MODULATION instability (MI) is a typical phenomenon

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 16, AUGUST 15, Modulation Instability in Dissipative Soliton Fiber Lasers and Its Application on Cavity Net Dispersion Measurement Junsong Peng, Li Zhan, Zhaochang Gu, Jinmei Liu, Shouyu Luo, Xuehao Shen, and Qishun Shen Abstract Modulation instability (MI) in passively mode-locked dissipative solitons lasers has been studied. The factors that affect MI, including the intensity of the nonlinear wave and the linear phase delay of the cavity, have been experimentally studied. It s found that MI induces sidebands in the spectrum of dissipative solitons. The sidebands can cause the pedestal on the pulse in time domain and thus limit the pulse duration. Additionally, a simple method to eliminate the sidebands is proposed and nearly pedestal free pulses are generated correspondingly. Finally, based on MI, a method to measure the cavity net dispersion is proposed and applied to two dissipative soliton lasers with different net dispersion. It s shown that the positions of the adjacent spectral sidebands can determine the intracavity net dispersion, and the measurement error is limited by the accuracy of the optical spectrum analyzer. This indicates it is a simple and precise method to measure the intracavity net dispersion. Index Terms Dissipative soliton, fiber laser, mode-locking, ultrafast. I. INTRODUCTION MODULATION instability (MI) is a typical phenomenon in physics such as plasma physics, fluid dynamics, and nonlinear optics. It refers to the process in which a weak perturbation of a uniform wave grows exponentially as a result of the interplay between dispersion and nonlinearity ruled by nonlinear Schrödinger equation. In nonlinear optics, anomalous dispersion is required for MI to occur. Due to MI, high repetition rate soliton pulses can be generated [1] [7]. In the presence of normal dispersion, MI can only be generated by additional degree of freedom provided by coupling with another wave of different polarizations or wavelengths [8], [9]. Interestingly, Haelterman et al. suggested theoretically that the extra coupling is not necessary to generate MI in the normal dispersion regime [10]. It was shown that MI can also be observed in the normal dispersion regime when the nonlinear wave is subjected to cavity boundary conditions. This study was demonstrated later by the experiment [11]. The experiment was based Manuscript received March 26, 2012; revised May 23, 2012; accepted June 05, Date of publication June 15, 2012; date of current version August 01, This work was supported in part by the National Natural Science Foundation of China under Grant / , the key project of the Ministry of Education of China under Grant , and the SMC Young Star Scientist Program of Shanghai Jiao Tong University. The authors are with Department of Physics, State Key Lab of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai , China ( Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT on a nonlinear passive ring cavity pumped by a mode-locked laser, with a fiber stretcher to adjust the synchronism between the cavity and the mode-locked laser. The system aims to provide boundary conditions to subject the nonlinear wave. It s expected that MI should also occur in lasers, in which boundary conditions exist automatically. It s well known that dissipative solitons exist in all normal dispersion cavities, and it can also be found in dispersion-managed cavities with net positive dispersion [12] [14], which can be explained by the stability of dissipative solitons under anomalous dispersion perturbations [15]. In this paper, we experimentally studied MI in a passively mode-locked dissipative soliton fiber laser with net positive dispersion. Dissipative solitons with spectral sidebands caused by MI are observed. Previous studies focused on the generation of MI [11], [16], and the dynamics were not studied. We study the dynamics of MI experimentally by tuning the pump power and the linear phase delay of the cavity in the laser, which can be well explained based on the theoretical work of Haelterman et al. [10], [11]. Dissipative solitons were developed extensively in recent years [17], [18]. They have much higher energy than the solitons or dispersion managed solitons. Disadvantages of dissipative solitons are not clear. Here, we show for the first time that that MI degrades the quality of dissipative soliton extensively, as MI induced sidebands cause large pedestal in the pulse wings and broaden the duration of dissipative