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CHARACTERIZING AND CONTROLLING EXTREME OPTICAL NONLINEARITIES IN PHOTONIC CRYSTAL FIBERS By SHENGBO XU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Shengbo Xu This dissertation is dedicated to my family, for their love and support. ACKNOWLEDGMENTS I would like to express sincere acknowledgements to my advisor, Dr. David Reitze, for leading me to this exciting field. His experience, expertise and constant wise guidance are invaluable to my doctoral research and the completion of this dissertation. I would also like to thank him for everything he did during my internship at Intel, without his help and encouragement, I could never have this chance to broaden my horizon and expand my expertise. I could not have imagined having a better mentor for my Ph. D. My thanks go to the fellow graduate students in the group: Xiaoming Wang, Vidya Ramanathan and Jinho Lee, for their open minds, all the supports and helpful discussions. You guys are a great fun team to work with. I would also like to thank the support stuffs in the department, especially Darlene Latimer, for making my life a lot easier outside research. Finally, I want to thank my parents and my wife Yuanjie Li for their love and encouragement all the way since the beginning of my studies, without their enthusiasm this dissertation could not have been possible. I love you. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ........... ........ ....... ....... .. ........... ......... .. vii LIST OF FIGURES ............... .......................... ........... .......... ..... viii ABSTRACT .............. ..................... .......... .............. xii CHAPTER 1 INTRODUCTION ............... .................................. ................... 1 2 PHOTONIC CRY STAL FIBERS ........................................ .......................... 8 2.1 Conventional Fiber Optics Propagation, Dispersion, and Optical N onlinearities ................................... ..................................... 2.2 Nonlinear Fiber Optics: Extended Nonlinear Schrodinger Equation .................20 2.3 Supercontinuum Generations in Photonic Crystal Fibers.............................. 25 2.3.1 Photonic C rystal Fibers ...................... .................. ............... .... 26 2.3.2 Supercontinuum Generation in Microstructured Fibers.............................31 2.3.3 Applications of PCFs and Supercontinuum Generation ..........................37 3 OVERVIEW OF EXPERIMENTAL TECHNIQUES .............. ............... 40 3 .1 U ltrashort L aser P ulses ......... ................. .........................................................40 3.2 Laser Pulse Term inology ............................. ............................41 3.2.1 Laser Pulse Time Bandwidth Product (TBP)..........................................41 3.2.2 L aser Pulse Phase and C hirp ........................................... .....................43 3.3 Femtosecond Lasers (Ti:Sapphire Lasers).......... .......................................45 3.3.1 Ti:Sapphire Crystal ........................................ ...................... ........ .... 45 3.3.2 Kerr Lens ModeLocking .... ......................................46 3.3.3 Cavities of Ti:Sapphire Lasers .............. ........................... .. ............. 48 3.3.3.1 Homebuild Ti:Sapphire oscillator..................... ................. 48 3.3.3.2 Coherent M ira 900 system .................................... ............... 50 3.4 FR O G and SPID E R ............................................................................. ............51 3.4.1 AutoCorrelation and CrossCorrelation .......................................... 52 3.4.2 Frequency Resolved Optical Gating (FROG) .........................................55 3.4.3 Spectral Phase Interferometry for Directelectricfield Reconstruction (S P ID E R ) ..................................................... ................ 5 9 3 .5 P u lse S h ap in g ................................................... ................ 6 1 3.5.1 Femtosecond Pulse Shaping ..... ................. .... ......... .................... 62 3.5.2 Fourier Domain Pulse Shaping Using Spatial Light Modulator ...............64 3.5.2.1 Fourier dom ain pulse shaping ................................. .. ............ 64 3.5.2.2 Liquid crystal spatial light modulator (LCSLM)..........................67 3.5.2.3 Experim ental considerations ................................. ................ 68 4 CONTROL OF SUPERCONTINUUM GENERATION IN PCFS USING OPTIMALLY DESIGNED PULSE SHAPES ...................................... ...............72 4.1 Control of Supercontinuum Generation in PCFs...................................... 73 4.2 Open Loop Control Experiment Setup and NLSE Simulation...........................77 4.3 Influences of Quadratic and Cubic Spectral Phase on Propagation Dynamics ....81 4.4 Open Loop Control of SelfSteepening Nonlinear Effect ..................................91 5 CLOSED LOOP CONTROL OF THE SUPERCONTINUUM GENERATION IN PCFS VIA ADAPTIVE PULSE SHAPING................................... .................97 5.1 Closed Loop Control and Genetic Algorithm s ..................................................97 5.2 Enhancement of Spectral Broadening via Adaptive Pulse Shaping ................... 103 5.3 Adaptive Control of Soliton SelfFrequency Shift ......................... ..........109 6 ROUTES TO PULSE COMPRESSION USING DISPERSIONFLATTENED MICROSTRUCTURED FIBER: SIMULATIONS USING THE NONLINEAR SCHRODINGER EQUATION ............... ... .................. .....................119 6.1 M otivation and Overview of Pulse Compression......... ....................... .........1. 19 6.2 Simulation of Pulse Compression Using Dispersion Flattened Microstructured F ib er ................ ............... ........ .... ........................ .......... .. ...... 124 6.2.1 Dispersion Flattened Microstructured Fibers .................................. 125 6.2.2 Simulations of Pulse Compression........................ ............... 126 6.3 Supercontinuum Pulse Compression at 800 nm............................................128 6.3.1 Supercontinuum Generation Using DFMF at 800 nm ...........................128 6.3.2 Validation of Fidelity of Supercontinuum Pulse Compression ...............131 6.3.3 Supercontinuum Pulse Compression Results at 800 nm ........................135 6.4 Supercontinuum Pulse Compression 1550 nm.............................................138 6.5 Supercontinuum Pulse Compression 1050 nm.............................................141 7 C O N C L U SIO N ......... .................................................................... ......... .. ..... .. 146 LIST OF REFEREN CES ....... .......... .. ........... .............................. ............... 150 BIOGRAPH ICAL SKETCH ....................... .......... ........................................... 162 LIST OF TABLES Table page 3.1 Time bandwidth products (K) for Gaussian and hyperbolic secant pulse shapes....43 3.2 Relations between pulse duration and autocorrelation function duration for Gaussian and Hyperbolic secant function. .................................... .................53 4.1 List of the microstructured fiber parameters used in the NLSE simulation model..81 5.1 Illustration of the crossover operator and mutation operator (binary chrom osom e).................... ...... ..... .............. ...... .......... ........... 100 6.1 List of DFMF parameters in the NLSE simulation model at 800 nm....................129 6.2 List of DFMF parameters in the NLSE simulation model at 1550 nm..................139 6.3 List of the nonlinear microstructured fiber parameters in the NLSE simulation m odel at 1050 nm ...................... .. .... ....................... .. ...... .... ...........142 LIST OF FIGURES Figure page 2.1 Illustrations of the group velocity for pulse envelope and phase velocity of the underlying field. .................................................... ................. 14 2.2 Illustration of material dispersion for fused silica (zero GVD at 1.3 [tm) as well as the waveguide dispersion contribution to the chromatic dispersion ..................15 2.3 An example of 210 nm supercontinuum generation in the conventional fiber (dashed line). ........................................................................ 18 2.4 Temporal evolution over one soliton period for the firstorder and the thirdorder soliton ............................................................................... 24 2.5 Illustrations of selfsteepening and selffrequency shift nonlinear effects...............25 2.6 Crosssection electron micrograph of microstructured fiber and photonic band g a p fib e r ...................................... .................................. ................ 2 7 2.7 Illustrations of tailorable dispersion properties of microstructured fibers ..............30 2.8 An ultra broadband supercontinuum generated in a 75cm section of m icrostructure fiber. ...................... .................. ................... ..... ...... 32 3.1 Illustration of Gaussian pulses with linear chirps. ............. ................................... 44 3.2 Normalized absorption and emission spectra of Ti:Sapphire for 7t polarized light..46 3.3 Kerr lens modelocking principle: selffocusing effect by the optical Kerr effect....47 3.4 Schematic diagram of the Ti:Sapphire Laser and the external phase compensator. ........................................................................................................ . 4 9 3.5 Second order intensity autocorrelation and spectrum of Ti:Sapphire laser pulse....49 3.6 Schematic diagram of a Coherent Mira 900 Ti:Sapphire laser .............................50 3.7 FROG measurement of a Coherent Mira 900 Ti:Sapphire laser pulse. .................51 3.8 Schematic diagram of a SHG autocorrelator. ................................. ...............52 3.9 Schematic diagram of a SHG FROG apparatus ................................................. 57 3.10 An example of experimental FROG trace and retrieved spectral intensity/phase profile correspond to the FROG trace ....................................................................58 3.11 Plot of an ideal interferogram ........................................................ ............... 59 3.12 Schematic diagram of a SPIDER apparatus .................................. ............... 60 3.13 Principle of acoustooptic programmable dispersive filter (AOPDF). ..................63 3.14 Schematic diagram of a Fourier domain pulse shaping apparatus using a LC S L M ...................................... ..................................... ................ 6 5 3.15 A sectional view of a liquid crystal layer between two glass plates ......................67 3.16 Illustration of LCSLM array that consists of 128 LC pixels. ............................. 68 3.17 CRI SLM phase shifting curves as a function of drive voltage for750nm, 800nm and 850nm center w avelength ..................................................................... ... ..70 4.1 Subnm supercontinuum feature fluctuations as a result of input pulse power flu ctu atio n ..................................................... ................ 7 5 4.2 An example of supercontinuum generation in experiment. ....................................79 4.3 Schematic diagram of the open loop control experiment.................... ........ 80 4.4 Simulation results of the pulse peak intensity as a function of pulse spectral phase .................... ...... ... .. .......... .............................................83 4.5 Experimental results of supercontinuum generation dependence on the input pulse quadratic and cubic spectral phase....................................... ............... 84 4.6 Experimental results of supercontinuum generation bandwidth from 5 cm and 70 cm microstructured fibers as a function of input pulse cubic spectral phase...........87 4.7 Experimental and simulation results of supercontinuum generation bandwidth from a 45 cm m icrostructured fiber...................................... ........................ 88 4.8 SHG FROG measurements of the spectral intensities and spectral phases of the pulses before and after the phase compensation using a pulse shaper ...................90 4.9 Spectral blueshifted asymmetry due to selfsteepening. ........................................91 4.10 SHG FROG measurement results of phase sculpted forward "ramp" pulse generation ............................................................................93 4.11 Experimental and simulation results of suppression of selfsteepening effect using preshaped forward ramp pulse. ........................................ ............... 94 4.12 Simulation results of the evolutions of the pulse temporal intensities as a function of propagation length in the microstructured fiber ................. ........ 95 5.1 Schematic diagram of the closed loop control (adaptive pulse shaping) experim ent........... ........ ........... ............. ............ ........... 103 5.2 Experimental results of closed loop control of optimization of supercontinuum generation band idth. ............................................... ....................................... 105 5.3 Evolutions of phase pattern for adaptive pulse shaping experiment of supercontinuum bandwidth enhancement. ................................... ...............107 5.4 Experimental results of closed loop control of soliton selffrequency shift...........111 5.5 An example of the experimentally determined input driving pulse in the frequency domain and time domain. ............. ..................................112 5.6 Simulation results of soliton central wavelength as a function of input pulse peak in te n sity ................................ ................................................... .. 1 14 5.7 Spectra comparison of experimentally optimized soliton generation and simulated soliton using experimentally determined driving input pulse..............115 5.8 Simulation results of the sensitivity of soliton generation to the input driving pulse fluctuation. .............................. .............. ......... .. ........ .. .. 116 5.9 Simulation investigation of the sensitivity of soliton selffrequency shift to the fi ber dispersion properties. .............................................................. ..... ............. 117 6.1 Schematics of pulse compression simulation using supercontinuum generation and dispersion flattened microstructured fiber................... ................... ................127 6.2 Dispersion profile for the DFMF around Ti:Sapphire laser wavelength at 800 nm. ...................................................................................................... . 1 2 8 6.3 Supercontinuum output and spectral phase with 30 fs 10 kW TL input pulse propagating through 9 cm DFM F.................................... .......................... 130 6.4 Simulation of output supercontinuum spectral intensity and phase variations w ith 1% input pulse noise. ............................................ ............................. 131 6.5 Calculated coherence as a function of wavelength from the electric field of continuum generation for 5% RMS input pulse power noise in 100 runs. ............133 6.6 Simulated continuum generation for DFMF fiber and a regular microstructured fi b er. ........................................................................... 13 4 6.7 Simulated supercontinuum pulse compression for DFMF fiber ..........................136 6.8 Simulated supercontinuum pulse compression results using DFMF for different fiber lengths and input pulse peak powers at 800 nm. .........................................137 6.9 Simulated supercontinuum generation and coherence for DFMF at 1550 nm. .....139 6.10 Simulated supercontinuum pulse durations after compression using DFMF at 1550 nm for different fiber lengths and different input pulse peak powers...........141 6.11 Dispersion curve for a nonlinear microstructured fiber. ......................................142 6.12 Supercontinuum generation and the coherence in the nonlinear fiber for 1 cm fiber length and 400 kW input pulse peak power........................ ............ 143 6.13 An example of continuum temporal pulse before and after phase compensation for the nonlinear fiber at 1050 nm ............................................................ ........ 