Abstract

The time delay (TD) signature concealment of optical feedback induced chaos in an external cavity semiconductor laser is experimentally demonstrated. Both the evolution curve and the distribution map of TD signature are obtained in the parameter space of external feedback strength and injection current. The optimum parameter scope of the TD signature concealment is also specified. Furthermore, the approximately periodic evolution relation between TD signature and external cavity length is observed and indicates that the intrinsic relaxation oscillation of semiconductor laser may play an important role during the process of TD signature suppression.

© 2010 OSA

1. Introduction

It is well known that an external cavity semiconductor laser (ECSL) under suitable operation conditions could display rich chaotic behavior. The time delay (TD) in ECSL system introduces an infinite number of degree of freedom into the dynamical system, and therefore could produce high-dimensional chaotic signals [1]. Such signals have attractive applications in various fields such as optical chaotic encryption communication [2], chaotic radar [3], high speed random bit generation [4,5] and so on. On one hand, the introduction of TD is necessary to produce high-dimensional chaotic signals. However, on the other hand, the chaotic output of ECSL system usually retains a obvious TD signature, which sometimes is undesirable in some applications such as high speed random bit generation and chaotic secure communications. For high speed random bit generation, the TD signature induces recurrence features and affects partly the statistical performance [4,5], and therefore the elimination of the TD signature is one of key points. As for chaotic cryptosystems, the security relies mainly on the identifying difficulty of the transmitter parameters and the sensitivity of synchronization to parameter mismatch. However, the TD signature provides one of possible clues to the chaos encryption attackers. Based on some recently developing chaos analysis techniques [6,7], the reconstruction of delayed system may be computationally feasible [8].

At present, single or double external cavity scheme has been proposed to complicate or suppress the TD signature of ECSL system. Lee et al firstly illustrated the possibility to complicate the TD signature by adopting double ECSL system [9]. We further experimentally demonstrated that there exist two types of TD signature suppression in such ECSL system [10]. As for the single external cavity scheme, it is usually divided into two case, namely incoherent optical feedback and coherent optical feedback. The chaotic output of the incoherent ECSL system origins from incoherent interaction between the external reflected light and intracavity carriers, while the chaotic output of coherent ECSL system origins from nonlinear interaction between the external feedback light field and the inner light field of SL. Compared with double ECSL system, a single ECSL system owns some unique virtues such as simpler structure, easily experimental realization and feasible parameter controllability. We have experimentally demonstrated that the TD signature can be suppressed in an incoherent single ECSL system [11]. Recently, theoretical investigation and related simulations to suppress the TD signature in a single coherent ECSL system were performed [12,13]. In this paper, we experimentally investigate the TD signature of chaotic output in a single coherent ECSL system. The TD signature of the chaotic signals is retrieved and measured by using self-correlation function and mutual information techniques. The TD signature concealment is experimentally demonstrated. Both the evolution and the distribution of the TD signature are specified by controlling the operation parameters such as injection current, external feedback strength and external cavity length.

2. Experimental setup

Figure 1 is the schematic of experimental setup. An InGaAsP/InP DFB-SL (Wuhan Telecommunication Devices, LDM5S515-005) is used in experiment and firstly biased at 11.81mA (about 1.09Ith) by an ultra-low-noise current source (ILX-Lightwave, LDX-3620) and stabilized at 20.14°C by a thermoelectric controller (ILX-Lightwave, LDT-5412). The lasing wavelength of the solitary SL is measured to be about 1549.6nm by an optical spectrum analyzer (Ando AQ6317C). The emission of SL is firstly collimated by an aspheric lens and about 43% of optical energy is incident on an external cavity mirror by a beam splitter (BS). The external feedback strength (Fext), defined as the ratio of reflected power and the solitary SL output power, is controlled by a variable neutral density filter and monitored at point T by an optical power meter. An optical isolator (OI) (isolation>55dB) is also used to avoid the unwanted reflected disturbance from the front face of signal detection part. In the signal detection part, the optical signal is firstly transformed into electronic signal by a wide bandwidth photodetector (PD, New Focus 1544-B, bandwidth 12 GHz) and then analyzed by a 6 GHz digital oscilloscope (Agilent 54855A, sample interval 50ps). During experiment, the external cavity length is firstly set as about Lcav = 300mm, which corresponds a TD (Tdelay) of about 2ns.

