Abstract

The features of spirally polarized light beams focused by high-numerical-aperture systems are investigated in the nonparaxial regime by means of Debye theory with a multipole expansion technique. General expressions of the expanding coefficients are given, as well as the electric field distributions across the focal plane. Numerical examples are presented for the case of spirally polarized beams of the donut type. Comparisons with recent experimental results are also shown.

© 2005 Optical Society of America

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References

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  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
    [CrossRef]
  2. K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef] [PubMed]
  3. R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901–233904 (2003).
    [CrossRef] [PubMed]
  4. D. Ganic, X. Gan, M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11, 2747–2452 (2003).
    [CrossRef] [PubMed]
  5. N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
    [CrossRef] [PubMed]
  6. N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
    [CrossRef] [PubMed]
  7. P. Török, P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
    [CrossRef]
  8. D. P. Biss, T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384–393 (2004).
    [CrossRef] [PubMed]
  9. P. R. Munro, P. Török, “Vectorial, high-numerical-aperture study of phase-contrast microscopes,” J. Opt. Soc. Am. A 21, 1714–1723 (2004).
    [CrossRef]
  10. L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef] [PubMed]
  11. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
    [CrossRef]
  12. S. J. van Enk, H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
    [CrossRef]
  13. S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004).
    [CrossRef]
  14. D. P. Biss, T. G. Brown, “Polarization-vortex-driven second-harmonic generation,” Opt. Lett. 28, 923–925 (2003).
    [CrossRef] [PubMed]
  15. M. Ohtsu, K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).
    [CrossRef]
  16. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
    [CrossRef]
  17. R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
    [CrossRef]
  19. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Series on Optics and Optoelectronics) (Hilger, Bristol, UK, 1986).
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    [CrossRef] [PubMed]
  21. A. S. van de Nes, P. R. T. Munro, S. F. Pereira, J. J. M. Braat, P. Török, “Cylindrical vector beam focusing through a dielectric interface: comment,” Opt. Express 12, 967–969 (2004).
    [CrossRef] [PubMed]
  22. D. P. Biss, T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface: reply to comment,” Opt. Express 12, 970–971 (2004).
    [CrossRef] [PubMed]
  23. C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
    [CrossRef]
  24. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
  25. A. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [CrossRef]
  26. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  27. C. J. R. Sheppard, “Electromagnetic field in focal region of wide-angular annular lens and mirrors,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).
    [CrossRef]
  28. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6.
  29. R. Borghi, “On the angular-spectrum representation of multipole wavefields,” J. Opt. Soc. Am. A 21, 1805–1810 (2004).
    [CrossRef]
  30. S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
    [CrossRef]
  31. D. H. Foster, J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. 234, 351383 (2004).
    [CrossRef]
  32. M. A. Alonso, G. W. Forbes, “Uncertainty products for nonparaxial wave fields,” J. Opt. Soc. Am. A 17, 2391–2401 (2000).
    [CrossRef]
  33. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
    [CrossRef]
  34. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  35. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
    [CrossRef]

2004 (11)

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004).
[CrossRef]

D. H. Foster, J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. 234, 351383 (2004).
[CrossRef]

D. P. Biss, T. G. Brown, “Primary aberrations in focused radially polarized vortex beams,” Opt. Express 12, 384–393 (2004).
[CrossRef] [PubMed]

A. S. van de Nes, P. R. T. Munro, S. F. Pereira, J. J. M. Braat, P. Török, “Cylindrical vector beam focusing through a dielectric interface: comment,” Opt. Express 12, 967–969 (2004).
[CrossRef] [PubMed]

D. P. Biss, T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface: reply to comment,” Opt. Express 12, 970–971 (2004).
[CrossRef] [PubMed]

N. Davidson, N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29, 1318–1320 (2004).
[CrossRef] [PubMed]

P. Török, P. R. T. Munro, “The use of Gauss–Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605–3617 (2004).
[CrossRef]

P. R. Munro, P. Török, “Vectorial, high-numerical-aperture study of phase-contrast microscopes,” J. Opt. Soc. Am. A 21, 1714–1723 (2004).
[CrossRef]

