Abstract

A cascadable residue arithmetic processor based on optical Fredkin gate arrays and page-oriented holographic memories is introduced. The implementations of residue functions and operations by this processor are described. Analytic expressions are derived for the number of holograms and waveguide channels required for the implementation of residue addition and multiplication. The practical cases of 16-bit addition and multiplication are analyzed as specific examples. It is shown that, using the proposed architecture, these operations can be implemented with state-of-the-art technologies in holography and integrated optics.

© 1987 Optical Society of America

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  1. For example, see these special issues on Digital Optical Computing: Proc. IEEE 72, No. 7 (1984);Appl. Opt. 25, Nos. 10, 14,18 (1986);Opt. Eng. 24, No. 1 (1985);Opt. Eng. 25, No. 1 (1986);Opt. Eng. 26, No. 1 (1987).
    [PubMed]
  2. A. W. Lohmann, “What Classical Optics Can Do for the Digital Optical Computer,” Appl. Opt. 25,1543 (1986).
    [CrossRef] [PubMed]
  3. J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical Interconnections for VLSI Systems,” Proc. IEEE 72, 850 (1984).
    [CrossRef]
  4. L. A. Bergman et al., “Holographic Optical Interconnects for VLSI,” Opt. Eng. 25, 1109 (1986).
    [CrossRef]
  5. J. Shamir, H. J. Caulfield, W. J. Miceli, R. J. Seymour, “Optical Computing and the Fredkin Gates,” Appl. Opt. 25, 1604 (1986).
    [CrossRef] [PubMed]
  6. J. Shamir, H. J. Caulfield, “High-Efficiency Rapidly Programmable Optical Interconnections,” Appl. Opt. 26, 1032 (1987).
    [CrossRef] [PubMed]
  7. A. Svoboda, M. Valach, “Rational Numerical System for Residue Classes,” in Stroje na Zpracovani Informaci, Sbornik V. (Nakl. CSAV, Prague, 1957), pp. 9–37 (in English).
  8. N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).
  9. S. A. Collins, “Numerical Optical Data Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).
  10. C. Y. Yen, S. A. Collins, “Operation of a Numerical Optical Digital Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 160 (1980).
  11. C. C. Guest, T. K. Gaylord, “Truth-Table Look-Up Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 1201 (1980).
    [CrossRef] [PubMed]
  12. M. M. Mirsalehi, T. K. Gaylord, “Truth-Table Look-Up Parallel Data Processing Using an Optical Content-Addressable Memory,” Appl. Opt. 25, 2277 (1986).
    [CrossRef] [PubMed]
  13. A. Huang, “The Implementation of a Residue Arithmetic Unit via Optical and other Physical Phenomena,” In Proceedings, International Optical Computing Conference (IEEE, New York, 1975), p. 14.
  14. A. Huang, Y. Tsunoda, J. Goodman, S. Ishihara, “Optical Computation Using Residue Arithmetic,” Appl. Opt. 18, 149 (1979).
    [CrossRef] [PubMed]
  15. A. Tai, I. Cindrich, J. R. Fienup, C. C. Aleksoff, “Optical Residue Arithmetic Computer with Programmable Computation Modules,” Appl. Opt. 18, 2812 (1979).
    [CrossRef] [PubMed]
  16. D. Psaltis, D. Casasent, “Optical Residue Arithmetic: a Correlation Approach,” Appl. Opt. 18, 163 (1979).
    [CrossRef] [PubMed]
  17. D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).
  18. S. Y. Huang, S. H. Lee, “Residue Arithmetic Circuit Design Based on Integrated Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 321, 122 (1982).
  19. For example, see R. Chen, C. S. Tsai, “Thermally Annealed Single-Mode Proton-Exchange Channel-Waveguide Cutoff Modulator,” Opt. Lett. 11, 546 (1986).
    [CrossRef] [PubMed]
  20. For example, see J. W. Gladden, R. D. Leighty, “Recording Media,” in Handbook of Optical Holography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 277–298.
  21. For example, see E. H. Young, S.-K. Yao, “Design Considerations for Acousto-Optic Devices,” Proc. IEEE 69, 54 (1981).
    [CrossRef]
  22. C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, “Residue Number System Truth-Table Look-Up Processing—Moduli Selection and Logical Minimization,” IEEE Trans. Comput. C-33, 927 (1984).
    [CrossRef]
  23. C. A. Papachristou, “The Recurrency Classes in Multi-Operand Addition and Multiplication Modulo M,” in Proceedings, 1981 Conference on Information Systems and Science (John Hopkins U., Baltimore, Mar. 1981), pp. 402–407.

