Abstract

Airborne laser-communication systems require special considerations in size, complexity, power, and weight. We reduce the variability of the received signal by implementing optimized multiple-transmitter systems to average out the deleterious effects of turbulence. We derive the angular laser-beam separation for various isoplanatic and uncorrelated (anisoplanatic) conditions for the phase and amplitude effects. In most cases and geometries, the angles ordered from largest to smallest are: phase uncorrelated angle (equivalent to the tilt uncorrelated angle), tilt isoplanatic angle, phase isoplanatic angle, scintillation uncorrelated angle, and scintillation correlation angle (θψind>θTA>θ0>θχind>θχc) . Multiple beams with angular separations beyond θχc tend to reduce scintillation variations. Larger separations such as θTA reduce higher-order phase and scintillation variations and still larger separations beyond θψind tend to reduce the higher and lower-order (e.g. tilt) phase and scintillation effects. Simulations show two-transmitter systems reduce bit error rates for ground-to-air, air-to-air, and ground-to-ground scenarios.

©2008 Optical Society of America

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References

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  1. S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Commun. 21, 1346–1357 (2003).
    [Crossref]
  2. P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.
  3. P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
    [Crossref] [PubMed]
  4. A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006).
    [Crossref]
  5. I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
    [Crossref]
  6. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).
  7. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestial free-space optical communication link” Appl. Opt. 466561–6571 (2007).
    [Crossref] [PubMed]
  8. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 72, 52–61 (1982).
    [Crossref]
  9. R. J. Sasiela, Electromagnetic wave propagation in turbulence. Evaluation and application of Mellin transforms, 2nd ed. (SPIE Publications, 2007).
  10. R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646660 (1993).
    [Crossref]
  11. J. W. Goodman, Statistical Optics (John Wiley & Sons, Hoboken, NJ, 1985).
  12. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  13. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures,” J. Opt. Soc. Am. A 56, 1372–1379 (1966).
    [Crossref]
  14. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Optical Engineering Press Bellingham, WA, 2005).
    [Crossref]
  15. A. D. Wheelon, Electromagnetic scintillation. 1, Geometrical optics (Cambridge Univ. Press, 2001).
  16. F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, NM VHF radar observations,” Radio Sci. 33, 859–903 (1998).
    [Crossref]
  17. J. A. Louthain and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” (MSS Active E-O Systems Proceedings, Military Sensing Information Analysis Center (SENSIAC), 2007).
  18. E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
    [Crossref]
  19. D. L. Fried, “Spectral and Angular Covariance of Scintillation for Propagation in a Randomly Inhomogeneous Medium,” Appl. Optics 10, 721–731 (1971).
    [Crossref]
  20. M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
    [Crossref] [PubMed]
  21. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proceedings of SPIE 3125, 327 (1997).
    [Crossref]
  22. J. A. Louthain, “Master’s Thesis: Atmospheric turbulence scintillation effects of wavefront tilt estimation,” (1997).
  23. J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proceedings of SPIE 3381, 286–296 (1998).
    [Crossref]
  24. S. Coy, “Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation,” in Advanced Wavefront Control: Methods, Devices, and Applications III, J. D. G. Mark, T. Gruneisen, and M. K. Giles, eds., vol. 5894 (SPIE Press, Bellingham, WA, 2005).
  25. Y. Dikmelik and F. M. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44, 4946–4952 (2005). URL http://ao.osa.org/abstract.cfm?URI=ao-44-23-4946.
    [Crossref] [PubMed]
  26. S. B. Alexander, Optical Communication Receiver Design, SPIE Tutorial Texts in Optical Engineering, vol. TT22; IEE Telecommunications Series, vol. 37 (SPIE Press, Bellingham, WA, 1997).
  27. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, 2nd ed. (SPIE Press, 2001).
    [Crossref]
  28. E. Dereniak and G. Boreman, Infrared Detectors and Systems (Wiley New York, 1996).

2007 (2)

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestial free-space optical communication link” Appl. Opt. 466561–6571 (2007).
[Crossref] [PubMed]

2006 (2)

A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006).
[Crossref]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).

