Abstract

Sample entropy is introduced as a versatile measure for analyzing chaos dynamics and chaos communications. This measure can be directly applied to distinguish different dynamic behaviors for their superior advantages, such as fast computation (available for short data), extreme simplicity, robustness, and even more efficiency in the presence of a certain amount of noise. For the unidirectional coupling master–slave configuration of semiconductor lasers, it is shown that sample entropy analysis can be an effective tool to reveal the chaos synchronization quality. More particularly, when employed to discriminate the encryption performance of a chaos modulation scheme, the sample entropy algorithm seems to be a powerful and complementary tool for detecting and quantifying the presence of a message within a chaotic carrier.

© 2011 Optical Society of America

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    [CrossRef]
  5. N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. S. Tang and J. M. Liu, “Effects of message encoding and decoding on synchronized chaotic optical communications,” IEEE J. Quantum Electron. 39, 1468–1474 (2003).
    [CrossRef]
  25. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
    [CrossRef]
  26. S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
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2010 (3)

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications,” IEEE J. Quantum Electron. 46, 1430–1435(2010).
[CrossRef]

S.-Y. Xiang, W. Pan, L.-S. Yan, B. Luo, N. Jiang, K.-H. Wen, X.-H. Zou, and L. Yang, “Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization,” J. Opt. Soc. Am. B 27, 476–483 (2010).
[CrossRef]

2009 (2)

S. Ramdani, F. Bouchara, and J. Lagarde, “Influence of noise on the sample entropy algorithm,” Chaos 19, 013123 (2009).
[CrossRef] [PubMed]

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

2008 (2)

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, “Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: an information theory approach,” Phys. Lett. A 372, 1018–1023(2008).
[CrossRef]

2007 (2)

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

F. Y. Lin and M. C. Tsai, “Chaotic communication in radio-over-fiber transmission based on optoelectronic feedback semiconductor lasers,” Opt. Express 15, 302–311 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (2)

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

2003 (1)

S. Tang and J. M. Liu, “Effects of message encoding and decoding on synchronized chaotic optical communications,” IEEE J. Quantum Electron. 39, 1468–1474 (2003).
[CrossRef]

2002 (6)

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

C. R. Mirasso, J. Mulet, and C. Masoller, “Chaos shift-keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14, 456–458 (2002).
[CrossRef]

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102(2002).
[CrossRef] [PubMed]

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38, 1141–1154 (2002).
[CrossRef]

J.-M. Liu, H.-F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit-rates,” IEEE J. Quantum Electron. 38, 1184–1196 (2002).
[CrossRef]

2000 (1)

J. S. Richman and J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Am. J. Physiol. Heart Circ. Physiol. 278, 2039–2049 (2000).

1998 (1)

G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

1991 (1)

S. M. Pincus, “Approximate entropy as a measure of system complexity,” Proc. Natl. Acad. Sci. USA 88, 2297–2301(1991).
[CrossRef] [PubMed]

1983 (1)

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

1982 (1)

J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Annovazzi-Lodi, V.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Argyris, A.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102(2002).
[CrossRef] [PubMed]

Bouchara, F.

S. Ramdani, F. Bouchara, and J. Lagarde, “Influence of noise on the sample entropy algorithm,” Chaos 19, 013123 (2009).
[CrossRef] [PubMed]

Chen, H. F.

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

Chen, H.-F.

J.-M. Liu, H.-F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit-rates,” IEEE J. Quantum Electron. 38, 1184–1196 (2002).
[CrossRef]

Colet, P.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Duden, J.

R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Eswaran, H.

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

Farmer, J. D.

J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
[CrossRef]

Figliola, A.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Fischer, I.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Foglyano, R.

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

Garavaglia, M.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Garcia-Ojalvo, J.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Govindan, R. B.

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

Grassberger, P.

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

Griffin, M. P.

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

Hwang, S. K.

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

Jacquot, M.

