Abstract

The angular spectrum representation of the electromagnetic Green’s tensor has a part that is a superposition of exponentially decaying waves in the +z and -z directions (evanescent part) and a part that is a superposition of traveling waves, both of which are defined by integral representations. We have derived an asymptotic expansion for the z dependence of the evanescent part of the Green’s tensor and obtained a closed-form solution in terms of the Lommel functions, which holds in all space. We have shown that the traveling part can be extracted from the Green’s tensor by means of a filter operation on the tensor, without regard to the angular spectrum integral representation of this part. We also show that the so-called self-field part of the tensor is properly included in the integral representation, and we were able to identify this part explicitly.

© 2002 Optical Society of America

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  9. J. E. Sipe, “New Green function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
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  13. A. Baños, “Dipole radiation in the presence of a conducting half-space,” International Series of Monographs in Electromagnetic Waves, Vol. 9, A. L. Cullen, V. A. Fock, J. R. Wait, eds. (Pergamon, Oxford, UK, 1966).
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  15. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.
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    [CrossRef]
  17. T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
    [CrossRef]
  18. A. V. Shchegrov, P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A 16, 2583–2584 (1999).
    [CrossRef]
  19. R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).
  20. V. M. Agranovich, D. L. Mills, Surface Polaritons (North-Holland, Amsterdam, 1982).
  21. R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
    [CrossRef]
  22. R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
    [CrossRef]
  23. E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
    [CrossRef]
  24. M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
    [CrossRef]
  25. P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).
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    [CrossRef]
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  28. O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
    [CrossRef]
  29. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [CrossRef]
  30. J. D. Jackson, Classical Electrodynamics, 2th ed. (Wiley, New York, 1975), p. 141.
  31. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Sec. 3.2.4.
  32. O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
    [CrossRef]
  33. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 275–276.
  34. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975), Chap. 3.
  35. M. V. Berry, “Asymptotics of evanescence,” J. Mod. Opt. 48, 1535–1541 (2001).
    [CrossRef]
  36. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Special Functions, Vol. 2 of Integrals and Series (Gordon & Breach, New York, 1986), p. 188, no. 2.12.10.3. It should be noted that this formula contains a misprint. Here, [exp(ia …)…] should read [-i exp(ia …)…].
  37. D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
    [CrossRef]
  38. D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
    [CrossRef]
  39. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1922), p. 537.
  40. Page 487 of Ref. 15.
  41. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), p. 847.

2001 (2)

2000 (2)

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

A. Lakhtakia, W. S. Weiglhofer, “Evanescent plane waves and the far field: resolution of a controversy,” J. Mod. Opt. 47, 759–763 (2000).
[CrossRef]

1999 (4)

T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
[CrossRef]

M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

A. V. Shchegrov, P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A 16, 2583–2584 (1999).
[CrossRef]

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[CrossRef]

1998 (1)

1996 (1)

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

1994 (1)

D. Courjon, C. Bainier, “Near-field microscopy and near-field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

1991 (3)

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

H. F. Arnoldus, T. F. George, “Phase-conjugated fluorescence,” Phys. Rev. A 43, 3675–3689 (1991).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

1981 (1)

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).

1977 (2)

1976 (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

1975 (1)

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

1974 (1)

R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, D. L. Mills, Surface Polaritons (North-Holland, Amsterdam, 1982).

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), p. 847.

Arnoldus, H. F.

Bainier, C.

D. Courjon, C. Bainier, “Near-field microscopy and near-field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

Baños, A.

A. Baños, “Dipole radiation in the presence of a conducting half-space,” International Series of Monographs in Electromagnetic Waves, Vol. 9, A. L. Cullen, V. A. Fock, J. R. Wait, eds. (Pergamon, Oxford, UK, 1966).

Berry, M. V.

M. V. Berry, “Asymptotics of evanescence,” J. Mod. Opt. 48, 1535–1541 (2001).
[CrossRef]

Bertilone, D. C.

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975), Chap. 3.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Special Functions, Vol. 2 of Integrals and Series (Gordon & Breach, New York, 1986), p. 188, no. 2.12.10.3. It should be noted that this formula contains a misprint. Here, [exp(ia …)…] should read [-i exp(ia …)…].

Carney, P. S.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

A. V. Shchegrov, P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A 16, 2583–2584 (1999).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
[CrossRef]

Courjon, D.

D. Courjon, C. Bainier, “Near-field microscopy and near-field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

Fisher, D. G.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

Fisher, T. D.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

Foley, J. T.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
[CrossRef]

Friberg, A. T.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
[CrossRef]

George, T. F.

