Abstract

We introduce a new method of estimating the coherence function of a Gaussian–Schell model beam in the inertial subrange of atmospheric turbulence. It is compared with the previously published methods based on either the quadratic approximation of the parabolic equation or an assumed independence between the source’s randomness and the atmosphere using effective beam parameters. This new method, which combines the results of the previous two methods to account for any random source/atmospheric coupling, was shown to more accurately estimate both the coherence radius and coherence functional shape across much of the relevant parameter space. The regions of the parameter space where one method or another is the most accurate in estimating the coherence radius are identified along with the maximum absolute estimation error in each region. By selecting the appropriate estimation method for a given set of conditions, the absolute estimation error can generally be kept to less than 5%, with a maximum error of 7%. We also show that the true coherence function is more Gaussian than expected, with the exponential power tending toward 9/5 rather than the theoretical value of 5/3 in very strong turbulence regardless of the nature of the source coherence.

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
    [CrossRef]
  3. T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
    [CrossRef]
  4. Y. Dikmelik and F. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44, 4946–4952 (2005).
    [CrossRef] [PubMed]
  5. M. Toyoshima, “Maximum fiber coupling efficiency and optimum beam size in the presence of random angular jitter for free-space laser systems and their applications,” J. Opt. Soc. Am. A 23, 2246–2250 (2006).
    [CrossRef]
  6. J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
    [CrossRef]
  7. S. Shaklan and F. Roddier, “Coupling starlight into single-mode fiber optics,” Appl. Opt. 27, 2334–2338 (1988).
    [CrossRef] [PubMed]
  8. C. Ruilier and F. Cassaing, “Coupling of large telescopes and single-mode waveguides: application to stellar interferometry,” J. Opt. Soc. Am. A 18, 143–149 (2001).
    [CrossRef]
  9. D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
    [CrossRef]
  10. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23, 986–988 (1998).
    [CrossRef]
  11. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126(2004).
    [CrossRef]
  12. D. J. Wheeler and J. D. Schmidt, “Coupling of Gaussian Schell-model beams into single-mode optical fibers,” J. Opt. Soc. Am. A 281224–1238 (2011).
    [CrossRef]
  13. M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  15. J. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–185 (1978).
    [CrossRef]
  16. 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, 1297–1304 (1979).
    [CrossRef]
  17. J. Wu, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
    [CrossRef]
  18. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598(2002).
    [CrossRef]
  19. 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]
  20. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [CrossRef]
  21. 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. 43, 330–341 (2004).
    [CrossRef]
  22. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2901 (2007).
    [CrossRef]
  23. X. Ji and X. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010).
    [CrossRef]
  24. K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).
  26. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
    [CrossRef]
  27. Because the fields used to calculate Γ are taken at the same instant in time, it can alternatively be referred to as the mutual intensity J instead of the MCF . However, to maintain consistency with other published literature , we are referring to it in this paper as the MCF. Consequently, the time dependence of the field E is ignored.
  28. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  29. D. A. de Wolf, “Saturation of irradiance fluctuations due to turbulent atmosphere,” J. Opt. Soc. Am. 58, 461–466 (1968).
    [CrossRef]
  30. M. Beran, “Propagation of a finite beam in a random medium,” J. Opt. Soc. Am. 60, 518–521 (1970).
    [CrossRef]
  31. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  32. J. Wilbur and P. Brown, “Second moment of a wave propagating in a random medium,” J. Opt. Soc. Am. 61, 1051–1059(1971).
    [CrossRef]
  33. K. Furutsu, “Statistical theory of wave propagation in a random medium and the irradiance distribution function,” J. Opt. Soc. Am. 62, 240–254 (1972).
    [CrossRef]
  34. R. L. Fante, “Mutual coherence function and frequency spectrum of a laser beam propagating through atmospheric turbulence,” J. Opt. Soc. Am. 64, 592–598 (1974).
    [CrossRef]
  35. R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
    [CrossRef]
  36. M. J. Beran, “Coherence equations governing propagation through random media,” Radio Sci. 10, 15–21 (1975).
    [CrossRef]
  37. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [CrossRef] [PubMed]
  38. A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. U.S.S.R. 30, 301–305 (1941).
  39. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  40. L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
    [CrossRef]
  41. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).
    [CrossRef]
  42. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
    [CrossRef]

