Fiber Misalignment Analysis Ansys

School Of Electrical & Electronic Engineering

in partial fulfillment of the requirement for the degree of

Statement Of Originality

I hereby certify that the work embodied in this thesis is the result of original research and has not been submitted for a higher degree to any

Summary

Solitons, as stable localized wave packets that can propagate long distance in dispersive media without changing their shapes, are ubiquitous in nonlinear in the anomalous dispersion single mode fibers (SMF) by Mollenauer et al. in 1980 and optical dark solitons in the normal dispersion SMFs by P. Emplit et al. in 1987, optical solitons in SMFs had been extensively investigated. In reality a SMF always supports two orthogonal polarization modes. Taking fiber birefringence into account, it was later theoretically predicted that various types of vector solitons, including the bright-bright vector solitons, dark-dark vector solitons and dark- bright vector solitons, could be formed in SMFs. However, except the bright-bright type of vector solitons, other types of vector solitons are so far lack of clear experimental evidence.

Optical solitons have been observed not only in the SMFs but also in mode locked fiber lasers. It has been shown that the passively mode-locked erbium- doped fiber lasers offer a promising experimental platform for studying the scalar optical solitons. Vector solitons can also be formed in mode locked fiber lasers. In this dissertation, the author presents results of a series of theoretical and experimental investigations on the vector solitons in fiber lasers.

- Ii -

First of all, passively mode-locked erbium-doped fiber lasers with the semiconductor saturable absorber mirror (SESAM) as a mode locker were designed and constructed. Formation of various vector bright-bright solitons in the fiber lasers was demonstrated, and whose features and dynamics were investigated.

There are either the coherently or incoherently coupled vector solitons in the fiber lasers. Under coherent coupling between the two orthogonal polarization components of the fiber lasers, a type of polarization-locked high-order vector soliton as well as the polarization-rotating vector solitons was observed. Worth of mentioning is the special features of the high-order vector soliton: besides that its two orthogonal polarization components are phase locked, the two polarization components also have different soliton profiles. While the stronger polarization component is a single hump pulse, the weaker component has a double-humped structure with 180° phase difference between the humps. The features of the experimentally observed high-order vector soliton well match those of the theoretical predictions. Moreover, we show that in the normal dispersion cavity fiber lasers dissipative vector solutions could be formed, and under stronger cavity birefringence multi-wavelength dissipative soliton operation of the fiber lasers is possible.

In addition of the experimental studies, numerical simulations on the vector soliton operation of the fiber lasers were also carried out. To closely simulate the vector soliton evolution in a fiber laser, a model that is based on the coupled extended Ginzburg-Landau equations that take into account not only the fiber dispersion and nonlinearity, laser gain and cavity losses, saturable absorber effect,

- Iii -

but also the cavity feedback and boundary condition was built up. The coupled extended GLEs were solved numerically using the standard split-step method under the experimental laser cavity conditions. It was found that all the experimentally observed vector soliton features could be numerically reproduced.

Special attention was paid on the soliton formation in all normal dispersion cavity fiber lasers. It was firstly revealed that scalar dark solitons described by the nonlinear Schrodinger equation could be formed in an all dispersion cavity fiber laser with a polarizer inserted in the cavity. If the polarizer was replaced with a graphene based saturable absorber, vector dark-dark soliton and trapping of vector dark-dark solitons were further experimentally obtained. Numerical simulations have also well confirmed the experimental observations of dark soliton formation in the fiber lasers.

Independent on the laser cavity dispersion, another novel type of optical solitons known as the optical domain wall solitons was also firstly experimentally identified in the experiments. Domain wall soliton is also a kind of vector soliton.

However, different from the vector solitons mentioned above, its formation is a result of the nonlinear coupling between two coexisting eign states in an optical system. Either the coherently coupled optical domain wall solitons, represented as a phase-locked dark-bright pulse pair or dark-dark pulse pair, or the incoherently coupled optical domain wall solitons, represented as vector dark solitons that separating two stable optical domains were first experimentally demonstrated and numerically confirmed.

- Iv -

Lastly, vector soliton operation of graphene mode locked fiber lasers was demonstrated. Graphene is a newly discovered 2D single atomic layer nano- material with unique electronic properties, whose applications in the nano- electronics have attracted enormous attention. Graphene also possesses a number of special optical properties including the wavelength independent ultrafast saturable absorption. The author has first experimentally demonstrated the saturable absorber of graphene, and further exploited the saturable absorption of graphene for mode locking fiber lasers. It was shown that using graphene as a saturable absorber to mode lock the erbium-doped fiber lasers, large energy mode locked pulses with pulse energy up to 7.3 nJ could be achieved. Moreover, wide-band tuning of the mode locked pulses and various types of the previously discussed vector solitons could also be obtained in the graphene mode locked fiber lasers. Comparing with other types of the conventionally used mode lockers, e.g. SESAM, graphene as a mode locker has the following merits: polarization independent broadband saturable absorption, ultrafast saturation recovery time, tunable saturation modulation deepth, lower non-saturable loss, cheaper fabrication cost, easy integration in a laser system. It is envisaged that other applications of graphene in photonics and optoelectronics could be further identified.

- V -

I am grateful to all the people who had given me support whilst researching and accomplishing this dissertation. First and foremost, I wish to express my sincere gratitude to my supervisor, Prof. Tang Dingyuan, for introducing me to the wonderful world of nonlinear optics. His wide knowledge, logical way of thinking, passion and enthusiasm for research, has been lifelong wealth for me. I appreciate all his contribution of efforts, ideas, time and fund to make my Ph. D experience fruitful and stimulating. His understanding, encouraging and personal guidance have provided a good basis for the present thesis.

recommending me as a Ph. D candidate to NTU and offering me continuous supports. I wish to thank Prof. Randall Knize in United States Air Force Academy, who has given me many exciting ideas. I warmly thank Dr. Zhao Luming, who taught me a lot of experimental skills. I am thankful to Dr. Bao Qiaoliang, with who I could extend my research from laser to graphene optoelectronics. Many Guoqiang, Mr. Tan Wei De, Mr. Lin Bo for helping me a lot.

I would like to acknowledge Prof. Shum Ping, director of NTRC of NTU, who provided first-class experimental facilities for fiber laser research. I also wish

- Vi -

to thank all my colleagues and friends in Photonics Research Center and NTRC of NTU. Without their friendship and kindly assistance it would have been much tougher to finally complete this dissertation. In particular, I would like to show my appreciation to the technicians in the two labs for their kindly assistance with the This dissertation is dedicated to my family. My dad Zhang Guoxin, my mum Chen Zhiling and my young sister Zhang Jiayi, I am forever indebted to you for your great love, considerate understanding, eternal patience and encouragement

All The Way Along. Deeply Appreciated!

The financial support of NTU is gratefully acknowledged!

Table Of Contents

SUMMARY........................................................................................................................................I ACKNOWLEDGEMENTS .............................................................................................................V TABLE OF CONTENTS ..............................................................................................................VII LIST OF FIGURES........................................................................................................................ IX

Hapter 1.

INTRODUCTION.................................................................................................1 BACKGROUND AND MOTIVATION ......................................................................................1 DEVELOPMENT OF MODE-LOCKED SOLITON FIBER LASERS................................................3 Fundamentals of Mode locking....................................................................................4 Conservative/dissipative soliton operation..................................................................7 Active laser mode locking............................................................................................9 Passive laser mode locking........................................................................................12 MOTIVATION AND OBJECTIVES........................................................................................28 MAIN CONTRIBUTIONS OF THE RESEARCH.......................................................................33 OVERVIEW OF THE DISSERTATION...................................................................................37

Theory Of Vector Soliton Propagation In A Fiber Laser

LINEAR PULSE PROPAGATION IN OPTICAL FIBERS ............................................................41 PULSE PROPAGATION IN DISPERSIVE MEDIA ....................................................................44 NONLINEAR PULSE PROPAGATION IN OPTICAL FIBERS .....................................................48 Nonlinear effects in optical fibers..............................................................................48 Nonlinear Schrödinger equation................................................................................51 Fundamental Soliton..................................................................................................52 PULSE PROPAGATION IN ERBIUM-DOPED FIBER................................................................53 Gain profile of erbium-doped fibers ..........................................................................53 Ginzburg-Landau equation........................................................................................56 Soliton solution..........................................................................................................58 PULSE PROPAGATION IN LINEARLY BIREFRINGENT FIBERS ..............................................59 Fiber birefringence....................................................................................................59 Coupled Ginzburg-Landau equations........................................................................61 PROCEDURE OF NUMERICALLY SOLVE THE COUPLED GINZBURG-LANDAU EQUATIONS..63

Solitons

COHERENT INTERACTIONS ..............................................................................................70 INCOHERENT INTERACTION.............................................................................................79

Hapter 5.

DISSIPATIVE VECTOR SOLITONS............................................................100 POLARIZATION LOCKED AND ROTATING DVS...............................................................101 MULTI-WAVELENGTH DISSIPATIVE SOLITON .................................................................111

Hapter 6.

DYNAMICS OF DARK SOLITON IN FIBER LASERS .............................125

- Viii -

SCALAR DARK SOLITON.................................................................................................126 DISPERSION MANAGED DARK SOLITONS........................................................................135 TRAPPING OF DARK VECTOR SOLITONS..........................................................................144

Hapter 7.

DYNAMICS OF DOMAIN WALL SOLITONS............................................155 COHERENTLY COUPLED DOMAIN WALL SOLITONS.........................................................157 VECTOR DARK DOMAIN WALL SOLITONS.......................................................................168 DUAL-WAVELENGTH DOMAIN WALL SOLITONS.............................................................177

Hapter 8.

GRAPHENE MODE LOCKED FIBER LASERS.........................................187 ATOMIC LAYER GRAPHENE AS SATURABLE ABSORBER................................................189 GRAPHENE-POLYMER COMPOSITE AS SATURABLE ABSORBER.......................................197 GRAPHENE AS A FULL-WAVEBAND SATURABLE ABSORBER...........................................205 GRAPHENE MODE LOCKED VECTOR DISSIPATIVE SOLITONS...........................................208 PROSPECT OF GRAPHENE BASED ULTRAFAST PHOTONICS ..............................................216

Hapter 9.

CONCLUSIONS AND FUTURE WORK ......................................................218 SUMMARY OF ACCOMPLISHMENTS ................................................................................218 Vector bright-bright soliton.....................................................................................218 Scalar/vector dark soliton........................................................................................220 Domain wall soliton.................................................................................................221 Existence domain of diverse vector solitons ............................................................222 Graphene based ultrafast saturable Absorbers .......................................................225 RECOMMENDATIONS FOR FUTURE WORK ......................................................................225 REFERENCES:.............................................................................................................................229 AUTHOR’S PUBLICATIONS.....................................................................................................255 JOURNALS:...................................................................................................................................255 CONFERENCES:............................................................................................................................257

Ist Of Figures

Figure 1.1: Typical experimental setup of an actively mode-locked fiber laser...............................10 Figure 1.2: (a) schematic of NPR mode locking principle; (b) configuration of a typical NPR mode locking fiber laser..............................................................................................................................15 Figure 1.3: (a) Schematic structure of the SESAM; (b) Schematic configuration of a fiber laser mode locked using a SESAM............................................................................................................18 Figure 1.4: graphene, the parent of all graphitic forms [70, 71]. ......................................................23 Figure 1.5: illustration of chiral vector. The (n, m) nanotube naming scheme can be thought of as a vector (Ch) in an infinite graphene sheet that describes how to “roll up” the graphene sheet to make the nanotube, T denotes the tube axis, and a1 and a2 are the unit vectors of graphene in real space . ...................................................................................................................................................25 Figure 1.6: (a) & (b) schematic illustration of energy band structure of single wall carbon nano-tube with different diameters and chiralities ; (c) graphene’s energy band structureand photon absorption. .........................................................................................................................................26 Figure 2.1: Group velocity dispersion β2 for fused silica as a function of wavelength (After Ref.

)...................................................................................................................................................44 Figure 2.2: Energy levels of Er3+ ions...............................................................................................54 Figure 2.3: Typical absorption and emission spectra of the erbium-doped fibers (After Ref. ).

...........................................................................................................................................................56 Figure 3.1: Schematic of the SESAM mode locked fiber laser. .......................................................72 Figure 3.2: Optical spectra of the phase locked vector solitons of the laser measured without passing and passing through a polarizer: (a) and (b) were measured under different linear cavity birefringence......................................................................................................................................75 Figure 3.3: Numerically calculated optical spectra of the vector solitons formed in fiber ring lasers.

...........................................................................................................................................................77 Figure 3.4: Polarization resolved optical spectra of the vector solitons experimentally observed. (a): Obtained under large cavity birefringence. (b) Obtained under relatively weak cavity birefringence.

...........................................................................................................................................................83 Figure 3.5: Numerically calculated optical spectra of the vector solitons. (a): Cavity beat length Lb = 0.1 m. (b) Cavity beat length Lb = 10 m. Pump strength G0 = 80......................................................86 Figure 4.1: Schematic of the vector soliton fiber laser.....................................................................91 Figure 4.2: Polarization resolved soliton spectra and autocorrelation traces of the vector soliton observed. (a) Soliton spectra. (b) Autocorrelation traces...................................................................94 Figure 4.3: Oscilloscope trace of a harmonically mode-locked high order phase locked vector soliton state. Lc: cavity roundtrip time. 8 vector solitons coexist in cavity.......................................95 Figure 4.4: A stable high order phase locked vector soliton state numerically calculated. (a) Soliton intensity profiles of the two orthogonally polarized components. (b) The corresponding optical spectra of (a)......................................................................................................................................98 Figure 5.1: Schematic of the fiber laser..........................................................................................103 Figure 5.2: (a) Spectrum and corresponding autocorrelation trace of a polarization rotating DVS emission state of the laser; (b) Oscilloscope trace of (a) after passing through a polarizer; (c)

- X -

Polarization resolved optical spectra of a phase locked DVS emission state of the laser; (d) Oscilloscope trace of (c) after passing through a polarizer..............................................................105 Figure 5.3: Oscilloscope trace of a harmonically mode-locked gain-guided vector soliton state. Lc: cavity roundtrip time. 8 DVS coexist in cavity................................................................................107 Figure 5.4: (a) Combined pulse intensity evolution; (b) corresponding optical spectra numerically calculated.........................................................................................................................................110 Figure 5.5: Schematic of the experimental setup............................................................................113 Figure 5.6: (a) Optical spectra of single wavelength dissipative soliton. Insert: the oscilloscope trace. (b) The corresponding autocorrelation trace..........................................................................114 Figure 5.7: (a) Optical spectrum of dual wavelength dissipative solitons. Insert: the normalized optical spectrum; (b) Oscilloscope trace of dual wavelength dissipative solitons...........................117 Figure 5.8: Oscilloscope traces of synchronized dual wavelength dissipative solitons..................118 Figure 5.9: (a) Optical spectra of polarization locked gain guided vector soliton and dual wavelength spectrum obtained through rotating PCs but kept the pump strength fixed: normalized unit. (b) Oscilloscope trace of polarization locked gain guided vector soliton after passing through a polarizer...........................................................................................................................................120 Figure 5.10: Single/dual/triple wavelength spectra obtained through rotating PCs but kept the pump strength fixed...................................................................................................................................121 Figure 6.1: Schematic of the fiber laser..........................................................................................127 Figure 6.2: Dark pulse emission of the laser. (a) Oscilloscope traces, upper: single dark pulse emission; down: multiple dark pulse emission. (b) Optical spectra of the laser emissions. Inset: RF spectrum of the single dark pulse emission. ....................................................................................129 Figure 6.3: a dark pulse state numerically calculated. (a) Evolution with cavity roundtrips. (b) Intensity and phase profile. (c) Optical spectrum. ...........................................................................134 Figure 6.4: Schematic of the vector dark soliton fiber laser. ..........................................................136 Figure 6.5: Optical spectrum of DM bright soliton. Insert: its autocorrelation trace and oscilloscope trace. ................................................................................................................................................137 Figure 6.6: (a) Spectra of DM dark soliton and CW. (b) Oscilloscope traces, upper: single dark soliton; down: multiple dark solitons. .............................................................................................139 Figure 6.7: Region of existence of DM dark solitons in the (Dispersion, Gain) plane...................142 Figure 6.8: Evolution of DM dark solitons in time domain with the net-cavity dispersion 0.343 ps2: (a) Gain = 485 km–1; (b) Gain = 510 km–1. .....................................................................................143 Figure 6.9: Schematic of the fiber laser..........................................................................................146 Figure 6.10: Polarization resolved (a): oscilloscope trace of single dark vector soliton in the cavity and (b) its corresponding optical spectra Insert: zoom in of (b) near the spectral center. (c) Polarization resolved oscilloscope trace of multiple dark vector soliton in the cavity. (d) Enlarge scale of (c) at different positions. ....................................................................................................150 Figure 6.11: Stable dark vector soliton state numerically calculated. (a) Evolution of the dark vector soliton with cavity roundtrips. (b) Zoom in of (a). (c) The corresponding spectra and insert: zoom-in of (c). Gain = 1500. L/Lb = 60. ........................................................................................................152 Figure 6.12: Stable dark vector soliton state numerically calculated. L/Lb = 40. ........................................................................................................154 Figure 7.1: Setup of the fiber lasers. WDM: wavelength division multiplexer. EDF: erbium doped fiber. PC: Polarization controller. PBS: Polarization beam splitter.................................................160 Figure 7.2: A typical dark-bright vector soliton emission of the lasers. (a) Oscilloscope traces. (b) Zoom-in of the dark-bright pulses. (c) Polarization resolved optical spectra; inset: Autocorrelation trace of the bright pulses..................................................................................................................163

- Xi -

Figure 7.3: Oscilloscope traces of a typical dark-dark vector soliton emission of the positive dispersion fiber laser........................................................................................................................164 Figure 7.4: The dark-bright and dark-dark vector soliton states numerically calculated. (a) Evolution of the dark-bright vector soliton with the cavity roundtrips. (b) Optical spectra of the dark and bright solitons. Inset: soliton profiles of the dark and bright solitons. Gain = 120. (c) Soliton profiles of the dark-dark vector soliton numerically calculated. Gain = 200, ∆n = nu–nv = 1.7× 10–9.

