 / r 2 2 r 2 1 / r 2 2 0 0 ϵ 1 2 ϵ 1 / r 2 3 ϵ 2 2 ϵ 2 / r 2 3 0 0 0 0 r 3 1 / r 3 2 r 3 0 0 0 ϵ 2 2 ϵ 2 / r 3 3 ϵ 3 ] [ B 0 A 1 B 1 A 2 B 2 A 3 ] = [ r 1 E 0 ϵ 0 E 0 0 0 0 0 ] .
p i = α · E 2 ( r i ) .
p i = α E 2 r ( r i ) r ^ + α E 2 θ ( r i ) θ ^ ,
E 2 r ( r i ) = ( A 2 + 2 B 2 / r i 3 ) cos θ i ,
E 2 θ ( r i ) = ( A 2 + B 2 / r i 3 ) sin θ i .
ψ i ( r ) = p i · ( 1 R ) = p i · i ( 1 R ) ,
1 R = 4 π l = 0 m = l l 1 2 l + 1 r i l r l + 1 Y l m * ( θ i , ϕ i ) Y l m ( θ , ϕ ) , for     r i < r ,
1 R = 4 π l = 0 m = l l 1 2 l + 1 r l r i l + 1 Y l m * ( θ i , ϕ i ) Y l m ( θ , ϕ ) , for     r < r i ,
Y l m ( θ , ϕ ) = [ ( 2 l + 1 ) ( l m ) ! 4 π ( l + m ) ! ] 1 / 2 P l m ( cos θ ) e i m ϕ ,
ψ i ( r ) = α [ E 2 r ( r i ) r i ( 1 R ) + E 2 θ ( r i ) 1 r i θ i ( 1 R ) ] .
ψ i ( r ) = l = 0 m = l l u l m 1 r l + 1 Y l m ( θ , ϕ ) , for     r i < r ,
ψ i ( r ) = l = 0 m = l l v l m r l Y l m ( θ , ϕ ) , for     r < r i ,
u l m = 4 π α r i l 1 ( 2 l + 1 ) [ E 2 r ( r i ) l Y l m * ( θ i , ϕ i ) + E 2 θ ( r i ) θ i Y l m * ( θ i , ϕ i ) ] ,
v l m = 4 π α ( 2 l + 1 ) r i l + 2 [ E 2 r ( r i ) ( l + 1 ) Y l m * ( θ i , ϕ i ) E 2 θ ( r i ) θ i Y l m * ( θ i , ϕ i ) ] .
ψ ˜ 0 = l = 0 m = l l B ˜ l m ( 0 ) 1 r l + 1 Y l m ( θ , ϕ ) ,
ψ ˜ 1 = l = 0 m = l l [ A ˜ l m ( 1 ) r l + B ˜ l m ( 1 ) 1 r l + 1 ] Y l m ( θ , ϕ ) ,
ψ ˜ 2 = l = 0 m = l l [ A ˜ l m ( 2 ) r l + B ˜ l m ( 2 ) 1 r l + 1 ] Y l m ( θ , ϕ ) + ψ i ,
ψ ˜ 3 = l = 0 m = l l A ˜ l m ( 3 ) r l Y l m ( θ , ϕ ) ,
B ˜ l m ( 0 ) / r 1 l + 1 = A ˜ l m ( 1 ) r 1 l + B ˜ l m ( 1 ) / r 1 l + 1 ϵ 0 B ˜ l m ( 0 ) ( l + 1 ) / r 1 l + 2 = ϵ ˜ 1 [ A ˜ l m ( 1 ) l r 1 l 1 B ˜ l m ( 1 ) ( l + 1 ) / r 1 l + 2 ] A ˜ l m ( 1 ) r 2 l + B ˜ l m ( 1 ) / r 2 l + 1 = A ˜ l m ( 2 ) r 2 l + B ˜ l m ( 2 ) / r 2 l + 1 + u l m / r 2 l + 1 ϵ ˜ 1 [ A ˜ l m ( 1 ) l r 2 l 1 B ˜ l m ( 1 ) ( l + 1 ) / r 2 l + 2 ] = ϵ ˜ 2 [ A ˜ l m ( 2 ) l r 2 l 1 B ˜ l m ( 2 ) ( l + 1 ) / r 2 l + 2 u l m ( l + 1 ) / r 2 l + 2 ] A ˜ l m ( 2 ) r 3 l + B ˜ l m ( 2 ) / r 3 l + 1 + v l m r 3 l = A ˜ l m ( 3 ) r 3 l ϵ ˜ 2 [ A ˜ l m ( 2 ) l r 3 l 1 B ˜ l m ( 2 ) ( l + 1 ) / r 3 l + 2 + v l m l r 3 l 1 ] = ϵ ˜ 3 A ˜ l m ( 3 ) l r 3 l 1 .
[ 1 / r 1 2 r 1 1 / r 1 2 0 0 0 2 ϵ 0 / r 1 3 ϵ ˜ 1 2 ϵ ˜ 1 / r 1 3 0 0 0 0 r 2 1 / r 2 2 r 2 1 / r 2 2 0 0 ϵ ˜ 1 2 ϵ ˜ 1 / r 2 3 ϵ ˜ 2 2 ϵ ˜ 2 / r 2 3 0 0 0 0 r 3 1 / r 3 2 r 3 0 0 0 ϵ ˜ 2 2 ϵ ˜ 2 / r 3 3 ϵ ˜ 3 ] [ B ˜ 1 m ( 0 ) A ˜ 1 m ( 1 ) B ˜ 1 m ( 1 ) A ˜ 1 m ( 2 ) B ˜ 1 m ( 2 ) A ˜ 1 m ( 3 ) ] = [ 0 0 u 1 m / r 2 2 2 ϵ ˜ 2 u 1 m / r 2 3 r 3 v 1 m ϵ ˜ 2 v 1 m ] .
