Abstract

A direct-search method for the computer design of holograms is demonstrated. Two-dimensional, three-dimensional, and containing holograms with different levels of intensity (gray scales) were designed, fabricated, and optically reconstructed. The number of effective gray levels and the image-intensity noise and contrast are discussed. A modification of the state-variables cost function used in the direct-search algorithm that permits reliable control of gray scales is presented. Optical reconstructions of two-dimensional, three-dimensional, and gray-scale binary phase computer-generated holograms are presented.

© 1999 Optical Society of America

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References

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  1. M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
    [CrossRef]
  2. A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.
  3. R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
    [CrossRef]
  4. N. C. Gallagher, D. Sweeney, “Computer generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
    [CrossRef]
  5. M. T. Eismann, A. M. Tai, J. N. Cedrquist, “Holographic beamformer designed by an iterative technique,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 10–18 (1989).
  6. B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [CrossRef]
  7. M. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
    [CrossRef]
  8. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  9. B. K. Jennison, J. P. Allebach, “Direct binary search computer generated hologram: an accelerated design technique and measurement of wave-front quality,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 2–9 (1989).
  10. M. Clark, “A direct search method for the computer design of holograms,” Ph.D. dissertation (Imperial College, London, 1997).
  11. M. Clark, “Enhanced direct-search method for the computer design of holograms using state variables,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 24–34 (1996).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1974), Chap. 8.8.

1998 (1)

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

1996 (1)

M. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

1989 (3)

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

N. C. Gallagher, D. Sweeney, “Computer generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

1987 (1)

Allebach, J. P.

B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

B. K. Jennison, J. P. Allebach, “Direct binary search computer generated hologram: an accelerated design technique and measurement of wave-front quality,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 2–9 (1989).

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1974), Chap. 8.8.

Burge, R. E.

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

Cedrquist, J. N.

M. T. Eismann, A. M. Tai, J. N. Cedrquist, “Holographic beamformer designed by an iterative technique,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 10–18 (1989).

Clark, M.

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

M. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

M. Clark, “A direct search method for the computer design of holograms,” Ph.D. dissertation (Imperial College, London, 1997).

M. Clark, “Enhanced direct-search method for the computer design of holograms using state variables,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 24–34 (1996).
[CrossRef]

Coleman, C.

A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.

Eismann, M. T.

M. T. Eismann, A. M. Tai, J. N. Cedrquist, “Holographic beamformer designed by an iterative technique,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 10–18 (1989).

Gallagher, N. C.

N. C. Gallagher, D. Sweeney, “Computer generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

Gmitrio, A. F.

A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.

Hall, T. J.

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

Hoptroff, R. G.

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

Hossack, W. J.

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

Jennison, B. K.

B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

B. K. Jennison, J. P. Allebach, “Direct binary search computer generated hologram: an accelerated design technique and measurement of wave-front quality,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 2–9 (1989).

Keller, P. E.

A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.

Linnane, F.

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

Maker, P. D.

A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.

McOwan, P. W.

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

Seldowitz, M. A.

Sharples, S. D.

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

Smith, R.

M. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

Somekh, M. G.

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

Sweeney, D.

N. C. Gallagher, D. Sweeney, “Computer generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

Sweeney, D. W.

B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

Tai, A. M.

M. T. Eismann, A. M. Tai, J. N. Cedrquist, “Holographic beamformer designed by an iterative technique,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 10–18 (1989).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1974), Chap. 8.8.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. Clark, F. Linnane, S. D. Sharples, M. G. Somekh, “Frequency control in laser ultrasound with computer generated holography,” Appl. Phys. Lett. 72, 1963–1965 (1998).
[CrossRef]

Opt. Commun. (2)

R. G. Hoptroff, P. W. McOwan, T. J. Hall, W. J. Hossack, R. E. Burge, “Two optimisation approaches to COHOE design,” Opt. Commun. 73, 188–194 (1989).
[CrossRef]

M. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

Opt. Eng. (2)

N. C. Gallagher, D. Sweeney, “Computer generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

B. K. Jennison, D. W. Sweeney, J. P. Allebach, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Other (6)

M. T. Eismann, A. M. Tai, J. N. Cedrquist, “Holographic beamformer designed by an iterative technique,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 10–18 (1989).

