Abstract

The statistics of the angle-of-arrival fluctuations are studied for the case of a laser beam reflected from a curved surface in a uniformly turbulent atmosphere. The variance for the direct beam and for the reflected beam and the covariance for the two beams are calculated for arbitrary divergence of the illumination and radius of curvature of the reflector. Experimental results support the conclusions that the reflected-beam angle-of-arrival fluctuations are very sensitive to small deviations from perfect collimation and that the correlation of the fluctuations in the direct and reflected beams is high.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. L. Mironov, V. V. Nosov, “Random image displacement at a telescope focus located in a turbulent atmosphere,” Radiophys. Quantum Electron. 20, 1054–1056 (1977).
    [CrossRef]
  2. H. T. Yura, M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
    [CrossRef]
  3. V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
    [CrossRef]
  4. V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).
  5. V. A. Banakh, O. V. Tikhomirova, “Variance and spatial correlation of the intensity of laser beams reflected in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 524–528 (1984).
  6. V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).
  7. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), pp. 9–43.
    [CrossRef]
  8. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [CrossRef]
  9. G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2instrument model II,”NOAA Tech. Memo No. ERL WPL-51 (National Oceanographic and Atmospheric Administration, Washington, D.C., 1979); (National Technical Information Service, Springfield, Va.1979).
  10. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef]

1985 (1)

1984 (3)

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).

V. A. Banakh, O. V. Tikhomirova, “Variance and spatial correlation of the intensity of laser beams reflected in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 524–528 (1984).

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).

1980 (1)

V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
[CrossRef]

1977 (1)

V. L. Mironov, V. V. Nosov, “Random image displacement at a telescope focus located in a turbulent atmosphere,” Radiophys. Quantum Electron. 20, 1054–1056 (1977).
[CrossRef]

1970 (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

1966 (1)

Aksenov, V. P.

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).

Banakh, V. A.

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).

V. A. Banakh, O. V. Tikhomirova, “Variance and spatial correlation of the intensity of laser beams reflected in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 524–528 (1984).

Cartwright, W. D.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2instrument model II,”NOAA Tech. Memo No. ERL WPL-51 (National Oceanographic and Atmospheric Administration, Washington, D.C., 1979); (National Technical Information Service, Springfield, Va.1979).

Chen, B. N.

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).

Clifford, S. F.

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), pp. 9–43.
[CrossRef]

Kogelnik, H.

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Li, T.

Lukin, V. P.

V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
[CrossRef]

Mironov, V. L.

V. L. Mironov, V. V. Nosov, “Random image displacement at a telescope focus located in a turbulent atmosphere,” Radiophys. Quantum Electron. 20, 1054–1056 (1977).
[CrossRef]

Nosov, V. V.

V. L. Mironov, V. V. Nosov, “Random image displacement at a telescope focus located in a turbulent atmosphere,” Radiophys. Quantum Electron. 20, 1054–1056 (1977).
[CrossRef]

Ochs, G. R.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2instrument model II,”NOAA Tech. Memo No. ERL WPL-51 (National Oceanographic and Atmospheric Administration, Washington, D.C., 1979); (National Technical Information Service, Springfield, Va.1979).

Russell, D. D.

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2instrument model II,”NOAA Tech. Memo No. ERL WPL-51 (National Oceanographic and Atmospheric Administration, Washington, D.C., 1979); (National Technical Information Service, Springfield, Va.1979).

Sazanovich, V. M.

V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
[CrossRef]

Slobodyan, S. M.

V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
[CrossRef]

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Tavis, M. T.

Tikhomirova, O. V.

V. A. Banakh, O. V. Tikhomirova, “Variance and spatial correlation of the intensity of laser beams reflected in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 524–528 (1984).

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Spectrosc. (USSR) (3)

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Variance of displacements of the image of objects during optical location in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 529–532 (1984).

V. A. Banakh, O. V. Tikhomirova, “Variance and spatial correlation of the intensity of laser beams reflected in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 524–528 (1984).

V. P. Aksenov, V. A. Banakh, B. N. Chen, “Displacement of object images during optical location in conditions of strong intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. (USSR) 56, 445–446 (1984).