solitons. This fact is similar to that of the solitons whose duration is broadened by Kelly sidebands. As Kelly sidebands can be removed by a filter in soliton lasers resulting in pedestal free pulses [19], the MI induced sidebands can also be eliminated by the filter which is formed by nonlinear polarization rotation (NPR) in the cavity [12], [20], [21]. The pedestal caused by MI induced sidebands in the wings can be removed correspondingly resulting in nearly pedestal free pulses. Dispersion is a crucial parameter in fiber optics, especially in passively mode-locked fiber lasers [22] [27]. It is inherently important to measure the dispersion of fiber components. Additionally, knowing the exact net dispersion of a laser cavity can also guideline the numerical study of ultrafast fiber lasers. Measuring the dispersion of fibers requires sophisticated techniques [28]. The system relies on a low coherence reflectometer and dispersive Fourier spectroscopy, which is a little complicated for practical applications. Then a new approach to measure dispersion was proposed based on mode-locked Ti: sapphire lasers [29], which is much simpler than previous measurements. This method was then applied to measure fiber dispersion in actively /$ IEEE

2 2708 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 16, AUGUST 15, 2012 Fig. 1. Experimental configuration of the laser. mode-locked fiber lasers [30]. However, this method requires mode-locked lasers with tunable wavelength, and a tunable filter has to be employed [30], which still shows complexity. Kelly sidebands were then found to be available to measure the net dispersion [31]in a simple way, as only spectral sidebands are needed to determine it. Kelly sidebands only generate in anomalous dispersion cavity, thus it s a challenge to measure net dispersion in normal dispersion cavity where Kelly sidebands do not exist. Up to date, there s no corresponding way to measure net dispersion of mode-locked lasers in normal dispersion regime, in contrast to their counterpart in anomalous dispersion regime using Kelly sidebands to determine the net dispersion. In this letter, we show that MI induced sidebands can be used to measure intracavity dispersion of mode-locked lasers in normal dispersion regime. It can be simply calculated from the positions of the adjacent spectral sidebands. As shown later, the measurement error only comes from the accuracy of optical spectrum analyzer (OSA), which can be neglected regarding the resolution of OSA (0.05 nm). To confirm the validity of this method, two mode-locked lasers of different net positive dispersion are constructed and measured, both of which show the feasibility of this method. Additionally, once the net dispersion is determined, the dispersion of other fiber components in the cavity can be deduced. II. EXPERIMENTAL SETUP AND PRINCIPLES Fig. 1 shows the configuration of our Erbium-doped fiber ring laser. The ring cavity is made of a 190 cm EDF (80 db/m peak absorption ratio at 1530 nm), which is forward pumped by a 976 nm laser diode through a 980/1550 wavelength division multiplexer (WDM). An optical coupler (7:93) is located after the EDF to output the signal, and the 7% port is used as the output port. A polarization dependent isolator (PDI) sandwiched with two polarization controllers (PC1 and PC2) is used as the mode-locking component in the cavity. The total length of single-mode fiber (SMF) including above fiber components is 220 cm. Group velocity delay (GVD) parameters of the EDF and SMF are and 18 ps/(nm km), respectively. The net dispersion of the cavity is ps. It s to note that EDF used here is normal dispersion, which is crucial for dissipative soliton generation. The output port is connected to a autocorrelator, an optical spectrum analyzer to monitor the characteristics of output pulses. MI exists in dissipative soliton lasers when the intracavity power is high enough. Spectrum sidebands in other dissipative soliton lasers may also imply MI generation [21], [32], [33], but no attention was paid. Fig. 2. Pulse train observed by the oscilloscope when the laser is mode-locked under the pump power of 745 mw. Mode-locking was initiated by NPR, which relies on intensity dependent rotation of an elliptical polarization state in a length of optical fiber [34]. With proper settings of the initial polarization ellipse and phase bias, pulse shortening occurs. In the experiment, the length of the anomalous dispersion fiber is cut stepwise, and the spectrum is monitored when the laser is mode-locked. As