Abstract

The problem of wave-front estimation from wave-front slope measurements has been examined from a least-squares curve fitting model point of view. It is shown that the slope measurement sampling geometry influences the model selection for the phase estimation. Successive over-relaxation (SOR) is employed to numerically solve the exact zonal phase estimation problem. A new zonal phase gradient model is introduced and its error propagator, which relates the mean-square wave-front error to the noisy slope measurements, has been compared with two previously used models. A technique for the rapid extraction of phase aperture functions is presented. Error propagation properties for modal estimation are evaluated and compared with zonal estimation results.

© 1980 Optical Society of America

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References

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  1. K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am. 66, 1236–1239 (1976).
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    [CrossRef]
  4. D. L. Fried, “Least-square fitting of a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
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    [CrossRef]
  6. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  7. R. L. Frost, C. K. Rushforth, and B. S. Baxter, “Fast FFT-based algorithm for phase estimation in speckel imaging,” Appl. Opt. 18, 2056–2061 (1979).
    [CrossRef] [PubMed]
  8. J. Herrmann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  9. D. M. Young, Iterative Solutions of Large Linear Systems (Academic, New York, 1972).
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    [CrossRef]

1980 (1)

1979 (3)

1978 (1)

1977 (2)

1976 (1)

1974 (1)

Baxter, B. S.

Cubalchini, R.

Fried, D. L.

Frost, R. L.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Knox, K. T.

Noll, R. J.

Rimmer, M. P.

Rushforth, C. K.

Young, D. M.

D. M. Young, Iterative Solutions of Large Linear Systems (Academic, New York, 1972).

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Figures (5)

FIG. 1
FIG. 1

Slope measurement sampling geometry and wave-front mesh points. The horizontal dashes indicate positions of x-slope sampling. The vertical dashes are the y-slope sampling positions. The dots are the estimated phase points. Configuration B has been considered previously by Hudgin and configuration C by Fried.

FIG. 2
FIG. 2

Noise coefficient for zonal estimation versus N for a square array containing N2 phase points. Curves A, B, and C correspond to the sampling geometries in Fig. 1.

FIG. 3
FIG. 3

Noise coefficient for modal estimation versus N for a square array containing N2 phase points. Curves A, B, and C correspond to the sampling geometries in Fig. 1.

FIG. 4
FIG. 4

Noise coefficient for zonal estimation on a reduced aperture. The phase reconstruction error was evaluated as in Fig. 2, except that the perimeter points are weighted by one-half and the corners by one-fourth.

FIG. 5
FIG. 5

Noise coefficient comparison between zonal estimation and modal estimation. The slope geometry used was Fig. 1, configuration A.

Tables (3)

Tables Icon

TABLE I rms wave-front reconstruction error in waves as a function of the number of iterations for three different iterative methods. The sampled wave front contained astigmatism of the form W = 2.3717 (x2y2)/a2 + 6xy/a2 over a square aperture of area 4a2

Tables Icon

TABLE II Wave-front reconstruction results using the recursive formula of Hudgin (which neglects edge effects). Shown here is the rms wave-front reconstruction error in waves from slope data sampled from a wave front containing astigmatism of the form W = 2.3717(x2y2)/a2 + 6xy/a2 over a square aperture of area 4a2. In each case the recursive algorithm quickly converqed, but did not reproduce the sampled wave front

Tables Icon

TABLE III Two-dimensional Legendre polynomials used to represent a wave front ϕ ( x , y ) = Σ k = 1 M a k n k F k ( x , y ) over a square aperture of width D. The constants gN = 3(D/2)2 [1 − 7/(3N2)] and dN = (D/2)2 (1 − 1/N2) are determined such that the polynomials are normalized over a square grid of N2 sample points

Equations (68)

