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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 22 — Oct. 26, 2009
  • pp: 20124–20133
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Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback

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


Optics Express, Vol. 17, Issue 22, pp. 20124-20133 (2009)
http://dx.doi.org/10.1364/OE.17.020124


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Abstract

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

© 2009 OSA

1. Introduction

In the last two decades, optical chaos has been studied extensively for its potential applications in secure communications, chaotic radar and lidar, fast physical random bit generation etc [1

1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005). [CrossRef]

7

7. 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]

]. Various frames such as optical feedback, optoelectronic feedback and optical injection, have been presented to produce chaotic output in semiconductor lasers (SLs) [1

1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005). [CrossRef]

10

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

]. Since high dimension broadband chaos can be relatively easily generated through external optical feedback (namely introducing time delay), optical feedback SL system has been served as one of good candidate source generators of high-speed long distance chaos secure communication [1

1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005). [CrossRef]

,9

9. 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]

,10

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

]. Generally, obvious time delay (TD) signature can be observed in typical single optical feedback (SOF) SL system [10

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

], which inevitably provides a possible clue to the chaos encryption attackers. Recently, some time series analysis techniques for delayed systems have been developed [11

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

,12

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

], and the reconstruction of the chaotic system may be realized based on the TD signature identification [13

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

]. As a result, the security of the chaotic encryption may be partly compromised. If TD signature could be shielded, it would be quite difficult to perform such an identified projection. Therefore, it is necessary to develop some strategies to weaken or pretend the TD signature for ensuring the system security.

2. Experimental setup

Figure 1
Fig. 1 Experimental setup. Laser: DFB semiconductor laser; AL: aspheric lens; BS: beam splitter; M: mirror; OI: optical isolator; NDF: neutral density filter; FC: fiber coupler; PD: photo detector; T: test point of reflected optical power.
is the schematic of experimental setup. An InGaAsP/InP DFB-SL is used in experiment. It is biased at 20.08mA (about 1.61 times of threshold) by an ultra-low-noise current source (ILX-Lightwave, LDX-3620) and stabilized at 22.05°C by a thermoelectric controller (ILX-Lightwave, LDT-5412). Under these conditions, the solitary SL output is about 2.6mW and the lasing wavelength is about 1548.3nm. The SL output is collimated by a collimator lens. After that, the beam is split into three parts by two beam splitters (BS1 and BS2) and individually incident upon two reflecting mirror (M1, M2) and an optical isolator (OI, isolation>55dB). The two reflected beams by M1 and M2 are re-injected into SL and then form the DOF configuration. By the way, the DOF configuration can be easily transformed into SOF of cavity 1 (SOF1) or SOF of cavity 2 (SOF2) configurations just by hiding M2 or M1. In the signal detection part, the optical wavelength is measured by optical spectrum analyzer (Ando AQ6317C). The optical signal is transformed into electronic signal by a wide bandwidth photodetector (PD, New Focus 1544-B, bandwidth 12 GHz) and then analyzed by a 6GHz digital oscilloscope (Agilent 54855A, sample speed 20GS/s).

3. Experimental results and discussion

Figure 2
Fig. 2 Recorded chaotic time series and associated power spectra of (A) SOF1, (B) SOF2, and (C) DOF configuration.
compares the typical chaotic time series and corresponding power spectra under SOF and DOF configurations. During this experiment, the two external cavity lengths are taken closed values as lcav1≈477mm, lcav2≈461mm, respectively. The reflected optical power is monitored at point “T” (see Fig. 1) by an optical power meter and controlled by a variable optical attenuator, and is set as 9.4μw for cavity 1 and 15.3μw for cavity 2 in order to obtain almost equal feedback power ratios (FPR, defined as the ratio of the feedback light power to the laser output power) for SOF1 and SOF2. Above operation conditions can assure that the SL under SOF and DOF cases can all be rendered into chaotic states. For the SOF case, both the chaotic time series of SOF1 and SOF2 as shown in Fig. 2(A1) and Fig. 2(B1) behave intricately and irregularly. However, as observed from corresponding power spectra (Fig. 2(A2) and Fig. 2(B2)), some uniform spacing frequency peaks relevant to the TD signatures obviously emerge upon continuous spectrum background. Following these clues, the TD signature of the SOF-SL chaotic system may be retrieved [13

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

]. For the DOF case, the chaotic time series (see Fig. 2 (C1)) behave more irregularly than those for the SOF case. Meantime, the corresponding power spectrum (see Fig. 2 (C2)) becomes relatively smooth and flat, and no obvious regular peaks appear upon background.

