Abstract

Influences of particle size on surface-enhancement processes are discussed in terms of a simple physical model. When the size of a silver sphere is increased, the magnitude of the enhancement exhibits a slight increase followed by a strong decrease. Simultaneously the plasmon resonance is shifted and severely broadened. To interpret these effects, a self-consistent calculation of the particle polarization is performed. Initial increase in magnitude and shift of the resonance are due to dynamic depolarization, whereas the decrease in magnitude and broadening are caused by radiation damping. The importance of higher-order multipoles is assessed by analyzing their contributions separately.

© 1983 Optical Society of America

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References

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  1. R. K. Chang, T. E. Furtak, eds., Surface Enhanced Raman Scattering (Plenum, New York, 1982).
    [CrossRef]
  2. S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
    [CrossRef]
  3. P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
    [CrossRef]
  4. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  6. M. Kerker, D.-S. Wang, H. Chew, Appl. Opt. 19, 4159 (1980).
    [CrossRef] [PubMed]
  7. A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
    [CrossRef]
  8. R. Gans, H. Happel, Ann. Phys. 29, 277 (1909).
    [CrossRef]
  9. W. T. Doyle, A. Agarwal, J. Opt. Soc. Am. 55, 305 (1965).
    [CrossRef]
  10. P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  11. Radial and tangential fields produced by a retarded dipole [p] = peikr are given by5ER=2cosθ([p]/r3+[p˙]/cr2) and Eθ=sinθ([p]/r3+[p˙]/cr2+[p¨]/c2r). By expanding eikr, retaining terms up to order k3, and using E∥ = ER cos θ − Eθ sin θ, one obtains Eq. (2).
  12. The term ∝q2 in the denominator of Eq. (4) is called “dynamic depolarization” because (1) it is obtained only in a dynamic calculation (q > 0) and (2) the coefficient of (∊ − 1) is real, corresponding to a change in the effective particle depolarization factor A [see, e.g., C. J. F. Böttcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973)], Vol. 1.
  13. Consider a fraction of the form of Eq. (5), f = (1 + a2q2 + a3q3)/(1 + b2q2 + b3q3). Expanding by (1 + c2q2), one obtains f = [1 + (a2 + c2)q2 + a3q3]/[1 + (b2 + c2)q2 + b3q3], correct to order q3.

1983

P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

1982

A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

1980

S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
[CrossRef]

M. Kerker, D.-S. Wang, H. Chew, Appl. Opt. 19, 4159 (1980).
[CrossRef] [PubMed]

1972

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

1965

1909

R. Gans, H. Happel, Ann. Phys. 29, 277 (1909).
[CrossRef]

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Agarwal, A.

Barber, P. W.

P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Chang, R. K.

P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

Chew, H.

Christy, R. W.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Doyle, W. T.

Gans, R.

R. Gans, H. Happel, Ann. Phys. 29, 277 (1909).
[CrossRef]

Gordon, J. P.

A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

Happel, H.

R. Gans, H. Happel, Ann. Phys. 29, 277 (1909).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Kerker, M.

Liao, P. F.

A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

Massoudi, H.

P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

McCall, S. L.

S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Platzman, P. M.

S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
[CrossRef]

Wang, D.-S.

Wokaun, A.

A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Wolff, P. A.

S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
[CrossRef]

Ann. Phys.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

R. Gans, H. Happel, Ann. Phys. 29, 277 (1909).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Phys. Rev. B

P. B. Johnson, R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Phys. Rev. Lett.

P. W. Barber, R. K. Chang, H. Massoudi, Phys. Rev. Lett. 50, 997 (1983).
[CrossRef]

A. Wokaun, J. P. Gordon, P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

Phys.Lett. A

S. L. McCall, P. M. Platzman, P. A. Wolff, Phys.Lett. A 77, 381 (1980); J. I. Gersten, A. Nitzan, J. Chem. Phys. 73, 3023 (1980).
[CrossRef]

Other

R. K. Chang, T. E. Furtak, eds., Surface Enhanced Raman Scattering (Plenum, New York, 1982).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Radial and tangential fields produced by a retarded dipole [p] = peikr are given by5ER=2cosθ([p]/r3+[p˙]/cr2) and Eθ=sinθ([p]/r3+[p˙]/cr2+[p¨]/c2r). By expanding eikr, retaining terms up to order k3, and using E∥ = ER cos θ − Eθ sin θ, one obtains Eq. (2).

The term ∝q2 in the denominator of Eq. (4) is called “dynamic depolarization” because (1) it is obtained only in a dynamic calculation (q > 0) and (2) the coefficient of (∊ − 1) is real, corresponding to a change in the effective particle depolarization factor A [see, e.g., C. J. F. Böttcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973)], Vol. 1.

Consider a fraction of the form of Eq. (5), f = (1 + a2q2 + a3q3)/(1 + b2q2 + b3q3). Expanding by (1 + c2q2), one obtains f = [1 + (a2 + c2)q2 + a3q3]/[1 + (b2 + c2)q2 + b3q3], correct to order q3.

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Figures (1)

Fig. 1
Fig. 1

Electric-field enhancement on the surface of silver spheres. The radial surface field ER is normalized by the incident field E0. The squared magnitude of ER/E0, averaged over the surface, is shown as a function of particle radius a and wavelength λ. (a) Total squared field 〈|ER|2〉 = 〈|ER,Dip|2〉 + 〈|ER,Q|2〉. The absolute maximum occurs at a = 12.5 nm and λmax = 357 nm (electrostatic limit a = 0: λmax = 355 nm). (b) Dipolar contribution 〈|ER,Dip|2〉. (c) Quadrupolar contribution 〈|ER,Q|2〉. The vertical scale has been expanded by a factor of 5 compared with that of (a) and (b).

Equations (5)

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4 π P = ( 1 ) ( E 0 + E dep ) ,
d E dep , = [ 1 r 3 ( 3 cos 2 θ 1 ) + k 2 2 r ( cos 2 θ + 1 ) + i 2 3 k 3 ] d p ( r ) ,
E dep = ( 4 π 3 + k 2 4 π 3 a 2 + i 2 3 k 3 4 π 3 a 3 ) P .
P = 3 4 π ( 1 ) ( + 2 ) ( + 1 ) q 2 ( 1 ) i q 3 E 0 ( q = k a ) .
e B 1 = i q 3 ( 1 ) ( 1 q 2 / 10 ) ( + 2 ) ( 7 10 2 ) q 2 ( 1 ) i q 3 .

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