Abstract

We show that a small-scale wave front can be reconstructed by an algebraic procedure from its intensity distribution in the focal plane, except for the ambiguities of the piston phase and the point-symmetrical solution of the complex conjugate. Details of the reconstruction procedure for a 3 × 3 wave front are presented, and the effectiveness of this procedure for a contaminated case is shown by computer simulation. A method for overcoming the ambiguity problem resulting from the point-symmetrical solution is also suggested.

© 1998 Optical Society of America

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References

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  1. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
    [CrossRef]
  2. H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
    [CrossRef]
  3. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  4. R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
    [CrossRef]
  5. J. R. Fienup, “Phase retrieval algorithm: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  6. R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991), Chap. 5, p. 164.
  7. D. A. Huffman, “The generation of impulse-equivalent pulse trains,” IRE Trans. IT IT8, S10–S16 (1962).
    [CrossRef]
  8. W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik (Stuttgart) 44, 45–59 (1975).
  9. W. J. Dallas, “Digital computation of image complex amplitude from intensities in two image-space planes,” Opt. Commun. 18, 317–320 (1976).
    [CrossRef]
  10. T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).
    [CrossRef]
  11. J. R. Fienup, “Reconstruction of objects having latent reference points,” J. Opt. Soc. Am. 73, 1421–1426 (1983).
    [CrossRef]
  12. M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.
  13. M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
    [CrossRef]
  14. J. R. Fienup, “Phase retrieval using boundary conditions,” J. Opt. Soc. Am. A 3, 284–288 (1986).
    [CrossRef]
  15. S. MacLane, G. Birkhoff, Algebra, 3rd ed. (Chelsea Publishing, New York, 1988), Chap. 13, p. 436.

1987 (2)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
[CrossRef]

T. R. Crimmins, “Phase retrieval for discrete functions with support constraints,” J. Opt. Soc. Am. A 4, 124–134 (1987).
[CrossRef]

1986 (1)

1983 (2)

1982 (1)

1979 (1)

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1976 (2)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

W. J. Dallas, “Digital computation of image complex amplitude from intensities in two image-space planes,” Opt. Commun. 18, 317–320 (1976).
[CrossRef]

1975 (1)

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik (Stuttgart) 44, 45–59 (1975).

1962 (1)

D. A. Huffman, “The generation of impulse-equivalent pulse trains,” IRE Trans. IT IT8, S10–S16 (1962).
[CrossRef]

Bates, R. H. T.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
[CrossRef]

Birkhoff, G.

S. MacLane, G. Birkhoff, Algebra, 3rd ed. (Chelsea Publishing, New York, 1988), Chap. 13, p. 436.

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Burge, R. E.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

Crimmins, T. R.

Dallas, W. J.

W. J. Dallas, “Digital computation of image complex amplitude from intensities in two image-space planes,” Opt. Commun. 18, 317–320 (1976).
[CrossRef]

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik (Stuttgart) 44, 45–59 (1975).

Ferwerda, H. A.

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

Fienup, J. R.

Fright, W. R.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
[CrossRef]

Greenaway, A. H.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

Hayes, M. H.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

Huffman, D. A.

D. A. Huffman, “The generation of impulse-equivalent pulse trains,” IRE Trans. IT IT8, S10–S16 (1962).
[CrossRef]

Lane, R. G.

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
[CrossRef]

MacLane, S.

S. MacLane, G. Birkhoff, Algebra, 3rd ed. (Chelsea Publishing, New York, 1988), Chap. 13, p. 436.

Quatieri, T. F.

M. H. Hayes, T. F. Quatieri, “Recursive phase retrieval using boundary conditions,” J. Opt. Soc. Am. 73, 1427–1433 (1983).
[CrossRef]

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

Ross, G.

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991), Chap. 5, p. 164.

Appl. Opt. (1)

IEEE Trans. ASSP (1)

R. G. Lane, W. R. Fright, R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. ASSP ASSP-35, 520–526 (1987).
[CrossRef]

IRE Trans. IT (1)

D. A. Huffman, “The generation of impulse-equivalent pulse trains,” IRE Trans. IT IT8, S10–S16 (1962).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

W. J. Dallas, “Digital computation of image complex amplitude from intensities in two image-space planes,” Opt. Commun. 18, 317–320 (1976).
[CrossRef]

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Optik (Stuttgart) (1)

W. J. Dallas, “Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography,” Optik (Stuttgart) 44, 45–59 (1975).

Proc. R. Soc. London A (1)

R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London A 350, 191–212 (1976).
[CrossRef]

Other (4)

H. A. Ferwerda, “The phase reconstruction problem for wave amplitudes and coherence functions,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), Chap. 2.
[CrossRef]

R. K. Tyson, Principles of Adaptive Optics (Academic, London, 1991), Chap. 5, p. 164.

M. H. Hayes, T. F. Quatieri, “The importance of boundary conditions in the phase retrieval problem,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1982), pp. 1545–1548.

S. MacLane, G. Birkhoff, Algebra, 3rd ed. (Chelsea Publishing, New York, 1988), Chap. 13, p. 436.

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Figures (4)

Fig. 1
Fig. 1

Definitions of the side and the corner elements. The elements corresponding to (1), (2), (3), and (4) are the side elements. P 1 and P 2 represent a pair of corner elements. A pair of corner elements must belong to the same side of a wave front.

