Abstract

We present a framework for the information theoretic analysis of slow-light devices. We employ a model in which the device input is a binary-valued data sequence and the device output is considered within a window of finite duration. We use the mutual information between these two quantities to measure information content. This approach enables the information theoretic definitions of delay and throughput. We use our new framework to analyze a delay device based on stimulated Brillouin scattering (SBS) and find good agreement with previous SBS delay bounds.

© 2008 Optical Society of America

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References

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  1. E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photonics News 18, 41 (2007).
    [CrossRef]
  2. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
    [CrossRef]
  3. J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
    [CrossRef]
  4. J. Mok, C. M. Sterke, and B. J. Eggleton, “Delay-tunable gap-soliton-based slow-light system,” Opt. Express 14, 11987-11996 (2006).
    [CrossRef] [PubMed]
  5. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022-12027 (2006).
    [CrossRef] [PubMed]
  6. R. Pant, M. D. Stenner, and M. A. Neifeld, “Limitations of self-phase modulation based tunable delay system for all-optical buffer design,” Appl. Opt. (to be published).
  7. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44-50 (2006).
  8. M. González Herráez, K. Y. Song, and L. Thévenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395-1400 (2006).
    [CrossRef] [PubMed]
  9. D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express 13, 6234-6249 (2005).
    [CrossRef] [PubMed]
  10. J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. 31, 948-950 (2006).
    [CrossRef] [PubMed]
  11. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046-4066 (2005).
    [CrossRef]
  12. R. S. Tucker, “The role of optics and electronics in high-capacity routers,” J. Lightwave Technol. 24, 4655-4673 (2006).
    [CrossRef]
  13. D. A. B. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24, 1-18 (2007).
    [CrossRef]
  14. D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
    [CrossRef]
  15. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).
  16. C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 623-656 (1948).
  17. C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 1999), pp. 99-107.
  18. Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, “Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378-2384 (2005).
    [CrossRef]
  19. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
    [CrossRef]
  20. M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
    [CrossRef]
  21. R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
    [CrossRef] [PubMed]

2008 (1)

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

2007 (5)

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photonics News 18, 41 (2007).
[CrossRef]

D. A. B. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24, 1-18 (2007).
[CrossRef]

D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

2006 (6)

2005 (3)

2004 (1)

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

2001 (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

1948 (2)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 623-656 (1948).

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44-50 (2006).

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, “Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Casella, G.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 1999), pp. 99-107.

Chang-Hasnain, C. J.

Dahan, D.

Eggleton, B. J.

J. Mok, C. M. Sterke, and B. J. Eggleton, “Delay-tunable gap-soliton-based slow-light system,” Opt. Express 14, 11987-11996 (2006).
[CrossRef] [PubMed]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Eisenstein, G.

Gaeta, A. L.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44-50 (2006).

Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14, 12022-12027 (2006).
[CrossRef] [PubMed]

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, “Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Gauthier, D. J.

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44-50 (2006).

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, “Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

González Herráez, M.

Heebner, J. E.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Jackson, D. J.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Khurgin, J. B.

Ku, P. C.

Lee, M.

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Lowell, J. R.

E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photonics News 18, 41 (2007).
[CrossRef]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Miller, D. A. B.

D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
[CrossRef]

D. A. B. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24, 1-18 (2007).
[CrossRef]

Mok, J.

Neifeld, M. A.

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

R. Pant, M. D. Stenner, and M. A. Neifeld, “Limitations of self-phase modulation based tunable delay system for all-optical buffer design,” Appl. Opt. (to be published).

Okawachi, Y.

Pant, R.

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

R. Pant, M. D. Stenner, and M. A. Neifeld, “Limitations of self-phase modulation based tunable delay system for all-optical buffer design,” Appl. Opt. (to be published).

Parra, E.

E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photonics News 18, 41 (2007).
[CrossRef]

Robert, C. P.

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 1999), pp. 99-107.

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 623-656 (1948).

Sharping, J. E.

Shi, Z.

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Song, K. Y.

Stenner, M. D.

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier, “Maximizing the opening of eye diagrams for slow-light systems,” Appl. Opt. 46, 6513-6519 (2007).
[CrossRef] [PubMed]

R. Pant, M. D. Stenner, and M. A. Neifeld, “Limitations of self-phase modulation based tunable delay system for all-optical buffer design,” Appl. Opt. (to be published).

Sterke, C. M.

Thévenaz, L.

Tucker, R. S.

Willner, A. E.

Wong, V.

