Abstract

The bandwidth requirements for tracking through turbulence have been studied for the case in which a closed-loop transfer function of the form H(f) = (1 + if/f3dB)−1 is used. The results illustrate that the one-axis one-sigma (rms) jitter σθ is given by the expression σθ = (fT/f3dB)(λ/D), where λ is the wavelength of light used, D is the diameter of the tracking aperture, and fT is the fundamental turbulence-tracking frequency that is determined so that, when the servo bandwidth f3dB is equal to fT, σθ is equal to the diffraction angle λ/D. In practice a tracking system may measure one of various kinds of tilt. As a consequence both G tilt (obtained from a centroid measurement) and Z tilt (the direction that is defined by the normal to the plane that minimizes the mean-square wave-front distortion) have been evaluated in detail. The fundamental turbulence-tracking frequencies fTG and fTZ corresponding to the G tilt and the Z tilt, respectively, are found to be almost identical and are given by the expressions fTG = 0.331D−1/6λ−1[∫ dzCn2(z)V2(z)]1/2 and fTz = 0.368D−1/6λ−1[∫ dzCn2(z)V2(z)]1/2, where z is the range coordinate, Cn2() is the refractive-index structure function, and V() is the wind-velocity profile. For turbulence models that are applied to systems of interest the fundamental turbulence-tracking frequency that is defined by these expressions is about one ninth of fG, the Greenwood frequency associated with higher-order wave-front distortion. This illustrates the important point that the bandwidth that is necessary to control the turbulence-induced tilt is significantly less than the bandwidth that is necessary to control the turbulence-induced higher-order wave-front distortion.

© 1994 Optical Society of America

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References

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  1. D. P. Greenwood, D. L. Fried, “Power spectra requirements for wave-front compensation systems,”J. Opt. Soc. Am. 66, 193–206 (1976).
    [CrossRef]
  2. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,”J. Opt. Soc. Am. 67, 390–392 (1977).
    [CrossRef]
  3. G. A. Tyler, “Turbulence-induced adaptive-optics performance degradation: evaluation in the time domain,” J. Opt. Soc. Am. A 1, 251–262 (1984).
    [CrossRef]
  4. At this point it is not obvious that σθis inversely proportional to f3dB. However, the research that is presented in Sections 3 and 4 will demonstrate that this is true.
  5. D. L. Fried, “Diffusion analysis for the propagation of mutual coherence,”J. Opt. Soc. Am. 58, 961–969 (1968).
    [CrossRef]
  6. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (11.4.24).
  7. Ref. 6, Eq. (22.3.15).
  8. Ref. 6, Eq. (6.2.1).
  9. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, Orlando, Fla., 1980), p. 692, Eq. 6.574.2.
  10. The HV5/7turbulence model was developed by the U.S. Air Force by modifying a conventional Hufnagel–Valley turbulence model so that at zenith r0= 0.05 m and ϑ0= 7 × 10−6rad when the wavelength that is being used is 0.5 μm. These conditions are thought to be valid in the region of Albuquerque, New Mexico.
  11. Handbook of Geophysics and Space Environments, Air Force Cambridge Research Laboratories (McGraw-Hill, New York, 1965), Table 4-14.

1984 (1)

1977 (1)

1976 (1)

1968 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (11.4.24).

Fried, D. L.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, Orlando, Fla., 1980), p. 692, Eq. 6.574.2.

Greenwood, D. P.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, Orlando, Fla., 1980), p. 692, Eq. 6.574.2.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (11.4.24).

Tyler, G. A.

J. Opt. Soc. Am. (3)

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

Other (7)

At this point it is not obvious that σθis inversely proportional to f3dB. However, the research that is presented in Sections 3 and 4 will demonstrate that this is true.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (11.4.24).

Ref. 6, Eq. (22.3.15).

Ref. 6, Eq. (6.2.1).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, Orlando, Fla., 1980), p. 692, Eq. 6.574.2.

