Abstract

Fourier-transform holograms are produced by using achromatic-fringe interferometer arrangements in which the beam splitting is achieved by offset Fresnel zone plates. The effects of the size and the spectral bandwidth of the spatially incoherent source are discussed. Coherence requirements of the source are the same as those for in-line holograms. Experimental examples with object transparencies and a high-pressure mercury-arc lamp are given.

© 1969 Optical Society of America

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References

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  1. R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
    [Crossref]
  2. J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
    [Crossref]
  3. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 57, 975 (1967).
    [Crossref]
  4. John T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 1362 (1966).
    [Crossref]
  5. L. Mandel, J. Opt. Soc. Am. 56, 1636 (1966).
    [Crossref]
  6. Albert V. Baez, J. Opt. Soc. Am. 42, 756 (1952).
    [Crossref]

1967 (1)

1966 (4)

John T. Winthrop and C. R. Worthington, J. Opt. Soc. Am. 56, 1362 (1966).
[Crossref]

L. Mandel, J. Opt. Soc. Am. 56, 1636 (1966).
[Crossref]

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
[Crossref]

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

1952 (1)

Baez, Albert V.

Brooks, R. E.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
[Crossref]

Burch, J. M.

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Gates, J. W.

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Heflinger, L. O.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
[Crossref]

Hills, R. G. N.

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Leith, E. N.

Mandel, L.

Tanner, L. H.

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

Upatnieks, J.

Winthrop, John T.

Worthington, C. R.

Wuerker, R. F.

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
[Crossref]

IEEE J. Quantum Electronics (1)

R. E. Brooks, L. O. Heflinger, and R. F. Wuerker, IEEE J. Quantum Electronics QE-2, 275 (1966).
[Crossref]

J. Opt. Soc. Am. (4)

Nature (1)

J. M. Burch, J. W. Gates, R. G. N. Hills, and L. H. Tanner, Nature 212, 1347 (1966).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Fresnel zone-plate achromatic-fringe interferometer. A Fresnel zone plate is set at H1 to produce diffracted wavefronts of different curvatures; the zero-order wave is focused at Hc and one of the first orders is focused at HR. Holograms are recorded at plane HM where the zone-plate is imaged by a lens at H2.

Fig. 2
Fig. 2

Modified Fresnel zone-plate achromatic-fringe interferometer. The zone plate is illuminated by a convergent beam to give the increased distance of D ¯ = 50 mm. The illuminated area on the object is increased by the use of lens L, while the other system parameters are the same as given in Fig. 3. The area of the light flux at the zone plate is reduced to one half of that at L.

Fig. 3
Fig. 3

Reconstruction of a Fourier-transform hologram. The hologram was recorded on a copy film, with a 200-μ pinhole source 750 mm from the collimating lens. Exposure of the order of 30 sec was required.

Fig. 4
Fig. 4

Production of quasi-Fourier-transform holograms with Fresnel zone-plate achromatic-fringe systems. Transparent film T is inserted to compensate the path difference produced by the object transparency O. The size of the zone-plate was 15 × 15 mm, lens focal lengths F1 = 750 mm and F2 = 50 mm, D ¯ = 10 mm, 2a = b = 150 mm.

Fig. 5
Fig. 5

Fourier-transform hologram (magnified 50X) and reconstruction of a transparency. The hologram was recorded on a copy film capable of recording about 150 lines/mm, with a 100-μ pinhole source 750 mm from the collimating lens and required 13-min exposure. The original object was a microphotograph of 5 × 3 mm.

Fig. 6
Fig. 6

Reconstruction of the same transparency with Fig. 4. The diameters of the sources were increased to 200 μ in (a) and 400 μ in (b).