the net dispersion is changed when the length of anomalous dispersion fiber is decreased, different kinds of spectrum corresponding to different mode-locked pulses are generated. For example, dispersion-managed solitons are generated when the net dispersion is close to zero, whose pulse spectrum shape is Gaussian type. We focus on dissipative solitons generation, and they are observed when the net dispersion is increased to ps. NPR needs enough nonlinear phase to act as a saturable absorber. The nonlinear phase is proportional to cavity length. Due to short cavity length, the threshold pump power for mode-locking was as high as 745 mw, which is much higher than other dissipative soliton lasers [35] with long cavity length. However, benefiting from the hysteresis property of mode-locked lasers, the mode-locking state was stable when the pump power was decreased from 745 mw to 393 mw. The pulse trains are shown in Fig. 2 with a repetition rate of MHz, corresponding to the cavity length of 4.44 m. The pulse spectrum was monitored when decreasing the pump power. As shown in Fig. 3(a), the sidebands appear in the spectrum. Obviously, the spectrum is the typical one of dissipative soliton [12], [14], [17], [20], [32], [36] if excluding the new sidebands. These sidebands are reminiscent of Kelly sidebands. However, it is to note that they are not Kelly sidebands in twofold. First, Kelly sidebands exist in net anomalous dispersion cavity. It is well known that cavities consisting of negative and positive dispersion i.e., a dispersion map reduce phase-matched conditions required by Kelly sidebands generation [22], [24], [37]. Our cavity is a typical dispersion map, thus the sidebands cannot be Kelly sidebands. Second, assuming they are Kelly sidebands, the positions of the sidebands can be used to determine the net dispersion in the cavity [31]. Following the method in [31], the net dispersion of our laser is calculated to be anomalous. However, the net dispersion in our cavity is positive (0.079 ps ). Thus, they are not Kelly sidebands. In contrast, as be shown later, sidebands caused

3 PENG et al.: MODULATION INSTABILITY IN DISSIPATIVE SOLITON FIBER LASERS AND ITS APPLICATION ON CAVITY NET DISPERSION MEASUREMENT 2709 Fig. 3. Pulses spectrum scaling with (a) pump power and (b) linear phase delay. by MI can also be used to calculate the net dispersion. Assuming they are MI induced sidebands, the calculated net dispersion (0.08 ps ) is close to ps calculated by adding dispersion parameters of fiber components in the cavity. In other words, the sidebands are caused by MI, and they are not Kelly sidebands. The properties of Kelly sidebands and MI induced sidebands are very different, which will be discussed later. III. MODULATION INSTABILITY IN THE LASER Dynamics of MI are then studied, based on the pump power scaling and the linear phase delay tuning. The locations of the sidebands are pump power dependent as shown in Fig. 3(a). The sidebands shift to longer wavelength continuously as pump power is increased. The first order sidebands under pump power of 393 mw and 745 mw are compared. One (393 mw) is 1582 nm and the other (745 mw) is 1590 nm, indicating 8 nm shift with pump power increments. We note that pump power dependent positions of the sidebands was also observed in [33]. However, the physical mechanism of it was unclear at that time. Here, we explain this mechanism based on MI well, which shows good agreement with the experiment. This property can be well understood based on theoretical work in [11], [38]. The position of the k order sideband is given by where is the GVD, is the frequency of the th-order sideband; is the total cavity round-trip phase shift including linear and nonlinear contributions. is the cavity length and is the central wavelength, is the nonlinear coefficient, and is the intracavity power. When the pump power is increased, the phase shift from nonlinear contribution increases. decreases according to (1) resulting in the red-shift in the wavelength. The wavelength shift due to pump power of the second order sideband is 7.5 nm and is 6.7 nm of the third order sideband as seen in the figure, and it seems that the wavelength shift of the higher order sideband is smaller than the lower order sideband. From (1), one can obtain the derivative of with respect to. Thus, it can be seen that the amplitude of decreases with increasing under the same pump power increment, in other words, the wavelength shift of the higher order (1) sidebands is smaller than the lower order sidebands, which is