Abstract

The feasibility of employing volume holographic techniques for the implementation of highly multiplexed weighted fan-out/fan-in interconnections is analyzed on the basis of interconnection fidelity, optical throughput, and complexity of recording schedule or implementation hardware. These feasibility criteria were quantitatively evaluated using the optical beam propagation method to numerically simulate the diffraction characteristics of volume holographic interconnections recorded in a linear holographic material. We find that conventional interconnection architectures (that are based on a single coherent optical source) exhibit a direct trade-off between interconnection fidelity and optical throughput on the one hand, and recording schedule or hardware complexity on the other. In order to circumvent this trade-off we describe and analyze in detail an incoherent/coherent double angularly multiplexed interconnection architecture that is based on the use of multiple-source array of individually coherent but mutually incoherent sources. This architecture either minimizes or avoids several key sources of cross talk, permits simultaneous recording of interconnection weights or weight updates, and provides enhanced fidelity, interchannel isolation, and throughput performance.

© 1993 Optical Society of America

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1992 (3)

1991 (3)

1990 (5)

1989 (6)

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[Crossref] [PubMed]

E. G. Paek, J. R. Wullert, J. S. Patel, “Holographic implementation of a learning machine based on a multicategory perceptron algorithm,” Opt. Lett. 14, 1303–1305 (1989).
[Crossref] [PubMed]

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 5312–5316 (1989).
[Crossref] [PubMed]

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

M. Cronin-Golomb, “Dynamically programmable self-aligning optical interconnect with fan-out and fan-in using self-pumped phase conjugation,” Appl. Phys. Lett. 54, 2189–2191 (1989).
[Crossref]

1988 (1)

1987 (3)

1986 (1)

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for biréfringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).

1985 (2)

1984 (1)

C. W. Slinger, L. Solymar, “Volume phase holograms reconstructed by the object wave,” Opt. Quantum Electron. 16, 369–372 (1984).
[Crossref]

1983 (1)

S. Kessler, R. Hild, “A new method for simultaneous complex addition and subtraction,” Opt. Quantum. Electron. 15, 65–70 (1983).
[Crossref]

1982 (1)

D. Yevick, L. Thylen, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. 72, 1081–1089 (1982).
[Crossref]

1981 (1)

1980 (1)

1979 (1)

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase gratings,” Opt. Acta 26, 271–278 (1979).
[Crossref]

1978 (1)

1977 (1)

L. Solymar, “Two-dimensional N-coupled-wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[Crossref]

1976 (2)

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1976).
[Crossref]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

1975 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967 (1)

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Anderson, D. Z.

Asthana, P.

G. P. Nordin, P. Asthana, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic interconnections,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 165–167.

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, “Volume holographic techniques for highly multiplexed interconnection applications,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1991).

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

Benlarbi, B.

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase gratings,” Opt. Acta 26, 271–278 (1979).
[Crossref]

Brady, D.

D. Brady, D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
[Crossref]

D. Psaltis, X.-G. Gu, D. Brady, “Fractal sampling grids for holographic interconnections,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 468 (1988).
[Crossref]

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. Arsenault, ed. (Academic, New York, 1988), pp. 251–276.

Brady, D. J.

Brady, G.

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Burke, W. J.

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1976).
[Crossref]

Campbell, S.

Case, S. K.

Chin, H.

G. C. Petrisor, B. K. Jenkins, H. Chin, A. R. Tanguay, “Dual-function adaptive neural networks for photonic implementation,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981), p. 212.

Cronin-Golomb, M.

M. Cronin-Golomb, “Dynamically programmable self-aligning optical interconnect with fan-out and fan-in using self-pumped phase conjugation,” Appl. Phys. Lett. 54, 2189–2191 (1989).
[Crossref]

Farhat, N. H.

Feit, M. D.

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[Crossref] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Fleck, J. A.

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[Crossref] [PubMed]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Gaylord, T. K.

Glytsis, E. N.

Goodman, J.

Goodman, J. W.

J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
[Crossref]

Gu, C. X.-G.

C. X.-G. Gu, “Optical neural networks using volume holograms,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1990).

Gu, X.-G.

