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We investigate the dynamics of two semiconductor lasers with separate optical feedback when they are driven by a common signal injected from a chaotic laser under the condition of non-identical drive and response. We experimentally and numerically show conditions under which the outputs of the two lasers can be highly correlated with each other even though the correlation with the drive signal is low. In particular, the effects of the phase of the feedback light on the correlation characteristics are described. The maximum correlation between the two response lasers is obtained when the phase of the feedback light is matched between the two response lasers, while the minimum correlation is observed when the difference in the optical phase is π. On the other hand, the correlation between the drive and response is not sensitive to the phase of the feedback light, unlike the previously studied case of identical drive and response. We numerically examine the difference between the maximum and minimum cross correlations over a wide range of parameters, and show that it is largest when there is a balance between the injection strength and the feedback strength.
©2009 Optical Society of America
1. Introduction
The synchronization of lasers in oscillating or chaotic states can occur with different mechanisms and characteristics. Unidirectional injection of signal from one laser (drive) into another (response) laser can result in either identical or generalized synchronization between the drive and response lasers [1–6]. Moreover, recently it has been shown that identical synchronization between two response lasers injected with a common drive signal can occur even when synchronization between drive and response lasers is not identical [7]. In this experiment, the response lasers are not chaotic in the absence of drive signal. However, it has not been known whether this type of synchronization relation can occur when the response lasers themselves are strongly chaotic without the drive signal. In this paper, we report the synchronization of two response lasers by injection of common signal from a chaotic drive laser under the following two conditions (1) the two response lasers have separate optical feedbacks which can makes them independently chaotic in the absence of drive signal, and (2) the drive and response lasers are mismatched so that they do not identically synchronize.
Generalized synchronization in lasers has been experimentally observed in NH3 gas lasers [8], He-Ne gas lasers [9], Nd:YVO4 microchip solid-state lasers [10] and semiconductor lasers [7]. Synchronization of two response lasers with common signal injected from a chaotic laser has been observed in [7,9]. In an experiment using chaotic semiconductor lasers in [7] it was shown that correlation between response lasers can be high even when the correlation between drive and response is low. However, in this experiment the drive laser was chaotic due to optical self-feedback, but the response laser did not have any self-feedback, so in the absence of injected signal the response laser exhibits stable steady-state laser oscillation. This is the simplest type of response laser. The case of two response lasers each with independent optical self-feedback as well as common injection signal has not been previously studied. In this case the two response lasers have uncorrelated chaotic oscillations before the injection of the common signal, so it can be expected that the conditions and properties of synchronization are more complex, because of the stronger self-oscillation dynamics [11,12]. In particular, in this case it was not clear whether it would be possible to observe the particular correlation relation reported in [7], namely strong correlation between two response lasers but weak correlation between drive and response lasers. In this paper we clarify these points. Furthermore, we show the dependence on the feedback phase. It has been previously shown that the optical phase of the feedback light has a significant effect on the quality of synchronization between drive and response lasers [12]. Some schemes for on-off phase shift keying have been proposed for chaos communications [13,14]. Hence the feedback phase is an important parameter in our analysis.
2. Experimental setup
Figure 1 shows our experimental setup for synchronization by injection of common chaotic signal in semiconductor lasers with optical self-feedback. We used three distributed-feedback (DFB) semiconductor lasers (NTT Electronics, NLK1555CCA, optical wavelength 1547 nm) developed for optical fiber communications [7]. 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 to within 0.001 nm. The lasing thresholds of the injection current for solitary lasers Ith were 8.7 mA (Drive), 7.6 mA (Response 1), and 9.2 mA (Response 2), respectively.
Fig. 1. Experimental setup for synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback. 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; PC, personal computer; PD, photodetector; PZT, piezo-transducer; SA, radio-frequency spectrum analyzer; SL, semiconductor laser.
