Experimental characterization of the bifurcation structure ... - IEEE Xplore

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 12, DECEMBER 2003

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Experimental Characterization of the Bifurcation Structure in an Erbium-Doped Fiber Laser With Pump Modulation Alexander N. Pisarchik, Yuri O. Barmenkov, and Alexander V. Kir’yanov

Abstract—The bifurcation structure of phase space is investigated experimentally in an erbium-doped fiber laser with a sinusoidal pump modulation. We demonstrate a rich variety of bifurcations and dynamical states which appear in primary saddle-node bifurcations and discuss their relation to the main laser resonance. The systematic organization of coexisting attractors allows one to predict the behavior of the fiber laser, when initial conditions are allowed to evolve to their final states. Index Terms—Erbium-doped fiber laser, generalized multistability, laser dynamics, pump modulation.

I. INTRODUCTION

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RBIUM-DOPED fiber lasers are widely used in many areas of science and technology, including communications, reflectometry, sensing, and medicine [1], [2], due to their exclusive advantages of high gain and a single transversal mode operation. These lasers are also interesting from a point of view of nonlinear dynamics because of their high sensitivity to any external perturbation which can destabilize their normal operation so that the laser oscillates in a nonlinear regime. Therefore, the knowledge of dynamic behavior of the erbium-doped fiber lasers under external modulation is of great importance not only for their technological application, but also for fundamental research in nonlinear dynamics. A fiber laser belongs to class-B lasers [3] because the polarization in this laser can be adiabatically eliminated and the dynamics can be ruled by two rate equations for field and population inversion. The oscillations in the class-B lasers can be observed only if an additional degree of freedom is added in the form of either saturable absorber, light injection, or external modulation [4]. Different dynamical regimes, bistability, and chaos have been observed in the fiber laser with modulated parameters [5]–[10]. Recently, we have demonstrated numerically the coexistence of multiple periodic and chaotic attractors in an erbium-doped fiber laser with loss modulation [11]. Each of the dynamic states is determined by initial conditions. The periodicity of a particular attractor is equal to subharmonic of the modulation frequency. Each subharmonic branch is born in a regular saddle-node bifurcation (SNB) [12] and the optimal condition Manuscript received March 26, 2003; revised September 12, 2003. This work was supported by Consejo Nacional de Ciencia y Tecnología de México (CONACyT) through Project 32195-E and Project 33769-E) and UC MEXUS—CONACyT. The authors are with Centro de Investigaciones en Optica, Leon, Guanajuato 37150, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/JQE.2003.819559

(minimal driving amplitude) for the appearance of the subsequent branch is achieved when the driving frequency is close to the corresponding harmonic of the relaxation oscillation frequency. The overlapping of the subharmonic branches results in generalized multistability (coexistence of attractors). This paper is devoted to a systematic experimental investigation of the creation of multiple attractors in the erbium-doped fiber laser subjected to harmonic pump modulation. From the experimental point of view, the pump modulation is easier to realize than the modulation of the cavity loss. However, the dynamics of the laser under the pump modulation is different from that of the laser with the loss modulation, although there are many common features for both types of modulation. An important difference of the heavily erbium-doped fiber laser from other class-B lasers is that the former laser represents self-pulsations [13], [14], i.e., this laser acts as an autonomous system. The self-pulsing behavior is usually attributed to the presence of the saturable loss due to erbium ion pairs or pump depletion in the fiber [15]. Recently, period-doubling and quasi-periodic routes to chaos have been observed in a self-pulsing dual-wavelength erbium-doped fiber laser [16], [17]. In this sense, the dynamics of the erbium-doped fiber laser under pump modulation is more sophisticated than the dynamics of other class-B lasers. The paper is organized as follows. In Section II, we describe our experimental setup of the erbium-doped fiber laser. The bifurcation diagrams demonstrating coexistence of multiple attractors are presented in Section III. We determine the parameter domain where different dynamical regimes exist and demonstrate the relevance of the main laser resonance to SNBs in the parameter space of the modulation frequency and amplitude. We investigate the origin of the multistability and its relation with the fundamental laser frequency. Finally, the main conclusions are given in Section IV. II. EXPERIMENTAL SETUP The experimental setup is shown schematically in Fig. 1. The erbium-doped fiber laser is pumped by a commercial laser diode (wavelength 976 nm, maximum pump power 300 mW) through a wavelength-division multiplexer (WDM) and a polarization controller (PC). The linear laser cavity with a 1.5-m length is formed by a piece of heavily doped erbium fiber (SCL110–01 from IPHT) having a 70-cm length and a core diameter of 2.7 m (NA, 0.27), and two Fiber Bragg Gratings (FBG1 and FBG2) with a 2-nm full-width at half-maximum (FWHM) bandwidth and reflectivity of 91% and 95% at a

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Fig. 1. Schematic of the erbium-doped fiber laser.