R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications,” IEEE J. Quantum Electron. 46, 1430–1435(2010).
[CrossRef]

Jiang, N.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

S.-Y. Xiang, W. Pan, L.-S. Yan, B. Luo, N. Jiang, K.-H. Wen, X.-H. Zou, and L. Yang, “Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization,” J. Opt. Soc. Am. B 27, 476–483 (2010).
[CrossRef]

Kaffashi, F.

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

Kane, D. M.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Lagarde, J.

S. Ramdani, F. Bouchara, and J. Lagarde, “Influence of noise on the sample entropy algorithm,” Chaos 19, 013123 (2009).
[CrossRef] [PubMed]

Lake, D. E.

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Larger, L.

R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications,” IEEE J. Quantum Electron. 46, 1430–1435(2010).
[CrossRef]

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Lavrov, R.

R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications,” IEEE J. Quantum Electron. 46, 1430–1435(2010).
[CrossRef]

Lee, M. W.

Li, N. Q.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

Li, X. F.

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42, 953–960 (2006).
[CrossRef]

Lin, F. Y.

Liu, J. M.

S. Tang and J. M. Liu, “Effects of message encoding and decoding on synchronized chaotic optical communications,” IEEE J. Quantum Electron. 39, 1468–1474 (2003).
[CrossRef]

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

Liu, J.-M.

J.-M. Liu, H.-F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit-rates,” IEEE J. Quantum Electron. 38, 1184–1196 (2002).
[CrossRef]

Loparo, K. A.

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

Lowery, C. L.

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

Luo, B.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

S.-Y. Xiang, W. Pan, L.-S. Yan, B. Luo, N. Jiang, K.-H. Wen, X.-H. Zou, and L. Yang, “Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization,” J. Opt. Soc. Am. B 27, 476–483 (2010).
[CrossRef]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42, 953–960 (2006).
[CrossRef]

Ma, D.

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42, 953–960 (2006).
[CrossRef]

Masoller, C.

M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete-chaos synchronization with zero time lag in laser diodes,” J. Opt. Soc. Am. B 23, 846–851 (2006).
[CrossRef]

C. R. Mirasso, J. Mulet, and C. Masoller, “Chaos shift-keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14, 456–458 (2002).
[CrossRef]

Mirasso, C. R.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, “Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: an information theory approach,” Phys. Lett. A 372, 1018–1023(2008).
[CrossRef]

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

C. R. Mirasso, J. Mulet, and C. Masoller, “Chaos shift-keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14, 456–458 (2002).
[CrossRef]

Moorman, J. R.

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

J. S. Richman and J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Am. J. Physiol. Heart Circ. Physiol. 278, 2039–2049 (2000).

Mulet, J.

C. R. Mirasso, J. Mulet, and C. Masoller, “Chaos shift-keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14, 456–458 (2002).
[CrossRef]

Ohtsubo, J.

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38, 1141–1154 (2002).
[CrossRef]

Pan, W.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

S.-Y. Xiang, W. Pan, L.-S. Yan, B. Luo, N. Jiang, K.-H. Wen, X.-H. Zou, and L. Yang, “Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization,” J. Opt. Soc. Am. B 27, 476–483 (2010).
[CrossRef]

X. F. Li, W. Pan, B. Luo, and D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42, 953–960 (2006).
[CrossRef]

Paul, J.

Perez, D. G.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Pesquera, L.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Pincus, S. M.

S. M. Pincus, “Approximate entropy as a measure of system complexity,” Proc. Natl. Acad. Sci. USA 88, 2297–2301(1991).
[CrossRef] [PubMed]

Pompe, B.

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102(2002).
[CrossRef] [PubMed]

Preissl, H.

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

Procaccia, I.

P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Phys. Rev. Lett. 50, 346–349 (1983).
[CrossRef]

Ramdani, S.

S. Ramdani, F. Bouchara, and J. Lagarde, “Influence of noise on the sample entropy algorithm,” Chaos 19, 013123 (2009).
[CrossRef] [PubMed]

Richman, J. S.

