Abstract

We study the propagation of the two lowest-order Gaussian laser beams with different wavelengths in weak atmospheric turbulence. Using the Rytov approximation and assuming a slow detector, we calculate the longitudinal and radial components of the scintillation index for a typical free-space laser communication setup. We find the optimal configuration of the two laser beams with respect to the longitudinal scintillation index. We show that the value of the longitudinal scintillation for the optimal two-beam configuration is smaller by more than 50% compared with the value for a single lowest-order Gaussian beam with the same total power. Furthermore, the radial scintillation for the optimal two-beam system is smaller by 35%–40% compared with the radial scintillation in the single-beam case. Further insight into the reduction of intensity fluctuations is gained by analyzing the self- and cross-intensity contributions to the scintillation index.

© 2006 Optical Society of America

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References

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  1. A. Ishimaru, Laser Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  2. R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985).
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  4. M. S. Belenkii, A. I. Kon, and V. L. Mironov, "Turbulence distortions of the spatial coherence of a laser beam," Sov. J. Quantum Electron. 7, 287-290 (1977).
    [CrossRef]
  5. 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).
  6. M. S. Belenkii 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]
  7. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).
  8. V. A. Banakh and V. M. Buldakov, "Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere," Opt. Spektrosk. 55, 757-762 (1983).
  9. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
  10. J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
  11. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
  12. A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
  13. 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).
  14. M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
    [CrossRef]
  15. X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).
  16. 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]
  17. 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).
  18. J. C. Ricklin and F. M. Davidson, "Atmospheric optical communication with a Gaussian Schell beam," J. Opt. Soc. Am. A 20, 856-866 (2003).
  19. T. J. Schulz, "Optimal beams for propagation through random media," Opt. Lett. 30, 1093-1095 (2005).
  20. L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
    [CrossRef]
  21. Coherent Incorporated, Santa Clara, California (personal communication, 2005).
  22. J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).
  23. R. L. Fante, "Effect of source bandwidth and receiver response time on the scintillation index in random media," Radio Sci. 12, 223-229 (1977).
  24. R. L. Fante, "The effect of source temporal coherence on light scintillation in weak turbulence," J. Opt. Soc. Am. 69, 71-73 (1979).
  25. R. L. Fante, "Intensity fluctuations of an optical wave in a turbulent medium, effect of source coherence," Opt. Acta 28, 1203-1207 (1981).
  26. 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).
  27. K. Kiasaleh, "Scintillation index of a multiwavelength beam in turbulent atmosphere," J. Opt. Soc. Am. A 21, 1452-1454 (2004).
    [CrossRef]
  28. K. Kiasaleh, "Impact of turbulence on multi-wavelength coherent optical communications," in Free-Space Laser Communications V, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5892, 58920R1-58920R11 (2005).
  29. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1968).

2006 (1)

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

2005 (2)

T. J. Schulz, "Optimal beams for propagation through random media," Opt. Lett. 30, 1093-1095 (2005).

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).

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]

K. Kiasaleh, "Scintillation index of a multiwavelength beam in turbulent atmosphere," J. Opt. Soc. Am. A 21, 1452-1454 (2004).
[CrossRef]

2003 (4)

2002 (2)

1991 (1)

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).

1990 (1)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).

1983 (3)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

V. A. Banakh and V. M. Buldakov, "Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere," Opt. Spektrosk. 55, 757-762 (1983).

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).

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).

1980 (1)

M. S. Belenkii 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)

1977 (2)

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

R. L. Fante, "Effect of source bandwidth and receiver response time on the scintillation index in random media," Radio Sci. 12, 223-229 (1977).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1968).

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. 43, 330-341 (2004).
[CrossRef]

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

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

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

V. A. Banakh and V. M. Buldakov, "Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere," Opt. Spektrosk. 55, 757-762 (1983).

Baykal, Y.

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).

Bedford, R.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Belenkii, M. S.

M. S. Belenkii 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]

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

Boardman, A. D.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

V. A. Banakh and V. M. Buldakov, "Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere," Opt. Spektrosk. 55, 757-762 (1983).

Butterworth, S. D.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Caprara, A. L.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Charles, J. P.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Chilla, J. L. A.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Davidson, F. M.

Dogariu, A.

Fallahi, M.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Fan, L.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

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).