Generally, numerous methods such as self-correlation function (SF) and mutual information (MI) [14

14. 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]

,20

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

], filling factor analysis [21

21. 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]

], global nonlinear models with neural networks [22

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

], local linear models [12

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

], can be used to identify the TD signature. In this paper, we adopt the SF and MI approaches. For a delay-differential system, the SF function can be defined as:
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. The TD signature can be retrieved from the peak location of SF curve. The MI between P (t) and P (t + Δt) is described by:
M(Δt)=P(t),P(t+Δt)δ(P(t),P(t+Δt))logδ(P(t),P(t+Δt))δ(P(t))δ(P(t+Δt))
(2)
Where δ(P(t), P(t + Δt)) is the joint probability, δ(P(t)) and δ(P(t + Δt)) are the marginal probability densities. The peak location of MI can also characterize the TD signature.

The corresponding SF and MI curves of Fig. 2(A1)-(C1) are shown in Fig. 3
Fig. 3 SF (A1-C1) and MI (A2-C2) corresponding to chaotic time series of Fig. 2 (A1-C1).
, respectively. For the SOF case, the external cavity time (τcav1≈3.25ns and τcav2≈3.15ns) can be clearly extracted from the peaks location of SF and MI curves in Fig. 3(A)-(B). However, for the DOF case as shown in Fig. 3(C1) and Fig. 3(C2), both SF and MI curves show no significant peaks at τcav1 and τcav2, and all the TD signatures have almost been depressed into background. Therefore, such a DOF configuration can afford a possible TD signature concealment scheme under certain conditions. In addition, there are some small troughs with a period of 0.2ns in these SF curves, which characterizes the relaxation oscillation period (τRO) of the SL as τRO≈0.2ns [14

14. 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]

].

Above experimental results show that such a DOF configuration can effectively suppress the TD signature of a chaotic system. In the following, based on the operation conditions to obtain Fig. 2, we will further investigate the influence of the external cavity lengths and feedback powers on such TDSS performances in the DOF-SL system. Figures 4
Fig. 4 Evolution of chaotic time series (1st Column), power spectra (2nd Column), SF (3rd Column) and MI (4th Column) under a fixed feedback strength of cavity 1 (FPR≈0.3%) and different feedback strengths of SOF2 for two cavity lengths are about lcav1≈477mm and lcav2≈461mm, respectively, where the feedback optical powers of cavity 2 are 7.83μw (FPR≈0.15%) (Row A), 11.1μw (FPR≈0.22%) (Row B), 14.3μw (FPR≈0.28%) (Row C), 15.3μw (FPR≈0.3%) (Row D) and 16.4μw (FPR≈0.32%) (Row E), respectively.
(A1)-(E1) are experimentally recorded chaotic time series corresponding to different feedback optical powers of SOF2 while the feedback level of SOF1 keeps unchanged. Combining the according power spectra, SF and MI, one can observe that the TD signature gradually attenuates when the monitored feedback optical power of SOF2 increases from 7.83μw to 16.4μw (corresponding FPR varies from 0.15% to 0.32%). In particular, the TD signature can be completely concealed as shown in Figs. 4(D) for the feedback power of SOF2 is 15.3μw, where both two cavities have almost equal FPR (about 0.3%). Under this circumstance, an eavesdropper would be quite difficult to accurately identify the TD signature. Further increasing FPR of cavity 2 to 0.32% as shown in Fig. 4(E), the TD signature arises again.

Figure 5
Fig. 5 Chaotic time series (A1-C1), associated power spectra (A2-C2), SF (A3-C3) and MI (A4-C4) for SOF 1 (Row A), SOF 2 (Row B) and DOF configuration (Row C).
further presents another type of TDSS in DOF-SL system, where the length of cavity 2 is shortened to 241mm (about 1/2 of length of cavity 1) and according feedback optical power is raised to 113μw (FPR≈2.2%) and the operation parameters of cavity 1 keep unchanged. Increasing feedback optical power of cavity 2 is mainly because more feedback is needed to obtain similar chaotic state as Fig. 2(A) when the length of cavity 2 is shortened [23

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

]. Columns (A), (B) and (C) of this diagram correspond to SOF1, SOF2 and DOF configuration, respectively. Interestingly, the measured frequency interval under the DOF configuration (Fig. 5 (C2)) is very close to that under the SOF2 (Fig. 5(B2)). For the DOF case, the SF (Fig. 5(C3)) and MI (Fig. 5 (C4)) curves show apparent envelope peaks at Δt≈1.7ns, which is approximately the TD signature of cavity 2. However, the time peak at Δt≈3.25ns, which corresponds to the TD signature of cavity 1, has been significantly attenuated. This feature is quit different from that in Fig. 4(D), where all TD signatures are attenuated seriously. This phenomenon may provide a pseudo TD signature chaos generation scheme, and an eavesdropper would be confused whether the chaos carrier is produced from a SOF-SL system or a DOF-SL system.

