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PAGE 1 1 IN THE QUEST OF MANIPULATING LIGHT MATTER INTERACTIONS: COHERENT CONTROL OF TWOPHOTON INDUCED PROCESSES IN SOLUTION By DANIEL GUSTAVO KURODA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Daniel Gustavo Kuroda PAGE 3 3 To the most important people in my life, my wife Fedra, my parents Laura and Yuji, my brother Marcelo, and my si ster Fernanda. PAGE 4 4 ACKNOWLEDGMENTS As I look back upon the years that have led to this dissertation, I have been lucky to be surrounded by such exceptional people. I would like to extend m y warmest thanks to my research advisor Professor Valeria D. Kleima n for her scientific gui dance and continuous motivation throughout my time in her group. She has been a great mentor and friend from whom I have learned a lot. It was her trust, understa nding, enthusiasm, and tuto ring that have led the Coherent Control project to completion. I wish to thank my supervisory committee me mbers, Prof. David Reti ze, Prof. Jeff Krause, Prof. Nicol Omenetto, and Prof. Phillip Brucat for their guidance, fruitful collaborations, valuable discussions throughout my graduate stud ies, and for accompanying me in the final stage of my graduate career. I extend my thanks to Prof. Adrian Roitberg for his contribution in the theoretical work, to Prof. Nicol Omenetto for teaching me the fundamentals of laser science, and to Prof. Sam Colgate for his teaching, trust, and support. I am also grateful to the Machine Shop people: Todd Prox and Joe Shalosky for the nu merous hours that they have spent teaching me the art of machining. I also thank Prof. Zhonghua Peng from University of MissouriKansas for providing the unsymmetric dendrimers. Many friends as well as coworkers have had an important role during my graduate studies in Gainesville with their discussions and companionship. The memb ers of Kleiman Group provide a fun work environment. Thanks go out to Dr. Jrgen Mller for his friendship and tutoring on my first two years of graduate schoo l. I thank Dr. Evrim Atas for her contagious energy and personal support. I would like to thank Dr. Linds ay Hardison for being a good friend/lab mate and for sharing both science and personal matters, and Mr. Derek LaMontagne for suffering to correct my English in this di ssertation. I thank Dr. Ch ad Mair and Ms. Aysun Altan for their support and chat s in the lab. The newest members of our group, Ms. Shiori PAGE 5 5 Yamazaki, Ms. Sevnur Komurlu, Dr. Chandra Pa l Singh should also be thanked for their friendship. I also thank Mr. Montgomery and Prof Damrauer for their co ntributions and help with the genetic algorithm. I especially tha nk Mr. Bob Letiecq and Mr. Mike Herrick from SpectraPhysics who provided day and night technical support. My time here would not have been the same without the social dive rsions provided by all my friends in Gainesville. I am particularly th ankful to Jorge, Sarah, zge, Ece, Georgeos and Giovanni for their continuous friendship. I woul d like to thank Adrian and Valeria for being wonderful friends and great supporte rs since the day I arrived in th e United States of America six years ago, and also for allowing me and my wife to teach them how to play burako. I reserve a very special thanks for Gustavo for his friendshi p and support, and for motivating me to improve my fitness. My particular thanks go out to my friends from Argentina: Pablo H., Pablo E., German, Tamara, Ale C., Andres, Marcos, Flor, Ale L., Roby, and all the members of BWIM for their long distance friendship. My most immense thanks go to my parents Laura and Yuji for their guidance and support in the decisions that I have made since I was a child. They always have believed and supported me in whatever I have been doing. I also thank my sister Fernanda and my brother Marcelo for sharing a very happy childhood together and for their long distance suppo rt and understanding. Last but not least, I give a special thanks to my wife, Fedra, for her true love and understanding me without words. You are my soul mate and you bring happiness and peace to my life. PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES .........................................................................................................................10 LIST OF FIGURES .......................................................................................................................11 ABSTRACT ...................................................................................................................... .............20 CHAP TER 1 COHERENT CONTROL ....................................................................................................... 22 Introduction .................................................................................................................. ...........22 Single Parameter Quantum Control Schemes ......................................................................... 23 Multiparameter Quantum Contro l Schem e: Optimal Control ................................................ 26 Optimal Control in Solution ................................................................................................... 29 Single Parameter Control ................................................................................................ 31 Wavelength control .................................................................................................. 31 Spectral interference .................................................................................................31 Dynamics interference and pump and dump ............................................................ 33 Linear chirp control .................................................................................................. 34 Multiparameter Control ................................................................................................... 37 Control of selective electronic excitation ................................................................. 37 Control of vibra tional excitation ..............................................................................39 Control of complex molecular processes ................................................................. 40 Optimal Control and Molecular Mechanism .......................................................................... 42 Outline of the Dissertation ................................................................................................... ...44 2 FEMTOSECOND LASER PULSES ...................................................................................... 47 Introduction .................................................................................................................. ...........47 Femtosecond Laser Pulses ......................................................................................................47 General Mathematical Description of a Femtosecond Laser Pulse ................................. 47 Pulse Propagation ............................................................................................................51 Nonlinear Optical Effects ................................................................................................ 54 Femtosecond Laser Generation ..............................................................................................56 Ti:Saphire Oscillator ....................................................................................................... 56 Femtosecond Amplifier ................................................................................................... 58 Femtosecond Pulse Characterization ...................................................................................... 59 Frequency Resolved Optical Gating Technique .............................................................. 59 PAGE 7 7 3 FEMTOSECOND PULSE SHAPING ................................................................................... 63 Introduction .................................................................................................................. ...........63 Femtosecond Laser Pulse Shaping w ith Spatial L ight Modulators ........................................ 63 Pulse Shaping in the Frequency Domain ......................................................................... 64 Zero Dispersion 4f Compressor .......................................................................................66 Spatial Light Modulator ..................................................................................................67 Liquid crystal mask characteristics .......................................................................... 68 The spatial light modulator ...................................................................................... 70 Pulse shaping with LC masks ................................................................................... 72 The Pulse Shaper ....................................................................................................................75 Geometrical Configurations ............................................................................................ 75 Reflective Mode Pulse Shaper ......................................................................................... 78 Design Characteristics .....................................................................................................79 Reflective Mode Experimental Realization ..................................................................... 81 Pulse Shaper Phase Calibration .......................................................................................84 Pulse Shaper Wavelength Dispersion Calibration ........................................................... 89 Residual Phase .................................................................................................................90 Frequency Nonlinearity Correction ................................................................................. 94 Modulation Examples ...................................................................................................... 95 Spatial Light Modulation Limitations ............................................................................. 97 Summary ....................................................................................................................... ........101 4 CLOSED LOOP OPTIMIZATION S AND GENETIC ALGORITHM S ............................ 102 Introduction .................................................................................................................. .........102 Genetic Algorithms ............................................................................................................ ...103 Genetic Algorithm Code ................................................................................................ 104 Genetic Algorithm Implementation ............................................................................... 105 Parent selection ......................................................................................................105 Genetic operators ....................................................................................................106 Our algorithm code .................................................................................................110 The role of the genetic algorithm parameters ......................................................... 112 Optimization of Genetic Algorithm Parameters ............................................................ 113 Mutation probability ...............................................................................................116 Number of elite parents ..........................................................................................117 Number of crossover children ................................................................................ 118 Number of cloned children .....................................................................................118 Optimized Genetic Algorithm Parameters and Noise ................................................... 119 Summary ....................................................................................................................... .120 Experimental Pulse Compression ......................................................................................... 121 Implementation of Genetic Algorithms in Pulse Shaping Experiments ........................ 121 Optimization Problem ................................................................................................... 121 Setup and Optimization Conditions ............................................................................... 121 Results ...........................................................................................................................123 Summary ....................................................................................................................... .125 PAGE 8 8 5 QUANTUM CONTROL OF TWOPHOTON INDUCED FLUORESCENCE OF RHODAMI NE 6G IN SOLUTION ......................................................................................126 Introduction .................................................................................................................. .........126 Molecular System and Control ............................................................................................. 126 Experimental .................................................................................................................. .......129 Results ...................................................................................................................................131 Fluorescence Efficiency Optimization .......................................................................... 132 Power Dependence ........................................................................................................136 Optimization Analysis ................................................................................................... 139 Fluorescence Spectrum Control ....................................................................................141 Relationship between Objective and the Genetic Algorithm Solutions ........................ 143 Discussion .................................................................................................................... .........148 Physical Interpretation of the Fitness ............................................................................ 148 Molecular Interpretatio n of the Fitness ......................................................................... 149 Perturbation Analysis ....................................................................................................149 Summary ....................................................................................................................... ........152 6 COHERENT CONTROL OF ENERGY TRANSFER ........................................................154 Introduction .................................................................................................................. .........154 Donor Acceptor System ........................................................................................................ 155 Experimental Section ............................................................................................................158 Results ...................................................................................................................................161 TwoPhoton Cross Section ............................................................................................161 Closed Loop Experiments .............................................................................................162 Open Loop Experiments ................................................................................................ 167 Discussion .................................................................................................................... .........172 Summary ....................................................................................................................... ........176 7 STATISTICAL ANALYSIS OF THE RE SULT S OF THE QUANTUM CONTROL OF ENERGY TRANSFER EXPERIMENTS ...................................................................... 178 Introduction .................................................................................................................. .........178 Experimental .................................................................................................................. .......180 Partial Least Squares Regression ..........................................................................................181 Genetic Algorithm Evolution Analysis ................................................................................ 184 Statistical Analysis ........................................................................................................ 184 Discussion .................................................................................................................... ..189 Summary ....................................................................................................................... .191 Variable Space Reduction .....................................................................................................191 Variable Space Size Effect ............................................................................................191 Statistical Methodology Effect ...................................................................................... 193 Theoretical Phase Unwrapping Effect ........................................................................... 194 Experimental Phase Unwrapping Effect ....................................................................... 196 Discussion .................................................................................................................... ..197 Summary ....................................................................................................................... .198 PAGE 9 9 8 AZOBENZENE EXCITATED STATE DYNAMICS W ITH TWOPHOTON EXCITATION .................................................................................................................... ..199 Introduction .................................................................................................................. .........199 Azobenzene .................................................................................................................... .......200 Experimental .................................................................................................................. .......204 Results and Discussion ........................................................................................................ .206 Summary ....................................................................................................................... ........208 APPENDIX A PULSE SHAPER ALIGNMENT ......................................................................................... 209 B TIME DEPENDENCE PERTURBATION THEORY ......................................................... 212 C PARTIAL LEAST SQUARES ............................................................................................. 215 D PARTIAL LEAST SQUARE S ANALYSIS RESULTS ...................................................... 217 Partial Least Squares Regression for The Autocorrelation Evolution ..................................217 Partial Least Squares Regression for the Spectral Evolution ............................................... 219 Partial Least Squares Regression for the Phase Evolution ................................................... 222 E SPATIAL LIGHT MODULATO R CHARACTERIZATION ............................................. 225 LIST OF REFERENCES .............................................................................................................230 BIOGRAPHICAL SKETCH .......................................................................................................245 PAGE 10 10 LIST OF TABLES Table page 31 Theoretical zero dispersion com pressor characteristics. .................................................... 81 41 Initial genetic algorithm parameters. ............................................................................... 116 42 Optimized genetic algorithm parameters. ........................................................................ 119 43 Fitness results for simulations with noise. Set 1 r epresents the initial and set 2 the optimized genetic algorithm parameters. ......................................................................... 120 44 Genetic algorithm parameters. ......................................................................................... 122 51 Power dependence linear fit results. ................................................................................ 137 71 Root mean square for the different statistical m ethodologies. ......................................... 194 72 Experimental signal and fitness values for multiple components. ................................... 197 73 Experimental normalized fitness values for multiple components. ................................. 197 81 Experimental and literatu re decay times. ......................................................................... 207 PAGE 11 11 LIST OF FIGURES Figure page 11 BrumerShapiro phase control scheme Adapted from work by Brumer et al.15 ...............24 12 TannorKosloffRice timedomain control scheme. .......................................................... 24 13 Adiabatic rapid passage as proposed by Bergm ann. Adapted from work by Gaubatz et al.22 .................................................................................................................................25 14 Spectral sinusoidal phase creating a pulse train. A) Spectrum (solid line) and spectral phase (dotted line). B) Second order power spectrum comparison between the modulated pulse (solid line) and a tr ansform limited pulse (dashed line). ........................32 15 Pump and dump scheme for linearly chirped pulses. Adapted from work by Cerullo et al.22 .................................................................................................................................35 21 Temporal electric field representation of a Gaussian pulse. Tem poral electric field, E(t) (dash line), real amplitude, A(t) (grey line), and intensity (black line). ..................... 49 22 Temporal electric field for phase modulat ed u ltrafast laser pulses. A) Pulse with positive chirp ( a>0). B) Pulse with negative chirp ( a<0). ................................................ 50 23 Experimental layout for a FROG appa ratus using second harm onic generation. .............. 60 24 Schematic of a generic FROG algorithm. Adapted from work by Trebino.100 ..................61 31 Basic layout for a zero dispersion compressor. G: grating, L: lens FP: Fourier plane. Adapted from work by Weiner et al.27 ...............................................................................67 32 Cartoon of a nematic phase liquid crystal. ea: extraordinary axis. .................................... 68 33 Side view of a single nematic crystal pi xel. A) W ithout electric field. B)With an external field applied. Adapted from work by Weiner et al.109 .........................................69 34 Schematic of spatial light modulator mas k. A) Front view. B) Lateral cut. Adapted from work by Weiner et al.27 .............................................................................................70 35 Setup for dual mask sp atial light m odulator. ..................................................................... 73 36 Basic layout of a pulse shaper. A) Tran sm ission mode configur ation. B) Reflective mode configuration. ...........................................................................................................75 37 Top view of different design types for pulse shapers based on an optical F ourier transform. .................................................................................................................... .......76 38 Tilt in the Fourier plane produced by a mi rror at 45 degrees with a vertical tilt. .............. 77 PAGE 12 12 39 Diagram of the experimental setup in a reflective mode configuration. ............................78 310 Grating groove density and beam waist versus f ocal length. Gr ating groove density effect (solid line). Beam wa ist effect (dashed line). .......................................................... 81 311 Top and side views of the built pulse sh aper. S M: spherical mi rror; G: grating, M: mirror; P: polarizer; WP: /2 waveplate; FR: Faraday rotator, CP: cube polarizer. .........82 312 3D Sketch of the built pulse shaper. SM: spherical m irror; G: grating, M: mirror; P: polarizer; WP: half waveplate; FR: Fa raday rotator, CP: cube polarizer. .........................82 313 Polarization splitting of the incoming and the outgoing pulse. Incom ing and outgoing polarizations are represented with a black and red arrow, respectively. ........................... 83 315 Experimental setup used for the m asks calibration. ........................................................... 86 316 Phase calibration curves of: A) front mask and B) back mask. ......................................... 87 317 Phase at 700 counts (1.71 V) for: A) front m ask and B) back mask. ................................ 87 318 Phase calibration of front mask. A) Corrected cu rves. B) Comparison between the average of all the curves a nd calibration curve at 799.66nm (i nset: correction factor). .... 88 319 Phase calibration of back mask. A) Corrected curves. B) Comparison between average of all the curves a nd calibration curve at 799.66nm (i nset: correction factor). .... 88 320 Dispersion calibration. A) Unm odulated and modulated sp ectra (red and black lines, respectively). B)Transimission produced by holes in the profile. C) Wavelength calibration curve. ................................................................................................................90 321 Pulse shaper setup using an external mi rror. A) W ithout the spatial light modulator. B) With the spatia l light modulator. .................................................................................. 91 322 Phase mask effect on the temporal prof ile of the pulse for A) Zero dispersion com pressor without the spatial light m odulator in place. B) Zero dispersion compressor with the spatial li ght modulator in place. .......................................................91 323 SHGFROG trace of the unm odulated pulse. A) Experi mental trace. B) Retrieved trace. C) Temporal intensity (black line ) and phase (red line). D) Spectral intensity (black line) and phase (red line). ........................................................................................ 92 324 SHGFROG trace of the phase corrected pul se. A) Experim ental trace. B) Retrieved trace. C) Temporal intensity (black line ) and phase (red line). D) Spectral intensity (black line) and phase (red line). ........................................................................................ 93 325 Effect of the nonlinear frequency calibrat ion in the pulse m odulation. A) Sinusoidal phase pulse without frequency nonlinear co rrection. B) Sinusoidal phase pulse with nonlinear frequency correction. .........................................................................................94 PAGE 13 13 326 Frequency calibration curve of the pulse shaper. Dots experim ental points, dotted line linear fit, a solid line: second order polynomial fit. .................................................... 