Abstract

A theory for making a disk-type multiplex hologram is derived. This theory relates the final image point, as seen by the observer, to a point on the original three-dimensional object through a set of equations. From these equations the distortion of the image and the wavelength as seen by each individual eye can be evaluated. Computer simulation shows the characteristics of this hologram. Some experimental results also confirm these characteristics. By reversing the process and specifying a desired image, we generated a set of distorted two-dimensional originals. The hologram fabricated with these distorted images can generate nearly distortion-free images.

© 1999 Optical Society of America

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References

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  1. S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545–1546 (1969).
  2. G. Saxby, Practical Holography, 2nd ed. (Prentice-Hall, New York, 1994), pp. 308–311.
  3. L. Huff, R. L. Fusek, “Color holographic stereograms,” Opt. Eng. 19, 691–695 (1980).
    [CrossRef]
  4. E. N. Leith, P. Voulgaris, “Multiplex holography: some new methods,” Opt. Eng. 24, 171–175 (1985).
    [CrossRef]
  5. N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
    [CrossRef]
  6. S. A. Benton, “Alcove holograms for computer-aided design,” in True Three-Dimensional Imaging Techniques and Display Technologies, D. F. McAllister, W. E. Robbins, eds., Proc. SPIE761, 1–9 (1987).
  7. K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
    [CrossRef]
  8. L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
    [CrossRef]

1995 (1)

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

1989 (1)

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

1987 (1)

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

1985 (1)

E. N. Leith, P. Voulgaris, “Multiplex holography: some new methods,” Opt. Eng. 24, 171–175 (1985).
[CrossRef]

1980 (1)

L. Huff, R. L. Fusek, “Color holographic stereograms,” Opt. Eng. 19, 691–695 (1980).
[CrossRef]

1969 (1)

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545–1546 (1969).

Benton, S. A.

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545–1546 (1969).

S. A. Benton, “Alcove holograms for computer-aided design,” in True Three-Dimensional Imaging Techniques and Display Technologies, D. F. McAllister, W. E. Robbins, eds., Proc. SPIE761, 1–9 (1987).

Fusek, R. L.

L. Huff, R. L. Fusek, “Color holographic stereograms,” Opt. Eng. 19, 691–695 (1980).
[CrossRef]

Honda, T.

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

Huff, L.

L. Huff, R. L. Fusek, “Color holographic stereograms,” Opt. Eng. 19, 691–695 (1980).
[CrossRef]

Leith, E. N.

E. N. Leith, P. Voulgaris, “Multiplex holography: some new methods,” Opt. Eng. 24, 171–175 (1985).
[CrossRef]

Minami, Y.

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

Murillo-Mora, L. M.

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

Ohyama, N.

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

Okada, K.

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

Saxby, G.

G. Saxby, Practical Holography, 2nd ed. (Prentice-Hall, New York, 1994), pp. 308–311.

Tsujiuchi, J.

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

Voulgaris, P.

E. N. Leith, P. Voulgaris, “Multiplex holography: some new methods,” Opt. Eng. 24, 171–175 (1985).
[CrossRef]

Watanabe, A.

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

Yamaji, Y.

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

Yoshii, S.

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545–1546 (1969).

Opt. Commun. (2)

N. Ohyama, Y. Minami, A. Watanabe, J. Tsujiuchi, T. Honda, “Multiplex holograms of a skull made of CT images,” Opt. Commun. 61, 96–99 (1987).
[CrossRef]

K. Okada, S. Yoshii, Y. Yamaji, J. Tsujiuchi, “Conical holographic stereograms,” Opt. Commun. 73, 347–350 (1989).
[CrossRef]

Opt. Eng. (3)

L. M. Murillo-Mora, K. Okada, T. Honda, J. Tsujiuchi, “Color conical holographic stereogram,” Opt. Eng. 34, 814–817 (1995).
[CrossRef]

