Abstract

Spatial and temporal coherence requirements for off-axis reference-beam holograms are reduced to those for in-line holograms by using interferometer arrangements producing achromatic fringes. Using this technique, we have produced high-quality holograms of back-lighted objects, using a high-pressure mercury-arc lamp.

© 1967 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. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
    [CrossRef]
  3. R. W. Ditchburn, Light (Blackie & Son, Ltd., London, 1963), 2nd ed., pp. 148–152.
  4. A. W. Lohmann, Optica Acta 9, 1 (1962).
    [CrossRef]
  5. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [CrossRef]
  6. W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
    [CrossRef]
  7. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [CrossRef]
  8. D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).
  9. The transition from Eq. (4) to Eq. (5) involves a rather great amount of algebra, as well as the evaluation of a Fresnel integral. A similar calculation is given in detail by Vander Lugt [A. Vander Lugt, Proc. IEEE 54, 1055 (1966)].
    [CrossRef]
  10. The holographic system is linear in the usual sense that, if a signal s(x3,y3) produces a recorded holographic signal s0(x4,y4), then a signal a1s1(x3,y3)+a2s2(x3,y3) produces the holographic signal a1s01(x4,y4)+a2s02(x4,y4), where s0, s01, s02 refer to the real or the virtual-image term, or both. The recording system, since it is a square-law detector, is, of course, nonlinear but that does not violate the criterion of linearity given above.
  11. L. Mandel, J. Opt. Soc. Am. 56, 1636 (1966).
    [CrossRef]

1966 (5)

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. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
[CrossRef]

The transition from Eq. (4) to Eq. (5) involves a rather great amount of algebra, as well as the evaluation of a Fresnel integral. A similar calculation is given in detail by Vander Lugt [A. Vander Lugt, Proc. IEEE 54, 1055 (1966)].
[CrossRef]

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

1963 (1)

1962 (2)

1949 (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

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. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

Carter, W. H.

W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
[CrossRef]

Ditchburn, R. W.

R. W. Ditchburn, Light (Blackie & Son, Ltd., London, 1963), 2nd ed., pp. 148–152.

Dougal, A. A.

W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
[CrossRef]

Engeling, P. D.

W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
[CrossRef]

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Gates, J. W.

J. M. Burch, J. W. Gates, R. G. N. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

Heflinger, L. O.

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

Hill, R. G. N.

J. M. Burch, J. W. Gates, R. G. N. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

Leith, E. N.

Lohmann, A. W.

A. W. Lohmann, Optica Acta 9, 1 (1962).
[CrossRef]

Mandel, L.

Tanner, L. H.

J. M. Burch, J. W. Gates, R. G. N. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

Upatnieks, J.

Vander Lugt, A.

The transition from Eq. (4) to Eq. (5) involves a rather great amount of algebra, as well as the evaluation of a Fresnel integral. A similar calculation is given in detail by Vander Lugt [A. Vander Lugt, Proc. IEEE 54, 1055 (1966)].
[CrossRef]

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 (2)

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

W. H. Carter, P. D. Engeling, and A. A. Dougal, IEEE J. Quantum Electronics QE-2, 44 (1966).
[CrossRef]

J. Opt. Soc. Am. (3)

Nature (1)

J. M. Burch, J. W. Gates, R. G. N. Hill, and L. H. Tanner, Nature 212, 1347 (1966).Burch has used the scatter-plate method to produce holograms with a mercury-arc source (Burch, private conversation, January1967);and he has also described a system in which a diffraction grating was imaged onto the hologram to provide a white-light image-plane hologram [J. M. Burch and A. E. Ennos, J. Opt. Soc. Am. 56, 541A (1966)].
[CrossRef]

Optica Acta (1)

A. W. Lohmann, Optica Acta 9, 1 (1962).
[CrossRef]

Proc. IEEE (1)

The transition from Eq. (4) to Eq. (5) involves a rather great amount of algebra, as well as the evaluation of a Fresnel integral. A similar calculation is given in detail by Vander Lugt [A. Vander Lugt, Proc. IEEE 54, 1055 (1966)].
[CrossRef]

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London) A197, 454 (1949).

Other (2)

R. W. Ditchburn, Light (Blackie & Son, Ltd., London, 1963), 2nd ed., pp. 148–152.

The holographic system is linear in the usual sense that, if a signal s(x3,y3) produces a recorded holographic signal s0(x4,y4), then a signal a1s1(x3,y3)+a2s2(x3,y3) produces the holographic signal a1s01(x4,y4)+a2s02(x4,y4), where s0, s01, s02 refer to the real or the virtual-image term, or both. The recording system, since it is a square-law detector, is, of course, nonlinear but that does not violate the criterion of linearity given above.

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Figures (4)

F. 1
F. 1

Grating achromatic-fringe interferometer. Grating spacing is 200 lines/mm over 3-cm square, lens focal length F=56 cm, D = 22 to 32 cm. Source spectral width about 100 Å.

