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We experimentally investigate phase synchronization between two electronically coupled diode laser pumped Nd:YAG lasers. As the coupling strength increases, the phase of the two chaotic laser outputs develops from a nonsynchronous state to a phase synchronous one through a phase jump state. We find that there are 2π phase jumps and a π/2 phase shift between the two laser outputs unlike in optically coupled Nd:YAG lasers. To clarify the transition to phase synchronization with a π/2 phase shift, we analyze the phenomenon of phase synchronization by using a phase portrait, phase difference dynamics, and frequency variation depending on the coupling strength and obtain the scaling rule of the average phase locking time in the intermittent phase jump state.
©2006 Optical Society of America
1. Introduction
Synchronous behaviors appearing in mutually coupled non-identical chaotic oscillators have attracted much attention since they were known to be readily found in physical [1, 2, 3], chemical [4, 5], and biological systems [6, 7]. Up to now, various types of synchronization, such as phase [8, 9, 10], lag [11], periodic phase [12], and generalized phase synchronizations [13], were found and their mechanisms were also studied. Among them, phase synchronization (PS) have been especially intensively studied because of its fruitful applicability to various fields in science since its first observation by Rosenblum et al in mutually coupled Rössler oscillators [8]. In PS, the phase difference of two oscillators are bounded within 2π above a critical coupling strength for PS, while the amplitudes remain chaotic and are, in general, weakly correlated. However when the coupling strength is below a critical value, the phase difference of two oscillators is intermittently unlocked because of 2π phase jumps.
PS was also extensively studied in lasers [14, 15, 16, 17, 18]. When the loss of a cw CO2 laser is modulated with an electro-optic modulator by the chaotic laser output, the laser exhibits periodic PS, intermittent PS, and PS between the modulation signal and the laser output depending on the modulation frequency [14]. When the laser output is fed to the acousto-optic modulator that is driven by two periodic signals with different frequencies, a diode laser pumped Nd:YAG laser exhibits ±2π phase jumps and PS state between the laser output and the driving signal [15]. In the case of coupled laser systems, PS and intermittent PS were observed in coupled diode lasers and vertical-cavity lasers [16]. Also intermittent PS and PS were observed in optically coupled arc lamp pumped Nd:YAG lasers [17, 18]. In these experiments, it was found of the transition that a nonsynchronization state develops to complete synchronization state through an intermittent phase locking state with ±2π phase jumps and PS state, as the coupling strength increases. In this transition, neither π/2 phase shift nor lag synchronization was found.
In this paper, we report the experimental results of PS with π/2 phase shift and the transition from nonsynchronization state to PS through intermittent PS with 2π phase jumps in electronically coupled diode laser pumped Nd:YAG lasers. We also analyze the phenomenon of PS by using a phase portrait, phase difference dynamics, and average frequency variation depending on the coupling strength and obtain the scaling rule of the average phase locking time in the intermittent phase jump state depending on the coupling strength when intermittent PS occurs.
2. Experimental Setup
To observe PS we couple two Nd:YAG lasers, whose YAG rod lengths are different (4.85 mm and 10.00 mm-long, respectively). Because the rods have different lengths, the operation conditions, such as threshold current and temporal behaviors, are not identical. Two commercial laser diodes LD1 and LD2 (Polaroid, POL-4100BW) which are driven by custum made low noise current sources (Master Technology, MSLD-10) are used as pumping sources of Nd:YAG lasers. Figure 1 shows a schematic diagram of the experimental setup. In the experiment, we set each output coupler whose transmittance is 97 percent at 1064 nm about 2.15 cm apart from each YAG rod. The back surfaces of the YAG rods are coated for total transmission at 808 nm and for total reflection at 1064 nm. The front surfaces are coated for total transmission at 1064 nm.
Fig. 1. Schematic diagram of the experimental setup in the coupled diode laser pumped Nd:YAG lasers. LD, PD, and PM are the diode laser, the Si P-I-N photo diode, and the potentio-meter, respectively.
Each laser signal is detected with two fast Si P-I-N photo diodes (ElectroOptic Technology, ET-2010) PD1 and PD2, and monitored with a 2-Mbyte memory digital storage oscilloscope (LeCroy, LC584AM). The output signals from the photo diodes are introduced to an electronic circuit to obtain the difference of the two laser signals. Buffers and operational amplifiers carry out the mathematical calculation of e(I 1 –I 2), where I 1 and I 2 are the unbiased output signals of the two lasers and ε is the coupling strength. For the calculation, the dc signals of the two laser outputs are removed with operational amplifiers and ε is adjusted precisely with a potentiometer with a step size of 0.2 percent. ε(I 1 –I 2) is applied to an LD2 controller and the inverted signal ε(I 2 – I 1) is applied to an LD1 controller so that the input powers are P 1 +ε(I 2 – I 1) and P 2 +ε(I 1 – I 2) for the first and the second laser diodes, respectively, where P 1,2 is the dc pumping power of the laser diodes. Finally the measured laser signals are transferred to a computer to be analyzed. This simple electronic coupling renders synchronization of the two chaotic lasers.