144 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARECTERIZING AND CONTROLLING EXTREME OPTICAL NONLINEARITIES IN PHOTONIC CRYSTAL FIBERS By Shengbo Xu May 2006 Chair: David H. Reitze Major Department: Physics The development of the photonic crystal fibers (PCFs or microstructured fibers) has been one of the most intellectually exciting events in the optics community within the past few years. The introduction of airhole structures in PCFs allows for new degrees of freedom to manipulate both the dispersion and optical nonlinearities of the fibers. Not only the zero groupvelocitydispersion of a PCF can be engineered from 500 nm to beyond 1500 nm, but the extremely high optical nonlinearities of PCFs also lead to ultrabroadband supercontinuum generation (>1000 nm) when pumped by nanoJoules femtosecond Ti:Sapphire laser pulses. Therefore, PCF is an ideal system for investigating nonlinear optics. In this dissertation, we present results of controlling nonlinear optical processes in PCFs by adjusting the input pulse properties and the fiber dispersion. We focus on supercontinuum, resulting from the extreme nonlinear processes. A simulation tool based on the extended nonlinear Schrodinger equation is developed to model our experiments and predict output spectra. To investigate the impact of input pulse properties on the supercontinuum generation, we perform open and closedloop control experiments. Femtosecond pulse shaping is used to change the input pulse properties. In the openloop (intuitively designed) control experiments, we investigate the effects of input pulse spectral phase on the bandwidth of supercontinuum generation. Furthermore, we use phasesculpted temporal ramp pulses to suppress the selfsteepening nonlinear effect and generate more symmetric supercontinuum spectrum. Using the genetic algorithm in closedloops (adaptive) control experiment to synthesize the appropriate temporal pulse shape, we enhance the supercontinuum generation bandwidth and perform control of soliton self frequency shift. For both the open and closedloop control, simulation results show good agreement compared with the experiment optimized spectra. To our knowledge, this is the first time that femtosecond pulse shaping has been used to control the pulse nonlinear propagations in PCFs. We also develop a pulse compression model to study how the microstructured fiber dispersion characteristics can affect the supercontinuum temporal compressions. Using numerically simulated dispersionflattened microstructured fibers at different wavelengths, simulation results show that it is possible to compensate the stable supercontinuum spectral phase and compress the pulse to the fewcycle regime. CHAPTER 1 INTRODUCTION Optical fibers have tremendous impact on the modern world. In 1979, progress in the fiber fabrication technology made it possible to manufacturer optical fibers with loss merely 0.2 dB/km in the 1550 nm wavelength region [1]. Conventional optical fibers are made of fused silica, which has zero group velocity material dispersion wavelength at 1300 nm. The design of the conventional fiber (e.g., core size, refractive index differences between core and cladding) can shift the zero GVD of the fiber to 1550 nm. Optical fibers at this wavelength with low loss and small dispersion have revolutionized the telecommunications industry [2]. Meanwhile, optical fibers have also rapidly progressed the field of nonlinear fiber optics in the last 30 years [3]. Various nonlinear effects, including selfphase modulation, stimulated Raman scattering, parametric four wavemixing, etc., have been studied extensively and the theory is well established by now. The emergence of photonic crystal fibers (PCFs) [4, 5] and their ability to easily generate supercontinuum have been one of the hottest topics in the optics community for the past several years. Photonic crystal fibers (e.g., microstructured fibers and photonic bandgap fibers) exhibit many special properties when compared to that of conventional fibers, mainly because of their unique design structures. Microstructured fibers [6] consist of a solid silica core surrounded by an array of air holes running along the fiber and its light guiding mechanisms is similar to that of conventional fibers. Photonic bandgap fibers (PBFs) [7] have a hollow core surrounded by the airholes array, and the guided light is confined to the low index core by the photonic bandgap effect. The air holes configuration of the PCFs gives a whole new level of freedom in the fiber design. Many PCFs that exhibit novel dispersion and nonlinear properties have been developed by carefully choosing the core size and airfilling fraction (airhole size and pitch) of the PCFs. In particular, the zero group velocity dispersion wavelength of the PCFs can be engineered at any wavelength from 500 nm to above 1500 nm, which evidently make the Ti:Sapphire laser wavelength in the anomalous dispersion region of the PCFs. The tailorable dispersion properties and high nonlinearities exhibit in the PCFs lead to the ultrabroadband supercontinuum generation. The impact of the supercontinuum generation on the optical community is phenomenal and it has led to a renewed interest in investigating nonlinear fiber optics. In principle, the nonlinear mechanisms that lead to the supercontinuum generation for PCFs are similar to that of conventional fibers. However, the unique dispersion properties and high nonlinearities of the PCFs determine that the magnitude of the nonlinear processes is large in magnitude and the required power for the input pulse to generate sufficient broadband spectrum is much less than that for the conventional fibers. As a result, over an octavespanning ultrabroadband supercontinuum can be generated using Ti:Sapphire laser pulses with only nJlevel energy [6]. The PCFs and the subsequent supercontinuum generation exhibit unparalleled properties, well beyond what conventional fiber can ever offer. These properties have distinguished themselves through a wide range of applications in the everwidening area of science and technology. For example, the most important application of the supercontinuum generation lies in the field of optical metrology. The ultrabroad bandwidth coherent light makes the measurement of relative frequency offsets possible. This led to the work of absolute optical frequency measurement which is one of the subjects of the 2005 Nobel price in Physics by Theodor Hansch and John L. Hall [8]. Because supercontinuum generation in PCFs is a nonlinear effect in the extreme sense of "nonlinear," researchers have put a great deal of effort to understand what kind of nonlinear interactions in the PCFs lead to the supercontinuum generation, what role and order and magnitude they are playing in the generation process. During this exploration, various simulation models have definitely played an important role. Simulation models based on the nonlinear Schrodinger equation are by far the most successful methods and have been well accepted by many researchers [9]. Now, it is commonly understood that supercontinuum generation is initialized by high order soliton generation, followed by the soliton splitting among with other nonlinear effects [10, 11]. Meanwhile, however, broadband noise of the supercontinuum generation in the PCFs has led to amplitude fluctuations as large as 50% for certain input pulse parameters [12]. Furthermore, supercontinuum generation is found to be extremely sensitive to the input pulse noise. Both the experiment and simulation have revealed that for the reasonable power fluctuations of the laser systems, subnanometer spectral structures of the supercontinuum vary from shot to shot [9, 13]. This problem greatly limits the application scope of supercontinuum generation. Although controlling supercontinuum generation (or any nonlinear process) is in general difficult because of the intrinsic fiber nonlinearity response to the input electric field, control of the supercontinuum generation is desirable and it will allow us to tailor the supercontinuum properties to suit for a specific application. It is also commonly understood that the supercontinuum generations depend critically on both the input pulse properties and PCF characteristics. In this dissertation we will present the results of controlling supercontinuum generations using both of these two approaches. In particular, we investigate how the femtosecond pulse shaping can be used to control the supercontinuum generation by controlling the evolution of the temporal and spectral structure of optical pulses propagating in the microstructured fibers. We also simulate the supercontinuum pulse compression using the dispersionflattened microstructured fibers. Femtosecond pulse shaping promises great advantages to the fields of fiber optics and photonics, ultrafast spectroscopy, optical communications and physical chemistry. In general, the pulse characteristics (e.g., pulse intensity modulation and pulse spectral chirp) one wants to use for a specific application may be different from what a laser system can directly offer. On the other hand, when optical pulses travel in a complex optical system, the optical components in the system will introduce significant amount of dispersion, which compromises the characteristics of ultrashort optical pulses. All these problems can be solved using femtosecond pulse shaping to alter the pulse intensity temporal profile for a specific application. Meanwhile, working with adaptive search algorithms (such as genetic algorithms) to efficiently search all the possible solutions for a direct target, adaptive pulse shaping has proven to be very useful when an intuitive driving pulse can not be directly derived. In this dissertation, we use femtosecond pulse shaping to change the input pulse properties and control the supercontinuum generations in the PCFs. Depending on whether an intuitive driving pulse is available, we present the results of both open and closedloop (adaptive) control experiments. Openloop control schematics utilize underlying physics to derive a suitable driving pulse. The experiment results of pulse shaping that change both the pulse spectral chirps and pulse intensity modulation (ramp pulse) are presented. Meanwhile, a simulation model based on the extended nonlinear Schrodinger equation using a splitstep Fourier method is developed. We use this simulation tool to model our experimental control results. We use the adaptive pulse shaping to control supercontinuum bandwidth and soliton selffrequency shift. Again, the nonlinear Schrodinger equation modeled results for the adaptive pulse shaping helps us to understand why the final derived driving pulse can interact with the PCF and achieve the desired supercontinuum. To our knowledge, this is the first demonstration on how femtosecond pulse shaping can be used to control the evolution of the temporal and spectral structure of optical pulses propagating in PCFs. Pulse compression is one of the most obvious, yet most challenging, applications of the supercontinuum generations in PCFs. Because of the ultrabroad bandwidth of the supercontinuum, pulse duration of the compression results is expected to be within the fewopticalcycle regime. However, the fluctuations in the spectral phase of the supercontinuum generation for the input pulse and propagation noise greatly increase the difficulties of performing the pulse compression. Experimentally, the coherence of the supercontinuum (used as a benchmark for the potential compressibility of the supercontinuum generation) can be severely compromised due to the inherent fluctuations of the input pulse. However, theoretical investigations have revealed that the coherence increases linearly with a shorter fiber propagation length [14] and there exists an optimum compressed distance at which compressed pulses with negligible fluctuation and time shift can be obtained [15]. Therefore all current pulse compression experiments involve a short piece of PCF, ranging from several mm to cm. Pulse characterization techniques such as FROG or SPIDER are used to retrieve the spectral phase of the supercontinuum. Current state of art compression experiment uses femtosecond pulse shaping (or even adaptive pulse shaping) with adequate phase compensation to generate compress pulses several femtoseconds in duration [16, 17]. Meanwhile, the dispersion of the PCFs is one of the main reasons that lead to the instabilities of the supercontinuum spectral phase, as evidenced by the increase in coherence when shortening the pulse propagation length. The unique PCFs structures provide another way to alter (minimize) the fiber dispersion even for a wide wavelength (over 300 nm, see Reeves et al. [18]). We use the simulation model based on the idea of dispersionflattened PCF to perform the pulse compression for the controlled supercontinuum generation. The layout of this dissertation is described as follows. An introduction of the PCFs will be given in chapter 2, including the discussion of both the dispersion properties and nonlinearities of the conventional fibers and PCFs, as well as a layout of the nonlinear fiber optics: the extended nonlinear Schrodinger equation. In chapter 3 we will give an overview of some relevant experimental techniques, which consist of the methods of generating, characterizing, and temporal tailoring ultrashort pulses. Openloop investigations of the influences of input pulse linear and quadratic chirp on the supercontinuum generation will be given in chapter 4, as well as the control results of suppressing the pulse selfsteepening nonlinear effect in the supercontinuum generations. Chapter 5 discusses the closedloop control experiments (adaptive pulse shaping), including the investigations of spectral broadening enhancement, soliton selffrequency 7 shift. We will discuss our supercontinuum generation pulse compression model and results using the dispersionflattened PCF in chapter 6. Finally, in chapter 7, we present our conclusions. CHAPTER 2 PHOTONIC CRYSTAL FIBERS The main work of this dissertation concerns the application of shaped femtosecond optical pulses in controlling the nonlinear interactions in photonic crystal fibers (PCFs). In this chapter, we will first summarize the propagation, dispersion and optical nonlinearities in conventional fibers in section 2.1. This will lead to the discussion of nonlinear fiber optics (the extended nonlinear Schrodinger equation), including various nonlinear effects that govern the nonlinear pulse propagation in the conventional fibers and PCFs. In section 2.3, we will discuss various configurations of PCFs that have different fiber properties (nonlinearities and dispersion) and applications, followed by the discussion of mechanisms of supercontinuum generations and its applications. 2.1 Conventional Fiber Optics Propagation, Dispersion, and Optical Nonlinearities Pulses propagating in the PCFs follows the same dispersion and nonlinear propagation principles as that in the conventional fibers, although the nonlinearities exhibited in the PCFs are in much larger scales because of the special configuration (smaller mode volume, zero dispersion point, dispersion profile) of PCFs. Nonlinear pulse propagation in the conventional fibers has been studied extensively over the past few decades and the theories are well established. Therefore, before beginning a discussion of PCFs, it is important to understand how conventional fibers work. Section 2.1 will give an overview of dispersion effect and optical nonlinearities in conventional fibers, followed by the wellestablished nonlinear pulse propagation theory (the nonlinear Schrodinger equation) in section 2.2. Uncladded glass fibers were first fabricated in the 1920s. It was not until the 1950s that people realized that using a cladding layer is extremely important to the fiber characteristics and fiber optics experienced a phenomenal rate of progress [19]. In the simplest form, conventional optical fibers are cylindrical dielectric waveguides (fiber core), made of low loss materials such as silica glass, surrounded by a cladding layer made of doped silica glass with slight lower constant refractive index than the core. Such fibers are generally referred as stepindex fibers. In gradedindex fibers, the refractive index of the core decreases gradually from center to the core boundary. Modem fabrication technology produces fiber optical loss close to the theoretical minimum [1] (0.2 dB/km at 1550 nm), a loss level determined by the fundamental Rayleigh scattering in silica. The availabilities of low loss silica fibers led to a revolution in the field of optical fiber communications as well as the emergence in the field of nonlinear fiber optics. Guidance of light in optical fibers based on total internal reflection (TIR); that is, an optical fiber consists of a central core of refractive index nl, where the light is guided, surrounded by an outer cladding area of a slightly lower refractive index n2, the light rays with incident angle on the corecladding interface greater than the critical angle 0~ = sin' (n2 In1) experience TIR. Therefore, the light rays incident on the fiber end satisfied the TIR can be guided without any refractive loss in the fiber core. The fiber can guide the light rays with different incident angles (different modes), as long as TIR is satisfied. In general, however, the TIR theory is too simplistic to explain fiber modes and propagation. One needs to start from Maxwell's equations and apply appropriate fiber boundary conditions to derive the mode solutions of the wave equation. A brief discussion will be given in this section. Interested readers can refer to Buck [20] and Agrawal [3] for more detailed discussion. Starting from Maxwell's equations in the optical fibers, V xE = a /t, VxH = Qd/Qt, X 0, /(2.1) VD = 0, V =B 0, where E and H are electric and magnetic field vectors, D and B are electric and magnetic flux densities. Using the constitutive relations in optical fibers given by D = sEo + P, Ef =' (2.2) B = ,H, one finds 1 72E i21 VxVxE2= I ato (2.3) C2 a2 10 t2 In general, the polarization P can be expanded in powers of the electric fieldE P = c0 ().+Z(3) EEE +..). (2.4) X(3) is the third order susceptibility that governs the third order nonlinear effects. In general, (2) (second order susceptibility) should also be present in equation 2.4. However, even order nonlinear effects disappear for silica fiber as required by symmetry and are not included in the equation. Solving equation 2.3 requires the nonlinear (3) effect to be included, which will lead to the derivation of the extended nonlinear Schrodinger equation discussed in the next section. For the mode propagation discussion, one can treat the nonlinear effect as a small perturbation to the total induced polarization and simply ignore it in the following discussion. This lead to the electric field wave equation 2 V2E + n 2(0) E = 0, (2.5) C where n(co) is the refractive index of the fiber and E(r, wo) is the Fourier transform of E(r, t). For an optical fiber, the natural symmetry is that of a cylinder. Thus, expressing the wave equation (2.5) in cylindrical coordinates p, q and z, and assuming E (r, m) = A(om)F(p) exp(+im ) exp(ifiz), (2.6) where A is a normalization constant, / is the propagation constant (/iz represents the phase of the field), and m is an integer, F(p) is the solution of the wellknown differential equation for Bessel functions d 2F 1 dF 22 m) 2 0, dp p dp p (2.7) with solutions taking the form F(p) = J (p), p a, (2.8) F(p) = Km (yp), p > a, where a is the fiber core radius and K =2 k0 _/ 2, 12 n22 2 (2.9) 2 =2 n2 k . Similar derivations for the magnetic field from equation 2.5 to 2.9 can be obtained. Simply applying the boundary conditions that the tangential components of the electric field and magnetic field are continuous across the corecladding interface, one can derive the eigenvalue equation as J. (Ka) K (y) Jn (a) n K(a) mf ko (n2 n10) cKJ (Ka) )K (ya) J. (aa) n 2 yK (a) an1K2 2 ) In general, the eigenvalue equation 2.10 has several solutions for / for each integer m It is customary to express these solutions by /mn, where both m and n are integers and represent mode indeces. Each eigenvalue iam corresponds to one specific mode supported by the fiber. The corresponding modal field distribution can be obtained from equation 2.6. There are two types of mode, designated as HEmn and EHm,. For m = 0, these modes corresponds to the fundamental transverseelectric (TE) and transversemagnetic (TM) mode; whereas for m > 0, all six components of the electromagnetic field are nonzero. The number of modes supported by the fiber depend on its design parameter, namely the core radius and the corecladding index difference n2 n1. It is useful to define a normalized frequency parameter and it has a straightforward way to determine how many eigenmodes an optical fiber can support. Normalized frequency parameter can be expressed as, V = ( ,, n1 n22, (2.11) where ko = 2/20A is wave number corresponds to the central wavelength of the incident light in the vacuum. Using the normalized parameter, a complete set of eigenmodes can be derived. When the normalized frequency parameter is smaller than 2.405 (from the solution of the first zero of the Bessel function), only the fundamental mode is allowed to propagate, in which case the fiber is called a singlemode fiber. Note that with fixed fiber parameters, there is a lowest wavelength (cutoff wavelength) that the fiber can sustain singlemode fiber. Singlemode fibers are widely used in pulse applications. The description of fibers includes two physical effects, fiber dispersion and nonlinear effects, both of which will be laid out in more detail in the next section. Below, we will take an example of optical fibers in telecommunications to briefly illustrate how these two aspects influence the performance of high speed optical