R. Borghi, “On the angular-spectrum representation of multipole wavefields,” J. Opt. Soc. Am. A 21, 1805–1810 (2004).
[CrossRef]

N. Bokor, N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
[CrossRef] [PubMed]

R. Borghi, M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004).
[CrossRef]

2003 (4)

2001 (5)

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. J. van Enk, H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
[CrossRef]

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[CrossRef]

D. P. Biss, T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9, 490–497 (2001).
[CrossRef] [PubMed]

2000 (3)

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

1981 (1)

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

1978 (1)

C. J. R. Sheppard, “Electromagnetic field in focal region of wide-angular annular lens and mirrors,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

1974 (1)

A. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Alonso, M. A.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Biss, D. P.

Bokor, N.

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Braat, J. J. M.

Brown, T. G.

Davidson, N.

Devaney, A.

A. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

Dirksen, P.

Dorn, R.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901–233904 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6.

Forbes, G. W.

Foster, D. H.

D. H. Foster, J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. 234, 351383 (2004).
[CrossRef]

Gan, X.

Ganic, D.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Gori, F.

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Gu, M.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).

Janssen, A. J. E. M.

Kimble, H. J.

S. J. van Enk, H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
[CrossRef]

Kobayashi, K.

M. Ohtsu, K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901–233904 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6.

Munro, P. R.

Munro, P. R. T.

Nöckel, J. U.

D. H. Foster, J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. 234, 351383 (2004).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

Ohtsu, M.

M. Ohtsu, K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).
[CrossRef]

Pereira, S. F.

Quabis, S.

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901–233904 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Santarsiero, M.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

C. J. R. Sheppard, “Electromagnetic field in focal region of wide-angular annular lens and mirrors,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Series on Optics and Optoelectronics) (Hilger, Bristol, UK, 1986).

Steinberg, S.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

Török, P.

van de Nes, A. S.

van Enk, S. J.

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004).
[CrossRef]

S. J. van Enk, H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
[CrossRef]

Wolf, E.

A. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

Wolf, K. B.

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth, T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
[CrossRef] [PubMed]

Appl. Phys. B Lasers Opt. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B Lasers Opt. 72, 109–113 (2001).
[CrossRef]

IEE J. Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, “Electromagnetic field in focal region of wide-angular annular lens and mirrors,” IEE J. Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

J. Math. Phys. (2)

A. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard, P. Török, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion,” J. Mod. Opt. 44, 803–818 (1997).
[CrossRef]

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

Opt. Commun. (2)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

D. H. Foster, J. U. Nöckel, “Methods for 3-D vector microcavity problems involving a planar dielectric mirror,” Opt. Commun. 234, 351383 (2004).
[CrossRef]

Opt. Express (7)

Opt. Lett. (3)

Phys. Rev. A (2)

S. J. van Enk, H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
[CrossRef]

S. J. van Enk, “Atoms, dipole waves, and strongly focused light beams,” Phys. Rev. A 69, 043813 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

L. Novotny, M. R. Beversluis, K. S. Youngworth, T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997).
[CrossRef]

R. Dorn, S. Quabis, G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901–233904 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (6)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I, Chap. 6.

M. Ohtsu, K. Kobayashi, Optical Near Fields (Springer, Berlin, 2004).
[CrossRef]

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Series on Optics and Optoelectronics) (Hilger, Bristol, UK, 1986).

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Figures (6)

Fig. 1
Fig. 1

Behavior function F l ( α ) as a function of the index l for a SPDB focused by an aplanatic (circles) and a Herschel (triangles) system. β = 1 . (a) α = π 12 , (b) α = π 4 , (c) α = π 3 , (d) α = π 2 .

Fig. 2
Fig. 2

Field lines of the transverse component of the focal electric field produced by focusing a SPDB through an aplanatic system for β = 1 , α = π 2 . (a) γ = 0 , (b) γ = π 6 , (c) γ = π 3 , (d) γ = π 2 .

