Abstract

With the use of the general beam formulation, the modulus of the complex degree of coherence for partially coherent cosh-Gaussian, cos-Gaussian, Gaussian, annular and higher-order Gaussian optical beams is evaluated in atmospheric turbulence. For different propagation lengths in horizontal atmospheric links, the moduli of the complex degree of coherence at the source and receiver planes are examined when reference points are taken on the receiver axis and off-axis. In the on-axis case, it is observed that in propagation, the moduli of the complex degree of coherence are symmetrical and look like the intensity profile of the related coherent beam propagating in a turbulent atmosphere. For all the beams considered, the moduli of the complex degree of coherence profiles turn into Gaussian shapes beyond certain propagation lengths. In the off-axis case, the moduli of complex degree of coherence patterns become drifted at the earlier propagation lengths. Among the beams investigated, the cos-Gaussian beam is found to be almost independent of the changes in the source partial coherence parameter, and the annular beam seems to be affected the most against the variations of the source partial coherence parameter.

© 2007 Optical Society of America

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References

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  1. T. L. Ho, "Coherence degradation of Gaussian beams in a turbulent atmosphere," J. Opt. Soc. Am. 60, 667-673 (1970).
    [CrossRef]
  2. A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187-1192 (1972).
    [CrossRef]
  3. J. C. Leader, "Atmospheric propagation of partially coherent radiation," J. Opt. Soc. Am. 68, 175-185 (1978).
    [CrossRef]
  4. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1307 (1979).
    [CrossRef]
  5. R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Opt. Acta 28, 1203-1207 (1981).
    [CrossRef]
  6. Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
    [CrossRef]
  7. Y. Baykal and M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. Am. A 2, 2124-2132 (1985).
    [CrossRef]
  8. L. C. Andrews, C. Y. Young, and W. B. Miller, "Coherence properties of a reflected optical wave in atmospheric turbulence," J. Opt. Soc. Am. A 13, 851-861 (1996).
    [CrossRef]
  9. H. Okayama and L. Z. Wang, "Measurement of the spatial coherence of light influenced by turbulence," Appl. Opt. 38, 2342-2345 (1999).
    [CrossRef]
  10. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  11. A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  12. O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
    [CrossRef]
  13. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).
  14. E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
    [CrossRef]
  15. Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
    [CrossRef]
  16. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]
  17. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
    [CrossRef]
  18. H. T. Eyyuboglu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
    [CrossRef] [PubMed]
  19. H. T. Eyyuboglu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
    [CrossRef] [PubMed]
  20. H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).
  21. Ç. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
    [CrossRef] [PubMed]
  22. Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
    [CrossRef]
  23. Y. Baykal and H. T. Eyyuboglu, "Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free space optics links," Opt. Eng. (Bellingham) 45, 056001 (2006).
    [CrossRef]
  24. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).
  25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
    [CrossRef]
  26. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.
  27. H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

2007 (1)

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

2006 (6)

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).

Y. Baykal and H. T. Eyyuboglu, "Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free space optics links," Opt. Eng. (Bellingham) 45, 056001 (2006).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

Ç. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (3)

H. T. Eyyuboglu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

2003 (1)

2002 (1)

1999 (1)

1996 (1)

1992 (1)

1988 (1)

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

1985 (1)

1983 (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

1981 (1)

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Opt. Acta 28, 1203-1207 (1981).
[CrossRef]

1979 (1)

1978 (1)

1972 (1)

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187-1192 (1972).
[CrossRef]

1970 (1)

Altay, S.

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).

Amarande, S.

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

L. C. Andrews, C. Y. Young, and W. B. Miller, "Coherence properties of a reflected optical wave in atmospheric turbulence," J. Opt. Soc. Am. A 13, 851-861 (1996).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

Arpali, Ç.

Arpali, S. A.

Baykal, Y.

Ç. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

Y. Baykal and H. T. Eyyuboglu, "Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free space optics links," Opt. Eng. (Bellingham) 45, 056001 (2006).
[CrossRef]

Y. Baykal, "Formulation of correlations for general-type beams in atmospheric turbulence," J. Opt. Soc. Am. A 23, 889-893 (2006).
[CrossRef]

H. T. Eyyuboglu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboglu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Y. Baykal and M. A. Plonus, "Intensity fluctuations due to a spatially partially coherent source in atmospheric turbulence as predicted by Rytov's method," J. Opt. Soc. Am. A 2, 2124-2132 (1985).
[CrossRef]

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Cai, Y.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Davidson, F. M.

Dogariu, A.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
[CrossRef] [PubMed]

Eyyuboglu, H. T.

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).

Y. Baykal and H. T. Eyyuboglu, "Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free space optics links," Opt. Eng. (Bellingham) 45, 056001 (2006).
[CrossRef]

Ç. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, "Simulator for general-type beam propagation in turbulent atmosphere," Opt. Express 14, 8918-8928 (2006).
[CrossRef] [PubMed]

H. T. Eyyuboglu and Y. Baykal, "Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere," Appl. Opt. 44, 976-983 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboglu and Y. Baykal, "Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere," Opt. Express 12, 4659-4674 (2004).
[CrossRef] [PubMed]

Fante, R. L.

