We report on the experimental observation of period-doubling bifurcation and period-doubling route to chaos in a femtosecond soliton fiber laser passively mode locked by using the nonlinear polarization rotation technique. Increasing energy of the solitons circulating in the laser cavity, it was revealed that the intensity pattern of the output solitons experiences a period doubling route to chaos. Period-doubling route to chaos is a universal property of the nonlinear dynamic systems transiting from a stable state to a chaotic state. This experimental result shows that the nonlinear propagation of soliton pulses in the laser cavity is an intrinsic dynamic process, which follows the universal laws of the nonlinear dynamic systems.
© 2004 Optical Society of America
Due to their excellent stability and compact size, soliton fiber lasers have attracted greatly attention as an alternative source to generate ultrashort optical pulses. Femtosecond optical pulses are now routinely generated in different types of fiber lasers passively mode-locked by various techniques, such as the stretched-pulse fiber lasers with a nonlinear optical loop mirror , figure of eight fiber lasers  and fiber ring lasers mode locked with the nonlinear polarization rotation (NPR) technique . Among the various passive mode-locking techniques, the NPR technique is widely used. The technique exploits the nonlinear birefringence of the single mode optical fibers and uses it to generate an artificial saturable absorber effect in the laser cavity for a self-started mode locking. To explain the operation principle of the technique, it is to note that generally the polarization state of light passing through a piece of birefringent fiber varies linearly with the fiber birefringence and length. However, when the light intensity is strong enough, the nonlinear optical Kerr effect existing in the fiber introduces extra changes to the light polarization. As the extra polarization changes are proportional to the light intensity, if a polarizer is placed behind the fiber, then through appropriately selecting the orientation of the polarizer, an effect that light with higher intensity experiences larger transmission through the polarizer could be obtained. Such an effect is known as an artificial saturable absorber effect. Incorporating such an artificial saturable absorber effect in the fiber laser cavity results in the passive mode locking of a laser. After mode locking a strong optical pulse is formed in a laser and circulates in the cavity. If the nonlinear phase modulation of the pulse could balance the pulse width broadening caused by the cavity dispersion, soliton is then obtained in the laser.
Suggested by the average soliton theory of lasers , it is generally believed that the output of the fiber soliton lasers is a uniform soliton pulse train. However, recently Kim et al. theoretically studied the pulse dynamics of the fiber lasers passively mode locked by the NPR technique [5, 6]. They found that depending on the strength of the fiber birefringence and the alignment of the polarizer with the fast- and slow-polarization axes of the fiber, the train of output pulses exhibits periodic fluctuations in intensity and polarization. Nevertheless, the nonuniformity of the pulse trains could be diminished by aligning the polarizer with either the fast or the slow axis of the fiber. We have experimentally investigated the output property of a fiber soliton ring laser passively mode-locked by using the NPR technique  and found that the soliton pulse nonuniformity is in fact an intrinsic feature of the laser, whose appearance is independent of the orientation of the polarizer in the cavity but closely related to the pump power. Based on numerical simulations we showed that depending on the linear cavity phase delay bias, the nonlinear polarization switching effect could play an important role on the soliton dynamics of the laser. When the linear cavity phase delay bias is set close to the nonlinear polarization switching point and the pump power is strong, the soliton pulse peak intensity could increase to so high that the generated NPR cross over the nonlinear polarization switching point, and consequently drive the laser cavity from the positive feedback regime to the negative feedback regime. Eventually the competition between the soliton pulses with the linear waves in the cavity such as the dispersive waves or CW laser emission then causes the amplitude of the soliton pulses to vary periodically. There are two methods to suppress the periodical intensity fluctuations: one is to reduce the pump power so that the peak intensity of the solitons is below the polarization switch threshold; the other is to increase the polarization switching power of a laser. However, the latter method needs to appropriately adjust the linear phase delay bias of the cavity.
