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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1741–1753
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Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system

Jia-Gui Wu, Zheng-Mao Wu, Guang-Qiong Xia, and Guo-Ying Feng  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1741-1753 (2012)
http://dx.doi.org/10.1364/OE.20.001741


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Abstract

In this paper, evolution of time delay (TD) signature of chaos generated in a mutual delay-coupled semiconductor lasers (MDC-SLs) system is investigated experimentally and theoretically. Two statistical methods, including self-correlation function (SF) and permutation entropy (PE), are used to estimate the TD signature of chaotic time series. Through extracting the characteristic peak from the SF curve, a series of TD signature evolution maps are firstly obtained in the parameter space of coupled strength and frequency detuning. Meantime, the influences of injection current on the evolution maps of TD signature have been discussed, and the optimum scope of TD signature suppression is also specified. An overall qualitative agreement between our theoretical and experimental results is obtained.

© 2012 OSA

1. Introduction

Optical chaos has attractive applications in various fields such as secure communications [1

1. G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]

, 2

2. 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]

], chaotic radar and lidar [3

3. 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]

, 4

4. F.-Y. Lin and J.-M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004). [CrossRef]

], rainbow refractometry [5

5. M. Peil, I. Fischer, W. Elsäßer, S. Bakić, N. Damaschke, C. Tropea, S. Stry, and J. Sacher, “Rainbow refractometry with a tailored incoherent semiconductor laser source,” Appl. Phys. Lett. 89(9), 091106 (2006). [CrossRef]

], and fast physical random bit generation [6

6. 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]

9

9. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010). [CrossRef] [PubMed]

] etc. In these applications, optical chaos generated from some delay-coupled semiconductor laser (SL) systems such as an external-cavity feedback SL (ECF-SL) system and a mutually delay-coupled SLs (MDC-SLs) system [2

2. 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]

18

18. T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011). [CrossRef] [PubMed]

], have attracted extensive attention due to their inherent wide bandwidth and dynamic complexity. For example, the optical chaos generated from an ECF-SL system has been successfully applied in Gbit/s unidirectional chaotic secure communication over 100km in a commercial metro-network [2

2. 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]

]. A MDC-SLs system is regarded as one of main candidates in bidirectional chaotic secure communication. Through digital sampling, the optical chaos output of an ECF-SL system can produce ultra-fast random bit sequences [6

6. 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]

8

8. 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]

]. Among above mentioned applications, the time delay (TD) term in delay-coupled SL systems plays an important role to modify the solitary SL becoming an infinite-dimensional dynamical system, which can export high-dimensional chaotic signals [11

11. R. Vicente, J. Dauden, 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]

].

Generally, TD signature of chaos is an important issue in nonlinear system analysis [19

19. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993). [CrossRef]

39

39. 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]

]. On one hand, many techniques such as the singular values fraction measure [22

22. A. C. Fowler and G. Kember, “Delay recognition in chaotic time series,” Phys. Lett. A 175(6), 402–408 (1993). [CrossRef]

], the filling factor analysis [24

24. 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]

], local linear models [25

25. 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]

], models with neural networks [26

26. 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]

], the self-correlation function (SF) and mutual information (MI) [27

27. 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(5), 373–377 (2005). [CrossRef]

31

31. 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]

] etc., have been proposed to identify TD signature of chaos. Recently, permutation entropy (PE) as one of information-theory-based methods is also formulated to perform the TD signature identification [32

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

34

34. 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]

]. On the other hand, obvious TD signature is not undesirable in some applications such as secure communication and random bit generation. For secure communication, the security of data encryption relies upon, to a large extent, the unpredictability of chaotic carrier. However, obvious TD signature offers a possible clue to the attackers, which may compromise the confidentiality of secure communication [35

35. C. Zhou and C. H. Lai, “Extracting messages masked by chaotic signals of time-delay systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 320–323 (1999). [CrossRef] [PubMed]

37

37. M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open and closed-loop receivers in all-optical chaos-based communications,” IEEE Photon. Technol. Lett. 21(7), 426–428 (2009). [CrossRef]

]. As for ultra-fast random bit generation, TD signature induces recurrence features and affects seriously the statistical performance of random bit sequences [6

6. 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]

, 7

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

]. Therefore, it is necessary to develop some effective strategies to suppress the TD signature.