Proc. IEEE (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
[CrossRef]

Radiophys. Quantum Electron. (2)

V. P. Lukin, V. M. Sazanovich, S. M. Slobodyan, “Random image shifts during ranging in a turbulent atmosphere,” Radiophys. Quantum Electron. 23, 484–490 (1980).
[CrossRef]

V. L. Mironov, V. V. Nosov, “Random image displacement at a telescope focus located in a turbulent atmosphere,” Radiophys. Quantum Electron. 20, 1054–1056 (1977).
[CrossRef]

Other (2)

G. R. Ochs, W. D. Cartwright, D. D. Russell, “Optical Cn2instrument model II,”NOAA Tech. Memo No. ERL WPL-51 (National Oceanographic and Atmospheric Administration, Washington, D.C., 1979); (National Technical Information Service, Springfield, Va.1979).

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, 1978), pp. 9–43.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometry used for angle-of-arrival fluctuation analysis.

Fig. 2
Fig. 2

Ratio of beam-wave to plane-wave angle-of-arrival variances σt2/σp2 as a function of the path length L divided by the focal range ft.

Fig. 3
Fig. 3

Ratio of reflected- to transmitted-beam angle-of-arrival variances σr2/σt2 for a collimated transmitter as a function of the path length L divided by the reflected-beam focal range fr.

Fig. 4
Fig. 4

Ratio of reflected to transmitted-beam angle-of-arrival variances σr2/σt2 for a plane reflector as a function of the path length L divided by the transmitted-beam focal range ft.

Fig. 5
Fig. 5

Normalized covariance of transmitted- and reflected-beam angle-of-arrival fluctuations Ctr/σtσr for a collimated transmitter as a function of the path length L divided by the reflected-beam focal range fr.

Fig. 6
Fig. 6

Normalized covariance of transmitted- and reflected-beam angle-of-arrival fluctuations Ctr/σtσr for a plane reflector as a function of the path length L divided by the transmitted-beam focal range ft.

Fig. 7
Fig. 7

Transmitter optics.

Fig. 8
Fig. 8

Reflector optics.

Fig. 9
Fig. 9

Variance of direct-beam angle-of-arrival fluctuations as a function of the turbulence level Cn2. Circles represent observed values, and the line represents collimated-beam theory.

Equations (39)