95 327 Odd pulse ( step). A) Spectr al intensity (black line), retrieved phase (red squares), and target phase (red line). B) Temporal intensity (black line) and phase (red squares). ..................................................................................................................... ........95 328 Sinusoidal phase modulation. A) Spectral in tensity (black line) retrieved phase (red squares), and target phase (red line). B) Tem poral intensity (black line) and phase (red squares). ......................................................................................................................96 329 Quadratic phase pulse (4607 fs2). A) Proposed phase (red line), and retrieved spectral intensity (black line) and phase (red squares). B) Temporal intensity (black line) and phase (red line). ............................................................................................................. .....96 330 Temporal double pulse with a 200 fs sepa ration. A) Proposed tem poral intensity (open circles) and phase (red line), retrie ved temporal intensity (black line) and phase (red line). B) Spectral intensit y (black line) and phase (red line). ........................... 97 331 A) Phase unwrapped. B) Phase wrapped. .......................................................................... 97 332 Quadratic phase of 103 pixel2/rad A) Sampled every 10 pixels B) Sample every 10 pixels (black squares) and every pixel (black line). ...........................................................98 41 Roulette wheel. Adapted fr om work by Xiufeng et al.129 ................................................105 42 Crossover operation. ........................................................................................................108 43 Cloning operator. ......................................................................................................... ....109 44 Mutation operation. ....................................................................................................... ...109 45 Implemented genetic algorithm. ...................................................................................... 111 46 Simulated experimental setup. The femtos econd pulse is m odulated in the glass rod (BK7) and its second harmonic intensity is produced in a nonlinear crystal (NLC)...... 114 47 Influence of probability of gene muta tion on the achieved f itness. Average best fitness (black squares). Overall best fitn ess (open circles). Overall worst fitness (open triangles). ...............................................................................................................116 48 Influence of the number of elite parents on the achieved fitness. Average best fitness (black squ ares). Overall best fitness ( open circles). Overall worst fitness (open triangles). ................................................................................................................... ......117 49 Influence of the number of individua ls generated by crossover on the achieved fitness. Average best fitn ess (black squa res). Overall best fitness (open circles). Overall worst fitness (open triangles). ............................................................................. 118 PAGE 14 14 410 Experimental setup used in pulse compression. .............................................................. 122 411 Evolution of the twophoton induced curre nt as a function of the generatio n. Average (open triangles), best (full squares), and wo rst (open circles) fitness values for each generation are plotted. ......................................................................................................123 412 Uncompressed pulse. A) Spectral intensity (black line) and phas e (red line). B) Tem poral intensity (black line) and phase (red line). ......................................................124 413 Compressed pulse. A) Spectral intensity (black line) and phase (red line). B) Tem poral intensity (black line) and phase (red line). ......................................................124 51 A) Rhodamine 6G molecular structure. B) Absorption spectrum (black line) and emission spectrum (dashed line). Inset contai ns a zoom of the absorption spectrum from 360 to 445 nm.......................................................................................................... 127 52 Jablonski diagram of Rhodam ine 6G. .............................................................................. 128 53 Experimental setup for controlling mol ecular fluorescence. L: lens. NLC: nonlinear crystal. PD: photodiode. PMT: photo multiplier tube. ...................................................... 129 54 A) Fitness evolution of typical expe rim ents of maximization of fluorescence efficiency. B) Experimental convergence reproducibility. .............................................. 132 55 A) Optimal autocorrelation for differe nt experiments. B) Typ ical SHGFROG trace of the optimum pulse. ...................................................................................................... 133 56 A) Fitness evolution of typical e xperim ents of minimization of fluorescence efficiency. B) Experimental convergence reproducibility. .............................................. 134 57 A) Optimal autocorrelation. B) Typical SHGFROG trace of the optim al pulse. ........... 135 58 Power dependence of the different pu lses. Top panel, fluorescence intensity dependence. Lower panel, second harmonic intensity dependence. ................................137 59 Power dependence in fluorescencesecond harm onic space. Fluorescence maximization (squares). Fluorescence e fficiency maximization (stars) and minimization (triangles). Zoom of the lowe r left corner of the plot (inset). ....................138 510 Variable space. A) All individuals of each experim ent: maximization of fluorescence (blue squares), maximization of fluorescen ce efficiency (green triangles), and minimization of fluorescence efficiency (g rey circles). B) Fitness for the best individuals on each generation of each expe riment and fitness for individuals with random phases. .................................................................................................................139 511 Variable space in using ratio and difference of signals. Inset shows a zoom of the left corner of the plot. Individuals corr esponding to the fluorescence efficiency m aximization (triangles) and minimization (circles). ...................................................... 140 PAGE 15 15 512 Spectrum measured with th e cutoff filter (grey area). .....................................................142 513 FROG traces for different experiments. A) Maxim ization of fluorescence efficiency when the whole spectrum is collected. B) Maximization of red si de fluorescence over second harmonic. ............................................................................................................. 142 514 FROG traces for different experiments. A) Min imization of fluorescence efficiency minimization when the whole spectrum is collected. B) Minimization of red side fluorescence over second harmonic. ................................................................................ 143 515 A) Fitness evolution of maximization of spectral control. B) Evolution of the cost function for minimization of spectral control. ................................................................. 143 516 A) Fitness evolution of the cost function (FRFB)FR/FB (black line) and simulated evolution of the cost function (FRFB)FR/FB with the data corresponding to maximization of (FBFR)FB/FR (red line). B) Fitness evolution of minimization of spectral control (black line). ............................................................................................145 517 Signal of fluorescence spectral control e xperim ents. A) Maximization, fluorescence (black line) and second harmonic signals (red line) B) Minimization, fluorescence (black line) and second harm onic signals (red line). .......................................................146 61 A)Chemical structures of phenylene ethynylene dendrim er: 2G2mPer. B) 3D model of the 2G2mPer. C) Absorption (gr een line) and emission (grey area) spectra of 2G2mPer, and absorption of 2G2mOH (black line) and ethylene perylene (red line) ......................................................................................................................... .........156 62 Energy diagram and dynamics of 2G2mPer. 1, 2, and 3 are ~0.4ps, ~0.5ps, and ~0.8ps, respectively. ........................................................................................................157 63 Coherent control setup. ....................................................................................................158 64 A) Twophoton cross section of 2G2mPe r (squares) and excitation laser spectrum (gray area). B) Sprectral windows used to measure the twophot on cross section. ......... 162 65 Best individual evolution for the different optim izations: maximization of fluorescence (red squares), maximization of the ratio (gr een squares) and minimization of the ratio (blue squares) Inset shows an e xpanded view of the maximization region. ....................................................................................................... 163 66 Fitness evolution for the maximization of fluorescence over twophoton induced current. Open squares and full circles repr es ent the best indi vidual for the ratio maximization and the calculated fitne ss using the values of the FL only maximization, respectively. ............................................................................................. 164 67 Retrieved (A) Temporal intensity and pha se and (B) Spectrum and spectral phase. ...... 165 PAGE 16 16 68 Autocorrelation of the pulse observed af ter the m inimization and of the transform limited pulse. ................................................................................................................ ....166 69 Variable space for acceptor and donoracceptor system. Full squares and black stars correspond to the optim ization experiment s on the donoracceptor system and using those phase modulated that optimize 2G 2mPer to induced excitation on the acceptor molecules alon e, respectively. ........................................................................... 167 610 A) Fluorescence second harmonic ratio versus the step size of the spectral phase for 2G2mPer (stars), 2G2mOH (open triangle) Perylene (crossed circle). Inset shows the applied phase. B) Experimental (s quares) and calculate d (line) TPIC. ..................... 168 611 Spectrally modulated pulses. A) Spectrum (red line) and spectral phase. B) Second harm onic spectrum (blue line). C) Temporal intensity (black line). From top to bottom: transform limited, 0.4 step, 1.4 step, linearly chirped 2000 fs2, and linearly chirped 2000 fs2 pulses. .....................................................................................170 612 A) Temporal autocorrelation of the puls e codified with a zero phase difference, where each color repres ent pulses separa ted by a different delay. B) Ratio of fluorescence over twophoton induced current versus the time delay between subpulses for zero phase re lation (black line) and phase relation (red line). ................ 171 613 FluorescenceTPIC ratio versus variable quadratic phase for 2G2mPer. ..................... 172 71 Variable space evolution and sampling for 2G2mPer energy transfer optimal control. A) Typical represen tation of the molecular signal versus twophoton signal. B) Ratio versus twophoton induced curre nt for each experiment. Red line (grey dots) and green line (dark grey dots) lines correspond to the best (all) individual for each generation of the ratio of fluorescence over twophoton signal and the fluorescence, respectively. ...............................................................................................184 72 Generational correlation between fluore scence and twophoton i nduced current (full squares) and between the ratio and tw ophoton induced current (open squares) for energy transfer efficiency maximiza tion (red) and fluorescence maximization (black). ...................................................................................................................... .......186 73 Pulse characteristics for the fluor escence for generation 1, 10, 20, 50, 100, and 200. A) Second harm onic spectra. B) Autocorrela tions. C) Fourier transforms of the autocorrelations. ............................................................................................................. ..187 74 Correlation between Fourier frequency amp litude and fitness. Three regions with correlation are observed: between 020 cm1 (negative), 40 to 130 cm1(positive), and 190 to 250 cm1(positive). ................................................................................................ 188 75 A) Latent variable 1 (black line). B) Latent variable 2( black line). Spectrum of the excitation (grey area) pulse. ............................................................................................. 189 PAGE 17 17 76 Latent variables for the best individuals of each of the 200 genera tions (red line) and total population (black line and squares). A) La tent variable 1. B) Latent variable 2. C) Latent variable 3. D) Latent variable 4. ...................................................................... 192 78 Comparison of the model predicted by different statistical m ethodologies and the optimal phase. A) Energy transfer optimi zation. B) Fluorescence maximization. Each graphs shows: original data (black line and squares), the partial least squares model (green line), and the principal comp onent analysis model (red line). .............................. 193 79 Comparison between the model of the optim al phase (red line) and the experim ental optimal phase (black line and points). A) Without phase unwrapping. B) With phase unwrapping. ................................................................................................................... ..195 710 Comparison among modeling of the optimal phase without unwrapping (red line) and with unwrapping (blue line), and the experim ental optimal phase (black line). .......196 81 A) Isomerization of azobenzene. B) Gr ound state absorption of azobenzene sam ples: trans (black line) and cis (red line). .................................................................................200 82 Rotation and inversion pathways of az obenzene isom erization. Adapted from work by Diau.176 ........................................................................................................................201 83 Experimental setup for tr ansient absorption experim ent probing with UV. nIR pump pulses are obtained from the residual fundamental of the OPA. A chopper wheel is used to compare the signal with and without pump. ........................................................ 205 84 Transient evolution of the absorption changes for azobenzene where open circles and black line represent the data and fitting m odel and the grey plots show the instrument response function. A) Probe at 440 nm. B) Probe at 480 nm. ......................................... 206 A1 Scheme of the setup with the apertures a nd translation stages. S1 and S2: translation stages; RS: rotation stage; A1, A2, A3, A4, and A5: iris apertures; M1, M2 steering m irrors; FM, folding mirror. ............................................................................................ 211 C1 Geometrical representation of partial least squares m odel. ............................................. 216 D1 A) Variance explained by each latent variab le. B) Cumulative variance explained. Autocorrelation model (Blue bars). Fitness model (red bars). ......................................... 217 D2 A) RMSE error and B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Autocorrelation model (Blue bars). Fitness model (red bars). .................................................................................................. 217 D3 Error of the fitness model: A) Linear correlation and B) Chi square. ............................. 217 D4 Chi square error of th e autocorrelation model. ................................................................218 D5 Autocorrelation latent variab les: A) F irst and B) Second. ............................................... 218 PAGE 18 18 D6 Autocorrelation latent variab les: A)T hird and B) Forth. ................................................. 218 D7 Fitness (blue line) and PLS linear model (gr een line). A) W ith one latent variable. B) With two latent variables. ................................................................................................ 219 D8 Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) W ith four latent variables. ........................................................................................... 219 D9 A) Variance explained by each latent variab le. B) Cumulative variance explained. Spectral model (Blue bars). Fitness model (red bars). ..................................................... 219 D10 A) RMSE error and B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Spectral model (Blue bars). Fitness model (red bars). ..............................................................................................................220 D11 Error of the fitness model: A) Li near correlation and B) Chi square. ............................. 220 D12 Chi square error of the spectral model. ............................................................................ 220 D13 Spectral latent variables: A) First and B) Second. ...........................................................221 D14 Spectral latent variables: A) Third and B) Forth. ...........................................................221 D15 Fitness (blue line) and PLS linear model (gr een line). A) W ith one latent variable. B) With two latent variables. ................................................................................................ 221 D16 Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) W ith four latent variables. ........................................................................................... 222 D17 A) Variance explained by each latent variab le. B) Cumulative variance explained. Phase model (Blue bars). F itness model (red bars). ......................................................... 222 D18 A) RMSE error and B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Phase model (Blue bars). Fitness model (red bars). .........................................................................................................................222 D19 Error of the fitness model: A) Li near correlation and B) Chi square. ............................. 223 D20 Chi square error of the phase model. ............................................................................... 223 D21 Phase latent variables: A) First and B) Second. ...............................................................223 D22 Phase latent variables: A) Third and B) Forth. ................................................................ 224 D23 Fitness (blue line) and PLS linear model (gr een line). A) W ith one latent variable. B) With two latent variables. ................................................................................................ 224 D24 Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) W ith four latent variables. ........................................................................................... 224 PAGE 19 19 E1 Spatial light modulator ma sk pixel inhom egeneity for ma sk 1 at level 4095 and mask 2 and level 600. .............................................................................................................. ..225 E2 Spatial light modulator ma sk pixel inhom egeneity for mask 1 at level 600 and mask 2 and level 4095. ............................................................................................................. .226 E3 Spatial light modulator ma sk pixel inhom egeneity for ma sk 1 at level 3500 and mask 2 and level 4095. ............................................................................................................. .227 E4 Spatial light modulator ma sk pixel inhom egeneity for ma sk 1 at level 4095 and mask 2 and level 3500. ............................................................................................................. .228 E5 Spatial light modulator ma sk pixel inhom egeneity for ma sk 1 at level 4095 and mask 2 and level 4095. ............................................................................................................. .229 PAGE 20 20 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IN THE QUEST OF MANIPULATING LIGHT MATTER INTERACTIONS: COHERENT CONTROL OF TWOPHOTON INDUCED PROCESSES IN SOLUTION By Daniel Gustavo Kuroda May 2008 Chair: Valeria D. Kleiman Major: Chemistry During the late 80s, BrumerShapiro1 and RiceTannor2 proposed the use of quantum interference of molecular proce sses to control the out come of chemical reactions. With the development of femtosecond lasers and pulse shaper s, it has become possible to experimentally select a specific reaction pathwa y from a pool of possibilities a nd consequently, to control the outcome of a photochemical process. This thes is focuses on controlling processes initiated by light and on the understandi ng of the dynamics that lead to the control. Many systems have been recently studied usi ng closed loop optimization techniques. Most of them show only the proof of principle. In he re the focus is rather on the understanding of the investigated process. The first part involves th e design, construction, and characterization of the apparatus for carrying out optimal control experi ments and the second part concerns the adaptive control of twophoton induced fluorescence, ener gy transfer, and molecular rearrangement in solution. In the first investigated topic, I present the adaptive control of the twophoton fluorescence of Rhodamine 6G in solution. The second topic of quantum control addre sses the energy transfer within a dendritic macromolecule with lighthar vesting properties in solution. In this study, we show the optimal control of energy flow be tween coupled energydonor and energyacceptor PAGE 21 21 chromophores. We utilize phase tailored near IR ex citation pulses to optimize the energy transfer process and amplitude tailored excitation pulses to learn about the na ture of the involved pathways. In addition, statistical correlation analysis is used to investigate control mechanisms based on coherent superposition of states presen t in the dendritic molecule. This analysis provides information regarding puls e parameters that are critical to optimal control of excitation and energy transfer. The last topic describes preliminary m easurements on azobenzene. It focuses on establishing if a twophoton excita tion of azobenzene is possible and on the possible changes in the excited state dynamics caused by multiphoton excitation. The ultimate goal of this study is to set the foundation for a twophoton excitation for coherently controlling the photoisomerization of azobenzene with a modulated 800 nm excitation source. PAGE 22 22 CHAPTER 1 COHERENT CONTROL Introduction Since ancient tim es, humans have been trying to control the transfor mation of matter; to some degree this has been achieved by chemis try. In order to produce the desired molecule, chemists know how to shift the equilibrium by varying thermodynamic properties of the system, such as temperature (average kinetic energy of the molecules) or solvent (interaction energy). These techniques involve synthesizing the propos ed molecule by modifying the system energy on a macroscopic level. However, the reacti on outcome depends upon the way in which energy is injected into the system, i.e. certain reactants attack specific molecular bonds. In principle, a similar methodology can be ut ilized in photochemical reactions. Vibrational spectroscopy reveals resonant frequencies associated w ith specific periodic motions of bonds in molecules. It is possible to think about select ively transferring energy to the molecules using these resonant frequencies. This concept, know as modeselective chemistry, represents the first proposed way to control photochemical phe nomena. This approach was experimentally realized by tuning a monochromatic laser light to the vibrational frequency associated with the bond where the rupture is expected.3, 4 Although the idea sounded plausible, those experimental trials only worked when the selected bond was the weakest in the molecule.57 The failure of the scheme was due to the coupling between the diff erent vibrational modes which lead to a very efficient and fast intramolecular vibrati onal redistribution of the supplied energy.8 Scientists then understood that to control photoc hemical reactions it would be essential to know all of the processes affecting the dynamics. Time resolved and steady state spectroscopi es are well established techniques that have been effectively applied to investigate the dyn amical behavior of many molecular systems. PAGE 23 23 Today, countless molecular systems have been dynamically characterized from its early dynamics times (femtosecond time scale) to th e end of the dynamics (microsecond/millisecond time scale). Thus, the paradigm of molecular sp ectroscopy has shifted from purely observing to observing and controlling the system dynamics. This perspective differs from the proposed mode selective chemistry because it takes into consid eration not only specific molecular degrees of freedom, but the system as a whole. Atoms and molecules, like photons, present a wavelike behavior. This implies that interference phenomena observed in Youngs doubl e slit experiments can be extended to molecules, opening the possibility to produce very interesting effects in matter, i.e. interference.9 For instance, the population of a quantum stat e created simultaneously with two connected pathways will depend on the phase difference be tween the pathways used to generate it.1 Recent efforts have focused on using the coherent properties of la ser radiation to produce these interference phenomena in matter. Single Parameter Quantum Control Schemes Based on spectraltem poral proper ties of coherent laser fields quantum interference can be used to adjust the system from the initial to the desired final quantum state, i.e. quantum control. Since the introduction of the concept in 1978 by Yablonovitch, many different schemes for achieving this goal have b een proposed and realized. One of the first schemes, as proposed by Brumer and Shapiro in 1986, is a frequency domain approach that uses the interference betw een two different excitations reaching the same excited state to manipulate the photoprocess.1, 10 Quantum control is experimentally produced by shining two continuous wave la sers with different center fr equencies which simultaneously induce different excitations on the quantum system Hence the population of the excited state has two dependent components, which are related through a phase term imposed by the phase PAGE 24 24 difference of the two excitation pr ocesses and the molecular phase a ssociated with the final state. The modification of the relative phase between the two excitation pathways affects via interference the final popul ation observed in the excited state (F igure 11). This scheme was first experimentally demonstrated by Elliot and coworkers with atomic mercury.11 Later, Gordons group used a similar concept to control the di ssociation product of pol yatomic molecules and ionization of polyatomic molecules.1214 Figure 11. BrumerShapiro phase control sche me. Adapted from work by Brumer et al.15 A different approach was proposed by Tannor, Kosloff, and Rice in 1985. This methodology exploits the wave pack et propagation in the time domain.2, 16 The scheme principle is based on creating a wave packet and modifyi ng its evolution on the excited state potential energy surface (Figure 12). Figure 12. TannorKosloffRice timedomain control scheme. S0 S1 A+BC ABC AB+C t2 t1 t0 Potential energy Reaction coordinate time t2 t1 t0 3 1 2 Potential energy PAGE 25 25 Tannor, Kosloff, and Rices scheme can be illu strated as follows. Upon excitation with a temporally short pulse, a wave packet is gene rated on an excited poten tial energy surface. After its formation, the wave packet starts to evolve in this potential well according to its specific shape. During its trajectory the wave packet covers many different molecular conformations. When the wave packet is localized in a point on the potential energy surface where the molecular conformation corresponds to that of the desired product, a second pulse is used to transfer the wave packet back to the ground state where no mo lecular rearrangement can occur. To achieve control with this scheme, the time delay between the two ultrafast laser pulses must be correctly timed. Gerbers, Zewails and Wilsons groups were the first groups able to demonstrate its feasibility for controlling small molecules in gas phase.1719 Tannor, Kosloff, and Rices scheme was further developed by Fleming and coworkers.20 Flemings methodology not only uses the time difference between the two pulses, but also maintains a fixed phase relationship between the pulses to coherently control photochemical processes. Using this improved scheme, Flemin g and coworkers demonstrated wave packet interference on molecular iodine.21 Figure 13. Adiabatic rapid passage as proposed by Bergmann. Adapted from work by Gaubatz et al.22 The third scheme is based on the adiabati c passage technique. Laser induced rapid adiabatic passage was first demonstrated in a population inversion experiment by Loy in 1974. 