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We experimentally investigate lag synchronization in mutually coupled Nd:YAG lasers. Two laser diode pumped Nd:YAG lasers, when they are electronically coupled, exhibit lag synchronization for a strong coupling strength region. We investigate the synchronous behaviors of the lasers by analyzing error dynamics, phase portraits, and similarity function. We confirm the observation of lag synchronization by obtaining the probability distribution of the intermittent lag synchronization length before lag synchronization.
©2006 Optical Society of America
1. Introduction
Synchronization in coupled chaotic oscillators has been one of the most intriguing subjects in nonlinear dynamics. The phenomenon is caused by the interaction between two chaotic oscillators as a result of coupling, and various features of synchronization have been found depending on the systems and coupling strength [1, 2] such as complete (CS) [3], phase (PS) [4], lag (LS) [5], generalized [6], periodic phase [7], and generalized phase synchronization [8]. These synchronous phenomena have been also studied extensively in coupled lasers [9, 10, 11, 12, 13, 14, 15, 16, 17]. Recently, synchronous phenomena that appear in non-identical chaotic systems have attracted much attention. When two chaotic oscillators are identical, the time series of corresponding dynamical variables of the subsystems completely coincide such that x 1(t) = x 2(t) for a strong coupling strength regime. However, when parameters are slightly mismatched, the time series almost coincide with a lag time τL such that x 1(t) ≈ x 2(t +τL) for a strong coupling strength regime. This phenomenon named LS is developed from PS when there is a phase difference between two oscillators for PS state. A LS state can be identified by the transition from PS to LS state through an intermittent LS state, which is characterized by on-off intermittency [5, 18] .
Comparatively LS has a much shorter history of studies than CS. What is more, because some kinds of coupled chaotic oscillators transit from PS to CS directly without LS even though there is a parameter mismatch [19, 13], it is not easy to observe LS in experiment. For example, in optically coupled Nd;YAG lasers, PS state directly develops to CS state without LS state [13]. So LS in laser systems has been observe mostly in coupled laser diodes with time-delay feedback [16, 17]. In this paper, we report the experimental investigation of lag synchronization in electronically coupled diode laser pumped Nd:YAG lasers and show the characteristics of LS in these laser systems. To analyze the phenomenon, we study similarity function, phase portraits, and phase differences of the two laser outputs. Also we obtain the probability distribution of the intermittent LS length to confirm the transition from PS to LS through intermittent LS state.
2. Experimental setup
The experimental setup is shown in Fig. 1. Each Nd:YAG laser pumped by a diode laser (LD1 and LD2) has a 5mm-long 3mm-diameter Nd:YAG rod and an output coupler whose reflectivity is 97% at 1064 nm. The output couplers (M1 and M2) are set about 10cm apart from each Nd:YAG rod. The back surfaces of the Nd:YAG rods are coated for total transmission at 808nm and for total-reflection at 1064nm, and the front surfaces are coated for total transmission at 1064nm. To pump the Nd:YAG lasers, the laser outputs from laser diodes are collimated with focusing lens (Polaroid, POL-4100BW). The laser diodes are controlled with a diode laser controller (Master Technology, MSLD-10). For electronic coupling, the Nd:YAG laser outputs are detected with silicon P-I-N photo diodes (Electro-Optics Technology, ET-2010), and the difference of the two laser signals are obtained with an electronic circuit and the difference of the signals are fed to the laser diode controllers.
Fig. 1. Schematic diagram of the experimental setup in two coupled diode laser pumped Nd:YAG lasers. CL1 and CL2 are collimating lenses, M1 and M2 output couplers, F1 and F2 1064nm optical band-pass filters, and PD1 and PD2 photo diodes.
For the calculation, the dc signals of the two laser outputs are removed with operational amplifiers (Filter) and each signal is amplified three times with an OP-Amp (Amplifier). After the amplification, the unbiased two signals of I 1(t) and I 2(t) are introduced to an electronic circuit to carry out the mathematical calculation of ε(I 1(t) - I 2(t)), where ε is the coupling strength. The amplitude of (I 1(t)-I 2(t)) is reduced with a potentiometer with a 0.2% step size. The reduced signal, ε(I 1(t) - I 2(t)), is applied to the second LD controller and the inverted signal ε(I 2(t) - I 1(t)) is applied to the first LD controller so that the total input powers are P 1 +ε(I 2(t)-I 1(t)) and P 2 +ε(I 1(t) -I 2(t)) for LD1 and LD2, respectively, where P 1 and P 2 are DC pumping powers. For the acquisition of the experimental data, the laser outputs from PD1 and PD2 are monitored with a 2-Mbyte memory digital storage oscilloscope (LeCroy, LC584AM). To analyze the data, the stored time series are transferred to a computer.
3. Experimental results
The Nd:YAG lasers are pumped at different pumping powers for parameter mismatch. In the experiment, the currents of the two laser diodes are set at P 1 = 433 and P 2 = 464 mA, so that the Nd:YAG lasers generate chaotic outputs with different characteristic frequencies, which are 57 and 62 kHz for LD1 and LD2, respectively, when they are not coupled. In our experiment, for a stronger coupling strength region ε > 0.69, we find LS. As is shown the time series in Fig. 2, we can see an almost coincidence of the two laser outputs with a lag time τL, such that I 2(t +τL) ≈ I 1(t) at ε = 0.7.
