Abstract

Polarization-resolved chaotic emission intensities from a vertical-cavity surface-emitting laser (VCSEL) subject to feedback from a fiber Bragg grating (FBG) are numerically investigated. Time-delay (TD) signatures of the feedback are examined through various means including self-correlations of intensity time-series of individual polarizations, cross-correlation of intensities time-series between both polarizations, and permutation entropies calculated for the individual polarizations. The results show that the TD signatures can be clearly suppressed by selecting suitable operation parameters such as the feedback strength, FBG bandwidth, and Bragg frequency. Also, in the operational parameter space, numerical maps of TD signatures and effective bandwidths are obtained, which show regions of chaotic signals with both wide bandwidths and weak TD signatures. Finally, by comparing with a VCSEL subject to feedback from a mirror, the VCSEL subject to feedback from the FBG generally shows better concealment of the TD signatures with similar, or even wider, bandwidths.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  36. F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012).
    [Crossref]
  37. Y. Hong, “Experimental study of time-delay signature of chaos in mutually coupled vertical-cavity surface-emitting lasers subject to polarization optical injection,” Opt. Express 21(15), 17894–17903 (2013).
    [Crossref] [PubMed]

2015 (2)

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).
[PubMed]

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. 99, 1 (2015).
[Crossref]

2014 (3)

2013 (5)

S. Priyadarshi, Y. H. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

Z. Q. Zhong, Z. M. Wu, J. G. Wu, and G. Q. Xia, “Time-delay signature suppression of polarization-resolved chaos outputs from two mutually coupled VCSELs,” IEEE Photon. J. 5(2), 1500409 (2013).
[Crossref]

K. Panajotov, M. Sciamanna, M. A. Arteaga, and H. Thienpont, “Optical feedback in vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700312 (2013).
[Crossref]

Y. Hong, “Experimental study of time-delay signature of chaos in mutually coupled vertical-cavity surface-emitting lasers subject to polarization optical injection,” Opt. Express 21(15), 17894–17903 (2013).
[Crossref] [PubMed]

2012 (4)

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012).
[Crossref]

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (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] [PubMed]

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photon. J. 4(5), 1930–1935 (2012).
[Crossref]

2011 (3)

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[Crossref]

2010 (1)

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[Crossref]

2009 (2)

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-Delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17(15), 12619–12626 (2009).
[Crossref] [PubMed]

2008 (1)

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]

2007 (2)

2006 (1)

2005 (4)

R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005).
[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] [PubMed]

R. Vicente, J. Daudén, 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(4), 541–548 (2005).
[Crossref]

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[Crossref]

2002 (2)

J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
[Crossref]

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

2000 (1)

K. Iga, “Surface-emitting laser—its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1201–1215 (2000).
[Crossref]

1997 (2)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

1995 (1)

M. S. Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).

1994 (1)

I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

1992 (1)

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[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]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[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]

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] [PubMed]

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] [PubMed]

Arteaga, M. A.

K. Panajotov, M. Sciamanna, M. A. Arteaga, and H. Thienpont, “Optical feedback in vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700312 (2013).
[Crossref]

Aviad, Y.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[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] [PubMed]

Chan, S. C.

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. 99, 1 (2015).
[Crossref]

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photon. J. 4(5), 1930–1935 (2012).
[Crossref]

Chao, Y. K.

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012).
[Crossref]

Chen, Y. C.

Cheng, C. H.

Citrin, D. S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-Delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007).
[Crossref] [PubMed]

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[Crossref]

Colet, P.

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] [PubMed]

R. Vicente, J. Daudén, 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(4), 541–548 (2005).
[Crossref]

Daudén, J.

R. Vicente, J. Daudén, 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(4), 541–548 (2005).
[Crossref]

Davis, P.

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]

Deng, T.

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Elsa, W.

I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997).
[Crossref]

Fan, L.

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

Feng, G. Y.

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] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Feng, Q.

M. S. Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).

Fischer, I.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[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] [PubMed]

I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

García-Ojalvo, J.

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] [PubMed]

Gatare, I.

Göbel, E.

I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Hess, O.

I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994).
[Crossref] [PubMed]

Hirano, 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]

Hong, Y.

Hong, Y. H.

H. Lin, Y. H. Hong, and K. A. Shore, “Experimental study of time-delay signatures in vertical-cavity surface-emitting lasers subject to double-cavity polarization-rotated optical feedback,” J. Lightwave Technol. 32(9), 1829–1836 (2014).
[Crossref]

Y. H. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with time-delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron. 50(4), 236–242 (2014).
[Crossref]

S. Priyadarshi, Y. H. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
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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).
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Jiang, L.

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

Jiang, N.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Ju, R.

R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005).
[Crossref]

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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
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Kurashige, T.

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).
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R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
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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).
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M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
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Li, N. Q.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
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Li, S. S.

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. 99, 1 (2015).
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S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photon. J. 4(5), 1930–1935 (2012).
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Li, Y.