 

Fig. 1 Experimental setup. Laser: DFB semiconductor laser; BS: Beam splitter; M: Mirror; OI: Optical isolator; NDF: Neutral density filter; FC: Fiber coupler; PD: Photodetector; OSA: Optical Spectrum Analyzer; T: Feedback power monitoring point.

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3. Experimental results and discussion

The experimentally recorded typical chaotic intensity time series and the TD signature suppressed chaotic intensity time series are comparatively displayed in Fig. 2 (A1) for Fext ≈0.038 and Fig. 2 (B1) for Fext ≈0.0013. Both time series behave intricately. However, from the power spectrum (Fig. 2 (A2)) corresponding to Fext≈0.038, some uniform spacing frequency peaks emerge upon the background and reveal the external cavity characteristic frequency fcav≈500MHz. Thus, the value of Tdelay could be estimated as Tdelay = 1/fcav≈2 ns. In contrast, for the TD suppressed intensity time series in Fig. 2 (B1), the according power spectrum (Fig. 2 (B2)) becomes relatively smooth and has no significant frequency peaks upon background.

 

Fig. 2 Recorded chaotic intensity time series (A1) and associated power spectrum (A2) for Fext ≈0.038, and TD signature suppressed chaotic intensity time series (B1) and associated power spectrum (B2) for Fext ≈0.0013. The total time length of the recorded data is 500ns.

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For a delay-differential system, the SF function can be defined as: Generally, two standard statistical functions namely the self-correlation function (SF) and the mutual information (MI) can be used to retrieve the TD signature of chaotic intensity time series [11]. In the following, we will use these two functions to quantitatively evaluate TD signature of chaotic intensity time series by following formulas in Ref [12].

To further investigated the influence of external feedback strength on the TD signature of SL system. Figure 3 summarizes the measured chaotic intensity time series, corresponding SF and MI curves under different external feedback strength. In Fig. 3(A2) and Fig. 3(A3), the TD signature (Tdelay≈2 ns) is clearly exhibited. Combining all SF and MI curves, one can observe that the TD signatures are gradually suppressed as Fext decreasing from 0.038 to 0.0013. Especially, for the case of Fext≈0.0013 in Figs. 3(D), the TD signature is almost shielded completely into the background fluctuations. In this situation, an eavesdropper would be quite difficult to accurately identify the Tdelay of the SL dynamical system. Further decreasing Fext to 0.0003 as shown in Fig. 3(E), the TD signature arise reversely again. In addition, the small periodical troughs of SF curves relate closely to the relaxation oscillation of SL [12]. When the external feedback is weakened to a certain degree, the period of these troughs would accord well with the relaxation oscillation period (τRO) of SL, and could be estimated as about 0.55ns from Fig. 3(E2).

 

Fig. 3 Recorded chaotic intensity time series (left column), SF curve (middle column) and MI curve (right column) under different external feedback strengths, where (row A) Fext≈0.038, (row B) Fext≈0.019, (row C) Fext≈0.0064, (row D) Fext≈0.0013, (row E) Fext≈0.0003, respectively.

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To show the evolution pattern of TD signature under different external feedback strength, Fig. 4(A) and Fig. 4(B) respectively give variation curves of the amplitude (ρ) and the location (τ) of the maximum SF peak with Fext in a time window of 1.0 ns around Tdelay. Such a time window is sufficiently large to capture the possible shift of the SF peak and also sufficiently narrow to measure only the SF peak. From these two curves in Fig. 4, the evolution of TD signature could be roughly divided into three regions. For Fext>0.019 (region I), the TD signature is obvious and the identified Tdelay well conforms to expected Tdelay≈2ns. For 0.0013<Fext<0.019 (region II), the TD signature attenuates significantly with the decrease of Fext, and the weakest TD signature is obtained for Fext0.003. Meantime, the identified Tdelay begins to deviate from the expected Tdelay≈2ns. As for Fext<0.0013 (region III), the TD signature booms again. However, the deviation between identified Tdelay and expected Tdelay becomes larger than above two regions. These results confirm qualitatively with the theoretical prediction [12]. This phenomenon may provide a pseudo TD signature scheme for some specific applications.

 

Fig. 4 The variation curve of amplitude ρ (A) and location τ (B) of the maximum SF peak in time window 1.5ns<Δt<2.5ns under different Fext values, where the time window has been marked by dashed lines in SF curves of Fig. 3.

