We investigate a nonlinear dynamical behavior of the stimulated Brillouin scattering (SBS) near the threshold in the presence of feedback. When a cw Nd:YAG laser is injected into a single mode optical fiber, the SBS signal exhibits intermittently appearing irregular self-pulsations near the threshold. By examining the time series of the SBS signals, we verify that spontaneous Brillouin scattering having a stochastic property is dramatically suppressed due to the feedback. We explain the chaotic phenomenon of SBS in the presence of feedback in comparison with that in the absence of feedback.
©2007 Optical Society of America
More than a decade ago, Harrison et al found a strong irregular stimulated Brillouin scattering (SBS) in an optical fiber for a cw laser input . The phenomenon is developed from limit cycles to chaos through quasi-periodicity when both facets are perpendicular to the fiber axis for reflection of the Stokes wave. An explanation about the phenomenon is that the optical Kerr effect introduces a nonlinear optical coupling between pump and Stokes waves and renders the dynamics of SBS chaotic due to the feedback . The chaotic behavior is also confirmed numerically . In the absence of feedback, however, the irregular behavior near the threshold of SBS has been a controversy whether it is chaotic or stochastic. While Harrison et al showed the irregular behavior as an example of chaotic behavior , Gaeta et al showed a stochastic behavior of SBS near the threshold . To verify the behavior, Dämmig et al analyzed the attractor dimension of an irregular signal and showed that the dimension diverges out as increasing the system dimension . They attributed the phenomenon to the noise-like spontaneous Brillouin scattering, which is solely responsible for the irregularity. After the refutations, Harrison et al also found that SBS in the absence of feedback has stochastic characteristics near the threshold , and concluded that SBS has a stochastic behavior near the threshold in the absence of feedback.
However, recently, it was reported that SBS in the absence of feedback is chaotic near the threshold. As an evidence, one of the chaotic behaviors appearing around a bifurcation point, named on-off intermittency, was demonstrated. In the analysis, it was shown that the underlying dynamics of the irregular SBS signal is chaos, but that additive spontaneous Brillouin scattering, whose property is stochastic, acts as an external noise make a SBS signal look stochastic . Because of this additive spontaneous Brillouin scattering, it was hard to find the chaotic characteristics of SBS signal from the analysis of the correlation dimension near the threshold. Then it is natural to raise a question of why SBS in the presence of feedback has a chaotic property in the correlation dimension analysis. And what makes the difference of the SBS signals in terms of being in the presence or absence of feedback?
To answer to these questions, in this paper, we investigate the nonlinear dynamical behavior of SBS in the presence of feedback near the threshold. We focus on the transition route from SBS-off to SBS-on as the input pump power increases. Through an analysis of our experimental results, we verify that, in the presence of feedback, spontaneous Brillouin scattering having a stochastic property is dramatically suppressed due to the feedback. This suppression is the reason for why SBS in the presence of feedback has a clear chaotic property differently from the non-feedback case.
2. Experimental setup
The experimental setup is shown in Fig.1. A cw Nd:YAG laser for pumping has a Brewster angled quartz plate between the rear mirror and the rod to stabilize the polarization state of the laser light. An aperture whose radius is 1.6 mm is placed between the rod and the output mirror to obtain a TEM00 mode. For the effective launching of the pump beam into an optical fiber, the beam width of the laser output is expanded twice using the lenses L1 and L2, whose focal lengths are 5 cm and 10 cm, each. The expanded laser beam is launched into the optical fiber through a 10× microscope lens. The input laser power is controlled with a half wave plate (HP) and a polarizer (P) before the input facet of the fiber. The laser is optically isolated from the fiber by a Faraday isolator (Optics for Research Model IO-12-NIR-HP) whose isolation factor is 40.5 dB. The pump and the back scattered signals are sampled with a beam splitter. The pump and the back scattered SBS powers are measured with power meters(Scientech AC2500) simultaneously. The time series of the pump, SBS, and transmitted signals are detected with photo-detectors D1, D2, D3 (Electro-Optics Technology, Inc., 2020B), which have a rise time of ≤ 1.5 ns. The detected signals are monitored with and stored into a LeCroy digital oscillo-scope(WaveRunner 6100), whose maximum sampling rate is 10 GS/s. A single mode optical fiber of standard telecom grade with length L ≃ 185 m is used. The diameters of the core and cladding are 6 μm and 125 μm, respectively. For feedback, both facets are cleaved to be perpendicular to the fiber axis, so that the estimated reflectivity is ∼ 4.0%. By observing the projected beam on the screen with an electro-viewer (Electrophysics 7215B), we confirm the spatial pattern of the transmitted signal.
3. Experimental results
In the experiment, we use an irregular Nd:YAG laser beam with ∼ 5% fluctuation as a pump beam. By injecting the irregular pump beam into the optical fiber we can investigate a nonlinear dynamical behavior of SBS near the threshold. Figure 2(a)-(d) are four kinds of time series of SBS depending on the input power. As is shown in Fig.2(a), when the pump beam intensity is Ip = 258 mW, we can see a quasi-periodic SBS output, which is similar to the one observed by Lu et al in experiment  and in numerical calculation . Here to show the signal clearly, the total length of the time series of 200Tr is shown when the coupling of the oscilloscope is 50 ohms, where Tr ∼ 0.9 μs is the time passing the fiber. The quasi-periodic signal consists of short quasi-regular pulses. The pulse width is ∼ 0.3Tr and the period is 2Tr, which corresponds to the round trip time of a light inside the fiber. As Ip decreases, SBS begins to exhibit intermittently appearing irregular bursts. Figure 2(b) is the time series during 1.0 × 105Tr at Ip = 174 mW. Here we can see short laser-off states between large chaotic bursts. Each chaotic burst is actually consists of short quasi-regular pulses like in Fig. 2(a). (One of the time series of a burst is shown in Fig. 3(a).) On a further decrease of Ip, we can see long SBS-off states between rarely appearing SBS bursts as shown in Fig. 2(c) at Ip = 155 mW. When Ip = 129 mW, the time series exhibits that long SBS-off states are intermittently interrupted by rarely appearing chaotic bursts as shown in Fig. 2(d).
