Abstract

An analysis of adaptive optics compensation for atmospheric-turbulence-induced scintillation is presented with the figure of merit being the laser communications bit-error rate. The formulation covers weak, moderate, and strong turbulence; on-off keying; and amplitude-shift keying, over horizontal propagation paths or on a ground-to-space uplink or downlink. The theory shows that under some circumstances the bit-error rate can be improved by a few orders of magnitude with the addition of adaptive optics to compensate for the scintillation. Low-order compensation (less than 40 Zernike modes) appears to be feasible as well as beneficial for reducing the bit-error rate and increasing the throughput of the communication link.

© 2002 Optical Society of America

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References

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  1. R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed. (Wiley, New York, 1995).
  2. H. T. Yura, W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22, 3353–3358 (1983).
    [CrossRef] [PubMed]
  3. H. Ansari, L. H. Roberts, “Charge-coupled device imaging system for precision beam steering in laser communications,” Opt. Eng. 34, 3261–3264 (1995).
    [CrossRef]
  4. L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742–7751 (1995).
    [CrossRef] [PubMed]
  5. L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system: errata,” Appl. Opt. 36, 6068 (1997).
    [CrossRef]
  6. R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
    [CrossRef]
  7. W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
    [CrossRef]
  8. L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
    [CrossRef]
  9. R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).
  10. J. Gourlay, T. Yang, M. Ishikawa, A. C. Walker, “Low-order adaptive optics for free-space optoelectronic interconnects,” Appl. Opt. 39, 714–720 (2000).
    [CrossRef]
  11. 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, 4553–4559 (1998).
    [CrossRef]
  12. G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
    [CrossRef]
  13. R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. 35, 3640–3646 (1996).
    [CrossRef] [PubMed]
  14. A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).
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    [CrossRef] [PubMed]
  17. P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 35, 5999–6009 (1996).
    [CrossRef] [PubMed]
  18. R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
  19. L. C. Andrews, R. L. Phillips, Laser Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).
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    [CrossRef]
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    [CrossRef]
  22. L. C. Andrews, R. L. Phillips, C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Wash., 2001).
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    [CrossRef]
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    [CrossRef]
  27. W. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 11.
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    [CrossRef]
  30. J. H. Churnside, “Aperture averaging of optical scintillation in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
    [CrossRef] [PubMed]
  31. R. F. Lutomirski, H. T. Yura, “Aperture-averaging factor of a fluctuating light signal,” J. Opt. Soc. Am. 59, 1247–1248 (1969).
    [CrossRef]
  32. R. K. Tyson, “Using the deformable mirror as a spatial filter: application to circular beams,” Appl. Opt. 21, 787–793 (1982).
    [CrossRef] [PubMed]
  33. R. J. Sasiela, 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, 646–659 (1993).
    [CrossRef]
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    [CrossRef]
  35. R. J. Sasiela, “Wave-front correction by one or more synthetic beacons,” J. Opt. Soc. Am. A 11, 379–393 (1994).
    [CrossRef]
  36. R. J. Sasiela, J. D. Shelton, “Guide star system considerations,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 3.
  37. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), trans. by R. A. Silverman.
  38. K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
    [CrossRef]
  39. H. Hemmati, K. Wilson, “Free-space optical communications at NASA,” Deep Space Communications and Naviagation Systems (DESCANSO) Symposium, Pasadena, Calif., September 21–23, 1999.

2001 (1)

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

2000 (2)

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
[CrossRef]

J. Gourlay, T. Yang, M. Ishikawa, A. C. Walker, “Low-order adaptive optics for free-space optoelectronic interconnects,” Appl. Opt. 39, 714–720 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (2)

1996 (2)

1995 (2)

H. Ansari, L. H. Roberts, “Charge-coupled device imaging system for precision beam steering in laser communications,” Opt. Eng. 34, 3261–3264 (1995).
[CrossRef]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742–7751 (1995).
[CrossRef] [PubMed]

1994 (3)

1993 (1)

1991 (1)

1987 (1)

J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 723–733 (1987).
[CrossRef]

1983 (1)

1982 (1)

1981 (3)

1976 (1)

1969 (1)

1967 (1)

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system: errata,” Appl. Opt. 36, 6068 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742–7751 (1995).
[CrossRef] [PubMed]

W. B. Miller, J. C. Ricklin, L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11, 2719–2726 (1994).
[CrossRef]

R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
[CrossRef]

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 Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

Ansari, H.

