Abstract

Evaluating entropy rate of high-dimensional chaos and shot noise from analog raw signals remains elusive and important in information security. We experimentally present an accurate assessment of entropy rate for physical process randomness. The entropy generation of optical-feedback laser chaos and physical randomness limit from shot noise are quantified and unambiguously discriminated using the growth rate of average permutation entropy value in memory time. The permutation entropy difference of filtered laser chaos with varying embedding delay time is investigated experimentally and theoretically. High-resolution maps of the entropy difference are observed over the range of the injection-feedback parameter space. We also clarify an inverse relationship between the entropy rate and time delay signature of laser chaos over a wide range of parameters. Compared to the original chaos, the time delay signature is suppressed up to 95% with the minimum of 0.015 via frequency-band extractor, and the experiment agrees well with the theory. Our system provides a commendable entropy evaluation and source for physical random number generation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

S. Kreinberg, X. Porte, D. Schicke, B. Lingnau, C. Schneider, S. Höfling, I. Kanter, K. Lüdge, and S. Reitzenstein, “Mutual coupling and synchronization of optically coupled quantum-dot micropillar lasers at ultra-low light levels,” Nat. Commun. 10(1), 1539 (2019).
[Crossref]

Y. K. Chembo, D. Brunner, M. Jacquot, and L. Larger, “Optoelectronic oscillators with time-delayed feedback,” Rev. Mod. Phys. 91(3), 035006 (2019).
[Crossref]

2018 (1)

X. M. Guo, R. P. Liu, P. Li, C. Cheng, M. C. Wu, and Y. Q. Guo, “Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generator,” Entropy 20(11), 819 (2018).
[Crossref]

2017 (2)

J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017).
[Crossref]

Y. Hong and S. Ji, “Effect of digital acquisition on the complexity of chaos,” Opt. Lett. 42(13), 2507–2510 (2017).
[Crossref]

2016 (3)

2015 (7)

A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9258–9263 (2015).
[Crossref]

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

C. Quintero-Quiroz, S. Pigolotti, M. C. Torrent, and C. Masoller, “Numerical and experimental study of the effects of noise on the permutation entropy,” New J. Phys. 17(9), 093002 (2015).
[Crossref]

S.-S. Li and S.-C. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
[Crossref]

C.-H. Cheng, Y.-C. Chen, and F.-Y. Lin, “Chaos time delay signature suppression and bandwidth enhancement by electrical heterodyning,” Opt. Express 23(3), 2308–2319 (2015).
[Crossref]

2014 (1)

2013 (4)

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7(1), 60–65 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

L. Xia, D. Huang, J. Xu, and D. Liu, “Simultaneous and precise fault locating in WDM-PON by the generation of optical wideband chaos,” Opt. Lett. 38(19), 3762–3764 (2013).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

2012 (4)

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, and G.-Y. Feng, “Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system,” Opt. Express 20(2), 1741–1753 (2012).
[Crossref]

2011 (1)

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

2010 (1)

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

2009 (1)

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

2008 (3)

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Y. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008).
[Crossref]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

2004 (1)

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004).
[Crossref]

2002 (2)

G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep. 356(6), 367–474 (2002).
[Crossref]

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref]

1998 (1)

P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W. Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimental evidence for microscopic chaos,” Nature 394(6696), 865–868 (1998).
[Crossref]

1988 (1)

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, “Dimension Increase in Filtered Chaotic Signals,” Phys. Rev. Lett. 60(11), 979–982 (1988).
[Crossref]

1987 (1)

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Aida, H.

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Andersen, U. L.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Aoyama, K.

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

Arai, K.

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

Arai, K.-i.

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Aviad, Y.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

Badii, R.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, “Dimension Increase in Filtered Chaotic Signals,” Phys. Rev. Lett. 60(11), 979–982 (1988).
[Crossref]

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref]

Baumgartner, G. B.

J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017).
[Crossref]

Boffetta, G.

G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep. 356(6), 367–474 (2002).
[Crossref]

Breuer, S.

Briggs, M. E.

