Abstract

We discuss error propagation in the slope-based and the difference-based wavefront estimations. The error propagation coefficient can be expressed as a function of the eigenvalues of the wavefront-estimation-related matrices, and we establish such functions for each of the basic geometries with the serial numbering scheme with which a square sampling grid array is sequentially indexed row by row. We first show that for the wavefront estimation with the wavefront piston value determined, the odd-number grid sizes yield better error propagators than the even-number grid sizes for all geometries. We further show that for both slope-based and difference-based wavefront estimations, the Southwell geometry offers the best error propagators with the minimum-norm least-squares solutions. Noll’s theoretical result, which was extensively used as a reference in the previous literature for error propagation estimates, corresponds to the Southwell geometry with an odd-number grid size. Typically the Fried geometry is not preferred in slope-based optical testing because it either allows subsize wavefront estimations within the testing domain or yields a two-rank deficient estimations matrix, which usually suffers from high error propagation and the waffle mode problem. The Southwell geometry, with an odd-number grid size if a zero point is assigned for the wavefront, is usually recommended in optical testing because it provides the lowest-error propagation for both slope-based and difference-based wavefront estimations.

© 2006 Optical Society of America

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References

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  1. H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics, B.E.Saleh, ed. (Wiley, 2003), pp. 368, 1036.
  2. W. Zou and Z. Zhang, "Generalized wave-front reconstruction algorithm applied in a Shack-Hartmann test," Appl. Opt. 39, 250-268 (2000).
    [CrossRef]
  3. R. J. Noll, "Phase estimates from slope-type wave front sensors," J. Opt. Soc. Am. 68, 139-140 (1978).
    [CrossRef]
  4. W. H. Southwell, "Wave front estimation from wave front slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  5. K. Freischlad, "Wave front integration from difference data," in Interferometry: Techniques and Analysis, G.M.Brown, O.Y.Kwon,M.Kujawinska, and G.T.Reid, eds., Proc. SPIE 1755, 212-218 (1992).
  6. R. H. Hudgin, "Wave front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-378 (1977).
    [CrossRef]
  7. D. L. Fried, "Least-square fitting a wave front distortion estimate to an array of phase-difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  8. W. Zou and J. P. Rolland, "Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils," J. Opt. Soc. Am. A 22, 938-951 (2005).
    [CrossRef]
  9. K. Freischlad and C. L. Koliopoulos, "Modal estimation of a wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-186l (1986).
    [CrossRef]
  10. B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979).
    [CrossRef]
  11. R. Kress, Numerical Analysis, Vol. 1818 Graduate Texts in Mathematics (Springer, 1998), pp. 22, 127.
  12. J. R. Schott, Matrix Analysis for Statistics (Wiley, 1997), pp. 138, 171, 177.
  13. J. Herrmann, "Least-square wave front errors of minimum norm," J. Opt. Soc. Am. 70, 28-35 (1980).
    [CrossRef]
  14. M. D. Oliker, "Sensing waffle in the Fried geometry," in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, 964-971 (1998).
    [CrossRef]

2005

2000

1998

M. D. Oliker, "Sensing waffle in the Fried geometry," in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, 964-971 (1998).
[CrossRef]

1986

1980

1979

1978

1977

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics, B.E.Saleh, ed. (Wiley, 2003), pp. 368, 1036.

Freischlad, K.

K. Freischlad and C. L. Koliopoulos, "Modal estimation of a wave front from difference measurements using the discrete Fourier transform," J. Opt. Soc. Am. A 3, 1852-186l (1986).
[CrossRef]

K. Freischlad, "Wave front integration from difference data," in Interferometry: Techniques and Analysis, G.M.Brown, O.Y.Kwon,M.Kujawinska, and G.T.Reid, eds., Proc. SPIE 1755, 212-218 (1992).

Fried, D. L.

Herrmann, J.

Hudgin, R. H.

Hunt, B. R.

Koliopoulos, C. L.

Kress, R.

R. Kress, Numerical Analysis, Vol. 1818 Graduate Texts in Mathematics (Springer, 1998), pp. 22, 127.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics, B.E.Saleh, ed. (Wiley, 2003), pp. 368, 1036.

Noll, R. J.

Oliker, M. D.

M. D. Oliker, "Sensing waffle in the Fried geometry," in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, 964-971 (1998).
[CrossRef]

Rolland, J. P.

Schott, J. R.

J. R. Schott, Matrix Analysis for Statistics (Wiley, 1997), pp. 138, 171, 177.

Southwell, W. H.

Zhang, Z.

Zou, W.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

M. D. Oliker, "Sensing waffle in the Fried geometry," in Adaptive Optical System Technologies, D. Bonaccini and R. K. Tyson, eds., Proc. SPIE 3353, 964-971 (1998).
[CrossRef]

Other

R. Kress, Numerical Analysis, Vol. 1818 Graduate Texts in Mathematics (Springer, 1998), pp. 22, 127.

J. R. Schott, Matrix Analysis for Statistics (Wiley, 1997), pp. 138, 171, 177.

H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics, B.E.Saleh, ed. (Wiley, 2003), pp. 368, 1036.

K. Freischlad, "Wave front integration from difference data," in Interferometry: Techniques and Analysis, G.M.Brown, O.Y.Kwon,M.Kujawinska, and G.T.Reid, eds., Proc. SPIE 1755, 212-218 (1992).

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Figures (8)

Fig. 1
Fig. 1

Wavefront estimation schemes: (a) Hudgin geometry, (b) Southwell geometry, (c) Fried geometry.

Fig. 2
Fig. 2

Previous results on error propagation.