Fig. 3. Bifurcation diagrams of peak-to-peak laser intensity with modulation frequency as a control parameter at 50%-depth modulation of pump laser diode current. The fundamental laser frequency f is shown by the dashed line. The dotted line bounds the frequency locking region.

between the frequencies of these two processes. This competition is clearly seen through the bifurcation diagrams presented in the next section. III. RESULTS AND DISCUSSION A. Codimensional-One Bifurcation Diagrams

Fig. 2. Time traces of output intensity showing self-pulsation of fiber laser. (a) Diode current of 40 mA, corresponding to 15-mW output power. (b) Diode current of 400 mA (150 mW).

1560-nm wavelength. The active fiber has high concentration of erbium ions (about 2300 ppm) corresponding to absorption of 18.5 dB/m at a 980-nm wavelength. The output power of the pumping laser diode can be modulated with a signal generator controlling the drive current. The output signals from the pump diode laser and from the fiber laser are recorded with photodetectors D1 and D2 and analyzed with an oscilloscope. nm The optical spectrum bandwidth of the laser is less than (resolution of our spectrum analyzer). In the absence of pump modulation, the laser generates periodic oscillations with the fundamental laser frequency (relaxation oscillations) owing to the presence of the saturable loss in the fiber (Fig. 2). The amplitude and frequency of the oscillations depend on the pump power. When the pump power is low, the response of the laser is sinusoidal [Fig. 2(a)], whereas at the high powers ( 20 mW) the fiber laser oscillates in a pulsed regime with a higher repetition rate [Fig. 2(b)]. In the presence of harmonic pump modulation, two competitive processes are involved in laser dynamics. These are the self-oscillations (discussed above) and the external modulation. Thus, the final dynamics depends basically on the relationship

In Fig. 3, we plot the bifurcation diagram of the peak-to-peak laser output intensity with the modulation frequency as a control parameter for a fixed modulation depth 50%. This diagram is constructed by a slow increase and decrease of the control parameter. The coexistence of two and three attractors is clearly seen in the figure. The laser dynamics is ruled mainly by the ratio of modulation frequency to fundamental laser frequency . The latter is defined by the saturable loss in the fiber and kHz pump power. In all of our experiments, we fix [Fig. 2(a)] which corresponds to the pump power of 15 mW. The general form of the bifurcation diagram shown in Fig. 3 is similar to that obtained recently by Sola et al. [10] in an erbium-doped fiber ring laser, where bistability and even multistability regions can be distinguished. However, their interesting experimental and numerical study suffers from a lack of any bifurcation analysis. Earlier, Luo et al. [8], [9] demonstrated the coexistence of two attractors (optical bistability) in a similar laser. The bifurcation analysis of the diagram shown in Fig. 3 as well as similar diagrams reported by other authors [8]–[10] allow us to reveal the following general features of the dynamics of the erbium-doped fiber laser subjected to pump modulation. 1) When the modulation frequency is lower than half of the ) (left-hand side fundamental laser frequency ( from the vertical dotted line in Fig. 3), a strong interaction of with can lead to frequency and phase locking of self-pulsations to the external modulation. Thus, in this frequency range, the frequency of the laser response can be controlled by the parameter modulation. In the case of frequency locking, the ratio of the two frequencies becomes constant at a rational number. The locked regions in a plane of modulation frequency versus modulation amplitude form the Arnold’s tongues and a transition to chaos via period doubling is observed. Moreover, once

