Abstract

The analytic frequency responses of the traditional wavefront reconstructors of Hudgin, Fried, and Southwell are presented, which exhibit amplification or attenuation of the original signal at high spatial frequencies. To overcome this problem, a reconstructor with unity frequency response is developed based on a band-limited derivative calculation. The algorithm is both numerically and experimentally confirmed.

© 2008 Optical Society of America

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References

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  1. R. H. Hudgin, J. Opt. Soc. Am. 67, 375 (1977).
    [CrossRef]
  2. D. L. Fried, J. Opt. Soc. Am. 67, 370 (1977).
    [CrossRef]
  3. W. H. Southwell, J. Opt. Soc. Am. 70, 998 (1980).
    [CrossRef]
  4. W. Zou and J. P. Rolland, J. Opt. Soc. Am. A 23, 2629 (2006).
    [CrossRef]
  5. R. A. Gonsalves, Opt. Eng. (Bellingham) 21, 829 (1982).
  6. K. R. Freischlad and C. L. Koliopoulos, J. Opt. Soc. Am. A 3, 1852 (1986).
    [CrossRef]
  7. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prentice Hall, 1989).
  8. L. A. Poyneer, D. T. Gavel, and J. M. Brase, J. Opt. Soc. Am. A 19, 2100 (2002).
    [CrossRef]
  9. F. Roddier and C. Roddier, Appl. Opt. 30, 1325 (1991).
    [CrossRef] [PubMed]

2006 (1)

2002 (1)

1991 (1)

1986 (1)

1982 (1)

R. A. Gonsalves, Opt. Eng. (Bellingham) 21, 829 (1982).

1980 (1)

1977 (2)

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Figures (3)

Fig. 1
Fig. 1

Frequency response of Hudgin, Fried, and Southwell reconstructors when applied to slopes measurements. The maximum value of Hudgin and Fried in a one-dimensional plot is π 2 = 1 sinc ( 0.5 ) .

Fig. 2
Fig. 2

Flowchart of modified iterative Frequency-domain algorithm. S x , 0 and S y , 0 are measured slopes, and Ω is inside the pupil. DFT, discrete Fourier transform; IDFT, inverse discrete Fourier transform; F, filter function [Eq. (2) with A x and A y equal to 1]; D x , D y , band-limited-derivative operators. For a convergence criterion, one might use a relative improvement in slopes inside the pupil.

Fig. 3
Fig. 3

Comparison of frequency response between the original Southwell reconstructor and the band-limited reconstructor for the Southwell geometry, using reconstruction from (a) simulated and (b) measured slopes.

Equations (11)

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LSE [ φ ̂ ] = { D x φ ̂ ̃ A x S x ̃ 2 + D y φ ̂ ̃ A y S y ̃ 2 } ,
φ ̂ ̃ = ( D x * A x S x ̃ + D y * A y S y ̃ ) ( D x 2 + D y 2 ) .
φ ̃ x = i k ¯ x φ ̃ D x , Southwell φ ̃ ,
φ ̃ x , Hudgin = i k ¯ x exp ( i k ¯ x Δ x 2 ) φ ̃ D x , Hudgin φ ̃ ,
φ ̃ x , Fried = i k ¯ x exp ( i k ¯ x Δ x 2 + i k ¯ y Δ y 2 ) φ ̃ D x , Fried φ ̃ .
D x , Hudgin or Southwell = 1 Δ x [ exp ( i k ¯ x Δ x ) 1 ] ,
D x , Fried = 1 2 Δ x [ exp ( i k ¯ x Δ x ) 1 ] [ exp ( i k ¯ y Δ y ) + 1 ] .
A x , Southwell = 1 2 [ exp ( i k ¯ x Δ x ) + 1 ] .
H Hudgin = α sin α + β sin β sin 2 α + sin 2 β ,
H Fried = α sin α cos β + β sin β cos α sin 2 α cos 2 β + sin 2 β cos 2 α ,
H Southwell = α sin ( 2 α ) + β sin ( 2 β ) 2 sin 2 α + 2 sin 2 β ,

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