Abstract

The polarization-encoded optical shadow casting (POSC) scheme is used to design a 2-digit by 2-digit trinary multiplier. Three design algorithms for identifying the appropriate source patterns, input patterns, and output mask for both serial and parallel operations are presented. The inclusion of analyzer pattern reduces the number of pixel subcells for the POSC inputs.

© 1990 Optical Society of America

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  1. J. Tanida, Y. Ichi⊙ka, “Optical Array Logic Processor Using Shadowgrams,” J. Opt. Soc. Am. 73, 800–809 (1983).
    [CrossRef]
  2. Y. Ichioka, J. Tanida, “Optical Parallel Logic Gates Using a Shadow-Casting System for Optical Digital Computing,” Proc. IEEE 72, 787–801 (1984).
    [CrossRef]
  3. M. M. Mirsalehi, C. C. Guest, T. K. Gaylord, “Residue Number System Holographic Truth-Table Look-Up Processing: Detector Threshold Setting and Probability of Error due to Amplitude and Phase Variations,” Appl. Opt. 22, 3583–3592 (1983).
    [CrossRef] [PubMed]
  4. Y. Li, G. Eichmann, R. R. Alfano, “Optical Computing Using Hybrid Encoded Shadow Casting,” Appl. Opt. 25, 2636–2638 (1986).
    [CrossRef] [PubMed]
  5. M. A. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Logic Units: Design,” Appl. Opt. 26, 2220–2725 (1987).
    [CrossRef]
  6. A. A. S. Awwal, M. A. Karim, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Scheme: Design of Multioutput Trinary Combinational Logic Units,” Appl. Opt. 26, 4814–4818 (1987).
    [CrossRef] [PubMed]
  7. A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Design of a J-K Flip-Flop,” Appl. Opt. 27, 3719–722 (1988).
    [CrossRef] [PubMed]
  8. A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting Programmable Logic Array: Simultaneous Generation of Multiple Outputs,” Appl. Opt. 27, 932–936 (1988).
    [CrossRef] [PubMed]
  9. A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Direct Implementation of a Carry-Free Adder,” Appl. Opt. 28, 785–790 (1989).
    [CrossRef] [PubMed]
  10. A. A. S. Awwal, M. A. Karim, “Associative Polarization-Encoded Optical Shadow Casting: Gray-Level Image Encoding for Serial and Parallel Operations,” Appl. Opt. 28, 284–290 (1989).
    [CrossRef] [PubMed]
  11. A. A. S. Awwal, M. A. Karim, “Multiprocessor Design Using Polarization-Encoded Optical Shadow-Casting,” Appl. Opt. 29, 2107–2112 (1990).
    [CrossRef] [PubMed]
  12. R. A. Rizvi, K. Zaheer, M. S. Zubairy, “Separate and Simultaneous Generation of Multioutputs in a Polarization-Encoded Optical Shadow-Casting Scheme: Design of Half- and Full Adders and Subtractors,” Appl. Opt. 27, 5176–5180(1988).
    [CrossRef] [PubMed]
  13. R. A. Rizvi, K. Zaheer, M. S. Zubairy, “Design of Optical Decoders Using a Polarization-Encoded Optical Shadow-Casting Scheme,” Appl. Opt. 27, 5181–5184 (1988).
    [CrossRef] [PubMed]
  14. S. L. Hurst, “Multi-Valued Threshold Logic: Its Status and Its Realization,” Opt. Eng. 25, 44–55 (1986).
  15. A. A. S. Awwal, M. A. Karim, “Design of Polarization-Encoded Optical Shadow-Casting Logic Units Using Truth Table Partitioning,” Can. J. Phys. 66, 841–843 (1988).
    [CrossRef]
  16. E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, 1987).
  17. M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, 1990).

1990

1989

1988

1987

M. A. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Logic Units: Design,” Appl. Opt. 26, 2220–2725 (1987).
[CrossRef]

A. A. S. Awwal, M. A. Karim, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Scheme: Design of Multioutput Trinary Combinational Logic Units,” Appl. Opt. 26, 4814–4818 (1987).
[CrossRef] [PubMed]

1986

Y. Li, G. Eichmann, R. R. Alfano, “Optical Computing Using Hybrid Encoded Shadow Casting,” Appl. Opt. 25, 2636–2638 (1986).
[CrossRef] [PubMed]

S. L. Hurst, “Multi-Valued Threshold Logic: Its Status and Its Realization,” Opt. Eng. 25, 44–55 (1986).

1984

Y. Ichioka, J. Tanida, “Optical Parallel Logic Gates Using a Shadow-Casting System for Optical Digital Computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

1983

Alfano, R. R.

Awwal, A. A. S.

Cherri, A. K.

A. A. S. Awwal, M. A. Karim, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Scheme: Design of Multioutput Trinary Combinational Logic Units,” Appl. Opt. 26, 4814–4818 (1987).
[CrossRef] [PubMed]

M. A. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Logic Units: Design,” Appl. Opt. 26, 2220–2725 (1987).
[CrossRef]

Eichmann, G.

