Abstract

Optically interconnected processor arrays are compared to conventional fully electronic processor arrays in terms of interconnect density capabilities. A complexity model is introduced that allows the calculation of the array area growth rate as an asymptotic function of the number of processing elements in the array. Lower bounds on the area growth rate of electrically interconnected processor arrays are compared to upper bounds for free-space optically interconnected circuits that employ computer generated holograms. Results indicate that for connection networks such as the hypercube, perfect shuffle and crossbar networks, that have a high minimum bisection width (a measure of the global nature of an interconnect topology) and contain some degree of spatial invariance, optically interconnected circuit area growth rates are below lower bounds on VLSI circuit growth rates.

© 1989 Optical Society of America

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References

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  1. Daniel Hillis, The Connection Machine (MIT Press, Cambridge, 1985).
  2. C. Seitz, “Concurrent VLSI Architectures,” IEEE Trans. Cornput. C-33, 1247–1265 (1984).
    [CrossRef]
  3. K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
    [CrossRef]
  4. J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
    [CrossRef]
  5. M. R. Feldman, S. C. Esener, C. C. Guest, H. Lee Sing, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
    [CrossRef] [PubMed]
  6. S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
    [CrossRef]
  7. R. E. Brooks, “Micromechanical Light Modulators on Silicon,” Opt. Eng. 24, 101–106 (1985).
    [CrossRef]
  8. G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
    [CrossRef]
  9. E. Bradley, P. K. L. Yu, “Proposed Modulator for Global VLSI Optical Interconnect Network,” Jpn. J. Appl. Phys. 26, L971–L973 (1987).
    [CrossRef]
  10. W. H. Wu et al., “Implementation of Optical Interconnections for VLSI,” IEEE Trans. Electron Devices, ED-34, 706–713 (1987).
    [CrossRef]
  11. M. R. Feldman, C. C. Guest, “Computer Generated Holographic Optical Elements for Optical Interconnection of Very Large Scale Integrated Circuits,” Appl. Opt. 26, 4377–4384 (1987).
    [CrossRef] [PubMed]
  12. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical Imaging Applied to Microelectronic Chip-to-Chip Interconnections,” Appl. Opt., 24, 2851–2858 (1985).
    [CrossRef] [PubMed]
  13. C. D. Thompson, “Area-Time Complexity for VLSI,” in Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, Atlanta, GA, 30 Apr. 1979, pp. 81–88.
  14. J. D. Ullman, Computational Aspects of VLSI, Computer Science Press, (Rockville, MD, 1984) Chap. 2.
  15. D. Gabor, “Light and Information,” Prog. Opt. 1, 109–000 (1961).
    [CrossRef]
  16. R. Barakat, J. Reif, “Lower Bounds on the Computational Efficiency of Optical Computing Systems,” Appl. Opt., 26, 1015–1018 (1987).
    [CrossRef] [PubMed]
  17. M. R. Feldman, C. C. Guest, “Interconnect Density Capabilities of Computer Generated Holograms for Optical Interconnection of Very Large Scale Integrated Circuits,” Appl. Opt. 28, 3134–3137 (1989).
    [CrossRef] [PubMed]
  18. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
    [CrossRef]
  19. J. W. Goodman, “Fan-in and Fan-out with Optical Interconnections,” Opt. Acta, 32, 1489–1496 (1985).
    [CrossRef]
  20. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuck, T. C. Strand, “Architectural Implications of a Digital Optical Process,” Appl. Opt., 23, 3465–3474 (1984).
    [CrossRef] [PubMed]
  21. K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).
  22. A. W. Lohmann, “What Classical Optics Can Do for the Digital Optical Computer,” Appl. Opt., 25, 1543–1549 (1986).
    [CrossRef] [PubMed]
  23. C. W. Stirk, R. A. Athale, M. W. Haney, “Folded Perfect Shuffle Optical Processor,” Appl. Opt., 27, 202–203 (1988).
    [CrossRef] [PubMed]
  24. S. C. Esener, “Silicon Device Development for Si/PLZT Spatial Light Modulators,” Ph.D. Thesis (U. California, San Diego, La Jolla, June1986), Chapter 1.
  25. L. A. Glasser, D. W. Dobberpuhl, The Design and Analysis of VLSI Circuits, (Addison-Wesley, Reading, MA, 1985), p. 196.
  26. M. R. Feldman, C. C. Guest, “Automated Design of Holographic Optical Elements for Interconnection of Electronic Circuits,” J. Opt. Soc. Am. A 3(13) P80 (1986).
  27. M. R. Feldman, C. C. Guest, S. L. Lee, “Design of Computer Generated Holograms for a Shared Memory Network,” Proc. Soc. Photo-Opt. Instrum. Eng. 28, 258–261 (1989).