PISARCHIK et al.: EXPERIMENTAL CHARACTERIZATION OF THE BIFURCATION STRUCTURE IN AN ERBIUM-DOPED FIBER LASER

the ratio of and is an irrational number, a quasi-periodic route to chaos is possible. Similar behavior has been observed in other class-B lasers with a saturable absorber and widely discussed in scientific literature (see, for example, [4] and references therein). The detailed study of the frequency-locking regions and possible transitions to chaos is beyond the scope of this paper and will be addressed elsewhere. , the dynamics of the erbium-doped 2) When laser is very similar to that of other class-B lasers with pump modulation, for example, CO [18] and semiconand a bit higher, the ductor [19] lasers. At laser oscillates in the period-1 regime. As is increased, a rich variety of bifurcations are involved in the laser dynamics that gives rise to the appearance of other periodic attractors resulting in bistability and multistability. Among them, we may distinguish two kinds of attractors: the attractors born and dead in SNBs and their subharmonic allies born in period-doubling bifurcations (PDBs) (all labeled as P in Fig. 3), and also the regular period-1 attractors (labeled as R1 and R1 ) and their subharmonic attractors (R2 and R2 ) born in subcritical PDBs. Thus, the R and P attractors have different origins; the former originate from the linear response of the system to the parameter modulation and hence exist at any small values of the modulation amplitude, while the latter appear due to the system nonlinearity. As seen from Fig. 3, the first . Once is decreased, subcritical PDB arises at the R1 and R1 attractors evolve through period-doubling cascades to boundary crises due to collisions with the corresponding unstable periodic orbits [20]. Note that the P2 branch includes a period bubbling, i.e., the period-4 branch (P4 ) born in the supercritical PDB and dead in the subcritical PDB. This regime differs from the ordinary period-4 attractor (P4) which is born in the SNB. At higher modulation depths the period-bubbling cascade of PDBs may appear on P2 and other (P3, P4, etc.) branches culminating chaos. The bifurcation structure can be also visualized with bifurcation diagrams using the modulation depth as a control parameter. Such diagrams for the erbium-doped fiber laser were not previously measured experimentally. In Fig. 4, we plot the dikHz agrams for two different modulation frequencies, [Fig. 4(a)] and 92 kHz [Fig. 4(b)]. These diagrams evidently display the coexistence of multiple attractors in our laser. The targeting of a particular state is realized by switching off and on the signal generator that is equivalent to changing the initial conditions. For small values of the modulation amplitude, the laser response asymptotically approaches the stable limit cycle R1, so that we can consider the response to be linear. As the modulation depth is increased, different periodic and chaotic attractors are born in the SNBs and destroyed together with their basins of attraction in boundary crises. Fig. 5 demonstrates the coexistence of different periodic attractors with the time series. To characterize the dynamical regimes observed, we show in the lower trace the pump modulation signal. One can see that three different regimes, period 1 (R1 ), period 3 (P3), and period 4 (P4), coexist for the same

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Fig. 4. Bifurcation diagrams with modulation depth as a control parameter for modulation frequencies (a) f = 55 kHz and (b) 92 kHz. The dashed line indicates the frequency at which the time series in Fig. 5 are recorded.

Fig. 5. Time series demonstrating coexistence of different dynamical regimes found for parameters marked in Fig. 4(b). The lower trace shows modulation signal.

laser parameters, when the 50% depth modulation is applied at kHz (dashed line in Fig. 4). B. Codimensional-Two Bifurcation Diagram The global structure of bifurcations in the pump-modulated erbium-doped laser can be better understood with the codimen-

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proper control of multistability is organized (see, for example, [21]–[23]). There are a few approaches to modeling an erbium-doped fiber laser [2], [10], but these do not address properly the experimental observation of multiple coexisting attractors in the laser with pump modulation. A new theoretical model of this laser that describes well the main experimental features discussed above will be reported elsewhere.

REFERENCES

Fig. 6. Saddle-node bifurcation lines and dynamical regimes in parameter space of modulation frequency and modulation amplitude. The dashed line indicates fundamental laser frequency f .

sional-two bifurcation diagram in the two-dimensional (2-D) parameter space of the modulation frequency and modulation amplitude (Fig. 6). As seen from this diagram, each subharmonic branch forms a tongue in the parameter space. The overlapping of these tongues at high modulation amplitudes results in generalized multistability. The boundaries between the states of different periodicity are the SNB lines where the corresponding branch is born. For the modulation amplitudes above these lines, we find a rich variety of period-doubling bifurcations which give rise to chaos and crisis of the attractors (not shown). One can see from Fig. 6 that, for modulation frequencies (marked by the below the fundamental laser frequency dashed line), two period-1 attractors coexist. These are the original attractor (R1) which arises as a linear laser response to the periodic excitation and the other period-1 attractor (P1) born in the SNB due to the nonlinearity of the system. The P1 , i.e., the P1 attractor is born in the 2-D branch exists for . Correspondingly, the 2-D SNB SNB, which appears at . However, the point for the P2 attractor is located at attractors of higher orders (P3, P4, etc.) do not obey this rule, because the nonlinearity of the system increases drastically with increasing , so that the SNB lines are crossed in the 2-D SNB points, in which two attractors are born simultaneously. IV. CONCLUSION Global analysis of bifurcation structure of phase space has been performed in a heavily doped erbium fiber laser with harmonic pump modulation. We have demonstrated experimentally a rich variety of bifurcations and coexistence of multiple attractors that appear in the primary saddle-node bifurcations and discussed their relation to the fundamental laser frequency. The systematic organization of coexisting attractors allows us to predict the laser behavior, when initial conditions are allowed to evolve to their final states. The bifurcation analysis of the presented state diagrams can be of interest to experimentalists in order to determine resonance frequencies at which the maximum laser response in different regimes is expected for a given amplitude of the pump modulation. The coexistence of different periodic and chaotic states in the erbium-doped fiber laser may also be prominent for secure optical communications [2] if a