In recent years, several TD signature suppression (TDSS) strategies have been proposed. One type of TDSS strategies is based on logical process such as exclusive-OR (XOR) operation and least significant bits (LSBs) algorithm [6

6. 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]

8

8. 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]

]. These strategies are designed specifically to eliminate the TD signature in generation of random bit sequence. Another type of TDSS strategies is based on pure physical mechanism. These strategies attempt to utilize the physical interaction of nonlinear dynamics to suppress the TD signature of chaos. The pure physical TDSS has been investigated in a double ECF-SL system [31

31. 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]

, 38

38. 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 generally coherent [28

28. 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]

30

30. J. G. Wu, G. Q. Xia, X. Tang, X. D. Lin, T. Deng, L. Fan, and Z. M. Wu, “Time delay signature concealment of optical feedback induced chaos in an external cavity semiconductor laser,” Opt. Express 18(7), 6661–6666 (2010). [CrossRef] [PubMed]

] and incoherent [39

39. 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]

] ECF-SL system. Also, the pure physical TDSS has been demonstrated in an opto-electronic feedback SL system [40

40. E. M. Shahverdiev and K. A. Shore, “Erasure of time-delay signatures in the output of an opto-electronic feedback laser with modulated delays and chaos synchronisation,” IET Optoelectron. 3(6), 326–330 (2009). [CrossRef]

]. More recently, we preliminarily experimentally observed pure physical TDSS phenomenon in a MDC-SLs system [41

41. 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]

]. However, many important issues such as the overall evolution pattern of TD signature, the detailed distribution of TDSS state in the parameter space and the possible physical mechanism etc., are still unclear in such a MDC-SLs system and should be further clarified.

In this paper, we propose a systematical investigation about the TD signature evolution in a MDC-SLs system. In Section 2, the structure of experimental setup is illustrated. In Section 3, with the help of two quantifiers SF and PE, the TD signature of the chaotic signals is retrieved under different operation parameter conditions. Furthermore, a series of evolution maps of TD signature are obtained in the parameter space of coupling strength and frequencies detuning between two SLs. The influence of injection current on the evolution maps of TD signature has also been investigated. In Section 4, the theoretical simulations and analysis are given. Finally, conclusions are presented in Section 5.

2. Experimental setup

The experimental setup is illustrated in Fig. 1
Fig. 1 Schematic of the MDC-SLs system. SL: Distributed Feedback Bragg semiconductor laser; PC: polarization controller; VA: variable attenuator; OI: optical isolator; PD: photodetector; OSA: optical spectrum analyzer; PM: optical power meter.
. Two similar 1550nm InGaAsP/InP distributed feedback Bragg semiconductor lasers (SL1 and SL2) are connected by a fiber to form a mutually coupling system. The two SLs are driven by two ultra-low-noise current sources (ILX-Lightwave, LDX-3620) and two temperature controllers (ILX-Lightwave, LDT-5412).The polarization states of two lasers are matched each other by using a polarization controller. The coupling strength (κ) is defined as the optical power ratio of external injection light and the solitary SL output light. κ is controlled with a variable attenuator (VA1) and monitored by an optical power meter. The optical spectra of SLs are monitored by an optical spectrum analyzer (Ando AQ6317C). The frequency detuning ∆f is defined as ∆f = f1-f2, where f1 and f2 are the optical frequencies of free running SL1 and SL2, respectively. Different ∆f is obtained by tuning the temperature of SL2 while the temperature of SL1 keeps constant. The temperature-frequency coefficient of SL2 is measured as ∆f/∆T≈22GHz/K. The fiber length between SL1 and SL2 is about 5.35m, which corresponds to a coupling round-trip time τ≈53.55ns. Four optical isolators (OIs) (isolation>55dB) are inserted to prevent from unwanted external disturbances. Under proper conditions, the MDC-SLs system can route into complex dynamics and export chaotic signals [13

13. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001). [CrossRef] [PubMed]

18

18. T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011). [CrossRef] [PubMed]

]. The chaotic signals are detected by two 12 GHz bandwidth photodetectors (PDs, New Focus 1544-B) and analyzed by a digital oscilloscope (Agilent 54855A, 6 GHz bandwidth and 50ps sampling interval). For convenience of observing two time series output from SL1 and SL2 simultaneously, the amplitude difference between two chaotic signals sent into the digital oscilloscope should not be too large. Then, another VA (VA2) is introduced to control the optical power of SL1 into the PD.