F. 2
F. 2

Reconstruction of transparent letters on opaque background. The hologram was recorded on Kodak SO-243 film, with a 50-μ pinhole source 115 cm from the collimating lens and required 4-min exposure. The linewidth of the lettering was 33 μ.

F. 3
F. 3

Reconstruction of a transparency, back illuminated with a uniform wavefront. The hologram was recorded on a Kodak 649F plate, with a 500-μ pinhole 115 cm from a collimating lens. The size of the object transparency was 17×21 mm.

F. 4
F. 4

Reconstruction of a silhouetted scene produced by placing opaque objects in front of a sheet of ground glass which was illuminated from behind. The figures were 22-mm high, the bicycle 31 cm from the hologram, and the other figure 13 cm from the hologram. The hologram was recorded on Kodak SO–243 film with 10-min exposure, 50-μ pinhole.

Tables (1)

Tables Icon

Table I Temporal-coherence requirements and spatial-carrier values for various arrangements.

Equations (16)

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V ( t ) = 1 2 π 0 υ ( ω ) e i ω t d ω ,
a 0 + a 1 cos α x 1 ,
u 1 = i 4 π λ F 0 υ ( ω ) ( a 0 + a 1 cos α x 1 ) exp { i ( π / 2 λ F ) [ ( x f x 1 ) 2 + ( y f y 1 ) 2 ] } e i ω t d x 1 d y 1 d ω ,
u 2 = i λ F u 1 exp [ i π λ F ( x f 2 + y f 2 ) ] exp { i ( π / λ F ) [ ( x 2 x f ) 2 + ( y 2 y f ) 2 ] } d x f d y f .
u 2 = C υ ( ω ) exp [ i π λ F ( x 2 2 + y 2 2 ) a 0 exp [ i ( 2 π / λ F ) ( x 2 x 1 + y 2 y 1 ) ] + ( a 1 / 2 ) exp { i [ ( 2 π / λ F ) ( x 2 x 1 + y 2 y 1 ) + α x 1 ] } + ( a 1 / 2 ) exp { i [ ( 2 π / λ F ) ( x 2 x 1 + y 2 y 1 ) α x 1 ] } e i ω t d x 1 d y 1 d ω ,
u 2 = C υ ( ω ) exp [ i π λ F ( x 2 2 + y 2 2 ) ] [ a 0 δ ( 2 π λ F x 2 , 2 π λ F y 2 ) + a 1 2 δ ] α + 2 π λ F x 2 , 2 π λ F y 2 [ + a 1 2 δ ( α 2 π λ F x 2 , 2 π λ F y 2 ) ] e i ω t d ω ,
r = C 1 υ ( ω ) exp ( i λ α 1 2 F 4 π ) exp { i ( π / λ F ) [ ( x 4 λ α 1 F / 2 π ) 2 + y 4 2 ] } e i ω t d ω .
s = C 2 υ ( ω ) exp ( i λ α 2 2 F 4 π ) exp { i ( π / λ D ) [ ( x 3 λ α 2 F / 2 π ) 2 + y 3 2 ] } d ω = C 2 υ ( ω ) exp { i [ k 1 k 3 ( x 3 k 2 ) 2 k 3 y 3 2 ] } e i ω t d ω ,
s exp ( i β x 3 ) = C 2 υ ( ω ) exp i [ k 1 k 3 ( x 3 k 2 ) 2 + β x 3 k 3 y 3 2 ] d ω = C 2 υ ( ω ) exp i { k 1 k 3 [ x 3 ( k 2 + β / 2 k 3 ) ] 2 k 3 y 3 2 + k 2 β + β 2 / 4 k 3 } e i ω t d ω .
θ = k 1 + k 2 β + β 2 / 4 k 3
x 2 = k 2 + β / 2 k 3 ,
s 0 = C 2 υ ( ω ) exp i ( k 1 + k 2 β + β 2 / 4 k 3 ) exp i ( π / λ F ) [ x 4 ( k 2 + β / 2 k 3 ) ] 2 e i ω t d ω ,
| r + s 0 | 2 = ( | C 1 | 2 + | C 2 | 2 ) ϕ ( ω ) d ω + | 2 C 1 C 2 | ϕ ( ω ) cos { k 1 + k 2 β + ( β 2 / 4 k 3 ) ( π / λ F ) [ x 4 ( k 2 + β / 2 k 3 ) ] 2 ( λ α 1 2 F / 4 π ) + ( π / λ F ) ( x 4 λ α 1 F / 2 π ) 2 } d ω ,
| r + s 0 | 2 = 2 | C 3 | 2 ϕ ( ω ) ( b 0 + b 1 cos { [ α 2 α 1 + ( D / F ) β ] x 4 + ( λ β / 2 π ) ( 1 D / F ) ( α 2 F + 1 2 β D ) } ) d ω ,
Δ λ = 2 π 2 / [ β 2 D ( 1 D F ) ( 1 + 2 α 2 F β D ) ] .
Δ λ = 2 π 2 D β 2 ( 1 D / F ) = 2 π 2 D β 2 ( 1 D / F ) ( 1 + 2 α F / β D )