SOLITON DYNAMICS IN PASSIVELY MODE-LOCKED FIBER LASERS ZHAO LUMING 2006

Transcription

1 SOLITON DYNAMICS IN PASSIVELY MODE-LOCKED FIBER LASERS ZHAO LUMING 2006 SOLITON DYNAMICS IN PASSIVELY MODE- LOCKED FIBER LASERS ZHAO LUMING SCHOOL OF ELECTRICAL AND ELECTRONIC ENGINEERING 2006

2 Soliton Dynamics in Passively Mode-locked Fiber Lasers Zhao Luming School of Electrical and Electronic Engineering A thesis submitted to the Nanyang Technological University in fulfilment of the requirement for the degree of Doctor of Philosophy 2006

3 ACKNOWLEDGEMENTS First and foremost, I would like to thank Dr. Tang Dingyuan, my supervisor, for being an excellent advisor and a friend. He led me into the world of ultrashort pulses and always guided me the right way to the truth. It is impossible to accomplish my work without his guidance. I hope to have learned from his non-scientific skills as well, in particular his communication and management skills. I thank him for staying late in the evening for discussions on my experimental results and his patience for listening to my immature thought. Finally, I thank Dr. Tang Dingyuan for showing me a wonderful world of various nonlinear phenomena. I thank A/P. Shum Ping, the director of Network Technology Research Center, who kindly allowed me to join the PhRC-NTRC Communication Joint Lab, which greatly facilitated my work. I thank to all past and present members of Dr. Tang s research group that I have overlapped with for their company. Firstly, I am grateful to Dr. Kong Jian and Dr. Ng SiangPing, who enhance my understanding on solid-state lasers. Secondly, I thank Dr. Zhao Bin for initiating my detailed knowledge on soliton fiber lasers. I am grateful to the technicians in Photonics Lab 1 and Network Technology Research Center who help me to locate various devices I need in my work. i

4 Lastly, I thank my wife, Zhao Jie, who takes good care of me and supports me with all her heart. Nanyang Technological University provided the primary support for this work. I am grateful for the support, without it this work would not have been possible. In grateful dedication to my parents, Zhao Youling & Chen Dongsheng and my grandparents, Zhao Zongli & Chen Sujuan. ii

5 Table of contents TABLE OF CONTENTS ACKNOWLEDGEMENTS... i TABLE OF CONTENTS... iii SUMMARY... vi LIST OF FIGURES... x 1. Introduction Motivation Objectives Major contributions of the research Organization of the thesis Theory of pulse propagation in optical fibers Pulse propagation in optical fibers Fiber dispersion and birefringence Fiber nonlinearity Pulse propagation in single mode fibers Gain and gain dispersion of Erbium-doped fiber Numerical simulation method Cavity transmission analysis and soliton operation regimes Nonlinear polarization rotation mode locking Cavity transmission analysis Experimental implementation of the NPR mode locking iii

6 Table of contents 3.4 Modeling of the soliton fiber lasers Cavity-determined operation regimes Multiple soliton generation and soliton energy quantization Experimental observations Numerical simulations and results Mechanism of the multiple soliton generation and soliton energy quantization Summary Multi-pulse solitons Experimental results Mechanism of the multi-pulse soliton formation Summary Period doubling bifurcations and period doubling route to chaos Experimental observations on the single-pulse solitons Numerical demonstrations on the single-pulse solitons Experimental observations on the bound solitons Numerical evidences on the bound solitons Period doubling bifurcations in non-dispersion-managed fiber lasers Summary Broadband noise-like pulses Soliton collapse and bunched noise-like pulse generation iv

7 Table of contents Experimental observations Numerical simulations Noise-like pulse in non-dispersion-managed fiber lasers Numerical prediction Experimental demonstrations Summary Self-started high-repetition-rate soliton sources Theoretical model and simulation results Experimental demonstration Summary Conclusion and Recommendations Conclusion Recommendations for future work Author s publications Bibliography v

8 Summary SUMMARY Generation of ultrashort optical pulses is a rapidly evolving field including many scientific fundamental researches and various applications. Due to its compact structure, convenient collimation, flexible design, and improving pulse intensity of the generated pulses, fiber lasers have attracted more and more attention as a simple inexpensive ultrashort pulse source. Although the fiber lasers have been intensively studied, many features of the lasers are still unrevealed. In this thesis, a comprehensive experimental and theoretical investigation on the femtosecond soliton fiber lasers has been done. The primary emphasis is the nonlinear dynamics of the fiber lasers. Since even shorter pulse width and higher peak intensity are the ultimate objective of the ultrafast laser development, nonlinear dynamics related to the mutual interaction between the cavity components and the formed pulses of strong peak power is unavoidable. A thorough understanding of these dynamics has been indispensable for possible applications and fundamental researches. First of all, a passively mode-locked Erbium-doped fiber laser (PMEFL) with dispersionmanaged cavity exploiting the nonlinear polarization rotation technique was set up. The laser is operated in the negative net cavity dispersion regime. Various soliton operations of the laser were extensively investigated. Self-started mode locking was obtained in the laser simply by increasing the pump power provided the polarization state of the light was appropriately set. Upon mode locking, stable soliton operations of the laser could be achieved. Comparing with the solitons generated in the non-dispersion-managed fiber lasers, the solitons formed in the laser could have narrower pulse width and higher pulse vi

9 Summary intensity, which consequently results in richer nonlinear dynamics: multi-pulse solitons, period doubling bifurcations, period doubling route to chaos and so on. A pulse tracing technique is adopted to describe the pulse evolution in the laser. The evolution of optical pulses in the presence of temporal dispersion and nonlinear response of various fibers and cavity components, coupled with the feedback nature of the laser, is numerically simulated. Various features of the fiber soliton lasers have been investigated both experimentally and numerically. Multiple soliton generation and soliton energy quantization are well known phenomena for the fiber soliton lasers. However, their physical mechanisms were unexplained. We have shown numerically that the formation of multiple solitons in the lasers is caused by a peak power limiting effect of the laser cavity. It is also the same effect that suppresses the soliton pulse collapse, an intrinsic feature of solitons propagating in the gain media, and makes the solitons stable in the laser. Furthermore, we have shown that the soliton energy quantization observed is a natural consequence of the gain competition between the multiple solitons. Multi-pulse solitons were experimentally observed in a dispersion managed fiber laser. It was found that although formed under different experimental conditions, the various multi-pulse solitons (twin-pulse, three-pulse, four-pulse etc.) have exactly the same fixed pulse separation. Bound states of the multi-pulse solitons were also experimentally obtained, which suggested that the binding force between the pulses is strong, so the multi-pulse entity can function as a unit. vii

10 Summary For the first time, we have experimentally observed soliton period doubling and perioddoubling route to chaos on the soliton fiber lasers. In a soliton period-2 state of the laser operation, the soliton parameters return to their original values after every second round of circulation in the cavity, instead of after every one round-trip in the period-1 state. As period doubling and route to chaos are typical features of the nonlinear dynamic systems, this experimental result shows that the soliton propagation in the laser cavity itself is a nonlinear dynamic process. Numerical simulations on the soliton propagation in the lasers have well reproduced experimental observations. Period doubling and route to chaos of the bound solitons have also been experimentally observed and numerically verified. Apart from the soliton operation of the fiber lasers, a kind of noise-like pulse emission was also revealed. The optical spectrum of the laser emission could be as broad as nm, and by purely change the linear cavity phase delay bias, the laser emission could be switched between soliton operation and the noise-like pulse emission. Numerical simulation of the phenomenon turned out that such a laser emission was caused by the combined effect of soliton collapse and positive cavity feedback. Guided by the numerical simulation, and also taking the advantage of the soliton self-frequency shift (SSFS), we have also specially designed a fiber laser that can generate large energy (15 nj) super-broadband (93nm) noise-like pulse emission. Ultrahigh repetition rate pulse source is useful for the optical communications. Based on a traditional PMEFL we have also designed a self-started high-repetition-rate soliton viii

11 Summary fiber laser and obtained soliton pulse train with 100GHz repetition rate. The technique we used exploits the resonant effects of the self-induced modulational-instability and the cavity spectral filter. In our fiber laser this was implemented by inserting a segment of high-birefringence fiber in the cavity, due to the birefringence filter effect of the cavity and modulational instability gain, a high-repetition-rate ultrashort pulse train is then formed. With this technique, repetition rate up to THz could be realized provided the gain peak can match the free spectral range introduced by the high-birefringence fiber. Numerical simulations agree well with our experimental observations. ix

12 List of figures LIST OF FIGURES Fig. 2.1 Schematic illustration of the symmetrical split-step Fourier method Fig. 3.1 Schematic of mode locking in fiber lasers Fig. 3.2 Cavity transmission versus the phase delay between the two orthogonal polarization components as the relative orientation between the polarization directions of the polarizer and the analyzer varies from 0 to π/2. θ is fixed as π/ Fig. 3.3 Schematic of the experiment setup. λ/4 represents the quarter-wave plate; λ/2 represents the half-wave plate; BS represents the polarization beam splitter; WDM represents the wavelength-division multiplexer Fig. 3.4 Solitons (a) and corresponding spectra (b) with different linear cavity phase delay bias Fig. 3.5 Period doubling of soliton pulse train Fig. 3.6 Noiselike pulse when the linear cavity phase delay bias is 1.7π Fig. 4.1 A schematic of the soliton fiber laser. PI: Polarization dependent isolator. PC: Polarization controller. DSF: Dispersion shifted fiber. EDF: Erbium-doped fiber. WDM: Wavelength-division-multiplexer Fig. 4.2 A typical experimentally measured oscilloscope trace of the multiple soliton operation of the laser Fig. 4.3 Numerically calculated multiple soliton operation state of the laser. δφl = 1.20π, G=350. Other parameters used are described in the text Fig. 4.4 Relationship between the soliton number in the simulation window and the pump strength. δφl = 1.20π Fig. 4.5 Soliton shaping of the mode-locked pulse in the laser. δφl = 1.20π. (a) Evolution of pulse profile with the pump strength. (b) Evolution of the optical spectra with the pump strength Fig. 4.6 Process of the new soliton generation in the laser. δφl = 1.20π. (a) G=255; (b) G=270; (c) G= Fig. 4.7 Soliton evolutions calculated with δφl = 1.80π. (a) G=470; (b) G=478; (c) G= x

13 List of figures Fig. 4.8 Multiple soliton operation of the laser calculated with linear cavity phase delay bias set at δφl = 1.55π, G = Fig. 5.1 Optical spectrum (a) and autocorrelation trace (b) of a soliton Fig. 5.2 Optical spectrum (a) and autocorrelation trace (b) of a two-pulse bound soliton Fig. 5.3 Autocorrelation trace of a three-pulse bound soliton (a) and a four-pulse bound soliton (b) Fig. 5.4 Autocorrelation trace of bound multipulse bound solitons: (a) bound 2- pulse-bound solitons, (b) bound 3-pulse-bound solitons Fig. 6.1 Period doubling bifurcation to chaos of the soliton trains. (a) Period-one state; (b) Period-two state; (c) Period-four state; (d) Chaotic state. From (a) to (d) the pump intensity is increased Fig. 6.2 Oscilloscope traces of a period-two state of the laser emission Fig. 6.3 Optical spectra of the laser measured in the states of period-one (a) and period-two (b) Fig. 6.4 RF spectra of the laser output corresponding to (a) period-one; (b) period-two states Fig. 6.5 Soliton pulse evolution and the corresponding optical spectra numerically calculated under different pump strength. The linear cavity phase delay bias is set as δφ=1.6π. (a)/(f) period-1 soliton state, G=800; (b)/(g) Period-2 soliton state, G=850; (c)/(h) Period-4 soliton state, G=902; (d)/(i) Period-8 soliton state, G=908; (e)/(j) Chaotic soliton state, G= Fig. 6.6 Bifurcation diagram computed for the pulse peak intensity versus the small signal gain G (The linear cavity phase delay bias is set as δφ=1.6π) Fig. 6.7 Soliton pulse profiles and corresponding optical spectra in a period-2 state. (a)/(c) the first round trip; (b)/(d) the second round trip Fig. 6.8 Soliton pulse profiles and corresponding optical spectra in a period-4 state. (a)/(e) the first round trip; (b)/(f) the second round trip; (c)/(g) the third round trip; (d)/(h) the fourth round trip xi

14 List of figures Fig. 6.9 Soliton profiles and corresponding optical spectra numerically calculated. (a)/(d) State of period-3, G=730; (b)/(e) State of period-6, G=735; (c)/(f) Chaotic state, G= Fig Oscilloscope trace and corresponding spectra of period doubling bifurcations of a bound-soliton pulse train. (a)&(d) period-one state; (b)&(e) period-doubled state; (c)&(f) period-quadrupled state Fig A typical measured autocorrelation trace of the bound-soliton Fig Period doubling route to chaos of bound solitons. (a)/(e) state of stable bound solitons, G=1149; (b)/(f) state of period-2 of the bound solitons, G=1300; (c)/(g) state of period-4 of the bound solitons, G=1353; (d)/(h) Chaotic state of the bound solitons, G= Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a)/(e) state of stable soliton operation (period one), G=315; (b)/(f) state of period-2, G=340; (c)/(g) state of period-4, G=366; (d)/(h) state of period-8, G= Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a)/(d) state of stable bound solitons operation (period one), G=370; (b)/(e) state of period-2 of bound solitons, G=380; (c)/(f) state of period-4 of bound solitons, G= Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a) state of stable two solitons operation (period one), G=332; (b) state of period-2 of two solitons, G=380; (c) state of period-4 of two solitons, G= Fig. 7.1 A typical state of the bunched noise-like pulse emission. (a) Optical spectrum of the state; (b) Autocorrelation trace with a scan span of 5 ps; (c) Autocorrelation trace with a scan span of 0.5 ps xii

15 List of figures Fig. 7.2 A typical bunched noise-like pulse state numerically calculated. (a) Time evolution of the noise-like pulses; (b) Autocorrelation trace numerical calculated Fig. 7.3 Typical bunched noise-like pulse emission numerically calculated. (a) G=800. (b) G= Fig. 7.4 Bunched noise-like pulse emission. (a) Optical spectrum; (b) Autocorrelation trace; (c) Oscilloscope trace; (d) RF spectrum around the fundamental cavity repetition rate Fig. 8.1 Pulse intensity and spectrum of the generated high-repetition-rate pulse train under different pump power. (a)/(d) G=270; (b)/(e) G=300; (c)/(f) G= Fig. 8.2 Pulse intensity of the generated double-high-repetition-rate pulse train when A=0.5, G= Fig. 8.3 Autocorrelation trace and spectrum of the generated high-repetition-rate pulse train of GHz Fig. 8.4 Autocorrelation trace of the generated high-repetition-rate pulse train of 1 THz Fig. 8.5 A typical soliton spectrum obtained in our experiment xiii

16 Chapter 1: Introduction Chapter 1 Introduction Ultrafast optics is a rapidly growing field. Generation of ultra-short optical pulses attracts not only scientists in fundamental researches but also engineers in various applications. Ultrashort pulses with high intensity make themselves suitable for widespread use in communication, sensing, machining, or medical applications and so on. Various nonlinearities caused by the high pulse intensity also make ultrafast optics an important tool for many branches of science. To date, generation of femtosecond optical pulses is a routine process in two major lasers: bulk solid-state lasers and fiber lasers [1-6]. In order to obtain higher peak power and shorter pulse, different laser designs have been carried out. Currently, bulk solid-state lasers (most notable Ti: sapphire lasers) provide the highest peak power and the shortest pulse with improving reliability [3]. However, they are more complex and expensive comparing with fiber lasers. Fiber lasers are very promising since they are constructed by inexpensive components (mostly various fibers), highly integrated with flexibly miniaturized cavities, and conveniently collimated output. As a significantly more compact, more robust, and more inexpensive ultrashort pulse source, fiber lasers can greatly proliferate the ultrafast optics applications in various branches of industry and science. The only disadvantage of fiber lasers is that they could not provide the best performance (pulse intensity and pulse width) as that of bulk solid-state lasers. However, 1

17 Chapter 1: Introduction extensive researches have revealed that it is possible to improve the performance of fiber lasers to the level of bulk solid-state laser technology without trading-off their practical advantages. When operated in the negative net cavity dispersion regime, solitons can be generated in fiber lasers through the balanced interaction between the anomalous group velocity dispersion and the nonlinear Kerr effect. Soliton here means a narrow, high-intensity optical pulse at a fixed position of the laser cavity, whose pulse profile is invariant with time. One of the most promising applications of the soliton theory is in the field of optical fiber communications [7]. Recently studies show that ultrashort pulses can also be generated in the positive net cavity dispersion regime [8]. In order to apply soliton or ultrashort optical pulses in certain areas, we need to know the properties of ultrashort optical pulses generated. There are many features of solitons have been revealed both experimentally and theoretically, such as the soliton energy quantization [9,10], the soliton bunching [11], bound state of solitons [12], twin-pulse soliton [13], and the quasiharmonic and harmonic mode locking [14]. However, they are only the part of an iceberg on the sea. This thesis reports theoretical and experimental studies regarding dynamics of femtosecond fiber lasers passively mode-locked using nonlinear polarization rotation (NPR) technique in the telecommunication window of 1550 nm. The ultimate goal is to understand various properties of fiber lasers and develop practical, high-performance fiber lasers. We at first numerically analyzed the operation regime of the passively mode- 2

18 Chapter 1: Introduction locked fiber lasers using nonlinear polarization rotation technique. We found that depending on the selection of the linear cavity phase delay bias the fiber soliton lasers can exhibit various regimes of operation, each with distinctively different features. Roughly speaking, we could classify it into three regimes according to the amplitude of the linear cavity phase delay bias. When the linear cavity phase delay bias is selected to be small, multiple mode-locked pulses with weak peak power are observed; increasing the linear cavity phase delay bias increases the peak power of the generated solitons, consequently results in stronger nonlinearity. The solitons propagating in the cavity would show nonlinear phenomena, such as period doubling bifurcations and period doubling route to chaos; when the linear cavity phase delay bias is chosen big, solitons collapse before they reach the peak power limitation, consequently noiselike pulses with broad bandwidth are obtained. Based on the numerical simulation results we both experimentally and numerically studied the three different operation regimes in detail. Firstly we numerically studied the mechanism of multiple soliton generation and soliton energy quantization in a soliton fiber ring laser passively mode-locked by using the NPR technique [15]. We identified that the multiple soliton generation in the laser is caused by the peak power clamping effect of the cavity. Depending on the linear cavity phase delay setting, the nonlinear phase delay generated by a soliton propagating in the fiber cavity could be so large that it switches the cavity feedback from the initially selected positive regime into the negative regime. And as a result of the cavity feedback change the maximum achievable soliton peak power is then limited. In this case increasing the laser pump power will not increase the peak power of the solitons, but generate a new soliton. Therefore, multiple solitons are formed in the laser. As the solitons share the same laser 3

19 Chapter 1: Introduction gain, gain competition between them combined with the cavity feedback feature further results in that in the steady state they have exactly the same soliton parameters. The parameters of solitons formed in the laser are not fixed by the laser configuration but vary with the laser operation conditions, which are determined by the soliton internal energy balance between the shared laser gain and the dynamical losses of each soliton. We experimentally obtained multi-pulse bound solitons with fixed pulse separation and the bound states of the multi-pulse bound solitons in a dispersion-managed passively modelocked fiber ring laser [16]. Combined with our previous work on twin-pulse soliton [13], it clearly shows that the multi-pulse solitons are formed by the direct soliton interaction and it is actually a new form of soliton. Therefore, we have completed the multi-pulse soliton theory. Basing on a dispersion-managed fiber laser, we discovered the novel phenomena of period doubling bifurcations and period doubling route to chaos [17-20], which verified that the phenomena of period doubling bifurcations and period doubling route to chaos are intrinsic properties of the nonlinear pulses in the fiber lasers. They are caused by the deterministic dynamics of the system. Apart from the observation of broadband noise-like pulses basing on a dispersion-managed fiber laser [21], we extended the research regarding the noise-like pulses to the conventional passively mode-locked Erbium-doped fiber lasers (PMEFLs) [22]. We proved that the noise-like pulse emission is a generic property of all PMEFLs and it is caused by the soliton collapse effect and the associated soliton self-frequency shift in the laser. Higher power and broader spectral bandwidth noise-like pulse emission is in principle possible to achieve provided that even stronger pump source and larger nonlinearity fibers are available. By incorporating a segment of high-birefringence fiber in a non-dispersion-managed soliton fiber ring laser, 4

20 Chapter 1: Introduction we achieved a self-started high repetition rate ultrashort pulse source [23] whose repetition rate is determined by the high-birefringence fiber inserted and resonant modulation instability frequency. This chapter is intended to provide a basic overview on the background of this research. Section 1.1 gives a brief review on fiber lasers and presents the motivation of the research. Section 1.2 presents some problems facing in the research of ultrashort pulses and the objectives of this research. Section 1.3 lists out the major contributions of the research. Outline of the dissertation is shown in Section Motivation In 1834, John Scott Russell discovered a bizarre wave that he named as the Great Wave of Translation, which we now call as Soliton, when he observed a heap of water in a canal that propagated undistorted over several miles. The term soliton refers to special kinds of waves that can propagate undistorted over long distances and remain unaffected after collision with each other [24]. The discovery of Russell was almost totally forgotten for a century, until 1960, scientist realized that how important Russell s discovery had been. They reasoned that if a wave in water could be made to travel very far without distortion, what would happen to other waves such as light? Hasegawa and Tappert suggested in 1973 the possibility of soliton propagation in optical fibers [25]. The use of solitons for transmission over transoceanic distances was first demonstrated in 1988 [26]. Today various types of solitons have been discovered in many areas of physics. More and 5

21 Chapter 1: Introduction more nonlinear dynamical models in different branches of science have now been found that possess soliton-like solutions. The definition of a soliton has expanded its narrow concept to a more general one that refers to a relatively stable nonlinear solitary wave. Optical solitons are currently under intensive study since they are expected to provide not only distortionless signal transmission in ultra-high speed telecommunications but also many interesting nonlinear optical applications in fibers. Mode-locked fiber lasers can generate not only so-called solitons in the negative net cavity dispersion regime but also ultrashort pulses in the positive net cavity dispersion area. They are suitable sources of ultrashort pulses for laboratory, telecommunication, and signal-processing applications. According to current optical communication technology, mode-locked, Erbium-doped fiber lasers (EDFLs) are foci since such lasers are capable of generating ultrashort optical pulses in the spectral range of 1.55 µm, which is regarded as telecommunication window in optical communications. Generally, it is believed that EDFLs generate uniform ultrashort optical pulse train that is natural information carrier to transmit digital information. However, fiber lasers are intrinsic nonlinear systems, especially when they are operated to produce ultrashort pulses with high intensity, which will cause various nonlinear phenomena that should be avoided in optical communications or have potential applications in other areas. Therefore, we need to understand different properties of fiber lasers especially when they are operated to generate ultrashort optical pulses. 6

