Abstract

We report a general framework capable of describing the focusing of electromagnetic waves with spatially varying coherence and polarization properties in optical systems of arbitrary numerical aperture and Fresnel number. We also investigate the reduction of the dimensionality of the requisite integrals by use of a coherent mode expansion. We find that coherent mode expansions treating each component of the electric field vector individually are unsuitable for describing focusing systems because of the inter-component mixing that can occur in high numerical aperture systems. In addition, we show that the assumption of harmonic angular dependence allows the azimuthal integration to be performed analytically, providing further simplification of the analysis. We also find that the effective degree of spectral coherence of an electromagnetic beam is unchanged upon focusing. Finally, as an illustration of the developed framework, we calculate the transverse and axial focal distributions for a partially coherent source formed by incoherent superposition of radially and azimuthally polarized Laguerre-Gauss modes.

© 2009 Optical Society of America

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  1. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).
  2. V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Pet. 1, 1-36 (1919).
  3. V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. Pet. 1, 1-30 (1920).
  4. E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
    [CrossRef]
  5. B. Richards and 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]
  6. W. Wang, A. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491-496 (1997).
    [CrossRef]
  7. D. Fischer and T. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097-2102 (2004).
    [CrossRef]
  8. C. Rydberg, “First- and second-order statistics of partially coherent, high-numerical-aperture optical fields,” Opt. Lett. 33, 104-106 (2008).
    [CrossRef]
  9. X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262-267 (2005).
    [CrossRef]
  10. K. Lindfors, T. Setälä, M. Kaivola, and A. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
    [CrossRef]
  11. Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49-51 (2008).
    [CrossRef]
  12. T. van Dijk, G. Gbur, and T. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575-581 (2008).
    [CrossRef]
  13. R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
    [CrossRef]
  14. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
    [CrossRef]
  15. W. Gao, “Effects of coherence and vector properties of the light on the resolution limit in stimulated emission depletion fluorescence microscopy,” J. Opt. Soc. Am. A 25, 1378-1382 (2008).
    [CrossRef]
  16. P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009-3015 (1998).
    [CrossRef]
  17. J. W. Goodman, Statistical Optics (Wiley, 2000).
  18. N. Wiener, “Coherency matrices and quantum theory,” J. Math. Phys. 7, 109-125 (1928).
  19. N. Wiener, “Harmonic analysis and the quantum theory,” J. Franklin Inst. 207, 525-553 (1929).
    [CrossRef]
  20. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165-1181 (1959).
    [CrossRef]
  21. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).
  22. P. Török, P. Varga, Z. Laczik, and G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325-332 (1995).
    [CrossRef]
  23. P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
    [CrossRef]
  24. D. J. Innes and A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).
  25. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343-351 (1982).
    [CrossRef]
  26. C. Palma and G. Cincotti, “Imaging of J0 correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
    [CrossRef]
  27. S. Flewett, H. Quiney, C. Tran, and K. Nugent, “Extracting coherent modes from partially coherent wavefields,” Opt. Lett. 34, 2198-2200 (2009).
    [CrossRef]
  28. K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fenn., Ser. A1: Math.-Phys. A137, 1-79 (1947).
  29. M. Loève, Probability Theory, 4th ed. (Springer-Verlag, 1978).
  30. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78-84 (2003).
    [CrossRef]
  31. J. Tervo, T. Setälä, and A. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205-2215 (2004).
    [CrossRef]
  32. D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
    [CrossRef]
  33. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, 1995).
  34. R. Dorn, S. Quabis and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  35. G. Guattari, C. Palma, and C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173-178 (1989).
    [CrossRef]
  36. P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605-3617 (2004).
    [CrossRef]

2009 (1)

2008 (5)

2005 (2)

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262-267 (2005).
[CrossRef]

K. Lindfors, T. Setälä, M. Kaivola, and A. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

2004 (3)

2003 (2)

1998 (2)

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009-3015 (1998).
[CrossRef]

1997 (2)

W. Wang, A. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491-496 (1997).
[CrossRef]

C. Palma and G. Cincotti, “Imaging of J0 correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

1995 (1)

1989 (1)

G. Guattari, C. Palma, and C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173-178 (1989).
[CrossRef]

1987 (1)

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
[CrossRef]

1984 (1)

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

1982 (1)

1966 (1)

D. J. Innes and A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

1959 (3)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165-1181 (1959).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and 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]

1947 (1)

K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fenn., Ser. A1: Math.-Phys. A137, 1-79 (1947).

1929 (1)

N. Wiener, “Harmonic analysis and the quantum theory,” J. Franklin Inst. 207, 525-553 (1929).
[CrossRef]

1928 (1)

N. Wiener, “Coherency matrices and quantum theory,” J. Math. Phys. 7, 109-125 (1928).

1920 (1)

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. Pet. 1, 1-30 (1920).

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Pet. 1, 1-36 (1919).

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Bloom, A. L.

D. J. Innes and A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Bodner, S. E.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
[CrossRef]

Booker, G.

Borghi, R.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

Brown, D. P.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

Brown, T. G.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

Cincotti, G.

C. Palma and G. Cincotti, “Imaging of J0 correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

Dorn, R.

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

Fischer, D.

Flewett, S.

Friberg, A.

Gao, W.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gori, F.

Guattari, G.

Higdon, P. D.

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. Pet. 1, 1-30 (1920).

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Pet. 1, 1-36 (1919).

Innes, D. J.

D. J. Innes and A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Kaivola, M.

Karhunen, K.

