Abstract

For atmospheric turbulence-induced wave-front distortion, the mean-square change in the measured wave-front distortion for two measurements made at times separated by an interval Δt is shown to be equal to (Δt/τG)5/3, where τG is called the Greenwood time constant. It is shown that τG is related to the Greenwood frequency fG by the equation τGfG = 0.134.

© 1990 Optical Society of America

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References

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  1. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–392 (1977).
    [Crossref]
  2. D. L. Fried, “Spectral and angular covariance of scintillation for propagation in a randomly inhomogeneous medium,” Appl. Opt. 10, 721–731 (1971), appendix.
    [Crossref] [PubMed]
  3. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
    [Crossref]
  4. W. Grobnev, N. Hofreiter, Integraltafel Zweiter Teil Bestimmte Integrate (Springer-Verlag, Vienna, 1961), p. 120, Eq. (19c).

1977 (1)

1976 (1)

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

1971 (1)

Fried, D. L.

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

D. L. Fried, “Spectral and angular covariance of scintillation for propagation in a randomly inhomogeneous medium,” Appl. Opt. 10, 721–731 (1971), appendix.
[Crossref] [PubMed]

Greenwood, D. P.

D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–392 (1977).
[Crossref]

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

Grobnev, W.

W. Grobnev, N. Hofreiter, Integraltafel Zweiter Teil Bestimmte Integrate (Springer-Verlag, Vienna, 1961), p. 120, Eq. (19c).

Hofreiter, N.

W. Grobnev, N. Hofreiter, Integraltafel Zweiter Teil Bestimmte Integrate (Springer-Verlag, Vienna, 1961), p. 120, Eq. (19c).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front-compensation systems,” J. Opt. Soc. Am. 66, 193–206 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Other (1)

W. Grobnev, N. Hofreiter, Integraltafel Zweiter Teil Bestimmte Integrate (Springer-Verlag, Vienna, 1961), p. 120, Eq. (19c).

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Equations (21)

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σ ϕ 2 = ( f G / f 3 dB ) 5 / 3 .
σ ϕ 2 = ( Δ t / τ G ) 5 / 3 ,
τ G f G 0.134 .
φ ( f ) = d t exp ( 2 π ift ) ϕ ( t )
ϕ ( t ) = d f exp ( 2 π ift ) φ ( f ) .
C ϕ ( t ) = ϕ ( t + t ) ϕ ( t ) ,
Φ ϕ ( f ) = d t exp ( 2 π ift ) C ϕ ( t ) .
d f φ * ( f ) φ ( f ) g ( f ) = Φ ϕ ( f ) g ( f ) ,
Φ ϕ ( f ) 0.0163 k 2 | f | 8 / 3 path d z C N 2 ( z ) [ υ ( z ) ] 5 / 3 ,
f G = { 0.1024 k 2 path d z C N 2 ( z ) [ υ ( z ) ] 5 / 3 } 3 / 5 .
Φ ϕ ( f ) ( 2 π ) 1 f G 5 / 3 | f | 8 / 3 ,
σ ϕ 2 = | ϕ ( t + Δ t ) ϕ ( t ) | 2 .
σ ϕ 2 = | d f exp [ 2 π i f ( t + Δ t ] φ ( f ) d f exp ( 2 π ift ) φ ( f ) | 2 = | d f exp ( 2 π ift ) φ ( f ) [ exp ( 2 π i f Δ t ) 1 ] | 2 = d f exp ( 2 π i f t ) φ * ( f ) [ exp ( 2 π i f Δ t ) 1 ] × d f exp ( 2 π ift ) φ ( f ) [ exp ( 2 π i f Δ t ) 1 ] = d f d f φ * ( f ) φ ( f ) { exp [ 2 π i ( f f ) t ] × [ exp ( 2 π i f Δ t ) 1 ] [ exp ( 2 π i f Δ t ) 1 ] } = d f Φ ϕ ( f ) { [ exp ( 2 π i f Δ t ) 1 ] [ exp ( 2 π i f Δ t ) 1 ] } .
[ exp ( i x ) 1 ] [ exp ( i x ) 1 ] = 1 exp ( x ) exp ( x ) + 1 = 2 [ 1 cos ( x ) ] = 4 sin 2 ( ½ x )
σ ϕ 2 = 2 π f G 5 / 3 d f | f | 8 / 3 sin 2 ( π f Δ t ) .
x = π f Δ t ,
σ ϕ 2 = 2 π f G 5 / 3 ( π Δ t ) 5 / 3 d x | x | 8 / 3 sin 2 ( x ) = 2 π 2 / 3 ( f G Δ t ) 5 / 3 d x | x | 8 / 3 sin 2 ( x ) .
0 d x x α sin 2 ( x ) = π 2 3 α Γ ( α ) cos ( ½ π α ) if 1 < α < 3
d x | x | 8 / 3 sin 2 ( x ) = 2 [ π 2 1 / 3 Γ ( 8 / 3 ) cos ( 4 π / 3 ) ] = 6.629 .
σ ϕ 2 = 2 π 2 / 3 ( f G Δ t ) 5 / 3 ( 6.629 ) = 28.44 ( f G Δ t ) 5 / 3 .
σ ϕ 2 = ( 7.45 f G Δ t ) 5 / 3 ,

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