In this paper we report on another type of pulse train nonuniformity existing in soliton fiber ring laser. We show experimentally that as a result of the nonlinear propagation of an intense soliton pulse in the laser cavity, the output soliton intensity pattern experiences period-doubling bifurcations and period-doubling route to chaos. Period-doubling route to chaos is a well-known universal feature of nonlinear dynamic systems transiting from a stable state to a chaotic state, which has been extensively investigated [8 and Ref. therein]. Although period doubling and quasi-periodicity have been previously observed in an additive-pulse mode-locked F-center laser , and Tamura et al. have also mentioned the observation of period doubling and tripling in a soliton fiber ring laser with a broad filter in cavity , to our knowledge so far no experimental observation on a complete route of period doubling to chaos in passively mode locked fiber soliton lasers was reported. Our experimental result suggests that in the case of an intense optical pulse circulating in the laser cavity, the nonlinear pulse propagation and interaction with the cavity components is intrinsically a nonlinear dynamic process, which follows the universal dynamics of the nonlinear dynamic systems.
Our experimental setup is shown in Fig. 1. The fiber laser has a ring cavity of about 12-meter long. The cavity comprises a 2-meter-long 8000ppm erbium doped fiber with a group velocity dispersion of about +70 ps/nm km to achieve dispersion-managed cavity, other fibers used are all standard single mode fiber (SM28). The NPR technique is used to achieve the self-started mode locking. An isolator is inserted in the cavity to allow the unidirectional operation of the laser. Two polarization controllers, one consisting of two quarter-wave plates and the other two quarter-wave plates and one half-wave plate, are used to adjust the polarization of the light. The polarization controllers and a beam splitter are mounted on a 7-cm-long fiber bench. The laser is pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. The soliton propagation in the laser cavity is monitored via the beam splitter and analyzed with an optical spectrum analyzer (Ando AQ-6315B). A 26.5 GHz rf spectrum analyzer (Agilent E4407B ESA-E SERIES) and a 350 MHz oscilloscope (Agilent 54641 A) together with a 7 GHz photodetector are used to simultaneously monitor the intensity spectrum and periodicity of the soliton train formed in the cavity. A commercial optical autocorrelator (Autocorrelator Pulsescope) has also been used to measure the pulse width of the soliton pulses.
Compared with the pulse propagation in an equivalent cavity of uniform dispersion, in a dispersion-managed cavity a pulse extends its pulse width in the positive group velocity dispersion fiber and therefore can be efficiently amplified . Therefore, the energy of solitons formed in a dispersion-managed cavity can be much greater than that of solitons in an equivalent uniform-dispersion cavity . Provided that the orientations of the polarization controllers are appropriately set, self-started soliton operation of the laser is automatically obtained by simply increasing the pump power beyond the mode-locking threshold. Multiple solitons are initially obtained. However, decreasing the pump power the state with only one soliton existing in the cavity can always be achieved. The fundamental repetition rate of the laser is 17.4 MHz.
Starting from a stable soliton operation state, experimentally it was noticed that turning the orientation of one of the quarter-wave plates to one direction, which theoretically corresponds to shifting the linear cavity phase delay bias away from the nonlinear polarization switching point, the peak power of the soliton pulse formed in the cavity increases. Consequently, the strength of nonlinear interaction of the soliton pulses with the cavity components such as the optical fiber and the gain medium increases. To a certain level of the nonlinear interaction, it was observed that the output soliton intensity pattern of the laser experiences period doubling bifurcation and period doubling route to chaos. Figure 2 shows as example an experimentally observed period doubling route to chaos. The results shown in Fig. 2 were obtained with fixed linear cavity phase delay bias but increasing pump power. At a relatively weak pump power, a stable soliton pulse train with uniform pulse intensity was obtained. The pulses repeat themselves with the cavity fundamental repetition rate (Fig. 2(a)). We have experimentally measured the laser output power when it is operating in such a state. With a pump power of about 26 mW an average output power of about 140 μW was obtained, which gives that the single soliton pulse energy is about 8.05 pJ. Carefully increasing the pump power further, the intensity of the soliton pulse becomes no longer uniform, but alternates between two values (Fig. 2(b)). Although the round trip time of the solitons circulating in the cavity is still the same, the pulse energy returns back only every two round-trips, forming a so-called period-doubled state as compared with that of Fig 2(a). Further slightly increasing the pump power, a period-quadrupled state then appears (Fig. 2(c)). Eventually the process ends up with a chaotic soliton pulse energy variation state (Fig. 2(d)).
