Abstract

Bit error rates are calculated in the presence of shot noise for threshold optical logic devices that employ direct detection and intensity modulation. The device parameters considered are fan-in, contrast ratio, and light output. With given values for these parameters, the paper derives expressions for the device thresholds that generate the optimal bit error rates. In addition, it examines the fundamental quantum limit on reliability. Finally, the paper finds a bound on the device reliability that is sufficient to guarantee correct system operation with high probability.

© 1992 Optical Society of America

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  1. W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
    [CrossRef]
  2. P. Wheatley, J. E. Midwinter, “Operating curves for optical bistable devices,” Proc. Inst. Electr. Eng. Part J 134, 345–350 (1987).
  3. M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
    [CrossRef]
  4. K. Wagner, R. T. Weverka, D. Psaltis, “Threshold device tolerance requirements in digital optical computers,” in Optical Bistability III, Vol. 8 of Springer Proceedings in Physics, H. M. Gibbs, ed. (Springer, New York, 1986), pp. 16–20.
    [CrossRef]
  5. P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
    [CrossRef] [PubMed]
  6. G. W. Gigioli, “Optimization and tolerancing of nonlinear Fabry–Perot étalons for optical computing systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988).
  7. B. S. Wherrett, J. F. Snowden, “Tolerance studies for digital optical computing circuitry, with application to nonlinear interference filters,” Int. J. Opt. Comput. 1, 41–70 (1990).
  8. J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
    [CrossRef]
  9. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 216–217.
  10. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 260–263.
  11. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 565–569.
  12. C. W. Stirk, D. Psaltis, “The reliability of optical logic,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 14–17.
  13. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 7–13.
  14. W. J. Bertram, “Yield and reliability,” in VLSI Technology, S. M. Sze, ed. (McGraw-Hill, New York, 1983), pp. 599–637.
  15. R. W. Keyes, “Physical Limits in Digital Electronics,” Proc. IEEE 63, 740–767 (1975).
    [CrossRef]
  16. Z. Q. Pan, M. Dagenais, “Switching power dependence on detuning and current in bistable diode-laser amplifiers,” Appl. Phys. Lett. 58, 687–689 (1991).
    [CrossRef]

1991 (1)

Z. Q. Pan, M. Dagenais, “Switching power dependence on detuning and current in bistable diode-laser amplifiers,” Appl. Phys. Lett. 58, 687–689 (1991).
[CrossRef]

1990 (2)

B. S. Wherrett, J. F. Snowden, “Tolerance studies for digital optical computing circuitry, with application to nonlinear interference filters,” Int. J. Opt. Comput. 1, 41–70 (1990).

J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

1989 (1)

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

1988 (1)

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

1987 (2)

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

P. Wheatley, J. E. Midwinter, “Operating curves for optical bistable devices,” Proc. Inst. Electr. Eng. Part J 134, 345–350 (1987).

1975 (1)

R. W. Keyes, “Physical Limits in Digital Electronics,” Proc. IEEE 63, 740–767 (1975).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 565–569.

Athale, R. A.

J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Bertram, W. J.

W. J. Bertram, “Yield and reliability,” in VLSI Technology, S. M. Sze, ed. (McGraw-Hill, New York, 1983), pp. 599–637.

Cathey, W. T.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

Dagenais, M.

Z. Q. Pan, M. Dagenais, “Switching power dependence on detuning and current in bistable diode-laser amplifiers,” Appl. Phys. Lett. 58, 687–689 (1991).
[CrossRef]

Downs, M. M.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Filipowicz, P.

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

Garrison, J. C.

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

Gigioli, G. W.

G. W. Gigioli, “Optimization and tolerancing of nonlinear Fabry–Perot étalons for optical computing systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988).

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 216–217.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 7–13.

Keyes, R. W.

R. W. Keyes, “Physical Limits in Digital Electronics,” Proc. IEEE 63, 740–767 (1975).
[CrossRef]

Lee, S. H.

J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Meystre, P.

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

Miceli, W. J.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

Midwinter, J. E.

P. Wheatley, J. E. Midwinter, “Operating curves for optical bistable devices,” Proc. Inst. Electr. Eng. Part J 134, 345–350 (1987).

Neff, J. A.

J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Pan, Z. Q.

Z. Q. Pan, M. Dagenais, “Switching power dependence on detuning and current in bistable diode-laser amplifiers,” Appl. Phys. Lett. 58, 687–689 (1991).
[CrossRef]

Prise, M. E.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Psaltis, D.

