Abstract

The design, fabrication, experimental characterization, and system-performance analysis of a diffractive optical implementation of an error-diffusion filter for use in digital image halftoning is reported. A diffractive optical filter was fabricated as an eight-level phase element that diffuses the quantization error nonuniformly in both the weighting and the spatial dimensions, according to a prescribed algorithm. Ten identical diffractive elements were fabricated on ten different wafers and subsequently characterized experimentally. A detailed error analysis including both fabrication and instrumentation errors was carried out to quantify the performance of the fabrication process as well as the expected system performance of the filters. Halftone system performance was evaluated by use of the experimental filter’s performance and both quantitative and qualitative performance metrics. The results of this analysis demonstrate that multiple identical copies of a diffractive optical filter can be produced with sufficient accuracy that no loss in the halftoning system performance results.

© 1999 Optical Society of America

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References

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  1. D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
    [CrossRef]
  2. J. J. Hopfield, D. W. Tank, “Neural computation of decisions in optimization problems,” Biol. Cybern. 52, 141–152 (1985).
  3. H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
    [CrossRef]
  4. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuck, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
    [CrossRef] [PubMed]
  5. P. E. Keller, A. F. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 31, 5517–5526 (1992).
    [CrossRef] [PubMed]
  6. D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
    [CrossRef]
  7. J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  8. C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.
  9. See, for example, Diffractive Optics and Micro-Optics, Vol. 5 of OSA 1996 Technical Digest Series (Optical Society of America, Washington, D.C., 1996).
  10. B. L. Shoop, E. K. Ressler, M. W. Farn, “An optical neural network approach to digital image halftoning,” presented at Optical Society of America Annual Meeting, Dallas, Tex., 2–7 October 1994.
  11. J. N. Mait, D. W. Prather, B. L. Shoop, “Design of diffractive elements for photonic processors,” in Proceedings of the Thirteenth Annual Asilomar Conference on Signals Systems and Computers (IEEE Signal Processing Society, Boston, Mass., 1996), pp. 496–500.
  12. J. N. Mait, “Diffractive design: two examples from an optical analog-to-digital converter,” . (U.S. Army Research Laboratory, Adelphi, Md., 1997).
  13. B. L. Shoop, E. K. Ressler, “An error diffusion neural network for digital image halftoning,” in Proceedings of the IEEE Workshop on Neural Networks (Institute of Electrical and Electronics Engineers, Signal Processing Society and Neural Networks Council, Boston, Mass., 1995), pp. 427–436.
  14. B. L. Shoop, A. H. Sayles, D. M. Litynski, “New devices for optoelectronics: smart pixels,” in Handbook of Fiber Optic Data Communications, C. DeCusatis, D. Clement, E. Maass, R. Lasky, eds. (Academic, San Diego, Calif., 1997), pp. 705–758.
  15. R. A. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).
  16. R. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray scale,” Dig. SID 36, 35–36 (1975).
  17. E. K. Ressler, B. L. Shoop, “High quality digital halftones from error diffusion networks,” in Proceedings of the Society for Information Display (Society for Information Display, Santa Ana, Calif., 1996), pp. 506–509.
  18. E. K. Ressler, B. L. Shoop, “Analysis and design of error diffusion networks for digital halftoning,” in Proceedings of the Twentieth Army Science Conference (Office of the Assistant Secretary of the Army, Research, Development, and Acquisition, Washington, D.C., 1996), pp. 191–195.
  19. B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  21. M. Stern, “Binary optics fabrication,” in Microoptics, H.-P. Herzig, ed. (Taylor & Francis, New York, 1997), pp. 53–85.
  22. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  23. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  24. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
    [CrossRef]
  25. F. Wyrowski, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  26. J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).
  27. M. S. Bartlett, An Introduction to Stochastic Processes with Special Reference to Methods and Applications (Cambridge U. Press, New York, 1955).
  28. R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 26, 56–79 (1988).
    [CrossRef]

1997

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

1995

1992

1991

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1989

H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
[CrossRef]

1988

R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 26, 56–79 (1988).
[CrossRef]

1986

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
[CrossRef]

1985

J. J. Hopfield, D. W. Tank, “Neural computation of decisions in optimization problems,” Biol. Cybern. 52, 141–152 (1985).

1984

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

1975

R. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray scale,” Dig. SID 36, 35–36 (1975).

1973

Ananthanarayanan, K.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

Bartlett, M. S.