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. Arsenault, ed. (Academic, New York, 1988), pp. 251–276.

D. Psaltis, X.-G. Gu, D. Brady, “Fractal sampling grids for holographic interconnections,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 468 (1988).
[Crossref]

Hartman, E.

C. Peterson, S. Redfield, J. D. Keeler, E. Hartman, “Optoelectronic implementation of multilayer neural networks in a single photorefractive material,” Opt. Eng. 29, 359–368 (1990).
[Crossref]

Hebb, D. O.

D. O. Hebb, Organization of Behavior (Wiley, New York, 1949).

Hesselink, L.

Hild, R.

S. Kessler, R. Hild, “A new method for simultaneous complex addition and subtraction,” Opt. Quantum. Electron. 15, 65–70 (1983).
[Crossref]

Hong, J. H.

Hsu, K.

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. Arsenault, ed. (Academic, New York, 1988), pp. 251–276.

Jenkins, B. K.

S. Piazzolla, B. K. Jenkins, A. R. Tanguay, “Single-step copying process for multiplexed volume holograms,” Opt. Lett. 17, 676–678 (1992).
[Crossref] [PubMed]

G. P. Nordin, P. Asthana, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic interconnections,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 165–167.

G. C. Petrisor, B. K. Jenkins, H. Chin, A. R. Tanguay, “Dual-function adaptive neural networks for photonic implementation,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

B. K. Jenkins, A. R. Tanguay, “Photonic implementations of neural networks,” in Neural Networks for Signal Processing, B. Kosko, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1992), Chap. 9, pp. 287–382.

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

Johnson, K. M.

E. S. Maniloff, K. M. Johnson, “Dynamic holographic interconnects using static holograms,” Opt. Eng. 29, 225–229 (1990).
[Crossref]

Johnson, R. V.

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for biréfringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).

Keeler, J. D.

C. Peterson, S. Redfield, J. D. Keeler, E. Hartman, “Optoelectronic implementation of multilayer neural networks in a single photorefractive material,” Opt. Eng. 29, 359–368 (1990).
[Crossref]

Kessler, S.

S. Kessler, R. Hild, “A new method for simultaneous complex addition and subtraction,” Opt. Quantum. Electron. 15, 65–70 (1983).
[Crossref]

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[Crossref]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kostuk, R.

Lagasse, P. E.

Lee, H.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Lin, S.

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

Lininger, D. M.

Madhukar, A.

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

Maniloff, E. S.

E. S. Maniloff, K. M. Johnson, “Dynamic holographic interconnects using static holograms,” Opt. Eng. 29, 225–229 (1990).
[Crossref]

Moharam, M. G.

Mok, F. H.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Nordin, G.

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

Nordin, G. P.

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

G. P. Nordin, “Volume diffraction phenomena for photonic neural network implementations and stratified volume holographic optical elements,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1992).

G. P. Nordin, P. Asthana, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic interconnections,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 165–167.

Paek, E.

Paek, E. G.

Patel, J. S.

Peterson, C.

C. Peterson, S. Redfield, J. D. Keeler, E. Hartman, “Optoelectronic implementation of multilayer neural networks in a single photorefractive material,” Opt. Eng. 29, 359–368 (1990).
[Crossref]

Petrisor, G. C.

G. C. Petrisor, B. K. Jenkins, H. Chin, A. R. Tanguay, “Dual-function adaptive neural networks for photonic implementation,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

Piazzolla, S.

S. Piazzolla, B. K. Jenkins, A. R. Tanguay, “Single-step copying process for multiplexed volume holograms,” Opt. Lett. 17, 676–678 (1992).
[Crossref] [PubMed]

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

Prata, A.

Psaltis, D.

D. Brady, D. Psaltis, “Control of volume holograms,” J. Opt. Soc. Am. A 9, 1167–1182 (1992).
[Crossref]

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

D. Psaltis, D. J. Brady, K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1758 (1988).
[Crossref]

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[Crossref] [PubMed]

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. Arsenault, ed. (Academic, New York, 1988), pp. 251–276.

