Abstract

A new class of partially coherent beams with a separable phase, which carry optical vortices, is introduced. It is shown that any member of the class can be represented as an incoherent superposition of fully coherent Laguerre–Gauss modes of arbitrary order, with the same azimuthal mode index. The free-space propagation properties of such partially coherent beams are studied analytically, and their M2 quality factor is investigated numerically.

© 2001 Optical Society of America

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References

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  1. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  2. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  3. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  4. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  5. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  6. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]
  7. K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
    [CrossRef]
  8. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  9. R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 95–109 (1998).
    [CrossRef]
  10. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [CrossRef]
  11. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
    [CrossRef]
  12. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
    [CrossRef]
  13. M. S. Zubairy, J. K. McIver, “Second harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
    [CrossRef] [PubMed]
  14. K. M. Iftekharuddin, M. A. Karim, “Heterodyne detection by using diffraction-free beam: tilt and offset effects,” Appl. Opt. 31, 4853–4856 (1992);S. Klewitz, F. Brinkmann, S. Herminghaus, P. Leiderer, “Bessel-beam-pumped tunable distribution-feedback laser,” Appl. Opt. 34, 7670–7673 (1995).
    [CrossRef] [PubMed]
  15. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  16. L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  17. M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  18. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
    [CrossRef]
  19. M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
    [CrossRef]
  20. F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
    [CrossRef]
  21. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  22. C. Patereson, R. Smith, “Helicon waves: propagation invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–139 (1996).
    [CrossRef]
  23. C. Patereson, “Diffractional elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
    [CrossRef]
  24. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  25. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  26. For brevity, in the rest of the paper the frequency dependence of the mode functions is omitted.
  27. I. S. Gradstein, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  28. A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).
  29. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.
  30. The definition of the Fresnel transform adopted here differs from the one given in Ref. 29by a constant factor.
  31. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
  32. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [CrossRef]
  33. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  34. S. Ramee, R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–94 (2000).
    [CrossRef]

2000 (1)

1998 (3)

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 95–109 (1998).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

1996 (2)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

C. Patereson, R. Smith, “Helicon waves: propagation invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–139 (1996).
[CrossRef]

1994 (3)

C. Patereson, “Diffractional elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1993 (5)

1992 (3)

K. M. Iftekharuddin, M. A. Karim, “Heterodyne detection by using diffraction-free beam: tilt and offset effects,” Appl. Opt. 31, 4853–4856 (1992);S. Klewitz, F. Brinkmann, S. Herminghaus, P. Leiderer, “Bessel-beam-pumped tunable distribution-feedback laser,” Appl. Opt. 34, 7670–7673 (1995).
[CrossRef] [PubMed]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1991 (1)

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

1988 (1)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

1987 (2)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

M. S. Zubairy, J. K. McIver, “Second harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1982 (2)

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

1978 (1)

Allen, L.

M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Bejersbergen, M. W.

M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Brichkov, Yu. A.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Collett, E.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

Farina, J. D.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

Friberg, A. T.

Gase, R.

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Gradstein, I. S.

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

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Harris, M.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Hill, C. A.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Iftekharuddin, K. M.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Karim, M. A.

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Mandel, L.

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

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

McIver, J. K.

M. S. Zubairy, J. K. McIver, “Second harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

Mukunda, N.

R. Simon, N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15, 95–109 (1998).
[CrossRef]

K. Sundar, R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10, 2017–2023 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Narducci, L. M.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Patereson, C.

C. Patereson, R. Smith, “Helicon waves: propagation invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–139 (1996).
[CrossRef]

C. Patereson, “Diffractional elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

Ramee, S.

Ryzhik, I. M.

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

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Simon, R.

Smith, R.

C. Patereson, R. Smith, “Helicon waves: propagation invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–139 (1996).
[CrossRef]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Sundar, K.

Tervonen, E.

Turunen, J.

van der Ween, H. E. L. O.

M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vaughan, J. M. R.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Woerdman, J. P.

M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wolf, E.

Zubairy, M. S.

M. S. Zubairy, J. K. McIver, “Second harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Mod. Opt. (5)

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

C. Patereson, “Diffractional elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (8)

C. Patereson, R. Smith, “Helicon waves: propagation invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–139 (1996).
[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from globally incoherent source,” Opt. Commun. 32, 203–207 (1980).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

M. W. Bejersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (4)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

L. Allen, M. W. Bejersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. S. Zubairy, J. K. McIver, “Second harmonic generation by a partially coherent beam,” Phys. Rev. A 36, 202–206 (1987).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Other (8)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

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

For brevity, in the rest of the paper the frequency dependence of the mode functions is omitted.

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

A. P. Prudnikov, Yu. A. Brichkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1992).

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

The definition of the Fresnel transform adopted here differs from the one given in Ref. 29by a constant factor.

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Figures (5)

Fig. 1
Fig. 1

Normalized spectral intensity S¯(ρ)=S(ρ)/0dρ ρS(ρ) versus dimensionless radial variable ρ/w for different values of the azimuthal phase index m (m=0, 1, 2). The coherence parameter ξ is taken to have the value ξ=0.5.