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Opt. Acta 28, 1203-1207 (1981).
[CrossRef]

Friberg, A. T.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

He, Q. S.

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

He, S.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Ho, T. L.

Kon, A. I.

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187-1192 (1972).
[CrossRef]

Korotkova, O.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

Leader, J. C.

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.

Miller, W. B.

Okayama, H.

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

Plonus, M. A.

Ricklin, J. C.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Salem, M.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

Sermutlu, E.

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

Tatarskii, V. I.

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187-1192 (1972).
[CrossRef]

Tervonen, E.

Turunen, J.

E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Wang, L. Z.

Wang, S. C. H.

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Wolf, E.

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

Yazicioglu, C.

Young, C. Y.

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium. Effect of source coherence," Opt. Acta 28, 1203-1207 (1981).
[CrossRef]

Opt. Commun. (4)

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, "Change in degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 271, 1-8 (2007).
[CrossRef]

H. T. Eyyuboglu, S. Altay, and Y. Baykal, "Propagation characteristics of higher-order annular Gaussian beams in atmospheric turbulence," Opt. Commun. 264, 25-34 (2006).

Q. S. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

H. T. Eyyuboglu, Y. Baykal, and E. Sermutlu, "Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere," Opt. Commun. 265, 399-405 (2006).

Opt. Eng. (Bellingham) (2)

Y. Baykal and H. T. Eyyuboglu, "Effect of source spatial partial coherence on the angle-of-arrival fluctuations for free space optics links," Opt. Eng. (Bellingham) 45, 056001 (2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, "Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom," Opt. Eng. (Bellingham) 43, 330-341 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Radio Sci. (1)

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for weak atmospheric turbulence using a partially coherent source," Radio Sci. 18, 551-556 (1983).
[CrossRef]

Radiophys. Quantum Electron. (1)

A. I. Kon and V. I. Tatarskii, "On the theory of the propagation of partially coherent light beams in a turbulent atmosphere," Radiophys. Quantum Electron. 15, 1187-1192 (1972).
[CrossRef]

Waves Random Complex Media (1)

M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, "Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere," Waves Random Complex Media 14, 513-523 (2004).

Other (3)

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), Chap. 5.

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Figures (12)

Fig. 1
Fig. 1

Three-dimensional view of the modulus of the complex degree of coherence for the cosh-Gaussian beam at discrete propagation lengths and at p 2 = ( 0 , 0 ) .

Fig. 2
Fig. 2

Three-dimensional view of the modulus of the complex degree of coherence for the cos-Gaussian beam at discrete propagation lengths and at p 2 = ( 0 , 0 ) .

Fig. 3
Fig. 3

Three-dimensional view of the complex degree of coherence for the Gaussian beam at discrete propagation lengths and at p 2 = ( 0 , 0 ) .

Fig. 4
Fig. 4

Three-dimensional view of the modulus of the complex degree of coherence for the annular Gaussian beam at discrete propagation lengths and at p 2 = ( 0 , 0 ) .

Fig. 5
Fig. 5

Three-dimensional view of the modulus of the complex degree of coherence for the higher-order Gaussian beam at discrete propagation lengths and at p 2 = ( 0 , 0 ) .

Fig. 6
Fig. 6

Three-dimensional view of the modulus of the complex degree of coherence for the cosh-Gaussian beam at discrete propagation lengths and at p 2 = ( 1 cm , 1 cm ) .

Fig. 7
Fig. 7

Three-dimensional view of the modulus of the complex degree of coherence for the cos-Gaussian beam at discrete propagation lengths and at p 2 = ( 1 cm , 1 cm ) .

Fig. 8
Fig. 8

Three-dimensional view of the modulus of the complex degree of coherence for the Gaussian beam at discrete propagation lengths and at p 2 = ( 1 cm , 1 cm ) .

Fig. 9
Fig. 9

Three-dimensional view of the modulus of the complex degree of coherence for the annular Gaussian beam at discrete propagation lengths and at p 2 = ( 1 cm , 1 cm ) .

Fig. 10
Fig. 10

Three-dimensional view of the modulus of the complex degree of coherence for the higher-order Gaussian beam at discrete propagation lengths and at p 2 = ( 1 cm , 1 cm ) .

Fig. 11
Fig. 11

Two-dimensional variation of the modulus of the complex degree of coherence for all beams versus propagation lengths at C n 2 = 0 , 1 × 10 16 , 1 × 10 15 , and 1 × 10 14 m 2 3 .

Fig. 12
Fig. 12

Two-dimensional variation of the modulus of the complex degree of coherence for all beams against the source partial coherence parameter at α s x = 1, 2, 4, and 10 cm .