D. Psaltis, X.-G. Gu, D. Brady, “Fractal sampling grids for holographic interconnections,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 468 (1988).
[Crossref]

Redfield, S.

C. Peterson, S. Redfield, J. D. Keeler, E. Hartman, “Optoelectronic implementation of multilayer neural networks in a single photorefractive material,” Opt. Eng. 29, 359–368 (1990).
[Crossref]

Sheng, P.

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1976).
[Crossref]

Slinger, C.

Slinger, C. W.

C. W. Slinger, “Weighted volume interconnects for adaptive networks,” Opt. Comput. Process. 1, 219–232 (1991).

C. W. Slinger, L. Solymar, “Volume phase holograms reconstructed by the object wave,” Opt. Quantum Electron. 16, 369–372 (1984).
[Crossref]

Solymar, L.

C. W. Slinger, L. Solymar, “Volume phase holograms reconstructed by the object wave,” Opt. Quantum Electron. 16, 369–372 (1984).
[Crossref]

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase gratings,” Opt. Acta 26, 271–278 (1979).
[Crossref]

L. Solymar, “Two-dimensional N-coupled-wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[Crossref]

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981), p. 212.

Stoll, H. M.

Tackitt, M. C.

Tamir, T.

Tanguay, A. R.

S. Piazzolla, B. K. Jenkins, A. R. Tanguay, “Single-step copying process for multiplexed volume holograms,” Opt. Lett. 17, 676–678 (1992).
[Crossref] [PubMed]

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for biréfringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

G. C. Petrisor, B. K. Jenkins, H. Chin, A. R. Tanguay, “Dual-function adaptive neural networks for photonic implementation,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

B. K. Jenkins, A. R. Tanguay, “Photonic implementations of neural networks,” in Neural Networks for Signal Processing, B. Kosko, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1992), Chap. 9, pp. 287–382.

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

G. P. Nordin, P. Asthana, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic interconnections,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 165–167.

Thylen, L.

D. Yevick, L. Thylen, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. 72, 1081–1089 (1982).
[Crossref]

Tu, K.-Y.

van der Donk, J.

Van Roey, J.

Wagner, K.

Wullert, J. R.

Yamamura, A. A.

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

Yeh, P.

Yevick, D.

D. Yevick, L. Thylen, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. 72, 1081–1089 (1982).
[Crossref]

Appl. Opt. (11)

See, for example, the feature on neural networks, Appl. Opt. 26, 4909–5111 (1987).
[PubMed]

R. Kostuk, J. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
[Crossref] [PubMed]

D. Z. Anderson, D. M. Lininger, “Dynamic optical interconnects: volume holograms as optical two-port operators,” Appl. Opt. 26, 5031–5038 (1987).
[Crossref] [PubMed]

D. Psaltis, D. J. Brady, K. Wagner, “Adaptive optical networks using photorefractive crystals,” Appl. Opt. 27, 1752–1758 (1988).
[Crossref]

J. H. Hong, S. Campbell, P. Yeh, “Optical pattern classifier with perceptron learning,” Appl. Opt. 29, 3019–3025 (1990).
[Crossref] [PubMed]

H. Lee, “Volume holographic global and local interconnecting patterns with maximal capacity and minimal first-order crosstalk,” Appl. Opt. 28, 5312–5316 (1989).
[Crossref] [PubMed]

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[Crossref] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media,” Appl. Opt. 28, 2401–2421 (1989).
[Crossref] [PubMed]

K.-Y. Tu, H. Lee, T. Tamir, “Analysis of cross talk in volume holographic interconnections,” Appl. Opt. 31, 1717–1729 (1992).
[Crossref] [PubMed]

M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
[Crossref] [PubMed]

T. K. Gaylord, M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271 (1981).
[Crossref] [PubMed]

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[Crossref]

Appl. Phys. Lett. (1)

M. Cronin-Golomb, “Dynamically programmable self-aligning optical interconnect with fan-out and fan-in using self-pumped phase conjugation,” Appl. Phys. Lett. 54, 2189–2191 (1989).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE Commun. Mag. (1)