K. Wagner, R. T. Weverka, D. Psaltis, “Threshold device tolerance requirements in digital optical computers,” in Optical Bistability III, Vol. 8 of Springer Proceedings in Physics, H. M. Gibbs, ed. (Springer, New York, 1986), pp. 16–20.
[CrossRef]

C. W. Stirk, D. Psaltis, “The reliability of optical logic,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 14–17.

Snowden, J. F.

B. S. Wherrett, J. F. Snowden, “Tolerance studies for digital optical computing circuitry, with application to nonlinear interference filters,” Int. J. Opt. Comput. 1, 41–70 (1990).

Stirk, C. W.

C. W. Stirk, D. Psaltis, “The reliability of optical logic,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 14–17.

Streibl, N.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Wagner, K.

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

K. Wagner, R. T. Weverka, D. Psaltis, “Threshold device tolerance requirements in digital optical computers,” in Optical Bistability III, Vol. 8 of Springer Proceedings in Physics, H. M. Gibbs, ed. (Springer, New York, 1986), pp. 16–20.
[CrossRef]

Weverka, R. T.

K. Wagner, R. T. Weverka, D. Psaltis, “Threshold device tolerance requirements in digital optical computers,” in Optical Bistability III, Vol. 8 of Springer Proceedings in Physics, H. M. Gibbs, ed. (Springer, New York, 1986), pp. 16–20.
[CrossRef]

Wheatley, P.

P. Wheatley, J. E. Midwinter, “Operating curves for optical bistable devices,” Proc. Inst. Electr. Eng. Part J 134, 345–350 (1987).

Wherrett, B. S.

B. S. Wherrett, J. F. Snowden, “Tolerance studies for digital optical computing circuitry, with application to nonlinear interference filters,” Int. J. Opt. Comput. 1, 41–70 (1990).

Wright, E. M.

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

Z. Q. Pan, M. Dagenais, “Switching power dependence on detuning and current in bistable diode-laser amplifiers,” Appl. Phys. Lett. 58, 687–689 (1991).
[CrossRef]

Int. J. Opt. Comput. (1)

B. S. Wherrett, J. F. Snowden, “Tolerance studies for digital optical computing circuitry, with application to nonlinear interference filters,” Int. J. Opt. Comput. 1, 41–70 (1990).

Opt. Quantum Electron. (1)

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Phys. Rev. A (1)

P. Filipowicz, J. C. Garrison, P. Meystre, E. M. Wright, “Noise-induced switching of photonic logic elements,” Phys. Rev. A 35, 1172–1180 (1987).
[CrossRef] [PubMed]

Proc. IEEE (3)

R. W. Keyes, “Physical Limits in Digital Electronics,” Proc. IEEE 63, 740–767 (1975).
[CrossRef]

W. T. Cathey, K. Wagner, W. J. Miceli, “Digital computing with optics,” Proc. IEEE 77, 1558–1572 (1989).
[CrossRef]

J. A. Neff, R. A. Athale, S. H. Lee, “2-Dimensional spatial light modulators,” Proc. IEEE 78, 826–855 (1990).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

P. Wheatley, J. E. Midwinter, “Operating curves for optical bistable devices,” Proc. Inst. Electr. Eng. Part J 134, 345–350 (1987).

Other (8)

K. Wagner, R. T. Weverka, D. Psaltis, “Threshold device tolerance requirements in digital optical computers,” in Optical Bistability III, Vol. 8 of Springer Proceedings in Physics, H. M. Gibbs, ed. (Springer, New York, 1986), pp. 16–20.
[CrossRef]

G. W. Gigioli, “Optimization and tolerancing of nonlinear Fabry–Perot étalons for optical computing systems,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1988).

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 216–217.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), pp. 260–263.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), pp. 565–569.

C. W. Stirk, D. Psaltis, “The reliability of optical logic,” in Optical Computing, Vol. 6 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), pp. 14–17.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 7–13.

W. J. Bertram, “Yield and reliability,” in VLSI Technology, S. M. Sze, ed. (McGraw-Hill, New York, 1983), pp. 599–637.

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Figures (11)

Fig. 1
Fig. 1

Incoherent optical modulator logic with intensity modulation and direct detection.

Fig. 2
Fig. 2

Optical logic device transfer function. mH, mean photon number of logic 1; mL, mean photon for logic 0.

Fig. 3
Fig. 3

Calculation of the BER of a detector in the presence of shot noise: (a) logic 0 with a mean photon number equal to 5, (b) logic 1 with mean photon number equal to 10. The sum of the height of the thick lines represents the BER at a threshold between 6 and 7.