3. Experimental results

3.1Influence of coupling strength on TD signature

At first, the injection currents are respectively biased at 9.28mA for SL1 (about 1.14Ith, Ith is the threshold current) and 9.7mA for SL2 (about 1.14Ith). The temperatures are respectively stabilized at 19.86°C for SL1 and 19.60°C for SL2. Under these conditions, both the optical wavelengths of the two SLs are about 1553.11nm and then the corresponding ∆f is about 0GHz. The fist column of Fig. 2
Fig. 2 Recorded chaotic time series (the first column), associated power spectra (the second column), SF curves (the third column) and PE curves (the fourth column) under fixed frequency detuning ∆f≈0GHz and different κ values, where κ is about 0.0003 (Aa1-Aa4, Ba1-Ba4), 0.0007 (Ab1-Ab4, Bb1-Bb4), 0.0068 (Ac1-Ac4, Bc1-Bc4), 0.023 (Ad1-Ad4, Bd1-Bd4), 0.145 (Ae1-Ae4, Be1-Be4), respectively. A corresponds to SL1, and B is for SL2. The power spectra, SF curves and PE curves are calculated from the recorded chaotic time series with a 1000ns length.
summarizes the measured chaotic time series under different coupling strength κ. It can be observed that all time series vary intricately and don’t show directly TD signature. To identify the TD signature, various analysis functions can be employed [20

20. 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–R3085 (1996). [CrossRef] [PubMed]

34

34. 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]

]. Here, the self-correlation (SF) function is employed to retrieve TD signature from chaotic time series. The formula of SF can be formed as [28

28. 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]

31

31. 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]

]:
C(Δt)=(P(t+Δt)P(t))(P(t)P(t))(P(t)P(t)2P(t+Δt)P(t)2)1/2
(1)
where P (t) represents chaotic time series, Δt is the time shift, <·> denotes time average. The TD signature can be retrieved from the location of characteristic peak in SF curve. Furthermore, we introduce PE, derived from the information theory, as another method to identify TD signature. The PE owns some unique advantages such as simplicity, fast calculation and robustness to noise. Following formulas in Ref [32

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

34

34. 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]

], the calculation of PE is carried out by modifying the given time series {x(t), t = 1, ···, N} into D-dimensional vectors Xt = [x(t), x(t + τemd), ···, x(t + (D-1)τemd)], where τemd is the embedding delay time and D is the embedding dimension. Through changing the value of τemd and fixing the value of D, different PE values can be obtained and form a function curve. The value of PE will minimize when τemd is close to the characteristic time τ. Then, the TD signature can be retrieved by identifying the minima location in such PE curve. In this paper, the value of D is fixed as D = 6 after taking into account the suggestion in Ref [32

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

34

34. 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]

].

The second column of Fig. 2 shows the power spectra which could also be used to retrieve the TD signature by identifying the periodic ripple [41

41. 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]

]. Among these power spectra, it can be observed that there is apparent distribution variation for different κ. For κ≈0.0003, as shown in Figs. 2 (Aa2 and Ba2), the frequency components surround intensively a specific frequency 2.6GHz, which is the relaxation oscillation frequency fRO of SL [10

10. J. Ohtsubo, “Semiconductor Lasers: Stability, Instability and chaos,” Second Ed., Springer-Verlag, Berlin Heidelberg (2008).

]. When the coupling strength increases to κ≈0.023, the strongest frequency component of spectra moves to a much higher frequency (about 3.5GH) as given in Figs. 2 (Ad2 and Bd2). Furthermore, the third column of Fig. 2 shows the calculated SF curves of chaotic time series. As shown in Figs. 2 (Aa3-Ab3, Ad3-Ae3, Ba3-Bb3, and Bd3-Be3), clear peaks emerge around Δt≈ ± 53.55ns in SF curves. Meantime, in Figs. 2 (Aa4-Ab4, Ad4-Ae4, Ba4-Bb4, and Bd4-Be4), sharp valleys are observed around τemd≈53.55ns in PE curves. All the locations of such SF peaks and PE valleys match well with the characteristic time τ≈53.55ns, and reveal the TD signature contained in chaotic time series. But for the special case κ≈0.0068, as shown in Figs. 2 (Ac3-Ac4, Bc3-Bc4), no significant SF peaks or PE valleys are observed around the characteristic time τ≈53.55ns. Almost perfect δ function profile in the SF curves indicates that the TDSS happens in such a MDC-SLs system. Based on the unique parallel scheme in MDC-SLs system, two sets of chaotic signals with weak TD signatures can be obtained simultaneously.