D. Psaltis, A. A. Yamamura, K. Hsu, S. Lin, X.-G. Gu, G. Brady, “Optoelectronic implementations of neural networks,” IEEE Commun. Mag. 27(11), 37–40 (1989).
[Crossref]

IEEE Trans. Sonics Ultrason. (1)

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[Crossref]

J. Appl. Phys. (2)

W. J. Burke, P. Sheng, “Crosstalk noise from multiple thick-phase holograms,” J. Appl. Phys. 48, 681–685 (1976).
[Crossref]

H. Lee, X.-G. Gu, D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. 65, 2191–2194 (1989).
[Crossref]

J. Opt. Soc. Am. (3)

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

Opt. Acta (2)

J. W. Goodman, “Fan-in and fan-out with optical interconnections,” Opt. Acta 32, 1489–1496 (1985).
[Crossref]

B. Benlarbi, L. Solymar, “The effect of the relative intensity of the reference beam on the reconstructing properties of volume phase gratings,” Opt. Acta 26, 271–278 (1979).
[Crossref]

Opt. Commun. (1)

L. Solymar, “Two-dimensional N-coupled-wave theory for volume holograms,” Opt. Commun. 23, 199–202 (1977).
[Crossref]

Opt. Comput. Process. (1)

C. W. Slinger, “Weighted volume interconnects for adaptive networks,” Opt. Comput. Process. 1, 219–232 (1991).

Opt. Eng. (3)

R. V. Johnson, A. R. Tanguay, “Optical beam propagation method for biréfringent phase grating diffraction,” Opt. Eng. 25, 235–249 (1986).

C. Peterson, S. Redfield, J. D. Keeler, E. Hartman, “Optoelectronic implementation of multilayer neural networks in a single photorefractive material,” Opt. Eng. 29, 359–368 (1990).
[Crossref]

E. S. Maniloff, K. M. Johnson, “Dynamic holographic interconnects using static holograms,” Opt. Eng. 29, 225–229 (1990).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

C. W. Slinger, L. Solymar, “Volume phase holograms reconstructed by the object wave,” Opt. Quantum Electron. 16, 369–372 (1984).
[Crossref]

Opt. Quantum. Electron. (1)

S. Kessler, R. Hild, “A new method for simultaneous complex addition and subtraction,” Opt. Quantum. Electron. 15, 65–70 (1983).
[Crossref]

Other (16)

G. C. Petrisor, B. K. Jenkins, H. Chin, A. R. Tanguay, “Dual-function adaptive neural networks for photonic implementation,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

D. Psaltis, D. Brady, X.-G. Gu, K. Hsu, “Optical implementation of neural computers,” in Optical Processing and Computing, H. Arsenault, ed. (Academic, New York, 1988), pp. 251–276.

D. Psaltis, X.-G. Gu, D. Brady, “Fractal sampling grids for holographic interconnections,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 468 (1988).
[Crossref]

P. Asthana, G. Nordin, S. Piazzolla, A. R. Tanguay, B. K. Jenkins, “Analysis of inter channel cross talk and throughput efficiency in highly multiplexed fan-out–fan-in holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 242.

D. O. Hebb, Organization of Behavior (Wiley, New York, 1949).

C. X.-G. Gu, “Optical neural networks using volume holograms,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1990).

B. K. Jenkins, G. C. Petrisor, S. Piazzolla, P. Asthana, A. R. Tanguay, “Photonic architecture for neural nets using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. P. Nordin, A. R. Tanguay, S. Piazzolla, B. K. Jenkins, “Photonic components for neural net implementations using incoherent/coherent holographic interconnections,” in OC’90 Technical Digest (ICO-90 Organizing Committee, Kobe, Japan, 1990).

P. Asthana, H. Chin, G. Nordin, A. R. Tanguay, G. C. Petrisor, B. K. Jenkins, A. Madhukar, “Photonic components for neural net implementations using incoherent-coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 57.

B. K. Jenkins, A. R. Tanguay, S. Piazzolla, G. C. Petrisor, P. Asthana, “Photonic neural-network architecture based on incoherent–coherent holographic interconnections,” in 1990 OSA Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 56.