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ϕ = c 0 + c 1 x + c 2 x 2 .
S x = c 1 + 2 c 2 x .
ϕ = ϕ i + S i x x .
S i x = ( ϕ i + 1 ϕ i ) / h i = 1 , N 1 ,
S i j x = ( ϕ i + 1 , j ϕ i j ) / h i = 1 , N 1 j = 1 , N ,
S i j y = ( ϕ i , j + 1 ϕ i j ) / h i = 1 , N j = 1 , N 1 ,
S i j x = [ ( ϕ i + 1 , j + ϕ i + 1 , j + 1 ) / 2 ( ϕ i j + ϕ i , j + 1 ) / 2 ] h ,
S i j y = [ ( ϕ i , j + 1 + ϕ i + 1 , j + 1 ) / 2 ( ϕ i j + ϕ i + 1 , j ) / 2 ] h .
( S i + 1 , j x + S i j x ) 2 = ( ϕ i + 1 , j ϕ i j ) h i = 1 , N 1 j = 1 , N ,
( S i + 1 , j x + S i j y ) 2 = ( ϕ i , j + 1 ϕ i j ) h i = 1 , N j = 1 , N 1 .
S = A ϕ ,
( A A ) ϕ = A S ,
DS = A ϕ ,
( A A ) ϕ = A DS .
g j k ϕ j k [ ϕ j + 1 , k + ϕ j 1 , k + ϕ j , k + 1 + ϕ j , k 1 ] = [ S j , k 1 y S j k y + S j 1 , k x S j k x ] h ,
g j k = 2 j = 1 or N ; k = 1 or N 3 { j = 1 or N ; k = 2 to N 1 k = 1 to N ; j = 2 or N 1 4 otherwise ,
ϕ j k = ϕ ¯ j k + b j k / g j k ,
ϕ ¯ j k = [ ϕ j + 1 , k + ϕ j 1 , k + ϕ j , k + 1 ] / g j k ,
b j k = [ S j , k 1 y S j k y + S j 1 , k x S j k x ] h ,
ϕ j k ( m + 1 ) = ϕ ¯ j k ( m ) + b j k / g j k .
ϕ j k ( m + 1 ) = ϕ j k ( m ) + ω [ ϕ ¯ j k ( m ) + b j k / g j k ϕ j k ( m ) ] .
ω = 2 1 + sin [ π / ( N + 1 ) ] .
ϕ = ( 2.17 0.43 0.43 2.17 0.43 1.30 1.30 0.43 0.43 1.30 1.30 0.43 2.17 0.43 0.43 2.17 ) .
ϕ ̂ = ( 2.17 1.01 1.01 2.17 1.01 0.72 0.72 1.01 1.01 0.72 0.72 1.01 2.17 1.01 1.01 2.17 ) .
g j k ϕ j k [ ϕ j + 1 , k + ϕ j 1 , k + ϕ j , k + 1 + ϕ j , k 1 ] = 1 2 [ S j , k + 1 y S j , k 1 y + S j + 1 , k x S j 1 , k x ] h ,
g j k = 2 j = 1 or N ; k = 1 or N 3 { j = 1 or N ; k = 2 to N 1 k = 1 or N ; j = 2 to N 1 4 otherwise ,
( A e A e ) ϕ = A S ,
S = S 0 ± σ .
ϕ = ϕ 0 ± ,
( A e A e ) ( ϕ 0 ± ) = A ( S 0 ± σ ) ,
( A e A e ) = A σ ,
= B σ ,
B = ( A e A e ) 1 A
B = ( A e A e ) 1 A D ,
i = k B i k σ k .
i j = k l B i k B j l σ k σ l ;
σ k σ l = σ 2 δ k l .
i j = σ 2 k B i k B j k ,
= σ 2 [ B B + ] i j .
E = 1 N 2 i i 2
= σ 2 N 2 i k B i k 2 .
E = C h 2 σ 2 ,
C = 1 D 2 i k B i k 2 .
E = k E k ,
E k = 1 N 2 i B i k 2 σ k 2 ,
ϕ p d = ( 2 π / λ ) h s ,
C p d = C N 2 / D 2 = C / h 2 ,
( δ Φ ) 2 = C p d σ p d 2 ,
ϕ ( x , y ) = k = 0 M a k n k F k ( x , y ) ,
σ ϕ 2 = k = 1 M a k 2 .
n k n l i = 1 N j = 1 N F k ( x i , y i ) F l ( x i , y j ) = N 2 δ k l ,
S x = k = 1 M a k n k F k x ,
S y = k = 1 M a k n k F k y .
S = Aa ,
A Aa = A S ,
a = ( A A ) 1 A S ,
A A = ( 51.2 0 0 0 0 0 51.2 0 0 0 0 0 32.0 0 0 0 0 0 32.0 0 0 0 0 0 102.4 ) .
[ A A ] 1 ( 0.0195 0 0 0 0 0 0.0195 0 0 0 0 0 0.0031 0 0 0 0 0 0.0031 0 0 0 0 0 0.0098 ) .
A A = ( 51.2 0 0 0 0 0 0 145.1 0 0 5.12 0 0 0 0 0 0 145.1 0 0 320. 0 0 0 0 0 0 0 0 0 320. 0 0 0 0 0 0 0 0 0 102.4 0 0 0 0 0 0 0 0 0 371.2 0 0 0 0 0 0 0 0 0 371.2 0 0 145.1 0 0 0 0 0 0 1691. 0 0 145.1 0 0 0 0 0 0 1691. ) ,
( A A ) 1 = ( 0.0258 0 0 0 0 0 0 0.00211 0 0 0.0258 0 0 0 0 0 0 0.00211 0 0 0.00312 0 0 0 0 0 0 0 0 0 0.00312 0 0 0 0 0 0 0 0 0 0.00977 0 0 0 0 0 0 0 0 0 0.00269 0 0 0 0 0 0 0 0 0 0.00269 0 0 −0.00211 0 0 0 0 0 0 0.00078 0 0 −0.00211 0 0 0 0 0 0 0.00078 )
a = a 0 ± α ,
i = l α l n l F l ( x i , y i ) .
E = 1 N 2 i i 2 = α l 2 = α α ,
α l = j [ ( A + A ) 1 A + ] l j σ j .
E = l j k [ ( A + A ) 1 A + ] l j [ A + ( A + A ) 1 ] k l σ j σ k .
σ j σ k = σ 2 δ j k ,
E = l j [ ( A A ) 1 A ] l j [ A ( A A ) 1 ] j l σ 2 = l [ ( A A ) 1 A A ( A A ) 1 ] ll σ 2 = σ 2 l ( A A ) l l 1 = σ 2 tr ( A A ) 1 .
C M R = 1 h 2 tr ( A A ) 1 , modal reconstruction ,