The HV5/7turbulence model was developed by the U.S. Air Force by modifying a conventional Hufnagel–Valley turbulence model so that at zenith r0= 0.05 m and ϑ0= 7 × 10−6rad when the wavelength that is being used is 0.5 μm. These conditions are thought to be valid in the region of Albuquerque, New Mexico.

Handbook of Geophysics and Space Environments, Air Force Cambridge Research Laboratories (McGraw-Hill, New York, 1965), Table 4-14.

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Figures (4)

Fig. 1
Fig. 1

HV5/7 turbulence model. For a wavelength of 0.5 μm the predicted value of r0 is 5 cm, and the predicted value of ϑ0 is 7 μrad.

Fig. 2
Fig. 2

Geophysics wind model. This wind model was taken from Ref. 11.

Fig. 3
Fig. 3

Power spectra for G tilt and Z tilt. The HV5/7 turbulence model that is illustrated in Fig. 1 and the geophysics wind model that is illustrated in Fig. 2, as well as Eqs. (95)(97), are used in Eqs. (38) and (56) to obtain the one-axis tilt power spectra for the G tilt and the Z tilt. The solid curve pertains to the G tilt, and the dotted curve pertains to the Z tilt. As predicted by Eqs. (44) and (59) both power spectra are identical at low frequencies and are proportional to f−2/3. At high frequencies the power spectra differ. As predicted by Eq. (48) the G-tilt power spectrum is proportional to f−11/3. The dotted curve illustrates that the Z-tilt power spectrum is proportional to f−17/3, as predicted by Eq. (62).

Fig. 4
Fig. 4

Jitter variance versus servo bandwidth. The one-axis one-sigma jitter variance σθ2 is plotted against the servo bandwidth f3dB for both the G tilt and the Z tilt. To prepare the figure, the HV5/7 turbulence model and the geophysics wind model as presented in Figs. 1 and 2, as well as Eqs. (95)(97), are used in Eqs. (70) and (87). The solid curve presents the results for the G tilt, and the dotted curve presents the results for the Z tilt. In the case of a small servo bandwidth both the G-tilt and the Z-tilt results asymptote to the values predicted by Eqs. (75) and (89). In the case of a large servo bandwidth both the G-tilt and the Z-tilt results are proportional to f3dB−2, which is consistent with the concept of a fundamental tracking frequency and is predicted by Eqs. (83) and (93).

Equations (101)