23 pump Stokes 1 2 3 Potential energy Stokes pump time PAGE 26 26 Later, Bergmann and coworkers app lied this finding to show theoretically and experimentally the possibility of transferring a whol e initial quantum state to a differe nt final state using stimulated Raman adiabatic passage in a three state type quantum system.22 This concept is illustrated in Figure 13. The aim is to dire ctly transfer all the population situated in quantum state 1 to final state 3. If the population of the initia l quantum state is first optically promoted to state 2, a dissipative process occurring in 2 will removed a part of the population decreasing the population that can be later transferred to state 3. Using Bergmann scheme, the problem is avoided because states 1 and 3 are optically coup led before transferring the population to any other state. When states 1 and 2 are optically couple d, the coupling directly promotes the population from 1 to 3 without any transient population in 2. This total population transfer only occurs if the adiabatic conditions are fulfilled and if the time ordering of the laser pulses is the appropr iate (Figure 13). Multiparameter Quantum Control Scheme: Optimal Control All these m ethodologies have been optimized and many different examples can be found in the literature.24 In a more refined version of the TannorKosloffRice scheme, the double femtosecond phase locked lasers are replaced with a single femtosecond laser including complex electric field features which enable control of a chemical reaction.2 Ideally, a more complex pulse can be constructed such that it can continuously interact with the wave packet and progressively modify its trajectory until it reaches the desired destination, or equivalently, the desired product. Moreover, this envisioned comp lex pulse could allow one to combine the wave packet interference with pump and dump or pump and pump sche mes through the specific time ordering and phase relationship of its components. To produce these intricate electric fields, the PAGE 27 27 different components of the el ectric field must be codified with many independent and controllable parameters. Considering the ambition of this seminal scheme it is not evident whether this control is possible in any complex molecular system and if so, how the electric field characteristics and correct femtosecond pulse parameterization can be obtained. The first indication of this approachs feasibility was introduced by Clark and coworkers in 1983.25 Theoretically studying a quantum system with discrete spectral features, this group showed the possi bility of transferring the quantum system to any proposed target state with a specifically de signed electric field. However, they did not provide a way for finding th e complex electric field necessary to achieve quantum control. The theoretical calculations of electric field components re quired to manipulate a wave packet during its evolution in the potential energy surfaces are not simple. They require a very precise knowledge of the molecular Hamiltonian. If the coupling to the envi ronment is negligible a theoretical calculation might be possible for small molecule, but for large molecules in condensed phase where strong coupling with the environment is evident, the possibility of making such a detailed calculations are, up to date, impossible. To overcome the problem of knowing in advanc e the complex electric field components, Judson and Rabitz suggested a novel scheme where the complex electric field needed to reach a particular product is obtained experimentally wi thout prior knowledge of the Hamiltonian of the system.26 Based on a closed loop optimization, this sc heme uses an optimization algorithm and molecular signals to find the electric field ar rangement capable of controlling the molecular system. By varying simultaneously many paramete rs of the laser field, the algorithm creates different electric fields whic h are tested on the quantum syst em. The signals produced by the PAGE 28 28 system for a given electric field, known as pulse parameterization, are sent back to the algorithm where they are evaluated. Following this eval uation, feedback is pr ovided by the algorithm which suggests new and improved el ectric fields. After applying th is process successively, the algorithm manipulates the many di fferent electric field compon ents to control the quantum system. This results in a shaped electric field with optimal pulse parameterization suitable for controlling the system under investiga tion. This concept is usually refe rred to in the literature as optimal control or adaptive control. Optimal c ontrol can be applied to any system as long as there is a signal characteristic of the desired product. The drawback of an input free optimization is the rather complicated arrang ement of the electric field components, one from which the type of control scheme applied during the optimization is not easy to infer. This methodology leaves the possibility of producing any of the previously mentioned single parame ter control scenarios and eventually new ones. To make optimal control possible, it is necessary to generate of numer ous different electric fields in a very flexible manner. This has beco me possible with the advent of pulse shapers. Available pulse shapers can modulate the spectral27 and/or amplitude27 and/or polarization28 characteristics of the electric field of laser puls es. The number of different fields that they can generate by current pulse sh apers is on the order of 101000, making this technique suitable for studying the coherent light matter interactions in a systematic way with single parameter schemes as well as in an the adaptive way with a multiparameter scheme. The first experimental demonstration of an optimal control study was performed by Wilson and coworkers in 1997.29 Later, different groups working in different areas demonstrated the control of other processes th at include, but are not limite d to, selective bond cleavage,30 PAGE 29 29 molecular emission in solution,31 twophoton transitions in atoms,32 and high harmonics generation.33 Optimal Control in Solution In the last d ecade many different coherent c ontrol experiments have been carried out for studying a variety of molecular systems in gas phase as well as in condensed phase.9, 15, 30, 3439 Despite the impressive number of controlled systems, the vast majority of them have been focused on molecules in gas phase. Therefore, so lution phase control expe riments are a natural area to investigate. Specific issues related to condensed phase limit the ap plication of different approaches successfully; princi pally among these problems is th e nonlinear response produced by the solvent in the presence of intense electric fields. In optimal control with strong laser fields, an intense electric field is employed to drastically change the potential energy surf aces of the quantum system with the goal of modifying its shape and therefore the wave packet dynamics. Several examples of this type of control in gas phase have been published.40 When this approach is used in the condensed phase the nonlinear susceptib ility of the solvent induc es effects that can hide or destroy the control quantum system signal. For instance, an intense laser pulse generates white light in condensed media. The white light generation threshold is given by the power required for selffocusing and by the material band gap, which in the case of wa ter corresponds to an in tensity on the range of 1013 to 1014 W/cm2. 41 Thus if any electric fiel d is applied to a water so lution, its intensity can not overpass the constraint previous ly mentioned. This problem limits the possibility of producing strong field effects to control the dynamics or photoproducts of molecules in condensed phase. White light supercontinuum gene ration also detrimentally a ffects detection techniques commonly used for feedback. PAGE 30 30 Another challenge that has to be overcome in condensed phase is the rapid decoherence produced by the coupling of the mo lecular system with the solven t. Molecules in the gas phase are coupled to the environment through intermol ecular collisions. Mean free paths on the order of meter, or equivalently milliseconds, are co mmonly found. In contrast, molecules in condensed phase are highly coupled to the environment th rough intra and intermolecu lar interactions and certain coherent processes can be rapidly destroyed by those inte ractions. Nevertheless, not all the coherences are quickly lost in solution phase For example, the time scale of rovibrational molecular coherences is greatly diminished with the presence of a chaotic environment, but studies have shown rovibrational coherences las ting more than few picoseconds even under those conditions.24 To achieve control over mol ecular dynamic processes in solution, the dephasing times of the photoinduced process have to be comparable to the temporal features produced in the modulated pulse. Systems undergoing very fast relaxations compared to the time scale of the pulse features will not be sensitive to the tempor al pulse shape. At the other extreme, a system which relaxes to a state with very long reside nce time will loose all the coherences initially imprinted on it. Since the introduction of the c oncept of quantum control, ma ny different studies have shown the possibility of contro lling the behavior of matter. Mo st experiments are based on the modulation of the excitation source, although the number of electric field parameters modified to achieve control (single or multiple parameters) varies. Single parameter experiments are very valuable and many times they can provide a more intuitive picture of the process being controlled. PAGE 31 31 Single Parameter Control Wavelength control Wavelength control was the first proposed m e thod to achieve selec tive bond cleavage, but because of the intramolecular energy redistribut ion this methodology did not work as expected.57 Although, in some cases the correct selection of the excitation wavelength can still be used to control the photoproducts.24 Wavelength excitation control is produced when the ground state population is transferred to diffe rent places in the manifold of the excited potential energy surfaces. Considering the distinct shape of each potential ener gy surface, a different initial positioning of the wave packet will affect the evolution of the system. The probability of producing the excitation to differe nt parts of the potential ener gy surfaces is given only by the FranckCondon coefficients; therefore one can no t acknowledge the change in wavelength as a quantum control scheme because no coherences are involved. The problem changes when the coherent properties of the exci tation are intentionally modified to produce an effect in the molecular response. Spectral interference Another possibility involves the creation of spectrally interf ered pulses. F or example, a pulse sequence created by applying a sinusoidal spectral phase to a broad band transform limited pulse will not create interference fringes in its sp ectrum (Figure 14(A)), but will visibly do so on its second order power spectrum (Figure 14(B)). Spectral modulation will affect the excitation processes initiated via twophoton absorption, as the second order power spectrum is changed by the new phase function (Figure 14(B)).42 The first experiment displayi ng spectral interference effects was performed by Silberbergs group in 1998.32 In this work the effect of an odd pulse with a step spectral phase function was investigated for the twophoton absorption process in cesium gas. In cesium, a system with PAGE 32 32 narrow absorption bands, the odd pulse with the step centered at half the frequency of the transition gives the same emission as a transf orm limited pulse in a system with narrow absorption lines. 2 0 2 2200225023002350240024502500 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Angular Frequency (THz)Phase (rad)A 4550460046504700475048004850 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Angular Frequency (THz)B Figure 14. Spectral sinusoidal phase creating a pulse train. A) Spectrum (so lid line) and spectral phase (dotted line). B) Second order power spectrum comparison between the modulated pulse (solid line) and a transform limited pulse (dashed line). This can be explained by observing the second order power spectrum. Positioning the step at half the frequency of the transition, the second order power spectrum has its maximum in resonance with the electronic tr ansition and consequently will produce the same effect as the transform limited pulse. In a later experiment, the same group showed that a similar pulse applied to a dye in solution doe s not induce the same effect as in the atom in gas phase.42 The difference between the two results was explained in terms of the width of the band involved in the transition. While in the cesium atoms there is only one defined re sonant transition, in solvated molecules the bands are broad, whic h implies that many transitions simultaneously contribute to the excita tion. Hence, in a system with broad absorption bands a change in the second order power spectrum will always lower emission compared to that observed for the transform limited pulse in which its second orde r power spectrum covers the maximum number PAGE 33 33 of possible transitions. These studies show that when the spectrum has broad inhomogeneous line shapes, the pulse intensity is the driving force in the twophoton initia ted process. These results were confirmed by Joffrees group in a study involving twophoton emission of different dyes in solution.43 By removing the intensity dependence of the excitation process, they showed that the fluores cence efficiency can be modulated by changing the second order power spectrum of the femtosecond pulse. Since th e excitation is decouple d from the intensity, the emission will be given by the molecular population reaching the ex cited state. Thus, maximization of the emission efficiency will imply a maximization of the excitation efficiency. Twophoton excitation efficiency can be optim ized by shaping the excitation source such that its second order power spect rum maximizes the overlap betw een the excitation source and the twophoton excitation spectrum of the molecule under investigation.43 This optimization scheme was used to explain the results obtained in an optimal control experiment involving the twophoton induced emission of a ruthenium complex in solution. 44 It is interesting to note that this was a multiparameter experiment, which was reduced to a single parameter optimization to understand the photophysical process. Dynamics interference and pump and dump Although the interference effect has been used in phase m odulation studies to shape the second order power spectrum of the pulse, most studies focus its application on modifying the excited state wave packet dynamics rather than matching the excita tion source to the shape of the potential energy surfaces. One way of modifying the system dynamics is by using a double pulse with a particular time separation. The parameter used to produce and modify the wave packet interference is the time delay between the pulses in the sequence. PAGE 34 34 Early ultrafast studies in stilbene,45 retinal,46 and yellow protein47 show that this manipulation is possible, but those studies we re intended for studying th e system dynamics and they did not emphasize the time delay as a s ource for controlling the outcome of the photochemical reaction. While this scheme has been extensively used in solids,4749 only a few examples have been observed in solution photoprocesses.50 One demonstration of coherent control usi ng a wave packet interference scheme was performed by Wohlleben et al. in 2002.51 In this case, parameterization of pulses in the time domain was used after an early multivariable control experiment suggested that the system could be controlled with a train of femtosecond pulses.51, 52 The phase of the pu lse was modulated to generate an equally spaced pulse sequence. After scanning the time delay and phase between pulses, they concluded that a 160 cm1 frequency of the sinusoidal function was necessary for matching the vibrational deactivation coordina tes in a natural light harvesting complex. Another example was performed by Gerber and coworker to study the control landscape of IR140.53 By searching the control landscape with colored double pulses, they show the possibility of controlling the molecular induc ed fluorescence in a similar manner as that observed with linear chirp. Linear chirp control Linear chirp has shown to be an im portan t parameter in modulating the excited state population of many molecular system s in solution. In early experiments the quadratic spectral chirp, linear chirp, was generated either by twograting compressors or simply by passing the femtosecond pulse through material with known index of refraction. The first experiments applying this effect to control a process in solution were performed by the Shank group in 1996.54 Using high intensity femtosecond pul ses, the authors showed that the excited state population can be substantially enhanced or s uppressed depending on the linear PAGE 35 35 chirp imposed on the pulse. This observation was interpreted within the pump and dump model. In molecules with different minimum positions on the ground state and ex cited state potential energy surfaces (Figure 15), the wave packet is created on the FranckCondon region, located on one side of the excited potential energy surface (S1). When the nonequilibrium population evolves on the excited state, a dynamics Stokes shift where the population migrates from higher to lower energies is produced. This temporal evolution of the population causes a timedependent change of the energy gap between the surfaces. An intense laser field exploits this energy shift by dynamically matching the frequencies components of the pulse with the temporal evolution of energy separation between the stat es. Since the energy gap between the potential energy surfaces decreases as the wave packet evolves on S1, the chirp necessary to produce this effect must be negative. Furthermore the matching of instanta neous energy implies that neither a transform limited nor a positively chirped pulse can exert this molecular response. Similar observations were obtained in many molecules as well as biomolecular complexes.24 Figure 15. Pump and dump scheme for linearly ch irped pulses. Adapted from work by Cerullo et al.22 Modulation of twophoton excitatio n with linearly chirped inte nse laser fields has also been shown to work. The emission enhancement obtained in those experiments is produced via self phase modulation of the pulse in the sample.55 S0 S1 Potential energy Vibrational coordinate time Frequency h h t2 t1 PAGE 36 36 Other controllable processes us ing this type of mechanism have been demonstrated in coherent antiStokes Raman scattering (CARS). Besides the resolution improvement using femtosecond pulses with nonzero second order phase,56, 57 chirped pulses also permit the selective excitation of vibrational modes in CARS experime nts. The Maternity and Kiefer groups presented a selective excitation study where the population in one mode of a diacetylene single crystal was varied by the amount of chirp present on the femtosecond laser pulse.58 Figure 16. Ladder climbing scheme w ith tailored midinfrared pulses. A totally different application of chirp control in solution involves selective excitation of vibrational modes in the ground electronic state to promote popul ation to excited vibrational states within the same electronic surface. Ladder climbing in the condensed phase was first introduced by Heilweils group on tungsten hexacarbonil in nhexane solution.59 Using chirped infrared pulses, they showed the modulation of population transfer up to the second vibrational level by climbing the vibrationa l overtone ladder. While tran sform limited pulses transfer a smaller population to the second vi brational level, positively chir ped pulses leav e the population in the ground vibrational level. Later experiments extended the ladder climbing up to the seventh level.60 In addition, it has been show n experimentally in the ga s phase that a selective bond cleavage is produced by populating the ladder vibrational levels above the dissociation barrier. 61, S0 Potential energy Vibrational coordinate time Frequency h t5 t1 h h t3 PAGE 37 37 62 Although dissociation using ladder climbing in solution samples with chirp pulses has not been demonstrated yet, these experiments leave open the possibility of produc ing this dissociation. Multiparameter Control After the proposal by Judson and Rabitz26 of using a multiparameterized electric field and closed loop optimization to control photochemical processes, the first multiparameter control experiment to study a molecular process in c ondensed phase was presented by the Wilson group in the late 1990s.29 This induced an avalanche of experime ntal studies in the following years. To simplify the description of these studies, they are divided according to their objective in three different groups: selective electronic excitation, vibrational dynamics, and complex molecular photoprocesses. Control of selective electronic excitation In the ve ry first multiparameter control experiment, Wilson and Warren studied the electric field phase and amplitude effect on the excitati on/emission of a dye molecule (IR125) using a closed loop feedback.29 The controllable parameters of th is experiment were amplitude, second order phase, third order phase, center frequenc y, and spectral width of the femtosecond laser pulse. This study aimed to control two different objectives: maximization of fluorescence over the excitation energy ratio and of the fluorescence intensity. The first objective, maximization of the fluorescence for a minimum laser excitati on power, was achieved by modulating the laser excitation source such that its spectrum matched th e absorption features of the dye spectrum. The second objective, maximization of fluorescence regardless of laser excitation power, was attained with a modulated pulse with a strong positive chirp without any noticeable amplitude modulation. While the first experiment is a de monstration of a selec tive excitation control scheme, the second experiment represents th e mechanism of intrapulse pump and dump. PAGE 38 38 A later study of twophoton induced emission in a ruthenium complex was carried out by Gerber and coworkers in 1996.31 They used intensity independe nt optimization objectives to perform the optimal control experiment and retr ieve molecular information about the system. These objectives were generated by dividing the emission intensity with a second order intensity dependent process, thus rem oving the intensity dependence inherent to the twophoton excitation. The optimizations acknowledged th e intensity decoupling by converging to modulated electric fields differe nt from the most effective fields obtained for a nonlinear excitation, a transform limited pulse.42 The particularities observed in the tailored femtosecond pulse were assigned to the molecular and dynami cal properties of the system. A recent study performed by Damrauers group showed that the actual control mechanism of those experiments is based on matching the twophoton excitation spectrum of the complex rather than on dynamically controlling the system. 44 In 2001, Gerbers group also examined the sele ctive excitation of tw o different dyes in solution.63 Their results indicated the po ssibility of selectively controlling a system consisting of components with very similar one and twophoton spectra using specially designed femtosecond pulses. Many different groups have exhibited in recent years the usage of this concept for many practical applications.64, 65 For example, Dantus group showed the possibility of modifying the emission ratio in biological tissue66 and in samples with different pH enviroments. 67 Another example of selective excitation was performed by Wada and coworkers. In this work, the twophoton excitation of perylene in a monocrystal, as well as an aggregated solution, was optimized.68, 69 A closer look of the twophoton exci tation spectrum re veals that the temporal response is the product of modifying the excitation second order power spectrum to match the excitation spectrum of the aggregated system. PAGE 39 39 Control of vibrational excitation The use of sim ple parameter schemes in the midinfrared region was successfully demonstrated in different quantum systems, i.e. W(CO)6,59 carboxyhemoglobin,70 and chloroform.71 Due to the lack of direct modulation in the midinfrared spectrum, early coherent control experiments aiming to manipulate the vi brational excitation were performed with midinfrared shaped pulses in which the tailoring of the pulse was pr oduced by transferring the chirp during the generation of the midinfrared pul se, which only permit single variable studies.59 In the late 1990s, Bucksbaums group performed multiparameter control experiments on stimulated Raman scattering of liquid methanol with a self phase modulated femtosecond source.72 Using a closed loop optimization, they e ffectively controlled the Raman scatter spectrum of liquid methanol co rresponding to symmetric and antisymmetric carbonhydrogen stretching modes. The optimization accomplished a high degree of control on the system, which permitted the suppression or enhancement of both modes simultaneously as well as individually. Following experiments on benzene, deuterated ben zene, and deuteraed methanol they concluded that the control mechanism was intramolecular coupling rather than a change of the self modulated spectrum.38, 73 The optimal pulse was a pulse se quence whose separation coincided with the beating period of the two modes. This suggested that the contro l can occur by initially exciting a beat of the two modes followed by the impulsive redistri bution of the energy. Materny and coworkers studied the improvement of the excitation step in coherent antiStokes Raman spectroscopy by multidimensional adaptation. These optimal control studies produced, among other results, modeselective ex citation and suppression of certain modes. 