M. S. Bartlett, An Introduction to Stochastic Processes with Special Reference to Methods and Applications (Cambridge U. Press, New York, 1955).

Brady, D.

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
[CrossRef]

Chavel, P.

Chen, C.-H.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

Cox, J. A.

J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).

Farn, M. W.

B. L. Shoop, E. K. Ressler, M. W. Farn, “An optical neural network approach to digital image halftoning,” presented at Optical Society of America Annual Meeting, Dallas, Tex., 2–7 October 1994.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

Floyd, R.

R. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray scale,” Dig. SID 36, 35–36 (1975).

Forchheimer, R.

Gallagher, N. C.

Gieske, J.

J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).

Gmitro, A. F.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Gu, X. G.

H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
[CrossRef]

Gu, X.-G.

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
[CrossRef]

Hall, D. A.

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

Hewitt, K.

J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
[CrossRef]

Hopfield, J. J.

J. J. Hopfield, D. W. Tank, “Neural computation of decisions in optimization problems,” Biol. Cybern. 52, 141–152 (1985).

Hsu, K.

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
[CrossRef]

Huang, C. C.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

Jenkins, B. K.

Keller, P. E.

Kuznia, C. B.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

Lee, H.

H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
[CrossRef]

Litynski, D. M.

B. L. Shoop, A. H. Sayles, D. M. Litynski, “New devices for optoelectronics: smart pixels,” in Handbook of Fiber Optic Data Communications, C. DeCusatis, D. Clement, E. Maass, R. Lasky, eds. (Academic, San Diego, Calif., 1997), pp. 705–758.

Liu, B.

Mait, J. N.

J. N. Mait, “Design of binary- and multi-phase Fourier gratings for array generation,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
[CrossRef]

J. N. Mait, D. W. Prather, B. L. Shoop, “Design of diffractive elements for photonic processors,” in Proceedings of the Thirteenth Annual Asilomar Conference on Signals Systems and Computers (IEEE Signal Processing Society, Boston, Mass., 1996), pp. 496–500.

J. N. Mait, “Diffractive design: two examples from an optical analog-to-digital converter,” . (U.S. Army Research Laboratory, Adelphi, Md., 1997).

Prather, D. W.

J. N. Mait, D. W. Prather, B. L. Shoop, “Design of diffractive elements for photonic processors,” in Proceedings of the Thirteenth Annual Asilomar Conference on Signals Systems and Computers (IEEE Signal Processing Society, Boston, Mass., 1996), pp. 496–500.

Psaltis, D.

H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
[CrossRef]

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
[CrossRef]

Ressler, E. K.

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

B. L. Shoop, E. K. Ressler, M. W. Farn, “An optical neural network approach to digital image halftoning,” presented at Optical Society of America Annual Meeting, Dallas, Tex., 2–7 October 1994.

E. K. Ressler, B. L. Shoop, “High quality digital halftones from error diffusion networks,” in Proceedings of the Society for Information Display (Society for Information Display, Santa Ana, Calif., 1996), pp. 506–509.

B. L. Shoop, E. K. Ressler, “An error diffusion neural network for digital image halftoning,” in Proceedings of the IEEE Workshop on Neural Networks (Institute of Electrical and Electronics Engineers, Signal Processing Society and Neural Networks Council, Boston, Mass., 1995), pp. 427–436.

E. K. Ressler, B. L. Shoop, “Analysis and design of error diffusion networks for digital halftoning,” in Proceedings of the Twentieth Army Science Conference (Office of the Assistant Secretary of the Army, Research, Development, and Acquisition, Washington, D.C., 1996), pp. 191–195.

Rumelhart, D. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
[CrossRef]

Sawchuck, A. A.

Sawchuk, A. A.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

Sayles, A. H.

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

B. L. Shoop, A. H. Sayles, D. M. Litynski, “New devices for optoelectronics: smart pixels,” in Handbook of Fiber Optic Data Communications, C. DeCusatis, D. Clement, E. Maass, R. Lasky, eds. (Academic, San Diego, Calif., 1997), pp. 705–758.

Shoop, B. L.

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

E. K. Ressler, B. L. Shoop, “High quality digital halftones from error diffusion networks,” in Proceedings of the Society for Information Display (Society for Information Display, Santa Ana, Calif., 1996), pp. 506–509.

B. L. Shoop, E. K. Ressler, M. W. Farn, “An optical neural network approach to digital image halftoning,” presented at Optical Society of America Annual Meeting, Dallas, Tex., 2–7 October 1994.

