Abstract

The significance of tilt anisoplanatism is established, and a measurement theory, based on aperture-averaging intensity scintillation, is developed. The theory is a direct extention of the technique currently used to determine the isoplanatic angle as defined by Fried [ J. Opt. Soc. am. 72, 52 ( 1982)]. By using this theory, a physically realizable binary aperture-weighting function is derived for a particular case of interest. It is noted that direct quantitative measurements of tilt anisoplanatism can also be made, under specific circumstances, by tracking the relative centroid motion of a binary star pair. Thus independent verification of the remote-sensing theory for tilt anisoplanatism, based on aperture-averaging scintillation measurements, should be possible.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. L. Fried, “Varieties of Isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 20–29 (1976).
    [Crossref]
  2. S. Pollaine, A. Buffington, F. S. Crawford, “Measurement of the size of the isoplanatic patch using a phase-correcting telescope,” J. Opt. Soc. Am. 69, 84–89 (1979).
    [Crossref]
  3. D. P. Karo, A. M. Schneiderman, “Laboratory simulation of stellar speckle interferometry,” Appl. Opt. 18, 828–833 (1979).
    [Crossref] [PubMed]
  4. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [Crossref]
  5. J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65–68 (1975).
    [Crossref]
  6. R. K. Buchheim, R. Pringle, H. W. Schaefgen, “Lead angle effects in turbulence compensating reciprocity tracking systems,” Appl. Opt. 17, 165–166 (1978).
    [Crossref] [PubMed]
  7. K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communications,” Appl. Opt. 25, 1817–1825 (1986).
    [Crossref] [PubMed]
  8. D. P. Greenwood, “Tracking turbulence-induced tilt errors with shared and adjacent apertures,” J. Opt. Soc. Am, 67, 282–290 (1977).
    [Crossref]
  9. D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
    [Crossref]
  10. D. L. Fried, “Statistics of a geometric representation of a wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [Crossref]
  11. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [Crossref]
  12. B. L. Ellerbroek, P. H. Roberts, “Turbulence-induced angular separation measurement errors: expected values for the SOR-2 experiment,” Rep. No. TR-613 (Optical Sciences Company, Placentia, Calif., December1984).
  13. A. Peskoff, “Theory of remote sensing of clear-air turbulence profiles,” J. Opt. Soc. Am. 58, 1032–1040 (1968).
    [Crossref]
  14. R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
    [Crossref]
  15. D. L. Fried, “Remote probing of the optical strength of atmospheric turbulence and of wind velocity,” Proc. IEEE 57, 415–420 (1969).
    [Crossref]
  16. R. W. Lee, “Remote probing using spatially filtered apertures,” J. Opt. Soc. Am. 64, 1295–1303 (1974).
    [Crossref]
  17. A. Rocca, F. Roddier, J. Vernin, “Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation,” J. Opt. Soc. Am. 64, 1000–1004 (1974).
    [Crossref]
  18. G. R. Ochs, T. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-turbulence profiles measured by one-dimensional spatial filtering of scintillations,” Appl. Opt. 15, 2504–2510 (1976).
    [Crossref] [PubMed]
  19. G. C. Loos, C. B. Hogge, “Turbulence of the upper atmosphere and isoplanatism,” Appl. Opt. 18, 2654–2661 (1979).
    [Crossref] [PubMed]
  20. S. F. Clifford, J. H. Churnside, “Refractive turbulence profiling using synthetic aperture spatial filtering of scintillation,” Appl. Opt. 26, 1295–1303 (1987).
    [Crossref] [PubMed]
  21. J. L. Cacia, M. Azouit, J. Vernin, “Wind and CN2 profiling by single-star scintillation analysis,” Appl. Opt. 26, 1288–1294 (1987).
    [Crossref]
  22. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
    [Crossref]
  23. D. L. Walters, “Saturation and zenith angle dependence of atmospheric isoplanatic angle measurements,” in Effects of the Environment on Systems Performance, R. B. Gomez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.547, 38–41 (1985).
    [Crossref]
  24. K. B. Stevens, “Remote measurement of the atmospheric isoplanatic angle and determination of refractive turbulence profiles by direct inversion of the scintillation amplitude covariance function with Tikhonov regularization,” Ph.D. dissertation (Naval Postgraduate School, Monterey, Calif., December1985).
  25. F. D. Eaton, W. A. Peterson, J. R. Hines, G. Fernandez, “Isoplanatic angle direct measurements and associated atmospheric conditions,” Appl. Opt. 24, 3264–3273 (1985).
    [Crossref] [PubMed]
  26. H. T. Yura, M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
    [Crossref]
  27. J. H. Churnside, M. T. Tavis, H. T. Yura, “Zernike-polynomial expansion of turbulence-induced anisoplanatism,” Opt. Lett. 10, 258–260 (1985).
    [Crossref] [PubMed]
  28. M. T. Tavis, H. T. Yura, “Scintillation effects on centroid anisoplanatism,” J. Opt. Soc. Am. A 4, 57–59 (1987).
    [Crossref]
  29. J. L. Bufton, “Comparison of vertical turbulence structure with stellar observations,” Appl. Opt. 12, 1785–1793 (1973).
    [Crossref] [PubMed]
  30. R. Barletti, G. Ceppatelli, “Mean vertical profile of atmospheric turbulence relevant for astronomical seeing,” J. Opt. Soc. Am. 66, 1380–1383 (1976).
    [Crossref]
  31. D. L. Walters, K. E. Kunkel, “Atmospheric modulation transfer function for desert and mountain locations: the atmospheric effects on r0,” J. Opt. Soc. Am. 71, 397–405 (1981).
    [Crossref]
  32. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1545 (1970).
    [Crossref]
  33. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.