R. L. Fante, "The effect of source temporal coherence on light scintillation in weak turbulence," J. Opt. Soc. Am. 69, 71-73 (1979).

R. L. Fante, "Effect of source bandwidth and receiver response time on the scintillation index in random media," Radio Sci. 12, 223-229 (1977).

R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985).

Gbur, G.

Hader, J.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Hopen, C. Y.

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

Ishimaru, A.

A. Ishimaru, Laser Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Ji, X.

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).

Kaneda, Y.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Kiasaleh, K.

K. Kiasaleh, "Scintillation index of a multiwavelength beam in turbulent atmosphere," J. Opt. Soc. Am. A 21, 1452-1454 (2004).
[CrossRef]

K. Kiasaleh, "Impact of turbulence on multi-wavelength coherent optical communications," in Free-Space Laser Communications V, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5892, 58920R1-58920R11 (2005).

Koch, S. W.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Kon, A. I.

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

Korotkova, O.

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]

Lü, B.

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

M. S. Belenkii 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]

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

Moloney, J. V.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Murray, J. T.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

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. 43, 330-341 (2004).
[CrossRef]

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

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

Plonus, M. A.

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).

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).

Reed, M. K.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Ricklin, J. C.

Salem, M.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
[CrossRef]

Schulz, T. J.

Shirai, T.

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
[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).

Spinelli, L.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1968).

Stolz, W.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Voelz, D. G.

K. Kiasaleh, "Impact of turbulence on multi-wavelength coherent optical communications," in Free-Space Laser Communications V, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5892, 58920R1-58920R11 (2005).

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).

Wolf, E.

Wu, J.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).

Zakharian, A. R.

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

Zeitschel, A.

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

Appl. Phys. Lett. (1)

L. Fan, M. Fallahi, J. T. Murray, R. Bedford, Y. Kaneda, A. R. Zakharian, J. Hader, J. V. Moloney, W. Stolz, and S. W. Koch, "Tunable high-power high-brightness linearly polarized vertical-external-cavity surface emitting lasers," Appl. Phys. Lett. 88, 021105 (2006).
[CrossRef]

J. Mod. Opt. (2)

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).

J. Wu and A. D. Boardman, "Coherence length of a Gaussian-Schell beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).

J. Opt. Soc. Am. (2)

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).

Opt. Commun. (2)

M. Salem, T. Shirai, A. Dogariu, and E. Wolf, "Long-distance propagation of partially coherent beams through atmospheric turbulence," Opt. Commun. 216, 261-265 (2003).
[CrossRef]

X. Ji and B. Lü, "Turbulence-induced quality degradation of partially coherent beams," Opt. Commun. 251, 231-236 (2005).

Opt. Eng. (1)

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. (2)

Opt. Spektrosk. (2)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, "Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere," Opt. Spektrosk. 54, 1054-1059 (1983).

V. A. Banakh and V. M. Buldakov, "Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere," Opt. Spektrosk. 55, 757-762 (1983).

Radio Sci. (2)

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).

R. L. Fante, "Effect of source bandwidth and receiver response time on the scintillation index in random media," Radio Sci. 12, 223-229 (1977).

Sov. J. Quantum Electron. (2)

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

M. S. Belenkii 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 (8)

A. Ishimaru, Laser Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985).

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

Coherent Incorporated, Santa Clara, California (personal communication, 2005).

J. L. A. Chilla, S. D. Butterworth, A. Zeitschel, J. P. Charles, A. L. Caprara, M. K. Reed, and L. Spinelli, "High-power optically pumped semiconductor lasers," in Nanobiophotonics and Biomedical Applications, A. N. Cartwright, ed., Proc. SPIE 5332, 143-150 (2004).

K. Kiasaleh, "Impact of turbulence on multi-wavelength coherent optical communications," in Free-Space Laser Communications V, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5892, 58920R1-58920R11 (2005).

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1968).

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Figures (7)

Fig. 1
Fig. 1

Longitudinal component of the scintillation index σ I , l 2 as a function of the initial beam separation d. The solid curve is the result obtained by using the Von Kármán spectrum. The dashed curve is the result obtained by using the Kolmogorov spectrum and the dotted curve corresponds to the result obtained by using the Kolmogorov spectrum and assuming that r j W j 1 . The square and circle stand for the longitudinal scintillation of a single beam with the same total power and the same initial spot size and amplitude, respectively.

