1. Introduction

Synchronization of chaos in lasers has attracted much interest from the point of view of potential applications in communications, such as optical secure communications and spread-spectrum communications [1–5]. In previously well studied applications of synchronized laser chaos to secure communications, a chaotic carrier is used to hide a data signal. In this type of application, high-quality synchronization of chaos in the transmitter and receiver lasers is required to recover the hidden data. It is also possible that synchronized chaotic lasers could be used for generating correlated random signals to obtain secret communication keys which are secure in the sense of information theoretic security [6–8]. Optimally correlated sources for key generation and their key generation capacity under sampling attack have been analyzed in [7]. In this type of application it is necessary to achieve a strong correlation between the outputs of two remote chaotic lasers driven by one common driving signal, while the correlation between drive signal and response laser output should be undetectable so that an eavesdropper cannot predict the response signal even if she can receive the same driving signal.

From the physics point of view, this situation of one drive and two synchronized response systems is similar to the auxiliary system approach for detecting generalized synchronization [9] or noise-induced synchronization [10]. In this case the two response systems are strongly synchronized, however, the drive and response systems behave differently in time. Generalized synchronization has been experimentally observed in He-Ne gas lasers [11] by using the auxiliary system approach and in NH₃ gas lasers [12] and Nd:YVO₄ microchip solid-state lasers [13] by using repeatedly injected drive signal. Recently synchronization in three mutually-coupled lasers has been studied in semiconductor lasers [14] and fiber laser model [15]. However, there has been no report on generalized synchronization in semiconductor lasers in the auxiliary system configuration.

In this Letter we report on the synchronization in two response semiconductor lasers subject to a common chaotic signal from one drive semiconductor laser. The chaotic signal from the drive laser is optically injected into the two response lasers, which are not coupled. We show that over a significant parameter range the common chaotic drive can induce synchronization between the two responses even though the cross correlation between the drive and response signals is relatively low.

2. Experimental setup

Figure 1 shows our experimental setup for common-chaotic-signal induced synchronization. We used three distributed-feedback (DFB) semiconductor lasers (NTT Electronics, NLK1555CCA, the optical wavelength of 1547 nm) developed for optical fiber communications. The three semiconductor lasers were fabricated from the same wafer, so they have similar laser parameter values. One laser was used for a drive laser (called Drive) and the other two lasers were used for response lasers (called Response 1 and Response 2). The injection current and the temperature of the semiconductor lasers were adjusted by a current-temperature controller (Newport, 8000-OPT-41-41-41-41). The optical wavelength of the lasers was precisely controlled by the temperature of the laser with a ratio of 0.097 nm/K. The resolution of the temperature control was 0.01 K. The lasing thresholds of the injection current for solitary lasers I_th were 8.7 mA (Drive), 7.6 mA (Response 1), and 9.2 mA (Response 2), respectively.

An external mirror was placed in front of the Drive at a distance of 0.60 m, corresponding to a feedback delay time (roundtrip) of 4.0 ns. A portion of the laser beam from the Drive was fed back to the laser cavity of the Drive to induce chaotic fluctuation of laser output. The feedback power was adjusted by a neutral-density filter (a variable attenuator). The threshold reduction due to the optical feedback was 1.5 %. The laser beam from the Drive was divided into two beams by a beam splitter and one beam was transmitted to each of Response 1 and 2. Two optical isolators and two half wave plates were used to achieve one-way coupling. The laser beam was divided into two beams again by a cube-type beam splitter and the two beams were injected into Response 1 and 2, respectively. The distance between the Drive and each of the Response lasers was 1.20 m, corresponding to a coupling delay time of 4.0 ns. The laser power was 0.39 mW (Drive with optical feedback), 0.78 mW (Response 1), and 0.77 mW (Response 2) (measured in front of each laser). A portion of each laser output was extracted by a beam splitter, injected into a fiber collimator through an optical isolator and propagated through an optical fiber to be detected by a photodetector (New Focus, 1554-B, 12 GHz bandwidth). The converted electronic signal at the photodetector was amplified by an electronic amplifier (New Focus, 1422-LF, 20 GHz bandwidth) and sent to a digital oscilloscope (Tektronix, TDS7404B, 4 GHz bandwidth, 20 GigaSamples/s) and a radio-frequency (RF) spectrum analyzer (Advantest, R3172, 26.5 GHz bandwidth) to observe temporal dynamics and corresponding RF spectrum, respectively. The optical wavelength of the lasers was measured by an optical spectrum analyzer (Advantest, Q8384).