An external mirror was placed in front of the Drive at a distance of 1.40 m, corresponding to the (roundtrip) feedback delay time of 9.33 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 laser beam from the Drive was transmitted to each of Response 1 and 2. Two optical isolators (total isolation: -60 dB) and two half wave plates were used to achieve one-way coupling. The laser beam was divided into two beams 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. Each of Response 1 and 2 has an external mirror to introduce optical feedback that causes chaotic oscillation of laser output. The distance from the Response laser to the external mirror was set to 0.6 m (one-way) for both Response 1 and 2, corresponding to the feedback delay time (roundtrip) of 4.0 ns. The external cavity lengths of Response 1 and 2 (0.6 m) were detuned from that of Drive (1.4 m) to reduce correlation between the Drive and Response lasers. The external mirror of Response 1 was displaced by a piezo transducer (PI Inc., P-752.11C) with a resolution of 10 nm by using a feedback position controller (PI Inc., E-665.CR), so that the optical feedback phase (i.e., the optical phase of the feedback light) in Response 1 could be precisely varied. 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 wavelengths were measured by an optical spectrum analyzer (Advantest, Q8384).
The relaxation oscillation frequencies of the lasers were adjusted by changing the laser injection currents. The relaxation oscillation frequency of the Drive was set to 2.5 GHz. The relaxation oscillation frequencies of Response 1 and 2 were matched at 2.0 GHz, but mismatched with the Drive frequency to reduce correlation between the Drive and Response lasers. The injection currents were 12.00 mA (1.38 Ith) for the Drive, 10.95 mA (1.44 Ith) for Response 1, and 11.90 mA (1.29 Ith) for Response 2, respectively. We set the optical wavelength to 1547.333 nm for the solitary Drive laser and 1547.308 nm for both the solitary Response 1 and 2 lasers, corresponding to wavelength detuning between the Drive and the Response lasers of -0.025 nm (-3.125 GHz), so that optical injection locking could be achieved. The injection strengths from the Drive to the two Response lasers are adjusted to achieve injection locking. Under conditions for injection locking, the wavelengths of the Response lasers were pulled to that of the Drive so all three optical wavelengths were matched at 1547.333 nm.
3. Experimental results
3.1 Temporal waveforms
We observe the temporal waveforms of Response 1 and 2 and their correlation plot when the optical phase of Response 1 is changed. We define the difference between optical feedback phases in Response 1 and 2 as Δϕ r1,r2=ϕ r1-ϕ r2, and the difference between optical feedback phases in Drive and Response 1 as Δϕ d,r1=ϕ d-ϕ r1. Figures 2(a) and 2(b) show the temporal waveforms of Response 1 and 2 and their correlation plot when Δϕ r1,r2=0, and Figs. 2(c) and 2(d) show the temporal waveforms of Response 1 and 2 and their correlation plot when Δϕ r1,r2=π. Accurate synchronization is achieved between Response 1 and 2 when the optical feedback phases are matched, Δϕ r1,r2=0. On the other hand, when the optical feedback phase difference is maximum, Δϕ r1,r2=π, synchronization is completely destroyed and the temporal waveforms are dissimilar as shown in Figs. 2(c) and 2(d). This shows that synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback is sensitive with respective to the optical phase of the feedback light in the two Response lasers, as in the case of identical synchronization between drive and response lasers shown in [12].
We quantitatively define the degree of synchronization as the cross correlation between two temporal waveforms normalized by the product of their standard deviations: i.e.,
where I 1,2 are the total intensities of the two temporal waveforms, IĪ1,2 are their mean values, and σ 1,2 are their standard deviations. The angle brackets denote time averaging. The best synchronization corresponds to cross-correlation value C=1. The cross correlation values corresponding to Figs. 2(b) and 2(d) are 0.903 and 0.015, respectively.
For comparison, we observe the temporal waveforms of Drive and Response 1 and their correlation when the optical phase of the feedback light for Response 1 is changed. Figures 3(a) and 3(b) show the temporal waveforms of Drive and Response 1 and their correlation plot for Δϕ d,r1=0, and Figs. 3(c) and 3(d) show the temporal waveforms of Drive and Response 1 and their correlation for Δϕ d,r1=π. The cross correlation values of Figs. 3(b) and 3(d) are 0.601 and 0.599, respectively. These results confirm that good synchronization is not achieved between Drive and Response even when optical feedback phases are matched. It can be understood that the Drive and Response waveforms do not synchronize because different external cavity lengths and different relaxation oscillation frequencies were used for Drive and Response lasers.