Fig. 2
Fig. 2

Self- and cross-intensity contributions to the longitudinal scintillation index σ I , l , s 2 and σ I , l , c 2 respectively, versus beam separation d. The dashed curve represents σ I , l 2 ( d ; L ) and the dotted curve stands for σ I , l , c 2 ( d ; L ) . The solid curve corresponds to σ I , l , s 2 ( d ; L ) + 1 2 .

Fig. 3
Fig. 3

Total scintillation index σ I 2 ( r , L ) for the two-beam system with the optimal configuration d 0 = 2.8 cm at a propagation distance L = 1 km . The figure shows an 8 cm × 8 cm domain centered about the z axis.

Fig. 4
Fig. 4

Fractional reduction of the total scintillation index R a ( r , L ) obtained by using the optimal two-beam system relative to a single Gaussian beam with the same total intensity and initial spot size.

Fig. 5
Fig. 5

Self-intensity contribution to the total scintillation index σ I , s 2 ( r , L ) for the optimal two-beam system.

Fig. 6
Fig. 6

Cross-intensity contribution to the total scintillation index σ I , c 2 ( r , L ) for the optimal two-beam system.

Fig. 7
Fig. 7

Circularly averaged radial scintillation index for the optimal two-beam system σ r r 2 as a function of radius r (circles). The squares correspond to the result obtained for a single beam with the same power and initial spot size.

Equations (59)