2005 (1)

2003 (2)

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Commun. 21, 1346–1357 (2003).
[Crossref]

1998 (2)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, NM VHF radar observations,” Radio Sci. 33, 859–903 (1998).
[Crossref]

J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proceedings of SPIE 3381, 286–296 (1998).
[Crossref]

1997 (2)

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proceedings of SPIE 3125, 327 (1997).
[Crossref]

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

1995 (1)

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

1993 (1)

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646660 (1993).
[Crossref]

1982 (1)

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 72, 52–61 (1982).
[Crossref]

1971 (1)

D. L. Fried, “Spectral and Angular Covariance of Scintillation for Propagation in a Randomly Inhomogeneous Medium,” Appl. Optics 10, 721–731 (1971).
[Crossref]

1966 (1)

D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures,” J. Opt. Soc. Am. A 56, 1372–1379 (1966).
[Crossref]

Adhikari, P.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

Alexander, S. B.

S. B. Alexander, Optical Communication Receiver Design, SPIE Tutorial Texts in Optical Engineering, vol. TT22; IEE Telecommunications Series, vol. 37 (SPIE Press, Bellingham, WA, 1997).

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, 2nd ed. (SPIE Press, 2001).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Optical Engineering Press Bellingham, WA, 2005).
[Crossref]

Anguita, J. A.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestial free-space optical communication link” Appl. Opt. 466561–6571 (2007).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).

Boreman, G.

E. Dereniak and G. Boreman, Infrared Detectors and Systems (Wiley New York, 1996).

Coy, S.

S. Coy, “Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation,” in Advanced Wavefront Control: Methods, Devices, and Applications III, J. D. G. Mark, T. Gruneisen, and M. K. Giles, eds., vol. 5894 (SPIE Press, Bellingham, WA, 2005).

Das, S. T.

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

Davidson, F. M.

DeLaRue, I. A.

P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.

Dereniak, E.

E. Dereniak and G. Boreman, Infrared Detectors and Systems (Wiley New York, 1996).

Dikmelik, Y.

Eaton, F. D.

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, NM VHF radar observations,” Radio Sci. 33, 859–903 (1998).
[Crossref]

Fried, D. L.

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 72, 52–61 (1982).
[Crossref]

D. L. Fried, “Spectral and Angular Covariance of Scintillation for Propagation in a Randomly Inhomogeneous Medium,” Appl. Optics 10, 721–731 (1971).
[Crossref]

D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures,” J. Opt. Soc. Am. A 56, 1372–1379 (1966).
[Crossref]

Fugate, R. Q.

P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley & Sons, Hoboken, NJ, 1985).

Haas, S. M.

S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Commun. 21, 1346–1357 (2003).
[Crossref]

Hakakha, H.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, 2nd ed. (SPIE Press, 2001).
[Crossref]

Kim, I. I.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

Klein, L.

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

Korevaar, E. J.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

Louthain, J. A.

J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proceedings of SPIE 3381, 286–296 (1998).
[Crossref]

J. A. Louthain, “Master’s Thesis: Atmospheric turbulence scintillation effects of wavefront tilt estimation,” (1997).

J. A. Louthain and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” (MSS Active E-O Systems Proceedings, Military Sensing Information Analysis Center (SENSIAC), 2007).

Lowrey, W. H.

P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.

Magee, E. P.

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

Majumdar, A. K.

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

Moloney, J.

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

Moloney, J. V.

Montera, D. A.

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

Nastrom, G. D.

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, NM VHF radar observations,” Radio Sci. 33, 859–903 (1998).
[Crossref]

Neifeld, M. A.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestial free-space optical communication link” Appl. Opt. 466561–6571 (2007).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).

Peleg, A.

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, 2nd ed. (SPIE Press, 2001).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Optical Engineering Press Bellingham, WA, 2005).
[Crossref]

Polynkin, P.

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

Rhoadarmer, T.

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

Roggemann, M. C.

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Ryan, P. T.

P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.

Sasiela, R. J.

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646660 (1993).
[Crossref]

R. J. Sasiela, Electromagnetic wave propagation in turbulence. Evaluation and application of Mellin transforms, 2nd ed. (SPIE Publications, 2007).

Schmidt, J. D.

J. A. Louthain and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” (MSS Active E-O Systems Proceedings, Military Sensing Information Analysis Center (SENSIAC), 2007).

Shapiro, J. H.

S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Commun. 21, 1346–1357 (2003).
[Crossref]

Shelton, J. D.