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

J. S. Richman and J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Am. J. Physiol. Heart Circ. Physiol. 278, 2039–2049 (2000).

Rosso, O. A.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

O. A. Rosso, R. Vicente, and C. R. Mirasso, “Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: an information theory approach,” Phys. Lett. A 372, 1018–1023(2008).
[CrossRef]

Roy, R.

G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Shore, K. A.

Soriano, M. C.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Syvrids, D.

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Tang, S.

S. Tang and J. M. Liu, “Effects of message encoding and decoding on synchronized chaotic optical communications,” IEEE J. Quantum Electron. 39, 1468–1474 (2003).
[CrossRef]

J.-M. Liu, H.-F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit-rates,” IEEE J. Quantum Electron. 38, 1184–1196 (2002).
[CrossRef]

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

Toomey, J. P.

Toral, R.

R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Tsai, M. C.

Van Wiggeren, G. D.

G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

Vicente, R.

O. A. Rosso, R. Vicente, and C. R. Mirasso, “Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: an information theory approach,” Phys. Lett. A 372, 1018–1023(2008).
[CrossRef]

R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Wen, K.-H.

Wilson, C. G.

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

Wilson, J. D.

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
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N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

Xiang, S.-Y.

Yan, L. S.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

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Yang, L.

S.-Y. Xiang, W. Pan, L.-S. Yan, B. Luo, N. Jiang, K.-H. Wen, X.-H. Zou, and L. Yang, “Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization,” J. Opt. Soc. Am. B 27, 476–483 (2010).
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N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

Zheng, D.

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

Zou, X.-H.

Zunino, L.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Am. J. Physiol. Heart Circ. Physiol. (1)

J. S. Richman and J. R. Moorman, “Physiological time-series analysis using approximate entropy and sample entropy,” Am. J. Physiol. Heart Circ. Physiol. 278, 2039–2049 (2000).

Am. J. Physiol. Regul. Integr. Comp. Physiol. (1)

D. E. Lake, J. S. Richman, M. P. Griffin, and J. R. Moorman, “Sample entropy analysis of neonatal heart rate variability,” Am. J. Physiol. Regul. Integr. Comp. Physiol. 283, 789–797 (2002).

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S. Ramdani, F. Bouchara, and J. Lagarde, “Influence of noise on the sample entropy algorithm,” Chaos 19, 013123 (2009).
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R. Vicente, J. Duden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38, 1141–1154 (2002).
[CrossRef]

J.-M. Liu, H.-F. Chen, and S. Tang, “Synchronized chaotic optical communications at high bit-rates,” IEEE J. Quantum Electron. 38, 1184–1196 (2002).
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[CrossRef]

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C. R. Mirasso, J. Mulet, and C. Masoller, “Chaos shift-keying encryption in chaotic external-cavity semiconductor lasers using a single-receiver scheme,” IEEE Photon. Technol. Lett. 14, 456–458 (2002).
[CrossRef]

IEEE Trans. Circuits Syst. I (1)

S. Tang, H. F. Chen, S. K. Hwang, and J. M. Liu, “Message encoding and decoding through chaos modulation in chaotic optical communications,” IEEE Trans. Circuits Syst. I 49, 163–169(2002).
[CrossRef]

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

Nature (1)

A. Argyris, D. Syvrids, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fiber-optic links,” Nature 437, 343–346 (2005).
[CrossRef]

Opt. Commun. (1)

L. Zunino, M. C. Soriano, A. Figliola, D. G. Perez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: a multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Lett. A (1)

O. A. Rosso, R. Vicente, and C. R. Mirasso, “Encryption test of pseudo-aleatory messages embedded on chaotic laser signals: an information theory approach,” Phys. Lett. A 372, 1018–1023(2008).
[CrossRef]

Phys. Rev. E (1)

N. Jiang, W. Pan, B. Luo, L. S. Yan, S. Y. Xiang, L. Yang, D. Zheng, and N. Q. Li, “Properties of leader/laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81, 066217 (2010).
[CrossRef]

