Abstract

Scattering of a plane wave by a two-dimensional particle (i.e., a rod) below a metal surface is examined by means of a volume-integral approach based on Green’s dyadic. A detailed study of the structure of the near field when surface plasmons are excited is reported. The role of image enhancement is described for p polarization. A method of measuring the dielectric constant of a metal based on the structure of the near field is proposed.

© 1994 Optical Society of America

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References

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  1. A. Sommerfeld, “Über die Ausbreitung der wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).
  2. A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).
  3. L. M. Brekhovskikh, Waves in Layered Media (Academic, San Diego, Calif., 1980), p. 245.
  4. L. B. Felsen, N. Marcuwitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  5. G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
    [CrossRef]
  6. S. Efrima, H. Metiu, “Classical theory of light scattering by an adsorbed molecule. I. Theory,” J. Chem. Phys. 70, 1602–1613 (1979).
    [CrossRef]
  7. F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
    [CrossRef]
  8. A. Dereux, “Théorie de l’optique de champ proche,” Ph.D. dissertation (Université de Namur, Namur, Belgium, 1991).
  9. D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–739 (1992).
    [CrossRef]
  10. C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
    [CrossRef]
  11. C. Girard, “Plasmon resonance and near-field optical microscopy: a self-consistent theoretical model,” Appl. Opt. 31, 5380–5387 (1992).
    [CrossRef] [PubMed]
  12. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
    [CrossRef]
  13. M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
    [CrossRef]
  14. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
    [CrossRef]
  15. A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
    [CrossRef]
  16. N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
    [CrossRef]
  17. A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
    [CrossRef]
  18. M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
    [CrossRef]
  19. F. Pincemin, A. Sentenac, J.-J. Greffet, “Backscattering enhancement by subsurface particles,” to be submitted to Opt. Commun.
  20. K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
    [CrossRef] [PubMed]
  21. J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
    [CrossRef]
  22. J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
    [CrossRef]
  23. A. Sentenac, J.-J. Greffet, “Scattering by deep inhomoge-neous gratings,” J. Opt. Soc. Am. A 9, 996–1006 (1992).
    [CrossRef]
  24. J. V. Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).
  25. S. M. Candel, C. Crance, “Direct Fourier synthesis in layered media and the method of stationary phase,” J. Sound Vib. 74, 447–498 (1981).
    [CrossRef]
  26. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  27. P. R. Hilton, D. W. Oxtoby, “Surface enhanced Raman spectra: a critical review of the image dipole description,” J. Chem. Phys. 72, 6346–6348 (1980).
    [CrossRef]
  28. F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
    [CrossRef]
  29. A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
    [CrossRef]

1993 (1)

M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
[CrossRef]

1992 (4)

1991 (3)

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

1990 (4)

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

1989 (1)

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

1988 (1)

F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
[CrossRef]

1985 (2)

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

1984 (1)

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

1981 (1)

S. M. Candel, C. Crance, “Direct Fourier synthesis in layered media and the method of stationary phase,” J. Sound Vib. 74, 447–498 (1981).
[CrossRef]

1980 (1)

P. R. Hilton, D. W. Oxtoby, “Surface enhanced Raman spectra: a critical review of the image dipole description,” J. Chem. Phys. 72, 6346–6348 (1980).
[CrossRef]

1979 (1)

S. Efrima, H. Metiu, “Classical theory of light scattering by an adsorbed molecule. I. Theory,” J. Chem. Phys. 70, 1602–1613 (1979).
[CrossRef]

1978 (1)

F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
[CrossRef]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1909 (1)

A. Sommerfeld, “Über die Ausbreitung der wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Arya, K.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Banos, A.

A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).

Barchiesi, D.

Birman, J. L.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Bladel, J. V.

J. V. Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, San Diego, Calif., 1980), p. 245.

Bruce, N. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

Candel, S. M.