3.2 Influence of frequency detuning on TD signature

Since the oscillation frequency of SL is sensitive to temperature fluctuation, it is valuable to investigate the influences of frequency mismatch (labeled as ∆f) between SL1 and SL2 on the TD signature. Figure 3
Fig. 3 Recorded chaotic time series (the first column), associated power spectra (the second column), SF curves (the third column) and PE curves (the forth column) under fixed κ≈0.008 and different frequency detuning ∆f≈9.0GHz (Aa1-Aa4, Ba1-Ba4), ∆f≈5.7GHz (Ab1-Ab4, Bb1-Bb4), ∆f≈-1.1GHz (Ac1-Ac4, Bc1-Bc4), ∆f≈-5.3GHz (Ad1-Ad4, Bd1-Bd4) and ∆f≈-10.8GHz (Ae1-Ae4, Be1-Be4). A corresponds to SL1, and B is for SL2. The power spectra, SF curves and PE curves are calculated from the recorded chaotic time series with a 1000ns length.
collectively shows the recorded chaotic time series, power spectra, SF curves and PE curves when ∆f varies from 9GHz to −10.8GHz and κ is fixed as 0.008. At first, for ∆f≈9.0GHz as shown in Figs. 3(Aa3-Aa4, Ba3-Ba4), obvious TD signatures are observed in both SF curves and PE curves. Further adjusting ∆f to 5.7GHz, the TD signatures are significantly suppressed close to the background in SF curves and in PE curves as shown in Figs. 3(Ab3-Ab4, Bb3-Bb4). Moreover, the TDSS state can be well maintained for ∆f varies from 5.7GHz to −5.3GHz as given in Figs. 3(Ac3-Ac4, Ad3-Ad4, Bc3-Bc4, and Bd3-Bd4). However, when ∆f is tuned to −10.8GHz, the TDSS state is destroyed and TD signatures re-appear in SF curves and PE curves as shown in Figs. 3(Ae3-Ae4, Be3-Be4). Therefore, one can reasonably deduce that the TDSS state owns certain tolerance to the frequency detuning between SL1 and SL2 under specific coupling strength.

3.3 Evolution maps of TD signature in parameter space

3.4 Influence of injection current on TD signature

To explore the influence of injection current of SLs on TD signature, Fig. 5
Fig. 5 The evolution maps of TD signature under different injection currents of SLs. The first row: the injection current is 9.64mA (1.18Ith) for SL1 and 10.2mA (1.20Ith) for SL2. The second row: the injection current is 10.53mA (1.29Ith) for SL1 and 11.1mA (1.31Ith) for SL2. The third row: the injection current is 12.2mA (1.49Ith) for SL1 and 12.8mA (1.51Ith) for SL2. The different colors represent different strength of TD signature. A corresponds to SL1, and B is for SL2.
further show the recorded evolution maps of TD signature when the injection currents of SLs vary from about 1.2Ith to about 1.5Ith. From these maps, one can observe that the ‘V’ shape of TDSS is maintained for different injection currents but there exist some differences under different injection currents. The higher injection current level, the wider parameter scope covered by TDSS. However, for a relatively high injection current level, the TDSS effect is slightly worse than that for a relatively low injection current level since the darkest blue color in Figs. 5(A3 and B3) is slightly brighter than that in Figs. 5(A1 and B1). Therefore, a comprehensive consideration is necessary by taking into account various performance requirements. For the case that the requirement for the effect of TDSS is not high, high injection current of SLs can be chosen for conveniently hunting TDSS state in parameter space. On the contrary, low injection current of SLs should be adopted if the effect of TDSS is highly required. Moreover, if one observes these diagrams carefully, a mirror symmetry relationship can be found between the evolution maps of SL1 and SL2. That is to say, the left part of SL1’s maps is similar to the right part of SL2’s maps meanwhile the right part of SL1’s maps is similar to the left part of SL2’s maps.