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E ( r , 0 , t ) = j = 1 N E j ( r j , 0 , t ) = j = 1 N U j ( r j , 0 ) exp [ i ω j t ] ,
U j ( r j , 0 ) = A 0 j exp [ ( 1 W 0 j 2 + i k j 2 F 0 j ) r d j 2 ] .
2 U j + k j 2 [ 1 + n 1 ( r , z ) ] 2 U j = 0 ,
E ( r , L , t ) = j = 1 N E j ( r j , L , t ) = j = 1 N U j ( r j , L ) exp [ i ω j t ] .
2 U j + k j 2 [ 1 + 2 n 1 ( r , z ) ] U j = 0 .
U j ( r j , L ) = U 0 j ( r j , L ) exp [ ψ j ( r j , L ; k j ) ] ,
Θ 0 j = 1 L F 0 j , Λ 0 j = 2 L k j W 0 j 2 .
Θ j = 1 L F j = Θ 0 j Θ 0 j 2 + Λ 0 j 2 , Λ j = 2 L k j W j 2 = Λ 0 j Θ 0 j 2 + Λ 0 j 2 .
I ( r , L , t ) = j = 1 N I j ( r j , L ) + j N m j N E j ( r j , L , t ) E m * ( r m , L , t ) ,
I det ( r , L ) I ( r , L , t ) det 1 τ 0 τ d t I ( r , L , t ) ,
I det ( r , L ) I ( r , L , t ) det j = 1 N I j ( r j , L ) .
σ I 2 ( r , L ) I det 2 ( r , L ) I det ( r , L ) 2 1 ,
σ I 2 ( r , L ) = j = 1 N I j 2 ( r j , L ) + 2 j N m > j N I j ( r j , L ) I m ( r m , L ) ( j = 1 N I j ( r j , L ) ) 2 1 .
σ I , l 2 ( L ) σ I 2 ( 0 , L ) ,
σ r 2 ( r , L ) σ I 2 ( r , L ) σ I , l 2 ( L ) .
I j ( r j , L ) = I 0 j ( r j , L ) exp [ H 1 j ( r j , L ) ] ,
I 0 j ( r j , L ) = A 0 j 2 W 0 j 2 W j 2 exp [ 2 r j 2 W j 2 ] ,
H 1 j ( r j , L ) = 4 π 2 k j 2 L 0 1 d ξ 0 d κ κ Φ n ( κ ) × [ I 0 ( 2 Λ j r j ξ κ ) exp ( Λ j L κ 2 ξ 2 k j ) 1 ] .
I j 2 ( r j , L ) = I j ( r j , L ) 2 exp [ H 2 j ( r j , L ) ] ,
H 2 j ( r j , L ) = 8 π 2 k j 2 L 0 1 d ξ 0 d κ κ Φ n ( κ ) exp ( Λ j L κ 2 ξ 2 k j ) { I 0 ( 2 Λ j r j ξ κ ) cos [ L κ 2 ξ ( 1 Θ ¯ j ξ ) k j ] } ,
I j ( r j , L ) I m ( r m , L ) = I j ( r j , L ) I m ( r m , L ) exp { E 2 j m ( r j , r m ; k j , k m ) + E 2 m j ( r m , r j ; k m , k j ) + 2 Re [ E 3 j m ( r j , r m ; k j , k m ) ] } ,
E 2 j m ( r j , r m ; k j , k m ) = 4 π 2 k j k m L 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ γ j r j γ m * r m ) × exp [ i 2 κ 2 L ( γ j k j γ m * k m ) ξ ] ,
E 2 m j ( r m , r j ; k m , k j ) = 4 π 2 k j k m L 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ γ m r m γ j * r j ) × exp [ i 2 κ 2 L ( γ m k m γ j * k j ) ξ ] ,
E 3 j m ( r j , r m ; k j , k m ) = 4 π 2 k j k m L 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ γ j r j γ m r m ) × exp [ i 2 κ 2 L ( γ j k j + γ m k m ) ξ ] .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 κ in 2 ) ( κ 2 + κ out 2 ) 11 6 ,
σ R 2 = 1.23 C n 2 k 7 6 L 11 6
Φ n ( κ ) = 0.033 C n 2 κ 11 3 .
σ I , l 2 = σ I , l , s 2 + σ I , l , c 2 1 ,
σ I , l , s 2 ( L ) I 1 2 ( d y ̂ 2 , L ) + I 2 2 ( d y ̂ 2 , L ) I ( 0 , L ) 2 ,
σ I , l , c 2 ( L ) 2 I 1 ( d y ̂ 2 , L ) I 2 ( d y ̂ 2 , L ) I ( 0 , L ) 2 .
R a ( r , L ) = σ I , a 2 ( r , L ) σ I 2 ( r , L ) σ I , a 2 ( r , L ) ,
R b ( r , L ) = σ I , b 2 ( r , L ) σ I 2 ( r , L ) σ I , b 2 ( r , L ) .
σ I 2 ( r , L ) = σ I , s 2 ( r , L ) + σ I , c 2 ( r , L ) 1 ,
σ I , s 2 ( r , L ) I 1 2 ( r 1 , L ) + I 2 2 ( r 2 , L ) I ( r , L ) 2 ,
σ I , c 2 ( r , L ) 2 I 1 ( r 1 , L ) I 2 ( r 2 , L ) I ( r , L ) 2 .
I j ( r j , L ) = I 0 j ( r j , L ) exp [ ψ j ( r j , L ; k j ) + ψ j * ( r j , L ; k j ) ] ,