2011 (2)

D. J. Wheeler and J. D. Schmidt, “Coupling of Gaussian Schell-model beams into single-mode optical fibers,” J. Opt. Soc. Am. A 281224–1238 (2011).
[CrossRef]

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
[CrossRef]

2010 (2)

X. Ji and X. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010).
[CrossRef]

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

2007 (2)

2006 (1)

2005 (4)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

Y. Dikmelik and F. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44, 4946–4952 (2005).
[CrossRef] [PubMed]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[CrossRef]

2004 (2)

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. 43, 330–341 (2004).
[CrossRef]

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126(2004).
[CrossRef]

2003 (1)

2002 (3)

2001 (2)

C. Ruilier and F. Cassaing, “Coupling of large telescopes and single-mode waveguides: application to stellar interferometry,” J. Opt. Soc. Am. A 18, 143–149 (2001).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

1998 (1)

1997 (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

1995 (2)

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

D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

1994 (1)

1990 (1)

J. Wu, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

1988 (1)

1985 (1)

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

1980 (1)

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

1979 (2)

1978 (1)

1975 (1)

M. J. Beran, “Coherence equations governing propagation through random media,” Radio Sci. 10, 15–21 (1975).
[CrossRef]

1974 (1)

1972 (2)

1971 (3)

1970 (1)

1968 (1)

1941 (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. U.S.S.R. 30, 301–305 (1941).

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[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. 43, 330–341 (2004).
[CrossRef]

L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence,” J. Opt. Soc. Am. A 11, 1653–1660 (1994).
[CrossRef]

Baykal, Y.

Belen’kii, M. S.

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

Beran, M.

Beran, M. J.

M. J. Beran, “Coherence equations governing propagation through random media,” Radio Sci. 10, 15–21 (1975).
[CrossRef]

Bifano, T. G.

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Brown, P.

Cai, Y.

Cassaing, F.

Dashen, R.

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

Davidson, F.

Davidson, F. M.

de Wolf, D. A.

Dikmelik, Y.

Dogariu, A.

Drexler, K.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
[CrossRef]

Duncan, B. D.

D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Eyyuboglu, H. T.

Fante, R. L.

Furutsu, K.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Gowens, J. W.

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Jacob, D. K.

D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Ji, X.

X. Ji and X. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010).
[CrossRef]

Kolmogorov, A. N.

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. U.S.S.R. 30, 301–305 (1941).

Korotkova, O.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[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. 43, 330–341 (2004).
[CrossRef]

Leader, J. C.

Leeb, W. R.

Li, X.

X. Ji and X. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010).
[CrossRef]

Lutomirski, R. F.

Ma, J.

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

Mandel, L.

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

Mark, M. B.

D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[CrossRef]

Miller, W. B.

Mironov, V. L.

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[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. 43, 330–341 (2004).
[CrossRef]

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Plonus, M. A.

Rhoadarmer, T. A.

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126(2004).
[CrossRef]

Ricklin, J. C.

Roddier, F.

Roggemann, M.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
[CrossRef]

Ruilier, C.

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).
[CrossRef]

Schmidt, J. D.

D. J. Wheeler and J. D. Schmidt, “Coupling of Gaussian Schell-model beams into single-mode optical fibers,” J. Opt. Soc. Am. A 281224–1238 (2011).
[CrossRef]

Shaklan, S.

Shirai, T.

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

Tan, L.

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

Toyoshima, M.

Voelz, D.

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
[CrossRef]

Vorontsov, M. A.

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[CrossRef]

Wang, S. C. H.

Weyrauch, T.

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[CrossRef]

Wheeler, D. J.

D. J. Wheeler and J. D. Schmidt, “Coupling of Gaussian Schell-model beams into single-mode optical fibers,” J. Opt. Soc. Am. A 281224–1238 (2011).
[CrossRef]

Wilbur, J.

Winzer, P. J.

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[CrossRef]

T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[CrossRef]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598(2002).
[CrossRef]

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

Wu, J.

J. Wu, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

Yang, Y.

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

Yu, S.