P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W. Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimental evidence for microscopic chaos,” Nature 394(6696), 865–868 (1998).
[Crossref]

Broggi, G.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, “Dimension Increase in Filtered Chaotic Signals,” Phys. Rev. Lett. 60(11), 979–982 (1988).
[Crossref]

Brunner, D.

Y. K. Chembo, D. Brunner, M. Jacquot, and L. Larger, “Optoelectronic oscillators with time-delayed feedback,” Rev. Mod. Phys. 91(3), 035006 (2019).
[Crossref]

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

Calabrese, R. V.

P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W. Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimental evidence for microscopic chaos,” Nature 394(6696), 865–868 (1998).
[Crossref]

Cencini, M.

G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep. 356(6), 367–474 (2002).
[Crossref]

Chan, S.-C.

X.-Z. Li, J.-P. Zhuang, S.-S. Li, J.-B. Gao, and S.-C. Chan, “Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction,” Phys. Rev. E 94(4), 042214 (2016).
[Crossref]

S.-S. Li and S.-C. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
[Crossref]

Chembo, Y. K.

Y. K. Chembo, D. Brunner, M. Jacquot, and L. Larger, “Optoelectronic oscillators with time-delayed feedback,” Rev. Mod. Phys. 91(3), 035006 (2019).
[Crossref]

Chen, Y.-C.

Cheng, C.

X. M. Guo, R. P. Liu, P. Li, C. Cheng, M. C. Wu, and Y. Q. Guo, “Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generator,” Entropy 20(11), 819 (2018).
[Crossref]

Cheng, C.-H.

Choi, D.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

Ciliberto, S.

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, “Dimension Increase in Filtered Chaotic Signals,” Phys. Rev. Lett. 60(11), 979–982 (1988).
[Crossref]

Citrin, D. S.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

Colet, P.

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Sciamanna, M.

Sengers, J. V.

P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W. Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimental evidence for microscopic chaos,” Nature 394(6696), 865–868 (1998).
[Crossref]

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Shore, K. A.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Y. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, “Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface-emitting semiconductor lasers with polarization-preserved injection,” Opt. Lett. 33(6), 587–589 (2008).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Soriano, M. C.

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Spencer, P. S.

Sunada, S.

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

Sych, D.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Tang, X.

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Terashima, Y.

J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017).
[Crossref]

Thienpont, H.

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7(1), 60–65 (2013).
[Crossref]

Toomey, J. P.

Torrent, M. C.

C. Quintero-Quiroz, S. Pigolotti, M. C. Torrent, and C. Masoller, “Numerical and experimental study of the effects of noise on the permutation entropy,” New J. Phys. 17(9), 093002 (2015).
[Crossref]

Tsuzuki, K.

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

Uchida, A.

J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Virte, M.

M. Virte, R. Pawlus, M. Sciamanna, K. Panajotov, and S. Breuer, “Energy exchange between modes in a multimode two-color quantum dot laser with optical feedback,” Opt. Lett. 41(14), 3205–3208 (2016).
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M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7(1), 60–65 (2013).
[Crossref]

Vulpiani, A.

G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep. 356(6), 367–474 (2002).
[Crossref]

Wang, A. B.

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wang, B. J.

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wang, Y. C.

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wittmann, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

Wolfersberger, D.

Wu, J.-G.

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, and G.-Y. Feng, “Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system,” Opt. Express 20(2), 1741–1753 (2012).
[Crossref]

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref]

Wu, M. C.

X. M. Guo, R. P. Liu, P. Li, C. Cheng, M. C. Wu, and Y. Q. Guo, “Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generator,” Entropy 20(11), 819 (2018).
[Crossref]

Wu, Z.-M.

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, and G.-Y. Feng, “Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system,” Opt. Express 20(2), 1741–1753 (2012).
[Crossref]

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Xia, G.-Q.

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, and G.-Y. Feng, “Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system,” Opt. Express 20(2), 1741–1753 (2012).
[Crossref]

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Xia, L.

Xiao, M.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref]

Xu, J.

Yoshimori, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Yoshimura, K.

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

Zhuang, J.-P.

X.-Z. Li, J.-P. Zhuang, S.-S. Li, J.-B. Gao, and S.-C. Chan, “Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction,” Phys. Rev. E 94(4), 042214 (2016).
[Crossref]

Zunino, L.