S. M. Candel, C. Crance, “Direct Fourier synthesis in layered media and the method of stationary phase,” J. Sound Vib. 74, 447–498 (1981).
[CrossRef]

Celli, V.

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Chen, J. S.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Courjon, D.

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Crance, C.

S. M. Candel, C. Crance, “Direct Fourier synthesis in layered media and the method of stationary phase,” J. Sound Vib. 74, 447–498 (1981).
[CrossRef]

Dainty, J. C.

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

Dereux, A.

A. Dereux, “Théorie de l’optique de champ proche,” Ph.D. dissertation (Université de Namur, Namur, Belgium, 1991).

Efrima, S.

S. Efrima, H. Metiu, “Classical theory of light scattering by an adsorbed molecule. I. Theory,” J. Chem. Phys. 70, 1602–1613 (1979).
[CrossRef]

Felsen, L. B.

L. B. Felsen, N. Marcuwitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Ford, G. W.

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

Girard, C.

C. Girard, “Plasmon resonance and near-field optical microscopy: a self-consistent theoretical model,” Appl. Opt. 31, 5380–5387 (1992).
[CrossRef] [PubMed]

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Greffet, J.-J.

A. Sentenac, J.-J. Greffet, “Scattering by deep inhomoge-neous gratings,” J. Opt. Soc. Am. A 9, 996–1006 (1992).
[CrossRef]

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

F. Pincemin, A. Sentenac, J.-J. Greffet, “Backscattering enhancement by subsurface particles,” to be submitted to Opt. Commun.

Hilton, P. R.

P. R. Hilton, D. W. Oxtoby, “Surface enhanced Raman spectra: a critical review of the image dipole description,” J. Chem. Phys. 72, 6346–6348 (1980).
[CrossRef]

Ishimaru, A.

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Keilmann, F.

F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
[CrossRef]

King, F. W.

F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
[CrossRef]

Kussmaul, K. W.

F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Marcuwitz, N.

L. B. Felsen, N. Marcuwitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Maystre, D.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

Mendez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Metiu, H.

S. Efrima, H. Metiu, “Classical theory of light scattering by an adsorbed molecule. I. Theory,” J. Chem. Phys. 70, 1602–1613 (1979).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Nieto-Vesperinas, M.

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

Oxtoby, D. W.

P. R. Hilton, D. W. Oxtoby, “Surface enhanced Raman spectra: a critical review of the image dipole description,” J. Chem. Phys. 72, 6346–6348 (1980).
[CrossRef]

Pincemin, F.

F. Pincemin, A. Sentenac, J.-J. Greffet, “Backscattering enhancement by subsurface particles,” to be submitted to Opt. Commun.

Saillard, M.

M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
[CrossRef]

M. Saillard, D. Maystre, “Scattering from metallic and dielectric surfaces,” J. Opt. Soc. Am. A 7, 982–990 (1990).
[CrossRef]

Sanchez-Gil, J. A.

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Light scattering from random rough dielectric surfaces,” J. Opt. Soc. Am. A 8, 1270–1286 (1991).
[CrossRef]

Schatz, G. C.

F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
[CrossRef]

Sentenac, A.

A. Sentenac, J.-J. Greffet, “Scattering by deep inhomoge-neous gratings,” J. Opt. Soc. Am. A 9, 996–1006 (1992).
[CrossRef]

F. Pincemin, A. Sentenac, J.-J. Greffet, “Backscattering enhancement by subsurface particles,” to be submitted to Opt. Commun.

Sommerfeld, A.

A. Sommerfeld, “Über die Ausbreitung der wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Su, Z. B.

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Szentirmay, Z.

F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
[CrossRef]

Van Duyne, R. P.

F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
[CrossRef]

Van Labeke, D.

Weber, W. H.

