Abstract

A partially coherent quasi-monochromatic Gaussian laser beam propagating in atmospheric turbulence is examined by using a derived analytic expression for the cross-spectral density function. Expressions for average intensity, beam size, phase front radius of curvature, and wave-front coherence length are obtained from the cross-spectral density function. These results provide a model for a free-space laser transmitter with a phase diffuser used to reduce pointing errors.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Belmonte, A. Comeron, J. A. Rubio, J. Bara, E. Fernandez, “Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver,” Appl. Opt. 36, 8632–8638 (1997).
    [CrossRef]
  2. I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
    [CrossRef]
  3. Y. E. Yenice, B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE3266, 221–230 (1998).
    [CrossRef]
  4. B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, T. L. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998).
    [CrossRef]
  5. R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. 35, 3640–3646 (1996).
    [CrossRef] [PubMed]
  6. M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).
  7. V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).
  8. J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
    [CrossRef]
  9. V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
    [CrossRef]
  10. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  11. T. Friberg, R. L. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  12. M. S. Belenkii, V. L. Mironov, “Coherence of the field of a laser beam in a turbulent atmosphere,” Sov. J. Quantum Electron. 10, 595–597 (1980).
    [CrossRef]
  13. M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
    [CrossRef]
  14. S. C. H. Wang, M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
    [CrossRef]
  15. R. L. Fante, “The effect of source temporal coherence on light scintillations in weak turbulence,” J. Opt. Soc. Am. 69, 71–73 (1979).
    [CrossRef]
  16. Y. Baykal, M. A. Plonus, S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source beam,” Radio Sci. 18, 551–556 (1983).
    [CrossRef]
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.
  18. A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.
  19. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
    [CrossRef]
  20. L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex optical systems,” Appl. Opt. 32, 5918–5929 (1993).
    [CrossRef] [PubMed]
  21. The notation zˆ for normalized distance is introduced in Ref. 22, and the symbol rˆ for the normalized focusing parameter is introduced here. Previously used symbols for these quantities include zˆ=Ω or zˆ=Λo, and rˆ=Ωo or rˆ=Θo.
  22. M. A. Vorontsov, Wavefront Control in Optics (Oxford U. Press, New York, to be published), Chap. 1.
  23. L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Wash., 2001).
  24. L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).
  25. Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 10, 68–73 (1967).
  26. R. Lutomirski, H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]
  27. A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).
  28. 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]
  29. J. C. Ricklin, F. M. Davidson, “Point-to-point wireless communication using partially coherent optical fields,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 156–165 (2001).
    [CrossRef]
  30. J. C. Ricklin, W. B. Miller, L. C. Andrews, “Effective beam parameters and the turbulent beam waist for convergent Gaussian beams,” Appl. Opt. 34, 7059–7065 (1995).
    [CrossRef] [PubMed]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1993 (2)

1983 (2)

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

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

1982 (1)

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

1980 (1)

M. S. Belenkii, 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)

1977 (1)

M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

1972 (1)

1971 (1)

1967 (1)

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 10, 68–73 (1967).

Adhikari, P.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Andrews, L. C.

Banach, V. A.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Bara, J.

Barbier, P. R.

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

Baykal, Y.

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

Belenkii, M. S.

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

M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Belmonte, A.

Bruno, T. L.

Buldakov, V. M.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Comeron, A.

Davidson, F. D.

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

Davidson, F. M.

J. C. Ricklin, F. M. Davidson, “Point-to-point wireless communication using partially coherent optical fields,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 156–165 (2001).
[CrossRef]

Evans, B. G.

Y. E. Yenice, B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE3266, 221–230 (1998).
[CrossRef]

Fante, R. L.

Feizulin, Z. I.

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 10, 68–73 (1967).

Fernandez, E.

Friberg, T.

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

Hakakha, H.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Wash., 2001).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.

Jankevics, A.

Kim, I. I.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Kon, A. I.

