Abstract

The treatment of the problem of deducing the phase of an optical wavefield from image- and diffraction-plane intensity measurements recently presented by Robinson [J. Opt. Soc. Am <b>68</b>, 87–92 (1968)] is shown to be valid only in the limit of geometrical optics. A simple proof and example of the nonuniqueness of the <i>phase retrieval</i> problem in this limit is presented. It is argued that although the phase retrieval is not unique in the geometrical optics <i>limit</i> it does appear to possess a unique solution within the framework of the diffraction theory of image formation.

© 1978 Optical Society of America

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  1. Stanley R. Robinson, "On the problem of phase from intensity measurements," J. Opt. Soc. Am. 68, 87–92 (1978).
  2. E. L. O'Neill and A. Walther, "The question of phase in image formation," Opt. Acta 10, 33–40 (1963).
  3. A. Walther, "The question of phase retrieval in optics," Opt. Acta 10, 41–49 (1963).
  4. R. W. Gerchberg and W. O. Saxton, "Phase determination from image and diffraction plane pictures in the electron microscope," Optik. 34, 275–283 (1971).
  5. 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).
  6. 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, 2220–2225 (1973).
  7. A. J. Drenth et al., "The problem of phase retrieval in light and electron microscopy of strong optics," Opt. Acta 22, 615–628 (1975).
  8. J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," J. Opt. Soc. Am. 67, 1389 (1977).
  9. A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).
  10. W. H. Southwell, "Wave-front analyzer using a maximum likelihood algorithm," J. Opt. Soc. Am. 67, 396–399 (1977).
  11. A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).
  12. B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math, Phys. 16, 1719–1725 (1975).
  13. V. P. Schiske, "Ein- und Mehrdeutigkeit der phasenbestimmung aud bild und beugeunsfigur," Optik 40, 261–275 (1974).
  14. W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 41, 45–59 (1975).
  15. R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 56, 961–964 (1976).
  16. J. S. Lomont and H. E. Moses, "The assignment of wave function to energy densities and probability densities," Nuovo Cimento 30, 1291–1297 (1963).
  17. See, for example, Sec II of John B. DeVelis, "Comparison of methods for image evaluation," J. Opt. Soc. Am. 55, 165–174 (1965) or Section 3 of K. Miyamoto, "Wave optics and geometrical optics in optical design" in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1961) Vol. I, pp 33–66.
  18. N. G. van Kampen, "An asymptotic treatment of diffraction problems," Physica 14, 575–589 (1949).
  19. That the approximation obtained by Robinson and given in Eq. (2) λ follows at once when it is realized that the argument ƒ of ψ(·) is a "reduced coordinate" and is related to the position coordinate x′ over the image plane via the equation ƒ = x′/(λƒo). If one makes use of the above expression to change the variable of integration in Eq. (1) from ƒ to x′ one finds that the phase of the integrand is proportional to 1/λ from which it follows that the method of stationary phase yields an asymptotic expansion of U(x) valid for small λ.
  20. F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. 61, 158–164 (1947).
  21. E. Wolf, "The diffraction theory of aberrations," Rep. Progr. Phys. 14, 95–120 (1951).
  22. See, for example, the article by Miyamoto quoted in Ref. 18. A more comprehensive treatment of the material contained in this article is presented in: K.Miyamoto, "On a comparison between wave optics and geometric optics by using Fourier analysis": Part I. "General theory", J. Opt. Soc. Am. 48, 57–63 (1958); Part II, "Astigmatism, Coma, Spherical aberration", ibid. 48, 567–575 (1958); Part III. "Image evaluation by spot diagram," ibid 49, 35–40 (1959).
  23. The expression for the geometrical optics OTF as given in Eq. (7) is formally identical to the so-called image-motion transfer function which characterizes photographic image motion. The example presented in Fig. 1 was constructed by J. D. Finley, T. N. Morrissey, and A. M. Silvestri in the unpublished paper "Structure functions and transfer functions in photographic image motion" to demonstrate the nonuniqueness in the problem of deducing image motion from an image-motion transfer function.

1978 (1)

1977 (3)

W. H. Southwell, "Wave-front analyzer using a maximum likelihood algorithm," J. Opt. Soc. Am. 67, 396–399 (1977).

J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," J. Opt. Soc. Am. 67, 1389 (1977).

A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).

1976 (2)

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).

R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 56, 961–964 (1976).

1975 (3)

A. J. Drenth et al., "The problem of phase retrieval in light and electron microscopy of strong optics," Opt. Acta 22, 615–628 (1975).

W. J. Dallas, "Digital computation of image complex amplitude from image- and diffraction-intensity: an alternative to holography," Optik 41, 45–59 (1975).

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math, Phys. 16, 1719–1725 (1975).

1974 (1)

V. P. Schiske, "Ein- und Mehrdeutigkeit der phasenbestimmung aud bild und beugeunsfigur," Optik 40, 261–275 (1974).

1973 (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, 2220–2225 (1973).

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).

1965 (1)

1963 (3)

E. L. O'Neill and A. Walther, "The question of phase in image formation," Opt. Acta 10, 33–40 (1963).

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10, 41–49 (1963).

J. S. Lomont and H. E. Moses, "The assignment of wave function to energy densities and probability densities," Nuovo Cimento 30, 1291–1297 (1963).

1951 (1)

E. Wolf, "The diffraction theory of aberrations," Rep. Progr. Phys. 14, 95–120 (1951).

1949 (1)

N. G. van Kampen, "An asymptotic treatment of diffraction problems," Physica 14, 575–589 (1949).

1947 (1)

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. 61, 158–164 (1947).