Fig. 4
Fig. 4

Log10[BER] versus threshold for an or gate with a mean photon number for a logic 1 mH of 1000 and a mean photon number for a logic 0 mL of 100 for fan-ins N of 1, 2, 4, 8, and 16.

Fig. 5
Fig. 5

Log10[BER] versus threshold for an and gate with a mean photon number for a logic 1 mH of 1000 and a mean photon number for a logic 0 mL of 100 for fan-ins N of 1, 2, 4, 8, and 16.

Fig. 6
Fig. 6

Optimum log10[BER] versus the mean photon number for a logic 1 (mH) for an or gate with contrast = 10 and fan-ins N of 2, 4, 8, and 16.

Fig. 7
Fig. 7

Log10[BER] versus the mean photon number for a logic high (mH) for an and gate with contrast = 10 for fan-ins N of 2, 4, 8, and 16.

Fig. 8
Fig. 8

Optimum log10[BER] versus fan-in N of an optical or with the mean photon number for a logic high mH of 1000 and contrast = 2, 4, 8, 16, and 32.

Fig. 9
Fig. 9

Optimum log10[BER] versus fan-in N of an optical and with the mean photon number for a logic high mH of 1000 and contrast = 2, 4, 8, 16, and 32.

Fig. 10
Fig. 10

Exact log10[FBER] of an optical or gate versus the mean photon number per channel for a logic high mH for fan-ins N of 2, 4, 16, and 32.

Fig. 11
Fig. 11

Log10[FBER] of an optical and versus the mean photon number for a logic high mH at several fan-ins.

Equations (28)

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BER ( T ) = x input set P ( y z , x , T ) P ( x ) ,
x input set P ( x ) = 1.
Pr ( detects k photons when mean is m ) = P m ( k ) = m k k ! exp ( - m ) .
P ( 0 1 , 1 , T ) = k = 0 T - 1 m H k k ! exp ( - m H ) = Γ ( T , m H ) Γ ( T ) = 1 - P ( T , m H ) .
P ( 1 0 , 0 , T ) = k T m L k k ! exp ( - m L ) = P ( T , m H )
BER OR ( T ) = q N P ( T , μ L ) + i = 0 N - 1 ( N i ) p N - i q i [ 1 - P ( T , μ H ) ] .
BER AND ( T ) = p N [ 1 - P ( T , μ H ) ] + i = 0 N - 1 ( N i ) p i q N - i P ( T , μ L ) ,
P ( ω i ) = p i q N - i ( N i ) .
P ( X = k ω i ) = m i k k ! exp ( - m i ) ,
λ ( α j ω i ) = { 1 , if j = 1 , i = 0 or j = 0 , i > 0 0 , otherwise ,
λ ( α j ω i ) = { 1 , if j = 1 , i < N or j = 0 , i = N 0 , otherwise .
R ( α j k ) = i = 0 N λ ( α j ω i ) P ( ω i k ) .
R = k = 0 j = 0 1 R ( α j k ) P ( k ) .
P ( ω i k ) = P ( k ω i ) P ( ω i ) P ( k ) .
T ln m 0 - m 0 = ln [ i = 1 N ( p q ) i ( N i ) m j T exp ( - m i ) ] .
T ln m N - m N = ln [ i = 0 N - 1 ( N i ) ( p q ) i - N m i T exp ( - m i ) ] .
T = m H - m L ln ( m H m L ) .
BER guess = P ( 0 ) P ( g = 1 ) + P ( 1 ) P ( g = 0 ) ,
BER guess = 1 2 N 2 N - 1 2 N + 2 N - 1 2 N 1 2 N = 2 N - 1 2 2 N - 1 .
BER OR ( T ) = i = 1 N ( N i ) p i q N - i Γ ( T , i m H ) ( T - 1 ) ! .
FBER OR = i = 1 N ( N i ) p i q N - i exp ( - i m H ) = [ p exp ( - m H ) + q ] N - q N .
FBER OR = 2 - N [ N exp ( - m H ) + N ( N - 1 ) 2 × exp ( - 2 m H ) + + exp ( - N m H ) ] .
FBER OR 2 - N N exp ( - m H ) .
FBER detect = ½ exp ( - m H ) .
BER AND ( T ) = p N [ 1 - P ( T , N m H ) ] + i = 0 N - 1 ( N i ) p i q N - i [ 1 - P ( T , i m H ) ] ,
T ln N m H - N m H = ln [ j = 0 N - 1 ( N j ) ( p q ) j - N ( j m H ) T exp ( - j m H ) ] .
Pr ( zero system failures ) exp [ - BER ( N M Y / T ) ] ,
δ BER ( N M Y / T ) .

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