1989 (2)

M. R. Feldman, C. C. Guest, “Interconnect Density Capabilities of Computer Generated Holograms for Optical Interconnection of Very Large Scale Integrated Circuits,” Appl. Opt. 28, 3134–3137 (1989).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, S. L. Lee, “Design of Computer Generated Holograms for a Shared Memory Network,” Proc. Soc. Photo-Opt. Instrum. Eng. 28, 258–261 (1989).

1988 (2)

1987 (8)

S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
[CrossRef]

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

E. Bradley, P. K. L. Yu, “Proposed Modulator for Global VLSI Optical Interconnect Network,” Jpn. J. Appl. Phys. 26, L971–L973 (1987).
[CrossRef]

W. H. Wu et al., “Implementation of Optical Interconnections for VLSI,” IEEE Trans. Electron Devices, ED-34, 706–713 (1987).
[CrossRef]

M. R. Feldman, C. C. Guest, “Computer Generated Holographic Optical Elements for Optical Interconnection of Very Large Scale Integrated Circuits,” Appl. Opt. 26, 4377–4384 (1987).
[CrossRef] [PubMed]

R. Barakat, J. Reif, “Lower Bounds on the Computational Efficiency of Optical Computing Systems,” Appl. Opt., 26, 1015–1018 (1987).
[CrossRef] [PubMed]

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).

1986 (2)

A. W. Lohmann, “What Classical Optics Can Do for the Digital Optical Computer,” Appl. Opt., 25, 1543–1549 (1986).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Automated Design of Holographic Optical Elements for Interconnection of Electronic Circuits,” J. Opt. Soc. Am. A 3(13) P80 (1986).

1985 (3)

J. W. Goodman, “Fan-in and Fan-out with Optical Interconnections,” Opt. Acta, 32, 1489–1496 (1985).
[CrossRef]

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical Imaging Applied to Microelectronic Chip-to-Chip Interconnections,” Appl. Opt., 24, 2851–2858 (1985).
[CrossRef] [PubMed]

R. E. Brooks, “Micromechanical Light Modulators on Silicon,” Opt. Eng. 24, 101–106 (1985).
[CrossRef]

1984 (3)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuck, T. C. Strand, “Architectural Implications of a Digital Optical Process,” Appl. Opt., 23, 3465–3474 (1984).
[CrossRef] [PubMed]

C. Seitz, “Concurrent VLSI Architectures,” IEEE Trans. Cornput. C-33, 1247–1265 (1984).
[CrossRef]

1980 (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
[CrossRef]

1961 (1)

D. Gabor, “Light and Information,” Prog. Opt. 1, 109–000 (1961).
[CrossRef]

Athale, R. A.

C. W. Stirk, R. A. Athale, M. W. Haney, “Folded Perfect Shuffle Optical Processor,” Appl. Opt., 27, 202–203 (1988).
[CrossRef] [PubMed]

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

Barakat, R.

Bowler, K. C.

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Boyd, G. D.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Bradley, E.

E. Bradley, P. K. L. Yu, “Proposed Modulator for Global VLSI Optical Interconnect Network,” Jpn. J. Appl. Phys. 26, L971–L973 (1987).
[CrossRef]

Brooks, R. E.

R. E. Brooks, “Micromechanical Light Modulators on Silicon,” Opt. Eng. 24, 101–106 (1985).
[CrossRef]

Bruce, A. D.

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Chavel, P.

Chemla, D. S.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Dobberpuhl, D. W.

L. A. Glasser, D. W. Dobberpuhl, The Design and Analysis of VLSI Circuits, (Addison-Wesley, Reading, MA, 1985), p. 196.

English, J. H.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Esener, S. C.

M. R. Feldman, S. C. Esener, C. C. Guest, H. Lee Sing, “Comparison Between Optical and Electrical Interconnects Based on Power and Speed Considerations,” Appl. Opt. 27, 1742–1751 (1988).
[CrossRef] [PubMed]

S. C. Esener, “Silicon Device Development for Si/PLZT Spatial Light Modulators,” Ph.D. Thesis (U. California, San Diego, La Jolla, June1986), Chapter 1.

Feldman, M. R.

Forchheimer, R.

Gabor, D.

D. Gabor, “Light and Information,” Prog. Opt. 1, 109–000 (1961).
[CrossRef]

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
[CrossRef]

Glasser, L. A.