[1] M. Digonnet, Ed., Rare Earth Doped Fiber Lasers and Amplifiers. New York: Marcel Dekker, 1993. [2] L. G. Luo and P. L. Chu, “Optical secure communications with chaotic erbium-doped fiber lasers,” J. Opt. Soc. Amer. B, vol. 15, pp. 2524–2530, 1998. [3] F. T. Arecchi, Instabilities and Chaos in Quantum Optics, F. T. Arecchi and R. G. Harrison, Eds. Berlin, Germany: Springer, 1987, vol. 34, Springer Series on Synergetics, pp. 9–48. [4] C. O. Weiss and R. Vilaseca, Dynamics of Lasers. Weinheim: VCH, 1991. [5] G. Boulant, M. Lefranc, S. Bielawski, and D. Derozier, “Horseshoe templates with global torsion in a driven laser,” Phys. Rev. E, vol. 55, pp. 5082–5091, 1997. [6] D. Dangoisse, J. C. Celet, and P. Glorieux, “Global investigation of the influence of the phase of subharmonic excitation of a driven laser,” Phys. Rev. E, vol. 56, pp. 1396–1406, 1997. [7] T. C. Newell, A. Gavrielides, V. Kovanis, D. Sukow, T. Erneux, and S. A. Glasgow, “Unfolding of the period-doubling bifurcation in a fiber laser pumped with two modulation tones,” Phys. Rev. E, vol. 56, pp. 7223–7231, 1997. [8] L. Luo, T. J. Tee, and P. L. Chu, “Chaotic behavior in erbium-doped fiber-ring lasers,” J. Opt. Soc. Amer. B, vol. 15, pp. 972–978, 1998. , “Bistability of erbium-doped fiber laser,” Opt. Commun., vol. 146, [9] pp. 151–157, 1998. [10] I. J. Sola, J. C. Martín, and J. M. Álvarez, “Nonlinear response of a unidirectional erbium-doped fiber ring laser to a sinusoidally modulated pump power,” Opt. Commun., vol. 212, pp. 359–369, 2002. [11] J. M. Saucedo-Solorio, A. N. Pisarchik, A. V. Kir’yanov, and V. Aboites, “Generalized multistability in a fiber laser with modulated losses,” J. Opt. Soc. Amer. B, vol. 20, pp. 490–496, 2003. [12] E. Eschenazi, H. G. Solari, and R. Gilmore, “Basins of attraction in driven dynamical systems,” Phys. Rev. A, vol. 39, pp. 2609–2627, 1989. [13] F. Sanchez, P. LeBoudec, P.-L. François, and G. Stephan, “Effects of ion pairs on the dynamics of erbium-doped fiber lasers,” Phys. Rev. A, vol. 48, pp. 2220–2229, 1993. [14] E. Lacot, F. Stoeckel, and M. Chenevier, “Dynamics of an erbium-doped fiber laser,” Phys. Rev. A, vol. 49, pp. 3997–4008, 1993. [15] S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett., vol. 21, pp. 1987–1989, 1996. [16] J. Daniel, J.-M. Costa, P. LeBoudec, G. Stephan, and F. Sanchez, “Generalized bistability in an erbium-doped fiber laser,” J. Opt. Soc. Amer. B, vol. 15, pp. 1291–1294, 1998. [17] P. Besnard, F. Ginovart, P. Le Boudec, F. Sanchez, and G. M. Stéphan, “Experimental and theoretical study of bifurcation diagrams of a dual-wavelength erbium-doped fiber laser,” Opt. Commun., vol. 205, pp. 187–195, 2002. [18] A. N. Pisarchik, “CO -laser dynamics by optical modulation of inversion,” Opt. Quantum Electron., vol. 20, pp. 313–321, 1988. [19] A. N. Pisarchik and B. F. Kuntsevich, “Control of multistability in a directly modulated diode laser,” IEEE J. Quantum Electron., vol. 38, pp. 1594–1598, Dec. 2002. [20] C. Grebogi, E. Ott, and J. A. Yorke, “Chaotic attractors in crisis,” Phys. Rev. Lett., vol. 48, pp. 1507–1510, 1982. [21] A. N. Pisarchik and B. K. Goswami, “Annihilation of one of the coexisting attractors in a bistable system,” Phys. Rev. Lett., vol. 84, pp. 1423–1426, 2000. [22] A. N. Pisarchik, “Controlling the multistability of nonlinear systems with coexisting attractors,” Phys. Rev. E, vol. 64, pp. 046 203-1–5, 2001. [23] A. N. Pisarchik, Y. O. Barmenkov, and A. V. Kir’yanov, “Experimental demonstration of attractor annihilation in a multistable fiber laser,” Phys. Rev. E, to be published.

