Abstract

We identify wave fronts that have passed through atmospheric turbulence as fractal surfaces from the Fractional Brownian motion family. The fractal character can be ascribed to both the spatial and the temporal behavior. The simulation of such wave fronts can be performed with fractal algorithms such as the Successive Random Additions algorithm. An important benefit is that wave fronts can be predicted on the basis of their past measurements. A simple temporal prediction reduces by 34% the residual error that is not corrected by adaptive-optics systems. Alternatively, it permits a 23% reduction in the measurement bandwidth. Spatiotemporal prediction that uses neighboring points and the effective wind speed is even more beneficial.

© 1994 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]

1992

1991

1990

1989

R. F. Voss, “Random fractals: self-affinity in noise, music, mountains and clouds,” Physica D 38, 362–371 (1989).
[CrossRef]

1987

1971

Aitken, G. J. M.

Barakat, R.

Beletic, J. W.

Clifford, S. F.

Gardner, C. S.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Hege, E. K.

Jorgensen, M. B.

Kane, T. J.

Lukin, V. P.

Mandelbrot, B. B.

B. B. Mandelbrot, Fractals. Forms, Chance and Dimension (Freeman, San Francisco, Calif., 1977).

Phillips, R.

Ribak, E.

E. Ribak, “Phase relations and imaging in pupil plane interferometry,” in NOAO-ESO Conference on High-Resolution Imaging by Interferometry, F. Merkle, ed., Vol. 29 of European Southern Observatory Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 271–280.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Chap. V, p. 307.

Steams, S. D.

B. Widrow, S. D. Steams, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Thompson, L. A.

Voss, R. F.

R. F. Voss, “Random fractals: self-affinity in noise, music, mountains and clouds,” Physica D 38, 362–371 (1989).
[CrossRef]

Welsh, G.

Widrow, B.

B. Widrow, S. D. Steams, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1985).

Zuev, V. E.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Physica D

R. F. Voss, “Random fractals: self-affinity in noise, music, mountains and clouds,” Physica D 38, 362–371 (1989).
[CrossRef]

Other

H. O. Peitgen, D. Saupe, eds., The Science of Fractal Images (Springer-Verlag, Berlin, 1987), Chap. 2, pp. 71–136.

E. Ribak, “Phase relations and imaging in pupil plane interferometry,” in NOAO-ESO Conference on High-Resolution Imaging by Interferometry, F. Merkle, ed., Vol. 29 of European Southern Observatory Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 271–280.

B. Widrow, S. D. Steams, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1985).

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics XIX, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Chap. V, p. 307.

B. B. Mandelbrot, Fractals. Forms, Chance and Dimension (Freeman, San Francisco, Calif., 1977).

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Figures (7)

Fig. 1
Fig. 1

Power spectrum of a one-dimensional FBm generated by the SRA algorithm (averaged over 2000 realizations) compared with the expected 8/3 power law. High frequencies are noisy because of numerical errors.

Fig. 2
Fig. 2

(a) Realization of a wave front generated by the SRA algorithm. (b) The speckle pattern of a monochromatic point source through the aperture and the wave front shown in (a).

Fig. 3
Fig. 3

Numerical simulation of two-point temporal extrapolation. The MSE is plotted versus the first extrapolation parameter, and it is normalized to unity in the simple-lag case.

Fig. 4
Fig. 4

Two spatiotemporal prediction schemes: (a) with nine points in one past layer and (b) with six points in two past layers.

Fig. 5
Fig. 5

MSE for three cases: (a) simple lag (no prediction), equal to the structure function, with the structure constant being taken as unity; (b) spatiotemporal nine-point prediction [Fig. 4(a)]; (c) spatiotemporal six-point prediction [Fig. 4(b)]. x and y units are vΔτ/l.

Fig. 6
Fig. 6

Cartesian cuts through the MSE surfaces (see Fig. 5) for simple lag, two-point prediction, six-point prediction, and nine-point prediction. Displacement units are again vΔτ/l, and the MSE is scaled by the structure constant.

Fig. 7
Fig. 7

Adaptive-mirror control schemes. (a) The mirror is held at the last sampled value until the wave front is sampled again. (b) The mirror is moved linearly to a predicted value. No lag is assumed in both cases, and the mirror is moved instantly to the next sampled point.