L. A. Glasser, D. W. Dobberpuhl, The Design and Analysis of VLSI Circuits, (Addison-Wesley, Reading, MA, 1985), p. 196.

Goodman, J. W.

J. W. Goodman, “Fan-in and Fan-out with Optical Interconnections,” Opt. Acta, 32, 1489–1496 (1985).
[CrossRef]

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Optical Imaging Applied to Microelectronic Chip-to-Chip Interconnections,” Appl. Opt., 24, 2851–2858 (1985).
[CrossRef] [PubMed]

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

Gossard, A. C.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Guest, C. C.

Haney, M. W.

Hesselink, L.

Hillis, Daniel

Daniel Hillis, The Connection Machine (MIT Press, Cambridge, 1985).

Huang, K. S.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).

Jenkins, B. K.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).

B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuck, T. C. Strand, “Architectural Implications of a Digital Optical Process,” Appl. Opt., 23, 3465–3474 (1984).
[CrossRef] [PubMed]

Kenway, R. D.

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Kostuk, R. K.

Kung, S. Y.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

Lee, S. L.

M. R. Feldman, C. C. Guest, S. L. Lee, “Design of Computer Generated Holograms for a Shared Memory Network,” Proc. Soc. Photo-Opt. Instrum. Eng. 28, 258–261 (1989).

Leonberger, F. I.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

Lohmann, A. W.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
[CrossRef]

McCall, S. L.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Miller, D. A. B.

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
[CrossRef]

Pawley, G. S.

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Reif, J.

Sakai, S.

S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
[CrossRef]

Sawchuck, A. A.

Sawchuk, A. A.

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).

Seitz, C.

C. Seitz, “Concurrent VLSI Architectures,” IEEE Trans. Cornput. C-33, 1247–1265 (1984).
[CrossRef]

Shiraishi, H.

S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
[CrossRef]

Sing, H. Lee

Stirk, C. W.

Strand, T. C.

Thompson, C. D.

C. D. Thompson, “Area-Time Complexity for VLSI,” in Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, Atlanta, GA, 30 Apr. 1979, pp. 81–88.

Ullman, J. D.

J. D. Ullman, Computational Aspects of VLSI, Computer Science Press, (Rockville, MD, 1984) Chap. 2.

Umeno, M.

S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
[CrossRef]

Wallace, D. J.

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Wu, W. H.

W. H. Wu et al., “Implementation of Optical Interconnections for VLSI,” IEEE Trans. Electron Devices, ED-34, 706–713 (1987).
[CrossRef]

Yu, P. K. L.

E. Bradley, P. K. L. Yu, “Proposed Modulator for Global VLSI Optical Interconnect Network,” Jpn. J. Appl. Phys. 26, L971–L973 (1987).
[CrossRef]

Appl. Opt. (8)

Appl. Phys. Lett. (1)

G. D. Boyd, D. A. B. Miller, D. S. Chemla, S. L. McCall, A. C. Gossard, J. H. English, “Multiple Quantum Well Reflection Modulator,” Appl. Phys. Lett. 50, 1119–1121 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Sakai, H. Shiraishi, M. Umeno, “AlGaAs/GaAs Stripe Laser Diodes Fabricated on Si Substrates by MOCVD,” IEEE J. Quantum Electron. QE-23, 1080–1084 (1987).
[CrossRef]

IEEE Trans. Cornput. (1)

C. Seitz, “Concurrent VLSI Architectures,” IEEE Trans. Cornput. C-33, 1247–1265 (1984).
[CrossRef]

IEEE Trans. Electron Devices (1)

W. H. Wu et al., “Implementation of Optical Interconnections for VLSI,” IEEE Trans. Electron Devices, ED-34, 706–713 (1987).
[CrossRef]

J. Opt. Soc. Am. A (2)

K. S. Huang, B. K. Jenkins, A. A. Sawchuk, “Programming a Digital Optical Cellular Image Processor,” J. Opt. Soc. Am. A 4(13), P87–P88 (1987).

M. R. Feldman, C. C. Guest, “Automated Design of Holographic Optical Elements for Interconnection of Electronic Circuits,” J. Opt. Soc. Am. A 3(13) P80 (1986).