We obtain time-delayed phase portraits of the backscattered SBS signals for two cases of the pump power at Ip = 174 and 309 mW. For the phase portraits, the signals of y-axis are delayed by τ= 0.1Tr. For Ip = 174 mW, when one of the the chaotic pulses shown in Fig. 2(b) is expanded, we can see a clear time series and its phase portraits of quasi-periodic output as shown in Fig. 3(a) and (b), respectively. When Ip = 309 mW, we can see a chaotic time series as shown in Fig. 3(c), which is developed from the quasi-periodic output of Fig. 2(a). Figure 3(d) shows the chaotic trajectory of Fig. 3(c). In this figure, we can see an irregular time-delayed phase portrait. From these results, it is understood that SBS in the presence of feedback transits to chaos through quasi-periodicity, which is similar to the route observed by Lu et al [2, 3].
The temporal behaviors of SBS near the threshold shown in Fig. 2(c) and (d) are the typical time series of on-off intermittency, which is characterized by an intermittent interruption of off states by a large amplitude of chaotic bursts [8, 9, 10]. The phenomenon appears when a parameter of a nonlinear dynamical system is forced to vary through a bifurcation point, whether randomly, chaotically, or quasi-periodically. In our experiment, to emphasize, because the pump beam is irregular as shown in Fig. 2(e), a parameter for SBS is irregularly forced through a bifurcation point. If the intermittent behavior of SBS is on-off intermittency, we can understand that the underlying structure of SBS is chaotic near the threshold in the presence of feedback. To confirm on-off intermittency, we let SBS bursts be “on” states whereas we let the period without SBS bursts be an “off” state, which is a laminar phase. Then the probability distribution of the laminar phase is given by that Λn ∞ n -3/2, where n is the laminar phase length. In the analysis, when the amplitude of SBS is less than 5% of the maximum amplitude of the irregular burst, we let the state be an SBS-off state. Because of the amplitude for SBS-on states, the other noise which does not affect the SBS signal can be removed.
From the definition of the probability distribution of laminar phases, we obtain the distribution Λn versus SBS-off lengths n on a logarithmic scale as shown in Fig. 4. The white circles show the probability distribution for the average input pump power of ∼ 155 mW. The solid line is the slope of -3/2. The data experimentally obtained agree well with the power-law of on-off intermittency. The -3/2 slope means that SBS near the threshold has a nonlinear dynamical property. Surprisingly, there is no shoulder. It is known that a shoulder appears when an external noise is applied. When a noise is applied to a chaotic system, the shoulder connects two curves of the -3/2 slope for shorter phases and the exponential fall off for longer phases . The slope of -3/2 without shoulder means that there is no external noise affecting SBS recursively. In the absence of feedback, it was shown that the probability distribution has a shoulder causing an external noise . The possible noise source is only spontaneous Brillouin scattering initiating SBS. Also it was reported that spontaneous Brillouin scattering having stochastic characteristics is solely responsible for the irregular SBS [4, 5]. However, from our experimental result, we can understand that in the presence of feedback the spontaneous Brillouin scattering is dramatically suppressed so that one can not observe a shoulder in the probability distribution of laminar phase length.
We have shown that the probability distribution of SBS in the presence of feedback has no shoulder. This means that spontaneous Brillouin scattering is suppressed due to the feedback even though it initiates SBS. In the previous studies, it was expected that while, in the case of SBS without feedback, the spontaneous Brillouin scattering mainly contributes to the irregularity of SBS, in the presence of feedback, the spontaneous Brillouin is suppressed due to the feedback so that chaos is dominant [3, 5]. The evidence of the suppression could not be found experimentally yet, though. On the contrary, first, our experimental results give an experimental evidence of the suppression of spontaneous Brillouin scattering in the presence of feedback. Second, our results show that the underlying dynamics of SBS in the presence of feedback is chaotic. The difference between characteristics of SBSes in the presence and absence of feedback is that while, in the presence of feedback, spontaneous Brillouin scattering is dramatically suppressed, in the absence of feedback, it recursively affects the generation of SBS. So we see much spontaneous Brillouin scattering. For this reason, while it is hard to know whether SBS in the absence of feedback is chaotic or stochastic through an analysis of the correlation dimension, SBS in the presence of feedback definitely has a chaotic behavior in the analysis of correlation dimension. The suppression of spontaneous Brillouin scattering can be clearly seen by observing the phenomenon of on-off intermittency with irregular pumping.
In conclusion, we investigate the characteristic of SBS in the presence of feedback by injecting a cw Nd:YAG laser with small amplitude fluctuation into a single mode optical fiber. We find that the SBS signals exhibit intermittently appearing chaotic bursts. The signal near the threshold has the typical power law of on-off intermittency such as the -3/2 slope without a shoulder on the logarithmic scale. Thus we can conclude that SBS in the presence of feedback has a deterministic chaotic behavior near the threshold. Also we can conclude that the spontaneous Brillouin scattering is suppressed due to the feedback. These results tell why we can observe clear chaotic behavior in the presence of feedback and why not in the absence feedback in a single mode optical fiber with cw laser pumping.
This study was supported by Creative Research Initiatives (Center for Controlling Optical Chaos) of MOST/KOSEF.
References and links
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