Abstract

The propagating four-dimensional mutual coherence function of a partially coherent Gaussian beam containing an arbitrarily positioned optical vortex is analytically determined. The dark intensity core becomes diffuse under low coherence and the vortex is only detectable by examining the cross-correlation function. This function contains a vortex dipole or a ring dislocation depending on the vortex position in the beam. The position of these robust propagating correlation phase singularities is described.

© 2008 Optical Society of America

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  32. R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  43. I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  44. W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227-244 (1981).
    [CrossRef]
  45. W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245-259 (1981).
    [CrossRef]
  46. E. Wolf and W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953-964 (1978).
    [CrossRef]
  47. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226-2234 (1998).
    [CrossRef]
  48. M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Sessions XXXV, R.Balian, M.Kleman, and J.-P.Poirier, eds. ( North Holland, 1981).
  49. I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: detection,” (personal communication, 2005).
  50. A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
    [CrossRef]
  51. A. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).
  52. Wavelength of a partially coherent vortex field may be different from the wavelength of a Gaussian beam, used to form a vortex. Resulting spectral effects in the partially coherent vortex fields is a fascinating subject, however, it is beyond the scope of this paper.
  53. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian-Schell model beams,” Opt. Commun. 41, 383-387 (1982).
    [CrossRef]
  54. A. T. Friberg and R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075-1097 (1983).
    [CrossRef]

2007

G. A. Swartzlander, Jr. and R. Hernandez, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef] [PubMed]

2006

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

2005

2004

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

2003

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

J. P. Martikainen and H. T. C. Stoof, “Quantum fluctuations of a vortex in an optical lattice,” Phys. Rev. Lett. 91, 240403 (2003).
[CrossRef] [PubMed]

G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117-125 (2003).
[CrossRef]

G. V. Bogatyryova, C. F. Fel'de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878-880 (2003).
[CrossRef] [PubMed]

I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
[CrossRef]

H. Schouten, G. Gbur, T. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young's interference pattern,” Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

2002

G. S. Agarwal and J. Banergi, “Spatial coherence and information entropy in optical vortex fields,” Opt. Lett. 27, 800-802 (2002).
[CrossRef]

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673-1689 (2002).
[CrossRef]

D. M. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

2001

1999

S. R. Seshadri, “Average characteristics of a partially coherent Bessel-Gauss optical beam,” J. Opt. Soc. Am. A 16, 2917-2927 (1999).
[CrossRef]

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

1998

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

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226-2234 (1998).
[CrossRef]

1997

1993

1992

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

1989

P. Coullet, L. Gill, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1987

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

1983

A. T. Friberg and R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

1982

Y. Li and E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256-258 (1982).
[CrossRef] [PubMed]

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

1981

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227-244 (1981).
[CrossRef]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245-259 (1981).
[CrossRef]

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

1978

1974

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

1967

A. C. Schell, “Technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

1958

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Sov. Phys. JETP 7, 858-861 (1958).

1955

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, II: Fields with a spectral range of arbitrary width,” Proc. R. Soc. London A230, 246-265 (1955).

1954

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, I: Fields with a narrow spectral range,” Proc. R. Soc. London A225, 96-111 (1954).

Agarwal, G. S.

Allen, L.

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

Banergi, J.

Baranova, N. B.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Sessions XXXV, R.Balian, M.Kleman, and J.-P.Poirier, eds. ( North Holland, 1981).

Bogatyryova, G. V.

Borghi, R.

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

Bouchal, Z.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673-1689 (2002).
[CrossRef]

Carter, W. H.

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227-244 (1981).
[CrossRef]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245-259 (1981).
[CrossRef]

E. Wolf and W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953-964 (1978).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.

Coullet, P.

P. Coullet, L. Gill, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Danner, M.

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

Deutsch, I. H.

R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.

Dogariu, A.

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

Fadeeva, T. A.

A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
[CrossRef]

Fel'de, C. F.

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

Foo, G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., “Optical vortex coronagraph,” Opt. Lett. 30, 3308-3310 (2005).
[CrossRef]

Friberg, A. T.

A. T. Friberg and R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

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

Garrison, J. C.

R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.

Gbur, G.

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

Gill, L.

P. Coullet, L. Gill, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Ginzburg, V. L.

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Sov. Phys. JETP 7, 858-861 (1958).

Gori, F.

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

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

Guattari, G.

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

Hernandez, R.