networks. Optical fibers have revolutionized the telecommunications industry within the last decade. Singlemode fiber allows for a higher capacity to transmit information via the light pulses because it can retain the fidelity of each light pulse over longer distances. Supercontinuum generation offers the possibility of generating over 1000 dense wavelengthdivision multiplexing channels (DWDM) using only one single laser source, using over 10 THz single mode fiber bandwidth [21]. However, fiber dispersion, apart from the optical loss, will damage the fidelity and cause intersymbol interference (ISI). Modem communication requires a high data transmission rate when using a singlemode fiber, thus short pulses are preferred. For propagating optical pulses, the modepropagation constant / solved in equation (2.10) can be expanded in a Taylor series at (o where the pulse spectrum is centered, C 1 f() =n(o) = /0o +1(0 )0+ /2 ( 0)2 + ' (2.12) c 2 where = d (m = 0, 1,2,...). (2.13) /8 m [ d o0) o COO 14 The parameters /1 and /2 can be expressed as 1 ng 1 ( dn ,8 (n + o) ,,) v (2.14) S1( dn d2n S= (2 + 2 c do do2, where ng is group index and Vg is group velocity. The envelope of an optical pulse moves at the group velocity Vg and the phase of underlying field evolves at phase velocity vp, as shown in figure 2.1. /2 is commonly referred as group velocity dispersion (GVD), and the convention is that /2 > 0 for normal dispersion and /2 < 0 for anomalous dispersion. GVD is mainly responsible for pulse broadening. The dispersion parameter D (in the unit ofps/km nm) is commonly used in the fiber optics literature in stead of f2. Its relation to /2 is d,1 2 zc D c= 2. (2.15) ccA A U) 4j Time Figure 2.1: Illustrations of the group velocity for pulse envelope and phase velocity of the underlying field. ,73 is the third order dispersion (TOD) and its inclusion is necessary when the pulse wavelength approaches the zero GVD point. Dispersion causes individual frequency components in a pulse to travel at a different velocity; thus, the individual frequency components making up the pulse separate and "diphase". As a result, the pulse duration increases as it propagates. For telecommunication applications, information distortion can be manifested in the temporal spreading and consequent overlap of individual pulses, thus ISI. This effect can be minimized by choosing the central wavelength of the pulse close to the zero dispersion point of the group velocity dispersion. S30 Material E 20 Dispersion S o10 Chromatic C Dispersion S0 Waveguide S20Dispersion S40 50 1 1 1 1 1100 1200 1300 1400 1500 1600 Wavelength (nm) Figure 2.2: Illustration of material dispersion for fused silica (zero GVD at 1.3 Lam) as well as the waveguide dispersion contribution to the chromatic dispersion. Refer to Goff and Hansen [71]. There are three contributions to the fiber dispersion (chromatic dispersion). The first is the material dispersion, which manifests itself through the frequency dependence of the material refractive index (n(co), as noted above). This intrinsic dispersion is determined by the material itself, and we have little control over this. The zero group velocity dispersion (GVD) for the fused silica is 1.3 jma, as shown in figure 2.2. Waveguide dispersion comes about because of the effective mode index of the dielectric waveguiding is wavelength dependent as well. The waveguide dispersion contribution can easily be manipulated. Changing the core size and the refractive index difference between the core and cladding can evidently change the waveguide dispersion. (As we will discuss below, the waveguide dispersion contribution of photonic crystal fibers is extremely larger compared to that of conventional fiber). Figure 2.2 illustrates that the zero chromatic dispersion of the fiber is shifted from the 1.3 jn zero material dispersion point to 1.55 jn because of the waveguide dispersion contribution, thus the name zero dispersionshifted fiber. Furthermore, some fiber designs allow the waveguide dispersion to be customized and fine tuned. For example, gradedindex fiber allows the refractive index of the core being designed in the form of a parabolic curve and decreasing toward the cladding [22]. Dispersioncompensating fiber (DCF) can be designed to have the exact opposite dispersion of the fiber that is used in a transmission system, therefore nullifying the dispersion caused by that fiber [23, 24]. However, all the designs and controls of the waveguide dispersion have limitations, with zero group velocity dispersion between 1.3 and 1.55 nm The last contribution to the fiber dispersion is the modal dispersion, which results from the group velocity differences between different guided modes in a multimode fiber. Modal dispersion cannot exist in a singlemode fiber, since by definition only one mode propagates in the fiber. Pulses propagating in optical fibers also experience nonlinear effects, apart from the dispersion effects. The magnitude of nonlinear effects is directly related to the input pulse peak intensity. These nonlinear effects can be generalized as generating new frequency components (colors), or characterized as effects corresponding to different nonlinear mechanisms such as selfphase modulation (SPM), stimulated Raman scattering (SRS), etc. All these different nonlinear effects will be discussed in details in the next section. Since SiO2 is a symmetric molecule, second order (2) effects such as secondharmonic generation and sumfrequency generation cannot happen in silica fiber as required by symmetry. The nonlinearities in optical fibers are small, but they accumulate as light passes through many kilometers of fiber, especially when dense wavelengthdivision multiplexing (DWDM) [25] packs many channels into a single fiber. Unfortunately, the optical nonlinearities in a fiber are also governed by a material dependent parameter, the nonlinear index of refraction n2, and therefore selection of the material forces us to live with the size of the nonlinearity. For fused silica, n2 = 2 x 10 16cm2 /W However, a fascinating manifestation of the fiber nonlinearity occurs through temporal optical solitons [26] (discussed in details in the next section), formed as a result of the perfect balance between the anomalous group velocity dispersion and SPM nonlinear effect. Soliton effects are evidently extremely useful for the telecommunication application. The shape of the fundamental soliton does not change during propagation as a result of GVD and SPM completely balancing each other at a critical power (the fundamental solitonpower), whereas higherorder solitons (providing certain higher input power levels) propagate in a periodic evolution pattern with original shape recurring at multiples of the soliton period. On the other hand, optical fibers allow high optical intensities to be maintained over relatively long fiber length; therefore can be used to enhance the nonlinear effects. For example, zerodispersionshifted fiber can have the zero GVD at 1550 nm where fiber has minimum loss (0.2 dB/km). As the intense pulse propagates in an optical fiber, SPM along with all other nonlinear effects can lead to significant broadening of the optical spectrum; this very broad and continuous spectrum is called supercontinuum (SC). SC generation was first observed in 1970 by focusing picosecond pulses into a glass sample as a nonlinear medium [27], SPM is the primary mechanism that led to spectral broadening. Since then, the nonlinear media used for SC generation has evolved from gases and liquids to optical fibers. Figure 2.3 is an example of supercontinuum generation in conventional fiber. A supercontinuum bandwidth of 210 nm was generated by using 500 fs pulse with over 4 m conventional fiber and the energy per pulse is 90 pJ [28]. Compensated 10 Spectrum 0(b) /, SCSp 30F 0' Filtered 40 [ 450 Spectrum 50 t 60 " 70 1450 1500 1550 1600 1650 1700 Wavelength [nm] ectru m Figure 2.3: An example of 210 nm supercontinuum generation in the conventional fiber (dashed line). Refer to Nowak et al. [28]. The high effective nonlinearities of optical fibers have led to a dramatic reduction in the pump power requirements compared with those for other nonlinear media. The high optical intensities maintained over the propagation length will trigger a variety of Cl 0 0 0o C j__F S other nonlinear effects other than SPM, generating a much broader SC with a relative low input pulse peak level. As a result, SC generation is much easier and has important applications in various fields such as telecommunication, optical metrology and medical science, some of which will be discussed in more details in section 3. The problem of wavelength mismatch between optical fiber's zero group velocity dispersion wavelength and femtosecond laser's operating wavelength great limits the application of conventional optical fibers being used for SC generation. Ti:sapphire lasers are the most predominately used tunable solidstate ultrafast laser, with a central wavelength around 800 nm and optical bandwidths of over 300 nm, producing pulses in duration from a few femtoseconds to several picoseconds and pulse energies from nJ for the oscillators to as much as 25J for amplified sources, which makes Ti:Sapphire a great candidate as the laser source for the SC generation. However, at this wavelength range, silica fiber has such a large material dispersion (100 ps/km.nm) that the contribution of waveguide dispersion can essentially be ignored [3]. As a result, the initial input pulse with a short duration and intensive peak power experiences massive fiber dispersion, the pulse duration broadens very fast and the peak intensity drops dramatically. Therefore, the breadth and strength of the significant nonlinear interactions as well as the abilities of generating a very broad supercontinuum for Ti: Sapphire ultrafast laser source are greatly diminished. The key technological advance which has revolutionized the ability to investigate fiber dispersion and more importantly optical nonlinearities is the invention of photonic crystal fibers. We will discuss this in great detail in section 2.3, but first will layout the theory of nonlinear fiber optics. 2.2 Nonlinear Fiber Optics: Extended Nonlinear Schr6dinger Equation For an understanding of the nonlinear phenomena in optical fibers as well as the mechanisms that leads to supercontinuum generation, it is necessary to consider the theory of electromagnetic wave propagation in dispersive nonlinear media, in particular, optical fibers. Before specifically focusing on photonic crystal fibers, in this section we review pulse propagation in optical fibers from a general perspective. This section follows closely to the Agrawal's "Nonlinear Fiber Optics" [3]. In a frame of reference moving at the group velocity of the pulse, an extended nonlinear Schrodinger equation (NLSE) that governs the optical pulse propagation in singlemode fibers can be derived under the slowly varying envelope approximation (Ao / a << 1) as OA a i in+ "nA T 2 +A =iy 1+ A(z,T) R(T)A(z,T ) dT' (2.16) 8z 2 n n! T" Co Ty where A(z, t) = A0 (z, t) e't) is the intensity temporal profile of the pulse, a is the fiber loss, ,/ terms correspond to the chromatic dispersion (CD) of the fiber. The mode propagation constant is fl )= no))= 2 ' (o (2.17) C m! where /m = / and describes the wavevector of the light in the fiber. y =  d m cAff w o 0eg is the nonlinear parameter, a very important parameter that determines the magnitude of the optical nonlinearity. The predominant nonlinearity for silica fiber is the third order nonlinearity, governed by (3.) since the secondorder nonlinearity dispears due to the inversion sysmmetry at the silica molecular level and the magnitudes of higherorder nonlinearities are too small for silica. The thirdorder nonlinear susceptibility is responsible for elastic nonlinear processes such as selfphase modulation (SPM), crossphase modulation (CPM), fourwave mixing (FWM) and thirdharmonic generation (THG). The nonlinear index coefficient n2 is related to (3) as n = Re(z3), (2.18) where no is the linear refractive index. Elastic nonlinear processes correspond to photonphoton interactions and no energy is exchanged between the electromagnetic field and the dielectric medium. THG and FWM are usually not efficient in optical fibers, unless special efforts are made to achieve phase matching. Nonlinear fiber optics also involves the stimulated inelastic scattering in which the optical field transfer part of its energy to the nonlinear medium via photon phonon interactions. These phenomena includes stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS). The right hand side of the extended NLSE (Eq. 2.16) accounts for the nonlinear response of the fiber. The response function R(t) can be written as R(t)= (1 fR)6(t)+ fRhR(t), (2.19) where fR =0.15 represents the fractional contribution of the delayed Raman response. The Raman response function hR (t) takes an approximate analytic form as ( r =12.2 fs and r2 =32 fs) 2 2 hR(t)= + 2 exp(t/r2)sin(t/rz). (2.20) ZT1 Z2 There are two very important length scales in nonlinear fiber optics: the dispersion length LD TO2 I 2 (2.21) and the nonlinear length LL = 1/(yo). (2.22) LD and L, provide the length scales over which dispersive or nonlinear effects become dominant for pulse evolution. For pulses shorter than 5 ps but wide enough to contain many optical cycles (width>>10 fs), the extended nonlinear Schrodinger equation can be simplified as AA a "+1 8" A 1 a'A,1, T A\ (2.23)i + A =Iy \AA +(A A)TA (2.23) 8z 2 ,, O T" cO OT R T. using a Taylorseries expansion such that \A(z,T T')2 & A(z, T) T' A(z, T)2 (2.24) and TR tR(t)dt. (2.25) The term proportional to 83 governs the effects of 3rd order dispersion and become important for untrashort pulses because of their wide bandwidth. The term proportional to ,82 is responsible for group velocity dispersion (GVD). It causes temporalpulse broadening when an unchirped pulse propagates in a singlemode fiber. Higher order dispersion become dominant when the input pulse central wavelength is near the zero GVD fiber dispersion or when the bandwidth of the pulse becomes a significant fraction of the central frequency. The term proportional to A1 A is responsible for SPM nonlinear effect, a phenomenon that leads to spectral broadening of optical pulses. In the anomalousdispersion regime of an optical fiber, interplay between GVD and SPM can cooperate in such a way that the pulse propagates as an optical soliton. In the normal dispersion regime, the combined effects of GVD and SPM can be used for pulse compression. The term proportional to o)0o is responsible for selfsteepening (SS) and shock formation. The last term proportional TR to is responsible for selffrequency shift (SFS) induced by intrapulse Raman scattering. A fascinating manifestation of the fiber nonlinearity occurs through optical solitons, formed as a result of the interplay between the nonlinear and dispersive effects, i.e., anomalous fiber dispersion. A soliton is a special kind of wave packet that can propagate undistorted over long distances. If we define a parameter N as N2 = LD NL = YPoT2 /,2 (2.26) the integer values of N are found to be related to the soliton order. In time domain, the solution to the nonlinear Schrodinger equation for pure soliton propagation produces a solution of the form A(t) = sech(t), as the solution given by Zakharov and Shabat using the inverse scattering method in 1971 [29]. The soliton order N is determined not only by the characteristics of input pulse (TO, P0 andy), but also the properties of the fiber itself (y and /2 ). Only a certain combination of these parameters can reach a soliton solution, given by an integer value ofN Note that to form a higher order soliton, the input peak power required increases quadratically, which is in turn more difficult. The shape of the fundamental soliton (N =1) does not change during propagation, whereas higherorder solitons propagate in a periodic evolution pattern with original shape recurring at multiples of the soliton period z0 = LD r / 2. As shown in figure 2.4 (a), fundamental soliton propagates in the fiber without distortion as GVD and SPM complete balance each other. For the higher order soliton (figure 2.4 (b)), SPM dominates initially but GVD soon catches up and leads to the pulse contraction; therefore higherorder solitons propagate in a periodic evolution pattern with original shape recurring at multiples of the soliton period. From a physical point of view, soliton is generated because of the interplay between selfphase modulation and anomalous group velocity dispersion. When the soliton pulse propagates through the fiber, selfphase modulation generates new frequency components that are redshifted near the redend and blue shifted near the blueend of the spectrum. As the red components travel slower than the blue components in the anomalous dispersion regime, selfphase modulation leads to a pulse narrowing effect which counteracts group velocity dispersion's pulse broadening effect. By carefully choosing selfphase modulation and anomalous group velocity dispersion parameters, the pulse itself can become a soliton by adjusting itself to a hyperbolicsecant shape during propagation to make such cancellation as complete as possible. 1.0 4 4  (a) (b) N3 E.0 i 1.0 S 0.20 543 2 1 0 1 2 3 4 5 . 4 3 .2 1 0 1 2 3 4 5 Time TITO Time TITO Figure 2.4: Temporal evolution over one soliton period for the firstorder and the third order soliton. (a) The first order soliton (fundamental soliton) propagates without distortion. (b) The third order soliton repeats itself over one soliton period. Two other important nonlinear effects occurring during pulse propagation are self steepening and selffrequency shift induced by intrapulse Raman scattering. Self steepening results from the intensity dependence of group velocity in such a way that the peak moves at a lower speed than the wings in time domain. As the pulse propagates along the fiber, the temporal shape becomes asymmetric, with its peak shifting toward the trailing edge [Fig. 2.5 (a)]. As a result of selfsteepening, the trailing edge becomes steeper and steeper as the pulse propagates which implies larger spectral broadening on the blue side as selfphase modulation generates blue components near the trailing edge. Intrapulse Raman scattering affects the pulse spectrum in such a way that the Raman gain amplifies the low frequency components of a pulse by transferring energy from the high frequency components of the same pulse. As a result, the pulse spectrum shifts toward the lowfrequency (red) side as the pulse propagates inside the fiber, a phenomenon referred as the selffrequency shift [Fig. 2.5 (b)]. 1.0 (a) Input pulse 200 (b) uspectm 0.8 Output pulse 2000 S 0.4 1000 0.2 oo 00 0.0 0 O.B 0.4 0.0 0.4 O.B 600 700 B00 900 1000 t (ps) Wavelength (nm) Figure 2.5: Illustrations of selfsteepening and selffrequency shift nonlinear effects. a) Selfsteepening effect in a nondispersive fiber propagation case. b) Spectrum showing the combined effects of selfsteepening and selffrequency shift, the spectral modulation is induced by selfphase modulation. 