M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Korevaar, E. J.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Kravtsov, Yu. A.

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 10, 68–73 (1967).

Landers, F.

Levine, B. M.

B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, T. L. Bruno, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998).
[CrossRef]

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

Lutomirski, R.

Majumdar, A. K.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Mandel, L.

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

Martinsen, E. A.

Miller, W. B.

Mironov, V. L.

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

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

M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Wash., 2001).

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

Plett, M. L.

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

Plonus, M. A.

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

S. C. H. Wang, M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
[CrossRef]

Polak-Dingels, P.

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

Polejaev, V. I.

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

Ricklin, J. C.

J. C. Ricklin, W. B. Miller, L. C. Andrews, “Effective beam parameters and the turbulent beam waist for convergent Gaussian beams,” Appl. Opt. 34, 7059–7065 (1995).
[CrossRef] [PubMed]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10, 661–672 (1993).
[CrossRef]

L. C. Andrews, W. B. Miller, J. C. Ricklin, “Geometrical representation of Gaussian beams propagating through complex optical systems,” Appl. Opt. 32, 5918–5929 (1993).
[CrossRef] [PubMed]

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

J. C. Ricklin, F. M. Davidson, “Point-to-point wireless communication using partially coherent optical fields,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 156–165 (2001).
[CrossRef]

Rubio, J. A.

Rush, D. W.

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

Schell, A. C.

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).

Sudol, R. L.

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

Toledo-Quinones, M.

Tyson, R. K.

Vorontsov, M. A.

M. A. Vorontsov, Wavefront Control in Optics (Oxford U. Press, New York, to be published), Chap. 1.

Wang, S. C. H.

Wang, S. J.

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

Weyrauch, T.

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

Wirth, A.

Wolf, E.

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

Yenice, Y. E.

Y. E. Yenice, B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE3266, 221–230 (1998).
[CrossRef]

Yura, H. T.

Appl. Opt. (7)

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

Z. I. Feizulin, Yu. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 10, 68–73 (1967).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

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

Opt. Spektrosk. (1)

V. A. Banach, V. M. Buldakov, V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spektrosk. 54, 1054–1059 (1983).

Radio Sci. (1)

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

Sov. J. Quantum Electron. (2)

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

M. S. Belenkii, A. I. Kon, V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977).
[CrossRef]

Other (14)

J. C. Ricklin, F. M. Davidson, “Point-to-point wireless communication using partially coherent optical fields,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE4489, 156–165 (2001).
[CrossRef]

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. I and II.

A. Ishimaru, “The beam wave case in remote sensing,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), Chap. 5.

The notation zˆ for normalized distance is introduced in Ref. 22, and the symbol rˆ for the normalized focusing parameter is introduced here. Previously used symbols for these quantities include zˆ=Ω or zˆ=Λo, and rˆ=Ωo or rˆ=Θo.

M. A. Vorontsov, Wavefront Control in Optics (Oxford U. Press, New York, to be published), Chap. 1.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Wash., 2001).

L. C. Andrews, R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

J. C. Ricklin, F. D. Davidson, T. Weyrauch, “Free space laser communication using a partially coherent source,” in Optics in Atmospheric Propagation and Adaptive Systems IV, A. Kohnle, J. D. Gonglewski, T. J. Schmugge, eds., Proc. SPIE4538, 13–23 (2001).
[CrossRef]

V. I. Polejaev, J. C. Ricklin, “Controlled phase diffuser for a laser communication link,” in Artificial Turbulence for Imaging and Wave Propagation, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3432, 103–107 (1998).
[CrossRef]

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

M. L. Plett, P. R. Barbier, D. W. Rush, P. Polak-Dingels, B. M. Levine, “Measurement error for a Shack–Hartmann wavefront sensor in strong scintillation conditions,” in Propagation and Imaging through the Atmosphere II, L. R. Bissonnette, ed., Proc. SPIE3433, 93–102 (1998).