Fig. 2
Fig. 2

Intensity distribution in the focal plane of the 3 × 3 pixel wave front defined by Eqs. (36) and (37).

Fig. 3
Fig. 3

Standard deviation of the error of the estimated wave-front phase for various noise values N r , as defined by Eq. (39).

Fig. 4
Fig. 4

Standard deviation of the error of the estimated wave-front phase for various detected total photons of the intensity distribution in the focal plane.

Equations (49)

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A uv = u v   P u v P u - u , v - v * ,
A u , n - 1 = u   P u , n P u - u , 1 * ,     - n + 1 u n - 1 ,
A n - 1 , v = v   P n , v P 1 , v - v * ,     - n + 1 v n - 1 ,
A u , n - 2 = u P u , n - 1 P u - u , 1 * + P u , n P u - u , 2 * , - n + 2 u n - 2 ,
A n - 2 , v = v P n - 1 , v P 1 , v - v * + P n , v P 2 , v - v * , - n + 2 v n - 2 .
f 1 ξ = 0 ,
f 2 η = 0 ,
ξ = P 2 * / P 1 * ,
η = P 2 P 1 * ,
phase ξ η = 0 ,
P = P 13 P 23 P 33 P 12 P 22 P 32 P 11 P 21 P 31 .
ξ = P 31 * / P 11 * ,
η = P 31 P 11 * .
A 22 ξ 2 - s ξ + A - 22 = 0 ,
η 2 - t η + A 22 A - 22 * = 0 ,
s 3 - A 02 s 2 + A 12 A - 12 - 4 A 22 A - 22 s + 4 A 22 A - 22 A 02 - A - 12 2 A 22 - A 12 2 A - 22 = 0 ,
t 3 - A 20 t 2 + A 21 A - 21 * - 4 A 22 A - 22 * t + 4 A 22 A - 22 * A 20 - A - 21 * 2 A 22 - A 21 2 A - 22 * = 0 ,
| P 11 | = | η / ξ | ,
ψ 11 = 0 ,
| P 31 | = | ξ η | ,
ψ 31 = phase η = - phase ξ ,
P 23 = 1 α A 12 A - 22 P 31 * - A - 12 A 22 P 11 * ,
P 32 = 1 β A 22 A - 21 * P 11 * - A 21 P 31 ,
P 12 = 1 β * A 21 * A - 22 P 31 * - A - 21 P 11 ,
P 21 = 1 α * A - 12 * P 11 - A 12 * P 31 ,
P 13 = A - 22 P 31 * ,
P 33 = A 22 P 11 * ,
α = A - 22 ξ - A 22 ξ ,
β = A 22 A - 22 * η - η ,
P 22 = Re A 11 - P 21 * P 32 - P 12 * P 23 / P 11 * - P 33 Re P 11 * + P 33 / P 11 * - P 33 + i   Im A 11 - P 21 * P 32 - P 12 * P 23 / P 11 * + P 33 Re P 11 * - P 33 / P 11 * + P 33 ,
P = P 13 P 23 O P 12 P 22 P 32 P 11 P 21 P 31 .
A 12 2 ξ 3 - A 12 A - 12 ξ 2 + A 02 A - 22 ξ - A - 22 2 = 0 ,
η 3 - A 20 η 2 + A 21 A - 21 * η - A 21 2 A - 22 * = 0 .
P 22 = P 23 A 11 - P 23 P 12 * - P 32 P 21 * - P 33 A 01 - P 33 P 32 * - P 13 P 12 * - P 32 P 31 * - P 12 P 11 * P 23 mh ; 9 uP 11 * - P 33 P 21 * .
E =   ı ˆ xy - ı ˜ xy 2   ı ˜ xy 2 1 / 2 ,
ψ = 0.59 1.93 - 2.08 - 1.05 0.92 1.69 0 1.56 2.20 rad .
| P | = c 1 1 1 1 1 1 1 1 1 ,
ı ˜ xy = i xy + n xy .
N r =   n xy 2   i xy 2 1 / 2 .
A 11 ξ 2 - A 01 ξ + A - 11 = 0 ,
η 2 - A 10 η + A 11 A - 11 * = 0 ,
A 22 3 ξ 6 - A 02 A 22 2 ξ 5 + A 12 A - 12 A 22 - A 22 2 A - 22 ξ 4 + 2 A 22 A - 22 A 02 - A 12 2 A - 22 - A - 12 2 A 22 ξ 3 + A 12 A - 12 A - 22 - A 22 A - 22 2 ξ 2 - A 02 A - 22 2 ξ + A - 22 3 = 0 ,
η 6 - A 20 η 5 + A 21 A - 21 * - A 22 A - 22 * η 4 + 2 A 22 A - 22 * A 20 - A 21 2 A - 22 * - A - 21 * 2 A 22 η 3 + A 21 A - 21 * A 22 A - 22 * - A 22 2 A - 22 * 2 η 2 - A 20 A 22 2 A - 22 * 2 η + A 22 3 A - 22 * 3 = 0 ,
P 21 = P 11 A 02 * - A - 22 * ξ - 1 * / A 12 * ,
P 12 = P 11 A 20 * - η * / A 21 * ,
P 22 = A 11 - P 21 * A 21 / P 11 * - P 12 * A 12 / P 11 * / P 11 * ,
P 32 = A 21 / P 11 * ,
P 13 = A - 22 / P 31 * ,
P 23 = A 12 / P 11 * .

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