Fig. 3
Fig. 3

Behavior of the angle, normalized to π, between the transverse electric field and the radial direction for γ = π 3 , for α = π 12 (solid curve), α = π 3 (dashed curve), and α = π 2 (dotted curve) as functions of the normalized variable r sin α λ . Gray lines correspond to values of the angles π 3 , 2 π 3 , and 5 π 3 .

Fig. 4
Fig. 4

Focal intensity distribution as a function of r λ pertinent to a SPDB with β = 1 , α = π 2 . (a) γ = 0 , (b) γ = π 4 , (c) γ = π 3 , (d) γ = π 2 . Dashed curve, transverse intensity; dotted curve, longitudinal intensity; solid curve, total intensity.

Fig. 5
Fig. 5

Contour plots of the total integrated intensity distribution, around the focal plane, as a function of the axial ( z ) and transverse ( x ) coordinates. λ = 632.8 nm , NA of 0.9, β 1.06 . (a) γ = 0 , (b) γ = π 4 , (c) γ = π 3 , (d) γ = π 2 . (a) and (d) could be compared with Figs. 2(a) and 2(b), respectively, of Ref. [3].

Fig. 6
Fig. 6

(b) and (d) Plots of the total intensity and (a) and (c) the integrated intensity distribution across the focal plane as functions of the transverse position. Circles are the experimental data, extracted from Fig. 2 of Ref. [3]; the solid curves are theoretical values obtained with the same parameters as in Fig. 5.

Equations (76)