With fixed pump power the states of period-one, period-two and chaos are very stable. Provided there are no great disturbances they can last for several hours. Careful control on the pump power is required to obtain the period-four state. It was also confirmed experimentally by combined use of the autocorrelator (PulseScope, scan range varies from 500 fs to 50 ps) and a high-speed oscilloscope (Agilent 86100A 50 GHz) that there is only one soliton existing in the cavity. Limited by the resolution of our autocorrelator the measured autocorrelation traces under different states show no unusual features of the soliton pulses and thus give no evidence of the behavior of period doubling bifurcations, which is similar as also observed by G. Sucha. et al. . In all states the average soliton duration measured was about 316±10 fs. Our experimental results demonstrate that contrary to the general understanding to the mode locked lasers, after one round-trip the mode locked pulse does not return to its original value, but does it in every two or four round trips in the stable periodic states. Depending on the strength of the nonlinear interaction between the pulse and the cavity components, the pulse could even never return back to its original state in the chaotic state. To exclude any possibility of artificial digital sampling effect of the oscilloscope, we have checked the pulse intensity alternation of the various periodic states by using a high-speed sampling oscilloscope (Agilent 86100A 50 GHz). Figure 3 shows the result corresponding to a period-2 state. In obtaining the figure we used the soliton pulse itself as the trigger for the oscilloscope and a high oscilloscope resolution (50ps/div). Due to the high scan speed of the oscilloscope, we could clearly see that the individual soliton pulse trace on the screen now becomes broader. Therefore, no sampling problem exists. Triggered by different pulses the oscilloscope traces formed have two distinct peak intensities, indicating that the solitons in the laser output have indeed two different pulse energies.
The optical spectra of the solitons corresponding to the period-one and period-two states are shown in Fig. 4(a) and Fig. 4(b), respectively. While the spectral curve shown in figure 4a is smooth, the spectral curve shown in Fig. 4(b) exhibits clear modulations. The spectral curve shown in Fig. 4(a) possesses typical features of the soliton spectra of the passively mode-locked fiber lasers, characterized by the existence of sidebands superposing on the soliton spectrum. As in a period-one state solitons are identical in the soliton train, the spectrum shown in figure 4a is also the optical spectrum of each individual soliton. In contrast, the optical spectrum shown in Fig. 4(b) is an average of the spectra of two different solitons, each with different energy and frequency chirps. Based on Fig. 4(b) it is to conclude that after a period-doubling bifurcation, the solitons possess different frequency property as that of the solitons before bifurcation.
We have also measured the intensity frequency distribution of the soliton train with a rf spectrum analyzer. If period doubling does occur, there should appear a new frequency component in the rf spectrum locating exactly at the half of the fundamental cavity repetition rate position. Fig. 5 shows the rf spectra of the laser output measured. As expected, after a period-doubling bifurcation a new frequency component of about 8.7 MHz appears in the spectrum. The amplitude of the new frequency component is quite strong compared to the fundamental frequency component, which vividly suggests that the soliton peak intensity alternates between two values with large difference. When period quadrupling occurs, in the rf spectrum we found that the amplitude of the new frequency component decreased, however, the frequency components correspond to the period quadrupling were too weak to be clearly distinguished from the background noise.