H. F. Arnoldus, T. F. George, “Phase-conjugated fluorescence,” Phys. Rev. A 43, 3675–3689 (1991).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 275–276.

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975), Chap. 3.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2th ed. (Wiley, New York, 1975), p. 141.

Kaivola, M.

T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
[CrossRef]

Keller, O.

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
[CrossRef]

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

Kunz, R. E.

Lakhtakia, A.

A. Lakhtakia, W. S. Weiglhofer, “Evanescent plane waves and the far field: resolution of a controversy,” J. Mod. Opt. 47, 759–763 (2000).
[CrossRef]

Lalor, É.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Lukosz, W.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Sec. 3.2.4.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Special Functions, Vol. 2 of Integrals and Series (Gordon & Breach, New York, 1986), p. 188, no. 2.12.10.3. It should be noted that this formula contains a misprint. Here, [exp(ia …)…] should read [-i exp(ia …)…].

Miller, A. H.

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

Mills, D. L.

V. M. Agranovich, D. L. Mills, Surface Polaritons (North-Holland, Amsterdam, 1982).

Moyer, P. J.

M. A. Paesler, P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications (Wiley, New York, 1996).

Paesler, M. A.

M. A. Paesler, P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications (Wiley, New York, 1996).

Pohl, D. W.

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. J. R. Sheppard, eds. (Academic, San Diego, Calif., 1991), p. 243.

Prock, A.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Special Functions, Vol. 2 of Integrals and Series (Gordon & Breach, New York, 1986), p. 188, no. 2.12.10.3. It should be noted that this formula contains a misprint. Here, [exp(ia …)…] should read [-i exp(ia …)…].

Setälä, T.

T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
[CrossRef]

Shchegrov, A. V.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

A. V. Shchegrov, P. S. Carney, “Far-field contribution to the electromagnetic Green’s tensor from evanescent modes,” J. Opt. Soc. Am. A 16, 2583–2584 (1999).
[CrossRef]

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
[CrossRef]

Sipe, J. E.

J. E. Sipe, “New Green function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

J. van Kranendonk, J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 25, pp. 246, ff.

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

van Kranendonk, J.

J. van Kranendonk, J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 25, pp. 246, ff.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1922), p. 537.

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), p. 847.

Weiglhofer, W. S.

A. Lakhtakia, W. S. Weiglhofer, “Evanescent plane waves and the far field: resolution of a controversy,” J. Mod. Opt. 47, 759–763 (2000).
[CrossRef]

Wolf, E.

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Sec. 3.2.4.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

Xiao, M.

M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 39, 1–65 (1978).

J. Chem. Phys. (2)

R. R. Chance, A. Prock, R. Silbey, “Lifetime of an emit-ting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744–2748 (1974).
[CrossRef]

R. R. Chance, A. H. Miller, A. Prock, R. Silbey, “Fluorescence and energy transfer near interfaces: the complete and quantitative description of the Eu+3/mirror systems,” J. Chem. Phys. 63, 1589–1595 (1975).
[CrossRef]

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Mod. Opt. (6)

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

P. S. Carney, D. G. Fisher, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Fisher, E. Wolf, “Comment: evanescent waves do contribute to the far field,” J. Mod. Opt. 47, 757–758 (2000).

A. Lakhtakia, W. S. Weiglhofer, “Evanescent plane waves and the far field: resolution of a controversy,” J. Mod. Opt. 47, 759–763 (2000).
[CrossRef]

M. V. Berry, “Asymptotics of evanescence,” J. Mod. Opt. 48, 1535–1541 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Opt. Lett. (1)

Phys. Rep. (1)

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

Phys. Rev. A (1)

H. F. Arnoldus, T. F. George, “Phase-conjugated fluorescence,” Phys. Rev. A 43, 3675–3689 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (1)

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

Phys. Rev. E (1)

T. Setälä, M. Kaivola, A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
[CrossRef]

Rep. Prog. Phys. (1)

D. Courjon, C. Bainier, “Near-field microscopy and near-field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

Surf. Sci. (1)

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Other (18)

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Special Functions, Vol. 2 of Integrals and Series (Gordon & Breach, New York, 1986), p. 188, no. 2.12.10.3. It should be noted that this formula contains a misprint. Here, [exp(ia …)…] should read [-i exp(ia …)…].