.........................................................................................................................................................167 Figure 7.5: Schematic of the experimental setup............................................................................171 Figure 7.6: Duration variation of the square pulses versus the orientation angle of one of the paddles of the intra-cavity PC..........................................................................................................173 Figure 7.7: Vector dark polarization domain wall soliton emission of the laser. (a) Total laser emission (upper trace) and one of the polarized laser emissions (lower trace). (b) The corresponding optical spectra..................................................................................................................................174 Figure 7.8: Polarization domain wall numerically calculated. (a) Evolution of the polarization domain wall with the cavity roundtrips. (b) Domain wall profiles at particular roundtrip. (c) The vector domain wall soliton and its ellipticity degree at particular roundtrip. ..................................177 Figure 7.9: (a) Spectrum; (b) oscilloscope traces of the dual-wavelength optical domain wall. (c): the wall duration as a function of the pump strength.......................................................................180 Figure 7.10: (a) Spectrum and (b) oscilloscope trace of the single dark soliton obtained through the paddles of PC. (For comparison).....................................................................................................181 Figure 7.11: Generation of the multiple dark solitons at each individual wavelength of the optical domain wall. (a) Spectra and (b) oscilloscope traces, upper (lower) trace corresponds to spectral broadening at longer (shorter) wavelength. .....................................................................................183 Figure 7.12: dual wavelength domain wall numerically calculated. Evolution of the dual wavelength domain wall with the cavity roundtrips: (a) one wavelength (shorter wavelength) (b) Another wavelength (longer wavelength). (c) Domain wall profiles at particular roundtrip (d) Its corresponding spectra......................................................................................................................185 Figure 8.1: (a) Schematic of the fiber laser. (b) 2D atomic layer of carbon atoms arranged in a hexagonal lattice..............................................................................................................................191 Figure 8.2: Characterization of graphene thin film covering on the fiber core. (a) Raman spectra of the grapheme film. (b) Raman image around the fiber core plotted by the intensity of the Raman peak of SiO2. The scale bar is 3 µm. (c) Raman images around the fiber core plotted by the intensity of G band of graphene. ................................................................................193 Figure 8.3: Pulse operation of the fiber laser. (a) Pulse spectra measured. Insert: long-term fluctuation of the FWHM. (b) Autocorrelation traces of the pulses. (c) An oscilloscope trace of the single pulse emission. Insert: pulse train of CW mode-locking in millisecond time scale. (d) The fundamental radio-frequency (RF) spectrum of the laser output. Insert: wideband RF spectrum up to 100 MHz..........................................................................................................................................195 Figure 8.4: the single pulse energy in respect to the pump power..................................................196 Figure 8.5: (a) SEM image of the graphene-polymer nanofiber networks. Inset: a photo of the free- standing graphene-polymer composite membrane. (b) Transmission electron microscopy (TEM) image of a graphene-PVDF nanofiber.............................................................................................199 Figure 8.6: (a) UV-VIS-NIR absorption spectra of graphene-based PVDF nanocomposites and pure PVDF. The inset shows the chemical structure of the functionalized graphene. b) Power dependent nonlinear saturable absorption of graphene-based PVDF nanocomposites.....................................201 Figure 8.7: Soliton operation of the fiber laser. (a) Soliton spectra measured. (b) Autocorrelation traces of the solitons. (c) An oscilloscope trace of the laser emission. (d) The RF spectrum of the laser output. Insert: RF spectrum up to 1 GHz ................................................................................204

- Xii -

Figure 8.8: Absorption of light in graphene. (a) Schematic excitation processes responsible for absorption of light in graphene. The arrow indicates optical interband transition. (b) The photogenerated carriers thermalize and cool down within subpicosecond to form a hot Fermi-Dirac distribution, an equilibrium electron and hole distribution could be finally approached through intraband phonon scattering and electron-hole recombination. (c) At enough high excitation intensity, the photogenerated carriers cause the states near the edge of the conduction and valence bands to fill, blocking further absorption.........................................................................................206 Figure 8.9: Wideband spectra tuning from 1570 to 1600 nm. ........................................................207 Figure 8.10: Illustration of optical conductivity of graphene: Incident light normal to the graphene layer (x-z plane)...............................................................................................................................209 Figure 8.11: Polarization resolved saturable absorption curve of graphene-based mode locker....210 Figure 8.12: Dissipative vector soliton operation of the fiber laser. (a) Optical spectrum measured.

(b) Pulse profile measured with a high speed oscilloscope. (c) Polarization resolved oscilloscope traces: the polarization of the soliton is rotating in the cavity. ........................................................213 Figure 8.13: Polarization locked dissipative soliton operation of the fiber laser. (a) Polarization resolved oscilloscope trace; (b) the corresponding optical spectra..................................................215

Background And Motivation

Solitons, as particle-like nonlinear localized waves due to an interior self- reinforcing against dispersion through nonlinearity, universally existed in a large amount of nonlinear physics scenarios from fluids and plasma physics to optics, biological and atmospheric systems . Optical solitons have attracted everlasting interest in the last three decades thanks to their theoretical values and attractive practical applications in the optical communication and signal processing intensity optical pulses that do not dissipate and temporally retain their shapes through the balance between the dispersion and nonlinearity. One of the most promising applications of the soliton theory could be credited to the field of optical fiber communications . Hasegawa and Tappert firstly predicted the existence of solitons in optical fibers and proposed the potential application of solitons for optic communications in 1973 . Seven years later, Mollenauer et al. at the Bell Laboratories experimentally demonstrated the propagation of optical bright solitons in anomalous dispersion fibers , and subsequently in 1987, P. Emplit et al. from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton in normal dispersion fibers, both of which heralded a new era of soliton and whipped up a storm of studying on optical

Hapter1: Introduction

solitons that continue to amaze . Optical bright solitons were characterized by temporally localized intensity peaks while the optical dark solitons were featured by temporally localized dips on a continuous background. Decades of study on optical solitons show that their dynamics could be well understood by the nonlinear Schrödinger equation (NLSE). However, in reality, an optical single mode fiber (SMF) is not rigorously single “mode” but always endorses two orthogonal polarization modes. Considering the fiber birefringence, it was later theoretically found that depending on the sign of fiber dispersion and the strength of fiber birefringence, three types of vector solitons: bright-bright, dark-dark, and dark- bright , could be yielded in SMFs. Although theoretical investigations on optical solitons had explosively progressed, experimental studies on the vector solitons seriously lagged.

A short pulse laser source is one of the prerequisites of optical soliton formation. Usually, “mode lock” is employed to generate ultra-short pulses in lasers . In the case of mode-locked fiber lasers, apart from the cavity components that are necessary for achieving mode locking, laser cavity is mainly made up of optical fibers. If the fiber laser cavity is free of any polarization dependent elements, particularly, passive mode lockers purely contributed by material based saturable absorbers other than nonlinear polarization rotation technique (NPR); it is natural to anticipate that under certain conditions optical vector solitons can also propagate along the laser cavity without polarizer limitations. Lately, research interests have been shifted from the anomalous to normal dispersion regime with the purpose of large energy formation. In normal dispersion domain, dissipative solitons are

Hapter1: Introduction

formed as a result of the mutual nonlinear interaction among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, laser gain saturation and gain bandwidth filtering . Since dissipative solitons manifested entirely different features from conventional solitons, it is meaningful to experimentally and theoretically study the kinetics of vector sides of dissipative solitons.

Except the soliton formation induced by mode locking, NLSEs automatically permit the dark soliton in normal dispersion regime and the gain competition among the two orthogonal polarizations allows the formation of domain wall pulse.

Besides mode locking operation, NLSE- or domain wall- type dark solitons could be experimentally realized under particular cavity parameters. Thus, we could naturally conclude that the fiber laser is indeed a well-controlled platform to reveal the dynamics of vector solitons from vector bright soliton, vector dark soliton to domain wall soliton.

This chapter is intended to provide a basic overview on soliton theory and mode- locked fiber soliton lasers. Section 1.1 gives a brief review on the development of mode-locked fiber soliton lasers. The basic theories and development of mode locking and soliton generation are provided in Section 1.2. Section 1.3 presents the motivation and objectives of the research. Section 1.4 discusses the main contribution in this thesis. An overview of the dissertation is summarized in Section 1.5.

Evelopment Of Mode-Locked Soliton Fiber Lasers

Since the low-loss silica fiber was available in 1970s, fiber lasers have gained incessant worldwide attention due to their flexible, simplicity, durability, high

Hapter1: Introduction

efficiency, compact size, modest energy and cost. Many different rare-earth ions, such as erbium (Er3+), neodymium (Nd3+), and ytterbium (Yb3+) have been used to dope the normal fibers. These rare-earth-doped fibers then can be used as the gain medium in fiber lasers or amplifiers depending on the required operating wavelength range.

With the rapid development of optical communications in the last three decades, the demand for stable ultrashort pulse laser sources has become the central goal of research. Mode-locked fiber lasers are capable of producing pulses with widths in a very wide range of from tens of fs to nanosecond and a wide repetition rate range of from less than 1 MHz to 1 THz. From the technical point of view, mode-locked, erbium-doped fiber lasers (EDFLs) stand out in that they are capable of generating ultra-short optical pulses in the spectral range of 1.5 µm, which allows use of readily available telecommunications components and can be easily frequency doubled to 760 nm to 820 nm, offering a promising alternative to expensive Ti:sapphire laser. In addition, in contrast with other wavelength, both single mode anomalous and normal dispersion fibers are effortlessly available in EDFLs, providing an attractive experimental setup to investigate solitons from anomalous-, zero- to normal- dispersion cavity. This is a two-way street; the developed soliton theory in EDFL also supports the advancement of laser operations at other wavelengths.

Fundamentals Of Mode Locking

Laser is essentially an optical oscillator requiring the two basic constituents of any oscillator, namely amplification and feedback. The stimulated emission in a gain

Hapter1: Introduction

medium provided the amplification while the laser cavity, composed by sets of mirrors reflecting light, supplied the feedback. The word “mode locking” firstly appeared in the paper of Hargroves in 1964 . The requirement that the electromagnetic field be unchanged after one round trip in the laser makes lasing only occurring for discrete frequencies such that the cavity length is an integer number of wavelengths. The “cavity” of a laser ensures that light is emitted at well- defined wavelengths, known as modes. By introducing a relatively weak modulation synchronous with the round trip time of the laser, the coherence between the phases of different modes could be realized, and pulsed radiation could be produced. The history of laser mode locking is a progression of new and better ways to generate shorter and shorter pulses, and of improvements in the understanding of mode-locking processes. Over the last three decades, mode locking has been used in all kinds of lasers and even ultra-short pulses as short as 47 fs were achieved . In the temporal domain, mode locking actually produces a pulse train, where the time interval between neighboring pulses equals the cavity round trip time. It should be mentioned that there are others types of lasers that generate pulses, the most common being called a Q-switched laser. Furthermore, the pulse evolution in those lasers is unrelated to soliton dynamics. Non-pulsed lasers are defined as “continuous wave” (CW).

The theory of mode locking is a little complicated. Here, we only give the basic physics of mode locking. A large number of longitude modes can be stimulated simultaneously in the gain bandwidth of a laser provided that the pump is strong.

(1.1.1)

where Lopt is the optical length during one round trip inside the cavity. Therefore,

M

φ and ωm are the amplitude, phase and frequency of the specific modes among (2M + 1) modes permitted by the laser gain bandwidth. If all these modes operate independently of each other without definite phase relationship between them, the interference terms in the total intensity

Averages Out To Zero. Under

this situation, the laser works in a multimode CW state. Mode locking occurs when the phases of various longitude modes are synchronized such that the phase difference between any two neighboring modes is locked to a constant value, viz.

, If

we assume for simplicity that all modes have the same amplitude E0, the total intensity can be analytically calculated and presented as:

(1.1.3)

The intensity shows as a periodic function with period

, Which Is Just The

cavity round trip time. Under mode locking, the laser output is in the form of a pulse train with a repetition rate equals to ∆ν. Under the modulation in the cavity, the initiated pulse is shortened every time it passes through the resonator. This shortening process continues until the pulse

Hapter1: Introduction

becomes so short and its spectrum so wide that the pulse lengthening or spectrum narrowing mechanisms, such as the finite bandwidth of the gain, spring to action.

+1

represents the total bandwidth of all mode-locked modes, the pulse width is inversely related to the spectral bandwidth over which phases of various modes can be synchronized. In practice, the exact relationship between the pulse width and the gain bandwidth depends on the nature of the gain broadening. In rare-earth doped fibers, the fiber characters such as the birefringence also affect the pulse width. In general, there are two sorts of mode locking: active mode locking, and passive mode locking. Both methods have been used in fiber lasers to achieve ultra-short optical pulses.

Onservative/Dissipative Soliton Operation

Conservative soliton operation is an intrinsic feature of mode-locked fiber lasers with anomalous dispersion and has been intensively studied . In general, a pulse propagating in optical fibers is affected by both the group velocity dispersion (GVD) and the nonlinear optical Kerr effect. The GVD broadens the pulse in time domain, while the nonlinear self-phase modulation (SPM) broadens the pulse in frequency domain, which corresponds to narrowing the pulse in time domain. If these two effects on a pulse totally compensate with each other, it will maintain its pulse shape and pulse width during the propagation in the fiber i.e. forming a soliton. The formation of solitons in optical fibers is described by the NLSE .

In a fiber laser, after mode locking, an ultra-short pulse is firstly formed in the laser.

Hapter1: Introduction

If the peak power of the mode-locked pulse is strong, due to the nonlinear optical Kerr effect in the cavity, SPM occurs, which narrows the pulse width. If the strength of the SPM is strong enough that it can balance the pulse width broadening caused by the cavity dispersion, the pulse will propagate in the fiber without changing its pulse width. In fact, since the SPM is the pulse intensity dependent, as a pulse propagates in anomalous-dispersion fiber, it can adjust its intensity so that the effects induced by the optical Kerr effect and the anomalous GVD are automatically balanced. Namely, optical solitons is actually a generic property of mode-locked fiber lasers with anomalous cavity dispersion. Due to the existence of gain and loss, the soliton formation in a fiber laser is also governed by the Ginzburg-Landau equation (GLE) .

Current soliton theory considers the nonlinear systems as a conservative system or takes the energy import-export dynamics as a small perturbation to the conservative Therefore, more appropriate description of the nonlinear systems is compulsory.

Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatio-temporal dissipative systems due to mechanisms of self- organization . They can be considered as an extension of the classical soliton insight into the concrete dynamics in nonlinear systems without approximation.

In addition, we point out that the balance between gain and loss is certainly a necessary condition for the dissipative soliton formation in a fiber laser, but the

Hapter1: Introduction

balance between the linear gain and linear loss is not a sufficient condition. It is inappropriate to employ the linear gain-loss balance as the sole condition to judge that all solitons formed in a laser are dissipative solitons. For a dissipative soliton in the anomalous dispersion fiber lasers, the spectral filtering effect, which is actually frequency dependent loss, must be strong enough (like the case of soliton formation in the normal dispersion fiber lasers), but in the practice so far in all anomalous dispersion fiber lasers, due to the pulse peak clamping effect this effect did not appear, consequently it did not contribute to the pulse shaping, indicating that dissipative soliton was hardly observed in anomalous dispersion fiber lasers.

The situation is completely different in the normal dispersion regime, where no NLSE pulse shaping exists and the spectral filtering could become a significant effect. Therefore, dissipative soliton is always formed.

Soliton operation can be achieved in both actively and passively mode-locked fiber lasers through the soliton shaping of the mode-locked pulses. Once the soliton operation is achieved in a laser, the pulse characteristics are no longer determined by the mode locking mechanism but by the soliton shaping. Compared with a conventional mode-locked pulse, solitons have narrower pulse width and therefore higher peak power.

Active Laser Mode Locking

A typical arrangement for an integrated actively mode-locked fiber laser is shown in Figure 1.1. An active modulator is necessary in actively mode-locked lasers to modulate either the amplitude or the phase of the intracavity optical field at a

Hapter1: Introduction

frequency that equals to integer multiples of the cavity longitude mode spacing . As far as the modulation frequency is matched to the cavity length, active mode locking of fiber lasers can be easily obtained. With this mode locking technique, high repetition rate pulses with good noise performance can be generated. For instance, optical pulses of less than 6 ps duration generated at repetition rate to 40 GHz have been reported in an actively mode-locked erbium- doped fiber ring laser .

Figure 1.1: Typical experimental setup of an actively mode-locked fiber laser Without soliton shaping an actively mode-locked fiber laser produces mode-locked pulses of Gaussian pulse profile, whose width is determined by the laser gain

Hapter1: Introduction

bandwidth, and the modulation frequency and strength . For the EDFLs this is normally in the range of several tens of picoseconds. If the peak power of the pulses is strong enough, soliton shaping happens and shortens the mode-locked pulse as well as changes the pulse profile from a Gaussian shape to a Sech2 shape [49, 50]. However, comparing with the passively mode-locked fiber lasers, soliton operation cannot be easily achieved in actively mode-locked fiber lasers. The main reason for the difficulty is that harmonic mode locking is normally implemented. In a mode locked state too many mode-locked pulses are generated in the laser cavity, and they share the laser cavity energy. The energy of each mode locked pulse is therefore weak. And determined by the mode-locking technique, the mode-locked pulse width is initially broad, leading to that the peak power of the pulse achievable is also low. Therefore, only very weak nonlinear SPM could be actually generated by the pulses, which limits the strength of their soliton shaping. To obtain ultra narrow soliton pulses in the laser either strong pumping and/or low repetition rate operation are necessary. Actively mode-locked fiber lasers also suffer from the pulse drop-out problem , which could also be traced back to the gain competition between the pulses. In short, although actively mode-locked fiber lasers have the advantages such as high repetition rates, narrow line-width, they also have the drawbacks of broad pulse width, low peak power, and expensive as a modulator is required to be inserted in the cavity. Moreover, as actively mode locked pulses have only weak nonlinearity, they are impossible to be shaped into optical solitons that possess the born preponderance: good stability, low time

Hapter1: Introduction

jittering, short pulse width and high peak power. Fortunately, passive mode locking might overcome those shortcomings.