consistent with the experimental observations. Furthermore, as shown in the figure, the number of sidebands is also pump power dependent. Up to five sidebands are observed under the maximum pump power of 745 mw, only four sidebands being observed when it is decreased to 584 mw, and three sidebands remain as the pump power is reduced to 425 mw. Spectrum broadening of dissipative solitons with pump power increment has been demonstrated theoretically [14] and experimentally [12], [20], [21], which can also be seen in Fig. 3(a). Under high pump power, the spectrum is so wide that it can excite high order sidebands [38]. As the pump power is decreased, the spectral width is decreased and the high order sidebands disappear. Additionally, as seen from (1), the linear phase is also important on the contribution to the total cavity round-trip phase shift, and it can also shift the position of the sidebands. Although the numerical study found that the positions of the sidebands vary with the linear phase delay [39], no experiments have demonstrated this up to date. It s well known that the linear phase delay can be tuned in a mode-locked laser through rotating polarization controllers [40]. The rotating angle should be as small as possible after the laser is mode-locked, to insure that the laser is still in the mode-locking regime. If the rotating angle is too large, mode-locking state breaks down. Fig. 3(b) shows the spectrum scaling with detuning one PC paddle angle clockwise under the pump power of 745 mw. As seen in the figure, the positions of the sidebands shift when rotating the paddle of PC i.e., changing the linear phase delay. Thus, we experimentally demonstrate that the linear phase delay can also shift the position of the sidebands, consistent with the theoretical study as well as numerical investigation [39]. Additionally, the number of sidebands also decreases with the linear phase delay, eventually, all sidebands disappearing. The elimination of sidebands is caused by a spectrum filter in the cavity. The spectrum filter formed by a birefringence plate and a polarizer has been demonstrated in a laser [41]. Here, a spectrum filter with tunable bandwidth exists in the laser automatically formed by PDI and fiber birefringence, and the filter bandwidth depends on the birefringence of the fiber resulting in tunable spectrum width of the pulses, which has been demonstrated experimentally [12], [20], [21]. In the mode-locked lasers based on NPR, fiber is wrapped around the two PCs, which generates bending- induced birefringence [42]. When tuning the PC, the birefringence magnitude is changed resulting in changing the bandwidth of the filter. It can also be seen in Fig. 3(b) that the spectrum bandwidth decrease with PC tuning, and the sidebands can be filtered when the bandwidth of the filter decreases to certain value, which is similar to the process through optimizing the bandwidth of the filter to eliminate Kelly sidebands [41]. Thus, the linear phase delay not only plays roles in (1) but also influences the bandwidth of the spectrum filter in the cavity, resulting in shift the positions of the sidebands and elimination of them respectively. The impact of MI on dissipative solitons is further experimentally studied. The sidebands induced by MI result in the pedestal on the pulse and broadening the pulse, similar to Kelly sidebands. The autocorrelation trace of the initial pulse is shown in Fig. 4(a). The pulse duration is 1.8 ps if assuming a Gaussian shape pulse. Compressor made of SMF was then added outside

4 2710 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 16, AUGUST 15, 2012 Fig. 4. Autocorrelation trace of the pulses with sidebands (black) and when sidebands are eliminated by a filter (red). The four figures correspond to different length of the compressor (SMF) outside of the cavity. Fig. 5. Optical spectrum of the dissipative soliton under the pump power of 500 mw. of the cavity to compress the pulse through cut-back method. Fig. 4(b) shows the autocorrelation trace after SMF (5 m) is added. The black curve is the trace of the pulse with MI induced sidebands and the red one is the pulse whose spectral sidebands are eliminated by the spectrum filter. Obviously, the black curve has two-stage pedestal and only single stage pedestal is present in the red curve. This indicates that MI induced sidebands cause additional pedestal on the pulse and can be removed through the filter. The single stage pedestal caused by the chirp from the over length compressor (SMF) can be removed by further