1987 (1)

1986 (5)

1984 (3)

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, “Residue Number System Truth-Table Look-Up Processing—Moduli Selection and Logical Minimization,” IEEE Trans. Comput. C-33, 927 (1984).
[CrossRef]

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

For example, see these special issues on Digital Optical Computing: Proc. IEEE 72, No. 7 (1984);Appl. Opt. 25, Nos. 10, 14,18 (1986);Opt. Eng. 24, No. 1 (1985);Opt. Eng. 25, No. 1 (1986);Opt. Eng. 26, No. 1 (1987).
[PubMed]

1982 (1)

S. Y. Huang, S. H. Lee, “Residue Arithmetic Circuit Design Based on Integrated Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 321, 122 (1982).

1981 (1)

For example, see E. H. Young, S.-K. Yao, “Design Considerations for Acousto-Optic Devices,” Proc. IEEE 69, 54 (1981).
[CrossRef]

1980 (3)

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

C. Y. Yen, S. A. Collins, “Operation of a Numerical Optical Digital Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 160 (1980).

C. C. Guest, T. K. Gaylord, “Truth-Table Look-Up Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 1201 (1980).
[CrossRef] [PubMed]

1979 (3)

1977 (1)

S. A. Collins, “Numerical Optical Data Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

Aleksoff, C. C.

Athale, R. A.

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

Bergman, L. A.

L. A. Bergman et al., “Holographic Optical Interconnects for VLSI,” Opt. Eng. 25, 1109 (1986).
[CrossRef]

Carlotto, M.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

Casasent, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

D. Psaltis, D. Casasent, “Optical Residue Arithmetic: a Correlation Approach,” Appl. Opt. 18, 163 (1979).
[CrossRef] [PubMed]

Caulfield, H. J.

Chen, R.

Cindrich, I.

Collins, S. A.

C. Y. Yen, S. A. Collins, “Operation of a Numerical Optical Digital Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 160 (1980).

S. A. Collins, “Numerical Optical Data Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

Fienup, J. R.

Gaylord, T. K.

Gladden, J. W.

For example, see J. W. Gladden, R. D. Leighty, “Recording Media,” in Handbook of Optical Holography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 277–298.

Goodman, J.

Goodman, J. W.

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

Guest, C. C.

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, “Residue Number System Truth-Table Look-Up Processing—Moduli Selection and Logical Minimization,” IEEE Trans. Comput. C-33, 927 (1984).
[CrossRef]

C. C. Guest, T. K. Gaylord, “Truth-Table Look-Up Processing Utilizing Binary and Residue Arithmetic,” Appl. Opt. 19, 1201 (1980).
[CrossRef] [PubMed]

Huang, A.

A. Huang, Y. Tsunoda, J. Goodman, S. Ishihara, “Optical Computation Using Residue Arithmetic,” Appl. Opt. 18, 149 (1979).
[CrossRef] [PubMed]

A. Huang, “The Implementation of a Residue Arithmetic Unit via Optical and other Physical Phenomena,” In Proceedings, International Optical Computing Conference (IEEE, New York, 1975), p. 14.

Huang, S. Y.

S. Y. Huang, S. H. Lee, “Residue Arithmetic Circuit Design Based on Integrated Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 321, 122 (1982).

Ishihara, S.

Kung, S.-Y.

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

Lee, S. H.