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E 2 = ( f G / f 3 dB ) 5 / 3
H ( f ) = 1 1 + i f / f 3 dB
f G = 2.31 λ - 6 / 5 [ d z C n 2 ( z ) V 5 / 3 ( z ) ] 3 / 5 ,
σ θ = ( f T / f 3 dB ) ( λ / D ) ,
θ G ( t ) = 1 π k R 2 d r W ( r / R ) ϕ ( r , t ) ,
W ( r ) = { 1 r 1 0 r > 1 .
ϕ ( r , t ) = k d z n ( r + V t , z ) ,
C G ( τ ) = θ G ( t 1 ) · θ G ( t 2 ) ,
τ = t 1 - t 2 .
C G ( τ ) = 1 π 2 R 4 d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) × 1 n ( r 1 + V 1 t 1 , z 1 ) · 2 n ( r 2 + V 2 t 2 , z 2 ) .
1 n ( r 1 + V t 1 , z 1 ) · 2 n ( r 2 + V t 2 , z 2 ) = - 12 2 n ( r 1 + V t 1 , z 1 ) n ( r 2 + V t 2 , z 2 ) ,
C G ( τ ) = 1 2 π 2 R 4 d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) × 12 2 [ n ( r 1 + V 1 t 1 , z 1 ) - n ( r 2 + V 2 t 2 , z 2 ) ] 2 .
[ n ( r 1 , z 1 ) - n ( r 2 , z 2 ) ] 2 = C n 2 ( z 1 + z 2 2 ) ( r 1 - r 2 2 + z 1 - z 2 2 ) 1 / 3 ,
C G ( τ ) = 1 2 π 2 R 4 d r 1 d r 2 d z 1 d z 2 C n 2 ( z 1 + z 2 2 ) × W ( r 1 / R ) W ( r 2 / R ) 12 2 [ ( r 1 - r 2 + V 1 t 1 - V 2 t 2 2 + z 1 - z 2 2 ) 1 / 3 - z 1 - z 2 2 / 3 ] ,
z + = 1 2 ( z 1 + z 2 ) ,
z - = z 1 - z 2
d x [ ( A 2 + x 2 ) 1 / 3 - x 2 / 3 ] = 2.91 A 5 / 3 .
C G ( τ ) = 2.91 2 π 2 R 4 d r 1 d r 2 d z C n 2 ( z ) W ( r 1 / R ) W ( r 2 / R ) × 12 2 r 1 - r 2 + V τ 5 / 3 .
r 5 / 3 = 6.65 × 10 - 3 d κ κ - 11 / 3 [ 1 - exp ( 2 π i κ · r ) ] .
C G ( τ ) = - 2.91 ( 6.65 × 10 - 3 ) 2 π 2 R 4 d r 1 d r 2 d z C n 2 ( z ) × W ( r 1 / R ) W ( r 2 / R ) 12 2 d κ κ - 11 / 3 × exp [ 2 π i κ · ( r 1 - r 2 + V τ ) ] .
d r W ( r / R ) exp ( 2 π i κ · r ) = R J 1 ( 2 π R κ ) κ .
α = R κ
C G ( τ ) = 2 ( 2.91 ) ( 6.65 × 10 - 3 ) R 1 / 3 d z d α C n 2 ( z ) α - 11 / 3 × J 1 2 ( 2 π α ) exp ( 2 π i α · V τ / R ) .
0 2 π d θ exp [ i β cos ( θ ) ] = 2 π J 0 ( β )
κ · V = κ V cos ( θ ) ,
D = 2 R ,
z = h sec ( ψ ) .
C G ( τ ) = 0.306 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) 0 d α α - 8 / 3 × J 1 2 ( 2 π α ) J 0 ( 4 π α υ τ ) ,
υ = V / D .
Φ G ( f ) = d τ C G ( τ ) exp ( - 2 π i f τ ) ,
Φ G ( f ) = 0.306 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) 0 d α α - 8 / 3 J 1 2 ( 2 π α ) × d τ J 0 ( 4 π α υ τ ) exp ( - 2 π i f τ ) .
d t J n ( t ) exp ( - i ω t ) = { 2 ( - i ) n T n ( ω ) / ( 1 - ω 2 ) 1 / 2 for ω 2 < 1 0 for ω 2 > 1 ,
T n [ cos ( θ ) ] = cos ( n θ ) .
t = 4 π α υ τ ,
ω = f 2 α υ ,
Φ G ( f ) = 0.306 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) f / ( 2 υ ) d α α - 8 / 3 × J 1 2 ( 2 π α ) 2 π α υ { 1 - [ f / 2 ( α υ ) ] 2 } 1 / 2 ,
