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 Optical Society of America
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, 2]. Lasers are typical such oscillatory systems and exhibit various synchronization phenomena [3, 4]. The synchronization of lasers has potential applications to secure communications, and many studies have been made on this issue [4–7]. 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 [8–15]. This type of synchronization has been experimentally demonstrated in semiconductor lasers driven by common optical injection with chaotically fluctuating amplitude and phase [16, 17]. 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, 19]. 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 . 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 [21–23]. 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 .
For the above cryptographic applications of generating correlated random bits with synchronized response lasers, it is necessary to achieve low correlation between the driving light and the outputs of response lasers so as to prevent an eavesdropper from estimating the output waveform from the intensity variation of driving light. In our previous experiments [16, 17], however, the cross correlation between the drive and response lasers is not sufficiently small, being around 0.6, since the intensity fluctuation of the driving light strongly affects the output intensities of response lasers. In order to overcome this problem, the use of a driving light with constant-amplitude and random-phase (CARP) was proposed, and it has been numerically found that injection of common CARP light also can induce synchronization of two semiconductor lasers, i.e., the phase information is sufficient for the synchronization . Recently, experimental demonstration of the synchronization with common CARP light has been reported . In addition, conditions for the synchronization have been numerically investigated in detail . However, experimental study on the nature of synchronization with common CARP light is still lacking.
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 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) 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, 4] 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 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 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 byFig. 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 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) 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 . 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 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, 27–29], 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 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 .
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 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) 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) 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.
We next change both the optical feedback strengths of Response 1 and 2 simultaneously and investigate the cross correlation characteristics. Figure 13(a) shows the maximum and minimum values of cross correlation between Response 1 and 2 as a function of the feedback strength of Response 1 and 2, normalized by the optical injection strength. The maximum value of cross correlation is obtained at zero phase difference (Δθr1,r2 = 0), whereas the minimum value is observed at Δθr1,r2 = π. The maximum value of cross correlation stays almost constant at ~0.9 up to κr ≈0.25, and then it decreases with increasing κr. On the other hand, the minimum value of cross correlation rapidly decreases as κr increases. Therefore, the difference between maximum and minimum values of cross correlation (ΔC = Cmax - Cmin) increases up to a certain value of the feedback strength, and then it decreases: the maximum difference of ΔC = 0.931 is obtained at the intermediate feedback strength κr = 0.10. For large feedback strength, the injection locking breaks down, and the maximum correlation value, as well as ΔC, are small.
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  with the CARP injection term:
As for the random phase modulation ϕ(t), we assumed the Ornstein-Uhlenbeck process defined by the stochastic differential equation
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)/2π = 0.2 GHz between Response 1 and 2 lasers. As for the CARP light, we set as and . 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 . The integration time step Δt = 1 ps was used. As for the initial conditions, Ej(t) was randomly given over the time interval 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 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, 4]. This feature is in agreement with the experimental result in Fig. 11(b).
Figure 15 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 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, 24, 31]. 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 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 Cmax≈Cmin 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 Cmax≈Cmin 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.
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.
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).
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.
11. D. S. Goldobin and A. Pikovsky, “Synchronization of self-sustained oscillators by common white noise,” Physica A 351(1), 126–132 (2005). [CrossRef]
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]
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]
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]