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

APL Photonics (1)

J. D. Hart, Y. Terashima, A. Uchida, G. B. Baumgartner, T. E. Murphy, and R. Roy, “Recommendations and illustrations for the evaluation of photonic random number generators,” APL Photonics 2(9), 090901 (2017).
[Crossref]

Chaos (1)

S. Sunada, T. Harayama, P. Davis, K. Tsuzuki, K. Arai, K. Yoshimura, and A. Uchida, “Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation,” Chaos 22(4), 047513 (2012).
[Crossref]

Entropy (1)

X. M. Guo, R. P. Liu, P. Li, C. Cheng, M. C. Wu, and Y. Q. Guo, “Enhancing Extractable Quantum Entropy in Vacuum-Based Quantum Random Number Generator,” Entropy 20(11), 819 (2018).
[Crossref]

IEEE J. Quantum Electron. (3)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

N. Q. Li, L. Zunino, A. Locquet, B. Kim, D. Choi, W. Pan, and D. S. Citrin, “Multiscale ordinal symbolic analysis of the Lang-Kobayashi model for external-cavity semiconductor lasers: a test of theory,” IEEE J. Quantum Electron. 51(8), 1–6 (2015).
[Crossref]

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S.-S. Li and S.-C. Chan, “Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback,” IEEE J. Sel. Top. Quantum Electron. 21(6), 541–552 (2015).
[Crossref]

IEEE Photonics Technol. Lett. (2)

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

J.-G. Wu, Z.-M. Wu, G.-Q. Xia, T. Deng, X.-D. Lin, X. Tang, and G.-Y. Feng, “Isochronous synchronization between chaotic semiconductor lasers over 40-km fiber links,” IEEE Photonics Technol. Lett. 23(24), 1854–1856 (2011).
[Crossref]

Nat. Commun. (2)

D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, “Parallel photonic information processing at gigabyte per second data rates using transient states,” Nat. Commun. 4(1), 1364 (2013).
[Crossref]

S. Kreinberg, X. Porte, D. Schicke, B. Lingnau, C. Schneider, S. Höfling, I. Kanter, K. Lüdge, and S. Reitzenstein, “Mutual coupling and synchronization of optically coupled quantum-dot micropillar lasers at ultra-low light levels,” Nat. Commun. 10(1), 1539 (2019).
[Crossref]

Nat. Photonics (4)

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4(10), 711–715 (2010).
[Crossref]

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[Crossref]

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7(1), 60–65 (2013).
[Crossref]

Nature (2)

P. Gaspard, M. E. Briggs, M. K. Francis, J. V. Sengers, R. W. Gammon, J. R. Dorfman, and R. V. Calabrese, “Experimental evidence for microscopic chaos,” Nature 394(6696), 865–868 (1998).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

New J. Phys. (1)

C. Quintero-Quiroz, S. Pigolotti, M. C. Torrent, and C. Masoller, “Numerical and experimental study of the effects of noise on the permutation entropy,” New J. Phys. 17(9), 093002 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (6)

Phys. Rep. (1)

G. Boffetta, M. Cencini, M. Falcioni, and A. Vulpiani, “Predictability: A way to characterize complexity,” Phys. Rep. 356(6), 367–474 (2002).
[Crossref]

Phys. Rev. E (2)

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E 85(1), 016211 (2012).
[Crossref]

X.-Z. Li, J.-P. Zhuang, S.-S. Li, J.-B. Gao, and S.-C. Chan, “Randomness evaluation for an optically injected chaotic semiconductor laser by attractor reconstruction,” Phys. Rev. E 94(4), 042214 (2016).
[Crossref]

Phys. Rev. Lett. (5)

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref]

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012).
[Crossref]

C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref]

R. Badii, G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi, and M. A. Rubio, “Dimension Increase in Filtered Chaotic Signals,” Phys. Rev. Lett. 60(11), 979–982 (1988).
[Crossref]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref]

Proc. Natl. Acad. Sci. U. S. A. (1)

A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Harvesting entropy and quantifying the transition from noise to chaos in a photon-counting feedback loop,” Proc. Natl. Acad. Sci. U. S. A. 112(30), 9258–9263 (2015).
[Crossref]

