Abstract

The functional dependence of irradiance moments with distance from the pupil plane is studied within the framework of Fresnel diffraction theory. The concept of analytic pupil function is introduced, and for such pupil functions it is shown that any finite-order irradiance moment exists, even in the presence of arbitrary continuous phase aberrations. The uniqueness of the relationship between pupil-plane phase and irradiance moments, when the moments are calculated over an orthogonal plane at a fixed point along the optical axis in image space, is obscure, and the relationship between phase and moments is generally nonlinear. However, by studying the behavior of irradiance moments throughout the neighborhood of a given axial point in image space, one may determine, for a large class of pupils, the pupil-plane phase uniquely (within an arbitrary additive constant), and only a linear problem need be solved for phase retrieval. In particular, unique phase retrieval may be accomplished by measuring moments in the neighborhood of either the pupil plane or the image plane. Examples of this technique are given.

© 1982 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 66, 961–964 (1976).
  2. A. J. Devaney and R. Chidlaw, "On the uniqueness question in the problem of phase retrieval from intensity measurements," J. Opt. Soc. Am. 68, 1352–1354 (1978).
  3. S. R. Robinson, "On the problem of phase from intensity measurements," J. Opt. Soc. Am. 68, 87–92 (1978).
  4. W. H. Southwell, N. A. Massie, and J. S. Hartlove, "Wavefront sensor using far-field irradiance measurements," J. Opt. Soc. Am. 69, 1468 (A) (1979).
  5. H. P. Baltes, ed., Inverse Source Problems in Optics (Springer- Verlag, New York, 1978). See also Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).
  6. B. J. Hoenders, "On the solution of the phase retrieval problem," J. Math. Phys. 16, 1719–1725 (1975).
  7. J. R. Fienup, "Reconstruction of an object from the modulus of its Fourier transform," Opt. Lett. 3, 27–29 (1978).
  8. J. G. Walker, "The phase retrieval problem; a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).
  9. J. M. Wood, M. A. Fiddy, and R. E. Burge, "Phase retrieval using two intensity measurements in the complex plane," Opt. Lett. 6, 514–516 (1981).
  10. J. T. Foley and R. B. Butts, "Uniqueness of phase retrieval from intensity measurements," J. Opt. Soc. Am. 71, 1008–1014 (1981).
  11. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  12. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  15. W. H. Southwell, "Validity of the Fresnel approximation in the near field," J. Opt. Soc. Am. 71, 7–14 (1981).
  16. N. Bareket, "Second moment of the diffraction point spread function as an image quality criterion," J. Opt. Soc. Am. 69, 1311–1312 (1979).
  17. R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, New York, 1978), Vol. I.
  18. M. R. Teague, "Image analysis via the general theory of moments," J. Opt. Soc. Am. 70, 9204–930 (1980).
  19. G. Strang, Linear Algebra and its Applications (Academic, New York, 1980).
  20. E. U. Condon and H. Odishaw, Handbook of Physics, 2nd ed. (McGraw-Hill, New York, 1967).
  21. F. J. Dyson, "Photon noise and atmospheric noise in active optical systems," J. Opt. Soc. Am. 65, 551–558 (1975).
  22. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).
  23. J. Herrmann, "Cross coupling and aliasing in modal wave-front estimation," J. Opt. Soc. Am. 71, 989–992 (1981).
  24. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1975), Chap. II.
  25. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

1981 (4)

1980 (1)

M. R. Teague, "Image analysis via the general theory of moments," J. Opt. Soc. Am. 70, 9204–930 (1980).

1979 (2)

W. H. Southwell, N. A. Massie, and J. S. Hartlove, "Wavefront sensor using far-field irradiance measurements," J. Opt. Soc. Am. 69, 1468 (A) (1979).

N. Bareket, "Second moment of the diffraction point spread function as an image quality criterion," J. Opt. Soc. Am. 69, 1311–1312 (1979).

1978 (3)

1976 (1)

1975 (2)

F. J. Dyson, "Photon noise and atmospheric noise in active optical systems," J. Opt. Soc. Am. 65, 551–558 (1975).

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

Bareket, N.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Burge, R. E.

Butts, R. B.

Chidlaw, R.

Condon, E. U.

E. U. Condon and H. Odishaw, Handbook of Physics, 2nd ed. (McGraw-Hill, New York, 1967).

Davis, P. J.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

Devaney, A. J.

Dyson, F. J.

Fiddy, M. A.

Fienup, J. R.

Foley, J. T.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hartlove, J. S.

W. H. Southwell, N. A. Massie, and J. S. Hartlove, "Wavefront sensor using far-field irradiance measurements," J. Opt. Soc. Am. 69, 1468 (A) (1979).

Herrmann, J.

Hoenders, B. J.

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

Massie, N. A.

W. H. Southwell, N. A. Massie, and J. S. Hartlove, "Wavefront sensor using far-field irradiance measurements," J. Opt. Soc. Am. 69, 1468 (A) (1979).

Odishaw, H.

E. U. Condon and H. Odishaw, Handbook of Physics, 2nd ed. (McGraw-Hill, New York, 1967).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Rabinowitz, P.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

Richtmyer, R. D.

R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, New York, 1978), Vol. I.

Robinson, S. R.

Southwell, W. H.

W. H. Southwell, "Validity of the Fresnel approximation in the near field," J. Opt. Soc. Am. 71, 7–14 (1981).

W. H. Southwell, N. A. Massie, and J. S. Hartlove, "Wavefront sensor using far-field irradiance measurements," J. Opt. Soc. Am. 69, 1468 (A) (1979).

Strang, G.

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980).

Teague, M. R.

M. R. Teague, "Image analysis via the general theory of moments," J. Opt. Soc. Am. 70, 9204–930 (1980).

von Neumann, J.

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1975), Chap. II.

Walker, J. G.

J. G. Walker, "The phase retrieval problem; a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Wood, J. M.

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. (10)

Opt. Acta (1)

J. G. Walker, "The phase retrieval problem; a solution based on zero location by exponential apodization," Opt. Acta 28, 735–738 (1981).

Opt. Lett. (2)

Other (11)

H. P. Baltes, ed., Inverse Source Problems in Optics (Springer- Verlag, New York, 1978). See also Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, New York, 1980).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. D. Richtmyer, Principles of Advanced Mathematical Physics (Springer-Verlag, New York, 1978), Vol. I.

G. Strang, Linear Algebra and its Applications (Academic, New York, 1980).

E. U. Condon and H. Odishaw, Handbook of Physics, 2nd ed. (McGraw-Hill, New York, 1967).

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1975), Chap. II.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.