Abstract

For optical neural networks implemented with computer-generated planar holograms the space–bandwidth product of the hologram is a major consideration. Off-axis holograms can be fabricated with a single binary transmission mask. However, the carrier frequency greatly increases the space–bandwidth product. On axis-holograms use a lower space–bandwidth product to encode interconnections but require a multilevel phase transmission profile. Significant errors can result during the fabrication of multilevel phase structures. With modification of the on-axis geometry the effects of the fabrication errors can be reduced while a lower space–bandwidth product per interconnection is retained. The interconnection accuracy, the diffraction efficiency, the and sensitivity to fabrication errors are compared for the off-axis, the on-axis, and the modified on-axis diffraction geometries.

© 1993 Optical Society of America

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References

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  1. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
    [CrossRef] [PubMed]
  2. H. J. Caulfield, “Parallel N4-weighted optical interconnections,” Appl. Opt. 26, 4039–4040 (1987).
    [CrossRef] [PubMed]
  3. P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
    [CrossRef]
  4. 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).
  5. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
  6. N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  7. M. Feldman, C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1988).
    [CrossRef]
  8. G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

1992 (1)

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

1989 (1)

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

1988 (1)

1987 (1)

1984 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

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).

1953 (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Caulfield, H. J.

Chavel, P.

Feldman, M.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

Forchheimer, 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).

Gmitro, A.

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

Guest, C.

Jenkins, B. K.

Keller, P.

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

Metropolis, N.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Sawchuk, A. A.

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).

Strand, T. C.

Swanson, G. J.

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Teller, A.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Veldkamp, W. B.

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Appl. Opt. (3)

J. Chem. Phys. (1)

N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, “Equations of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).

Opt. Lett. (1)

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).

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Figures (7)

Fig. 1
Fig. 1

Architecture of the optoelectronic neural network. Each neuron output in the source plane illuminates a single subhologram, which forms weighted connections to the other neuron inputs in the detector plane.

Fig. 2
Fig. 2

Diffraction-plane layout of the three diffraction geometries: (a) the off-axis, (b) the on-axis, and (c) the modified on-axis. Each white square represents a valid detector location. The boundaries of each diffraction pattern are set by the Nyquist frequencies of the holograms.

Fig. 3
Fig. 3

Reconstruction error of the interconnection weights as a function of the SBWP per interconnection for the three diffraction geometries. The fundamental SBWP per connection is 4 for the on-axis holograms and 16 for the off-axis holograms. Additional SBWP is used for subhologram replication.

Fig. 4
Fig. 4

Reconstruction error of the interconnection weights as a function of mask misalignment for the three diffraction geometries. Misalignment is measured in fractional units of a pixel dimension.

Fig. 5
Fig. 5

Diffraction efficiency as a function of mask misalignment for holograms designed for the three diffraction geometries.

Fig. 6
Fig. 6

Reconstruction error of connection weights as a function of phase-step error for the three diffraction geometries; +90° and −90° of phase-step error represent 100% error in pixel phase value.

Fig. 7
Fig. 7

Diffraction efficiency as a function of phase-step error for holograms designed for the three diffraction geometries.

Equations (8)

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rms = ( i , j N ( w ˆ i j w i j ) 2 N ) 1 / 2 .
η d = ROI I h ( x , y ) d x d y I 0 .
M n ( ξ, η ) = i , j pixels bit ( H i j , n ) rect ( ξ j Δ x Δ x , η i Δ y Δ y ) , ( n = 1 , , m ) ,
H ( ξ, η ) = n = 1 m 1 + [ exp ( i π 2 n m ) 1 ] M n ( ξ , η ) .
H ˜ ( ξ , η ) = H ( ξ , η ) * G e - beam ( ξ , η ) * G photo ( ξ , η ) * G etch ( ξ , η ) ,
h ˜ ( x , y ) = h ( x , y ) g e - beam ( x , y ) g photo ( x , y ) g etch ( x , y ) = h ( x , y ) g fab ( x , y ) .
H ( ξ , η ) = n = 1 m 1 + [ exp ( i π 2 n m ) 1 ] M n ( ξ ξ n , η η n ) ,
H ( ξ , η ) = n = 1 m 1 + { exp [ i ( π 2 n m + n ) ] 1 } M n ( ξ , η ) ,

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