Abstract

A new kind of partially coherent beam with non-conventional correlation function named elliptical Laguerre-Gaussian correlated Schell-model (LGCSM) beam is introduced. Analytical propagation formula for an elliptical LGCSM beam passing through a stigmatic ABCD optical system is derived. The elliptical LGCSM beam exhibits unique features on propagation, e.g., its intensity in the far field (or in the focal plane) displays an elliptical ring-shaped beam profile, being qualitatively different from the circular ring-shaped beam profile of the circular LGCSM beam. Furthermore, we carry out experimental generation of an elliptical LGCSM beam with controllable ellipticity, and measure its focusing properties. Our experimental results are consistent with the theoretical predictions. The elliptical LGCSM beam will be useful in atomic optics.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [CrossRef] [PubMed]
  2. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [CrossRef]
  3. H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [CrossRef] [PubMed]
  4. Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated random media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [CrossRef] [PubMed]
  5. Z. Tong, O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012).
    [CrossRef] [PubMed]
  6. Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
    [CrossRef] [PubMed]
  7. S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
    [CrossRef] [PubMed]
  8. O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
    [CrossRef] [PubMed]
  9. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
    [CrossRef] [PubMed]
  10. S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
    [CrossRef]
  11. Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
    [CrossRef]
  12. Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
    [CrossRef]
  13. C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
    [CrossRef] [PubMed]
  14. Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
    [CrossRef]
  15. Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
    [CrossRef] [PubMed]
  16. F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [CrossRef] [PubMed]
  17. R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
    [CrossRef] [PubMed]
  18. Y. Chen, Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
    [CrossRef] [PubMed]
  19. Y. Chen, F. Wang, C. Zhao, Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
    [CrossRef] [PubMed]
  20. J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119–204.
  21. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
    [CrossRef] [PubMed]
  22. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
    [CrossRef] [PubMed]
  23. M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
    [CrossRef] [PubMed]
  24. X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
    [CrossRef]
  25. J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
    [CrossRef]
  26. T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
    [CrossRef]
  27. F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [CrossRef]
  28. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [CrossRef] [PubMed]
  29. Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28(13), 1084–1086 (2003).
    [CrossRef] [PubMed]
  30. Z. Mei, D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005).
    [CrossRef] [PubMed]
  31. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
    [CrossRef] [PubMed]
  32. Y. Cai, Z. Wang, Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008).
    [CrossRef] [PubMed]
  33. Y. Cai, Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21(6), 1058–1065 (2004).
    [CrossRef] [PubMed]
  34. Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
    [CrossRef] [PubMed]
  35. Z. Mei, D. Zhao, “Controllable elliptical dark-hollow beams,” J. Opt. Soc. Am. A 23(4), 919–925 (2006).
    [CrossRef] [PubMed]
  36. J. C. Gutiérrez-Vega, “Characterization of elliptical dark hollow beams,” Proc. SPIE 7062, 706207 (2008).
    [CrossRef]
  37. C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008).
    [CrossRef]
  38. H. Li, J. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011).
    [CrossRef] [PubMed]
  39. R. Chakraborty, A. Ghosh, “Generation of an elliptical hollow beam using Mathieu and Bessel functions,” J. Opt. Soc. Am. A 23(9), 2278–2282 (2006).
    [CrossRef]
  40. Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004).
    [CrossRef]
  41. Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
    [CrossRef]
  42. X. Lü, Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
    [CrossRef]
  43. C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
    [CrossRef] [PubMed]
  44. L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University, 1995).
  45. S. A. Collins., “Lens-system diffraction integral written in terms ofmatrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [CrossRef]
  46. Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
    [CrossRef] [PubMed]
  47. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  48. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
    [CrossRef]

2014 (7)

2013 (5)

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

2012 (4)

2011 (2)

2009 (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

2008 (5)

2007 (3)

2006 (3)

2005 (1)

2004 (2)

Y. Cai, Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21(6), 1058–1065 (2004).
[CrossRef] [PubMed]

Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (2)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

1998 (1)

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

1997 (1)

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

1995 (1)

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

1987 (1)

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

1979 (1)

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

1970 (1)

Anderson, D. Z.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Baykal, Y.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Cai, Y.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

Y. Chen, Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[CrossRef] [PubMed]

Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

Y. Chen, F. Wang, C. Zhao, Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
[CrossRef]

C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[CrossRef] [PubMed]

Y. Cai, Z. Wang, Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008).
[CrossRef] [PubMed]

X. Lü, Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
[CrossRef]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
[CrossRef] [PubMed]

Y. Cai, S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A 21(6), 1058–1065 (2004).
[CrossRef] [PubMed]

Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28(13), 1084–1086 (2003).
[CrossRef] [PubMed]

Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Chakraborty, R.

Chávez-Cerda, S.

Chen, H.