Gaylord, T. K.

Guest, C. C.

Hurst, S. L.

S. L. Hurst, “Multi-Valued Threshold Logic: Its Status and Its Realization,” Opt. Eng. 25, 44–55 (1986).

Ichi?ka, Y.

Ichioka, Y.

Y. Ichioka, J. Tanida, “Optical Parallel Logic Gates Using a Shadow-Casting System for Optical Digital Computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

Johnson, E. L.

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, 1987).

Karim, M. A.

A. A. S. Awwal, M. A. Karim, “Multiprocessor Design Using Polarization-Encoded Optical Shadow-Casting,” Appl. Opt. 29, 2107–2112 (1990).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Direct Implementation of a Carry-Free Adder,” Appl. Opt. 28, 785–790 (1989).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Associative Polarization-Encoded Optical Shadow Casting: Gray-Level Image Encoding for Serial and Parallel Operations,” Appl. Opt. 28, 284–290 (1989).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting Programmable Logic Array: Simultaneous Generation of Multiple Outputs,” Appl. Opt. 27, 932–936 (1988).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Design of Polarization-Encoded Optical Shadow-Casting Logic Units Using Truth Table Partitioning,” Can. J. Phys. 66, 841–843 (1988).
[CrossRef]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Design of a J-K Flip-Flop,” Appl. Opt. 27, 3719–722 (1988).
[CrossRef] [PubMed]

M. A. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Logic Units: Design,” Appl. Opt. 26, 2220–2725 (1987).
[CrossRef]

A. A. S. Awwal, M. A. Karim, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Scheme: Design of Multioutput Trinary Combinational Logic Units,” Appl. Opt. 26, 4814–4818 (1987).
[CrossRef] [PubMed]

M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, 1990).

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, 1987).

Li, Y.

Mirsalehi, M. M.

Rizvi, R. A.

Tanida, J.

Y. Ichioka, J. Tanida, “Optical Parallel Logic Gates Using a Shadow-Casting System for Optical Digital Computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

J. Tanida, Y. Ichi⊙ka, “Optical Array Logic Processor Using Shadowgrams,” J. Opt. Soc. Am. 73, 800–809 (1983).
[CrossRef]

Zaheer, K.

Zubairy, M. S.

Appl. Opt.

M. M. Mirsalehi, C. C. Guest, T. K. Gaylord, “Residue Number System Holographic Truth-Table Look-Up Processing: Detector Threshold Setting and Probability of Error due to Amplitude and Phase Variations,” Appl. Opt. 22, 3583–3592 (1983).
[CrossRef] [PubMed]

Y. Li, G. Eichmann, R. R. Alfano, “Optical Computing Using Hybrid Encoded Shadow Casting,” Appl. Opt. 25, 2636–2638 (1986).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Scheme: Design of Multioutput Trinary Combinational Logic Units,” Appl. Opt. 26, 4814–4818 (1987).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting Programmable Logic Array: Simultaneous Generation of Multiple Outputs,” Appl. Opt. 27, 932–936 (1988).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Design of a J-K Flip-Flop,” Appl. Opt. 27, 3719–722 (1988).
[CrossRef] [PubMed]

R. A. Rizvi, K. Zaheer, M. S. Zubairy, “Separate and Simultaneous Generation of Multioutputs in a Polarization-Encoded Optical Shadow-Casting Scheme: Design of Half- and Full Adders and Subtractors,” Appl. Opt. 27, 5176–5180(1988).
[CrossRef] [PubMed]

R. A. Rizvi, K. Zaheer, M. S. Zubairy, “Design of Optical Decoders Using a Polarization-Encoded Optical Shadow-Casting Scheme,” Appl. Opt. 27, 5181–5184 (1988).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Associative Polarization-Encoded Optical Shadow Casting: Gray-Level Image Encoding for Serial and Parallel Operations,” Appl. Opt. 28, 284–290 (1989).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Polarization-Encoded Optical Shadow-Casting: Direct Implementation of a Carry-Free Adder,” Appl. Opt. 28, 785–790 (1989).
[CrossRef] [PubMed]

A. A. S. Awwal, M. A. Karim, “Multiprocessor Design Using Polarization-Encoded Optical Shadow-Casting,” Appl. Opt. 29, 2107–2112 (1990).
[CrossRef] [PubMed]

M. A. Karim, A. A. S. Awwal, A. K. Cherri, “Polarization-Encoded Optical Shadow-Casting Logic Units: Design,” Appl. Opt. 26, 2220–2725 (1987).
[CrossRef]

Can. J. Phys.

A. A. S. Awwal, M. A. Karim, “Design of Polarization-Encoded Optical Shadow-Casting Logic Units Using Truth Table Partitioning,” Can. J. Phys. 66, 841–843 (1988).
[CrossRef]

J. Opt. Soc. Am.

Opt. Eng.