x = f 2 α υ .
1 2 Φ G ( f ) = 0.155 D - 1 / 3 sec ( ψ ) f - 8 / 3 d h C n 2 ( h ) υ 5 / 3 F G ( f / υ ) ,
F G ( y ) = 0 1 d x x 5 / 3 1 - x 2 J 1 2 ( π y / x ) .
J ν ( z ) [ ( 1 / 2 z ) ] ν ν ! ,
1 2 Φ G ( f ) = 0.155 ( π 2 4 ) D - 1 / 3 sec ( ψ ) f - 2 / 3 × d h C n 2 ( h ) υ - 1 / 3 0 1 d x x - 1 / 3 1 - x 2 .
Γ ( z ) Γ ( w ) Γ ( z + w ) = 0 1 d t t z - 1 ( 1 - t ) w - 1
0 1 d x x - 1 / 3 1 - x 2 = Γ ( ) Γ ( ½ ) 2 Γ ( 5 / 6 ) .
1 2 Φ G ( f ) = 0.804 D - 1 / 3 sec ( ψ ) f - 2 / 3 d h C n 2 ( h ) υ - 1 / 3 .
J ν ( z ) = 2 π z cos ( z - 1 2 ν π - 1 4 π ) .
1 2 Φ G ( f ) = 0.155 ( 2 π 2 ) D - 1 / 3 sec ( ψ ) f - 11 / 3 d h C n 2 ( h ) υ 8 / 3 × 0 1 d x x 8 / 3 1 - x 2 cos 2 ( π f υ x - 3 4 π ) .
0 1 d x x 8 / 3 1 - x 2 = Γ ( ¹¹ / ) Σ ( ½ ) 2 Γ ( / ) .
1 2 Φ G ( f ) = 0.0110 D - 1 / 3 sec ( ψ ) f - 11 / 3 d h C n 2 ( h ) υ 8 / 3 .
θ z ( t ) = 4 π k R 4 d r W ( r / R ) ϕ ( r , t ) r .
C Z ( τ ) = θ Z ( t 1 ) · θ Z ( t 2 ) .
C Z ( τ ) = - 8 π 2 R 8 d r 1 d r 2 d z 1 d z 2 W ( r 1 / R ) W ( r 2 / R ) r 1 · r 2 × [ n ( r 1 + V 1 t 1 , z 1 ) - n ( r 2 + V 2 t 2 , z 2 ) ] 2 ,
C Z ( τ ) = ( 2.91 ) ( 6.65 × 10 - 3 ) 2 π 2 R 8 d r 1 d r 2 d z C n 2 ( z ) × W ( r 1 / R ) W ( r 2 / R ) r 1 · r 2 d κ κ - 11 / 3 × exp [ 2 π i κ · ( r 1 - r 2 + V τ ) ] .
d r W ( r / R ) exp ( 2 π i κ · r ) r = i R 2 κ 2 J 2 ( 2 π R κ ) κ .
C Z ( τ ) = 0.124 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) 0 d α α - 14 / 3 × J 2 2 ( 2 π α ) J 0 ( 4 π α υ τ ) .
Φ Z ( f ) = d τ C Z ( τ ) exp ( - 2 π i f τ ) .
Φ Z ( f ) = 0.124 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) 0 d α α - 14 / 3 J 2 2 ( 2 π α ) × d τ J 0 ( 4 π α υ τ ) exp ( - 2 π i f τ ) .
1 2 Φ Z ( f ) = 0.251 D - 1 / 3 sec ( ψ ) f - 14 / 3 d h C n 2 ( h ) υ 11 / 3 F Z ( f / υ ) ,
F Z ( y ) = 0 1 d x x 11 / 3 1 - x 2 J 2 2 ( π y / x ) ,
1 2 Φ Z ( f ) = 0.155 ( π 2 4 ) D - 1 / 3 sec ( ψ ) f - 2 / 3 × d h C n 2 ( h ) υ - 1 / 3 0 1 d x x - 1 / 3 1 - x 2 .
1 2 Φ Z ( f ) = 0.804 D - 1 / 3 sec ( ψ ) f - 2 / 3 d h C n 2 ( h ) υ - 1 / 3 .
Φ Z ( f ) = 0.0509 D - 1 / 3 sec ( ψ ) f - 17 / 3 × d h C n 2 ( h ) υ 14 / 3 0 1 d x x 14 / 3 1 - x 2 ,
0 1 d x x 14 / 3 1 - x 2 = Γ ( ¹⁷ / ) Γ ( ½ ) 2 Γ ( ¹⁰ / ) .
1 2 Φ Z ( f ) = 0.0140 D - 1 / 3 sec ( ψ ) f - 17 / 3 d h C n 2 ( h ) υ 14 / 3 .
σ θ 2 = d f Φ ( f ) 1 - H ( f ) 2 ,
σ θ 2 = d f Φ ( f ) ( f / f 3 dB ) 2 1 + ( f / f 3 dB ) 2 .
σ θ 2 = d f [ 1 2 Φ G ( f ) ] ( f / f 3 dB ) 2 1 + ( f / f 3 dB ) 2 .