22 Chapter 1: Introduction 1.2 Objectives Self-starting passively mode-locked Erbium-doped fiber lasers, as one of the attractive sources for ultrashort optical pulses, have been intensively studied due to their simplicity and tunability. Although the basic operation principle of the fiber lasers is generally understood, and they have been used in lab for the long-distance optical fiber communications, many features of the lasers still remain unexplained and need to be investigated, especially when the nonlinearity become very strong in the lasers. When operated in the negative net cavity dispersion regime, due to the balanced interaction between the group velocity dispersion and nonlinear Kerr effect, soliton operation is obtained after the mode locking of the fiber lasers. Generally, solitons generated in fiber lasers exhibit characteristics such as harmonic mode locking [14], multiple soliton pulsation [12,13], and energy quantization [9,10]. What s the physical mechanism for the multiple soliton generation and energy quantization? Why the individual soliton seems identical in the state of multiple soliton operation? Since increasing pump power would generally result in the generation of multiple solitons, instead of increasing of soliton peak power of the solitons in the laser cavity, how to suppress the multiple soliton generation if we want to obtain a soliton with higher peak power. Recently, a new form of solitary waves twin-pulse soliton has been experimentally observed and numerically verified in passively mode-locked fiber lasers [13]. Similar features between twin-pulse solitons and single-pulse solitons strongly suggested that the twin-pulse solitons are in fact another form of solitary waves in the 7

23 Chapter 1: Introduction laser. In this thesis, we reported further research on multi-pulse solitons with fixed pulse separations in a dispersion-managed fiber laser. The property and formation mechanism of this state are investigated. Shorter pulse width is always the pursuing goal of scientists for ultrafast optics. Shorter pulse width often means higher pulse intensity and consequently various nonlinearities. An optical pulse circulating in a fiber laser is equivalent to propagating in a periodical gain-and-loss system. It periodically experiences the amplification of the amplifier and the output loss. Suggested by the average soliton theory of lasers [27], 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 [28,29]. 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, aligning the polarizer with either the fast or the slow axis of the fiber could diminish the nonuniformity of the pulse trains. Our group has experimentally investigated the output property of a fiber soliton ring laser passively mode-locked by using the NPR technique [30] 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 it is shown 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 8

24 Chapter 1: Introduction 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 and the linear waves in the cavity, such as the dispersive waves or CW laser emission, causes the amplitude of the soliton pulses to vary periodically. As a representative of nonlinear systems, fiber lasers should also present some universal properties of nonlinear systems, for example, phenomena of period doubling bifurcations and period doubling route to chaos. Although period doubling and quasi-periodicity have been previously observed in an additive-pulse mode-locked F-center laser [31], 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 [32], to our knowledge, so far no experimental observation on a complete period doubling route to chaos in passively mode-locked fiber soliton lasers was reported. How does the period doubling bifurcation occur? What is the relationship between the occurrence of period doubling and pulse intensity? Whether there exists a period-doubling route to chaos in fiber lasers also needs to be confirmed. To clarify these questions is not only important for understanding the ultrashort pulse operation of the passively mode-lock fiber lasers, but also fundamental for future applications of ultrashort pulses in the ultra-high-bit-rate optical communication systems since we need to avoid the emergence of the pulse intensity fluctuation. It is well known, single-pulse solitons could form bound solitons where several single-pulse solitons bound together to form a bunch and propagate in the laser. The number of single-pulse solitons in the bunch and the separations between the 9

25 Chapter 1: Introduction adjacent solitons vary with pump power and laser parameters. Twin-pulse soliton and multi-pulse soliton in a sense could be treated as bound solitons except that now the pulse separation is fixed even under the pump power changing. Since multi-pulse soliton is another form of solitary waves in fiber lasers, whether the phenomena of period doubling bifurcations and period doubling route to chaos would also appear on the twinpulse/multi-pulse soliton, and even on the bound solitons? Another interesting topic is the broadband noise-like pulse generation. A wide range of potential applications require optical sources with broad spectrum, for example, optical sources for coherence metrology, for sensors and for gyroscopes. LEDs that emit broadband noise are commonly used for such applications. However, the power of LEDs is limited. M. Horowitz et al. have shown a pulsed Erbium-doped fiber laser that generates a train of high-intensity, broadband, noiselike pulses [33]. In their experiments the overall laser cavity dispersion was significantly positive and their laser cannot support short pulses because of the strong positive dispersion and the significant birefringence. Since traditional soliton fiber lasers are operated in the negative net cavity dispersion regime, an immediate question would be whether the broadband noiselike pulses observed in the positive dispersion regime could also appear? To the best of our knowledge, so far no report has addressed this question. What is the physical mechanism of the broadband noise-like pulse generation? How broad spectrum can be achieved for such nosie-like pulses? All these questions need answers. 10

26 Chapter 1: Introduction Ultrahigh-repetition-rate optical pulse sources play a key role in ultrahigh speed optical communication systems. Based on self-starting passively mode-locked Erbium-doped fiber lasers, generation of optical pulse train at repetition rate of 100 GHz has been demonstrated [34-36]. It is attributed to the modulational-instability (MI) of the continuous-wave field resonating in the laser cavity, as firstly theoretically proposed by Akira Hasegawa [37]. However, T. Sylvestre et al. found that the generation of selfstarted high-repetition-rate pulses in a laser is not intrinsically linked to MI since they obtained similar high-repetition-rate pulse train in a fiber laser with totally normal dispersion, which lacks the anomalous dispersion necessary for getting MI [38]. So what is the true mechanism for the generation of ultrahigh-repetition-rate pulse train in a laser? What are the factors influence the generated repetition rate? Are there any limitations of the maximum repetition rate achievable in the fiber laser? How to achieve possibly higher repetition rate pulses in a laser? All these questions have so far not been clearly addressed. 1.3 Major contributions of the research First of all, a passively mode-locked Erbium-doped fiber laser with dispersion-managed cavity using the NPR technique has been set up; various operation features of the laser have been examined. The laser is operated in the negative net cavity dispersion regime. Three operation regimes of the laser have been obtained and investigated: solitons with low peak power where multiple solitons are easily obtained, solitons with higher peak 11

27 Chapter 1: Introduction power where nonlinear phenomena of period doubling bifurcations and period doubling route to chaos can be observed, and noise-like pulse operation. A pulse tracing technique is adopted to describe the pulse evolution in the soliton fiber lasers. The evolution of optical pulses in the presence of temporal dispersion and nonlinear response of various fibers and cavity components, coupled with the feedback nature of the laser, is numerically simulated. The features of the laser operation have been precisely numerically simulated and compared with the experimental observations. Multiple soliton generation and soliton energy quantization in a soliton fiber ring laser passively mode-locked by using the NPR technique are numerically studied. We found numerically that the formation of multiple solitons in the laser is caused by a peak power limiting effect of the laser cavity. It is also the same effect that suppresses the soliton pulse collapse, an intrinsic feature of solitons propagating in the gain media, and makes the solitons stable in the laser. Furthermore, we show that the soliton energy quantization observed in the lasers is a natural consequence of the gain competition between the multiple solitons. Enlightened by the numerical result we speculate that the multi-soliton formation and soliton energy quantization observed in other types of soliton fiber lasers could have similar mechanism. Multi-pulse solitons including twin-pulse soliton were experimentally observed in the dispersion managed fiber laser. The formation mechanism of these new forms of soliton was intensively studied. It was found that although formed under different experimental 12

28 Chapter 1: Introduction conditions, various multi-pulse solitons (twin-pulse, three-pulse, four-pulse et al.) have exactly the same fixed pulse separation. Bound states of the multi-pulse solitons were also experimentally obtained, which suggested that the multiple pulses in the solitons have strong binding force. They together could be treated as a unit. Phenomena of period doubling bifurcations and period doubling route to chaos of solitons have been observed in the fiber laser. 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 an equilibrium state to a chaotic state. This experimental result shows that the nonlinear propagation of soliton pulses in the laser cavity is an intrinsic nonlinear dynamic process, which follows the universal laws of the nonlinear dynamic systems. Numerical studies on the soliton dynamics confirmed our experimental observations. In addition, our numerical simulations also verified that the period doubling bifurcations and period doubling route to chaos are intrinsic properties of the solitons in the lasers. The period doubling route to chaos of the single-pulse soliton but also the bound solitons was obtained and numerically confirmed. The dispersion-managed fiber laser emits not only solitons but also intense bunched noise-like pulses including the transform-limited pulses. The optical spectrum of the laser emission has a bandwidth as broad as nm. It was found that purely depending on the linear cavity phase delay the laser could be switched between the soliton operation and the noise-like pulse emission. Numerical simulations showed that the laser emission 13

29 Chapter 1: Introduction was caused by the combined effect of soliton collapse and positive cavity feedback in the laser. According to our numerical model, broadband noise-like pulses should be a generic property of all PMEFLs, whose appearance is independent of the cavity dispersion management. To confirm these, we built up a conventional PMEFL without dispersion management and indeed observed the noise-like pulse generation in the laser. In particular, we show that by taking advantage of the soliton collapse effect and the soliton self-frequency shift (SSFS), high power superbroad spectral bandwidth noise-like pulses can also be generated, and the spectral bandwidth of the laser emission is practically only limited by the effective laser gain available. Finally, we achieved a self-started high-repetition-rate soliton train source based on a traditional PMEFL by inserting a segment of high-birefringence fiber in the cavity. The source can produce a high-repetition-rate ultrashort pulse train output. The repetition rate of the ultrashort pulse train is determined by the separation of the transmission peaks of the cavity spectral filter introduced by the high-birefringence fiber and the intracavity polarizer. Repetition rate up to THz can be realized provided the modulational-instability gain peak can match the free spectral range of the cavity filter. 1.4 Organization of the thesis The thesis is organized as follows: Chapter 1 provides a brief introduction of the research. 14

30 Chapter 1: Introduction Chapter 2 presents briefly the background knowledge on the pulse propagation in optical fibers. Extended coupled complex nonlinear Schrödinger equations that describe the pulse propagation in optical fibers are introduced. Numerical method used to simulate the pulse propagation is illustrated. Chapter 3 describes the mode locking mechanism of NPR technique and analyzes the cavity transmission. Experimental implementation of NPR is explained. In addition, the pulse tracing technique that describes the pulse propagation in the cavity is presented. Numerically, it was shown that depending on the linear cavity phase delay bias, there are three different operation regimes for the mode-locked fiber lasers. Chapter 4 shows the study of numerical simulations regarding the multiple soliton generation and soliton energy quantization in soliton fiber ring lasers passively modelocked by using the NPR technique. Basing on our simulation results, the physical mechanism of multiple soliton generation and soliton energy quantization is illustrated. Chapter 5 presents the experimental observation on the multi-pulse soliton generation in a passively mode-locked Erbium-doped fiber laser with dispersion-managed cavity. Chapter 6 describes novel phenomena of soliton period doubling bifurcation and period doubling route to chaos in a fiber laser. Numerical simulations confirm our observations. The mechanism of the soliton pulse nonuniformity is also studied. 15

31 Chapter 1: Introduction Chapter 7 shows the laser operation in the broadband noise-like pulsation regime. The mechanism of the broadband noise-like pulse generation is discussed in details. Numerical simulations also predict similar operation in the fiber lasers with nondispersion-managed cavity, which then be confirmed experimentally. Chapter 8 presents a self-started high-repetition-rate soliton train source based on a traditional PMEFL by inserting a segment of high-birefringence fiber in the cavity. The factors limiting the repetition rate achievable are discussed. Finally, we conclude the thesis in Chapter 9. Recommendations for future research are also given. 16

32 Chapter 2: Theory of pulse propagation in optical fibers Chapter 2 Theory of pulse propagation in optical fibers Generation of optical pulses in fiber lasers is a result of mode locking in the cavity comprising of fibers and other components. After mode locking a pulse will propagate repeatedly along the cavity. The propagation of optical pulses in optical fibers is affected by the dispersive and nonlinear properties of the fibers. Also the gain and gain dispersion introduced by the amplified media such as Erbium-doped fiber will complicate the pulse propagation in the fiber lasers. In addition, the fiber modal birefringence caused by the imperfect cylindrical symmetry due to random variations in fiber core shape and stressinduced anisotropy along the fiber length should be taken into account when analyzing the pulse propagation in practical fiber systems. The objective of this chapter is to present the theoretical background and to describe the basic equations that govern the nonlinear pulse propagation in single mode fibers. Numerical method to simulate the pulse propagation in fibers is also introduced. Section 2.1 presents a brief introduction of the pulse propagation in optical fibers. Section 2.2 shows the numerical method that we used to simulate the pulse propagation. 2.1 Pulse propagation in optical fibers 17

33 Chapter 2: Theory of pulse propagation in optical fibers Pulse propagation in nonlinear dispersive media under the slowly varying envelope approximation has been extensively investigated [24]. Equations that describe the pulse propagation in optical media, which start from the wave equation obtainable from Maxwell s equations, have been derived. Here, we present a brief summary Fiber dispersion and birefringence Dispersion is an intrinsic property of optical fibers. It describes that light of different wavelengths has different propagation speeds in fibers. In single mode optical fibers dispersion results from the interplay of two underlying effects: material dispersion and waveguide dispersion. Material dispersion arises from the wavelength dependence of the refractive index of the materials made up of a fiber; Waveguide dispersion is rooted in the wavelength dependence of the propagation constant of a mode to the fiber core diameter and the difference in refractive index between the fiber core and cladding. Although the material dispersion of a fiber is normally difficult to be changed, through specially designing the fiber refraction index structure, the total dispersion of a single mode fiber can be altered. Based on this technique, single mode optical fibers of different total dispersion have been made and are now commercially available. While the standard single mode fibers have a zero dispersion wavelength at around 1.3 µm, the dispersion shifted single mode fibers shifted the zero dispersion to about 1.55 µm. Dispersion is a linear optical property. Therefore, it always plays a role on the pulse propagation in single mode fibers. Mathematically, the effect of fiber dispersion on the 18

34 Chapter 2: Theory of pulse propagation in optical fibers propagation of an optical pulse is described by expanding the mode-propagation constant in a Taylor series about the pulse s carrier frequency ω 0: β ( ω) = β 0 + β1( ω ω0 ) + β 2 ( ω ω0 ) + β3( ω ω0 ) +... (2.1) 2 6 where β m m d β ω d = m ω = ω 0 m=0,1,2,3 (2.2) The first term in the β(ω) expansion is the propagation constant at the carrier frequency, the second term is the inverse of the group velocity of the pulse, and the third term represents the dispersion of the group velocity. As the third term is responsible for the optical pulse broadening when it propagates in a single mode fiber, β 2 is also called the group velocity dispersion (GVD). A single mode fiber is generally not perfectly symmetric everywhere along the fiber, either due to the intrinsic factors such as the geometric irregularities of the fiber core and internal stresses, or external factors such as fiber bending, twisting etc. Practically, a single mode fiber has two polarization eigenmodes, or in another words a certain birefringence. The polarization mode beat length L b describes the strength of birefringence of a fiber, which is defined as: 19

35 Chapter 2: Theory of pulse propagation in optical fibers L b = n x λ n y (2.3) where n x and n y are the effective indices of the two polarization modes of the fiber. The propagation of a weak CW light in a linearly birefringent single mode fiber is described by: da dz da dz x y = iβ A x y x = iβ A y (2.4) Where β x =kn x and β y =kn y (k=2π/λ) are the propagation constants along the two polarization axes of the fiber, respectively, A x and A y are the projections of the light field amplitude on the two polarization axes. After propagating a distance of L, the two polarization components will gain a phase difference of ( n n ) 2π x y L 2πL Φ = =, λ L which means that the polarization of the light varies along the fiber. At a multiple distance of L b, a multiple phase difference of 2π will be generated between the two polarization components and the polarization of the light returns to its initial state. Therefore, physically, L b is the length of light propagation in a fiber over which the same polarization state appears. b 20

36 Chapter 2: Theory of pulse propagation in optical fibers The speed of light propagation is directly related to the refractive index. Light polarized along different polarization axes has different phase as well as group velocities. In a birefringent fiber if n x >n y, the light polarized along the X axis moves slower than that of light polarized along the Y axis. The X-axis is called the slow axis and the Y-axis the fast axis. Different single mode fibers could have very different beat length, for the normal standard single mode optical fibers, beat lengths are in the range of several meters, while for the polarization maintaining fibers, as a strong birefringence is deliberately made into the fibers, they have a beat length in the range of several millimeters Fiber nonlinearity If a strong light beam propagates in a single mode fiber, the effect of fiber nonlinearity must be considered. Fiber nonlinearity is caused by the nonlinear response of polarization to strong light field. As optical fibers are made of SiO 2 glass material, which has a central symmetric molecular structure, the lowest order of nonlinear effect in the single mode fibers is the third order effect. There are different third-order nonlinear effects, these include the third-harmonic generation, four-wave mixing, stimulated Raman and Brillouin scattering, and optical Kerr effect. As some of the effects require the fulfillment of the phase matching condition, they are not efficient in the single mode fibers. Effects that are automatically phase matched can always appear provided that the light intensity reaches their appearance thresholds. One of these effects is the optical Kerr effect, which refers to the phenomenon that the refractive index of a fiber is the light intensity dependent: 21

37 Chapter 2: Theory of pulse propagation in optical fibers n 0 2 ( I) = n + n I (2.5) where I is the light intensity inside a fiber, n 0 is the refractive index of the fiber without the light. n 2 is the nonlinear refractive index coefficient related to the third-order nonlinear property of the fiber. The nonlinear refractive index coefficient n 2 of optical fibers has a very small value, generally it is in the range of 2.2 ~ m 2 /W [39]. However, the efficiency of the nonlinear optical process can become very high in optical fibers due to the possible long distance of light-matter interaction, the small fiber core of the single mode fibers (normally a diameter < 10µm), and the confined light propagation in the fiber core with almost ignorable loss Pulse propagation in single mode fibers Different to a CW laser beam, whose spectral linewidth is very narrow, therefore could be roughly considered as a monochromatic wave, an optical pulse consists of many frequency components. The shorter an optical pulse, the broader is its frequency bandwidth. When a strong optical pulse propagates in a single mode optical fiber, both the fiber dispersion and the nonlinear optical Kerr effect must be considered. Optical pulse propagation in a single mode fiber is described by the nonlinear Schrödinger equation (NLSE), which takes account of the effects generated by both the fiber dispersion and fiber Kerr nonlinearity on a pulse [24]: 22

38 Chapter 2: Theory of pulse propagation in optical fibers A iβ 2 = β1 z t 2 A 2 2 A iγ A 2 t A (2.6) where the high order dispersion effect is omitted, A is the slowly varying field amplitude of the optical pulse, and the nonlinear parameter γ is defined as n 0 γ = 2ω (2.7) ca eff A eff is the effective fiber core area; c is the velocity of light in vacuum. Propagation of optical pulses in single mode fibers depends crucially on the fiber dispersion property and the intensity of light. Although in the linear case pulse propagation in single mode fibers always broadens its pulse width, and associated with the pulse width broadening, the pulse also becomes linearly frequency chirped, under the existence of the optical Kerr effect, different complicated optical pulse propagation could be resulted in. However, a special situation exists if the fiber dispersion is anomalous. The nonlinear optical Kerr effect always generates a positive linear frequency chirp in the central part of a pulse, while the frequency chirp generated by the GVD can be either positive or negative depending on the sign of the GVD parameter. As a fiber with anomalous dispersion generates a negative linear frequency chirp, under the simultaneous action of both the nonlinear optical Kerr effect and the GVD, it is possible to eliminate the generation of frequency chirp on a pulse when it propagates in the fiber, and consequently no pulse broadening will be generated. In fact, as the chirp induced by the 23

39 Chapter 2: Theory of pulse propagation in optical fibers nonlinear optical Kerr effect is the pulse intensity dependent, when a strong pulse propagates in anomalous dispersion fibers, it can adjust its intensity so that the instantaneous frequency chirps induced by the two effects are automatically balanced. An optical soliton is formed when the frequency chirp generated by the fiber dispersion is balanced by that generated by the fiber nonlinear effect. It is obvious that in order to form an optical soliton, not only the optical pulse must be strong enough so that the nonlinear phase modulation could be generated. It must also propagate in the single mode fibers of anomalous dispersion. A soliton pulse also possesses a special Sech 2 -form pulse shape. Soliton formation is a generic property of nonlinear optical pulse propagation in single mode optical fibers. Even an optical pulse has initially an arbitrary pulse shape, when it propagates in a nonlinear fiber it automatically evolves into a soliton through emitting dispersive waves. Depending on the pulse width, a fundamental soliton in a fiber could have different peak power and pulse energy. Generally, the larger the fiber GVD, the larger is the soliton pulse energy. If the birefringence of a fiber is considered, the optical pulse propagation in it is described by the coupled NLSEs [40,41]: A x z A y z = iβa x = iβa Ax δ t y Ay + δ t i A β 2 2 t 2 x 2 2 i A β t + iγ A y x 2 + iγ A y A 2 3 y A 2 A 2 x x A iγ + A 3 y 2 y iγ + A 3 A 2 x * x A * y (2.8) 24

40 Chapter 2: Theory of pulse propagation in optical fibers where A x and A y are the two normalized slowly varying pulse envelopes along the slow and the fast axes, A x * and A y * are their conjugates, respectively. 2β = 2π ( n n ) / λ = 2π / L is the wave-number difference, 2 δ = 2βλ / 2πc is the x y b inverse group-velocity difference. The equations are written in the coordinate system that moves with the average group velocity β + β 1 1x 1y v g = β1 = 2. When an optical pulse propagates in a birefringent single mode fiber, not only that the fiber dispersion would broaden the pulse width, but also that the pulses along different polarizations will have different group velocities, which broadens the pulse width or even splits the pulse. As there exist two polarization components, the nonlinear fiber Kerr effect will not only generate self-phase modulation (SPM) of each component, but also generate cross phase modulations (XPM) between the two polarization components. The coherent coupling between the two polarization components also generates degenerated four-wave-mixing (FWM), whose strength is related to the strength of the linear fiber birefringence [40] Gain and gain dispersion of Erbium-doped fiber In fiber laser systems, one of the essential components is the active media, which generally are rare earth doped fibers. When an optical pulse propagates in the rare earth doped fibers, the effect of light amplification must be considered. A fiber amplifier is characterized by its small signal gain, gain bandwidth, gain saturation, and noise feature. For a homogeneously broadened gain medium, the gain coefficient can be written as [42]: 25