H. Ansari, L. H. Roberts, “Charge-coupled device imaging system for precision beam steering in laser communications,” Opt. Eng. 34, 3261–3264 (1995).
[CrossRef]

Araki, K.

K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
[CrossRef]

Arimoto, Y.

K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
[CrossRef]

Barakat, R.

Bennet, W. R.

M. Schwartz, W. R. Bennet, S. Stein, Communications Systems and Techniques (IEEE Press, New York, 1996).

Bracher, C.

Bruno, T. L.

Capron, B. A.

Castellanos, D. C.

Churnside, J. H.

J. H. Churnside, “Aperture averaging of optical scintillation in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
[CrossRef] [PubMed]

J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 723–733 (1987).
[CrossRef]

Costello, T. P.

Flatté, S. M.

Frehlich, R. G.

R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
[CrossRef]

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (NOAA Environmental Research Laboratories, Boulder, Colo., 1996).

Fried, D. L.

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed. (Wiley, New York, 1995).

Gatt, P.

Gourlay, J.

Harney, R. C.

Heimmermann, D. A.

Hemmati, H.

H. Hemmati, K. Wilson, “Free-space optical communications at NASA,” Deep Space Communications and Naviagation Systems (DESCANSO) Symposium, Pasadena, Calif., September 21–23, 1999.

Hill, R. J.

R. J. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
[CrossRef]

J. H. Churnside, R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 723–733 (1987).
[CrossRef]

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (NOAA Environmental Research Laboratories, Boulder, Colo., 1996).

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
[CrossRef]

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

Ishikawa, M.

Jankevics, A.

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed. (Wiley, New York, 1995).

Landers, F.

Lesh, J. R.

K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
[CrossRef]

Levine, B. M.

Lutomirski, R. F.

Marcum, J. I.

J. I. Marcum, A Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix, (The Rand Corporation, Santa Monica, Calif., 1948).

Martinsen, E. A.

McKinley, W. G.

Miller, W. B.

Nakagami, M.

M. Nakagami, “The m distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

Noll, R. J.

Otto, W. D.

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (NOAA Environmental Research Laboratories, Boulder, Colo., 1996).

Parry, G.

G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
[CrossRef]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system: errata,” Appl. Opt. 36, 6068 (1997).
[CrossRef]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742–7751 (1995).
[CrossRef] [PubMed]

R. L. Phillips, L. C. Andrews, “Measured statistics of laser-light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71, 1440–1445 (1981).
[CrossRef]

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 Propagation through Random Media (SPIE Press, Bellingham, Wash., 1998).

Pratt, W.

W. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 11.

Ricklin, J. C.

Roberts, L. H.

H. Ansari, L. H. Roberts, “Charge-coupled device imaging system for precision beam steering in laser communications,” Opt. Eng. 34, 3261–3264 (1995).
[CrossRef]

Sasiela, R. J.

Schwartz, M.

M. Schwartz, W. R. Bennet, S. Stein, Communications Systems and Techniques (IEEE Press, New York, 1996).

Shapiro, J. H.

Shelton, J. D.

R. J. Sasiela, 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, 646–659 (1993).
[CrossRef]

R. J. Sasiela, J. D. Shelton, “Guide star system considerations,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 3.

Stein, S.

M. Schwartz, W. R. Bennet, S. Stein, Communications Systems and Techniques (IEEE Press, New York, 1996).

Stickley, C. M.

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), trans. by R. A. Silverman.

Toledo-Quinones, M.

Tyson, R. K.

Walker, A. C.

Wang, G.-Yu.

Weeks, A. R.

Wilson, K.

H. Hemmati, K. Wilson, “Free-space optical communications at NASA,” Deep Space Communications and Naviagation Systems (DESCANSO) Symposium, Pasadena, Calif., September 21–23, 1999.

Wilson, K. E.

K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
[CrossRef]

Wirth, A.

Yang, T.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).

Yu, P. T.

Yura, H. T.