Fig. 3
Fig. 3

Grid array with serial numbering scheme (SNS).

Fig. 4
Fig. 4

WFD-based error propagators for the Hudgin geometry.

Fig. 5
Fig. 5

WFD-based error propagators for the Southwell geometry.

Fig. 6
Fig. 6

WFD-based error propagators for the Fried geometry.

Fig. 7
Fig. 7

Comparison of the WDF-based error propagators.

Fig. 8
Fig. 8

Comparison of the slope-based error propagators.

Tables (1)

Tables Icon

Table 1 Qualitative Comparisons of the WFD-Based Error Propagators

Equations (57)

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CW = S ,
S = μ a MG ,
C T CW = C T S ,
2 W = S = f ( y , z ) ,
σ w 2 = ε 2 2 m ,
η = σ w 2 σ n 2 .
η Fried = 0.6558 + 0.3206 ln ( t ) ,
η Hudg = 0.561 + 0.103 ln ( t ) .
η South = 0.10447 + 0.2963 ln ( t ) .
η Noll = 0.1072 + 0.318 ln ( t ) .
η Frei = 0.09753 + 1 π ln ( t ) .
C ε = N .
ε = C + N ,
N = μ a MN .
ε 2 ( j = 1 m ε j 2 ) 1 2 = ( tr ε ε T ) 1 2 ,
ε ε T = C + NN T ( C + ) T = C + μ a MN N T M T μ a ( C + ) T ,
N N T = [ n 1 2 n 1 n 2 n 1 n m n 2 n 1 n 2 2 n 2 n m n m n 1 n m n 2 n m 2 ] .
n i n j = σ s 2 δ i j = { 0 , when i j σ s 2 , when i = j .
N N T = σ s 2 I ,
ε ε T = μ 2 a 2 σ s 2 C + MM T ( C + ) T .
σ w 2 = 1 m ε 2 2 = 1 m tr { ε ε T } = μ 2 a 2 σ s 2 m tr [ C + MM T ( C + ) T ] .
σ n 2 = 1 m N 2 2 = 1 m tr { NN T } = μ 2 a 2 σ s 2 m tr MM T .
η = σ w 2 σ n 2 = tr [ C + M ( C + M ) T ] tr [ MM T ] .
η = C + M F 2 M F 2 .
S x i + 1 , i = g x i + 1 , i a ,
S y i , i + t = g y i , i + t a ,
w i + 1 w i = S x i + 1 , i i = 1 , 2 , , m , i k t , k integer ,
w i w i + t = S y i , i + t , i = 1 , 2 , , m t .
HW = S ,
η H = tr H + ( H + ) T tr [ I ] = 1 m tr [ ( H T H ) + ] = 1 m H + F 2 .
η H = 1 m tr [ ( H T H ) + ] = 1 m ( i = 1 γ λ H , i 1 ) .
η H , odd = 0.3797 + 0.3171 ln ( t 0.6672 ) 0.3222 + 0.3316 ln ( t ) ( t is odd ) ,
η H , even = 0.3294 + 0.479 ln ( t 0.2136 ) 0.3049 + 0.4856 ln ( t ) ( t is even ) .
η H , LSMN = 0.3252 + 0.1593 ln ( t 1.1208 ) 0.2605 + 0.1764 ln ( t ) .
w i + 1 w i = a 2 ( g x i + g x i + 1 ) ,
w i w i + t = a 2 ( g y i + g y i + t ) ,
HW = S H ,
η S = tr [ H + C s ( H + C s ) T ] tr [ C s C s T ] = tr [ C s T ( H + ) T H + C s ] tr [ HH T ] ,
η S = tr [ C s T ( H + ) T H + C s ] 4 ( m t ) = tr [ C s T H [ ( H T H ) + ] 2 H T C s ] 4 ( m t ) .
η S = 1 4 ( m t ) ( i = 1 γ λ s , i ) ,
η S , odd = 0.1489 + 0.2936 ln ( t 0.06186 ) 0.1428 + 0.2952 ln ( t ) ( t is odd ) ,
η S , even = 0.04941 + 0.4662 ln ( t + 2.7673 ) 0.2861 + 0.41 ln ( t ) ( t is even ) .
η S , LSMN = 0.205 + 0.1487 ln ( t + 0.2562 ) 0.217 + 0.1455 ln ( t ) .
w i + 1 w i + w i + t + 1 w i + t = 2 s x j ,
w i + w i + 1 w i + t w i + t + 1 = 2 s y j ,
FW = S ,
η F = tr [ F + ( F + ) T ] tr [ II T ] = 1 m tr [ ( F T F ) + ] .
η F = 1 m ( i = 1 γ λ F , i 1 ) .
η F , odd = 0.4461 + 0.2 ln ( t 0.8805 ) = 0.4146 + 0.207 ln ( t ) ( t is odd ) ,
η F , even = 0.4933 + 0.2866 ln ( t 0.6547 ) = 0.4338 + 0.3022 ln ( t ) ( t is even ) .
η F , LSMN = 0.475 + 0.114 ln ( t 1.821 ) 0.4076 + 0.1303 ln ( t ) .
η = σ w 2 σ s 2 = μ 2 a 2 m tr [ C + M ( C + M ) T ] ,
η = K η ,
K = μ 2 A 0 tr [ MM T ] m ,
η H = A 0 m tr [ ( H T H ) + ] = K H η H ,
η S = A 0 4 m tr [ C s T ( H + ) T H + C s ] = K S η S ,
η F = 4 A 0 m tr [ ( F T F ) + ] = K F η F ,

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