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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 11813–11829
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Experiment on synchronization of semiconductor lasers by common injection of constant-amplitude random-phase light

Hiroki Aida, Masaya Arahata, Haruka Okumura, Hayato Koizumi, Atsushi Uchida, Kazuyuki Yoshimura, Jun Muramatsu, and Peter Davis  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 11813-11829 (2012)
http://dx.doi.org/10.1364/OE.20.011813


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Abstract

We experimentally and numerically observe the synchronization between two semiconductor lasers induced by common optical injection with constant-amplitude and random-phase modulation in configurations with and without optical feedback. Large cross correlation (~0.9) between the intensity oscillations of the two response lasers can be achieved although the correlation between the drive laser and either one of the two response lasers is very small (~0.2). High quality synchronization is achieved in the presence of optical feedback in response lasers with matched feedback phase offset. We investigate the dependence of synchronization on parameter values over wide parameter ranges.

© 2012 OSA

1. Introduction

A variety of physical systems exhibit oscillatory dynamics in the real world. Such systems are as diverse as electrical circuits, chemical reaction systems, and neuronal networks. It is well known that these systems can exhibit various types of synchronization phenomena [1

1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, 1984).

, 2

2. A. S. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, 2001).

]. Lasers are typical such oscillatory systems and exhibit various synchronization phenomena [3

3. J. Ohtsubo, Semiconductor Lasers - Stability, Instability and Chaos (Springer-Verlag, 2008).

, 4

4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

]. The synchronization of lasers has potential applications to secure communications, and many studies have been made on this issue [4

4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

7

7. A. Uchida, F. Rogister, J. Garcia-Ojalvo, and R. Roy, “Synchronization and communication with chaotic laser systems,” Progress in Optics, E. Wolf, ed. (Elsevier, 2005), Vol. 48, Chap. 5, pp. 203–341.

]. It is important to clarify the nature of synchronization phenomena in laser systems from the point of view of these applications.

Recently, it has been theoretically revealed that a common random input could give rise to synchronization between two uncoupled nonlinear dynamical systems [8

8. R. Toral, C. R. Mirasso, E. Hernandez-Garcia, and O. Piro, “Analytical and numerical studies of noise-induced synchronization of chaotic systems,” Chaos 11(3), 665–673 (2001). [CrossRef] [PubMed]

15

15. K. Yoshimura, J. Muramatsu, and P. Davis, “Conditions for common-noise-induced synchronization in time-delay systems,” Physica D 237(23), 3146–3152 (2008). [CrossRef]

]. This type of synchronization has been experimentally demonstrated in semiconductor lasers driven by common optical injection with chaotically fluctuating amplitude and phase [16

16. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. I. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers,” Opt. Express 15(7), 3974–3980 (2007). [CrossRef] [PubMed]

, 17

17. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]

]. In particular, it was shown that the output intensities of two lasers driven by a common signal from a chaotic drive laser can have irregularly fluctuating waveforms which are highly correlated with each other even when correlation between the output and injection lights is relatively low. In addition, the dynamics of semiconductor lasers subject to random signal has been numerically investigated [18

18. S. Wieczorek, “Stochastic bifurcation in noise-driven lasers and Hopf oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036209 (2009). [CrossRef] [PubMed]

, 19

19. S. Wieczorek and W. W. Chow, “Bifurcations and chaos in a semiconductor laser with coherent or noisy optical injection,” Opt. Commun. 282(12), 2367–2379 (2009). [CrossRef]

]. This phenomenon is expected to be useful for applications of hardware-oriented security systems. For example, recently, Buskila et al. have proposed a scheme for generating common random bit sequences between two users who share a common secret key [20

20. O. Buskila, A. Eyal, and M. Shtaif, “Secure communication in fiber optic systems via transmission of broad-band optical noise,” Opt. Express 16(5), 3383–3396 (2008). [CrossRef] [PubMed]

]. The scheme makes use of highly correlated outputs of two identical optical scramblers driven by common random light. The generated common random bit sequences can be used for secure communications between the users. We propose that it is possible to use lasers that are synchronized by injection of common random light as such scrambler devices. A secure key distribution scheme using correlated random bit sequences is known in the field of information theory [21

21. U. M. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory 39(3), 733–742 (1993). [CrossRef]

23

23. J. Muramatsu, K. Yoshimura, and P. Davis, “Information theoretic security based on bounded observability,” Lect. Notes Comput. Sci. 5973, 128–139 (2010). [CrossRef]

]. We showed that it is possible to implement this type of scheme for secure key distribution by using the common-random-light-induced synchronization phenomenon in lasers to generate the correlated random bits [24

24. K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012). [CrossRef] [PubMed]

].

In this paper, we experimentally study the nature of synchronization of two semiconductor lasers subject to a common CARP light in detail. We investigate the dependence of synchronization on the laser parameters such as the injection strength and the optical wavelength detuning. Moreover, we carry out numerical simulations to fully clarify the parameter dependence, complementing the experimental results.

2. Experimental setup

Figure 1
Fig. 1 Experimental setup for the synchronization of semiconductor lasers subject to a common constant-amplitude random-phase (CARP) light. Amp, electronic amplifier; ISO, optical isolator; PD, photodetector; SL, semiconductor laser.
shows our experimental setup for the synchronization of two semiconductor lasers by the injection of a common CARP light. We use three distributed-feedback (DFB) semiconductor lasers (optical wavelength 1547 nm). One laser is used for a Drive laser and the other two lasers are used for Response 1 and 2 lasers. The injection current and temperature of the semiconductor lasers are adjusted by a controller. The optical wavelength of the semiconductor lasers is precisely controlled by the temperature of the laser with a ratio of 0.097 nm/K. The resolution of the temperature control is 0.01 K. The lasing thresholds of injection current Ith are 10.57 mA (Drive), 9.38 mA (Response 1), and 9.49 mA (Response 2), respectively. In our experiment, we used polarization maintaining optical fibers for all the fiber components. Therefore, linear polarization is maintained and active light polarization control is not necessary.