G. A. Swartzlander, Jr. and R. Hernandez, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef] [PubMed]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

Jeng, C.-C.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Johnson, E. G.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Kivshar, Y.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Konshu, E. A.

A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
[CrossRef]

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Law, C. T.

Lee, J. H.

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

Li, Y.

Maleev, I. D.

Mamaev, A. V.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

Mandel, L.

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

Marathay, A.

A. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).

Marathay, A. S.

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: detection,” (personal communication, 2005).

Martikainen, J. P.

J. P. Martikainen and H. T. C. Stoof, “Quantum fluctuations of a vortex in an optical lattice,” Phys. Rev. Lett. 91, 240403 (2003).
[CrossRef] [PubMed]

Motzek, K.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Padgett, M. J.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Padovani, C.

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

Palacios, D. M.

G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., “Optical vortex coronagraph,” Opt. Lett. 30, 3308-3310 (2005).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

D. M. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: detection,” (personal communication, 2005).

Perina, J.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673-1689 (2002).
[CrossRef]

Pilipetski, N. F.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

Pitaevskii, L. P.

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Sov. Phys. JETP 7, 858-861 (1958).

Polyanskii, P. V.

Ponomarenko, S. A.

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

Rocca, F.

P. Coullet, L. Gill, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Rozas, D.

Sacks, Z. S.

Santarsiero, M.

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

Schell, A. C.

A. C. Schell, “Technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

A. C. Schell, “The multiple plate antenna,” Doctoral dissertation (Massachusetts Institute of Technology, 1961), (unpublished), Sec. 7.5.

Schmit, J.

G. A. Swartzlander, Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

Schouten, H.

Schwartz, C.

Seshadri, S. R.

Shih, M.-F.

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

Shinkarev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Shkunov, V. V.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

Shvedov, V. G.

A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
[CrossRef]

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Soskin, M. S.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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]

Stoof, H. T. C.

J. P. Martikainen and H. T. C. Stoof, “Quantum fluctuations of a vortex in an optical lattice,” Phys. Rev. Lett. 91, 240403 (2003).
[CrossRef] [PubMed]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

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

Swartzlander, G. A.

G. A. Swartzlander, Jr. and R. Hernandez, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef] [PubMed]

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

G. Foo, D. M. Palacios, and G. A. Swartzlander, Jr., “Optical vortex coronagraph,” Opt. Lett. 30, 3308-3310 (2005).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices in partially coherent light: theory,” J. Opt. Soc. Am. B 21, 1895-1900 (2004).
[CrossRef]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169-1176 (2003).
[CrossRef]

D. M. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497-499 (2001).
[CrossRef]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226-2234 (1998).
[CrossRef]

D. Rozas, C. T. Law, and G. A. Swartzlander, Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054-3065 (1997).
[CrossRef]

C. T. Law and G. A. Swartzlander, Jr., “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586-588 (1993).
[CrossRef] [PubMed]

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: detection,” (personal communication, 2005).

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

Vicalvi, S.

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

Visser, T.

Volyar, A. V.

A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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.

H. Schouten, G. Gbur, T. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young's interference pattern,” Opt. Lett. 28, 968-970 (2003).
[CrossRef] [PubMed]

G. V. Bogatyryova, C. F. Fel'de, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878-880 (2003).
[CrossRef] [PubMed]

Y. Li and E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256-258 (1982).
[CrossRef] [PubMed]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227-244 (1981).
[CrossRef]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245-259 (1981).
[CrossRef]

E. Wolf and W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am. 68, 953-964 (1978).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, II: Fields with a spectral range of arbitrary width,” Proc. R. Soc. London A230, 246-265 (1955).

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, I: Fields with a narrow spectral range,” Proc. R. Soc. London A225, 96-111 (1954).

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

Wright, E. M.

R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.

Zel'dovich, B. Y.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

IEEE Trans. Antennas Propag.

A. C. Schell, “Technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187-188 (1967).
[CrossRef]

J. Mod. Opt.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[CrossRef]

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

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673-1689 (2002).
[CrossRef]

C. K. Hitzenberger, M. Danner, W. Drexler, and A. F. Fercher, “Measurement of the spatial coherence of superluminescent diodes,” J. Mod. Opt. 46, 1763-1774 (1999).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

A. T. Friberg and R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. I. General theory,” Opt. Acta 28, 227-244 (1981).
[CrossRef]

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, primary, scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245-259 (1981).
[CrossRef]

Opt. Commun.

