Abstract

In this paper we propose an alternative technique for producing digital Fresnel holograms. The evaluation of a diffraction pattern in a wide region is implemented in such a way as to avoid redundant calculations and preserve the precision. Because of the symmetry of the kernel, the complex amplitude is calculated at four points in the registration plane simultaneously. This algorithm decreases the required CPU time 4 times with respect to direct calculation. The digital Fresnel hologram is numerically and optically reconstructed, and some qualitative comparisons are made.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. Toyohiko Yatagai, “Three-dimensional display using computer-generated holograms,” Opt. Commun. 12, 43–45 (1974).
    [CrossRef]
  3. J. P. Waters, “Three-dimensional Fourier-transform method for synthesizing binary holograms,” J. Opt. Soc. Am. 58, 1284–1288 (1968).
    [CrossRef]
  4. D. Leseberg, “Sizable Fresnel-type hologram generated by computer,” J. Opt. Soc. Am. A 6, 229–233 (1989).
    [CrossRef]
  5. D. Leseberg, C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planes segments,” Appl. Phys. 27, 3020–3024 (1988).
  6. D. Leseberg, “Computer-generated three-dimensional image holograms,” Appl. Phys. 31, 223–229 (1992).
  7. D. Leseberg, “Computer generated holograms: cylindrical, conical and helical waves,” Appl. Phys. 26, 4385–4390 (1987).
  8. A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
    [CrossRef]
  9. M. Lucente, “Iterative computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
    [CrossRef]
  10. R. J. Collier, Optical Holography (Academic, San Diego, Calif., 1993).
  11. P. Harihara, Optical Holography, Principles, Techniques and Applications (Cambridge U. Press, Cambridge, UK, 1984).

1993 (1)

M. Lucente, “Iterative computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

1992 (2)

D. Leseberg, “Computer-generated three-dimensional image holograms,” Appl. Phys. 31, 223–229 (1992).

A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

1989 (1)

1988 (1)

D. Leseberg, C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planes segments,” Appl. Phys. 27, 3020–3024 (1988).

1987 (1)

D. Leseberg, “Computer generated holograms: cylindrical, conical and helical waves,” Appl. Phys. 26, 4385–4390 (1987).

1974 (1)

Toyohiko Yatagai, “Three-dimensional display using computer-generated holograms,” Opt. Commun. 12, 43–45 (1974).
[CrossRef]

1968 (1)

Collier, R. J.

R. J. Collier, Optical Holography (Academic, San Diego, Calif., 1993).

Frere, C.

D. Leseberg, C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planes segments,” Appl. Phys. 27, 3020–3024 (1988).

Goodman, J. W.

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Harihara, P.

P. Harihara, Optical Holography, Principles, Techniques and Applications (Cambridge U. Press, Cambridge, UK, 1984).

Leigh, J. S.

A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Leseberg, D.

D. Leseberg, “Computer-generated three-dimensional image holograms,” Appl. Phys. 31, 223–229 (1992).

D. Leseberg, “Sizable Fresnel-type hologram generated by computer,” J. Opt. Soc. Am. A 6, 229–233 (1989).
[CrossRef]

D. Leseberg, C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planes segments,” Appl. Phys. 27, 3020–3024 (1988).

D. Leseberg, “Computer generated holograms: cylindrical, conical and helical waves,” Appl. Phys. 26, 4385–4390 (1987).

Lucente, M.

M. Lucente, “Iterative computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

Stein, A. D.

A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Wang, Z.

A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

Waters, J. P.

Yatagai, Toyohiko

Toyohiko Yatagai, “Three-dimensional display using computer-generated holograms,” Opt. Commun. 12, 43–45 (1974).
[CrossRef]

Appl. Phys. (3)

D. Leseberg, C. Frere, “Computer-generated holograms of 3-D objects composed of tilted planes segments,” Appl. Phys. 27, 3020–3024 (1988).

D. Leseberg, “Computer-generated three-dimensional image holograms,” Appl. Phys. 31, 223–229 (1992).

D. Leseberg, “Computer generated holograms: cylindrical, conical and helical waves,” Appl. Phys. 26, 4385–4390 (1987).

Comput. Phys. (1)

A. D. Stein, Z. Wang, J. S. Leigh, “Computer-generated hologram: a simplified ray-tracing approach,” Comput. Phys. 6, 389–392 (1992).
[CrossRef]

J. Electron. Imag. (1)

M. Lucente, “Iterative computation of holograms using a look-up table,” J. Electron. Imag. 2, 28–34 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Toyohiko Yatagai, “Three-dimensional display using computer-generated holograms,” Opt. Commun. 12, 43–45 (1974).
[CrossRef]

Other (3)

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. J. Collier, Optical Holography (Academic, San Diego, Calif., 1993).