The light from the Drive laser passes unidirectionally through an optical isolator (ISO) and a phase modulator (PM). The optical phase of the Drive laser light is randomly modulated by the PM with a random signal from an electric noise generator (Noisecom, UFX7110), and CARP light is generated. The CARP light from the Drive laser is divided by a fiber coupler. One light is injected into an optical isolator and a photodetector (PD). The other light is divided by another fiber coupler and injected into the Response 1 and 2 lasers unidirectionally through an optical isolator. The light power is adjusted by using an attenuator. The Response 1 and 2 lasers are set to have as similar parameter values as possible. The lights from the two Response lasers are injected into PDs through fiber couplers and converted into electric signals. The electric signals are amplified by electric amplifiers (Amp), connected to a digital oscilloscope (Tektronix, DPO71604B, 16 GHz bandwidth, 50 GigaSamples/s) and a radio-frequency (RF) spectrum analyzer (Agilent, N9010A-526, 26.5 GHz bandwidth) to observe temporal waveforms and RF spectra, respectively. The optical spectra are observed by using an optical spectrum analyzer (Advantest, Q8384).

3. Experimental results for open-loop configuration

3.1 Common-signal-induced synchronization

First, we used the open-loop configuration, where no optical feedback was introduced to the two Response lasers. We set the relaxation oscillation frequencies of the Drive and Response lasers by adjusting the injection current of the lasers. The relaxation oscillation frequencies between the Response 1 and Response 2 lasers are matched at 2.0 GHz, whereas that betweenthe Drive (2.5 GHz) and Response lasers are mismatched. At this condition, the injection current is 14.00 mA (1.32 Ith) for the Drive, 12.30 mA (1.31 Ith) for the Response 1, and 12.68 mA (1.34 Ith) for the Response 2 lasers, respectively. The optical phase of the Drive laser is randomly modulated by the noise generator whose bandwidth is 1.5 GHz.

We set the optical wavelength of the Drive and Response lasers by adjusting the temperature of the lasers. Figure 2(a)
Fig. 2 Experimental result of optical spectra (a) without and (b) with optical injection from the Drive to Response lasers. Solid black curve: Drive laser, dashed red curve: Response 1 laser, dotted blue curve: Response 2 laser. The injection strengths from the Drive to Response 1 lasers and from the Drive to Response 2 lasers are 20.1 and 18.4 μW, respectively (the maximum values obtained in the experiment). The initial optical wavelength detunings between the Drive and Response 1 lasers and between the Drive and Response 2 lasers are ΔλR1D = −0.018 nm and ΔλR2D = −0.019 nm, respectively.
shows the optical spectra of the solitary three lasers without optical injection from the Drive laser. We set the optical wavelength of 1546.954 nm for the Drive, 1546.936 nm for the Response 1, and 1546.935 nm for the Response 2 lasers, respectively. The optical wavelength detuning between Drive and Response 1 is ΔλR1D = λResponse 1 - λDrive = −0.018 nm (−2.3 GHz) and that between Drive and Response 2 is ΔλR2D = λResponse 2 - λDrive = −0.019 nm (−2.4 GHz). Figure 2(b) shows the optical spectra of the three lasers in the presence of optical injection from the Drive laser to the Response 1 and 2 lasers. In this case with optical injection, the dominant peak in the optical spectra of each of the Response 1 and 2 lasers matches that of the Drive laser, 1546.954 nm. The matching is due to injection locking [3

3. J. Ohtsubo, Semiconductor Lasers - Stability, Instability and Chaos (Springer-Verlag, 2008).

, 4

4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

] of the main oscillation component of the Response laser. However the whole spectrum is not matched and in particular there is a second peak in the spectrum of the Response lasers, visible in Fig. 2(b).

Figure 3
Fig. 3 Experimental result of (a) temporal waveforms and (b) corresponding correlation plots for the outputs of the Drive and Response 1 lasers. The injection strengths and the initial optical wavelength detunings for the three lasers are the same as shown in Fig. 2. (b) The cross correlation value is C = 0.158.
shows the temporal waveforms and the correlation plot of the Drive and Response 1 in the case where there is optical injection from the Drive to the Response lasers under the same conditions as Fig. 2(b). As seen in Fig. 3(a), there is very little variation of the intensity of the Drive laser output because only phase modulation is applied to the Drive signal. The small fluctuation of the Drive intensity is due to the intrinsic relaxation oscillation fluctuations of the semiconductor laser used for the Drive laser. The temporal waveforms of the Drive and Response 1 lasers are very different, as can be seen in Fig. 3(a). The correlation plot of Fig. 3(b) also shows that the correlation is low between the Drive and Response laser intensities.

Figure 4
Fig. 4 Experimental result of (a) temporal waveforms and (b) corresponding correlation plots for the outputs of the Response 1 and Response 2 lasers. The injection strengths and the initial optical wavelength detunings for the three lasers are the same as shown in Fig. 2. (b) The cross correlation value is C = 0.929.
shows the temporal waveforms and the correlation plot of the Response 1 and 2 lasers under the same conditions as Figs. 2(b) and 3 - that is in the case where there is optical injection from the Drive to the Response lasers and the Response lasers experience injection locking. The temporal waveforms of the Response 1 and Response 2 lasers are almost the same fluctuation, indicating high-quality synchronization. Synchronization can be clearly seen in the correlation plot of Fig. 4(b).

We use the cross correlation coefficient to quantitatively evaluate the quality of synchronization. It is defined by
C=(I1(t)I¯1)(I2(t)I¯2)σ1σ2
(1)
where I1,I2 are temporal waveforms of the output intensities of Response 1 and 2 lasers, respectively, I¯1,I¯2 are their mean values, σ1,σ2 are their standard deviations, and the angle brackets denote time averaging. When the cross correlation value is 1, the best synchronization is obtained. The cross correlation value between the Drive and Response 1 lasers shown in Fig. 3(b) is 0.158 and low correlation is observed. On the other hand, the cross correlation value between the Response 1 and Response 2 lasers shown in Fig. 4(b) is 0.929 and high correlation is observed.