1987 (3)

1986 (1)

1985 (3)

1982 (1)

1981 (1)

1979 (3)

1978 (1)

1977 (1)

D. P. Greenwood, “Tracking turbulence-induced tilt errors with shared and adjacent apertures,” J. Opt. Soc. Am, 67, 282–290 (1977).
[Crossref]

1976 (2)

1975 (2)

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[Crossref]

J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65–68 (1975).
[Crossref]

1974 (2)

1973 (1)

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]

1969 (2)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

D. L. Fried, “Remote probing of the optical strength of atmospheric turbulence and of wind velocity,” Proc. IEEE 57, 415–420 (1969).
[Crossref]

1968 (1)

1967 (1)

1966 (1)

1965 (1)

Azouit, M.

Barletti, R.

Buchheim, R. K.

Buffington, A.

Bufton, J. L.

Cacia, J. L.

Ceppatelli, G.

Churnside, J. H.

Clifford, S. F.

Crawford, F. S.

Eaton, F. D.

Ellerbroek, B. L.

B. L. Ellerbroek, P. H. Roberts, “Turbulence-induced angular separation measurement errors: expected values for the SOR-2 experiment,” Rep. No. TR-613 (Optical Sciences Company, Placentia, Calif., December1984).

Fernandez, G.

Fried, D. L.

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

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[Crossref]

D. L. Fried, “Remote probing of the optical strength of atmospheric turbulence and of wind velocity,” Proc. IEEE 57, 415–420 (1969).
[Crossref]

D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
[Crossref]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[Crossref]

D. L. Fried, “Statistics of a geometric representation of a wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[Crossref]

D. L. Fried, “Varieties of Isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 20–29 (1976).
[Crossref]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.

Greenwood, D. P.

D. P. Greenwood, “Tracking turbulence-induced tilt errors with shared and adjacent apertures,” J. Opt. Soc. Am, 67, 282–290 (1977).
[Crossref]

Harp, J. C.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

Hines, J. R.

Hogge, C. B.

Karo, D. P.

Kunkel, K. E.

Lawrence, R. S.

G. R. Ochs, T. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-turbulence profiles measured by one-dimensional spatial filtering of scintillations,” Appl. Opt. 15, 2504–2510 (1976).
[Crossref] [PubMed]

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

Lee, R. W.