2.3 Supercontinuum Generations in Photonic Crystal Fibers Supercontinuum generations in photonic crystal fibers results from high nonlinearities and low fiber dispersion properties of the photonic crystal fibers. In this section we will first discuss various types of PCFs that have different dispersion and nonlinear properties, followed by the fascinating supercontinuum generation and applications. 2.3.1 Photonic Crystal Fibers The idea of photonic crystal fibers (PCFs) can be traced back to early 1970s [30], when it was suggested that a cylindrical Bragg waveguide with a central core surrounded by rings of high and lowrefractive index might be produced. However, it was not until 1990s when the advances in the technology enabled the fabrication of these complex waveguide structures [4]. It is not an overstatement to say that the development of photonic crystal fibers (PCFs) is one of the most exciting events in optics for the past few years. Its impact on the optics community has been phenomenal and has led to a renewed interested in investigating nonlinear optical phenomena. PCFs may be divided into two categories: microstructured fibers, in which light is guided in a solid core by the similar principle as that of conventional fibers, and photonic bandgap fibers where the guided light is confined to the low index (hollow) core by the photonic bandgap (PBG) effect. Microstructured fibers consist of a solid silica core surrounded by an array of air holes running along the fiber, as shown in figure 2.6 (a). Since the microstructured cladding area (the microstructured airfilled region) is a mixture of silica and air holes, an effective refractive index is used to calculate the modal properties [32]. Light can still be guided inside the core according to the principle of total internal reflection as in standard optical fibers because the cladding has a lower effective refractive index than the solid core. Photonic bandgap fibers exhibit a hollow core or a core made of a dielectric whose refractive index is lower than the silica refractive index [Fig. 2.6 (b)]. The guiding mechanism of the photonic bandgap fibers differs dramatically from the microstructured Figure 2.6: Crosssection electron micrograph of microstructured fiber and photonic bandgap fiber. (a) Microstructured fiber, refer to Ranka et al. [5]. (b) Photonic bandgap fiber, refer to Russell [31]. The darker spots with different shapes in both figures are air holes. fiber and conventional fibers. In photonic bandgap fibers, the periodicity of the airhole lattice enables to trap the light in the core by twodimensional photonic bandgaps [33]. The term photonic crystal comes from the analogy between electrons in the periodic potential of a semiconductor crystal and photons in a periodic index profile (formed by a regular air hole pattern). Similarly, photonic bandgap results from the wellknown Bloch Theorem, which in turn means no light propagation modes are allowed in the photonic bandgap. As a result, only light with a given wavelength range can be guided in the hollow or dielectric core, as the PBG effect makes propagation in the microstructured cladding region impossible [7]. The fabrication process of the PCFs is a basic stackanddraw method [31]. First, capillary tubes and rods made of silica are stack together. In this step, an arrangement of the capillary tubes allows changing the air silica structure, therefore, providing the control flexibility of effective index of the cladding area. The PCFs are then fabricated by feeding the stack into a hot furnace with a certain heating temperature at a proper speed, much in the same manner as conventional fibers. Various PCFs with complex air hole structures can be fabricated with different dispersion and nonlinear properties that are well suited for intended applications. Photonic crystal fibers exhibit many unique features when compared to standard optical fibers. The introduction of airholes structure allows new degrees of freedom to manipulate both the dispersive and nonlinear properties of the PCFs. Many parameters can affect properties of the PCFs dramatically. These parameters include the size and shape of airholes, air filling fraction, the choices of material used to fabricate PCFs as well as the dielectric core of photonic bandgap fibers. A detailed discussion of photonic bandgap crystals and fibers is beyond the scope of this dissertation, and can be found in Cregan et al. [7] and Knight et al. [34]. Only the properties of microstructured fiber will be discussed in the following section. The fiber mode theory describe in section 3.1 has found to be quite useful for describing mode propagation properties of the microstructured fiber. However, due to the intrinsic structure differences between the microstructured fibers and conventional fibers, it is crucial to be able to model the microstructured fiber in more rigorous ways, i.e., it requires field analysis using the exact boundary conditions of the microstructured air holes configuration. These microstructured fiber modal methods include beam propagation method [35], effective index modal [36], scalar beam propagation method [37] and vectorial planewave expansion method [38], multiple expansion method [39] and finiteelement method [40, 41], etc. As a result, a large number of new microstructured fibers have been designed with novel waveguide properties suitable for a wide range of applications. For instance, the effective index method allows rigorous calculation of effective refractive index of the cladding, which in this case is usually wavelengthdependent. The normalized frequency parameter Vfor a conventional fiber can eventually exceed singlemode limit of 2.405, providing a short wavelength and a large core radius. However, properly choosing the airfilling fraction of the microstructured fiber (defined as the ratio of the hole diameter d to the pitch of the lattice A), the normalized frequency parameter V can stay constant even at short wavelengths [42] or even with a large core radius [43, 44]. Making this constant V smaller than 2.405, a microstructured fiber can sustain the singlemode property over a very broad wavelength range, thus the name of endless singlemode microstructured fiber [42]. Other microstructured fibers that exhibit interesting properties include singlepolarization singlemode PCF [45], highly birefringent PCF [46] and cobweb microstructured fiber [47], etc. As mentioned in section 2.1, conventional fibers have large normal material dispersion at the Ti:Sapphire laser operating wavelength, with modifications of the waveguide properties having little impact. This large normal dispersion greatly diminishes the magnitude of nonlinear effects and impairs the possibilities of supercontinuum generation. The invention of microstructured fibers has opened new opportunities for exploring nonlinearities in microstructured fibers. The strong wavelength dependence of the effective refractive index of the cladding of microstructured fibers leads to a new range of dispersion properties that cannot be achieved with conventional fibers. The waveguide dispersion properties of the microstructured fibers strongly depend on the airfilling fraction and core size. For instance, increasing the airfilling fraction as well as decreasing the core size can dramatically increase the waveguide dispersion, allowing compensation of the silica material dispersion at any wavelengths range from 500 nm to beyond 1500 nm [48]. As a result, the zero group velocity dispersion wavelength can be pushed far below 800 nm [Fig. 2.7 (a)], making the Ti:Sapphire laser operating wavelength in the anomalous dispersion region and soliton propagation, for the first time, available for the visible 300 Silica strand b) in pu computed 20 0 ... ... ... ................. 2000 100 PCF measured . 08 .1oo 200 Bulk Silica  300 4  0.5 0.6 0.7 0.8 0.9 1.0 26 1,100 1.200 1,300 1.400 1.w00 1.000 1.700 Wavelength um wavW ungt (nm) Figure 2.7: Illustrations of tailorable dispersion properties of microstructured fibers. (a) Measured dispersion curve of a microstructured fiber with zero GVD below 600nm, refer to Knight et al. [48]. (b) Microstructured fiber with a flat dispersion (curve B) by adjusting pitch size as 2.41 pm and airfilling fraction as 0.22, refer to Reeves et al. [18]. wavelength range. Furthermore, by choosing the proper fiber parameters such as air hole size and pitch, one can easily tailor the dispersion characteristics of the microstructured fiber, such as fabricating fibers with very low and flat dispersion over a relatively broad wavelength range [18], e.g., dispersionflattened microstructured fiber (DFMF), as we can see in curve B in Fig. 2.7 (b). In general, a proper choice of airhole sizes and pitches enable to engineer a variety of dispersion profiles. This precise control of the fiber dispersion greatly expands the horizon of fiber applications and allows customized fiber with tailored dispersion characteristics being used in a desired application. The interest in microstructured fibers is not only because of their special waveguide properties, but more importantly, lies in the fact of the significant nonlinearities that exhibit in the microstructured fibers. Actually, the high efficiency of various nonlinear effects is directly related to the unique waveguide properties that allow the engineering of the zero dispersion point and tailoring the dispersion profile, in addition to the extremely small core area. When laser pulses with their central wavelength near the zero dispersion point propagate in microstructured fibers, the high peak intensities dominate the pulse evolution and LNL << LD. In particular, the pulse can still maintain a short duration and high intensity over a much longer fiber propagation length, inducing an extraordinary larger nonlinearity when compared to conventional fibers. As a result, this greater nonlinearity reduces the threshold pulse energy for observing nonlinear effects and nonlinear optical processes take place on a grand scale in microstructured fibers. These different nonlinear processes, including SPM, SRS, FWM and THG, have been observed in the microstructured fiber [4952]. Furthermore, combining with the tailorable dispersion characteristics of the microstructured fibers, a number of dramatic nonlinear optical effects are observed at 800 nm that were not previously possible or have been severely limited [6]. On the other hand, large core singlemode microstructured fibers can also be fabricated to minimize the nonlinear optical effects. Now let us turn to the most dramatic and amazing of the nonlinearities, ultra broadband supercontinuum generation. 2.3.2 Supercontinuum Generation in Microstructured Fibers Supercontinuum generation is definitely the most fascinating outcome of the invention of microstructured fibers. In particular, it results directly from the combination of microstructured fibers' enhanced nonlinearities and unique engineerable dispersion properties. Ranka's postdeadline talk at Conference on Laser and ElectroOptics (CLEO) in 1999 as well as the paper published in early 2000 have a tremendous impact on the research conducted on fiber design probabilities, fiber dispersions and nonlinearities, as well as many important applications in the past few years. In this paper [6], an ultra broadband supercontinuum ranges from 400 nm to 1600 nm (figure 2.8) is generated in a S I I I I t I I 100 2 102 , S .4 o 10. 6 101 400 600 800 1000 1200 1400 1600 Wavelength (nm) Figure 2.8: An ultra broadband supercontinuum generated in a 75cm section of microstructure fiber. The dashed curve shows the spectrum of the initial 100 fs pulse. Refer to Ranka et al. [6]. specially designed "microstructured fiber" using Ti:Sapphire laser 100 fs pulses with 790 nm central wavelength and only 0.8 nJ energy. The microstructured fiber consists of a silica core of 1.7 /m surrounded by an array of 1.3 /m diameter air holes in a hexagonal closepacked arrangement, see Fig. 2.6 (a), sustaining singlemode laser pulse propagation for wavelengths range from 500 nm to 1600 nm. It is quite surprising that the mode is well confined within the first air hole ring next to the core; the outer rings of air holes do not affect the fiber waveguide properties at all. With this air hole and pitch configuration, zero group velocity dispersion wavelength is pushed down to 767 nm, making the Ti:Sapphire input pulse central wavelength in the anomalous fiber dispersion region. Nonlinear optical effects, include pulse compression, soliton propagation and efficient fourwave mixing are observed at 800 nm. All these nonlinear effects have been severely limited at this wavelength for conventional fibers. The intensive interest in the supercontinuum generations is well justified. First, performing the experiment of supercontinuum generation is not hard at all, considering the popularization of the Ti:Sapphire laser all over the world and the advantage of low power requirement. Numerous papers have been published on this subject, taking up a large portion of microstructured fiber literature. Many of the papers are about the applications of supercontinuum generation, which will be presented in details in the next section. The more significant reason for the interest in the supercontinuum generations lies in its complex process, that is, how the ultra broadband supercontinuum is generated? From a general point of view, there are two contributions which lead to ultra broadband supercontinuum generation. The first one comes from geometry. A microstructured fiber has an extremely small core area, nearly two orders of magnitude smaller when compared to that of conventional singlemode fiber. The tighter confinement of the mode propagation leads to an increased power density and enhanced effective nonlinearities. The second contribution is the tailorable dispersion characteristics of the microstructured fibers, resulting from both the ability to precisely engineer the air hole and pitch size as well as the high refractive index difference between the core and cladding. As a result, the zero GVD dispersion can be pushed down to visible wavelength range and the input Ti:Sapphire laser pulse that experiences a low fiber dispersion maintains a high peak power while propagating along. The enhanced fiber nonlinearities combining with the intensive peak power of the laser pulse trigger many nonlinear processes, generating the ultra broadband supercontinuum. In principle, the supercontinuum generation process in microstructured fibers is not different from that in conventional fibers. Well established theories (see, for example Broderick et al [49] and references therein) explaining many nonlinear processes in optical fibers has been developed over years. However, supercontinuum generation is a nonlinear process in the extreme. It is evident that supercontinuum generation is not resulting from or dominant by one individual nonlinear process; on the contrary, it results from the complex interplay between various nonlinear optical processes and dispersion characteristics of microstructured fibers. The complexity also lies in the fact that the significant pulse spectral broadening process usually happens within the first several millimeters of the microstructured fiber at sufficiently high power due to the enhanced nonlinearities and low dispersion. It is crucial to understand the supercontinuum generation dynamics; in particular, the roles of different nonlinear optical processes play in a certain pulse propagation stages and their contributions to the final supercontinuum characteristics. The interpretations of the supercontinuum generation dynamics from the experimental data are sometimes misleading, as it is usually hard to separately study the individual nonlinear effect in the experiment when various nonlinear effects are interacting together in a tangled state. The extended NLSE model using the splitstep Fourier transform method has proven to be a wellsuited numerical technique that can truly simulate the pulse nonlinear propagation processes in the microstructured fibers. Besides the ability of investigating individual nonlinear effect and its contribution to the supercontinuum generation, this simulation model has also predicted some new supercontinuum properties that have been experimentally verified. However, the complexity and ultra broadband properties of the supercontinuum generation require a large number of data points and an extremely small propagation step when implement the extended NLSE simulation model, necessitating the use of high speed computers only available to general researchers in the last five to ten years. Many papers have been published from the extensive research on the formation and evolution of the supercontinuum. A consentaneous conclusion has been formed based on the rigorous simulation and experimental results [10, 11]. It is believed that the development and evolvement of the supercontinuum is related to the formation and fission of higher order solitons in the microstructured fibers. As we mentioned in the section 2.2, in the anomalous dispersion region, higher order solitons are generated in the microstructured fiber due to the interplay between the SPM and GVD, where the soliton order is determined by equation 2.14. Without any perturbation, these higher order solitons will propagate in a periodic evolution pattern and supercontinuum generation would never occur. However, a small perturbation will affect a higher order (N>1) soliton's relative group velocities and, subsequently, trigger the soliton fission: a higher order soliton with soliton order N breaks up to N fundamental solitons. These small perturbations include higher order dispersion, selfsteepening nonlinear effect and soliton selffrequency shift. As a result of soliton fission and soliton selffrequency shift, the fundamental solitons continuously shift toward the longer wavelength of the broadened spectrum, causing a considerable spectral expansion on the red side. The blue side the supercontinuum is developed due to the blueshifted nonsolitonic radiation (NSR) [3, 10, 53] or phasematched radiation; that is, the blue dispersive wave component satisfying the resonance condition gets amplified by the energy transferred from the phasematching soliton under the influence of higher order dispersion [54]. Assuming the soliton frequency o0, and the nonsolitonic radiation component o,, the phase matching condition can be written as [53], AK = /fP() P(o) (a) c,) / Vg 7Po = 0, (2.27) where / is propagation constant, vg is the group velocity, y is the nonlinear parameter and Po is the input peak power. Consequently, an ultra broadband supercontinuum is generated as the fundamental solitons continuously shift to the red side and the blue phasematching components are developed and amplified. It is worth pointing out that supercontinuum is the outcome from interactions of the most complex nonlinear optical processes with fiber dispersion. Supercontinuum generation usually occurs within the first several millimeters of the propagation length, followed by the increasingly fine interference features from the interplay between the individual soliton and the related dispersive wave as supercontinuum continues to evolve along the microstructured fibers. It turns out to be true that any small input pulse power fluctuation can substantially change the features of this fine structure, as verified by both the simulation results and experimental evidences [9, 13]. The fine structure which tends to vary from shot to shot can be a huge problem for many applications. For example, stable spectral phases are required for supercontinuum pulse compression application. Researchers have put a lot of efforts in finding out the stability range for various applications. Consequently, different stability fiber lengths range from 1cm to tens of centimeters corresponding to different input power levels have been found [15, 55]. Meanwhile, changing the microstructured fiber dispersion properties can evidently minimize the instability of the supercontinuum. Detailed simulation results following this idea will be discussed in chapter 6. When the central wavelength of the input pulse is in the normal dispersion region of the microstructured fiber, supercontinuum can also be generated. Before the spectral broadened pulse reaching the anomalous dispersion region, due to the high peak power of the input pulse, SPM nonlinear effect is mainly responsible for the pulse spectral broadening at this stage. Meanwhile, the peak intensity of the pulse is still intense because of the low fiber dispersion. After the spectrum broadening into the anomalous dispersion region, higher order solitons are generated and the mechanisms similar to what have been described above lead to the supercontinuum generation. 