Figure 5
Fig. 5 Experimental result of RF spectra for (a) Drive, (b) Response 1, and (c) Response 2 lasers. The injection strengths and the initial optical wavelength detunings for the three lasers are the same as shown in Fig. 2.
shows the RF spectra of the Drive, Response 1, and Response 2 lasers. The RF spectra of the Response 1 and 2 (Figs. 5(b) and 5(c)) are very similar, whereas those of the Drive and Response 1 (Figs. 5(a) and 5(b)) are different. The peak frequencies of the Response 1 and Response 2 are 1.5 GHz and 5.3 GHz. The peak frequency of 1.5 GHz corresponds to the bandwidth of the noise signal used for random phase modulation. On the contrary, the peak frequency of 5.3 GHz roughly corresponds to the difference between the optical carrier frequency of the Drive laser and the secondary peak in the spectrum of the Response lasers with optical injection, as seen in Fig. 2(b) (see Section 3.3 for details). On the other hand, the peak of the RF spectrum of the Drive laser corresponds to its relaxation oscillation, whose frequency is 2.5 GHz.

These results confirm that high-quality synchronization of the output intensity signal between the Response 1 and Response 2 lasers is achieved. Thus we have experimentally confirmed that common-signal-induced synchronization in semiconductor lasers can be achieved with constant-amplitude random-phase (CARP) drive signal. We emphasize that the synchronization between the two Response lasers is achieved even though the correlation between the Drive and Response is very low. We also note that the synchronization between the two Response lasers was achieved even though the optical spectra of the Response contains components that do not exist in the Drive signal as seen in Fig. 2(b). The optical-carrier frequencies of the three lasers are matched by injection locking due to the optical injection from the drive to the response lasers. For strong injection strength, the two response lasers are synchronized by the common drive laser light, and the outputs of the response lasers are very similar. Therefore, the correlation between the two response lasers is high. In general, synchronization by injection locking has some correlation between the drive and response lasers. However, in our work we used constant-amplitude random-phase (CARP) light as a drive light, and phase fluctuation of the drive light is converted into intensity fluctuation inside the response lasers. Thus, new dynamics are generated in the response lasers, and the correlation between the drive and one of the response lasers is very low.

3.2 Parameter dependence of synchronization

We investigate the dependence of synchronization on laser parameter values. First, the change in the cross correlation is observed while changing the strength of the optical injection from the Drive to each of the Response lasers. Figure 6(a)
Fig. 6 Experimental results of the cross correlation between the Response 1 and Response 2 lasers (solid orange curve), between the Drive and Response 1 (solid black curve), and the optical wavelength detuning between the Drive and Response 1 lasers under optical injection (dashed blue curve) as a function of (a) the injection strength and (b) the initial optical wavelength detuning between the Drive and Response 1 lasers (ΔλRD). (a) The injection strength is normalized by the maximum injection strength obtained in the experiment. The initial optical wavelength detuning is fixed at −0.025 nm. (b) The injection strength is fixed at 1. The green dotted vertical lines indicate the injection locking range, where the average optical wavelengths of the Drive and Response lasers are matched due to optical injection.
shows the cross correlation values between the Response 1 and 2 lasers and between the Drive and Response 1 lasers as a function of the optical injection strength. As the injection strength is increased, the cross correlation between the Response 1 and 2 lasers becomes larger and reaches ~0.9. The common-signal-induced synchronization with CARP light is achieved with large optical injection strength. On the contrary, the cross correlation between the Drive and Response lasers remains at low values of ~0.2.

In Fig. 6(b), the cross correlation between the Response 1 and 2 lasers and that between the Drive and Response 1 lasers are shown as a function of the initial optical wavelength detuning, that is the detuning between the Drive and Response 1 lasers without optical injection (ΔλRD = λResponse - λDrive). When ΔλRD is increased from negative to positive values, the cross correlation increases rapidly at ΔλRD = −0.080 nm. The maximum cross correlation is obtained at ΔλRD = −0.030 nm, and the correlation value decreases as ΔλRD is increased further. The synchronization with large cross correlation is achieved in the range −0.080 nm < ΔλRD < 0.010 nm as shown in Fig. 6(b).

The blue dashed curves in Figs. 6(a) and 6(b) represent the average optical wavelength detuning between the Drive and Response 1 lasers with optical injection. The injection locking range can be defined as the region in which the absolute value of the optical wavelength detuning with optical injection is small. The green vertical lines in Fig. 6(b) show the range where the detuning is less than 0.005 nm with optical injection. The injection locking occurs for the injection strength larger than 0.08 in Fig. 6(a) and in the range −0.080 nm < ΔλRD < 0.010 nm in Fig. 6(b), respectively. It should be noted that these injection locking ranges coincide with the conditions for large cross correlation. Therefore, it can be concluded that synchronization with common CARP light occurs with injection locking.

3.3 Effect of optical wavelength detuning

We observe the RF spectra and the corresponding optical spectra for the Response 1 and 2 lasers when the initial optical wavelength detuning is set to 0.000 nm, as shown in Fig. 7
Fig. 7 Experimental result of (a) the RF spectra and (b) the corresponding optical spectra for the Response 1 and 2 lasers under optical injection from the Drive laser. The initial optical wavelength detuning is set to 0.000 nm. The difference between the first and second peaks of the optical spectra is 0.044 nm (5.5 GHz), corresponding to the peak frequency of the RF spectra. The maximum injection strength is used in the experiment.
. In the optical spectra of Fig. 7(b), there are two main peaks, corresponding to the optical wavelength of the Drive laser (the first main peak) and the wavelength of the Response laser, slightly shifted due to the optical injection (the second main peak). The difference between these two peaks corresponds to the component frequency at 5.5 GHz in the intensity modulation spectrum, as seen in Fig. 7(a). This fact indicates that synchronization of the output intensity signals of the Response lasers can be achieved even though there are large components in the optical spectrum of the Drive and Response lasers which do not exist in the optical spectrum of the Drive laser, since the value 0.000 nm of initial optical wavelength detuning in Fig. 7 is inside the synchronization region from Fig. 6(b).