J. van Kranendonk, J. E. Sipe, “Foundations of the macroscopic electromagnetic theory of dielectric media,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1977), Vol. 25, pp. 246, ff.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 275–276.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, New York, 1975), Chap. 3.

V. M. Agranovich, D. L. Mills, Surface Polaritons (North-Holland, Amsterdam, 1982).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1922), p. 537.

Page 487 of Ref. 15.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), p. 847.

M. A. Paesler, P. J. Moyer, Near-Field Optics: Theory, Instrumentation, and Applications (Wiley, New York, 1996).

M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, Berlin, 1998).

K. T. V. Grattan, B. T. Meggitt eds., Optical Fiber Sensor Technology: Fundamentals (Kluwer Academic, Boston, Mass., 2000).

A. Baños, “Dipole radiation in the presence of a conducting half-space,” International Series of Monographs in Electromagnetic Waves, Vol. 9, A. L. Cullen, V. A. Fock, J. R. Wait, eds. (Pergamon, Oxford, UK, 1966).

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. J. R. Sheppard, eds. (Academic, San Diego, Calif., 1991), p. 243.

D. W. Pohl, D. Courjon, eds., Near Field Optics, Vol. 242 of Proceedings of the NATO Advanced Research Workshop on Near Field Optics, Series E, Applied Sciences (Kluwer Academic, Dordrecht, The Netherlands, 1993).

J. D. Jackson, Classical Electrodynamics, 2th ed. (Wiley, New York, 1975), p. 141.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, England, 1995), Sec. 3.2.4.

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Figures (3)

Fig. 1
Fig. 1

Evanescent part of the auxiliary function M0(ρ, ζ) (curve a), as a function of |ζ| for ρ=5. Curve b is the asymptotic approximation J0(ρ)/|ζ|, and curve c is the asymptotic approximation with both terms from Eq. (51). The exact value remains finite for |ζ|0, but the approximations diverge near the xy plane.

Fig. 2
Fig. 2

Illustration of the splitting of the function M0(ρ, ζ) into its evanescent part (curve a) and traveling part. The real and imaginary parts of the traveling part are shown as curves b and c, respectively. The value of the traveling part at the origin is equal to i, whereas the evanescent part diverges.

Fig. 3
Fig. 3

This graph shows the filter function F1(ρ, ρ) as a function of ρ, for ρ=100. The pronounced peak is located near ρ=100, and the peak height is approximately 1/π, as indicated by Eq. (101).

Equations (106)