41 Chapter 2: Theory of pulse propagation in optical fibers G g( ω ) = (2.9) ( ω ωa ) T + P 2 P s where G is the small signal gain, ω a is the resonant frequency of the dopant energy levels, T 2 is the dipole relaxation time of the dopants, P is the power of the light being amplified, P s is the saturation power of the gain medium, which depends on the stimulated emission cross section of the dopants and the upper energy level lifetime. In the case of three-level pumping, the saturation power is also the pumping strength dependent. The gain bandwidth is defined as the full width at half maximum (FWHM) of the gain spectrum g(ω). In the case of homogeneous broadening the gain bandwidth is inversely related to the dipole relaxation time T 2 of the dopants. When an optical pulse propagates in the amplifier such as Erbium-doped fibers, the effect of light amplification must be considered. The role of Erbium ions is to provide gain for the light pulse. Generally, the interaction of doped ions with light is governed by the Maxwell-Bloch equations [42], however, in the case that the polarization relaxation time T 2 of the Erbium ions is shorter than the pulse width, the rate equation approximation can be made and consequently the gain term can be simply added into the NLSE. As the gain dispersion is to reduce the gain for spectral components away from the gain peak ω 0, therefore, the frequency dependence of the gain can be approximated by 2 g 1 g 2 g ( ω ) = g( ω 0 ) + ω = ω ( ω ω 0 ) + ( 0 ) ω = ω ω ω (2.10) 0 ω 2 ω 26

42 Chapter 2: Theory of pulse propagation in optical fibers Without loss of the generality assuming that the pulse carrier frequency coincides with the gain peak, the equation describing optical pulse propagation in the doped fiber becomes: A A i A g g A = β 1 (2.11) z t β 2 + iγ A A + A t 2 2Ω g t where g is the peak gain coefficient, Ω g is the gain bandwidth. The time dependence of the peak gain coefficient varies with time due to the gain saturation and is determined by [43] g t G g = T 1 g A E s 2 (2.12) where G is the small signal gain, T 1 is the population decay time, E s is the saturation energy. As for the Erbium doped ions T 1 is in the time scale of 10 ms, if the pulse width is far narrower than it, which is normally the case, the T 1 term is negligible during pulse amplification, and g(t) becomes: g t 1 2 ( t) G A dt = exp (2.13) Es Typical values of E s for Erbium-doped fibers are about 10 µj. As the pulse energies are normally much smaller than the saturation energy Es, gain saturation is negligible over 27

43 Chapter 2: Theory of pulse propagation in optical fibers 28 the duration of a single pulse. However, in the case of a pulsed fiber laser, the pulse circulating in the laser cavity, the average power of the light will saturate the gain and determines the saturated gain value. The equation (2.11) is known as the Ginzburg-Landau equation (GLE). Different to the NLSE, which is integrable and has an exact soliton solution, the GLE is non-integrable. However, it was found that in the case of anomalous fiber dispersion the equation also has solitary wave solutions in the sense of optical pulses whose shape does not change during propagation [44]. Due to the influence of the optical gain, the solitons in the doped fibers become frequency chirped. In addition, while in the case of the undoped fiber, the fundamental soliton of a fiber can have different pulse widths and peak powers, as far as the GVD effect is balanced by the self-phase-modulation effect, all the fundamental solitons in a doped fiber have exactly the same pulse width and peak power, which are uniquely determined by the gain bandwidth and the gain of the doped fiber [45]. An arbitrary pulse incident to a doped fiber will evolve eventually to the soliton. In doped linearly birefringent fiber, the equations (2.8) then become * * t A g A g A A i A A A i t A i t A A i z A t A g A g A A i A A A i t A i t A A i z A y g y y x y x y y y y y x g x x y x y x x x x x Ω = Ω = γ γ β δ β γ γ β δ β (2.14) ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library

44 Chapter 2: Theory of pulse propagation in optical fibers In this case the gain saturation results from light along both polarizations, so the saturated gain coefficient is calculated by 2 2 ( Ax + Ay ) dt + 1 g = G exp (2.15) Es 2.2 Numerical simulation method Fiber lasers are typically comprised of several segments of fibers with different optical properties. The propagation of light in the fiber is determined by either the equation (2.8) if the fiber is undoped or (2.14) if the fiber is doped. Due to the essence of nonlinear partial differential equations, there are no explicit analysis solutions except some special cases. Therefore, in order to understand the dynamics of pulse generation, numerical simulation is the only way even though it is a time-consuming process. Here we use the standard split-step Fourier method [24] that has been used extensively to solve the pulse propagation problem in nonlinear dispersive media. To understand the physical meanings behind the split-step Fourier method, it is useful to write NLSE formally in the form: E z = (Dˆ + Nˆ )E (2.16) 29

45 Chapter 2: Theory of pulse propagation in optical fibers where Dˆ is a propagation operator that accounts for dispersion and absorption in a linear medium and Nˆ is a nonlinear operator that governs the effect of fiber nonlinear on pulse propagation. The formal exact solution of Eq. (2.16) is given by E (z + h,t) = exp[h(dˆ + Nˆ )]E(z,T) (2.17) Generally, dispersion and nonlinear effects act together along the length of the fiber. The split-step Fourier method obtains an approximate solution by assuming that in propagating the optical field over a small distance h, the dispersive and nonlinear effects can be assumed to act independently. More specifically, propagation from z to z+h is carried out in two steps. In the first step, the dispersion acts alone and Nˆ = 0 in Eq. (2.16). In the second step, the nonlinear acts alone and Dˆ = 0 in Eq. (2.16). Mathematically, E(z + h,t) exp(hdˆ )exp(hnˆ )E(z, T) (2.18) By adopting a different procedure to propagate the optical pulse over one segment from z to z+h, the accuracy of the split-step Fourier method can be improved. In this procedure Eq. (2.18) is replaced by 30

46 Chapter 2: Theory of pulse propagation in optical fibers z+ h h h E(z + h,t) exp( Dˆ )exp[ Nˆ (z')dz']exp( Dˆ )E(z, T) 2 (2.19) 2 z The main difference is that the effect of nonlinear is included in the middle of the segment rather than at the segment boundary. This scheme is known as the symmetrical split-step Fourier method because of the symmetric form of the exponential operators in Eq. (2.19). If the step size h is small enough, the integral in the middle exponential can be approximated by exp (hnˆ ). The most important advantage of using the symmetrical form of Eq. (2.19) is that the leading error term is of third order in the step size h. Fig. 2.1 Schematic illustration of the symmetrical split-step Fourier method. The implementation of the split-step Fourier method is relatively straightforward. As shown in Fig 2.1, the fiber length is divided into a large number of segments that need not be spaced equally. The optical pulse is propagated from segment to segment using the prescription of Eq. (2.19). 31

47 Chapter 2: Theory of pulse propagation in optical fibers Although the method is relatively straightforward to implement, it requires that step sizes in z and T be selected carefully to maintain the required accuracy. The optimum choice of step sizes depends on the complexity of the problem. Sometimes it may be necessary to repeat the calculation by reducing the step size to ensure the accuracy of numerical simulations. The time window should be wide enough to ensure that the pulse energy remains confined within the window. Typically, window size is times of the pulse width. 32

48 Chapter 3: Cavity transmission analysis and soliton operation regimes Chapter 3 Cavity transmission analysis and soliton operation regimes Mode locking of a laser generates optical pulses circulating in the cavity. In the case of a fiber laser, apart from the cavity components that are necessary for achieving mode locking and for laser output, its cavity is mainly made up of optical fibers. It is natural to expect that under certain conditions optical soliton can also be formed in the lasers. However, in addition to the propagation in optical fibers, an optical pulse propagating in the laser cavity also subjects to actions caused by the other cavity components and by the cavity. As the pulse is circulating in the cavity, these actions occur periodically. Therefore, solitons obtained in a fiber laser are of new characteristics and features. In this chapter we first analyze the nonlinear polarization rotation (NPR) mode locking technique in Section 3.1. Section 3.2 introduces cavity transmission analysis. The experimental implementation of the NPR mode locking is presented in Section 3.3. The modeling of the soliton fiber lasers is illustrated in Section 3.4. Following that three soliton operation regimes of the mode-locked fiber lasers are numerically outlined in Section Nonlinear polarization rotation mode locking 33

49 Chapter 3: Cavity transmission analysis and soliton operation regimes Soliton operation of laser systems is closely related to the mode locking of a laser. Through mode locking an optical pulse is formed in the laser cavity. If the peak power of the pulse is strong, it is then shaped into a soliton under the mutual action of the laser cavity dispersion and the nonlinear phase modulation effect. There are various techniques that can be used to achieve mode locking in a laser, either actively such as the active amplitude or phase modulation [45-47], or passively such as the semiconductor saturable absorber method [48,49], the additive pulse methods [50,51]. A nice overview of active and passive mode locking and its historical development could be found in Ref. [52]. From the laser mode locking point of view, different techniques have different operation principles, therefore, even with the same laser gain medium, could have different modelocked pulse widths and pulse stabilities. However, if ignoring these differences of the specific mode locking techniques, after a soliton is formed in a laser the influence of each of the mode locking techniques on the soliton is subtle. Extensive experimental studies have shown that the soliton operations of lasers mode-locked with different techniques exhibit similar general properties. For this reason and also for the easiness of description, we will focus our discussions on the soliton fiber lasers passively mode-locked by the NPR technique. Based on the soliton operation of the lasers we elaborate the general properties of the fiber soliton lasers, the effect of laser cavity and gain medium on the soliton dynamics of a laser. The NPR mode locking technique exploits the nonlinear birefringence of the single mode optical fibers and uses it to realize an artificial saturable absorber effect with a polarization dependent element in the laser cavity for a self-started mode locking. The 34

50 Chapter 3: Cavity transmission analysis and soliton operation regimes same effect of optical fibers was used previously for the polarization switching and optical pulse shaping [45,53,54]. Hoffer et al. were the first one who used the effect as a self-sustaining mechanism for passive mode locking of fiber lasers [55]. However, owing to the short length and relatively high birefringence of the fiber used, mode locking of their laser could not self-start. The first successful demonstration of the effect for a selfstarted mode locking was shown by Matsas et al. [56]. The technique was then widely used for the self-started mode locking of the passively mode-locked ultrashort pulse fiber lasers. In order to get insight into the NPR mode locking technique, it is worth briefly introducing the saturable absorber mode locking at first since the NPR mode locking is in a sense regarded as an artificial saturable absorber effect. Saturable absorber mode locking is a well-known technique extensively investigated and widely used [57-59]. Here we only present a very simple explanation. The saturable absorbers in the cavity are generally made of semiconductor with quantum-well structures, which exhibits an intensity-dependent transmission or reflection. The physical mechanism responsible for this behavior can be simply explained as following: When light with low intensity passes through the saturable absorber, the saturable absorber absorbs photons. An electron in the absorber is consequently excited to a higher energy level then quickly non-radiatively relaxes to its original energy level, where it is able to absorb another incident photon. In the high intensity case, we may envisage that there are many photons incident simultaneously so some of them are absorbed and saturate the absorber, at the same time other incident photons are able to transmit the medium without being absorbed. Therefore, 35

51 Chapter 3: Cavity transmission analysis and soliton operation regimes for randomly initially formed pulses in the cavity, the saturable absorber will attenuate the pulse wing and allow the center of the pulse to propagate in the cavity with minimal loss. Such repeated process shortens the pulse, whose final width is ultimately determined by the properties of the saturable absorber as well as other cavity parameters and operation conditions. Fig. 3.1 Schematic of mode locking in fiber lasers. It is better to use a configuration as shown in Fig. 3.1 to illustrate the mode locking principle of the NPR technique. A segment of linearly birefringent optical fiber is placed between two linear polarizers. Light of arbitrary polarization incident to the setup is transferred into linearly polarized light by the polarizer before the fiber. When the light further propagates in the fiber, it splits into two components along the two polarization axes of the fiber, respectively. The propagation of light in the birefringent fiber is determined by either the equation (2.8) if the fiber is undoped or (2.14) if the fiber is doped. After passing through the fiber, generally the polarization of the light becomes elliptically polarized. If the light intensity is weak, then the ellipticity of the light polarization will be fully determined by the linear fiber birefringence and the orientation of the polarizer. However, if the intensity of the light is strong, the nonlinear effects of the fiber must be considered. Then due to the nonlinear optical Kerr effect of the fiber, 36

52 Chapter 3: Cavity transmission analysis and soliton operation regimes the polarization of the light after passing through the fiber will become light intensity dependent. Consequently, the transmission of the light through the polarizer after the fiber (also called as analyzer) will as well become the light intensity dependent. Through appropriately selecting the orientation of the analyzer, a situation could be achieved with the setup that the stronger the light, the larger is the light intensity transmission through the analyzer. Physically, such a result is equivalent to that generated by a saturable absorber. Therefore, if incorporating such a setup in a fiber laser, under the action of the artificial saturable absorber effect generated by the setup, mode locking of a laser should be possible to achieve. As the artificial saturable absorption is generated based on the optical Kerr effect, which has a recovery time in the order of several femtoseconds, from the laser mode-locking point of view, ultrashort mode-locked pulses could be generated by the technique. 3.2 Cavity transmission analysis To explain the mode locking mechanism mathematically, we analyze the transmission of the laser cavity as shown in Fig The light transmission through the setup can be calculated by using the Jones matrix method [60]. Assuming that the two principal polarization axes of the birefingent fiber are the X (horizontal) and Y (vertical) axis; the polarizer and the analyzer are set in such positions that their polarization directions have 37

53 Chapter 3: Cavity transmission analysis and soliton operation regimes an angle of θ and ϕ to the Y axis of the fiber polarization, respectively. Then the intensity transmission T of light through the setup is [61]: I out T = = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ ) cos( Φ F ) (3.1) I 2 in where Φ F is the phase delay generated between the two light polarization components when they traverse the birefringent fiber. The phase delay actually consists of two parts: linear phase delay Φ l caused by the linear birefringence of the fiber, this part always exists; and nonlinear part Φ nl generated by the nonlinear effects of the fiber. If the light intensity is weak, the nonlinear part nearly becomes zero. Hence one can divide the light transmission into the linear transmission and the nonlinear transmission. The linear intensity transmission can be written as: T l = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ ) cos( Φ l ) (3.2) 2 Once the orientations of the polarizer and analyzer with respect to the fiber polarization principal axes are fixed, the linear transmission of the setup is a sinusoidal function of the linear phase delay between the two polarization components. Fig. 3.2 shows curves of the cavity linear transmission versus the total linear phase delay between the two polarization components, for the difference between the polarization directions of the polarizer and the analyzer varying from 0 to π/2, and when θ is equal to π/4. 38

54 Chapter 3: Cavity transmission analysis and soliton operation regimes Fig. 3.2 Cavity transmission versus the phase delay between the two orthogonal polarization components as the relative orientation between the polarization directions of the polarizer and the analyzer varies from 0 to π/2. θ is fixed as π/4. The figure shows that the maximum linear system loss occurs at positions where the total linear phase delay between the two polarization components is an integer multiple of 2π when the difference between the polarization directions of the polarizer and the analyzer varying from π/4 to π/2. When the phase delay is away from this value, the system transmission increases. Therefore, if one sets the polarization controllers such that the light at particular wavelength λ is within the gain bandwidth of the EDFA and satisfies this condition, mode locking can occur because the nonlinear refractive index changes the phase delay between the two polarization components. Fig. 3.2 also shows that when the difference between the polarization directions of the polarizer and the analyzer is offset from π/2, the difference between the maximum and minimum transmission of the 39

55 Chapter 3: Cavity transmission analysis and soliton operation regimes system is decreased. This means that the mode-locked pulse will be less stable because the cw is not effectively rejected. Therefore, although the relative orientation between the polarizer and the analyzer could be arbitrarily selected, in order to possibly achieve a stable mode-locked operation, it is preferred to choose the relative orientation between the polarizer and the analyzer π/2. Under such selection the linear transmission is possible to be set to zero. To illustrate the effect of the NPR on the intensity transmission, we assume that the nonlinear fiber birefringence only introduces small extra phase delay Φ nl. Then the intensity transmission can be written as: T = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ ) cos( Φ l + Φ n l ) = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ )(cos Φ l cos Φ n l sin Φ l sin Φ n l ) = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ )cos Φ l cos Φ n l 2 1 sin(2θ) sin(2ϕ )sin Φ l sin Φ n l sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ )cos Φ l sin(2θ) sin(2ϕ )(sin Φ l ) Φ = Tl sin(2θ )sin(2ϕ )sin( Φ l ) Φ nl (3.3) 2 n l The nonlinear phase delay introduces a transmission change relative to the value of the linear transmission T l. The change is not only the nonlinear phase delay Φ nl, therefore, the light intensity dependent, but also the orientations of the polarizer and analyzer, as well as the linear phase delay dependent. As a result, in order to generate an artificial 40

56 Chapter 3: Cavity transmission analysis and soliton operation regimes saturable absorber effect by the nonlinear fiber birefringent effect, it is necessary to appropriately select the combination of all these parameters. It is instructive to consider a special case to illustrate it. Consider that the polarizer and the analyzer are set orthogonal. In this case the linear intensity transmission is T l [1 cos( Φ )] = sin 2 2θ l (3.4) 2 Therefore, the maximum linear intensity transmission is limited by the factor of The intensity transmission under the existence of nonlinear phase delay Φ nl is sin 2 2θ. T 1 2 = T l + sin (2θ )sin( Φ l ) Φ nl (3.5) 2 The orientation of the polarizer determines the projection of light on the two polarization axes of the fiber, therefore, determines whether positive Φ nl or negative Φ nl would be generated. Assuming that negative Φ nl is generated, then further depending on the linear phase delay, the NPR could cause either an increase or decrease in the intensity transmission. The magnitude of Φ nl is always proportional to the intensity of light. This means that with the increase of the light intensity, either an increase or a decrease in the transmission could be generated, which is purely determined by the linear phase delay selection. 41

57 Chapter 3: Cavity transmission analysis and soliton operation regimes Strictly speaking, the nonlinear phase delay between the two polarization components of an optical pulse traversing a segment of birefringent fiber can only be determined by numerically solving the coupled NLSEs (2.8) or (2.14). In the practice it is difficult to set the linear phase delay just within the range where the saturable absorption effect is achieved. Therefore, a polarization controller (PC) is normally inserted in the setup, as shown in Fig. 3.1, to efficiently control the value of the linear phase delay Φ l. Mathematically, this is equivalent to add a variable linear phase delay bias term in the intensity transmission formula. Hence the intensity transmission can be further written as: I out T = = sin θsin ϕ + cos θcos ϕ + sin(2θ) sin(2ϕ ) cos( Φ PC + Φ F ) (3.6) I 2 in Where Φ pc is the phase delay bias introduced by the polarization controller and it is continuously tunable. With the insertion of a polarization controller in the setup, it is always possible to achieve an artificial saturable absorber effect through changing the linear birefringence in the cavity. 3.3 Experimental implementation of the NPR mode locking To achieve the self-started mode locking using the NPR technique, in this research, a fiber ring laser as shown in Fig. 3.3 is built up. Compared with linear cavity design [62], ring cavity structure needs less components and collimation of the light path is also easier. The use of fiber based components makes the setup more stable and simple. 42

58 Chapter 3: Cavity transmission analysis and soliton operation regimes Fig. 3.3 Schematic of the experiment setup. λ/4 represents the quarter-wave plate; λ/2 represents the half-wave plate; BS represents the polarization beam splitter; WDM represents the wavelength-division multiplexer. The fiber laser cavity comprises most fiber segments. According to different experimental requirement, various fiber amplifiers such as Erbium-doped fiber, Ybdoped fiber, and Nd-doped fiber and different fiber-based components such as wavelength-division multiplexers, couplers, and isolators are used. Here we use a segment of Erbium-doped fiber with positive group velocity dispersion as the gain media. The NPR technique is used to achieve the self-started mode locking. To this end, two sets of 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 light in the cavity. A fiber-based isolator is inserted in the cavity to allow the unidirectional operation of the laser. The laser is pumped by a high power Fiber Raman Laser source (BWC-FL ) of wavelength 1480 nm. The output of the laser is taken via a polarization beam splitter and analyzed with measurement devices such as optical 43

59 Chapter 3: Cavity transmission analysis and soliton operation regimes spectrum analyzer, autocorrelator, sampling oscilloscope, and RF spectrum analyzer. The beam splitter and the polarization controllers are mounted on a 76-mm-long fiber bench. Self-started mode locking is obtained in the laser by simply increasing the pump power and adjusting the orientations of the polarization controllers. When the polarization states of the polarization controllers are appropriately set, soliton operation of the laser is readily obtained after mode locking. The detailed evolution of the light in the laser can be roughly understood as following: any incident light propagates in the cavity, it will be transformed into elliptically polarized light when it incidents to the first PC (two quarterwave plates). After the PC the light is still elliptically polarized and incidents to the beam splitter, however, the orientation of the light polarization could be modified by the two quarter-wave plates over the whole 2π range. The polarization beam splitter will reflect most S/P linearly polarized component of the elliptically polarized light and transmit most P/S linearly polarized component of the elliptically polarized light. The second PC (two quarter-wave plates sandwiching a half-wave plate) will transform the linearly polarized light into another linearly polarized light with different polarization orientation. The insertion of the half-wave plate is to make the tuning simpler. The linearly polarized light incident to the fiber will then be transformed into elliptically polarized light when it propagates in the fiber. Generally, elliptically polarized light can be treated as a superposition of a left-hand and a right-hand circularly polarized mode of different intensity. The two polarizations will experience different linear and nonlinear phase shifts through the fiber. Therefore, the 44