4. Theoretical simulation and analysis

The MDC-SLs system can be modeled by the well known Lang–Kobayashi rate equations [13

13. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001). [CrossRef] [PubMed]

]. The slowly varying complex electric field E and the average carrier number N in the active region of SLs can be expressed as:
E˙1(t)=12(1+iβ1)[g1(N1(t)N0)1+εE1(t)21τp1]E1(t)+ηE2(tτ2)ei2π(f2τ2+Δft)+F1(t)
(2)
E˙2(t)=12(1+iβ2)[g2(N2(t)N0)1+εE2(t)21τp2]E2(t)+ηE1(tτ2)ei2π(f1τ2Δft)+F2(t)
(3)
N˙1,2(t)=JN1,2(t)/τNg1,2(N1,2(t)N0)1+εE1,2(t)2|E1,2(t)|2
(4)
where the subscripts 1, 2 stand for SL1 and SL2, respectively. η is the coupling rate and labels the coupling strength, τ is the coupling round-trip time, β is the linewidth-enhancement factor, g is differential gain coefficient, ε is the gain saturation coefficient and N0 is the transparency carrier number. f1 and f2 are the optical frequencies of free running SL1 and SL2, respectively. Δf is defined as Δf = f1-f2, where different Δf is obtained by varying f2 while f1 keeps a constant. τp is the photon lifetime, F(t) is the spontaneous-emission noise, J is the injection carrier rate, τN is the carrier lifetime. The relaxation oscillation period τRO of SLs could be estimated by τRO≈2π(gE2/τp)-1/2.

Above coupled nonlinear rate-Eqs. (2)-(4) can be numerically solved by using the fourth-order Runge-Kutta algorithm. During the calculations, the parameters are set as: β = 4, f1 = 1.9355 × e14Hz, τp = 4.2ps, τN = 1.6ns, g = 2 × 104s−1, N0 = 1.25 × 108, ε = 1.5 × 10−7 and τ = 8ns. Considering the weak influence of the noise on the TD signature, the spontaneous-emission noise F1(t) and F2(t) are set to zero for simplicity. J is set initially to 1.15Jth. Under these circumstances, τRO is estimated to be about 0.4ns. η, Δf and J are variable for different considerations. The initial conditions of SL1 and SL2 are also set different to accord with the real physical situation. Additionally, it should be pointed out that during experiments, the detection bandwidth is limited by the used digital oscilloscope whose bandwidth is 6GHz in our experimental system. However, the observed peak frequency in power spectra of the obtained chaotic signal (shown as the second column in Figs. 2 and 3) is always smaller than 6GHz of the detection bandwidth. After considering this fact, the finite detection bandwidth of the experimental system does not be taken into account during the simulations.

Next, we will investigate the influence of frequency detuning on the TD signature. Figure 7
Fig. 7 Combination of the simulated temporal traces of SLs (the first column), power spectra (the second column), SF curves (the third column) and PE curves (the forth column) under fixed η≈7.5ns−1 and different values of ∆f, where ∆f≈9.5GHz (Aa1-Aa4, Ba1-Ba4), ∆f≈7GHz (Ab1-Ab4, Bb1-Bb4), ∆f≈0GHz (Ac1-Ac4, Bc1-Bc4), ∆f≈-7GHz (Ad1-Ad4, Bd1-Bd4), ∆f≈-9.5GHz (Ae1-Ae4, Be1-Be4), respectively. A corresponds to SL1, and B is for SL2. The power spectra, SF curves and PE curves are calculated from the chaotic time series with a 1000ns length.
gives the simulated temporal traces, power spectra, SF curves and PE curves under fixed η≈7.5ns−1 and different values of ∆f. As shown in Figs. 7(Ab3- Ab4, Ac3- Ac4, Ad3- Ad4, Bb3- Bb4, Bc3- Bc4, and Bd3- Bd4), the TD signatures always keep at a low level for |∆f|<7GHz. The TD signatures will be obvious if |∆f| are relatively large as shown in Figs. 7 (Aa3-Aa4, Ae3-Ae4, Ba3-Ba4, and Be3-Be4). Furthermore, the transient temporal evolutions of SL are also given in the first column of Fig. 7. Through observing these temporal traces, one could find that obvious TD signatures are always companied with periodically temporal oscillation, while the TDSS is occurred for the case that the time series rapidly moves into complex oscillation.