P. Asthana, “Volume holographic techniques for highly multiplexed interconnection applications,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1991).

B. K. Jenkins, A. R. Tanguay, “Photonic implementations of neural networks,” in Neural Networks for Signal Processing, B. Kosko, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1992), Chap. 9, pp. 287–382.

G. P. Nordin, “Volume diffraction phenomena for photonic neural network implementations and stratified volume holographic optical elements,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1992).

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1981), p. 212.

G. P. Nordin, P. Asthana, A. R. Tanguay, B. K. Jenkins, “Analysis of weighted fan-out/fan-in volume holographic interconnections,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 165–167.

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Figures (22)

Fig. 1
Fig. 1

Schematic representation of fan-out/fan-in interconnections between input and output planes of neuron units.

Fig. 2
Fig. 2

Schematic diagram of a single-source holographic interconnection architecture in which diffraction gratings in a volume hologram connect pixels in the input plane to pixels in the output plane. Interconnection gratings are formed by recording the interference between light from pixels in the training plane and light from pixels in the input plane. L1–L3 are lenses; f1f3 are focal lengths.

Fig. 3
Fig. 3

Layout of the 2-D single-source architecture used in the modeling studies: R is the beam splitter ratio.

Fig. 4
Fig. 4

Schematic representation of the plane waves generated by the pixels in the input and training planes for a 10-to-10 single-source architecture

Fig. 5
Fig. 5

Single-source architecture simulation results for the simultaneous recording method: (a) diffracted outputs, (b) ratio of each diffracted output to an arbitrarily chosen output, and (c) percentage error of each ratio. Readout is performed with mutually coherent beams. The horizontal axis represents the grating strength of the largest interconnection grating recorded in the holographic medium.

Fig. 6
Fig. 6

Simulation results showing the optical throughput (i.e., the amount of incident power diffracted into the desired outputs) for several combinations of recording and readout methods for the single-source architecture. Clarification of the legend is as follows: (1) simultaneous recording method with coherent readout (see Fig. 5), (2) pagewise-sequential recording method with a beam splitter ratio of 100 and coherent readout, (3) pagewise-sequential recording method with a beam splitter ratio of 1000 and coherent readout (see Fig. 7), (4) fully sequential recording with coherent readout (see Fig. 8), and (5) fully sequential recording with incoherent readout (see Fig. 9). The horizontal scale is the same for all single-source-architecture 10-to-10 simulation results, shown in Figs. 5, 79, and 11.

Fig. 7
Fig. 7

Same as Fig. 5 but for the pagewise-sequential recording method with a beam splitter ratio R of 1000.

Fig. 8
Fig. 8

Single-source architecture simulation results for the pagewise-sequential recording method for a beam splitter ratio, R, of 1000: (a) diffracted outputs, (b) ratio of each diffracted output to an arbitrarily chosen output, and (c) percentage error of each ratio. Readout is performed with mutually coherent beams.

Fig. 9
Fig. 9

Single-source architecture simulation results for the fully sequential recording method: (a) diffracted outputs, (b) ratio of each diffracted output to an arbitrarily chosen output, and (c) percentage error of each ratio. Readout is performed with mutually coherent beams.

Fig. 10
Fig. 10

(a) Schematic diagram of the recording geometry of a 4-to-4 interconnection system; (b) schematic diagram of the readout geometry of the 4-to-4 interconnection system with a single beam (x3) and the resulting outputs; (c) simulation results for readout of the 4-to-4 interconnection system, in which the power cross coupled by beam degeneracy from the desired outputs to the x1′, x2′, and x4′ beams can be significant.

Fig. 11
Fig. 11

Simulation results showing the rms fidelity error [as defined in Eq. (35)] for several combinations of recording and readout methods for the single-source architecture. An explanation of the legend is provided in Fig. 6.

Fig. 12
Fig. 12

Angular response characteristics of two Bragg gratings that have the same grating period and slightly different slant angles. Although the main lobes of the angular responses are well separated, the sidelobes and the main lobes overlap.