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

Ann. Phys. (Leipzig) (1)

A. Sommerfeld, “Über die Ausbreitung der wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–737 (1909).

Ann. Phys. (N.Y.) (1)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

F. Keilmann, K. W. Kussmaul, Z. Szentirmay, “Imaging of optical wavetrains,” Appl. Phys. B 47, 169–176 (1988).
[CrossRef]

J. Chem. Phys. (3)

P. R. Hilton, D. W. Oxtoby, “Surface enhanced Raman spectra: a critical review of the image dipole description,” J. Chem. Phys. 72, 6346–6348 (1980).
[CrossRef]

S. Efrima, H. Metiu, “Classical theory of light scattering by an adsorbed molecule. I. Theory,” J. Chem. Phys. 70, 1602–1613 (1979).
[CrossRef]

F. W. King, R. P. Van Duyne, G. C. Schatz, “Theory of Raman scattering by molecules adsorbed on electrode surfaces,” J. Chem. Phys. 69, 4472–4481 (1978).
[CrossRef]

J. Mod. Opt. (1)

N. C. Bruce, J. C. Dainty, “Multiple scattering from random rough surfaces using the Kirchhoff approximation,” J. Mod. Opt. 38, 579–590 (1991).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Sound Vib. (1)

S. M. Candel, C. Crance, “Direct Fourier synthesis in layered media and the method of stationary phase,” J. Sound Vib. 74, 447–498 (1981).
[CrossRef]

Opt. Commun. (3)

M. Saillard, “Numerical evidence of Anderson localization for electromagnetic surface waves,” Opt. Commun. 96, 1–7 (1993).
[CrossRef]

J.-J. Greffet, “Scattering of s-polarized electromagnetic waves by a 2D obstacle near an interface,” Opt. Commun. 72, 274–278 (1989).
[CrossRef]

A. Lakhtakia, “Macroscopic theory of the coupled dipole approximation method,” Opt. Commun. 79, 1–5 (1990).
[CrossRef]

Phys. Rep. (1)

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

Phys. Rev. B (4)

A. R. McGurn, A. A. Maradudin, V. Celli, “Localization effects in the scattering of light from a randomly rough grating,” Phys. Rev. B 31, 4866–4871 (1985).
[CrossRef]

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self-consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

J. A. Sanchez-Gil, M. Nieto-Vesperinas, “Resonance effects in multiple light scattering from statistically rough metallic surfaces,” Phys. Rev. B 45, 8623–8633 (1992).
[CrossRef]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

K. Arya, Z. B. Su, J. L. Birman, “Localization of the surface plasmon polariton caused by random roughness and its role in surface-enhanced optical phenomena,” Phys. Rev. Lett. 54, 1559–1562 (1985).
[CrossRef] [PubMed]

Waves Random Media (1)

A. Ishimaru, J. S. Chen, “Scattering from very rough metallic and dielectric surfaces: a theory based on the modified Kirchhoff approximation,” Waves Random Media 1, 21–34 (1991).
[CrossRef]

Other (6)

A. Dereux, “Théorie de l’optique de champ proche,” Ph.D. dissertation (Université de Namur, Namur, Belgium, 1991).

A. Banos, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966).

L. M. Brekhovskikh, Waves in Layered Media (Academic, San Diego, Calif., 1980), p. 245.

L. B. Felsen, N. Marcuwitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973).

F. Pincemin, A. Sentenac, J.-J. Greffet, “Backscattering enhancement by subsurface particles,” to be submitted to Opt. Commun.

J. V. Bladel, Singular Electromagnetic Fields and Sources (Clarendon, Oxford, 1991).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Square modulus of the electric field along a line z = −Δz/2: (a) p polarization, (b) s polarization. εm = −3.472, εp = 1, Δx = Δz = λ/40, θi = 20°.

Fig. 3
Fig. 3

Square modulus of the electric field below a lossy metal. The parameters are the same as in Fig. 2, except that εm = −3.472 + i.