 

Fig. 1. Experimental setup for common-chaotic-signal induced synchronization in three semiconductor lasers. Amp, electronic amplifier; BS, beam splitter; FC, fiber collimator; ISO, optical isolator; L, lens; λ/2, half wave plate; M, mirror; NDF, neutral density filter; OSC, digital oscilloscope; PD, photodetector; SA, radio-frequency spectrum analyzer; SL, semiconductor laser.

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3. Results

3.1 Temporal waveforms and RF spectra

We tuned the relaxation oscillation frequencies of the lasers by adjusting the injection current of the lasers. We matched the relaxation oscillation frequency between Response 1 and 2 at 1.90 GHz, but mismatched it between the Drive (1.50 GHz) and the Response lasers. At this tuning condition, the injection currents were 9.93 mA (1.14 I_th) for the Drive, 10.86 mA (1.43 I_th) for Response 1, and 11.62 mA (1.26 I_th) for Response 2, respectively. We set the optical wavelengths of 1547.376 nm for the Drive and 1547.356 nm for both Response 1 and 2 by controlling the temperature of the lasers so that injection locking could be achieved between the Drive and the Response lasers. The optical wavelength detuning between the Drive and the Response lasers was set to -0.020 nm (-2.5 GHz). In this condition the optical frequency of the Response lasers was pulled into that of the Drive and all the three optical wavelengths were completely matched at 1547.376 nm under injection locking.

Figure 2 shows a comparison of temporal waveforms of Drive, Response 1, and Response 2, and their corresponding correlation plots. For Figs. 2(a)–2(d), we shifted the temporal waveforms of Response 1 and 2 with respect to Drive by the coupling delay time (4.0 ns) to obtain the optimal synchronization (a type of generalized synchronization), whereas there is no time shift for Figs. 2(e) and 2(f). Drive and Response 1 (or 2) are not well synchronized since the relaxation oscillation frequencies are mismatched (1.50 and 1.90 GHz, respectively). The faster oscillation component is observed in the temporal waveforms of Response 1 and 2 compared with Drive due to the faster relaxation oscillation frequencies, as shown in Figs. 2(a) and 2(c). The cross correlation values are relatively small, i.e., 0.711 for Fig. 2(b) and 0.659 for Fig. 2(d), respectively. On the contrary, Response 1 and 2 are strongly synchronized as shown in Figs 2(e) and 2(f). The cross correlation of Fig. 2(f) is 0.947, showing high-quality synchronization between Response 1 and 2. Therefore good synchronization is observed between Response 1 and 2, even though the two Responses are not themselves well synchronized with the commonly injected drive signal.

 

Fig. 2. Experimental result of temporal waveforms and corresponding correlation plots for (a), (b) Drive and Response 1, (c), (d) Drive and Response 2, and (e), (f) Response 1 and Response 2. The coupling delay time (4.0 ns) between the two temporal waveforms is compensated in (a)-(d). The cross correlation values are (b) 0.711, (d) 0.659, and (f) 0.947.

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Fig. 3. Experimental result of RF spectra for (a) Drive, (b) Response 1, and (c) Response 2.

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Fig. 4. Experimental result for the cross correlation between each pair of lasers as a function of (a) the relaxation oscillation frequency of the Response lasers and (b) the optical wavelength detuning between Drive and Response lasers. Solid black curve: Drive and Response 1, dotted blue curve: Drive and Response 2, dashed red curve: Response 1 and Response 2.