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, A. K. Majumdar, “Scintillation reduction using multiple apertures,” in Free-Space Laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 102–113 (1997).
[CrossRef]

Y. E. Yenice, B. G. Evans, “Adaptive beam-size control for ground-to-space laser communications,” in Free-Space Laser Communication Technologies X, G. Mecherle, ed., Proc. SPIE3266, 221–230 (1998).
[CrossRef]

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 (8)

Fig. 1
Fig. 1

Propagation geometry.

Fig. 2
Fig. 2

Normalized beam size wζ2(z)/wo2 as a function of normalized distance zˆ=z/zˆd for different values of the source coherence parameter ζS.

Fig. 3
Fig. 3

Normalized beam size wζ2(z)/wo2 as a function of normalized distance zˆ=z/zˆd for a coherent (ζS=1) and a partially coherent (ζS=26) collimated beam (rˆ=1) and for a coherent and a partially coherent beam with focusing (rˆ=0.001).

Fig. 4
Fig. 4

Comparison of the beam size for a coherent collimated beam with beam sizes for two partially coherent collimated beams with different source coherences in moderate turbulence after propagating 2 km.

Fig. 5
Fig. 5

Normalized radius of curvature Rζ(z)/(0.5wo2k) as a function of normalized distance zˆ for different values of ζS.

Fig. 6
Fig. 6

Normalized radius of curvature Rζ(z)/(0.5wo2k) as a function of normalized distance zˆ for varying strengths of atmospheric turbulence. The source coherence parameter ζS=1 for each curve, so that any partial coherence effects are due to atmospheric turbulence.

Fig. 7
Fig. 7

Wave-front coherence length ρC as a function of the refractive-index structure parameter Cn2 for a slightly divergent beam (rˆ=2) showing the effects of having a partially coherent source beam: ζS=1 (coherent source beam), 3, 10, and 1000. The hypothetical line ζS=0 is shown as an upper bound.

Fig. 8
Fig. 8

Propagation geometry for the cross-spectral density function.

Equations (47)