Appl. Opt. (10)

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system,” Appl. Opt. 34, 7742–7751 (1995).
[CrossRef] [PubMed]

L. C. Andrews, R. L. Phillips, P. T. Yu, “Optical scintillations and fade statistics for a satellite-communication system: errata,” Appl. Opt. 36, 6068 (1997).
[CrossRef]

H. T. Yura, W. G. McKinley, “Optical scintillation statistics for IR ground-to-space laser communication systems,” Appl. Opt. 22, 3353–3358 (1983).
[CrossRef] [PubMed]

J. Gourlay, T. Yang, M. Ishikawa, A. C. Walker, “Low-order adaptive optics for free-space optoelectronic interconnects,” Appl. Opt. 39, 714–720 (2000).
[CrossRef]

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, 4553–4559 (1998).
[CrossRef]

J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3313 (1981).
[CrossRef] [PubMed]

P. Gatt, T. P. Costello, D. A. Heimmermann, D. C. Castellanos, A. R. Weeks, C. M. Stickley, “Coherent optical array receivers for the mitigation of atmospheric turbulence and speckle effects,” Appl. Opt. 35, 5999–6009 (1996).
[CrossRef] [PubMed]

R. K. Tyson, “Adaptive optics and ground-to-space laser communications,” Appl. Opt. 35, 3640–3646 (1996).
[CrossRef] [PubMed]

J. H. Churnside, “Aperture averaging of optical scintillation in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
[CrossRef] [PubMed]

R. K. Tyson, “Using the deformable mirror as a spatial filter: application to circular beams,” Appl. Opt. 21, 787–793 (1982).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (4)

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

Opt. Acta (1)

G. Parry, “Measurements of atmospheric turbulence-induced intensity fluctuations in a laser beam,” Opt. Acta 28, 715–728 (1981).
[CrossRef]

Opt. Eng. (3)

H. Ansari, L. H. Roberts, “Charge-coupled device imaging system for precision beam steering in laser communications,” Opt. Eng. 34, 3261–3264 (1995).
[CrossRef]

L. C. Andrews, R. L. Phillips, C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39, 3272–3280 (2000).
[CrossRef]

M. A. Al-Habash, L. C. Andrews, R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

Other (14)

W. Pratt, Laser Communications Systems (Wiley, New York, 1969), Chap. 11.

J. I. Marcum, A Statistical Theory of Target Detection by Pulsed Radar: Mathematical Appendix, (The Rand Corporation, Santa Monica, Calif., 1948).

R. M. Gagliardi, S. Karp, Optical Communications, 2nd ed. (Wiley, New York, 1995).

R. J. Hill, R. G. Frehlich, W. D. Otto, “The probability distribution of irradiance scintillation,” (NOAA Environmental Research Laboratories, Boulder, Colo., 1996).

A. Yariv, Optical Electronics in Modern Communications (Oxford U. Press, New York, 1997).

M. Schwartz, W. R. Bennet, S. Stein, Communications Systems and Techniques (IEEE Press, New York, 1996).

R. J. Sasiela, J. D. Shelton, “Guide star system considerations,” in Adaptive Optics Engineering Handbook, R. K. Tyson, ed. (Marcel Dekker, New York, 2000), Chap. 3.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), trans. by R. A. Silverman.

K. E. Wilson, J. R. Lesh, K. Araki, Y. Arimoto, “Overview of the ground-to-orbit lasercom demo,” in Free-Space laser Communication Technologies IX, G. Mecherle, ed., Proc. SPIE2990, 23–30 (1997).
[CrossRef]

H. Hemmati, K. Wilson, “Free-space optical communications at NASA,” Deep Space Communications and Naviagation Systems (DESCANSO) Symposium, Pasadena, Calif., September 21–23, 1999.

R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic, Boston, Mass., 1998).

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

M. Nakagami, “The m distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. C. Hoffman, ed. (Pergamon, New York, 1960), pp. 3–36.

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

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Figures (9)

Fig. 1
Fig. 1

BER improvement with adaptive optics. The analysis uses the downlink formulation with a 10-cm aperture and λ=1.06 μm. Calculations for two zenith angles, representing weak and strong turbulence, are shown along with variation of the SNR of 21 and 24. Even for the strong-turbulence cases, low-order adaptive optics can provide up to four-orders-of-magnitude improvement in the BER.

Fig. 2
Fig. 2

BER improvement with adaptive optics. The analysis uses the downlink formulation with a 10-cm aperture and λ=1.06 μm for a fixed SNR of 21. The zenith angle is varied from vertical propagation (0°) to near the horizon (80°). Dramatic improvement in BER is possible with as low as ten Zernike modes removed.

Fig. 3
Fig. 3

Beneficial effect of aperture averaging, beyond the benefit from adaptive optics. This analysis also uses the downlink formulation with λ=1.06 μm, Ψ=0°, and a fixed SNR of 21.