3. Experimental Results
In the experiment, to observe PS, we obtain the temporal behaviors of the two laser outputs as the coupling strength increases. Figure 2 shows the temporal behaviors of the two Nd:YAG laser outputs for the three cases of the coupling strength. When the two lasers are not coupled, that is, ε is zero, they exhibits independent time series as shown in Fig. 2(a). Figure 2(b) is the temporal behaviors when the coupling strength ε = 0.428. As is shown in the time series in the box, the phases slip by 2π. This is called a 2π phase jump. Figure 2(c) shows that when the coupling strength increases up to ε= 0.468, the phases of the two laser outputs are slightly mismatched occasionally. However, the phase difference between the two lasers is bounded within 2π, although the amplitudes are not locked to each other; this is the very phenomenon of PS as defined by Rosenblum et al. [8]. From these results, we can understand that the phase difference of the coupled laser outputs develops from a nonsynchronous state to a PS state through an intermittent 2π phase jump state as the coupling strength increases.
Fig. 2. (Color online). Temporal behaviors of the two laser outputs. Coupling strength (a) ε= 0.000, (b) ε= 0.428, and (c) ε= 0.468.
In order to show the difference between nonsynchronization state and PS state, we obtain the power spectra of the two laser outputs through the fast Fourier transformation of the time series. The results are shown in Fig. 3. When the coupling strength is zero, ε= 0, we can see apparently different spectra as shown in Figs. 3(a) and (b). The two Nd:YAG lasers generate different frequencies, whose peaks are 128 and 146 kHz for lasers 1 and 2, respectively, at different pumping currents of the two laser diodes, P 1 = 513mA and P 2 = 657 mA. What is interesting is that as we increases the coupling strength, the center frequencies of the two laser begin to be close to each other. That is, while the frequency of laser1 increases that of laser 2 decreases due to mutual coupling. Finally, the frequencies come to coincide at the middle frequency when PS occurs. Figures 3(c) and (d) show the spectra of the two lasers at ε = 0.428. The figures show that the center frequencies are 132 and 140 kHz for lasers 1 and 2, respectively. When the coupling strength is 0.468, where we can find PS, the center frequencies come to coincide as shown in Figs. 3(e) and (f). The frequency of 137 kHz is the middle of the two frequencies when the two lasers are not coupled. In our investigation, the distance between the two frequenceis is almost consistant until ε ~ 0.38. After the value, the distance decreases almost linearly as we increase the coupling strength. From these features, we can understand that the two independent lasers come to coincide as the coupling strength increases.
Fig. 3. Power spectra of the two laser outputs. (a) and (b) are the spectra of laser 1 and 2 at coupling strength ε= 0.000, (c) and (d) at ε= 0.436, and (e) and (f) at ε= 0.468. This shows that ε= 0.000, (c) and (d) at ε= 0.436, and (e) and (f) at ε= 0.468. This shows that, in the synchronous state, their center frequencies coincide with each other.
Another evidence of the transition from nonsynchronization state to PS can be found in the phase portrait on the I1 versus I2 phase space. When the coupling strength is ε = 0.0 the trajectory is scattered on the whole space as shown in Fig. 4(a). This means that the intensities of the two laser outputs are neither locked to nor correlated with each other. When the coupling ε reaches 0.468, PS occurs. Here we can find that the trajectory moves circularly. This result implies that there is a phase shift between the two laser outputs when their phases are locked. The trajectory in Fig. 4(b) is quite similar to that of the coupled Rössler oscillators. In the coupled Rössler oscillators, there is a π/2 phase shift in between.
To observe PS and phase jumps, we define the phase increment of each laser output as follows:
for t(Iin) ≤ t ≤ t(I i n+1), where Iin is the n-th local maximum of the signal of the laser i. We interpret a PS state as a difference between the two phases of the laser signals satisfying the condition, ∣ϕ1(t) - ϕ2(t)∣ < 2π. Following this definition of the phase, we first calculate the phase difference ϕ1(t) - ϕ2(t) between the two lasers for the four cases of the coupling strength. Fig. 5 shows the phase difference according to the coupling strength. As the coupling is increased, we can find that the number of intermittent phase jumps is decreased gradually. This jumping behavior is the typical one observed in coupled Rössler oscillators when their characteristic frequencies are slightly mismatched. In Fig. 5(d), there are no more intermittent 2π phase jumps, and phases are locked within 2π. The inset in Fig. 5 clearly shows a π/2 phase shift when the phases of two laser outputs are locked, which correspond to the phase portrait of Fig. 4(b). These phase jumps and π/2 phase shift are different from those in optically coupled Nd:YAG lasers, which exhibit ±2π phase jumps and no phase shift.