To analyze the LS phenomenon, first, we explicitly obtain the lag time by using a similarity function (synchronization error) and confirm the lag time by comparing the time with what is obtained from a correlation function. The similarity function S(τ) is given as follows [5]:
Since the similarity function is a time averaged difference between the laser outputs of I 1(t) and I 2(t +τ) depending on delay time τ, we can obtain the lag time τL by measuring the time when S(τ) is minimum. In the case of CS, S(τ) approaches zero for τ= 0 because I 1(t) = I 2(t). In the case of LS, however, S(τ) has a minimum value which is nonzero for τ = τL because I 1(t) ≈ I 2(t +τL). We present the obtained similarity function as shown in Fig. 3(a). In the figure, we find that S(τ) has a periodic dip and the minimum dip appears at τ = 12.04 μs, which is the lag time. The lag time is also confirmed from the cross correlation function, which is equivalent to the similarity function, as shown in Fig. 3(b).
Fig. 2. Temporal behavior of the two laser outputs when the coupling strength is ε = 0.7, where (a) is I 1(t) and (b) I 2(t).
Fig. 3. The similarity function (a) and the correlation function (b) of two laser outputs at ε= 0.7, where τL = 12.04 μs as shown the arrow in (b).
To show the LS behavior, we obtain the phase portraits on the space of I 1(t) versus I 2(t +τ). As is shown in Fig. 4(a), when τ = 0 the phase portraits show a ring-like trajectory with a narrow band width. However, when τ = 12.04 μs, which is the lag time obtained from the similarity function, the distribution of the trajectory is localized on the diagonal line as shown in Fig. 4(b). This implies that the lag time obtained from the similarity and the correlation function is the real lag time between the two laser outputs. And the localized trajectory implies that the states of the two systems are almost identical, which is the typical feature of LS.
To confirm the observation of LS, we analyze the transition to LS. The difference of the two signals is shown in Fig. 5. Before the transition at ε = 0.65, the difference of the two signals I 2(t +τL)-I 1(t) exhibits small amplitude chaotic fluctuations between the large chaotic bursts as shown in Fig. 5(a), which is a typical feature of intermittent LS. The small amplitude fluctuations are the temporal LS states. After the transition, no large chaotic bursts can be observed for e= 0.7 as shown in Fig. 5(b). This is the evidence of the transition to LS.
Fig. 4. Phase diagram for lag synchronization at ε = 0.7. (a) is for τ = 0.0 and (b) for τ= 12.04 μs.
Fig. 5. Temporal behavior of the difference of the two signals, I 2(t +τL) - I 1(t) at (a) ε= 0.65 and (b) ε= 0.7, where τL = 12.04 μs is the lag time obtained from the similarity function.
Before the transition to LS, the phenomenon of the large chaotic bursts interrupting the small amplitude chaotic fluctuations shown in Fig. 4(a) is known as the phenomenon of on-off inter-mittency. While the large chaotic burst is the “on” state, the small amplitude chaotic fluctuation is the “off” state, which is the laminar phase. To analyze the phenomenon, we obtain the probability distribution of laminar phase Λ(n) depending on the laminar length at ε= 0.65. To obtain the distribution, we let a chaotic fluctuation less than 0.03 to be the “off” state and measure the laminar length by counting the number of the oscillations of I 1(t). As is shown in Fig. 6, we obtain a -3/2 slope on the logarithmic scale. This slope well agrees with the the probability distribution of laminar phase Λ(n) ~ n -3/2 [20] This statistical analysis of an intermittent LS state indicates that the phenomenon we have observed is lag synchronization [21].
4. Discussion
It is known that there are two routes in coupled chaotic oscillators with slight parameter mismatch depending on the systems: One is the direct transition from PS without a phase shift to CS [19] and the other from PS with a phase shift to LS [5]. In the previous studies of synchronous behavior in mutually coupled solid states lasers, two lasers are optically coupled and only the direct transition to CS was found even though the parameter mismatch is unavoidable in real situation. In two arc lamp pumped Nd:YAG lasers are optically coupled, they exhibits direct transition from PS state to CS without LS [12]. In an Nd:YAG crystal pumped by two argon laser beams with different paths, the two laser outputs from the crystal due to the different paths exhibit CS when the distance between the two path is close [9]. As one more example, in a LiNdP4O12 crystal is pumped with two argon laser beams with different paths, the two laser outputs also exhibit CS when the scattered beams of the two lasers are injected into the two lasers [10]. However, the route of our coupled Nd:YAG lasers is the latter case exhibiting LS. When we investigated the PS state, there was a π/2 phase shift in phase dynamics [14]. This phase shift in PS state leads to LS, so that the output signals of the two chaotic Nd:YAG lasers almost coincide with a lag time. When there is no phase shift in the PS state, the PS state should directly transit to CS as the coupling strength increases [13, 19]. Hence we can understand that the transition from PS to LS is caused by the electronic coupling.
Fig. 6. Probability distribution of laminar phase for intermittent lag synchronization state at ε = 0.65.
5. Conclusions
We have experimentally investigated lag synchronization in electronically coupled laser diode pumped Nd:YAG lasers. We have shown that the two laser outputs almost coincide with a lag time τL = 12.04 μs for the strong coupling strength regime ε > 0.69. The lag time is obtained from the similarity and the correlation function, respectively. The phase portraits of I 2(t + τL) versus I 1(t) with the obtained lag time clearly demonstrate LS in the mutually coupled Nd:YAG lasers. We have also confirmed the observation of LS by obtaining the probability distribution of intermittent lag synchronization length, before the transition to LS. As a consequence, we can understand that when Nd:YAG lasers are electronically coupled, they transit from PS state to LS state as the coupling strength increases.
This work was supported by the Creative Research Initiatives (Controlling Optical Chaos) of MOST/KOSEF.
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