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

E ( r ) = i λ Ω 1 A ( θ , ϕ ) exp ( i k s ̂ r ) d Ω ,
A ( θ , ϕ ) = A 1 ( θ ) A 2 ( θ ) a ̂ ( θ , ϕ ) ,
Y l , ± m ( θ , ϕ ) = ( ± 1 ) m [ 2 l + 1 4 π ( l m ) ! ( l + m ) ! ] 1 2 exp ( ± i m ϕ ) P l ( m ) ( cos θ ) ,
Y l , m ( θ , ϕ ) = L Y l , m ( θ , ϕ ) ,
L = i r × .
L = L + u + L u + + L z z ̂ ,
L ± Y l , m ( θ , ϕ ) = a Y l , m ± 1 ( θ , ϕ ) ,
L z Y l , m ( θ , ϕ ) = m Y l , m ( θ , ϕ ) ,
a ± = [ ( l ± m ) ( l m + 1 ) ] 1 2 = [ l ( l + 1 ) m ( m 1 ) ] 1 2 .
L = e ̂ θ i sin θ ϕ + e ̂ ϕ i θ ,
L = 1 sin θ L z e ̂ θ + exp ( i ϕ ) L + exp ( i ϕ ) L 2 i e ̂ ϕ .
Z l , m ( θ , ϕ ) = s ̂ × Y l , m ( θ , ϕ )
4 π Y l , m ( θ , ϕ ) Y l , m * ( θ , ϕ ) d Ω = δ l , l δ m , m l ( l + 1 ) ,
4 π Z l , m ( θ , ϕ ) Y l , m * ( θ , ϕ ) d Ω = δ l , l δ m , m l ( l + 1 ) ,
4 π Z l , m ( θ , ϕ ) Y l , m * ( θ , ϕ ) d Ω = 0 .
A ( θ , ϕ ) = l = 1 m = l + l [ v l , m Y l , m ( θ , ϕ ) + w l , m Z l , m ( θ , ϕ ) ] ,
v l , m = 1 l ( l + 1 ) Ω 1 A Y l , m * d Ω ,
w l , m = 1 l ( l + 1 ) Ω 1 A Z l , m * d Ω = 1 l ( l + 1 ) Ω 1 A ( s ̂ × Y l , m * ) d Ω = 1 l ( l + 1 ) Ω 1 ( A × s ̂ ) Y l , m * d Ω ,
N l , m ( r ) = i k × [ r Λ l , m ( r ) ] ,
M l , m ( r ) = × { × [ r Λ l , m ( r ) ] } ,
Λ l , m ( r ) = j l ( k r ) Y l , m ( θ , ϕ ) ,
N l , m ( r ) = ( i ) l k 4 π 4 π Y l , m ( θ , ϕ ) exp ( i k s ̂ r ) d Ω ,
M l , m ( r ) = ( i ) l k 4 π 4 π Z l , m ( θ , ϕ ) exp ( i k s ̂ r ) d Ω .
E ( r ) = l , m 2 i l 1 [ v l , m N l , m ( r ) + w l , m M l , m ( r ) ] .
c B ( r ) = i λ Ω 1 s ̂ × A ( θ , ϕ ) exp ( i k s ̂ r ) d Ω ,
s ̂ × A ( θ , ϕ ) = l , m [ v l , m Z l , m ( θ , ϕ ) w l , m Y l , m ( θ , ϕ ) ] ,
c B ( r ) = l , m 2 i l 1 [ v l , m M l , m ( r ) w l , m N l , m ( r ) ] .
E = Ω 1 A ( θ , ϕ ) 2 d Ω ,
E = l , m l ( l + 1 ) ( v l , m 2 + w l , m 2 ) .
J = 1 E Re Ω 1 { ( L A ) * A + [ L ( s ̂ × A ) ] * ( s ̂ × A ) } d Ω ,
J = 1 E Re { l , m , l , m [ ( v l , m * v l , m + w l , m * w l , m ) × 4 π ( L Y l , m ) * Y l , m d Ω + ( v l , m * w l , m w l , m * v l , m ) × 4 π ( L Y l , m ) * ( s ̂ × Y l , m ) d Ω ] } .
E x ( r ) = cos ( ϕ + γ ) f ( r ) ,
E y ( r ) = sin ( ϕ + γ ) f ( r ) ,
r = r 0 exp ( ϕ tan γ ) ,
E ( r ) = f ( r ) ( cos γ u ̂ r + sin γ u ̂ a ) .
A 1 ( θ ) = f ( F sin θ ) ,
a ̂ ( r ) ( θ , ϕ ) = e ̂ θ ,
a ̂ ( a ) ( θ , ϕ ) = e ̂ ϕ ,
s ̂ × a ̂ ( r ) ( θ , ϕ ) = a ̂ ( a ) ( θ , ϕ ) ,
s ̂ × a ̂ ( a ) ( θ , ϕ ) = a ̂ ( r ) ( θ , ϕ ) ,
E ( r ) ( r ) = c B ( a ) ( r ) ,
E ( a ) ( r ) = c B ( r ) ( r ) .
v l , m ( r ) = 1 l ( l + 1 ) [ Ω 1 A 1 ( θ ) A 2 ( θ ) a ̂ ( r ) L Y l , m d Ω ] * ,
v l , m ( r ) = 1 l ( l + 1 ) [ 0 α d θ A 1 ( θ ) A 2 ( θ ) 0 2 π m Y l , m d ϕ ] * ,