60 Chapter 3: Cavity transmission analysis and soliton operation regimes polarization ellipse is rotated after propagating through the fiber. And due to the nonlinearity of the fiber, the ellipticity and the azimuth of the light polarization depend on the pulse intensity, which in turn causes that the cavity transmission depends on the light intensity. By carefully choosing the linear phase delay of the two PCs, one can achieve an artificial saturable absorber effect in the laser cavity, where the light with higher intensity experiences larger transmission, which furthermore leads to the passively mode locking. 3.4 Modeling of the soliton fiber lasers Generally, fiber lasers comprise of mainly optical fibers and other necessary cavity components for achieving mode locking and for laser output. A realistic description on the soliton operation of a practical laser can be obtained from the so-called pulse tracing technique. The basic idea of the technique is similar to that used by Fox and Li in calculating the transverse mode of a laser cavity [63]. It regards a certain laser output observed experimentally as a natural result of both the light propagation in the cavity and interaction with the various cavity components including the nonlinear interaction with the gain medium. Through appropriately modeling the light propagation in the cavity and interaction with the cavity components, virtually any state of laser output should be able to be reproduced. A trade-off of the technique is that generally no analytical solutions but the numerical solutions are possible. To obtain a certain output of a laser, one starts with an arbitrary light field and let it circulate in the cavity. In each cavity round trip the light will pass through the various cavity components in the same order of the real cavity 45

61 Chapter 3: Cavity transmission analysis and soliton operation regimes configuration. The light propagation in the optical fibers is described by the coupled extended NLSEs of Eq. (2.8) or (2.14). Whenever the light meets a discrete cavity component, such as the polarizer and the polarization controller, the light field is multiplied by the transfer-matrix of the discrete cavity component. The result of the previous round of calculation is then used as the input of the next round of calculation, until a steady state of calculation is reached. For example, in the experimental setup corresponding to our simulation, the laser cavity comprises optical fibers with different dispersion properties, the isolator, the polarization controllers and polarization-dependent beam splitter. The latter three items have discrete effects on the light beam circulating in the cavity. In order to model the fiber soliton laser more accurately, we numerically circulate the light in the cavity and multiply the transformation matrix of each cavity component to the light field. Generally but not necessarily, we started the simulation with a small hyperbolic-secant pulse and let it circulate in the laser system. Within one round trip in the laser system, the pulse will in turn go through each optical component including fibers, fiber amplifiers, the polarization controllers, the beam splitter, and the isolator and so on as same turn in the experiment. After finishing one round trip in the laser system, the final state of the optical pulse will be used as the initial state in the next round trip calculation. This process is repeated until an equilibrium state of optical pulse is obtained. Another traditional model that describes the soliton operation of lasers was the master equation [64-66], which takes into account the cavity dispersion, the optical Kerr 46

62 Chapter 3: Cavity transmission analysis and soliton operation regimes nonlinearity, gain, and saturable absorption in a laser. An advantage of the model is that it can give an explicit analytical solution of the mode-locked pulse of a laser, which therefore provides a clear understanding on the soliton nature of the pulse and its dependence on various laser operation parameters. However, since the analytical solution is only possible under the assumption that the pulse change within one cavity round trip is small, and the model also ignores the discrete actions of the cavity components, this master-equation model has limited application in simulating a real fiber soliton laser. The fundamental difference of the numerical model used in this thesis work to those used by other groups is that the author did not make the so-called master-equation simplification. Therefore, there is no limitation on the action of cavity components to the soliton pulse when it circulates in the cavity. The pulse tracing technique [67,68] has several advantages in comparison with the master equation model. Since the calculation is made following the pulse propagation in the cavity, therefore, pulse evolution within one round trip can be studied. Within each step of calculation the pulse s variation is always small, even if the change of a pulse within one round trip could be big. As there is no limitation on the pulse change within one round trip, dynamical process of soliton evolution, such as the process of a new soliton generation in the cavity, soliton interaction etc. can be investigated. In addition, the effect of discrete cavity components on the soliton, the influence of the dispersive waves and the different order of the cavity components on the soliton properties are automatically included in the calculation. 47

63 Chapter 3: Cavity transmission analysis and soliton operation regimes 3.5 Cavity-determined operation regimes The theoretical model used in our simulation for the pulse propagation in the fiber segments is based on the coupled extended nonlinear Schrödinger equations (2.8) or (2.14) that explicitly take into account the birefringence of the fiber, the gain and loss of the laser cavity. Here we combine Eqs. (2.8) and (2.14), then write them in a uniform format as shown in Eq. (3.7). u z v z u ik = iβu δ t 2 v ik = iβv + δ t 2 '' 2 u ik + 2 t 6 '' ''' 2 v ik + 2 t 6 3 u + iγ ( u 3 t ''' 2 3 v + iγ ( v 3 t iγ 2 v )u + v u iγ 2 u )v + u v 3 + g g u + 2 2Ω + g g g v + 2 2Ω 2 u 2 t g 2 v 2 t (3.7) where u and v are the two normalized slowly varying pulse envelopes along the slow and the fast axes, respectively. 2β = 2π n λ is the wave number difference. 2δ = 2βλ 2πc is the inverse group velocity difference. '' k is the dispersion parameter, ''' k is the third order dispersion coefficient. γ represents the nonlinear coefficient of the fiber. g is the laser gain coefficient and for doped fiber, g is represented by Eq. (2.15). Ω g is the bandwidth of the laser gain. For undoped fiber, g=0; Based on the pulse tracing technique the soliton operations of the passively mode-locked fiber ring lasers by using the NPR technique were numerically simulated. Properties of the soliton pulses formed in the lasers, influence of different laser cavity parameters on the soliton properties, as well as the intrinsic laser cavity effect on the soliton operation of 48

64 Chapter 3: Cavity transmission analysis and soliton operation regimes a laser were numerically studied. In all of the numerical simulations if it is not explicitly pointed out, the same simulation parameters as the following were used. The fiber nonlinearity γ=3 W -1 km -1, gain bandwidth Ω g =25 nm, saturation energy Psat =1000 pj and the orientations of the polarizer and the analyzer with the fast axis of the birefringent fiber are θ=π/8 and ϕ= π/2+π/8. The detailed simulating parameters will be given out in individual chapters according to different experimental conditions. Numerical simulations show that depending on the selection of the linear cavity phase delay bias the fiber soliton lasers can exhibit various regimes of operation, each with distinctively different features. Fig. 3.4 shows for example the laser operation versus the linear cavity phase delay (here denoted with Ph) in a laser cavity of L=1m SMF +4m EDF +1m SMF =6m, where " k EDF = -10 ps/nm/km, " k SMF = -18 ps/nm/km. Since The birefringence n / n for communication fibers is in the range of 10-6 ~10-7 [69], we chose beat length L b =3m, which corresponds to n = n x n y = When the linear cavity phase delay is fixed, increasing the gain would increase the peak power of the generated solitons with narrower pulse width. However, the maximum soliton peak power achievable is limited by the linear cavity phase delay. Detailed analysis of the peak power limitation effect will be illustrated in Chapter 4. Therefore, when the linear cavity phase delay bias is set small, increasing pump power only increases the number of pulses instead of continuously increasing of the peak power of the generated pulses as the peak is clamped by the peak power limitation effect. Within the mode-locked regimes, the bigger the linear cavity phase delay, the higher the maximum soliton peak power achievable. 49

65 Chapter 3: Cavity transmission analysis and soliton operation regimes (a) (b) Fig. 3.4 Solitons (a) and corresponding spectra (b) with different linear cavity phase delay bias. As it is well known, solitons in fiber lasers are generated due to the balanced interaction between the cavity dispersion effect and the nonlinear Kerr effect. Solitons are nonlinear wave propagating in the cavity. Higher soliton peak power results in stronger nonlinearity 50

66 Chapter 3: Cavity transmission analysis and soliton operation regimes when the soliton propagates in the cavity. Numerically we found that when the linear cavity phase delay bias is beyond 1.4π, phenomena of period timing (period doubling, tripling etc.) occur under certain operation parameters. Fig. 3.5 shows for example a state of period doubling. With carefully selection of the operation parameters, we could even obtain a complete route of period doubling to chaos in our lasers. Period doubling route to chaos is a well-known universal feature of nonlinear dynamical systems transiting from an equilibrium state to a chaotic state, which has been extensively investigated [70]. However, in fiber lasers, so far as we know, it is the first time that a complete route of period doubling to chaos in passively mode-locked fiber soliton lasers was discovered. Detailed experimental and numerical research on the novel nonlinear phenomena will be represented in Chapter 6. Fig. 3.5 Period doubling of soliton pulse train. When the linear cavity phase delay bias is set to be large, which means the maximum soliton peak power achievable is very high, solitons collapse before they reach the peak power limitation with increasing pump power. In this case a soliton is first slowly built up 51

67 Chapter 3: Cavity transmission analysis and soliton operation regimes to a very high peak power and very narrow pulse width and then quickly collapses. This process repeats continuously. When the pump power is strong, several of such pulses generate and disappear randomly in the cavity forming a state of so-called noise-like pulse operation as shown in Fig Detailed discussion about noise-like pulse generation and their properties will be shown in Chapter 7. Fig. 3.6 Noiselike pulse when the linear cavity phase delay bias is 1.7π. Based on the above discussions, it is obvious that the fundamentals of soliton formation in a laser is still the balance between the cavity dispersion and nonlinear phase modulation, however, the property of the cavity (represented by the linear cavity phase delay bias) strongly affect the soliton properties and dynamics. Therefore, the understanding of the cavity effect on the soliton generation and propagation in fiber lasers is fundamental to the comprehension of the soliton dynamics in passively modelocked fiber lasers. Chapter 4 to Chapter 7 will describe in detail the experimental and numerical studies on the soliton dynamics in passively mode-locked fiber lasers using NPR technique under different linear cavity phase delay bias. 52

68 Chapter 4: Multiple soliton generation and soliton energy quantization Chapter 4 Multiple soliton generation and soliton energy quantization Passively mode-locked fiber lasers as a simple and economic ultrashort pulse source have been extensively investigated in the past decade [11,71-78]. By implementing the soliton pulse shaping technique in the lasers it was demonstrated that optical pulses in the subpicosecond range could be routinely generated. Various passive mode locking techniques, such as the nonlinear loop mirror method [11,74], the nonlinear polarization rotation (NPR) technique [71,75,76] and the semiconductor saturable absorber method [77,78], have been used to mode-lock the lasers. Independent of the specific mode locking techniques it was found that the soliton operation of all the lasers exhibited a common feature, namely under strong pumping strength multiple soliton pulses are always generated in the laser cavity, and in the steady state all the solitons have exactly the same pulse properties: the same pulse energy and pulse width when they are far apart. The latter property of the solitons was also called as the soliton energy quantization effect [9]. The multiple soliton generation and the soliton energy quantization effect limit the generation of optical pulses with larger pulse energy and narrower pulse width in the lasers. Therefore, in order to further improve the performance of the lasers it is essential to have a clear understanding on the physical mechanism responsible for these effects. In this chapter we present results of numerical simulations on the soliton formation and soliton energy quatization in a fiber ring laser passively mode-locked by using the NPR 53

69 Chapter 4: Multiple soliton generation and soliton energy quantization technique. We show that soliton formation is actually a natural consequence of a modelocked pulse under strong pumping if a laser is operating in the anomalous total cavity dispersion regime. Especially we will show how the parameters of a laser soliton, such as the peak power and pulse width, vary with the laser operation conditions. Based on our numerical simulations we further show that, for the first time to our knowledge, the multiple soliton formation in the laser is caused by a peak power limiting effect of the laser cavity. It is also the effect of the cavity that suppresses the soliton collapse and makes the solitons stable in the laser even when the laser gain is very strong. Furthermore, we demonstrate numerically that the soliton energy quantization of the laser is a natural consequence of the gain competition between the solitons in the cavity. In Section 4.1, we show some typical experimental observations of the multiple soliton operation and soliton energy quantization of the soliton fiber lasers. In Section 4.2, we discuss simulation results under various parameter sets and compare them with experimental observations. In Section 4.3, basing on experimental observation and numerical simulation, physical mechanism of the multiple soliton generation and soliton energy quantization is concluded. Section 4.4 is the summary. 4.1 Experimental observations For the purpose of comparison and a better understanding of our numerical simulations, we present here some of the typical experimental results on the multiple soliton operation and soliton energy quantization of the soliton fiber lasers. We note that although the results presented here were obtained from a particular soliton fiber ring laser as described 54

70 Chapter 4: Multiple soliton generation and soliton energy quantization below, similar features were also observed in other lasers [11,74,75,77], which are in fact independent of the specific laser systems. Fig. 4.1 A schematic of the soliton fiber laser. PI: Polarization dependent isolator. PC: Polarization controller. DSF: Dispersion shifted fiber. EDF: Erbium-doped fiber. WDM: Wavelengthdivision-multiplexer. A schematic of the fiber soliton laser we used in our experiments is shown in Fig It contains 1-meter-long dispersion shifted fiber with group velocity dispersion (GVD) of about 2 ps/nm/km, 4-meter-long Erbium doped fiber (EDF) with GVD of about 10 ps/nm/km and 1-meter-long standard single mode fiber (SMF) with GVD of about 18 ps/nm/km. Two polarization controllers (PCs), one consisting of two quarter-wave plates and the other one two quarter-wave plates and one half-wave plate, were used to control the polarization of the light in the cavity. A polarization-dependent isolator was used to enforce the unidirectional operation of the laser and also determine the polarization of the light at the position. A 10% output- coupler was used to outlet the light. The soliton pulse width of the laser was measured with a commercial autocorrelator, and the average 55

71 Chapter 4: Multiple soliton generation and soliton energy quantization soliton output power was measured with a power meter. The soliton pulse evolution inside the laser cavity was monitored with a high-speed detector and a sampling oscilloscope. The soliton operation of the laser was extensively investigated previously [10,79,80], various features such as the pump power hysteresis, multiple soliton generation and various modes of multiple soliton operation, bound states of solitons were observed. Worth of mentioning here is the pump hysteresis effect of the soliton operation. It was found experimentally that the laser always started mode locking at a high pump power level, and immediately after the mode locking multiple solitons were formed in the cavity. After the soliton operation was obtained, the laser pump power could then be reduced to a very low level while the laser still maintained the soliton operation. This phenomenon of the laser soliton operation was known as the pump power hysteresis [81]. It was later turned out that the pump power hysteresis effect is related to the multiple soliton operation of the laser. Once multiple solitons are generated in the cavity, decreasing the pump power the number of solitons is reduced. However, as far as one soliton is remained in the cavity, the soliton operation state (and therefore the mode locking of the laser) is maintained. Not only the soliton operation of the laser exhibited pump power hysteresis, but also the generation and annihilation of each individual soliton in the laser exhibited pump power hysteresis [10]. Experimentally it was observed that if there were already solitons in the cavity, carefully increasing the pump power, new solitons could be generated one by one in the cavity. As in this case the laser is already 56

72 Chapter 4: Multiple soliton generation and soliton energy quantization mode-locked, the generation of a new soliton only requires a small increase of the pump power. Fig. 4.2 A typical experimentally measured oscilloscope trace of the multiple soliton operation of the laser. An important characteristic of the multiple soliton operation of the laser is that, as far as the solitons are far apart in the cavity, they all have exactly the same soliton parameters: the same pulse width, pulse energy and peak power. To demonstrate the property we have shown in Fig. 4.2 the oscilloscope trace of a typical experimentally measured multiple soliton operation state of our laser. The cavity round trip time of the laser is about 26ns. There are 6 solitons coexisting in the cavity. It can be clearly seen that each soliton has exactly the same pulse height in the oscilloscope trace. Although with the electronic detection system the detailed pulse profile of the solitons cannot be resolved, nevertheless, the measured pulse height in the oscilloscope trace is directly related to the energy of each individual soliton. Based on the measured autocorrelation traces and 57

73 Chapter 4: Multiple soliton generation and soliton energy quantization optical spectra it was further identified that all the solitons indeed have exactly the same soliton parameters. 4.2 Numerical simulations and results To find out the physical mechanism of the multiple soliton formation in our laser, we have conducted numerical simulations to study the soliton evolution in the laser and compared with those typically observed experimental results of the fiber soliton lasers. The simulations are based on the numerical model described in Section 3.4. To possibly close to the experimental conditions of the laser, we have used the following fiber parameters for our simulations: Ω g = 20 nm, cavity length L=6m and the beat length of the fiber birefringence L b =L/4. To simulate the cavity effect, we let the light circulate in the cavity. Starting from the intra-cavity polarizer (PI in Fig. 4.1), which has an orientation of θ = π to the fiber s fast axis, the light then propagates in the various fibers, first through the 1m dispersion shifted fiber (DSF), which has a GVD coefficient of '' k = -2 ps/nm/km, then the 4m EDF whose GVD coefficient ps/nm/km, and finally the 1m standard single mode fiber whose GVD coefficient '' k = -10 '' k = -18 ps/nm/km. Subsequently the light passes through the waveplates, which cause a fixed polarization rotation of the light. Note that changing the relative orientations of the waveplates is physically equivalent to adding a variable linear cavity phase delay bias to the cavity. Certainly the principal polarization axes of the waveplates are not aligned with those of the fibers, and in general the different fibers used in the laser cavity could also 58

74 Chapter 4: Multiple soliton generation and soliton energy quantization have different principal polarization axes. However, for the simplicity of numerical calculations, we have treated them all having the same principal polarization axes, and considered the effect caused by the principal polarization axis change by assuming that the polarizer has virtually a different orientation to the fast axis of the fiber when it acts as an analyzer. Fig. 4.3 Numerically calculated multiple soliton operation state of the laser. δφ l = 1.20π, G=350. Other parameters used are described in the text. We found that by appropriately setting the linear cavity phase delay bias of the cavity, so that an artificial saturable absorber effect can be generated in the laser, self-started mode locking can always be generated in our simulations through simply increasing the small signal gain coefficient, which corresponds to increasing the pump power in the experiments. Exactly like the experimental observations, multiple soliton pulses are formed in the simulation window immediately after the mode locking. In the steady state and when the solitons are far separated, all the solitons obtained have exactly the same pulse parameters such as the peak power and pulse width. Fig. 4.3 shows for example a 59

75 Chapter 4: Multiple soliton generation and soliton energy quantization numerically calculated multiple soliton operation of the laser. Like the experimental observations, the soliton operation of the laser and the generation and annihilation of each individual soliton in the cavity exhibit pump hystersis. Decreasing numerically the pump power, the soliton number in the simulation window reduces one by one, while carefully increasing the pump strength, with at least one soliton already existing in the cavity, solitons can also be generated one by one as shown in Fig All these numerically calculated results are in qualitative agreement with the experimental observations [10]. Fig. 4.4 Relationship between the soliton number in the simulation window and the pump strength. δφ l = 1.20π. In a practical laser due to the existence of laser output, fiber splices etc, the linear cavity losses are unavoidable. However, in the numerical simulations we could artificially reduce the linear cavity losses and even make it to zero. We found numerically that the weaker the linear cavity loss, the smaller is the pump hysteresis of the soliton operation. With a very weak linear cavity loss we found numerically that a single soliton pulse 60

76 Chapter 4: Multiple soliton generation and soliton energy quantization could even be directly formed from a mode-locked pulse through increasing the pump strength. This numerical result clearly shows that the large pump hysteresis of the soliton operation of the laser is caused by the existence of large linear cavity loss of a practical laser. A large linear cavity loss makes the mode locking threshold of a laser very high, which under the existence of cavity saturable absorber effect, causes that the effective gain of the laser after mode locking is very large. As will be shown below, when the peak power of a pulse is clamped, this large effective laser gain will then results in the formation of multiple solitons immediately after the mode locking of the laser. We noted that the theoretical work of A. Komarov et al. on the multistability and hysteresis phenomena in passively mode-locked fiber lasers [82]. In the framework of their model they have explained these phenomena as caused by the competition between the positive nonlinear feedback and the negative phase modulation effect [82]. It is noted that in their model in order to obtain the multiple soliton operation, the cavity loss term caused by the frequency selective filter has to be added in, which from another aspect confirms our numerical result shown above. 61

77 Chapter 4: Multiple soliton generation and soliton energy quantization (a) (b) Fig. 4.5 Soliton shaping of the mode-locked pulse in the laser. δφ l = 1.20π. (a) Evolution of pulse profile with the pump strength. (b) Evolution of the optical spectra with the pump strength. By making the linear cavity loss small, we have numerically investigated the process on how a soliton is formed in the laser cavity. Fig. 4.5 shows the results of numerical simulations. In obtaining the result the linear cavity phase delay bias is set to δφ l =1.2π. When G is less than 251, there is no mode locking. In the experiment this corresponds to the case that the laser is operating below the mode-locking threshold. When G is equal to 62

78 Chapter 4: Multiple soliton generation and soliton energy quantization 252, a mode-locked pulse emerges in the cavity. The mode-locked pulse has weak pulse intensity and broad pulse width. Due to the action of the mode locker, which in the laser is the artificial saturable absorber, the mode-locked pulse circulates stably in the cavity, just like any mode-locked pulse in other lasers. Although such a mode-locked pulse has stable pulse profile during circulation in the cavity, we emphasize that it is not a soliton but a linear pulse. The linear nature of the pulse is also reflected by that its optical spectrum has no sidebands. When G is further increased, the peak power of the pulse quickly increases. Associated with the pulse intensity increase the nonlinear optical Kerr effect of the fiber also becomes strong and eventually starts to play a role. An effect of the pulse self-phase-modulation (SPM) is to generate a positive frequency chirp, which in the anomalous cavity dispersion regime counterbalances the negative frequency chirp caused by the cavity dispersion effect and compresses the pulse width. When the pulse peak power has become so strong that the nonlinear SPM effect alone can balance the pulse broadening caused by the cavity dispersion effect, even without the existence of the mode-locker, a pulse can propagate stably in the dispersive laser cavity. In this case a mode-locked pulse then becomes a soliton. In the case of our simulation, this corresponds to the state of G=253. A soliton in the laser is also characterized by the appearance of the sidebands in the optical spectrum as shown in Fig. 4.5b. Once the laser gain is fixed, a soliton with fixed peak power and pulse width will be formed, which are independent of the initial conditions. The states shown in Fig. 4.5 are stable and unique. This result confirms the auto-soliton property of the laser solitons [45]. However, if the pump power is continuously increased, solitons with even higher peak 63