R. W. Lee, “Remote probing using spatially filtered apertures,” J. Opt. Soc. Am. 64, 1295–1303 (1974).
[Crossref]

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

Loos, G. C.

Ochs, G. R.

Peskoff, A.

Peterson, W. A.

Pollaine, S.

Pringle, R.

Roberts, P. H.

B. L. Ellerbroek, P. H. Roberts, “Turbulence-induced angular separation measurement errors: expected values for the SOR-2 experiment,” Rep. No. TR-613 (Optical Sciences Company, Placentia, Calif., December1984).

Rocca, A.

Roddier, F.

Schaefgen, H. W.

Schneiderman, A. M.

Shapiro, J. H.

Stevens, K. B.

K. B. Stevens, “Remote measurement of the atmospheric isoplanatic angle and determination of refractive turbulence profiles by direct inversion of the scintillation amplitude covariance function with Tikhonov regularization,” Ph.D. dissertation (Naval Postgraduate School, Monterey, Calif., December1985).

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.

Vernin, J.

Walters, D. L.

D. L. Walters, K. E. Kunkel, “Atmospheric modulation transfer function for desert and mountain locations: the atmospheric effects on r0,” J. Opt. Soc. Am. 71, 397–405 (1981).
[Crossref]

D. L. Walters, “Saturation and zenith angle dependence of atmospheric isoplanatic angle measurements,” in Effects of the Environment on Systems Performance, R. B. Gomez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.547, 38–41 (1985).
[Crossref]

Wang, T.

Winick, K. A.

Yura, H. T.

Appl. Opt. (9)

D. P. Karo, A. M. Schneiderman, “Laboratory simulation of stellar speckle interferometry,” Appl. Opt. 18, 828–833 (1979).
[Crossref] [PubMed]

R. K. Buchheim, R. Pringle, H. W. Schaefgen, “Lead angle effects in turbulence compensating reciprocity tracking systems,” Appl. Opt. 17, 165–166 (1978).
[Crossref] [PubMed]

K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communications,” Appl. Opt. 25, 1817–1825 (1986).
[Crossref] [PubMed]

G. R. Ochs, T. Wang, R. S. Lawrence, S. F. Clifford, “Refractive-turbulence profiles measured by one-dimensional spatial filtering of scintillations,” Appl. Opt. 15, 2504–2510 (1976).
[Crossref] [PubMed]

G. C. Loos, C. B. Hogge, “Turbulence of the upper atmosphere and isoplanatism,” Appl. Opt. 18, 2654–2661 (1979).
[Crossref] [PubMed]

S. F. Clifford, J. H. Churnside, “Refractive turbulence profiling using synthetic aperture spatial filtering of scintillation,” Appl. Opt. 26, 1295–1303 (1987).
[Crossref] [PubMed]

J. L. Cacia, M. Azouit, J. Vernin, “Wind and CN2 profiling by single-star scintillation analysis,” Appl. Opt. 26, 1288–1294 (1987).
[Crossref]

F. D. Eaton, W. A. Peterson, J. R. Hines, G. Fernandez, “Isoplanatic angle direct measurements and associated atmospheric conditions,” Appl. Opt. 24, 3264–3273 (1985).
[Crossref] [PubMed]

J. L. Bufton, “Comparison of vertical turbulence structure with stellar observations,” Appl. Opt. 12, 1785–1793 (1973).
[Crossref] [PubMed]

J. Opt. Soc. Am (1)

D. P. Greenwood, “Tracking turbulence-induced tilt errors with shared and adjacent apertures,” J. Opt. Soc. Am, 67, 282–290 (1977).
[Crossref]

J. Opt. Soc. Am. (11)

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

J. H. Shapiro, “Point-ahead limitation on reciprocity tracking,” J. Opt. Soc. Am. 65, 65–68 (1975).
[Crossref]

S. Pollaine, A. Buffington, F. S. Crawford, “Measurement of the size of the isoplanatic patch using a phase-correcting telescope,” J. Opt. Soc. Am. 69, 84–89 (1979).
[Crossref]