K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fenn., Ser. A1: Math.-Phys. A137, 1-79 (1947).

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Laczik, Z.

Lehmberg, R. H.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
[CrossRef]

Leuchs, G.

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

Lindfors, K.

Liu, X.

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262-267 (2005).
[CrossRef]

Loève, M.

M. Loève, Probability Theory, 4th ed. (Springer-Verlag, 1978).

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Munro, P.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Nugent, K.

Padovani, C.

G. Guattari, C. Palma, and C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173-178 (1989).
[CrossRef]

Palma, C.

C. Palma and G. Cincotti, “Imaging of J0 correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

G. Guattari, C. Palma, and C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173-178 (1989).
[CrossRef]

Piquero, G.

Pu, J.

Z. Zhang, J. Pu, and X. Wang, “Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective,” Opt. Lett. 33, 49-51 (2008).
[CrossRef]

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262-267 (2005).
[CrossRef]

Quabis, S.

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

Quiney, H.

Richards, B.

B. Richards and 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]

Rydberg, C.

Santarsiero, M.

Schmitt, A. J.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
[CrossRef]

Setälä, T.

Simon, R.

Spilman, A. K.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

Tervo, J.

Török, P.

Tran, C.

van Dijk, T.

Varga, P.

Visser, T.

Volkov, S. N.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

Wang, W.

Wang, X.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, 1995).

Wiener, N.

N. Wiener, “Harmonic analysis and the quantum theory,” J. Franklin Inst. 207, 525-553 (1929).
[CrossRef]

N. Wiener, “Coherency matrices and quantum theory,” J. Math. Phys. 7, 109-125 (1928).

Wilson, T.

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

Wolf, E.

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

W. Wang, A. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491-496 (1997).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343-351 (1982).
[CrossRef]

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165-1181 (1959).
[CrossRef]

B. Richards and 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]

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Zhang, Z.

Ann. Acad. Sci. Fenn., Ser. A1: Math.-Phys. (1)

K. Karhunen, “Über lineare Methoden in der Wahrscheinlichkeitsrechnung,” Ann. Acad. Sci. Fenn., Ser. A1: Math.-Phys. A137, 1-79 (1947).

IEEE J. Quantum Electron. (1)

C. Palma and G. Cincotti, “Imaging of J0 correlated Bessel-Gauss beams,” IEEE J. Quantum Electron. 33, 1032-1040 (1997).
[CrossRef]

J. Appl. Phys. (1)

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, “Theory of induced spatial incoherence,” J. Appl. Phys. 62, 2680-02701 (1987).
[CrossRef]

J. Franklin Inst. (1)

N. Wiener, “Harmonic analysis and the quantum theory,” J. Franklin Inst. 207, 525-553 (1929).
[CrossRef]

J. Math. Phys. (1)

N. Wiener, “Coherency matrices and quantum theory,” J. Math. Phys. 7, 109-125 (1928).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

P. Török, “Focusing of electromagnetic waves through a dielectric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009-3015 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78-84 (2003).
[CrossRef]

W. Wang, A. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491-496 (1997).
[CrossRef]

P. Török, P. Varga, Z. Laczik, and G. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

T. van Dijk, G. Gbur, and T. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A 25, 575-581 (2008).
[CrossRef]

W. Gao, “Effects of coherence and vector properties of the light on the resolution limit in stimulated emission depletion fluorescence microscopy,” J. Opt. Soc. Am. A 25, 1378-1382 (2008).
[CrossRef]

D. Fischer and T. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097-2102 (2004).
[CrossRef]

J. Tervo, T. Setälä, and A. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205-2215 (2004).
[CrossRef]

K. Lindfors, T. Setälä, M. Kaivola, and A. Friberg, “Degree of polarization in tightly focused optical fields,” J. Opt. Soc. Am. A 22, 561-568 (2005).
[CrossRef]

Nuovo Cimento (1)

E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Nuovo Cimento 13, 1165-1181 (1959).
[CrossRef]

Opt. Commun. (4)

P. Török, P. D. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300-315 (1998).
[CrossRef]

X. Liu and J. Pu, “Focal shift and focal switch of partially coherent light in dual-focus systems,” Opt. Commun. 252, 262-267 (2005).
[CrossRef]

D. P. Brown, A. K. Spilman, T. G. Brown, R. Borghi, S. N. Volkov, and E. Wolf, “Spatial coherence properties of azimuthally polarized laser modes,” Opt. Commun. 281, 5287-5290 (2008).
[CrossRef]

G. Guattari, C. Palma, and C. Padovani, “Cross-spectral densities with axial symmetry,” Opt. Commun. 73, 173-178 (1989).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

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

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, and C. Yamanaka, “Random phasing of high-power lasers for uniform target acceleration and plasma-instability suppression,” Phys. Rev. Lett. 53, 1057-1060 (1984).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and 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]

Spectra-Phys. Laser Tech. Bull. (1)

D. J. Innes and A. L. Bloom, “Design of optical systems for use with laser beams,” Spectra-Phys. Laser Tech. Bull. 5, 1-10 (1966).

Trans. Opt. Inst. Pet. (2)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Pet. 1, 1-36 (1919).

V. S. Ignatowsky, “The relationship between geometrical and wave optics and diffraction of homocentrical beams,” Trans. Opt. Inst. Pet. 1, 1-30 (1920).

Other (5)

J. W. Goodman, Statistical Optics (Wiley, 2000).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

M. Loève, Probability Theory, 4th ed. (Springer-Verlag, 1978).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, 1995).

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