Abstract

We propose a unified approach to the multicriteria design of diffractive optics. A multicriteria version of the direct binary search that allows the user to adjust the compromise between the diffraction efficiency and the signal-to-noise ratio already exists. This technique has proved to be extremely powerful but also very time consuming. We extend this multicriteria approach to the iterative Fourier transform algorithm, which helps to reduce the computation time dramatically, especially for multilevel domains. Simulations as well as experimental validations are provided.

© 2001 Optical Society of America

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References

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  1. 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]
  2. D. C. Youla, “Image restoration by the method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  3. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  4. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  5. L. Legeard, P. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer-generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
    [CrossRef]
  6. P. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  7. F. Wyrowski, “Diffraction efficiency of analog and quantized digital amplitude holograms: analysis and manipulation,” J. Opt. Soc. Am. A 7, 383–393 (1990).
    [CrossRef]
  8. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  9. H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.
  10. J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
    [CrossRef]
  11. F. Fetthauer, C. Stroot, O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136, 7–10 (1997).
    [CrossRef]
  12. K.-H. Brenner, “Method for designing arbitrary two-dimensional continuous phase elements,” Opt. Lett. 25, 31–33 (2000).
    [CrossRef]
  13. L. Legeard, “Étude et optimisation d’hologrammes synthétisés par ordinateur: application au stockage d’informations et aux éléments diffractifs,” Ph.D. dissertation (Université de Haute Alsace, Alsace, France, 1995).
  14. R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578 (1991).
    [CrossRef]
  15. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  16. P. Ambs, L. Bigué, C. Stolz, “Dynamic computer generated hologram displayed on a spatial light modulator for information processing,” in Euro American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1999), pp. 151–170.
  17. I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

2000 (1)

1997 (2)

L. Legeard, P. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer-generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
[CrossRef]

F. Fetthauer, C. Stroot, O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136, 7–10 (1997).
[CrossRef]

1995 (1)

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

1991 (1)

1990 (3)

1988 (1)

1987 (1)

1982 (1)

1978 (1)

D. C. Youla, “Image restoration by the method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Allebach, J. P.

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]

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Ambs, P.

L. Legeard, P. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer-generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
[CrossRef]

P. Ambs, L. Bigué, C. Stolz, “Dynamic computer generated hologram displayed on a spatial light modulator for information processing,” in Euro American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1999), pp. 151–170.

Bigué, L.

P. Ambs, L. Bigué, C. Stolz, “Dynamic computer generated hologram displayed on a spatial light modulator for information processing,” in Euro American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1999), pp. 151–170.

Bräuer, R.

Brenner, K.-H.

Bryngdahl, O.

Campos, J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

Fetthauer, F.

F. Fetthauer, C. Stroot, O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136, 7–10 (1997).
[CrossRef]

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gorecki, C.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

Legeard, L.

L. Legeard, P. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer-generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
[CrossRef]

L. Legeard, “Étude et optimisation d’hologrammes synthétisés par ordinateur: application au stockage d’informations et aux éléments diffractifs,” Ph.D. dissertation (Université de Haute Alsace, Alsace, France, 1995).

Moreno, I.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

Réfrégier, P.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Seldowitz, M. A.

Sezan, M. I.

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

Stark, H.

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

Stolz, C.

P. Ambs, L. Bigué, C. Stolz, “Dynamic computer generated hologram displayed on a spatial light modulator for information processing,” in Euro American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1999), pp. 151–170.

Stroot, C.

F. Fetthauer, C. Stroot, O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136, 7–10 (1997).
[CrossRef]

Sweeney, D. W.

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]

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Wyrowski, F.

Youla, D. C.