Passive Laser Mode Locking

The distinction between the actively and passively mode locking technique is that active mode-lockers are based on externally modulated media or device while passive mode-lockers are using an optical effect in a material without any time varying intervention. In this dissertation, we mainly discuss the passive elements because they are more fundamental. Passive mode lockers could be divided into two categories: artificial saturable absorber based on nonlinear light interference and real saturable absorber based on material’s nonlinear optical absorption property. Basically, nonlinear polarization rotation (NPR), semiconductor saturable absorber mirror (SESAM), carbon nano-tube and graphene based saturable absorbers are the mainly recognized passive mode lockers. Their operation principles can be generalized: making use of a nonlinear device whose response to the entering optical pulse is intensity dependant so that the optical pulse is shortened every time passing through it. Comparing with actively mode-locked fiber lasers, passively mode-locked fiber lasers can produce much shorter and more intense optical pulses as well as keep a relatively low component count simultaneously. As a result, soliton operation can be easily obtained after mode locking in a passively mode-locked fiber laser.

Type 1. Nonlinear polarization rotation mode-locked fiber laser The NPR mode locking technique exploits the nonlinear birefringence of the single

Hapter1: Introduction

mode optical fibers for the generation of an artificial saturable absorber effect in the laser cavity. The same effect of optical fibers was used previously for the polarization switching and optical pulse shaping [52, 53]. Hoffer et al. were the first who used the effect as a self-sustaining mechanism for passive mode locking of fiber lasers . However, owing to the short length and relatively high birefringence of the fiber used, mode locking of their fiber laser could not self-start.

The first successful demonstration of the effect for a self-started mode locking was shown by Matsas et al . The technique was then widely used for the self-started mode locking of the passively mode-locked ultra-short pulse fiber lasers. We use a configuration to illustrate the mode locking principle of the technique, as shown in Figure 1.2a. A piece of linearly birefringent optical fiber is placed between two linear polarizers. Light of arbitrary polarization incident to the setup is transferred into a linearly polarized light by the polarizer before the fiber. When the light propagates in the fiber, it splits into two components along the two polarization axes of the fiber, respectively. After passing through the fiber, generally the polarization of the light becomes elliptically polarized. If the light intensity is weak, then the ellipticity and azimuth of the light polarization are fully determined by the linear birefringence of the fiber and the orientation of the polarizer. However, if the intensity of the light is strong, the nonlinear effects of the fiber must be considered.

Due to the nonlinear optical Kerr effect of the fiber, the polarization of the light after passing through the fiber will depend on the light intensity. Furthermore, the transmission of the light through the analyzer will also depend on the light intensity.

Through appropriately selecting the orientations of the polarizer and the analyzer, a

Hapter1: Introduction

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. Saturable absorber mode locking is a well-known technique that has been extensively investigated and widely used . Therefore, through incorporating the setup in a fiber laser, an artificial saturable absorber effect is automatically generated, which results in mode locking of the laser. As the artificial saturable absorption is generated based on the optical Kerr effect, which has a recovery time in the order of several femto-seconds, from the laser physics point of view, ultra-short mode-locked pulses can be generated by the technique. Figure 1.2 shows the cavity configuration of a typical soliton fiber laser mode locked by the NPR technique. To achieve the self-started mode locking, a ring cavity configuration is normally used. A polarization independent isolator is inserted in the cavity to force the unidirectional operation of the ring. A polarizer is put in the cavity to set the polarization of light at the cavity position. As the cavity is a ring, the polarizer plays both the roles of the polarizer and the analyzer shown in Figure 1.2. To provide gain for the laser operation, a segment of erbium-doped fiber is incorporated in the cavity, which can be directly pumped either by a 980 nm or a 1480 nm laser diode. The pump light is coupled into the cavity by a wavelength-division-multiplexing (WDM) coupler, and the laser output at the 1550 nm is coupled out of the cavity by a fiber coupler. With above basic cavity components, if the cavity length is appropriately selected, mode locking of the laser could always self start. However, in the practice one or two polarization controllers are inserted in the cavity to fine-tune the linear phase delay

Hapter1: Introduction

between the two polarization components. For the purpose of accurate control on the linear cavity phase delay, frequently a set of two quarter-wave-plates is used. An advantage of using the set of quarter-wave plates is that changing the relative orientations between the quarter-wave plates could not only generate a linear phase delay with a value between 0 and 2π, but also introduces no influence on the other parts of the cavity.

Figure 1.2: (a) schematic of NPR mode locking principle; (b) configuration of a typical NPR mode locking fiber laser. Experimentally, erbium-doped fiber lasers of similar cavity configuration but with

Hapter1: Introduction

different cavity lengths, fiber birefringence and fiber dispersion properties have been setup and investigated [57, 58]. As far as the linear cavity phase delay is appropriately selected, and the pump power is set beyond the mode-locking threshold, self-started mode locking has always been obtained. Depending on the concrete laser parameters and operation condition, various soliton features have been experimentally observed. However, free-space elements must be incorporated into the laser cavity, which additionally introduces a considerable amount of loss, breaks the all-fiber integrated format and also makes the lasers environmentally unstable. In order to sidestep this drawback and fullfill the all-fiber requirement, a real passive mode locker should be used.

Type 2. Semiconductor saturable absorber mode-locked fiber laser Semiconductor saturable absorbers are also widely used for passively mode locking lasers. The mode locking technique bases on the mutual interactions of light with the laser gain medium and the saturable absorber. Therefore, properties of the saturable absorber such as the recovery time, saturable absorption strength play an important role on the mode locking quality. For mode-locked fiber lasers using this technique, the most suitable saturable absorbers are the specially designed semiconductor structures. One typical structure of SESAM was grown on n-GaAs (1 0 0) substrates by means of solid-source molecular beam epitaxy (MBE), as schematically shown in Figure 1.3a. The nominal structure has 25 pairs of GaAs (77 nm)/AlAs (91 nm) distributed Bragg reflector grown on the GaAs substrate, followed by GaAs (90 nm) and GaAs0.43P0.57 (10 nm) spacer layers, and five 6 nm Ga0.69In0.31As QWs. In between the QWs were four 22 nm GaAs0.43P0.57 barrier

Hapter1: Introduction

layers. On top of the last QW was a cap formed by 10 nm GaAs0.43P0.57 and 90 nm GaAs layers. The growth temperatures were 6000C for GaAs/AlAs DBRs, 5100C for GaInAs/GaAs(P) QWs/barriers, and 5900C for GaAs buffers and caps.

Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. Essential parameters of the SESAM are the recovery time, the modulation depth, the bandwidth, the saturation intensity and the non-saturable losses. Usually, the Bragg stack layer can be chosen to be either anti-resonant or resonant. The only difference is: SESAMs based on resonant Bragg stacks can have quite large modulation depths, but with the limited bandwidth of the resonant structure while anti-resonant SESAMs can have quite large bandwidths (e.g. 100 nm) but at the expense of a smaller modulation depth. A larger modulation depth can be obtained from an anti-resonant design at the expense of higher non-saturable losses. In solid state lasers where the single pass gain is low, the non-saturable losses of the SESAM are very crucial and must be as low as possible, but in fiber lasers where the single pass gain is much higher, non- saturable losses are less significant.

Hapter1: Introduction

Figure 1.3: (a) Schematic structure of the SESAM; (b) Schematic configuration of a fiber laser mode locked using a SESAM. Ideally, smaller recovery time is always desirable provided that mode locking operation is still ensured. If the pulse is chirped, pulse will develop asymmetric spectra at the impact of SESAMs when their recovery times are the same orders of magnitude as the pulse duration, indicating that recovery times of SESAM could strongly affect the pulse dynamics inside the cavity. Even larger recovery times can limit the obtainable pulse duration. Since the relaxation time due to the spontaneous photon emission in a semiconductor is at nanosecond scale, some

Hapter1: Introduction

precautions must be taken to shorten the relaxation time drastically. There are two technologies adopted to introduce lattice defects in the absorber layer for fast non- irradiative relaxation of the carriers: low-temperature molecular beam epitaxial growth and ion implantation, which could reduce the relaxation time from nano- second scale down to a few picoseconds.

SESAMs are well known to show a bi-temporal recovery time with the shortest time in the picosecond or sub-picosecond range. This bi-temporal recovery time is suitable for mode-locked lasers in that the short recovery time enables ultrafast pulses generation while the longer recovery time initiates mode-locking. For fast saturable absorbers with recovery times much faster than the pulse duration, the

The Recovery Time. To More Accurately Describe

the saturable absorption, the following rate equation was usually used:

∂

This differential equation can be numerically integrated to give q (t), and from q (t) the reflection from the SESAM can be determined as:

−

Here, l0 is the linear non-saturable loss. The saturation energy can be calculated as the product of the saturation fluency and the effective area on the SESAM. Figure 1.3b shows a typical laser configuration of an erbium-doped fiber soliton laser mode locked by using a SESAM. SESAM mode locked lasers have been extensively studied by a lot of researchers . As the relaxation times of semiconductor saturable absorbers are generally quite long, limited by the saturable absorber recovery time, the conventional mode-locked pulses have broad pulse width. However, with the soliton shaping, pulse width is no longer determined by the recovery time of the absorber, but by the soliton effect in the laser. In this case the pulse width could be significantly narrower than the absorber recovery time.

After a soliton is formed in the cavity, the function of the saturable absorber turns to stabilize the soliton through suppressing the background noise. Like the soliton operation of other fiber lasers, here again the saturable absorber initiates the mode locking in the lasers.

Apart from mode locking, semiconductor saturable absorbers were also used for the timing stabilization of harmonically mode-locked fiber lasers . It was suggested that the phase effects in semiconductor saturable absorber could lead to pulse repulsion, which provides self-organization of the pulse repetition rate.

However, for the applications, the recovery time and saturation energy density of the saturable absorbers must be carefully designed. Although appreciable success has been achieved with SESAMs, yet their

Hapter1: Introduction

fabrication involves MBE growth and treatment such as low temperature growth or post-growth ion-implantation is needed to reduce the relaxation time. Problems such as lattice-mismatch and poor thermal properties have seriously limited the restricting the potential applications of SESAMs as wideband saturable absorbers operating at other wavelengths. Consequently, there is always strong demand of seeking new materials to replace SESAMs, achieving the goal of perfect saturable absorbers with wider operation range, faster response time and lower cost.

Type 3. Carbon material based saturable absorber mode-locked fiber laser Nowadays, the ability to extensively control/tune photonic property of material through different physical, chemical and nano-technological approaches is now at the center of modern photonics. Researchers gradually diverted their research focuses from the previous III-V semiconductors to IV materials in that silicon is silicon is the second most abundant element after oxygen in Earth's crust while carbon is the materia prima for life and the basis of all organic chemistry. Due to the flexibility of its bonding, carbon-based materials show an unlimited amount of different structures with an equally large variety of optical properties. These optical properties are, in great part, the result of the dimensionality of these structures.

Among systems with only carbon atoms, graphene—a two-dimensional allotrope of carbon—plays a central role since it is the basis for the understanding of the electronic/optical properties in other allotropes. Graphene is made out of carbon atoms arranged on a honeycomb structure made out of hexagons, also well known as chicken-wire array, as shown in Figure 1.4, and can be thought of as composed

Hapter1: Introduction

of view, fullerenes are molecules where graphene layer are arranged spherically and are zero-dimensional with discrete energy levels; carbon nano-tubes are produced by rolling graphene along a given direction and reconnecting the carbon bonds and hence carbon nano-tubes have only hexagons and can be thought of as one- dimensional object; graphite is generated by stacking graphene layers that are weakly coupled by van der Waals forces and can be recognized as three- dimensional allotrope of carbon. Although graphene is the parent for all these different allotropes, it was only recently isolated by A. K. Geim and co-workers at .

Hapter1: Introduction

Figure 1.4: graphene, the parent of all graphitic forms [70, 71]. Since fullerene, graphene, carbon nano-tubes and graphite have different dimensionalities from 0-D to 3-D, they show distinctive electronic band structures, respectively, leading to unique optical property. Briefly, fullerene was experimentally verified to have optical limiting property (or reversed saturable absorption) while graphite completely absorbs any light with trivial features, indicating that both fullerene and graphite might have little potential applications as saturable absorbers to mode lock lasers. A single wall carbon nanotube (SWCNT) is a hexagonal network of carbon atoms rolled into a cylinder with each end capped with half of a fullerene molecule [72, 73]. The diameter of SWCNTs is normally

Hapter1: Introduction

close to one nanometre while their lengths can be up to orders of centimetres. The chirality of SWNTs is defined by a single vector called chiral vector.

+

, where, a1 and a2 are unit vectors of the hexagonal lattice, as shown in Figure 1.5. This vector connects two crystallographically equivalent sites on a two-dimensional graphene sheet and different chiral vectors are differentiated using two integers (n, m). Depending on the value of the chiral vector, SWCNTs are classified as armchair, zigzag and chiral. Armchair type of SWCNTs corresponds to the case where n = m while zigzag type satisfy either n = 0 or m = 0. All other cases correspond to chiral type. A SWCNT may be semiconducting or metallic depending on its chirality. Metallic nano-tube arises when n–m is an integer multiple of three while for all other arrangements of (n, m) the corresponding nano- tubes exhibit semiconductor properties. Because current synthesis methods for SWCNTs cannot yield accurately controlled and desired diameter or chirality, it is a usually mixer of both semiconductor and metallic tubes. Statistically, 1/3 of them are metallic nano-tubes and the others are semiconducting nano-tubes.

Hapter1: Introduction

Figure 1.5: illustration of chiral vector. The (n, m) nanotube naming scheme can be thought of as a vector (Ch) in an infinite graphene sheet that describes how to “roll up” the graphene sheet to make the nanotube, T denotes the tube axis, and a1 and a2 are the unit vectors of graphene in real space .

Semiconducting SWCNTs are direct band-gap materials having series of van-Hove singularities in the density of states (Figure 1.6) . The band gap of SWCNTs depends on the tube diameter , by growing of SWCNTs with a proper diameter distribution, it is possible to set the absorption peak positions in a broad spectral range between visible and near infrared . Figure 1.6 illustrated that SWCNTs

Hapter1: Introduction

have different absorption peaks. It should be noted that, this spectral tuning capability for SWNTs growing is very useful for the operation of saturable absorber devices.

Figure 1.6: (a) & (b) schematic illustration of energy band structure of single wall carbon nano-tube with different diameters and chiralities ; (c) graphene’s energy band structureand photon absorption.

By setting the absorption peak of SWNTs-based saturable absorbers coincident at

Hapter1: Introduction

the lasers’ operating wavelengths, passive mode-locking operation has been achieved for a wide range of wavelengths between 1 µm and 1.55 µm for both solid state lasers and fiber lasers . It has been shown that SWCNT mode lockers have the advantages such as intrinsically ultrafast recovery time, large saturable absorption, easy to fabricate, and low cost. In particular, as SWCNTs are direct- bandgap materials with a gap that depends on the nano-tubes’ diameter and chirality, through mixing SWCNTs with different diameters, a broadband saturable absorption mode locker could be made. A wideband wavelength tunable erbium- doped fiber laser mode locked with SWCNTs was experimentally demonstrated . However, the broadband SWCNT mode locker suffers intrinsic drawbacks: SWCNTs with a certain diameter only contribute to the saturable absorption of a particular wavelength of light, and SWCNTs tend to form bundles that finish up as scattering sites. Therefore, coexistence of SWCNTs with different diameters introduces extra linear losses to the mode locker, making mode locking of a laser difficult to achieve.

Quite recently, we experimentally found that these drawbacks could be circumvented if graphene is used as a broadband saturable absorber. Unlike the conventional semiconductor saturable absorbers, the energy band diagram of graphene has zero band-gap and a linear dispersion relation. These unique energy band properties combined with the Pauli blocking principle renders graphene a full band ultrafast saturable absorber . Because of the Pauli Exclusion Principle, when pumping of electrons in the excited state is quicker than the rate at which they relax, the absorption saturates. Figure 1.6c shows a schematic

Hapter1: Introduction

illustration of the energy band structure and photon absorption of graphene. Li et al. had experimentally shown that atomic layer graphene could absorb a considerable amount of infrared light without any bandwidth limitation . Due to interactions between electrons and lattice vibration modes of the carbon atoms, or mutual interactions among electrons, graphene could absorb a considerable amount of infrared light without any bandwidth limitation. Through the absorption of photon with energy hv and energy transference from photon towards electron,

Lower Than Fermi Energy Can Be

excited to its corresponding conduction band with energy

Higher Than Fermi

energy. But such absorption is photon number dependent. When the photons number is low, photons could be continuously depleted through the excitation of the electrons from valence band to conduction band. When the photons number is large enough, owing to the Pauli blocking principle, the newly generated carriers fill the valence bands, preventing further excitation of electrons at valence band and thus allowing photons transmitted without absorption, interpreting graphene’s wide-band saturable absorption. By the virtue of graphene’s broadband nonlinear optical property, graphene mode locked fiber lasers at 1 µm or 1.55 µm with widely tunable central wavelength were successfully achieved.