It is to note that with the selection of linear cavity phase delay bias close to the nonlinear polarization switching point of the cavity, single soliton operation of the laser can still be obtained. However, because of the peak intensity of the soliton pulses is limited by the nonlinear polarization switching power, which with the linear cavity phase delay bias selection is weak, no matter how strong the pump power is, no period-doubling bifurcation could be observed. There also exists a threshold for the occurrence of the period doubling bifurcation. Only when the linear cavity phase delay bias, which determines the stable soliton peak intensity, is appropriately set so that the stable soliton peak intensity exceeds a certain value, period doubling bifurcation can be achieved. The experimental result further confirms that the appearance of the period-doubling bifurcations and period-doubling route to chaos is the soliton pulse intensity dependent, and it is a nonlinear dynamic feature of the laser. The detailed mechanism for the observed period doubling bifurcations in the femtosecond fiber soliton laser needs to be further investigated, possibly in combination of appropriate theoretical modeling. One thing confirmed in our experiment is that for the occurrence of the effect, the soliton pulse energy or peak power must be strong. It is to imagine that in this case the nonlinear interaction between light and the gain medium, light and the nonlinear laser cavity will also become strong. It is well-known that as a result of strong nonlinear interaction between light and gain medium in the laser cavity, a laser operating in the CW or Q-switched mode can exhibit period-doubling route to chaos. Our experimental result now further demonstrated that this phenomenon can even appear in a mode locked soliton laser. Finally, we point out that Daniel Côté et al. have reported period doubling of a femtosecond Ti:sapphire laser by total mode locking of the TEM00 and TEM01 modes in an effective confocal cavity . They believe that the gain saturation is a likely mechanism to support the transverse mode locking and the period-doubling. However, in our laser there only exists one mode due to the characteristic of the single mode fibers.
In conclusion, we have experimentally observed period-doubling bifurcations and period-doubling route to chaos on the output intensity pattern of a fiber soliton laser passively mode-locked by using the NPR technique. Through selecting the linear cavity phase delay bias far away from the nonlinear polarization switching point of the cavity, soliton with large pulse energy could be obtained in the laser. Furthermore, as a consequence of the strong nonlinear interaction between the intense optical pulse and the laser cavity components, period doubling bifurcation and period-doubling route to chaos occur in the laser. This experimental result demonstrates the even the nonlinear pulse propagation in the laser cavity is a dynamic process, which follows the universal dynamics of the nonlinear dynamic systems.
References and Links
1. F. Ö. Ilday, F. W. Wise, and T. Sosnowski, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Opt. Lett. 27, 1531–1533 (2002). [CrossRef]
2. M. J. Guy, D. U. Noske, and J. R. Taylor, “Generation of femtosecond soliton pulses by passive mode locking of an ytterbium-erbium figure-of-eight fiber laser,” Opt. Lett. 18, 1447–1449 (1993). [CrossRef] [PubMed]
3. D. Y. Tang, W. S. Man, H. Y. Tam, and M. S. Demokan, “Modulational instability in a fiber siliton ring laser induced by periodic dispersion variation,” Phys. Rev. A 61, 023804 (2000). [CrossRef]
5. A. D. Kim, J. N. Kutz, and D. J. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000). [CrossRef]
6. Kristin M. Spaulding, Darryl H. Yong, Arnold D. Kim, and J. Nathan Kutz, “Nonlinear dynamics of mode-locking optical fiber ring lasers,” J. Opt. Soc. Am. B 19, 1045–054 (2002). [CrossRef]
7. B. Zhao, D. Y. Tang, L. M. Zhao, and P. Shum, “Pulse-train nonuniformity in a fiber soliton ring laser mode-locked by using the nonlinear polarization rotation technique,” Phys. Rev. A 69, 043808 (2004). [CrossRef]
8. R. Braun, F. Feudel, and P. Guzdar, “Route to chaos for a two-dimensional externally driven flow,” Phys. Rev. E 58, 1927–1932 (1998). [CrossRef]
10. K. Tamura, C. R. Doerr, H. A. Haus, and E. P. Ippen, “Soliton fiber ring laser stabilization and tuning with a broad intracavity filter,” IEEE Phot. Tech. Lett. 6, 697–699 (1994). [CrossRef]
12. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996). [CrossRef]
13. Daniel Côté and Henry M. van Driel, “Period doubling of a femtosecond Ti:sapphire laser by total mode locking,” Opt. Lett. 23, 715–717 (1998). [CrossRef]