Rev. Mod. Phys. (2)

M. C. Soriano, J. García-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[Crossref]

Y. K. Chembo, D. Brunner, M. Jacquot, and L. Larger, “Optoelectronic oscillators with time-delayed feedback,” Rev. Mod. Phys. 91(3), 035006 (2019).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Experimental setup. TC, temperature controller; CS, current source; LD, distributed feedback laser diode; PC, polarization controller; VOA, variable optical attenuator; OC, optical coupler; PD, photodetector; M, mixer; SG, signal generator; Filter, low-pass filter; DAS, data acquisition system containing an oscilloscope and a spectrum analyzer; L, lens; PM, power meter; HWP, half-wave plate; PBS, polarizing beamsplitter; BD, balanced-homodyne detector.
Fig. 2.
Fig. 2. (a) Measured original chaotic laser when $J=1.5J_{th}$ , $\gamma =4.5$ dB and (c) shot noise power spectrum; (b) filtered chaotic laser and (d) filtered shot noise power spectrum with 100 MHz frequency bandwidth.
Fig. 3.
Fig. 3. (a), (c) Theoretical and (b), (d) experimental results for $H_{d}-H_{d-1}$ of filtered chaotic laser at four $J$ (a), (b): $1.1J_{th}$ , $1.3J_{th}$ , $1.5J_{th}$ , and $1.7J_{th}$ ; four $\kappa (\gamma )$ (c), (d): $44$ $ns^{-1}$ ( $1.5$ dB), $22$ $ns^{-1}$ ( $4.5$ dB), $11$ $ns^{-1}$ ( $7.5$ dB), $9.1$ $ns^{-1}$ ( $8.5$ dB). The embedding dimension $d$ is chosen as 5.
Fig. 4.
Fig. 4. Measured maps of the $H_{d}-H_{d-1}$ ( $d=5$ ) of filtered chaotic laser versus embedding delay time with varying injection currents (a1) $J=1.5J_{th}$ , (b1) $J=1.75J_{th}$ , (c1) $J=2.0J_{th}$ and feedback attenuations (a2) $\gamma =1.5$ dB, (b2) $\gamma =4.5$ dB, (c2) $\gamma =10.5$ dB.
Fig. 5.
Fig. 5. Measured difference $H_{d}-H_{d-1}$ of filtered chaotic laser as functions of (a) embedding delay time $t$ and (b) embedding dimension $d$ . The shot noise versus $d$ are indicated with the solid gray line.
Fig. 6.
Fig. 6. (a) ACF of original chaotic laser, filtered shot noise and filtered chaotic laser. The inset shows the peak value $C_{p}$ at feedback delay time of 86.7 ns; (b) Entropy rate $R_{d}$ of filtered shot noise and filtered chaotic laser as a function of the dimension $d$ . The entire embedding delay time is $t_{m}=4.7$ ns. The solid line indicates the $R_{d}$ of ideal noise limit. The inset shows the difference $H_{d}-H_{d-1}$ for $d=5$ .
Fig. 7.
Fig. 7. Measured entropy rate and TDS of filtered chaotic laser as functions of the injection currents for (a1) $\gamma =1.5$ dB, (b1) $\gamma =4.5$ dB, (c1) $\gamma =10.5$ dB and feedback attenuations for (a2) $J=1.5J_{th}$ , (b2) $J=1.75J_{th}$ , (c2) $J=2.0J_{th}$ , with $d=5$ and $t_{m}=4.7$ ns.
Fig. 8.
Fig. 8. (a) Theoretical and experimental TDS of the filtered chaotic laser as a function of RF frequency for $J=1.5J_{th}$ and $\gamma =4.5$ dB ( $\kappa =22$ $ns^{-1}$ ); (b) theoretical results of the maxima and the minima of $C_{p}$ with varying the RF frequency.
Fig. 9.
Fig. 9. Experimental results of the maximum $C_{p}$ in the TDS suppression and the corresponding $R_{d}$ of filtered chaotic laser as a function of LPF bandwidths.

Equations (1)

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R d = H d H d 1 t m ,

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