Jpn. J. Appl. Phys. (1)

E. Bradley, P. K. L. Yu, “Proposed Modulator for Global VLSI Optical Interconnect Network,” Jpn. J. Appl. Phys. 26, L971–L973 (1987).
[CrossRef]

Opt. Acta (1)

J. W. Goodman, “Fan-in and Fan-out with Optical Interconnections,” Opt. Acta, 32, 1489–1496 (1985).
[CrossRef]

Opt. Commun. (1)

M. G. Moharam, T. K. Gaylord, R. Magnusson, “Criteria for Raman-Nath Regime Diffraction by Phase Gratings,” Opt. Commun., 32, 19–23 (1980).
[CrossRef]

Opt. Eng. (1)

R. E. Brooks, “Micromechanical Light Modulators on Silicon,” Opt. Eng. 24, 101–106 (1985).
[CrossRef]

Phys. Today (1)

K. C. Bowler, A. D. Bruce, R. D. Kenway, G. S. Pawley, D. J. Wallace, “Exploiting Highly Concurrent Computers for Physics,” Phys. Today 40, 40–48 (1987).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE, 72, 850–866 (1984).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. R. Feldman, C. C. Guest, S. L. Lee, “Design of Computer Generated Holograms for a Shared Memory Network,” Proc. Soc. Photo-Opt. Instrum. Eng. 28, 258–261 (1989).

Prog. Opt. (1)

D. Gabor, “Light and Information,” Prog. Opt. 1, 109–000 (1961).
[CrossRef]

Other (5)

C. D. Thompson, “Area-Time Complexity for VLSI,” in Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, Atlanta, GA, 30 Apr. 1979, pp. 81–88.

J. D. Ullman, Computational Aspects of VLSI, Computer Science Press, (Rockville, MD, 1984) Chap. 2.

Daniel Hillis, The Connection Machine (MIT Press, Cambridge, 1985).

S. C. Esener, “Silicon Device Development for Si/PLZT Spatial Light Modulators,” Ph.D. Thesis (U. California, San Diego, La Jolla, June1986), Chapter 1.

L. A. Glasser, D. W. Dobberpuhl, The Design and Analysis of VLSI Circuits, (Addison-Wesley, Reading, MA, 1985), p. 196.

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Figures (5)

Fig. 1
Fig. 1

Illustration of a general VLSIO circuit.

Fig. 2
Fig. 2

Schematic diagrams of several types of crossbar interconnection schemes. Fig. (2a) illustrates a dedicated link crossbar interconnection network. Fig. (2b) shows a shared link crossbar network. Fig. (2c) illustrates a crossbar switch network.

Fig. 3
Fig. 3

Reflective system utilizing a multi-element double-pass HOE.

Fig. 4
Fig. 4

Transmissive architecture for implementing interconnections with a basis set or space-invariant CGH.

Fig. 5
Fig. 5

A space-invariant superset of all connection patterns for a shared link hypercube for a 16 × 16 element array (N = 256). Hypercube connections only involve PEs separated by 2j nodes where j = 0 , 1 , , log ( N / 2 ).

Tables (3)

Tables Icon

Table I Lower Bounds on Order of Area Dependence on Number of PEs

Tables Icon

Table II Upper Bounds on Area Growth Rates for WSIO

Tables Icon

Table III Growth Rates of Cost Per Performance for Linear Speedup

Equations (29)

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

A E = Ω ( ω 2 ) .
A O = Ω ( ω ) .
O = f ( S ; I 1 , I 2 , I F ) ,
O i = f i ( S ; I 1 , I 2 , I F ) ,
D 1.2 π z / d λ .
D = Ω ( z / d ) .
A I = Ω ( N F ) .
ω N ,
A E = Ω { max ( ω F , ω 2 ) } .
w = θ { h 2 / ( A S cos 3 ϕ ) } , w = θ { h 2 / ( A S cos ϕ ) } ,
A D = θ { ( h 2 2 + L 2 ) 2 / ( h 2 2 A S ) } .
A D = O ( L max 2 / A S ) .
A I = N A S + N F A D .
L = θ { L max A S + F A D } .
A I = O ( N F L 2 ) .
A I = N F A S + N F A D ,
L = θ { L max F A S + F A D } .
A I = O ( N F 2 L 2 ) .
A I = O { max ( N I M , M N O ) } ,
$ = N A PE $ P + A I $ I ,
$ = Ω ( A E ) ,
$ = N A PE $ P + A I $ H + A L $ L ,
$ = O ( A O ) ,
P = S / ( τ P + τ I ) ,
τ I = Ω ( L max ) ,
$ / P = Ω ( A E ( τ P + τ I ) / S ) .
$ / P = Ω ( A E τ I / S ) = Ω ( A E L max / S ) ,
$ / P = O ( A O / S ) .
E SW = p τ I .

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