cutting the length of compressor, as shown in Fig. 4(c) (red curve). In contrast, the black curve still shows a large pedestal in the figure, which is caused by the spectral sidebands. Further cutting the SMF can get the shortest pulse as shown in Fig. 4(d). The shortest pulse duration is 182-fs if assuming a Gaussian shape pulse. The little pedestal in the red curve is caused by nonlinear chirp [43]. Large pedestal is still present in the black curve, and the pulse duration is 228-fs. Apparently, MI induced sidebands cause the pedestal in the pulse and can be removed through a spectrum filter to generate cleaner and shorter pulses. IV. CAVITY NET DISPERSION MEASUREMENT Net dispersion is a critical parameter in designing modelocked fiber lasers. Kelly sidebands can be used to measure the net dispersion of mode-locked fiber lasers with anomalous dispersion [31], but Kelly sidebands don t exist in all normal or large net normal dispersion cavities, and new methods are needed in such cavities. We propose a corresponding way to measure the net dispersion of net normal dispersion cavities based on MI induced sidebands. From (1), one can obtain a relation between the adjacent sidebands Finally, net dispersion can be represented by, and one can see that net dispersion can be measured when the positions of adjacent sidebands are determined. This was examined experimentally. Fig. 5 is chosen from Fig. 3(a). The central wavelength, first order, second order, (2) Fig. 6. Optical spectrum of the dissipative soliton when the net dispersion is increased under the pump power of 1.3 W. and third order sideband are marked by in the figure. The locations of them are 1565 nm, 1582 nm, nm, and nm, respectively. The net dispersion can be calculated with adjacent sidebands such as first and second order sidebands, second order and third order sidebands. The net dispersion is 0.08 ps deduced from first and second order sidebands, and it is ps deduced from the second order and third order sideband. The final result is ps, which is close to ps estimated by adding the dispersion parameters of the different components in the cavity. To further demonstrate the validity of this method, we construct another mode-locked dissipative soliton laser with different net dispersion. 2.5 m EDF with GVD of ps/(nm km) at 1550 nm was employed to insure large net normal dispersion. The total length of the cavity is 7.2 m optimized by cut-back methods, and dissipative solitons are observed as shown in Fig. 6, the spectrum of which is a typical one of dissipative solitons. The central wavelength, k order, order, order sideband are marked by in the figure. The locations of them are 1565 nm, nm, 1595 nm, and nm as seen in the figure. The net dispersion is ps calculated from and

5 PENG et al.: MODULATION INSTABILITY IN DISSIPATIVE SOLITON FIBER LASERS AND ITS APPLICATION ON CAVITY NET DISPERSION MEASUREMENT 2711 order sidebands, and it is ps deduced from and order sidebands. The final result is ps, which is close to 0.12 ps estimated from the GVD of different components in the cavity. It is to note that the instrument measurement error of this method only comes from the accuracy of OSA as only the positions of the adjacent sidebands are needed to calculate the net dispersion. The resolution of OSA in our lab is 0.05 nm, as a result, the measurement error of the net dispersion in the first laser (0.08 ps )is ps, and in the other laser (0.13 ps )is ps. The measurement error is so small that can be neglected, indicating high accuracy of this method. Note that based on this cavity net dispersion measurement method, measuring the dispersion parameter of fiber components becomes much more convenient than previous studies [28]. A fiber component with unknown dispersion parameter can be added to a laser cavity in which the dispersion parameters of the other components are known, and the net dispersion is managed until dissipative solitons are generated. The net cavity dispersion can be calculated from the sidebands of the dissipative solitons. Once the net dispersion is known, the unfixed dispersion parameter the fiber component can be determined correspondingly. It is necessary to compare Kelly sidebands with MI induced sidebands. First, the formation mechanisms are different. The former is a linear effect caused by dispersive waves interfering shed by solitons, which has been extensively studied [19], [41], [44] [46]. The later is a nonlinear effect called MI, and the sidebands are generated by nonlinear gain other than interfering. Second, Kelly