Fig. 2. Experimental result of temporal waveforms and their correlation plots for Response 1 and 2 at (a), (b) Δϕr1,r2=0, and (c), (d) Δϕr1,r2=π, where Δϕr1,r2=ϕr1-ϕr2 is the difference between optical feedback phases in Response 1 and 2. The cross correlation values are (b) 0.903 and (d) 0.015.
Fig. 3. Experimental result of temporal waveforms and their correlation plots for Drive and Response 1 at (a), (b) Δϕd,r1=0, and (c), (d) Δϕd,r1=π, where Δϕd,r1=ϕd-ϕr1 is the difference between optical feedback phases in Drive and Response 1. The cross correlation values are (b) 0.601 and (d) 0.599.
Finally, the results shown in Fig. 2(a), (b) together with the results shown in Fig. 3(a), (b) confirm that it is possible to achieve identical synchronization between the response lasers even though the drive and response are not identically synchronized.
3.2 Cross correlation characteristics
We investigate the cross correlation as the optical feedback phase difference is continuously varied. Figure 4(a) shows the cross correlation between Response 1 and 2 as a function of the optical phase difference between Response 1 and 2 at the maximum feedback strength of the two Response lasers. The cross correlation changes periodically as the optical feedback phase difference is varied continuously. The period of the correlation curve is 1.5 µm, corresponding to the optical wavelength of the semiconductor lasers. The maximum and minimum values of cross correlation are 0.958 and 0.032, respectively. On the contrary, Fig. 4(b) shows the cross correlation between Drive and Response 1 as a function of the position of the external mirror for Response 1 (optical phase difference) at the maximum feedback strength. The cross correlation value stays between 0.45 and 0.65 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 5(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. The maximum value of cross correlation is obtained at zero phase difference (Δϕ r1,r2=0), whereas the minimum value is observed at Δϕ r1,r2=π. The maximum value of cross correlation stays almost constant at 0.95, while the minimum value decreases as the feedback strength is increased. Therefore, the difference between maximum and minimum values of cross correlation (ΔC=Cmax-Cmin) increases as the feedback strength is increased, and the maximum difference of ΔC=0.880 is obtained at the maximum feedback strength which can be achieved in the experimental setup. We could not further increase the feedback strength due to limitations of our experimental setup. However, we can anticipate that at some larger value of feedback strength, the injection locking will breakdown. So we should expect the maximum correlation value, and hence the difference between maximum and minimum correlation values, to decrease again at larger feedback strength.
Figure 5(b) shows the maximum and minimum values of cross correlation between Drive and Response 1 as a function of feedback strength of Response 1. The values of maximum and minimum cross correlation gradually decrease as the feedback strength is increased. However, only relatively small change in ΔC is observed at different optical feedback strengths and the maximum value of ΔC is 0.105. This shows that the cross correlation between Drive and Response 1 is not sensitive to the optical feedback phase even at strong feedback strengths.
Fig. 4. 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 position of the external mirror for Response 1 (optical phase difference) at the maximum feedback strength of the two Response lasers. The powers of the injection and feedback lights are 4.0 and 3.0 µW, respectively (measured in front of the Response 1 laser). (a) The cross correlation changes periodically as the optical feedback phase difference is changed. The maximum and minimum cross correlation values are 0.958 and 0.032, respectively. The period of the correlation curve is 1.5 µm, corresponding to the optical wavelength of the laser. (b) The cross correlation values stay between 0.45 and 0.65 and do not show periodical change.
Fig. 5. Experimental result of the maximum and minimum cross correlation values (Cmax and Cmin) between (a) Response 1 and 2, and between (b) Drive and Response 1, as a function of the feedback strength of the two Response lasers. The maximum value of cross correlation is obtained at zero phase difference (Δϕr1,r2=0), whereas the minimum value is observed at Δϕr1,r2=π. (a) The difference between maximum and minimum values of cross correlation (ΔC=Cmax-Cmin) increases as the feedback strength is increased, and the maximum value of ΔC=0.880 is obtained at the maximum feedback strength which could be achieved in the experiment. (b) The values of maximum and minimum cross correlation gradually decrease as the feedback strength is increased. However, only relatively small change in ΔC is observed at different optical feedback strengths. The maximum value of ΔC is 0.105.