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Figure 5 integrate the identified TD amplitudes and TD locations from SF curves to form two distribution maps about TD signature in the operation parameter space of injection current and external feedback strength. From Fig. 5(A), for different injection currents, it is clear that there always exists a weak TD signature region under certain Fext values (colored as the dark blue region). We could name it as TD signature suppression (TDSS) region. Additionally, with the increasing of injection current, the TDSS region gradually expands and the minimum TD signature point has a tendency to monotonically move towards a higher Fext level. From Fig. 5(B), it could be observed that the identified Tdelay always deviates from the expected Tdelay for Fext bellow a key value about 0.01. Based on these two diagrams, one can specify the optimal parameter scope of chaotic output with weak and pseudo TD signature. Moreover, above results show that weak TD signature region usually locates around relatively low Fext level. However, from the viewpoint of the information dimension of chaotic attractor, low Fext may decrease the Kaplan-Yorke dimension [1]. Therefore, a comprehensive consideration is necessary after taking into account various performance requirements.

 

Fig. 5 The distribution map of the amplitude ρ (A) and the location τ (B) under different injection currents and different Fext, where the black dots represent the measured points and the smooth color surfaces are obtained by triangle-based linear interpolation.

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To illustrate the influence of the external cavity length Lcav on TD signature, Fig. 6 integrates SF curves to form an evolution map of TD signature when Lcav varies from 122mm to 477mm and Fext is fixed at about 0.00013 since the TD signature is disturbed significantly by the τRO related fluctuations around this Fext value. As shown in Fig. 6, the oblique black dashed line Δt = 2Lcav/C (C is the speed of light) reveals the TD signatures for different Lcav. Meanwhile, some weak parallel oblique lines periodically accompany by the TD signature line and own a common Δt interval of about 0.55ns (≈τRO). Figures 6(A) to Figs. 6(E), which respectively correspond to Lcav1≈167mm, Lcav2≈247mm, Lcav≈287mm, Lcav3≈337mm and Lcav4≈417mm, show the detailed chaotic features, and demonstrate that the external cavity length has a significant influence on the TD signature of an ECSL system. Interestingly, another periodic evolution pattern of TD signatures could be also observed at four cavity lengths which correspond to Figs. 6(A), Figs. 6(B), Figs. 6(D) and Figs. 6(E), respectively. Compared these four figures with Figs. 6(C), the TD signatures are relatively weak. Moreover, if one observes above four figures carefully, it could be found that the TD signature weakens gradually with the decrease of the external cavity length. Especially at cavity lengths Lcav1≈167mm (Figs. 6(E)) and Lcav2≈247mm (Figs. 6(D)), the TD signatures are significantly suppressed. Furthermore, after transforming these Lcav into Tdelay = 2Lcav/C, it can be obtained that Tdelay1≈1.11ns, Tdelay2≈1.65ns, Tdelay3≈2.25ns and Tdelay4≈2.78ns. If one analyzes these Tdelay, it can be found that these time intervals among these four Tdelay are roughly close to the characteristic time τRO (about 0.55ns). This may imply that the characteristic time τRO will take an important role for the evolution and the distribution of TDSS in ECSL system, which has also been revealed in other related theoretical calculations and experimental results [1013]. Then one can reasonably deduce that the TDSS may origin from the nonlinear interaction between intrinsic relaxation oscillation and external delay feedback disturbance.

 

Fig. 6 Evolution map of TD signature for Fext is fixed at about 0.00013 and external cavity length varies from 122mm to 477mm (corresponding Tdelay changes from 0.81ns to 3.18ns), where the measurement step of external cavity length is 5mm and the different colors represent different SF values. The detailed chaotic time series (A1-E1), associated SF curves (A2-E2) and MI curves (A3-E3) are for Lcav4≈417mm (Row A), Lcav3≈337mm (Row B), Lcav = 287mm (Row C), Lcav2≈247mm (Row D) and Lcav1≈167mm (Row E), respectively.

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4. Conclusion

In summary, the suppression of TD signature in ECSL system is demonstrated experimentally in an ECSL system. The variation curve and the distribution map of TD signature are both obtained in the operation parameter space of external feedback strength and injection current. As a result, the optimum scope for weakest TD signature could be determined. Furthermore, the approximately periodic evolution of TD signature is found by varying the external cavity length. Our observations indicate that the intrinsic relaxation oscillation of SL may play an important role during the process of TDSS. We hope this work would offer a pure physical method to suppress the TD signature for high speed random bit generation and the SL based chaos cryptosystems. Also, this work may offer a useful insight to the nonlinear dynamics of an ECSL system.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 60978003, the Open Fund of the State Key Lab of Millimeter Waves of China and the Special Funds of Southwest University for Basic Scientific Research in Central Universities.