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E(r, t)=1πRe0dωEˆ(r, ω)exp(-iωt),
Eˆ(r, ω)=iωμo4πd3rg(r-r, ω)·jˆ(r, ω).
Eˆ(r, ω)=iωμo4πd3rg(r-r, ω)jˆ(r, ω)+iωμo4πko2·d3rg(r-r, ω)jˆ(r, ω),
g(r, ω)=exp(ikor)/r.
Eˆ(r, ω)=-i3oωjˆ(r, ω)+iωμo4πd3rd(r-r, ω)·jˆ(r, ω).
d(r, ω)=I+1ko2g(r, ω),
g(r, ω)=-4π3ko2δ(r)I+d(r, ω).
g(r, ω)=i2πd2k1βexp(ik·r+iβ|z|),
β=ko2-k2,k<koik2-ko2,k>ko,
g(r, ω)=i2πd2k1βI+1ko2×exp(ik·r+iβ|z|).
g(r, ω)=-4πko2δ(r)ezez+i2πd2k1β×I-1ko2KKexp(iK·r),
K=k+β sgn(z)ez.
γ(r, ω)=-4πko3δ(r)ezez+12(I+ezez)M0(ρ, ζ)+12sgn(ζ)(rˆez+ezrˆ)M1(ρ, ζ)+12(I-ezez-2rˆrˆ)M2(ρ, ζ)+12(I-3ezez)N(ρ, ζ).
M0(ρ, ζ)=i0dααβˆJ0(αρ)exp(iβˆ|ζ|),
M1(ρ, ζ)=20dαα2J1(αρ)exp(iβˆ|ζ|),
M2(ρ, ζ)=-i0dαα3βˆJ2(αρ)exp(iβˆ|ζ|),
N(ρ, ζ)=i0dααβˆJ0(αρ)exp(iβˆ|ζ|).
M0(ρ, ζ)ev=1dαα(α2-1)1/2J0(αρ)×exp(-|ζ|α2-1),
γ(r, ω)=-4πko3δ(r)ezez+γ(r, ω)tr+γ(r, ω)ev,
M0(ρ, ζ)ev=0duJ0(ρ1+u2)exp(-u|ζ|),
q=kor=ρ2+ζ2,
γ(r, ω)=-4π3ko3Iδ(r)+I1+iq-1q2exp(iq)q-rˆrˆ1+3iq-3q2exp(iq)q,
rˆ=1q(ρrˆ+ζez).
M0(ρ, ζ)=exp(iq)q,
M1(ρ, ζ)=-2ρ|ζ|q31+3iq-3q2exp(iq),
M2(ρ, ζ)=ρ2q31+3iq-3q2exp(iq),
N(ρ, ζ)=-8π3ko3δ(r)+1q21q-iexp(iq)+ζ2q31+3iq-3q2exp(iq).
M0(ρ, ζ)=g(r, ω)/ko,
M2(ρ, ζ)=-ρ2|ζ|M1(ρ, ζ).
N(ρ, ζ)=-8π3ko3δ(r)+13M0(ρ, ζ)-|ζ|ρM1(ρ, ζ)-M2(ρ, ζ),
M1(ρ, ζ)ev=21dαα2J1(αρ)exp(-|ζ|α2-1),
M2(ρ, ζ)ev=-1dαα3(α2-1)1/2J2(αρ)×exp(-|ζ|α2-1).
M2(ρ, ζ)ev=-1|ζ|J2(ρ)-1|ζ|1dα exp(-|ζ|α2-1)×ddα[α2J2(αρ)].
M2(ρ, ζ)ev=-1|ζ|J2(ρ)-ρ2|ζ|M1(ρ, ζ)ev.
M2(ρ, ζ)tr=1|ζ|J2(ρ)-ρ2|ζ|M1(ρ, ζ)tr.
δ(r)=1(2π)3d3k exp(ik·r).
δ(r)=12πδ(z)0dkkJ0(kr).
δ(r)=ko32πδ(ζ)0AdααJ0(αρ),A.
δ(r)=ko32πδ(ζ)AρJ1(Aρ),A.
N(ρ, ζ)ev=-1dααα2-1J0(αρ)×exp(-|ζ|α2-1).
N(ρ, ζ)ev=-AρJ1(Aρ)A2-1 exp(-|ζ|A2-1)+1ρ1AdααJ1(αρ)ddαα2-1×exp(-|ζ|α2-1),
A.
δ(ζ)=12A2-1 exp(-|ζ|A2-1),A,
N(ρ, ζ)ev=-4πko3δ(r)-|ζ|2ρM1(ρ, ζ)ev+1ρ1dαα2(α2-1)1/2J1(αρ)×exp(-|ζ|α2-1).
M2(ρ, ζ)ev=M0(ρ, ζ)ev-N(ρ, ζ)ev-2ρ1dαα2(α2-1)1/2J1(αρ)×exp(-|ζ|α2-1).
N(ρ, ζ)ev=-8π3ko3δ(r)+13M0(ρ, ζ)ev-|ζ|ρM1(ρ, ζ)ev-M2(ρ, ζ)ev.
N(ρ, ζ)tr=13M0(ρ, ζ)tr-|ζ|ρM1(ρ, ζ)tr-M2(ρ, ζ)tr,
1dααn+1(α2-1)1/2Jn(αρ)exp(-|ζ|α2-1)