Furthermore, we will investigate the influence of the feedback power on the second type TDSS corresponding to Fig. 5. Figure 6
Fig. 6 Evolution of chaotic time series (1st Column), power spectra (2nd Column), SF (3rd Column) and MI (4th Column) under a fixed feedback strength of cavity 1 (FPR≈0.3%) and different feedback strengths of cavity 2 for two cavity lengths are about lcav1≈477mm and lcav2≈241mm, respectively, where the feedback powers of cavity 2 are 59μw (FPR≈1.15%) (Row A), 77μw (FPR≈1.5%) (Row B), 80μw (FPR≈1.56%) (Row C), 113μw (FPR≈2.2%) (Row D) and 135μw (FPR≈2.63%) (Row E), respectively.
shows chaotic time series, power spectra, SF and MI for different feedback powers of cavity 2 while the feedback power of cavity 1 keeps unchanged. When monitored feedback power of cavity 2 is about 59μw (FPR≈1.15%) as shown in the first row of Fig. 6, TD signature of cavity 1 appears distinctively while the TD signature of cavity 2 is weak. Increasing the feedback power of cavity 2 from 59μw to 113μw (corresponding FPR varies from 1.15% to 2.2%), the TD signature of cavity 1 gradually attenuates, but TD signature Δt≈1.7ns of cavity 2 enhances significantly. Particularly, for 2.2% FPR as shown in Fig. 6 (D), the TD signature of cavity 1 is suppressed seriously. Further increasing FPR to 2.63% as Fig. 6 (E), the TD signature of cavity 1 recovers again.

4. Theoretical simulation and analysis

The DOF-SL system can be modeled by well known Lang–Kobayashi rate equations. The slowly varying complex electric field E and the average carrier number N in the active region, can be expressed as:
E˙(t)=12(1+iβ)(G(t)γp)E(t)+κcav1/τLE(tτcav1)ei2πω0τcav1+κcav2/τLE(tτcav2)ei2πω0τcav2+F(t)N˙(t)=JN(t)/τNG(t)|E(t)|2
(3)
where κcav1 and κcav1 denote the feedback strengths of external cavity 1 and external cavity 2, τcav1 andτcav2 are the delay times corresponding to the round-trip time of the external cavity 1 and the external cavity 2, β is the linewidth-enhancement factor. G(t) = g(N(t)−N0)/(1 + εE(t)2) is the gain coefficient, where g is differential gain coefficient, ε is the gain saturation coefficient and N0 is the transparency carrier number. ω0 is the angular frequency of SL, γp is the photon loss rate. γp = 1/τp, where τp is the photon lifetime. τL is the round-trip time in SL intracavity, F(t) is the spontaneous-emission noise, J is the injection carrier rate, τN is the carrier lifetime. The relaxation oscillation period τRO is an intrinsic damping time of the free-running laser and could be estimated by τRO≈2π(gE 2/τp)-1/2. Equation (3) can be solved by using the fourth-order Runge-Kutta algorithm, and the parameters are set as: β = 4, ω0 = 1.216 × e15rad/s, τp = 4.2ps, τL = 8.5ps, τN = 1.6ns, g = 2 × 104s−1, F(t) = 0, N0 = 1.25 × 108, and ε = 1 × 10−7. For comparison with experimental results, J is set as 1.6J th with τRO≈0.2ns, τcav1 is fixed as 3.2ns and κcav1 = 0.04. τcav2 and κcav2 are variable for different considerations.

Figures 8 (A)-(C)
Fig. 8 Simulated chaotic temporal waveforms (A1-C1), associated power spectra (A2-C2), SF (A3-C3) and MI (A4-C4) of the first type TDSS for SOF 1 (Row A), SOF 2 (Row B) and DOF configuration (Row C), where τcav1 = 3.2ns, τcav2 = 3.12ns, κcav1 = 0.04, κcav2 = 0.05.
simulated the chaotic temporal waveforms, and corresponding power spectra, SF and MI curves of the first type TDSS where τcav1 = 3.2ns, τcav2 = 3.12ns, κcav1 = 0.04, κcav2 = 0.05, respectively. For the SOF situation ((A) and (B)), clear TD peaks can be seen. But for the DOF configuration ((C)), the entire TD signature is attenuated significantly. Comparing this diagram with Figs. 2-3, the numerical calculations relatively successfully demonstrate the experimental results. Further calculations show that by reducing τcav2 and increasing κcav2 to suitable value, the second type TDSS as shown in Fig. 5 can also be simulated.

