Abstract

As a finite cross-section laser beam propagates through the atmosphere, the beam spreads due to both diffraction and atmospheric turbulence effects. Using turbulence theory valid in both weak and strong optical turbulence regimes, a relationship between atmospheric beam spread and the resulting return power for an optical system and the refractive-index structure parameter or Cn2 can be established. A technique for estimating the path-averaged Cn2 using a laser-and-corner-cube system based on this relationship is described. Experimental results using near-infrared laser wavelengths show good agreement between theoretical predictions and scintillometer-measured Cn2 values for near-ground line-of-sight propagation paths.

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2009 (1)

2008 (4)

Y. Jiang, J. Ma, L. Tan, S. Yu, and W. Du, “Measurement of optical intensity fluctuation over an 11.8 km turbulent path,” Opt. Express 16, 6963-6973 (2008).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng 47, 026003(2008).
[CrossRef]

W. P. Cole, M. A. Marciniak, and M. B. Haeri, “Atmospheric-turbulence-effects correction factors for the laser range equation,” Opt. Eng. 47, 126001 (2008).
[CrossRef]

A. Tunick, “Optical turbulence parameters characterized via optical measurements over a 2.33 km free-space laser path,” Opt. Express 16, 14645-14654 (2008).
[CrossRef]

2007 (4)

2006 (4)

F. S. Vetelino, B. Clare, K. Corbett, C. Young, K. Grant, and L. Andrews, “Characterizing the propagation path in moderate to strong optical turbulence,” Appl. Opt. 45, 3534-3543 (2006).
[CrossRef]

A. MacDonald, S. C. Cain, and E. E. Armstrong, “Maximum a posteriori image and seeing condition estimation from partially coherent two-dimensional light detection and ranging images,” Opt. Eng. 45, 086201 (2006).
[CrossRef]

S. Zamek and Y. Yitzhaky, “Turbulence strength estimation from an arbitrary set of atmospherically degraded images,” J. Opt. Soc. Am. A 23, 3106-3113 (2006).
[CrossRef]

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

2005 (2)

2004 (2)

F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

S. Bendersky, N. S. Kopeika, and N. Blaunstein, “Atmospheric optical turbulence over land in Middle East coastal environments: prediction modeling and measurements,” Appl. Opt. 43, 4070-4079 (2004).
[CrossRef]

2003 (2)

2002 (3)

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

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).
[CrossRef]

2000 (1)

1999 (1)

D. L. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38, 1288-1295 (1999).
[CrossRef]

1991 (3)

1988 (3)

1975 (2)

M. L. Wesely and Z. I. Derzko, “Atmospheric turbulence parameters from visual resolution,” Appl. Opt. 14, 847-853 (1975).
[CrossRef]

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669-1692 (1975).
[CrossRef]

1971 (2)

1970 (2)

R. S. Lawrence, G. R. Ochs, and S. F. Clifford, “Measurements of atmospheric turbulence relevant to optical propagation,” J. Opt. Soc. Am. 60, 826-830 (1970).
[CrossRef]

V. I. Klyatskin and V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828-833 (1970).
[CrossRef]

1968 (1)

J. W. Strohbehn, “Line of sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Andreas, E. L.

Andrews, L.

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng 47, 026003(2008).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mean irradiance and beam spread,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 237-238.

L. C. Andrews and R. L. Phillips, “Backscatter amplification factor: strong fluctuations,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 577-581.

L. C. Andrews and R. L. Phillips, “Weak and strong fluctuation conditions,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 140-141.

Armstrong, E. E.

A. MacDonald, S. C. Cain, and E. E. Armstrong, “Maximum a posteriori image and seeing condition estimation from partially coherent two-dimensional light detection and ranging images,” Opt. Eng. 45, 086201 (2006).
[CrossRef]

Bar-Sagi, Y.

Y. Glick, R. Zaibel, G. Bar-Tal, and Y. Bar-Sagi, “Atmospheric turbulence measurements by angle of arrival fluctuations and intensity scintillations with large aperture,” in 2000 Conference on Lasers and Electro-Optics Europe (CLEO 2000) (Optical Society of America, 2000), p. 244.

Bar-Tal, G.