Tables (1)

Tables Icon

Table 1 Sets of Coefficients and Mean-Square Errors for Four Linear-Predictor Lengths

Equations (52)

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D φ ( r ) = [ φ ( R + r ) + φ ( R ) ] 2 = 6.88 ( r r 0 ) 5 / 3 ,
P φ ( κ ) κ - 11 / 3 .
[ B ( T + t ) - B ( T ) ] 2 t .
[ B h ( T + t ) - B h ( T ) ] 2 t 2 H ,
P B ( κ ) κ - ( 2 H + E ) ,
F = E + 1 - H .
- B h ( - t ) B h ( t ) [ B h ( t ) ] 2 = 1 2 [ B h ( t ) - B h ( - t ) ] 2 - 2 [ B h ( t ) ] 2 [ B h ( t ) ] 2 = ½ ( 2 t ) 2 H - t 2 H t 2 H = 2 2 H - 1 = 1.
φ ˜ ( x , y , t ) = i j k r i j k φ ( x + i Δ x , y + j Δ y , t - k Δ t ) .
φ ˜ ( t ) = i N r i φ ( t - i Δ t ) .
2 = φ ( t ) - φ ˜ ( t ) 2 = | φ ( t ) - i N r i φ ( t - i Δ t ) | 2 = Γ ( 0 ) + i N r i 2 Γ ( 0 ) - 2 i N r i Γ ( i Δ t ) + 2 i > j N r i r j Γ ( i - j Δ t ) ,
Γ ( τ ) φ ( t ) φ ( t - τ )
Γ ( τ ) = Γ ( 0 ) - 1 2 c τ 2 H .
2 = Γ ( 0 ) ( 1 - i N r i ) 2 + c [ i N r i ( i Δ t ) 2 H - i > j N r i r j ( i - j Δ t ) 2 H ] .
i N r i = 1.
φ ˜ ( x , y , t ) = i j k r i j k φ ( x + i Δ x , y + j Δ y , t - k Δ t ) R T Φ ,
2 = [ φ ( x , y , t ) - φ ˜ ( x , y , t ) ] 2 = [ φ x , y , t - R T Φ ] 2 = φ 2 - 2 R T φ Φ + R T Φ Φ T R φ 2 - 2 R T P + R T MR .
R m = M - 1 P .
2 m = φ 2 - P T R m .
φ ( x + i Δ x , y + j Δ y , t - k Δ t ) φ ( x , y , t ) .
φ ( x + i Δ x + k v x Δ t , y + j Δ y + k v y Δ t , t ) φ ( x , y , t ) Γ ( r ) .
r = [ ( i Δ x + k v x Δ t ) 2 + ( j Δ y + k v y Δ t ) 2 ] 1 / 2 .
P φ ( κ ) ( κ 2 + κ 0 2 ) - 11 / 6 .
γ ( r ) 1 - 1.864 ( κ 0 r ) 5 / 3 + 1.25 ( κ 0 r ) 2 + O ( κ 0 r ) 11 / 3
γ ( r ) 1 - 1.864 ( κ 0 r ) 5 / 3 + 1.5 ( κ 0 r ) 2 + O ( κ 0 r ) 11 / 3 .
φ ˜ ( x , y , t ) = r 1 φ ( x , y , t - Δ t ) + r 2 φ ( x , y , t - 2 Δ t ) ,
M = [ 1 1 - G Δ t 5 / 3 1 - G Δ t 5 / 3 1 ] ,
P = [ 1 - G Δ t 5 / 3 1 - G ( 2 Δ t ) 5 / 3 ] ,
G = 1.864 ( κ 0 v ) 5 / 3 .
R m = [ 2 2 / 3 1 - 2 2 / 3 ] = [ 1.58740 - 0.58740 ] .
[ x - Δ x , y - Δ y x , y - Δ y x + Δ x , y - Δ y x - Δ x , y x , y x + Δ x , y x - Δ x , y + Δ y x , y + Δ y x + Δ x , y + Δ y ]