C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008).
[CrossRef]

Chen, R.

Chen, Y.

Y. Chen, F. Wang, C. Zhao, Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Chen, Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[CrossRef] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Collins, S. A.

Cornell, E. A.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Du, S.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Eyyuboglu, H. T.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Y. Cai, H. T. Eyyuboğlu, Y. Baykal, “Scintillation of astigmatic dark hollow beams in weak atmospheric turbulence,” J. Opt. Soc. Am. A 25, 1497–1503 (2008).
[CrossRef]

Gbur, G.

Ghosh, A.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Gu, Y.

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Gutiérrez-Vega, J. C.

He, S.

Hirano, T.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Iturbe-Castillo, M. D.

Jhe, W.

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

Korotkova, O.

Kuga, T.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Lajunen, H.

Lee, K.

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

Li, H.

Liang, C.

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Lin, Q.

Liu, L.

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

R. Chen, L. Liu, S. Zhu, G. Wu, F. Wang, Y. Cai, “Statistical properties of a Laguerre-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Express 22(2), 1871–1883 (2014).
[CrossRef] [PubMed]

Liu, X.

Lu, X.

Lü, X.

X. Lü, Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Mei, Z.

Minogin, V. G.

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

Montgomery, D.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Qu, J.

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Renn, M. J.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Saastamoinen, T.

Sahin, S.

Santarsiero, M.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

Sasada, H.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shchepakina, E.

Shimizu, Y.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

Tong, Z.

Torii, T.

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Vdovin, O.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Wang, F.

Wang, L.

C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008).
[CrossRef]

Wang, Y.

C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[CrossRef] [PubMed]

Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004).
[CrossRef]

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

Wang, Z.

Y. Cai, Z. Wang, Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008).
[CrossRef] [PubMed]

Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004).
[CrossRef]

Wieman, C. E.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Wu, G.

Xu, X.

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

Yin, J.

H. Li, J. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011).
[CrossRef] [PubMed]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

Yuan, Y.

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

Zhang, Y.

Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Y. Chen, F. Wang, C. Zhao, Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[CrossRef] [PubMed]

C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008).
[CrossRef]

C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[CrossRef] [PubMed]

Zhao, D.

Zhu, S.

Zhu, Y.

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (4)

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of Collet-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Y. Yuan, X. Liu, F. Wang, Y. Chen, Y. Cai, J. Qu, H. T. Eyyuboğlu, “Scintillation index of a multi-Gaussian Schell-model beam in turbulent atmosphere,” Opt. Commun. 305, 57–65 (2013).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[CrossRef]

Z. Wang, Q. Lin, Y. Wang, “Control of atomic rotation by elliptical hollow beam carrying zero angular momentum,” Opt. Commun. 240(4-6), 357–362 (2004).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (2)

C. Zhao, X. Lu, L. Wang, H. Chen, “Hollow elliptical Gaussian beams generated by a triangular prism,” Opt. Laser Technol. 40(3), 575–580 (2008).
[CrossRef]

S. Du, Y. Yuan, C. Liang, Y. Cai, “Second-order moments of a multi-Gaussian Schell-model beam in a turbulent atmosphere,” Opt. Laser Technol. 50, 14–19 (2013).
[CrossRef]

Opt. Lett. (16)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
[CrossRef] [PubMed]

Q. Lin, Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Y. Cai, X. Lu, Q. Lin, “Hollow Gaussian beam and its propagation,” Opt. Lett. 28(13), 1084–1086 (2003).
[CrossRef] [PubMed]

H. Li, J. Yin, “Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber,” Opt. Lett. 36(4), 457–459 (2011).
[CrossRef] [PubMed]

H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[CrossRef] [PubMed]

S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[CrossRef] [PubMed]

Z. Tong, O. Korotkova, “Non-uniformly correlated beams in uniformly correlated random media,” Opt. Lett. 37(15), 3240–3242 (2012).
[CrossRef] [PubMed]

Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett. 32(21), 3179–3181 (2007).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[CrossRef] [PubMed]

C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014).
[CrossRef] [PubMed]

C. Zhao, Y. Cai, F. Wang, X. Lu, Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008).
[CrossRef] [PubMed]

Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[CrossRef] [PubMed]

Y. Gu, G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38(9), 1395–1397 (2013).
[CrossRef] [PubMed]

F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
[CrossRef] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[CrossRef] [PubMed]

Y. Chen, Y. Cai, “Generation of a controllable optical cage by focusing a Laguerre-Gaussian correlated Schell-model beam,” Opt. Lett. 39(9), 2549–2552 (2014).
[CrossRef] [PubMed]

Phys. Lett. A (2)

Y. Zhang, L. Liu, C. Zhao, Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[CrossRef]

X. Lü, Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A 369(1-2), 157–166 (2007).
[CrossRef]

Phys. Rev. A (3)