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

Yura, H. T.

Zhao, F.

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

Appl. Opt. (4)

C. R. Acad. Sci. U.S.S.R. (1)

A. N. Kolmogorov, “The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. U.S.S.R. 30, 301–305 (1941).

J. Math. Phys. (1)

R. Dashen, “Path integrals for waves in random media,” J. Math. Phys. 20, 894–920 (1979).
[CrossRef]

J. Mod. Opt. (1)

J. Wu, “Propagation of a Gaussian–Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990).
[CrossRef]

J. Opt. (1)

X. Ji and X. Li, “Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through atmospheric turbulence,” J. Opt. 12, 035403 (2010).
[CrossRef]

J. Opt. A (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. A 3, 1–9 (2001).
[CrossRef]

J. Opt. Soc. Am. (7)

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

Opt. Commun. (1)

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246, 35–43 (2005).
[CrossRef]

Opt. Eng. (4)

K. Drexler, M. Roggemann, and D. Voelz, “Use of a partially coherent transmitter beam to improve the statistics of received power in a free-space optical communication system: theory and experimental results,” Opt. Eng. 50, 025002(2011).
[CrossRef]

J. Ma, F. Zhao, L. Tan, S. Yu, and Y. Yang, “Degradation of single-mode fiber coupling efficiency due to localized wavefront aberrations in free-space laser communications,” Opt. Eng. 49, 045004 (2010).
[CrossRef]

D. K. Jacob, M. B. Mark, and B. D. Duncan, “Heterodyne ladar system efficiency enhancement using single-mode optical fiber mixers,” Opt. Eng. 34, 3122–3129 (1995).
[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. 43, 330–341 (2004).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” Proc. SPIE 5553, 112–126(2004).
[CrossRef]

T. Weyrauch, M. A. Vorontsov, J. W. Gowens, II, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002).
[CrossRef]

Radio Sci. (1)

M. J. Beran, “Coherence equations governing propagation through random media,” Radio Sci. 10, 15–21 (1975).
[CrossRef]

Sov. J. Quantum Electron. (1)

M. S. Belen’kii and V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
[CrossRef]

Other (9)

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

D. J. Wheeler and J. D. Schmidt, “Coupling of Gaussian Schell-model beams into single-mode optical fibers,” J. Opt. Soc. Am. A 281224–1238 (2011).
[CrossRef]

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, 1971).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).

Because the fields used to calculate Γ are taken at the same instant in time, it can alternatively be referred to as the mutual intensity J instead of the MCF . However, to maintain consistency with other published literature , we are referring to it in this paper as the MCF. Consequently, the time dependence of the field E is ignored.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Difference in the absolute error between various methods of estimating the coherence radius. Lines are for the q c values of 0, 0.001, 0.01, 0.1, 1, 10, 100, and 1000. The solid curves are the difference between the absolute error using the effective pa rameters as given in Eq. (25) and using modified effective parameters as given in Eq. (29). The dotted curves are the difference between the absolute error using the quadratic approximation as given in Eq. (13) and using the modified effective parameters as given in Eq. (29). The difference is positive when using the modified effective parameters is more accurate, and it is negative when using the modified effective parameters is less accurate.

Fig. 2
Fig. 2

Absolute error of different methods of estimating the coherence radius. Lines are for the q c values of 0, 0.001, 0.01, 0.1, 1, 10, 100, and 1000. The solid lines are the absolute error of the modified effective parameters method given by Eq. (29), and the dotted lines are the absolute error of the quadratic approximation given by Eq. (13).

Fig. 3
Fig. 3

Difference in the absolute error between various methods of estimating the coherence radius. Lines are for the q c values of 0, 0.001, 0.01, 0.1, 1, 10, 100, and 1000. The solid curves are the difference between the absolute error using the effective param eters approximate expression as given in Eq. (28) and using the modified effective parameters approximate expression as given in Eq. (31). The dotted curves are the difference between the absolute error using the quadratic approximation as given in Eq. (13) and using the modified effective parameters approximate expression as given in Eq. (31). The difference is positive when using the modified effective parameters is more accurate, and it is negative when using the modified effective parameters is less accurate.