5. Conclusions

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 60978003, 61078003, 11004161 and 61178011, the Natural Science Foundation of Chongqing City, the Fundamental Research Funds for the Central Universities under Grant No. XDJK2010C021, and the Open Fund of the State Key Lab of Millimeter Waves.

References and links

1.

G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]

2.

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]

3.

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]

4.

F.-Y. Lin and J.-M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004). [CrossRef]

5.

M. Peil, I. Fischer, W. Elsäßer, S. Bakić, N. Damaschke, C. Tropea, S. Stry, and J. Sacher, “Rainbow refractometry with a tailored incoherent semiconductor laser source,” Appl. Phys. Lett. 89(9), 091106 (2006). [CrossRef]

6.

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]

7.

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

8.

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]

9.

A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010). [CrossRef] [PubMed]

10.

J. Ohtsubo, “Semiconductor Lasers: Stability, Instability and chaos,” Second Ed., Springer-Verlag, Berlin Heidelberg (2008).

11.

R. Vicente, J. Dauden, 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]

12.

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]

13.

T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett. 86(5), 795–798 (2001). [CrossRef] [PubMed]

14.

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74, 046201 (2006). [CrossRef] [PubMed]

15.

R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett. 32(4), 403–405 (2007). [CrossRef] [PubMed]

16.

W. L. Zhang, W. Pan, B. Luo, X. H. Zou, M. Y. Wang, and Z. Zhou, “Chaos synchronization communication using extremely unsymmetrical bidirectional injections,” Opt. Lett. 33(3), 237–239 (2008). [CrossRef] [PubMed]

17.

J. F. Martinez Avila and J. R. Rios Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express 17(24), 21442–21451 (2009). [CrossRef] [PubMed]

18.

T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express 19(9), 8762–8773 (2011). [CrossRef] [PubMed]

19.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993). [CrossRef]

20.

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–R3085 (1996). [CrossRef] [PubMed]

21.

H. Voss and J. Kurths, “Reconstruction of nonlinear time-delayed feedback models from optical data,” Chaos Solitons Fractals 10, 805–809 (1999).

22.

A. C. Fowler and G. Kember, “Delay recognition in chaotic time series,” Phys. Lett. A 175(6), 402–408 (1993). [CrossRef]

23.

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]

24.

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]

25.

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]

26.

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]

27.

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(5), 373–377 (2005). [CrossRef]

28.

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]

29.

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]

30.

J. G. Wu, G. Q. Xia, X. Tang, X. D. Lin, T. Deng, L. Fan, and Z. M. Wu, “Time delay signature concealment of optical feedback induced chaos in an external cavity semiconductor laser,” Opt. Express 18(7), 6661–6666 (2010). [CrossRef] [PubMed]

31.

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]

32.

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

33.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(4), 046212 (2010). [CrossRef] [PubMed]

34.

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]

35.

C. Zhou and C. H. Lai, “Extracting messages masked by chaotic signals of time-delay systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60(1), 320–323 (1999). [CrossRef] [PubMed]

36.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption system ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003). [CrossRef]

37.

M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open and closed-loop receivers in all-optical chaos-based communications,” IEEE Photon. Technol. Lett. 21(7), 426–428 (2009). [CrossRef]

38.

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]

39.

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]

40.

E. M. Shahverdiev and K. A. Shore, “Erasure of time-delay signatures in the output of an opto-electronic feedback laser with modulated delays and chaos synchronisation,” IET Optoelectron. 3(6), 326–330 (2009). [CrossRef]

41.

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]

42.

R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011). [CrossRef] [PubMed]

OCIS Codes
(140.5960) Lasers and laser optics : Semiconductor lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 15, 2011
Revised Manuscript: December 25, 2011
Manuscript Accepted: January 2, 2012
Published: January 11, 2012

Citation
Jia-Gui Wu, Zheng-Mao Wu, Guang-Qiong Xia, and Guo-Ying Feng, "Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system," Opt. Express 20, 1741-1753 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1741