2.3.3 Applications of PCFs and Supercontinuum Generation The diversity of new or improved performance, beyond what conventional fiber can offer, means that PCF is finding an increasing number of applications in everwidening areas of science and technology [31]. Photonic crystal fibers have important applications in the optical telecommunications. The enhanced waveguide nonlinearities in the microstructured fibers potentially make the telecommunication optical functions to be achieved within a much shorter fiber length comparing to that of conventional fibers [56]. Microstructured fibers also find applications in fiber lasers and amplifiers. Large mode area singlemode microstructured fiber allows one to obtain a high power output with relatively low power density. This way nonlinear phenomena and fiber damage due to overheating can be avoided, making large core microstructured fibers predestinated for high power operations [57]. Photonic bandgap fibers can be used in the sensor technology [58]. They can also be used for atom and particle guidance, which has a potential application in biology, chemistry and atomic physics. Particle levitation in hollowcore fiber over 150 mm distance with 80 mW laser power has been reported [59]. Furthermore, the core of the photonic bandgap fibers can be filled with a variety of gases to study gas based nonlinear optics. In particular, stimulated Raman scattering in hydrogenfilled photonic bandgap fiber has been extensively studied [60]. On the other hand, photonic bandgap fibers filled with argon gas can be potentially used for high harmonic generation, a phenomenon occurs when the gas election experiencing the recombination process after being ionized by the ultrashort highenergy pulses [61]. Supercontinuum generation, the most fascinating outcome of the PCFs, has a far more important application scopes. First, it has become a nature candidate for telecommunication applications. To construct flexible and robust photonic networks, it is essential to control, manage and fully utilize the vast optical frequency resources available. Indeed, supercontinuum generation offers the possibility of generating over 1000 densewavelengthdivisionmultiplexing (DWDM) channels using only one single light source [21] while maintaining its coherent characteristics. Supercontinuum generation also has application in medical imaging [6264]. For optical coherence tomography (OCT), longitudinal resolution in a biological tissue is inversely proportional to the bandwidth of the light source. Supercontinuum generation with high spatial and spectral coherence has increased the OCT longitudinal resolution to a micron level (1.3 jmn, see Wang et al. [64]). The most important application of supercontinuum generation is in the field of optical metrology [6567]. It makes the measurement of absolute optical frequency possible by establishing a direct link between the repetition rate of a modelocked laser and optical frequencies. Using a carrierenvelope phase locked laser, an optical clock with accuracy 12 orders of magnitude better than the 39 currently used cesium atom clocks is demonstrated [6870]. This work led to 2005 Nobel prize for two of the pioneers of the field, John Hall of JILA at the University of Colorado and Ted Hansch of the Max Planck Institute for Quantum Optics [8]. CHAPTER 3 OVERVIEW OF EXPERIMENTAL TECHNIQUES To understand how ultrashort laser pulses interact with PCFs, it is important to understand the methods of generating, characterizing, and temporally tailoring ultrashort pulses. In this chapter, we will lay out the experimental tools used in our research. A brief introduction of ultrashort laser pulse and terminology will be given in section 3.1 and 3.2. We discuss the operating principles and performance of our homebuild Ti: Sapphire laser oscillator and a commercial Coherent Mira 900 laser oscillator in section 3.3. A description and comparison of different ultrafast laser pulse characterization tools will be given in section 3.4. Finally, in section 3.5, we present the Fourier domain pulse shaping using the liquid crystal spatial light modulators. 3.1 Ultrashort Laser Pulses Lasers are the basic building block of the technologies for the generation of short light pulses. Only four decades after the laser had been invented, the duration of the shortest produced pulse has been reduced nine orders of magnitude, going from the nanosecond (109 s) regime to the attosecond (1018 s) regime [72]. One reason for generating ultrashort pulses is to challenge the physics limit: what is the shortest pulse mankind can generate? Another question may arise here: what kind of measurements can we make with these ultrashort laser pulses? One important application domain of ultrashort laser pulses is the behavior analysis (return to equilibrium) of a sample perturbed by the laser pulses [7375]. If some processes in the sample are very fast compared to the duration of the perturbation, these processes will be hidden during the perturbation so that only those processes which are slower than the pulse duration will be observed. Therefore, it will be easier to understand the faster processes if one can use a shorter perturbation. The unique property of ultrashort laser pulses makes them ideally suited for not only the initial perturbation of sample but also subsequent probing of the sample. On the other hand, ultrafast lasers can also be used to produce laser pulses with extremely high peak powers and power densities, which have applications such as multiphoton imaging [76, 77], generation of electromagnetic radiation at unusual wavelengths [7880] and laser machining and ablation [81]. Meanwhile, the large peak intensities associated with ultrashort laser pulses make them well suited to various nonlinear wavemixing processes, allowing the generation of new frequency components. Such processes include secondharmonic generation, sumfrequency generation, parametric oscillation and amplification, and continuum generation. 3.2 Laser Pulse Terminology To better understand ultrafast laser system, ultrashort laser pulse characterization and shaping, we begin with some terminology [82]. 3.2.1 Laser Pulse Time Bandwidth Product (TBP) A continuous wave laser generates continuous wave (CW) that is an electromagnetic wave of constant amplitude and frequency; and in mathematical analysis, of infinite duration. The time representation of the field (real part) is an unlimited cosine function. E(t) = Eoe""~ (3.1) On the other hand, constructing a light pulse implies multiplying (3.1) by a bell shaped function, e.g., a Gaussian function. Modelocked lasers generate this kind of pulses. In general, a pulse can be represented as E(t) = A(t)e"'Vote'(t), (3.2) where A(t) is the bell shaped function and 0(t) is the pulse phase. The physical field is Re[E(t)]. The general time and frequency Fourier transforms are E(o) = A(o)e'I()) = rE(t)e'"tdt 1 ,(3.3) E(t) = E(tm)edt where E(o)) is the pulse representation in the frequency domain. In particular, a Gaussian pulse can be written as E(t) = Eoe (t')2 el) (3.4) A limited duration caused by the Gaussian envelope of the light pulse indicates a limited frequency bandwidth, which can be clearly seen after the Fourier transformation of the original temporal pulse (Eq. 3.4). E(c) = JE(t)eddt = ET eL. 2 (3.5) The pulse duration and spectral width are defined as the full width half maximum (FWHM) of the pulse intensity (I(t) = E(t)12 and I(o) E(o))l2) in time domain and frequency domain, respectively. In the case of a Gaussian pulse (Eq. 3.4 and 3.5), the pulse duration and spectral width are At = r,21n2 2 (3.6) AO = 21n2 7 Note that time bandwidth product (TBP, K = At Af ) is a dimensionless number which depends on the shapes of the pulses, e.g., the shape of a Gaussian pulse is represented as function e (t/)2 Table 3.1 gives values of K for Gaussian shape and hyperbolic secant shape which are the most commonly used pulse envelopes. Table 3.1: Time bandwidth products (K) for Gaussian and hyperbolic secant pulse shapes. Shape E(t) K = At Af Gaussian function e (t/'r 0.441 Hyperbolic secant function 1 0.315 cosh(t / r) 3.2.2 Laser Pulse Phase and Chirp The TBP value in table 3.1 can only be reached when the instantaneous angular frequency is constant and equals the central angular frequency o0, which can be seen from Eq. 3.4 and ai(t) = o0. The pulse is called a Fourier transformlimited pulse. Now let's consider a more general case, that is, the instantaneous angular frequency is a function of time. Suppose the phase of the pulse obeys a quadratic law in time, E(t) = A(t)e'`ter'(t) = A(t)e'te'12, (3.7) then the instantaneous angular frequency varies linearly with time )(t) = oo + 9q//t = o)0 + 2at. (3.8) Depending on the sign of a, the pulse is positive or negative "chirped", i.e., linear chirp. It is very clear from Fig. 3.1, i.e., chirped Gaussian pulses, that the instantaneous frequency is more red in the leading part of the pulse and more blue in the trailing part when a positive chirped pulse is presented. (b) w a>O0 a<0 t t Figure 3.1: Illustration of Gaussian pulses with linear chirps. (a) A positive linear chirp. (b) A negative linear chirp. The pulse duration and spectral width in the linearly chirped Gaussian pulse are At = r 21n2, 1 T 2" (3.9) Aw = 2 1n2 [1 +ar2 (9 The TBP will always be greater than the value listed in table 2.1. This discussion is also true if we start from the frequency domain with a chirped pulse. So the conclusion is that a Fouriertransformlimited pulse (the shortest pulse) is generated when the pulse with a fixed spectrum has no spectral phase. Unfortunately, when ultrashort pulses propagate in an optical system, many optical components (lenses, gratings, crystals) are dispersive and can generate some kind of chirp (linear and/or higher order chirp) that causes pulse temporal broadening. It is common practice to expand the spectral phase in a Taylor Series as 0(2) )(3) (0) = (0) + (1)(0 0) (0 002 (0) + 0 ,) (3.10) 2 6 where 0 0)) (3.11) The linear and quadratic chirp are represented as 0(2) and 0(3), respectively. A pulse compressor [83] (a pair of parallel diffraction gratings and a retroreflector) can compensate most of the linear chirp, but can do nothing about the higher order chirps. To obtain the shortest pulse, the overall spectral phase (chirp) has to be measured and compensated. This will be one of the applications of pulse characterization and pulse shaping which will be discussed later in this chapter. 3.3 Femtosecond Lasers (Ti:Sapphire Lasers) Lasers are the basic building blocks to generate short pulses. Ti:Sapphire femtosecond laser has been studied extensively because its high performance. In order to understand how laser pulse can be temporally tailored and interact with the fibers, it is important to understand how femtosecond laser pulses are generated in the Ti:Sapphire lasers. In this section we will describe the properties of the Ti:Sapphire crystal, followed by the modelocking discussion. We will also lay out the schematics and performance parameters of the Ti:Sapphire lasers used in our experiments. 3.3.1 Ti:Sapphire Crystal In the past decade, the most spectacular advances in laser physics and, particularly, in the field ofultrashort light pulse generation [84, 85] have been based on the development of titaniumdoped aluminum oxide (Ti:A1203, Ti:Sapphire) laser. Ti:Sapphire laser has been investigated extensively and today it is the most widely used tunable solidstate laser. Ti:Sapphire possesses a favorable combination of properties which are up to now the best broadband laser materials. First, the active medium is solid state, that means long operational time and laser compactness. Second, Sapphire has high thermal conductivity, exceptional chemical inertness and mechanical resistance. Third, Ti:Sapphire crystal has the largest gain bandwidth and is therefore capable of producing the shortest pulse; it also provides the widest wavelength tunability [Fig. 3.2]. 1.0 0.8 0.6 S0.4 0.2 S  Absorption Fluorescence 0.0 I 400 500 600 700 800 900 Wavelength (nm) Figure 3.2: Normalized absorption and emission spectra of Ti:Sapphire for 7t polarized light. Refer to Rulliere [82]. From the fluorescence curve, the estimated the FWHM of the theoretical broadest spectrum is 190 nm, which corresponds to a 4.5 fs pulse if a Gaussian transformlimited pulse is assumed. In fact, custom build Ti:Sapphire laser that can generate 4.8 fs ultrashort pulse has been reported [86], merely two cycles of the optical field considering 800 nm center wavelength. 3.3.2 Kerr Lens ModeLocking A large number of modelocking techniques have been developed to generate short pulse with Ti:Sapphire as a gain medium: active modelocking, passive modelocking and selfmodelocking. Selfmodelocking (or Kerr lens modelocking) has proven to be the best way to achieve the modelocking. Selflocking of the modes utilizes the nonlinear properties of the amplifying medium to favor strong intensity maxima at the expense of weak ones. The nonlinear effect called selffocusing is due to the fact that the refractive index of Ti:Sapphire is a function of input pulse intensity: n = no + n2I(r, t). Because of the nonuniform power density distribution in the cavity Gaussian beam I(r, t) the refractive index changes across the beam profile and the phase delay experienced by the beam is greater in the center of the beam than at the edge for n2 > 0 [Fig. 3.3]. Therefore, Intensity Kerr medium CW pulsed Aperture Figure 3.3: Kerr lens modelocking principle: selffocusing effect by the optical Kerr effect. An example of hard aperture refers to Wikipedia [87]. the Ti:Sapphire crystal works like a nonlinear lens (Kerr lens) for high intensity light, with the focusing effect increasing with optical intensity. In the laser cavity, short noise bursts of light (pulses), which have higher peak intensities, are focused more tightly and are transmitted through the aperture, whilst lower intensities experience greater losses. By aligning the cavity in a way such that the resonator is lossier for CW beam than for pulses, the pulsed regime is favored and the laser will turn to modelocked regime. The favoring of pulsed regime over CW regime can be achieved by the cavity design, but is often supported by a hard aperture (shown in Fig. 3.3), that can simply cut off part of the CW beam at the focal region of the pulsed beam. 3.3.3 Cavities of Ti:Sapphire Lasers Depending on the experiment requirement and consideration, we use two different types of Ti:Sapphire oscillators in our experiments, all of which will be discussed in details in the rest of this section. 3.3.3.1 Homebuild Ti:Sapphire oscillator For most of our experiments, we use a homebuilt Ti:Sapphire oscillator. The design of the oscillator is shown schematically in Figure 3.4. The intracavity is defined by prism pairs and four mirrors, Ml, M2, M3 and the output coupler M4. The Ti:Sapphire crystal is located at the common focal plane of two 10cm spherical mirrors M2 and M3. The laser system is pumped by focusing a multiline ArgonIon CW laser (Coherent Innova 310). The output coupler M4 is an 85% wedged mirror. The intracavity prism pairs are the key components for selfmodelocking (Kerr lens modelocking). Hard apertures are placed in front of the cavity mirror M1 and output coupler M4. In this oscillator, modelocking is initiated by instantaneously moving the first intracavity prism P1 and M3. The dispersion caused by intracavity prism pairs and other optical component in the cavity (crystal and OC) is partially compensated with the external prism pairs, which is designated in the Figure 3.4 as the phase compensator. With 4.9 W pumping power, our homebuild Ti:Sapphire oscillator generates 300400 mW average pulse power. The repetition rate is 91 MHz and energy per pulse is around 3 nJ. 49 PHASE COMPENSATOR P P4 3 OC LASER SLIT P 2 t M4 M1 Pi 2 ,A L PULP P LASER SLIT Figure 3.4: Schematic diagram of the Ti:Sapphire Laser and the external phase compensator. P: prism. OC: output coupler The output spectrum is shown in Fig (b). It has a bandwidth of 65 nm and central wavelength of 805 nm. Fig (a) is the second order intensity autocorrelation, using a 100 pm KDP crystal. Assuming a Gaussian pulse shape, the measured FWHM of the laser pulse is 18 fs. a) b) 80 40 0 40 80 700 800 900 Time (fs) Wavelength (nm) Figure 3.5: Second order intensity autocorrelation and spectrum of Ti:Sapphire laser pulse. (a) Temporal autocorrelation of the pulse. (b) Measured spectrum of the pulse. 3.3.3.2 Coherent Mira 900 system For the experiment described in section 4.3, we use the commercial Coherent Mira 900 (Fig. 3.6) pumped by Ion CW Coherent Innova 400 laser. The Mira has a tunable wavelength range from 700 nm to 900 nm. From the schematics we can see there are several extra mirrors comparing with our oscillator. The only purpose is for the commercial compact design. Note that M8 and M9 are for alignment only and they are not in the oscillator cavity. However, Mira 900 does consist of a birefringent filter (BRF) which makes Mira 900 a central wavelength tunable laser. Such filters take advantage of the phase shifts between orthogonal polarizations to obtain narrow band outputs. Furthermore, the commercial design of a butterfly starter (not shown in the figure) between M3 and M4 makes modelocking a breeze. The wavelength used in our experiment is 763 nm. With 8 W pumping power, the average output pulse power is about 450 mW. The repetition rate is 76 MHz and energy per pulse is around 6 nJ. M8 Mg  M7 PUMP LASER M4 TiAl203   SLIT ZMS Mi M6 M2 BRF Oc M3 Figure 3.6: Schematic diagram of a Coherent Mira 900 Ti:Sapphire laser. Refer to Coherent Mira 900 [88]. The spectral bandwidth of Mira 900 output pulse is very narrow comparing to our homebuild Ti:Sapphire oscillator due to the BRF, in this case, 4.2 nm [Fig. 3.7 (b)]. The temporal pulse duration is measured by FROG (see next section for details). The calculated pulse duration for transformlimited Gaussian pulse with measured spectral intensity is 215 fs. The measured pulse duration is 225 fs [Fig. 3.7 (a)]. [ a) I b) 400 200 0 200 400 756 758 760 762 764 766 768 770 Time (fs) Wavelength (nm) Figure 3.7: FROG measurement of a Coherent Mira 900 Ti:Sapphire laser pulse. a) Temporal intensity. b) Spectral intensity. 