S. L. Hurst, “Multi-Valued Threshold Logic: Its Status and Its Realization,” Opt. Eng. 25, 44–55 (1986).

Proc. IEEE

Y. Ichioka, J. Tanida, “Optical Parallel Logic Gates Using a Shadow-Casting System for Optical Digital Computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

Other

E. L. Johnson, M. A. Karim, Digital Design: A Pragmatic Approach (PWS-Kent, Boston, 1987).

M. A. Karim, Electro-Optical Devices and Systems (PWS-Kent, Boston, 1990).

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Figures (8)

Fig. 1
Fig. 1

POSC system: (a) with regular inputs only; (b) with inputs and an analyzer.

Fig. 2
Fig. 2

Pixel subcell assignments for the trinary multiplier using serial algorithm I.

Fig. 3
Fig. 3

Trinary multiplier encodings using serial algorithm I for. (a) input and (b) source.

Fig. 4
Fig. 4

Pixel subcell assignments for the trinary multiplier using serial algorithm II.

Fig. 5
Fig. 5

Trinary multiplier encodings using serial algorithm II for. (a) input (b) analyzer, and (c) source.

Fig. 6
Fig. 6

Composite address map for the trinary multiplier using a parallel algorithm.

Fig. 7
Fig. 7

Trinary multiplier encodings using parallel multiplier for. (a) input and (b) source.

Fig. 8
Fig. 8

Output mask for the trinary multiplier having parallel outputs.

Tables (2)

Tables Icon

Table I Analyzer Control Mechanism

Tables Icon

Table II Truth Table for a 2-Digit by 2-Digit Trinary Multiplier

Equations (9)

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

[ 4 , 11 5 , 12 6 , 13 7 , 14 8 , 15 9 , 16 10 , 17 23 , 18 24 , 19 25 , 20 26 , 29 27 , 30 28 , 31 37 , 33 38 , 34 47 , 39 48 , 40 50 , 41 51 , 42 54 , 47 55 , 52 59 , 53 60 , 57 61 , 58 - - - - - - - - - - - - - - - - - - - - - ]
[ - - 6 7 8 - 17 18 19 20 - - - - - - - - - 54 55 , 52 53 - 61 1 , 2 15 , 3 16 , 9 21 , 10 22 , 27 29 , 28 30 , 43 31 , 44 45 , 47 46 , 48 49 , 62 64 , 63 - - - - - - - - - ]
[ - - - - - - - 23 24 25 26 27 28 - - 39 40 41 42 54 , 49 55 - - - - - - 21 22 29 30 , 43 31 , 44 - - - 64 32 , 35 33 , 36 34 , 37 56 , 38 57 , 59 58 , 60 61 - - ]
[ - - - - - - - - - - - - - - - 47 48 50 51 54 55 59 60 61 - - - - - - - - 45 46 62 63 - - - 56 57 58 - 52 , 64 52 ]
A 1 ^ A 0 ^ B ¯ 1 ^ B 0 = V ,
A ¯ 1 ^ A 0 ^ B ¯ ¯ 1 ^ B 0 = H ,
C 1 :             V { 4 , 5 , 6 , 7 , 8 , 9 , 10 , 23 , 24 , 25 , 26 , 27 , 28 , 37 , 38 , 47 , 48 , 50 , 51 , 54 , 55 , 59 , 60 , 61 } , H { 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 29 , 30 , 31 , 33 , 34 , 39 , 40 , 41 , 42 , 49 , 52 , 53 , 57 , 58 } .
C 0 : V { 1 , 21 , 22 , 45 , 46 , 64 } , H { 2 , 3 , 43 , 44 , 62 , 63 } ; C 2 : V { 32 , 56 } ,             H { 35 , 36 } .
C 1 = N { 4 , 5 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 29 , 30 , 31 , 33 , 34 , 37 , 38 , 57 , 58 } , S { 6 , 7 , 8 , 17 , 18 , 19 , 20 , 23 , 24 , 25 , 26 , 27 , 28 , 39 , 40 , 41 , 42 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 59 , 60 , 61 } ; C 0 = N { 1 , 2 , 3 , 9 , 10 , 15 , 16 , 27 , 28 , 47 , 48 , 49 } , S { 6 , 7 , 8 , 17 , 18 , 19 , 20 , 21 , 22 , 29 , 30 , 31 , 43 , 44 , 45 , 46 , 52 , 53 , 54 , 55 , 61 , 62 , 63 , 64 ) ; C 2 = N { 32 , 33 , 34 , 35 , 36 , 37 , 38 , 59 , 60 , 61 } , S { 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 39 , 40 , 41 , 42 , 43 , 44 , 49 , 54 , 55 , 56 , 57 , 58 , 64 } ; C 3 = N { 52 , 53 } , S { 45 , 46 , 47 , 48 , 50 , 51 , 54 , 55 , 56 , 57 , 58 , 59 , 60 , 61 , 62 , 63 , 64 } .

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