74, 75 Very recently Zannis group presented an acous tic optical modulator specifically designed for shaping the midinfrared region of the spectrum.76 They used this pulse shaper to perform the first closed loop learning control experiment s using shaped midinfrared femtosecond pulses.77 PAGE 40 40 The group reported the selective population of the excited vibr ational levels in tungsten hexacarbonyl. The multidimensional searching itera tion was successful in finding optimal pulses for populating specific vibrational levels by following the stimulated emission peaks as parameters of popul ation inversion. Control of complex molecular processes W ith the advance of the optimal control fiel d, the degree of complexity of the systems under study has increased too. In th e last five years many optimal control experiments have been performed in very complex system s, i.e. biomolecular complexes. In 2002, Motzkus and coworkers published a study presenting the control of the energy flow in the bacterial photosynthetic system LH2.51 LH2 antenna complex presents well known dynamics:7880 upon excitation with 525 nm light three different channels are opened. While two channels are responsible for the energy transfer from the periphery to the reaction center, the third is a deactivation channel given by internal conversion. M onitoring the transient absorption signals corresponding to internal conversion and energy transfer, shaped pulses obtained in a closed loop optimization changed the internal co nversion to energy transfer ratio by more than 30 %. This change in the ratio was produced with a modulated femtosecond excitation consisting of equispaced pulse sequences. Based on th ese results, an equispaced multipulse parameterization of the excitation revealed an an ticorrelation between the energy transfer and the internal conversion channels when the phase relationship among the subpulses was varied. Other studies involving different biological complexes can also be found in the literature, e.g. green fluorescence protein81 and living organisms.64 Besides the intense attention that biomolecule s have drawn from the optimal control field, the quest of producing new photoproducts with shaped pulses is still to be demonstrated. Recent PAGE 41 41 demonstrations of control of mo lecular rearrangement in differ ent system seri ously support the possibility of achieving this goal. The first adaptive control experiment targe ting isomerization in condensed phase was demonstrated in Gerbers group on a dye molecule (NK88) in 2005.82 To modulate the isomerization, the excitation s ource was produced by doubling the ne ar infrared tailored pulse. Detecting two different absorption signals, this experiment showed that the correct pulse modulation can enhance or suppress the isomeriz ation efficiency. These experiments were done under intense laser fields inevitably inducing mu ltiphoton interactions duri ng the excitation; thus preventing the interpretation of the pulse temporal shape in terms of the induced dynamics on the system. A later computational study of this is omerization process produced similar modulated electric fields as in the experiment.83 In 2006, a study on the isomerization control of a cyanine dye was performed by the Yartsev group.84 This study verified the controllability of the absolute quantum yield of isomerization which was achieved by modifying th e initial momentum distribution of the wave packet on an excited state potential energy su rface. The optimal pul se presented Fourier components of the torsional coordinate which is responsible for keeping the wave packet evolution parallel to the conica l intersection. This coordinate i nduces the isomerization process and avoids the wave packet relaxation to the ground state before the isomerization, maximizing the isomerization. To decrease th e isomerization the wave packet was pushed to evolve in the coordinates that drives the system to the coni cal intersection, avoidi ng its passage through the torsional coordinate. Another example of optimal c ontrol of isomerization was performed by Millers group, who investigated the reti nal in bacteriorhodopsin.85 Using a closed loop control scheme with PAGE 42 42 phase and amplitude modulation, the group succeeded in enhancing or dimi nishing the amount of formed photoproduct by 20 %. These experiments were performed in a linear regime (low field intensities) in which the obs erved change corresponds to the optimization isomerization efficiency. The same system was also studied by Gerber and coworkers.86 Employing an unmodulated pump and a modulated dump, they investigated the deactivation rather than the excitation process. The study focused on modifying the wa ve packet dynamics close to the conical intersection since the tailored dump had a lower energy than the corresponding transition from the ground state of either isomer. Their results sh ow that a delay of 200 fs in the tailored dump pulse was the most effective way to bring the system back to the ground state, preventing the isomerization. However, a question that the auth ors did not address is whether the time delay between excitation and dumping pulses or both dela y and phase are the critical parameters for exerting control. All the presented studies show the significant progress achieved in the optimal control field in condensed phase. Now it is clear that the electric field can be obtained with a multidimensional optimization, but the challenge of understanding the process under control in terms of its molecular properties still remains. Optimal Control and Molecular Mechanism Closed loop experim ents have been shown to be suitable techni ques to control any molecular systems that generate a signal for the desired product, but they have a major drawback: only a few of those experiments lead to results that are explained in terms of the molecular properties of the system under investig ation. The main reasons for this disadvantage are: a) the intricate relationship between the variables to be optimized and the molecular response, b) the arbitrary a nd random pathway followed by the optimal closed loop to produce PAGE 43 43 the manipulation, and c) the large number of parameters used for the generation of arbitrarily modulated pulses. In addition to the vast amount of data produced by the searching algorithm, optimization data can contain redund ancies, i.e. same pulses tested more than once in the closed loop optimization, and/or physically meaningless variables, i.e. phase parameter without any important weight in the fitness. In addition, if we include the experiments performed using high intensity fields, an unknown number of multiphoton interactions prevent the correct correlation of the molecular response with the char acteristic magnitude s of the field. An ultimate goal in optimal control is to develop a procedure for gaining molecular information from the molecules under study, especi ally in processes where the photophysical or photochemical mechanism is unknown. A few resear ch groups (including us) have taken the initial steps towards the development of general tools for inferring the control mechanism using the closed loop optimization data.33, 8790 One proposed way to extract the molecular in formation encoded on the data is by finding the parameters directly involved in the optimizat ion procedure. The idea is to express all the independent electric field parameters used in the closed loop optimizatio n as a collective set directly related to the studied molecular process and at the same time filter out or discard those variables with negligible or no effect on the mo lecular response. This is a commonly established problem in many different research areas and it is usually solved by statistical analysis. Bucksbaums group used principal component an alysis to study the pulse parameterization space produced in the optimal control experiment of stimulated Raman scattering of liquid methanol.90 The authors demonstrated the possibility of simplifying the representation of the electric field by removing variab les with no connection to the ach ieved control. They concluded PAGE 44 44 that statistical analysis can be used to extract the effective de grees of freedom optimized during the experiment. Damrauers group used partial least square s technique to explain the observations produced in a previously performed experime nt of twophoton induced emission in solution.8890 With this methodology they extracted the collecti ve variables and they linked them to the observed molecular response. In addition, they were able to reduce a 208 phase parameter problem into 7 fundamental dimensions in the phase space,88 and to simplify the multivariable control experiment into a singl e control parameter experiment.89 Taking a completely different approach, Weinachts group has published two recent studies showing the viability of the para meter space reduction using simple algebra transformations.87, 89, 91 The first study focuses on reducing the complexity of the problem by including the parameters variability in the fitness of the experiment.87 The authors showed that a nonlinear change of basis can sign ificantly reduce the dimension of the search space. The second study presented a simple linear transformati on of used variables to transform the multidimensional space into a set of globally indepe ndent variables. The authors test successfully the transformation in closedloop molecular fragmentation experiments.91 Rabitzs group presented a statistical corre lation analysis to explain the attosecond dynamics of high harmonic generation.33 By examining the correlation behavior between the different harmonics produced and the optimiza tion objective, the resear ches show that the experimental evidence corroborates the proposed theoretical mechanism for this phenomenon. Outline of the Dissertation The m ain scope of this work is to inves tigate coherent light matter interactions of molecular systems in solution. For this resear ch study, we had devel oped and built a phase modulator apparatus and a pulse ch aracterization apparatus. Using this novel apparatus, we have PAGE 45 45 conducted experimental studies to show the possibility of controlling different twophoton induced process in solution. These studies are not intended to be only a proof of principles, but a new way to understand the dynamics and elec tronic structure of the molecules under investigation. Chapter 2 summarizes the theoretical de scription of a femtosecond laser, its implementation in our laboratory, and the tec hniques necessary for its characterization. Femtosecond pulse shaping is described in Ch apter 3. This chapter provides a theoretical description as well as the expe rimental implementation of fe mtosecond pulse shaping using a spatial light modulator. Differe nt examples of shaped pulse s are presented to show the capabilities of our design. Chapter 4 describes the closed loop optimizati on and the genetic algor ithm used in all the optimal control studies presented in this diss ertation. Besides describi ng the genetic algorithm and its implementation, this chapte r evaluates the best algorithm pa rameters that can be used to optimize a 128 phase parameter space. In additio n, a simple closed loop experiment involving pulse compression is presented. In Chapter 5, we study the phase effect on the twophoton induced emission on Rhodamine 6G. In this particular system, a f itness parameterization is examined. Chapter 6 focuses on the control of energy tran sfer in dendritic macromolecular structures with lightharvesting properties. This chapter not only includes a closed loop optimization, but also a series of single parameter studies to discern which coherent control mechanisms are produced in the optimization. Chapter 6 presen ts an understanding of the energy transfer photophysics that could not be obtained via any other method. PAGE 46 46 The first part of Chapter 7 describes how variable space analysis, in conjunction with statistical analysis (correlation and multivariate), is used to extract a linear regression model of the genetic algorithm evolution during th e optimization. The second part focuses on understanding what requisites are necessary for using statistical techniques to reduce the optimization variable space dimensionality. Chapter 8 addresses feasibility of the twophoton excitation of azobenzene, and outlines possible changes in the excited state dyna mics caused by this type of excitation. PAGE 47 47 CHAPTER 2 FEMTOSECOND LASER PULSES Introduction Since the first realization of a laser system by Maiman in 1960,92 scientists have been interested in producing the shorte st laser pulse for studying the dyna mics of processes that occur in very fast times scales, being femtosec onds the time scale of nuclear motion. Recent technologic and scientific break throughs, such as modelocking,93 have changed laser technology. Today, it is possible to generate and char acterize high power, high energy, and tunable femtosecond laser pulses. Hence, a broad spectrum of new spectroscopic a pplications has been opened in different fields of science including chemistry, physics, and biology. This chapter provides a brief introduction to the theory of femtosecond laser pulses and their realization followed by a short description of the femtosecond laser system used in our laboratory. At the end of this chapter, the tw o main characterization techniques of femtosecond pulses and our implementation are presented. Femtosecond Laser Pulses General Mathematical Description of a Femtosecond Laser Pulse The tem poral and spatial dependence of a fe mtosecond laser pulse can be represented by its electric field, given by,,, E xyzt.94 To simplify the description of an ultrafast pulse, the spatial dependence of the electric field is assumed to be polarized in one dimension. ,,,, E xyztEztEtzEt (21) The electric field is a real physical quantity, but in many calculations a complex representation simplifies considerably the math ematical treatment. Thus, the complex spectral PAGE 48 48 intensity (E ) is defined as the complex Fourier transf orm of the real temporal electric field ( E t) i itEEtedtEe (22) Here, () E denotes the spectral amplitude and () its spectral phase. By definition of the Fourier pair,() Et is computed through the inverse Fourier transform ofE 1 2itEtEed (23) The trade off of modeling the el ectric field in a complex form is that the complex spectrum (() E ) has positive and negative frequencies with nonzero values. Negativ e frequencies have not physical meaning, so a more convenient representation (() Et) is employed, 01 () 2itEtEed (24) Note that in this representation the Fourie r transform is restricted to only positive frequencies. To further simplify the electric field description (() Et) is usually expressed in terms of amplitude and phase terms (Equation 25). ()1 2it E tAte (25) where A(t) is the envelope and (t) is the phase term. Generally the spectral amplitude is centered around the mean frequency, 0, and it has nonzero values in a small frequency interv al compared to the center frequency (0 ). This allows one to decompose the electric field in terms of an envelope and a carrier frequency (Equation 26). PAGE 49 49 00()11 22it it itEtAteeAte (26) where 0 is the carrier frequency, (t) the timedependent phase, and (t) and At are the real and complex field envelop, respectively. An ex ample of this represen tation can be found in Figure 21. 15105051015 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Electric Field (a.u.)time(fs) Figure 21. Temporal electric fiel d representation of a Gaussian pulse. Temporal electric field, E(t) (dash line), real amplitude, A(t) (grey line), and intensity (black line). This description is only valid when 0 or equivalently, 01 It is important to highlight that in pulses with few optical cycles this approxima tion might not be valid, and the electric field will not be described correctly in this representation. For pulses with carrier frequencies in the visible region of the spectru m, this assumption breaks down when their time duration is on the order of 2 fs.94 Another important property to describe the el ectric filed is its inst antaneous frequency. Instantaneous frequency is defined as th e first derivative of the phase term, (t) (Equation 25), 0dtdt dtdt (27) With this definition the carrier frequency is uniquely defined as the frequency that minimizes the time variation of the tempor al phase. The time dependence of the phase PAGE 50 50 component, (t) defines whether the frequency has a ti me evolution or not. For example, if dtdtft, the carrier frequency changes with time and the pulse is said to be frequency modulated or chirped. In Figure 22, examples of the electri c field modulation produced when 2dtdtat are depicted. 30252015105051015202530 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Electric Field (a.u.)time (fs)A30252015105051015202530 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Electric Field (a.u.)time (fs)B Figure 22. Temporal electric fi eld for phase modulated ultrafast laser pulses. A) Pulse with positive chirp ( a>0). B) Pulse with negative chirp ( a<0). A similar decomposition of the electric field in terms of carrier frequency and a field enveloped can be performed in the frequency doma in. The spectral envelope of the electric field can be calculated from its Fourier pair ( At ) it A Atedt (28) Many times the measured observable is not the electric field, but the intensity and spectrum. These physical quantities are related to field envelopes with th e following expressions: 2 0 04 cn SA (29) 02 cn I tAtAt (210) PAGE 51 51 In femtosecond pulses, the most commonly used envelope is a Gau ssian profile (Equation 211). 20()GtAtAe (211) In this representation, G is defined as the full width at half maximum (FWHM) of the intensity profile (Equation 210). Similarly, G represents the FWHM of its corresponding spectral intensity (Equation 29). For a given laser pulse, its temporal and spectra l fields are related to each other through Fourier transforms. Because of this interconversion, the bandwidth, p and the pulse duration,p cannot change independently. This rela tion is expressed by the timebandwidth product. 2ppBc (212) Here,Bc is a constant that depends on the shape of the pulse. The lower limit for this expression is observed in pulses without frequency modulation, 0dtdt, which for Gaussian pulses corresponds to cB = 0.441. A very important consequence of the timeba ndwidth product is that temporally short pulses will have very broad spectra. Since ma tter has different refraction index values for different wavelengths, a spectrally broad ultrafast pulse propa gating through matter will suffer changes in its electric field characteristics. Pulse Propagation To describe the effect of the pulse propagation through matter, sp atial and temporal dependence of the electric field are necessary. Considering the el ectric field linearly polarized PAGE 52 52 and propagating in the z direction as a plane wave, the follo wing wave equation for the electric field can be derived from Maxwell equations94 2 22 0 2222, 1 zt Ezt zct t P. (213) Here 0 is the magnetic permeability on vacuum, and P is the polarization. The implication of this equation is that induced polarization is responsible for th e effect of the medium over the incident electric field as well as for the medium s response. Polarization can be decomposed in two terms: linear and nonlinear polarization.94 LNLPPP (214) While the first term (LP) is responsible for effects such as diffraction and dispersion, the second tern (NLP) is responsible for nonlinear phenom ena such as harmonic generation. The pulse propagation through a medium is described by the linear term of the polarization, PL. The general solution of the wave e quation (Equation 213) constrained to a linear polarization term is ,,0ikzEzEe (215) Here, k() is the wave vector and co ntains properties of the material in which the pulse is propagating, i.e. index of refraction, n. 22 2 2n k c (216) The wave propagation vector, k(), can be expressed as a Tayl or series around the central frequency, 0, 0 02 0000 2... dk dk kk kk dd (217) PAGE 53 53 where k contains all the Taylor terms except the constant k0. The temporal counter part of ,Ezcan be calculated thr ough Fourier transform, assuming that the envelope slow ly varies in time and space. 00011 ,, 0 e x pe x p 2 ,itkzEtzEikzitde Atz (218) From this result (Equation 218) and using the Fourier transform it is easy to demonstrate that if only the linear term of the Taylor expansi on of the wave vector is included, the resulting electric field will have the following representation, 00 01 ,, 0 2itkzdk EtzAtze d (219) Therefore in the time domain, the presence of a linear term in the wave vector propagation implies a temporal delay in th e pulse which is proportional to the distance of propagation. The term responsible for the delay is the group velocity, 01 gdk v d (220) In the case where the quadratic term in the Taylor expansion of the wave vector is not zero, the group velocity will not be th e same for all the frequencies. Hence, the different frequency components will be travelling at different speeds with respect to the group velocity producing a broadening in the pulse temporal profile, i.e. ch irping the pulse (Figure 22). This effect, group velocity dispersion, is defined as 22 22 2 32 22 2222gg gdvv v dk dn dcdccd (221) PAGE 54 54 and represents the second or higher order terms effect on the change in the group velocity. Since the wave propagation vector and the index of refraction are related (Equation 216), the group velocity dispersion can be interp reted as the change produced in the group velocity of the pulse due to a non constant change in the refraction index of the medium. Equivalently, the group velocity dispersion can be expressed in the frequency domain of the electric field as previously done. Discarding the first term of the Taylor series of the propagation vector (the dela y in the pulse) we obtain 0023 23 00 23,,0exp 11 ,0exp ... 2! 3! EzEikz dk dk Ei z dd (222) The group velocity dispersion produces a quadratic term in the spectral phase, which in the time domain gives also a quadratic term. This explains why temporal pulse broadening is observed when an initially transform lim ited pulse travels in a material where 220 dnd or equivalently group velocity dispersion is not zero. Nonlinear Optical Effects In the last section, we analyze the intera ction of the linear polarization term with the electric field. However, when short and intens e pulses propagate through material, the nonlinear polarization produces very interes ting effects, such as Kerrs effect. These the nonlinear terms must be taken into consideration when solving the wave eq uation (Equation 213). 12 2 0...LNLEEPPP (223) The second order induced polarizatio n can be written in terms of E(t,z) 22 2 0, EtzP, (224) and since E(t,z) is PAGE 55 55 00 00*11 ,,, 22itkz itkzEtztze tze (225) The effect of the second order induced polarization in a nonlinear medium can be seen as: 00 002 2 2 22 00 0 2 0 00, 11 ,, 22 ,cos 1cos2 2itkz itkzEtz tzetze tztkz tz tkz P (226) This result implies that the nonlinear polari zation induced by two wa ves interacting in a medium produces two terms: a constant term and a term with double the frequency of the original waves. This second term is called the second harmonic of the incident wave. A similar derivation can be done for highe r order nonlinear effects. The derivation of the second harmonic intensity in terms of the nonlinear polarization and phase matching conditions is not a trivial demonstrati on, but it can be f ound in any nonlinear optics book.95 The intensity of the se cond harmonic is given by 2 2 2 22 0 22 23 0sinc 22IL kL I cn (227) This result exhibits the sec ond harmonics dependence on the intensity of the fundamental wave, I, the second order susceptibility, (2), the optical path, L and the wave vector change k Since the dependence of the wave vector change is through a sinc function square, the maximum second harmonic conversion that can be produced in a given material is obtained when k is close to zero. In second harmonic genera tion produced by two collinear beams, k can be expressed as 3010202kkk (228) PAGE 56 56 Using its relation with the index of refr action (Equation 216), the condition can be rewritten as 002nn (229) To produced second harmonic, the fundamental wave and its second harmonic must travel with their relative phase constant inside the nonlinear medium. In most materials, this condition can not be fulfilled. However, birefringent materials can meet this constraint. Birefringent crystals are anisotropic mediums with two different optical axes, each characterized with its own index of refraction. If fundamental and sec ond harmonic waves travel through the different optical axis of the materials, the phase match condition is fulfilled and second harmonic will be generated. Femtosecond Laser Generation Ultrafast laser pulses are gene rated with a modelocked lase r. Since its introduction in 1986, Ti:Sapphire lasers have repl aced dye lasers as source of femtosecond laser pulses because of their tunability, stability, excellent mode quality, high output power, and ultrashort laser generation.96 Today it is possible to buy commerciall y available Ti:Sapphire femtosecond laser sources with pulses as short as 6 fs and several watts of power output.97 Two main components are necessary to produce amplified femtosec ond laser pulses: a femtosecond laser seed (Ti:Sapphire oscillator) and an amplifier (in our case, a chirped re generative amplifier). Ti:Saphire Oscillator Ti:Saphire oscillators can produce pulses as s hort as 4.5 fs. Ultrashort pulses are generated by keeping all the longitudinal mode s of a cavity with a fixed phase relationship, i.e. phaselock or modelocked. Modelocked can be attained with two different techniques: active and passive mode locking. Active modelocking is produced by acoustooptical or el ectrooptical modulation PAGE 57 57 of the longitudinal modes inside the cavity.93 Passive modelocking can be achieved with saturable absorbers or by Kerrs effect.93 Briefly, modelocking with an acoustoopt ical modulator consists on producing phase locked modes by introducing an acoustooptic tr ansducer into the cavity. This component has a small fused silica element with a piezoelectric transducer attached to it. The piezoelectric transducer produces a standing wave in the crysta l, creating a transmissi on grating perpendicular to the longitudinal modes of the cavity. A fundamental mode passing through the optical resonator is amplitude modulated such that ne w modes appear in the cavity. These new modes have frequencies of the fundamental mode freque ncy plus/minus the frequency introduced by the transducer, typically MHz. The interaction of these new modes with the crystal produces more modes. After many passes, the different modes ar e phase locked and they add up coherently to form the femtosecond pulse. This modelocking technique adjusts the acousto optical modulator frequency continuously to match the cavity round trip time, avoi ding the necessity of a fixed cavity size.98 However, as the number of modes increases their relative phases start to change due to the intrinsic dispersion of the cavity elements. To avoid this problem, oscillator cavities also included a prism pair compressor and negatively chirped mirrors to k eep all the longitudinal modes phase locked. In our laboratory, the Ti:Sapphire oscillator is a commercially available system (Tsunami Spectra Physics ). The Ti:Sapphire rod is pum ped with the second harmonic of a continuous wave Nd:YVO4 laser (Millennia Spectra Physics). The oscillator cavity is actively modelocked with a regenerative modelocking mechanism. Th e Tsunami provides ultrashort pulses (>35 fs) in the near infrared region (tuning range from 710 to 980 nm) with a 80 MHz repetition rate. Since PAGE 58 58 the oscillator pulses have low energy (~5 nJ per pulse ), its output is used as a seed for the chirped femtosecond amplifier. Femtosecond Amplifier Laser amplification is usually achieved by pumping an optically active material with a pump source. In the case of femtosecond laser am plification this process cannot be produced by direct amplification because the gain medium of the amplifier will be damaged by the peak intensities of an amplified femtosecond pulse. To produce the amplification of a pulse from the oscillator, a chirped regenerati ve pulse amplifier is used.99 The chirped pulse amplifier uses a Ti:Sapphire medium pumped with the second harmonic of a Nd:YLF Qswitched laser. The oscillator serves as a seed fo r the amplification process. To avoid damaging the active medium, the oscillat or femtosecond pulse is temporally broadened (chirped) in a stretcher, before coupling it with the amplifier cavity In a regenerative amplifier, a pair of Pockels cells are used to trap the seed pulse it in the amplif ier cavity until it reaches maximum amplification. Once inside the cavity, the amplification of the seed occurs with each pass through the Ti:Sapphire crystal. Upon reachi ng saturation, the pulse is released from the amplification process with the second Pockel cell and a polarization beam splitter. After the pulse