ψ ˜ 0 = 1 r 2 m = 1 1 B ˜ 1 m ( 0 ) Y 1 m ( θ , ϕ ) ,
B ˜ 1 m ( 0 ) = ( T 13 / r 2 2 2 ϵ ˜ 2 T 14 / r 2 3 ) u 1 m ( r 3 T 15 + ϵ ˜ 2 T 16 ) v 1 m .
p T 13 / r 2 2 2 ϵ ˜ 2 T 14 / r 2 3 ,
q r 3 T 15 ϵ ˜ 2 T 16 ,
B ˜ 1 m ( 0 ) = p u 1 m + q v 1 m .
Y 10 ( θ i , ϕ i ) = ( 3 4 π ) 1 / 2 cos θ i ,
Y 11 ( θ i , ϕ i ) = ( 3 8 π ) 1 / 2 sin θ i e i ϕ i ,
Y 1 , 1 ( θ i , ϕ i ) = ( 3 8 π ) 1 / 2 sin θ i e i ϕ i .
u 10 = α ( 4 π 3 ) 1 / 2 [ A 2 + B 2 r i 3 3 B 2 r i 3 cos 2 θ i ] ,
v 10 = α ( 4 π 3 ) 1 / 2 1 r i 3 [ A 2 + B 2 r i 3 + 3 ( B 2 r i 3 A 2 ) cos 2 θ i ] ,
u 11 = 3 α 2 ( 8 π 3 ) 1 / 2 B 2 r i 3 sin θ i cos θ i e i ϕ i ,
v 11 = 3 α 2 ( 8 π 3 ) 1 / 2 1 r i 3 ( A 2 B 2 r i 3 ) sin θ i cos θ i e i ϕ i ,
u 1 , 1 = 3 α 2 ( 8 π 3 ) 1 / 2 B 2 r i 3 sin θ i cos θ i e i ϕ i ,
v 1 , 1 = 3 α 2 ( 8 π 3 ) 1 / 2 1 r i 3 ( A 2 B 2 r i 3 ) sin θ i cos θ i e i ϕ i .
p x = ( 3 8 π ) 1 / 2 ( B ˜ 11 ( 0 ) B ˜ 1 , 1 ( 0 ) ) ,
p y = i ( 3 8 π ) 1 / 2 ( B ˜ 11 ( 0 ) + B ˜ 1 , 1 ( 0 ) ) ,
p z = ( 3 4 π ) 1 / 2 B ˜ 10 ( 0 ) .
p x = 3 α r i 3 ( p B 2 + q A 2 q B 2 r i 3 ) sin θ i cos θ i cos ϕ i ,
p y = 3 α r i 3 ( p B 2 + q A 2 q B 2 r i 3 ) sin θ i cos θ i sin ϕ i ,
p z = α ( p + q r i 3 ) ( A 2 + B 2 r i 3 ) + 3 α r i 3 ( p B 2 + q A 2 q B 2 r i 3 ) cos 2 θ i .
σ d ( n ^ i , n ^ s ) = | f ( n ^ i , n ^ s ) | 2 ,
f ( n ^ i , n ^ s ) = k 2 4 π ϵ 0 [ p n ^ s ( n ^ s · p ) ] ,
σ d ( n ^ i , n ^ s ) = k 4 ( 4 π ϵ 0 ) 2 [ | p | 2 | n ^ s · p | 2 ] .
σ d = k 4 ( 4 π ϵ 0 ) 2 [ | p y | 2 + | p z | 2 ] .
σ ¯ d = 1 V 2 V 2 σ ( r i ) d 3 r i = 1 V 2 r 3 r 2 r i 2 d r i 0 2 π d ϕ i 0 π d θ i sin θ i σ d ( r i , θ i , ϕ i ) ,
J y ( r i ) 0 2 π d ϕ i 0 π d θ i sin θ i | p y | 2 ,
J z ( r i ) 0 2 π d ϕ i 0 π d θ i sin θ i | p z | 2 ,
σ ¯ d = k 4 V 2 ( 4 π ϵ 0 ) 2 r 3 r 2 [ J y ( r i ) + J z ( r i ) ] r i 2 d r i .
J y ( r i ) = 12 π α 2 5 r i 6 | p B 2 + q A 2 q B 2 r i 3 | 2 ,
J z ( r i ) = 4 π α 2 { | ( p + q r i 3 ) ( A 2 + B 2 r i 3 ) | 2 + 9 5 r i 6 | p B 2 + q A 2 q B 2 r i 3 | 2 2 r i 3 Re [ ( p + q r i 3 ) ( A 2 + B 2 r i 3 ) ( p B 2 + q A 2 q B 2 r i 3 ) * ] } .
ϵ ( ω ) = 1 + k = 1 6 Δ ϵ k a k ω 2 i b k ω + c k ,
ϵ = ϵ b + i Δ ϵ ,
Δ ϵ = ω p 2 v f ω 3 L .
L = 4 r 1 3 ( 1 q 3 ) ( 1 + q 2 ) ,

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