D. C. Youla, “Image restoration by the method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Yzuel, M. J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

Appl. Opt. (3)

IEEE Trans. Circuits Syst. (1)

D. C. Youla, “Image restoration by the method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

F. Fetthauer, C. Stroot, O. Bryngdahl, “On the quantization of holograms with the iterative Fourier transform algorithm,” Opt. Commun. 136, 7–10 (1997).
[CrossRef]

Opt. Eng. (1)

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 31, 3520–3525 (1995).

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (4)

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

L. Legeard, “Étude et optimisation d’hologrammes synthétisés par ordinateur: application au stockage d’informations et aux éléments diffractifs,” Ph.D. dissertation (Université de Haute Alsace, Alsace, France, 1995).

P. Ambs, L. Bigué, C. Stolz, “Dynamic computer generated hologram displayed on a spatial light modulator for information processing,” in Euro American Workshop on Optoelectronic Information Processing, P. Réfrégier, B. Javidi, eds., Vol. CR74 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1999), pp. 151–170.

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Figures (8)

Fig. 1
Fig. 1

Typical OCC for the criteria (η, Err a ) for the trade-off parameter μ as it varies from 0 to 1. The most interesting part is highlighted.

Fig. 2
Fig. 2

Principle of the IFTA for the design of binary-amplitude holograms. βout is applied in the object plane and in ℜ̅ (i.e., outside the ROI ℜ) and is less than 1; the scaling factor β j applies to ℜ only. In the classical IFTA, β j is fixed to 1, whereas it varies in the multicriteria IFTA, as explained in Section 3. FT, Fourier transformation.

Fig. 3
Fig. 3

Diffraction efficiency η plotted versus the scaling factor β j for the case of a binary-amplitude coding domain.

Fig. 4
Fig. 4

Loci of points (η, error) for the variation of μ from 0 to 1 for various techniques and for binary-amplitude DOEs. For both multicriteria IFTA curves, i.e., with and without an equalizer, the points corresponding to the nonmulticriteria case are highlighted with a larger symbol.

Fig. 5
Fig. 5

Various reconstructions produced by the Model SVGA1 SLM operated in the binary mode. Simulations for different values of μ and their corresponding optical reconstructions are shown: (a) μ = 0 and (b) its reconstruction; (c) μ = 0.01 and (d) its reconstruction; (e) μ = 0.05 and (f) its reconstruction; (g) μ = 0.999 and (h) its reconstruction. When μ increases a degradation in image accuracy as well as an increase in the diffraction efficiency can be noted.

Fig. 6
Fig. 6

All three windows used for our criteria evaluation: W(s, i) is the signal window (ROI); W(n, i) is used for the SSNR and is a noise window inside the signal window; W(n, o) is used for the NRR and is a noise window outside the signal window.

Fig. 7
Fig. 7

Loci of points (η, 1/NRR) and (η, 1/SSNR) for the variation of μ from 0 to 1, corresponding to the experimental reconstructions shown in Fig. 5. The experimental efficiency η is defined as the mean energy expressed in terms of the gray-level value in the ROI ℜ. As is highlighted, using the NRR may result in the incorrect evaluation of the reconstruction accuracy.

Fig. 8
Fig. 8

Loci of points (η, Err a ) for the various values of μ that correspond to the simulated reconstructions shown in Fig. 5.

Tables (2)

Tables Icon

Table 1 Typical Computation Times for 128 × 128 Pixel DOEs on a DEC Model AlphaStation 500 at 500 MHz by Use of Various Techniques

Tables Icon

Table 2 Comparison of Simulated and Experimental Data

Equations (11)

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

ηh=k |gk|2MN,
Errah=k|fk|-α|gk|2k |fk|2,
α=k |fk||gk|k |gk|2,
ηh0ηh,  Errah0Errah.
Eμh=μ 1ηh+1-μErrah,
βj=MNμ,
hk=hkβj  if khkβout  else.
hk=hk  if khkβout=hkβoutβj  else
gi+1k=1-γgik+γfik  kgik  else,
NRR=EWn,oEWs,i,
SSNR=EWs,iEWn,i.

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