Fig. 13
Fig. 13

Simulation results for a 4-to-4 single-source architecture. Shown are (a) the rms fidelity error for various recording and readout combinations and (b) the optical throughput for the various recording and readout combinations. Clarification of the legend [in (b), from top to bottom] is as follows: simultaneous recording method with coherent readout, pagewise-sequential recording method with a beam splitter ratio of 16 and coherent readout, pagewise-sequential recording method with a beam splitter ratio of 160 and coherent readout, fully sequential recording with coherent readout, and fully sequential recording with incoherent readout.

Fig. 14
Fig. 14

Schematic diagram of the full-aperture configuration of the incoherent/coherent double angularly multiplexed architecture showing (a) general layout, (b) recording, and (c) reconstruction. M1 and M2 are mirrors; L1–L5 are lenses; BS2 is a second beam splitter.

Fig. 15
Fig. 15

Simulation results for the 10-to-10 incoherent/coherent double angularly multiplexed architecture (full-aperture configuration) for readout with mutually incoherent beams. Shown as functions of the grating strength of the largest grating are (a) the diffracted outputs, (b) the ratios of the diffracted outputs, (c) the percentage error using Rel. (28) for the dependence of each weight on grating strength, and (d) the percentage error using Rel. (36) for the dependence of each weight on grating strength.

Fig. 16
Fig. 16

Simulation results for the 10-to-10 incoherent/coherent double angularly multiplexed architecture (full-aperture configuration). Shown are (a) the rms fidelity error for two different functional dependencies of the weights on the grating strength and (b) the optical throughput.

Fig. 17
Fig. 17

Schematic diagram of the subhologram configuration of the incoherent/coherent double angularly multiplexed architecture.

Fig. 18
Fig. 18

Schematic diagram of a subhologram array. Each subhologram is shown as spatially separate in this case.

Fig. 19
Fig. 19

Simulation results for the 10-to-10 incoherent/coherent double angularly multiplexed architecture (subhologram configuration) for readout with mutually incoherent beams. Shown as functions of the grating strength of the largest grating are (a) the diffracted outputs, (b) the ratios of the diffracted outputs, and (c) the percentage error of each ratio.

Fig. 20
Fig. 20

(a) The rms fidelity error and (b) the throughput for various beam splitter ratios in the 10-to-10 incoherent–coherent double angularly multiplexed architecture (subhologram configuration). When R = 100, the fidelity error and throughput approach the case for which there are no cross gratings. Readout is performed with mutually incoherent beams.

Fig. 21
Fig. 21

(a) The rms fidelity error and (b) the throughput for various beam splitter ratios in the 4-to-4 incoherent/coherent double angularly multiplexed architecture (subhologram configuration). Comparison with Fig. 20 indicates how fidelity and throughput variations scale with the number of interconnection nodes.

Fig. 22
Fig. 22

Rms error as a function of the throughput for the single-source interconnection architecture (parameterized by recording method) and for the two configurations of the incoherent/coherent double angularly multiplexed architecture. In all cases, the single-source architecture is read out with mutually coherent beams and the incoherent/coherent double angularly multiplexed architecture is read out with mutually incoherent beams. The curve for the full-aperture configuration of the incoherent/coherent double angularly multiplexed architecture lies almost directly on the horizontal axis.

Equations (37)