Y. Glick, R. Zaibel, G. Bar-Tal, and Y. Bar-Sagi, “Atmospheric turbulence measurements by angle of arrival fluctuations and intensity scintillations with large aperture,” in 2000 Conference on Lasers and Electro-Optics Europe (CLEO 2000) (Optical Society of America, 2000), p. 244.

Barton, D. K.

D. K. Barton, Modern Radar System Analysis (Artech, 1988).

Bendersky, S.

Berman, G. P.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606-7 (2007).
[CrossRef]

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Bernhardt, M.

Blaunstein, N.

Brown, W. P.

Bucaille, S.

F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

Burris, H. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Cain, S. C.

A. MacDonald, S. C. Cain, and E. E. Armstrong, “Maximum a posteriori image and seeing condition estimation from partially coherent two-dimensional light detection and ranging images,” Opt. Eng. 45, 086201 (2006).
[CrossRef]

Carter, D. A.

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

Cernius, J.

C. Cooke, J. Cernius, and A. J. LaRocca, “Ranging, Communications, and Simulation Systems,” in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, eds. (IRIA Center, Environmental Research Institute of Michigan, for the Office of Naval Research, Department of the Navy, 1985), Chap. 23, p. 23-6.

Chan, K. P.

Chumak, A. A.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606-7 (2007).
[CrossRef]

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

Clare, B.

Clifford, S. F.

Cole, W. P.

W. P. Cole, M. A. Marciniak, and M. B. Haeri, “Atmospheric-turbulence-effects correction factors for the laser range equation,” Opt. Eng. 47, 126001 (2008).
[CrossRef]

Collins, S.

Conan, J.

Cooke, C.

C. Cooke, J. Cernius, and A. J. LaRocca, “Ranging, Communications, and Simulation Systems,” in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, eds. (IRIA Center, Environmental Research Institute of Michigan, for the Office of Naval Research, Department of the Navy, 1985), Chap. 23, p. 23-6.

Corbett, K.

Dainty, C.

Davidson, F.

Davidson, F. M.

F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

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).
[CrossRef]

Derzko, Z. I.

Dikmelik, Y.

Du, W.

Eaton, F. D.

F. D. Eaton, “Recent developments of optical turbulence measurement techniques,” Proc. SPIE 5793, 68-77 (2005).
[CrossRef]

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669-1692 (1975).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng 47, 026003(2008).
[CrossRef]

Frehlich, R. G.

Gage, K. S.

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

Gilbreath, G. C.

F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

Gimmestad, G. G.

G. G. Gimmestad and D. W. Roberts, “Laser remote sensing of atmospheric turbulence,” Proc. SPIE 5087, 167-172 (2003).
[CrossRef]

Glick, Y.

Y. Glick, R. Zaibel, G. Bar-Tal, and Y. Bar-Sagi, “Atmospheric turbulence measurements by angle of arrival fluctuations and intensity scintillations with large aperture,” in 2000 Conference on Lasers and Electro-Optics Europe (CLEO 2000) (Optical Society of America, 2000), p. 244.

Gorshkov, V. N.

G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606-7 (2007).
[CrossRef]

Grant, K.

Grayshan, K.

Haeri, M. B.

W. P. Cole, M. A. Marciniak, and M. B. Haeri, “Atmospheric-turbulence-effects correction factors for the laser range equation,” Opt. Eng. 47, 126001 (2008).
[CrossRef]

Hartten, L. M.

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

Hill, R. J.

Holmes, J. F.

Hutt, D. L.

D. L. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38, 1288-1295 (1999).
[CrossRef]

Izumi, Y.

Jiang, Y.

Johnston, P. E.

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

Johnston, R. A.

Killinger, D. K.

Klyatskin, V. I.

V. I. Klyatskin and V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828-833 (1970).
[CrossRef]

Kopeika, N. S.

LaRocca, A. J.

C. Cooke, J. Cernius, and A. J. LaRocca, “Ranging, Communications, and Simulation Systems,” in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, eds. (IRIA Center, Environmental Research Institute of Michigan, for the Office of Naval Research, Department of the Navy, 1985), Chap. 23, p. 23-6.

Lawrence, R. S.

Love, C. H.

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

Lukin, V. P.