P. Coullet, L. Gill, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403-408 (1989).
[CrossRef]

G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117-125 (2003).
[CrossRef]

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

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

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R., J. C. Spreeuw, and 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]

Phys. Rev. Lett.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, and S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749-3752 (1991).
[CrossRef] [PubMed]

D. M. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

C.-C. Jeng, M.-F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr. and R. Hernandez, “Optical Rankine vortex and anomalous circulation of light,” Phys. Rev. Lett. 99, 163901 (2007).
[CrossRef] [PubMed]

J. H. Lee, G. Foo, E. G. Johnson, and G. A. Swartzlander, Jr., “Experimental verification of an optical vortex coronagraph,” Phys. Rev. Lett. 97, 053901 (2006).
[CrossRef] [PubMed]

J. P. Martikainen and H. T. C. Stoof, “Quantum fluctuations of a vortex in an optical lattice,” Phys. Rev. Lett. 91, 240403 (2003).
[CrossRef] [PubMed]

Pis'ma Zh. Eksp. Teor. Fiz.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. F. Pilipetski, and V. V. Shkunov, “Speckle-inhomogeneous field wavefront dislocations,” Pis'ma Zh. Eksp. Teor. Fiz. 33, 206-210 (1981).

Proc. R. Soc. London

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, I: Fields with a narrow spectral range,” Proc. R. Soc. London A225, 96-111 (1954).

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources, II: Fields with a spectral range of arbitrary width,” Proc. R. Soc. London A230, 246-265 (1955).

Proc. R. Soc. London, Ser. A

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165-190 (1974).
[CrossRef]

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Sov. Phys. JETP

V. L. Ginzburg and L. P. Pitaevskii, “On the theory of superfluidity,” Sov. Phys. JETP 7, 858-861 (1958).

Tech. Phys. Lett.

A. V. Volyar, V. G. Shvedov, T. A. Fadeeva, and E. A. Konshu, “Nonparaxial Gaussian beams: 3. Optical vortices,” Tech. Phys. Lett. 27, 525-528 (2001).
[CrossRef]

Other

A. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).

Wavelength of a partially coherent vortex field may be different from the wavelength of a Gaussian beam, used to form a vortex. Resulting spectral effects in the partially coherent vortex fields is a fascinating subject, however, it is beyond the scope of this paper.

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

M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Sessions XXXV, R.Balian, M.Kleman, and J.-P.Poirier, eds. ( North Holland, 1981).

I. D. Maleev, D. M. Palacios, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation vortices: detection,” (personal communication, 2005).

A. C. Schell, “The multiple plate antenna,” Doctoral dissertation (Massachusetts Institute of Technology, 1961), (unpublished), Sec. 7.5.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. M. Wright, “Solitons in quantum nonlinear optics,” in Frontiers in Nonlinear Optics: The Serge Akhmanov Memorial Volume, H.Walther, ed. (Hilger, 1992), pp. 151-182.

M.Vasnetsov and K.Staliunas, eds., Optical Vortices. Horizons in World Physics, Vol. 228 (Nova Science, 1999).

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Intensity and (b) phase profiles of an optical vortex displaced by s from the Gaussian beam center. Coordinates ( r , ϕ ) and ( r , ϕ ) are measured with respect to the beam and vortex core centers, respectively. The vectors are related by r = r s .

Fig. 2
Fig. 2

(a) Transverse real space (r space) and transverse momentum space (k space) geometric notation. Vectors r 1 and r 2 are drawn from the origin, and r 1 and r 2 are drawn from the vortex center at ( x , y ) = ( s , 0 ) . The k-space coordinates describe the Fourier transformed (FT) space. Vectors k 1 and k 2 define two arbitrary points in k space. The points ( 0 , ± k V ) demark the location of correlation vortices when s 0 .

Fig. 3
Fig. 3

Relative far-field separation between spatial correlation vortices, 2 k V w 0 , as a function of both the relative near-field vortex displacement, s w 0 , and the relative coherence length, l c w 0 , where w 0 is the size of the initial Gaussian envelope in the near field. Lines a b ¯ , b d ¯ , and a c ¯ represent the on-axis vortex, the coherent case, and the incoherent case, respectively.