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Fig. 5. Numerical result of the cross correlation between two of three lasers as a function of (a) the relaxation oscillation frequency of two Responses and (b) the optical wavelength detuning between Drive and one of Responses. Solid black curve: Drive and Response 1, dotted blue curve: Drive and Response 2, dashed red curve: Response 1 and Response 2.

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Figure 3 shows the RF spectra of Drive, Response 1, and Response 2. The RF spectra of Response 1 and 2 look very similar, whereas those of Drive and Response 1 (or 2) are different. A peak of the high-frequency component (the shifted relaxation oscillation frequency due to the optical injection) is observed at around 3.5 GHz in both Response 1 and 2, but not in Drive. These results confirm that synchronization between Response 1 and 2 is strong, but synchronization between the Drive and each of the Response lasers is weak.

3.2 Parameter dependence of synchronization

We investigated the parameter dependence of common-chaotic-signal induced synchronization when one of the parameters in Response 1 and 2 was changed. We quantitatively define the accuracy of synchronization as the cross correlation of two temporal waveforms normalized by the product of their standard deviations [13]. Figure 4(a) shows the cross correlation for two temporal waveforms of the three lasers as a function of the relaxation oscillation frequencies of Response 1 and 2. The relaxation oscillation frequency of the Drive is fixed at 1.5 GHz. As the relaxation oscillation frequencies of two Responses are increased, the cross correlation between Response 1 and 2 [dashed red curve in Fig. 4(a)] is larger than that between the Drive and each of the Responses (solid black and dotted blue curves) over a wide range up to 2.4 GHz. This range is where strong synchronization is obtained between Response 1 and 2 and although the correlation is not large between the Drive and the Response lasers. Figure 4(b) shows the cross correlation between temporal waveforms for each pair of lasers as a function of the optical wavelength detuning Δλ = λ _Response - λ _Drive. The cross correlation between Response 1 and 2 (dashed red curve in Fig. 4(b)) is larger than that between the Drive and each of the Response lasers (solid black and dotted blue curves) in the injection locking range (-0.047 nm (-6.0 GHz) < Δλ < -0.010 nm (-1.25 GHz)).

To confirm our experimental observations we considered a numerical model for the synchronization of semiconductor lasers. We used three sets of the Lang-Kobayashi equations for semiconductor lasers subject to optical feedback and injection. We used the fourth-order Runge-Kutta method in our numerical calculation. We numerically obtained temporal waveforms, correlation plots, and RF spectra which were similar to our experimental results shown in Figs. 2 and 3. We also investigated the parameter dependence of the cross correlation when one of the Response parameters was changed. Figure 5 shows our numerical results for the cross correlation between pairs of temporal waveforms of the three lasers as a function of the relaxation oscillation frequency of Response 1 and 2 [Fig. 5(a)] and as a function of the optical wavelength detuning between the Drive and the Response lasers [Fig. 5(b)]. For both cases, the cross correlation between Response 1 and 2 (dashed red curve) is larger than that between the Drive and each of the Response lasers (solid black and dotted blue curves) over a wide parameter region. These results agree well with our experimental results shown in Fig. 4. The large difference in correlation values shown by these curves could be a useful property, for example, in secret key generation schemes which require two Responses to be synchronized with high accuracy while the correlation between Drive and Responses is as low as possible.

4. Conclusion

We have experimentally and numerically observed synchronization of two semiconductor lasers commonly driven by a chaotic semiconductor laser subject to optical feedback. The drive and response lasers are coupled unidirectionally, whereas there is no coupling between the two response lasers. Under the condition that the relaxation oscillation frequency is matched between the two response lasers, but mismatched between the drive and the two response lasers, we showed that it is possible to observe strongly correlated synchronization between the two response lasers even when the correlation between the drive and response lasers is low. We also investigated the parameter dependence of synchronization and showed that the cross correlation between the two responses is larger than that between drive and responses over a wide parameter region.