Otivation And Objectives

Due to their intrinsic stability in propagation, optical solitons have been proposed as information carriers for the long-distance optic fiber communications. Self- starting passively mode-locked erbium-doped fiber lasers, as attractive sources for ultrashort soliton pulses for their simplicity, tunability and ultrashort pulse

Hapter1: Introduction

operation, have been intensively studied. Soliton operation is a natural consequence of mode-locked fiber lasers in the anomalous cavity dispersion regime under strong pumping. Due to the balance effect between the GVD and SPM, soliton operation is ready to be obtained after the mode locking of the fiber laser. However, as in a fiber laser, besides the optical fibers, there also exist other components e.g. the gain medium and the output coupler, whose existence further affects the dynamics of the formed solitons. Therefore, compared with the solitons formed in single-mode fibers, the solitons obtained in a fiber laser are of new characteristics, which required further investigations. However, a real long-distance soliton fiber optic communication system also involves in the soliton losses and periodic amplifications. Therefore, the dynamics of solitons in the system is no longer described by the NLSE, which is integral and only describes the conservative systems, but by the GLE, where soliton is formed not only as a result of the balanced interaction between the fiber Kerr nonlinearity and dispersion, but also as a result of the balance between fiber losses and gain generated by the fiber amplifiers. As a soliton propagating in the system periodically experiences loss and amplification, dispersive waves with discrete spectra are generated, which resonantly draw energy from the soliton. A fiber laser is also periodical gain-and- loss system, in which, an optical pulse periodically experience the amplification of the fiber amplifier and the output loss. In this sense, a passively mode-locked soliton fiber ring laser can be regarded as a miniature of a soliton optic fiber communication system. Study on the soliton propagation and interaction in the laser cavity therefore gives a direct insight into the soliton interactions in the long-

Hapter1: Introduction

locked fiber lasers has been extensively investigated . Generally, the soliton formation in optical fibers is governed by the GLE, and the soliton pulses have a single-peak bell-shape intensity profile. Although the fiber birefringence always existed, due to the polarization limitation from NPR mode locking technique, the formed solitons had fixed polarization state once they passed the polarizer. Consequently, the polarization states of solitons were unable to evolve under the actions of fiber birefringence and the formed solitons could only be termed as scalar solitons. Extensive theoretical and experimental investigations on scalar solitons in NPR mode-locked fiber lasers have been carried out including soliton bunching, stable randomly spaced soliton distribution, bound scalar solitons, long range or direct interaction of scalar solitons, gain-guided or dissipative scalar solitons, modulation instability of scalar solitons, period-doubling scalar solitons and multi-pulse formation mechanism of scalar solitons . Although the basic principles of the operation of a scalar soliton fiber laser are generally understood, many features of the vector soliton laser still remain uninvestigated.

To establish the vector soliton emission in a fiber laser, any polarization dependent element must be removed so that the polarization states of the formed soliton could freely evolve. In contrast with scalar soliton, due to additional polarization freedom, vector soliton manifested richer and more fascinating features than scalar soliton.

How do the two-orthogonal polarizations of the vector soliton interact together and bound together? How does the cavity birefringence influence the vector soliton dynamics? What are the cavity parameters for the generation of polarization

Hapter1: Introduction

rotating or locked vector soliton? In normal dispersion cavity, could the dissipative nature affect the polarization state of dissipative vector soliton? How about the interaction forces among the conservative/dissipative vector soliton? Could the two-orthogonal polarizations of the vector soliton show completely different soliton features? For example, one polarization was fundamental soliton while the other polarization was high-order (bound) soliton.

In the field of optical soliton, although many soliton theoretists had predicted the existence of dark soliton family including vector dark-dark soliton, vector bright- dark soliton and domain wall soliton, those patterns widely existing in other physics systems, no direct experimental evidences were provided. So, it should be fundamentally interesting to study the dynamics of those novel dark solitons. All these questions have so far not been clearly addressed. To clarify them is not only important for understanding the vector soliton in fiber lasers, but also potentially useful for the future application of those novel vector solitons in the ultra-high-bit- Another interesting topic is whether it is achievable to extend the family of saturable absorbers for passively mode locking. Although SESAMs were widely require complex and costly clean-room-based fabrication systems , an additional substrate removal process is needed in some cases; high-energy heavy-ion implantation required to introduce defect sites in order to reduce the device recovery time (typically a few nanoseconds) to the picosecond regime, the reflection properties of SESAM limted the structure of fiber lasers, and low optical

Hapter1: Introduction

damage threshold. Quite recently, by the virtue of the carbon material based saturable absorption property, researchers have successfully produced a new type of effective saturable absorber, leading to the demonstration of mode-locked pico- or subpicosecond erbium-doped fiber (EDF) lasers. SWCNT films onto flat glass substrates, mirror substrates, or end facets of optical fibers were fabricated as the new saturable absorber devices . But the non-uniform chiral properties of SWNTs present inherent problems for precise control of the properties of the saturable absorber. Furthermore, the emergence of bundled and entangled SWNTs, catalyst particles, and the formation of bubbles induced high nonsaturable losses in the cavity. Under large energy ultrashort pulses multi-photon effect induced oxidation occurs, which degrades the long term stability of the saturable absorbers.

In order to overcome the above disadvantages, graphene, a single two-dimensional atomic layer of carbon atom arranged in a hexagonal lattice, was successfully veried as another effective saturable absorbers.

solitons in fiber lasers. In experiment, an erbium-doped fiber ring laser will be intentionally designed for experimental investigation. In theory, the pulse propagation and vector soliton formation in a fiber laser will be investigated.

Numerical simulations on the laser system will also be carried out to get a better understanding on the operation of the laser. In particular, various types of vector solitons including: high-order vector bright-bright soliton, vector dark-dark soliton, vector bright-dark soliton and domain wall soliton were firstly experimentally discovered in our fiber lasers. Moreover, to extend the family of saturable

Hapter1: Introduction

absorbers, we successfully fabricated graphene based saturable absorber which has the widest operation range, fastest response time and polarization insensitive optical properties, paving the way to studying the vector soliton in graphene mode- locked fiber lasers.

Ain Contributions Of The Research

First of all, fiber lasers passively mode-locked by SESAMs have been built up; various operation features of the laser have been examined. Different operation regimes of the lasers, from purely anomalous to normal dispersion, have been obtained and investigated. Vector soliton operations, as an intrinsic feature of the fiber laser have been achieved and comprehensively studied.

In an all-anomalous dispersion cavity with weak birefringence, one novel type of spectral sideband generation on the soliton spectra of the phase locked vector solitons in a passively mode-locked fiber ring laser was experimentally and numerically identified. The polarization resolved study on the soliton spectrum demonstrated that coherence energy exchange between the two orthogonal polarization components of the vector solitons accounted for the generation of such new spectral sidebands. When the cavity birefringence was strong enough, we experimentally and numerically found that the induced temporal solitons were formed by the XPM between the two orthogonal polarization components of the birefringence laser, and the induced solitons could either have the same or different soliton frequency to the inducing soliton. As the induced solitons always have the same group velocity as that of the inducing soliton, they form vector solitons in the laser. To our knowledge, this is the first experimental observation of temporal

Hapter1: Introduction

induced solitons. Moreover, we also study the effect of XPM on the vector solitons including the induced vector solitons, trapping of vector solitons both in purely negative dispersion cavity and dispersion managed cavity with net positive dispersion. Under relatively higher pumping and sufficiently weak cavity birefringence, a new type of high order phase locked vector soliton in a passively mode-locked fiber laser was experimentally observed and numerically verified.

The high order vector soliton is characterized by that its two orthogonal polarization components are phase locked, while the stronger polarization component is a single hump pulse, the weaker component has a double-humped structure with 180° phase difference between the humps. Our experimental result firstly confirmed the theoretical prediction on the high order phase locked vector solitons in birefringent dispersive media. In normal dispersion cavity, dissipative vector solitons (DVSs) have been experimentally demonstrated in a dispersion- managed fiber laser passively mode locked by a SESAM. It was found that despite of their large frequency chirp of the gain-guided solitons, polarization rotating and polarization locked DVSs could still be formed. In addition, formation of multiple DVSs with identical soliton parameters and stable harmonic DVSs mode-locking are also experimentally obtained. Except the stable existence of vector solitons, group interactions of vector solitons have been both experimentally and numerically observed; our experimental observation shows that two groups of vector soliton traveling at different group velocities because their polarization states are orthogonal to each other. Correspondingly, they collide with each other endlessly. Polarization rotating and locking of DVSs have been both

Hapter1: Introduction

experimentally and numerically obtained in either a dispersion-managed or purely normal dispersion fiber laser cavity. The period of DVSs polarization rotation could be still locked to integer multiple of the cavity roundtrip. We have also experimentally shown than despite of the existence of the laser gain competition, the angle of polarization ellipse orientation between two sets of DVSs can be varied from 00 to 900 because larger chirp and the broader pulse separation could partially counteract the influence of laser gain competition. Numerical simulations have confirmed such polarization rotation. If the cavity birefringence was further increased and therefore an artificial birefringence filter was deliberately introduced into the cavity, multi-wavelength dissipative solitons in an all normal dispersion fiber laser passively mode-locked with SESAM were firstly experimentally observed. Depending on the strength of the cavity birefringence, stable single-, dual- and triple-wavelength DVSs can be formed in the laser. The multi- wavelength soliton operation of the laser was experimentally investigated, and the formation mechanisms of the multi-wavelength DVSs are discussed.

Secondly, in the non-mode-locking regime where SESAM is replaced with a polarization dependent isolator, scalar dark soliton emission in a fiber laser was firstly observed. We anticipated that within a narrow operation regime, the mode locking behavior could be suppressed and the fiber laser could be operated in the non-mode-locking regime. Correspondingly, dark pulse rather than bright pulse was established. Through nonlinearity accumulation and continuous pulse shaping, eventually, dark solitons could be formed. Moreover, if a weak saturable absorber instead of the polarization dependent isolator was used, we could observe the

Hapter1: Introduction

vector dark-dark soliton and even the trapping of vector dark-dark soliton under moderate cavity birefringence. Numerical simulations confirmed the above experimental observations, and further proved that the observed dark soliton could be a genetic feature of NLSE. In both the normal and anomalous dispersion cavity, optical domain walls, characterized by topological structures separating different components (different polarization or wavelength), were experimentally observed in a fiber laser. It is noticed that the optical domain walls are irrelative to the cavity dispersion but dependent on the cavity birefringence. Particularly, similar to the bright-bright vector soliton, when the cavity is weakly birefringent, the two polarization components of the optical domain wall solitons are coherently coupled and phased locked. Moreover, they could be interpreted as dark-bright vector soliton, which is a special case of domain wall solitons. When the cavity is largely birefringent, the two components of the optical domain wall solitons are incoherently coupled and the wall widths are strongly related to the cavity birefringence. The numerical simulation could well reproduce the above experimental observations.

Lastly, we proposed and achieved another novel saturable absorber made of graphene. Although it is a zero bandgap semiconductor, like the SWCNTs, graphene also holds the feature of saturable absorption due to Pauli blocking effect.

As a single 2D atomic layer of carbon atom arranged in a hexagonal lattice, graphene based saturable absorbers exibit superior advantages including: (i) Controllable saturable absorption strength through controlling the number of graphene layers or chemical functionalization; (ii) Super broadband saturable

Hapter1: Introduction

absorption; (iii) Ultrafast saturation recovery time; (iv) Easy to be fabricated. Correspondingly, graphene was fabricated as a wideband saturable absorber for ultrafast mode-locking fiber lasers. The polarization indepdence of graphene’s saturable absorption property also guaranteed graphene as a polarization insenstive time of graphene also facilitates ultrashort pulse generation. The optical modulation depth can be tuned in a wide range by using single to multilayer graphene or doping/intercalating with other materials. We believe those features of the graphene can also find other interesting photonic applications.

The Dissertation Is Organized As Follows:

Chapter 1 serves as an introduction of the research. Chapter 2 presents the experimental and theoretical background of the ultra-short in optical fibers. Major fiber characteristics that affect the pulse evolution in an optical fiber including the GVD, fiber nonlinearity, and fiber birefringence are discussed. As an important factor, the gain and gain dispersion in erbium-doped fibers are also introduced. Equations that describe the pulse evolution in a real gain and loss fiber system, e.g. a fiber laser, are derived out step by step in Chapter 2.

Chapter 3 demonstrates the experimental and numerical investigations on the mutual interaction forces among the two orthogonal polarizations of a vector soliton. Specifically, how the Four-wave mixing (also called as coherent energy exchange) and cross polarization coupling influence the dynamics of vector soliton.

Hapter1: Introduction

We found that a new kind of spectral sideband could be generated due to the coherent energy exchange between these two polarizations of vector solitons and a weak soliton in one polarization of a vector soliton could be induced by its perpendicular strong component owing to the cross polarization coupling.

Chapter 4 introduces one novel vector soliton: high order vector soliton. The high order vector soliton is characterized by that its two orthogonal polarization components are phase locked, and while the stronger polarization component is a single hump pulse, the weaker component has a double-humped structure with 180° phase difference between the humps. Our experimental result firstly confirmed the theoretical predictions on the high order phase locked vector solitons in birefringent dispersive media.

Chapter 5 addresses the generation of dissipative vector soliton in all-normal- dispersion cavity or a dispersion managed cavity with net normal dispersion. Different from the conventional vector solitons which only survives in the anomalous dispersion regime, the prerequisite of dissipative vector soliton generation is a natural balance among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, laser gain saturation and gain bandwidth filtering. Both polarization-locked and rotating dissipative vector solitons have been obtained. In the end, multi-wavelength dissipative soliton is also generated due to the existence of an artificial birefringent filter when cavity birefringence is very strong.

Chapter 6 provides the observation of dark soliton in a fiber laser. Both scalar and vector dark soliton have been experimentally and numerically obtained. Dark

Hapter1: Introduction

solitons in all normal dispersion and dispersion managed cavity possess different soliton dynamics, which will be experimentally and numerically studied. Trapping of vector dark solitons will also be studied under very strong cavity birefringence.

Chapter 7 discusses the experimental and numerical observation of two types of domain wall solitons: polarization domain wall and dual wavelength domain wall. When cavity birefringence was small enough, dark-bright and dark-dark type polarization domain wall soliton could be formed; while in large cavity birefringence, incoherent interaction of two wavelengths with the same polarizations could lead to the generation of dual wavelength domain wall.

Chapter 8 demonstrates a novel type of saturable absorber: graphene and its application for ultra-fast mode locked lasers. Graphene as an ideal two-dimension nano-material has unique optical and photonic properties. We show that graphene possesses wavelength independent ultrafast saturable absorption due to Pauli blocking, which can be exploited as a super broadband saturable absorber for passive mode locking of lasers. Various graphene-based saturable absorbers for ultrafast lasers have summarized in Chapter 8.

Chapter 9 concludes the dissertation, summarizes the achievements of the research and gives recommendations for future work. Chapter 2: Theory of vector soliton propagation in a fiber laser Chapter 2. Theory of vector soliton propagation in

A Fiber Laser

The envelope of a light wave in a single-mode optical fiber is deformed by the dispersion (variation of the group velocity as a function of the frequency) and nonlinearity (variation of the phase velocity as a function of the wave intensity) of the optical fibers. The dispersive property of the light wave envelope is decided by the GVD while the nonlinear properties of the light wave envelope are characterized by a combination of Kerr effect (an effect of the increase in the refractive index n in proportion to the light intensity). In a fiber laser, the gain and gain dispersion introduced by the erbium-doped fiber amplifier will also complicate the pulse propagation. However, “single” mode optical fiber actually supports two orthogonal polarization modes while the term “single” mode refers only to the transverse profile. The polarization modes could only be completely degenerate provided that optical fiber was perfectly isotropic. In reality, manufacturing imperfections, externally applied stress or bending easily breaks down the degeneracy between the modes. Thus fiber supports two orthogonally-polarized modes with differing propagation constants; i.e., fiber is birefringent . The difference in phase velocities of the two modes causes the polarization state of a pulse to evolve as it propagates. Fortunately, although being exposed to mediate fiber birefringence, theoretical analysis showed that the orthogonally-polarized components of the soliton could stick together and propagate as one non-dissipative entity through shift their center frequency slightly, forming the state of vector soliton [40, 109].

The fiber birefringence induced polarization evolution is very crucial aspect Chapter 2: Theory of vector soliton propagation in a fiber laser that must be taken into account when analyzing the vector soliton propagation operation in fiber lasers, it is necessary to firstly consider the theory of wave propagation in fibers in the presence of these effects.

The objective of Chapter 2 is to derive the basic equations that govern the vector soliton pulse propagation in a fiber laser cavity. Section 2.1 introduces the group velocity dispersion in fibers and its effect on the pulse propagation.

Section 2.2 provides the theory of nonlinear pulse propagation in optical fibers. In Section 2.2, the NLSE is derived and the fundamental soliton solution of this equation is given. Section 2.3 discusses the gain property introduced by the erbium-doped fibers. By including the gain and gain dispersion, the pulse propagation is then described by the Ginzburg-Landau equation. Section 2.4 considered the polarization characters and derivation of the coupled Ginzburg- Landau equations, paradigm equation for vector soliton generation. Finally, various vector solitons including vector conservative/dissipative bright-bright soliton, vector dark-dark soliton, vector dark-bright soliton and domain wall solitons will be summarized in the Section 2.5.

Inear Pulse Propagation In Optical Fibers

When an optical pulse propagates within an optical fiber, it becomes distorted due to intramodal (i.e. chromatic) dispersion and intermodal delay effects . The distortion effects can be explained by analyzing the group velocities of the guided modes. Group velocity is the speed at which energy in a particular mode travels along the fiber. For an optical signal that normally contains a finite number of frequency modes, a phenomenon known as the group velocity Chapter 2: Theory of vector soliton propagation in a fiber laser dispersion (GVD) occurs. GVD also referred to as the chromatic dispersion, exhibit as pulse spreading that occurs within a single-mode fiber (SMF), which arises due to the finite spectral width of the optical signal. This phenomenon is called GVD since the dispersion is a result of the group velocity being a function of the wavelength.

Since the refractive index of the medium is frequency dependent, which can be

N

, chromatic dispersion always exists with the propagation of light in fibers. For CW light or pulse with big pulsewidth (usually T0 > 100 ps), this dispersion can be ignored. But in the case of short pulses, especially for the ultrashort pulses as those presented in this dissertation, fiber dispersion plays a critical role because different spectral components associated with the pulse

，Where C Is The Speed Of Light In

vaccum. Because of dispersion, a pulse will broaden with the propagation in On a fundamental level, the origin of dispersion is related to the characteristic resonance frequencies at which the medium absorbs the electromagnetic radiation through oscillations of bound electrons. The refractive index can be well approximated by the Sellmeier equation at frequencies far from the

N

, (2.1.1) where ωj is the resonance frequency and Bj is the strength of jth resonance. The sum in Eq. (2.1.1) extends over all material resonances that contribute to the Chapter 2: Theory of vector soliton propagation in a fiber laser frequency range of interest. For silica, a three-term Sellmeier equation is typically used, accounting for the resonances in the ultraviolet and infrared.

