Abstract

Evanescent- and propagating-field distributions from a strongly-focused-wave beam with subwavelength waist waλ as a function of polar angle and distance are investigated. Exact amplitudes and intensities of evanescent Eev and propagating Ep fields, including interference terms, are presented both in near- and far-field regions. It is shown that the amplitude of Eev decays as exp(-r/wa) in the near-field region and that evanescent waves do not contribute to the far field in the forward direction as they do in the transverse directions θ=π/2, even though the oscillating evanescent field of the same strength but opposite in sign to the propagating field exists in the transverse plane.

© 2003 Optical Society of America

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References

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  1. 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]
  2. G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
    [CrossRef]
  3. T. Setala, A. T. Friberg, M. Kaivola, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E 59, 1200–1206 (1999).
    [CrossRef]
  4. 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]
  5. E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
    [CrossRef]
  6. M. Xiao, “Evanescent fields do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
    [CrossRef]
  7. P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” J. Mod. Opt. 47, 757–758 (2000).
  8. A. Lakhtakia, W. S. Weiglhofer, “Evanescent plane waves and the far field: resolution of a controversy,” J. Mod. Opt. 47, 759–763 (2000).
    [CrossRef]
  9. A. Rahmani, G. W. Bryant, “Contribution of evanescent waves to the far field: the atomic point of view,” Opt. Lett. 25, 433–435 (2000).
    [CrossRef]
  10. T. Setala, M. Kaivola, A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures,” J. Opt. Soc. Am. A 18, 678–688 (2001).
    [CrossRef]
  11. C. J. R. Sheppard, F. Aguilar, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177–180 (2001); comment on M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
    [CrossRef]
  12. M. V. Berry, “Asymptotics of evanescence,” J. Mod. Opt. 48, 1535–1541 (2001).
    [CrossRef]
  13. M. Xiao, “On the evanescent field of dipole (reply),” J. Mod. Opt. 47, 765–768 (2000).
  14. M. Xiao, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures: comment,” J. Opt. Soc. Am. A 19, 1447–1448 (2002).
    [CrossRef]
  15. T. Setala, M. Kaivola, A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures: reply to comment,” J. Opt. Soc. Am. A 19, 1449–1451 (2002).
    [CrossRef]
  16. O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16, 835–847 (1999).
    [CrossRef]
  17. H. F. Arnoldus, J. T. Foley, “Traveling and evanescent parts of the electromagnetic Green’s tensor,” J. Opt. Soc. Am. A 19, 1701–1711 (2002).
    [CrossRef]
  18. J. Durnin, “Exact solutions for nondifracting beams. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  19. Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
    [CrossRef]
  20. N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001); www.opticsexpress.org .
    [CrossRef] [PubMed]
  21. S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
    [CrossRef]

2002 (3)

2001 (4)

N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001); www.opticsexpress.org .
[CrossRef] [PubMed]

T. Setala, M. Kaivola, A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures,” J. Opt. Soc. Am. A 18, 678–688 (2001).
[CrossRef]

C. J. R. Sheppard, F. Aguilar, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177–180 (2001); comment on M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

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

2000 (4)

M. Xiao, “On the evanescent field of dipole (reply),” J. Mod. Opt. 47, 765–768 (2000).

P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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]

A. Rahmani, G. W. Bryant, “Contribution of evanescent waves to the far field: the atomic point of view,” Opt. Lett. 25, 433–435 (2000).
[CrossRef]

1999 (4)

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

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

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

1998 (1)

1991 (1)

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]

1990 (1)

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

1987 (1)

1986 (1)

Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
[CrossRef]

1976 (1)

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

Aguilar, F.

C. J. R. Sheppard, F. Aguilar, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177–180 (2001); comment on M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

Arnoldus, H. F.

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]

Bryant, G. W.

Carney, P. S.

P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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]

Durnin, J.

Fischer, D. G.

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

Foley, J. T.

H. F. Arnoldus, J. T. Foley, “Traveling and evanescent parts of the electromagnetic Green’s tensor,” J. Opt. Soc. Am. A 19, 1701–1711 (2002).
[CrossRef]

P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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.

T. Setala, M. Kaivola, A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures: reply to comment,” J. Opt. Soc. Am. A 19, 1449–1451 (2002).
[CrossRef]

T. Setala, M. Kaivola, A. T. Friberg, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures,” J. Opt. Soc. Am. A 18, 678–688 (2001).
[CrossRef]

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

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

Kaivola, M.

Keller, O.

Kino, G. S.

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

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, E.

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

Leviatan, Y.

Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
[CrossRef]

Mansfield, S. M.

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

Petrov, N. I.

Rahmani, A.

Setala, T.

Shchegrov, A. V.

P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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]

Sheppard, C. J. R.

C. J. R. Sheppard, F. Aguilar, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177–180 (2001); comment on M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

Sherman, G. C.

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

Stamnes, J. J.

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

Visser, T. D.

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

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. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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]

Xiao, M.

M. Xiao, “Evanescent and propagating electromagnetic fields in scattering from point-dipole structures: comment,” J. Opt. Soc. Am. A 19, 1447–1448 (2002).
[CrossRef]

M. Xiao, “On the evanescent field of dipole (reply),” J. Mod. Opt. 47, 765–768 (2000).

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

Appl. Phys. Lett. (1)

S. M. Mansfield, G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
[CrossRef]

J. Appl. Phys. (1)

Y. Leviatan, “Study of near-zone fields of a small aperture,” J. Appl. Phys. 60, 1577–1583 (1986).
[CrossRef]

J. Math. Phys. (1)

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

J. Mod. Opt. (7)

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

P. S. Carney, D. G. Fischer, J. T. Foley, A. T. Friberg, A. V. Shchegrov, T. D. Visser, E. Wolf, “Evanescent waves do contribute to the far field (comment),” 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]

C. J. R. Sheppard, F. Aguilar, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 48, 177–180 (2001); comment on M. Xiao, “Evanescent waves do contribute to the far field,” J. Mod. Opt. 46, 729–733 (1999).
[CrossRef]

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

M. Xiao, “On the evanescent field of dipole (reply),” J. Mod. Opt. 47, 765–768 (2000).

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]

J. Opt. Soc. Am. A (6)

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (1)

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

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Figures (5)

Fig. 1
Fig. 1

Amplitude of the evanescent field in three dimensions as a function of direction for kr=10 and kr=50.

Fig. 2
Fig. 2

Amplitude of the evanescent field in two dimensions as a function of direction for kr=3 and kr=50.

Fig. 3
Fig. 3

Variation with distance of the field intensities along the axial line for λ=0.65 μm and wa=λ/50.

Fig. 4
Fig. 4

Distributions of the fields in a transverse plane at various distances from the initial plane (a) z=0, (b) z=λ/2, and (c) z=1 μm at λ=0.65 μm, wa=λ/10, and n0=1.

Fig. 5
Fig. 5

Ratio of the amplitudes of the evanescent wave Eev (dashed curves) and the propagating wave Ep (solid curves) of wavelength 0.65 µm to the total amplitude Esum as a function of spot size wa for different values of refractive index: 1-n0=1.0, 2-n0=1.5, 3-n0=3.87.

Equations (5)

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

E(ρ, 0)=n=0cnψn(ρ),
cn=E(ρ, 0)ψn(ρ, ϕ)ρdρdϕ,
n2(ρ)=n02-ω2ρ2,
cp,0=2w0waw02+wa2·w02-wa2w02+wa2p,p=0, 1, 2,,
E(ρ, z)=p=0cpψp exp(iβpz),

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