Fig. 4
Fig. 4

Comparison between the exact near-field calculation (curve) and the sum of the zero-order field and the surface plasmon (markers) for a lossy and a nonlossy metal. The parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Influence of the depth of the particle, (a) The square modulus of the electric field in the particle, (b) The modulus of the amplitude of the surface plasmon. The parameters are the same as in Fig. 2, except that Δx = Δz = λ/200.

Fig. 6
Fig. 6

Influence of the angle of incidence, (a) The square modulus of the electric field in the particle, (b) The modulus of the amplitude of the surface plasmon. The parameters are the same as in Fig. 2, except that the depth of the particle is −2.5Δz.

Fig. 7
Fig. 7

Square modulus of the total field scattered by a particle located at −Δz/2 (i.e., a groove), represented on the (x, y) plane by a linear gray scale. The geometry is shown in Fig. 1; p polarization, θi = 28°, ε = 3.47 + i0.18, Δx = Δz = λ/16.

Fig. 8
Fig. 8

Square modulus of the total field scattered by a particle located at −Δz/2 (i.e., a groove); s polarization, with the same parameters as Fig. 7.

Fig. 9
Fig. 9

Square modulus of the field scattered by a particle located at −Δz/2: (a) s polarization, (b) p polarization. The parameters are the same as in Fig. 7. The value in the particle is 0.47 and 20 for s and p polarization, respectively.

Fig. 10
Fig. 10

Interference between the surface plasmons scattered by two particles. The square modulus of the field is shown. Δx = Δz = λ/16.5, θi = 28°, with the distance between the two particles being equal to 4.85λ. (a) Scattered field only, (b) total field.

Fig. 11
Fig. 11

Interference between the surface plasmons. The figure shows the square modulus of the scattered field along a line below the surface at a depth of −Δz/2. Δx = Δz = λ/15.7, θi = 0°, d = 6.35λ.

Equations (71)