Fig. 2
Fig. 2

Modulus of the spectral degree of coherence μ of the axially symmetric field m=0 at a pair of points with polar coordinates (ρ, ϕ) and (ρ, ϕ), respectively. The coherence parameter ξ is taken to have the value ξ=0.5.

Fig. 3
Fig. 3

Same as in Fig. 2 but with the field possessing the azimuthal phase index m=10.

Fig. 4
Fig. 4

Radiant intensity of the field with the azimuthal phase index m=1, plotted as a function of the dimensionless spatial frequency kw|s| for two cases: (a) very coherent source (ξ=0.01) and (b) nearly incoherent source (ξ=0.91). The radiant intensity is normalized to the total power.

Fig. 5
Fig. 5

Dependence of the M2 beam factor on the coherence parameter ξ. The solid curve corresponds to a vortex-free beam with the angular index m=0, whereas the dashed and dotted–dashed curves correspond to beams with topological charges m=2 and m=4, respectively.

Equations (43)

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ψnm(ρ)=2ρwmLnm2ρ2w2exp(-imϕ)exp-ρ2w2,
W(ρ, ρ, ω)=sλsψs*(ρ, ω)ψs(ρ, ω).
d2ρ W(ρ, ρ, ω)ψs(ρ)=λsψs(ρ),
λs0.
n=0n!(m+n)! znLnm(x)Lnm(y)
=(xyz)-m/21-zexp-z(x+y)1-zIm4xyz1-z.
Aξ-m/21-ξexp[-im(ϕ-ϕ)]
×exp-1+ξ1-ξ(ρ2+ρ2)w2Im4ξ1-ξρρw2
=l=-n=0Aλnlψnl*(ρ)ψnl(ρ),
λnl=n!(n+l)! ξlδml,
W(ρ, ρ)=Aξ-m/21-ξexp[-im(ϕ-ϕ)]×exp-1+ξ1-ξ(ρ2+ρ2)w2Im4ξ1-ξρρw2.
S(ρ)=Aξ-m/21-ξexp-2ρ2w21+ξ1-ξIm4ξ1-ξρ2w2.
μ(ρ, ρ)=W(ρ, ρ)W(ρ, ρ)W(ρ, ρ).
μ(ρ, ρ)=exp[-im(ϕ-ϕ)]×Im(2ρρ/σc2)Im(2ρ2/σc2)Im(2ρ2/σc2).
1σc2=2ξ(1-ξ)w2.
ξ=σc4w41+w4σc41/2-12.
Im(x)1m!x2m,
W(ρ, ρ)exp[im(ϕ-ϕ)]ρwmρwm×exp-ρ2w2exp-ρ2w2.
J(sˆ)=W˜(-ks, ks),
W˜(f, f )=d2ρd2ρ(2π)2 W(ρ, ρ)exp[-i(fρ+fρ)].
W˜(f, f )=n,lλnlψ˜nl*(f)ψ˜nl(f ),
Jm(x)=02πdϕ2πexp(imϕ-ix cos ϕ),
0dx xα/2exp(-px)Jα(bx)Lnα(cx)
=b2α(p-c)npn+α+1exp-b24pLnαb2c/4pc-p.
W˜(f, f)=A(w2/2)2ξ-m/21-ξexp[im(θ-θ)]×exp-1+ξ1-ξ(f2+f2)w24Imξffw21-ξ.
J(sˆ)=Bξ-m/21-ξ×exp-1+ξ1-ξk2w2|s|22Imξk2w2|s|21-ξ,
W˜(-f, f )0unless|f |k.
(fw)mexp(-f 2w2/2)0unless|f |k.
1/w2k2.
Im(x)12πxexp(x).
1f 2w2exp-1-ξ(1+ξ)2 f 2w20unless|f |k.
(1+ξ)22ξ1σc2k2.
W(ρ, ρ, z, z)=k24π2zzexp[ik(z-z)]d2ρ1×d2ρ2 W(ρ, ρ, 0)expik2z (ρ-ρ1)2×exp-ik2z (ρ-ρ2)2.
W(ρ, ρ, z, z)=nmλnmψnm*(ρ, z)ψnm(ρ, z),
ψnm(ρ, z)=wwz2ρwzmLnm2ρ2w2×exp-ρ2w2exp(-imϕ)×expikz-(m+1)Φz+kρ22Rz,
wz=w2+4z2k2w22,
Rz=z+k2w44z,
Φz=arctan2zkw2.
W(ρ, ρ, z, z)
=Aξ-m/21-ξw2wzwzexp[im(ϕ-ϕ)]×exp{i[k(z-z)-(m+1)(Φz-Φz)]}×exp[i(kρ2/2Rz-kρ2/2Rz)]×exp-1+ξ1-ξρ2wz2+ρ2wz2Im4ξ1-ξρρwzwz.
M2=2πσρσ,
σρ=d2ρ ρ2S(ρ)d2ρ S(ρ),
σ=d2(ks)(ks)2J(ks)d2(ks)J(ks),

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