Abstract

To model elastic and inelastic light scattering on a metal nanosphere with spatially dispersive permittivity, an extension of the Lorenz–Mie theory is applied. The theory takes into account longitudinal vector spherical harmonics inside the sphere and determines the generalized Mie coefficients using a condition of vanishing electron flow through the sphere surface. In general, this condition is distinct from the conventional additional boundary condition of the continuity of the normal component of the electric field. Therefore, contrary to the common belief, the problem of divergence of the local density of electromagnetic states at the surface of an absorbing sphere is not solved by considering the spatial dispersion of the permittivity. We illustrate the theory by a study of the optical properties of a silver nanosphere using a hydrodynamic model for the dielectric function of the electron gas. Predictions of the nonlocal theory differ markedly from those of the local one if the sphere’s radius or the distance to the surface is smaller than a few nanometers. In particular, we demonstrate a shift of the Fröhlich resonance of nanometer-sized Ag particles caused by the spatial dispersion. Excitation of high-order spherical harmonics in larger particles is discussed. We show how the spatial dispersion decreases the rate of fluorescence quenching in close proximity to the particle surface.

© 2011 Optical Society of America

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2011

V. V. Datsyuk, “A generalization of the Mie theory for a sphere with spatially dispersive permittivity,” Ukr. J. Phys. 56, 122–129(2011).

2010

2009

V. Giannini, J. A. Sánchez-Gil, O. L. Muskens, and J. G. Rivas, “Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced fluorescence,” J. Opt. Soc. Am. B 26, 1569–1577(2009).
[CrossRef]

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
[CrossRef]

J. B. Khurgin and G. Sun, “Enhancement of optical properties of nanoscaled objects by metal nanoparticles,” J. Opt. Soc. Am. B 26, B83–B95 (2009).
[CrossRef]

C. M. Galloway, P. Etchegoin, and E. C. Le Ru, “Ultrafast nonradiative decay rates on metallic surfaces by comparing surface-enhanced Raman and fluorescence signals of single molecules,” Phys. Rev. Lett. 103, 063003 (2009).
[CrossRef] [PubMed]

H. Mertens and A. Polman, “Strong luminescence quantum-efficiency enhancement near prolate metal nanoparticles: dipolar versus higher-order modes,” J. Appl. Phys. 105, 044302(2009).
[CrossRef]

2008

2007

S. Foteinopoulou, J. P. Vigneron, and C. Vandenbem, “Optical near-field excitations on plasmonic nanoparticle-based structures,” Opt. Express 15, 4253–4267 (2007).
[CrossRef] [PubMed]

T. Härtling, P. Reichenbach, and L. M. Eng, “Near-field coupling of a single fluorescent molecule and a spherical gold nanoparticle,” Opt. Express 15, 12806–12817 (2007).
[CrossRef] [PubMed]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[CrossRef] [PubMed]

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

Z. E. Goude and P. T. Leung, “Surface enhanced Raman scattering from metallic nanoshells with nonlocal dielectric response,” Solid State Commun. 143, 416–420 (2007).
[CrossRef]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

E. C. Le Ru, P. G. Etchegoin, J. Grand, N. Félidj, J. Aubard, and G. Lévi, “Mechanisms of spectral profile modification in surface-enhanced fluorescence,” J. Phys. Chem. C 111, 16076–16709(2007).
[CrossRef]

2006

S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

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

R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[CrossRef]

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

2005

A. Moroz, “Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell,” Ann. Phys. 315, 352–418 (2005).
[CrossRef]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

K. Aslan, I. Gryczynski, J. Malicka, E. Matveeva, J. R. Lakowicz, and C. D. Geddes, “Metal-enhanced fluorescence: an emerging tool in biotechnology,” Curr. Opin. Biotechnol. 16, 55–62 (2005).
[CrossRef] [PubMed]

2004

L. A. Blanco and F. J. García de Abajo, “Spontaneous light emission in complex nanostructures,” Phys. Rev. B 69, 205414 (2004).
[CrossRef]

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef] [PubMed]

I. A. Larkin, M. I. S. M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: giant enhancement of relaxation in microscopic theory,” Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

2002

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometer scale,” Nature 418, 159–162 (2002).
[CrossRef] [PubMed]

M. H. Hider and P. T. Leung, “Nonlocal electrodynamic modeling of fluorescence characteristics for molecules in a spherical cavity,” Phys. Rev. B 66, 195106 (2002).
[CrossRef]

2001

A. Pack, M. Hietschold, and R. Wannemacher, “Failure of local Mie theory: optical spectra of colloidal aggregates,” Opt. Commun. 194, 277–287 (2001).
[CrossRef]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Decay of an excited atom near an absorbing microsphere,” Phys. Rev. A 64, 013804 (2001).
[CrossRef]

V. V. Klimov, M. Ducloy, and V. S. Letokhov, “Spontaneous emission of an atom in the presence of nanobodies,” Quantum Electron. 31, 569–586 (2001).
[CrossRef]

J. R. Lakowicz, “Radiative decay engineering: biophysical and biomedical applications,” Anal. Biochem. 298, 1–24 (2001).
[CrossRef] [PubMed]