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

ɛ f ( z ) = ɛ 1 for z > 0 , ɛ f ( z ) = ɛ 3 for z < 0 .
i ( x , z , t ) = Re [ E i ( x , z ) exp ( i ω t ) ] = Re { ( E inc x u x + E inc z u z ) × exp [ i ( ω / c ) ( sin θ i x + cos θ i z ) i ω t ] } ,
f ( x , z , t ) = Re [ E f ( x , z ) exp ( i ω t ) ]
E d = E E f ,
curl curl E ɛ ( x , z ) k 0 2 E = 0 ,
curl curl E f ɛ f ( z ) k 0 2 E f = 0 ,
curl curl E d ɛ f ( z ) k 0 2 E d = [ ɛ ( x , z ) ɛ f ( z ) ] k 0 2 E .
curl curl G ( x x , z , z ) ɛ f ( z ) k 0 2 G ( x x , z , z ) = I δ ( x x ) δ ( z z ) ,
G E = ( G x x E x + G x z E z ) u x + ( G z x E x + G z z E z ) u z ,
E ( x , z ) = E f ( x , z ) + Ω 2 k 0 2 [ ɛ ( x , z ) ɛ f ( z ) ] × G ( x x , z , z ) E ( x , z ) d x d z ,
E x j = E f x j + M x x j i E x i + M x z j i E z i ,
E z j = E f z j + M z x j i E x i + M z z j i E z i ,
E x i = E ( X i , Z i ) ,
E f x i = E f ( X i , Z i ) ,
M u υ j i = [ ɛ ( X i , Z i ) ɛ f ( Z i ) ] k 0 2 G u υ j i ,
G u υ j i = X i Δ x / 2 X i + Δ x / 2 d x Z i Δ z / 2 Z i + Δ z / 2 G u υ ( X j x , Z j z ) d z ,
G ( x x , z , z ) = i 4 π g ( κ , z , z ) exp [ i κ ( x x ) ] d κ ,
E ( r ) = e ( κ ) exp [ i ( κ x γ 3 z ) ] d κ ,
E ( r ) ( 2 π n 3 k 0 r ) 1 / 2 γ 3 e ( κ ) exp [ i ( κ x + γ 3 z + π / 4 ) ] , κ γ 3 = x z , r = ( x 2 + z 2 ) 1 / 2 .
d σ d θ = 2 π n 3 γ 3 2 k 0 | e ( κ ) | 2 .
κ sp 2 = ω 2 c 2 ɛ 1 ɛ 3 ɛ 1 + ɛ 3 .
E sp = E x sp ( 1 , 0 , κ sp / γ 1 sp ) exp [ i ( κ sp x + γ 1 sp z ) ] for z > 0 ,
E sp = E x sp ( 1 , 0 , κ sp / γ 3 sp ) exp [ i ( κ sp x γ 3 sp z ) ] for z < 0 ,
f ( κ ) ɛ 1 γ 3 + ɛ 3 γ 1 exp ( i κ x ) d κ = l ( κ ) ( κ κ sp ) ( κ + κ sp ) exp ( i κ x ) d κ = l ( κ ) 2 κ sp [ 1 ( κ κ sp ) 1 ( κ + κ sp ) ] exp ( i κ x ) d κ ,
f ( κ ) ɛ 1 γ 3 + ɛ 3 γ 1 exp ( i κ x ) d κ = { f ( κ ) ɛ 1 γ 3 + ɛ 3 γ 1 1 2 κ sp [ l ( κ sp ) ( κ κ sp ) l ( κ sp ) ( κ + κ sp ) ] } × exp ( i κ x ) d κ + i π l ( κ sp ) κ sp H ( x ) × exp ( i κ sp x ) + i π l ( κ sp ) κ sp H ( x ) exp ( i κ sp x ) .
l ( ± κ sp ) = 2 f ( ± κ sp ) γ 1 ( ± κ sp ) γ 3 ( ± κ sp ) ɛ 1 γ 1 ( ± κ sp ) + ɛ 3 γ 3 ( ± κ sp ) .
G ( x x , z , z ) = G reg ( x x , z , z ) + G sp ( x x , z , z ) ,
E = E f + ( ɛ m ɛ p ) k 0 2 A G sp E p = E f + E sp ,
| E | 2 = | E f + E s p | 2 = | E f | 2 + | E s p | 2 + 2 Re ( E s p E f * ) .
E f = E f x ( 1 , 0 , κ inc / γ 1 inc ) exp [ i ( κ inc x + γ 1 inc z ) ] ,