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

d α = Δ n ( z ) w ( L ) d z ,
α = 1 w ( L ) 0 L Δ n ( z ) d z .
σ t 2 = 1 w 2 ( L ) 0 L 0 L d z 1 d z 2 Δ n ( z 1 ) Δ n ( z 2 ) ,
Δ n ( z ) = n [ z , w ( z ) 2 ] - n [ z , - w ( z ) 2 ] ,
Δ n ( z 1 ) Δ n ( z 2 ) = ½ { n [ z 1 , ½ w ( z 1 ) ] - n [ z 2 , - ½ w ( z 2 ) ] } 2 + { n [ z 1 , - ½ w ( z 1 ) ] - n [ z 2 , ½ w ( z 2 ) ] } 2 - { n [ z 1 , ½ w ( z 1 ) ] - n [ z 2 , ½ w ( z 2 ) ] } 2 - { n [ z 1 , - ½ w ( z 1 ) ] - n [ z 2 , - ½ w ( z 2 ) ] } 2 .
Δ n ( z 1 ) Δ n ( z 2 ) = C n 2 ( { ( z 2 - z 1 ) 2 + ¼ [ w ( z 2 ) + w ( z 1 ) ] 2 } 1 / 3 - [ ( z 2 - z 1 ) 2 + ¼ [ w ( z 2 ) - w ( z 1 ) ] 2 ] 1 / 3 ) ,
w ( z ) = D | 1 - z f t | ,
D = D t for D r D t | 1 - L f t | , D = D r | 1 - L f t | for D r < D t | 1 - L f t | ,
σ t 2 = C n 2 D - 4 / 3 ( 1 - L f t ) - 2 0 L d z 1 | 1 - z 1 f t | 2 / 3 × 0 L d z 2 { [ 1 + ( z 2 - z 1 D | 1 - z 1 f t | ) 2 ] 1 / 3 - ( z 2 - z 1 D | 1 - z 1 f t | ) 2 / 3 } .
σ t 2 = 2.92 C n 2 L D - 1 / 3 ³ / f t L 1 - ( 1 - L f t ) | 1 - L f t | 5 / 3 ( 1 - L f t ) 2 .
σ p 2 = 2.92 C n 2 L D - 1 / 3 ,
σ s 2 = ³ / ( 2.92 C n 2 L D r - 1 / 3 ) .
| 1 - L f t | D λ L ,
d α = Δ n t ( z ) + Δ n r ( z ) w r ( 0 ) w z ,
α = 1 w r ( 0 ) 0 L [ Δ n t ( z ) + Δ n r ( z ) ] d z .
σ r 2 = 1 w r 2 ( 0 ) 0 L 0 L d z 1 d z 2 [ Δ n t ( z 1 ) Δ n t ( z 2 ) + Δ n r ( z 1 ) Δ n r ( z 2 ) + 2 Δ n t ( z 1 ) Δ n r ( z 2 ) ] ,
Δ n t ( z ) = n [ z , w t ( z ) 2 ] - n [ z , - w t ( z ) 2 ]
Δ n r ( z ) = n [ z , w r ( z ) 2 ] - n [ z , - w r ( z ) 2 ] .
w t ( z ) = D | 1 - z f t | ,             w r ( z ) = D | 1 - L - z f r | ,
D = D t | 1 - L f r | ,
σ r 2 = 2.92 C n 2 L D - 1 / 3 [ 3 8 ( D D ) 2 f t L × 1 - ( 1 - L f t ) | 1 - L f t | 5 / 3 ( 1 - L f r ) 2 + 3 8 ( D D ) 1 / 3 f r L × 1 - ( 1 - L f r ) | 1 - L f r | 5 / 3 | 1 - L f r | 2 + ( D D ) 2 2 2 / 3 ( 1 - L f r ) 2 × 0 1 d u ( 1 - u L f t ) ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | ( 1 - u L f t ) ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | × ( { | 1 - u L f t | + D D | [ 1 - ( 1 - u ) L f r ] | } 5 / 3 - | | 1 - u L f t | - D D | [ 1 - ( 1 - u ) L f r ] | | 5 / 3 ) ] .
1 - u L f t > 0 for 0 u 1 , 1 - ( 1 - u ) L f r > 0 for 0 u 1 ,
D = D t , D = D t ( 1 - L f t ) .
σ r 2 = 2.92 C n 2 L D t - 1 / 3 3 8 ( 1 - L f t ) 2 ( 1 - L f r ) 2 × { f t L [ 1 - ( 1 - L f t ) 8 / 3 ] + f r L ( 1 - L f t ) 5 / 3 [ 1 - ( 1 - L f t ) 8 / 3 ] - 4 ( 1 - L f t ) 8 / 3 - 2 - 2 / 3 [ 1 + ( 1 - L f t ) ( 1 - L f r ) ] 8 / 3 1 - ( 1 - L f t ) ( 1 + L f r ) - 2 - 2 / 3 | 1 - ( 1 - L f t ) ( 1 - L f r ) | 5 / 3 } ,
f r = ± ( f t - L ) .