79 Chapter 4: Multiple soliton generation and soliton energy quantization power and narrower pulse width will be generated. Associated with the soliton pulse width narrowing, the spectrum of the soliton broadens, and consequently more sidebands become visible (stronger). However, the positions of the sidebands are almost fixed. The physical mechanism of sideband generation of laser solitons was extensively investigated previously and is well understood now [83]. It is widely believed that the sideband generation is a fundamental limitation to the soliton pulse narrowing in a laser [84]. However, our numerical simulations clearly show that the sideband generation is just an adaptive effect, whose existence does not limit the soliton pulse narrowing. As far as the pump power could balance the loss caused by the sidebands, soliton pulse width can still be narrowed. Based on our numerical simulation and if there is no other limitation as will be described below in Section 4.3, the narrowest soliton pulse that can be formed in a laser should be ultimately only determined by the laser cavity dispersion property, including the net dispersion of all the cavity components and the dispersion of the gain medium. 64

80 Chapter 4: Multiple soliton generation and soliton energy quantization Fig. 4.6 Process of the new soliton generation in the laser. δφl = 1.20π. (a) G=255; (b) G=270; (c) G=

81 Chapter 4: Multiple soliton generation and soliton energy quantization With already one soliton in the simulation window, we then further increased the pump strength. Depending on the selection of the linear cavity phase delay bias, we found that the mechanism of the new soliton generation and the features of the multiple soliton operation in the laser are different. With the laser parameters as described above, we found that when the linear cavity phase delay bias is set small, say at about δφ l =1.2π, further increasing the pump power, initially the soliton pulse peak power will be increased and its pulse width narrowed as expected. However, to a certain fixed value these will stop, instead the background of the simulation window becomes unstable and weak background pulses become visible as shown in Fig. 4.6b. Further slightly increasing the pump power, a new soliton is quickly formed in the cavity through the soliton shaping of one of the weak background pulses. As the weak background pulses are always initiated from the dispersive waves of the solitons, W. S. Man et al. have called the new soliton generation soliton shaping of dispersive waves [85]. In the steady state both solitons have exactly the same pulse width and peak power as shown in Fig. 4.6c. When the pump power is further increased, new solitons are generated one by one in the simulation window in exactly the same way and eventually a multiple soliton state as shown in Fig. 4.3 is obtained. This numerically simulated result is well in agreement with the experimental observations [10]. Because of the new soliton generation, the solitons formed in the laser cannot have large pulse energy and high peak power through simply increasing the pump power. The larger the laser gain, the more solitons would be formed in the cavity. 66

82 Chapter 4: Multiple soliton generation and soliton energy quantization Fig. 4.7 Soliton evolutions calculated with δφ l = 1.80π. (a) G=470; (b) G=478; (c) G=

83 Chapter 4: Multiple soliton generation and soliton energy quantization When the linear cavity phase delay bias is set at a very large value, say at about δφ l =1.8π, which is still in the positive cavity feedback range but close to the other end, no stable propagation of the solitons in cavity can be obtained. With the linear cavity phase delay bias selection, there is a big difference between the linear cavity loss and the nonlinear cavity loss. Therefore, if the gain of the laser is smaller than the dynamical loss that a soliton experienced, the soliton quickly dies out as shown in Fig. 4.7a. While if the gain of the laser is even slightly larger than the dynamical loss that a soliton experienced, the soliton peak power will increase. Higher soliton peak power results in smaller dynamical loss and even larger effective gain, therefore, the soliton peak will continuously increase. Associated with the soliton peak increase the soliton pulse width decreases, eventually the soliton breaks up into two solitons with weak peak power and broad pulse width as described by Kärtner et al. [86]. Once a soliton is broken into two solitons with weak peak power, the dynamical loss experienced by each of the solitons becomes very big. Consequently the gain of the laser cannot support them. The new solitons are then immediately destroyed as shown in Fig. 4.7b. If very large gain is available in the laser, the new solitons may survive in the cavity temporally and each of them repeats the same process as shown in Fig. 4.7b, and eventually a state as shown in Fig. 4.7c is formed. Therefore, no stable soliton propagation is possible with too large linear cavity phase delay setting in the laser. Even in the cases of stable multiple soliton operation, depending on the selection of the linear cavity phase delay, the solitons obtained have different parameters. Fig. 4.8 shows for comparison the multiple soliton operation obtained with the linear cavity phase delay 68

84 Chapter 4: Multiple soliton generation and soliton energy quantization bias set at δφ l =1.55π. It is obvious that solitons with higher peak power and narrower pulse width can be formed with the linear cavity phase delay setting comparing with Fig Extensive numerical simulations have shown that the larger the linear cavity phase delay setting, the higher the soliton peak and the narrower of the soliton pulse achievable. Fig. 4.8 Multiple soliton operation of the laser calculated with linear cavity phase delay bias set at δφ l = 1.55π, G = Mechanism of the multiple soliton generation and soliton energy quantization Apparently, depending on the laser linear cavity phase delay bias setting, there exist two different mechanisms of new soliton generation in the laser. One is the soliton shaping of the unstable dispersive waves or the CW components, and the other one is the wellknown mechanism of pulse splitting. It is obvious that in the laser the process of soliton 69

85 Chapter 4: Multiple soliton generation and soliton energy quantization splitting occurs only in the regime where the new solitons formed are practically unstable. W. S. Man et al. have already reported previously the phenomenon of soliton generation through unstable background in the lasers [85]. Here we further explain its physical origin. The laser setup shown in Fig. 4.1 could also be simplified to the laser cavity shown in Fig In Section 3.2, we have analyzed the transmission of the laser cavity. Here we further analyze how and to what extent the NPR effects affect the soliton parameters and soliton dynamics. As mentioned in Section 3.2, if we consider the nonlinear fiber birefringence, the intensity transmission can be written as [61]: T = sin ( θ )sin ( ϕ) + cos ( θ ) cos ( ϕ) + sin(2θ )sin(2ϕ ) cos( Φ l + Φ nl ) (4.1) 2 C. J. Chen et al. [61] and R.P. Davey et al. [87] have already shown how to select the orientations of the polarizer and the analyzer so that the cavity would generate efficiently saturable absorption effect. In a previous paper [88] W. S. Man et al. have shown that the linear cavity transmission of the laser is a sinusoidal function of the linear cavity phase delay Φ l with a period of 2π. We should point out that within one period of the linear cavity phase delay change, the laser cavity can provide positive (the saturable absorber type) cavity feedback only in half of the period, in the other half of the period it actually has negative feedback. 70

86 Chapter 4: Multiple soliton generation and soliton energy quantization As shown in Eq. 4.1, the actual cavity transmission for an optical pulse is also the nonlinear phase delay Φ nl dependent, which has been literally explained in Section 3.2. Here we use our simulations as an example to illustrate functions of this part. In our simulations the orientation of the polarizer has an angle of θ = 0.125π to the fast axis of the fiber, so light propagation in the fiber will generate a negative nonlinear phase delay. The linear cavity beat length is ¼ of the cavity length, therefore, the maximum linear cavity transmission is at the positions of (2n+1)π linear cavity phase delays, where n=0,1,2. Furthermore, when the linear cavity phase delay is biased within the range between the (2n+1)π to 2(n+1)π, the cavity will generate a positive feedback, as under the effect of the NPR the actual cavity transmission increases. While if the linear cavity phase delay is located in the range from 2nπ to (2n+1)π, the cavity will generate a negative feedback. The maximum linear cavity transmission point also marks the switching position of the two feedbacks. For the soliton operation the laser is always initially biased in the positive cavity feedback regime. It is clear that depending on the selection of the linear cavity phase delay and the strength of the nonlinear phase delay, the cavity feedback is possible to be dynamically switched from the positive feedback regime to the negative feedback regime. For the soliton operation of a laser this cavity feedback switching has the consequence that the peak of a soliton formed in the cavity is limited. We found that it is this soliton peak limiting effect that results in the multiple soliton generation in the soliton fiber laser and the soliton energy quantization. To explain these, we assume that the peak power of a soliton is so strong that it switches the cavity from the positive to the negative feedback regime. In this case although 71

87 Chapter 4: Multiple soliton generation and soliton energy quantization increasing the pump strength will still cause the peak power of the soliton to increase, the higher the soliton peak power increases, the smaller the actual cavity transmission becomes. To a certain fixed value of the soliton peak power, which depends on the linear cavity phase delay setting, further increase of the soliton peak power would results in that the actual cavity transmission that the soliton experiences becomes smaller than the linear cavity transmission. At this point the soliton peak will be clamped. Further increasing the laser gain will not amplify the soliton but the background noise such as the dispersive waves. If the background noise of a certain frequency fulfills the lasing condition, it could also start to lase and form a CW component in the soliton spectrum. We note that coexistence of solitons with CW is a generic effect of the soliton fiber lasers, and the phenomenon was reported by several authors [89,90]. Linear waves are intrinsically unstable in the cavity due to the modulation instability. When they are strong enough, they become modulated. And under the effect of saturable absorption, the strongest background pulse will be amplified and shaped into a new soliton. This was exactly what we have observed in the experiments on how a new soliton was generated. The two solitons in the cavity share the same laser gain. As the cavity generates a positive feedback for the weak soliton and a negative one for the strong soliton, under the gain competition the two solitons have to adjust their strength so that the stronger one becomes weaker, and the weaker one becomes stronger, eventually they will stabilize at a state that both solitons have exactly the same peak power. The soliton internal energy balance further determines their other parameters. Except that there are interactions between the solitons, they will always have identical parameters in the equilibrium state. 72

88 Chapter 4: Multiple soliton generation and soliton energy quantization It turns out that the multiple soliton formation in the laser is in fact caused by the peak power clamping effect of the cavity. In addition, the soliton energy quantization observed is also a natural consequence of the gain competition between the solitons in the laser. Obviously the maximum achievable soliton peak power in the laser is the linear cavity phase delay dependent. When the linear cavity phase delay is set close to the cavity feedback switching point, solitons with relatively lower peak power could already dynamically switch the cavity feedback. Therefore, solitons obtained at this linear cavity phase delay setting have lower peak power and broader pulse width as shown in Fig. 4.6a. While if the linear cavity phase delay is set far away from the switching point, soliton peak power is clamped at a higher value, solitons with higher peak and narrower pulse width would then be obtained as shown in Fig In particular, if the linear cavity phase delay is set too close to the switching point, as the peak power of the pulse is clamped to too small value, except the mode-locked pulses, no soliton could be formed in the laser. While when the linear cavity phase delay is set too far away from the switching point as demonstrated numerically in Fig. 4.7, before the soliton peak reaches the switching point, it has already become so high and so narrow that it splits, no stable soliton propagation could be obtained in the laser. Instead only the state of so-called noise-like pulse emission will be observed [33]. Finally we note that the multiple soliton operation and soliton energy quantization effect have also been observed in other passively mode-locked soliton fiber lasers, such as in the figure-of-eight lasers and the lasers passively mode-locked with semiconductor saturable absorbers [77,78]. Even in the actively mode-locked fiber lasers [73] these 73

89 Chapter 4: Multiple soliton generation and soliton energy quantization phenomena have also been observed. Despite of the fact that those soliton lasers are not mode-locked with the NPR technique, therefore, their detailed cavity transmission could not have the same feature as described by Eq. 4.1, enlightened by the result obtained in our laser, we speculate that there must also have a certain pulse peak power limiting mechanism in the lasers, which causes their multiple pulse formation. Indeed, we found that for the figure-of-eight lasers, if the fiber birefringence of the nonlinear loop were further considered, it would also generate a similar pulse peak clamping effect in the laser. However, birefringence of fibers in the lasers is normally ignored. It was also reported that due to the two-photon absorption effect the SESAM used for the passive mode locking of fiber lasers has a pulse peak power limiting effect [91]. It is therefore not surprising that soliton laser mode-locked with the material could also exhibit multiple solitons. For the actively mode-locked laser, in most cases the multiple soliton generation is due to the harmonic mode locking. In this case as too many solitons share the limited cavity gain, the energy of each pulse is weak. Therefore, even when the net cavity dispersion is negative, solitons are normally difficult to form. We point out that for an actively mode-locked fiber laser if the cavity is not carefully designed, the cavity birefringence combined with the modulator, which is a polarizing device, could form a birefringence filter and further limit the peak power of the pulses formed in the lasers. 4.4 Summary In this chapter we have numerically studied the mechanism of multiple soliton generation and soliton energy quantization in a soliton fiber ring laser passively mode-locked by 74

90 Chapter 4: Multiple soliton generation and soliton energy quantization using the NPR technique. We identified that the multiple soliton generation in the laser is caused by the peak power clamping effect of the cavity. Depending on the linear cavity phase delay setting, the nonlinear phase delay generated by a soliton propagating in the fiber cavity could be so large that it switches the cavity feedback from the initially selected positive regime into the negative regime. And as a result of the cavity feedback change the maximum achievable soliton peak power is then limited. In this case increasing the laser pump power will not increase the peak power of the solitons, but generate a new soliton. Therefore, multiple solitons are formed in the laser. As the solitons share the same laser gain, gain competition between them combined with the cavity feedback feature further results in that in the steady state they have exactly the same soliton parameters. The parameters of solitons formed in the laser are not fixed by the laser configuration but vary with the laser operation conditions, which are determined by the soliton internal energy balance between the shared laser gain and the dynamical losses of each soliton. 75

91 Chapter 5: Multi-pulse solitons Chapter 5 Multi-pulse solitons This chapter reports on the experimental evidence of multi-pulse solitons in a dispersionmanaged passively mode-locked fiber ring laser. We show experimentally that the multipulse solitons are formed by the direct soliton interaction, and they are strongly stable. Although formed under different experimental conditions, multi-pulse solitons with different pulse number were found to have exactly the same fixed pulse separations. Bound states of multi-pulse solitons were also experimentally obtained, which suggested that the multiple pulses in the solitons behave as a unit. They together could be treated as a new form of solitary wave multi-pulse soliton. Section 5.1 presents the experimental observation on the multi-pulse solitons and the bound states of the multi-pulse solitons. The physical mechanism of the generation of multi-pulse solitons is illustrated in Section 5.2. And Section 5.3 is the summary. 5.1 Experimental results The solitons formed in a fiber laser are known as the average solitons [27] and generally have single peak profile. Single-pulse solitons generated in lasers could form bound states of solitons, where several solitons bind together, and propagate in the laser cavity. Generally, the soliton separation in the bound states may vary with laser operation conditions. However, bound states of solitons with fixed soliton separation have also 76

92 Chapter 5: Multi-pulse solitons been theoretically predicted [92] to exist in nonlinear dynamical systems such as the Ginzburg-Landau equation [92-94], and the coupled nonlinear Schrödinger equations [95]. The formation mechanism of these bound states is due to a direct soliton interaction between the solitons [96], and the propagation of them is characterized by the discrete, fixed separations. Our group first experimentally observed the stable bound states of solitons with discrete, fixed soliton separations in a passively mode-locked fiber soliton laser [97]. Further experimental research and theoretical studies revealed that the boundsoliton pair behaves as a unit, which now we call twin-pulse soliton. The properties of the bound solitons have exactly the same features as those of the single-pulse soliton in the laser, such as soliton energy quantization [9,10], soliton bunching [11,13], and quasiharmonic and harmonic mode locking [14,98]. Experimental observation and numerical simulation strongly suggest that the twin-pulse soliton is in fact another form of solitary waves in the laser. However, apart from the twin-pulse solitons, multi-pulse solitons (pulse number greater than 2) have not been experimentally observed. Since twin-pulse solitons are actually a bound soliton pair with fixed soliton separation, it is plausible that under suitable condition three-, four-, or even multi-pulse solitons should also be able to be formed. Therefore, we have experimentally studied the multi-pulse soliton formation in fiber lasers. The experiments were conducted on a dispersion-managed fiber ring laser that has a similar cavity configuration as that one in Chapter 3 (Fig. 3.3). The laser cavity has a length of about 12 meters. In order to achieve the dispersion management, a 2-meter-long EDF with positive GVD of about 70 ps/nm/km is used as the gain medium, other fiber 77

93 Chapter 5: Multi-pulse solitons segments are all standard SMFs, whose GVD is about -18 ps/nm/km. The nonlinear polarization rotation technique is used to achieve the self-started mode locking. The use of the EDF with positive GVD partially compensates the dispersion of the cavity. As the pulse circulating in the cavity is alternately stretched and compressed, the average intracavity peak power is reduced, which consequently lower the possibility of overdriving the nonlinearity that limits the pulse width and the pulse energy by saturating the artificial saturable absorber or causing soliton instability. Solitary pulses with large pulse energy were possibly obtained [99-101]. Spectra Intensity (a) Wavelength Intensity (arb. units) (b) Time (ps) Fig. 5.1 Optical spectrum (a) and autocorrelation trace (b) of a soliton. 78

94 Chapter 5: Multi-pulse solitons Spectra Intensity (a) Wavelength Intensity (arb. units) (b) Time (ps) Fig. 5.2 Optical spectrum (a) and autocorrelation trace (b) of a two-pulse bound soliton. Self-started soliton operation was obtained in the laser by simply increasing the pump power and adjusting the orientations of the polarization controllers. Fig. 5.1 shows the optical spectrum (resolution of 0.5nm) and autocorrelation trace of a typical soliton observed in the laser. Due to the dispersion management of the cavity, the soliton spectrum differs from those obtained from the non-dispersion managed lasers, e.g. no sharp sidebands are visible. The autocorrelation trace also shows a strong pedestal underneath the main pulse, indicating that there exists a broad secondary structure of the solitary pulse. So far the physical mechanism of the secondary structure formation is 79

95 Chapter 5: Multi-pulse solitons unclear and under investigation. Nevertheless, similar feature has also been observed in other dispersion managed fiber lasers [6,101,102], which suggests that it could be an inherent property of the soliton in this type of fiber lasers. The soliton pulse width of the laser is about 136 fs if a sech-form pulse profile is assumed. Apart from the soliton operation, various bound states of solitons were routinely observed under different experimental conditions, e.g. different pump power and orientations of the polarization controllers. Fig. 5.2 shows as example the spectrum and autocorrelation trace of a typical two-pulse bound soliton. The two solitons in the state have a peak-topeak separation of about 2.46 ps and the same pulse height. As now two solitons are bound closely, comparing with that of Fig. 5.1 the measured soliton spectrum becomes modulated. The modulation peaks have a separation of about 3.6 nm. If the central wavelength of the soliton pulse is assumed to be 1560 nm, this corresponds to a pulse separation of 2.25 ps in the time domain. Due to the frequency chirp of the solitons and the large separation between the soliton peaks in the current case, the phase relationship between the bound solitons becomes less obvious from the measured spectrum. Nevertheless, we believe that the phase difference between the two bound solitons is about π since there is an obvious concavity in the center of the spectrum envelope. Checked with the high-speed oscilloscope and the autocorrelator we also confirmed that in the cavity there was no other solitons. Two-pulse bound solitons with a pulse separation of 3.86 ps have also been observed. However, its appearance was less frequent. Probably because it has bigger pulse separation, therefore, the binding force between the solitons is weaker. Like the two-pulse bound solitons observed in the non-dispersion 80

96 Chapter 5: Multi-pulse solitons managed fiber lasers [97], the pulse separations are fixed. Whenever a bound soliton is obtained, it has either of the two separations. Intensity(arb. units) (a) Time (ps) Intensity(arb. units) (b) Time (ps) Fig. 5.3 Autocorrelation trace of a three-pulse bound soliton (a) and a four-pulse bound soliton (b). Apart from the two-pulse bound solitons, three-pulse, four-pulse bound solitons and bound soliton arrays were also observed. Surprisingly all the pulse separations between the adjacent soliton pulses in the multipulse bound solitons are the same, it is fixed at the value of 2.46 ps, which is also the one observed for the two-pulse bound soliton. Fig. 5.3 shows for example the autocorrelation traces of the three-pulse and the four-pulse bound solitons. Again we confirmed by using the high-speed oscilloscope and the autocorrelator 81

97 Chapter 5: Multi-pulse solitons that the bound soliton was the only pulse in the cavity. In Fig. 5.3a, all the solitons have almost the same pulse height and equal separation since the different peaks observed in the autocorrelation trace are in ratio of 1:2:3:2:1. The same also occurs in Fig. 5.3b with ratio of 1:2:3:4:3:2:1. We emphasize the same soliton separations between the adjacent solitons in a bound soliton and between the various multiple-pulse bound solitons. Similar overall soliton spectral profile as that of Fig. 5.1 were observed for the three- and four-pulse bound solitons, however, as there are now more than two solitons in the bound state, extra weak spectral modulation corresponding to the extra soliton separations appear on the spectra. As all these spectral modulations have a static structure on the measured optical spectral trace, we believe that the phase difference between consecutive pulses is still locked. Experimentally, it was found that the larger the number of solitons in a bound state, the more difficult to generate it. The multipulse bound solitons are very stable in the laser. With stronger pumping power multiple multipulse bound solitons of the same structure can coexist in the cavity. Surprisingly, it seems that no bound solitons of different structures could coexist under the experimental conditions. In this case each bound soliton behaves as a unit. Bound states of the multipulse bound solitons were also experimentally observed as shown in Fig Fig. 5.4a shows the bound state of the two-pulse bound solitons, Fig. 5.4b shows the bound state of the three-pulse bound solitons. Due to the limited scan range of the autocorrelator, only part of the whole autocorrelation trace could be recorded for the state. However, we should point out that these bound states of the multipulse bound solitons are only weakly stable and the separations between the multipulse bound solitons are not 82

98 Chapter 5: Multi-pulse solitons fixed. This property shows that they were formed based on a different mechanism to that of the multipulse bound solitons described above. We note that the pulse separation between the adjacent pulses within a multipulse bound soliton is still fixed at the value of 2.46 ps. This ultrastability of the multipulse bound solitons clearly differs from those reported by Ph. Grelu et al., as to our understanding the soliton separations of the multipulse solitons reported in [67-68,99, ] vary from case to case. Bound solitons with similar properties to [68,99,104,105] were reported previously also by Kim et al. [12]. Intensity (arb. units) (a) Time (ps) Intensity (arb. units) (b) Time (ps) Fig. 5.4 Autocorrelation trace of bound multipulse bound solitons: (a) bound 2-pulse-bound solitons, (b) bound 3-pulse-bound solitons. 83