Figure 8
Fig. 8 Experimental result of the difference between the first and second peak frequencies of the optical spectra of the Response laser 1 (open circles), and the peak frequency of the RF spectra (solid curve with solid dots), as a function of the initial optical wavelength detuning between the Drive and Response 1 lasers without optical injection. The difference between the two peak frequencies of the optical spectra cannot be measured in the range −0.080 nm < ΔλRD < −0.030 nm due to the limited resolution of the optical spectrum analyzer. The green dotted vertical lines indicate the injection locking range, where the average optical wavelengths of the Drive and Response lasers are matched due to optical injection.
shows the correspondence between the difference between the first and second peaks of the optical spectrum corresponds to the peak value of the RF spectra as the initial optical wavelength detuning is changed. The optical frequency difference matches very well with the peak value of the RF spectra both within and outside of the injection locking range.

4. Experimental results for closed-loop configuration

4.1 Common-signal-induced synchronization with various feedback phases

We observe common-signal-induced synchronization in the configuration where both of the Response lasers have optical feedback, the so-called closed-loop configuration [17

17. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]

, 27

27. R. Vicente, T. Pérez, and C. R. Mirasso, “Open-versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1197–1204 (2002). [CrossRef]

29

29. M. Peil, T. Heil, I. Fischer, and W. Elsäßer, “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. 88(17), 174101 (2002). [CrossRef] [PubMed]

], as shown in Fig. 1. Each of Response 1 and 2 has a variable fiber reflector to introduce optical feedback (see Fig. 1). The distance from the Response laser to the fiber reflector was set to 3.67 m (one-way) for both Response 1 and 2, corresponding to the feedback delay time (roundtrip) of 35.3 ns. The phase of the feedback light from the fiber reflector is modulated by an electro-optic phase modulator. We observe the temporal dynamics and correlation plots when the difference in the optical feedback phase for the two Response lasers is varied. The difference between the optical feedback phases in Response 1 and 2 is defined as Δθr1,r2 = θr1 – θr2.

Figure 9
Fig. 9 Experimental result of temporal waveforms and their correlation plots for Response 1and 2 in the closed-loop configuration at (a), (b) the parameter mismatching condition (Δθr1,r2 = π) and (c), (d) the parameter matching condition (Δθr1,r2 = 0), where Δθr1,r2 = θr1 – θr2 and θ is the optical feedback phase. The cross correlation values are (b) 0.002 and (d) 0.949. The injection strengths from the Drive to Response 1 lasers and from the Drive to Response 2 lasers are 20.3 and 18.8 μW, respectively (the maximum values obtained in the experiment). The feedback strengths for the Response 1 and 2 lasers are 1.96 and 1.87 μW, respectively (0.10 and 0.10, normalized by the injection strengths). The initial optical wavelength detunings between the Drive and Response 1 lasers and between the Drive and Response 2 lasers are ΔλR1D = −0.021 nm and ΔλR2D = −0.025 nm, respectively.
shows the temporal waveforms of Response 1 and 2 and their correlation plot for Δθr1,r2 = π (Figs. 9(a) and 9(b)) and Δθr1,r2 = 0 (Figs. 9(c) and 9(d)). For the condition of Δθr1,r2 = π, the temporal waveforms of the two Response lasers are dissimilar and no synchronization is observed as shown in Figs. 9(a) and 9(b). On the other hand, accurate synchronization is achieved between Response 1 and 2 when the optical feedback phases are matched (Δθr1,r2 = 0). The cross correlation values of Figs. 9(b) and 9(d) are 0.002 and 0.949, respectively. This result shows that synchronization by injection of a common CARP signal in semiconductor lasers with optical feedback is sensitive with respect to the optical phase of the feedback light in the two Response lasers, as in the case of injection of a common chaotic signal, as shown in [17

17. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]

].

For comparison, we observe the temporal waveforms of the Drive and Response 1 lasers and their correlation when the optical phase of the feedback light for Response 1 is changed. Figure 10
Fig. 10 Experimental result of temporal waveforms and their correlation plots for Drive and Response 1 in the closed-loop configuration at (a), (b) the parameter mismatching condition (Δθr1,r2 = π) and (c), (d) the parameter matching condition (Δθr1,r2 = 0), where Δθr1,r2 = θr1 – θr2 and θ is the optical feedback phase. The cross correlation values are (b) 0.124 and (d) 0.123. The injection strengths, the feedback strengths, and the initial optical wavelength detunings are the same as shown in Fig. 9.
shows the temporal waveforms of Drive and Response 1 and their correlation plot for Δθr1,r2 = π (Figs. 10(a) and 10(b)) and Δθr1,r2 = 0 (Figs. 10(c) and 10(d)). The cross correlation values of Figs. 10(b) and 10(d) are 0.124 and 0.123, respectively. These results confirm that low correlation is achieved between the Drive and Response lasers due to the CARP injection. In addition, the correlation value between the Drive and Response lasers is not dependent on the optical feedback phase of the Response laser.

4.2 Parameter dependence of common-signal-induced synchronization in the closed-loop configuration

We investigate the dependence of synchronization on laser parameter values in the closed-loop configuration. The cross correlation is measured when changing the strength of the optical injection from the Drive to each of the Response lasers. Figure 11(a)
Fig. 11 Experimental result of the cross correlation between the Response 1 and Response 2 lasers (solid orange curve), between the Drive and Response 1 (solid black curve), and the optical wavelength detuning between the Drive and Response 1 lasers under optical injection (dashed blue curve) as a function of (a) the injection strength and (b) the initial optical wavelength detuning between the Drive and Response 1 lasers (ΔλRD) in the closed-loop configuration. (a) The injection strength is normalized by the maximum injection strength obtained in the experiment. The initial optical wavelength detuning is fixed at −0.025 nm. (b) The injection strength is fixed at 1. The green dotted vertical lines indicate the injection locking range, where the average optical wavelengths of the Drive and Response lasers are matched due to optical injection.
shows the values of the cross correlations between the Response 1 and 2 lasers, between the Drive and Response 1 lasers, and the optical wavelength detuning between the Drive and Response 1 lasers under optical injection, as a function of the optical injection strength in the closed-loop configuration. As the injection strength is increased, the cross correlation between the Response 1 and 2 lasers becomes larger and reaches ~0.9. Comparing Fig. 11(a) with the case of the open-loop configuration of Fig. 6(a), we experimentally found that larger injection strength is required to achieve high correlation, due to the existence of the optical feedback in the Response lasers.