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Physica A (1)

R. B. Govindan, J. D. Wilson, H. Eswaran, C. L. Lowery, and H. Preissl, “Revisiting sample entropy analysis,” Physica A 376, 158–164 (2007).
[CrossRef]

Physica D (2)

F. Kaffashi, R. Foglyano, C. G. Wilson, and K. A. Loparo, “The effect of time delay on approximate and sample entropy calculations,” Physica D 237, 3069–3074 (2008).
[CrossRef]

J. D. Farmer, “Chaotic attractors of an infinite-dimensional dynamical system,” Physica D 4, 366–393 (1982).
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[CrossRef] [PubMed]

Science (1)

G. D. Van Wiggeren and R. Roy, “Communication with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Evolution of SampEn values as a function of the dimension m for logistic map and a semiconductor laser system, where different lengths of data are included, namely N = 500 and N = 5000 , respectively.

Fig. 2
Fig. 2

(a) Bifurcation diagram, (b) Lyapunov exponent (LE), (c) normalized SampEn for the logistic map as a function of the parameter λ with step Δ λ = 10 4 . (c) Noise-free case (dark curve) and noisy case (gray curve) included.

Fig. 3
Fig. 3

Normalized SampEn for the logistic map as a function of the parameter λ with step Δ λ = 10 4 , and (bottom to top) different noise levels 0.2 (dark curve), 0.25 (gray curve) and 0.3 (blue curve).

Fig. 4
Fig. 4

Information related to chaos synchronization plots when the cross correlation coefficient is 1. (a) SampEn values versus the dimension m, (b) SampEn values versus the tolerance r, (c) cross correlation coefficient as a function of the time shift.

Fig. 5
Fig. 5

Same as Fig. 4 when the cross correlation coefficient is 0.7.

Fig. 6
Fig. 6

SampEn boxplots for sampling period Ω s = 1 ps and different message strengths. Lower and upper rows correspond to B s = 1 Gb / s and B s = 2.5 Gb / s in CM encryption scheme.

Fig. 7
Fig. 7

Same as Fig. 6 for Ω s = 10 ps .

Fig. 8
Fig. 8

Same as Fig. 6 for Ω s = 100 ps .

Equations (11)

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X i ( m ) = ( x i , x i + τ , , x i + ( m 1 ) τ ) , i S ,
ϕ i m ( N , τ , r ) = 1 N m τ j = 1 , | j i | > τ N m τ θ ( r X j ( m ) X i ( m ) ) ,
B m ( N , τ , r ) = 1 N m τ i = 1 N m τ ϕ i m ( N , τ , r ) .
φ i m ( N , τ , r ) = 1 N m τ j = 1 , | j i | > τ N m τ θ ( r X j ( m + 1 ) X i ( m + 1 ) ) ,
A m ( N , τ , r ) = 1 N m τ i = 1 N m τ φ i m ( N , τ , r ) .
SampEn ( N , m , τ , r ) = 1 τ log ( A m ( N , τ , r ) B m ( N , τ , r ) ) .
x i + 1 = λ x i ( 1 x i ) ,
d E 1 , 2 ( t ) d t = 1 2 ( 1 + i α ) [ G 1 , 2 ( t ) 1 τ p ] E 1 , 2 ( t ) + 2 β N 1 , 2 ( t ) χ 1 , 2 + k f E 1 ( t τ f ) exp ( i 2 π f 1 τ f ) + σ E 1 ( t τ D ) exp ( i 2 π f 1 τ D ) exp ( i 2 π Δ f 12 t ) ,
d N 1 , 2 ( t ) d t = I 1 , 2 q N 1 , 2 ( t ) τ s G 1 , 2 ( t ) | E 1 , 2 ( t ) | 2 ,
G 1 , 2 ( t ) = g [ N 1 , 2 ( t ) N 0 ] 1 + s | E 1 , 2 ( t ) | 2 ,
E CM ( t ) = E 1 ( t τ D ) [ 1 + A m ( t τ D ) ] ,

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