4. Numerical results
We carried out numerical simulations to verify our experimental observations and moreover to clarify parameter dependences of the difference ΔC between the maximum and minimum cross correlation values (ΔC=Cmax-Cmin). The difference ΔC characterizes the influence of feedback phase on synchronization. We used the Lang-Kobayashi equations for semiconductor lasers with optical feedback, assuming the configuration for synchronization by injection of common chaotic signal. The equations for Drive are given by
where E and N are the complex electric field and the carrier density, respectively. The equations for Response j (j=1, 2) are given as follows:
where Δωj=ω0-ωj is the detuning of optical angular frequency between Drive and Response j and the last term in the first equation represents the optical injection from Drive. In what follows, we also use the optical frequency detuning Δf defined by Δf=Δω1/2π. The optical phase of the feedback light is represented by ϕj. Parameter values used in our simulations are as follows: α=3, GN=8.4×10-13 m3s-1, N0=1.4×1024 m-3, N th=2.018×1024 m-3, τ in=8.0 ps, τs=2.04 ns, κD=0.00745, τD=9.333 ns, JD=1.3 J th, τ=4.0 ns, J=1.19 J th, where J th=N th/τs is the threshold of the injection current. Drive and Response have different relaxation oscillation frequencies of 2.5 and 2.0 GHz, respectively, which coincide with the values in our experiments. The relaxation oscillation frequency for Drive is detuned from that for Response 1 and 2. For these parameter values, Drive is in a chaotic regime. Response 1 and 2 are assumed to have a small mismatch in their optical frequencies (ω1-ω2)/2π=0.25 GHz. As for the optical phases of the feedback light, ϕ 1 is fixed as ϕ 1=0 while ϕ2 is varied. We use ϕ2=0 to obtain Cmax and ϕ 2=π to obtain Cmin. Using these Cmax and Cmin, we calculate ΔC (=Cmax-Cmin).
Figure 6 shows the contour plot of ΔC as a function of the optical frequency detuning between Drive and Response 1 (Δf) and the injection strength from Drive to Response (κinj). In the calculation, the feedback strengths of Response 1 and 2 are fixed as κr=0.08946. The synchronization region, in which the maximum cross correlation Cmax between Response 1 and 2 is close to unity, is shown by black dashed line. There exists a wide parameter region for synchronization. The synchronization region is asymmetric with respect to the line Δf=0: the region is shifted to the negative frequency detuning side due to the α-parameter of semiconductor lasers. It should be noted that ΔC strongly depends on the parameters Δf and κinj in the synchronization region. Large values of ΔC, say ΔC>0.75, are achieved for relatively small κinj while ΔC is small for large κinj. In the region of large κinj, ΔC decreases with increasing κinj. This is a natural behavior because the injection light comes to dominate the response laser dynamics and thus the influence of the optical feedback becomes relatively weak. Our experimental observation in Fig. 2 corresponds to the conditions Δf≈-3.125 GHz and κinj≈0.1~0.2. The injection strength inside the laser cannot be measured directly, so the correspondence with the simulation value is roughly estimated by the following procedure. The injection strength is measured as a ratio of the threshold power required for the onset of injection locking in the absence of external feedback, and then the correspondence between the threshold powers in experiment and simulation is used to estimate the corresponding value of κinj.
Fig. 6. Numerical result of the contour plot of the cross correlation difference ΔC (=Cmax-Cmin) between Response 1 and 2 as a function of the optical frequency detuning between Drive and Response (Δf) and the injection strength from Drive to Response (κinj). Synchronization between Response 1 and 2 is achieved in the region above the dashed curve.