S. Y. Huang, S. H. Lee, “Residue Arithmetic Circuit Design Based on Integrated Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 321, 122 (1982).

Leighty, R. D.

For example, see J. W. Gladden, R. D. Leighty, “Recording Media,” in Handbook of Optical Holography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 277–298.

Leonberger, F. I.

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

Lohmann, A. W.

Miceli, W. J.

Mirsalehi, M. M.

M. M. Mirsalehi, T. K. Gaylord, “Truth-Table Look-Up Parallel Data Processing Using an Optical Content-Addressable Memory,” Appl. Opt. 25, 2277 (1986).
[CrossRef] [PubMed]

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, “Residue Number System Truth-Table Look-Up Processing—Moduli Selection and Logical Minimization,” IEEE Trans. Comput. C-33, 927 (1984).
[CrossRef]

Neft, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

Papachristou, C. A.

C. A. Papachristou, “The Recurrency Classes in Multi-Operand Addition and Multiplication Modulo M,” in Proceedings, 1981 Conference on Information Systems and Science (John Hopkins U., Baltimore, Mar. 1981), pp. 402–407.

Psaltis, D.

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

D. Psaltis, D. Casasent, “Optical Residue Arithmetic: a Correlation Approach,” Appl. Opt. 18, 163 (1979).
[CrossRef] [PubMed]

Seymour, R. J.

Shamir, J.

Svoboda, A.

A. Svoboda, M. Valach, “Rational Numerical System for Residue Classes,” in Stroje na Zpracovani Informaci, Sbornik V. (Nakl. CSAV, Prague, 1957), pp. 9–37 (in English).

Szabo, N. S.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

Tai, A.

Tanaka, R. I.

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

Tsai, C. S.

Tsunoda, Y.

Valach, M.

A. Svoboda, M. Valach, “Rational Numerical System for Residue Classes,” in Stroje na Zpracovani Informaci, Sbornik V. (Nakl. CSAV, Prague, 1957), pp. 9–37 (in English).

Yao, S.-K.

For example, see E. H. Young, S.-K. Yao, “Design Considerations for Acousto-Optic Devices,” Proc. IEEE 69, 54 (1981).
[CrossRef]

Yen, C. Y.

C. Y. Yen, S. A. Collins, “Operation of a Numerical Optical Digital Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 160 (1980).

Young, E. H.

For example, see E. H. Young, S.-K. Yao, “Design Considerations for Acousto-Optic Devices,” Proc. IEEE 69, 54 (1981).
[CrossRef]

Appl. Opt. (8)

Digital Optical Computing: Proc. IEEE (1)

For example, see these special issues on Digital Optical Computing: Proc. IEEE 72, No. 7 (1984);Appl. Opt. 25, Nos. 10, 14,18 (1986);Opt. Eng. 24, No. 1 (1985);Opt. Eng. 25, No. 1 (1986);Opt. Eng. 26, No. 1 (1987).
[PubMed]

IEEE Trans. Comput. (1)

C. C. Guest, M. M. Mirsalehi, T. K. Gaylord, “Residue Number System Truth-Table Look-Up Processing—Moduli Selection and Logical Minimization,” IEEE Trans. Comput. C-33, 927 (1984).
[CrossRef]

Opt. Eng. (1)

L. A. Bergman et al., “Holographic Optical Interconnects for VLSI,” Opt. Eng. 25, 1109 (1986).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (2)

For example, see E. H. Young, S.-K. Yao, “Design Considerations for Acousto-Optic Devices,” Proc. IEEE 69, 54 (1981).
[CrossRef]

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

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

D. Psaltis, D. Casasent, D. Neft, M. Carlotto, “Accurate Numerical Computation by Optical Convolution,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 151 (1980).

S. Y. Huang, S. H. Lee, “Residue Arithmetic Circuit Design Based on Integrated Optics,” Proc. Soc. Photo-Opt. Instrum. Eng. 321, 122 (1982).

