Abstract

Phase retrieval implies extraction of the phase of a complex signal f from its modulus |f|. We give examples where an additional constraint is imposed: knowledge of the modulus of F, the Fourier transform of f. The retrieval is accomplished by computer processing of samples of |f| and |F|. The problems of noisy data and nonuniqueness are addressed.

© 1976 Optical Society of America

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  1. T. T. Taylor and J. R. Whinnery, “Applications of Potential Theory to the Design of Linear Arrays,” J. Appl. Phys. 22, 19–29 (1951).
    [Crossref]
  2. E. Wolf, “Is a Complete Determination of the Energy Spectrum of Light Possible from Measurements of the Degree of Coherence?”, Proc. Phys. Soc. Lond. 80, 1269–1272 (1962).
    [Crossref]
  3. D. A. Huffman, “The Generation of Impulse-Equivalent Pulse Trains,” I.R.E. Trans IT 8, S10–S16 (1962).
    [Crossref]
  4. E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
    [Crossref]
  5. A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
    [Crossref]
  6. H. B. Voelcker, “Toward a Unified Theory of Modulation Part I: Phase-Envelope Relationships,” Proc. IEEE 54, 340–353. (1966).
    [Crossref]
  7. C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
    [Crossref]
  8. D. Dialetis, “Some Theorems Concerning the Phase Problem of Coherence Theory,” J. Math. Phys. 8, 1641–1649 (1967).
    [Crossref]
  9. H. M. Nussenzveig, “Phase Problem in Coherent Theory,” J. Math. Phys. 8, 561–572 (1967).
    [Crossref]
  10. R. W. Gerchberg and W. O. Saxton, “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik. 34, 275–283 (1971).
  11. D. Kohler and L. Mandel, “Source Reconstruction from the Modulus of the Correlation Function: A Practical Approach to the Phase Problem of Optical Coherence Theory,” J. Opt. Soc. Am. 63, 126–134 (1973).
    [Crossref]
  12. D. L. Misell, “An Examination of an Iterative Method for the Solution of the Phase Problem in Optics and Electron Optics,” J. Phys. D 6, 2200–2225 (1973).
    [Crossref]
  13. A. J. Drenth and et al., “The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects,” Opt. Acta 22, 615–628 (1975).
    [Crossref]
  14. B. J. Hoenders, “On The solution of the Phase Retrievel Problem,” J. Math. Phys. 16, 1719–1725 (1975).
    [Crossref]
  15. R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).
  16. V. P. Schiske, “Ein- und Mehrdeutigkeit der Phasenbestimmung aud Bild und Beugeunsfigur,” Optik 40, 261–275 (1974).
  17. W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image- and Diffraction-Intensity: An Alternative to Holography,” Optik 41, 45–59 (1975).
  18. R. Gonsalves and et al., “On Optical Holographic Filters,” Proc. SPIE 45, 293–297 (1974).
    [Crossref]
  19. R. Gonsalves and P. Considine, “Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery,” Opt. Eng. 15, 64–67 (1976).
    [Crossref]
  20. S. L. S. Jacoby and et al., Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).
  21. Our earlier discussion centered on extraction of β(x) in the spatial domain. But we find it more intuitive to present examples where one determines θ(γ) in the frequency domain.

1976 (1)

R. Gonsalves and P. Considine, “Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery,” Opt. Eng. 15, 64–67 (1976).
[Crossref]

1975 (3)

A. J. Drenth and et al., “The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects,” Opt. Acta 22, 615–628 (1975).
[Crossref]

B. J. Hoenders, “On The solution of the Phase Retrievel Problem,” J. Math. Phys. 16, 1719–1725 (1975).
[Crossref]

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image- and Diffraction-Intensity: An Alternative to Holography,” Optik 41, 45–59 (1975).

1974 (2)

R. Gonsalves and et al., “On Optical Holographic Filters,” Proc. SPIE 45, 293–297 (1974).
[Crossref]

V. P. Schiske, “Ein- und Mehrdeutigkeit der Phasenbestimmung aud Bild und Beugeunsfigur,” Optik 40, 261–275 (1974).

1973 (2)

D. Kohler and L. Mandel, “Source Reconstruction from the Modulus of the Correlation Function: A Practical Approach to the Phase Problem of Optical Coherence Theory,” J. Opt. Soc. Am. 63, 126–134 (1973).
[Crossref]

D. L. Misell, “An Examination of an Iterative Method for the Solution of the Phase Problem in Optics and Electron Optics,” J. Phys. D 6, 2200–2225 (1973).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

1971 (1)

R. W. Gerchberg and W. O. Saxton, “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik. 34, 275–283 (1971).

1967 (2)