Fig. 4
Fig. 4

For the same conditions as in Fig. 3, the BER improvement effects are not sensitive to wavelength. Remember that laser power and detector response are adjusted to maintain a SNR of 21.

Fig. 5
Fig. 5

BER improvement with adaptive optics for a spherical-wave uplink. The analysis uses a 10-cm aperture and λ=1.06 μm. Calculations for two zenith angles, representing weak and strong turbulence, are shown along with variation of the SNR between 21 and 24. Low-order adaptive optics can provide up to four-orders-of-magnitude improvement in the BER.

Fig. 6
Fig. 6

BER improvement with adaptive optics for a spherical wave uplink. The zenith angle is varied from vertical propagation (0°) to near the horizon (80°). Dramatic improvement in BER is possible with as low as ten Zernike modes removed.

Fig. 7
Fig. 7

BER improvement with adaptive optics for a horizontal path spherical wave with a fixed Cn2. Three ranges—1, 5, and 10 km—capture both weak and strong turbulence.

Fig. 8
Fig. 8

BER improvement with adaptive optics for a horizontal path with variation in aperture size and range.

Fig. 9
Fig. 9

ASK modulation shows improvement in the BER with adaptive optics for a horizontal path. The analysis uses λ=1.06 μm, a 10-cm or 20-cm aperture, SNR=18, and Cn2=10-16 m-2/3.

Equations (20)

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

P(s, SNR)=12 erfcis22σN,
BER(OOK)=12 0pl(s)erfcSNRs22isds.
pI(s)=2(αβ)(α+β)/2Γ(α)Γ(β)is sis[(α+β)/2]-1Kα-β2αβsis.
BER(ASK)=1-Q(2 CNR, TNR),
Cn2(h)=5.94×10-23h10 exp(-h)(W/27)2+2.7×10-16 exp(-2h/3)+A exp(-10h),
h=(Re2+2zRe cos Ψ+z2)1/2-Re,
σI2=exp [σln x2+σln y2]-1=exp0.49σ12(1+1.11σ112/15)7/6+0.51σ12(1+0.69σ112/5)5/6-1.
σ12=2.6062πk2 h0HCn2(z, Ψ) 02π01-coszκ2k×κ-8/31-i=1NFi(κ, D, ϕ)dκdϕdz.
Feven m,n(κ, D, ϕ)=2(n+1)2Jn+1(κD/2)κD/22 cos2(mϕ),
Fodd m,n(κ, D, ϕ)=2(n+1)2Jn+1(κD/2)κD/22 sin2(mϕ),
Fm=0,n(κ, D, ϕ)=(n+1)2Jn+1(κD/2)κD/22.
σI2=expσln x2+σln y2-1.
σln x2=2.6062πk2 h0HCn2(z, Ψ)Az(z, Ψ)×02π01-coszκ2kξ(1-ξ)×exp-34 κκx5/3ξ5/3(1-ξ)5/3×κ-8/31-i=1NFi(κ, D, ϕ)dκdϕdz,
Az(z, Ψ)=9.31 h0HCn2(z, Ψ)ξ5/6(1-ξ)5/6 dzh0HCn2(z, Ψ)ξ-1/3(1-ξ)-1/3dz,
ξ=z-h0zmax-h0.
σln x2=2.6062πk2 h0HCn2(z, Ψ)Bz(z, Ψ)×02π01-coszmaxκ2kξ(1-ξ)×κ(κ2+κy2)11/6 1-i=1N Fi(κ, D, ϕ)dκdϕdz,
Bz(z, Ψ)=4.535 h0HCn2(z, Ψ)ξ5/6(1-ξ)5/6 dzh0HCn2(z, Ψ)dz.
σ12=2.6062πk2Cn2 02π0L-kκ2 sinLκ2k×κ-8/31-i=1NFi(κ, D, ϕ)dκdϕ,
σln x2=2.6062πk2Cn2 0H02π01-coszκ2kξ(1-ξ)×exp-34κκx5/3ξ5/3(1-ξ)5/3×κ-8/31-i=1N Fi(κ, D, ϕ)dκdϕdz
σln x2=2.6062πk2Cn2×0H02π01-coszmaxκ2kξ(1-ξ)×κ(κ2+κy2)11/6 1-i=1N Fi(κ, D, ϕ)dκdϕdz.

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