References and links

1. 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]  

2. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]  

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

4. 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]  

5. 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]  

6. 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]  

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

8. 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]  

9. 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]  

10. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009). [CrossRef]   [PubMed]  

11. J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, “Z. and 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]  

12. 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]  

13. 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]  

References

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  1. 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]
  2. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]
  3. F. Y. Lin and J. M. Liu, “chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40(6), 815–820 (2004).
    [CrossRef]
  4. 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]
  5. 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]
  6. 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]
  7. R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
    [CrossRef]
  8. 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]
  9. 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]
  10. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
    [CrossRef] [PubMed]
  11. J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]
  12. 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]
  13. 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]

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 (3)

J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
[CrossRef] [PubMed]

J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]

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]

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 (1)

2005 (4)

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]

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]

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. Garcıa-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 (1)

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

1998 (1)

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

1996 (1)

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]

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. Garcıa-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]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

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]

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]

Bünner, M. J.

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

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. Garcıa-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]

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]

Fischer, I.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

Garcia-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

Giaquinta, A.

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

Hegger, R.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “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]

Kanter, 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]

Kantz, H.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “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, 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. Garcıa-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]

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]

Lin, F. Y.

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. M.

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.

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]

Meyer, T.

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. Garcıa-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.

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]

Parisi, J.

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]

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

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]

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]

Sciamanna, M.

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]

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. Garcıa-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]

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]

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

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]

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.

J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
[CrossRef] [PubMed]

J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]

Wu, Z. M.

Wu, Z.-M.

J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]

Xia, G. Q.

J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]

J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
[CrossRef] [PubMed]

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.

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]

IEE Proc., Optoelectron. (1)

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]

IEEE J. Quantum Electron. (3)

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]

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

Nat. Photonics (2)

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]

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]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcıa-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]

Opt. Commun. (1)

J. G. Wu, G. Q. Xia, L. P. Cao, and Z.-M. Wu, Z. Wu, and 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]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

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]

Phys. Rev. Lett. (1)

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

Physica D (1)

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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Figures (6)

Fig. 1
Fig. 1

Experimental setup. Laser: DFB semiconductor laser; BS: Beam splitter; M: Mirror; OI: Optical isolator; NDF: Neutral density filter; FC: Fiber coupler; PD: Photodetector; OSA: Optical Spectrum Analyzer; T: Feedback power monitoring point.

Fig. 2
Fig. 2

Recorded chaotic intensity time series (A1) and associated power spectrum (A2) for Fext ≈0.038, and TD signature suppressed chaotic intensity time series (B1) and associated power spectrum (B2) for Fext ≈0.0013. The total time length of the recorded data is 500ns.

Fig. 3
Fig. 3

Recorded chaotic intensity time series (left column), SF curve (middle column) and MI curve (right column) under different external feedback strengths, where (row A) Fext ≈0.038, (row B) Fext ≈0.019, (row C) Fext ≈0.0064, (row D) Fext ≈0.0013, (row E) Fext ≈0.0003, respectively.

Fig. 4
Fig. 4

The variation curve of amplitude ρ (A) and location τ (B) of the maximum SF peak in time window 1.5ns<Δt<2.5ns under different Fext values, where the time window has been marked by dashed lines in SF curves of Fig. 3.

Fig. 5
Fig. 5

The distribution map of the amplitude ρ (A) and the location τ (B) under different injection currents and different Fext , where the black dots represent the measured points and the smooth color surfaces are obtained by triangle-based linear interpolation.

Fig. 6
Fig. 6

Evolution map of TD signature for Fext is fixed at about 0.00013 and external cavity length varies from 122mm to 477mm (corresponding Tdelay changes from 0.81ns to 3.18ns), where the measurement step of external cavity length is 5mm and the different colors represent different SF values. The detailed chaotic time series (A1-E1), associated SF curves (A2-E2) and MI curves (A3-E3) are for Lcav4 ≈417mm (Row A), Lcav3 ≈337mm (Row B), Lcav = 287mm (Row C), Lcav2 ≈247mm (Row D) and Lcav1 ≈167mm (Row E), respectively.

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