D. Dialetis, “Some Theorems Concerning the Phase Problem of Coherence Theory,” J. Math. Phys. 8, 1641–1649 (1967).
[Crossref]

H. M. Nussenzveig, “Phase Problem in Coherent Theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

1966 (2)

H. B. Voelcker, “Toward a Unified Theory of Modulation Part I: Phase-Envelope Relationships,” Proc. IEEE 54, 340–353. (1966).
[Crossref]

C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
[Crossref]

1963 (2)

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

1962 (2)

E. Wolf, “Is a Complete Determination of the Energy Spectrum of Light Possible from Measurements of the Degree of Coherence?”, Proc. Phys. Soc. Lond. 80, 1269–1272 (1962).
[Crossref]

D. A. Huffman, “The Generation of Impulse-Equivalent Pulse Trains,” I.R.E. Trans IT 8, S10–S16 (1962).
[Crossref]

1951 (1)

T. T. Taylor and J. R. Whinnery, “Applications of Potential Theory to the Design of Linear Arrays,” J. Appl. Phys. 22, 19–29 (1951).
[Crossref]

Balachadran, A. P.

C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
[Crossref]

Considine, P.

R. Gonsalves and P. Considine, “Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery,” Opt. Eng. 15, 64–67 (1976).
[Crossref]

Dallas, W. J.

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image- and Diffraction-Intensity: An Alternative to Holography,” Optik 41, 45–59 (1975).

Dialetis, D.

D. Dialetis, “Some Theorems Concerning the Phase Problem of Coherence Theory,” J. Math. Phys. 8, 1641–1649 (1967).
[Crossref]

Drenth, A. J.

A. J. Drenth and et al., “The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects,” Opt. Acta 22, 615–628 (1975).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik. 34, 275–283 (1971).

Gonsalves, R.

R. Gonsalves and P. Considine, “Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery,” Opt. Eng. 15, 64–67 (1976).
[Crossref]

R. Gonsalves and et al., “On Optical Holographic Filters,” Proc. SPIE 45, 293–297 (1974).
[Crossref]

Hoenders, B. J.

B. J. Hoenders, “On The solution of the Phase Retrievel Problem,” J. Math. Phys. 16, 1719–1725 (1975).
[Crossref]

Huffman, D. A.

D. A. Huffman, “The Generation of Impulse-Equivalent Pulse Trains,” I.R.E. Trans IT 8, S10–S16 (1962).
[Crossref]

Jacoby, S. L. S.

S. L. S. Jacoby and et al., Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Kohler, D.

Mandel, L.

Mehta, C. L.

C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
[Crossref]

Misell, D. L.

D. L. Misell, “An Examination of an Iterative Method for the Solution of the Phase Problem in Optics and Electron Optics,” J. Phys. D 6, 2200–2225 (1973).
[Crossref]

Nussenzveig, H. M.

H. M. Nussenzveig, “Phase Problem in Coherent Theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

O’Neill, E. L.

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik. 34, 275–283 (1971).

Schiske, V. P.

V. P. Schiske, “Ein- und Mehrdeutigkeit der Phasenbestimmung aud Bild und Beugeunsfigur,” Optik 40, 261–275 (1974).

Taylor, T. T.

T. T. Taylor and J. R. Whinnery, “Applications of Potential Theory to the Design of Linear Arrays,” J. Appl. Phys. 22, 19–29 (1951).
[Crossref]

Voelcker, H. B.

H. B. Voelcker, “Toward a Unified Theory of Modulation Part I: Phase-Envelope Relationships,” Proc. IEEE 54, 340–353. (1966).
[Crossref]

Walther, A.

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

Whinnery, J. R.

T. T. Taylor and J. R. Whinnery, “Applications of Potential Theory to the Design of Linear Arrays,” J. Appl. Phys. 22, 19–29 (1951).
[Crossref]

Wolf, E.

E. Wolf, “Is a Complete Determination of the Energy Spectrum of Light Possible from Measurements of the Degree of Coherence?”, Proc. Phys. Soc. Lond. 80, 1269–1272 (1962).
[Crossref]

Wolfe, E.