V. P. Lukin, “Influence of the source spectrum on the optical measurements of turbulence,” Appl. Opt. 48, A93-A97 (2009).
[CrossRef]

V. P. Lukin, “Statistical characteristics of the phase of specularly reflected optical waves,” in Atmospheric Adaptive Optics (SPIE, 1995), pp. 69-76.

Ma, J.

MacDonald, A.

A. MacDonald, S. C. Cain, and E. E. Armstrong, “Maximum a posteriori image and seeing condition estimation from partially coherent two-dimensional light detection and ranging images,” Opt. Eng. 45, 086201 (2006).
[CrossRef]

Marciniak, M. A.

W. P. Cole, M. A. Marciniak, and M. B. Haeri, “Atmospheric-turbulence-effects correction factors for the laser range equation,” Opt. Eng. 47, 126001 (2008).
[CrossRef]

Michau, V.

Moore, C.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Ochs, G. R.

Oh, E.

F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng 47, 026003(2008).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mean irradiance and beam spread,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 237-238.

L. C. Andrews and R. L. Phillips, “Backscatter amplification factor: strong fluctuations,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 577-581.

L. C. Andrews and R. L. Phillips, “Weak and strong fluctuation conditions,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 140-141.

Reavell, F. C.

Reed, A. E.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Ricklin, J. C.

Robert, C.

Roberts, D. W.

G. G. Gimmestad and D. W. Roberts, “Laser remote sensing of atmospheric turbulence,” Proc. SPIE 5087, 167-172 (2003).
[CrossRef]

Scharpf, W. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Steinvall, O.

Strohbehn, J. W.

J. W. Strohbehn, “Line of sight wave propagation through the turbulent atmosphere,” Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Sugimoto, N.

Suite, M. R.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Tan, L.

Tatarskii, V. I.

V. I. Klyatskin and V. I. Tatarskii, “On the theory of the propagation of light beams in a medium having random inhomogeneities,” Radiophys. Quantum Electron. 13, 828-833 (1970).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng 47, 026003(2008).
[CrossRef]

Tunick, A.

Védrenne, N.

Vetelino, F. S.

Vilcheck, M. J.

M. J. Vilcheck, A. E. Reed, H. R. Burris, W. J. Scharpf, C. Moore, and M. R. Suite, “Multiple methods for measuring atmospheric turbulence,” Proc. SPIE 4821, 300-309 (2002).
[CrossRef]

Wang, J.

J. Wang, “Radar range equation for meteorological targets,” in Proceedings of 1996 Chinese Institute of Electronics (CIE) International Conference of Radar, ICR'96 (IEEE, 1996), pp. 561-565.

Wesely, M. L.

Wood, S.

V. E. Zuev, translated by S. Wood, Laser Beams in theAtmosphere (Consultants Bureau, 1982).

Wooder, N. J.

Wyngaard, J. C.

Yitzhaky, Y.

Young, C.

Young, C. Y.

Yu, S.

Zaibel, R.

Y. Glick, R. Zaibel, G. Bar-Tal, and Y. Bar-Sagi, “Atmospheric turbulence measurements by angle of arrival fluctuations and intensity scintillations with large aperture,” in 2000 Conference on Lasers and Electro-Optics Europe (CLEO 2000) (Optical Society of America, 2000), p. 244.

Zamek, S.

Zuev, V. E.

V. E. Zuev, translated by S. Wood, Laser Beams in theAtmosphere (Consultants Bureau, 1982).

Appl. Opt. (12)

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[CrossRef]

R. A. Johnston, N. J. Wooder, F. C. Reavell, M. Bernhardt, and C. Dainty, “Horizontal scintillation detection and ranging Cn2(z) estimation,” Appl. Opt. 42, 3451-3459 (2003).
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R. G. Frehlich, “Estimation of the parameters of the atmospheric turbulence spectrum using measurements of the spatial intensity covariance,” Appl. Opt. 27, 2194-2198 (1988).
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[CrossRef]

S. Bendersky, N. S. Kopeika, and N. Blaunstein, “Atmospheric optical turbulence over land in Middle East coastal environments: prediction modeling and measurements,” Appl. Opt. 43, 4070-4079 (2004).
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O. Steinvall, “Effects of target shape and reflection on laser radar cross sections,” Appl. Opt. 39, 4381-4391 (2000).
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Y. Dikmelik and F. Davidson, “Fiber-coupling efficiency for free-space optical communication through atmospheric turbulence,” Appl. Opt. 44, 4946-4952 (2005).
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J. F. Holmes, “Enhancement of backscattered intensity for a bistatic lidar operating in atmospheric turbulence,” Appl. Opt. 30, 2643-2646 (1991).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