Equations (12)

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

u s ( s ) = u s ( s x , s y ) = = 1 N A exp ( j θ ) H n ( a x s x + b x ) exp [ ( 0.5 k α x s x 2 + j V x s x ) ] H m ( a y s y + b y ) exp [ ( 0.5 k α y s y 2 + j V y s y ) ] ,
α x = 1 ( k α s x 2 ) + j F x , α y m = 1 ( k α s y 2 ) + j F y .
Γ s ( s 1 , s 2 ) = u s ( s 1 ) u s * ( s 2 ) exp [ 0.25 ( s 1 s 2 ) 2 ρ s 2 ] ,
Γ r ( p 1 , p 2 , L ) = k 2 ( 2 π L ) 2 d 2 s 1 d 2 s 2 Γ s ( s 1 , s 2 ) exp { j k [ p 1 s 1 2 p 2 s 2 2 ] ( 2 L ) } exp { [ s 1 s 2 2 + ( s 1 s 2 ) ( p 1 p 2 ) + p 1 p 2 2 ] ρ 0 2 }
Γ r ( p 1 , p 2 , L ) = b 2 exp [ j b ( p 1 x 2 p 2 x 2 + p 1 y 2 p 2 y 2 ) ] exp [ ( p 1 x p 2 x + p 1 y p 2 y ) 2 ρ 0 2 ] 1 = 1 N 2 = 1 N A 1 A 2 * exp [ j ( θ 1 θ 2 ) ] ρ s 2 ρ 0 2 ( P x P y ) 0.5 E y E x S x S y ,
S x = x 1 = 0 [ n 1 2 ] n x 1 = 0 n 1 2 x 1 k x 1 = 0 [ n x 1 2 ] n x 11 = 0 n x 1 2 k x 1 x 2 = 0 [ n 2 2 ] n x 2 = 0 n 2 2 x 2 k x 2 = 0 [ ( n x 11 + n x 2 ) 2 ] ( 1 ) x 1 + x 2 2 n 1 + n 2 x 1 x 2 n x 1 n x 2 T x 1 T x 2 ( n 1 2 x 1 ) ( n 2 2 x 2 ) ( n 1 2 x 1 n x 1 ) ( n x 1 2 k x 1 n x 11 ) ( n 2 2 x 2 n x 2 ) ( n x 1 ) ! ( n x 1 2 k x 1 ) ! ( k x 1 ) ! ( n x 11 + n x 2 ) ! ( n x 11 + n x 2 2 k x 2 ) ! ( k x 2 ) ! ( a x 1 ) n x 1 ( a x 2 * ) n x 2 ( b x 1 ) n 1 2 x 1 n x 1 ( b x 2 * ) n 2 2 x 2 n x 2 ( ρ s 2 ) n x 1 k x 1 n x 11 ( ρ s 2 + 0.25 ρ 0 2 ) n x 11 ( ρ 0 2 ) k x 1 n x 11 n x 2 + 2 k x 2 ( p 1 x + p 2 x 2 j b ρ 0 2 p 1 x j ρ 0 2 V x 1 ) n x 1 2 k x 1 n x 11 ( D x ) n x 1 + k x 1 + k x 2 ( P x ) n x 11 n x 2 + k x 2 ( Q x ) n x 11 + n x 2 2 k x 2 ,
D x = 0.5 k α x 1 ρ s 2 ρ 0 2 + ρ s 2 + 0.25 ρ 0 2 j b ρ s 2 ρ 0 2 ,
P x = 0.5 k [ 0.5 k α x 1 α x 2 * ρ s 2 ρ 0 2 + ( α x 1 + α x 2 * ) ( ρ s 2 + 0.25 ρ 0 2 ) + j b ρ s 2 ρ 0 2 ( α x 1 α x 2 * ) + 2 b 2 ρ s 2 ρ 0 2 k ] ,
Q x = ( p 1 x p 2 x ) ( 3 j b ρ s 2 ρ 0 2 0.5 j b ρ 0 4 + 0.5 k α x 1 ρ s 2 ρ 0 2 ) + j b ρ s 2 ρ 0 4 ( k α x 1 2 j b ) p 2 x + j ρ 0 2 V x 2 * D x j ρ 0 2 V x 1 ( ρ s 2 + 0.25 ρ 0 2 ) ,
E x = exp { [ ρ s 2 ρ 0 2 P x ( p 1 x + p 2 x 2 j b ρ 0 2 p 1 x j ρ 0 2 V x 1 ) 2 + Q x 2 ] ( 4 ρ 0 4 D x P x ) } .
μ ( p 1 , p 2 , L ) = Γ r ( p 1 , p 2 , L ) Γ r ( p 1 , p 1 , L ) Γ r ( p 2 , p 2 , L ) .
μ ( s 1 , s 2 , 0 ) = Γ s ( s 1 , s 2 , 0 ) Γ s ( s 1 , s 1 , 0 ) Γ s ( s 2 , s 2 , 0 ) = u s ( s 1 ) u s * ( s 2 ) exp [ 0.25 ( s 1 s 2 ) 2 ρ s 2 ] u s ( s 1 ) 2 u s ( s 2 ) 2 = exp [ 0.25 ( s 1 s 2 ) 2 ρ s 2 ] .

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