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646660 (1993).
[Crossref]

Vasic, B. V.

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Spatial correlation and irradiance statistics in a multiple-beam terrestial free-space optical communication link” Appl. Opt. 466561–6571 (2007).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).

Welsh, B. M.

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proceedings of SPIE 3381, 286–296 (1998).
[Crossref]

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proceedings of SPIE 3125, 327 (1997).
[Crossref]

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Wheelon, A. D.

A. D. Wheelon, Electromagnetic scintillation. 1, Geometrical optics (Cambridge Univ. Press, 2001).

Whiteley, M. R.

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

Appl. Opt (1)

M. C. Roggemann, B. M. Welsh, D. A. Montera, and T. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt 34, 4037–4051 (1995).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Optics (1)

D. L. Fried, “Spectral and Angular Covariance of Scintillation for Propagation in a Randomly Inhomogeneous Medium,” Appl. Optics 10, 721–731 (1971).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

S. M. Haas and J. H. Shapiro, “Capacity of Wireless Optical Communications,” IEEE J. Sel. Areas Commun. 21, 1346–1357 (2003).
[Crossref]

J. Opt. Soc. Am. A (4)

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. A 72, 52–61 (1982).
[Crossref]

R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646660 (1993).
[Crossref]

D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures,” J. Opt. Soc. Am. A 56, 1372–1379 (1966).
[Crossref]

A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006).
[Crossref]

Opt. Lett (1)

P. Polynkin, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett 32, 885–887 (2007).
[Crossref] [PubMed]

Proceedings of SPIE (5)

I. I. Kim, H. Hakakha, P. Adhikari, E. J. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proceedings of SPIE 2990, 102 (1997).
[Crossref]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Multi-beam space-time coded systems for optical atmospheric channels,” Proceedings of SPIE 6304, 1–9 (2006).

J. A. Louthain and B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” Proceedings of SPIE 3381, 286–296 (1998).
[Crossref]

E. P. Magee, M. R. Whiteley, S. T. Das, and B. M. Welsh, “Tilt anisoplanatism in extended turbulence propagation,” Proceedings of SPIE 4976, 13–21 (2003).
[Crossref]

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proceedings of SPIE 3125, 327 (1997).
[Crossref]

Radio Sci. (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range, NM VHF radar observations,” Radio Sci. 33, 859–903 (1998).
[Crossref]

Other (12)

J. A. Louthain and J. D. Schmidt, “Anisoplanatic Approach to Airborne Laser Communication,” (MSS Active E-O Systems Proceedings, Military Sensing Information Analysis Center (SENSIAC), 2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Optical Engineering Press Bellingham, WA, 2005).
[Crossref]

A. D. Wheelon, Electromagnetic scintillation. 1, Geometrical optics (Cambridge Univ. Press, 2001).

J. W. Goodman, Statistical Optics (John Wiley & Sons, Hoboken, NJ, 1985).

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

R. J. Sasiela, Electromagnetic wave propagation in turbulence. Evaluation and application of Mellin transforms, 2nd ed. (SPIE Publications, 2007).

J. A. Louthain, “Master’s Thesis: Atmospheric turbulence scintillation effects of wavefront tilt estimation,” (1997).

S. Coy, “Choosing Mesh Spacings and Mesh Dimensions for Wave Optics Simulation,” in Advanced Wavefront Control: Methods, Devices, and Applications III, J. D. G. Mark, T. Gruneisen, and M. K. Giles, eds., vol. 5894 (SPIE Press, Bellingham, WA, 2005).

P. T. Ryan, W. H. Lowrey, I. A. DeLaRue, and R. Q. Fugate, “Scintillation characterization for multiple beams,” in Propagation and Imaging through the Atmosphere III, M. C. Roggemann and L. R. Bissonnette, eds., vol. 3763, (SPIE Press, Bellingham, WA, 1999) pp. 210–217.

S. B. Alexander, Optical Communication Receiver Design, SPIE Tutorial Texts in Optical Engineering, vol. TT22; IEE Telecommunications Series, vol. 37 (SPIE Press, Bellingham, WA, 1997).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, 2nd ed. (SPIE Press, 2001).
[Crossref]

E. Dereniak and G. Boreman, Infrared Detectors and Systems (Wiley New York, 1996).

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Figures (8)

Fig. 1.
Fig. 1. The phase and amplitude structure functions are plotted for a 100 km horizontal propagation at 10 km altitude, with angularly separated beams. The strength of turbulence, L 0/r 0=286 and L 2 0/(λL)=23225.