3.4 FROG and SPIDER Ultrashort laser pulses are the shortest events mankind ever generated. Before ultrashort laser pulses are used in experiments, pulse diagnostics are necessary. Since the femtosecond time scale is beyond the range of the fastest electronics, the pulse measurement techniques have to be redesigned in order to fully characterize the amplitude and the phase of the electric field. Most techniques are based on the idea "measuring pulse using the pulse itself!" Section 3.4.1 will briefly describe the standard techniques that determine the temporal profile of the pulse, such as autocorrelation and crosscorrelation. Other techniques, which are more sophisticated giving both information of frequency and time, such as FROG and SPIDER, will be presented in the section 3.4.2 and 3.4.3. 3.4.1 AutoCorrelation and CrossCorrelation Maybe the most widely used technique for measuring femtosecond laser pulses is the secondorder autocorrelation, which was first demonstrated in 1966 by Maier and co workers [89]. This method takes advantage of the second harmonic generation in nonlinear crystals. In Figure 3.8 the basic principle of SHG autocorrelation is displayed. This is typically used to measure the time duration of femtosecond pulses. LASER BEAM 0) I LENS FILTER <, / POWER I I _____ DECTECTOR \cZ~ !SHG Xtal I. / ....... .... S,"' DELAY Figure 3.8: Schematic diagram of a SHG autocorrelator. BS: beam splitter. SHG Xtal: second harmonic generation crystal. The incident laser pulse is split in two pulses by a 50/50 beamsplitter. Similar to a Michelson interferometer, the two pulses are reflected in each arm and subsequently focused onto a frequencydoubling crystal. The resulting secondharmonicgeneration (SHG) autocorrelation trace is detected by a photomultiplier as a function of the time delay r between the two pulses. The SHG crystals can be type I phasematching or type II, and the two pulses can recombine collinearly or noncollinearly in the frequency doubling medium. Figure 3.8 is an example of noncollinear recombination of the two pulses. The SHGs of each individual pulses are filtered by a hard aperture and a BG39 filter may also be used to eliminate the original laser pulse signal that pass through the SHG crystal. If the two fields are of intensity I(t) and I(t At), the autocorrelation of the two pulses is I(At)= I(t)I(t At)dt (3.12) As can be seen from the equation, the autocorrelation is always a symmetric function in time. Therefore, autocorrelation gives very little information about the shape of the pulse. The most widely used procedure to determine the pulse duration is to "preassume" a pulse shape (usually hyperbolic secant or a Gaussian shape, for chirpfree and linear chirped pulses) and to calculate the pulse duration from the known ratio between the FWHM of the autocorrelation and of the pulse. Thus the autocorrelation function depends on the assumed shape of the pulse. Table 3.2 lists the relevant parameters for various shapes. Table 3.2: Relations between pulse duration and autocorrelation function duration for Gaussian and Hyperbolic secant function. Shape Ia(t) TBP: K Pulse duration At/r Gaussian function ( 41n2t2) 0.441 52 e At2 Hyperbolic secant function 1.76t 0.315 1.5 Sech ( ) At In crosscorrelation, the ultrashort pulse is not correlated with itself. The cross correlation implies the use of a reference pulse of a known shape I,(t) in order to determine the temporal profile of an unknown laser pulse I, (t). The intensity cross correlation to be measured is IA Icc ()At) = I,((3.13) The convolution with the intensity of the reference pulse leads to a smooth cross correlation shape. The fundamental problem of autocorrelation and crosscorrelation is that it does not uniquely determine the pulse characteristics. It cannot even accurately determine the pulse duration because a priori information about the pulse shape is required, which is sometimes impossible to obtain. Furthermore, a systematic analysis given in Chung et al. [90] shows that very similar autocorrelation traces and power spectra can be produced by pulses with drastic different shapes and durations. In fact, to fully characterize a pulse, including the intensity and phase in time domain or frequency domain, one needs a total number of 2N points. The correlation function only gives N points, meaning the pulse characterization for this technique is indeed quite underdetermined. Of course, to understand the pulse itself, which includes what time a color occurs in a pulse, or equivalently, pulse spectral phase, full pulse intensity and phase characterization techniques are required. The development of several such kinds of technique occurs in the early 1990s. Among which, SHG frequencyresolved optical gating (FROG) and spectral phase interferometry for direct electricfield reconstruction (SPIDER) are the most wellknown techniques. Frequency resolved optical gating uses two dimensional representation of the one dimensional electric field, while SPIDER uses one dimensional spectral interferometry. All of which will be discussed in details in this section. 3.4.2 Frequency Resolved Optical Gating (FROG) Frequency resolved optical gating (FROG) was first developed in 1993 [91, 92]. Frequency resolved optical gating is a method that can acquire both intensity and phase of a pulse without making prior assumptions about pulse shape. Frequency resolved optical gating is based on spectrally resolving autocorrelation function; therefore generating the spectrogram of the pulse to be measured (refer to Fig. 3.10 (a) as an example). A spectrogram, S(wo, r), is a twodimensional representation of the pulse as a function of time delay and frequency (or wavelength), 2 S(co, r) = Eg(t, )e ,dt (3.14) where E,,,(t,r) is simply the autocorrelation signal and has several forms depending on different version of FROG techniques, e.g., selfdiffraction (SD) FROG, polarization gating (PG) FROG, SHG FROG and third harmonic generation (THG) FROG. Among all these versions, only SHG FROG utilizes the second order nonlinearities while all the other versions rely on the third order nonlinearities to perform the autocorrelation. This makes SHG FROG one of the most widely used FROG techniques. For the SHG FROG, the E,, (t,r) is simply a second order autocorrelation function, EsHG~ g (t, r) = E(t)E(t r). (3.15) Experimentally, a SHG FROG apparatus is an autocorrelator followed by a spectrometer. Therefore, instead of just measuring the intensity of the nonlinear optical signal generated by the two variably delayed pulses, the nonlinear optical signal S(Co, ) is spectrally resolved into a delay dependent spectrum, i.e., spectrogram. The resulting data is a two dimensional timefrequency representation of the pulse, that is, a 2D function of time delay and optical frequency. The spectrogram obtained is referred to as the measured "FROG trace", as show in Fig 3.10 (a). In some cases, the shape of the FROG trace can be interpreted to give an overview of the shape of the pulse. In order to "retrieve" the original pulse intensity and phase, a sophisticated iterative inversion algorithm needs to be applied to the 2D FROG trace. The twodimensional phase retrieval algorithm has been well established in the field of image science. The electric field and phase of the pulse that created the FROG trace can indeed be uniquely determined from the FROG trace, e.g., Fig 3.10 (b) is an example of retrieved spectral intensity and phase of the pulse that created the FROG trace in Fig. 3.10 (a) (save a few trivial ambiguities). One reason that the FROG iterative retrieval algorithm converges well despite the absence of an absolute guarantee of convergence with the retrieval algorithm is that the FROG trace is "overdetermined". The FROG trace, unlike the auto correlation using N points to calculate the input pulse 2Npoints field, has N2 points. Because of this buildin data redundancy, FROG technique guarantees a solution for an experimental trace; in addition, it also gains the ability of retrieving from the FROG trace that has a large amount of random noise [93]. In the FROG retrieval algorithm, convergence is always determined by calculating the root mean square difference (FROG error) between the measured FROG trace and the trace computed from the retrieved pulse field. Indeed, FROG error gives an estimate on how reliable the retrieved spectral intensity and phase are. Figure 3.9 is the schematic of our homebuild FROG setup. We use this FROG for all our pulse characterization experiments. A 100 [tm thick freestanding KDP crystal is used for second harmonic generation. We use an image system after the crystal for imaging optimization purpose and providing a 10 times image magnification. We used single shot and multishot FROG techniques in our research. A multishot FROG trace consists of a series of 1D spectra taken by manually change the delay stage LASER BEAM 03 /S LENS FILTER A< A 2D SPEC S_2co TROMETER \ ir! KDP Xtal .' DELAY Figure 3.9: Schematic diagram of a SHG FROG apparatus. BS: beam splitter. Xtal: crystal. position, whereas singleshot FROG trace itself is a 2D image (delay vs. wavelength). Multishots FROG trace has a higher signaltonoise ratio, but the whole acquisition time is much longer. To retrieve the FROG trace, we send the singleshot and multishot FROG trace to the retrieval algorithms (Femtosoft software). To get a reasonable FROG error, accurate delay axis and wavelength axis calibrations are required, as well as appropriate data correction procedures (background subtraction and flatfield correction). Fig. 3.10 is an example of using FROG technique in our example, with the original FROG trace shown in Fig. 3.10 (a) and retrieved spectral intensity and phase shown in 58 Fig. 3.10 (b). Note that the retrieved spectral phase shows an example of linear chirp (quadratic spectral phase) in the original pulse. The retrieved FROG error is 0.025 using FemtoSoft FROG V1.5. FROG is a very useful technique, particularly as temporal and spectral information are simultaneously measured, yielding a phase reconstruction. The downside is that sometimes spatial chirp and pulsefront tilt can be an issue when reliable FROG traces are desired. A beam with spatial chirp has color varying spatially across the beam and it will tilt the FROG trice by a small angle. The pulsefront tilt means the pulse intensity front is not perpendicular to the pulse propagation vector and this effect will shift the zero delay line in the FROG trace. Furthermore, the FROG technique is not well suited for real time pulse monitoring and adaptive control experiments due to the timeconsuming retrieval algorithms and occasionally expertise in the selection of reconstruction parameters. For other pulse characterization applications, FROG is an excellent choice. .4I 0)bo) C O Sn a, (U ) 0.6 n * d a d c 1 1700 0 1700 Delay (fs) 754 756 58 760 762 764 766 768 Wavelength (nm) Figure 3.10: An example of experimental FROG trace and retrieved spectral intensity/phase profile correspond to the FROG trace. The spectral intensity is an example of Gaussian pulse and the spectral phase indicates a residual linear chirp for the original pulse. a)  I 3.4.3 Spectral Phase Interferometry for Directelectricfield Reconstruction (SPIDER) The technique, spectral phase interferometry for direct electricfield reconstruction (SPIDER), is a specific implementation of spectral shearing interferometry [94, 95]. This interferometric technique is based on the measurement of the interference between two pulses with a certain delay in time. These two pulses are spectrally sheared, i.e., they are identical except for their central frequencies. The spectrum of this pulse pair is an interferogram signal [Fig. 3.11] S(o)) = I(o) + Q) + I(m) + 2 I(o0 + Q)I(o0) cos[(m + Q) (o)) + Co r] (3.16) where Q is the spectral shear (the difference between the central frequencies of the pulse pair), r is the time delay between the pulses, I(o) and q(co) are the pulse spectral intensity and phase. Signal Intensity Frequency (Hz) Figure 3.11: Plot of an ideal interferogram. The interferogram is of the form shown in Eq. (3.16). The nominal spacing between the fringes is 1/T. Refer to Shuman et al. [96]. As mentioned in this section before, the full pulse characteristics in the frequency domain require determining both the ()) and w(co). The measurement of the electric field is quite straightforward and the spectral phase can be extracted from the SPIDER interferogram. Evidently from the Eq. 3.16, the spectral shear Q determines discrete spectral phase sampling resolution, as only the spectral phase at a series of spectral frequency separated by Q can be determined. To utilize the SPIDER technique, the generation of spectrally sheared pulse pair is required. This is usually done [Fig. 3.12] by frequency mixing two pulse replicas, that is separated by delay time with a chirped pulse in a nonlinear crystal. A pulse stretcher, e.g., a glass block in the Figure 3.12, is used to generate the chirped pulse. The chirped pulse needs to be much longer than the delay time r to satisfy the condition that each pulse replica can frequency mix with different frequency in the chirped pulse in the crystal. As a result, the unconverted pulses are spectrally sheared. The resulting interferogram is resolved with a spectrometer. After the interferogram is sampled properly, the spectral phase information can be extracted from the data. BS PR TS1 SFG HA TS2 FM Figure 3.12: Schematic diagram of a SPIDER apparatus. BS: beam splitter. GDD: SF10 glass block. TS: translational stage for delay adjustment. FM: focusing mirror. OMA: optical multichannel analyzer. SFG: upconversion crystal (BBO). Refer to Gallmann et al. [107]. The SPIDER trace is onedimensional, and the inversion is algebraic. The SPIDER pulse retrieval algorithm uses a direct inversion procedure [97], which is much faster comparing with the FROG's iterative algorithms. As a consequence, realtime measurements (with a 20Hz refresh rate) from the output of a regeneratively amplified laser system have been obtained [96]. Meanwhile, combination with SPIDER and adaptive pulse control makes realtime ultrashort coherent control possible. However, SPIDER requires a different optical setup for measurement of different optical pulses, which limits its flexibility because prior knowledge of the pulse chirp is required. Furthermore, the accuracy of SPIDER is reliant solely on the initial calibration, which cannot be checked in the same way as FROG, in which marginals of the FROG trace can be used to highlight any calibration irregularities [98]. 3.5 Pulse Shaping Femtosecond pulse shaping [99] promises great advantages to the fields of fiber optics and photonics, ultrafast spectroscopy, optical communications and physical chemistry. Femtosecond pulse shaping also plays a major role in the field of coherent or quantum control [100], in which quantum mechanical wave packets or quantum state can be manipulated via the phase and intensity profiles of the input laser pulse. Changing the laser intensity temporal profiles can only be achieved by femtosecond pulse shaping. Meanwhile, several schemes have been proposed in which pulse shaping would favorably enhance laserelectron interactions. One application involves laser generation of large amplitude, relativistic plasma waves [101]. In recent years, the combination of the femtosecond pulse shaping with the adaptive control technique, i.e., adaptive pulse shaping, has led to several interesting demonstrations [102104]. In this section we will lay out the schematics of pulse shaping techniques in frequency and time domain, the practical consideration for implementing pulse shaping technique, as well as configurations of the pulse shaper used in our experiments. 3.5.1 Femtosecond Pulse Shaping Femtosecond pulse shaping has passive and active control methods. Passive methods use grating pairs or prism pairs to give a relatively fixed and limited control of the pulse, such as the pulse stretcher used in the SPIDER apparatus and the pulse compressor in CPA systems. Active control methods use programmable modulators, which perform a much powerful and robust control over pulse phase and/or amplitude. Active methods can be performed in two domain, frequency domain or direct intime. Frequency domain femtosecond pulse shaping has been studied extensively and the technique is welldeveloped by now. The method is based on liquid crystal (LC) device or acousticoptic modulator (AOM) placed in the Fourier plan of a grating based zero dispersion 4fconfiguration (Figure 3.14) to experimentally control the spectral phase and/or intensity of the pulse. The different wavelengths are spatially separated and can then be addressed individually in this case. This turns out to be a great advantage in conjunction with genetic algorithm for adaptive pulse shaping, because the desired control pulse can have a rather random phase pattern. However, the setup requires careful realignment after changing the wavelength and its large size can be a limitation in some applications. The detailed description of the 4f Fourier domain pulse shaping using LC SLM will be given later in this section. Direct intime femtosecond pulse shaping can be achieved using an acoustooptic programmable dispersive filter (AOPDF). Acoustooptic programmable dispersive filters are based on the propagation of light in an acoustooptic birefringent crystal [Fig. 3.13]. The interaction of an incident ordinary optical wave with a collinear acoustic wave leads to an extraordinary wave. The acoustic wave is modulated to achieve phase matching for the incident pulse to be scattered into the perpendicular polarization direction at different depth for different wavelength. Spectral phase and amplitude pulse shaping of a femtosecond optical pulse can then be achieved by controlling the frequency modulation and acoustic wave magnitude through the modulation of the RF voltage for the piezoelectric transducer. The optical output Eo,(co) can be related to the optical input E,,(o) and electric driving signal S(o) as [105] Eo, (c) oc E,, (o) S(ao), (3.17) with the scaling factor defined as a= An(V/c), (3.18) where An is the index difference between the ordinary optical wave and extraordinary wave and V/c is the ratio of speed of sound to the speed of light. S(Co) is numerically calculated and sent to the crystal. Fast Ordinary A Acoust wave (mode 1) Compressed pulse Chirped pulse Slow Extraordinary Axis (mode 2) Figure 3.13: Principle of acoustooptic programmable dispersive filter (AOPDF). Compressed pulse is an example that AOPDF can be used as a pulse shaper to compensate the original chirped pulse phase. Refer to Verluise et al. [105]. The collinear acoustooptic interaction and the reduced size result in an easyto align device. Acoustooptic programmable dispersive filter have proven to be very useful to correct the time aberrations introduced in Chirped Pulse Amplifiers, for amplitude and phase control of ultrashort pulses [105], or even in characterization setups [106]. 