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

Xu, C.

Zhu, Z.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

Z. Zhu, D. J. Gauthier, Y. Okawachi, J. E. Sharping, A. L. Gaeta, R. W. Boyd, and A. E. Willner, “Numerical study of all-optical slow-light devices via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378-2384 (2005).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379-423 (1948).

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 623-656 (1948).

IEEE J. Quantum Electron. (2)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characterisitics of fiber ring resonators,” IEEE J. Quantum Electron. 40, 726-730 (2004).
[CrossRef]

J. Lightwave Technol. (2)

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

Opt. Commun. (1)

M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, “SBS gain-based slow light system with a Fabry-Perot resonator,” Opt. Commun. 281, 2975-2984 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Opt. Photonics News (1)

E. Parra and J. R. Lowell, “Toward applications of slow light technology,” Opt. Photonics News 18, 41 (2007).
[CrossRef]

Photonics Spectra (1)

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Slow light: from basics to future prospects,” Photonics Spectra 40, 44-50 (2006).

Phys. Rev. Lett. (2)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2007).
[CrossRef]

D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
[CrossRef]

Other (2)

C. P. Robert and G. Casella, Monte Carlo Statistical Methods (Springer, 1999), pp. 99-107.

R. Pant, M. D. Stenner, and M. A. Neifeld, “Limitations of self-phase modulation based tunable delay system for all-optical buffer design,” Appl. Opt. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Block diagram representing the information theoretic model of a slow-light delay device. X is a binary-valued discrete input sequence; the “OOK GP” block uses X to modulate, via OOK, a sequence of truncated Gaussian pulses; H S L is the slow-light operator; AWGN represents additive white Gaussian noise; and Y is the continuous-valued output signal.

Fig. 2
Fig. 2

Application of the information theoretic analysis to an ideal delay device. (a) Example signals at the input (dashed curve) and output (solid curve) of the slow-light operator. The input window (dotted-dashed vertical lines) along with two candidate output windows (solid and dotted vertical lines) are shown. (b) Mutual information as a function of output window offset. Note that WO represents the window offset and OW represents the output window.

Fig. 3
Fig. 3

Example signals used in the information theoretic analysis of SBS delay. (a) 6 - bit modulated signal used as input to H S L , (b) output of H S L for g L = 5 , and (c) the signal shown in (b) normalized to have total energy equal to that of (a).

Fig. 4
Fig. 4

Mutual information versus window offset for some example SBS systems. (a) Mutual information for σ = 0.45 and three values of g L = 1 , 5 , 10 . (b) Mutual information for g L = 10 and three values of σ = 0.45 , 0.59 , 0.71 .

Fig. 5
Fig. 5

Summary of information theoretic results for SBS. (a) IT on the left axis (solid curves) and ID on the right axis (dashed curve). (b) Delay results computed using three different methods (ID, T g , and T e ).

Fig. 6
Fig. 6

(a) Maximum delay versus bandwidth for SBS slow-light devices under real-world operating constraints on maximum gain ( g L 10 ) and output signal fidelity: solid curve corresponds to constraint on IT > 5.4   bits and dashed curve corresponds to constraint on EO > 0.65 . (b) Optimal gain. (c) IT constraint (solid curve) and IT limit (dotted line). (d) EO constraint (dashed curve) and EO limit (dotted line).

Fig. 7
Fig. 7

Mutual information versus window offset for some example SBS systems. This data is obtained from unnormalized SBS output signals. (a) Mutual information for σ = 10 and five values of g L . (b) Summary of information theoretic results. Information throughput (IT) on left axis (solid curve) and information delay (ID) right axis (dashed curve).

Fig. 8
Fig. 8

Comparison of information theoretic results for 6- and 10 - bit input sequences for σ = 0.45 . (a) IT and (b) ID.

Equations (5)

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

I ( X ; Y ) = i = 1 M p ( x i ) log 2 p ( x i ) + i = 1 M p ( x i , Y ) log 2 p ( x i Y ) d Y ,
= n + i = 1 M p ( x i ) p ( Y x i ) log 2 p ( Y x i ) p ( x i ) j = 1 M p ( x j ) p ( Y x j ) d Y ,
p ( Y x i ) 1 ( 2 π σ 2 ) L n exp ( 1 2 σ 2 Y H S L x i 2 ) ,
k ( f ) = γ g L 4 π ( f f p + F B + i γ ) ,
T g g L 4 π γ ( 1 3 ( f f p + F B ) 2 γ 2 ) .

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