Abstract

The simulation of beam propagation is used to examine the uncertainty inherent to the process of optical power measurement with a practical heterodyne receiver because of the presence of refractive turbulence. Phase-compensated heterodyne receivers offer the potential for overcoming the limitations imposed by the atmosphere by the partial correction of turbulence-induced wave-front phase aberrations. However, wave-front amplitude fluctuations can limit the compensation process and diminish the achievable heterodyne performance.

©2008 Optical Society of America

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References

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  1. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
    [Crossref]
  2. J. H. Shapiro, “Imaging and Optical Communication through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., (Springer Verlag, Berlin, 1978) pp. 210–220.
  3. H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
    [Crossref]
  4. J. Y. Wang and J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [Crossref]
  5. G. -m. Dai, “Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 12, 2182–2193 (1995).
    [Crossref]
  6. A. Belmonte and B. J. Rye, “Heterodyne lidar returns in turbulent atmosphere: performance evaluation of simulated systems,” Appl. Opt. 39, 2401–2411 (2000).
    [Crossref]
  7. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39, 5426–5445 (2000).
    [Crossref]
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    [Crossref] [PubMed]
  9. A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).
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    [Crossref]
  11. N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).
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    [Crossref]
  16. M. C. Roggemann and A. C. Koivunen, “Branch-point reconstruction in laser beam projection through turbulence with finite-degree-of-freedom phase-only wave-front correction,” J. Opt. Soc. Am. A 17, 53–62 (2000).
    [Crossref]
  17. G. A. Tyler, “Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,” J. Opt. Soc. Am. A 17, 1828–1839 (2000).
    [Crossref]
  18. J. Y. Wang, “Phase-compensated optical beam propagation through atmospheric turbulence,” Appl. Opt. 17, 2580–2590 (1978).
    [PubMed]
  19. J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of Hartmann Sensors in Strong Scintillation,” Appl. Opt. 41, 1012–1021 (2002).
    [Crossref] [PubMed]
  20. J. D. Barchers, D. L. Fried, and D. J. Link, “Evaluation of the Performance of a Shearing Interferometer in Strong Scintillation in the Absence of Additive Measurement Moise,” Appl. Opt. 41, 3674–3684 (2002).
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  21. J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite-resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002).
    [Crossref]
  22. G. A. Tyler, “Adaptive optics compensation for propagation through deep turbulence: initial investigation of gradient descent tomography,” J. Opt. Soc. Am. A 23, 1914–1923 (2006).
    [Crossref]
  23. L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
    [Crossref]
  24. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
    [Crossref]

2007 (1)

2006 (1)

2002 (3)

2000 (4)

1998 (1)

1995 (1)

1992 (1)

L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

1988 (1)

N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

1982 (1)

1979 (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[Crossref]

1978 (2)

1976 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[Crossref]

1965 (1)

J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Andrews, L. C.

L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

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

Barchers, J. D.

Belmonte, A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Dai, G. -m.

Fried, D. L.

Goodman, J. W.

J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Hopen, C. Y.

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

Jelalian, A. W.

A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).

Koivunen, A. C.

Link, D. J.

Markey, J. K.

Noll, R. J.

Perlot, N.

Phillips, R. L.

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

Roggemann, M. C.

Rye, B. J.

Shapiro, J. H.

N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

J. H. Shapiro, “Imaging and Optical Communication through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., (Springer Verlag, Berlin, 1978) pp. 210–220.

Tyler, G. A.

Tyson, R. K.

Wang, J. Y.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Yura, H. T.

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[Crossref]

Zirkind, N. E.

N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

Appl. Opt. (7)

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive-index power spectrum and its application to optical scintillation in the atmosphere,” J. Mod. Opt. 39, 1849–1853 (1992).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Acta (1)

H. T. Yura, “Signal-to-noise ratio of heterodyne lidar systems in the presence of atmospheric turbulence,” Opt. Acta 26, 627–644 (1979).
[Crossref]

Proc. IEEE (2)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–67 (1967).
[Crossref]

J. W. Goodman, “Some effects of Target-induced Scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700 (1965).
[Crossref]

Proc. SPIE (1)

N. E. Zirkind and J. H. Shapiro, “Adaptive optics for large aperture coherent laser radars,” Proc. SPIE 999, paper 13 (1988).

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

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

J. H. Shapiro, “Imaging and Optical Communication through Atmospheric Turbulence,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed., (Springer Verlag, Berlin, 1978) pp. 210–220.

A. W. Jelalian, Laser Radar Systems (Artech House, Boston, 1995).

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Figures (4)

Fig. 1.
Fig. 1. Heterodyne efficiency gain (in decibels) as a function of the number of modes J removed by adaptive optics systems and several receiver aperture diameters by use of the simulation of realistic beam propagation in turbulent atmosphere. The levels of turbulence considered in these plots are typical daytime values of strong scintillation. In this plots, strong turbulence means a Fried’s coherent length r0=3 cm and a Rytov variance σ1=2; it is denoted as C2 n=10-13m-2/3 (right). We define a second, weaker turbulence level, denoted as as C2 n=10-14 m-2/3, where r0=6 cm and σ1=1 (left).

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Fig. 2.
Fig. 2. Heterodyne efficiency gain (in decibels) as a function of the receiver aperture diameter and several number of modes corrected by the adaptive optics system. The no-scintillation case is also shown (dased lines). Here, the compensating phases are expansions through 2rd-order (astigmatism, J=6), 5th-order (J=20), and a high-order case (J=80). The levels of turbulence considered in this plots are similar to those described in Fig. 1.

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Fig. 3.
Fig. 3. Optical heterodyne power uncertainty (in decibels) as a function of the number of modes J removed by adaptive optics systems and several receiver aperture diameters. Levels of turbulence are similar to those in Fig. 1.

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Fig. 4.
Fig. 4. Optical heterodyne power uncertainty (in decibels) as a function of the receiver aperture diameter and several number of modes corrected by the adaptive optics system. Again, the compensating phases are expansions through 2rdorder (astigmatism or J=6), 5th-order (J=20), and a high-order case (J=80). The no-correction case (J=0) is also considered. The no-scintillation situation (dashed line), where irradiance fluctuations have been canceled, helps us to clearly identify the amplitude effects on the uncertainty. Levels of turbulence are similar to those described in Fig. 1.

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Equations (11)

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Φ ( v ) = ∑ j = 1 ∞ a j Z j ( v )
Φ c ( v ) = ∑ j = 1 J a j Z j ( v )
Φ J ( v ) = Φ ( v ) − Φ c ( v ) = ∑ j = J ∞ a j Z j ( v )
∂ ∂ a j ∫ W ( v ) [ Φ ( v ) − ∑ j = 1 J a j Z j ( v ) ] 2 d v = 0
∫ W ( v ) Z i ( v ) Z j ( v ) d v = δ ij
a j = ∫ W ( v ) Z j ( v ) Φ ( v ) d v ∫ W ( v ) Z j 2 ( v ) d v
P = 〈 [ ∫ W ( v ) U S ( v ) U LO * ( v ) d v ] 2 〉
P = ∬ W ( v 1 ) W ( v 2 ) M S ( v 1 , v 2 ) M LO * ( v 1 , v 2 ) d v 1 d v 2
M S ( v 1 , v 2 ) = 〈 U s ( v 1 ) U S * ( v 2 ) 〉
M LO ( v 1 , v 2 ) = U LO ( v 1 ) U LO * ( v 2 ) .
U S ( v ) = A S ( v ) exp [ − j Φ ( v ) ] exp [ j Φ c ( v ) ] = A S ( v ) exp [ − j Φ J ( v ) ]

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