Lin, F. Y.

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).
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F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012).
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Lin, H.

Lin, X. D.

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
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Liu, J.

Liu, Q.

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photon. J. 4(5), 1930–1935 (2012).
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Locquet, A.

Luo, B.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Mao, S.

Mark, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
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Martin-Regalado, J.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
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M. S. Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).

Mirasso, C. R.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[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).
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Moloney, J. V.

M. S. Miguel, Q. Feng, and J. V. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52(2), 1728–1739 (1995).

Mork, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

Mu, P.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
[Crossref]

Naito, 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).
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J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002).
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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]

Ortin, S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-Delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[Crossref]

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[Crossref]

Pan, W.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Panajotov, K.

K. Panajotov, M. Sciamanna, M. A. Arteaga, and H. Thienpont, “Optical feedback in vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700312 (2013).
[Crossref]

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007).
[Crossref] [PubMed]

Pesquera, L.

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[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] [PubMed]

Pierce, I.

S. Priyadarshi, Y. H. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
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Prati, F.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Priyadarshi, S.

S. Priyadarshi, Y. H. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

Rees, P.

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[Crossref]

Reidler, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[Crossref]

Rontani, D.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-Delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007).
[Crossref] [PubMed]

Rosenbluh, M.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
[Crossref]

Rosso, O. A.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[Crossref]

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33(5), 765–783 (1997).
[Crossref]

Sciamanna, M.

K. Panajotov, M. Sciamanna, M. A. Arteaga, and H. Thienpont, “Optical feedback in vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700312 (2013).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-Delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[Crossref]

I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, “Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers,” Opt. Lett. 32(12), 1629–1631 (2007).
[Crossref] [PubMed]

D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007).
[Crossref] [PubMed]

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.

H. Lin, Y. H. Hong, and K. A. Shore, “Experimental study of time-delay signatures in vertical-cavity surface-emitting lasers subject to double-cavity polarization-rotated optical feedback,” J. Lightwave Technol. 32(9), 1829–1836 (2014).
[Crossref]

Y. H. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with time-delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron. 50(4), 236–242 (2014).
[Crossref]

S. Priyadarshi, Y. H. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[Crossref]

R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005).
[Crossref]

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[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] [PubMed]

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.

M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. 47(2), 252–261 (2011).
[Crossref]

Spencer, P. S.

Y. H. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with time-delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron. 50(4), 236–242 (2014).
[Crossref]

R. Ju, P. S. Spencer, and K. A. Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 41(12), 1461–1467 (2005).
[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] [PubMed]

Tang, X.

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Thienpont, H.

K. Panajotov, M. Sciamanna, M. A. Arteaga, and H. Thienpont, “Optical feedback in vertical-cavity surface-emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700312 (2013).
[Crossref]

Toral, R.

R. Vicente, J. Daudén, 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(4), 541–548 (2005).
[Crossref]

Tromborg, B.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992).
[Crossref]

Uchida, A.

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]

Valle, A.

M. W. Lee, P. Rees, K. A. Shore, S. Ortin, L. Pesquera, and A. Valle, “Dynamical characterisation of laser diode subject to double optical feedback for chaotic optical communications,” IEE Proc., Optoelectron. 152(2), 97–102 (2005).
[Crossref]

Vicente, R.

R. Vicente, J. Daudén, 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(4), 541–548 (2005).
[Crossref]

Wu, J. G.

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

Z. Q. Zhong, Z. M. Wu, J. G. Wu, and G. Q. Xia, “Time-delay signature suppression of polarization-resolved chaos outputs from two mutually coupled VCSELs,” IEEE Photon. J. 5(2), 1500409 (2013).
[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] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

Wu, T. C.

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48(8), 1010–1014 (2012).
[Crossref]

Wu, Z. M.

Y. Li, Z. M. Wu, Z. Q. Zhong, X. J. Yang, S. Mao, and G. Q. Xia, “Time-delay signature of chaos in 1550 nm VCSELs with variable-polarization FBG feedback,” Opt. Express 22(16), 19610–19620 (2014).
[Crossref] [PubMed]

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[Crossref]

Z. Q. Zhong, Z. M. Wu, J. G. Wu, and G. Q. Xia, “Time-delay signature suppression of polarization-resolved chaos outputs from two mutually coupled VCSELs,” IEEE Photon. J. 5(2), 1500409 (2013).
[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] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17(15), 12619–12626 (2009).
[Crossref] [PubMed]

Xia, G. Q.