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

U(r, 0)=exp-1wo2+jk2Ror2,
U(ρ, z)=exp(jkz)rˆ+jzˆ exp-1rˆ+jzˆ 1wo2+jk2Roρ2,
rˆ(z)=Ro-zRo,zˆ=zzˆd,
w(z)=wo(rˆ2+zˆ2)1/2,R(z)=z(rˆ2+zˆ2)rˆ(1-rˆ)-zˆ2.
w(z)=wo[1+(λz/πwo2)]1/2,
R(z)=z[1+(πwo2/λz)]1/2.
I(ρ, z)=wo2w2(z) exp-2ρ2w2(z).
U(ρ, z)=d2rG(r, ρ, z)U(r, 0),
G(r, ρ, z)=-jk2πz expjkz+jk2z|ρ-r|2+Ψ(r, ρ),
U(ρ, z)=-jk2πz exp(jkz)d2rU(r, 0)×expjk2z|ρ-r|2+Ψ(r, ρ).
|ρ2-ρ1|c1Δν,
W(ρ1, ρ2, z)=U(ρ1, z)U*(ρ2, z)=1(λz)2 d2r1d2r2W(r1, r2, 0)×exp[Ψ(r1, ρ1)+Ψ*(r2, ρ2)]×expjk2z[(ρ1-r1)2-(ρ2-r2)2].
U˜(r, 0)=U(r, 0)exp[jφd(r)],
W(r1, r2, 0)=U˜(r1, 0)U˜*(r2, 0)=U(r1, 0)U*(r2, 0)×exp[jφd1(r1)]exp[-jφd2(r2)]=U(r1, 0)U*(r2, 0)exp-(r1-r2)22σg2.
rS=12(r1+r2),rd=r1-r2,
ρS=12(ρ1+ρ2),ρd=ρ1-ρ2,
W(rS, rd, 0)=exp-1wo2 12(rd2+4rS2)-jk2Ro(2rd·rS)-rd22σg2,
expjk2z[(ρ1-r1)2-(ρ2-r2)2]=expjkz[(rS-ρS)·(rd-ρd)].
exp(Ψ(r1, ρ1)+Ψ*(r2, ρ2))exp-1ρo2(rd2+rd·ρd+ρd2),
W(ρS, ρd, z)=1(λz)2 d2rdd2rS exp-2rS2wo2×exp-jkrS·rdRo+jkrS·(rd-ρd)z×exp-rd22wo2-rd22σg2-rd2+rd·ρd+ρd2ρo2-jkρS·(rd-ρd)z.
W(ρS, ρd, z)=wo2wζ2(z) exp-ρd21ρo2+12wo2 zˆ2+2jρS·ρdwo2 zˆexp-2ρS2wζ2(z)×exp-(jϕ)2ρd22wζ2(z)exp-2jϕ ρS·ρdwζ2(z),ϕrˆzˆ-zˆ wo2ρo2.
wζ(z)=wo(rˆ2+ζ zˆ2)1/2,ζ=1+wo2σg2+2wo2ρo2.
ζS=1+wo2σg2.
wζ(z)=wb(1+ζSzˆ2)1/2=wb1+2zkδwb21/2,
wζ(z)=wbΔ(z)=2σSΔ(z),
Δ(z)=1+2zkδwb21/2,
I(ρ)=wo2wζ2(z) exp-2ρ2wζ2(z).
ρ¯S(z)=wζ(z)2=2σSΔ(z)2=2σSΔ(z),
{W(ρS, ρd, z)}=wo2wζ2(z) exp-jkρS·ρdRζ(z),
Rζ(z)=z(rˆ2+ζzˆ2)ϕzˆ-ζzˆ2-rˆ2,ϕrˆzˆ-zˆ wo2ρo2.
μ(ρd, z)=W(0, ρd, z)W(0, 0, z)=exp-ρd21ρo2+12wo2 zˆ2exp-(jϕ)2ρd22wζ2(z).
μ(ρd, z)=exp-ρd2ρo2 1+ρo22wo2 zˆ2-ϕ2ρo22wζ2(z)=exp-ρd2ρC2.
ρC=ρo1+ρo22wo2zˆ2-ϕ2ρo22wζ2(z)-1/2.
1ρ¯μ2(z)=12wb2 zˆ2-ϕ22wζ2(z)=12wb2 zˆ2-12wζ2(z).
ρ¯μ(z)=2δΔ(z),
ρ¯S(z)ρ¯μ(z)=σSδ=ζS2;
ρ¯S(z)ρ¯μ(z)=σSσg.
ρ¯S(z)ρ¯μ(z)=wζ2(z)2ρo2+wζ2(z)4wo2 zˆ2-ϕ24.
W(ρ1, ρ2; ν)=k2π2d2r1 d2r2 W(r1, r2; ν)×exp[jk(R2-R1)]R2R1,
0dν exp(-2πjντ)W(ρ1, ρ2; ν)=Γ(ρ1, ρ2; τ)=0dν exp(-2πjντ)×k2π2d2r1 d2r2 W(r1, r2; ν)×exp[jk(R2-R1)]R2R1.
Γ(ρ1, ρ2; τ)=1R2R1 d2r1 d2r20dννc2×W(r1, r2; ν)exp(-2πjντ)×exp2πjν(R2-R1)c.
νck¯2π,
0dν W(r1, r2; ν)exp(-2πjντ)exp2πjν(R2-R1)c=Γr1, r2; τ-R2-R1c
Γr1, r2; τ-R2-R1cΓ(r1, r2; τ)exp[jk¯(R2-R1)].
Γ(ρ1, ρ2; τ)=k¯2π2d2r1 d2r2 Γ(r1, r2; τ)×exp[jk¯(R2-R1)]R2R1.
R2-R1c1Δν
|ρ2-ρ1|c1Δν.

Metrics