D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57, 169–175 (1967).
[Crossref]

D. L. Fried, “Statistics of a geometric representation of a wave-front distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
[Crossref]

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[Crossref]

A. Peskoff, “Theory of remote sensing of clear-air turbulence profiles,” J. Opt. Soc. Am. 58, 1032–1040 (1968).
[Crossref]

R. Barletti, G. Ceppatelli, “Mean vertical profile of atmospheric turbulence relevant for astronomical seeing,” J. Opt. Soc. Am. 66, 1380–1383 (1976).
[Crossref]

D. L. Walters, K. E. Kunkel, “Atmospheric modulation transfer function for desert and mountain locations: the atmospheric effects on r0,” J. Opt. Soc. Am. 71, 397–405 (1981).
[Crossref]

R. W. Lee, “Remote probing using spatially filtered apertures,” J. Opt. Soc. Am. 64, 1295–1303 (1974).
[Crossref]

A. Rocca, F. Roddier, J. Vernin, “Detection of atmospheric turbulent layers by spatiotemporal and spatioangular correlation measurements of stellar-light scintillation,” J. Opt. Soc. Am. 64, 1000–1004 (1974).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Proc. IEEE (3)

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

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote probing,” Proc. IEEE 57, 375–406 (1969).
[Crossref]

D. L. Fried, “Remote probing of the optical strength of atmospheric turbulence and of wind velocity,” Proc. IEEE 57, 415–420 (1969).
[Crossref]

Radio Sci. (1)

D. L. Fried, “Differential angle of arrival: theory, evaluation, and measurement feasibility,” Radio Sci. 10, 71–76 (1975).
[Crossref]

Other (5)

D. L. Fried, “Varieties of Isoplanatism,” in Imaging through the Atmosphere, J. C. Wyant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.75, 20–29 (1976).
[Crossref]

B. L. Ellerbroek, P. H. Roberts, “Turbulence-induced angular separation measurement errors: expected values for the SOR-2 experiment,” Rep. No. TR-613 (Optical Sciences Company, Placentia, Calif., December1984).

D. L. Walters, “Saturation and zenith angle dependence of atmospheric isoplanatic angle measurements,” in Effects of the Environment on Systems Performance, R. B. Gomez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.547, 38–41 (1985).
[Crossref]

K. B. Stevens, “Remote measurement of the atmospheric isoplanatic angle and determination of refractive turbulence profiles by direct inversion of the scintillation amplitude covariance function with Tikhonov regularization,” Ph.D. dissertation (Naval Postgraduate School, Monterey, Calif., December1985).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 389.

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 (5)

Fig. 1
Fig. 1

Anisoplanatic tilt weighting function.

Fig. 2
Fig. 2

Anisoplanatic tilt weighting function versus altitude.

Fig. 3
Fig. 3

Approximate anisoplanatic tilt weighting function.

Fig. 4
Fig. 4

Anisoplanatic tilt angle contribution versus altitude.

Fig. 5
Fig. 5

Error in tilt weighting function approximation.

Tables (1)

Tables Icon

Table 1 Measured Atmospheric Cn2(h) Profilea

Equations (66)