99 Chapter 5: Multi-pulse solitons 5.2 Mechanism of the multi-pulse soliton formation To understand the physical mechanism of the multipulse bound solitons formation, first we want to emphasize the close similarity in property between the bound solitons of the laser and the two-pulse bound solitons observed in the non-dispersion managed fiber lasers. As that two-pulse bound solitons have a pulse separation of about 3.5 times of the soliton pulse width, it doubtlessly shows that direct soliton-soliton interaction played an essential role on its formation and stability. The direct soliton-soliton interaction results in a strong binding force between the solitons and makes the bound solitons very stable. The soliton peak separation of the current multipulse bound solitons is about 19.5 times of the soliton pulse width. At the first sight it seems there is no direct soliton interaction between the solitons. However, we note that the fundamental soliton of the dispersion management fiber laser has an inherent secondary structure as shown in Fig We have experimentally investigated the structure and found that independent of the states of bound solitons, the peak of the secondary structure has almost a fixed separation of 527 fs to the peak of the soliton. If taking the peak separation between the secondary structure and the soliton peak roughly as the effective soliton pulse width, then the pulse separation of the multipulse bound solitons would become about 4.7 times of this effective pulse width. Certainly, in a real fiber laser due to the existence of other effects, such as gain recovery and depletion, dispersive wave emission, other forms of soliton interaction could also exist. For example, Buryak has shown that mutual trapping of radiative waves of solitons could form bound solitons [106]. Socci and Romagnoli have shown bound soliton formation by the dispersive waves mediated long-range soliton interaction [107]. 84

100 Chapter 5: Multi-pulse solitons Nevertheless, bound solitons formed by these mechanisms have weak stability. They could be easily destroyed. We believe that the bound states of the multipulse bound solitons shown in Fig. 5.4 could be formed by these mechanisms. However, the multipulse bound solitons observed here are very stable, especially when we bear in mind that the multipulse bound solitons can function as a unit to form bound states of their own. Ph. Grelu et al. have also reported the experimental observations of three and multiple soliton bound states [68,104]. Nevertheless, their results on the multiple pulse bound solitons were not characteristic as the separations between the adjacent pulses in the bound solitons vary from case to case, and in some cases unequal soliton separations and even soliton separations with more than five times of the soliton pulse width were obtained, where the direct soliton interaction is little so the soliton separation is changeable. Here we demonstrate further experimental evidence of multi-pulse solitons with fixed pulse separations in a dispersion-managed fiber soliton laser. Not only the twin-pulse solitons as described previously [97] have been obtained in our current laser, but also the three-pulse, four-pulse and pulse array of bound multi-pulse solitons. Different to the results reported by Ph. Grelu et al. [68,99,104,105], the pulse separations between the adjacent pulses in all the multi-pulse solitons observed in our laser are identical, which suggested that it could be an intrinsic feature of the soliton interaction. These characteristics of the multipulse bound solitons are exactly the same as those of the two-pulse bound solitons observed in non-dispersion managed fiber lasers. It suggests that the bound soliton formed in either laser could have the same physical mechanism, namely they are formed through the direct soliton-soliton interaction in the laser cavity, 85

101 Chapter 5: Multi-pulse solitons and the formation of the bound solitons could be a general feature of the soliton fiber lasers. 5.3 Summary The experimental results clearly show that the multi-pulse soliton generation in the dispersion-managed fiber ring laser. Differing from the multisoliton states reported in [68,99,104,105], the multi-pulse solitons we observed have the characterizations that all the solitons have identical fixed internal pulse separations between the adjacent pulse, and the bound structures are very stable, in particular the multi-pulse solitons can function as a unit to form various bound states. These characterizations of the multi-pulse solitons are exactly the same as those of the twin-pulse solitons observed in the nondispersion managed fiber lasers [80]. It suggests that the bound soliton formed in either laser could have the same physical mechanism, namely they are formed through the direct soliton-soliton interaction in the laser cavity, and the formation of the multi-pulse solitons could be a general feature of the soliton fiber lasers. Intuitively, multi-pulse solitons could also be formed in the non-dispersion managed fiber lasers. However, so far as we know, only the twin-pulse solitons were reported [97]. Compared with the non-dispersion managed cavity, dispersion management can diminish the total cavity GVD while still maintain a large GVD in each piece of the fibers forming the cavity. Therefore, soliton pulses with large pulse energy and small pulsewidth can be generated in the laser and strong soliton-soliton interaction could be created. We believe 86

102 Chapter 5: Multi-pulse solitons it is probably because of the lack of this strong pulse interaction the multi-pulse solitons could be hardly formed in the non-dispersion managed fiber lasers. 87

103 Chapter 6: Period doubling bifurcations and period doubling route to chaos Chapter 6 Period doubling bifurcations and period doubling route to chaos Suggested by the average soliton theory of lasers [27], it is generally believed that the output of the fiber soliton lasers is a uniform soliton pulse train. Recently Kim et al. theoretically studied the pulse dynamics of the fiber lasers passively mode-locked by the NPR technique [28,29]. However, 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, aligning the polarizer with either the fast or the slow axis of the fiber could diminish the nonuniformity of the pulse trains. Our group has experimentally investigated the output property of a fiber soliton ring laser passively mode-locked by using the NPR technique [30] 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 it is 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 88

104 Chapter 6: Period doubling bifurcations and period doubling route to chaos 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. Apart from the soliton peak fluctuation caused by the soliton background perturbations, we have also revealed an intrinsic soliton peak variation caused by the dynamic feature of the system. This chapter presents the experimental observations and numerical evidences 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 (NPR) technique. Section 6.1 and 6.2 presents the experimental observations and numerical demonstrations of period doubling bifurcation and period doubling route to chaos of single-pulse soliton, respectively. Similar phenomena basing on bound solitons are reported in Section 6.3 and numerically verified in Section 6.4. Section 6.5 numerically extends our research into the case of non-dispersion-management. Section 6.6 is the summary. 89

105 Chapter 6: Period doubling bifurcations and period doubling route to chaos 6.1 Experimental observations on the single-pulse solitons The experimental setup used is the same as that shown in the last Chapter. Compared with the pulse propagation in an equivalent cavity of uniform dispersion, in a dispersionmanaged cavity a pulse extends its pulse width in the positive group velocity dispersion fiber and therefore can be efficiently amplified [71]. Therefore, the energy of solitons formed in a dispersion-managed cavity can be much stronger than that of solitons in an equivalent uniform-dispersion cavity [108]. 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 tuning 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. Fig. 6.1 shows as example an experimentally observed period doubling route to 90

106 Chapter 6: Period doubling bifurcations and period doubling route to chaos chaos. The results shown in Fig. 6.1 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. 6.1a). 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. 6.1b). 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 6.1a. Further slightly increasing the pump power, a period-quadrupled state then appears (Fig. 6.1c). Eventually the process ends up with a chaotic soliton pulse energy variation state (Fig. 6.1d). Intensity (2 mv/div) Intensity (2 mv/div) Time (50 ns/div) Time (100 ns/div) (a) (c) Intensity (2 mv/div) Intensity (2 mv/div) Time (50 ns/div) Time (100 ns/div) Fig. 6.1 Period doubling bifurcation to chaos of the soliton trains. (a) Period-one state; (b) Periodtwo state; (c) Period-four state; (d) Chaotic state. From (a) to (d) the pump intensity is increased. (b) (d) 91

107 Chapter 6: Period doubling bifurcations and period doubling route to chaos Intensity (15 mv/div) Time (50 ps/div) Fig. 6.2 Oscilloscope traces of a period-two state of the laser emission. With fixed pump power all the states shown are stable. Provided there are no great disturbances they can last for several hours. 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 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. [31]. In all states the average soliton duration measured was about 316±10 fs. The 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 92

108 Chapter 6: Period doubling bifurcations and period doubling route to chaos states by using a high-speed sampling oscilloscope (Agilent 86100A 50 GHz). Fig. 6.2 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. (a) (b) Spectral Intensity Spectral Intensity Wavelength Wavelength Fig. 6.3 Optical spectra of the laser measured in the states of period-one (a) and period-two (b). The optical spectra of the solitons corresponding to the period-one and period-two states are shown in Fig. 6.3a and Fig. 6.3b, respectively. While the spectral curve shown in Fig 6.3a is smooth, the spectral curve shown in Fig 6.3b exhibits clear modulations. The spectral curve shown in Fig. 6.3a 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 Fig 6.3a is also the optical spectrum of each 93

109 Chapter 6: Period doubling bifurcations and period doubling route to chaos individual soliton. In contrast, the optical spectrum shown in Fig. 6.3b is an average of the spectra of two different solitons, each with different energy and frequency chirps. Based on Fig. 6.3b it is concluded that after a period doubling bifurcation, the solitons possess different frequency property as that of the solitons before bifurcation. (a) (b) Intensity (5 db/div) Intensity (5 db/div) 0 MHz Frequency 60 MHz 0 MHz Frequency 60 MHz Fig. 6.4 RF spectra of the laser output corresponding to (a) period-one; (b) period-two states. 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. 6.4 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. 94

110 Chapter 6: Period doubling bifurcations and period doubling route to chaos It is noted 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 will be explained in the next section. 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 expected 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 [ , ,120]. Our experimental result now further demonstrated that this phenomenon could 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 TEM 00 and TEM 01 modes in an effective confocal cavity [109]. 95

111 Chapter 6: Period doubling bifurcations and period doubling route to chaos 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 intrinsic characteristic of the single mode fibers. 6.2 Numerical demonstrations on the single-pulse solitons To understand the soliton feature in the laser, especially to determine the soliton properties in each of the period-doubled states, we have further numerically simulated the soliton operation of the laser. Again we used the model illustrated in Chapter 3. In accordance with our experimental condition we have used the following parameters for the simulations: '' k EDF =50 ps/nm/km, '' k SMF = -20 ps/nm/km, gain saturation energy P sat =250, cavity length L=6 SMF +2 EDF +4 SMF =12 m, beat length L b =L/2, and the orientation of the intracavity polarizer to the fiber fast birefringent axis Ψ= 0.125π. Our simulation procedure is exactly the same as reported previously [ ]. The advantage of the simulation is that it can simulate any state of the soliton operation of the lasers, even when the soliton pulse change caused by a specific cavity component is very big. However, the disadvantage is that it is too complicated, only numerical solutions are feasible. By properly choosing the linear cavity phase delay bias, which corresponds in the experiment to appropriately selecting the orientations of the polarization controllers, soliton operation can be always obtained in our simulations. With a fixed linear cavity 96

112 Chapter 6: Period doubling bifurcations and period doubling route to chaos phase delay bias but different values of gain, as long as the generated soliton pulse peak power is weaker than that of the polarization switching power of the cavity [15], stable uniform soliton pulse train can always be obtained. The exact soliton parameters, such as the pulse width and peak power, are determined by the laser parameter settings and the laser operation condition such as the gain value. Under the larger pump power the solitons generated have higher peak power and narrower pulse width. When the linear cavity phase delay bias is chosen as δφ=1.6π, which corresponds to set the nonlinear polarization switching threshold of the cavity high so that the soliton pulse formed could have high peak power, stable soliton operation can still be obtained. As now the soliton peak power can reach to a very high value under strong pumping, it was observed that when the peak power of the soliton increased to a certain value, period doubling bifurcations and period doubling route to chaos of the solitons as observed experimentally automatically appeared. Fig. 6.5 shows for example a numerically calculated period doubling route to chaos of the lasers. When the gain coefficient was set at G=800, a stable and uniform high-intensity soliton train was obtained (Fig. 6.5a). Increasing the value of G and keeping all the other parameters fixed, the soliton repetition period in the cavity was then doubled at G=850 (Fig. 6.5b), at G=902 it doubled again (Fig. 6.5c), and further doubled at G=908 (Fig. 6.5d). Eventually the soliton repetition in the laser became chaotic (Fig. 6.5e). Fig. 6.5f to Fig. 6.5j shows the corresponding optical spectra of the solitons in Fig. 6.5a to Fig. 6.5e. Associated with the soliton intensity variation the soliton spectrum also exhibited period doubling changes. 97

113 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (f) (b) (g) (c) (h) (d) (i) (e) (j) Fig. 6.5 Soliton pulse evolution and the corresponding optical spectra numerically calculated under different pump strength. The linear cavity phase delay bias is set as δφ=1.6π. (a)/(f) period-1 soliton state, G=800; (b)/(g) Period-2 soliton state, G=850; (c)/(h) Period-4 soliton state, G=902; (d)/(i) Period-8 soliton state, G=908; (e)/(j) Chaotic soliton state, G=

114 Chapter 6: Period doubling bifurcations and period doubling route to chaos In order to describe the soliton dynamics versus pump power, a numerically simulated bifurcation diagram is also shown in Fig. 6.6 with the linear cavity phase delay bias δφ=1.6π. Fig. 6.6 Bifurcation diagram computed for the pulse peak intensity versus the small signal gain G (The linear cavity phase delay bias is set as δφ=1.6π). Based on the numerical model we have further investigated the soliton features in each of the period-doubled states. Fig. 6.7 shows for example the soliton profiles and corresponding optical spectra of the laser output in the period-2 state. In this state the soliton energy alternates between two values, which were also demonstrated experimentally. Numerical simulations further show that actually all the parameters of the soliton, such as the pulse profile, pulse width and peak power, vary within one period in the laser cavity. 99

115 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (c) (b) (d) Fig. 6.7 Soliton pulse profiles and corresponding optical spectra in a period-2 state. (a)/(c) the first round trip; (b)/(d) the second round trip. It is well known that the soliton formed in a laser is an average soliton [27]. A soliton circulating in the cavity periodically changes its parameters due to the action of gain and output loss. In the equilibrium state the gain experienced by the soliton within one round trip balances the losses, therefore, after one round trip the soliton returns to its previous state. Soliton laser operating in the state emits stable uniform soliton pulse train. However, soliton in the state also has weak pulse energy and peak power. Comparing with the period-1 state, the soliton in the period-2 state still has the feature: within one period the soliton parameters vary, however, the period no longer equals the natural cavity length but twice of it. The ultimate soliton peak power and pulse width that a soliton can reach in the state is also substantially higher and narrower than those of the soliton in the period-1 state. Probably it is because of this higher peak power to reach that the soliton needs to take double passages in the cavity. It is interesting to compare the soliton spectra of the laser output after each round trip. While after one round trip (or half of the period) the soliton spectrum has reasonably the same feature as that of the soliton 100

116 Chapter 6: Period doubling bifurcations and period doubling route to chaos in the period-1 state, after another round trip the soliton spectrum then becomes very different to it. Extra soliton sidebands clearly appeared in the spectrum. How these extra sidebands link to the soliton parameter variation and the cavity periodicity needs to be further studied. Nevertheless, the relation of the new sideband generation with the intrinsic soliton dynamics is obvious. Fig. 6.8 shows the soliton pulse profile and spectrum variation within the period-4 state. In this state after every four cavity-roundtrips the soliton intensity and profile return to the original value and shape. As it is essentially a further period doubling of the period-2 state, therefore, overall the soliton spectra look like a duplication of those of the period-2 state. However, we point out that careful comparison with those of the period-2 state shows that further new spectral sidebands have been generated. The spectra shown in Fig. 6.8f and Fig. 6.8h are also different. Corresponding to the soliton of the highest peak power, the soliton spectrum (Fig. 6.8f) also exhibits the strongest sidebands and spectral modulations, indicating the existence of strong nonlinear self-phase modulation (SPM) on the pulse. The change from one soliton operation state to the other is abrupt. At the bifurcation point when the gain coefficient is slightly increased, the soliton quickly jumps to another state with the doubled periodicity, exhibiting the universal characteristic of the period doubling bifurcation and route to chaos of the nonlinear dynamics systems. 101

117 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (e) (b) (f) (c) (g) (d) (h) Fig. 6.8 Soliton pulse profiles and corresponding optical spectra in a period-4 state. (a)/(e) the first round trip; (b)/(f) the second round trip; (c)/(g) the third round trip; (d)/(h) the fourth round trip. 102

118 Chapter 6: Period doubling bifurcations and period doubling route to chaos The appearance of the period doubling bifurcation is independent of the specific laser cavity design. Under different cavity parameter settings, as far as the linear cavity phase delay bias is big enough, that is, the soliton generated could have strong peak power, we could always obtain the phenomenon in our simulations. Even the period doubling route to chaos of a period-three state, which in the nonlinear dynamics theory is known as a periodic window within the chaotic regime, has also been numerically revealed, as shown in Fig To obtain the state we have used '' k EDF = 70 ps/nm/km, '' k SMF = -20 ps/nm/km, and the linear cavity phase delay bias δφ=1.5π. However, we should point out that due to the coexistence of other effects in the laser, e.g. the soliton peak non-uniformity [30] and soliton collapse [21], in order to obtain a full period doubling route to chaos in the lasers, the cavity parameters must be appropriately selected. In some of our numerical simulations frequently only certain period doubling bifurcations e.g. the period-one to period-two bifurcation and then to chaos, or the period-one to chaotic state could be obtained. If the linear cavity phase delay bias was not set large enough, the cavity NPS effect could also limit the peak power of the solitons. Consequently only bifurcations to a certain periodic state e.g. the period-four state could be reached. Further increase of the gain would cause the generation of a new soliton rather than the further bifurcation to chaos, which clearly shows the direct relation of the period doubling bifurcation to the soliton peak power. 103

119 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (d) (b) (e) (c) (f) Fig. 6.9 Soliton profiles and corresponding optical spectra numerically calculated. (a)/(d) State of period-3, G=730; (b)/(e) State of period-6, G=735; (c)/(f) Chaotic state, G=750. Based on the results of our numerical simulations we could now take a close look into the formation mechanism of the soliton period doubling in the lasers. Previous studies on the synchronously pumped passive ring cavities have also revealed period doubling cascade to chaos in the sequence of pulses emerging from the cavities [113,114]. It was shown that the bifurcations and route to chaos of the system is caused by the repetitive interference between the input pulse and the pulse that has completed a round trip in the cavity [113]. As the pulse traveling in the cavity suffers nonlinear phase shift, which 104

120 Chapter 6: Period doubling bifurcations and period doubling route to chaos itself is the pulse intensity dependent, the transmission of the cavity is a nonlinear function of the pulse intensity. It is an intrinsic property of such a nonlinear cavity that under larger nonlinear phase shift of the pulse, its output exhibits the period doubling route to chaos [115]. We note that a similar repetitive interference process exists in the fiber soliton lasers. Due to the birefringence of the laser cavity, the pulse propagation in the laser actually comprises two orthogonal polarization components. Although there are nonlinear couplings between the two orthogonally polarized pulses as can be seen from the Eqs. 3.7, after one round-trip they experience different linear and nonlinear phase shifts. The interference between them at the intracavity polarizer results in that the effective cavity transmission is a nonlinear function of the soliton intensity. Based on the studies to the synchronously pumped passive ring cavities, it is imagined that under strong soliton peak intensity, the period doubling bifurcation and route to chaos could also appear in the lasers. Indeed, our numerical simulation show that the effect could only occur in the lasers when the linear cavity phase delay bias is set away from the nonlinear polarization switching point. In this case the soliton peak power is hardly to be clamped by the cavity and can increase to a very high value with the increase of the pump strength. A high soliton peak power generates large nonlinear phase shift difference between the two polarization components, which causes the intrinsic instability of the system. Finally, we emphasize the excellent agreement of the soliton dynamics of the laser with the universal laws of the deterministic chaos theory. It suggests that the passively modelocked soliton fiber lasers are a nonlinear dynamical system. They could be used as a test bed to investigate the dynamical instabilities of the optical solitons. 105

121 Chapter 6: Period doubling bifurcations and period doubling route to chaos 6.3 Experimental observations on the bound solitons For many applications it is important to know the exact shape of an optical pulse and the pulse uniformity of a pulse train. To this end the detailed generation mechanism of the ultrashort optical pulses and the interaction between pulses with the gain medium and cavity components of a laser need to be studied. A common feature of all soliton fiber lasers is the multiple soliton generation under strong pumping. And in the steady states all the solitons generated have identical properties. It was known as the soliton energy quantization effect [9]. Interaction between the multiple solitons has been extensively investigated previously. It was found that various modes of the multiple soliton operation could be formed [11]. A special situation also experimentally observed is that under appropriate laser cavity conditions, the solitons in the cavity could bind automatically together and form states of bound solitons [97,104]. Depending on the strength of the soliton binding a certain state of the bound solitons can even become the only equilibrium state in a laser. In such a state the bound solitons as an entity exhibit features that are in close similarity to those of the single pulse solitons [97,98,116]. When a laser is operating in such a bound state of solitons, an immediate question would be whether the nonuniformities observed for the single pulse soliton could also appear? To partially answer the question we showed in this section the experimental observation of period doubling, quadrupling, and period doubling route to chaos of bound solitons in the passively mode-locked fiber ring laser. We demonstrate experimentally that, like the conventional single-pulse solitons, the bound solitons as an 106

122 Chapter 6: Period doubling bifurcations and period doubling route to chaos entity can also exhibit complicated deterministic dynamics. This experimental result further shows that the bound solitons in the lasers have indeed very strong binding energy, as an entity they can behave just like an ordinary single-pulse soliton. The laser setup is the same as that presented in Section 6.1. It is well known that selfstarted soliton operation is always obtained in the laser by simply increasing the pump power and appropriately setting the orientations of the polarization controllers. Multiple solitons can be easily obtained in the laser by increasing the pump power. Generally, depending on the laser operation conditions the multiple solitons generated could form various soliton patterns as also observed in other soliton fiber lasers. Under appropriate cavity conditions bound states of solitons could be obtained in the laser. We found that even the bound states of solitons could also exhibit the period doubling bifurcations and period doubling route to chaos. Fig. 6.10a, Fig. 6.10b and Fig. 6.10c show for example an experimentally observed period doubling bifurcation route of a bound-soliton. In the current case several solitons are tightly bound and move together at the cavity fundamental repetition frequency. When the peak intensity of the bound solitons is strong, the total intensity of the bound solitons exhibits the period doubling bifurcation. The response time of the photo-detector used in our experiment is about 140 ps, which can clearly resolve the pulse train but not the intensity profile of the bound solitons. Therefore, the measured oscilloscope traces shown in Fig have no distinctions to those of the period doubling bifurcations of the single-pulse solitons [17]. Fig. 6.10d, Fig. 6.10e and Fig. 6.10f show the simultaneously measured optical spectra corresponding to the Fig. 6.10a, Fig. 6.10b and Fig. 6.10c. 107