The cross correlation is measured when the initial optical wavelength detuning is changed in the closed-loop configuration. The solid orange curve of Fig. 11(b) shows the cross correlation between the Response 1 and 2 lasers as a function of the initial optical wavelength detuning. Comparing Fig. 11(b) with the case of the open-loop configuration of Fig. 6(b), the region for high correlation becomes narrow. This is due to the existence of optical feedback in the Response lasers, causing smaller injection locking range.

Based on the above experimental observation, we can conclude that synchronization region in the closed-loop configuration (Fig. 11) is smaller than that in the open-loop configuration (Fig. 6). In addition, Figs. 11(a) and 11(b) show that the region for high correlation (C ≈1) of the two response lasers is matched to the region for zero optical wavelength detuning with optical injection (ΔλRD with injection ≈0). Therefore, it is also concluded that the synchronization is accompanied by the injection locking, as in the case of open-loop configuration.

4.3 Effect of optical feedback phase

We investigate the cross correlation as the optical feedback phase difference is continuously varied. Figure 12(a)
Fig. 12 Experimental result of the cross correlation value between (a) Response 1 and 2, and between (b) Drive and Response 1, as a function of the difference in the optical feedback phase in the external cavity between Response 1 and 2 lasers at the maximum injection strength of the two Response lasers. (a) The cross correlation changes periodically as the optical feedback phase difference is changed. The period of the correlation curve is 2π. (b) The cross correlation values stay around 0.2 and do not show periodical change. The feedback strengths normalized by the optical injection strength are 0.09 and 0.11 for Response 1 and 2, respectively. The initial optical wavelength detunings between the Drive and Response 1 lasers and between the Drive and Response 2 lasers are ΔλR1D = −0.030 nm and ΔλR2D = −0.031 nm, respectively.
shows the cross correlation between Response 1 and 2 as a function of the optical phase difference between Response 1 and 2. The cross correlation changes periodically as the optical feedback phase difference is varied continuously. The period of the correlation curve is 2π in terms of the phase shift, corresponding to the optical wavelength of the semiconductor lasers. The maximum and minimum values of cross correlation are 0.935 and 0.013, respectively. On the contrary, Fig. 12(b) shows the cross correlation between Drive and Response 1 as a function of the optical feedback phase difference. The cross correlation value stays around 0.2 and does not show periodical change. This shows that the change in optical feedback phase has little influence on the correlation between Drive and Response 1.

Figure 13(b) shows the maximum and minimum values of cross correlation between Drive and Response 1 as a function of feedback strength of Response 1. Only small change in ΔC is observed at different optical feedback strengths, even though the values of maximum and minimum cross correlation gradually decrease as the feedback strength is increased. This result shows that the cross correlation between Drive and Response 1 is not sensitive to the optical feedback phase even at strong feedback strengths.

5. Numerical simulation

We carried out numerical simulations to verify our experimental observations and moreover to clarify parameter dependences of the synchronization phenomenon in detail. To model our experimental system in Fig. 1, we use the Lang-Kobayashi equation [30

30. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

] with the CARP injection term:
E˙j(t)=12(1+iα)GN(Nj(t)Nth)Ej(t)+κrτinEj(tτ)exp[iθj]+κinjτinE0exp[i(Δωjt+ϕ)],N˙j(t)=J1τsNj(t)GN(Nj(t)N0)|Ej(t)|2,
(2)
where Ej and Nj are the complex electric field and the carrier number density, respectively, and their index j (j = 1, 2) indicates Response j laser. τ is the delay time of the optical self feedback, κr is the feedback strength, and θj is the optical phase shift of the feedback light. The last term in the first equation represents the CARP injection, where E0 is a real constant, κinj is the injection strength, and ϕ represents the random phase modulation. The detuning parameter Δωj is defined by Δωj = ω0- ωj, where ω0 is the optical angular frequency of Drive laser and ωj is that of Response j laser. For later use, we introduce the optical wavelength detuning Δλ defined by Δλ = 2πc(1/ω1−1/ω0)≈2πcΔω102 = λ02Δω1/2πc, where c is the light speed in vacuum and λ0 = 1548 nm is the optical wavelength of Drive laser.

As for the random phase modulation ϕ(t), we assumed the Ornstein-Uhlenbeck process defined by the stochastic differential equation
ϕ.(t)=1τmϕ(t)+2τmσξ(t),
(3)
Where τm and σ are positive constants and ξ(t) is the normalized white Gaussian noise with the properties ξ(t)=0 and ξ(t)ξ(s)=δ(ts), where δ is Dirac's delta function and denotes the ensemble average. It can be shown that ϕ(t) has the properties ϕ(t)=0 and ϕ(t)ϕ(s)=σ2exp[|ts|/τm]. The latter property indicates that the correlation time and standard deviation of ϕ(t) are given by τm and σ, respectively.