σ θ 2 = 0.310 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) υ 5 / 3 × 0 d f f - 8 / 3 F G ( f / υ ) ( f / f 3 dB ) 2 1 + ( f / f 3 dB ) 2 .
α = f / υ ,
γ = υ / f 3 dB .
σ θ 2 = 0.310 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) × 0 d α ( α γ ) 2 1 + ( α γ ) 2 α - 8 / 3 F G ( α ) .
( α γ ) 2 1 + ( α γ ) 2 = 1             for γ .
σ θ 2 = 0.310 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) × 0 1 d x x 5 / 3 1 - x 2 0 d α α - 8 / 3 J 1 2 ( π α / x ) .
0 d t t - λ J ν ( a t ) J μ ( a t ) = a λ - 1 Γ ( λ ) Γ [ ( ν + μ - λ + 1 ) / 2 ] 2 λ Γ [ ( - ν + μ + λ + 1 ) / 2 ] Γ [ ( ν + μ + λ + 1 ) / 2 ] Γ [ ( ν - μ + λ + 1 ) / 2 ] .
0 1 d x ( 1 - x 2 ) - 1 / 2 = π 2 .
σ θ 2 = 2.84 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) .
( 1 r 0 ) 5 / 3 = 2.91 6.88 ( 2 π λ ) 2 sec ( ψ ) d h C n 2 ( h ) .
σ θ 2 = 0.170 ( λ / D ) 2 ( D / r 0 ) 5 / 3 .
( α γ ) 2 1 + ( α γ ) 2 = ( α γ ) 2             for γ 0.
σ θ 2 = 0.310 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) γ 2 0 1 d x x 5 / 3 1 - x 2 × 0 d α α - 2 / 3 J 1 2 ( π α / x ) .
0 1 d x x 2 1 - x 2 = π 4 .
σ θ 2 = 0.110 D - / 3 sec ( ψ ) d h C n 2 ( h ) γ 2 .
σ θ 2 = ( λ / D ) 2 ( 1 / f 3 dB ) 2 [ 0.110 D - 1 / 3 λ - 2 sec ( ψ ) d h C n 2 ( h ) V 2 ] ,
σ θ 2 = ( f T G / f 3 dB ) 2 ( λ / D ) 2 ,
f T G = 0.331 D - 1 / 6 λ - 1 sec 1 / 2 ψ [ d h C n 2 ( h ) V 2 ] 1 / 2 .
σ θ 2 = d f [ 1 2 Φ Z ( f ) ] ( f / f 3 dB ) 2 1 - ( f / f 3 dB ) 2 .
σ θ 2 = 0.502 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) υ 11 / 3 × 0 d f f - 14 / 3 F Z ( f / υ ) ( f / f 3 dB ) 2 1 + ( f / f 3 dB ) 2 .
σ θ 2 = 0.502 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) × 0 d α ( α γ ) 2 1 + ( α γ ) 2 α - 14 / 3 F Z ( α ) .
σ θ 2 = 0.502 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) × 0 1 d x x 11 / 3 1 - x 2 0 d α α - 14 / 3 J 2 2 ( π α / x ) ,
σ θ 2 = 3.04 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) .
σ θ 2 = 0.182 ( λ / D ) 2 ( D / r 0 ) 5 / 3 .
σ θ 2 = 0.502 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) γ 2 0 1 d x x 11 / 3 1 - x 2 × 0 d α α - 8 / 3 J 2 2 ( π α / x ) ,
σ θ 2 = 0.135 D - 1 / 3 sec ( ψ ) d h C n 2 ( h ) γ 2 .
σ θ 2 = ( f T Z / f 3 dB ) 2 ( λ / D ) 2 ,
f T Z = 0.368 D - 1 / 6 λ - 1 sec - 1 / 2 ( ψ ) [ d h C n 2 ( h ) V 2 ] 1 / 2 .
D = 3.5 m ,
λ = 0.5 μ m ,
ψ = .
f T G = 7.44 Hz .
f 3 dB = 29.76.
σ θ = 35.7 nrad .
f G = 66.4 Hz .

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