Abstract

For many applications in adaptive optics the eigenfunctions of the power spectrum of turbulence-induced distortions on the actual telescope aperture constitute the ideal basis for modeling wave fronts. The mathematical derivation of these functions is reviewed, and a procedure for their calculation, allowing for a circular central obscuration and leaving considerable freedom in the form of the structure function, is described. Examples of wave-front simulations are presented for a range of power-law structure functions and for Kolmogorov structure functions with an outer scale smaller than the diameter of the region being modeled.

© 1996 Optical Society of America

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References

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  1. R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
    [Crossref] [PubMed]
  2. R. C. Cannon, “Global wave-front reconstruction using Shack–Hartmann sensor,” J. Opt. Soc. Am. A 12, 2031–2039 (1995).
    [Crossref]
  3. N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
    [Crossref]
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  5. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [Crossref]
  6. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [Crossref]
  7. L. Fox, Numerical Linear Algebra (Oxford U. Press, London, 1964).

1995 (1)

1994 (1)

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
[Crossref]

1992 (1)

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

1978 (1)

1976 (1)

Cannon, R. C.

Fox, L.

L. Fox, Numerical Linear Algebra (Oxford U. Press, London, 1964).

Iye, M.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
[Crossref]

Lane, R. G.

Markey, J. K.

Noll, R. J.

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Takato, N.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
[Crossref]

Tallon, M.

Wang, J. Y.

Yamaguchi, I.

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[Crossref]

Publ. Astron. Soc. Pacific (1)

N. Takato, M. Iye, I. Yamaguchi, “Wavefront reconstruction errors of Shack–Hartmann sensors,” Publ. Astron. Soc. Pacific 106, 182–188 (1994).
[Crossref]

Other (1)

L. Fox, Numerical Linear Algebra (Oxford U. Press, London, 1964).

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Figures (6)

Fig. 1
Fig. 1

Structure functions of simulated Kolmogorov wave fronts. Two simulations, separated vertically for clarity, are shown. In each case the solid line is the analytic structure function, and the crosses are points on the structure function derived from a simulation of 100 wave fronts. For the upper line the first 500 KL functions were used; for the lower one, the first 2000. The simulated structure functions were calculated after interpolation onto a Cartesian grid: the numbers next to the crosses are the displacements in points on this grid corresponding to the value of r/d on the ordinate.

Fig. 2
Fig. 2

Structure functions of simulated wave fronts (dashed curves) and the original structure function (solid lines) used to compute the basis. The original structure functions are power laws with indices, as marked on the right-hand side. The vertical displacements are arbitrary and were chosen simply to separate the lines. The first thousand basis elements were used for each wave front.

Fig. 3
Fig. 3

Same as Fig. 2, but for Kolmogorov structure functions showing an outer scale of down to a third of the aperture. The structure functions are given by relation (23) with α = 3 and the quantity rc, as marked next to each curve.

Fig. 4
Fig. 4

Eigenvalues corresponding to the first thousand KL functions for the structure functions exhibited in (a) Fig. 2 and (b) Fig. 3. The plots show a steady shift with power-law index and outer scale, respectively, so only the two extreme values in each case are shown.

Fig. 5
Fig. 5

First two radial functions for each of the first three azimuthal orders for the KL functions without a central obscuration (solid curves) and with a third of the aperture by radius obscured (dotted curves). The curves are labeled by the pairs (azimuthal order, radial order). The underlying structure function is a power law of index 5/3.

Fig. 6
Fig. 6

Comparison of the KL functions for a Kolmogorov structure function (dotted curves) and for a structure function that is Kolmogorov at small scales but saturates at a third of the aperture (solid curves). In each case there is a circular central obscuration of radius one third that of the aperture. The orders shown are the same as in Fig. 5.

Equations (24)

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r i < r < 1
s · s = 1 π ( 1 - r i 2 ) A s ( r ) s ( r ) d r ,
v i · v j = δ i j ,
w · v i w · v j = δ i j e i 2 ,
w ( r ) = ( w · v i ) v i ( r ) .
1 π ( 1 - r i 2 ) A w ( r ) w ( r ) v i ( r ) d r = e i 2 v i ( r ) .
D ( x ) = [ w ( r ) - w ( r + x ) ] 2 ,
w ( r ) w ( r ) = - 1 / 2 D ( r - r ) + 1 / 2 w ( r ) 2 + 1 / 2 w ( r ) 2 .
v k p e = v k p ( r ) cos 2 π p θ ,             v k p o = u k p ( r ) sin 2 π p θ .
1 π ( 1 - r i 2 ) r i 1 0 2 π C ( r , r , θ , θ ) u k p ( r ) { cos 2 π p θ sin 2 π p θ } r d θ d r = u k p ( r ) { cos 2 π p θ sin 2 π p θ } ,
C ( r , r , θ , θ ) = - 1 / 2 D [ r 2 + r 2 - 2 r r cos ( θ - θ ) ] 1 / 2 + Φ 2 ,
r i 1 r K p ( r , r ) u ( r ) d r = e p u ( r ) ,
K p ( r , r ) = - 1 2 π ( 1 - r i 2 ) 0 2 π D ( r 2 + r 2 - 2 r r cos θ ) 1 / 2 × cos 2 π p θ d θ ,
K 0 ( r , r ) = - 1 2 π ( 1 - r i 2 ) 0 2 π D ( r 2 + r 2 - 2 r r cos θ ) 1 / 2 × d θ - π Φ 2 .
s = r 2 ,             t p ( s ) = u p ( s ) ,             L p ( s , s ) = 1 / 2 K p ( s , s )
r i 2 1 L p ( s , s ) t p ( s ) d s = e p t p ( s ) .
L p t = e t ,
L i j p = 1 - r i 2 n L p ( s i , s j )             t i = t ( s i ) .
1 π ( 1 - r i 2 ) r i 1 0 2 π u k p ( r ) u l p ( r ) { cos 2 sin 2 } 2 π p θ r d r d θ 1 2 n i t k p ( s i ) t l p ( s i ) 1 2 n t l p · t k p .
U i , j = { 0 if i > j + 1 - [ j / ( j + 1 ) ] 1 / 2 if i = j + 1 1 / ( j 2 + j ) 1 / 2 if i j , j n 1 / n if j = n ,
t k 0 = U ( t k 0 0 )
w ( r , θ ) = p cos 2 π p θ [ k x k p o e k p u k p ( r ) ] + p sin 2 π p θ [ k x k p e e k p u k p ( r ) ] .
D ( r ) = 6.88 ( r / r 0 ) 5 / 3 ,
D ( r ) [ r ( r c α + r α ) 1 / α ] 5 / 3 ,

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