L. Huff, R. L. Fusek, “Color holographic stereograms,” Opt. Eng. 19, 691–695 (1980).
[CrossRef]

E. N. Leith, P. Voulgaris, “Multiplex holography: some new methods,” Opt. Eng. 24, 171–175 (1985).
[CrossRef]

Other (2)

G. Saxby, Practical Holography, 2nd ed. (Prentice-Hall, New York, 1994), pp. 308–311.

S. A. Benton, “Alcove holograms for computer-aided design,” in True Three-Dimensional Imaging Techniques and Display Technologies, D. F. McAllister, W. E. Robbins, eds., Proc. SPIE761, 1–9 (1987).

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Figures (10)

Fig. 1
Fig. 1

Object point A 0 rotates about the z axis. The CCD camera takes the image for every unit rotation of the object.

Fig. 2
Fig. 2

Anamorphic optical system for multiplex hologram recording. The image magnification from the CCD detector to the LCTV is k l . From the LCTV to the horizontal image plane the magnification ratio is k u . From the LCTV to the plane of the slit, the magnification ratio is k v [R/(R + h)]. D is the horizontal image of the object point on the LCTV. The fan-shaped slit is utilized to limit the recording area of the individual hologram.

Fig. 3
Fig. 3

Geometry showing the orientation of the object ray DAE¯ with respect to the recording film at the slit plane. D is the horizontal image, and A is the vertical image of the object point. The object ray is characterized by the three parameters b, θ, and ϕ a .

Fig. 4
Fig. 4

Geometric relationship among object ray ã, reference ray r̃, reconstruction reference ray r̃′, and reconstruction object ray ã′.

Fig. 5
Fig. 5

Disk-type multiplex hologram is placed on the uv plane with its center I at the top of the figure. The object ray ã′ is reconstructed by the reference wave from s′ (s′, reconstruction reference source; s, reference source). In the recording of hologram the object ray and the reference wave from s produce interference pattern at position A. When the hologram is rotated by an angle δ and when this interference pattern is moved to point B, the reconstructed object ray ã′ is seen by the left eye (E l ) of the observer.

Fig. 6
Fig. 6

Configuration showing the image point A0 seen by the left eye (a l , 0, R) and the right eye (a r , 0, R) through different holograms at A l and A r . Both eyes are situated on the uw plane. The hologram is placed on the uv plane.

Fig. 7
Fig. 7

(a) The object taken by the CCD camera at two different directions. (b) These two photographs are recorded as hologram IAf¯ and IBf¯, which are seen by two eyes of the observer correspondingly. For the left eye of the observer, the image A0 is seen through hologram IAf¯ and B1 is seen through hologram IBf¯. The two image points A0 and B1 correspond to the object points A 0 and B 1. Hence a distorted image is seen by an individual eye of the observer.

Fig. 8
Fig. 8

(a) Reconstructed image for α0 = 0° that shows that the horizontal lines curve downward. (b) Reconstructed image for α0 = 90° in which the horizontal lines curve upward. (c) and (d) Two views of the least distorted image for α0 ≈ 34.3°.

Fig. 9
Fig. 9

Two views of the distorted object generated by specification of a desired cube image.

Fig. 10
Fig. 10

Experimental reconstruction of a distortion corrected image from the disk-type multiplex hologram fabricated with the 2D distorted originals. The vertical lines are the images of individual holograms.