(2.1.2)

Mathematically, the effects of fiber dispersion are accounted for by expending the mode-propagation constant β in a Taylor series about the frequency ω0 at

(2.1.4)

The parameters β1 and β2 are related to the refractive index n and its derivatives

(2.1.6)

where ng is the group index and vg is the group velocity. Physically speaking, the envelope of an optical pulse moves at the group velocity while parameter β2 represents dispersion of the group velocity and is responsible for pulse broadening. β2 is referred to as the group velocity dispersion (GVD) parameter.

Figure 2.1 shows how β2 vary with wavelength λ in fused silica . Chapter 2: Theory of vector soliton propagation in a fiber laser Figure 2.1: Group velocity dispersion β2 for fused silica as a function of wavelength (After Ref. ).

In the figure, β2 vanishes at the wavelength of about 1.27 µm, which is referred to as the zero-dispersion wavelength and is denoted as λD. For wavelengths such that λ < λD, the fiber is said to exhibit normal dispersion as β2 > 0. In the normal dispersion regime, the higher frequency (blue shifted) components of an optical pulse travel slower than lower frequency (red-shifted) components. By contrast, the fiber shows anomalous dispersion as β2 < 0 when the light wavelength exceeds the zero-dispersion wavelength (λ > λD). And in this regime, the blue-shifted components of an optical pulse travel faster than red- shifted components. The anomalous-dispersion regime is of considerable interest for the study of solitons. Because it is in this regime that optical fibers support solitons through a balance between the dispersive and nonlinear effects.

Pulse Propagation In Dispersive Media

Basically, like all electromagnetic phenomena, the propagation of optical fields in the media is governed by Maxwell’s equations. Then deriving from the Chapter 2: Theory of vector soliton propagation in a fiber laser Maxwell’s equations, one can get the fundamental wave equation

(2.1.7)

where E and P are electric field vector and the induced electric polarization, respectively. And c is the speed of light in vacuum and has a relation

,

where µ0 is the vacuum permeability and ε0 is the vacuum permittivity. Before solving the wave equation, two simplifications are made. First, by just considering linear polarized light propagation in the media, the vectors in Eq.

(2.1.7) can be replaced by scalars. Second, by ignoring the nonlinear effects temporarily in this section, Eq. (2.1.7) is linear and can be easily converted to the frequency domain.

Using the two simplifications and also adopting the operation

∇

, (2.1.8) One can easily get the wave propagating equation in frequency domain:

(2.1.10)

In addition, the paraxial approximation is also implied in Eq. (2.1.9), which is applicable and reasonable in laser optics by replace r with z along which the light propagates.

Chapter 2: Theory of vector soliton propagation in a fiber laser

,Z

is assumed to be a slowly varying function of z.

(2.1.11)

Therefore, the wave equation (2.1.9) can be further simplified as:

=

is the propagation constant varying with frequencyω. In

,Z

is assumed to be a slowly varying function of z. Now let us go on to study the propagation equation of an optical pulse. The electric field of an optical pulse can be generally written as:

, (2.1.13)

E (z, t) is the pulse envelop, which contains both the amplitude and phase information. ω0 is the central frequency of the pulse and

Is The

propagation constant at the frequency of ω0. Correspondingly, in the frequency domain, an optical pulse can be described by the Fourier transform of (

,Z

obeys the conventional transmission relation of Eq. (2.1.12) If the pulse width is big enough so that the dispersion effect can be ignored, each spectral component of the pulse at ω can be well approximated Chapter 2: Theory of vector soliton propagation in a fiber laser

Fraction Of

ω , one must account for the fact that each spectral component of

Β

. In this condition, by substituting Eq. (2.1.14) and Eq. (2.1.3) into Eq. (2.1.12), one can finally obtain an equation

(2.1.15)

Since a perturbation induced by the amplitude of the field itself usually exists (i.e. n = n (ω) + ∆n), the factor δβ is imported to denote the perturbation it induced on β (i.e. β = β (ω) + δβ).

At this point, one can go back to the time domain by taking inverse Fourier transform of Eq. (2.1.15), and then obtain the propagation equation for (

(2.1.16)

In practice, a synchronous coordinate system moving at the group velocity vg is always used to make Eq. (2.1.16) even simpler in form. The transformation between the original and synchronous coordinate systems is as:

(2.1.17)

Finally, the propagation equation of an optical pulse can be reformed as: Chapter 2: Theory of vector soliton propagation in a fiber laser

(2.1.18)

Eq. (2.1.18) describes the propagation of a short pulse in dispersive media. It will be further used to acquire pulse propagation in single-mode fibers in the next section of Chapter 2.

Based on Eq. (2.1.18), one can get some features of pulse propagation in

(2.1.19)

Therefore, any envelope function can propagate any distance without distortion.

Β

, then the pulse shapes become distorted as they propagate in the media. In practice, GVD always plays an important role in pulse propagation, especially in ultrashort pulse propagation.

Nonlinear Effects In Optical Fibers

Besides the frequency dependence, the refractive index of the optical media is always intensity dependent. Hence when the peak power of a pulse is very strong, an interesting manifestation of that occurs through self–phase modulation (SPM) arising from the Kerr nonlinearity. This phenomenon leads to the spectral broadening of an optical pulse .

Chapter 2: Theory of vector soliton propagation in a fiber laser On a fundamental level, the origin of nonlinearity is related to the nonlinear

(2.2.1)

where, ε0 is the vacuum permittivity, and χ (j) (j = 1, 2…) is the jth order susceptibility. The linear susceptibility χ (1) represents the dominant contribution to P, whose effects are included through the refractive index n. The second- order susceptibility χ (2) is caused by the asymmetry in molecular structure, which is responsible for nonlinear effects such as the second-harmonic generation and sum-frequency generation. Since SiO2 has a symmetric molecular structure, χ (2) vanishes for silica glasses. Therefore, in the case of a silica fiber, the lowest-order nonlinear effects originate from the third-order susceptibility χ (3). These third-order nonlinear effects include the third-order harmonic generation, four-wave mixing, and nonlinear refraction. However, as the third-harmonic generation and the four-wave mixing both needs strict phase matching condition, they are not efficient in single-mode fibers. Most of the nonlinear effects in optical fibers such as the optical Kerr effect, stimulated Raman and Brillouin scattering, originate from nonlinear refraction, which can automatically occur provided that the light intensity reach some thresholds. In its simplest form, the nonlinear refractive index can be written as:

N

is the linear part given by Eq. (2.1.1), |E|2 is the optical intensity inside the fiber, and n2 is the nonlinear-index coefficient determined by χ (3). Chapter 2: Theory of vector soliton propagation in a fiber laser Normally, n2 of optical fibers has a very small value, which is in the range of

×

. However, because the core of SMFs is very small, which is normally < 10 µm in diameter, and the light is confined to propagate in the fiber core with almost ignorable loss, a very long distance of light-matter interaction is thus possible, which in consequence makes the efficiency of the nonlinear optical process very high in optical fibers. The intensity dependence of the refractive index leads to a large number of interesting nonlinear effects.

The two most widely studied are self-phase modulation (SPM) and cross-phase modulation (XPM). Self-phase modulation refers to the self-induced phase shift experienced by an optical field during its propagation in optical fibers. Its magnitude can be obtained by noting that the phase of an optical field changes

K

, and L is the fiber length. The intensity-dependent nonlinear

Spm Is Responsible For The Spectral

broadening of ultrashort pulses and the formation of optical solitons in the anomalous-dispersion regime of fibers. Cross-phase modulation refers to the nonlinear phase shift of an optical field induced by another field having a different wavelength, direction, or polarization state. Its origin can be understood by noting for example that the

Ω , Polarized Along The Same Axis,

propagate simultaneously inside the fiber. The nonlinear phase shift introduced

Therefore In This Case The

total nonlinear phase shift for the field E1 would be

, (2.2.5)

The two terms on the right side are due to SPM and XPM, respectively. Among other things, XPM is responsible for the asymmetric spectral broadening of the co-propagating optical pulses. XPM also gives an anticipation of soliton interaction and the existence of paired solitons in fiber soliton lasers.

Nonlinear Schrödinger Equation

By including the nonlinear part of refractive index into δβ of Eq. (2.1.16)

=

. (2.2.6) When introducing the normalized time τ, distance ξ and wave envelope U by: Chapter 2: Theory of vector soliton propagation in a fiber laser

P

τ is the pulse width, Eq. (2.2.5) is then reduced to the standard NLSE:

(2.2.8)

Eq. (2.2.8) gives a simple description over the propagation of an optical pulse in nonlinear dispersive media, which is affected by both the GVD and nonlinear effects denoted respectively by the second and third term in the equation.

Fundamental Soliton

From Eq. (2.2.5) or (2.2.8), it is easy to understand that the propagation of an optical pulse is the combinative effect of both GVD and nonlinear effects. The nonlinear optical Kerr effect always generates a positive frequency chirp in the center part of a pulse, while the frequency chirp induced by the GVD can be either positive or negative depending on the sign of the GVD parameter. Since in anomalous dispersion regimes (β2 < 0), the fiber generates a negative linear frequency chirp, under the simultaneous action of the nonlinear Kerr effect, the frequency chirp on a pulse may be eliminated and no pulse distortion will happen as it propagating in the fiber. In fact, as the chirp induced by the nonlinear Kerr effect is the pulse intensity dependent, as a pulse with strong intensity propagates in anomalous-dispersion fiber, it can adjust its intensity so that the chirp induced by the two effects is automatically balanced. Namely, Chapter 2: Theory of vector soliton propagation in a fiber laser optical solitons is actually a generic property of fibers in the anomalous dispersion regime.

The fundamental soliton solution can be derived theoretically from NLSE by NSLE using the inverse scattering method , one can get the solution of

(2.2.9)

where α is the fiber loss. z0 represents the soliton period that is defined as:

=

. (2.2.10) The wave packet in Eq. (2.2.9) is referred to as the fundamental soliton because its shape does not change with propagation.

Gain Profile Of Erbium-Doped Fibers

Rare-earth doped fibers manufactured by doping the rare-earth ions in SMF Various doping ions are chosen to provide gain for light in different wavelength regions. Among them, erbium-doped fiber amplifiers (EDFAs) have attracted especial attention, as they have a broadband gain in the 1.55 µm optical fiber communication window.

By pumping the ions from their ground state to the exited states, population inversion could be achieved between the two energy levels of the ions. Then the Chapter 2: Theory of vector soliton propagation in a fiber laser stimulated emission happens and amplifies the light incident to the fiber whose frequency is within the gain profile. The optical spectrum of a fiber amplifier is mainly determined by the property of the dopants in the fiber. Based on the rate equations, the gain coefficient of an optical amplifier can be calculated

=Σ

, where σ is the transition cross-section, and N1 and N2 are the atomic densities in the two energy levels of the ions. In practice, there are many factors that affect the gain coefficient of doped fiber amplifiers. Different rare- earth doped fiber amplifier or even the fiber amplifiers doped with the same rare-earth ion but with different doping densities and fiber properties could have different gain profiles. The pumping scheme of the Er3+ ions can be regarded as a three-level system, which is schematically illustrated as Figure 2.2. Efficient pumping is possible using semiconductor lasers operating near 980 nm and 1480 nm wavelengths.

Figure 2.2: Energy levels of Er3+ ions. Generally, an amplifier is characterized by its small signal gain, gain bandwidth, gain saturation power and the noise figure. For a homogeneously broadened gain medium as Er3+ ions, the gain coefficient can be written as

, (2.3.1)

g0 is the small signal gain, ω is the frequency of the incident light, ω0 is the resonant frequency of the dopants, T2 is the dipole relaxation time of the dopants, P is the power of the incident light, and PS is the saturation power of the gain medium. The saturation power depends on the stimulated emission cross-section and the upper energy level lifetime of the dopants. In the case of three-level pumping scheme, it is also the pump strength dependent. In practice, the gain saturation can be negligible for most doped fiber amplifier for single pulse amplification, i.e. P/PS << 1. By neglecting the P/PS term, the gain

(2.3.2)

This equation shows that the gain is governed by a Lorentzian profile with the maximum value happening when the signal frequency coincides with the atomic transition frequency ω0. The gain bandwidth Ωg is defined as the full width at half maximum (FWHM) of the gain spectrum g (ω). For a Lorentzian profile, Ωg is inversely related to the dipole relaxation time T2 of the dopants.

Figure 2.3 shows a typical absorption and emission spectrum of erbium-doped fiber amplifiers . The shapes of the absorption and emission spectrum of erbium-doped fiber amplifiers can be associated with the energy-level transitions of Er3+ ions shown in Figure 2.2. Details of the spectrum also depend on the fiber composition and the co-dopants. Worthy of mentioning here, the gain of isolated Er3+ ions is homogenously broadened, which exhibit a Lorentzian profile. However, as Er3+ ions are doped into silica fiber, due to the Chapter 2: Theory of vector soliton propagation in a fiber laser interaction of the ions with the silica host and other co-dopants such as germania and alumina within the fiber core, their energy levels are no longer discrete, but consist of a number of closely spaced levels. As a result, the gain and absorption spectrum is broadened with a double peak structure as shown in Figure 2.3. For erbium-doped fiber, the gain bandwidth could reach approximately 40 nm, which can correspondingly support as short to 60 fs pulse.

But in practice, not the whole part of the gain can be efficiently used due to some effects such as the nonuniform gain profile, the fiber birefringence, and also the cavity effect in fiber lasers.

Figure 2.3: Typical absorption and emission spectra of the erbium-doped fibers (After Ref. ).

Ginzburg-Landau Equation

When an optical pulse propagates in the erbium-doped fibers, the effect of the light amplification must be considered. Generally, the interaction of the doped ions with the light is governed by the Maxwell-Bloch equations . However, in the case of that the pulse widths is larger comparing with the dipole Chapter 2: Theory of vector soliton propagation in a fiber laser relaxation time T2 of the Er3+ ions (T2 < 0.1 ps), 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 ω0, therefore, the frequency dependence of the gain can be

(2.3.3)

Without loss of the generality, assuming that the pulse carrier frequency coincides with the gain peak, the equation describing optical pulse propagation

(2.3.4)

where g is the peak gain coefficient, Ωg is the gain bandwidth. The time dependence of the peak gain coefficient varies with time due to the gain

, (2.3.5)

where g0 is the small signal gain, T1 is the population decay time, and Es is the saturation energy. As for the erbium doped ions T1 is in the time scale of 10 ms, if the pulse width is far narrower than it, which is normally the case, the T1 term is negligible during pulse amplification, and g (t) becomes:

(2.3.6)

Chapter 2: Theory of vector soliton propagation in a fiber laser Typical values of Es 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 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 may still saturate the gain and determines the saturated gain value.

In reality, one may also need to take into account the fiber loss (~0.2 dB/km) for long distance light propagation in fibers.

= 0 And G = 0, The Equation

reduces back to NLSE. Eq. (2.3.7) can be also written in a domainless form by making the

Soliton Solution

Different to the NLSE, which is integrable and has an exact soliton solution, the Chapter 2: Theory of vector soliton propagation in a fiber laser GLEQ is non-integrable. However, it is 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. When N = 1, fundamental soliton solution can be obtained in the case of a single eigen-value

 (2.3.11)

where p, q and Γ are constants, comparing with the case in un-doped fiber, due to the influence of the fiber gain, the solitons in the doped fibers become frequency chirped. And for a SMF of GVD β2, the peak power required to form

(2.3.12)

N > 1 corresponds to high-order solitons. However, although theoretically predicted, by now no high-order soliton has been ever experimentally observed to our knowledge due to its intrinsic instability.

Fiber Birefringence

In all the above processes, we have ignored the polarization characteristic of light in the fiber. Actually, even a SMF is not truly single mode because it can support two degenerate modes that polarized in two orthogonal directions.

Under ideal conditions, the two polarization states would not couple to each other. But in practice, all fibers exhibit some modal birefringence because of small departures from cylindrical symmetry due to random variations in core Chapter 2: Theory of vector soliton propagation in a fiber laser shape and/or stress-induced anisotropy along the fiber length. Mathematically, the strength of modal birefringence is defined as:

=

is the difference in the propagation constant, and nx and ny are the respective modal refractive indices for the two orthogonally polarized components. Here it is assumed the fiber is linearly birefringent, i.e. the fiber has two principle axes along which it is capable of maintaining the state of linear polarization of the incident light in the absence of nonlinear effects. The axis along which the mode index is smaller, namely the group velocity is larger for light propagating in this direction, is called the fast axis. And similarly, the axis with the larger mode index is called the slow axis. This assumption is ideally the case for polarization-maintaining fibers (PMF), where the built-in birefringence is made much larger than random changes occurring due to stress and core-shape variations.

For a given value of B, after propagating a length of L, the two polarization components of the incident light will gain a phase difference of

=

is called the beat length. From Eq. (2.4.2), it can be seen that the light varies its polarization periodically along the fiber. At a multiple distance of LB, a multiple phase difference of 2 π will be generated between the two polarization components, which mean that the light returns to its initial polarization state. From a viewpoint of physics, the Chapter 2: Theory of vector soliton propagation in a fiber laser two polarization modes exchange their powers as propagating in the fiber with a period of the beat length LB. Different single-mode fibers could have very different beat lengths. For a standard SMF, LB is in the range of several meters, while for the PMF, as a strong birefringence is deliberately built in the fibers, they could have a beat length of several millimeters.

When the nonlinear effects in optical fibers become dominant, a sufficiently intense optical field can induce nonlinear birefringence whose magnitude is intensity dependent. Consequently, the refractive index is described as

=

(j = x, y) (2.4.3)

Jn

is the linear part of the refractive index. And the nonlinear

, (2.4.4)

Normally one takes the value of n2 the same for the two polarizations, which is also practically just the case. In Eq. (2.4.4), the first term is responsible for SPM. The second term results in XPM since the nonlinear phase shift acquired by one polarization component dependent on the intensity of the other polarization component. The presence of this term induces a nonlinear coupling between the two polarization eigen-modes Ex and Ey.

