Abstract

A new method for the synthesis of holograms, based on a Fourier transformation, has been theoretically derived and experimentally verified. The method extends the familiar two-dimensional optical Fourier-transform analysis to include objects located in a volume centered about the focal point of the transform lens, and is specifically developed to provide an easily synthesized binary hologram for the reconstruction of three-dimensional objects. The synthetic holograms produced display, in addition to three-dimensionality of the reconstructed image, all other properties associated with photographic holograms. This technique for synthesizing holograms has the potential advantage of computational simplicity and increased image resolution over the Fresnel-transform technique, and permits the synthesis of a variety of objects composed of discrete points, continuous lines, and continuous surfaces.

© 1968 Optical Society of America

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References

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  1. B. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [Crossref] [PubMed]
  2. A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).
  3. J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
    [Crossref]
  4. J. P. Waters, J. Opt. Soc. Am. 57, 563A (1967).
  5. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).
  6. G. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
    [Crossref]
  7. G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
    [Crossref]
  8. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  9. J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
    [Crossref]
  10. J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley Publishing Company, Reading, Mass., 1967), p. 80.
  11. G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
    [Crossref]

1968 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).

1967 (1)

J. P. Waters, J. Opt. Soc. Am. 57, 563A (1967).

1966 (3)

B. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
[Crossref]

1965 (1)

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

1964 (3)

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

G. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

1953 (1)

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[Crossref]

Brown, B.

DeVelis, J. B.

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley Publishing Company, Reading, Mass., 1967), p. 80.

Falconer, D. G.

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).

Lohmann, A. W.

A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

B. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
[Crossref] [PubMed]

Paris, D. P.

A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

Parrent, G.

G. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

Reynolds, G. O.

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley Publishing Company, Reading, Mass., 1967), p. 80.

Rhodes, J. E.

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[Crossref]

Stroke, G. W.

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Thompson, B. J.

G. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

Upatnieks, J.

Waters, J. P.

J. P. Waters, J. Opt. Soc. Am. 57, 563A (1967).

J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
[Crossref]

Am. J. Phys. (1)

J. E. Rhodes, Am. J. Phys. 21, 337 (1953).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Letters (2)

G. W. Stroke, Appl. Phys. Letters 6, 201 (1965).
[Crossref]

J. P. Waters, Appl. Phys. Letters 9, 405 (1966).
[Crossref]

J. Opt. Soc. Am. (4)

J. P. Waters, J. Opt. Soc. Am. 57, 563A (1967).

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, J. Opt. Soc. Am. 58, 729A (1968).

A. W. Lohmann and D. P. Paris, J. Opt. Soc. Am. 56, 537A (1966).

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
[Crossref]

Opt. Acta (1)

G. Parrent and B. J. Thompson, Opt. Acta 11, 183 (1964).
[Crossref]

Phys. Letters (1)

G. W. Stroke and D. G. Falconer, Phys. Letters 13, 306 (1964).
[Crossref]

Other (1)

J. B. DeVelis and G. O. Reynolds, Theory and Applications of Holography (Addison–Wesley Publishing Company, Reading, Mass., 1967), p. 80.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram illustrating the method for (a) theoretical construction of a synthetic hologram, H1, by transforming the object, O, positioned a distance z′ from the reference-beam point source, P, through the lens, L1; and (b) experimental reconstruction of the two first-order images, I, a distance ±z″ from the zero-order beam, P″. The reconstruction is formed by passing a collimated coherent light beam through the synthetic hologram H2 and lens L2.

Fig. 2
Fig. 2

Schematic diagram illustrating the extent and characteristics of the object space available for the construction of a synthetic hologram, H1, using a lens L1 of focal length f1. The object space (x′,z′) consists of an unvignetted region (cross hatched) and a vignetted region (dotted). The unvignetted region terminates at the point z′ = lλ1f12/xmax, where l is the spatial frequency on the hologram and xmax the lateral extent of the hologram.

Fig. 3
Fig. 3

Schematic illustrations of (a) the construction process, showing the physical relationships between the object (letters U and A), lens L1, and hologram H1; and (b) the image space in the reconstruction process, showing the positions of the two first-order images of both letters in relationship to the point at which the zero-order beam comes to focus, P″.

Fig. 4
Fig. 4

Photograph of the reconstruction from a Fourier-transform synthetic hologram.

Fig. 5
Fig. 5

A portion of a synthetic hologram, shown 7.5× actual size, and constructed using the Fourier-transform technique to represent the letters U and A.

Fig. 6
Fig. 6

Photograph of the reconstruction from a synthetic Fourier-transform hologram constructed with an auxiliary amplitude-weighting function.

Equations (13)

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U ( x , y ) = ( i k 2 π z ) U ( x , y , z ) × exp { i k [ ( x - x ) 2 + ( y - y ) 2 ] 2 z } ,
U s ( x , y ) = ( i k 2 π f 1 ) V U ( x , y , z ) × exp { - i k 2 [ z f 1 2 ( x 2 + y 2 ) + 2 f 1 ( x x + y y ) ] } d V ,
I = U 0 2 + U 0 U s * + U 0 * U s + U s 2 ,
z = ± z ( λ 2 / λ 1 ) ( f 2 / f 1 ) 2
x = ± x ( f 2 λ 2 ) / ( f 1 λ 1 )
y = ± y ( f 2 λ 2 ) / ( f 1 λ 1 ) .
M long = ( λ 1 / λ 2 ) M 2 lat ,
x = ± [ l λ 1 f 1 - ( z x max / f 1 ) ] ,
x = ± [ l λ 1 f 1 + ( z x max / f 1 ) ] .
S ( x , y ) = sin [ k ( x 2 + y 2 ) / 2 f ]
U s ( x , y ) = V U ( x , y , z ) sin [ k ( x 2 + y 2 ) 2 f 1 ] × exp { - i k 2 [ z f 1 2 ( x 2 + y 2 ) + 2 f 1 ( x x + y y ) ] } d V ;
S ( x , y ) = sin k ( x 2 + y 2 ) 2 f S ˜ ( x , y ) = sin [ k ( x 2 + y 2 ) 2 f - π 4 ] .
S ( x , y ) = exp [ ( i k / 2 f ) ( x 2 + y 2 ) ] ,