In our numerical simulations, the following parameter values were used: α = 3, GN = 8.4 × 10−13 m3s−1, N0 = 1.4 × 1024 m−3, Nth = 2.018 × 1024 m−3, τin = 8.0 ps, τs = 2.04 ns, τ = 35.3 ns, J = 1.19Jth, where Jth = Nth/ τs is the threshold of the injection current. For this value of J, Response lasers have the relaxation oscillation frequency 2.0 GHz. We assumed a slight detuning (ω1- ω2)/ = 0.2 GHz between Response 1 and 2 lasers. As for the CARP light, we set as E0=0.29Jth/GN(NthN0) and σ=π/2. This value of E0 leads to Drive laser’s relaxation oscillation frequency of 2.5 GHz. The relaxation oscillation frequencies of Drive and Responses coincide with the values in our experiment. The other parameters κr, κinj, Δλ and τm were varied in the simulations. The numerical calculation method used here is described in [14

14. K. Yoshimura, I. Valiusaityte, and P. Davis, “Synchronization induced by common colored noise in limit cycle and chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026208 (2007). [CrossRef] [PubMed]

]. The integration time step Δt = 1 ps was used. As for the initial conditions, Ej(t) was randomly given over the time interval τt<0 for each response laser.

We measure the synchronization, using the correlation C defined by Eq. (1). It was found that the phase shifts θ1 and θ2 are important parameters, which significantly affect the degree of synchronization. They were set as θ1 = 0 and θ2 = (ω1- ω2)τ to maximize C. We will use these θ1 and θ2 in what follows, except for the calculation in Fig. 17. Figure 14
Fig. 14 Numerical result of the contour plot of the cross correlation C between Response 1 and 2 as a function of the optical wavelength detuning between Drive and Response (Δλ) and the injection strength from Drive to Response (κinj). Paremeters are κr = 0.05 and τm = 1 ns.
shows contour plot of C as a function of (Δλ,κinj) for κr = 0.05 and τm = 1 ns. The condition C>0.8 is satisfied inside the wedge-shaped region bounded by red line. It was observed that C is very close to unity over most part of this region. Hereafter, we use C>0.8 as the criterion for synchronization. Figure 14 indicates that synchronization by common CARP injection is possible in a wide wedge-shaped parameter region. The synchronization region is asymmetric with respect to the line Δλ = 0: the region is shifted to the negative wavelength detuning side due to the α-parameter of semiconductor lasers [3

3. J. Ohtsubo, Semiconductor Lasers - Stability, Instability and Chaos (Springer-Verlag, 2008).

, 4

4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

]. This feature is in agreement with the experimental result in Fig. 11(b).

Figure 15
Fig. 15 Numerical result for synchronization region with C>0.8 in (Δλ,κinj) plane for τm = 10 ns (solid line), τm = 1 ns (dashed line), and τm = 0.1 ns (dotted line), where κr = 0.05.
shows how the synchronization region depends on the time scale τm of random phase modulation in the CARP light. The regions for C>0.8 are shown in (Δλ,κinj) plane for different values of τm. This result indicates that the synchronization is robust against a change in τm and it is possible over a wide range of τm values. The shape of the synchronization region slightly depends on τm: for relatively large κinj, the range of Δλ for synchronization becomes larger as τm decreases. The time scale of random phase modulation in our experiments is fixed to the order of τm = 1 ns, corresponding to the bandwidth 1.5 GHz of the noise generator. However, the numerical result in Fig. 15 indicates that the synchronization is still possible for the random phase modulation with much larger or smaller bandwidth. Hereafter, we use τm = 1 ns for our simulations.

We examine the effects of the feedback strength κr. Figure 16
Fig. 16 Numerical result for synchronization region with C>0.8 in (Δλ,κinj) plane for κr = 0 (solid line), 0.05 (dashed line), 0.1 (dotted line), and 0.2 (dash-dotted line), where τm = 1 ns.
shows the synchronization regions in (Δλ,κinj) plane for different values of κr, where τm = 1 ns. The contour lines of C = 0.8 are shown for four different values of κr. The synchronization region becomes smaller as the feedback strength κr increases: it is necessary to supply stronger injection light to achieve the synchronization for larger κr. This result coincides with the experimental observations in Figs. 6 and 11.

In some applications to secure communications, it is important to switch synchronized and desynchronized states by changing parameters of Response lasers [4

4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

, 24

24. K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012). [CrossRef] [PubMed]

, 31

31. T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, “ON/OFF phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1162–1170 (2002). [CrossRef]

]. So, we discuss effects of the phase shift parameter θj on the cross correlation C. As for the dependence of C on Δθ = θ2-θ1, it was found that C takes a maximum Cmax and a minimum Cmin at Δθ = (ω1- ω2)τ and Δθ = (ω1- ω2)τ + π, respectively. We introduce the quantity ΔC = Cmax-Cmin, which measures the quality of synchronization-desynchronization switching: the switching is well achieved when ΔC is close to unity. Figure 17
Fig. 17 Numerical result of the contour plot of the cross correlation difference ΔC ( = Cmax - Cmin) between Response 1 and 2 as a function of (κrinj), where Δλ = −0.025 nm and τm = 1 ns. Synchronization between Response 1 and 2 is achieved in the region above the dashed curve.
shows a contour plot of ΔC as a function of (κr,κinj), where Δλ = −0.025 nm and τm = 1 ns. The boundary of the synchronization region, above which C>0.8 holds, is also shown by a gray dashed line. It is clearly observed that ΔC strongly depends on κr and κinj. A region of large ΔC, for example ΔC >0.8, forms an elongated region appearing just above the boundary curve of synchronization region. It should be noted that the ratio κinj/κr is found to be around 2.5 in this region of large ΔC. This indicates that ΔC takes a large value when κinj and κr balance with each other in an appropriate ratio. This fact can be qualitatively explained as follows. The injection light dominantly affects the output and CmaxCmin holds when κr is relatively smaller than κinj. On the other hand, chaotic dynamics of the laser induced by the feedback light prevents the synchronization and Cmax≈0 holds when κr is relatively larger than κinj. Thus, ΔC becomes large when κinj and κr are balanced. In particular, Fig. 17 shows that if κr is increased for a fixed value of κinj, then ΔC increases up to a certain value of κr and then decreases. This numerical result and the above qualitative explanation are consistent with the experimental results in Fig. 13(a). Indeed, it is clearly observed in Fig. 13(a) that CmaxCmin holds for small κr, Cmax≈0 for large κr, and thus ΔC takes a maximum at an intermediate value of κr.