Oupled Ginzburg-Landau Equations

The NLSE remains universal due to two reasons. The first reason is it predicts the behavior of the observed pulses and secondly it can be manually derived Chapter 2: Theory of vector soliton propagation in a fiber laser

(2.4.6)

The NLSE above only describe the propagation in one spatial dimension. Pulse propagation in a fibre laser needs to consider coupling between the two orthogonally polarized components. Therefore, a coupled version of the NLSE is required.

*

where Ax and Ay are the two normalized slowly varying pulse envelopes along

* Represent Their Conjugates And ∆Β Is The

wave-number difference. The NLSE can be further developed into the coupled GLEs, which describes the pulse propagation within the whole cavity including the amplification effect of the erbium doped fiber gain.



where, u and v are the normalized envelopes of the optical pulses along the two orthogonal polarized modes of the optical fiber. k″ is the second order dispersion coefficient; k′′′ is the third order dispersion coefficient and γ represents the nonlinearity of the fiber. g is the saturable gain coefficient of the fiber and Ωg represent the bandwidth of the laser gain.

(2.4.9)

Chapter 2: Theory of vector soliton propagation in a fiber laser In this thesis, a numerical approach is applied to understand the concept of the nonlinear pulse propagation in the soliton fiber laser. To this end the Coupled Ginzburg-Landau equations are numerically solved.

Procedure of numerically solve the Coupled Ginzburg-Landau

Equations

Nonlinear pulse propagation in optical fibers can be represented by the coupled

Step 1–Transform Into A Retarded Reference Frame

In practice, a coordinate system (so called “retarded frame”) moving at the

Mean Group- Velocity

gv (the average of the group-velocities of the two polarization components, which may differ from the mutual group-velocity

(2.4.11)

Chapter 2: Theory of vector soliton propagation in a fiber laser taken by the two polarization modes) is employed. The transformation/mapping between the original coordinate system and the retarded frame is defined as:

And Equations (2.4.12), (2.4.13), (2.4.14) And

(2.4.15) into equations (2.4.7) and (2.4.8) while ignoring fiber attenuation (α =

=

. Step 2–Eliminate the exponential components in the FWM terms

∂

Step 3–Take into account the amplification by the EDF The gain provided by the Erbium-doped fiber (EDF) and its gain dispersion need to be considered as gain medium is an indispensable part of a laser. Thus,

F In Equation

(2.4.21) account for the gain of the EDF. Whereas, the last term in equation

(2.4.19)

Chapter 2: Theory of vector soliton propagation in a fiber laser (2.4.21) is due to the gain dispersion of the EDF. gp (t) in equation is defined as :

P

where g0 refers to the small signal gain and Es is the saturation energy, which has a typical value of 1 µJ. Equations (2.4.20) and (2.4.21) symbolize the coupled Ginzburg-Landau equation (GLE).

Step 4–Symmetrized Split-Step Fourier Method

The above coupled Ginzburg-Landau equations (GLE) can be solved using the symmetrized Fourier split-step method. The procedures of transforming the

J

. Perform Fourier Transform on Equations (2.4.22) and (2.4.23), we

Can Then Be Obtained By Performing Inverse

Fourier Transform on equations (2.4.26) and (2.4.27).

In The Coupled

GLE, a so-called Circular Transformation is employed.

=

Substitute them into equations (2.4.28) and (2.4.29), we obtain:

Γ

(2.4.30) + (2.4.31) and divide the sum by 2, we obtain:

Γ

(2.4.30) – (2.4.31) and divide the difference by 2, we obtain:

Can Then Be Obtained By Performing Inverse

Circular Transform on equations (2.4.34) and (2.4.35).

Ector Solitons

Among all the features of vector soliton phenomena, probably, interactions of vector solitons are the most fascinating ones. Generally, the interaction forces could be divided into two types: coherent and incoherent interaction. Coherent interaction occurs when the nonlinear medium can respond to interference effects that take place when solitons overlap . This interaction depends on the relative phases of the interaction fields. To maximize the strength of coherent interaction, phase matching condition must be fulfilled. Corresponding, it only occurs when the nonlinear medium is weakly anisotropic or weakly birefringent.

However, incoherent interactions, on the other hand, occur when the relative phase between the soliton varies much faster than relaxation time of the nonlinear medium. It arises from the third order susceptibility: propagation refractive index of one light beam could be modified/modulated by another light beam. This process is phase insensitive and only determined by the relative strength of individual light beam.

I presented, in Chapter 3, the investigation of these two kinds of interactions among the two polarizations of a vector soliton. Specifically, the Four-wave- mixing effect (also known as coherent energy exchange) would be discussed in the Section 3.1. To obviously observe this effect, the cavity birefringence must be kept as weak as possible in order to fulfill the coherent interaction requirement. The Section 3.2 addressed the incoherent interaction, termed as

Hapter 3: Coherent And Incoherent Interaction

cross phase modulation (XPM) in fiber laser and reported the first experimental observation of induced vector soliton in a fiber laser.

Oherent Interactions

Passive mode-locking of erbium-doped fiber lasers with SESAM has been extensively investigated . In contrast to the NPR mode-locking, mode- locking incorporating a SESAM does not require any polarization element inside the laser cavity, thereby under suitable condition of the cavity birefringence, vector solitons could be formed in the lasers . Recently, it was reported that even the polarization-locked vector solitons (PLVSs) could be formed in the mode-locked fiber lasers . Formation of a PLVS requires not only that the group velocities of the two orthogonal polarization components of a vector soliton are locked but also that their phase velocities are also locked. It is well known that through the SPM and XPM between the two polarization-modes of a fiber, the group velocity locked vector solitons (GVLVSs) could be formed . Although it was also pointed out that the four-wave-mixing (also called coherent energy exchange) coupling between the polarization components of a vector soliton could have contributed to the formation of the PLVSs , so far no experimental evidence on the soliton internal FWM has been shown.

In this section, we report on the experimental observation of coherent interaction between the two orthogonal polarization components of a vector soliton formed in a fiber laser passively mode locked with a SESAM. A new type of spectral sidebands was first experimentally observed on the polarization

Hapter 3: Coherent And Incoherent Interaction

resolved soliton spectra of the PLVSs of the fiber lasers. The new spectral sidebands are characterized by that their positions on the soliton spectrum vary with the strength of the linear cavity birefringence, and while on one vector soliton polarization component the sideband appears as a spectral peak, then on the orthogonal polarization component it is a spectral dip, indicating the energy exchange between the two orthogonal polarization components of the vector solitons. Numerically we confirmed that the formation of the new type of spectral sidebands was formed by the FWM between the two polarization components of the vector solitons.

The fiber laser is illustrated in Figure 3.1. It has a ring cavity consisting of a piece of 4.6 m EDF with GVD parameter of 10 ps/km/nm and a total length of 5.4 m standard SMF with GVD parameter of 18 ps/km/nm. The cavity has a length of 4.6EDF + 5.4SMF = 10 m. Note that within one cavity round-trip the signal propagates twice in the SMF between the circulator and the SESAM. A circulator is used to force the unidirectional operation of the ring and simultaneously to incorporate the SESAM in the cavity. An intra cavity polarization controller is used to change the cavity’s linear birefringence. The laser is pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. A 10% fiber coupler is used to output the signals. The laser operation is monitored with an optical spectrum analyzer (Ando AQ- 6315B), a 26.5 GHz RF spectrum analyzer (Agilent E4407BESA-E SERIES) and a 350 MHz oscilloscope (Agilent 54641A) together with a 5 GHz photodetector. A commercial autocorrelator (Femtochrome FR-103MN) is used to measure the pulse width of the soliton pulses. The SESAM used is made

Hapter 3: Coherent And Incoherent Interaction

based on a GalnNAs quantum well and has a saturable absorption modulation depth of 5%, a saturation fluence of 180 µJ/cm2 and 10 ps relaxation time. The central absorption wavelength of the SESAM is at 1550 nm.

Figure 3.1: Schematic of the SESAM mode locked fiber laser. Experimentally, it was noticed that after mode-locking multiple mode locked pulses were always initially formed in the cavity. Depending on the net cavity birefringence, they were either the GVLVSs, characterized by the rotation of soliton polarization state along the cavity, or the PLVSs, characterized by the fixed polarization at the laser output. With multiple vector solitons in cavity, as a result of mutual soliton interaction complicated relative soliton movement or vector soliton bunches with random fixed soliton separations were observed. To exclude the complications caused by soliton interactions, we have always reduced the number of solitons in cavity through carefully decreasing pump

Hapter 3: Coherent And Incoherent Interaction

power so that only one or a few widely separated solitons exist in cavity. Figure 3.2 shows typical measured optical spectra of the PLVSs of the laser. The soliton feature of the mode-locked pulses is confirmed by the existence of soliton sidebands. However, apart from the existence of the conventional Kelly soliton sidebands, on the vector soliton spectrum there are also extra sets of spectral sidebands. Experimentally it was noticed that different from the Kelly sidebands whose positions are almost independent of the laser operation conditions, such as the pump strength and polarization controller orientation change (linear cavity birefringence change), the positions of the new spectral sidebands varied sensitively with the linear cavity birefringence. We note that Cundiff et al. have reported a new type of spectral sidebands on the GVLVS spectrum and interpreted their formation as caused by the vector soliton polarization evolution in the cavity . However, in our experiment the sidebands were observed on the PLVSs, whose polarization remains unchanged as they propagate along the laser cavity.

To determine the physical origin of the extra sideband formation, we then conducted polarization resolved measurement of the vector soliton spectrum. To this end the laser output was first passed through a rotatable external cavity linear polarizer. To separate the two orthogonal polarization components of a vector soliton, we always first located the orientation of the polarizer to the maximum soliton transmission, which sets the long axis of an elliptically polarized vector soliton; we then rotated the polarizer by 180 degree to determine the soliton polarization component along the short axis of the polarization ellipse. Through separating the two orthogonal polarization

Hapter 3: Coherent And Incoherent Interaction

components of the vector solitons, it turned out that the formation of the extra spectral sidebands was due to the coherent energy exchange between the two soliton polarization components. As can be clearly seen from the polarization resolved spectra, at the positions of extra spectral sidebands, while the spectral intensity of one soliton polarization component has a spectral peak, the orthogonal polarization component then has a spectral dip, indicting coherent energy exchange between them. We note that the energy flow between the two polarization components is not necessarily from the strong one to the weak one.

Energy flow from the weak component to the strong component was also observed. In addition, it is to see from the polarization resolved spectra that the extra sidebands are symmetric with respect to the soliton peak frequency and at different wavelength positions the peak-dip can also alternate, suggesting that the energy exchange is the relative phase of the coupled components dependent.

Figure 3.2a and Figure 3.2b were obtained from the same laser but under different intra cavity polarization controller orientations. Obviously, the positions of the extra sidebands are the cavity birefringence dependent.

Hapter 3: Coherent And Incoherent Interaction

Figure 3.2: Optical spectra of the phase locked vector solitons of the laser measured without passing and passing through a polarizer: (a) and (b) were measured under different linear cavity birefringence.

To verify our experimental observations and determine the extra sideband

Hapter 3: Coherent And Incoherent Interaction

formation mechanism, we further numerically simulated the effects of the FWM between the two polarization-components of a vector soliton formed in the laser. We used a round-trip model as described in for the simulations.

Briefly, we used the following coupled extended Ginzburg-Landau equations to describe the pulse propagation in the weakly birefringent fibers:

(1)

u and v are the normalized envelopes of the optical pulses along the two orthogonal polarized modes of the optical fiber. 2β = 2π∆n/λ is the wave number difference between the two modes and Lb = λ/∆n is the beat length.

2δ = 2βλ/2πc is the inverse group velocity difference. k″ is the second order dispersion coefficient; k′′′ is the third order dispersion coefficient and γ represents the nonlinearity of the fiber. g is the saturable gain coefficient of the fiber and Ωg is the bandwidth of the laser gain. For undoped fibers g = 0; for erbium doped fiber, we considered its gain saturation as

(2)

where G is the small signal gain coefficient and Psat is the normalized saturation energy. The saturable absorption of the SESAM is described by the rate

Hapter 3: Coherent And Incoherent Interaction

where Trec is the absorption recovery time, l0 is the initial absorption of the absorber, and Esat is the absorber saturation energy. To make the simulation possibly close to the experimental situation, we used the following parameters: γ = 3 W–1km–1, Ωg = 24 nm, Psat = 100 pJ, k″SMF = –23 ps2/km, k″EDF = –13 ps2/km, k′′′ = –0.13 ps3/km, Esat = 1 pJ, l0 = 0.15, and Trec = 6 ps, Cavity length L = 10 m.

Figure 3.3: Numerically calculated optical spectra of the vector solitons formed in fiber ring lasers. Figure 3.3 shows the results obtained. Extra spectral sidebands appeared clearly on the vector soliton spectrum of the laser. In particular, the extra sidebands of the orthogonal soliton polarization components exhibited out-of- phase intensity variations. To verify that the extra spectral sidebands were caused by the FWM between the orthogonal soliton polarization components, we deliberately removed the coherent coupling terms in our simulations.

Hapter 3: Coherent And Incoherent Interaction

Without the FWM terms no extra spectral sidebands were observed. Numerically, it was also noticed that the appearance of the extra sidebands is related to the small linear cavity birefringence. When the linear cavity birefringence is set zero, although strong energy exchange exists between the two polarization components, no extra sidebands were observed, instead the overall soliton spectrum exhibits “peak-dip” alternation as the soliton propagates in cavity. Moreover, numerical simulations have also exhibited the dependence of extra sideband positions with the linear cavity birefringence.

The numerical simulations well reproduced the extra spectral sidebands and confirmed that their appearance is indeed caused by the FWM between the orthogonal soliton components. Based on the numerical model we further investigated the FWM interaction between the soliton polarization components and its impact on the vector soliton. Numerically it was observed that as a result of the weak linear cavity birefringence, FWM between the two vector soliton components occurred. The FWM caused an antiphase type of periodic pulse intensity variation between the two orthogonally polarized soliton components, and the stronger the linear cavity birefringence the weaker the periodic pulse intensity variation. However, independent of the cavity birefringence the pulse intensity of the vector soliton always remained constant. The observed features of the vector solitons could be easily understood. Due to small linear cavity birefringence, coherent coupling between the two polarization components of a vector soliton can no longer be neglected. Its existence causes coherent energy exchange between the two orthogonal soliton polarization components.

Nevertheless, as far as the linear cavity birefringence is not zero, energy

Hapter 3: Coherent And Incoherent Interaction

exchange does not occur at whole soliton spectrum, but only at certain wavelengths where the phase matching condition is fulfilled under the aid of the laser cavity, which then leads to the formation of the discrete extra spectral sidebands. However, as the FWM is a parametric process and occurs between the internal components of a vector soliton, its appearance only causes an antiphase periodic intensity variation between the coupled soliton components but not the intensity of the vector soliton.

In conclusion, we have experimentally observed the evidence of coherent interaction among the two orthogonal polarizations of a vector soliton. A novel type of spectral sideband generation on the soliton spectra of the phase locked vector solitons in a passively mode-locked fiber ring laser has been generated due to such coherent interaction. Further polarization resolved study on the soliton spectrum revealed that the new sidebands were caused by the coherence energy exchange between the two orthogonal polarization components of the vector solitons. Numerical simulations have confirmed our experimental observation.

Ncoherent Interaction

Optical solitons were first experimentally observed in SMF by Maulenauer et al. in 1980 . The formation of the solitons was a result of the balanced interaction between the effects of anomalous fiber dispersion and the pulse SPM. To observe the solitons it is necessary that the intensity of a pulse be above a threshold where the nonlinear length of the pulse becomes comparable with the dispersion length. Apart from SPM, theoretical studies have also

Hapter 3: Coherent And Incoherent Interaction

shown that the incoherent interaction, or specifically XPM, could lead to soliton formation. Soliton formation through XPM was known as the induced soliton formation. An important potential application of the effect is the light controlling light. Various cases of soliton formation in SMF caused by XPM were predicted, these include the formation of a bright soliton in the normal fiber dispersion regime supported by a dark soliton in the anomalous dispersion regime , and bright solitons formation in anomalous fiber dispersion regime supported by each other through XPM . Spatial soliton formation through XPM has been experimentally observed . However, to the best of our knowledge, no induced temporal soliton formation experiments have been reported. In Chapter 3, we report on the experimental observation of induced solitons in a passively mode-locked fiber laser. Using a birefringence cavity fiber laser, we observed that due to the cross coupling between the two orthogonal polarization components, if a strong soliton is formed along one principal polarization axis, a weak soliton can always be induced along the orthogonal polarization axis. Especially, the intensity of the weak soliton could be so weak that it alone cannot form a soliton by the SPM. Numerical simulations have well supported the experimental observations.

Our fiber laser has nearly the same experimental parameters with the Figure 3.1. However, the cavity birefringence is no longer weak but at a moderation value through adjusting the polarization controllers. As no polarizer was used in the cavity, due to the weak birefringence of the fibers the cavity exhibited obvious birefringence features, e. g. varying the linear cavity birefringence we could observe the various types of vector solitons in the laser . In order to

Hapter 3: Coherent And Incoherent Interaction

identify features of the vector solitons, we explicitly investigated their polarization resolved spectra under various experimental conditions. To measure their polarization resolved spectra, we let the laser output first pass through a rotatable external cavity polarizer, based on the measured soliton intensity change with the orientation of the polarizer we then identify the long and short polarization ellipse axes of the vector solitons. In our measurements we found that apart from vector solitons with comparable coupled orthogonal polarization components, vector solitons with very asymmetric component intensity also exist. Figure 3.4 shows for example two cases experimentally observed. Figure 3.4a shows a case that was measured under laser operation with a relatively large cavity birefringence. In this case the spectral intensity difference between the two orthogonal polarization directions at the center soliton wavelength is more than 30 dB. The soliton nature of the strong polarization component is obvious as characterized by the existence of the Kelly sidebands. Kelly sidebands have also appeared on the weak polarization component. We emphasize the different locations of the Kelly sidebands along different polarization directions. It excludes the possibility that the sidebands on the spectrum of the weak component were produced due to an experimental artifact. The first order Kelly sidebands of the strong component are located at 1547.4 nm and 1569.5 nm, respectively; those of the weak component are at 1546.8 nm and 1568.9 nm. The separations of both sets of sidebands are the same. The appearance of Kelly sidebands on spectrum of the weak component suggests that it is also a soliton. In particular, due to the large cavity birefringence the solitons formed along the two orthogonal polarization

Hapter 3: Coherent And Incoherent Interaction

directions have different center wavelengths. Therefore, their Kelly sidebands have different locations.