Y. Li, Z. M. Wu, Z. Q. Zhong, X. J. Yang, S. Mao, and G. Q. Xia, “Time-delay signature of chaos in 1550 nm VCSELs with variable-polarization FBG feedback,” Opt. Express 22(16), 19610–19620 (2014).
[Crossref] [PubMed]

Z. Q. Zhong, Z. M. Wu, J. G. Wu, and G. Q. Xia, “Time-delay signature suppression of polarization-resolved chaos outputs from two mutually coupled VCSELs,” IEEE Photon. J. 5(2), 1500409 (2013).
[Crossref]

P. Xiao, Z. M. Wu, J. G. Wu, L. Jiang, T. Deng, X. Tang, L. Fan, and G. Q. Xia, “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun. 286, 339–343 (2013).
[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] [PubMed]

J. G. Wu, Z. M. Wu, X. Tang, X. D. Lin, T. Deng, G. Q. Xia, and G. Y. Feng, “Simultaneous generation of two sets of time delay signature eliminated chaotic signals by using mutually coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 23(12), 759–761 (2011).
[Crossref]

J. Liu, Z. M. Wu, and G. Q. Xia, “Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection,” Opt. Express 17(15), 12619–12626 (2009).
[Crossref] [PubMed]

Xiang, S.

N. Q. Li, W. Pan, B. Luo, L. S. Yan, X. Zou, M. Xu, N. Jiang, S. Xiang, and P. Mu, “Numerical characterization of time delay signature in chaotic vertical-cavity surface-emitting lasers with optical feedback,” Opt. Commun. 285(18), 3837–3848 (2012).
[Crossref]

Xiang, S. Y.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Jiang, L. Yang, and H. Zhu, “Conceal time-delay signature of chaotic vertical-cavity surface-emitting lasers by variable-polarization optical feedback,” Opt. Commun. 284(24), 5758–5765 (2011).
[Crossref]

Xiao, P.

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of a polarization-resolved monitoring of a VCSEL under FBG feedback. VCSEL: vertical-cavity surface-emitting laser; BS: beamsplitter; NDF: neutral density filter; FBG: fiber Bragg grating; PBS: polarization beam splitter.
Fig. 2
Fig. 2 Polarization-resolved outputs of the VCSEL under FBG feedback. Columns A and C: time-series for the individual polarizations. Columns B and D: power spectra for the individual polarizations. The feedback rate is set to η = 0 (Row 1), 0.45 ns−1 (Row 2), and 12 ns−1 (Row 3). The feedback has no detuning.
Fig. 3
Fig. 3 Polarization-resolved time-delay signatures at zero detuning. Columns A and C: self-correlation functions (SFs) for the individual polarizations. Columns B and D: permutation entropies (PEs) for the individual polarizations. Column E: cross-correlation function (XF) between both polarizations. The feedback rate is set to η = 40 ns−1 (Row 1), 30 ns−1 (Row 2), and 20 ns−1 (Row 3). The feedback is from the FBG with κB = 100 m−1 for the red curves, whereas it is from a mirror for the green curves.
Fig. 4
Fig. 4 Polarization-resolved time-delay signatures (A-E) under η = 20 ns−1. Row 1 and row 2 are for fm = −15 GHz, 15 GHz, respectively. The reflection spectra of FBG under fm is −15 GHz and 15 GHz are given in F1 and F2, respectively.
Fig. 5
Fig. 5 Polarization-resolved maps of the TD signatures σ from self-correlations (Row 1) and effective bandwidths (Row 2) of the chaotic outputs of the VCSEL under FBG feedback. Column A: x-PC. Column B: y-PC. The maps are shown in the parameter space of η and fm.
Fig. 6
Fig. 6 TD signatures (Row 1) and effective bandwidths (Row 2) of polarization-resolved outputs from the VCSEL. Column A: x-PC. Column B: y-PC. The VCSEL is subject to feedback from mirror (solid lines), FBG under fm = 0 (circles) and fm = 15 GHz (triangles), respectively. The magnitude of the coupling coefficient κB is varied for the FBG. The feedback rate is fixed at η = 20 ns−1.

Equations (9)

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d E x dt =κ( 1+iα )[ ( N1 ) E x +in E y ]( γ a +i γ p i2π f m ) E x +η e iθ r( t ) E x ( tτ )
d E y dt =κ( 1+iα )[ ( N1 ) E y in E x ]+( γ a +i γ p +i2π f m ) E y +η e iθ r( t ) E y ( tτ )
dN dt =γ( μN )γN( | E x | 2 + | E y | 2 )+iγn( E x E y E y E x )
dn dt = γ s nγn( | E x | 2 + | E y | 2 )+iγN( E x E y E y E x )
r( Ω )= κ B sinh( κ B 2 δ 2 L ) δsinh( κ B 2 δ 2 L )+i κ B 2 δ 2 cosh( κ B 2 δ 2 L )
C xx ( Δt )= ( I x ( t+Δt ) I x )( I x ( t ) I x ) ( I x I x ) 2
C yy ( Δt )= ( I y ( t+Δt ) I y )( I y ( t ) I y ) ( I y I y ) 2
C xy ( Δt )= ( I x ( t+Δt ) I x )( I y ( t ) I y ) ( I x I x ) 2 ( I y I y ) 2
H( τ e )= p( π ) log D! p( π )

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