[ 0 1 2 1 2 5 2 5 8 1 0 1 2 1 2 5 2 5 2 1 0 5 2 1 8 5 2 1 2 5 0 1 2 1 2 5 2 1 2 1 0 1 2 1 2 5 2 1 2 1 0 5 2 1 2 2 8 1 2 5 0 1 2 5 2 5 2 1 2 1 0 1 8 5 2 5 2 1 2 1 0 ] .
[ ( Δ x - v x Δ t ) 2 + ( Δ y - v y Δ t ) 2 ( v x Δ t ) 2 + ( Δ y - v y Δ t ) 2 ( Δ x + v x Δ t ) 2 + ( Δ y - v y Δ t ) 2 ( Δ x - v x Δ t ) 2 + ( v y Δ t ) 2 ( v x Δ t ) 2 + ( v y Δ t ) 2 ( Δ x - v x Δ t ) 2 + ( v y Δ t ) 2 ( Δ x - v x Δ t ) 2 + ( Δ y + v y Δ t ) 2 ( v x Δ t ) 2 + ( Δ y + v y Δ t ) 2 ( Δ x + v x Δ t ) 2 + ( Δ y + v y Δ t ) 2 ] .
[ 0.004 ( v x - θ y ) Δ t / l 0.518 v y Δ t / l 0.004 ( - v x + v y ) Δ t / l 0.518 v x Δ t / l 1 - 0.518 v x Δ t / l 0.004 ( v x - v y ) Δ t / l - 0.518 v y Δ t / l - 0.004 ( v x + v y ) Δ t / l ] .
[ 0.004 v x Δ t / l 0 - 0.004 v x Δ t / l 0.518 v x Δ t / l 1 - 0.518 v x Δ t / l 0.004 v x Δ t / l 0 - 0.004 v x Δ t / l ] .
[ 0.525 v x Δ t / l 1 - 0.525 v x Δ t / l ] .
[ 0.5 v x Δ t / l 1 - 0.5 v x Δ t / l ] .
f s = 3.1 ( v e r 0 ) ,
v e = [ C n 2 ( ξ ) v 5 / 3 ( ξ ) d ξ C n 2 ( ξ ) d ξ ] 3 / 5 .
f s = 2.4 ( v e r 0 ) .
P φ ( co ) ( κ ) = C 1 κ - 11 / 3 ,             κ κ 0 ,
P φ ( VK ) ( κ ) = C 2 ( κ 2 + κ 0 2 ) - 11 / 3 ,             κ 0.
Γ ( r ) = 2 π 0 d κ κ P φ ( κ ) J 0 ( κ r )
Γ 1 ( r ) = 2 π C 1 ( 1 κ 0 ) 5 / 3 { 0.6 F 1 2 [ - 5 6 , ( 1 6 , 1 ) , - κ 0 2 r 2 4 ] - 1.11833 ( κ 0 r ) 5 / 3 } ,
Γ 1 ( r ) 2 π C 1 [ 0.6 κ 0 - 5 / 3 - 1.11833 r 5 / 3 + 0.75 κ 0 - 1 / 3 r 2 + O ( r ) 11 / 3 ] .
γ 1 ( r ) 1 - 1.86389 ( κ 0 r ) 5 / 3 + 1.25 ( κ 0 r ) 2 + O ( r ) 11 / 3 .
Γ 2 ( r ) = 2 π C 2 0.59664 κ 0 - 5 / 6 r 5 / 6 K 5 / 6 ( κ 0 r ) ,
Γ 2 ( r ) 2 π C 2 [ 0.6 κ 0 - 5 / 3 - 1.11833 r 5 / 3 + 0.9 κ 0 1 / 3 r 2 + O ( r ) 11 / 3 ] ,
γ 2 ( r ) 1 - 1.86389 ( κ 0 r ) 5 / 3 + 1.5 ( κ 0 r ) 2 + O ( r ) 11 / 3 .
C 2 = 0.4896 r 0 - 5 / 3 .
2 = 1 Δ t 0 Δ t 6.88 ( t τ 0 ) 5 / 3 d t = 0.375 × 6.88 ( Δ t τ 0 ) 5 / 3 .
2 = 1 Δ t 0 Δ t ( φ ( t ) - { φ ( 0 ) + t Δ t [ r φ ( 0 ) + ( 1 - r ) φ ( - Δ t ) ] } ) 2 d t .
2 = 6.88 ( Δ t τ 0 ) 5 / 3 [ 0.375 + 0.333 ( 1 - r ) 2 + 0.412 ( 1 - r ) ] .

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