X. Xu, V. G. Minogin, K. Lee, Y. Wang, W. Jhe, “Guiding cold atoms in a hollow laser beam,” Phys. Rev. A 60(6), 4796–4804 (1999).
[CrossRef]

J. Yin, Y. Zhu, W. Jhe, Y. Wang, “Atom guiding and cooling in a dark hollow laser beam,” Phys. Rev. A 58(1), 509–513 (1998).
[CrossRef]

Y. Chen, F. Wang, L. Liu, C. Zhao, Y. Cai, O. Korotkova, “Generation and propagation of a partially coherent vector beam with special correlation functions,” Phys. Rev. A 89(1), 013801 (2014).
[CrossRef]

Phys. Rev. Lett. (2)

T. Kuga, T. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, “Novel optical trap of atoms with a doughnut beams,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. C. Gutiérrez-Vega, “Characterization of elliptical dark hollow beams,” Proc. SPIE 7062, 706207 (2008).
[CrossRef]

Science (2)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283(5408), 1689–1695 (1999).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[CrossRef] [PubMed]

Other (3)

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 2003), Vol. 44, pp. 119–204.

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University, 1995).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Density plot of the square of the modulus of the degree of coherence of the elliptical LGCSM beam for different values of δ 0x and δ 0y with beam order n = 5.

Fig. 2
Fig. 2

Density plot of the intensity distribution of the elliptical LGCSM beam in the geometrical focal plane for different values of δ 0x and δ 0y with beam order n = 5.

Fig. 3
Fig. 3

Experimental setup for generating an elliptical LGCSM beam, measuring the square of the modulus of its degree of coherence and its focused intensity. RM, reflecting mirror; BE, beam expander; SLM, spatial light modulator; CA, circular aperture; RGGD, rotating ground-disk; L, L 1 , L 2 , L 3 , thin lenses; GAF, Gaussian amplitude filter; CCD, charge-coupled device; BPA, beam profile analyzer; PC 1 , PC 2 , personal computers.

Fig. 4
Fig. 4

Phase gratings for generating an elliptical DH beams (n = 5) of different values of ω 0x / ω 0y with ω 0x =0.8mm . (a) ω 0x / ω 0y =0.4, (b) ω 0x / ω 0y =0.8, (c) ω 0x / ω 0y =1, (d) ω 0x / ω 0y =1.2, (e) ω 0x / ω 0y =2.5.

Fig. 5
Fig. 5

Experimental results of (a) the intensity distribution and (b) the corresponding cross line (dotted curve) of the generated elliptical LGCSM beam (n = 5) just behind the GAF. The solid curve is a result of the theoretical fit.

Fig. 6
Fig. 6

Experimental results of the square of the modulus of the degree of coherence and the corresponding cross lines (dotted curves) of the generated elliptical LGCSM beam (n = 5) just behind the GAF for different values of coherence widths δ 0x and δ 0y . The solid curve is a result of the theoretical fit.

Fig. 7
Fig. 7

Experimental results of the intensity distribution of the generated elliptical LGCSM beam (n = 5) and the corresponding cross lines in the geometrical focal plane for different values of coherence widths δ 0x and δ 0y . The solid curves denote the theoretical results calculated by Eqs. (17)-(20) and (27).

Equations (27)