B. L. Shoop, E. K. Ressler, “An error diffusion neural network for digital image halftoning,” in Proceedings of the IEEE Workshop on Neural Networks (Institute of Electrical and Electronics Engineers, Signal Processing Society and Neural Networks Council, Boston, Mass., 1995), pp. 427–436.

J. N. Mait, D. W. Prather, B. L. Shoop, “Design of diffractive elements for photonic processors,” in Proceedings of the Thirteenth Annual Asilomar Conference on Signals Systems and Computers (IEEE Signal Processing Society, Boston, Mass., 1996), pp. 496–500.

B. L. Shoop, A. H. Sayles, D. M. Litynski, “New devices for optoelectronics: smart pixels,” in Handbook of Fiber Optic Data Communications, C. DeCusatis, D. Clement, E. Maass, R. Lasky, eds. (Academic, San Diego, Calif., 1997), pp. 705–758.

E. K. Ressler, B. L. Shoop, “Analysis and design of error diffusion networks for digital halftoning,” in Proceedings of the Twentieth Army Science Conference (Office of the Assistant Secretary of the Army, Research, Development, and Acquisition, Washington, D.C., 1996), pp. 191–195.

Steinberg, L.

R. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray scale,” Dig. SID 36, 35–36 (1975).

Stern, M.

M. Stern, “Binary optics fabrication,” in Microoptics, H.-P. Herzig, ed. (Taylor & Francis, New York, 1997), pp. 53–85.

Strand, T. C.

Tank, D. W.

J. J. Hopfield, D. W. Tank, “Neural computation of decisions in optimization problems,” Biol. Cybern. 52, 141–152 (1985).

Ulichney, R. A.

R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 26, 56–79 (1988).
[CrossRef]

R. A. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

Werner, T. R.

J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).

Williams, R. J.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
[CrossRef]

Wyrowski, F.

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Appl. Opt.

Biol. Cybern.

J. J. Hopfield, D. W. Tank, “Neural computation of decisions in optimization problems,” Biol. Cybern. 52, 141–152 (1985).

Dig. SID

R. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray scale,” Dig. SID 36, 35–36 (1975).

Int. J. Optoelectron. Smart Pixels

B. L. Shoop, A. H. Sayles, D. A. Hall, E. K. Ressler, “A smart pixel implementation of an error diffusion neural network for digital halftoning,” Int. J. Optoelectron. Smart Pixels 11, 217–228 (1997).

J. Appl. Phys.

H. Lee, X. G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal crosstalk,” J. Appl. Phys. 65, 2919–2194 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Nature

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning representations by back-propagating error,” Nature 323, 533–536 (1986).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

Opt. Lett.

Proc. IEEE

R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 26, 56–79 (1988).
[CrossRef]

Rep. Prog. Phys.

F. Wyrowski, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other

J. A. Cox, J. Gieske, K. Hewitt, T. R. Werner, “CO-OP DOE foundry process error summary report,” 30September1996 (Honeywell Technology Center, Minneapolis, Minn., 1996).

M. S. Bartlett, An Introduction to Stochastic Processes with Special Reference to Methods and Applications (Cambridge U. Press, New York, 1955).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

M. Stern, “Binary optics fabrication,” in Microoptics, H.-P. Herzig, ed. (Taylor & Francis, New York, 1997), pp. 53–85.

E. K. Ressler, B. L. Shoop, “High quality digital halftones from error diffusion networks,” in Proceedings of the Society for Information Display (Society for Information Display, Santa Ana, Calif., 1996), pp. 506–509.

E. K. Ressler, B. L. Shoop, “Analysis and design of error diffusion networks for digital halftoning,” in Proceedings of the Twentieth Army Science Conference (Office of the Assistant Secretary of the Army, Research, Development, and Acquisition, Washington, D.C., 1996), pp. 191–195.

C. B. Kuznia, C. C. Huang, K. Ananthanarayanan, C.-H. Chen, A. A. Sawchuk, “Micro diffractive optical elements for smart pixel fanout interconnections,” in Organic Thin Films, Vol. 21 of OSA 1995 Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 149.

See, for example, Diffractive Optics and Micro-Optics, Vol. 5 of OSA 1996 Technical Digest Series (Optical Society of America, Washington, D.C., 1996).