amplification, the pulse is temporally co mpressed to the femtosecond time duration with a double pass grating compressor. In our laboratory, the Ti:Sapphi re amplifier is a commercially available system (Spectra Physics, Spitfire). The gain medium, a Ti:Sa pphire rod, is pumped with a 7 W Nd:YLF Qswitched laser (Spectra Physic s, Evolution) and seeded w ith the Ti:Sapphire oscillator previously described. The amplifier delivers ultrashort amplified pulses (45 fs) in the near infrared region (25 nm FWHM at 800 nm) with 1 KHz repetition rate. The amplification in this PAGE 59 59 chirped regenerative amplifier is more than 6 or ders of magnitude yielding pulse energies up to 850 J. Femtosecond Pulse Characterization To study lightmatter interactions, it is necessary to have a correct description of the laser electric field, i.e. amplitude and phase. Tw o different types of di agnostic techniques are commonly employed to characterize completely the femtosecond electric fiel d. The first type is the frequency resolved optical gating (FROG) technique which involves a frequency resolved autocorrelation.100 The second type is spect ral phase interferometry for direct electric field reconstruction (SPIDER), and as it s name indicates the technique is based on interferometry to measure the timefrequency profile of the laser pulse.101 The main advantages of each technique are: SPIDER is a real time char acterization technique, and FROG is a technique with a simple implementation. Thus if the priority is to ch aracterize in real time th e ultrashort pulse SPIDER should be chosen and when the real time charact erization is not a experimental priority, the FROG technique should be selected. The FROG technique was implemented since in our experiments the realtime characterizati on of the laser pulse is not mandatory. Frequency Resolved Optical Gating Technique Frequency resolved optical gating is based on measuring the spectrally resolved signal of an autocorrelation. As in the autocorrelation, th e pulse is divided into two identical replicas which are recombined in a nonlinear medium. Th e signal produced in the nonlinear medium is later spectrally resolved with a spectrometer. Ma ny different nonlinear properties of materials can be use, e.g. Kerrs effect, second harmonic, third harmonic, self diffraction, etc. The FROG technique based on second harmonic generation, SHGFROG, is the most widely used because unlike any other nonlinear proces ses the signal depends on th e second order susceptibility PAGE 60 60 coefficient of the nonlinear medium, which is usually large. This makes SHGFROG an extremely sensitive technique that can be used to characterize pulses down to 1 pJ of energy. The main limitation of SHGFROG is that due to the time symmetry of the traces the time direction is unknown. In cases where the electric field consists of tw o or more pulses, the relative phase of the pulses has an ambiguity. Specifi cally, double pulses with a phase relationship between them of or will have the same FROG trace. In its important to note that the FROG technique was created to analyze well behaved pulses, close to transform limited pulses. For comple x electric fields one should be careful in the interpretation of the phase retrieved by this technique. The usual realization of this te chnique is shown in Figure 23.100 Figure 23. Experimental layout for a FROG apparatus using second harmonic generation. The FROG trace corresponds to 2,e x pSHG FROG I EtEtitdt (230) A phase retrieval algorithm is used to retr ieve the phase out of this trace. The phase retrieval problem is not a simple problem because it does not have an analytical solution and it gets even more complicated when the pulse ga tes itself, like in SHGFROG. This problem is generated because the re trieval algorithm needs the gate f unction to retrieve the phase. Many Spectrometer Nonlinear crystal lens Delay line Beam Splitter PAGE 61 61 different algorithms have been developed to solve this problem. In fact, the commercially available program used to retrieve the phase a nd amplitude of a pulse out of its SHGFROG trace implements various different algorithms. The basic idea of the two dimensional phase retr ieval algorithm can be seen in Figure 24. This algorithm uses two constrai nts for solving the problem, i.e. it uses mathematical and data constraints.100 The mathematical constraint is related to the nonlinear property used to generate the trace. In SHGFROG this constraint is represented by ,sigEtEtEt (231) The data constraint is implemented by replacin g the square root of the amplitude of the experimental trace with the new field trace without modifying its phase, generating a new electric field. Figure 24. Schematic of a generic FROG algorithm. Adapted from work by Trebino.100 The FROG algorithm starts with an initial gue ss of the electric field in the time domain. Using the experimental FROG trace and the Fourie r transform of the guess into the frequency domain, a new and improved electric field is genera ted. This field is back transformed to the time domain. The last step of the cycle consists on using this new electric fi eld to generate a new guess. The process is repeated until convergence ha s been reached, which is evaluated in terms of the root mean square difference of the meas ure traced and the trace obtained by the algorithm. ,sigE ,sigEt ,new sigE ,new sigEt sigEt Start (1) (2) (3) (4) (5) ,FROGI PAGE 62 62 The program used for retrieving the phase of the SHGFROG in our laboratory is the commercially available software FROG 3.2.2 by Femtosoft Technologies. PAGE 63 63 CHAPTER 3 FEMTOSECOND PULSE SHAPING Introduction Envisioned two decades ago, one of the latest applications of femtosecond lasers is to control photochemical process.24 The manipulation of the tempor al and spectral components of the ultrafast laser electric field is requir ed to produce quantum control of molecular photophysics. Due to the lack of fast electroni cs, it is currently im possible to modulate a femtosecond pulse in the time domain. Mode rn technology takes advantage of the timefrequency inversion by modulating pulses in the frequency domain Optical devices capable of synthesizing different ultrafas t pulse waveforms in the frequency domain have been proposed and realized, e.g. spatial light modulators, acoustics optics modulators, deformable mirrors, etc.27, 102, 103 Originally develop by Weiner and coworkers, pulse shaping with spatial light modulators is a widespread technique because of its high tran smission and its capability of shaping amplified laser pulses. Femtosecond Laser Pulse Shaping with Spatial Light Modulators The first spatial light modulator was experi mentally realized by Weiner and coworker using a zero dispersion pulse compressor w ith a fixed mask in its Fourier plane.104 Later, the same group developed the first programmable spa tial light modulator usi ng a voltage controlled multielement liquid crystal mask.105 Nowadays, pulse shaping with liquid crysta l masks has became a major methodology for tailoring femtosecond pulses because it allows for the arbitrary modulation of the individual frequency components of an ultrafast laser field. PAGE 64 64 In this section, a brief descrip tion of the basic principles of pulse shaping with spatial light modulator is provided, followed by the experiment al realization of the pulse shaper in our laboratory and examples of its capabilities. Pulse Shaping in the Frequency Domain The concept of femtosecond pulse shaping is based on the application of a linear, timeinvariant filter.27 Linear filtering can be described either in the time or frequency domain. In the frequency domain, the filters frequency response function, H() characterizes the effect produced by the filter on the input electric field Ein() so the output electric field Eout() is defined as out inEEH (31) The time domain filter response can be computed using the Fourier tran sform of the filters frequency response function, H() itHHtedt (32) consequently 1 2itHtHed (33) The modulated output of the electri c field in the time domain is out in in E tEtHtEuHutdu (34) where the denotes the convolution of the two functions In the case where th e input pulse is a delta function in the time domain the input spectrum is unity 1it it in inEEtedttedt (35) and the output spectrum is equal to the frequency response of the filter. PAGE 65 65 1out inEEHHH (36) For short pulses, specific temporal output pulse s can be generated by ca lculating the linear filter with the proposed impulse response. A comple x frequency response of the filter is given by iHTe (37) where T() and () are the real and imaginary parts of the filter response, respectively. The filtered electric field is then expout inEETi (38) In the previous section, it was shown that a pulse propagating in a medium suffers a change in its electric fi eld (Equation 215) given by ,,0exp n EzEiz c (39) If we compare the effect of propagation (Equatio n 39) with the output of the electric field when a filter is imposed on it (Equation 38) ,0inETE (310) n z c (311) This result is the basis for pulse shaping with spatial light modulator s. It shows that by controlling the filters refractiv e index for the input frequency components it is possible to arbitrarily modify its phase, and hence its tempor al shape. Moreover, changing the real part of the filter function, T(), will produce spectral amplitude modulation on the input field. The phase of the filter can be expressed as a Taylor series around the pulse central frequency of the pulse (0) PAGE 66 66 00 01 !n n n nd nd (312) An arbitrary pulse in the time domain can be calculated from the electric field in the spectral domain (Equation 23) and filtered electric field (Equation 38). 1 2i it outinEtETeed (313) Replacing the imaginary part of the filters response with the Taylor series, we obtain 00 011 exp 2!n n it out in n nd EtETi ed nd (314) This expression shows that many different te mporal pulses can be created depending on the function applied to filter the electric field in the frequency domain. Spectral filters capable of producing totally arbitrary functions, not necessarily smooth filters functions like the Taylor series, can produce pulses with the mo st unthinkable temporal shapes. To experimentally realize the pulse shapi ng of a femtosecond pulse in the frequency domain two important components are necessary : a zero dispersion compressor and a spatial light modulator. Zero Dispersion 4f Compressor A zero dispersion compressor consists of two gr atings and two lenses arranged in a line (Figure 31).105 Each grating is located in the front focal plane of one lens and the distance between both lenses is twice thei r focal length. An incident beam is dispersed in its frequency components and then focused into the back focal plan e of the first lens. In this plane, the Fourier plane, all the frequency components are spread along the horizontal axis. A second lens grating pair recombines all the pulse components into a single pulse. This spat ial arrangement can be understood as the first lensgrati ng pair performing the spatial F ourier transform on the incident PAGE 67 67 pulse and the second gratingle ns pair performing the inve rse Fourier transform on the components located at the Fourier plane. Any phase or amplitude filter placed at th e Fourier plane will modulate the individual frequency components of the input pulse. As a result, after the inverse Fourier transform the output pulse will be modulated in phase and/or amplitude. Figure 31. Basic layout for a zero dispersion compressor. G: grating, L: lens, FP: Fourier plane. Adapted from work by Weiner et al.27 Without a filter in the Fourier plane, this t ype of optical arrangement produces an output pulse with the same temporal shape as the input pulse (zer o dispersion compressor). This configuration is the standard setup used to co mpress ultrafast pulses because the displacement of the second grating compensates for the second order phase component of an input chirped pulse.106 Spatial Light Modulator The spatial light modulator used in this wo rk is based on utilizi ng a liquid crystal to produce a filter function in the Fourier plane of the zero dispersion compressor. To understand the basic functioning characteristics of our spa tial light modulator a br ief introduction on liquid crystal mask properties is provided in the next subsection. FP f f f f L L G G PAGE 68 68 Liquid crystal mask characteristics The liquid crystal mask consists of a linear ar ray of independent pixe ls. Each pixel has its own control which permits the generation of tota lly arbitrary filter res ponses. One of the most important characteristics of these devices is their operation over an extensive range of temperatures without the need of a temperature controller.107 A liquid crystal can present different pha ses, e.g. smetic, nematic, and chiral.107 Each different phase is characterized by the type of ordering in its cell volume. Liquid crystal masks use the properties of the nematic phase to co ntrol the index of refraction of the pixels. Liquid crystal molecules are elongated rods. In a nematic liquid crystal phase, all the rods are vertically oriented without any type of ordering in the othe r two axes, i.e. distance between rods is not constant (Figure 32). Due to its elongated shape, these nematic materials are optically anisotropic leading to birefringence. These birefringent materials are optically characterized by their anisotropic dielectric tensor, in which the ex traordinary index of refraction coincides with the long axis of the rod. Their di electrically anisotropic tensor makes the nematic molecules orientation susceptible to the presence of an external electric field.108 Hence, liquid crystals are materials with a controllable birefringence. Figure 32. Cartoon of a nematic phase liq uid crystal. ea: extraordinary axis. Liquid crystal masks are fabricated by rubbi ng a window substrate w ith a fine polishing compound along one specific direction. When liquid crystal molecule s are sandwiched in between one rubbed and one flat surface, they automatically align in the rubbing direction.107 By ea x y z PAGE 69 69 coating the two surfaces of the containing cell with a thin, tr ansparent, and electrically conductive film of indium tin oxide (ITO), the liquid crystal orientat ion can be controlled with an externally applied field between the electrodes. AB Figure 33. Side view of a single nematic crysta l pixel. A) Without elect ric field. B)With an external field applied. Adapte d from work by Weiner et al.109 The liquid crystal mask operates as follows (Fig ure 33): when no electric field is applied, the liquid crystal molecules are oriented with their long axis parallel to the grooves ( z axis) created in the window (Figure 33(A) ). When the electric field is turned on, the molecules rotate so that their long axis is aligne d parallel the direction of the fiel d (Figure 33(B)). This rotation changes the refraction inde x along the rotation axis ( z axis). For light pr opagating through the mask with a polarization at 45 degrees with respect to the li quid crystal el ongated axis, the component along the extraordinary optical axis w ill be delayed with respect to the component propagating parallel to the ordinary axis. The change in the index of refr action, or retardation, can be calculated using the following relation:110 22 22 2 0cossin 1eVV nn nV (315) ITO coating z E ITO coating PAGE 70 70 where, represents the angle of in clination of the liquid crystal which is a function of the external applied field, V .111 A series of consecutive pixels form a discre te variable retarder perfectly suitable for filtering the electric field of an ultrafast pul se in the Fourier plane of a zero dispersion compressor. In addition, the use of a double ma sk with a special configuration permits the modulation of the electric field in phase and amplitude. The spatial light modulator The liquid crystal based spatia l light modulator used in this work is a commercially available model (CRI Inc. SLM640DVN). The ma sk consists of two linear arrays with 640 pixels each (Figure 34(A)). Each liquid crystal array is composed of two glass substrates coated with ITO. While one substrate acts as ground, th e other is lithographically patterned into a linear array of 640 elec trodes each with 5 mm height and 97 m width. Each electrode is connected to its own voltage controller, which produces 640 i ndependent pixels on th e mask. The separation between pixels is 3 m and as no electric connector is pres ent in this region the refraction index can not be controlled, leadi ng to an interpixel gap. AB Figure 34. Schematic of spatial light modulator mask. A) Front view. B) Lateral cut. Adapted from work by Weiner et al.27 0 1 2 3 64 mm 3 m 100 m 5 mm Quartz LC +45 LC 45 ITO coating Epoxy PAGE 71 71 To make the phase and amplitude spatial light modulator, two liquid crystal arrays are glued together such that the pixels of the first array spatiall y coincide with the pixels in the second array. The space between mask s is held constant (1.02 mm) by adding little quartz rods in between the arrays (Figure 34(B)). The glue used is a refraction index matching epoxy to minimize loses due to surface reflections. The spatial light modulator is controlled w ith an electronic driver located inside the housing of the apparatus, which provides an indepe ndent voltage control for each of the pixels in the mask. This driver is computer controll ed through a USB port. Th e voltage adjustment dynamic range of the driver is 12 bit or 4096 discrete levels, where the maximum level corresponds to 10 V (level 4095) and the minimum to 0 V (level 0). To avoid electromigration effects produced by DC sources, the controller source is an alte rnating current source (3.3 kHz square waveform).112 The usual response time of the liquid crystal mask in our system is close to 50 milliseconds. In each side of the double mask there is a thin polarizer limiting the use of the mask to the 488900 nm region. The damage threshold for 900 nm is 200 J/cm2 (50 fs pulse, repetition rate 1kHz). If the mask is used without the polariz ers the spectral working range extends up to 1620 nm. In addition, by replacing the output polariz er with a specially designed mirror the spatial light modulator can be used in a reflective m ode, which presents many important advantages. This spatial light modulator operates onto the polarized incident light. In our system, the extraordinary index of refraction of each mask is +45 degrees and 45 degrees with respect to the entrance horizontal polarizer, wh ich permits phase and amplitude modulation of the incident light. PAGE 72 72 Pulse shaping with LC masks Introduced in previous chapters, the phase produced by ultrafast pul ses propagating in a material is (Equation 222) 2 nn zor z c (316) The index of refraction in a liquid crystal can be modulated by applying an external voltage, thus 2, nV Vz (317) The difference in phase, produced by a change in voltage ( V ) is 2, nV Vz (318) where n is the change in inde x of refraction due to a V change in voltage. This change in refraction index of a liquid crysta l can be expressed in terms of its ordinary and extraordinary components ,,, 0o LC LCnVnVn (319) where nLC is 22 222cossin 1 ,oe LC LC LCVV nVnn (320) The change in phase, or retardation, is only applied to the polariza tion component of the incident wave parallel to the opti cal axis of the liquid crystal pixe l. In the spatial light modulator, the liquid crystal optical axis on each mask is at +45 and 45 degrees with respect to the vertical axis of the array (Figure 35).113 PAGE 73 73 Figure 35. Setup for dual mask spatial light modulator. The extraordinary and ordinary axes of the li quid crystal for each mask can be expressed as 1111 and 22LC LCeXYoXY (321) and 2211 and 22LC LCeXYoXY (322) where e and o are the extraordinary and ordinary axis, and LC1 and LC2 are the first and second liquid crystal masks, respectively For an incident electric field, 0E polarized in the X direction, its electric field can be expressed in terms of the extraordinary axis of both masks 0 12 002LCLCE EEXee (323) The retardation and the electric field observed after passing both masks will be 120 12 12LC LCii LCLCE Eeeee (324) where LCi is the retardation produced by the ith mask. Expressing the electric field back in terms of the X and Y axis 12120 12LC LC LC LCiiiiE EXeeYee (325) L C2 E x y E 1 o LC2 o LC1 e LC1 e LC2 +45 45E 2 E 1 E 2 E 1 E 2 L C1 PAGE 74 74 By Eulers formula, it can be expressed as 12 12 12 10exp cos sin 222LCLC LCLC LCLCEEi X iY (326) In the spatial light modulator the input and output polarizer set the polarization in the x axis, thus 1212exp cos 22LCLC LCLC xEEi (327) The output electric field, which is tailored after propagating th rough both masks, has two terms: the first one, an exponential function, is imaginary, while the second one, a cosine function, is real. While the imaginary term will only change th e phase of the transmitted electric field, the real term will produce amplitude modulation ove r the spectrum of the pulse. Phase and/or amplitude modulation will be imprinted on the pulse depending on the applied voltage on each mask. This modulation can be expressed in term s of the filters freque ncy response (Equation 38) expout inEETi (328) where is the phase filter function and T() is the transmission filt er function. Comparing with the above result (Equation 336) we obtain 122LCLC (329) 12cos 2LCLCT (330) Note that this result present two special cases,110LCLC and 120LCLC While the former case corresponds to amplitude m odulation only, the latter corresponds to phase modulation only. PAGE 75 75 If the transmission and phase filter functions to produce a certain pulse are known, the phase applied to each mask can be calculated as 1arccosarccosLCTA (331) 2arccosarccosLCTA. (332) The Pulse Shaper Geometrical Configurations Pulse shapers can be implemented in several different configurati ons. The first proposed configuration is depicted in Figure 36(A). In this configura tion, the beam propagates through the Fourier plane to the other side of the pulse shaper, which is a mirror image of the first side. If a mirror is positioned at the Fourier plane of th e pulse shaper, another possible configuration is obtained, Figure 36(B). The former implementati on, transmission mode, is the most standard methodology found in the literature ; the later, reflective mode, provides unmatched advantages such as smaller footprint size and lower cost. AB Figure 36. Basic layout of a pul se shaper. A) Transmission mode configuration. B) Reflective mode configuration. Figure 37 shows many pr eviously implemented27, 114117 and newly proposed pulse shaper geometries.I Each of these configurations has its adva ntages and drawbacks. A detailed analysis of each of the possible implementations was used to determine the most suitable realization for our pulse shaper. I All configurations can be converted to reflective mode by setting a mirror in their corresponding Fourier plane. PAGE 76 76 The first setup (Figure 37(A)) has the adva ntage of having all the optics positioned along a line, but the use of lenses introduces chromatic aberrations especially in pulses with very broad spectrums (temporally short), thus a configuration with reflective focusing optics is preferred. Figure 37. Top view of differe nt design types for pulse shaper s based on an optical Fourier transform. The remaining configurations (Figure 37(B)( E)) can be categorized in two groups: those with (B) and those without (B,C) a folding mi rror. A pulse shaper without a folding mirror creates spatial aberrations since its focusing element reflects the beam in a direction out of its optical axis, producing astigmatism. This type of aberration is very di fficult to compensate. The remaining configurations introduce at least one folding mirror (Figure 37(C)(E)). When the dispersed beam is reflected such that the angle between incident and reflected beam is A B D C E PAGE 77 77 non zero and the height of the reflected beam is also changed (Figure 38( C),(E)), a tilt on the beam in the Fourier plane is observed (Figure 38). Figure 38. Tilt in the Fourier plane produced by a mirror at 45 degrees with a vertical tilt. The other two possibilities (Fi gure 37(D)(E)) are very simila r. In both cases the tilt of the Fourier plane is minimized either because th e reflection angle is zero and only the height is changed (Figure 38(D)) or a very small angle is used in the folding mirro r (Figure 38(E)). The apparatus depicted in Figure 37(E) needs a bigger folding mirror than the one observed in Figure 37(D). Large good quality mirrors are usually more expensive and difficult to get. For these reasons, we chose the design in Figure 37(D) for constructing our experimental setup. To increase the input power in the pulse sh aper, cylindrical optics are preferable to spherical optics. In cases where the output power of the pulse shaper is not critical or is not required, the use of spherical lenses is pref erable because they have smaller fabrication aberrations than cylindrical optics. We choose to use spherical mirrors because of the above reasons and because our studies do not need high energy pulses. h a b a b a b a b PAGE 78 78 Reflective Mode Pulse Shaper Due to the difficulties in its implementation, only the transmission mode configuration had been successfully realized with liquid crystal mask based pulse shapers. With the recent development of large mirrors (needed for a 64 mm spatial light modulator) the experimental realization of the reflective mode geometry setups is possible. We demons trate the feasibility of a reflective mode configurati on using the new 640 pixels CRIs spatial light modulator. One limitation of the reflective mode geometry is the required separation between input and output beams. Since both beams must travel through the sa me optics, it is necessary to develop a way to separate the beams. One possibility is to change the height of the inpu t and output beams. As it was mentioned before, this can lead to aberrations on the beam focusing. To minimize beam aberrations and front wave tilt, we have de veloped a new geometry that uses collinear input/output beams. This innovative geometry us es polarization to separate input and output beams. To our knowledge, it is the first reflec tive mode collinear phase and amplitude modulator setup constructed to date (Figure 39). Figure 39. Diagram of the e xperimental setup in a refl ective mode c onfiguration. Our setup presents important advantages over the transmission mode. First, the number of optical components is reduced to half: one spherical mirror, one fo lding mirror, one back mirror, and one grating. Second, the to tal phase achieved in our setup is higher than any transmission mode setup because the beam travels twice through each phase mask. Third, the collinear geometry simplifies the alignment reducing the distortions produced by misalignment. Fourth, PAGE 79 79 there is only one grating angle to align. Last but not least, the footprint size of this setup is half of that observed in a transmission pulse shaper whic h is a critical constraint for a large mask. Design Characteristics In the design of the pulse shaper, the objectiv e is to have the maximum spectral resolution so more complex filters can be produced without significantly cutting the spectrum of the laser pulse. To find what are the best components for constructing the pul se shaper, we will assum that the mask needs to hold at least three times the spectral FWHM maximum of our laser source and that the incident and diffracted angles are the same (Littrow configuration). While fitting three times the spectral FWHM of the laser spectrum warrants that the mask can accept 98.8% of the spectrum, it loses only frequencies with intensities of less than 0.2% of the maximum intensity. The Littrow configuration keeps the diffracted beam centered on the spherical optic, avoiding astigmatism. Our femtosecond laser source delivers pulses with a temporal duration (FWHM) of 42 fs and a bandwidth of 25 nm (FWHM), thus the necessary spatial dispersion is given by 75 1.1 64dnmnm dxmmmm (333) where 75 nm corresponds to the three spectral FHWM and the 64 mm to the width of the mask. The grating equation establishes that for Littr ow configuration the angle of diffraction is111 arcsin 2G o (334) where G is the groove density in mm1, is the wavelength diffracted, and o is the diffraction order. The linear spatial di spersion in the Fourier plane of the compressor is given by105 cosdd dxGf (335) PAGE 80 80 where d is the diffraction angle and f is the focal length of the focusing optic. The relationship between the grating groove density and the focal length becomes cosarcsin 2 G o f d G dx (336) An additional important parameter that needs be taken into consideration in this design is the spot size of each wavelength at the Fourier plane. Each wavelength of the pulse spectrum produces an image on the Fourier plane of the comp ressor. The size of this spot must be similar to the pixel size, neither smaller nor larger. If th e spot size of a given wa velength is larger than the pixel size (97 m), the pulse shaper will lose resolu tion as each wavelength will be spread over more than one pixel. If the spot size is comparable to the gap between pixels (3 m) some wavelengths will not be modulated. For a spot size of 40 m only 7 % of its intensity will not be modulated. It is therefore critic al to consider this parameter when choosing the focusing optics. For a Gaussian beam, the spot size at the focus plane as a function of the focal length for a given optic is110 0 s f (337) where s is the waist spot at the focusing optic, f is the focal length, and the wavelength. Both effects, resolution and spot size, must be considered in the design. We find that using the relationship between grating groove density and beam waist versus focal length gives the possible combinations fo r a grating lens pair. For a Gaussian input beam with 3 mm beam diameter a beam waist size from 40 to 70 m is produced with a focal length of 200 to 400 cm (see Figure 310). Th e corresponding grating groove density necessary to keep the spatial dispersion close to 1.1nm/mm is between 1750 and PAGE 81 81 2100 mm1. A combination of grating lens in that ra nge is suitable for bu ilding a high resolution pulse compressor. 