The optical phase of the CARP light is randomly modulated at 1.5 GHz in our experiments, which is much slower than the optical carrier frequency. Therefore, the slowly varying envelope approximation is reasonable and the Lang-Kobayashi equation is valid in our simulation. In addition, our experimental and numerical results are well matched, as shown in Figs. 11(b) and 14. This is evidence for the validity of our simulation.

In addition, we have also performed numerical simulations of the Lang-Kobayashi equations with a Langevin noise term that represents spontaneous emission noise. We found that the result was almost unchanged from the results with just random initial conditions. This is due to the robustness of the synchronization dynamics. Therefore, the spontaneous emission term is not essential in our simulation.

6. Conclusions

We have experimentally investigated common-signal-induced synchronization with a constant-amplitude random-phase (CARP) drive signal in semiconductor lasers. It was found that common-signal-induced synchronization is achieved under the condition of the optical wavelength matching by injection locking between the Drive and two Response lasers. The cross correlation between the Drive and Response 1 lasers is small (~0.2), while the cross correlation between the two Response lasers is large (~0.9). This is a significant characteristic of synchronization by CARP drive signal. We have also investigated the dependence of synchronization on laser parameter values. The parameter region for synchronization becomes smaller as the optical feedback strength increases in the case of Response lasers with optical feedback, the so-called closed loop configuration. The quality of synchronization depends largely on the optical feedback phase. It was found that the best synchronization-desynchronization switching by using this phase parameter can be achieved when the optical injection and feedback strengths balance with each other. Numerical simulation confirms the experimental observations.

Acknowledgments

We would like to thank Rajarshi Roy and Sebastian Wieczorek for fruitful discussions and comments. We acknowledge support from Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology in Japan and TEPCO Research Foundation. We also would like to thank Naonori Ueda, Eisaku Maeda, Junji Yamato, and Atsushi Nakamura at NTT Communication Science Laboratories for their continuous encouragement and support.

References and links

1.

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, 1984).

2.

A. S. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, 2001).

3.

J. Ohtsubo, Semiconductor Lasers - Stability, Instability and Chaos (Springer-Verlag, 2008).

4.

A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).

5.

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

6.

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

7.

A. Uchida, F. Rogister, J. Garcia-Ojalvo, and R. Roy, “Synchronization and communication with chaotic laser systems,” Progress in Optics, E. Wolf, ed. (Elsevier, 2005), Vol. 48, Chap. 5, pp. 203–341.

8.

R. Toral, C. R. Mirasso, E. Hernandez-Garcia, and O. Piro, “Analytical and numerical studies of noise-induced synchronization of chaotic systems,” Chaos 11(3), 665–673 (2001). [CrossRef] [PubMed]

9.

C. Zhou and J. Kurths, “Noise-induced phase synchronization and synchronization transitions in chaotic oscillators,” Phys. Rev. Lett. 88(23), 230602 (2002). [CrossRef] [PubMed]

10.

J. N. Teramae and D. Tanaka, “Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators,” Phys. Rev. Lett. 93(20), 204103 (2004). [CrossRef] [PubMed]

11.

D. S. Goldobin and A. Pikovsky, “Synchronization of self-sustained oscillators by common white noise,” Physica A 351(1), 126–132 (2005). [CrossRef]

12.

H. Nakao, K. Arai, and Y. Kawamura, “Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators,” Phys. Rev. Lett. 98(18), 184101 (2007). [CrossRef] [PubMed]

13.

K. Yoshimura, P. Davis, and A. Uchida, “Invariance of frequency difference in nonresonant entrainment of detuned oscillators induced by common white noise,” Prog. Theor. Phys. 120(4), 621–633 (2008). [CrossRef]

14.

K. Yoshimura, I. Valiusaityte, and P. Davis, “Synchronization induced by common colored noise in limit cycle and chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026208 (2007). [CrossRef] [PubMed]

15.

K. Yoshimura, J. Muramatsu, and P. Davis, “Conditions for common-noise-induced synchronization in time-delay systems,” Physica D 237(23), 3146–3152 (2008). [CrossRef]

16.

T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. I. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers,” Opt. Express 15(7), 3974–3980 (2007). [CrossRef] [PubMed]

17.

I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]

18.

S. Wieczorek, “Stochastic bifurcation in noise-driven lasers and Hopf oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036209 (2009). [CrossRef] [PubMed]

19.

S. Wieczorek and W. W. Chow, “Bifurcations and chaos in a semiconductor laser with coherent or noisy optical injection,” Opt. Commun. 282(12), 2367–2379 (2009). [CrossRef]

20.

O. Buskila, A. Eyal, and M. Shtaif, “Secure communication in fiber optic systems via transmission of broad-band optical noise,” Opt. Express 16(5), 3383–3396 (2008). [CrossRef] [PubMed]

21.

U. M. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory 39(3), 733–742 (1993). [CrossRef]

22.

J. Muramatsu, K. Yoshimura, K. Arai, and P. Davis, “Some results on secret key agreement using correlated sources,” NTT Tech. Rev. 6(2), 1–7 (2008).

23.

J. Muramatsu, K. Yoshimura, and P. Davis, “Information theoretic security based on bounded observability,” Lect. Notes Comput. Sci. 5973, 128–139 (2010). [CrossRef]

24.

K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012). [CrossRef] [PubMed]

25.

S. Goto, P. Davis, K. Yoshimura, and A. Uchida, “Synchronization of chaotic semiconductor lasers by optical injection with random phase modulation,” Opt. Quantum Electron. 41(3), 137–149 (2009). [CrossRef]

26.

K. Yoshimura, A. Uchida, P. Davis, J. Muramatsu, T. Harayama, and S. Sunada, “Synchronization of semiconductor lasers by common optical injection with constant-amplitude and random-phase modulation,” Rev. Laser Eng. 39, 520–524 (2011) (in Japanese).

27.

R. Vicente, T. Pérez, and C. R. Mirasso, “Open-versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1197–1204 (2002). [CrossRef]

28.

