Abstract

We experimentally and numerically demonstrate the time delay (TD) signature suppression of chaotic output in a double optical feedback semiconductor laser (DOF-SL) system. Two types of TD signature suppression are demonstrated by adjusting the lengths and the feedback power ratios of the two external cavities. One can significantly eliminate all TD signatures of the DOF-SL system and the corresponding power spectrum distribution becomes quite smooth and flat, the other suppresses one of two TD signatures and remains another one.

© 2009 OSA

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    [CrossRef]
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    [CrossRef]
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  6. F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004).
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    [CrossRef]
  10. J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (2005).
    [CrossRef]
  11. M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
    [CrossRef] [PubMed]
  12. R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
    [CrossRef]
  13. M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (2005).
    [CrossRef]
  23. C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. 128(4-6), 363–376 (1996).
    [CrossRef]

2009

2008

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]

A. Többens and U. Parlitz, “Dynamics of semiconductor lasers with external multicavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(1), 016210 (2008).
[CrossRef] [PubMed]

2007

2006

2005

S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (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]

V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay- differential equations,” J. Opt. Technol. 72, 373–377 (2005).
[CrossRef]

G. Q. Xia, Z. M. Wu, and J. G. Wu, “Theory and simulation of dual-channel optical chaotic communication system,” Opt. Express 13(9), 3445–3453 (2005).
[CrossRef] [PubMed]

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
[CrossRef]

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]

J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (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 437(7066), 343–346 (2005).
[CrossRef]

2004

F. Y. Lin and J. M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. 40(6), 682–689 (2004).
[CrossRef]

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

2000

1998

R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

1997

M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997).
[CrossRef]

1996

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. 128(4-6), 363–376 (1996).
[CrossRef]

1994

V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. 30(7), 1537–1541 (1994).
[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 437(7066), 343–346 (2005).
[CrossRef]

V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. 30(7), 1537–1541 (1994).
[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 437(7066), 343–346 (2005).
[CrossRef]

Bezruchko, B. P.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
[CrossRef]

Blondel, M.

Bünner, M. J.

R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997).
[CrossRef]

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

Cao, L. P.

J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282(15), 3153–3156 (2009).
[CrossRef]

Citrin, D. S.

Colet, P.

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]

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 437(7066), 343–346 (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]

Deparis, O.

Donati, S.

V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. 30(7), 1537–1541 (1994).
[CrossRef]

Fischer, I.

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 437(7066), 343–346 (2005).
[CrossRef]

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 437(7066), 343–346 (2005).
[CrossRef]

Gavrielides, A.

Goedgebuer, J. P.

Gutierrez, J. M.

S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (2005).
[CrossRef]

Hegger, R.

R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

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]

Inoue, 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]

Kantz, H.

R. Hegger, M. J. Bünner, and H. Kantz, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

Karavaev, A. S.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
[CrossRef]

Kittel, A.

M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997).
[CrossRef]

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

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).
[CrossRef]

Larger, L.

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 437(7066), 343–346 (2005).
[CrossRef]

V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay- differential equations,” J. Opt. Technol. 72, 373–377 (2005).
[CrossRef]

Lee, M. W.

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]

J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (2005).
[CrossRef]

Lin, F. Y.

F. Y. Lin and J. M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. 40(6), 682–689 (2004).
[CrossRef]

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

Liu, J.

Liu, J. M.

F. Y. Lin and J. M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. 40(6), 682–689 (2004).
[CrossRef]

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

Locquet, A.

Manna, M.

V. Annovazzi-Lodi, S. Donati, and M. Manna, “Chaos and locking in a semiconductor laser due to external injection,” IEEE J. Quantum Electron. 30(7), 1537–1541 (1994).
[CrossRef]

Masoller, C.

C. Masoller, “Effect of the external cavity length in the dynamics of a semiconductor laser with optical feedback,” Opt. Commun. 128(4-6), 363–376 (1996).
[CrossRef]

Mégret, P.

Meyer, T.

M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997).
[CrossRef]

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

Mirasso, C. R.

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 437(7066), 343–346 (2005).
[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).
[CrossRef]

Oowada, I.

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.

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]

S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (2005).
[CrossRef]

Parisi, J.

M. J. Bünner, T. Meyer, A. Kittel, and J. Parisi, “Recovery of the time evolution equation of time-delay systems from time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(5), 5083–5089 (1997).
[CrossRef]

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

Parlitz, U.

A. Többens and U. Parlitz, “Dynamics of semiconductor lasers with external multicavities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 78(1), 016210 (2008).
[CrossRef] [PubMed]

Paul, J.

J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (2005).
[CrossRef]

Pesquera, L.

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 437(7066), 343–346 (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]

S. Ortin, J. M. Gutierrez, L. Pesquera, and H. Vasquez, “Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction,” Physica A 351(1), 133–141 (2005).
[CrossRef]

Pisarchik, A. N.

Ponomarenko, V. I.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
[CrossRef]

Popp, M.