S. A. Collins, “Numerical Optical Data Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 128, 313 (1977).

C. Y. Yen, S. A. Collins, “Operation of a Numerical Optical Digital Processor,” Proc. Soc. Photo-Opt. Instrum. Eng. 232, 160 (1980).

Other (5)

A. Huang, “The Implementation of a Residue Arithmetic Unit via Optical and other Physical Phenomena,” In Proceedings, International Optical Computing Conference (IEEE, New York, 1975), p. 14.

A. Svoboda, M. Valach, “Rational Numerical System for Residue Classes,” in Stroje na Zpracovani Informaci, Sbornik V. (Nakl. CSAV, Prague, 1957), pp. 9–37 (in English).

N. S. Szabo, R. I. Tanaka, Residue Arithmetic and Its Applications to Computer Technology (McGraw-Hill, New York, 1967).

For example, see J. W. Gladden, R. D. Leighty, “Recording Media,” in Handbook of Optical Holography, H. J. Caulfield, Ed. (Academic, New York, 1979), pp. 277–298.

C. A. Papachristou, “The Recurrency Classes in Multi-Operand Addition and Multiplication Modulo M,” in Proceedings, 1981 Conference on Information Systems and Science (John Hopkins U., Baltimore, Mar. 1981), pp. 402–407.

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Figures (10)

Fig. 1
Fig. 1

Fredkin gate.

Fig. 2
Fig. 2

Waveguide coupler implementation of the Fredkin gate. I is the interaction region where coupling is switched on or off.

Fig. 3
Fig. 3

Fredkin gate array of four channels and four switching layers.

Fig. 4
Fig. 4

Example implementations of functions in residue arithmetic by interconnecting systems: (a) addition of 2 to a residue number modulo 4; (b) raising a residue modulo 4 number by power 3. The input is entered from the left, and the output is obtained from the right.

Fig. 5
Fig. 5

Schematic diagram of the proposed processor: POHM, page-oriented holographic memory; OFGA, optical Fredkin gate array.

Fig. 6
Fig. 6

Interconnections corresponding to residue addition (N1 + N2) modulo 4. The interconnections (a), (b), (c), and (d) correspond to N2 = 0, 1, 2, and 3, respectively. The input N1 is entered from the left, and the output N1 + N2 is obtained from the right.

Fig. 7
Fig. 7

Required switching states for implementing residue addition (N1 + N2) modulo 4. The hatched switching elements are on. The four interconnections realized in (a), (b), (c), and (d) correspond to N2 = 0, 1, 2, and 3, respectively.

Fig. 8
Fig. 8

Interconnections corresponding to residue multiplication (N1 × N2) modulo 4. The interconnections (a), (b), (c), and (d) correspond to N2 = 0, 1, 2, and 3, respectively. The input N1 is entered from the left, and the output N1 × N2 is obtained from the right.

Fig. 9
Fig. 9

Required switching states for implementing residue multiplication (N1 × N2) modulo 4. The hatched switching elements are on. The four interconnections realized in (a), (b), (c), and (d) correspond to N2 = 0, 1, 2, and 3, respectively.

Fig. 10
Fig. 10

Cascaded system for evaluating P(x) = a4x4 + a3x3 + a2x2 + a1x + a0 using Horner's rule.

Equations (9)

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

M = i = 1 n m i .
C = C , A = A and B = B , if C = 0 , A = B and B = A , if C = 1 .
N h = ( n + 1 ) p n n p n 1 2 .
N c = ( n + 1 ) p n n p n 1 .
N l = N c m + 1 ,
N s = [ ( N c 1 ) N l / 2 ] .
P ( x ) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = { [ ( a 4 x + a 3 ) x + a 2 ] x + a 1 } x + a 0 .
N h = ( p n 1 ) + ( n 1 ) ( p 1 ) p n 1 + ( p n p n 1 1 ) = ( n + 1 ) p n n p n 1 2 .
N c = p n + ( n 1 ) ( p n p n 1 ) + ( p n p n 1 ) = ( n + 1 ) p n n p n 1 .

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