Hapter 3: Coherent And Incoherent Interaction

Figure 3.4: Polarization resolved optical spectra of the vector solitons experimentally observed. (a): Obtained under large cavity birefringence. (b) Obtained under relatively weak cavity birefringence.

Hapter 3: Coherent And Incoherent Interaction

Using a commercial auto-correlator we measured the soliton pulse width. Assuming a Sech2 pulse profile it is about 1 ps. If only the SPM is considered, we estimate that the peak power of the fundamental solitons in our laser is about 24 W. This is well in agreement of the experimentally measured strong component soliton peak power of about 25 W. The experimentally measured weak component soliton peak power is only 0.5 W. Obviously with the intensity of the weak pulse it is impossible to form a soliton. The weak soliton should be an induced soliton.

Through adjusting the intra cavity polarization controller the net cavity birefringence could be changed. Another situation as shown in Figure 3.5b where both solitons have the same center wavelength was also obtained. Even in the case the two solitons have different Kelly sidebands. In particular, the first order Kelly sidebands of the weak soliton have slightly larger separation than that of the strong soliton, indicating that the induced soliton has a narrower pulse width than that of the strong soliton [122, 123].

To confirm the experimental observation, we numerically simulated the operation of the laser. We used the coupled Ginzburg-Landau equations to describe the pulse propagation in the weakly birefringent fibers. To make the simulation possibly close to the experimental situation, we used the following parameters γ = 3 W–1km–1, Ωg = 24 nm, Psat = 100 pJ, k″SMF = –23 ps2/km, k″EDF = –13 ps2/km, k′′′ = –0.13 ps3/km, Esat = 0.6 nJ, l0 = 0.15, and Trec = 6 ps.

Figure 3.5 shows the simulations obtained under different cavity birefringence. Figure 3.5a shows the case of the laser with a beat length Lb = 0.1 m. In this

Hapter 3: Coherent And Incoherent Interaction

case the strong soliton can either be formed along the slow or the fast axis of the cavity. Associated with the strong soliton there is always a weak soliton induced in the orthogonal polarization direction. The induced soliton has different central wavelengths, which causes that the Kelly sidebands of them have different locations. However, the wavelength shifts of their sidebands to the central soliton wavelength are the same, just like the experimental observation. Figure 3.5b shows a case of the cavity with a beat length of Lb = 10 m. Due to the small cavity birefringence, the induced soliton has always exactly the same central wavelength as the strong soliton. Numerically we found that as Lb changed from 10 m to 100 m, the strong soliton swapped from the slow axis to the fast axis of the cavity as a result of the polarization instability . Nevertheless, in both cases the first order Kelly sidebands of the weak soliton have slightly larger separation than that of the strong soliton.

To verify that the weak soliton is induced by the strong soliton through XPM, the XPM terms were deliberately removed from the simulations. It was found that in this case the weak component kept continuously fading away, no stable soliton pulse could be formed. We note that apart from the Kelly sidebands, in Figure 3.5, there are some other discrete sharp spectral peaks. We have also numerically identified their formation as caused by the four-wave-mixing (FWM) between the two polarization-components . By removing the FWM terms from Equation (1), these spectral sidebands completely disappeared from the numerically calculated soliton spectra. Like the experimental observations, the FWM sidebands are more pronounced on the spectra of the induced solitons due to their weak intensity. Especially, the first

Hapter 3: Coherent And Incoherent Interaction

order of the FWM sidebands appeared between the first order Kelly sidebands and the central soliton wavelength, and their exact positions varied with the cavity linear birefringence.

Figure 3.5: Numerically calculated optical spectra of the vector solitons. (a): Cavity beat length Lb = 0.1 m. (b) Cavity beat length Lb = 10 m. Pump strength

Hapter 3: Coherent And Incoherent Interaction

In conclusion, formation of induced temporal solitons has been experimentally observed in a passively mode-locked fiber laser with birefringence cavity. It was found that the induced solitons were formed by the XPM between the two orthogonal polarization components of the birefringence laser, and the induced solitons could either have the same or different soliton frequency to the inducing soliton. As the induced solitons always have the same group velocity as that of the inducing soliton, they form vector solitons in the laser. To our knowledge, this is the first experimental observation of temporal induced solitons.

Solitons

Actually, vector solitons discussed in Chapter 3 belong to the family of fundamental order vector soliton, i.e., each polarization component intensity distribution having the shape of Sech2 profile with single hump. Previous studies on coupled NLSEs and the quintic complex Ginzburg-Landau equation have predicted that higher order soliton (also called as bound solitons) could be formed as a result of direct soliton interaction . In contrast with the fundamental order soliton, the intensity profile high order soliton has more than one hump. Although the dynamics of high order scalar soliton in a fiber laser has been well-known, whether high order vector solitons exist or how they evolve was a question. In Chapter 4, a novel form of high order vector soliton is experimentally investigated and numerically confirmed. Moreover, similar to the fundamental order vector soliton, its polarization could be still locked under suitable condition. Finally, the concept of high order vector soliton could be extended to the dissipative/non-Hamiltonian system where the gain/loss effect dominates over the dispersion/nonlinearity.

Soliton as a stable localized nonlinear wave has been observed in various physical systems and been extensively studied . Optical solitons were first experimentally observed in SMF by Mollenauer et al. in 1980 . It was shown that dynamics of the solitons could be well described by the NLSE, a paradigm equation governing optical pulse propagation in ideal SMFs. However, in reality a SMF always supports two orthogonal polarization modes. Taking fiber

Hapter 4: High-Order Vector Soliton

birefringence into account, it was later found that depending on the strength of fiber birefringence, different types of vector solitons, such as the group velocity locked vector solitons, the rotating polarization vector solitons, and the phase locked vector solitons , could also be formed in SMFs.

Optical solitons were also observed in mode-locked fiber lasers. Pulse propagation in a fiber laser cavity is different from that in a SMF. Apart from propagating in the fibers that form the laser cavity, a pulse propagating in a laser also subjects to actions of the laser gain and other cavity components.

Dynamics of solitons formed in a fiber laser is governed by the Ginzburg- Landau equation, which takes account of not only the fiber dispersion and Kerr nonlinearity, but also the laser gain and losses. However, it was shown that under suitable conditions solitons formed in fiber lasers have analogous features to those of solitons formed in SMFs. Furthermore, vector solitons were also predicted in mode-locked fiber lasers and confirmed experimentally recently .

Among the various vector solitons formed in mode-locked fiber lasers or SMFs, the phase locked one has attracted considerable attention. Back to 1988 Christodoulides and Joseph first theoretically predicted a novel form of phase locked vector soliton in birefringent dispersive media , which is now known as a high order phase locked vector soliton in SMFs. The fundamental form of the phase locked temporal vector solitons was recently experimentally observed . However, to the best of our knowledge, no high order temporal vector solitons have been demonstrated. Numerical studies have shown that the high order phase locked vector solitons are unstable in SMFs . In Chapter

Hapter 4: High-Order Vector Soliton

4, we report on the experimental observation of a stable phase-locked high order vector soliton in a mode-locked fiber laser. Multiple high order vector solitons with identical soliton parameters coexisting in laser cavity and harmonic mode-locking of the high order vector solitons were also observed.

Moreover, based on a coupled Ginzburg-Landau equation model we show numerically that phase locked high order vector solitons are stable in mode- locked fiber lasers.

The experimental setup is shown in Figure 4.1. The fiber laser has a ring cavity consisting of a piece of 4.6 m EDF with GVD parameter 10 ps/km/nm and a total length of 5.4 m standard SMF with GVD parameter 18 ps/km/nm. Mode- locking of the laser is achieved with SESAM. Note that within one cavity round-trip the pulse propagates twice in the SMF between the circulator and the SESAM. A polarization independent circulator was used to force the unidirectional operation of the ring and simultaneously to incorporate the SESAM in the cavity. The laser was pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. A 10% fiber coupler was used to output the signals. The SESAM used is made based on GalnNAs quantum wells. It has a saturable absorption modulation depth of 5%, a saturation fluence of 180 µJ/cm2 and a recovery time of 10 ps. The central absorption wavelength of the SESAM is at 1550 nm.

Hapter 4: High-Order Vector Soliton

Figure 4.1: Schematic of the vector soliton fiber laser. As no polarizer was used in the cavity, depending on the net cavity linear birefringence, various types of vector solitons such as the group velocity locked vector solitons, the polarization rotating vector solitons, and the fundamental phase locked vector solitons were obtained in the laser. Especially, we found that the experimentally observed features of these vector solitons could be well described by an extended coupled Ginzburg-Landau equation model, which also considered effects of the saturable absorber and the laser cavity .

Encouraged by the results we had further searched for the high order phase locked vector solitons theoretically predicted. Through splicing a fiber pigtailed

Hapter 4: High-Order Vector Soliton

optical isolator between the output port and the external cavity measurement apparatus, which serves as suppressing the influence of spurious back reflection on the laser operation, we could indeed obtain one of such vector solitons.

Figure 4.2a shows for example the optical spectra and autocorrelation traces of the soliton. Polarization locking of the soliton is identified by measuring the polarization evolution frequency (PEF) of the soliton pulse train . No PEF could be detected. As the vector soliton has a stationary elliptic polarization, we could use an external polarizer to separate its two orthogonal polarization components. The optical spectra of the components are shown in Figure 4.2a.

The spectra have the same central wavelength and about 10 dB peak spectral intensity differences. Both spectra display soliton sidebands. It shows that both of the components are optical soliton. In addition, coherent energy exchange between the two soliton components, represented by the appearance of spectral peak-dip sidebands , is also visible on the spectra. Different from the polarization resolved spectra of the fundamental phase locked vector solitons, there is a strong spectral dip at the center of the soliton spectrum of the weak component, while no such dip on the spectrum of the strong soliton component.

To identify the formation mechanism of the spectral dip, we further measured the autocorrelation traces of each of the soliton components. It turned out that the weak component of the vector soliton had a double-humped intensity profile as shown in Figure 4.2b. The pulse width of the humps is about 719 fs if the Sech2 profile is assumed, and the separation between the humps is about 1.5 ps.

The strong component of the vector soliton is a single-hump soliton. It has a pulse width of about 1088 fs if the Sech2 profile is assumed. The components of

Hapter 4: High-Order Vector Soliton

the vector soliton have the pulse intensity profiles exactly as those predicted by Akhmediev et al. and Christodoulides for a high order phase locked vector soliton. Furthermore, the spectral dip at the center of the spectrum indicates that the two humps have 180° phase difference, which is also in agreement with the theoretical prediction.

Once the laser operation conditions were appropriately selected, the high order phase locked vector soliton operation was always obtained in the laser. Experimentally, multiple such vector solitons with identical soliton parameters were also obtained. Through carefully changing the pump strength one could even control the number of the vector solitons in cavity, and it did not change the structure of the vector solitons. Like the scalar solitons observed in the conventional soliton fiber lasers, harmonic mode locking of the high order phase locked vector solitons was also observed, as shown in Figure 4.3, where 8 such vector solitons were equally spaced in the cavity. All our experimental results show that formation of the high order phase locked vector solitons is an intrinsic feature of the fiber laser.

Hapter 4: High-Order Vector Soliton

Figure 4.2: Polarization resolved soliton spectra and autocorrelation traces of the vector soliton observed. (a) Soliton spectra. (b) Autocorrelation traces.

Time (20 Ns/Div)

Figure 4.3: Oscilloscope trace of a harmonically mode-locked high order phase locked vector soliton state. Lc: cavity roundtrip time. 8 vector solitons coexist in cavity.

To confirm our experimental observations, we also numerically simulated the operation of the laser with the model used before . In order to more accurately reflect the observation, we used the following parameters for our simulations for possibly matching the experimental conditions: γ = 3 W–1km–1, Ωg = 24 nm, Psat = 50 pJ, k″SMF = –23 ps2/km, k″EDF = –3 ps2/km, k′″ = –0.13 ps3/km, Esat = 10 pJ, l0 = 0.3, and Trec = 2 ps, cavity length L = 10 m.

We used the standard split-step Fourier technique to solve the equations and a so-called pulse tracing method to model the effects of laser oscillation . We have always started our simulations with an arbitrary weak light input.

Figure 4.4 shows one of the typical results obtained. With a cavity linear

Hapter 4: High-Order Vector Soliton

birefringence of Lb = 3L, a stable high order phase locked vector soliton state was obtained. The weak polarization component of the vector soliton consists of two bound solitons with pulse separation of about 1 ps, while the strong polarization component of the vector soliton is a single-hump soliton. It is interesting to see that the pulse of the strong component is only temporally overlapped with one of the two pulses of the weak component. Due to the strong cross-phase coupling between the temporally overlapped pulses, the two pulses of the weak components have different pulse widths and intensities.

Propagating along the cavity, obvious coherent energy exchange between the two temporally overlapped solitons is visible. Figure 4.4b further gives the calculated spectra of the vector soliton components, which also show that the phase difference between the two bound solitons of the weak component is 180°.

Depending on the laser parameter selections, other high order phase locked vector soltions, such as the one with both soliton components having a double- humped structure, were also numerically obtained. We note that similar high order vector solitons were also predicted for pulse propagation in weakly birefringent fibers, but they are unstable. However, we found that all the numerically obtained high order phase locked vector solitons were stable in the laser. We believe that the different stability feature of the high order phase locked vector solitons in fiber and in fiber lasers could be traced back to their different soliton nature. While the soliton formed in a SMF is essentially a Hamiltonian soliton, the one formed in a fiber laser is a dissipative soliton, which is in fact a strong attractor of the laser system. The formation of multiple

Hapter 4: High-Order Vector Soliton

identical high-order vector solitons in the fiber laser clearly shows the dissipative nature of the formed vector solitons.

Hapter 4: High-Order Vector Soliton

Figure 4.4: A stable high order phase locked vector soliton state numerically calculated. (a) Soliton intensity profiles of the two orthogonally polarized components. (b) The corresponding optical spectra of (a).

Hapter 4: High-Order Vector Soliton

In conclusion, we have first experimentally observed a novel type of high order phase locked vector soliton in a passively mode-locked fiber laser. The high order vector soliton is characterized by that its two orthogonal polarization components are phase locked, and while the stronger polarization component is a single hump pulse, the weaker component has a double-humped structure with 180° phase difference between the humps. Our experimental result firstly confirmed the theoretical predictions on the high order phase locked vector solitons in birefringent dispersive media.

Hapter 5. Dissipative Vector Solitons

Although vector solitons investigated in Chapter 3 and Chapter 4 have different soliton profile, they have one common feature: nearly transform limited in that the fiber laser cavities are made of purely anomalous dispersion fibers and belonging to conservative NLSE solitons are formed. Their generations only require the balance between the anomalous dispersion and nonlinearity.

However, in the normal dispersion regime, the formed solitons are much different in that they are largely chirped and more importantly, their formations need the mutual balance between gain, loss, normal dispersion and nonlinearity.

Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatially extended dissipative systems due to mechanisms of self- organization. They exist for an extended period of time, even though parts of the structure experience gain and loss of energy and/or mass. Normal dispersion fiber laser is a natural dissipative system as the gain/loss effect must be taken into consideration.

Recently, formation of DSs in pure normal-dispersion-cavity fiber lasers passively mode locked by the NPR technique was reported . It was shown that formation of the solitons is a result of the mutual nonlinear interaction among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, laser gain saturation and gain bandwidth filtering. Although the NPR mode-locking process of the lasers unavoidably affected the detailed soliton features, it was found that its effect on the soliton shaping was minor . In

Hapter 5: Dissipative Vector Solitons

addition, both the experimental and theoretical studies have shown that the dissipative solitons formed in the large net positive cavity GVD regime have very different characteristics from those formed in the net negative cavity GVD fiber lasers, and the differences can be traced back to the different soliton shaping mechanisms in the two cavity dispersion regimes. Mode-locking of a fiber laser can also be achieved with other techniques, e.g. using SESAM [61, 62]. Different from the NPR mode locking, mode-locking with a SESAM requires no polarizer in the laser cavity, which potentially allows the formation of a dissipative vector soliton (DVS) in the laser cavity.

As discussed in Chapter 3 and Chapter 4, the dynamics of vector soliton is strongly related with the cavity birefringence. Under different birefringence regime, the polarization effect and coupling strength might be much different.

Thus, various types of vector solitons might appear. Chapter 5 is organized as follows. When the cavity birefringence is weak, both polarizations locked and rotating DVS could be experimentally observed, which is discussed firstly in Section 5.1. When the cavity birefringence is in moderate value, dual- wavelength dissipative solitons could be obtained; while the cavity birefringence is further strong, triple- wavelength dissipative solitons would appear. The generation mechanism of multi-wavelength dissipative soliton would be elaborately discussed in Section 5.2.

Polarization Locked And Rotating Dvs

Formation of both the frequency locked and phase locked DVSs in the negative cavity GVD regime were theoretically predicted by Akhmediev et al. , and

Hapter 5: Dissipative Vector Solitons

experimentally observed in a fiber laser mode-locked with a SESAM . Different from a dissipative soliton formed in the net negative cavity GVD regime, a dissipative soliton formed in the net large positive cavity GVD regime has different soliton shaping mechanism, and furthermore is strongly frequency chirped. It would be interesting to find out whether a phase-locked DVS could be formed in a fiber laser with large net positive cavity dispersion or not. In this section we will address this question. We show experimentally that either the polarization rotating or the phase-locked DVSs can be formed in the fiber lasers.