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

2 π λ ( a x i x + a y i y ) ,
P 0 ( r ) = { 1 , | r | 0.5 D 0 , | r | > 0.5 D ,
a x i = λ 2 π ( π D 4 64 ) 1 x P 0 ( r ) ϕ i ( r ) d r ,
a y i = λ 2 π ( π D 4 64 ) 1 y P 0 ( r ) ϕ i ( r ) d r ,
θ 0 2 = ( a x 1 a x 2 ) 2 + ( a y 1 a y 2 ) 2 ,
θ 0 2 = 2.91 ( 64 π ) 2 D 1 / 3 0 C n 2 ( z ) W ( v + 0.5 u ) × W ( v 0.5 u ) ( υ 2 0.25 u 2 ) [ 0.5 | u + z δ α D | 5 / 3 + 0.5 | u z δ α D | 5 / 3 u 5 / 3 ( z δ α D ) 5 / 3 ] d v d u d z ,
u = | u | , υ = | v | , δ α = | δ α | , W ( r ) = { 1 , | r | 0.5 0 , | r | > 0.5 ,
W ( v + 0.5 u ) W ( v 0.5 u ) ( υ 2 0.25 u 2 ) d v = 1 16 [ cos 1 ( u ) ( 3 u 2 u 3 ) ( 1 u 2 ) 1 / 2 ] .
θ 0 2 = 2.91 ( 16 π ) 2 D 1 / 3 0 C n 2 ( z ) f α ( z ) d z ,
f α ( z ) = 0 2 π 0 1 u [ cos 1 ( u ) ( 3 u 2 u 3 ) ( 1 u 2 ) 1 / 2 ] × { 0.5 [ u 2 + 2 u s cos ( ω ) + s 2 ] 5 / 6 + 0.5 [ u 2 2 u s cos ( ω ) + s 2 ] 5 / 6 u 5 / 3 s 5 / 3 } d u d ω ,
s = z δ α / D .
α = 0.5 ( α 1 + α 2 ) .
θ 0 2 = 2.91 ( 16 π ) 2 D 1 / 3 sec ( α ) 0 C n 2 ( h ) f α ( h ) d h ,
h = z cos ( α ) .
S ( t ) = P ( x ) I ( x , t ) d x .
σ S 2 = [ S ( t ) S ( t ) ] 2 [ S ( t ) ] 2 ,
σ S 2 = 4 ( 2 π ) 4 0.033 ( 2 π λ ) 2 A 2 sec ( β ) 0 C n 2 ( h ) G β ( h ) d h ,
A = P ( x ) d x ,
N ( L ) = | 0 ρ J 0 ( L ρ ) P ( ρ ) d ρ | 2 ,
G β ( h ) = 0 N ( L ) L 8 / 3 sin 2 [ L 2 h sec ( β ) λ 4 π ] d L .
G β ( h ) c f α ( h ) ,
average difference = 0 20 km [ G β ( h ) c f α ( h ) ] d h ,
θ 0 = 0.0964 c 1 / 2 A D 1 / 6 λ ( sec α sec β ) 1 / 2 σ S .
P [ ρ cos 1 / 2 ( β ) / cos 1 / 2 ( β ) ] .
c f α ( h ) = G β ( h ) = 0 | 0 ρ J 0 ( L ρ ) P ( ρ ) d ρ | 2 × L 8 / 3 sin 2 [ L 2 h sec ( β ) λ 4 π ] d L .
P ( ρ ) = { 1 , | ρ | R 0 = 2.35 c m 0 , | ρ | > R 0 = 2.35 c m .
f α ( h ) G β ( h ) = 0 R 0 4 J 1 2 ( L R 0 ) ( L R 0 ) 2 L 8 / 3 sin 2 [ L 2 h sec ( β ) λ 4 π ] d L .
percent error = f α ( h ) c 1 G β ( h ) f α ( h ) ,
W ( v + 0.5 u ) W ( v 0.5 u ) ( υ 2 0.25 u 2 ) d v = g ( u ) + s ( u ) ,
g ( u ) = W ( v + 0.5 u ) W ( v 0.5 u ) υ 2 d v ,
s ( u ) = W ( v + 0.5 u ) W ( v 0.5 u ) ( 0.25 u 2 ) d v .