Equations (55)

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lcos α0 cos αϕ+sin α0 sin αθ cos αϕ, l sin αθ sin αϕ, lsin α0 cos αθ-cos α0 sin αθ cos αϕ.
x, y=zil sin αθ sin αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ,×zilsin α0 cos αθ-cos α0 sin αθ cos αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ.
u, v=zik1kul sin αθ sin αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ,×zik1kvlsin α0 cos αθ-cos α0 sin αθ cos αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ.
b=GC¯R/R+h,
GC¯=zik1kvlsin α0 cos αθ-cos α0 sin αθ cos αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ.
ϕa=tan-1b/R,
θ=tan-1CD¯/AC¯,
CD¯=zik1kul sin αθ sin αϕz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕ,
AC¯=hcos ϕa.
ã=a exp-iku1 sin θ0+ikw cos θ0,
u1=u cos ϕc+v sin ϕc.
r˜=r expikv sin η+ikw cos η.
fu=sin θ0λcos ϕc,
fv=sin θ0λsin ϕc+sin ηλ,
r˜=r expikv sin η+ikw cos η,
ã=a exp-iku2 sin θ+ikw cos θ,
v=v cos ϕ-u sin ϕ,
u2=u cos ψ+v sin ψ.
sin θ0 cos ϕc=sin θ,
sin θ0 sin ϕc=cos θ sin ϕa,
sin θ=λλsin θ cos ψ-sin η sin ϕ,
cos θ sin ϕa+sin η=λ/λsin θ sin ψ+sin η cos ϕ.
m-bsin δ=t-p-m-bcos δtan ϕ,
ϕ=δ+ϕ=δ+tan-1×m-bsin δt-p-m-bcos δ,
n sin ϕ=m-bsin δ,
n=m-bsin δsin ϕ=m-bsin δsinϕ-δ,
η=tan-1t-m+bg,
η=tan-1ng-q=tan-1m-bsin δg-qsinϕ-δ,
ψ=tan-1σ m-m-bcos δ-a+m-bsin δ,
θ=tan-1×m-bcos δ-m2+σa-m-bsin δ21/2R,
sin θj=σ λλjsin θj cos ψj-sin ηj sin ϕj,
cos θj sin ϕaj+sin ηj=λλjsin θj sin ψj+sin ηj cos ϕj
bj=RR+hzik1kvlsin α0 cos αθ-cos α0 sin αθ cos αϕjz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕj,
ϕaj=tan-1bj/R,
θj=tan-1zik1kul sin αθ sin αϕj1/hz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕj,
ϕj=-σδj-σ tan-1m-bjsin δjt-p-m-bjcos δj,
ηj=tan-1t-m+bjg,
ηj=tan-1-σm-bjsin δjg-qsinϕj+δj,
ψj=tan-1m-m-bjcos δj-σa+m-bjsin δj,
θj=tan-1×m-bjcos δj-m2+σa-m-bjsin δj21/2R.
-δr+δl=αϕl-αϕr,
Alm-blsin δl, m-m-blcos δl, 0 and Arm-br×sin δr, m-m-brcos δr, 0,
Ll: u-alal-m-blsin δl=v-m+m-blcos δl=w-RR,
Lr: u-arar-m-brsin δr=v-m+m-brcos δr=w-RR.
δB1l-δA0l=αϕA0l-αϕB1l+π/2,
sin θA0j=σ λλA0jsin θA0j cos ψA0j-sin ηA0j sin ϕA0j,
cos θA0j sin ϕaA0j+sin ηA0j=λ/λA0jsin θA0j sin ψA0j+sin ηA0j cos ϕA0j,
bA0j=RR+h×zik1kvlsin α0 cos αθ-cos α0 sin αθ cos αϕA0jz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕA0j,
ϕaA0j=tan-1bA0j/R,
θA0j=tan-1×zik1kul sin αθ sin αϕA0j1/hz0-lcos α0 cos αθ+sin α0 sin αθ cos αϕA0j,
ϕA0j=-σδA0l-σ tan-1m-bA0jsin δA0jt-p-m-bA0jcos δA0j,
ηA0j=tan-1t-m+bA0jg,
ηA0j=tan-1-σm-bA0jsin δA0jg-qsinϕA0j+δA0j,
ψA0j=tan-1m-m-bA0jcos δA0j-σa+m-bA0jsin δA0j,
θA0j=tan-1{m-bA0j)cos δA0j-m2+σa-m-bA0jsin δA0j2}1/2R,

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