A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361–372(2001).
[CrossRef]

2000

R. Chang, P. T. Leung, S. H. Lin, and W. S. Tse, “Surface-enhanced Raman scattering at cryogenic substrate temperatures,” Phys. Rev. B 62, 5168–5173 (2000).
[CrossRef]

1997

B. Labani, M. Boustimi, and J. Baudon, “van der Waals interaction between a molecule and a spherical cavity in a metal: nonlocality and anisotropy effects,” Phys. Rev. B 55, 4745–4750 (1997).
[CrossRef]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

1995

P. T. Leung and W. S. Tse, “Nonlocal electrodynamic effect on the enhancement factor for surface enhanced Raman scattering,” Solid State Commun. 95, 39–44 (1995).
[CrossRef]

1993

P. T. Leung and M. H. Hider, “Nonlocal electrodynamic modeling of frequency shifts for molecules at rough surfaces,” J. Chem. Phys. 98, 5019–5022 (1993).
[CrossRef]

1992

C. Girard, “Multipolar propagators near a corrugated surface: implication for local-probe microscopy,” Phys. Rev. B 45, 1800–1810 (1992).
[CrossRef]

1990

P. T. Leung, “Decay of molecules at spherical surfaces: nonlocal effects,” Phys. Rev. B 42, 7622–7625 (1990).
[CrossRef]

1987

R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987).
[CrossRef]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

1984

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287(1984).
[CrossRef]

1983

G. S. Agarwal and S. V. O’Neil, “Effect of hydrodynamic dispersion of the metal on surface plasmons and surface-enhanced phenomena in spherical geometries,” Phys. Rev. B 28, 487–493(1983).
[CrossRef]

1982

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

1981

B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
[CrossRef]

1979

R. Ruppin, “Mie theory with spatial dispersion,” Opt. Commun. 30, 380–382 (1979).
[CrossRef]

1977

J. D. Eversole and H. P. Broida, “Size and shape effects in light scattering from small silver, copper, and gold particles,” Phys. Rev. B 15, 1644–1655 (1977).
[CrossRef]

1975

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975).
[CrossRef]

1965

V. B. Gil’denburg and I. G. Kondrat’ev, “Diffraction of electromagnetic waves by a bounded plasma in the presence of spatial dispersion,” Radiotekh. Elektron. (Moscow) 10, 658–664(1965).

1963

A. Yildiz, “Scattering of a plasma wave from a plasma sphere,” Nuovo Cimento 30, 5740–5765 (1963).
[CrossRef]

1946

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Achermann, M. I. S. M.

I. A. Larkin, M. I. S. M. Achermann, and V. I. Klimov, “Dipolar emitters at nanoscale proximity of metal surfaces: giant enhancement of relaxation in microscopic theory,” Phys. Rev. B 69, 121403(R) (2004).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and S. V. O’Neil, “Effect of hydrodynamic dispersion of the metal on surface plasmons and surface-enhanced phenomena in spherical geometries,” Phys. Rev. B 28, 487–493(1983).
[CrossRef]

Anger, P.

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

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

Aslan, K.

K. Aslan, I. Gryczynski, J. Malicka, E. Matveeva, J. R. Lakowicz, and C. D. Geddes, “Metal-enhanced fluorescence: an emerging tool in biotechnology,” Curr. Opin. Biotechnol. 16, 55–62 (2005).
[CrossRef] [PubMed]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Aubard, J.

E. C. Le Ru, P. G. Etchegoin, J. Grand, N. Félidj, J. Aubard, and G. Lévi, “Mechanisms of spectral profile modification in surface-enhanced fluorescence,” J. Phys. Chem. C 111, 16076–16709(2007).
[CrossRef]

Ausloos, M.

J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Baudon, J.

B. Labani, M. Boustimi, and J. Baudon, “van der Waals interaction between a molecule and a spherical cavity in a metal: nonlocality and anisotropy effects,” Phys. Rev. B 55, 4745–4750 (1997).
[CrossRef]

Bharadwaj, P.

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
[CrossRef]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[CrossRef] [PubMed]

P. Bharadwaj, P. Anger, and L. Novotny, “Nanoplasmonic enhancement of single-molecule fluorescence,” Nanotechnology 18, 044017 (2007).
[CrossRef]

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

Blanco, L. A.

L. A. Blanco and F. J. García de Abajo, “Spontaneous light emission in complex nanostructures,” Phys. Rev. B 69, 205414 (2004).
[CrossRef]

Boffety, M.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bouhelier, A.

Boustimi, M.

B. Labani, M. Boustimi, and J. Baudon, “van der Waals interaction between a molecule and a spherical cavity in a metal: nonlocality and anisotropy effects,” Phys. Rev. B 55, 4745–4750 (1997).
[CrossRef]

Broida, H. P.

J. D. Eversole and H. P. Broida, “Size and shape effects in light scattering from small silver, copper, and gold particles,” Phys. Rev. B 15, 1644–1655 (1977).
[CrossRef]

Cai, W.

Carminati, R.