sidebands are only observed in anomalous dispersion regime and MI induced sidebands can be observed both in anomalous dispersion [16] and large net normal dispersion regime. Third, the positions of Kelly sidebands are mainly determined by dispersion. Theoretically, the positions of Kelly sidebands also depend on pump power. The position of the sideband depends on the pulse duration, but the pulse duration varies with energy i.e., the pump power. However, no experiment has demonstrated this up to date. MI induced sidebands not only depend on dispersion but also pump power and linear phase delay. Both of the two sidebands can be used to measure the net dispersion of mode-locked lasers, and limit the pulse duration, which are something in common. Mode-locked fiber lasers are excellent tools to study nonlinear phenomena. In contrast to solid lasers, nonlinearity, dispersion, and boundary conditions coexist in mode-locked fiber lasers simultaneously, which helps to generate kinds of nonlinear phenomenon. Through parameters management in the cavity such as dispersion, nonlinearity as well, desired nonlinear effects can be observed. In this work, we aim to observe MI in dissipative soliton lasers, and benefiting from dispersion management and nonlinearity increment by large pump power, MI is observed. Dynamics of MI accompanying with pump power and the linear phase delay is shown experimentally, which can be explained well based on the theoretical work in [11], [38]. Furthermore, we find that though dissipative solitons progress extensively nowadays, MI ultimately degrades the pulse quality with large pedestal in the wings. A method to eliminate the pedestal is also demonstrated, which relies on a spectrum filter in the laser. V. CONCLUSION In conclusion, we have studied MI in dissipative soliton lasers experimentally. MI induces sidebands in the spectrum of dissipative solitons, and the sidebands show different properties from Kelly sidebands. The positions of the sidebands depend on the pump power as well as the linear phase delay. Furthermore, MI induced spectral sidebands broaden the pulse duration, and the pedestal can be removed by a spectrum filter in the laser resulting in nearly pedestal free pulses. Additionally, the sidebands can be used to calculate the net dispersion of the laser. The different net dispersion of two dissipative soliton lasers is determined with high accuracy, which shows the possibility of this new method. The method can be further used to measure the dispersion parameter of other fiber components, which is much more convenient to operate. Such study is under investigation. REFERENCES [1] E. Yoshida and M. 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Eickhoff, Bending-induced birefringence in single-mode fibers, Opt. Lett., vol. 5, pp , [43] X. Liu, Numerical and experimental investigation of dissipative solitons in passively mode-locked fiber lasers with large net-normal- dispersion and high nonlinearity, Opt. Exp., vol. 17, pp , [44] K. Tamura, L. E. Nelson, H. A. Haus, and E. P. Ippen, Soliton versus nonsoliton operation of fiber ring lasers, Appl. Phys. Lett., vol. 64, pp , [45] S. Kelly, Characteristic sideband instability of periodically amplified average soliton, Electron. Lett., vol. 28, pp , [46] M. L. Dennis and I. N. Duling, III, Experimental study of sideband generation in femtosecond fiber lasers, IEEE J. Quantum Electron., vol. 30, no. 6, pp , Jun Junsong Peng received the B.S. degree in the science and technology of optics information from Anhui University, Hefei, China, in He is currently working toward the Ph.D. degree in optics engineering at Shanghai Jiao Tong University, Shanghai, China. His current research focuses on mode-locked fiber lasers, nonlinear fiber optics and optical communications. Li Zhan received the B.S. and M.S. degrees from Shanghai Jiao Tong University, Shanghai, China, in 1990 and 1993, respectively, and the Ph.D. degree from City University of Hong Kong in He is currently a Professor with the Department of Physics, Shanghai Jiao Tong University, Shanghai, China. His research interests include fiber lasers, fast/slow light, nonlinear optics, and plasmonics. Zhaochang Gu, biography not available at the time of publication. Jinmet Liu, biography not available at the time of publication. Shouyu Luo, biography not available at the time of publication. Xuehao Shen, biography not available at the time of publication. Qishun Shen, biography not available at the time of publication.

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