3.5.2 Fourier Domain Pulse Shaping Using Spatial Light Modulator Spatial light modulator (SLM) has been widely used in the Fourier domain pulse shaping technique because of its programmable shaping ability of addressing both the phase and amplitude of each individual frequency component of the input pulse. Combining with the Genetic Algorithm's powerful abilities of efficiently searching the parameter space, adaptive pulse shaping using spatial light modulator has shown great potential to locate the right driving pulse for a specific application. In this section we will lay out the schematics of Fourier domain pulse shaping, followed by the discussion on how the SLM can be used to phase and amplitude control of the input pulse, as well as some experiment considerations. 3.5.2.1 Fourier domain pulse shaping Fourier domain pulse shaping is the most successful and widely adopted pulse shaping method. Therefore, a setup consideration and design details will be discussed in this section. Figure 3.14 shows the basic Fourier domain pulse shaping apparatus, which consists of a pair of diffraction gratings and mirrors (or lenses), and a pulse shaping mask (LCSLM). The individual frequency components contained within the incident ultrashort pulse are angularly dispersed by the first diffraction grating, and then focused to small diffraction limited spots at the back focal plane of the first mirror, where the frequency components are spatially separated along one dimension. Essentially the first mirror performs a Fourier transform which coverts the angular dispersion from the grating to a spatial separation at the back focal plane. Spatially patterned amplitude and phase masks are placed in this plane in order to manipulate the spatially dispersed optical Fourier components. The second mirror and grating recombine all the frequencies into a single collimated beam. Grating Grating I SLMI /Input pulse Shaped pulse N f 1H f  Figure 3.14: Schematic diagram of a Fourier domain pulse shaping apparatus using a LC SLM. In this setup, dl+d2=f to maintain the 4f configuration. LCSLM: Liquid Crystal Spatial Light Modulator. In the 4f Fourier domain pulse shaper, the field immediately after the SLM mask M(x) is [99] Em(, co) oc E,, (co)e 'oM(x), (3.19) where a is a frequency to space parameter written as 27r cf a = ~ ,(3.20) co d cos(Od)' and w0 is the radius of the focused beam at the mask plan cos() fA (3.21) Wo = COO f (3.21) cos(d) TW,, In these equations, 0,, and Od are the incident angle and diffraction angle for the grating, d is the grating parameter, f is the focal length of the focusing mirror (or the lens), w,, is the radius of the incident beam before the first grating, co0 is the central frequency and c is the speed of light. For the discrete N pixel SLM device, the mask M(x) can be written as n=N/2 1 M )=(x= (x) 5(x p n ) rect(x / w), (3.22) n=N/2 where H(x) is the continuous mask that predefined for specific frequency filter, w, is the physical width of each pixel of the SLM (see next section for details), the rectangular function is 1 for x < 1/2 and 0 otherwise, 0 is for convolution. Assuming a linear space to frequency mapping x = a(co co), the final electric field after the shaper can be expressed as n=N/21 1 eot(t)~ emn(t) H(tn)sinc( itf) (3.23) n=N/2 5J where H(t) is the Fourier transform of the function H(x), 6f is the frequency bandwidth of each SLM pixel. The setup of the Fourier domain pulse shaping requires that the output pulse should exactly reproduce the input pulse if no mask is presented in the back focal plan of the first mirror [99]. In another words, the grating and mirror configuration can not introduce extra dispersion. There are several considerations. First, the second grating has to be carefully placed so that the incident angle on the second grating equals the output angle of the first grating. Second, 4f configuration must be maintained, which means the grating pairs must locate in the outside mirror focal planes and two mirrors are separated by 2f. Note that sometimes lenses are used in the pulse shaper; however, using mirrors instead of lenses can minimize the material dispersion. 3.5.2.2 Liquid crystal spatial light modulator (LCSLM) The pulse shaper in our experiments uses LCSLM (CRI SLM128NIR) as the pulse phase modulation component. The LC consists of long, thin, rodlike molecules which are aligned with their long axes along the y direction without the external electrical field (see Fig. 3.15). When the voltage V is applied on the LC cell in the z direction, electric dipoles are induced and the electric forces tilt the LC molecules along z direction; causing a change in the refractive index for pulse polarized in y direction, while the refractive index in the x direction remains constant [108]. Therefore, the polarization of the incoming electric field must be parallel to the orientation of the long axis (y axis) of the liquid crystal molecules for use of the cell as a phase modulator. (a) (b) / 00 00000 I / \ Glass ITO Figure 3.15: A sectional view of a liquid crystal layer between two glass plates. (a) No voltage applied. (b) Voltage is applied on the ITO electrodes in the zdirection. Refer to Weiner et al. [108]. The LCSLM in our pulse shaper consists one LC layer, so it can only act as a pure phase modulator. However, SLM that consists two LC layers, in which their long axis are phase modulator. However, SLM that consists two LC layers, in which their long axis are in perpendicular directions, can be used as both a phase modulator and an amplitude modulator. The LCSLM used in our experiments has 128 pixels that are 2 mm high, 97 pm wide, 15 pm thick, and separated by 3 pm gaps [Fig. 3.16]. Therefore, the phase shift applied to pixel "i" by the change of its refractive index is equal to S= 27r~An, 15m A =, = (3.24) ITO ground plate LC layer ITO "electrodes 2mm . . Figure 3.16: Illustration of LCSLM array that consists of 128 LC pixels. The dimension and positions of the pixels are given as well. Refer to Weiner et al. [108]. 3.5.2.3 Experimental considerations For the most part, we perform frequency domain femtosecond pulse shaping using liquid crystal spatial light modulators in our research. Two identical gratings (1200 lines/mm) and spherical mirrors (f = 25 cm) are used in the setup. Depending on the experiments for different input central wavelength, the alignment is changed. If the gratings and the SLM are designed for different pulse polarization directions, two half waveplates have to be used before and after the SLM. The overall power convert efficiency is above 50% (comparing the pulse power launch into and after the pulse shaper). The SLM (CRI SLM128NIR) positioned in the masking plane consists of 128 individually addressable elements 100 Am wide with a 3 pm gap between pixels. Configuration of the shaper, i.e. positions of optical components and input pulse angle, can be calculated by applying grating equation and geometrical optics. The goal is utilizing a large number of SLM pixels to achieve a high pulse shaping resolution. Combining pulse shaper and open and/close close loop control (see chapter 4 for details) can yield great advantages in coherent pulse control over pulse interaction with PCFs. There are a few practical experimental considerations we need discuss before jumping into the next chapter. 1. High pulse shaping resolution Using x as the coordinate in the transverse direction parallel to the front of the SLM, f for the focal length of the lens, 60 for the diffraction angle and D for the grating line spacing, the pulse shaping resolution can be written as dA/dx = (D/f) cos O (3.25) With pulse bandwidth (baseline) and SLM pixel size (x direction) available, diffraction angle 0d and incident angle 0, for the first grating can be determined. 2. Liquid crystal voltage and phase shift calibration curve A given phase shift caused by the refractive index shift on an individual SLM pixel is a function of both wavelength and LC voltage. Even for a fixed wavelength, the change 70 in refractive index shift and voltage response of the LC is not a linear function. So LC voltage and phase shift calibration curves are required. Calibration curves can be established with the SLM placed between two crossed polarizers, each polarizer is placed 45 degrees to the long axis (y) of the LC molecules. Measuring the transmission coefficient as a function of driving voltage, for different wavelength and different SLM pixel (if necessary), and applying the equation T = 0.5(1 cos )), LC voltage and phase shift calibration curves can be measured [Fig. 3.22]. For values that do not lie on one of the three curves, interpolation is used. 26 24 850 nm 22   800 nm ........ 750 nm 20 18 16 D 14 ". 12 S10 8 6 4  0i 0 500 1000 1500 2000 2500 3000 3500 4000 Drive Level (mV) Figure 3.17: CRI SLM phase shifting curves as a function of drive voltage for750nm, 800nm and 850nm center wavelength. Linear interpolation can be applied to derive the phase shifting values for other wavelengths. Curves are measured by Anatoly Efimov. 3. SLM alignment It is critical to perform transverse SLM alignment. Since phase shift is a function of driving voltage and wavelength, it is important to know the exact frequency component for each individual SLM pixel. Fortunately, this calibration is fast and easy. Applying a large phase jump to several SLM pixels across the SLM array, watching the spectrum change can easily determine the corresponding wavelengths. Note that this wavelength and SLM pixel number calibration should be a linear function, according to equation 3.25. 4. limitations of SLM The most important limitation of SLM is the Nyquist limit. The Nyquist sampling theorem states that the phase difference between two adjacent SLM pixels should not be greater than z Therefore, the phase jump across the SLM array should be limited within 128 z Else wise, it will cause an aliens problem. The pixelation effect is due to the gaps between two neighboring pixels. The spectral components that pass through the gap, although a small portion, will not experience the refractive index change. If an exact pulse shape is desired, this effect is observable as small imperfections between the desired and the measured shape. CHAPTER 4 CONTROL OF SUPERCONTINUUM GENERATION IN PCFS USING OPTIMALLY DESIGNED PULSE SHAPES This chapter will focus on the control of the supercontinuum generation in microstructured fibers using femtosecond pulse shaping. Specifically, we will present results of our experiments on manipulating the bandwidth and shape of supercontinuum using optimally designed intensity profiles. In section 4.1, an overview of the two different types of control schemes will be presented, including both "openloop" control and "closedloop" control methods. In the ultrafast coherent control community, open loop refers to control processes in which the optimal pulse shape for achieving a specific goal is determined by intuitive methods through careful consideration of the underlying physics. In closedloop control, a learning algorithm is applied experimentally to synthesize the optimal pulse. We will discuss the considerations and setup of our open loop control experiment, as well as the NLSE simulation tools we used to model the experimental results in section 4.2. We investigate the effect of input pulse second order and third order spectral phase variations on the supercontinuum generation in section 4.3. In particular, effects of input pulse third order spectral phase on the supercontinuum with input pulse center frequency near the microstructured fiber zero dispersion (GVD) point are studied. Moreover, in section 4.4, we perform open loop control of pulse propagation selfsteepening nonlinear effect in microstructured fiber using a preshaped "ramp" pulse to counteract the selfsteepening. Simulation results based on the extended NLSE model corresponding to open loop control will also be presented and compared to experimental results. A standard splitstep Fourier algorithm [3] is used in the simulation model. Closed loop control experiments and results will be discussed in details in the next chapter. 4.1 Control of Supercontinuum Generation in PCFs Continuum generation results from the nonlinear interactions between the input pulse and nonlinear media. Due to the intrinsic properties of nonlinear interactions, the resulting bandwidth of the continuum is much larger than that of the input pulse. Continuum generation has been studied over decades. Continuum generation of 400 nm bandwidth was first observed in the glass in 1970 [109]. Since then, the nonlinear media used for continuum generation has evolved from bulk glass and liquid to optical fiber. The low loss, tightly guided modes, and high nonlinearities of the optical fibers make them natural candidates for the supercontinuum generation. As such, optical fibers lead to a dramatic reduction in the pump power for the continuum generation. Various nonlinear effects such as SPM, SRS and FWM subsequently lead to the continuum generation in the optical fibers. Due to its high material dispersion at Ti:Sapphire wavelengths, continuum generation in the optical fibers has been mainly in the telecommunication wavelength region (1550 nm) before the invention of PCFs. In addition to their extremely small core sizes, PCFs revolutionized continuum generation by introducing a large amount of waveguide dispersion contribution, which can cancel the fiber material dispersion and yield zero dispersion for as low as 500 nm [48]. The advantages of PCFs manifest themselves through a continuum generation range from 400 nm to 1600 nm (supercontinuum) using only a Ti:Sapphire oscillator [6]. Since then, supercontinuum generation has quickly found applications in various fields such as telecommunication, optical metrology and medical science. In fact, the nonlinear mechanisms that lead to the supercontinuum generation in the PCFs are in no way different from that of continuum generation in conventional fibers. But in PCFs, the nonlinear interactions usually happen within the first several millimeters of the fiber length and in a much larger scale. A large amount of research in the area of supercontinuum generation in PCFs have been performed to explain the active roles of various nonlinear interactions that lead to the supercontinuum generation, including soliton generation and splitting, SPM, SRS, etc. Supercontinuum generates in PCFs as a result of high order soliton generation and splitting, followed by various other nonlinear interactions slight changing the supercontinuum envelope as propagating along the PCFs. Supercontinuum generation in PCFS is nonlinear processes in the extreme. Large modulation structures are easily seen in the continuum envelope due to the soliton splitting. Meanwhile, sharp subnm continuum features vary drastically from shot to shot [9, 13], as both the experimental and simulation results show that supercontinuum generation is extremely sensitive to the input pulse power fluctuation. In figure 4.1, left figure shows the experimental supercontinuum fine structure variation from three different shot, measured by XFROG. Simulation result shown in the right figure 4.1 also reveal that for 1% input pulse power fluctuation, although the continuum envelope does not show dramatic change, the fine structures of the supercontinuum change drastically. This drawback greatly affects some of the supercontinuum generation applications, such as pulse compression and high precision optical metrology. Various research activities have been carried out to study the stabilities of the supercontinuum generation. In general, these research interests include variation of the input pulse properties, microstructured fiber propagation length effects [15] and tailorable dispersion properties. Our approach in Chapter 6 will focus on the simulation of utilizing dispersion flattened microstructured fiber to study the pulse compression application. 520 560 600 waveleng(e)th (nm) 40sim io tin ti i S p er varies by r s to G a [].01 2.0 V V 0.01 0.0 I I 600 601 602 603 604 605 520 560 600 wavelength (nrm) Figure 4.1 : Subnm supercontinuum feature fluctuations as a result of input pulse power fluctuation. Left figure, experimental result of supercontinuum fine structure variation from three different shots, refers to Gu et al. [13]. Right figure, simulation of subnm supercontinuum feathers fluctuation as the input peak power varies by 1%, refers to Gaeta [9]. Controlling nonlinear optical processes in microstructured fiber is in general difficult, resulting from their intrinsic "nonlinear" responses to the input fields. Meanwhile, controlling the supercontinuum generation process in microstructured fibers is important for a number of applications. It allows us to extensively study various nonlinear processes and dispersion effects that lead directly to the fascinating supercontinuum generation, as well as gain the ability to generate the supercontinuum which properties are suited for one particular application. It is evident that the supercontinuum generation and pulse nonlinear propagation in the microstructured fibers strongly depend on both the input pulse parameters and microstructured fiber properties. The "control knobs" for supercontinuum generation in the microstructured fiber can therefore be characterized as the following two categories. "Control knobs" related to the microstructured fiber properties include simple controls such as fiber length and more advanced controls such as fiber dispersion and nonlinear properties. As mentioned in the chapter 2, fiber dispersion properties can be engineered by choosing the proper fiber parameters such as air hole size and pitch; meanwhile changing the core diameter can subsequently alter the nonlinear response of the microstructured fiber. The input pulse parameters, on the other hand, are much easier to handle and have a larger number of "control knobs". One can easily enumerate many input pulse parameters such as pulse power, wavelength, polarization or even the control of pulse chirp of some kind. Apolonski's results published in 2002 examined the influences of these control "knobs" on the supercontinuum generations [110]. More generally, Fourier domain pulse shaping [99], in which Fourier synthesis methods are used to generate nearly arbitrarily shaped ultrafast optical waveforms has been proven to be a powerful tool and have applications in many optical fields such as high power laser amplifiers, quantum control and optical communications. When coupled with adaptive or learning algorithms [111], Fourier domain pulse shaping has shown to be very effective in exercising control over nonlinear optical processes and producing a specific "target" nonlinear output state. Two distinct control methods, open loop control and closed loop control, can be utilized to perform the Fourier domain pulse shaping. In general terms, open loop control is defined as the application of specific pulse or sequence of pulses and is carried out irrespective of the outcome of the experiment. In particular, for the pulse shaping technique, open loop control method utilizes knowledge of the underlying physics to intuit or derive a suitable control pulse. For example, the fundamental dark soliton was observed in optical fibers utilizing specially shaped and asymmetric input pulses, which is in quantitative agreement with numerical solution to the nonlinear Schrodinger equation [112]. Open loop pulse shaping shows great advantage in the coherent control regime. Ultrafast coherent control of excitons in quantum wells use intuitively shaped pulse trains to investigate the generation process and intermediate virtual states in quantum structures [113]. Coherent control of Bloch oscillations using open loop pulse shaping provides a way to control the emitted THz radiation [114]. These are but a few of the recent applications of coherent control with shaped pulses. For pulse nonlinear propagation in the microstructured fiber, the chirp (or the spectral phase) of the input pulse can be altered with much higher resolution in the Fourier pulse shaping comparing to that of Apolonski where multiple pairs of chirp mirrors were used; therefore gaining better abilities on how the input pulse spectral phase can affect the supercontinuum generation. Meanwhile, using the phase sculpted "ramp" pulse to counteract self steepening nonlinear effect allows us to suppress the blueshifted continuum generation. Closed loop control, also referred as adaptive control, involves "feedback" through the repetitive application of optical waveforms synthesized using learning algorithms to ensure that the physical goals are met. Closed loop control coupled with Fourier domain pulse shaping have become a powerful means for optimizing a particular physical process. Closed loop control experiments and results will be discussed in details in the next chapter. To our knowledge, this dissertation and its related papers investigate for the first time how Fourier domain pulse shaping can be used to control the evolution the nonlinear pulse propagation in microstructured fibers. 