Chidlaw, R.

A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).

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).

Devaney, A. J.

A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).

DeVelis, B.

Drenth, A. J.

A. J. Drenth et al., "The problem of phase retrieval in light and electron microscopy of strong optics," Opt. Acta 22, 615–628 (1975).

Drenth, A. J. J.

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).

Ferwerda, H. A.

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).

Fienup, J. R.

J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," J. Opt. Soc. Am. 67, 1389 (1977).

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. A.

A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).

R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 56, 961–964 (1976).

Hoenders, B. J.

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math, Phys. 16, 1719–1725 (1975).

Huiser, A. M. J.

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).

Lomont, J. S.

J. S. Lomont and H. E. Moses, "The assignment of wave function to energy densities and probability densities," Nuovo Cimento 30, 1291–1297 (1963).

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, 2220–2225 (1973).

Moses, H. E.

J. S. Lomont and H. E. Moses, "The assignment of wave function to energy densities and probability densities," Nuovo Cimento 30, 1291–1297 (1963).

O’Neill, E. L.

E. L. O'Neill and A. Walther, "The question of phase in image formation," Opt. Acta 10, 33–40 (1963).

Robinson, Stanley R.

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).

Southwell, W. H.

van Kampen, N. G.

N. G. van Kampen, "An asymptotic treatment of diffraction problems," Physica 14, 575–589 (1949).

Walther, A.

E. L. O'Neill and A. Walther, "The question of phase in image formation," Opt. Acta 10, 33–40 (1963).

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10, 41–49 (1963).

Wolf, E.

E. Wolf, "The diffraction theory of aberrations," Rep. Progr. Phys. 14, 95–120 (1951).

Zernike, F.

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. 61, 158–164 (1947).

J. Math, Phys. (1)

B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math, Phys. 16, 1719–1725 (1975).

J. Opt. Soc. Am. (6)

See, for example, Sec II of John B. DeVelis, "Comparison of methods for image evaluation," J. Opt. Soc. Am. 55, 165–174 (1965) or Section 3 of K. Miyamoto, "Wave optics and geometrical optics in optical design" in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1961) Vol. I, pp 33–66.

W. H. Southwell, "Wave-front analyzer using a maximum likelihood algorithm," J. Opt. Soc. Am. 67, 396–399 (1977).

Stanley R. Robinson, "On the problem of phase from intensity measurements," J. Opt. Soc. Am. 68, 87–92 (1978).

J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," J. Opt. Soc. Am. 67, 1389 (1977).

A. J. Devaney, R. A. Gonsalves, and R.Chidlaw, "Application of phase retrieval techniques to adaptive imaging systems," J. Opt. Soc. Am. 67, 1422 (1977).

R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 56, 961–964 (1976).

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, 2220–2225 (1973).

Nuovo Cimento (1)

J. S. Lomont and H. E. Moses, "The assignment of wave function to energy densities and probability densities," Nuovo Cimento 30, 1291–1297 (1963).

Opt. Acta (3)

A. J. Drenth et al., "The problem of phase retrieval in light and electron microscopy of strong optics," Opt. Acta 22, 615–628 (1975).

E. L. O'Neill and A. Walther, "The question of phase in image formation," Opt. Acta 10, 33–40 (1963).

A. Walther, "The question of phase retrieval in optics," Opt. Acta 10, 41–49 (1963).

Optik (3)

A. M. J. Huiser, A. J. J. Drenth, and H. A. Ferwerda, "On phase retrieval in electron microscopy from image and diffraction pattern," Optik 45, 303–316 (1976).

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. (2)

R. W. Gerchberg and W. O. Saxton, "Phase determination from image and diffraction plane pictures in the electron microscope," Optik. 34, 275–283 (1971).

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).

Physica (1)

N. G. van Kampen, "An asymptotic treatment of diffraction problems," Physica 14, 575–589 (1949).

Proc. Phys. Soc. (1)

F. Zernike, "Diffraction and optical image formation," Proc. Phys. Soc. 61, 158–164 (1947).

Rep. Progr. Phys. (1)

E. Wolf, "The diffraction theory of aberrations," Rep. Progr. Phys. 14, 95–120 (1951).

Other (3)

See, for example, the article by Miyamoto quoted in Ref. 18. A more comprehensive treatment of the material contained in this article is presented in: K.Miyamoto, "On a comparison between wave optics and geometric optics by using Fourier analysis": Part I. "General theory", J. Opt. Soc. Am. 48, 57–63 (1958); Part II, "Astigmatism, Coma, Spherical aberration", ibid. 48, 567–575 (1958); Part III. "Image evaluation by spot diagram," ibid 49, 35–40 (1959).

The expression for the geometrical optics OTF as given in Eq. (7) is formally identical to the so-called image-motion transfer function which characterizes photographic image motion. The example presented in Fig. 1 was constructed by J. D. Finley, T. N. Morrissey, and A. M. Silvestri in the unpublished paper "Structure functions and transfer functions in photographic image motion" to demonstrate the nonuniqueness in the problem of deducing image motion from an image-motion transfer function.

That the approximation obtained by Robinson and given in Eq. (2) λ follows at once when it is realized that the argument ƒ of ψ(·) is a "reduced coordinate" and is related to the position coordinate x′ over the image plane via the equation ƒ = x′/(λƒo). If one makes use of the above expression to change the variable of integration in Eq. (1) from ƒ to x′ one finds that the phase of the integrand is proportional to 1/λ from which it follows that the method of stationary phase yields an asymptotic expansion of U(x) valid for small λ.

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