M. J. Bünner, M. Popp, T. Meyer, A. Kittel, and J. Parisi, “Tool to recover scalar time-delay systems from experimental time series,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(4), R3082–3085 (1996).
[CrossRef] [PubMed]

Prokhorov, M. D.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3-4), 209–223 (2005).
[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]

Rogister, F.

Rontani, D.

Ruiz-Oliveras, F. R.

Sciamanna, M.

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.

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 437(7066), 343–346 (2005).
[CrossRef]

J. Paul, M. W. Lee, and K. A. Shore, “3.5-GHz signal transmission in an all-optical chaotic communication scheme using 1550-nm diode lasers,” IEEE Photon. Technol. Lett. 17(4), 920–922 (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]

Someya, H.

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Figures (10)

Fig. 1
Fig. 1

Experimental setup. Laser: DFB semiconductor laser; AL: aspheric lens; BS: beam splitter; M: mirror; OI: optical isolator; NDF: neutral density filter; FC: fiber coupler; PD: photo detector; T: test point of reflected optical power.

Fig. 2
Fig. 2

Recorded chaotic time series and associated power spectra of (A) SOF1, (B) SOF2, and (C) DOF configuration.

Fig. 3
Fig. 3

SF (A1-C1) and MI (A2-C2) corresponding to chaotic time series of Fig. 2 (A1-C1).

Fig. 4
Fig. 4

Evolution of chaotic time series (1st Column), power spectra (2nd Column), SF (3rd Column) and MI (4th Column) under a fixed feedback strength of cavity 1 (FPR≈0.3%) and different feedback strengths of SOF2 for two cavity lengths are about lcav1 ≈477mm and lcav2 ≈461mm, respectively, where the feedback optical powers of cavity 2 are 7.83μw (FPR≈0.15%) (Row A), 11.1μw (FPR≈0.22%) (Row B), 14.3μw (FPR≈0.28%) (Row C), 15.3μw (FPR≈0.3%) (Row D) and 16.4μw (FPR≈0.32%) (Row E), respectively.

Fig. 5
Fig. 5

Chaotic time series (A1-C1), associated power spectra (A2-C2), SF (A3-C3) and MI (A4-C4) for SOF 1 (Row A), SOF 2 (Row B) and DOF configuration (Row C).

Fig. 6
Fig. 6

Evolution of chaotic time series (1st Column), power spectra (2nd Column), SF (3rd Column) and MI (4th Column) under a fixed feedback strength of cavity 1 (FPR≈0.3%) and different feedback strengths of cavity 2 for two cavity lengths are about lcav1 ≈477mm and lcav2 ≈241mm, respectively, where the feedback powers of cavity 2 are 59μw (FPR≈1.15%) (Row A), 77μw (FPR≈1.5%) (Row B), 80μw (FPR≈1.56%) (Row C), 113μw (FPR≈2.2%) (Row D) and 135μw (FPR≈2.63%) (Row E), respectively.

Fig. 7
Fig. 7

Superimposed SF curves of two types TDSS, where (A) (integration of Fig. 3 (A1) and Fig. 3 (B1)) and (B) (integration of Fig. 5 (A3) and Fig. 5 (B3)) correspond to the first type and the second type TDSS, respectively, and blue line, red line and black line represent SOF 1, SOF 2 and DOF configurations, respectively.

Fig. 8
Fig. 8

Simulated chaotic temporal waveforms (A1-C1), associated power spectra (A2-C2), SF (A3-C3) and MI (A4-C4) of the first type TDSS for SOF 1 (Row A), SOF 2 (Row B) and DOF configuration (Row C), where τcav1 = 3.2ns, τcav2 = 3.12ns, κcav1 = 0.04, κcav2 = 0.05.

Fig. 9
Fig. 9

Calculated maps of TD signature for κcav1 = κcav2 = 0.04 and τcav1 = 3.2ns, where τcav2 varies from 0.4ns to 4ns. Here, different colors represent different SF function values.

Fig. 10
Fig. 10

The evolution map (A1, B1) and the variation curve of amplitude ρ (A2, B2) for τcav1 = 3.2ns, τcav2 = 3.12ns and κcav1 = 0.04 under different κcav2 . A1 and A2 relate to the type I TDSS; B1 and B2 relate to the type II TDSS. The amplitude ρ is the maximum of the SF peak in the time window, which is marked by dashed lines in (A1) and (B1).

Equations (3)

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C(Δt)=(P(t+Δt)P(t))(P(t)P(t))(P(t)P(t)2P(t+Δt)P(t)2)1/2
M(Δt)=P(t),P(t+Δt)δ(P(t),P(t+Δt))logδ(P(t),P(t+Δt))δ(P(t))δ(P(t+Δt))
E˙(t)=12(1+iβ)(G(t)γp)E(t)+κcav1/τLE(tτcav1)ei2πω0τcav1+κcav2/τLE(tτcav2)ei2πω0τcav2+F(t)N˙(t)=JN(t)/τNG(t)|E(t)|2

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