Equations (25)

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a 0 + a 1 exp [ - i π λ 0 F 3 ( r 1 - d ) 2 ] + a 1 exp [ i π λ 0 F 3 ( r 1 - d ) 2 ] ,
u = i 2 λ F 1 H 1 { a 0 + a 1 exp [ - i π λ F 3 ( r 1 - d ) 2 ] } × exp [ - i π λ F 1 ( r 1 - r s ) 2 ] exp ( i π λ F 1 r 1 2 ) × exp [ - i π 2 λ F 2 ( r 2 - r 1 ) 2 ] d r 1
u z = i λ Z H 2 u exp ( i π λ F 2 r 2 2 ) exp [ - i π λ Z ( r z - r 2 ) 2 ] d r 2 .
- exp ( - i α 2 x 2 ) d x = ( 1 - i ) π / ( 2 α )
1 2 π - exp ( i k x ) d x = δ ( k ) ,
u c = C 1 exp { i π / λ [ 2 F 2 ( r s / F 1 ) 2 - r s 2 / F 1 - r c 2 / F 2 ] } × δ ( r s / F 1 F 2 + r c ) ,
1 / Z 0 + 1 / ( F 3 + 2 F 2 ) = 1 / F 2 ,
u R = C 2 exp { i π λ [ 2 F 2 F 3 2 F 2 + F 3 ( r s F 1 + d F 3 ) - r s 2 F 1 - d 2 F 3 - F 3 + F 2 F 2 ( F 3 + 2 F 2 ) r R 2 ] } δ [ F 2 F 3 F 3 + F 2 ( r s F 1 + d F 3 ) + r R ] ,
D ¯ = F 2 2 / ( F 2 + F ¯ 3 ) ,
u 0 = i λ D ¯ H c u c exp [ - i π λ D ¯ ( r 0 - r c ) 2 ] d r c .
u s = i λ ( F 2 - D ¯ ) H 0 u 0 t ( r 0 ) × exp [ - i π λ ( F 2 - D ¯ ) ( r - r 0 ) 2 ] d r 0 .
u s = C 3 exp ( - i π λ F 1 r s 2 ) exp [ - i π λ ( r 2 F 2 + 2 r s F 1 r ) ] × τ ( r , r s ) λ ,
τ ( r , r s ) λ = H 0 t ( r 0 ) exp { - i π B [ r 0 - ( r M - F 2 - D ¯ F 1 r s ) ] 2 } d r 0 ,
u r = i λ ( F 2 - D ) H R u R × exp [ - i π λ ( F 2 - D ) ( r - r R ) 2 ] d r R
u r = C 4 exp { - i π λ F 2 [ F 3 + F 2 F 3 r 2 + 2 F 2 ( r s F 1 + d F 3 ) r + F 2 F 1 r s 2 + F 2 F 3 d 2 ] } .
u s + u r 2 = u s 2 + u r 2 + u s u r * + u s * u r ,
u s u r * = C τ ( r , r s ) λ exp [ i π λ 0 F 3 ( r + d ) 2 ] .
ϕ ( ω ) u s + u r 2 d ω d r s = ϕ ( ω ) ( u s 2 + u r 2 ) d ω d r s + ϕ ( ω ) u s * u r d ω d r s + ϕ ( ω ) u s u r * d ω d r s ,
τ ( r , r s ) λ H 0 t ( r 0 ) exp [ - i π M ( F 2 - D ¯ ) λ ¯ ( r 0 - r M ) 2 ] × exp [ i π M ( F 2 - D ¯ ) λ ¯ 2 Δ r 2 Δ λ ] exp ( i 2 π λ ¯ F 1 Δ r r s ) d r 0 ,
2 Δ λ λ ¯ 2 M ( F 2 - D ¯ ) Δ r m - 2 ,
2 Δ r s ( 1 / 2 ) ƛ F 1 Δ r m - 1 ,
d A = λ ¯ / ( 2 sin γ m ) = λ ¯ / [ 2 Δ r m ( F 2 - D ¯ ) - 1 ] .
2 Δ λ = 4 M ( F 2 - D ¯ ) - 1 d A 2 ,
2 Δ r s = F 1 ( F 2 - D ¯ ) - 1 d A .
2 Δ r c = F 2 ( F 2 - D ¯ ) - 1 d A .