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

J 0 ( r 1 , r 2 )= I( v ) H * ( r 1 ,v )H( r 2 ,v ) d 2 v,
J 0 ( r 1 , r 2 )= J i ( v 1 , v 2 ) H * ( r 1 , v 1 )H( r 2 , v 2 ) d 2 v 1 d 2 v 2 ,
J i ( v 1 , v 2 )= I( v 1 )I( v 2 ) δ( v 1 v 2 ).
H( r,v )= i λf T( r )exp[ iπ λf ( v 2 2rv ) ],
I(v)= ( v x 2 ω 0x 2 + v y 2 ω 0y 2 ) n exp( 2 v x 2 ω 0x 2 2 v x 2 ω 0y 2 ),
T( r )=exp( r 2 4 σ 0 2 ),
J 0 ( r 1 , r 2 )= G 0 exp[ r 1 2 + r 2 2 4 σ 0 2 ]γ( r 1 , r 2 ),
γ( r 2 r 1 )=exp[ ( x 2 x 1 ) 2 2 δ 0x 2 ( y 2 y 1 ) 2 2 δ 0y 2 ] L n 0 [ ( x 2 x 1 ) 2 2 δ 0x 2 + ( y 2 y 1 ) 2 2 δ 0y 2 ],
J( ρ 1 , ρ 2 ,z )= 1 (λB) 2 exp[ ikD 2B ( ρ 1 2 ρ 2 2 ) ] × J 0 ( r 1 , r 2 )exp[ ikA 2B ( r 1 2 r 2 2 ) ]exp[ ik B ( r 1 ρ 1 r 2 ρ 2 ) ] d 2 r 1 d 2 r 2 ,
r s = r 1 + r 2 2 ,Δr= r 1 r 2 , ρ s = ρ 1 + ρ 2 2 ,Δρ= ρ 1 ρ 2 .
J( ρ 1 , ρ 2 ,z )= G 0 ( λB ) 2 exp[ ikD B ρ s Δρ ] × exp[ 1 2 σ 0 2 r s 2 +( ikA B Δr+ ik B Δρ ) r s ] d 2 r s ×exp[ 1 8 σ 0 2 Δ r 2 + ik B ρ s Δr Δ x 2 2 δ 0x 2 Δ y 2 2 δ 0y 2 ] L n 0 [ Δ x 2 2 δ 0x 2 + Δ y 2 2 δ 0y 2 ] d 2 Δr.
J( ρ 1 , ρ 2 ,z )= 2π G 0 σ 0 2 ( λB ) 2 exp[ ikD B ρ s Δρ σ 0 2 2 ( k B ) 2 ( AΔr+Δρ ) 2 ] ×exp[ 1 8 σ 0 2 Δ r 2 + ik B ρ s Δr Δ x 2 2 δ 0x 2 Δ y 2 2 δ 0y 2 ] L n 0 [ Δ x 2 2 δ 0x 2 + Δ y 2 2 δ 0y 2 ] d 2 Δr.
L n (x)= p=0 n ( n p ) (1) p p! x p ,
( x 2 + y 2 ) p = m=0 p ( p m ) x 2( pm ) y 2m .
J( ρ 1 , ρ 2 ,z )= 2π G 0 σ 0 2 ( λB ) 2 exp[ ikD B ρ s Δρ ]exp[ σ 0 2 2 ( k B ) 2 Δ ρ 2 ] p=0 n m=0 p ( n p ) ( p m ) (1) p p! ( 1 2 δ 0x ) 2( pm ) ( 1 2 δ 0y ) 2m exp[ ( σ 0 2 2 ( Ak B ) 2 1 8 σ 0 2 1 2 δ 0x 2 )Δ x 2 ]exp[ ( σ 0 2 ( k B ) 2 AΔρ+ ik B ρ s )Δr ] exp[ ( σ 0 2 2 ( Ak B ) 2 1 8 σ 0 2 1 2 δ 0y 2 )Δ y 2 ]Δ x 2( pm ) Δ y 2m dΔxdΔy.
x n exp[ ( xβ ) 2 ]dx= ( 2i ) n π H n ( iβ ),
J( ρ 1 , ρ 2 ,z )= 2 G 0 π 2 σ 0 2 ( λB ) 2 exp[ ikD B ρ s Δρ ]exp[ σ 0 2 2 ( k B ) 2 Δ ρ 2 ] × p=0 n m=0 p ( n p ) ( p m ) (1) p p! ( 1 2 δ 0x ) 2( pm ) ( 1 2 δ 0y ) 2m × ( 2i ) 2p a x a y ( a x ) ( pm ) ( a y ) m H 2( pm ) ( i b x 2 a x ) H 2m ( i b y 2 a y ) ×exp[ b x 2 2 a x + b y 2 2 a y ],
a x = σ 0 2 2 ( Ak B ) 2 + 1 8 σ 0 2 + 1 2 δ 0x 2 , a y = σ 0 2 2 ( Ak B ) 2 + 1 8 σ 0 2 + 1 2 δ 0y 2 ,
b x =( σ 0 2 ( k B ) 2 AΔ ρ x + ik B ρ sx ), b y =( σ 0 2 ( k B ) 2 AΔ ρ y + ik B ρ sy ).
I(ρ,z)=J( ρ,ρ,z ).
( A B C D )=( 1 f 0 1 )( 1 0 1/f 1 )=( 0 f 1/f 1 ).
g ( 2 ) ( r 1 , r 2 )= I( r 1 ,t )I( r 2 ,t ) I( r 1 ,t ) I( r 2 ,t ) ,
g ( 2 ) ( r 1 , r 2 )=1+ | γ( r 1 , r 2 ) | 2 .
| γ( r 1 , r 2 =0 ) | 2 = 1 M m=1 M I ( m ) ( x 1 , y 1 ) I ( m ) ( 0,0 ) I ¯ ( x 1 , y 1 ) I ¯ ( 0,0 ) 1,
I ¯ ( x 1 , y 1 )= m=1 M I ( m ) ( x 1 , y 1 )/M,
I ¯ ( 0,0 )= m=1 M I ( m ) ( 0,0 )/M.
( A B C D )=( 0 f 3 1/ f 3 0 ).

Metrics