Fig. 4
Fig. 4

Comparison of the exponential power n as different methods of estimating the coherence function are fit to Eq. (33). Lines are for the q c values of 0, 1/1000, 1/100, 1/10, 1, 10, 100, and 1000. Solid curves are from the numerically derived coherence function, the dashed–dotted curves are from the modified effective parameters method as given in Eq. (29), and the dotted curves are from the effective parameters method as given in Eq. (25).

Equations (36)

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Γ ( ρ 1 , ρ 2 , L ) = E ( ρ 1 , L ) E * ( ρ 2 , L ) ,
Γ ( ρ 1 , ρ 2 , 0 ) = exp [ | ρ 1 | 2 + | ρ 2 | 2 w 0 2 | ρ 1 ρ 2 | 2 l c 2 j k 2 F 0 ( | ρ 1 | 2 | ρ 2 | 2 ) ] ,
| μ ( ρ 1 , ρ 2 , L ) | = | Γ ( ρ 1 , ρ 2 , L ) | [ Γ ( ρ 1 , ρ 1 , L ) Γ ( ρ 2 , ρ 2 , L ) ] 1 / 2 .
| μ ( ρ 1 , ρ 2 , 0 ) | = exp ( | ρ 1 ρ 2 | 2 l c 2 ) .
Γ ( ρ 1 , ρ 2 , L ) = 1 ( λ L ) 2 S S d 2 ρ 1 d 2 ρ 2 Γ ( ρ 1 , ρ 2 , 0 ) exp [ j k 2 L ( | ρ 1 ρ 1 | 2 | ρ 2 ρ 2 | 2 ) ] × exp ( 4 π 2 k 2 L 0 1 d t 0 d κ κ Φ n ( κ ) { 1 J 0 [ | t ( ρ 1 ρ 2 ) + ( 1 t ) ( ρ 1 ρ 2 ) | κ ] } ) ,
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
Γ ( ρ s , ρ d , L ) = w 0 2 8 π u d 2 u exp [ A | ρ d | 2 B | u | 2 + ( C ρ d + j ρ s ) · u 1 ρ p 5 / 3 0 1 d t | ρ d L t k u | 5 / 3 ] ,
A = 1 2 w 0 2 + 1 l c 2 + k 2 w 0 2 8 F 0 2 , B = L 2 2 k 2 w 0 2 + L 2 k 2 l c 2 + w 0 2 8 ( 1 L F 0 ) 2 , C = L k w 0 2 + 2 L k l c 2 k w 0 2 4 F 0 ( 1 L F 0 ) ,
Θ 0 = 1 L F 0 , Λ 0 = 2 L k w 0 2 , q c = L k l c 2 , and q = L k ρ p 2 ,
Γ ( ρ ˜ s , ρ ˜ d , L ) = q 4 π Λ 0 u ˜ d 2 u ˜ exp [ A ˜ | ρ ˜ d | 2 B ˜ | u ˜ | 2 + ( C ˜ ρ ˜ d + j ρ ˜ s ) · u ˜ 0 1 d t | ρ ˜ d t q u ˜ | 5 / 3 ] ,
A ˜ = 1 4 q Λ 0 [ ( 1 Θ 0 ) 2 + Λ 0 2 + 4 q c Λ 0 ] , B ˜ = q 4 Λ 0 ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) , C ˜ = 1 2 Λ 0 ( Θ 0 2 + Λ 0 2 Θ 0 + 4 q c Λ 0 ) .
Γ ( ρ ˜ s , ρ ˜ d , L ) = q 4 π Λ 0 u ˜ d 2 u ˜ exp [ A ˜ | ρ ˜ d | 2 B ˜ | u ˜ | 2 + ( C ˜ ρ ˜ d + j ρ ˜ s ) · u ˜ 3 α 0 1 d t | ρ ˜ d t q u ˜ | 2 ] ,
Γ ( ρ 1 , ρ 2 , L ) = w 0 2 w q 2 exp [ | ρ 1 | 2 + | ρ 2 | 2 w q 2 | ρ 1 ρ 2 | 2 ρ q 2 j k 2 F q ( | ρ 1 | 2 | ρ 2 | 2 ) ] ,