P. E. Johnston, L. M. Hartten, C. H. Love, D. A. Carter, and K. S. Gage, “Range errors in wind profiling caused by strong reflectivity gradients,” J. Atmos. Ocean. Technol. 19, 934 (2002).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Eng (1)

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[CrossRef]

Opt. Eng. (4)

W. P. Cole, M. A. Marciniak, and M. B. Haeri, “Atmospheric-turbulence-effects correction factors for the laser range equation,” Opt. Eng. 47, 126001 (2008).
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A. MacDonald, S. C. Cain, and E. E. Armstrong, “Maximum a posteriori image and seeing condition estimation from partially coherent two-dimensional light detection and ranging images,” Opt. Eng. 45, 086201 (2006).
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F. M. Davidson, S. Bucaille, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980 nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689-2695 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

G. P. Berman and A. A. Chumak, “Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere,” Phys. Rev. A 74, 013805 (2006).
[CrossRef]

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G. P. Berman, A. A. Chumak, and V. N. Gorshkov, “Beam wandering in the atmosphere: the effect of partial coherence,” Phys. Rev. E 76, 056606-7 (2007).
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Other (10)

C. Cooke, J. Cernius, and A. J. LaRocca, “Ranging, Communications, and Simulation Systems,” in The Infrared Handbook, W. L. Wolfe and G. J. Zissis, eds. (IRIA Center, Environmental Research Institute of Michigan, for the Office of Naval Research, Department of the Navy, 1985), Chap. 23, p. 23-6.

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V. E. Zuev, translated by S. Wood, Laser Beams in theAtmosphere (Consultants Bureau, 1982).

Y. Glick, R. Zaibel, G. Bar-Tal, and Y. Bar-Sagi, “Atmospheric turbulence measurements by angle of arrival fluctuations and intensity scintillations with large aperture,” in 2000 Conference on Lasers and Electro-Optics Europe (CLEO 2000) (Optical Society of America, 2000), p. 244.

L. C. Andrews and R. L. Phillips, “Weak and strong fluctuation conditions,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 140-141.

L. C. Andrews and R. L. Phillips, “Backscatter amplification factor: strong fluctuations,” in Laser Beam Propagation through Random Media (SPIE, 2005), pp. 577-581.

V. P. Lukin, “Statistical characteristics of the phase of specularly reflected optical waves,” in Atmospheric Adaptive Optics (SPIE, 1995), pp. 69-76.

User Manual BLS-900, Scintec Boundary Layer Scintillometer, Revision 1.43 (Scintec AG, 2007), http://www.scintec.com/.

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Figures (3)

Fig. 1
Fig. 1

Normalized return power as a function of C n 2 for three different sets of corner-cube (cc) and receiver (rec) diameter sizes, where testing conditions are as shown in Table 2.

Fig. 2
Fig. 2

1 in. ( 2.5 cm ) PLX corner cube.

Fig. 3
Fig. 3

Comparison of C n 2 measurements using laser-and-corner-cube technique (experiment) and a commercial scintillometer for (a) summer and (b) winter.

Tables (3)

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Table 1 Summer 860 nm Testing Parameters

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Table 2 Winter 1064 nm Testing Parameters

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Table 3 Calculated and Measured Experimental Data Set

Equations (5)

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P r = P laser D rec 2 θ T 2 R 4 σ tar τ a 2 τ T τ rec ,
τ turb = P turb P fs = A rec E turb ( r , z = 2 R ) d A A rec E fs ( r , z = 2 R ) d A = [ 1 exp ( 2 r rec 2 W LT 2 ) ] [ 1 exp ( 2 r rec 2 W rec 2 ) ] ,
W LT W rec 1 + 1.63 ( 1.23 C n 2 k 7 / 6 z 11 / 6 ) 6 / 5 ( 2 z k W rec 2 ) ,
W rec 1.22 R λ D cc ,
P r = P laser P shape D rec 2 θ T 2 R 4 σ tar τ a 2 τ T τ rec τ turb ,

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