C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
[Crossref]

I.R.E. Trans IT (1)

D. A. Huffman, “The Generation of Impulse-Equivalent Pulse Trains,” I.R.E. Trans IT 8, S10–S16 (1962).
[Crossref]

J. Appl. Phys. (1)

T. T. Taylor and J. R. Whinnery, “Applications of Potential Theory to the Design of Linear Arrays,” J. Appl. Phys. 22, 19–29 (1951).
[Crossref]

J. Math. Phys. (4)

B. J. Hoenders, “On The solution of the Phase Retrievel Problem,” J. Math. Phys. 16, 1719–1725 (1975).
[Crossref]

C. L. Mehta, E. Wolfe, and A. P. Balachadran, “Some Theorems on the Unimodular-Complex Degree of Optical Coherence,” J. Math. Phys. 7, 133–138 (1966).
[Crossref]

D. Dialetis, “Some Theorems Concerning the Phase Problem of Coherence Theory,” J. Math. Phys. 8, 1641–1649 (1967).
[Crossref]

H. M. Nussenzveig, “Phase Problem in Coherent Theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. D (1)

D. L. Misell, “An Examination of an Iterative Method for the Solution of the Phase Problem in Optics and Electron Optics,” J. Phys. D 6, 2200–2225 (1973).
[Crossref]

Opt. Acta (3)

A. J. Drenth and et al., “The Problem of Phase Retrieval in Light and Electron Microscopy of Strong Objects,” Opt. Acta 22, 615–628 (1975).
[Crossref]

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta 10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

Opt. Eng. (1)

R. Gonsalves and P. Considine, “Spot Shaping and Incoherent Optical Smoothing for Raster Scanned Imagery,” Opt. Eng. 15, 64–67 (1976).
[Crossref]

Optik (3)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

V. P. Schiske, “Ein- und Mehrdeutigkeit der Phasenbestimmung aud Bild und Beugeunsfigur,” Optik 40, 261–275 (1974).

W. J. Dallas, “Digital Computation of Image Complex Amplitude from Image- and Diffraction-Intensity: An Alternative to Holography,” Optik 41, 45–59 (1975).

Optik. (1)

R. W. Gerchberg and W. O. Saxton, “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik. 34, 275–283 (1971).

Proc. IEEE (1)

H. B. Voelcker, “Toward a Unified Theory of Modulation Part I: Phase-Envelope Relationships,” Proc. IEEE 54, 340–353. (1966).
[Crossref]

Proc. Phys. Soc. Lond. (1)

E. Wolf, “Is a Complete Determination of the Energy Spectrum of Light Possible from Measurements of the Degree of Coherence?”, Proc. Phys. Soc. Lond. 80, 1269–1272 (1962).
[Crossref]

Proc. SPIE (1)

R. Gonsalves and et al., “On Optical Holographic Filters,” Proc. SPIE 45, 293–297 (1974).
[Crossref]

Other (2)

S. L. S. Jacoby and et al., Iterative Methods for Nonlinear Optimization Problems (Prentice-Hall, Englewood Cliffs, N.J., 1972).

Our earlier discussion centered on extraction of β(x) in the spatial domain. But we find it more intuitive to present examples where one determines θ(γ) in the frequency domain.

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Figures (6)

FIG. 1
FIG. 1

Polynomial-phase reconstruction with space-frequency iterations (NF = 9, NX = 64, NI = 27).

FIG. 2
FIG. 2

Random-phase reconstruction with space-frequency iterations (NF = 9, NX = 256, NI = 27).

FIG. 3
FIG. 3

Asymmetric modulus and phase of F(γ).

FIG. 4
FIG. 4

Reconstruction with space-frequency iterations based on noisy data (NF = 9, NX = 64, NI = 9).

FIG. 5
FIG. 5

Polynomial-phase reconstruction by parametric search (NF = 11, NP = 4, NI = 50).

FIG. 6
FIG. 6

Random-phase reconstruction by parametric search (NF = 10, NP = 10, NP = 10, NI = 50).

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

F ( γ ) = d x f ( x ) exp ( - i 2 π x γ ) ,
f ( x ) = d γ F ( γ ) exp ( i 2 π x γ ) .
f ( x ) = f ( x ) exp [ i β ( x ) ] .
ϕ ( x ) = f ( x ) 2
Φ ( γ ) = - d v F * ( v ) F ( v + γ ) .
f 1 ( x ) = f ( x ) e i β 1 ( x ) ,
F 2 ( γ ) = F ( γ ) [ F 1 ( γ ) / F 1 ( γ ) ] .
f 3 ( x ) = f ( x ) [ f 2 ( x ) / f 2 ( x ) ] .
F ( γ ) = { 1 , γ < 1 0 , γ > 1.
F ( γ ) = F ( γ ) e i θ ( γ ) ,
σ 2 ( x ) = 1 12 ( 0.2 ) 2 f ( x ) .
θ 0 ( γ ) = θ 0 - θ ( - γ ) .
e i θ 0 F 0 * ( - γ ) = F ( γ ) e i θ 0 e - i θ ( - γ )
θ p ( γ ) = p 1 γ + p 2 γ 2 + p 3 γ 3 + p 4 γ 4
p = ( p 1 , p 2 , p 3 , p 4 )
M = - d γ Φ ( γ ) - Φ p ( γ ) ,
F p ( γ ) = F ( γ ) e i θ p ( γ )