σ r 2 = 2.92 C n 2 L D t - 1 / 3 3 8 ( 1 - 2 L f t ) 2 { f t L [ 1 - ( 1 - 2 L f t ) 8 / 3 ] + 16 3 ( 1 - L f t ) 5 / 3 - 2 | L f t | 5 / 3 } .
| 1 - L f r | D λ L .
σ r 2 σ t 2 = 2 2 / 3 + 5 6 2.42 ,
D = D t for f t > L , D = D t 1 - L f t for f t < 0 ,
D = D t for f t < 0 ,             f r > L , D = D t ( 1 - L f t ) for f t > L ,             f r < L - f t or f t > L ,             f r > L , D = D t 1 - L f t for f t < 0 ,             f r < 0 or f t > L ,             L - f t < f r < 0 ,
σ r 2 = 2.92 C n 2 L D t - 1 / 3 { 1 + 3 8 1 - ( 1 - L f r ) 8 / 3 L f r ( 1 - L f r ) 5 / 3 + 3 8 ( 2 - L f r ) 8 / 3 - ( 2 - 2 L f r ) 8 / 3 + ( - L f r ) 8 / 3 2 2 / 3 L f r ( 1 - L f r ) 5 / 3 } .
σ r 2 = 3 8 2.92 C n 2 L ( D t 1 - L f t ) - 1 / 3 ( 1 - ( 1 - L f t ) 8 / 3 L f t ( 1 - L f t ) 2 + ( 1 - L f t ) 8 / 3 - ( 1 - 2 L f t ) 8 / 3 L f t ( 1 - L f t ) 1 / 3 ( 1 - 2 L f t ) 5 / 3 + 1 - 2 L f t 2 2 / 3 ( L f t ) 2 ( 1 - L f t ) 2 × { [ ( 2 - 3 L f t ) ( 1 - L f t ) 1 - 2 L f t ] 8 / 3 - ( 2 - L f t ) 8 / 3 } - 1 - 2 L f t 2 2 / 3 L f t ( 1 - L f t ) 2 ( 3 L f t - 2 ) × { [ L f t ( L f t - 1 ) 1 - 2 L f t ] 8 / 3 + ( - L f t ) 8 / 3 } ) .
C t r = 1 w t ( L ) w r ( 0 ) 0 L 0 L d z 1 d z 2 Δ n t ( z 1 ) [ Δ n t ( z 2 ) + Δ n r ( z 2 ) ] .
C t r = 2.92 C n 2 L D - 1 / 3 [ 3 8 D D f t L 1 - ( 1 - L f t ) | 1 - L f t | 5 / 3 | 1 - L f t | | 1 - L f r | + D D 2 5 / 3 | 1 - L f t | 2     | 1 - L f r | × 0 1 du ( 1 - u L f t ) ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | ( 1 - u L f t ) ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | × ( { | 1 - u L f t | + | ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | } 5 / 3 - | | 1 - u L f t | - | ( 1 - L f t ) [ 1 - ( 1 - u ) L f r ] | | 5 / 3 ) ] .
C t r = 2.92 C n 2 L D t - 1 / 3 3 8 ( 1 - L f t ) 2 ( 1 - L f r ) × { f t L [ 1 - ( 1 - L f t ) 8 / 3 ] - 2 ( 1 - L f t ) 8 / 3 - 2 - 5 / 3 [ 1 + ( 1 - L f t ) ( 1 - L f r ) ] 8 / 3 1 - ( 1 - L f t ) ( 1 + L f r ) - 2 - 5 / 3 | 1 - ( 1 - L f t ) ( 1 - L f r ) | 5 / 3 } ,
f r = ± ( f t - L ) .
C t r = 2.92 C n 2 L D - 1 / 3 3 8 ( 1 - L f t ) ( 1 - 2 L f t ) × { f t L [ 1 - ( 1 - L f t ) 8 / 3 ] + 8 3 ( 1 - L f t ) 5 / 3 - | L f t | 5 / 3 }
C t r = 2.92 C n 2 L D t - 1 / 3 × [ 1 + 3 8 ( 2 - L f r ) 8 / 3 - ( 2 - 2 L f r ) 8 / 3 + ( - L f r ) 8 / 3 2 5 / 3 L f r ( 1 - L f r ) 5 / 3 ] .
C t r = 3 8 2.92 C n 2 L ( D t 1 - L f t ) - 1 / 3 ( 1 - ( 1 - L f t ) 8 / 3 L f t ( 1 - L f t ) 2 + 1 - 2 L f t 2 5 / 3 ( L f t ) 2 ( 1 - L f t ) 2 × { [ ( 2 - 3 L f t ) ( 1 - L f t ) 1 - 2 L f t ] 8 / 3 - ( 2 - L f t ) 8 / 3 } - 1 - 2 L f t 2 5 / 3 L f t ( 1 - L f t ) 2 ( 3 L f t - 2 ) × { [ L f t ( L f t - 1 ) 1 - 2 L f t ] 8 / 3 + ( - L f t ) 8 / 3 } ) .

Metrics