P. Harihara, Optical Holography, Principles, Techniques and Applications (Cambridge U. Press, Cambridge, UK, 1984).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the reference system.

Fig. 2
Fig. 2

Object planes used in this study: (a) INAOE’s logo (binary image). (b) Girl’s face (gray-level image).

Fig. 3
Fig. 3

Computer time employed to make the digital Fresnel holograms.

Fig. 4
Fig. 4

Setup employed in the reconstruction of the digital Fresnel–Kirchhoff hologram.

Fig. 5
Fig. 5

Optical reconstruction of the INAOE logo in the three diffraction planes: (a) pseudoscopic plane, (b) focal plane, and (c) orthoscopic plane.

Fig. 6
Fig. 6

Optical and numerical reconstruction without the lens of the hologram of girl’s face. Both kinds of reconstruction show three diffraction orders: the primary image (reconstructed object), the zero order (square form), and the conjugate image (diffuse form)

Equations (31)

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Ox, y=--aX, YexpikrrcosηdXdY,
Ox, y=sp=1Pq=1QaXp, Yqexpikrx, y, Xp, Yqr2x, y, Xp, Yq,
rx, y, Xp, Yq=Xp-x2+Yq-y2+s21/2
Ax0, y0=p=1Pq=1QaXp, YqKx0, y0, Xp, Yq, A-x0, y0=p=1Pq=1Qa-Xp, YqKx0, y0, Xp, Yq, Ax0, -y0=p=1Pq=1QaXp, -YqKx0, y0, Xp, Yq, A-x0, -y0=p=1Pq=1Qa-Xp, -YqKx0, y0, Xp, Yq,
Kx0, y0, Xp, Yq=expikrx0, y0, Xp, Yqrx0, y0, Xp, Yq2.
Rxn, ym=A expik·rn,m,
k=2π/λcosθxiˆ+cosθyjˆ+cosθzkˆ,
rn,m=xniˆ+ymjˆ+skˆ.
Hxn, ym=A expik·rn,m+sp=1Pq=1QaXp, Yq×expikrxn, ym, Xp, Yqr2xn, ym, Xp, Yq.
Ixn, ym=H xn, ym2.
Ixn, ym=log(Ixn, ym+1,
Ixn, ym=NgIxn, ym-IminImax-Imin,
tx, y=t0+βTIx, y,
f=1λsinθ2.
A=n=1Nm=1MOxn, ym.
rx, y,-X, Y=-X-x2+Y-y21/2=X+x2+Y-y21/2=X--x2+Y-y21/2=r-x, y, X, Y,
rx, y, X, -Y=rx, -y, X, Y, rx, y, -X, -Y=r-x, -y, X, Y.
Kx, y, -X, Y=K-x, y, X, Y,Kx, y, X, -Y=Kx, -y, X, Y,Kx, y, -X, -Y=K-x, -y, X, Y.
S=Xp=2cP/p=1,,P.
S+=Xi/i=1,,P2, S-=Xj/j=P2+1,,P,
X1=-XPX2=-XP-1   XP/2-1=-XP/2+2 XP/2=-XP/2+1.
Xk=-XP-k+1,
XP/2-k+1=-XP/2+k, k=1,,P2.
A-x0=p=1PaXpK-x0, Xp.
A-x0=p=1PaXpKx0, -Xp.
A-x0=k=1P/2aXkKx0, -Xk+k=P/2+1PaXkKx0, -Xk,
A-x0=k=1P/2aXkKx0, -Xk+k=1P/2aXP/2+kKx0, -XP/2+k.
A-x0=i=1P/2aXP-k+1Kx0,-XP-k+1+k=1P/2a-XP/2-k+1Kx0, XP/2-k+1.
l=P-k+1, k=1l=P, k=P2l=P2+1, l=P2-k+1 k=1l=P2 k=P2l=1,
A-x0=l=PP/2+1a-XlKx0, Xl+l=P/21a-XlKx0, Xl.
A-x0=l=1Pa-XlKx0, Xl.

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