PISARCHIK et al.: EXPERIMENTAL CHARACTERIZATION OF THE BIFURCATION STRUCTURE IN AN ERBIUM-DOPED FIBER LASER

Alexander N. Pisarchik was born in Minsk, U.S.S.R. (now Belarus), in 1954. He received the M.Sc. degree in physics from the Byelorussion State University in 1976 and the Ph.D. degree in physics and mathematics from the Institute of Physics, Belarus Academy of Sciences, Minsk, Belarus, in 1990. He was with the Institute of Physics during 1979–1992. In 1992, he was a Visiting Scientist at the Université Libre de Bruseles, Belgium, a Visiting Professor at the Universitat Autónoma de Barcelona, Spain, from 1993 to 1999, a Research Associate at the University of Iceland, Reykjavik, in 1995, and a Research Engineer at the Monocrom: Laser Applications Company, Villanova, Barcelona, Spain, in 1999. Since 1999, he has been a Research Professor at the Centro de Investigaciones en Optica, Leon, Guanajuato, Mexico, where he is engaged in research and student education in nonlinear laser dynamics. His current research interests are synchronization of coupled oscillators, generalized multistability and dynamical control in lasers. Dr. Pisarchik is a Member of the System of National Researches (SNI, level II), the Society for Industrial and Applied Mathematics (SIAM), the Mexican Academy of Optics, and the National System of Evaluators in Science and Technology (SINECYT). In 1999, he was awarded a First Prize from the Belarus Academy of Sciences for the best research in nonlinear dynamics.

Yuri O. Barmenkov was born in Leningrad, U.S.S.R. (now St. Petersburg, Russia), in 1961. He received the M.Sc. and Ph.D. degrees from St. Petersburg State Technical University, St. Petersburg, Russia, in 1984 and 1991, respectively, all in radiophysics and electronics. His research is dedicated properties of photorefractive media, fiber-optic current, and mechanical sensors. Since 1997, he has been a Research Professor at the Centro de Investigaciones en Optica, Leon, Guanajuato, Mexico. He is a coauthor of 50 scientific publications and two patents. His main research activity includes fiber optic sensors, nonlinear optical media, and fiber lasers. Dr. Barmenkov is a Member of the System of National Researches (SNI, level I).

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Alexander V. Kir’yanov was born in Moscow, U.S.S.R. (now Russia), in 1962. He received the M.Sc. degree from the M.V. Lomonosov State University, Moscow, Russia, in 1986 and the Ph.D. degree in optics and laser physics from the A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Russia, in 1995. He was with the A.M. Prokhorov General Physics Institute during 1987–1998. In 1988, he was a Visiting Scientist/Lecturer at the Central Research Institute for Physics, Hungary Academy of Sciences, Budapest, at the Institute of Material Chemistry, University of Technology, Tampere, Finland, in 1996, at the Lawrence Livermore National Laboratory, Livermore, CA, in 1997, and at the Imperial College, London, U.K., during 2001–2003. Since 1998, he has been a Research Professor at the Centro de Investigaciones en Optica, Leon, Guanajuato, Mexico. He is a coauthor of more than 75 scientific papers, a presenter of contributions at more than 40 international conferences, and a holder of three patents. His research interests are infrared solid-state lasers with passive -switching and passive mode-locking, phase conjugating in solid-state lasers, and nonlinear properties of organic films. Dr. Kir’yanov is a member of the System of National Researches (SNI, level II). In 1993, he was awarded a “Special Individual Grant” from the International Science Foundation (“Soros Foundation”) and in 1998 “The Best Young Scientist Project Award” from the Russian Academy of Sciences.

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