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References

  1. G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science279(5354), 1198–1200 (1998). [CrossRef] [PubMed]
  2. 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,” Nature438(7066), 343–346 (2005). [CrossRef] [PubMed]
  3. 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]
  4. F.-Y. Lin and J.-M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron.10(5), 991–997 (2004). [CrossRef]
  5. M. Peil, I. Fischer, W. Elsäßer, S. Bakić, N. Damaschke, C. Tropea, S. Stry, and J. Sacher, “Rainbow refractometry with a tailored incoherent semiconductor laser source,” Appl. Phys. Lett.89(9), 091106 (2006). [CrossRef]
  6. 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. Photonics2(12), 728–732 (2008). [CrossRef]
  7. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett.103(2), 024102 (2009). [CrossRef] [PubMed]
  8. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010). [CrossRef]
  9. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express18(18), 18763–18768 (2010). [CrossRef] [PubMed]
  10. J. Ohtsubo, “Semiconductor Lasers: Stability, Instability and chaos,” Second Ed., Springer-Verlag, Berlin Heidelberg (2008).
  11. R. Vicente, J. Dauden, 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]
  12. 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]
  13. T. Heil, I. Fischer, W. Elsässer, J. Mulet, and C. R. Mirasso, “Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers,” Phys. Rev. Lett.86(5), 795–798 (2001). [CrossRef] [PubMed]
  14. E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74, 046201 (2006). [CrossRef] [PubMed]
  15. R. Vicente, C. R. Mirasso, and I. Fischer, “Simultaneous bidirectional message transmission in a chaos-based communication scheme,” Opt. Lett.32(4), 403–405 (2007). [CrossRef] [PubMed]
  16. W. L. Zhang, W. Pan, B. Luo, X. H. Zou, M. Y. Wang, and Z. Zhou, “Chaos synchronization communication using extremely unsymmetrical bidirectional injections,” Opt. Lett.33(3), 237–239 (2008). [CrossRef] [PubMed]
  17. J. F. Martinez Avila and J. R. Rios Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express17(24), 21442–21451 (2009). [CrossRef] [PubMed]
  18. T. Deng, G. Q. Xia, Z. M. Wu, X. D. Lin, and J. G. Wu, “Chaos synchronization in mutually coupled semiconductor lasers with asymmetrical bias currents,” Opt. Express19(9), 8762–8773 (2011). [CrossRef] [PubMed]
  19. H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys.65(4), 1331–1392 (1993). [CrossRef]
  20. 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. Topics54(4), R3082–R3085 (1996). [CrossRef] [PubMed]
  21. H. Voss and J. Kurths, “Reconstruction of nonlinear time-delayed feedback models from optical data,” Chaos Solitons Fractals10, 805–809 (1999).
  22. A. C. Fowler and G. Kember, “Delay recognition in chaotic time series,” Phys. Lett. A175(6), 402–408 (1993). [CrossRef]
  23. M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D203(3-4), 209–223 (2005). [CrossRef]
  24. 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. Topics56(5), 5083–5089 (1997). [CrossRef]
  25. 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]
  26. 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 A351(1), 133–141 (2005). [CrossRef]
  27. 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(5), 373–377 (2005). [CrossRef]
  28. 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]
  29. 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]
  30. J. G. Wu, G. Q. Xia, X. Tang, X. D. Lin, T. Deng, L. Fan, and Z. M. Wu, “Time delay signature concealment of optical feedback induced chaos in an external cavity semiconductor laser,” Opt. Express18(7), 6661–6666 (2010). [CrossRef] [PubMed]
  31. 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. Express17(22), 20124–20133 (2009). [CrossRef] [PubMed]
  32. C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett.88(17), 174102 (2002). [CrossRef] [PubMed]
  33. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.82(4), 046212 (2010). [CrossRef] [PubMed]
  34. 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]
  35. C. Zhou and C. H. Lai, “Extracting messages masked by chaotic signals of time-delay systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics60(1), 320–323 (1999). [CrossRef] [PubMed]
  36. V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption system ruled by nonlinear time delay differential equations,” Phys. Lett. A308(1), 54–60 (2003). [CrossRef]
  37. M. C. Soriano, P. Colet, and C. R. Mirasso, “Security implications of open and closed-loop receivers in all-optical chaos-based communications,” IEEE Photon. Technol. Lett.21(7), 426–428 (2009). [CrossRef]
  38. 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]
  39. 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]
  40. E. M. Shahverdiev and K. A. Shore, “Erasure of time-delay signatures in the output of an opto-electronic feedback laser with modulated delays and chaos synchronisation,” IET Optoelectron.3(6), 326–330 (2009). [CrossRef]
  41. 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]
  42. R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett.36(22), 4332–4334 (2011). [CrossRef] [PubMed]

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