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]

29.

M. Peil, T. Heil, I. Fischer, and W. Elsäßer, “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. 88(17), 174101 (2002). [CrossRef] [PubMed]

30.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]

31.

T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, “ON/OFF phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1162–1170 (2002). [CrossRef]

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(140.1540) Lasers and laser optics : Chaos
(140.5960) Lasers and laser optics : Semiconductor lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 24, 2012
Revised Manuscript: April 13, 2012
Manuscript Accepted: May 6, 2012
Published: May 10, 2012

Citation
Hiroki Aida, Masaya Arahata, Haruka Okumura, Hayato Koizumi, Atsushi Uchida, Kazuyuki Yoshimura, Jun Muramatsu, and Peter Davis, "Experiment on synchronization of semiconductor lasers by common injection of constant-amplitude random-phase light," Opt. Express 20, 11813-11829 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11813


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References

  1. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, 1984).
  2. A. S. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, 2001).
  3. J. Ohtsubo, Semiconductor Lasers - Stability, Instability and Chaos (Springer-Verlag, 2008).
  4. A. Uchida, Optical Communication with Chaotic Lasers - Applications of Nonlinear Dynamics and Synchronization (Wiley-VCH, 2012).
  5. G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]
  6. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-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] [PubMed]
  7. A. Uchida, F. Rogister, J. Garcia-Ojalvo, and R. Roy, “Synchronization and communication with chaotic laser systems,” Progress in Optics, E. Wolf, ed. (Elsevier, 2005), Vol. 48, Chap. 5, pp. 203–341.
  8. R. Toral, C. R. Mirasso, E. Hernandez-Garcia, and O. Piro, “Analytical and numerical studies of noise-induced synchronization of chaotic systems,” Chaos 11(3), 665–673 (2001). [CrossRef] [PubMed]
  9. C. Zhou and J. Kurths, “Noise-induced phase synchronization and synchronization transitions in chaotic oscillators,” Phys. Rev. Lett. 88(23), 230602 (2002). [CrossRef] [PubMed]
  10. J. N. Teramae and D. Tanaka, “Robustness of the noise-induced phase synchronization in a general class of limit cycle oscillators,” Phys. Rev. Lett. 93(20), 204103 (2004). [CrossRef] [PubMed]
  11. D. S. Goldobin and A. Pikovsky, “Synchronization of self-sustained oscillators by common white noise,” Physica A 351(1), 126–132 (2005). [CrossRef]
  12. H. Nakao, K. Arai, and Y. Kawamura, “Noise-induced synchronization and clustering in ensembles of uncoupled limit-cycle oscillators,” Phys. Rev. Lett. 98(18), 184101 (2007). [CrossRef] [PubMed]
  13. K. Yoshimura, P. Davis, and A. Uchida, “Invariance of frequency difference in nonresonant entrainment of detuned oscillators induced by common white noise,” Prog. Theor. Phys. 120(4), 621–633 (2008). [CrossRef]
  14. K. Yoshimura, I. Valiusaityte, and P. Davis, “Synchronization induced by common colored noise in limit cycle and chaotic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026208 (2007). [CrossRef] [PubMed]
  15. K. Yoshimura, J. Muramatsu, and P. Davis, “Conditions for common-noise-induced synchronization in time-delay systems,” Physica D 237(23), 3146–3152 (2008). [CrossRef]
  16. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. I. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers,” Opt. Express 15(7), 3974–3980 (2007). [CrossRef] [PubMed]
  17. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17(12), 10025–10034 (2009). [CrossRef] [PubMed]
  18. S. Wieczorek, “Stochastic bifurcation in noise-driven lasers and Hopf oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036209 (2009). [CrossRef] [PubMed]
  19. S. Wieczorek and W. W. Chow, “Bifurcations and chaos in a semiconductor laser with coherent or noisy optical injection,” Opt. Commun. 282(12), 2367–2379 (2009). [CrossRef]
  20. O. Buskila, A. Eyal, and M. Shtaif, “Secure communication in fiber optic systems via transmission of broad-band optical noise,” Opt. Express 16(5), 3383–3396 (2008). [CrossRef] [PubMed]
  21. U. M. Maurer, “Secret key agreement by public discussion from common information,” IEEE Trans. Inf. Theory 39(3), 733–742 (1993). [CrossRef]
  22. J. Muramatsu, K. Yoshimura, K. Arai, and P. Davis, “Some results on secret key agreement using correlated sources,” NTT Tech. Rev. 6(2), 1–7 (2008).
  23. J. Muramatsu, K. Yoshimura, and P. Davis, “Information theoretic security based on bounded observability,” Lect. Notes Comput. Sci. 5973, 128–139 (2010). [CrossRef]
  24. K. Yoshimura, J. Muramatsu, P. Davis, T. Harayama, H. Okumura, S. Morikatsu, H. Aida, and A. Uchida, “Secure key distribution using correlated randomness in lasers driven by common random light,” Phys. Rev. Lett. 108(7), 070602 (2012). [CrossRef] [PubMed]
  25. S. Goto, P. Davis, K. Yoshimura, and A. Uchida, “Synchronization of chaotic semiconductor lasers by optical injection with random phase modulation,” Opt. Quantum Electron. 41(3), 137–149 (2009). [CrossRef]
  26. K. Yoshimura, A. Uchida, P. Davis, J. Muramatsu, T. Harayama, and S. Sunada, “Synchronization of semiconductor lasers by common optical injection with constant-amplitude and random-phase modulation,” Rev. Laser Eng. 39, 520–524 (2011) (in Japanese).
  27. R. Vicente, T. Pérez, and C. R. Mirasso, “Open-versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1197–1204 (2002). [CrossRef]
  28. 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]
  29. M. Peil, T. Heil, I. Fischer, and W. Elsäßer, “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. 88(17), 174101 (2002). [CrossRef] [PubMed]
  30. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
  31. T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, “ON/OFF phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 38(9), 1162–1170 (2002). [CrossRef]

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