01020304050 1200 1400 1600 1800 2000 2200 2400 2600 0 10 20 30 40 50 60 70 80 90 100 Grating groove density (mm1)Focal length (mm) Beam waist (m) Figure 310. Grating groove density and beam wais t versus focal length. Grating groove density effect (solid line). Beam wa ist effect (dashed line). In our designed, we selected a grating of 1800 mm1 groove density in combination with a focusing optic of 300 mm. The selection took into consideration the commercial availability of the parts. The summary of the theoretical pulse sh aper characteristics for this configuration is presented in Table 31. Table 31. Theoretical zero dispersion compressor characteristics. Parameter Value Grating groove density 1800 mm1 Grating Littrow angle 46.05 degrees Mirror focus 300 mm Theoretical spatial dispersion 1.285 nm/mm Theoretical spectral window accepted by the SLM 82.24 nm Raleigh length 32 mm Beam waist 56.6 m Reflective Mode Experimental Realization Our reflective mode phase and amplitude pulse shaper is presented in Figures 311 and 312. To obtain phase modulation in the frequency do main the optical setup is arranged in a folded zero dispersion compressor (Figure 311). This ty pe of configuration reduces the setup size by half. PAGE 82 82 In our folded configuration, the zero di spersion compressor is composed of: a gold spherical mirror (Edmund Optics, f =300 mm), one folding gold mirror (half of a two inch diameter mirror, CVI Inc), one inch diameter gold steering mirror, one goldcoated holographic grating (Richardson Gratings, 1800 groves/mm 30x30 mm), and the mask back mirror (CRI Inc.). B Figure 311. Top and side views of the built pulse shaper. SM: spherical mirror; G: grating, M: mirror; P: polarizer; WP: /2 waveplate; FR: Faraday rotator, CP: cube polarizer. Figure 312. 3D Sketch of the built pulse shaper. SM : spherical mirror; G: grating, M: mirror; P: polarizer; WP: half waveplate; FR: Fa raday rotator, CP: cube polarizer. SM G M M M M P SLM WP FR CP CP FR WP M M SM M P M SM G G M Side view Top view PAGE 83 83 This optical configuration pr ovides a spectral resolution of 0.129 nm per pixel with a bandwidth window of approximately 83 nm (s ee wavelength calibration section). All the compressor elements are arranged in a line with the beam traveling slightly out of the plane parallel to the optical table to avoid spatial di stortions in the mask (F igure 311, side view). While the folding mirror is mounted on a standard mirror mount, the rest of the components are mounted on special custommade mounts to m eet the necessary degrees of freedom (see alignment procedure, Appendix A). A critical component of our a pparatus is its collinear optic al path. This configuration requires a polarization method to separate the unmodulated (incoming) from the modulated (outgoing) pulse. Based on the same principle of pulsed regenerative amplifier cavities, the experimental arrangement uses a cube polarizer (Optics for Research, PH8), a Faraday rotator (ElectroOptics Technology, BB88R), and a /2 waveplate (Karl Lambretch, MWPAA222HEAR800) (Figure 313). The cube polarizer acts as a polarizati on beam splitter, the Faraday rotator as a cumulative retarder, and the /2 waveplate as a polarization corrector. Figure 313. Polarization split ting of the incoming and the outgoing pulse. Incoming and outgoing polarizations are represented with a black and red arrow, respectively. Its operation can be described as follows: th e incoming beam (black arrow, +90 degrees) crosses the Faraday rotator and its polarizati on is tilted by +45 degrees. Since the waveplate CP FR WP x y z PAGE 84 84 produces the same effect over the pulse as the Fa raday rotator, the electric field accumulates +45 degrees yielding an electric field with a 0 de gree orientation inside the pulse shaper. The outgoing beam (gray arrow, +0 degree) crosses the waveplate and its polar ization changes to +45 degrees. However, when the pulse propagates through the Faraday rotator the polarization changes 45 degrees to 0 degrees and the cube polarizer splits the different polarization, separating the incoming (unmodulated, 90 degree polarization) from the outgoing (modulated, 0 degree polarization) beam. This arrangement allows a collinear path, simplifies the alignment procedure, minimizes temporal dispersions, and provides a very compact and less expensive setup instead of the large setup usually used in highresolution pulse shapers. Our dual mask system has a high average transm ission (>94%). Our setup is configured to produce amplitude and phase modulation using an entrance polarizer (transmission <80% from 750 to 850nm) and a returning mirror (transmissi on <95%). Due to the double pass, these components reduce the overall transmission to less than 55 %. The overall pulse shaper transmission is 30%, which is mainly determined by all the polarization optics and the e fficiency of the grating. Pulse Shaper Phase Calibration The spatial light modulator should be placed at the Fourier plane of the 4f compressor to modify the temporal and spectral characteristics of the incident pulse. Th e phase control in the modulator is obtained by changing the applied vo ltage to each pixel in the individual mask. Before an experiment, a calibration of the pixel response using applied voltage must be performed to control precisely the phase imposed by each pixel. The index of refraction of the liquid crystal is voltage and wavelength dependent. A pixel by pixel calibration can take too long to be pe rformed as a daily routine, since there are 1280 PAGE 85 85 (640 in each mask) independent pixels. Instead, all pixels of each mask of the spatial light modulator are calibrated simultaneously with the corresponding wavele ngth by measuring the spectrum of the modulated pulse with a spectro meter, greatly reducing the time of the phase calibration and avoiding the individual pixel by pi xel calibrations. The phase introduced by a change in the a pplied voltage is given by Equation 318. Assuming that the dependence on wavelength and voltage can be treated independently, the change of refractive i ndex can be written as ,,0. nVnfV, (338) where the first term (n(,0)) represents the change in the in dex of refraction as a function of wavelength alone and the second term ( f(V)) models the voltage response for a liquid crystal pixel. If the mask pixel inhom ogeneity is neglegible, a calibrati on of one pixel as a function of wavelength can be used to extrapol ate to the rest of the pixels in the array. The change of the refraction index as a function of wavelength can be calculated from the corresponding ordinary and extraordinary index of refraction (Equation 320). While characterizing the spatial light modulator we found that the pixels in both masks are not truly uniform. From one pixel to the next there are changes on their refractive index even for the same wavelength when no voltage is applied. This complicates the cali bration of our spatial light modulator. The same model (Equation 338) can be used to calibrate the birefringence of the mask, considering that n(,0) is an effective refraction index. To achieve the spatial light modulator calibration a spectromete r is set at the output of th e pulse shaper and the whole spectrum of the beam is measured after pass ing through the masks. This calibration not only includes the wavelength dependence of the refractive index, but also takes into consideration the pixel inhomogeneity. The cal ibration setup is depict ed in Figure 315. PAGE 86 86 To perform the calibration a home written pr ogram in Labview (National Instrument TM) is used. This program measures the output pulse spect rum while the voltage of one of the masks is varied every 10 bit (2.5 mV) and the other mask voltage is kept constant at the value of maximum transmission. Figure 315. Experimental setup used for the masks calibration. When the spectrum is measured, only the real part of the spectral filter response is measured: 2 outinSST (339) where T() is the real part of the filter re sponse function (Equati ons 37 and 330). Under these conditions, the transmission A() is 2 out inS AT S (340) and the difference in phase applied by the mask is 122arccosLCLCT (341) Since only one mask is modulated during the phase calibration, the phase produced by the this mask is Laser Spectrometer PAGE 87 87 max2arccosLCi LCjT (342) where LCi is the change in phase produced at each wavelength of the spectrum for a given voltage, and max L Cj is a phase constant. This phase cons tant will be discarded by setting the phase of the calibrated mask at zero when the voltage applied is 10 V. The phase curve for each wavelength is calc ulated using Equation 342. The calibration results are shown in Figures 316 and 317. 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Phase (counts) A 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Phase (counts)B Figure 316. Phase calibration curves of : A) front mask and B) back mask. 760770780790800810820830 18.5 19.0 19.5 20.0 20.5 21.0 Phase (rad)Wavelength (nm)A 760770780790800810820830 18.5 19.0 19.5 20.0 20.5 21.0 Phase (rad)Wavelength (nm)B Figure 317. Phase at 700 counts (1.71 V) for: A) front mask and B) back mask. PAGE 88 88 Since the phases corresponding to different wavelengths at a given voltage do not follow the expected Sellmeirs refractive index behavior (Figure 317), the phase values for each curve at 700 counts (1.71 V) are normalized with resp ect to the phase of 799.666 nm and used as a correction factor for the phase curves. The phase at 799.666 nm is used as the only phase calibration curve and its phase corr ection factor is used to correct the deviation to the calibration curve of other pixels due to pixel inhomogeneity and wavelength index of refraction dependence. 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Voltage (counts)A 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Voltage (counts) 760780800820 1.00 1.05 1.10 Wavelength (nm)B Figure 318. Phase calibration of front mask. A) Corrected curves. B) Comparison between the average of all the curves a nd calibration curve at 799.66nm (inset: correct ion factor). 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Voltage (counts)A 0 1000 2000 3000 4000 0 5 10 15 20 25 Phase (rad)Voltage (counts) Avg 799.666nm780800820 1.00 1.05 1.10 B Figure 319. Phase calibration of back mask. A) Corrected curves. B) Comparison between average of all the curves a nd calibration curve at 799.66nm (inset: correct ion factor). PAGE 89 89 The corrected phase calibration sh ows a maximum retardation of 8(25 radians) for both masks. The maximum dynamic range for a 2 modulation is produced between 1250 and 4095 driver counts (corresponding to 2.5 V and 10 V, respectively). The wavelength correction factor does not show the expected dependence for the index of refraction versus wavelength (Sellmeiers equation). This difference can be a ttributed to the inherent inhomogeneity of the mask. The maximum change in the retardation fo r a 1.71 V (700 counts) observed on either mask is on the order of 10 % with respect to the normalization wavelength (center wavelength, 799.666 nm). The inhomogeneity of the mask has e normous consequences on the performance of the pulse shaper setup because it imposes a residu al phase on the input pulse which reduces the dynamic range of the spatial light modulator. In addition, this re sidual phase must be compensated to produce transform limited pulses and to relate the output pulse modulation of an arbitrary pulse to the applied voltage. Pulse Shaper Wavelength Dispersion Calibration A wavelength calibration of the spatial light modulator must be performed to assign the wavelength correction factor to each pixel. The way to produce this calibration is similar to the voltage calibration, but in this cas e instead of varying the voltage of both masks, the pulse shaper is set to produce amplitude modulation. By im posing in some pixels of the first mask a retardation of and leaving the other entire mask at a re tardation of 0 radians (10 V), holes will be observed in the spectrum of the output pulse. Since the position of the pixels with phase is known, the holes in the spectrum are directly assigned to them. The wavelength versus pi xel calibration curve is 0d pp dp (343) PAGE 90 90 where the observed experimental values are: 0 = 843.36 nm and d/dp = 0.12896 nm/pixel. Since the pixel size is 100 m, the spatial dispersion in the Fourier plane is 1.29 nm/mm which agrees with the theoretical spatial dispersion fo r this zero dispersion compressor apparatus (Table 31). 750760770780790800810820830840850 0 500 1000 1500 2000 2500 3000 3500 Intensity (a.u.)Wavelength (nm)A 750760770780790800810820830840850 0.7 0.8 0.9 1.0 TransmissionWavelength (nm)B 100150200250300350400450500550 780 790 800 810 820 830 Wavelength (nm)PixelC Figure 320. Dispersion calibration. A) Unmodulated and modulated spectra (red and black lines, respectively). B)Transimission produced by holes in the profile. C) Wavelength calibration curve. Residual Phase The pixel inhomogeneity in each mask produces a residual phase in the modulator. This background phase is amplified in our double pa ss implementation of a reflective mode configuration setup. To observe characterize, and correct this effect, the zero dispersion PAGE 91 91 compressor was modified by replacing the spatial light modulator back mirror with an external, independent mirror (Figure 321). The effect pr oduced by putting the doubl e mask in the zero dispersion compressor could be obs erved with this modified c onfiguration. The modulators residual phase was measured by collecting the puls e autocorrelation with and without the mask. AB Figure 321. Pulse shaper setup using an external mirror. A) Without the spatial light modulator. B) With the spatia l light modulator. 6004002000200400600 0.20 0.40 0.60 0.80 1.00 Intenisity (a.u.)Time (fs)A 6004002000200400600 0.90 1.00 Intenisity (a.u.)Time (fs)B Figure 322. Phase mask effect on the temporal profile of the pulse for A) Zero dispersion compressor without the spatial light m odulator in place. B) Zero dispersion compressor with the spatial light modulator in place. Figure 322 shows that the spat ial light modulator has an im portant residual phase that should be compensated to modulat e pulses. One way to compensate for this residual phase is by measuring the phase of the output pulse and im posing the same phase, but with opposite sign on the pulse shaper. Laser Laser PAGE 92 92 We use multishot SHGFROG and commercially available software (Femtosoft FROG 3.2) to retrieve the phase out of the generated FROG trace. Figure 323 shows the SHGFROG trace measured for the output pulse shaper pulse with no phase applied. The retrieve d trace of this pulse presen ts a good agreement with the experimental SHGFROG trace (Fig ure 323(A) and (B)). From the retrieved electric field we can extract the phase needed to achieve the re sidual phase correction in the spatial light modulator. Knowing the phase need ed to generate a transform limited pulse, we use the previous calibration to generate the voltages required at ea ch pixel. After applying the phase opposite in sign to the one measured, we collect the SHGFROG traces again (Figure 324). A B 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 1000 500 0 500 1000 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)C2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 760770780790800810820830 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)D Figure 323. SHGFROG trace of the unmodulated pulse. A) Experimental trace. B) Retrieved trace. C) Temporal intensity (black line ) and phase (red line). D) Spectral intensity (black line) and phase (red line). PAGE 93 93 Figure 324 shows that the phase can be successf ully reduced from 20 radians to less than 1 radian by applying the opposite phase in the pulse shaper. The new SHGFROG trace and the retrieved electric field show pulses very close to the transform limited pulse with an almost flat phase across the spectrum. The lack of zero residual phase in the co rrected pulse is a product of the phase determination via FROG. To improve this phase retrieval a technique like MIIPS can be used.118 This technique has been shown to produce transf orm limited pulses with a phase error of less than 0.01 rad. The other possibili ty is to compress the pulse with a closed loop optimization. Implementation of this technique will be shown in the following chapter. A B 0 2 4 6 8 10 12 30025020015010050050100150200250300 0.0 0.2 0.4 0.6 0.8 1.0 IntensityTime (fs) Phase (rad)C1 0 1 2 760 780 800 820 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)D Figure 324. SHGFROG trace of the phase corrected pulse. A) Experimental trace. B) Retrieved trace. C) Temporal intensity (black line ) and phase (red line). D) Spectral intensity (black line) and phase (red line). PAGE 94 94 Frequency Nonlinearity Correction For the sake of simplicity, a linear change of frequency from one pi xel to the next is assumed in the pulse shapers (Figure 326, dotte d line). For noncomplex pulses, this assumption works well. However, when the output phase m odulated pulses are very complicated, this assumption fails and leads to spatiotemporal di stortions which deviate the results from the theoretical prediction. Figure 325(A) show the SHGFROG trace of a sequence of pulses generated by applying a sine function to the sp ectral phase. This type of modulation creates equispaced subpulses. The SHGFROG trace in panel A shows an additional distortion with tilted traces that are also broa der in time (phase modulated). A B Figure 325. Effect of the nonlinear frequency cali bration in the pulse m odulation. A) Sinusoidal phase pulse without frequenc y nonlinear correction. B) Si nusoidal phase pulse with nonlinear frequency correction. The pixel versus frequency plot presented in Figure 326 shows a small nonlinearity that needs to be taken into account. The periodic ity in time is no longer constant for all the frequencies, and the output pulse presents br oader replica pulses with more time delay. PAGE 95 95 When this correction is accounted for and the pu lses are codified linearly in frequency, the same modulated pulse does not present any distortions on its replica pulses. This can be observed in Figure 325(B) where all the replicas are narrower and the tilt is removed. 100150200250300350400450500550 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 Angular Frequnecy (x103 THz)Pixel Figure 326. Frequency calibration curve of the pulse shaper. Dots experimental points, dotted line linear fit, a solid line : second order polynomial fit. Modulation Examples Finally, we chose some masking operations using amplitude and phase modulation to check the performance of our experimental setup. An example of a pulse with step phase modulation is presented in Figure 327. 0 1 2 3 4 760 780 800 820 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)A1 0 1 2 3 4 5 6 3002001000100200300 0.0 0.2 0.4 0.6 0.8 1.0 IntensityTime (fs) Phase (rad)B Figure 327. Odd pulse ( step). A) Spectral intensity (black line), retrieved phase (red squares), and target phase (red line). B) Tempor al intensity (black line) and phase (red squares). PAGE 96 96 In Figure 327, red squares correspond to the experimental results, whereas the red line is the proposed target phase. In all cases, the calc ulated waveforms agree with the experimental pulses. Nevertheless, the difference between th e analytical waveform and the experimental results shows some temporal pulse distortions. This difference can be attributed to uncertainties in the compensation of the residual phase in our modulator and/or ma sk calibration errors. 760 780 800 820 0.0 0.2 0.4 0.6 0.8 1.0 4 2 0 2 4 6 8 IntensityWavelength (nm)Spectral Phase (rad)A500 0 500 0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 IntensityTime (fs)Temporal Phase (rad)B Figure 328. Sinusoidal phase modula tion. A) Spectral inte nsity (black line) re trieved phase (red squares), and target phase (re d line). B) Temporal intens ity (black line) and phase (red squares). Other examples using only phase modulation were successfully im plemented: sinusoidal spectral phase pulse, quadratic phase pulse, and zero phase double pulse. 760 780 800 820 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 IntensityWavelenght (nm)Temporal Phase (rad)A1000 500 0 500 1000 0.0 0.2 0.4 0.6 0.8 1.0 10 0 10 20 30 40 50 60 70 IntensityTime (fs)Temporal Phase (rad)B Figure 329. Quadratic phase pulse (4607 fs2). A) Proposed phase (red line), and retrieved spectral intensity (black line) and phase (red squares). B) Temporal intensity (black line) and phase (red line). PAGE 97 97 4 2 0 2 4003002001000100200300400 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Time (fs) Phase (rad)A0 2 4 6 8 10 12 14 16 770780790800810820830 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)B Figure 330. Temporal double pulse with a 200 fs separation. A) Proposed temporal intensity (open circles) and phase (red line), retrieved temporal intensity (b lack line) and phase (red line). B) Spectral intensity (black line) and phase (red line). Spatial Light Modulation Limitations The physical limit of the maximum phase modul ation of the spatial light modulator is given by the thickness of the liquid crystal used in the mask (Equation 318). In our pulse shaper, this limit is ~25 radians (see phase calibration curves, Figures 316 and 317). To overcome this physical limitation, phase wrapping is used. Due to the definition of phase, it is possible to arbitrarily add or subt ract any multiple of 2 without physically aff ecting its value (phase wrapping). An example of phase wra pping is presented in Figure 331. 0100200300400500600 0 10 20 30 40 50 Phase (rad)pixelA 0100200300400500600 0 1 2 3 4 5 6 Phase (rad)pixelB Figure 331. A) Phase unwrapped. B) Phase wrapped. PAGE 98 98 Figure 331(A) shows phase valu es that cannot be achieved w ith our pulse shaper. Using the phase wrapping treatment, the new phase (Fi gure 331(B)) is physically equivalent to the unwrapped phase (Figure 331(A)) which then can be applied with our pulse shaper because its values do not pass the pulse shaper phase limits. Phase wrapping solves the limited dynamic range produced by birefringent materials utilized in spatia l light modulators. Another limitation of the spatia l light modulator is given by the discrete nature of the spatial light modulator masks. This constrains the accessible phase modulations to the Nyquists limit. The Nyquist theorem states:119 If a function f(t) contains no frequencies higher than W cycles per second, it is completely determined by giving its ordinates at a series of points spaced 1/(2W) seconds apart This theorem implies that the sampling must be made twice per period, or twice in the working phase interval. Since the phase working interval for the sp atial light modulator is from 0 to 2, the maximum change phase that can be produced with out aliasing is (344) An example of the aliasing effect is presented in Figure 332. 