## Evolution of time delay signature of chaos generated in a mutually delay-coupled semiconductor lasers system |

Optics Express, Vol. 20, Issue 2, pp. 1741-1753 (2012)

http://dx.doi.org/10.1364/OE.20.001741

Acrobat PDF (2528 KB)

### 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

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]

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. 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]

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]

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. 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]

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]

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. 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]

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

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. 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]

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]

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]

**2**(12), 728–732 (2008). [CrossRef]

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]

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. 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]

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. 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]

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]

*etc*., are still unclear in such a MDC-SLs system and should be further clarified.

## 2. Experimental setup

*∆f*is defined as

*∆f*=

*f*-

_{1}*f*, where

_{2}*f*and

_{1}*f*are the optical frequencies of free running SL1 and SL2, respectively. Different

_{2}*∆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. 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]

## 3. Experimental results

### 3.1Influence of coupling strength on TD signature

*I*,

_{th}*I*is the threshold current) and 9.7mA for SL2 (about 1.14

_{th}*I*). 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

_{th}*∆f*is about 0GHz. The fist column of Fig. 2 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. 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]

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. 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]

*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. 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]

*x*(t), t = 1, ···, N} into

*D*-dimensional vectors

*X*

_{t}= [

*x*(t),

*x*(t +

*τ*

_{emd}), ···,

*x*(t + (

*D*-1)

*τ*

_{emd})], where

*τ*

_{emd}is the embedding delay time and

*D*is the embedding dimension. Through changing the value of

*τ*

_{emd}and fixing the value of

*D*, different PE values can be obtained and form a function curve. The value of PE will minimize when

*τ*

_{emd}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]

**47**(2), 252–261 (2011). [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]

*f*of SL [10]. 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

_{RO}*Δt*≈ ± 53.55ns in SF curves. Meantime, in Figs. 2 (Aa4-Ab4, Ad4-Ae4, Ba4-Bb4, and Bd4-Be4), sharp valleys are observed around

*τ*

_{emd}≈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

*∆f*) between SL1 and SL2 on the TD signature. Figure 3 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

*I*to about 1.5

_{th}*I*. 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.

_{th}## 4. Theoretical simulation and analysis

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]

*E*and the average carrier number

*N*in the active region of SLs can be expressed as: 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

*N*is the transparency carrier number.

_{0}*f*and

_{1}*f*are the optical frequencies of free running SL1 and SL2, respectively.

_{2}*Δf*is defined as

*Δf*=

*f*-

_{1}*f*, where different

_{2}*Δf*is obtained by varying

*f*while

_{2}*f*keeps a constant.

_{1}*τ*is the photon lifetime,

_{p}*F*(t) is the spontaneous-emission noise,

*J*is the injection carrier rate,

*τ*is the carrier lifetime. The relaxation oscillation period

_{N}*τ*of SLs could be estimated by

_{RO}*τ*≈2π(

_{RO}*gE*

^{2}/

*τ*)

_{p}^{-1/2}.

*β*= 4,

*f*= 1.9355 × e

_{1}^{14}Hz,

*τ*= 4.2ps,

_{p}*τ*= 1.6ns,

_{N}*g*= 2 × 10

^{4}s

^{−1},

*N*= 1.25 × 10

_{0}^{8},

*ε*= 1.5 × 10

^{−7}and

*τ*= 8ns. Considering the weak influence of the noise on the TD signature, the spontaneous-emission noise

*F*(

_{1}*t*) and

*F*(

_{2}*t*) are set to zero for simplicity.

*J*is set initially to 1.15

*J*. Under these circumstances,

_{th}*τ*is estimated to be about 0.4ns.

_{RO}*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.

*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.

*η*and

*∆f*under different

*J*. As shown in this diagram, for low

*η*, the optimum scope of TDSS locates at the center part, such as the white dashed lines surrounded region in Figs. 8 (A1 and B1). But for relative high

*η*, the TDSS region splits into two branches. Thus, the ‘V’ shape of TDSS region is formed. Moreover, with the increase of injection current, the scope of TDSS gradually broadens while the ‘V’ shape is basically maintained. All these characteristics of simulated maps agree qualitatively with previous experimental results shown in Fig. 4 and Fig. 5. In addition, the mirror symmetry relationship can also be observed in Fig. 8. The reason about such symmetry could be found from the rate Eqs. (2-3), where there is a structural similarity between Eq. (2) and Eq. (3). The role of negative

*∆f*in Eq. (2) is similar to that of positive

*∆f*in Eq. (3). As a result, the left part (corresponding negative

*∆f*) of evolution maps of SL1 is similar to the right part (corresponding positive

*∆f*) of evolution maps of SL2. Likewise, the role of positive

*∆f*in Eq. (2) is similar to that of negative

*∆f*in Eq. (3). Then, the right part of evolution maps of SL1 is similar to the left part of evolution maps of SL2. Therefore, the mirror symmetry relationship is formed.

_{RO}, is desirable in an ECF-SL system [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]

_{RO}≈0.4ns).

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]

## 5. Conclusions

## Acknowledgments

## References and links

1. | G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science |

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 |

3. | F.-Y. Lin and J.-M. Liu, “Diverse waveform generation using semiconductor lasers for radar and microwave applications,” IEEE J. Quantum Electron. |

4. | F.-Y. Lin and J.-M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. |

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. |

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 |

7. | I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. |

8. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics |

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21. | H. Voss and J. Kurths, “Reconstruction of nonlinear time-delayed feedback models from optical data,” Chaos Solitons Fractals |

22. | A. C. Fowler and G. Kember, “Delay recognition in chaotic time series,” Phys. Lett. A |

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 |

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 |

25. | R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. |

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 |

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. |

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. |

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. |

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 |

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 |

32. | C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. |

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. |

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. |

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 |

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 |

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. |

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. |

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. |

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. |

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. |

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. |

**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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