E sp = | E x sp | exp ( i ϕ ) ( 1 , 0 , η κ sp / γ 1 sp ) exp [ i ( η κ sp x + γ 1 sp z ) ] ,
| E | 2 = ( | E f | 2 + | E s p | 2 ) { 1 + V cos [ ( κ inc η κ s p ) x + Φ ( z ) ] } ,
V = 2 | E x s p | E f x exp [ η κ s p x ( γ 1 s p + γ 1 inc ) z ] | E f | 2 + | E s p | 2 × | 1 + η κ s p κ inc γ 1 * inc γ 1 s p | .
Φ ( z ) = ϕ + ( γ 1 s p γ 1 inc ) z + arg ( 1 + κ inc η κ s p γ 1 inc γ 2 * s p ) .
| E | 2 = | E 0 + E s p | 2 = | E 0 | 2 + | E s p | 2 + 2 Re ( E s p * E 0 ) .
E 0 = E x 0 { 1 + r x exp ( i 2 γ 3 inc z ) , 0 , [ 1 r x × exp ( 2 i γ 3 inc z ) ] κ inc / γ 3 inc } exp [ i ( κ inc x + γ 3 inc z ) ] ,
E s p = | E x s p | exp ( i ϕ ) ( 1 , 0 , η κ s p / γ 1 s p ) exp [ i ( η κ s p x + γ 1 s p z ) ] ,
| E | 2 = ( | E 0 | 2 + | E s p | 2 ) { 1 + V ( x , z ) × cos [ ( κ inc η κ s p ) x + Φ ( z ) ] } .
V ( x , z ) = 2 | E x s p | E x 0 exp [ η κ s p x + γ 3 s p z ] | E 0 | 2 + | E s p | 2 | K | ,
Φ = ϕ + ( γ 3 s p + γ 3 inc ) z + arg ( K ) ,
K = 1 + r x exp ( 2 i γ inc z ) η κ s p * κ inc γ s p * γ inc [ 1 r x exp ( 2 i γ inc z ) ] .
κ s p 2 = ω 2 c 2 ɛ m ɛ m + 1 .
curl curl G ( x x , z , z ) ɛ f ( z ) k 0 2 G ( x x , z , z ) = I δ ( x x ) δ ( z z ) ,
G ( x x , z , z ) = i 4 π g ( κ , z , z ) exp [ i κ ( x x ) ] d κ ,
g x x ( κ , z , z ) = γ 1 ɛ 1 k 0 2 { exp ( i γ 1 | z z | ) + r x ( κ ) exp [ i γ 1 ( z + z ) ] } for z > 0 and z > 0 , g x x ( κ , z , z ) = γ 1 ɛ 1 k 0 2 t x ( κ ) exp [ i ( γ 1 z γ 3 z ) ] for z < 0 and z > 0 ,
g z z ( κ , z , z ) = κ 2 ɛ 1 k 0 2 γ 1 { exp ( i γ 1 | z z | ) + r z ( κ ) × exp [ i γ 1 ( z + z ) ] } + 2 i ɛ 1 k 0 2 δ ( z z ) for z > 0 and z > 0 , g z z ( κ , z , z ) = κ 2 ɛ 1 γ 1 k 0 2 t z ( κ ) exp [ i ( γ 1 z γ 3 z ) ] for z < 0 and z > 0 ,
g x z ( κ , z , z ) = κ ɛ 1 k 0 2 { sgn ( z z ) exp ( i γ 1 | z z | ) + r x ( κ ) × exp [ i γ 1 ( z + z ) ] } for z > 0 and z > 0 , g x z ( κ , z , z ) = κ ɛ 1 k 0 2 t x ( κ ) exp [ i ( γ 1 z γ 3 z ) ] for z < 0 and z > 0 ,
g z x ( κ , z , z ) = κ ɛ 1 k 0 2 { sgn ( z z ) exp ( i γ 1 | z z | ) + r z ( κ ) × exp [ i γ 1 ( z + z ) ] } for z > 0 and z > 0 , g z x ( κ , z , z ) = κ ɛ 1 k 0 2 t z ( κ ) exp [ i ( γ 1 z γ 3 z ) ] for z < 0 and z > 0 .
r x ( κ ) = ɛ 1 γ 3 ɛ 3 γ 1 ɛ 1 γ 3 + ɛ 3 γ 1 ,
r z = r x ,
t x ( κ ) = 2 ɛ 1 γ 3 ɛ 1 γ 3 + ɛ 3 γ 1 ,
t z ( κ ) = 2 ɛ 1 γ 1 ɛ 1 γ 3 + ɛ 3 γ 1 .
γ j = ( ɛ j k 0 2 κ 2 ) 0.5 , Im ( γ j ) > 0 , Re ( γ j ) > 0 , j = 1 , 2 , 3 .