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

y i = f ( ρ i ) ,
ρ i = j = 1 N W i j x j ,
Δ W i j = αδ i ( m ) x j ( m ) ,
ρ i = j = 1 N W i j x j ,
ρ i = | j = 1 N ( W i j x j ) 1 / 2 exp [ i ( ϕ j + Φ i j ) ] | 2 ,
ρ i = | j = 1 N ( W i j x j ) 1 / 2 | 2 .
ρ i = | j = 1 N W i j amp x i amp exp [ i ( ϕ j + Φ i j ) ] | 2 ,
ψ ( θ m ) = 2 π n d B cos θ m / λ ,
I ( r ) = I 0 | j = 1 N ( x j ) 1 / 2 exp ( i k j · r ) + i = 1 N ( δ i R ) 1 / 2 exp ( i k i · r ) | 2 = I 0 R [ R j = 1 N x j + 1 R j = 1 N δ j + 2 i = 1 N j = 1 N ( x j δ i ) 1 / 2 cos ( K i j · r ) + 2 1 R j = 1 N j = j + 1 N ( x j x j ) 1 / 2 cos ( K j j · r ) + 2 1 R i = 1 N i = i + 1 N ( δ i δ i ) 1 / 2 cos ( K i i · r ) ] ,
Δ n ( r ) = i = 1 N j = 1 N Δ n i j cos ( K i j · r ) + j = 1 N j = j + 1 N Δ n j j cos ( K j j · r ) + i = 1 N i = i + 1 N Δ n i i cos ( K i i · r ) ,
Δ n i j = C 1 ( x j δ i ) 1 / 2 ,
Δ n j j = C 1 ( R x j x j ) 1 / 2 ,
Δ n i i = C 1 ( δ i δ i / R ) 1 / 2 ,
ν = 2 π Δ n D / λ ,
ν i j = ( 2 π C 1 D / λ ) ( x j δ i ) 1 / 2 ,
ν j j = ( 2 π C 1 D / λ ) ( R x j x j ) 1 / 2 ,
ν i i = ( 2 π C 1 D / λ ) ( δ i δ i / R ) 1 / 2 ,
Δ n ( r ) = m = 1 N [ i = 1 N j = 1 N Δ n i j ( m ) cos ( K i j · r ) + j = 1 N j = j + 1 N Δ n j j ( m ) cos ( K j j · r ) + i = 1 N i = i + 1 N Δ n i i ( m ) cos ( K i i · r ) ] ,
Δ n i j ( m ) = C 1 [ x j ( m ) δ i ( m ) ] 1 / 2 ,
Δ n j j ( m ) = C 1 [ R x j ( m ) x j ( m ) ] 1 / 2 ,
Δ n i i ( m ) = C 1 [ δ i ( m ) δ i ( m ) / R ] 1 / 2 .
Δ n ( r ) = i = 1 N j = 1 N Δ n i j ( m ) cos ( K i j · r ) + j = 1 N j = j + 1 N Δ n j j ( m ) cos ( K j j · r ) + i = 1 N i = i + 1 N Δ n i i ( m ) cos ( K i i · r ) ,
Δ n μ ν ( M ) = m = 1 M Δ n μ ν ( m ) ,
ν i j ( M ) = ( 2 π C 1 D / λ ) m = 1 M [ x j ( m ) δ i ( m ) ] 1 / 2 ,
ν j j ( M ) = R ( 2 π C 1 D / λ ) m = 1 M [ x j ( m ) x j ( m ) ] 1 / 2 ,
ν i i ( M ) = ( 1 / R ) ( 2 π C 1 D / λ ) m = 1 M [ δ i ( m ) δ i ( m ) ] 1 / 2 ,
W i j x j δ i
W i j ν i j 2 .
W i j [ ν i j ( M ) ] 2 { m = 1 M [ x j ( m ) δ i ( m ) ] 1 / 2 } 2 .
Γ i i = 100 ( ρ i / ρ i ρ i / ρ i ρ i / ρ i ) ,
Δ n ( r ) = m = 1 N { i = 1 N [ j = 1 N Δ n i j ( m ) cos ( K i j · r ) + i = 1 N i = i + 1 N Δ n i i ( m ) cos ( K i i · r ) ] }
Δ n ( r ) = i = 1 N j = 1 N Δ n i j ( M ) cos ( K i j · r ) + N i = 1 N i = i + 1 N Δ n i i ( M ) cos ( K i i · r ) .
ν i i ( M ) = ( N / R ) ( 2 π C 1 D / λ ) m = 1 M [ δ i ( m ) δ i ( m ) ] 1 / 2 .
Δ n ( r ) = i = 1 N j = 1 N Δ n i j ( M ) cos ( K i j · r ) ,
= | u ˆ u ˆ | = [ j = 1 N ( ρ j | ρ | ρ j | ρ | ) 2 ] 1 / 2 ,
W i j sin 2 [ ν i j ( M ) / 2 ] ,
W i j [ ν i j ( M ) ] 2 ,

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