w q = w 0 [ Θ 0 2 + Λ 0 2 + 4 ( q c + α q ) Λ 0 ] 1 / 2 ,
ρ q = ρ p { q [ Θ 0 2 + Λ 0 2 + 4 ( q c + α q ) Λ 0 ] q c ( 1 + 4 α q Λ 0 ) + α q ( 1 + Θ 0 2 + Λ 0 2 + Θ 0 + 3 α q Λ 0 ) } 1 / 2 ,
F q = L [ Θ 0 2 + Λ 0 2 + 4 ( q c + α q ) Λ 0 ] Θ 0 2 + Λ 0 2 Θ 0 + 2 ( 2 q c + 3 α q ) Λ 0 .
| μ q ( ρ 1 , ρ 2 , L ) | = exp ( | ρ 1 ρ 2 | 2 ρ q 2 ) .
E ( ρ , L ) = E 0 ( ρ , L ) exp [ Ψ S ( ρ , L ) ] exp [ ψ ( ρ , L ) ] ,
Γ ( ρ 1 , ρ 2 , L ) = Γ c ( ρ 1 , ρ 2 , L ) Γ a ( ρ 1 , ρ 2 , L ) ,
| μ c ( ρ 1 , ρ 2 , L ) | = exp ( | ρ 1 ρ 2 | 2 ρ c 2 ) ,
ρ c = l c ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) 1 / 2 .
| μ R ( ρ 1 , ρ 2 , L ) | = exp { 3 8 [ a ( | ρ 1 ρ 2 | ρ p ) 5 / 3 + 0.618 Λ 11 / 6 q 1 / 6 ( | ρ 1 ρ 2 | ρ p ) 2 ] } ,
a = { 1 Θ 8 / 3 1 Θ for     Θ 0 1 + | Θ | 8 / 3 1 Θ for     Θ < 0 , Θ = 1 + L F , Λ = 2 L k w 2 ,
Θ a = 1 + L F a , Λ a = 2 L k w a 2 ,
| μ a ( ρ 1 , ρ 2 , L ) | = exp { 3 8 [ a a ( | ρ 1 ρ 2 | ρ p ) 5 / 3 + 0.618 Λ a 11 / 6 q 1 / 6 ( | ρ 1 ρ 2 | ρ p ) 2 ] } ,
a a = { 1 Θ a 8 / 3 1 Θ a for     Θ a 0 , 1 + | Θ a | 8 / 3 1 Θ a for     Θ a < 0.
| μ e ( ρ 1 , ρ 2 , L ) | = exp [ ( | ρ 1 ρ 2 | ρ a ) 5 / 3 ( | ρ 1 ρ 2 | ρ c ) 2 ] ,
ρ a = ρ p ( 8 3 a a ) 3 / 5 ,
ρ c = ρ p [ q c q ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) + 0.232 Λ a 11 / 6 q 1 / 6 ] 1 / 2 .
ρ e = ρ p [ q c q ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) + 0.232 Λ a 11 / 6 q 1 / 6 + ( 3 a a 8 ) 6 / 5 ] 1 / 2 .
| μ m ( ρ 1 , ρ 2 , L ) | = exp { 3 a q 8 ( | ρ 1 ρ 2 | ρ p ) 5 / 3 [ q c q ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) + 0.232 Λ q 11 / 6 q 1 / 6 ] ( | ρ 1 ρ 2 | ρ p ) 2 } ,
a q = { 1 Θ q 8 / 3 1 Θ q for     Θ q 0 1 + | Θ q | 8 / 3 1 Θ q for     Θ q < 0 , Θ q = 1 + L F q , Λ q = 2 L k w q 2 .
ρ m = ρ p [ q c q ( Θ 0 2 + Λ 0 2 + 4 q c Λ 0 ) + 0.232 Λ q 11 / 6 q 1 / 6 + ( 3 a q 8 ) 6 / 5 ] 1 / 2 .
| μ ( ρ , L ) | = | Γ ( ρ , 0 , L ) | [ Γ ( ρ , ρ , L ) Γ ( 0 , 0 , L ) ] 1 / 2 .
| μ ( ρ ) | = exp [ ( ρ ρ 0 ) n ] ,
err x = 100 | ρ x ρ 0 1 | ,

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