G x x j i = 1 2 π ɛ 1 k 0 2 S 1 [ exp ( i γ 1 Δ z ) 1 ] { r x exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] + exp [ i γ 1 ( | Z j Z i | Δ z 2 ) ] } exp [ i κ ( X j X i ) ] d κ for Z i Z j , Z i and Z j > 0 ,
G x x j i = δ j i ɛ 1 k 0 2 + 1 2 π ɛ 1 k 0 2 d κ S 1 { r x exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] [ exp ( i γ 1 Δ z ) 1 ] + 2 exp ( i γ 1 Δ z 2 ) } exp [ i κ ( X j X i ) ] for Z i = Z j , Z i and Z j > 0 ,
G z z j i = 1 2 π ɛ 1 k 0 2 κ sin ( κ Δ x / 2 ) S 2 { 2 ɛ 3 ɛ 1 γ 3 + ɛ 3 γ 1 exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] + exp [ i γ 1 ( | Z j Z i | Δ z 2 ) ] exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] γ 1 } exp [ i κ ( X j X i ) ] d κ for Z i Z j , Z i and Z j > 0 ,
G z z j i = 1 2 π ɛ 1 k 0 2 d κ κ sin ( κ Δ x / 2 ) ( S 2 2 ɛ 3 ɛ 1 γ 3 + ɛ 3 γ 1 exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] + 1 γ 1 { 2 γ 1 [ exp ( i γ 1 Δ z / 2 ) ɛ 1 k 0 2 κ 2 ] S 2 exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] } ) exp [ i κ ( X j X i ) ] for Z i = Z j , Z i and Z j > 0 ,
G z x j i = 1 2 π ɛ 1 k 0 2 S 2 sin ( κ Δ x / 2 ) { r z exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] + sgn ( Z j Z i ) exp [ i γ 1 ( | Z j Z i | Δ z 2 ) ] } × exp [ i κ ( X j X i ) ] d κ for Z i Z j , Z i and Z j > 0 ,
G z x j i = 1 2 π ɛ 1 k 0 2 S 2 sin ( κ Δ x / 2 ) r z exp [ i γ 1 ( Z j + Z i Δ z 2 ) ] exp [ i κ ( X j X i ) ] d κ for Z i = Z j , Z i and Z j > 0 ,
G x x j i = 1 2 π ɛ 1 k 0 2 S 1 [ exp ( i γ 1 Δ z ) 1 ] t x exp [ i ( γ 1 Z j γ 3 Z i γ 1 Δ z 2 ) ] exp [ i κ ( X j X i ) ] d κ for Z i > 0 and Z j < 0 ,
G z z j i = 1 2 π ɛ 1 k 0 2 κ 2 γ 1 S 1 S 2 t z exp [ i ( γ 1 Z i γ 3 Z j γ 1 Δ z 2 ) ] exp [ i κ ( X j X i ) ] d κ for Z i > 0 and Z j < 0 ,
G z x j i = 1 2 π ɛ 1 k 0 2 sin ( κ Δ x / 2 ) S 2 t z exp [ i ( γ 1 Z i γ 3 Z i γ 1 Δ z 2 ) ] exp [ i κ ( X j X i ) ] d κ for Z i > 0 and Z j < 0 ,
S 2 = [ exp ( i γ 1 Δ z ) 1 ] γ 1 ,
S 1 = sin ( κ Δ x 2 ) κ .
V ( M ) = q 2 π ɛ 1 ln ( A M ) ɛ 1 ɛ 3 ɛ 1 + ɛ 3 q 2 π ɛ 1 ln ( A M ) , z > 0 ,
V ( M ) = 2 ɛ 3 ɛ 1 + ɛ 3 q 2 π ɛ 3 ln ( A M ) , z < 0 ,
V dip ( M ) = 1 2 π ɛ 1 ( p AM A M 2 + ɛ 1 ɛ 3 ɛ 1 + ɛ 3 p A M A M 2 ) , z > 0 ,
E ( A ) = 1 2 π ɛ 1 ɛ 1 ɛ 3 ɛ 1 + ɛ 3 [ p A A 2 2 ( p A A ) A A A A 4 ]
p = α [ E ext + E ( A ) ] .
α eff = α 1 + α 2 π ɛ 1 ɛ 1 ɛ 3 ɛ 1 + ɛ 3 1 d 2 .
d 2 = α 2 π ɛ m 1 ɛ m 1 + ɛ m .

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