A. F. Gmitrio, P. E. Keller, C. Coleman, P. D. Maker, “Design and fabrication of multi-level phase holograms for on-axis optical interconnects,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 239–242.

B. K. Jennison, J. P. Allebach, “Direct binary search computer generated hologram: an accelerated design technique and measurement of wave-front quality,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 2–9 (1989).

M. Clark, “A direct search method for the computer design of holograms,” Ph.D. dissertation (Imperial College, London, 1997).

M. Clark, “Enhanced direct-search method for the computer design of holograms using state variables,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 24–34 (1996).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1974), Chap. 8.8.

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Figures (8)

Fig. 1
Fig. 1

Flowchart of the simple direct-search algorithm.

Fig. 2
Fig. 2

Schematic diagram showing the parameters used to determine the sampling conditions. The hologram is at the left-hand side of the figure, and the light travels from the left, through the hologram, and then propagates a distance F to the image.

Fig. 3
Fig. 3

Three two-dimensional continuous images reconstructed optically from three separate CGH’s. The diagram on the left-hand side shows the reconstruction geometry. The circular patch visible in the centers of the images is the residual zero order.

Fig. 4
Fig. 4

Three two-dimensional images reconstructed optically at different focal lengths from the same CGH. The diagram on the left-hand side shows the reconstruction geometry. The images are arranged such that the in-focus images are not covered by the out-of-focus images.

Fig. 5
Fig. 5

Three two-dimensional images reconstructed optically at different focal lengths from the same CGH. The diagram on the left-hand side shows the reconstruction geometry. The images are all centered on the optical axis so that the in-focus image is partially obscured by the out-of-focus images.

Fig. 6
Fig. 6

Three-dimensional continuous image reconstructed optically from a CGH. The diagram on the left-hand side shows the reconstruction geometry. The upper images on the right-hand side are optical reconstructions of a CGH designed to produce a three-dimensional image; the lower images are optical reconstructions of a CGH designed to produce a two-dimensional image. On the left-hand side the images are reconstructed on a plane tilted by 75° to the normal image plane; on the right-hand side the images are on the normal image plane (perpendicular to the optical axis). The depth of the image is approximately 5 times the depth of focus of the CGH.

Fig. 7
Fig. 7

Gray-level image reconstructed optically from a CGH. This CGH contains three intensity levels with a contrast ratio between the least and the most bright levels of 1:10. (a) An image of the outline of a motor vehicle gasket. The images in (b) and (c) side show details of the image in (a). In (b) and (c) the intensity has been reduced and the image magnified to show that the central feature is made up of discrete points. The reconstruction geometry is the same as that shown in Fig. 3.

Fig. 8
Fig. 8

Diagram showing the construction used to estimate the area of the -1 order.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

C=-aI¯+bσI,
ΔUI-i2λ Usexp-iϕr+ϕd+ϕsR OΔS,
dH<λFDI,
dI<12λFDH.
dZ<λF2DH2.
C=-aI¯W+bσIW,
C=-aI¯WW+bσIWW=-aI¯+bσIWW,
I¯WW=j=1N WjIj/Wjj=1N Wj=I¯,  j=1N Wj=N,
σIWW=j=1N WjIj/Wj-I¯WW2j=1N Wj1/2=1Nj=1N WjIj/Wj-I¯21/2.
C=I¯+1I¯-1A-1A+1π2DH+DI21/2Nλ/2NA2=8π2DH+DI2DH2NF2λ2.
δII2 U-1U+12A+1A-11/221C1/2,  SNRC2,

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