Fig. 4. Phase portraits of the two laser outputs. (a) uncoupled lasers at ε= 0.000, (b) phase synchronization state at ε= 0.468. Units are arbitrary.
Fig. 5. Phase difference of the two laser outputs when (a) ε = 0.456, (b) ε = 0.460, (c) ε = 0.464, and (d) ε = 0.468.
We are interested in the average locking time of temporal phase synchronous states depending on the coupling strength. Fig. 6 is the average phase locking time depending on the coupling strength at the border of the synchronization region. To show the exponent of the scalings, we plot the average phase locking time according to the coupling strength on ln 〈l〉 versus (εc - ε)1/2 space, where 〈l〉 is the average locking time and εc is the critical coupling strength for PS. In Fig. 6, the straight line shows a linear dependence that was already discussed by Pikovsky et al. [9]. From this result we can understand that what we have observed in our laser system is phase synchronization.
Fig. 6. Scaling behavior of average phase locking time depending on the coupling strength. The strait line implies that the experimental result is well matched to the scaling of eyelet intermittency.
4. Discussions
In the study of chaos synchronization in lasers, generally, optical coupling is used. In this case, the frequencies [19, 20] and the polarizations [21] of the operating laser modes of two lasers should be matched with each other for synchronization. Electronic coupling does not require any consideration of the frequencies and the polarizations of the operating laser modes. So when we aim to couple two Nd:YAG lasers unidirectionally, we can easily couple the lasers electronically without considering any cavity detuning for the frequency matching and any polarizer for the polarization matching of the laser modes. So electronic coupling is more convenient than optical coupling for synchronization.
Another aspect we want to address here is the underlying structure of the intermittent phase jump state and π/2 phase shift in the PS state. It is known that there are two kinds of PS: PS with a π/2 phase shift between two oscillators [8] and PS without a phase shift [22]. In the former case, a nonsynchronization state transits to the PS state through an intermittent 2π phase jump state. On further increase of the coupling strength PS transits to lag synchronization. In the latter case, a nonsynchrnization state transits to PS through an intermittent ±2π phase jump state. On further increase of the coupling strength, PS transits to complete synchronization. The two transitions were explained with the local Poincaré map of the phase dynamics. While that of the intermittent 2π phase jump state has the structure of type-I intermittency in the presence of noise [23], the local Poincaré map of the intermittent ±2π phase jump state has the structure of type-II intermittency in the presence of noise [22]. Since the electronically coupled Nd:YAG lasers have 2π phase jumps in the intermittent PS state and a π/2 phase shift in the PS state between two laser outputs, the mechanism can be understood as type-I intermittency in the presence of noise. This is not the case in optically coupled Nd:YAG lasers since they exhibit ±2π phase jumps and no phase shift in their phase dynamics [17]. Also the PS state transits to complete synchronization state [18]. From this π/2 phase shift and 2π phase jumps, we can distinguish the underlying structure of this electronically coupled lasers with that of optically coupled ones.
The PS state with a π/2 phase shift is observed in the coupled Rössler oscillators, while one without a phase shift is observed in the coupled Lorenz and the coupled Hyper-Rössler oscillators. So we can understand that our experimental results can be considered to be the typical features of PS appearing in the coupled Rössler oscillators. All these results correspond to what is expected of the experimental observation of PS through a phase jump state in coupled Nd:YAG laser systems.
5. Conclusions
In conclusion we have experimentally investigated PS via phase jumps in electronically coupled Nd:YAG lasers. When the coupling strength is below the critical value, the outputs of the Nd:YAG lasers show intermittent 2π phase jumps, and then rare phase jumps occur as the coupling strength approaches the critical value. Once it exceeds the critical value, the laser outputs exhibit a PS phenomenon with a π/2 phase shift. We also observe the frequency variation of the two lasers. As the coupling strength increases, the different center frequencies of the two laser spectra come to be close and finally to coincide when PS occurs. The phase portraits and the phase difference observed in our experiment are similar to those obtained for coupled Rössler oscillators and the scaling relation agrees well with the theoretical results.
This work was supported by Creative Research Initiatives of the Korean Ministry of Science and Technology.
References and links
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