Fig. 7. Numerical result of the contour plot of the cross correlation difference ΔC (=Cmax-Cmin) between Response 1 and 2 as a function of the feedback strength of Response (κr) and the injection strength from Drive to Response (κinj). Synchronization between Response 1 and 2 is achieved in the region above the dashed curve.
Figure 7 shows the contour plot of ΔC as a function of the feedback strength of Response (κr) and the injection strength from Drive to Response (κinj), where the detuning Δf is fixed as Δf=-3.125GHz, which coincides with the experimental value. The boundary curve, above which synchronization between Response 1 and 2 occurs, is also shown by black dashed line. It is clearly observed that ΔC strongly depends on the parameters κr and κinj. The difference ΔC takes large values (ΔC>0.9) in a region, which is roughly located as 0.10<κr<0.16 and 0.18<κinj<0.33. This region of large ΔC is an elongated region parallel to the boundary curve of the synchronization region. This figure shows that ΔC takes a large value when κinj and κr balance with each other: as κr increases, κinj also has to increase for ΔC to remain large.
The numerical results in Fig. 7 are consistent with our experimental results for the feedback strengths achievable in the experimental setup. According to our experimental results shown in Fig. 5(a), for an injection strength above the onset of synchronization ΔC increases monotonically up to a value of ΔC=0.88 as the feedback strength is increased up to the maximum feedback strength possible in the experimental setup. In Fig. 7 this corresponds to values of κinj and κr in the ranges κinj≈0.1~0.2 and κr≈0.0~0.1. The value of κr corresponding to the experiment is estimated from the estimate of the injection strength (as explained above) and the power ratio between the injection and feedback lights. We note that the numerical results confirm the possibility of this scenario. Further, the numerical results show additional features in parameter regions which could not be accessed in the experiment due to technical setup limitations. In particular, they confirm the prediction that the maximum correlation difference should eventually decrease again as the injection-locking boundary is approached. Further, the numerical simulations show that the effect of the feedback phase can be large (ΔC>0.9) over a significantly wider range of feedback strength and injection strength than could be achieved in the experiment. The reason for this dependence of ΔC on injection strength and feedback strength is as follows. When the injection effect is dominant both the maximum and minimum correlations are high. When the feedback effect is dominant, both the maximum and minimum correlations are low. The difference between maximum and minimum correlation ΔC is largest when there is a balance between these two effects.
5. Conclusion
We have observed synchronization between two semiconductor lasers subjected to a common injection signal from a chaotic drive laser under the following two conditions (1) the two response lasers have separate optical feedbacks which can makes them independently chaotic in the absence of drive signal, and (2) the drive and response lasers have mismatched parameters so that they do not identically synchronize. In particular, we demonstrated that it is possible to achieve identical synchronization between the response lasers under these conditions. That is, we showed that it is possible to achieve strong correlation between two response lasers even though there is only weak correlation between drive and response lasers.
We experimentally examined the correlations between the laser output waveforms and their dependence on parameters, in particular, the effects of the phase of the feedback light on the correlation over a range of injection strengths and optical feedback strengths. It was observed that the correlation between the response lasers strongly depends on the match between the phases of their optical feedbacks. The maximum correlation between the two response lasers is obtained when the phase of the feedback light is matched between the two response lasers, while the minimum correlation is observed when the difference in the optical phase is π. On the other hand, it was shown that the correlation between the drive and response is always low, and not sensitive to the phase of the feedback light.
A numerical investigation of the laser dynamics was consistent with the experimental results and clarified the dependence of correlation characteristics on parameters over a wider range of parameters than was possible in the experiment. In particular, we examined the difference between the maximum and minimum cross correlations, which characterizes the effect of the feedback phase on the synchronization. It was confirmed that there is a region where the difference is largest when there is a balance between the values of injection strength and the feedback strength.
Acknowledgments
We thank Yoshinobu Tonomura, Naonori Ueda, Kenji Nakazawa, and Masato Miyoshi for their support and encouragement. A.U. acknowledges support from NTT Corporation, JGC-S Scholarship Foundation, The Mazda Foundation, CASIO Science Promotion Foundation, TEPCO Research Foundation, and Grants-in-Aid for Young Scientists from the Japan Society for the Promotion of Science.
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