Fig. 4
Fig. 4

Expansion of the correlation ring dislocation as a function of the normalized propagation distance for various initial transverse coherence lengths, l c , where w 0 is the initial Gaussian beam size. A low coherence beam may exhibit different slopes in the near and far fields.

Fig. 5
Fig. 5

Propagated 2 × 2 mm cross-sectional profiles of amplitude [(a),(c),(e)] and phase [(b),(d),(e)] for a beam having an off-axis vortex at s = 0.2 mm , where w 0 = 1.0 mm , λ = 632 nm , and l c = 0.1 mm . The top, middle, and bottom rows correspond to propagation distances of z = 1 , 10, and 400 mm and K w 0 2 2 = 4.97 [ m ] . Equal phase isolines 3 π 4 , π 4 , π 4 , and 3 π 4 are shown as contrast lines.

Fig. 6
Fig. 6

Position of one of the pair of vortices in the cross correlation function showing the linear variation when the field vortex is translated along the x axis by a distance, s, ranging in value from 0 to 3. Cases are shown for fixed normalized propagation distances, q, and transverse coherence length, l c . Transverse distances have been normalized by the initial beam size, w 0 .

Fig. 7
Fig. 7

Position of one of the pair of vortices in the cross correlation function as the beam propagates from q = 0 to 3.0. The initial position of the correlation vortex coincides with the initial position of the field vortex at the point ( s , 0 ) , where s = 0.2 in this example. Cases are shown for different transverse coherence lengths l c . All transverse distances have been normalized by the initial beam size w 0 .

Tables (1)

Tables Icon

Table 1 Relative Coherence Parameters Relating the Coherence Length l c to the Beam Size w 0

Equations (30)