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Fig. 2.
Fig. 2. (a) Scenarios for plots b-e are shown pictorially. Phase isoplanatic angle (θ 0), scintillation correlation angle ( θ χ c ) , tilt isoplanatic angle (θTA ), and phase uncorrelated angle ( θ ψ ind ) are shown for a receiver diameter of DR =20 cm. (b) Horizontal propagation: altitude h=1 m, L 0=40 cm, C 2 n =10-14 m-2/3, and L=0 to 10 km. (c) Horizontal propagation: altitude h=10 km, L 0=100 km, C 2 n =10-17 m-2/3, and L=0 to 300 km. (d) Air-to-ground path: Transmitter height HTx =4 to 20 km, zenith angle ξ=70°, and receiver height HRx =0 km for HV-57 profile. (e) Ground-to-air path: HTx =0 km, zenith angle ξ=70°, and HRx =4 to 20 km for HV-57 profile.

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Fig. 3.
Fig. 3. In turbulence when D>r 0, the spot size is determined by 2.44λf/r 0. Where as when r 0>D, the spot size is limited primarily by diffraction, leading to the tighter spot size of 2.44λf/D.

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Fig. 4.
Fig. 4. Differential irradiance variance for two angularly separated beams. Irradiance is taken from the center of the untracked beams, separately tracked beams, and a single combined beam tracker. The solid blue line is two times the variance of on-axis irradiance of a single beam. The differential variance approaches two times this value as the separation increases. (a) Air-to-air path angular separation (b) Ground-to-air path angular separation.

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Fig. 5.
Fig. 5. These plots show the BER for a ground-to-ground link. In plot (a) the beams were angularly separated and in plot (b) the beams were separated, but traveled in parallel.

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Fig. 6.
Fig. 6. Bit error rate for a ground-to-air link with angularly separated beams with various tracking systems (a) ideal centroid tracker, (b) σj =λ/(8D), (c) σj =λ/(4D), and (d) σj =3λ/(8D).

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Fig. 7.
Fig. 7. Bit error rate for a ground-to-air link for parallel separated beams. Center beam tracker. Various tracking systems (a) ideal centroid tracker, (b) σj =λ/(8D), (c) σj =λ/(4D), and (d) σj =3λ/(8D).

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Fig. 8.
Fig. 8. Air-to-air 100km path at 10km altitude. Various tracking systems (a) ideal centroid tracker, (b) σj =λ/(8D), (c) σj =λ/(4D), and (d) σj =3λ/(8D).

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Tables (1)

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Table 1. Atmospheric Parameters for the scenarios used in the BER calculations.

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Equations (37)