123 Chapter 6: Period doubling bifurcations and period doubling route to chaos Intensity (5mV/div) Intensity (5mV/div) Intensity (5mV/div) (a) (b) (c) Time (50ns/div) Time (50ns/div) Intensity Intensity Intensity (d) (e) (f) Time (100ns/div) Wavelength Fig Oscilloscope trace and corresponding spectra of period doubling bifurcations of a bound-soliton pulse train. (a)&(d) period-one state; (b)&(e) period-doubled state; (c)&(f) period-quadrupled state. Clear optical spectral modulations exist on the spectra, which show that more than one soliton are actually in the cavity and they are closely spaced. The oscillating peaks of the optical spectra have a peak separation of 0.70 nm. Assuming the central wavelength of the soliton pulse is 1550 nm, based on the Fourier transformation, the solitons have a peak separation of about 11.5 ps in the time domain. The bound soliton nature of the state is also confirmed by the simultaneous autocorrelation trace measurement as shown in Fig Limited by our autocorrelator resolution, the measured autocorrelation traces show no distinguishable features among different period-doubled states. The measured single soliton duration of the state is about 326±12fs, and the soliton separation between 108

124 Chapter 6: Period doubling bifurcations and period doubling route to chaos neighboring solitons is about 11.8 ps, which well agrees with the 0.70 nm period of optical spectral modulation within the experimental errors. The soliton separations under different period-doubled states are also the same. From the autocorrelation traces it is obvious that there are at least 3 solitons with exactly the same soliton separations binding together in the bound-soliton state. Due to the limited scan range of our autocorrelator, we could not identify how many solitons are actually in the state. Nevertheless, it is confirmed that all the solitons in the state are equally spaced, and the bound soliton as an entity exhibits period doubling bifurcations. We note that the bound soliton shown in Fig is the only pulse existing in the cavity, which is confirmed by the combined monitoring of the autocorrelation trace and a high-speed oscilloscope trace (Agilent 86100A 50 GHz). Intensity (arb. unit) Fig Time (ps) A typical measured autocorrelation trace of the bound-soliton. Fig. 6.10a, Fig. 6.10b, and Fig. 6.10c are obtained with continuously increased pump power while keeping all the other cavity parameters fixed. Chaotic state of the bound solitons could be obtained with further increasing pump power from the state Fig. 6.10c. However, the chaotic state was not stable. It quickly evolved into a chaotic state of the single-pulse soliton. The change of the laser emission from a chaotic state of bound solitons to that of single-pulse solitons was experimentally identified by the change of the 109

125 Chapter 6: Period doubling bifurcations and period doubling route to chaos optical spectra. It was observed that whenever the bound-soliton became chaotic, the modulations on the optical spectrum, which is a direct indication of close soliton separation in the time domain, disappeared, and subsequently the optical spectrum had exactly the same profile as that of the chaotic state of the single-pulse solitons. Why the bound solitons become unstable in the chaotic state is so far not well understood, which needs to be further studied. However, considering that in a chaotic state the energy of solitons varies randomly, which may affect the binding energy between the solitons and destroy their binding, this result seems also plausible. The states of period-doubled and quadrupled bound solitons are stable. Nevertheless, compared with those states of the single-pulse solitons [17,22], they are more sensitive to the environment perturbations. Analyzing the optical spectra of the period-doubled bound solitons shown in Fig. 6.10, we found that the overall spectral profile in each state is similar to those of the singlepulse soliton undergone the period doubling, except that it is now modulated [17]. The feature of the optical spectra indicates that the individual solitons within a bound-soliton are still the same as the single-pulse soliton of the laser. RF spectra of the bound-soliton pulse train also disclose different period-doubled states. The emergence of a new frequency component at the position of half of the fundamental cavity repetition frequency clearly shows that the repetition rate of the bound solitons is doubled. The amplitude of the new frequency component is nearly half of that of the fundamental frequency component, which vividly suggests that the total peak intensity of the bound solitons alternates between two values with large difference. Same as observed in Section 6.1, the new frequency component corresponding to the period-quadrupling is 110

126 Chapter 6: Period doubling bifurcations and period doubling route to chaos not distinguishable from the background noise. Limited by the resolution of our measurement system, the detailed intensity variations of each of the solitons under period doubling bifurcations could not be resolved. There are two possible ways for a two-pulse bound-soliton exhibiting a period doubling intensity pattern. Either the two solitons simultaneously experience the period doubling, or only one soliton experiences the period doubling while the other one still remains stable. With more solitons binding together the process could become more complicated. Experimentally, we have also observed period doubling bifurcations of bound solitons with different soliton separations. Bound solitons with different pulse separations can be easily distinguished by their optical spectra as different spectral modulation periods correspond to different pulse separations in the time domain. This experimental result suggests that the appearance of the phenomenon should be a generic feature of the laser, which is independent of the specific property of the optical pulses. Period doubling route to chaos is a well-known nonlinear dynamic phenomenon widely investigated. Period doubling route to chaos of the CW and the Q-switched lasers as a result of strong nonlinear interaction between the light field and the gain medium have already been reported [ ]. Except that the laser modes are phase locked, physically the interaction between the light and the gain medium in a mode-locked laser is still the same. Therefore, it is not surprising that under existence of strong mode-locked pulse, period doubling route to chaos on the pulse repetition rate could still be obtained. Nevertheless, it was a little bit unexpected that a bound-soliton can exhibit period doubling bifurcations as intuitively the dynamic bifurcation of the laser state could easily 111

127 Chapter 6: Period doubling bifurcations and period doubling route to chaos damage the binding between the solitons. In our experiment a dispersion managed laser cavity was used. The purpose of using a dispersion-managed cavity is to possibly make the energy of the formed solitons strong, so in average strong nonlinear interaction between the pulses with the cavity components could be achieved. Our experimental results show that the solitons in a fiber laser can not only bind together and form equilibrium states of bound solitons, but the formed bound solitons as an entity can exhibit the deterministic nonlinear dynamics under strong interaction with the laser gain medium and the cavity components. In addition, this experimental result also shows that the period doubling route to chaos is a generic property of the laser system, whose appearance is independent of the specific mode-locked pulse profile. 112

128 Chapter 6: Period doubling bifurcations and period doubling route to chaos 6.4 Numerical evidences on the bound solitons (a) (e) (b) (f) (c) (g) (d) (h) Fig Period doubling route to chaos of bound solitons. (a)/(e) state of stable bound solitons, G=1149; (b)/(f) state of period-2 of the bound solitons, G=1300; (c)/(g) state of period-4 of the bound solitons, G=1353; (d)/(h) Chaotic state of the bound solitons, G=

129 Chapter 6: Period doubling bifurcations and period doubling route to chaos The phenomenon of period doubling route to chaos has been observed on the bound solitons, where the total energy of the bound solitons exhibited period doubling bifurcations and route to chaos. In the case of bound solitons several solitons bind together with close soliton separations. Although the state of bound solitons could be easily identified by their optical spectra and autocorrelation traces, experimentally it is difficult to monitor the energy variation of each individual soliton within the bound solitons. Therefore, no detailed information on the individual soliton property during the period doubling bifurcation process could be experimentally obtained. With multiple solitons coexisting in the laser cavity, experimentally it was shown that the solitons could interact with each other and form the so-called bound states of solitons [12,97]. In our numerical simulations bound states of solitons have also been obtained. In fact our previous numerical studies on the soliton interaction in the passively modelocked fiber lasers have shown that, due to the existence of mode locking force in the laser, solitons formed in the laser cavity have the tendency of forming bound solitons under direct soliton interaction [96]. Period doubling bifurcations and route to chaos have also been numerically revealed for the bound solitons. Fig shows for example one of such results obtained. In calculating the state we have used the same laser parameters as those for obtaining the single-pulse soliton period doubling route to chaos in Section 6.2, only the pump strength and the initial state were selected different. Fig. 6.12a shows that two solitons coexist in the cavity and bind together. Note that due to the close separation between the solitons, their optical spectra have strong intensity modulations, but from round to round the modulation patterns do not change, which indicates that the 114

130 Chapter 6: Period doubling bifurcations and period doubling route to chaos phase difference between the solitons is fixed as well. The binding nature of the solitons is represented by the fixed soliton separation and phase difference even under the existence of soliton interaction between them. Increasing the pump strength from Fig. 6.12a, both solitons experienced simultaneously period doubling bifurcations and route to chaos. Associated with the soliton period doubling the soliton separation between the bound solitons also changed slightly. However, after the period doubling bifurcation the soliton separation then remained constant again. This soliton separation change suggests that the dynamic bifurcation of the system could affect the soliton interaction. 6.5 Period doubling bifurcations in non-dispersion-managed fiber lasers Pulse formation in a femtosecond laser is typically dominated by the interplay between dispersion and nonlinearity. Soliton operation could always be achieved as long as the total group velocity dispersion of the cavity is negative. Compared with the pulse propagation in an equivalent non-dispersion-managed cavity, in a dispersion-managed cavity a pulse could extend its pulse width in the positive group velocity dispersion fiber segment and therefore can be efficiently amplified [71,119]. Hence the energy of solitons formed in a dispersion-managed cavity can be much greater than that of solitons in an equivalent non-dispersion-managed cavity [108]. However, solitons formed in nondispersion-managed cavity could still have high pulse intensity only if the linear polarization rotation phase bias is correctly set. So it is possible to observe the same phenomena of period doubling bifurcations and routes to chaos in fiber ring laser with 115

131 Chapter 6: Period doubling bifurcations and period doubling route to chaos non-dispersion-managed cavity. We numerically simulated the pulse propagation in the fiber ring laser with non-dispersion-managed cavity and indeed found the same nonlinear phenomena of period doubling bifurcations and routes to chaos. The structure of the fiber ring laser with non-dispersion-managed cavity is as same as that one in Section 4.1, the cavity length L=1 DSF +4 EDF +1 SMF =6 m, '' k DSF = -2 ps/nm/km, '' k EDF = -10 ps/nm/km, '' k SMF = -18 ps/nm/km. In order to obtain solitons with pulse intensity as high as possible, we chose the gain saturation energy Psat=1000, other parameters are same as the case of dispersion-managed cavity (see Section 6.2). Similar to the case of the fiber ring laser with dispersion-managed cavity, when the linear cavity phase delay bias is chosen as δφ=1.5π, it was observed that when the peak power of the soliton increases to a certain value, period doubling bifurcations and period doubling route to chaos appear in the laser. Fig shows for example a numerically calculated period doubling route to chaos of the laser. When the gain coefficient is set at G=315, a stable and uniform high-intensity soliton train is obtained (Fig. 6.13a). Increasing the value of G and keeping all the other parameters fixed, the soliton repetition period in the cavity is then doubled at G=340 (Fig. 6.13b), to G=366 it doubled again (Fig. 6.13c), and further doubled at G=369 (Fig. 6.13d). Fig. 6.13e to Fig. 6.13h shows the corresponding spectra of the solitons in Fig. 6.13a to Fig. 6.13d. 116

132 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (e) (b) (f) (c) (g) (d) (h) Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a)/(e) state of stable soliton operation (period one), G=315; (b)/(f) state of period-2, G=340; (c)/(g) state of period-4, G=366; (d)/(h) state of period-8, G=

133 Chapter 6: Period doubling bifurcations and period doubling route to chaos (a) (d) (b) (e) (c) (f) Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a)/(d) state of stable bound solitons operation (period one), G=370; (b)/(e) state of period-2 of bound solitons, G=380; (c)/(f) state of period-4 of bound solitons, G=395. Different from the case of the fiber ring laser with dispersion-managed cavity, when we continue to increase the value of G, instead of the emergence of chaos state, a stable highintensity pulse train of two bound solitons with little intensity fluctuation is obtained (G=370, Fig. 6.14a). Increasing the value of G and keeping all the other parameters fixed, the repetition period of the bound solitons in the cavity is then doubled at G=380 (Fig. 6.14b), to G=395 it doubled again (Fig. 6.14c), further the chaotic state of the two bound solitons occurs, then three solitons would be formed with further increasing pump power. 118

134 Chapter 6: Period doubling bifurcations and period doubling route to chaos The separation between the two bound solitons is less than 4 ps, which is about 20 times of the pulse width of the single soliton in Fig. 6.13a. Although the soliton separation is large, we could clearly see the coupled part between the two bound solitons, especially when period doubling bifurcation occurs, as shown in Fig. 6.14b and Fig. 6.14c, which clearly suggests that these two solitons are bound together. We note that on the route to chaos of the two bound solitons, the pulse separation between these two bound solitons will change among different period bifurcation states, same as the case of dispersionmanaged cavity. Experimentally multiple solitons could coexist in cavity with large pulse spacing. In our simulation, we also calculated such case and obtained the complete period doubling route to chaos of two solitons with pulse spacing equaling to 64 ps, as shown in Fig Different from the route to chaos of the bound solitons as shown in Fig. 6.14, the soliton separation between the two solitons fixed at the value of 64 ps in spite of the occurrence of period doubling bifurcation and route to chaos, which strongly suggests that there is no strong binding force between these two solitons. Each soliton evolves as if there is only itself existing in the simulation window. We also obtained another state of period doubling of two coexisting solitons under other parameter set, which showed that only one soliton experiences the period doubling while the other one still remains uniform. Experimentally, the general method to detect period bifurcation phenomena is to use a photodetector to transform the optical signal into electrical signal then use an oscilloscope to monitor it. Limited by the performance of the photodetector and the oscilloscope, generally it is impossible to resolve the detailed dynamics of individual soliton among coexisting solitons with separation less than 1 ns. Our simulation results 119

135 Chapter 6: Period doubling bifurcations and period doubling route to chaos show that with more than one soliton coexisting in the cavity with large separation, period doubling bifurcation and route to chaos could still occur, but the dynamics of individual soliton may be very different. (a) (b) (c) Fig Soliton pulses and the corresponding optical spectra numerically calculated under different pump strength with linear polarization rotation phase bias equals to 1.5π. (a) state of stable two solitons operation (period one), G=332; (b) state of period-2 of two solitons, G=380; (c) state of period-4 of two solitons, G=

136 Chapter 6: Period doubling bifurcations and period doubling route to chaos 6.6 Summary To understand the soliton features observed, we note that when the soliton energy becomes strong, cavity nonlinear effect would become important [118, ]. A soliton circulating in the nonlinear laser cavity is in essence a kind of nonlinear mapping. It is well known that the nonlinear mapping can generate deterministic chaos including the period doubling bifurcations and period doubling route to chaos. It is therefore no surprise that under appropriate cavity conditions, the solitons in the laser can experience period doubling route to chaos. As the soliton dynamics is physically a manifestation of the nonlinear cavity effect, the appearance of the period doubling phenomenon itself is independent of whether the light circulating in the cavity is a soliton or not, and whether the soliton is a single-pulse soliton or bound solitons. Finally, we point out that based on the same numerical model we have also obtained period doubling bifurcations and period doubling route to chaos in the non-dispersion managed fiber lasers. Similar properties and features of the solitons under the dynamical bifurcation have been obtained. This numerical result further supports that the period doubling bifurcation and period doubling route to chaos is an intrinsic property of the laser. Its appearance is independent of the details of laser cavity design and the pulse profile provided the peak power of the pulse generated is high enough. In this chapter we have shown the experimental observation of the soliton period doubling bifurcations and period doubling route to chaos of the single-pulse soliton or the bound solitons in a passively mode-locked fiber soliton ring laser with dispersion- 121

137 Chapter 6: Period doubling bifurcations and period doubling route to chaos managed cavity. The laser was mode-locked by using the NPR technique. Experimentally, we found that when the peak power of solitons in the cavity is strong, after one round-trip in the cavity the soliton energy does not return to its previous value, but it returns to the previous value after two or four round trips, and under strong pumping the soliton energy does not return to its previous value at all. This soliton energy variation follows exactly the so-called period doubling route to chaos of the nonlinear dynamics systems, which suggests that it is an intrinsic dynamics of the solitons in the lasers. Based on the wellknown pulse tracing technique we have numerically confirmed the soliton dynamics of the laser and identified its conditions of appearance. Our numerical simulation also revealed that in a period-doubled state, the soliton peak power can become very high and soliton pulse width very narrow. This property of the solitons may be exploited to get even narrower pulse width and higher peak power optical pulses. In addition, our simulation result shows that the dynamic variation of the soliton energy introduces new sidebands on the soliton spectra and the dynamic bifurcation of the system can slightly change the separation of the bound solitons. Independent of detailed cavity design, we numerically confirmed the period doubling routes to chaos of either single-pulse soliton or the bound solitons not only in non-dispersion-managed cavity but also in dispersionmanaged cavity, which strongly suggest that the routes to chaos are intrinsic properties of all passively mode-locked fiber ring lasers using NPR technique. Our simulation results accurately reproduce all the experimental details we obtained in [17,18] and predict that the nonlinear properties of period doubling bifurcation and route to chaos should also exist in fiber soliton ring laser with non-dispersion-managed cavity. 122

138 Chapter 7: Broadband noise-like pulses Chapter 7 Broadband noise-like pulses Generation of noise-like pulses from fiber laser is now a well-documented phenomenon [21,33,123]. In [33], Horowitz et al. have shown that a train of long noiselike 100-ps pulses with a unique broad spectrum as wide as 44 nm can be produced in an Erbiumdoped fiber laser. They attributed it to the internal birefringence of the laser cavity. Michael L. Dennis et al. also set up a broadband pulse source with a dispersion-managed cavity and an Er: Yb-codoped fiber amplifier [124]. However, most of previous researches are focused in the positive net cavity dispersion regime. Since the birefringence always exists in fibers, does the same phenomenon occur in the negative net cavity dispersion area? In this chapter we report on the noiselike pulse operation in the negative net cavity dispersion fiber laser passively mode-locked by using the nonlinear polarization rotation (NPR) technique. We show experimentally that by simply shifting the linear cavity phase delay, the soliton operation of the laser can be tuned into a mode-locked state where the mode-locked pulse consists of a bunch of noise-like pulses while its optical spectrum has a super-broad bandwidth. Based on the numerical simulations we further show that this kind of laser emission is caused by the combined effect of soliton collapse and positive cavity feedback in the laser. Our theoretical model has no limitation on the detailed cavity design, that is, it can be applied to the case of non-dispersion-managed cavity, 123

139 Chapter 7: Broadband noise-like pulses which triggers us to study the noise-like operation in the non-dispersion-managed cavity. Numerically we obtained the noise-like pulses in the non-dispersion-managed cavity, following experiments verified it. The state of noise-like emission is stable. With a fixed laser cavity configuration the exact duration of the overall bunch, the 3-dB optical spectral bandwidth and energy vary with the pump strength. Detailed experimental investigations show that the maximum achievable spectral bandwidth is closely related to the EDF length. Provided that the pump strength is strong enough, the longer the EDF the broader the spectral bandwidth can be obtained. With the pump power available in our laser the optimized EDF length is 17.6m, which generates the maximum 3-dB bandwidth of nm. This optimized EDF length corresponds to the length that the pump power is just fully absorbed by the EDF. Our experimental results not only confirmed that the noise-like pulse emission is a generic property of all PMEFLs, but also suggests that even higher power and broader spectral bandwidth noise-like pulse emission is in principle possible to achieve provided that even stronger pump source and larger nonlinear fibers are available. 7.1 Soliton collapse and bunched noise-like pulse generation Experimental observations The fiber soliton laser used is similar as that shown in former two chapters. As the total cavity GVD of the laser is in the negative dispersion regime, despite the fact that the cavity is dispersion managed, soliton operation could still be obtained. Experimentally 124

140 Chapter 7: Broadband noise-like pulses we found that the overall soliton operation of the laser is similar to that of the nondispersion managed soliton fiber lasers, e.g. effects such as the pump power hysteresis [125], multiple soliton generation and various modes of the multiple soliton operation [4], bound states of solitons [12] were observed in the laser. Under normal conditions the single pulse soliton of the laser has a pulse width of about 300 fs and an optical spectral bandwidth of about 10 nm. The exact soliton parameters vary within a small range with the detailed laser operation conditions such as the orientations of the waveplates and the pump power strength. A striking feature of the laser is that by simply changing the orientation of one of the waveplates, the soliton operation could be tuned into a mode-locked state whose optical spectrum has super broad bandwidth, while the corresponding pulse profile consists of a bunch of intense noise-like pulses. Fig. 7.1 shows a typical result observed experimentally. Fig. 7.1a shows the optical spectrum and Fig. 7.1b and Fig. 7.1c the corresponding autocorrelation traces under different scan ranges. The optical spectrum is smooth. It has a 3-dB bandwidth of about nm. The unchanged broadband spectrum suggests that the laser is still mode-locked. However, its spectral distribution is very different to that of a stable soliton of the laser. Once the state is obtained, decreasing the pump power does not reduce the spectral bandwidth. 125

141 Chapter 7: Broadband noise-like pulses Fig. 7.1 A typical state of the bunched noise-like pulse emission. (a) Optical spectrum of the state; (b) Autocorrelation trace with a scan span of 5 ps; (c) Autocorrelation trace with a scan span of 0.5 ps. 126

142 Chapter 7: Broadband noise-like pulses The autocorrelation trace of Fig. 7.1b has a profile of narrow pulse riding on a broad pedestal, which suggests that the actual mode-locked pulse consists of a series of coherent narrow random spikes. Changing the pump strength, the width of the pedestal could vary significantly, even extending beyond the scan range of the autocorrelator (50ps). However, the width of the narrow pulse has almost a constant value, indicating that the number of coherent spikes varies while the average pulse width of the spikes remains the same. By shortening the scan range of the autocorrelator we could obtain an autocorrelation trace as shown in Fig. 7.1c, which shows that the narrow intensity modulation pulses have an averaged pulse width of about 75fs, which corresponds roughly to a transform-limited pulse determined by the measured spectral bandwidth. With the help of the high-speed oscilloscope it was further confirmed that independent of the pump power, there is always only one such mode-locked pulse circulating in the cavity. We note that a similar kind of fiber laser emission was also reported by M. Hororwitz et al. [33] and explained as caused by the polarization-dependent delay (PDD) effect as the laser used had a large cavity birefringence. However, our laser cavity has only weak birefringence, which obviously rules out the possibility of the PDD effect Numerical simulations To find out the physical mechanism we have numerically simulated the laser operation by using the pulse tracing technique as before. We have used the following parameters for the simulations: '' k = -20 ps/nm/km (SMF); '' k = +70 ps/nm/km (EDF); cavity length L = 127

143 Chapter 7: Broadband noise-like pulses 5(SMF)+2(EDF)+5(SMF)=12m; cavity beat length L b =L/2. The polarizer orientation to the fiber fast axis ψ = 0.152π. (a) (b) Fig. 7.2 A typical bunched noise-like pulse state numerically calculated. (a) Time evolution of the noise-like pulses; (b) Autocorrelation trace numerical calculated. We found numerically that the model could reproduce all the features of the soliton operation of the laser. In particular, by simply shifting the linear cavity phase delay the soliton operation of the laser can be tuned into a mode-locked state as described in last 128