To show the influence of the external cavity delay time on TDSS, Fig. 9
Fig. 9 Calculated maps of TD signature for κcav1 = κcav2 = 0.04 and τcav1 = 3.2ns, where τcav2 varies from 0.4ns to 4ns. Here, different colors represent different SF function values.
integrates SF curves to form a map of TD signature distribution for τcav1 = 3.2ns and κcav1 = κcav2 = 0.04 and τcav2 varies from 0.4ns to 4ns. As shown in this diagram, the diagonal line Δt≈τcav2 and the vertical line Δt≈3.2ns reveal the TD signatures of cavity 2 and of cavity 1. Meantime, there exist numerous relatively weak peaks and valleys with a period of about 0.1ns≈1/2τRO. Two obvious twisted regions can be observed around τcav2≈1.6ns and τcav2≈3.2ns. For the region around τcav2≈3.2ns, the TD signatures become very weak at several separated sub-regions such as τcav2≈2.92ns, τcav2≈3.12ns, τcav2≈3.28ns and τcav2≈3.48ns, which indicates possible TDSS sub-regions. Interestingly, these TDSS sub-regions does not locate at the point where τcav2 equals to τcav1 strictly, but periodically distribute around τcav1 and roughly follow relations: τcav2-τcav1≈-3/2τRO, −1/2τRO, 1/2τRO, 3/2τRO, respectively. Here, we name them as type I TDSS. As for the region around τcav2≈1.6ns, there also appear some slight TDSS sub-regions around τcav2≈1.44ns, τcav2≈1.54ns, τcav2≈1.64ns and τcav2≈1.74ns, respectively, and these sub-regions roughly satisfy following relations: 2τcav2-τcav1≈-3/2τRO, −1/2τRO, 1/2τRO, 3/2τRO. Here, we name them as type II TDSS. Further calculations show that by selecting suitable feedback strength, the TDSS effect, which suppresses the TD signature of one external cavity and remains TD signature of another external cavity, can be strengthened. Above theoretical results reveal that there exist some special regions where two types of TDSS happens just as observed in our experiments.

5. Conclusion

In summary, chaotic dynamics of a DOF-SL system are both experimentally and numerically investigated, and the TD signatures of chaotic time series are retrieved and analyzed by using SF function and the MI technique. Two types of the TD signature suppression are demonstrated. One is that all TD signatures of two external cavities are depressed significantly and the corresponding power spectrum becomes smooth and flat when the two cavities have roughly equal lengths and feedback power ratios. The other scheme is that one of two external cavities TD signatures is attenuated seriously and another one is remained when one cavity length is about half of another cavity length. In addition, our experimental observations and simulations also illustrate that the intrinsic characteristic time τRO plays an important role during TDSS. We hope that this work will be helpful for enhancing security in chaotic cryptosystems. Also, this work may offer some useful insights for the nonlinear dynamical properties of such DOF-SL system.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 60978003 and the Open Fund of the State Key Lab of Millimeter Waves.

References and links

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A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005). [CrossRef]

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X. F. Wang, G. Q. Xia, and Z. M. Wu, “Theoretical investigations on the polarization performances of current-modulated VCSELs subject to weak optical feedback,” J. Opt. Soc. Am. B 26(1), 160–168 (2009). [CrossRef]

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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]

10.

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

11.

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

12.

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

13.

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

14.

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]

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F. R. Ruiz-Oliveras and A. N. Pisarchik, “Phase-locking phenomenon in a semiconductor laser with external cavities,” Opt. Express 14(26), 12859–12867 (2006). [CrossRef] [PubMed]

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F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external-cavity laser diode,” Opt. Lett. 25(11), 808–810 (2000). [CrossRef]

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20.

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

21.

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]

22.

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

23.

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

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: September 9, 2009
Revised Manuscript: October 15, 2009
Manuscript Accepted: October 16, 2009
Published: October 20, 2009

Citation
Jia-Gui Wu, Guang-Qiong Xia, and Zheng-Mao Wu, "Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback," Opt. Express 17, 20124-20133 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20124


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References

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