=1|ζ|Jn(ρ)+ρ|ζ|1dααnJn-1(αρ)
×exp(-|ζ|α2-1),
n=0, 1, 2,,
M0(ρ, ζ)ev=1|ζ|J0(ρ)-ρ|ζ|1dαJ1(αρ)×exp(-|ζ|α2-1).
M0(ρ, ζ)ev=1|ζ|J0(ρ)-ρ|ζ|0du×exp(-u|ζ|)u(1+u2)1/2J1(ρ1+u2).
M0(ρ, ζ)ev=1|ζ|J0(ρ)-ρ|ζ|3J1(ρ)-ρ|ζ|30du exp(-u|ζ|)d2du2u(1+u2)1/2J1(ρ1+u2),
M0(ρ, ζ)ev=1|ζ|J0(ρ)-ρ|ζ|3J1(ρ)+O(|ζ|-5),
M1(ρ, ζ)ev=2|ζ|2J1(ρ)+6ρ|ζ|4J0(ρ)+O(|ζ|-6).
M2(ρ, ζ)ev=-1|ζ|J2(ρ)-ρ|ζ|3J1(ρ)+O(|ζ|-5),
N(ρ, ζ)ev=-2|ζ|3J0(ρ)+O(|ζ|-5).
M0(0, ζ)ev=1|ζ|,
M1(0, ζ)ev=M2(0, ζ)ev=0,
N(0, ζ)ev=-2|ζ|3,
γ(r, ω)ev=12|ζ|(I+ezez)-1|ζ|3(I-3ezez),
γ(r, ω)ev12|ζ|(I+ezez).
Jn(ρ)2πρ1/2cosρ-12nπ-14π.
γ(r, ω)ev1q3/2(I-rˆrˆ)1|cos θ|2π sin θ1/2×cosπ4-q sin θ,
M0(ρ, ζ)ev=1q[2U0(q-|ζ|, ρ)-J0(ρ)],
U(a, b)=m=0(-1)mab+2mJ+2m(b),
M1(ρ, ζ)ev=2 sgn(ζ)2ρζM0(ρ, ζ)ev.
Uρ=ρq12ab-1J-1-U+1,
Uζ=-ζqU+1-a2qsgn(ζ)ab-1J-1.
U+U+2=abJ,
M1(ρ, ζ)ev=2q2ρJ0+ρ|ζ|q(J0-2U0)1-3q2+1q2[6ρ|ζ|U1+(ζ2-2ρ2)J1].
M2(ρ, ζ)ev=1ρ-ρρM0(ρ, ζ)ev,
M2(ρ, ζ)ev=1q2|ζ|J0-|ζ|J12ρ+3ρq2-6ρ2q2U1-ρ2q(J0-2U0)1-3q2,
N(ρ, ζ)ev=-8π3ko3δ(r)+1q3-2ζ2U0-|ζ|(q+|ζ|)J0+3ρ|ζ|qJ1+(2qU1+2U0-J0)×1-3ζ2q2.
Im[γ(r, ω)tr]=Im[γ(r, ω)],
Im[M0(ρ, ζ)tr]=sin qq,
Im[M1(ρ, ζ)tr]=-2ρ|ζ|q33 cos qq+1-3q2sin q,
Im[M2(ρ, ζ)tr]=ρ2q33 cos qq+1-3q2sin q,
Im[N(ρ, ζ)tr]=ζ2q3sin q+1q21-3ζ2q2×sin qq-cos q.
Re[γ(r, ω)tr]=Re[γ(r, ω)]-γ(r, ω)ev,
M0(0, ζ)tr=-1|ζ|[1-exp(i|ζ|)],
M1(0, ζ)tr=M2(0, ζ)tr=0,
N(0, ζ)tr=2|ζ|3[1-exp(i|ζ|)]+1+2i|ζ|exp(i|ζ|)|ζ|.
M0(0, 0)tr=i,
N(0, 0)tr=i/3.
γ(0)tr=2i3I,
U0(ρ, ρ)=12[J0(ρ)+cos ρ],
U1(ρ, ρ)=12 sin ρ.
M0(ρ, 0)ev=cos ρρ,
M1(ρ, 0)ev=-2ρJ2(ρ),
M2(ρ, 0)ev=cos ρρ-3ρ2sin ρ+cos ρρ,
N(ρ, 0)ev=1ρ2sin ρ+cos ρρ.
Re[M0(ρ, 0)tr]=Re[M2(ρ, 0)tr]=Re[N(ρ, 0)tr]=0,
Re[M1(ρ, 0)tr]=2ρJ2(ρ).
M0(ρ, 0)=exp(iρ)ρM2(ρ, 0),
γ(r, ω)ζ=0(I-rˆrˆ)exp(iρ)ρ,ρlarge.
Mn(ρ, ζ)=0dαα{}Jn(αρ),
Mn(ρ, ζ)tr=01dαα{}Jn(αρ).
{}=0dρρMn(ρ, ζ)Jn(αρ).
Mn(ρ, ζ)tr=0dρFn(ρ, ρ)Mn(ρ, ζ),n=0, 1, 2,
Fn(ρ, ρ)=ρ01dααJn(αρ)Jn(αρ),n=0, 1,.
Fn(ρ, ρ)=ρρ2-(ρ)2[ρJn(ρ)Jn+1(ρ)-ρJn(ρ)Jn+1(ρ)].
Fn(ρ, ρ)=12ρ[Jn(ρ)]2+1-n2ρ2Jn(ρ)2,
Fn(ρ, ρ)1πsin(ρ-ρ)ρ-ρ,
Im[Mn(ρ, ζ)]=0dρFn(ρ, ρ)Im[Mn(ρ, ζ)].

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