4.2 Open Loop Control Experiment Setup and NLSE Simulation We use two laser systems in the open loop control pulse shaping experiments, depending on the experiment of interest. The Mira 900 Ti:Sapphire laser system generates pulses of 76 MHz repetition rate, with 6 nJ pulse energies, 200 fs pulse duration and pulse central wavelength is tunable from 700 nm to 900 nm. The bandwidth of the output pulse is 5 nm, and the Mira 900 is used primarily to study the effect of third order input pulse phase on the supercontinuum generation when the input pulse's central wavelength is close to zero GVD point the microstructured fiber ( 763 nm). For other open loop control experiments, we use a 82 MHz, 30 fs Ti:sapphire laser producing 3 nJ pulses that centered at 800 nm. The bandwidth of this laser system is 65 nm, a much larger bandwidth comparing to that of Mira 900 as one can easily see by comparing the pulse durations of these two laser systems. The output pulse train is phaseonly shaped (pulse shaper that only modulates the pulse spectral phase) by an allreflective 4f Fourier domain pulse shaper using a 128 pixel programmable liquid crystal spatial light modulator (LCSLM) as described in Chapter 3. The focal length of the mirror used in the pulse shaper is 25 cm. Because the two laser systems have different central wavelengths and bandwidths, two considerations are taking into account with the design of pulse shaper to get high pulse shaping resolutions. 300 lines/mm grating pairs are used for Ti:Sapphire laser with 65 nm bandwidth and 1500 lines/mm grating pairs are used for Mira 900 laser with 5 nm bandwidth; the grating line spacings correspond to 3.3 un and 0.67 /n, respectively. Input pulse incident angles for pulse shapers can be easily calculated according to equation 3.16 to utilize large number SLM pixels and yield high pulse shaping resolutions. The power transmission through the pulse shaper is approximately 40%. After the pulse shaper, the pulse is coupled into a piece of microstructured fiber using a 100x objective (Vickers w4017) of a numerical aperture 1.3. Meanwhile, the shaped pulse is characterized by the second harmonic generation frequency resolved optical gating (SHG FROG) to verify the pulse temporal structure. Some dispersion is introduced by the glass in the objective; this is measured separately and compensated for in our experiments. The microstructured fiber consists of a 1.7 pn diameter silica core surrounded by an array of 1.3 pn diameter air holes in a hexagonal closepacked arrangement. The microstructured fiber has a zero GVD wavelength 760 nm and the coupling efficiency using the objective is estimated to be 20%. The supercontinuum generated in the microstructured fiber is collimated using a 50x objective and the spectrum is recorded using a 0.25 m spectrometer with a CCD detector. An example of the supercontinuum generation in our experiment is show in figure 4.2. Figure 4.3 shows the schematics of open loop control experiment setup. 500 600 700 800 900 1000 1100 Wavelength (nm) Figure 4.2: An example of supercontinuum generation in experiment. Inset: a dramatic picture of supercontinuum after a grating, taken by Anatoly. Gradng Gramdnag dl d dl=f dZ~ / ~dl+d2=f k  /  ^  /  r Figure 4.3: Schematic diagram of the open loop control experiment. SLM: Spatial light modulator. SHG FROG: Second harmonic generation frequency resolved optical gating. We follow the experimental considerations discussed in section 3.5.2.3 to setup and characterize the pulse shaper. For pulse shapers corresponding to different input central wavelengths, we use LC voltage and phase shift calibration curves (Fig. 3.16) and linearly interpolate the values that do not lie on the curves. Meanwhile, we perform wavelengthtopixel calibrations for the pulse shaper setups to verify the phase shaping resolutions. In both cases, 100 pixels over the whole 128 pixels are used for the pulse shaping. In addition, both autocorrelation and FROG are used to verify the output pulse is the replica of the input pulse when imposing zero additional phases on the pulse shaper. To modal our experiments, we use an extended nonlinear Schrodinger equation [3] (NLSE, described in details in Chapter 2) with a standard splitstep Fourier algorithm to compute the propagation of shaped pulses in the microstructured fiber. The NLSE has been proven to be valid for femtosecond pulses in the limit that the optical frequency bandwidth approaching the central frequency of the pulse [3, 9, 115]. A higher order split step method [115] is used to insure the stability for high input peak powers. In the simulation model, the time resolution is 0.5 fs, the total sampling points is 214=16384 and the propagation step size is chosen as 0.5 mm. The frequency resolution is therefore 111.1 GHz. The input temporal profile of the pulse for the NLSE is calculated from the inverse Fourier transform of the experimentally determined input pulse spectrum and spectral phase, in which the latter is specified from the additional phase induced by the pulse shaper in the condition that the phase of the original unshaped pulse has been well compensated (transformlimited pulse). For the fiber characteristics including the fiber dispersion parameters, we use the parameters taken from the literature for input pulse central wavelength of 770 nm [9]. Here is the list of fiber characteristics that we use in our simulations. Table 4.1: List of the microstructured fiber parameters used in the NLSE simulation model. Parameter (unit) Value n2 (cm2W1) 2.0 x1016 7 (km'W1) 96.3 Aef (/2m) 1.63 f2 (ps'km 1) 0.72 fi3 (pskm 1) 2.16x102 /4 (ps4km 1) 1.296 x105 4.3 Influences of Quadratic and Cubic Spectral Phase on Propagation Dynamics As the first example of open loop pulse shaping experiment, we examine the influences of quadratic and cubic spectral phase to the output supercontinuum generation when the input central wavelength is in the anomalous dispersion region of the microstructured fiber. For the Ti:Sapphire 800 nm laser system used in this experiment, the output pulse peak power is 100 kW. Considering 40% power transmission through the pulse shaper and 20% fiber coupling efficiency, it corresponds to a nonlinear length of1 mm (defined in equation 3.25) calculating from the fiber parameters list in table 4.1), which is 103 times shorter than the dispersion length for this microstructured fiber. This ratio manifests itself through the drastic nonlinear effects in the microstructured fiber which consequently lead to the supercontinuum generation. The magnitudes of various nonlinear effects (i.e., supercontinuum bandwidth) taken place in the microstructured fiber critically depend on the input pulse peak power, which is determined by the input pulse spectral phase given a fixed spectrum. Figure 4.4 illustrates this point, in which simulation results using the experimentally determined Ti:Sapphire laser spectrum show pulse peak intensity variations as a function of pulse spectral phase. Note that the pulse spectral phase is determined by the following equation. d(2) d(3) () = (1)(0 o0) + (0 00)2 + ( 0)3 +..., (4.1) 2 6 where wo0 is the central angular frequency and the e'0 are the higher order phase terms defined in equation 3.11. Note that the linear phase shift does not affect the pulse peak intensity, therefore is not considered in the discussion. In Figure 4.4 (a) and (b), the pulse peak intensity drops continuously with the increases of both cubic and quadratic spectral phase. Since pulse peak intensity changes do not depend on the sign of the introduced spectral phases, only positive cubic and quadratic spectral phase are shown in the figures. It is quite obvious that the peak intensity dropping is much faster for the quadratic spectral phase coefficients than the cubic phase coefficients, as the dropping ratio is over 50% for the quadratic phase of 2000 fs2 and less than 1% for the cubic phase of 2000 fs3, although it also applies for the phase changes according to equation 4.1. In addition, the temporal intensity profiles showing in figure 4.4 (c) illustrate this huge dropping ratio difference between the quadratic phase and cubic phase. Note that for both quadratic and cubic phase variation, acquiring maximum pulse peak intensity implies a zero spectral phase (transformlimited) for the input pulse. 100,0 10* . S99.8 ) zem phase 99 6 a c Phi(2)2000 fs2 99.4 Phi(3)2000 fWs3 99.4 (a) 0 500 1000 1500 2000 60 o2000 60 IGO _______________ 00 40 90 S80 20 70 2 S40 200 100 0 100 200 0 500 1000 1500 2000 TIme (fs) 0(Z (f/.) Figure 4.4: Simulation results of the pulse peak intensity as a function of pulse spectral phase. In (a) and (b), the pulse peak intensity verses pulse cubic and quadratic spectral phase, respectively. (c) Comparison of the pulse temporal intensity with transformlimited pulse, pulse with quadratic spectral phase of 2000 fs2 and cubic spectral phase of 2000 fs3. Based on previous discussions, we investigate the influences of input pulse quadratic and cubic spectral phase on the supercontinuum generations. The experimental results showing in figure 4.5 nicely authenticate the theoretical explanations. The spectra of supercontinuum generation change dramatically upon variations of input pulse quadratic spectral phase. The bandwidth of the supercontinuum is greatly suppressed when introducing extra quadratic spectral phase, reducing from 450 nm for the transform limited input pulse to 250 nm for the induced quadratic spectral phase of the largest magnitude. There are only slight (but experimentally measurable) influences of the cubic spectral phase on the supercontinuum generation. Corresponding to the pulse peak intensity variations for introduced pulse quadratic and cubic phase, the latter shows a much smaller supercontinuum generation influences, although the maximum supercontinuum bandwidth is still generated by a transformlimited input pulse. There is also an added discovery when looking into the experimental data. We expect to see the U" 2000 2000 600 700 800 900 1000 600 700 800 900 1000 Wavelength (nm) Wavelength (nm) Figure 4.5: Experimental results of supercontinuum generation dependence on the input pulse quadratic and cubic spectral phase. In both (a) quadratic phase variation and (b) cubic phase variation, the supercontinuum obtains the broadest bandwidth for transformlimited input pulse. soliton generation from the supercontinuum since the input central wavelength is in the anomalous dispersion region and soliton generation results from the interplay between the fiber anomalous dispersion and selfphase modulation effect. Both figures show the evidence of soliton generation, as evidenced by the spilling off of a large spectral component on the long wavelength side of the supercontinuum and its subsequent shift to the red side of the spectrum as the phase is minimized. The first interesting observation is that soliton generation disappears when large magnitudes of the spectral quadratic phase are presented; resulting from the steep peak intensity drop for the input pulse and that soliton order is proportional to the square root of input pulse peak power. Meanwhile, whereas the spectral quadratic phase acts as a coarse control knob for the positions of soliton generation, the spectral cubic phase shows the fine tunability. Therefore, it is understandable that the peak intensities of the input pulse actually perform the control over soliton generation, as further illustrated by both the closed loop experimental and simulation results discussed in the next chapter. To further investigate the influence of the input pulse spectral phase on the pulse propagation dynamics, we study the case that the input pulse frequency is near the zero GVD point of the microstructured fiber. It is known that finite bandwidth pulses centered at the zero dispersion wavelength experience both normal and anomalous dispersion. Near the zero GVD point of the microstructured fiber the dispersion length of the third order fiber dispersion (TOD), L3d, is comparable to or even smaller than that of second order fiber dispersion (SOD) Ld Here L = _' /,I where r is the input pulse duration and 8, is the ith order fiber dispersion parameter. Therefore TOD plays an important role for the fiber dispersion property as well as the interplay between the fiber dispersion and selfphase modulation nonlinear effect when L3d is comparable toLd In addition, for the input pulses presenting a cubic spectral phase, the intrinsic fiber TOD can compensate the cubic spectral phase and compress the input pulse, providing the sign of the cubic spectral phase is opposite to the sign of the intrinsic TOD; therefore enhancing the magnitude of nonlinear effects (i.e., SPM) by increasing the pulse peak power as pulse propagates along the fiber and generating a broader supercontinuum. When the signs of input pulse cubic spectral phase and fiber TOD are the same, nonlinear effects are suppressed due to the fact that the fiber TOD induces extra spectral phase and lower the pulse peak power along the propagation length. The zero GVD point of the microstructured fiber listed in the table 4.1 is estimated to be 760 nm. To make the input central wavelength close to the zero GVD point of the microstructured fiber, in the first set of experiments, we use our homebuild Ti:Sapphire oscillator and push the laser pulse central wavelength down to 770 nm. At this particular wavelength, L3d L2d, as one can easily see by the calculation using the listed parameters. Upon changing only the cubic spectral phase of the input pulses, the supercontinuum maximum bandwidth should occur for a ,a3 = 0(33/ Liber where ,A is the fiber TOD parameter and 0(3) corresponds to the input pulse cubic spectral phase. Note that in this equation the signs of ,3 and 0(3) should be opposite, as only in this case the fiber TOD can interact with the propagating pulse to compensate the pulse intrinsic 0(3). We use two pieces of microstructured fibers with different fiber length of 5 cm and 70 cm to study the influence of input pulse cubic spectral phase on supercontinuum generation bandwidth. As we can see from figure 4.6 (a), the experimental output spectra bandwidths from the 5 cm microstructured fiber show a dramatic asymmetry upon examining the sign change of the input pulse cubic spectral phases, indicating fiber TOD dispersion has a different effects on the positive and negative pulse cubic spectral phases. The maximum bandwidth occurs when the input pulse 0(3)is 500 fs3. This corresponds to the fiber TOD parameter ,3 of a value +0.01 ps3/km, which is in reasonable agreement with the reported A value above. According to the equation, for the 70 cm microstructured fiber, proportionally more 0(3) is needed and the GVD broadening will play a larger role, thus the effects of pulse cubic phase should be minimal when imposing same amount of 0(3) as for the short fiber, which can be seen in figure 4.6 (b). 87 400 (a) (b) 390 100 Z 380 S370 2000 0 360 I i 2000 4000 2000 0 2000 4000 6000 8000 600 800 1000 1200 1400 D(3) (fs3) Wavelength (nm) Figure 4.6: Experimental results of supercontinuum generation bandwidth from 5 cm and 70 cm microstructured fibers as a function of input pulse cubic spectral phase. (a) Bandwidth changes of SC generation from 5 cm MSF with variation of input pulse n(3). Inset: output spectra from a 5 cm MSF with variation of input pulse ((3). (b) Bandwidth of SC generation from 70 cm MSF with variation of input pulse (3). In order to further exam this effect and move closer to the zero GVD point of the microstructured fiber, we use the Mira 900 wavelengthtunable laser oscillator. It has a wavelengthtunable range of 700 nm to 900 nm. The drawback is that the bandwidth of the Mira 900 laser pulse (5 nm) is rather small compare to the homebuild Ti:Sapphire laser. Therefore, to see an enhanced effect of the fiber TOD comparing to the first set of experiments, the central wavelength of the Mira system has to be tuned very closely to the absolute zero GVD point of the fiber. However, the fiber parameters provided in table 4.1 are only estimated values at this precision level, and the pulse shaper also needs to be adjusted each time when the tuned central wavelength is changed. In order to find the absolute zero GVD point of the fiber, we first bypassed the pulse shaper and projected the laser beam directly into the microstructured fiber and find that output spectra have the maximum bandwidth at 763 nm. Since input pulses have a small bandwidth and experience the minimal GVD at zero GVD point, thus generating maximum bandwidth supercontinuum at this wavelength, 763 nm is the zero GVD point of the fiber. Based on 