E. Castanié, M. Boffety, and R. Carminati, “Fluorescence quenching by a metal nanoparticle in the extreme near-field regime,” Opt. Lett. 35, 291–293 (2010).
[CrossRef] [PubMed]

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

Castanié, E.

Chang, R.

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Figures (4)

Fig. 1
Fig. 1

Normalized decay rate f ( d ) close to a silver sphere calculated with different number of terms, l c , in the expansion of Eq. (1), l c = 1 (dotted line), l c = 40 (dashed-dotted line), l c = 200 (dashed line), and l c = 1000 (solid line).

Fig. 2
Fig. 2

Frequencies of dipolar plasmon resonance in elastic light scattering spectra found from Fröhlich condition [Eq. (23)] (dashed line), local and nonlocal Mie theories (dotted and solid lines, respectively). The circles indicate the resonant frequencies in the experimental extinction spectra [64]. The rectangle shows the experimental data of [65].

Fig. 3
Fig. 3

The factor f for a silver nanosphere in water calculated with the local (dash lines) and nonlocal (solid lines) electromagnetic theories at d = 0 , 1, and 10 nm , Figs. (a), (b), and (c), respectively. Expansion of Eq. (1) has l c = 1 , 4, and 40 terms; the larger l c the higher the curve in each figure. The dotted line in Fig. 3c shows the normalized extinction cross section Q ext .

Fig. 4
Fig. 4

Relative quantum yields ϕ / ϕ 0 of emission of an electric dipole with ϕ 0 = 10 2 near a silver nanosphere in water calculated with the local (a), (c), (e) and nonlocal (b), (d), (f) electromagnetic theories as functions of ω and d. Figures (a) and (b), (c) and (d), and (e) and (f) correspond to different numbers of terms l c = 1 , 4, and 40 in the expansions for R and A . Symbols ⋆ and • indicate positions of maxima and minima of Φ, respectively. The dotted, dashed and solid lines depict the coordinates of the values of Φ equal to 2, 1, and 0.5, respectively. The minimum values of Φ in Figs. (e) and (f) are 1.1 × 10 4 and 2.6 × 10 4 , respectively.

Equations (25)

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f = A A 0 = 1 + 1 2 y 2 l = 1 ( 2 l + 1 ) { a l ζ l 2 ( y ) + b l [ ( ζ l ( y ) ) 2 + l ( l + 1 ) y 2 ζ l 2 ( y ) ] } .
f = 1 + 1 2 y 3 l = 1 ( 2 l + 1 ) ( l + 1 ) α l r 2 l + 1 ,
α l = l ( ϵ ϵ μ ) l ϵ + ( l + 1 ) ϵ μ a 2 l + 1 .
l = 1 l ( l + 1 ) ( a r ) 2 l + 1 = 2 ( a r r 2 a 2 ) 3 .
ε l = [ 2 π ( 2 l + 1 ) a 0 j l 2 ( k a ) ϵ ( ω , k ) d k ] 1 .
ϵ ( ω , k ) = ϵ g + ϵ h ( ω , k ) ,
ϵ h ( ω , k ) = ω p 2 / ( ω 2 + i ω Γ β 2 k 2 ) ,
ϵ g = 1 + 2.2 ω i b 2 ω i b 2 ω 2 i Γ i b ω ,
ε l = ϵ 0 [ 1 + ( ϵ 0 ϵ g - 1 ) ( 2 l + 1 ) I l + 1 / 2 ( u ) K l + 1 / 2 ( u ) ] - 1 ,
( 2 l + 1 ) I l + 1 / 2 ( u ) K l + 1 / 2 ( u ) 1 .
lim l α l a 2 l + 1 = ϵ g ϵ μ ϵ g + ϵ μ .
k T = ( ω / c ) ϵ ( ω ) ,
ϵ L ( ω , k L ) = 0 ,
ϵ L ( ω , k ) = ϵ g + ϵ h ( ω , k ) .
ϵ T ( ω ) = ϵ g + ϵ h ( ω , 0 ) .
k 2 - ( ω / c ) 2 ϵ T ( ω , k ) = 0 ,
( jn ) | r S = 0 ,
D t = ϵ 0 ϵ g E t + j ,
b l = ( 1 + δ l ) x ψ l ( x ) ψ l ( z ) z ψ l ( z ) ψ l ( x ) ( 1 + δ l ) x ζ l ( x ) ψ l ( z ) z ψ l ( z ) ζ l ( x ) ,
δ l = ϵ T ϵ g ϵ g l ( l + 1 ) j l ( z ) j l ( z L ) ψ l ( z ) z L j l ( z L ) ,
a l = j l ( x ) ψ l ( z ) j l ( z ) ψ l ( x ) h l ( x ) ψ l ( z ) j l ( z ) ζ l ( x ) .
Q ext = 2 x 2 l = 1 ( 2 l + 1 ) ( a l b l ) .
ϵ T = 2 ϵ μ .
| b 1 ( ω res ) / x | = max | b 1 ( ω ) / x | .
φ = R R 0 ( A A 0 + 1 φ 0 1 ) 1 .

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