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Φ n ( κ , z ) = 0.033 C n 2 ( z ) ( κ 2 + κ 0 2 ) 11 6 ,
D ψ ( Δ x ) = E { [ ψ ( x ) − ψ ( x + Δ x ) ] 2 }
= 2 Γ ψ ( 0 ) − 2 Γ ψ ( Δ x ) , for stationary random processes ,
= 2.91 k 2 ( Δ x ) 5 3 ∫ 0 L C n 2 ( z ) d z .
D ψ ( θ , L ) = 2.91 k 2 [ sin ( θ ) ] 5 3 ∫ 0 L ( L − z ) 5 3 C n 2 ( z ) d z .
D ψ ( θ 0 , L ) = 1 rad 2
θ 0 = [ 2.91 k 2 ∫ 0 L C n 2 ( z ) ( L − z ) 5 3 d z ] − 3 5 ,
D ψ ( θ ψ ind , L ) = 2 σ ψ , p l . 2
θ ψ ind = 2 σ ψ , p l 2 θ 0 .
σ ψ , p l 2 ≅ 4 π 2 k 2 ∫ 0 L ∫ 0 ∞ κ Φ n ( κ , z ) d κ d z
= 0.78 k 2 κ 0 − 5 3 ∫ 0 L C n 2 ( z ) d z .
θ ψ ind = 0.7402 k 4 5 C n 4 5 L − 3 5 κ 0 − 5 3 .
L 0 = L 0 vert cos ξ ,
D ψ ( Δ x , L ) = 2.91 k 2 ( Δ x ) 5 3 ∫ 0 L C n 2 ( z ) d z .
Δ x 0 = [ 2.91 k 2 ∫ 0 L C n 2 ( z ) d z ] − 3 5 = 0.6611 ρ 0 = 0.3148 r 0 ,
Δ x 0 = 0.5268 k − 6 5 C n − 6 5 L − 3 5 = 0.555 L θ 0 .
Δ x ind = 2 σ ψ , pl 2 Δ x 0 = 0.4109 κ 0 − 5 3 [ k 2 ∫ 0 L C n 2 ( z ) d z ] 2 5 .
Δ x ind = 0.4109 k 4 5 C n 4 5 L 2 5 κ 0 − 5 3 = 0.555 L θ ψ ind .
[ σ ∥ 2 σ ⊥ 2 ] = [ σ ∥ 2 σ ⊥ 2 ] L + [ σ ∥ 2 σ ⊥ 2 ] U .
[ σ ∥ 2 σ ⊥ 2 ] = [ σ ∥ 2 σ ⊥ 2 ] L = 6.08 C n 2 D 1 3 × { [ 1.316 0.439 ] [ ( θ D ) 2 ( L 3 3 ) − [ 2.2955 1.377 ] ( θ D ) 4 ( L 5 5 ) + . . . ]
+ [ 2.195 0.388 ] [ ( θ D ) 14 3 ( 3 17 ) L 17 3 − [ 0.1756 0.1298 ] ( θ D ) 20 3 ( 3 23 ) L 23 3 + . . . ] } ,
σ T 2 = σ ∥ 2 + σ ⊥ 2 ≈ 6.08 C n 2 D 1 3 × [ 1.755 ( θ T A D ) 2 ( L 3 3 ) ] = ( 0.61 λ D ) 2 .
θ T A = 0.323 λ D 1 6 C n L 3 2 , θ < D L .
[ σ ∥ 2 σ ⊥ 2 ] ≈ 6.08 C n 2 D 1 3 { L [ 1 1 ] − [ 0.7801 0.9057 ] D θ − [ 0.797 1.197 ] ( D θ ) 1 3 [ L 2 3 − ( D θ ) 2 3 ] } .
σ T 2 = σ ∥ 2 + σ ⊥ 2 ≈ 6.08 C n 2 D 1 3 [ 2 L + 0.3077 D θ − 1.9935 ( D θ ) 1 3 L 2 3 ] θ > D L .
ρ c = L k .
𝓡 sph = 0.5631 k 7 6 ∫ 0 L C n 2 ( z ) ( L − z ) 5 6 ( z L ) 5 6 d z
≈ σ χ 2 𝓡 sph ≲ 0.25
≳ σ χ 2 𝓡 sph ≳ 0.25 .
D χ ( d ) = 3.089 ( L 0 r 0 ) 5 3 ∫ 0 ∞ [ 1 − J 0 ( κ d L 0 ) ] [ 1 − 2 π L 0 2 λ L κ 2 sin ( λ L κ 2 2 π L 0 2 ) κ d κ ( κ 2 + 4 π 2 ) 11 6 , ]
D ψ ( d ) = 3.089 ( L 0 r 0 ) 5 3 ∫ 0 ∞ [ 1 − J 0 ( κ d L 0 ) ] [ 1 + 2 π L 0 2 λ L κ 2 sin ( λ L κ 2 2 π L 0 2 ) κ d κ ( κ 2 + 4 π 2 ) 11 6 , ]
ϕ ̂ ( x ) = ∑ n = − ( N − 1 ) N − 1 ∑ n ′ = − ( N − 1 ) N − 1 c n , n ′ ϕ exp { j 2 π ( n x D p + n ′ y D p ) } ,
E { i shot 2 } = σ shot 2 = 2 q i S B = 2 η q 2 P B h v ,
E { i elec 2 } = σ elec 2 = 4 K T B R ,
E { i ASE 2 } = 4 q 2 n s p η in η out 2 G ( G − 1 ) P h v B
σ ASE 2 = 4 n s p η o u t q ( G − 1 ) i s B ,
σ j 2 = σ T A 2 + σ T T 2 + σ P J 2 + σ T M 2

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