I j ( r j , L ) I m ( r m , L ) = I 0 j ( r j , L ) I 0 m ( r m , L ) exp [ ψ j m ( tot ) ( r , L ) ] ,
Ψ j m tot ( r , L ) = ψ j ( r j , L ; k j ) + ψ j * ( r j , L ; k j ) + ψ m ( r m , L ; k m ) + ψ m * ( r m , L ; k m ) .
exp [ Ψ j m ( tot ) ( r , L ) ] = exp { Ψ j m ( tot ) ( r , L ) + [ Ψ j m ( tot ) 2 ( r , L ) Ψ j m ( tot ) ( r , L ) 2 ] } .
Ψ j m ( tot ) ( r , L ) Ψ j m 1 ( tot ) ( r , L ) + Ψ j m 2 ( tot ) ( r , L ) [ ψ j 1 ( r j , L ; k j ) + ψ j 1 * ( r j , L ; k j ) + ψ m 1 ( r m , L ; k m ) + ψ m 1 * ( r m , L ; k m ) ] + [ ψ j 2 ( r j , L ; k j ) + ψ j 2 * ( r j , L ; k j ) + ψ m 2 ( r m , L ; k m ) + ψ m 2 * ( r m , L ; k m ) ] ,
Ψ j m ( tot ) ( r , L ) Ψ j m 2 ( tot ) ( r , L ) = ψ j 2 ( r j , L ; k j ) + ψ j 2 * ( r j , L ; k j ) + ψ m 2 ( r m , L ; k m ) + ψ m 2 * ( r m , L ; k m ) ,
Ψ j m ( tot ) ( r , L ) 2 Ψ j m 1 ( tot ) ( r , L ) 2 = 0 .
Ψ j m ( tot ) 2 ( r , L ) Ψ j m 1 ( tot ) 2 ( r , L ) = ψ j 1 2 ( r j , L ; k j ) + ψ j 1 * 2 ( r j , L ; k j ) + ψ m 1 2 ( r m , L ; k m ) + ψ m 1 * 2 ( r m , L ; k m ) + E 2 j j ( r j , r j ; k j , k j ) + E 2 m m ( r m , r m ; k m , k m ) + E 2 j m ( r j , r m ; k j , k m ) + E 2 m j ( r m , r j ; k m , k j ) + 2 Re [ E 3 j m ( r j , r m ; k j , k m ) ] ,
E 2 j m ( r j , r m ; k j , k m ) = ψ j 1 ( r j , L ; k j ) ψ m 1 * ( r m , L ; k m ) ,
E 3 j m ( r j , r m ; k j , k m ) = ψ j 1 ( r j , L ; k j ) ψ m 1 ( r m , L ; k m ) .
E 1 j ( r j ; k j ) = ψ j 2 ( r j , L ; k j ) + 1 2 ψ j 1 2 ( r j , L ; k j ) ,
exp [ Ψ i j ( tot ) ( r , L ) ] = exp { 2 E 1 j ( r j ; k j ) + 2 E 1 m ( r m ; k m ) + E 2 j j ( r j , r j ; k j , k j ) + E 2 m m ( r m , r m ; k m , k m ) + E 2 j m ( r j , r m ; k j , k m ) + E 2 m j ( r m , r j ; k m , k j ) + 2 Re [ E 3 j m ( r j , r m ; k j , k m ) ] } .
I j ( r j , L ) = I 0 j ( r j , L ) exp [ 2 E 1 j ( r j ; k j ) + E 2 j j ( r j , r j ; k j , k j ) ]
I j ( r j , L ) I m ( r m , L ) = I j ( r j , L ) I m ( r m , L ) exp { E 2 j m ( r j , r m ; k j , k m ) + E 2 m j ( r m , r j ; k m , k j ) + 2 Re [ E 3 j m ( r j , r m ; k j , k m ) ] } ,
ψ j 1 ( r j , L ; k j ) = i k j 0 L d z d ν ( K , z ) exp [ i γ j ( z ) K r j i κ 2 γ j ( z ) ( L z ) 2 k j ] ,
γ j ( z ) = 1 + α j z 1 + α j L ,
α j = 2 k j W 0 j 2 + i F 0 j .
n 1 ( r , z ) = d ν ( K , z ) exp ( i K r ) ,
ν ( K , z ) ν * ( K , z ) = F n ( K , z z ) δ ( K K ) d 2 K d 2 K ,
E 2 j m ( r j , r m ; k j , k m ) = k j k m 0 L d z 0 L d z d 2 K F n ( K , z z ) exp { i K [ γ j ( z ) r j γ m * ( z ) r m ] i κ 2 2 [ γ j ( z ) ( L z ) k j γ m * ( z ) ( L z ) k m ] } .
E 2 j m ( r j , r m ; k j , k m ) = k j k m 0 L d η d 2 K d μ F n ( K , μ ) exp { i K [ γ j ( η ) r j γ m * ( η ) r m ] i κ 2 2 [ γ j ( η ) k j γ m * ( η ) k m ] ( L η ) } .
Φ n ( K ) = 1 2 π d μ F n ( K , μ ) .
E 2 j m ( r j , r m ; k j , k m ) = 4 π 2 k j k m 0 L d η 0 d κ κ Φ n ( κ ) J 0 ( κ γ j r j γ m * r m ) × exp { i κ 2 2 [ γ j ( η ) k j γ m * ( η ) k m ] ( L η ) } .
E 2 j m ( r j , r m ; k j , k m ) = 4 π 2 k j k m 0 1 d ξ 0 d κ κ Φ n ( κ ) J 0 ( κ γ j r j γ m * r m ) exp [ i 2 κ 2 L ( γ j k j γ m * k m ) ξ ] ,

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