w l , m ( r ) = 1 l ( l + 1 ) [ Ω 1 A 1 ( θ ) A 2 ( θ ) a ̂ ( a ) L Y l , m d Ω ] *
= 1 l ( l + 1 ) { 0 α d θ A 1 ( θ ) A 2 ( θ ) sin θ 2 i 0 2 π [ a exp ( i ϕ ) Y l , m + 1 a + exp ( i ϕ ) Y l , m 1 ] d ϕ } * = 1 l ( l + 1 ) { 0 α d θ A 1 ( θ ) A 2 ( θ ) sin θ 2 i 0 2 π [ a exp ( i ϕ ) Y l , m + 1 a + exp ( i ϕ ) Y l , m 1 ] d ϕ } * ,
w l , m ( r ) = i F l ( α ) δ m , 0 ,
F l ( α ) = [ π ( 2 l + 1 ) l ( l + 1 ) ( l 1 ) ! ( l + 1 ) ! ] 1 2 0 α A 1 ( θ ) A 2 ( θ ) P l ( 1 ) ( cos θ ) sin θ d θ .
E ( r ) ( r ) = 2 l = 1 i l F l ( α ) M l , 0 ( r ) ,
c B ( r ) ( r ) = 2 l = 1 i l F l ( α ) N l , 0 ( r ) ,
E ( r ) = 2 l = 1 i l F l ( α ) [ cos γ M l , 0 ( r ) sin γ N l , 0 ( r ) ] ,
c B ( r ) = 2 l = 1 i l F l ( α ) [ cos γ N l , 0 ( r ) + sin γ M l , 0 ( r ) ] .
N l , 0 ( r ) = e ̂ ϕ i j l ( k r ) ( 2 l + 1 4 π ) 1 2 p l ( 1 ) ( cos θ ) ,
M l , 0 ( r ) = e ̂ r j l ( k r ) k r l ( l + 1 ) ( 2 l + 1 4 π ) 1 2 P l ( 0 ) ( cos θ ) e ̂ θ T l ( k r ) ( 2 l + 1 4 π ) 1 2 P l ( 1 ) ( cos θ ) ,
E z ( r ) focal plane = i cos γ π q = 0 F 2 q + 1 ( α ) ( 2 q + 1 ) ! ! 2 q q ! ( 4 q + 3 ) 1 2 T 2 q + 1 ( k r ) ,
E r ( r ) focal plane = cos γ π q = 1 2 q F 2 q ( α ) ( 2 q + 1 ) ! ! 2 q q ! ( 4 q + 1 ) 1 2 j 2 q ( k r ) k r ,
E ϕ ( r ) focal plane = sin γ π q = 0 F 2 q + 1 ( α ) ( 2 q + 1 ) ! ! 2 q q ! ( 4 q + 3 ) 1 2 j 2 q + 1 ( k r ) ,
f ( r ) = A ( r w ) exp ( r 2 w 2 ) ,
A 2 ( θ ) = { cos 1 2 θ for aplanatic systems 1 for Herschel systems } .
E = { π sin 2 α 4 β 2 G ( β ) for aplanatic systems π sin 2 α 2 β 2 G ( β ) for Herschel systems } ,
G ( β ) = [ 1 ( 1 + 2 β 2 ) exp ( 2 β 2 ) ] ,
× [ r Λ l , m ( r ) ] = e ̂ θ 1 sin θ ϕ Λ l , m e ̂ ϕ θ Λ l , m ,
L + = exp ( i ϕ ) ( θ + i cot θ ϕ ) ,
L + = exp ( i ϕ ) ( θ + i cot θ ϕ ) ,
L z = i ϕ ,
θ = exp ( i ϕ ) L + exp ( i ϕ ) L 2 ,
ϕ = i L z ,
× [ r λ l , m ( r ) ] = N l , m ( r ) i k = e ̂ θ j l ( k r ) i m sin θ Y l , m ( θ , ϕ ) + e ̂ ϕ j l ( k r ) G l , m ( θ , ϕ ) ,
G l , m ( θ , ϕ ) = a + exp ( i ϕ ) Y l , m 1 ( θ , ϕ ) a exp ( i ϕ ) Y l , m + 1 ( θ , ϕ ) 2 .
M l , m ( r ) = × { × [ r Λ l , m ( r ) ] } ,
M l , m ( r ) = e ̂ r j l ( k r ) r sin θ L 2 Y l , m e ̂ θ Y l , m θ + 1 r d d r [ r j l ( k r ) ] + e ̂ ϕ 1 sin θ Y l , m ϕ 1 r d d r [ r j l ( k r ) ] ,
L 2 = [ 1 sin θ θ ( sin θ θ ) + 1 sin 2 θ 2 ϕ 2 ]
L 2 Y l , m = l ( l + 1 ) Y l , m ,
1 r d d r [ r j l ( k r ) ] = k T l ( k r ) ,
T l ( x ) = l j l ( x ) x j l 1 ( x ) ,
M l , m ( r ) = e ̂ r k j l ( k r ) k r l ( l + 1 ) Y l , m ( θ , ϕ ) + e ̂ θ k T l ( k r ) G l , m ( θ , ϕ ) e ̂ ϕ k T l ( k r ) i m sin θ Y l , m ( θ , ϕ ) .

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