B. L. Shoop, E. K. Ressler, M. W. Farn, “An optical neural network approach to digital image halftoning,” presented at Optical Society of America Annual Meeting, Dallas, Tex., 2–7 October 1994.

J. N. Mait, D. W. Prather, B. L. Shoop, “Design of diffractive elements for photonic processors,” in Proceedings of the Thirteenth Annual Asilomar Conference on Signals Systems and Computers (IEEE Signal Processing Society, Boston, Mass., 1996), pp. 496–500.

J. N. Mait, “Diffractive design: two examples from an optical analog-to-digital converter,” . (U.S. Army Research Laboratory, Adelphi, Md., 1997).

B. L. Shoop, E. K. Ressler, “An error diffusion neural network for digital image halftoning,” in Proceedings of the IEEE Workshop on Neural Networks (Institute of Electrical and Electronics Engineers, Signal Processing Society and Neural Networks Council, Boston, Mass., 1995), pp. 427–436.

B. L. Shoop, A. H. Sayles, D. M. Litynski, “New devices for optoelectronics: smart pixels,” in Handbook of Fiber Optic Data Communications, C. DeCusatis, D. Clement, E. Maass, R. Lasky, eds. (Academic, San Diego, Calif., 1997), pp. 705–758.

R. A. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. H. Arsenault, T. Szoplik, B. Macukow, eds. (Academic, San Diego, Calif., 1989), Chap. 8, pp. 251–276.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Smart-pixel-based error-diffusion neural-network architecture that incorporates optical interconnect weighting and distribution. BS, beam splitter; PBS, polarizing beam splitter; M, mirror.

Fig. 2
Fig. 2

Block diagram of a 2-D error-diffusion architecture.

Fig. 3
Fig. 3

SEM images of the fabricated diffractive optical filter: (a) Two periods of the unit cell and (b) magnification of a portion of the unit cell showing increased etch detail and the eight etch levels.

Fig. 4
Fig. 4

Experimental configuration for characterizing the performance of the diffractive optical filter.

Fig. 5
Fig. 5

Comparison of the measured and the IFTA-predicted percent error from the design (unique) coefficients.

Fig. 6
Fig. 6

Images of the Cadet Chapel at West Point, New York: (a) gray-scale image, (b) halftoned gray-scale image obtained by use of the designed intensity distribution for error diffusion, (c) halftoned gray-scale image obtained by use of the experimentally measured distribution.

Fig. 7
Fig. 7

Radially averaged frequency spectra obtained by use of (a) the designed coefficients (0.25% gray) and (b) the measured coefficients (0.25% gray). Radially averaged anisotropy curves obtained by use of (a) the designed coefficients (0.25% gray) and (d) the measured coefficients (0.25% gray).

Tables (5)

Tables Icon

Table 1 Coefficient Weighting wx,y for the 7 × 7 Error-Diffusion Filter

Tables Icon

Table 2 Designed Phase Values for the 7 × 7 Diffractive Optical Filter

Tables Icon

Table 3 Predicted Intensity Distribution for the 7 × 7 Diffractive Optical Filter

Tables Icon

Table 4 IFTA-Predicted Coefficient Distribution That Accounts for the Fabrication Etch-Depth Errors

Tables Icon

Table 5 Normalized Power Measurements for the 7 × 7 Diffractive Optical Filter Averaged Across the 10-Sample Ensemble

Equations (18)

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

u=Wq-u+g.
a div b=ab,  if ab0=ab,  if ab<0.
Eg, q=qTAq-2qTAg+gTAg=Bq-gerrorTBq-gerror,
k : AWk,k0.
Pu, v=Au, vexpjΘu, v,
px, y= Pu, vexpj 2πλfux+vydudv,
qx, y=n=1N qnδx-xn, y-yn,
Pu, v=P˜u, v * * replu, v,
px, y=p˜x, ysampx, y.
η=x,yX |px, y|2=n=1N |pn|2.
eint=x,yX|px, y|2-α|qx, y|22dxdy=n=1N|pn|2-α|qn|22.
αmin=x,yX |px, y|2|qx, y|2dxdyx,yX |qx, y|4dxdy=n=1N |pn|2|qn|2n=1N |qn|4.
eub=n=1N|pn|2-αub|qn|22,
W=λfd.
η=x=17y=17 px,ypincident.
Ptft=1Nrfri=1Nrfr Pˆf,
s2ft=1Nrfr-1i=1NrfrPˆf-Prfr2.
s2frPr2fr,

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