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

E ( r , z , t ) = A ( r , z ) g ( r , z ) exp i ( m ϕ + β + k z z ω t ) ,
Γ ( r 1 , r 2 , z ) = E 1 ( r 1 , z , t ) E 2 * ( r 2 , z , t ) ,
C ( r 1 r 2 ) = exp ( r 1 r 2 2 l C 2 ) ,
Γ m ( r 1 , r 2 , 0 ) = [ ( x 1 s + i y 1 ) ( x 2 s i y 2 ) w 0 2 ] m Γ 0 ( r 1 , r 2 , 0 ) ,
Γ 0 ( r 1 , r 2 , 0 ) = E 0 2 exp ( [ r 1 2 + r 2 2 ] w 0 2 ) C ( r 1 r 2 ) ,
Γ ̃ m ( k 1 , k 2 , 0 ) = Γ m ( r 1 , r 2 , 0 ) exp ( i k 2 r 2 i k 1 r 1 ) d r 1 d r 2 ,
Γ ̃ 0 ( k 1 , k 2 , 0 ) = ( E 0 2 π 2 w 0 4 p ) exp ( α k 1 2 α k 2 2 + 2 δ k 1 k 2 )
I ̃ 0 ( k , 0 ) = Γ ̃ 0 ( k , k , 0 ) = ( E 0 2 π 2 w 0 2 k C 2 ) exp ( k 2 2 k C 2 ) ,
X ̃ 0 ( k , 0 ) = Γ ̃ 0 ( k , k , 0 ) = ( E 0 2 π 2 w 0 2 k c 2 ) exp ( w 0 2 k 2 2 ) .
i u 1 + v 1 u 2 v 2 u 1 + v 1 + u 2 + v 2 k 1 x u 1 k 2 x u 2 k 1 y v 1 k 2 y v 2 Γ ̃ m ( k 1 , k 2 , 0 ) = x 1 u 1 x 2 u 2 y 1 v 1 y 2 v 2 Γ m ( r 1 , r 2 , 0 ) exp ( i k 2 r 2 i k 1 r 1 ) d r 1 d r 2 ,
Γ ̃ 1 ( k 1 , k 2 , 0 ) = ( 1 w 0 2 ) ( k 1 x + i s + i k 1 y ) ( k 2 x i s i k 2 y ) Γ ̃ 0 ( k 1 , k 2 , 0 ) .
Γ ̃ 1 ( k 1 , k 2 , 0 ) = ( 1 w 0 2 ) { 4 δ + c 1 k 1 k 2 c 2 ( k 1 2 + k 2 2 ) + i c 3 ( k 2 x k 1 y k 1 x k 2 y ) 2 s [ p ( k 1 y + k 2 y ) + i w 0 2 ( k 2 x k 1 x ) ] + s 2 } Γ ̃ 0 ( k 1 , k 2 , 0 ) ,
I ̃ 1 ( k , 0 ) = ( w 0 2 4 ) { 4 k x 2 l C 4 σ 2 + ( l C 2 k y σ 2 s w 0 2 ) 2 + 4 σ } I 0 ( k , 0 ) ,
X ̃ 1 ( k , 0 ) = ( w 0 2 4 ) { k 2 + 4 i s k x w 0 2 + k V 2 } X 0 ( k , 0 ) ,
k V = ( k R 2 + 4 s 2 w 0 4 ) 1 2 .
X ̃ 1 ( k , 0 ) ( w 0 2 4 ) ( i s k x ± k y k V ) X 0 ( k , 0 ) .
k j z K ( 1 k j 2 2 K 2 ) .
Γ m ( r 1 , r 2 , z ) = ( 1 2 π ) 4 Γ ̃ m ( k 1 , k 2 , 0 ) exp [ i z ( k 2 2 k 1 2 ) 2 K ] exp ( i k 1 r 1 i k 2 r 2 ) d k 1 d k 2 .
Γ 0 ( r 1 , r 2 , z ) = ( E 0 2 w 0 2 Δ 2 ) exp { [ p ( r 1 + r 2 ) 2 + ( r 1 r 2 ) 2 + 4 i q ( r 2 2 r 1 2 ) ] 2 p Δ 2 } ,
I 0 ( r , z ) = ( E 0 2 w 0 2 Δ 2 ) exp ( 2 r 2 Δ 2 ) ,
X 0 ( r , z ) = ( E 0 2 w 0 2 Δ 2 ) exp ( 2 r 2 p Δ 2 ) ,
i + u 1 + v 1 u 2 v 2 ( u 1 + v 1 + u 2 + v 2 x 1 u 1 x 2 u 2 y 1 v 1 y 2 v 2 ) Γ m ( r 1 , r 2 , z ) = ( 1 2 π ) 4 k 1 x u 1 k 2 x u 2 k 1 y v 1 k 2 y v 2 Γ ̃ m ( k 1 , k 2 , 0 ) exp [ i z ( k 2 2 k 1 2 ) 2 K ] exp ( i k 1 r 1 i k 2 r 2 ) d k 1 d k 2 .
Γ 1 ( r 1 , r 2 , z ) = ( 1 w 0 2 ) { 4 δ + c 1 ( 2 x 1 x 2 + 2 y 1 y 2 ) + c 2 ( 2 x 1 2 + 2 y 1 2 + 2 x 2 2 + 2 y 2 2 ) + i c 3 [ ( 2 y 1 x 2 2 x 1 y 2 ) ] ( s 2 ) [ w 0 2 ( x 2 + x 1 ) i p ( y 1 y 2 ) ] + s 2 } Γ 0 ( r 1 , r 2 , z ) .
Γ 1 ( r 1 , r 2 , z ) = { a w 0 2 σ + [ b ( r 1 + r 2 ) 2 c ( r 1 r 2 ) 2 ] ( w 0 2 4 Δ 4 ) ( i q w 0 4 σ p Δ 4 ) ( r 1 2 r 2 2 ) + i ( x 2 y 1 x 1 y 2 ) Δ 2 + ( s Δ 2 ) [ ( i q p ) ( x 1 x 2 ) ( x 2 + x 1 ) i ( y 1 y 2 ) q ( y 2 + y 1 ) ] + s 2 w 0 2 } Γ 0 ( r 1 , r 2 , z ) ,
I 1 ( r , z ) = { a w 0 2 σ + b r 2 w 0 2 Δ 4 ( 2 s Δ 2 ) [ x + q y ] + s 2 w 0 2 } I 0 ( r , z ) .
X 1 ( r , z ) = { a w 0 2 σ c r 2 w 0 2 Δ 4 + ( 2 i s Δ 2 ) [ ( q p ) x y ] + s 2 w 0 2 } X 0 ( r , z ) .
r V = Δ 2 a c σ = ( w 0 2 q l c ) ( 1 + q 2 p ) 1 2 ( 1 + q 2 p 2 ) 1 2 ,
x V = R V ( 1 + q 2 p 2 ) 1 2 ,
y V = q x V p ,
R V = ( s 2 ( 1 + q 2 p ) + w 0 4 q 2 l c 2 ) 1 2 ( 1 + q 2 p 1 + q 2 p 2 ) 1 2 .

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