Abstract

A method is proposed for the recovery of phase information from a single intensity distribution. The proposal requires the use of a specially designed spatial filter and is analyzed for a one-dimensional case using the theory of entire functions. The only, and unlikely, ambiguity is detectable a posteriori.

© 1977 Optical Society of America

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References

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  1. E. N. Leith, J. Upatnieks, “Reconstructed Wavefronts and Communication Theory,” J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  2. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  3. E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
    [CrossRef]
  4. B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
    [CrossRef]
  5. R. E. Burge, M. A. Fiddy, A. H. Greenaway, G. Ross, “The phase problem,” Proc. R. Soc. London Ser. A 350, 191–212 (1976).
    [CrossRef]
  6. R. P. Boas, Entire Function (Academic Press, New York, 1954).
  7. E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
    [CrossRef]
  8. A. W. Lohmann, “Another approach to holography in the electron microscope,” Optik 41, 1–9 (1974).

1976 (1)

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

1975 (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

1974 (1)

A. W. Lohmann, “Another approach to holography in the electron microscope,” Optik 41, 1–9 (1974).

1963 (2)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

1962 (1)

1926 (1)

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[CrossRef]

Boas, R. P.

R. P. Boas, Entire Function (Academic Press, New York, 1954).

Burge, R. E.

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

Fiddy, M. A.

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

Greenaway, A. H.

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

Hoenders, B. J.

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

A. W. Lohmann, “Another approach to holography in the electron microscope,” Optik 41, 1–9 (1974).

O’Neill, E. L.

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Ross, G.

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

Titchmarsh, E. C.

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[CrossRef]

Upatnieks, J.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

J. Math. Phys. (1)

B. J. Hoenders, “On the solution of the phase retrieval problem,” J. Math. Phys. 16, 1719–1725 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

E. L. O’Neill, A. Walther, “The question of phase in image formation,” Opt. Acta 10, 33–40 (1963).
[CrossRef]

Optik (1)

A. W. Lohmann, “Another approach to holography in the electron microscope,” Optik 41, 1–9 (1974).

Proc. London Math. Soc. (1)

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

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

Other (1)

R. P. Boas, Entire Function (Academic Press, New York, 1954).

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Figures (1)

Fig. 1
Fig. 1

Schematic representation of the particular f(ξ) under consideration, showing g(ξ), h(ξ), and the position of the gap (c, d).

Equations (10)

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F ( x ) = a b d ξ f ( ξ ) e - i ξ x .
I ( x ) = F ( x ) 2 = a b d ξ d ξ f ( ξ ) f * ( ξ ) exp - i x ( ξ - ξ ) ,
F ( z ) = a b d ξ f ( ξ ) e - i ξ z .
F ( z ) = F ( z ) ( z - τ j * z - τ j ) .
F ( x ) = F ( x ) j = 1 N ( x - τ j * x - τ j ) ,
- d x j = 1 N ( x - τ j * x - τ j ) e i ξ x = δ ( ξ ) - A ( ξ ) j = 1 N c j y j e i τ j ξ ,
c j = 2 k = 1 k j N ( 1 - i 2 y k z j - z k ) , A ( ξ ) = 1 , ξ 0 = 0 , ξ < 0.
f ( ξ ) = f ( ξ ) - f ( θ ) * A ( θ ) j = 1 N c j y j e i τ j ξ ,
f ( ξ ) = - j = 1 N c j y j - d ω { g ( ω ) + h ( ω ) } A ( ξ - ω ) exp [ i τ j ( ξ - ω ) ] ,             ξ ( c , d ) .
f ( ξ ) = - j = 1 N c j y j e i τ j ξ G ( τ j ) , ξ ( c , d ) .

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