Abstract

A mathematical identity allows us to express the product of two complex light fields as a sum of the intensities of four different superpositions of the two. One common holographic scheme and one proposed by Gabor and Goss turn out to be the experimental realizations of the first term and the sum of the first two terms respectively of this identity. One scheme that we propose, using the identity, bears a strong resemblance to a proposal of Wiener for producing an operational equivalent of a nonlinear transducer.

© 1978 Optical Society of America

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References

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  1. E. L. O’Neill Walther, “The question of phase in image formation,” Opt. Acta 210, 33–40 (1963).
    [Crossref]
  2. A. H. Greenaway, “Proposal for phase recovery from a single intensity distribution,” Opt. Lett. 1, 10–12 (1977).
    [Crossref] [PubMed]
  3. A. W. Lohmann, “Another approach to holography in the electron microscope,” Optik 41, 1–9 (1974).
  4. G. Bachman, L. Narici, Functional Analysis (Academic, New York, 1966), p. 10.
  5. D. Gabor, W. P. Goss, “Interference microscope with total wavefront reconstruction,” J. Opt. Soc. Am. 56, 849–858 (1966).
    [Crossref]
  6. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [Crossref]
  7. See Ref. 4, p. 155.
  8. N. Wiener, “The mathematics of self-organizing systems,” in Recent Developments in Information and Decision Processes, R. E. Machol, P. Gray, eds. (Macmillan, New York, 1962).

1977 (1)

1974 (1)

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

1968 (1)

1966 (1)

1963 (1)

E. L. O’Neill Walther, “The question of phase in image formation,” Opt. Acta 210, 33–40 (1963).
[Crossref]

Bachman, G.

G. Bachman, L. Narici, Functional Analysis (Academic, New York, 1966), p. 10.

Gabor, D.

Goss, W. P.

Greenaway, A. H.

Lohmann, A. W.

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

Montgomery, W. D.

Narici, L.

G. Bachman, L. Narici, Functional Analysis (Academic, New York, 1966), p. 10.

O’Neill Walther, E. L.

E. L. O’Neill Walther, “The question of phase in image formation,” Opt. Acta 210, 33–40 (1963).
[Crossref]

Wiener, N.

N. Wiener, “The mathematics of self-organizing systems,” in Recent Developments in Information and Decision Processes, R. E. Machol, P. Gray, eds. (Macmillan, New York, 1962).

J. Opt. Soc. Am. (2)

Opt. Acta (1)

E. L. O’Neill Walther, “The question of phase in image formation,” Opt. Acta 210, 33–40 (1963).
[Crossref]

Opt. Lett. (1)

Optik (1)

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

Other (3)

G. Bachman, L. Narici, Functional Analysis (Academic, New York, 1966), p. 10.

See Ref. 4, p. 155.

N. Wiener, “The mathematics of self-organizing systems,” in Recent Developments in Information and Decision Processes, R. E. Machol, P. Gray, eds. (Macmillan, New York, 1962).

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Figures (1)

Fig. 1
Fig. 1

Arrangement for the superposition of an object beam O and reference beam R onto the plane P.

Equations (9)

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ϕ ψ * = ¼ α α ϕ + α ψ 2 = ¼ ( ϕ + ψ 2 - i ϕ - i ψ 2 - ϕ - ψ 2 + i ϕ + i ψ 2 ) ,
( ϕ , ψ ) = ¼ α α ϕ + α ψ 2 ,
( ϕ , ψ ) = R 2 ϕ ( x ) ψ * ( x ) dx
ϕ + α ψ 2 = R 2 ϕ ( x ) + α ψ ( x ) 2  dx
ϕ ( x ) + 1 2 = ϕ ( x ) 2 + ϕ ( x ) + ϕ * ( x ) + 1.
ϕ + 1 2 - i ϕ - i 2 = ( 1 - i ) ( ϕ 2 + 1 ) + 2 ϕ .
( ψ n , ψ m ) = { 0 if m n 1 if m n .
ϕ ( x ) = n ( ϕ , ψ n ) ψ n ( x ) .
u ( t ) v n ( t ) = ¼ { [ u ( t ) + v n ( t ) ] 2 - [ u ( t ) - v n ( t ) ] 2 } .

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