144 Chapter 7: Broadband noise-like pulses section. Fig. 7.2 shows for example a typical such mode-locked state numerically calculated when the linear cavity phase delay bias is set to 1.7π. Fig. 7.2a shows the time evolution of the mode-locked pulses with the cavity round trips and the pulse profiles in the cavity. The noise-like feature of the pulse profiles is clear to see. Careful examination of the state shows that the laser emission actually is a bunch of pulses with random varying pulse width, peak power and pulse number. The average number of pulse within the bunch increases with the pump strength, which also determines the overall length of the bunch. A pulse in the bunch grows up and simultaneously its pulse width narrows down until it becomes a transform-limited pulse, and then the pulse collapses. Fig. 7.2b is the calculated autocorrelation trace of the mode-locked pulse. Considering that the experimentally measured autocorrelation trace is actually time averaged, the result is then well in agreement with the experimental measurement. Based on the numerical simulations we further figured out that the state was actually formed by the combined effect of the soliton collapse and the positive cavity feedback in the laser. To explain it we note that depending on the linear cavity phase delay there exist two cavity feedback regimes in the laser, e.g. with our current laser parameter selection the cavity provides a positive feedback if the linear cavity phase delay ϕ is biased in the range of π < ϕ < 2π and a negative feedback in the range of 0 < ϕ < π. The ϕ = π is the feedback switching point. The laser can always be mode-locked if the cavity provides a positive feedback. A generic property of solitons formed in gain media is the explosively increase in amplitude and decrease in pulse width. It eventually leads to soliton collapse [21,126,127]. This property also applies for solitons formed in lasers. However, in the 129

145 Chapter 7: Broadband noise-like pulses case of the current laser when the linear cavity phase delay ϕ is biased close to the cavity feedback switching point, the cavity feedback switching generates a soliton peak power limitation mechanism, which stabilizes the soliton operation. While when ϕ is biased far away from the cavity feedback switching point, a soliton collapses before its peak power reaches the feedback switching point. Consequently solitons in the laser are constantly destroyed and generated. Under strong pumping multiple of such process coexist unsynchronized, therefore, it forms the experimentally observed noise-like pulses. It is noted that as the linear cavity loss under such a linear cavity phase delay is very big, solitons can only be formed in the cavity at the position where a soliton was collapsed, which leads to the result that all the noise-like pulses are bunched in the cavity. 7.2 Noise-like pulse in non-dispersion-managed fiber lasers Numerical prediction We have experimentally observed noise-like pulse emission in a passively mode-locked fiber ring laser with dispersion-managed cavity and numerically verified it. However, contrary to the previous understanding we found that the laser emission could be caused by the soliton collapse effect in the lasers. Soliton collapse is a general property of the gain-guided solitons [44]. In fact solitary waves also exist in amplified nonlinear dispersive media independent of the medium dispersion [44]. However, these solitons were found unstable. After initial undistorted soliton propagation they experience a sudden explosive temporal compression and then collapse [127]. Provided that our 130

146 Chapter 7: Broadband noise-like pulses understanding is correct, it would suggest that solitons observed in the lasers are essentially gain-guided solitons. Therefore, the formation of the noise-like pulses should also be a generic property of all passively mode-locked Erbium-doped fiber lasers, whose appearance is independent of the cavity dispersion management. Fortunately, our theoretical model has no limitation on the cavity dispersion management, which means we could also simulate the case of non-dispersion-managed cavity. Numerically, we confirmed our claim. (a) (b) Fig. 7.3 Typical bunched noise-like pulse emission numerically calculated. (a) G=800. (b) G=

147 Chapter 7: Broadband noise-like pulses We numerically simulated the laser operation and confirmed the noise-like pulse emission. Typical numerical results are shown in Fig We used the following parameters for the simulations: '' k = -18 ps/nm/km (SMF); '' k = -18 ps/nm/km (EDF); '' k = -2 ps/nm/km (DSF); gain saturation intensity P sat =500; cavity length L = 1 SMF +5 EDF +2.5 DSF =8.5 m; cavity beat length L b =L/2. The polarizer orientation to the fiber fast axis ψ = 0.125π. The linear cavity phase delay is chosen as 1.7π. Our numerical simulations show again that purely depending on the linear cavity phase delay bias setting, the laser can either operate in the conventional soliton emission state or the noise-like pulse emission state. In particular, numerical simulations turned out that the solitons formed in the lasers have the intrinsic feature of soliton collapse due to the existence of saturable absorption. Stable solitons can only be observed in the lasers when the soliton peak power is clamped, e.g. in the laser by the cavity nonlinear polarization switching effect, which could occur before the soliton collapse depending on the linear cavity phase delay bias setting [21]. If the pulse peak power is unclamped, the soliton peak will continuously increase, and associated with it soliton pulse width decreases and spectral bandwidth broadens. When the soliton spectral bandwidth becomes so broad that it is comparable to the gain bandwidth of the EDF, the gain dispersion then imposes a loss mechanism on the soliton, consequently the soliton collapses. However, due to the self-starting property of the laser, a new soliton will build up slowly from the collapsed soliton, and evolves eventually into collapse again, and the whole process repeats as shown in Fig. 7.3a. Depending on the laser gain, generally many of this soliton collapse and generation process coexist in the cavity, and under the gain competition, they are 132

148 Chapter 7: Broadband noise-like pulses always unsynchronized. Therefore, at any instance pulses of randomly varying amplitudes and pulse widths coexist in the cavity. As new pulses are always generated from the collapsed solitons, they are correlated to each other and form the so-called bunched noise-like pulses in the cavity as shown in Fig. 7.3b. We note that just before a soliton is collapsed it has not only an optical spectrum comparable to the EDF gain bandwidth, but also very high peak power. In this case the soliton Raman effect can no longer be ignored [128]. The soliton self-frequency shift further broadens the spectral bandwidth of the noise-like pulses, and results in that the noise-like pulse emission has optical spectrum broader than the EDF gain bandwidth Experimental demonstrations To confirm our numerical prediction, in this section we report further on the experimental observation of the noise-like pulse generation in a conventional PMEFL. In particular, we show that by taking advantage of the soliton collapse effect and the soliton self-frequency shift (SSFS), high power superbroad spectral bandwidth noise-like pulses can also be generated, and the spectral bandwidth of the laser emission is practically only limited by the effective laser gain available. The fiber soliton laser used is similar to the one shown in Fig. 4.1 except that now the length of the DSF is about 2.5m, the length of the EDF is variable, and the laser emission is outlet through a 70% fiber output coupler. 133

149 Chapter 7: Broadband noise-like pulses Spectral Intensity (db) (a) Wavelength Intensity (arb. unit) (b) Intensity (100mV/div) Time (ps) Time (50 ns/div) (c) (d) Intensity (db) Frequency Fig. 7.4 Bunched noise-like pulse emission. (a) Optical spectrum; (b) Autocorrelation trace; (c) Oscilloscope trace; (d) RF spectrum around the fundamental cavity repetition rate. 134

150 Chapter 7: Broadband noise-like pulses Stable soliton operation could be easily obtained in the laser provided that the orientations of the polarization controllers are appropriately set. Under normal conditions the single-pulse soliton of the laser has a pulse width of about 300fs and a 3-dB optical spectral bandwidth of about 10 nm. After the single-pulse soliton is obtained, tuning one of the wave plates while remaining all the other experimental conditions unchanged will shift the laser operation from the conventional soliton emission into the noise-like pulse emission, exactly as that reported in Sec Fig. 7.4 shows the optimized noise-like pulse emission state observed when the length of EDF is 17.6 m. The optical spectrum is smooth. It has a 3-dB bandwidth of about nm, far broader than the EDF gain bandwidth. Fig. 7.4b shows the corresponding autocorrelation trace. Again the autocorrelation trace consists of a narrow spike riding on a broad and smooth shoulder that extended over the entire width of the measurement window of 50 ps. With a small range scan we have determined that the narrow spike corresponds to an average pulse width of about 80 fs (sech 2 profile assumed). Fig. 7.4c shows the measured oscilloscope trace and Fig. 7.4d the RF spectrum of the laser emission. Under the noise-like pulse operation there is always one mode-locked pulse in the cavity and it circulates with the fundamental cavity repetition rate. Based on the combined information provided by the autocorrelation trace and the oscilloscope trace, we confirmed that the mode-locked pulse consists of a bunch of pulses with randomly varying pulse widths and peak powers, namely the noise-like pulse [33]. The noise-like feature is also represented by its RF spectrum as shown in Fig. 7.4d. Around the fundamental cavity repetition frequency, there are also two broad symmetrical spectral sidebands, which indicate the existence of random peak modulation of the mode-locked pulses. The output power of the laser 135

151 Chapter 7: Broadband noise-like pulses emission is about 21.6 dbm, which gives that the energy of the mode-locked pulse is about 15 nj. The state of noise-like emission is stable. With a fixed laser cavity configuration the exact duration of the overall bunch, the 3-dB optical spectral bandwidth and energy vary with the pump strength. Detailed experimental investigations show that the maximum achievable spectral bandwidth is closely related to the EDF length. Provided that the pump strength is strong enough, the longer the EDF the broader the spectral bandwidth can be obtained. Experimentally we changed EDF length while keeping other cavity and experimental parameters fixed. Although noise-like pulse operation always exists regardless of the EDF lengths, the 3-dB spectral bandwidth achievable varies. Under fixed pump strength, there exists an optimized EDF length for achieving the broadest noise-like pulse spectral bandwidth. In our experiments, the optimized EDF length is about 17.6 m. With even longer EDF, e.g. 21m, the maximum 3-dB bandwidth achievable reduces. Comparing the optical spectra obtained with different EDF lengths when the laser is operating in the noise-like pulse mode, we found that the peak of 1480 nm decreased with the increase of EDF length. When the EDF length is equal to 17.6m, the intensity of the peak becomes already very weak as shown in Fig. 7.4a. It then totally disappeared when the EDF length is further increased. This result indicates that the pump energy is fully absorbed at the EDF length. If the EDF length is further increased, the generated noise-like pulses will be absorbed by the EDF. With the pump power available in our laser the optimized EDF length is 17.6m, which generates the maximum 3-dB bandwidth of nm as shown in Fig This optimized EDF length corresponds to the length that the pump power is just fully absorbed by the EDF. 136

152 Chapter 7: Broadband noise-like pulses 7.3 Summary Apart from the soliton operation in the fiber lasers, a kind of noise-like pulse emission was revealed both experimentally and numerically in passively mode-locked fiber lasers. It was found that purely depending on the linear cavity phase delay bias setting, the fiber lasers could be switched between the states of soliton operation and the noise-like pulse emission. Numerical simulations show that the bunched noise-like pulse emission is a natural state of the laser operation and it is caused by the combined effect of the soliton collapse and cavity positive feedback. By taking the advantage of the soliton selffrequency shift effect, we purposely designed a conventional PMEFL and obtained a high power (15nJ) super broadband (93nm) noise-like pulse. Our experimental results not only confirm that noise-like pulse emission is a generic property of all PMEFLs, but also suggest that even higher power and broader spectral bandwidth noise-like pulse emission is in principle possible to achieve provided that even stronger pump source and larger nonlinear fibers are available. We believe such a superbroad spectrum light source could have applications in optical metrology, wavelength-division multiplexing and optical sensing systems. It is however born in mind that the nature of the light source is a bunch of pulses with randomly varying pulse widths, peak powers, and central wavelengths. Although the spectral bandwidth of the light is broad, it could be difficult to obtain narrow pulses with constant pulse width by spectrally slicing the spectrum. Therefore, it could not be suitable for applications where stable and narrow pulse width is important. 137

153 Chapter 8: Self-started high-repetition-rate soliton sources Chapter 8 Self-started high-repetition-rate soliton sources Ultrahigh-repetition-rate optical pulse sources play a key role in ultrahigh speed optical communication systems. Mode-locked fiber lasers offer a potential candidate for such sources. Recently, generation of optical pulse train at repetition rate of 100 GHz has been achieved, based on the use of superimposed fiber Bragg gratings [129]. However, this method actually only multiplies the input pulse train. Litchinistser et al. proposed a highrepetition-rate soliton-train source based on adiabatic compression of a dual-frequency optical signal in nonuniform fiber Bragg gratings [130]. It can reshape the signal propagating through the grating into a train of Bragg solitons, but with the same repetition rate of the input signal. Tadakuma et al. have studied high-repetition-rate pulse generation by short length comb-like dispersion profiled fiber using high nonlinearity dispersion shifted fiber and realized a 100 GHz 380 fs pulse train [131]. However, it needs two wavelength tunable light sources to produce input beat pulse sequence and the structure of fibers is complex. All the above methods need an input pulse sequence which means more complex and costly. Recently, several authors have reported the spontaneous generation of pulse trains with repetition rates in excess of 100 GHz from passively mode-locked fiber ring lasers [34-36]. They attribute it to the modulational-instability (MI) of the continuous-wave field resonating in the laser cavity, which is firstly theoretically proposed by Akira Hasegawa 138

154 Chapter 8: Self-started high-repetition-rate soliton sources [37]. MI is a process in which the amplitude and phase modulations of a wave grow as a result of interplay between the nonlinearity and anomalous dispersion [132]. Anomalous dispersion is essential for getting MI, however, T. Sylvestre et al. found that the selfstarted high-repetition-rate is not intrinsically linked to MI [38]. They used a laser with total normal dispersion and still obtained similar high-repetition-rate pulse train. Their result agrees with the theoretical study of M. Quiroga-Teixeiro et al. [133], who considered a fiber laser with a gain profile of symmetric lateral frequency bands, and explained the mechanism for the pulse formation as dissipative four-wave mixing. Here we presented a simple technique for the generation of self-started high-repetitionrate pulse train in fiber ring lasers. Our approach further supports Quiroga-Teixeiro and Sylvestre s work. It was numerically found that soliton-like pulses at repetition rate of 125 GHz could be obtained by inserting a spectral filter in the fiber lasers. Optimizing parameters of the filter and the laser cavity, we can further compress the pulse width of the obtained ultrashort pulse train. By directly incorporating a segment of high birefringence (HiBi) fiber in the fiber lasers mode-locked with NPR technique, due to the birefringence filter effect of the cavity, a spectral filter is induced in the cavity [ ]. Our experimental observation supports our proposal. Depending on the length of the incorporating HiBi fiber, pulse train with repetition rate up to THz could be achievable. However, only when the MI gain peaks well match the free spectral range of the filter, uniform pulse train could be obtained. 139

155 Chapter 8: Self-started high-repetition-rate soliton sources This chapter presented the experimental and theoretical research on the self-started highrepetition-rate soliton sources. Section 8.1 presents the basic theory model and the numerical simulation results. The experimental observation of the self-started highrepetition-rate soliton train is presented in Section 8.2. And Section 8.3 is the summary. 8.1 Theoretical model and simulation results It is well known that an individual pulse can be spread into a burst of pulses by a combination of spectral amplitude and phase filtering [137]. Fig. 8.1a shows the generated high-repetition-rate pulse train when we introduced a periodical spectral filter into our simulations. Experimentally, this corresponds to that we insert a filter in the laser cavity. The filter will spectrally modulate light pulse circulating in the cavity. Here we again use the pulse tracing technique to simulate pulse evolution in the cavity. The spectral modulation can be expressed as: U ' ( ω) = F U( ω) F = A(1 + sinωt) exp( iφ) (8.1) where U (ω) and U(ω) are the spectra intensity function of the light traveling in the cavity after and before filtering, respectively, F is the transmission function of the spectral filter. Here A is the amplitude factor of the spectral filter; Φ is the phase change introduced by the filter; for simplification, we chose sinusoidal function as the filter profile. 140

156 Chapter 8: Self-started high-repetition-rate soliton sources The parameters for our simulations are: κ = -2 ps/nm/km (DSF); κ = -10 ps/nm/km (EDF); Cavity length L = 1 DSF +4 EDF +1 DSF =6m; Beat length L b = L/2; Passive polarizer orientation to the fiber fast birefringent axis ψ= 0.125π; Linear polarization rotation phase bias ϕ=1.2π. (a) (d) (b) (e) (c) (f) Fig. 8.1 Pulse intensity and spectrum of the generated high-repetition-rate pulse train under different pump power. (a)/(d) G=270; (b)/(e) G=300; (c)/(f) G=

157 Chapter 8: Self-started high-repetition-rate soliton sources Fig. 8.1b shows a pulse train with repetition rate of 125 GHz. It is obtained when we chose that G=300 and F = 0.49(1 + sinω t)exp(i 0.4π ). The free spectral range (FSR) of the filer is chosen to be 1 nm. When we fixed all the other parameters and varied the value of G, which practically corresponds to change the pump power, we found that the operation of high-repetition-rate pulse train can be sustained. However, the profile of the repeated pulse would change. Fig. 8.1a and Fig. 8.1c show the cases of G=270 and G=350, respectively. When pump power is not high enough (Fig. 8.1a), only broad sinewave-like pulse train could be obtained, and there is a DC component. In this case, MI is small and negligible. With higher pump power, soliton-like pulse train is formed as shown in Fig. 8.1b. There is small intensity modulation on the pulse train, which was found to be caused by MI and the MI gain peaks do not perfectly match the free spectral range of the spectral filter. Since the MI frequency that gives the maximum gain in the MI process is proportional to the square root of the average power in the cavity, it is possible to tune the MI frequency to perfectly match the modulation frequency introduced by the spectral filter. Then a stable uniform high-repetition-rate pulse train could be obtained. Higher pump power results in narrower pulse width. Numerically we found that further increasing the pump power, a new pulse will emerge to accompany the existing pulse as shown in Fig. 8.1c. The new pulse and the original pulse bind as a unit, and the repetition rate of the laser remains at 125 GHz. The pulse intensity of the newly formed pulse and the separation between the two pulses vary with the pump power. Apart from the formation of high-repetition-rate pulse train, whose repetition rate is determined by the FSR of the inserted spectral filter, high-repetition-rate double pulse train could also be obtained if the pump power is appropriately set. Fig. 8.1d, Fig. 8.1e, and Fig. 8.1f 142

158 Chapter 8: Self-started high-repetition-rate soliton sources show the spectrum corresponding to Fig. 8.1a, Fig. 8.1b, and Fig. 8.1c. The 1 nm peak-topeak spacing of the spectral peaks in Fig. 8.1d is determined by the FSR of the spectral filter and it also corresponds to pulse separation of 8 ps in the time domain, which can be clearly seen in Fig. 8.1a. The contour of Fig. 8.1e clearly shows the profile of a soliton spectrum, which undoubtedly suggests that each individual pulse in Fig. 8.1b is a soliton indeed. Numerically we also studied the influence of the filter loss on the formed pulse train. Filter with form of A(1 + sinω t) exp( iφ) can be regarded as a filter of ( 1+ sinω t) exp( iφ) plus an attenuator A (amplitude factor), so the modulation depth of this type of filters is large. When the equivalent attenuation is too large, no pulse can be formed in the cavity. We found that different amplitude factor does not change the repetition rate of the generated pulse train but it is critical for the formation of the pulse train. In our simulations, for example, for Psat=1000 and G=350, we found that only for amplitude factor with value greater than 0.42, 125 GHz ultrashort pulse train can be achieved. If the amplitude factor is less than 0.42, too high attenuation of spectra of the pulses will make the pulses die out eventually. As shown in Fig. 8.2, When A=0.5, G=350, we obtained a nearly perfect 125 GHz double pulse train as predicted above. Numerically we can also achieve high-repetition-rate pulse train such as 250 GHz by choosing 2nm FSR of the filter. However, it needs higher G that means it requires much higher pump power. 143

159 Chapter 8: Self-started high-repetition-rate soliton sources Fig. 8.2 Pulse intensity of the generated double-high-repetition-rate pulse train when A=0.5, G= Experimental demonstration There are many methods to introduce a spectral filter in the cavity, such as FP filter [35], grating pairs [137] and optical loop [36]. Here we directly introduce a segment of HiBi fiber in our cavity, combined with the polarizer in the cavity, to function as a Lyot-type filter [ ]. The free spectral range of the Lyot-type filter is determined by the beat length of the HiBi fiber and the HiBi fiber length [136], as shown in Eq. (8.2), where is the free spectral range, L = λ n n ) is the beat length, n x and n y are the refractive b ( x y index of the two polarization eigenmodes of the fiber, L is the fiber length. As λ ν λ / λ 2 = c, it was found that ν ( x y is proportional to the reciprocal of L since c and n n ) are constant in our experiment as shown in Eq. (8.3). Actually ν is also the repetition rate of the generated pulse train because the energy in the cavity is selectively confined in the longitudinal modes of the spectral filter that is introduced by the HiBi 144

160 Chapter 8: Self-started high-repetition-rate soliton sources fiber. Depending on the length of the HiBi fiber, we could determine the repetition rate of the generated pulse train. The beat length of the HiBi fiber is about 7 mm at 1550 nm wavelength, which corresponds to birefringence of about We note that the birefringence of communication fibers is in the range of 10-6 ~10-7 [69]. The insertion of the HiBi fiber is equivalent to increase the averaged cavity linear birefringence. However, it would result in a small difference between the calculated FSR and the real FSR introduced by the HiBi fiber. L b λ λ = L (8.2) c ν = ( nx ny) L (8.3) The experiment setup is similar to our previous one [22] in Section 7.2 except that we inserted a segment of HiBi fiber between the output coupler and the Erbium-doped fiber to achieve the spectral modulation. The 70% output coupler is replaced by a 10% output coupler, which enhances the laser power in the cavity. We have measured the MI in the laser without inserting the HiBi fiber. The wavelength shift is about 1 nm, which corresponds to about 125 GHz if the central wavelength is assumed to be 1550 nm. The birefringent coefficient of the HiBi fiber we used is So we chose a fiber length of 11.5 m to match the MI frequency. 145

161 Chapter 8: Self-started high-repetition-rate soliton sources Intensity (arb. unit) (a) Spectral Intensity (db) (e) Intensity (arb. unit) Time (ps) (b) Spectral Intensity (db) Wavelength (f) Intensity (arb. unit) Time (ps) (c) Spectral Intensity (db) Wavelength (g) Intensity (arb. unit) Time (ps) (d) Spectral Intensity (db) Wavelength (h) Time (ps) Wavelength Fig. 8.3 Autocorrelation trace and spectrum of the generated high-repetition-rate pulse train of GHz. 146