g ( u ) = 4 0 ( 1 u ) / 2 d x 0 [ 0.25 ( 0.5 u + x ) 2 ] 1 / 2 ( x 2 + y 2 ) d y
= 4 0 ( 1 u ) / 2 { x 2 [ 0.25 ( 0.5 u + x ) 2 ] 1 / 2 + 1 3 [ 0.25 ( 0.5 u + x ) 2 ] 1.5 } d x .
x = 0.5 ( s u ) ,
g ( u ) = 0.25 u 1 ( s 2 2 u s + u 2 ) ( 1 s 2 ) 1 / 2 d s + 1 12 u 1 ( 1 s 2 ) 3 / 2 d s .
u 1 ( 1 s 2 ) 1 / 2 d s = 0.5 u ( 1 u 2 ) 1 / 2 + 0.5 cos 1 u ,
u 1 s ( 1 s 2 ) 1 / 2 d s = 1 3 ( 1 u 2 ) 3 / 2 ,
u 1 s 2 ( 1 s 2 ) 1 / 2 d s = 0.25 u ( 1 u 2 ) 3 / 2 + 0.125 cos 1 u 0.125 u ( 1 u 2 ) 1 / 2 ,
u 1 ( 1 s 2 ) 3 / 2 d s = 3 8 cos 1 u 5 8 u ( 1 u 2 ) 1 / 2 + 1 4 u 3 ( 1 u 2 ) 1 / 2 .
s ( u ) = 0.25 u 2 W ( v + 0.5 u ) W ( v 0.5 u ) d v .
s ( u ) = 0.25 u 2 [ 0.5 cos 1 u 0.5 u ( 1 u 2 ) 1 / 2 ] .
W ( v + 0.5 u ) W ( v 0.5 u ) ( υ 2 0.25 u 2 ) d v = 1 16 [ cos 1 u ( 3 u 2 u 3 ) ( 1 u 2 ) 1 / 2 ] .
I ( x ) = I 0 exp [ 2 χ ( x ) ] .
C χ ( ρ ) = [ χ ( x ) χ ] [ χ ( x ) χ ] ,
ρ = | ρ | = | x x | .
C I ( ρ ) = [ I ( x ) I 0 ] [ I ( x ) I 0 ] .
C I ( ρ ) = I 0 2 { exp [ 4 C χ ( ρ ) ] 1 } .
C I ( ρ ) I 0 2 4 C χ ( ρ ) .
C χ ( ρ ) = 4 π 2 k 2 0 C n 2 ( z ) 0 L J 0 ( L ρ ) Φ n 0 ( L ) sin 2 ( L 2 z 2 k ) d L d z ,
Φ n 0 ( L ) = 0.033 L 11 / 3 .
S = P ( x ) I ( x ) d x ,
σ S 2 = ( S S ) 2 / S 2 .
σ S 2 { P ( x ) [ I ( x ) I 0 ] d x } { P ( x ) [ I ( x ) I 0 ] d x } / S 2 .
σ S 2 4 A 2 P ( x ) P ( x ) C χ ( ρ ) d x d x ,
A = P ( x ) d x .
ρ = x x
ρ = 0.5 ( x + x )
σ S 2 = 4 A 2 P ( ρ + 0.5 ρ ) P ( ρ 0.5 ρ ) d ρ C χ ( ρ ) d ρ .
M ( ρ ) = M ( ρ ) = P ( ρ + 0.5 ρ ) P ( ρ 0.5 ρ ) d ρ .
σ S 2 = 4 A 2 M ( ρ ) C χ ( ρ ) d ρ = 4 A 2 0 2 π ρ M ( ρ ) C χ ( ρ ) d ρ .
σ S 2 = 4 ( 2 π ) 2 0.033 A 2 k 2 0 C n 2 ( z ) × 0 [ 0 2 π ρ J 0 ( L ρ M ( ρ ) d ρ ] L 8 / 3 sin 2 ( L 2 z 2 k ) d L d z .
4 π 2 N ( L ) = 2 π 0 ρ J 0 ( L ρ ) M ( ρ ) d ρ .
N ( L ) = | 0 ρ J 0 ( L ρ ) P ( ρ ) d ρ | 2 .
σ S 2 = 4 ( 2 π ) 4 0.033 k 2 A 2 0 C n 2 ( z ) 0 N ( L ) L 8 / 3 × sin 2 ( L 2 z 2 k ) d L d z .
z = h sec β
σ S 2 = 4 ( 2 π ) 4 0.033 ( 2 π λ ) 2 A 2 sec ( β ) 0 C n 2 ( h ) 0 N ( L ) L 8 / 3 × sin 2 [ L 2 h sec ( β ) λ 4 π ] d L d h .

Metrics