In addition, we also show that multiple vector solitons with identical soliton parameters and harmonic mode locking of the vector solitons can be formed in the fiber lasers. Numerical simulations are found in agreement of the experimental observations, which confirm the DVS formation in the fiber lasers.

Our experimental setup is shown in Figure 5.1. The fiber laser has a ring cavity consisting of a piece of 1.5 m EDF with a GVD of about 40.8 ps2/km, a total length of 3.5 m standard single mode fiber SMF with GVD of about –23 ps2/km and 18.2 m DCF with GVD of about 2.55 ps2/km. The total cavity length is 23.2 m. Mode-locking of the laser is achieved with a SESAM. A polarization independent circulator was used to force the unidirectional operation of the ring and simultaneously incorporate the SESAM in the cavity. Note that within one cavity round-trip the pulse propagates twice in the SMF between the circulator and the SESAM. The laser was pumped by a high power Fiber Raman Laser source (BWC-FL-1480-1) of wavelength 1480 nm. A 10% fiber coupler was used to output the signals. The SESAM used is made based on GalnNAs quantum wells. It has a saturable absorption modulation depth of 5%, a

Hapter 5: Dissipative Vector Solitons

saturation fluence of 90 µJ/cm2 and a recovery time of 10 ps. The central absorption wavelength of the SESAM is at 1550 nm. Figure 5.1: Schematic of the fiber laser.

This fiber laser has a typical dispersion-managed cavity with net normal cavity GVD of about 0.027 ps2. To control the net cavity GVD, DCF with different lengths were inserted between the circulator and the SESAM. When the net cavity GVD was anomalous, it was observed that the spectrum of the mode- locked pulses had the typical features of those of the DVS reported in .

Increasing the length of the DCF, the net cavity GVD shifted to large positive values, correspondingly, the soliton operation of the laser shifted to a new dissipative soliton regime. Experimentally, the self-started mode locking of the laser could still be easily achieved at a large positive cavity GVD.

Hapter 5: Dissipative Vector Solitons

Figure 5.2 shows a typical optical spectrum of the dissipative solitons of the laser obtained at a large net positive cavity GVD. The soliton spectrum has characteristic steep spectral edges. However, different from the dissipative solitons formed in the fiber lasers mode-locked with the NPR technique, the soliton consists of two orthogonal polarization components. To highlight the vector nature of the soliton, we let the soliton pulse train pass through a rotatable external cavity polarizer and compared the features of the pulse train before and after passing through the polarizer, either with a high speed oscilloscope or an RF-spectrum analyzer. We found that the soliton, shown in Figure 5.2a, was a polarization rotating vector soliton. Polarization rotation of the soliton could be easily identified e.g. by the oscilloscope trace measurement.

Without passing through the external polarizer, the soliton pulse had identical pulse intensity on the oscilloscope trace for each cavity roundtrip, while after passing through the polarizer it became varying with the cavity roundtrips as shown in Figure 5.2b. It indicates that the polarization of the soliton rotated along the cavity. Figure 5.2a also shows the autocorrelation trace of the vector soliton. It has a width (FWHM) of 40 ps. If a Sech2 pulse shape is assumed, the soliton pulse width is 24 ps. The 3 dB spectrum bandwidth of the soliton is 6.5 nm, which gives a time-bandwidth product about 18.9, indicating that the soliton is strongly chirped.

Hapter 5: Dissipative Vector Solitons

Figure 5.2: (a) Spectrum and corresponding autocorrelation trace of a polarization rotating DVS emission state of the laser; (b) Oscilloscope trace of (a) after passing through a polarizer; (c) Polarization resolved optical spectra of a phase locked DVS emission state of the laser; (d) Oscilloscope trace of (c) after passing through a polarizer.

Controlling the linear cavity birefringence through an intra-cavity polarization controller, polarization locked vector solitons were obtained. A polarization locked vector soliton has the characteristic that it has a fixed polarization during circulation in the laser cavity. Correspondingly, after passing through an external cavity polarizer the pulse height of such vector solitons on the oscilloscope trace would have identical value as shown in Figure 5.2d. For a phase locked vector soliton we could also measure its optical spectra along the long and the short polarization ellipse axes, as shown in Figure 5.2c. Different

Hapter 5: Dissipative Vector Solitons

from the polarization resolved dissipative vector solitons formed in the net negative cavity GVD fiber lasers, no four-wave-mixing spectral sidebands could be identified. We believe their absence could be traced back to the large frequency chirp of the soliton. The two orthogonal polarization components shown in Figure 5.2c have comparable spectral intensity and the same center wavelength, but clearly different spectral distributions. Experimentally, the phase locking between the two orthogonal polarizations was further confirmed by the polarization evolution frequency measurement as described in .

Apart from the above near circularly polarized polarization locked DVS, polarization locked DVSs with significant spectral intensity difference between the two orthogonal polarization components were also observed. In one case the spectral intensity difference at the center soliton wavelength was as large as 20 dB. We experimentally measured the peak power of the weak soliton component in the case. It was about 0.1 W. With the pulse peak power it is impossible to form a soliton in the laser. Therefore, we believe it could be an induced soliton formed by the cross coupling with the strong soliton component .

Adjusting the polarization controller, the central wavelength of the DVSs could be altered in a wide range from 1560 nm to 1575 nm. However, as the central soliton wavelength is increased, the formed DVS becomes less stable. This could be due to that the central absorption wavelength of the SESAM is about 1550 nm, which favors the formation of the DVS centered near 1550 nm.

Multiple DVSs with identical soliton parameters were also obtained under strong pumping. In this case, harmonic mode locking was frequently obtained.

Hapter 5: Dissipative Vector Solitons

Figure 5.3 illustrates a case where 8 DVSs were equally spaced in the cavity with a separation of about 14.5 ns. Other harmonic mode locking states of the DVS were also obtained by simply varying the pumping strength.

Figure 5.3: Oscilloscope trace of a harmonically mode-locked gain-guided vector soliton state. Lc: cavity roundtrip time. 8 DVS coexist in cavity. To gain an insight into the DVS formation, we also numerically simulated the operation of the laser. We used a round-trip model to include the laser cavity effects as well as the saturable absorber effect in our simulations. To make the simulation possibly close to the experimental situation, we used the following parameters: γ = 3 W–1km–1, Ωg = 16 nm, Psat = 50 pJ, k″SMF = –23 ps2/km, k″EDF = 41 ps2/km, k″DCF = 2.6 ps2/km, k″′ = –0.13 ps3/km, Esat = 35 nJ, l0 = 0.2, Trec = 2 ps, and cavity length L = 23.2 m.

Numerically, it was found that under the current saturable absorber parameter selection, the formation of DVS in the laser is strongly the gain bandwidth dependent. When the gain bandwidth is large, i.e. 24 nm, DVS is unable to form.

In this case the spectrum of the mode-locked pulses has a Gaussian profile.

Hapter 5: Dissipative Vector Solitons

Stable DVS is formed when the gain bandwidth is narrower than 16 nm. The narrower the gain bandwidth used for simulation, the smaller the spectrum bandwidth of the obtained DVS. Figure 5.4a shows a typical case of the calculated DVS evolution with the cavity roundtrips. Numerically we found that the total pulse intensity is unchanged with the cavity roundtrips, but intensity of the horizontal and the vertical component exhibits out-of-phase intensity variation, indicating that coherent energy exchange between them still exists . After removing the four-wave-mixing terms from Equation (1), such an out-of-phase intensity variation between the vector soliton components then disappeared. The calculated pulse width is about 20 ps and its spectrum bandwidth is about 7 nm, which indicates that the formed DVS is strongly frequency chirped.

The result shown in Figure 5.4 was obtained for a laser cavity with a beat length of Lb = 100 m. The DVSs formed have two orthogonal polarization components with comparable intensity and coincident central wavelength.

Numerically, as Lb changed from 100 m to 0.1 m, i.e., increasing the cavity birefringence, intensity difference between the two orthogonal polarization components becomes larger, in accord with our experimental observations. We have also numerically investigated polarization evolutions of the formed DVSs by using the method reported in . Depending on the cavity birefringence, either the polarization locked or polarization rotating DVSs were numerically obtained. Calculated based on the peak point of the DVSs, a polarization locked DVS has its polarization ellipse orientation fixed as it propagates along the cavity, while the polarization ellipse of a rotating DVS changes its orientation

Hapter 5: Dissipative Vector Solitons

along the cavity. The relative phase between the components of the polarization locked DVS at the pulse peak was always fixed at π /2. However, at the other points of the pulse it deviated from π / 2, demonstrating the effect of frequency chirp. To determine how the frequency chirp of the solitons affects their polarization, we also numerically calculated the polarization states at different points of a pulse, e.g. at the pulse’s peak and FWHM points. For the polarization locked DVSs it was found that the polarization ellipses calculated at different points had slightly different orientations, nevertheless, the difference remained the same along the propagation. This numerical result suggests that despite the strong frequency chirp of the DVSs, the temporal variation of their two orthogonally polarized components is always phase locked.

In conclusion, DVSs have been experimentally demonstrated in a dispersion- managed fiber laser with large net normal cavity GVD. It was found that despite of the large frequency chirp of the dissipative solitons formed in the lasers, the polarization rotating and locked DVSs could still be formed. In addition, formation of multiple DVSs with identical soliton parameters and stable harmonic DVS mode-locking are also experimentally obtained.

Hapter 5: Dissipative Vector Solitons

Figure 5.4: (a) Combined pulse intensity evolution; (b) corresponding optical spectra numerically calculated.

Ulti-Wavelength Dissipative Soliton

Multi-wavelength mode-locked fiber lasers have versatile applications, including fiber optic sensing, photonic component characterization and WDM optical communications. Several methods for achieving multi-wavelength mode locking have been studied. Li et al. reported the generation of triple-wavelength picosecond mode locked pulses using a self-seeded Fabry–Perot laser diode with fiber Bragg gratings . Multiwavelength actively mode-locked fiber lasers incorporating either a single sampled fiber Bragg grating or a biased semiconductor optical amplifier in cavity was shown by Yao et al.

By virtue of the NPR effect, simultaneous dual- and five-wavelength actively mode-locked erbium-doped fiber lasers at 10 GHz were demonstrated by Pan et al. . Although multiwavelength actively mode-locked fiber lasers have the advantages such as high repetition rates, narrow linewidth, they also have the drawbacks of broad pulse width, low peak power, and expensive as a modulator is required to be inserted in the cavity. Moreover, as actively mode locked multiwavelength pulses have only weak nonlinearity, they are impossible to be shaped into optical solitons that possess the born preponderance: good stability, low time jittering, short pulse width and high peak power.

Back in 1992, Matsas et al. reported the experimental observation of dual- wavelength soliton emission in a fiber laser exploiting the NPR technique for mode locking . A long laser cavity made of anomalous dispersion fibers was adopted in their experiment. However, no explanation on the formation mechanism of dual-wavelength solitons was given. In this section, we report on the experimental observation of multiple wavelength dissipative soliton

Hapter 5: Dissipative Vector Solitons

operation of an erbium-doped fiber laser. Formation of dissipative solitons in normal dispersion fiber lasers has recently attracted considerable attention [20- 25, 144, 145]. Although in an all-normal dispersion fiber laser there is no natural balance between the actions of the fiber dispersion and fiber nonlinear optical Kerr effect, therefore, no natural NLSE soliton is formed. It was shown that as a result of the mutual interactions among the normal cavity dispersion, cavity fiber nonlinear Kerr effect, and the effective laser gain bandwidth, an optical soliton can still be formed in the laser. The formed solitons were known as the dissipative solitons . Occasionally they were also called as the gain- guided solitons to distinguish from those solitons formed in the anomalous dispersion fiber lasers . Single wavelength dissipative solitons have been observed in various fiber lasers [144, 145]. In a previous paper we have also reported the observation of dissipative vector solitons in a dispersion managed cavity fiber laser . However, to the best of our knowledge, no multi- wavelength dissipative solitons operation of a fiber laser has so far been reported.

Our fiber laser is schematically shown in Figure 5.5a. It has a ring cavity made of pure normal dispersion fibers. A piece of 5.0 m EDF with GVD of –32 (ps/nm)/km was used as the gain medium, and all the other fibers are the DCF with GVD of –4 (ps/nm)/km. The cavity has a length of 13.5 m. Mode-locking of the laser is achieved with a SESAM. A polarization independent circulator was used to force the unidirectional operation of the ring and simultaneously incorporate the SESAM in the cavity. Note that within one cavity round-trip the pulse propagates twice in the DCF between the circulator and the SESAM. A

Hapter 5: Dissipative Vector Solitons

50% fiber coupler was used to output the signal, and the laser was pumped by a high power Fiber Raman Laser source (KPS-BT2-RFL-1480-60-FA) of wavelength 1480 nm. The maximum pump power can be as high as 5 W. All the passive components used (WDM, Coupler, and Circulator) were made of the DCF. An optical spectrum analyzer (Ando AQ-6315B) and a 350 MHz oscilloscope (Agilen 54641A) together with a 2 GHz photo-detector were used to simultaneously monitor the spectra and the mode locked pulse train, respectively. The SESAM used was made based on GalnNAs quantum wells. It has a saturable absorption modulation depth of 30%, a saturation fluence of 90 µJ/cm2 and a recovery time of 10 ps. The SESAM was pigtailed with 0.5 m

F.

Figure 5.5: Schematic of the experimental setup.

Hapter 5: Dissipative Vector Solitons

Figure 5.6: (a) Optical spectra of single wavelength dissipative soliton. Insert: the oscilloscope trace. (b) The corresponding autocorrelation trace Mode locking of the laser self-started as the pump power was increased above

Hapter 5: Dissipative Vector Solitons

the mode locking threshold. Immediately after the mode locking, multiple soliton pulses were generally formed in the cavity. However, through carefully decreasing the pump power, the number of soliton pulses could be reduced and eventually a single soliton operation state could be obtained. Figure 5.6 shows a typical single soliton operation state of the laser. The optical spectrum of the pulse has the characteristic steep spectral edges, which shows that it is a dissipative soliton [20-25, 144, 145]. Based on the measured autocorrelation trace, the soliton pulse width is estimated 28.8 ps if a Sech2-shape pulse is assumed. Experimentally we confirmed that the soliton pulse was linearly polarized. The center wavelength of the dissipative soliton shown in Figure 5.6 is located at 1576.2 nm. Experimentally, adjusting the orientations of the paddles of the PC, which corresponds to changing the linear birefringence of the cavity, the central wavelength of the soliton could be varied in a wide range from 1570 nm to 1590 nm. Nevertheless, solitons with central wavelengths close to 1570 nm or 1590 nm were less stable.

Under stronger pumping, a new soliton was formed in the cavity. It was found that the features of the new soliton sensitively depended on the cavity birefringence. Figure 5.7 shows a soliton operation state experimentally observed. Figure 5.7a is the optical spectrum of the laser emission. Compared with the spectrum shown in Figure 5.6, another steep-edge shaped spectrum with different central wavelength also appeared. Figure 5.7b shows the oscilloscope trace of the laser emission. There are two solitons in the cavity.

Each soliton has different pulse energies as represented by the two different pulse heights in the oscilloscope trace. The two solitons also propagated with

Hapter 5: Dissipative Vector Solitons

different group velocities in the cavity. Therefore, when triggered with one soliton the other soliton then moved randomly on the oscilloscope screen, indicating that the two solitons have different group velocities in the cavity. We had further experimentally studied the polarization features of the pulses using an external cavity polarizer. Through rotating the external cavity polarizer it was identified that one soliton could be completely suppressed while the other still remained on the oscilloscope trace, associated with the suppression of one soliton pulse on the oscilloscope one squared-shaped spectrum also disappeared on the optical spectrum. Figure 5.7 actually shows a state of the dual wavelength dissipative soliton operation of the fiber laser. Specifically, the dissipative soliton with the large pulse energy has a center wavelength of 1576.2 nm, and the one with the weak energy has a center wavelength of 1579.6 nm, and the two dissipative solitons have orthogonal polarizations.

Hapter 5: Dissipative Vector Solitons

Figure 5.7: (a) Optical spectrum of dual wavelength dissipative solitons. Insert: the normalized optical spectrum; (b) Oscilloscope trace of dual wavelength dissipative solitons.

The relative strength of the solitons varied with the cavity birefringence. Slightly tuning the orientation of the PC, one could continuously change the relative soliton pulse energy. At a certain soliton intensity relation, it was found that two solitons could even have the same group velocity despite of the fact that they have different central wavelengths. Figure 5.8 shows the oscilloscope of such a case, where the two solitons have fixed soliton separation as they

Hapter 5: Dissipative Vector Solitons

circulated in the cavity. However, such a state was unstable, after a short time the solitons lost their synchronization and moved independently again. Figure 5.8: Oscilloscope traces of synchronized dual wavelength dissipative solitons.

Through Careful control of the cavity birefringence, experimentally we found that the dual wavelength solitons could also be transformed into a single wavelength vector dissipative soliton, as shown in Figure 5.9. To obtain the result we have significantly changed the orientation of the PC paddles while the pump strength was kept fixed. On the oscilloscope trace the previous two solitons now merged together, consequently only one pulse could be observed in the cavity. Checked with a high speed oscilloscope (50 GHz) combined with a commercial autocorrelator (FR-103MN), no fine structures were detected within the pulse, confirming it is vector soliton. Note that the central

Hapter 5: Dissipative Vector Solitons

wavelength of the vector dissipative soliton has now shifted to a value between those of the linearly polarized solitons shown in Figure 5.7a. The polarization resolved study on the state further confirmed that the soliton pulse was elliptically polarized, and its two orthogonal components have comparable spectral intensity and the same central wavelength .

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

Related Journal Articles & DOI Links

Selected peer-reviewed publications relevant to 12 Lead ECG Acquisition. Click the DOI to access the full paper (may require institutional access).

Why Choose Us?

Bangalore guidance for robotics, Spectre and autonomous systems projects.