0100200300400500600 0 1 2 3 4 5 6 Phase (rad)PixelA 0100200300400500600 0 1 2 3 4 5 6 Phase (rad)PixelB Figure 332. Quadratic phase of 103 pixel2/rad A) Sampled every 10 pixels B) Sample every 10 pixels (black squares) and every pixel (black line). PAGE 99 99 Figure 332(A) shows that for a sampling of 10 pixels a quadratic phase of 103 pixel2/rad starts to be undefined below the pixel 200 and above the pixel 440. A comparison with the phase sampled at every pixel (Figure 332(B) black line) reveals that outside the 200440 pixel range only one point is used for defining the phase within each 2 phase interval, producing aliasing on the phase. Using the limits imposed by the Nyquist limit, it is possible to calculate the maximum phase coefficients that can be applied if the pha se is codified using a Taylor expansion. The different terms of a Taylor expansio n of the phase in the frequency are 0!i i i nb i (345) where is the frequency, 0 is the frequency at the center of the array, and i is the Taylor polynomial order. Using the frequency calibration of the pulse shaper at each pixel the Taylor coefficients are expressed in te rms of the discrete pixel number !2i i i i nb N n i (346) where is the linear frequency change in each pixel, n is the pixel number, and N is the total number of pixels (640 in our spatial light modulator). Th e phase difference between two consecutive pixels is 11 !22ii i iii i nnnb NN nn i (347) For the linear coefficient of the expans ion, the Nyquist limited coefficient is 1 max max1b (348) where PAGE 100 100 2 max 0 12 b c (349) Similarly the second order phase coefficient is max 2 2 2 max21 2 b nN. (350) Since discrete polynomial functions present th eir maximum changes fo r the highest values of the independent variable, the maximum difference will be obtained for pixels N and N1, thus the Nyquist limited coefficient is max 2 2 2 max1 2 b N, (351) 4 max 0 2 2222 121 b NcN (352) All the other Nyquist limited coefficients can be calculated in a similar manner. The calculated Nyquists limits for the linear and quadratic phase for our setup configuration parameters (Table 1) are fs and fs2, respectively. By applying a linear phase to the incident pulse, pulses are shifted in time (see Ch apter 2). Within the Nyquists limit of ~8ps the linear modulations will not distort the original pulse structure. This limit constrains the modulator to an effective delay range of 16 ps ( ps) with a re solution of 18 fs (minimum slope created by changing the minimum phase step in our phase modulator). This time resolution (18 fs or 5.4 m) is comparable with the time step reso lution of any commercially available stepper motor translation stage used for optical delay lines. As a result, the pulse shaper can be used as a delay generator for pump and/or probe experiments with the capabilities of phase and amplitude modulation of its pump pulse. PAGE 101 101 In addition, the Nyquists th eoretical predictions state th at the designed pulse shaping apparatus can stretch pulses from 42 fs (FWHM) to close to 4.5ps (maximum quadratic phase possible) without producing a ny measurable distortion. The Nyquist theoretical limits in our pulse shaper apparatus are constrained by the residual phase of the mask, thus the real Nyquist limits must be calculated numerically including this residual phase term. Summary We have constructed a state of the art pul se shaper with many important improvements with respect to literature implementations. Among those improvements, it is important to highlight its small footprint and easy ali gnment achieved without compromising its high resolution. In the following chapters this appara tus coupled to a genetic algorithm is used for scientifically relevant problems. PAGE 102 102 CHAPTER 4 CLOSED LOOP OPTIMIZATION S AND GE NETIC ALGORITHMS Introduction Femtosecond laser pulse shaping has been show n to be the most promising technique to produce quantum control because it can create elect ric fields capable of affecting the molecular excited state wave packets duri ng their evolution in their molecular potential energy surfaces.2, 120 Since molecular evolution is governed by th e system Hamiltonian, the possibility of knowing which laser field parameters can control a certain molecular system is very limited for all but few very simple systems. In particular for strongly coupled systems, such as large molecules in condensed phase, the complete Ham iltonian is either unknown or too complicated to use for electric field calcula tions. Researchers have been usi ng optimal control experiments to control photoinduced processes with out prior knowledge of the mo lecular quantum properties. First proposed by Judson and Rabitz, optimal control involves an it erative closed loop optimization with feedback signals arising from the molecular process to be controlled (see chapter 1).26 Ideally, closed loop optimization will optim ize the electric field without requiring any prior knowledge of the molecular system, requiring only a measurable signal from the desired photoproduct. Closed loop optimizations require an optimization algorithm capable of optimizing many independent variables simultaneously. This algorithm should be able to find the optimum solution in surfaces with many differe nt local optima and in the presence of experimental noise. Genetic algorithms can fulfill these requirements because they are non local searching algorithms capable of handling rough variable spaces in the presence of noise.121 In this section, an introduction to genetic algorithm is provided, followed by a description of its implementation to compress chirped femtosecond pulses. PAGE 103 103 Genetic Algorithms Genetic algorithms are an heuristic search technique used to find approximate or exact solutions to given problems. Th ese searching procedures are suitable for problems where the solution space is too large for exhaustive search. They have b een successfully applied in many different research areas to solve a great variet y of problems involving a large number of variable parameters.122 Based on the principle of biol ogical evolution, genetic algo rithms rely on inheritance, natural selection, and variability to search the solution space.123, 124 In nature, a given species has a certain number of individuals from which new offspring is born. The natural selection principle states that only those individua ls who by random variations on th eir genome become adapted to the environment will, on average, leave more o ffspring. Less apt individuals will not survive long enough to be involved in th e reproduction process. Natural se lection does not provide all the necessary ingredients for biological evolution, inheritance and variability are also needed. Inheritance is the individuals capacity of transmitting part of the individual characteristics to his offspring. As demonstrat ed by Mendel and Watson and Crick,125 the characteristics of each individual that pass from genera tion to generation are encoded in their genome. Variability is responsible for maintaining the diversity of th e population genome, otherwise, inbreeding will take place and the population will be stacked in terms of evolution. There are two sources of variability: crossover and mutation. Crossover re presents descendants genome as combination of pieces from both parents genome. Mutation symbolizes the variability in the genome of individuals due to errors in the duplication and occasional random alterations of the genome. PAGE 104 104 Genetic Algorithm Code The basic idea of a genetic algorithm is to mimic natural evolu tion in a computer algorithm. During the optimization, the algorithm simulates natural selection, inheritance, and variability. A pseudo code of this algorithm is presented below: 0 Begin genetic algorithm; 1 If generation=0; 2 Initialize population P(generation); 3 While not optimized do; 4 Get population P(generation) fitness; 5 generation = generation +1; 6 Select parents from population P(generation 1); 7 Generate offspring P(generation); 8 End while; 9 End GA; This algorithm can be summarized as follows ; the solutions (individuals) to particular problems are represented as a onedimensional arra y of parameters (genes). An initial population of random solutions with random parameters is generated and the fitness (adaptability) of each individual is measured. Those so lutions (individuals) with the highest fitnest (adaptability) will be selected for reproduction (nat ural selection) and they will pass to the next iteration loop (natural selection). Since the new set of solutions (offspring) is generated from the solution of the previous round, the new solutions will have better optimization parameters for a particular trait (phenotype) than its predecessors. Each new solu tion is generated by recombination (crossover) of the fittest solutions. Since cloning is allowed, the new solutions will have to compete with the best solutions of the previous generations to be allowed to reproduce and/or to pass to the next PAGE 105 105 optimization loop step. The new set of solutions is passed through a mutation step, where some of the parameters of each soluti on are changed. In principle, this new solution set has the best variables for solving the problem; and in terms of fitness, they will probably outperform the old solutions (evolution). This new generation is ev aluated and the cycle continues until a certain termination criterion is met, e.g. maximum number of generations, a given value for the measured fitness, etc. Genetic Algorithm Implementation Since the initial implementation of ge netic algorithm for quantum control,29 several different optimization algorithms have been implemented.126128 Our genetic algorithm code is based on previously published algorithms.129 It uses roulette wheel pa rent and elitism for parent selection, random initial population, and special floating point genetic operators for reproduction. Among the various advantages, these genetic operators yield a fast convergence with a small number of individuals per generation. Parent selection In genetic algorithms, the parent selection provi des individuals with hi gher fitness a greater probability of reproducing. There are several differ ent ways to implement this selection. In our algorithm, the roulette wheel parent selection is used to select from the elite population those individuals apt for reproduction. Figure 41. Roulette wheel. Adapte d from work by Xiufeng et al.129 f1f2f3f4f5f6f7f8 PAGE 106 106 The roulette wheel is composed of slots size d according to the fitne ss of each individual (Figure 41). Elitism is achieved by including only the best individuals in the roulette wheel. In Figure 41, fi represents the fitness of each indivi dual and the size of the wheel slots is proportional to the fitness value. The roulette wheel is constructed as follows, 1. Determine the fitness, fi,, of the ith individual in one generation. 2. Calculate the total fitnes s of the elite population: epop i iFf. (41) 3. Calculate the probability of select ion for each individual according to, i i f p F. (42) 4. Calculate the cumulative probability of selection for each individual, qi, of the elite pool. epop ii iqp (43) The selection process is based on spinning the wheel as many times as there are individuals in the elite population; each time one pare nt is selected in the following manner, 1. Generate a random number, r from 0 to 1. 2. If r < q1 then select the first individual; otherwise select the ith individual such that qi1< r < q1. Genetic operators Genetic operators represent the different fo rms of reproduction. While numerous genetic operators are found in the literature,122 only a few of them have been demonstrated to be highly PAGE 107 107 efficient. The classical genetic operators can be divided in three: cloning, crossover, and mutation. The first operator, cloning, represents an exact copy of the individual and it is mainly used because of its robustness ag ainst experimental noise. Crossover involves the creation of a new individu al with the genetic code of two parents. In our algorithm, two types of crossover operators are used: tw opoint crossover and twopoint arithmetical crossover. The last operator, mutation, is utilized to provide new i ndividuals to the population avoiding the inbreeding problem found in a closed population. Inbreeding leads to unwanted convergence towards a local minimum. Three di fferent mutation operators are used: uniform mutation, nonuniform mutation, and nonuniform arithmetical muta tion. While the first type of mutation produces a real change of the population, the other two operators are important for the fine tuning capabilities of the algorithm. An individual Gi in generation k is represented with a vector of n parameters (genes) 123,,,...,ind g eneration nGgggg (44) Here, each gene is represented with a nu mber from the domain of the corresponding parameter [ L ,U ], where L and U represent the lower and upper lim it of the domain, respectively. The number of parameters codified in one individual depends on the pr oblem itself and not on the algorithm. Each genetic operator can be represented as follows: 1Crossover: It occurs when parts of th e parents genomes are swapped in randomly selected points, generating two ne w individuals with a mixture of the parents genotypes (Figure 42). PAGE 108 108 Figure 42. Crossover operation. Twopoint crossover involves the selection of two random numbers from 1 to the size of the parameter space ( n). To avoid copying the whole genome from a single parent (cloning) the random numbers are selected su ch they are not equal and their sum is bigger than n +1. The individuals are generated as follows, 11111 123,,,...,old parmGgggg (45) 22222 123,,,...,old parmGgggg (46) for 1ijn, where i and j are random numbers delimiting the position of gene swapping in each individual. This leads to tw o new individuals of the form 1112211 11 1,...,,,...,,,...,new iijjnGgggggg (47) 2221122 111,...,,,...,,,...,new iijjnGgggggg. (48) Twopoint arithmetical crossover is similar to the twopoint crossover, but instead of directly replacing the ge nes, the genes are calculated as a linear combination of the parental genes. Starting from the same parent (Equation 45), the individuals ar e generated as follows: 11111 11 1,...,,,...,,,...,aa new iijjnGgggggg (49) 22222 111,...,,,...,,,...,bb new iijjnGgggggg (410) where each element a igand b ig is calculated as a linear combination of the old genes, 12 ,,1a iiold ioldgagag (411) 21 ,,1b iioldioldgagag (412) Crossover PAGE 109 109 The parameter a is calculated as in the nonuniform mutation 0.8t a T (413) 2Cloning: this operator is responsible for passing old indivi duals to the new generation by directly copying the individua l without changing its geneti c material (Figure 43). Figure 43. Cloning operator. A mathematical description is 312,,,...,oldnew nGGgggg. (414) 3Mutation: when this operator is applied ove r an individual, one of its genes is changed to a different number (Figure 44). Figure 44. Mutation operation. Mathematically it can be represented as follows: Individual before mutation: 1,2, ,,,...,,...,oldoldoldioldnoldGgggg (415) Individual after mutation: 1,2, ,,,...,,...,newoldoldinewnoldGgggg (416) While all mutation operators consist of repl acing a gene with a new one, each mutation operator produces a different new gene. Uniform mutation consists on replacing th e old gene with a new random number, r, belonging to the domain of the parameter, where ,, and [,]ioldinewggrrLU (417) Cloning Mutation PAGE 110 110 This operator is responsible for searching vast regions of the variab le space, since this mutation produces a stochastic operation. A secondary type of mutation is the nonuni form mutation in which a random number, r, from [0,1] is selected, followed by the calculation of the new gene, ,, ,,, if 0.5 if 0.5,iold iold inew iold ioldgtUgr g gtgLr (418) where the function (t,y) returns a value within [0,y] such that the probability of a smaller change of the gene increases with the generation number. This function, (t,y), is defined as (1/),1tTtyyr, (419) where r is a new random nu mber from [0,1], t is the generation number, and T is the total number of generations used in the optimization. Since its action is linked to the generation number, at the beginning of the optimization this mutation will act as a random operator. As the optimization proceeds, this operator will only produce small ch anges in the genome leading to fine tuning of the overall solution. Finally, non uniform arithmetical mutation is also designed for local tuning. The gene mutation is generated as follows: ,,1inewioldgagar (420) where r is a random number from the domain [L,U] and a is a generation variable parameter 0.8t a T (421) Our algorithm code Our adapted genetic algorithm has been impl emented in a LabView (National Instrument) programming environment. A representation of its code can be observed in Figure 45. PAGE 111 111 Figure 45. Implemented genetic algorithm. Some special rules are included in this algorithm: The total number of generations is a user defined parameter. The initial random population is produced by ra ndomly selecting the initial genes for each individual, although a starting poin t with a particularity design ed individual can also be used. Random population Evaluate fitness Cloning Order by fitness Roulette wheel Two point crossover Two point arithmetical crossover Uniform mutation Non uniform mutation Non uniform arithmetical mutation New population Finish Initiate Mutation Reproduction PAGE 112 112 The population size is a us er defined parameter. The number of crossover children is a user defined parameter Parent selection is performed with elitist roulette wheel. Two parents form two children via crossover, but only one, randomly selected, will form the offspring kept for the new generation. The number of cloned individuals is calcu lated from the difference between population size and number of crossover children. All the individuals are subject to mutation. The probability of mutating one of the genes is defined by the user. The mutation operator is randomly sele cted among the th ree possibilities. The termination of the algorithm is defined by the number of generations. The role of the genetic algorithm parameters Our adapted algorithm presents the followi ng controllable parameters: number of generations, number of individua ls, number of elite parents, number of cloned kids, and probability of gene mutation. The number of generation depe nds upon how long experimental ly it takes to run each optimization loop and how long it takes to reach convergence. Since ge netic algorithms are nothing but an optimized variable space sampler, the higher the number of trials the higher the possibility of finding a true maximum. Thus, th e number of generations should neither be too small nor too big. During the optimization loop, the time for findi ng the fitness of one individual is limited by the time that it takes to tr ansfer a phase from the computer to the spatial light modulator (~300 ms), and the number of average laser shots used for the feedback signals (~150 ms). It takes approximately 450 ms to evaluate each indivi dual. Since the reproducibility of the result is very important in these experiments, many successi ve repetitions have to be performed for each PAGE 113 113 optimization. The stability of our laser system can be maintained for approximately 24 hours straight, limiting the number of experiments that can be perfor med. An optimization lasting for 3 hours, 180 min, implies that 8 to 9 experiments can be performed while the laser system remains stable. This 3 hour limit is a reasonable time only if the achieved answer is reproducible. With a 180 min optimization, the algorithm can try a pproximately 24000 individuals. Usually, the number of individuals used in one population is 80. Thus in 180 min, 300 generations can be evaluated. To restrict the problem to shorter times, in this work the selected number of generations is 200 or 120 min. This time constraint can be modified for special cases where noisy signals are used or when more laser shots need to be averaged. For this selected time, it is possible to optimize the remaining parameters to produce maximum achievable fitness. As stated before each experimental optimization will take 120 minutes, making it unfeasible to optimize the rema ining parameters experimentally. Therefore, the rest of the parameters are optimized using a theoretical simulation of the experiment. The advantages of the optimization si mulations are: short run time, us ually between 1 or 2 min, and independence from environmental co nditions, e.g. temperature change. Optimization of Genetic Algorithm Parameters The aim of this simulation is to find the optim al genetic algorithm pa rameters to perform real optimization experiments. A simulation of pulse compression using our genetic algorithm with different algorithm paramete rs was performed to find the best set of conditions yielding the best optimization performance of the algorithm within an experimentally reasonable time.II The experiment consists of compressing a chirped puls e back to its transform limited form (Figure 46). II The software to perform this simulation can be freely obtained in www.lab2.de PAGE 114 114 Figure 46. Simulated experiment al setup. The femtosecond pulse is modulated in the glass rod (BK7) and its second harmonic intensity is produced in a nonlinear crystal (NLC). The chirped pulse is obtained by passing a transform limited pulse with a known Gaussian temporal shape through 10 cm of glass (BK7). As presented in Chapter 3 (Equation 39), propagation through this glass produces a modulation of the incident electric field: 0,0expitn EEeiz c (422) consequently the pulse gets broader in time. Assuming that the BK7 glass only impose quadratic chirp, the following expression can be derived130 22 0 4 016ln2 1tt t (423) where t0 is the FWHM pulse duration of the transform limited pulse, is the second order spectral coefficient, and t is the FWHM of the pulse afte r propagating through the glass. The signal used for feedback is the second harm onic generation produced by the pulse. Since second harmonic generation is proportional to the integrated intensity s quared (Equation 226 and 423) BK7 rod NLC Detector Feedback Feedback Laser Pulse shaper PAGE 115 115 4 2 2 2 2 02 exp 1.177SHGt ItItEtE t (424) A chirped pulse generates less second harm onic signal than a transform limited pulse. After the BK7 glass, the pulse is sent into the ideal pulse shaper where the quadratic phase can be compensated. Considering Equation 38 for the pulse shaper as a filter mask in a zero dispersion compresso r, the output is 07 7,0expit BK out BKappliedn EEeiz c (425) where applied is the phase applied by the filter mask, and nBK7 and zBK7 are the refractive index and thickness of the glass, respectively. In the case where 7 7BK applied BKn z c (426) the output pulse will have the same second ha rmonic signal as the transform limited pulse. The idea of this simulation is to find the best phase variables that maximize second harmonic generation, or equivalently find th e necessary phase to compensate the glass dispersion. The spectral phase is codified in 128 discrete variab les that can vary from 0 to 2. Genetic algorithms can produce similar but not identical solutions for consecutive optimizations; thus each optimization is performed five times a nd the fitness values of the best individual fitness are recorded. For compar ison, the average, best, and wo rst fitness corre sponding to the best five individuals in each se t of simulations are presented. The initial conditions for this optimi zation are summarized in Table 41. PAGE 116 116 Table 41. Initial genetic algorithm parameters. Parameter Value Number of individuals 80 Number of elite parents 16 Number of crossover kids 75 Number of cloned kids 5 Probability of gene mutation 0.03 Mutation probability The mutation effect on the optimization of second harmonic is studied. This parameter is optimized by changing the mutation probability while keeping all the other initial parameters constant. The effect produced by the different mutation probabili ties on the achieved fitness is illustrated in Figure 47. 0.010.020.030.040.050.06 0.94 0.95 0.96 0.97 0.98 0.99 Achieved fitness (normalized)Gen mutation probability Figure 47. Influence of probability of gene mutation on the achieved fitness. Average best fitness (black squares). Overall best fitne ss (open circles). Overal l worst fitness (open triangles). As observed in Figure 47, the optimal mutati on probability is found to be close to 0.02. The achieved fitness is very similar (>98 %) fo r the mutation rates from 0.01 to 0.03 and is greatly reduced when the mutation probability becomes larger. These results agree well with the expected behavior for the opera tor. When the mutation probability is too small only a few numbers of genes will be changed in each indi vidual and the population variability will rapidly PAGE 117 117 decrease, leaving the algorithm stacked in a local minimum. In contrast a large gene mutation probability produces too many new individuals in each generation removing the population inheritance and transforming the genetic algorithm in a trial and error search. Number of elite parents We performed similar simulations varying this parameter while keeping all the others constant to choose the optimal number of elite parents. Figure 48 presen ts the effect produced on the best individual fitness by selecting different numbers of elite parents. The plot shows that the fitness is a decreasing function of the number of elite parents. This dependence is produced because increasing the numb er of elite parents in cludes individuals with lower fitness values in the roulette wheel of pare nt selection, resulting in a less apt offspring. 04812162024 0.95 0.96 0.97 0.98 0.99 1.00 Achieved fitness (normalized)Number of elite parents Figure 48. Influence of the number of elite parents on the achieved fitness. Average best fitness (black squares). Overall best fitness (open circles). Overall worst fitness (open triangles). For less than 12 elite parents, the algor ithm recovers 99 % of the second harmonic independently of the number of selected parameter. Thus, any number of elite parents of less than 12 produces almost equal results. If the num ber of elite parents is too small, the roulette wheel selection will be formed with individuals of very similar genomes. This type of elitism will transform some operators like twopoi nt crossover into a cloning operator. PAGE 118 118 Number of crossover children In this optimization the number of crossover kids is changed while keeping all the other genetic algorithm parameters at their initial values. Figure 49 shows the fitness as a function of the number of children for this simulation. 556065707580 0.96 0.97 0.98 0.99 1.00 Achieved fitness (normalized)Number of crossovered kids Figure 49. Influence of the number of individuals generated by cr ossover on the achieved fitness. Average best fitness (black squa res). Overall best f itness (open circles). Overall worst fitness (open triangles). Comparing with other genetic algorithm parameters, the effect on achieved fitness for different numbers of crossover children is not very pronounced (Figure 49). We conclude that this parameter does not change substantially the result of the optimization and that the specific chosen value is not critical. The optimal number of crossover children is 70, but its difference with the fitness obtained using 75 elite parents is not considerable (<0.2 %). Therefore either one of them can be used. Number of cloned children Since the number of cloned children is obtained from the difference between the population size and the number of crossover children, this parame ter is optimized simultaneously with the number of crossover children. PAGE 119 119 Optimized Genetic Algorithm Parameters and Noise From the previous results, it is possible to extr act the best genetic algorithm parameters for compressing a pulse (Table 42) with an initial population of 80 individuals. Table 42. Optimized genetic algorithm parameters. Parameter Initial values Optimal values Number of individuals 80 80 Number of elite parents 16 4 Number of crossover kids 75 70 Number of cloned kids 5 10 Probability of gen mutation 0.03 0.02 Average Fitness 98.7 99.8 An optimization with the chosen values is pe rformed to confirm the correct selection of parameters. In this case the achieved fitness is mo re than 99.8 % (Table 2), which shows that the chosen values are excellent for phase compensati on. In general the algor ithm has a remarkable performance because even for initially selected parameters, such as those of the above experiments, the optimization always compresse s the pulse to 95 % of its transform limited pulse. The limitation with these simulations is the lack of noise. Noise is one the biggest constraints in closed loop optimizations.131 Rabitz and coworkers studied the noise effect in several twophoton processes. The main conclusion of that work is that noise can significantly change the quality of the achieved objective. We decided to investigate th e noise effect produced on the achieved fitness with the parameters chosen before by performing optimizations in which the fitness values have noise (as it would be fo r experimental data). Th e fitness is calculated using 1noise f frnoise (427) where f is second harmonic signal and r is a random number from 0 to 1. The investigated noise values are: 5 % and 10 %. PAGE 120 120 Each optimization is performed five times a nd the fitness value of the best individual fitness is recorded. For comparison, the average, best, and worst fitness co rresponding to the best five individuals in each set of simulations are presented. Table 43. Fitness results for simulations with no ise. Set 1 represents the initial and set 2 the optimized genetic algorithm parameters. Fitness Noise 0.00 0.05 0.10 Average Best Fitness Average Best Fitness Average Best Fitness Set 1 98.7 96.8 94.4 Set 2 99.8 97.0 96.1 ANOVA testIII Statistically no different Statistically no different Statistically no different Table 43 shows the results obt ained in the pulse compressi on simulation with noise. The noise in this simulation does not change significantly the perfor mance of the algorithm: for a 10 % noise the recovered pulse has ~95 % of th e second harmonic observed for the transform limited pulse (Table 43). However it is important to notice that when the noise is included with the first set of coefficients (set 1 for initially chosen values) we have the same performance as with the optimized parameters. This is confirmed with the ANOVA statistical test which shows that with 99.0 % confidence the two results are not statistically significantly different. Summary From this part we can conclude that this gene tic algorithm is very impressive in terms of efficiency and robustness in the presence of noise. Laser amplifiers usually have a noise less than 2 %, thus the intensity squared noi se is approximately 4 %, either set of parameters (1 or 2) found in this optimization can be used in a closed loop optimization. It is important to note that from the number obtained in Table 43, the larger the noise the more critical is the correct III This test was performed using the best five fitness for each set of parameters. PAGE 121 121 selection of the optimization parameters. Thus for experimental situations with large noise these optimization parameters should be reevaluated. Experimental Pulse Compression Implementation of Genetic Algorit hms in Pulse Shaping Experiments In pulse shaping experiments, the individual genetic code is repres ented by an array of spatial light modulator voltages or phases. Given the large size of the parameter space (640 pixels per mask) simultaneous optimization of al l parameters becomes impossible to pursue (and provides no physical insight into th e dynamics of the molecular system). Instead we parameterize each array with a function of 128 variables. Th e same parameterized function (linear or spline interpolation among 128 pixels) is applied to both masks to produce a phase only modulated pulses. The selection of feedback signals used to generate the fitness and the fitness function chosen in each optimization procedure depends on the desired experimental objective. Optimization Problem The spatial light modulator in our laboratory produces temporal dist ortions to the input pulse because of the inhomegeneity of the pixels. To test our genetic algo rithm, we performed an optimization experiment with the aim of comp ensating this residual phase and achieving a transform limited pulse. As we showed in the previous section, the pulse compression problem is equivalent to the maximization of a sec ond order dependent si gnal (Equation 424). Setup and Optimization Conditions The used experimental setup is shown in Fi gure 410. In this expe riment, the feedback signal is a twophoton photoinduced current, which is obtained by so ftly focusing a laser beam on a GaAsP photodiode with a 100 mm focal lens. Since GaAsP ba nd gap is 680nm and our laser beam has a spectrum centered at 790 nm, the phot ocurrent generate is a twophoton intensity dependent process.132 In contrast to the second harmonic generation in doubling crystals, the PAGE 122 122 generated electric signal does not have the constr aints observed in nonlinear materials, such as short pulses bandwidth phase matching and acceptance angles.110 Twophoton induced current signal is processed with a boxcar integrator and acquired with a PC based data acquisition card. Figure 410. Experimental set up used in pulse compression. An average of 150 laser shots is used to gene rate the fitness value of each individual in each generation, and since the noise to signal ratio for our laser system is better than 3 %, no corrections are made for energy fluctuation. The genetic algorithm parameters used in this experiment are presented in Table 44. Table 44. Genetic algorithm parameters. Parameter Value Number of generations 200 Number of individuals 80 Number of elite parents 16 Number of crossover kids 75 Number of cloned kids 5 Probability of gen mutation 0.03 2h Photodiode Feedback Feedback Laser Pulse shaper PAGE 123 123 Multishot SHGFROG is used to fully charact erize the optimization results. The electric field is retrieved from the experimental traces with commercially available software (FROG Femtosoft Technologies). Results Figure 411 shows the fitness evolution curves obtained by optimizing the twophoton process at the photodiode. 0 50 100 150 200 0.0 0.5 1.0 1.5 2.0 2.5 3.0 TPIC (arb. units)Generation Figure 411. Evolution of the twophoton induced current as a function of the generation. Average (open triangles), best (full squares) and worst (open circles) fitness values for each generation are plotted. Three different evolution curves are presented corresponding to the best, average and worst individuals fitness for each ge neration. All these fitness curv es show a monotonic increase of the fitness value during the optim ization. Starting from a photocu rrent signal of 0.5 the best fitness of the individuals evol ves to a signal value of 3 after 200 genera tion. This change represents an increment of th e twophoton current intensity of approximately 600 %. From these plots we can also observe that at ca. 100 generations the best i ndividual fitness reaches 83 % of the optimized value suggesting that the genetic op erators providing the fine tuning are critically important on the 100 final generations of the optimization. PAGE 124 124 To confirm the quality of the optimization re sults, SHGFROG traces of the initial and final pulses are analyzed. The in itial pulse is shown in Figure 412 in which a strong modulation is observed. Figure 413 shows the temporal and spectral characteristic of the pulse retrieved from the analysis of th e experimental traces. 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 760770780790800810820830 0.0 0.2 0.4 0.6 0.8 1.0 IntensityWavelength (nm) Phase (rad)A2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 1000 500 0 500 1000 0.0 0.2 0.4 0.6 0.8 1.0 IntensityTime (nm) Phase (rad)B Figure 412. Uncompressed pulse. A) Spectral intensity (black line) and phase (red line). B) Temporal intensity (black line) and phase (red line). 0 1 2 3 4 760 780 800 820 840 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Wavelength (nm) Phase (rad)A1.0 0.5 0.0 0.5 1.0 100 50 0 50 100 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Wavelength (nm) Phase (rad)B Figure 413. Compressed pulse. A) Spectral intensity (black line) and phase (red line). B) Temporal intensity (black line) and phase (red line). As seen in the temporal and spectral intensitie s of the optimized pulse, the phase change in both domains in less than half a radian for the spectral and temporal FWHM. This indicates that the algorithm is capable of compressing the pulse to a transform limited one, starting from a very PAGE 125 125 far position in the phase variable space (Figure 412). Some residual phase is observed after the optimization. This residual phase is probably produced by the lack of compensation of the different dispersive optics presented in the FROG and second order intensity signal setups. The pulse temporal FWHM is 47 fs, which is only 5 fs away from the theoretical limit for a 25 nm FWHM spectral bandwidth pulse. Summary From all the above results we conclude that our homewritten al gorithm is suitable for performing experiments involving a large number of variables in closed loop optimizations. Moreover, this compression is substantially better than the phase compensation presented in the previous chapter. PAGE 126 126 CHAPTER 5 QUANTUM CONTROL OF TWOPHOTON INDUCED FLUORESCENCE OF RHODAMI NE 6G IN SOLUTION Introduction Many systems have recently been studied us ing the closed loop optimization technique.24 Most of them have been focused on a proof of principle rather than on an understanding of the process investigated. In this chapter, we present the adapti ve control of the twophoton fluorescence of Rhodamine 6G in solution. Rhodamine 6G is a relatively simple quantum system in which both molecular dynamics and electronic structure133 are well known, making it a good candi date to find the components of the optimal laser field directly related to its molecular properties. The motivation for these experiments is not only to show control over a simple system, but also to gain molecular information about the possible ex cited state processe s occurring in Rhodamine 6G. To achieve these goals, adaptive control e xperiments are performed in our homebuilt femtosecond pulse shaping setup. Molecular System and Control Rhodamine 6G is a yellow emitting dye with a xanthane core structure (Figure 51(A)). Among its various molecular properties, it has hi gh photostability, high extinction coefficient, and high fluorescence quantum yield. Due to thes e properties, Rhodamine 6G is commonly used as a laser dye134 and as a biomolecular pr obe in microscopy studies.135 Steady state spectroscopy of R hodamine 6G is presented in Figure 51(B). The absorption spectrum shows two distinct ab sorption maxima at 350 and 530 nm, and a third absorption band at 390 nm with a low extinction coefficient (inset Figure 51(B)). The emission spectrum manifests the typical mirror image observed in mo lecules with similar potential energy surfaces in the ground and first electronic excited stat es. A fluorescence excita tion anisotropy study PAGE 127 127 performed by Eggeling et al. reve als the presence of three distinc tive electronic transitions at 450 to 550 nm, 375 to 425 nm, and 325 to 375 nm, confirming the band assignment made in the absorption spectrum.136 This assignment shows that the first two bands, 350 and 530 nm, are one photon allowed and the third, 390 nm, is one photon forbidden (twophoton allowed). O N H N H CH3 CO2CH2CH3 CH3 H3C H3C Cl A0.0 0.2 0.4 0.6 0.8 1.0 300 400 500 600 700 0.0 0.2 0.4 0.6 0.8 1.0 Absorption (normalized)Wavelength (nm) Fluorescence (normalized)360380400420440 0.00 0.01 0.02 0.03 B Figure 51. A) Rhodamine 6G mo lecular structure. B) Absorption spectrum (black line) and emission spectrum (dashed line). Inset contai ns a zoom of the absorption spectrum from 360 to 445 nm. Rhodamine 6G twophoton cross section in the 700 to 850 nm spectral region has been measured by different authors.135, 137 All these studies show a larg e and featureless cross section in that region of the spectrum. Fluorescence studies show that the fluorescence spectrum shape and quantum yield are excitation wavelength independent,137 indicating a strong coupling between different excited states, Sn, and the first excited state, S1. No evidence of phosphorescen ce is found in the emission spectrum, thus ruling out the possibility the presen ce of intersystem crossing leading to triplet states. Rhodamine 6Gs quantum yi eld of triplet state formation was measured to be less than 0.005.138 PAGE 128 128 A Jablonski diagram of Rhodamine 6G is repr esented in Figure 52. Disregarding triplet states, this diagram shows four electronic states, S0 to S3, with fluorescence arising exclusively from the S1 state. Timeresolved experiments show a simple deactivation mechanism (Figure 52).139145 Upon excitation to Sn>1, the system relaxes to the vibrational ground state of S1 in a subpicosecond time scale. Strong coupling among the ex cited states leads to an ultrafast internal conversion.139 Once in the first excited stat e, the system relaxes from S1 to S0 via emission or internal conversion.139 The quantum yield of the emission from S1 is 95 %,146 and its life time is 3.9 ns.136 Figure 52. Jablonski diagram of Rhodamine 6G. The emission quantum yield from Sn>1 depends on the coupling between the Sn>1 and S1 states and on the nonradiative deactivation pathwa ys that directly transf er the system to the ground state. This study aims to control the coupling between S2S1 using the phasemodulation of the twophoton excitation pulses t hus controlling the quantum yiel d. If quantum control of the process is achieved, it will indicate a control over the coupling between S2S1 and/or S1S0 states. The motivation for this experiment is not only to show control over a simple system but also to gain molecular information of the various deac tivation processes occurring in Rhodamine 6Gs excited states. S0S1S2S3Excitation IC IC 520 nm 540 nm PAGE 129 129 Experimental The experiment setup used in the optimal cont rol experiments is depicted in Figure 53. This apparatus consists of a femtosecond laser sour ce (described in Chapter 2), a pulse shaper to produce arbitrary phasemodulation (described in Chapter 3), a de tection system to reveal the molecular response to the tailored pulses, and a genetic algorithm to modify the pulses according to the molecular response (described in Chapter 4). Figure 53. Experimental setup for controlling molecular fluore scence. L: lens. NLC: nonlinear crystal. PD: photodiode. PMT: photomultiplier tube. Rhodamine 6G sample was purchased from Ex citon. Samples are prepared in methanol (spectroscopy grade) with concentrations below 105 M (OD of 0.25 mm1 at 520 nm) to avoid any aggregation.147 The samples photostability is checked by steady state absorption before and after each set of experiments. PD Feedback Feedback Laser NLC L PMT Sample PMT PAGE 130 130 In this study two signals ar e monitored simultaneously, namely the twophoton induced emission from the sample and the second harmonic generated in a nonlinear crystal. The phase modulated beam is split into two new beams with one beam used to excite the chromophore and the other one to generate the second harmonic. To induce excitation the sample, pulses with 300 nJ of energy are focused on the sample with a 100 mm lens to c.a. a 40 m diameter spot size. The sample is held in a 2 mm path length quartz cell with magnetic stirring to ensure identical experime ntal conditions for each laser excitation pulse. Two different photomultiplier tubes (Hamamatsu H5783), each at 45 degrees with respect to the ex citation beam, collect the twophoton induced emission of Rhodamine 6G. Twophoton absorption is a nonlinear process that w ill occur at the focus of the excitation beam., The sample is correctly placed at the focus of the lens by maximizing the fluorescence signal. BG39 color filters are used to avoid having st raight excitation light hit in the detector. Simultaneously, second harmonic signal is produ ced with a collimated 30 nJ beam on a 100 m BBO crystal and detected with a photodiode (Thorlabs DET210). As with the other detectors, a BG39 filter is used to eliminate residual fundamental light. The two signals, fluorescence and second harmon ic, are used as feedback in the closedloop optimization. A boxcar integrator (Stanfor d Research System, SR250) and PC based data acquisition card (National Instrument, PCIMIO1 6E4) are use to integrate and digitalize the signals, allowing coupling between the signals and the genetic al gorithm. The genetic algorithm uses the average of the signa l produced by 150 laser shots. In the genetic algorithm, the pulse shaper phase is codified by 128 independent parameters, which are transformed into th e corresponding 1280 mask pixel pha ses by linear interpolation between adjacent points. The algorithm starts the iterative search with a population of 80 random PAGE 131 131 individuals, each consisting of 128 genes, and finishes the search optimization after 200 iterative loops. Due to the large size of the search space, experi mental reproducibility is used as a way of demonstrating that the system evolved to a glob al extreme of the searched space. Optimization experiments are reproduced at least three times; also for each experiment the laser stability is followed by simultaneously measuring the raw fe edback signals for th e unmodulated pulses (phase coefficients in zero phase). In order to fully characterize the modulated pu lses obtained in the cl osed loop experiments, a second harmonic FROG trace is collected after each experiment This trace is acquired in a homebuilt FROG setup with a 12bit resolution fiber optic spectrometer (Ocean Optics). Results Emission maximization can always be produced with a transf orm limited pulse. In samples with broad absorption bands this effect occu rs because the transform limited pulse has the broadest spectrum leading to the maximum ove rlap between its second order power spectrum and the molecular absorption coeffi cients for the twophoton transition.42 In addition, twophoton processes depend on the intensity squared: the maximum value of which is achieved with transform limited pulses. In this work, the aim is to control the twophoton induced emission of Rhodamine 6G in methanol beyond this trivia l solution. The intensity dependence of the excitation is removed by experimentally working with the ratio of two signals with the same intensity dependence.63 The process utilized to repr esent the twophoton excitation in tensity dependence is second harmonic generation. As any second order inte nsity process, second harmonic generation is sensitive to the change of the phase of the elect ric field. In addition, second harmonic conversion PAGE 132 132 efficiency is wavelength independe nt for many nonlinear materials in the spectral window of our laser excitation (770830 nm).148 Fluorescence Efficiency Optimization The first set of experiments aims to optimi ze fluorescence efficiency, which is defined as the ratio of fluorescence over second harmonic si gnals. This ratio physic ally represents the fluorescence intensity normalized by the intensity used to induce it. 0255075100125150175200 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 FL/SHG (a.u.)GenerationA 0255075100125150175200 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 FL/SHG (a.u.)GenerationB Figure 54. A) Fitness evolution of typical experiments of maximization of fluorescence efficiency. B) Experimental convergence reproducibility. Figure 54(A) shows typical evolution curves obtained during the maximization of fluorescence efficiency, where solid black squares, open triangles, and open circles correspond to the fitness measured for the best individual, the average of all individuals, and the worst individual fitness in each generati on, respectively. Starting with low ratio values (for pulses with randomly generated phases), the curves exhi bit an improvement of the ratio after 200 generations. In contrast to the smooth changes observed for the individuals with the average and best fitness values as a functi on of generation, fitness values for the worst i ndividuals show strong oscillations. This is a nor mal effect exhibited in gene tic algorithms where very unfit individuals are created by the different reproduction operators in each generation. PAGE 133 133 In heuristic searches, the convergence of the optimization is given by the stability of the solution. This stability is repr esented by the evolution of the average of the fitness for all individuals, because ideally, the average fitness values should as ymptotically reach the solution. Figure 54(A) shows that stab ility is reached in the 125th generation indicating the presence of a possible maximum in the fluorescence efficiency. This maximum reaches a value for the fitness 35 % higher than that obtained when exciting with a transform limited pulse (FL/SHG~1.4). Figure 54(B) shows the convergence reproducib ility. Three different experiments show a good reproducibility in terms of the achieved obj ective, but a forth experiment deviates by 10 % from the average. Since the stab ility of the laser did not show any important fluctuation, this difference can not easily be explained. The autocorrelation of the best excitation pulse for all the experiments is also examined as an independent check of convergence reproducibility (Figure 55). 432101234 Time (ps) 432101234 0.2 0.4 0.6 0.8 1.0 Time (ps) Intensity (a.u.)432101234 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Time (ps)432101234 Time (ps) A B Figure 55. A) Optimal autocorrelation for di fferent experiments. B) Typical SHGFROG trace of the optimum pulse. All autocorrelations pr esent similar complex temporal st ructures (pulses with a subpulse structure) indicating a convergence towards a nontrivial solution. The similar resemblance of the pulse temporal shapes reveals that the outlier optimization did not reach a new solution, but a PAGE 134 134 solution similar to the rest. Th is is also observed in their corresponding SHGFROG traces (not shown), confirming the reproducibil ity of the global solution. The second experiment presented here focu ses on the minimization of fluorescence efficiency. As in the previous experiment, the ratio shows an evolution of the fitness of the individual pulses with the gene ration. A typical fitness evoluti on plot with the FL/SHG ratio of the best individual (black full squares), the one for the worst individual (open circles), and the ratio average over all individuals of a given generation (ope n triangles) as a function of generation is presented in Figure 56(A). From the initial random phase to the optimized pulse, the efficiency ratio for the best individual of each generation decreases to a value close to 1.15, which compared to the transform limited puls e (~1.4) represents a reduction of 20 %. 0255075100125150175200 1.10 1.15 1.20 1.25 1.30 1.35 1.40 FL/SHG (a.u.)GenerationA 0255075100125150175200 1.15 1.20 1.25 1.30 1.35 FL/SHG (a.u.)GenerationB Figure 56. A) Fitness evolution of typical experiments of minimization of fluorescence efficiency. B) Experimental convergence reproducibility. In the evolution curves, the average ratio becomes constant after approximately 50 generations, indicating that the algorithm found a minimum. When compared to the previous experiment, the curve corresponding to the ratio values of the be st individuals shows a larger noise to signal ratio. This coul d suggest a solution trapped in a local minimum. However, the analysis of the convergence reproducibility (Figure 56 (B)) for experime nts starting with very PAGE 135 135 different pulses (different fitn ess) clearly shows that anothe r optimization produces the same effect excluding the previous possibilities. Agai n, one experiment did not converge to the same objective, although no laser stability issues were observed during the optimization. 432101234 0.2 0.4 0.6 0.8 1.0 Time (ps) Intensity (a.u.)432101234 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Time (ps) A Time (fs)Wavelength (nm) 3000 2000 1000 0 1000 2000 3000 390 392 394 396 398 400 402 404 406 408 410 200 400 600 800 1000 1200 1400 1600 B Figure 57. A) Optimal autocorrelation. B) Typical SHGFROG trace of the optimal pulse. Agreeing with the observed convergence, the autocorrelation of the optimal pulse shows that the minimization of the fluorescence efficien cy is not achieved by mo dulating the pulse to a very long featureless pulse, instead, by modulat ing it such it converge to a relatively short complex pulse. Similar to the fluorescence maximization experiments, the optimal pulses present a subpulse structure. Interestingly, a comparison between the optimal pulse for the maximization or minimization of fluorescence efficiency reveal s a counter intuitive result: the pulses that maximizes efficiency is significan tly longer than the one that mi nimizes the process, suggesting solutions specifically designed to match molecula r properties of Rhodamine 6G. A similar effect has been observed in an analogous study performed by Brixner et al.31 The optimal control of the fluorescence e fficiency manipulates the photophysics of Rhodamine 6G, but it does not prov ide direct information about th e process(es) being controlled. To understand where or how the quantum control is achieved, it is necessary to investigate the PAGE 136 136 nature of the states involved in the photoprocess, which can be inferred from power dependence studies. Power Dependence Since this study is based on populating an ex cited state via multiphoton absorption, a power dependence study will reveal which singlet st ate is initially populated It is important to note that even in the case of third interaction with the electric field, the power dependence will not show this interaction, if the 3rd interaction follows a twophoton resonant process. The intensity of the excitation source is enough to produce high order interact ions, considering that excited electronic state populations usually ha ve a large and reachable manifold of higher electronic excited states where they can be further transfered. In this case the excitation energy is not in resonance with the S1 to S0 transition, so a third interac tion will only promote the already excited state to a higher excite d state, but it will not increase or decrease the population in the initially excited state. Another case, contemplat es the possibility of creat ing two or more wave packets in the same excited state. As stated be fore the formation of those wave packets depends on the square of the excitation and so does the emission produced after their interaction. The twophoton stimulated emission is not considered to play a major roll in this process because it has only been observed at very high field intensities.55 The states that can be reached with this sour ce are only those whose energies are resonant with two or three photon transitions since the n ear infrared source can only induce the emission of Rhodamine 6G through a multiphoton excitation. The laser intensity on the sample is high enough to produce a three photon ex citation, therefore we meas ure the power dependence of each of the optimized pulses. For comparison purposes, the power dependence corresponding to a transform limited pulse is also investigated. PAGE 137 137 Figure 58 shows the dependence of the fluor escence and the second harmonic signal to power of the excitation pulse for the experiments performed with a transform limited pulse (open triangles), the pulse obtained af ter maximization of fluorescence efficiency (solid black squares), and the pulse obtained after minimization of fluorescence minimization (black stars). 1.92.02.12.22.32.4 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 log(SHG) (arb. units)log(Intensity (arb. units)) log(FL) arb. units) Figure 58. Power dependence of the different pulses. Top panel, fluorescence intensity dependence. Lower panel, second harmonic intensity dependence. A linear fitting of the intensity dependence tr end for each curve exhibits slopes always close to 2 (Table 51), confirming the twophoton behavior in both fluorescence and second harmonic signals for each particular pulse. Table 51. Power dependence linear fit results. Linear fit Pulse Slope y intercept log10ab FL SHG FL (a) SHG (b) Transform limited 1.97 1.81 4.14 4.09 0.89 Maximization experiment 1.97 1.97 4.35 4.79 2.75 Minimization experiment 2.29 2.01 5.49 5.09 0.25 This second order intensity dependence confir ms that the state reached with the 800 nm excitation corresponds to a state located at 25000 cm1 above S0, which is the dark state S2. PAGE 138 138 In contrast to the slope values, yinterceptions of the linear fittings (Table 51) have different values for each curve, showing a cha nge in the response be tween the investigated signals of fluorescence and second harmonic. A second order response can be expressed as 2 2raI (51) where a represent a proportionality constant which is strictly related to the process and the detection, and I is the intensity of the excitation pulse. In the loglog plot, the intercept is the log(a) and is represented by the y in terception. When the two respons es are plotted against each other, the obtained slope is gi ven by the ratio of these proportiona lity constants. Those slopes can be evaluated using the intercepts from the linear fit of th e power dependence. The different y intercepts indi cates that the optimum pulses s hould have different slope in a fluorescencesecond harm onic representation. 0.00.51.0 0.00 0.25 0.50 01234 0.0 0.5 1.0 1.5 2.0 Second Harmonic (arb. units)Fluorescence (arb. units) Figure 59. Power dependence in fluores cencesecond harmonic sp ace. Fluorescence maximization (squares). Fluorescence e fficiency maximization (stars) and minimization (triangles). Zoom of the lo wer left corner of the plot (inset). The representation of the optimal pulses in th e [SHG, FL] space is shown in Figure 59. It shows that each optimization of a different goal presents a different slope, and that th ey evolve in different zones of this variable space. Since these signals represent the molecular and field responses, the magnitude of the slope provides insight into the optim ization process. For PAGE 139 139 instance, comparison of maximization and minimi zation of the fluorescence efficiency reveals that the former explores a region of higher intensity of the laser fi eld while the latter stays within regions of lower electric fi eld intensities and emission. Optimization Analysis As shown in the previous section, a good re presentation to study the evolution of the optimization process is the fluorescence versus second harmonic signal space. Figure 510 shows the second harmonic signal and fluorescence intensities for all the tailored pulses tested in the optimization of the fluorescence efficiency. In addition to theses two optimizations, the experiment in which only the fluorescence is optimi zed (equivalent to pulse compression) is also included. 0.20.40.60.81.01.21.4 0.2 0.4 0.6 0.8 1.0 second harmonic (arb. units) fluorescence (arb. units)0123 0 1 2 A 0.30.40.50.60.70.8 0.3 0.4 0.5 0.6 fluorescence (arb. units)second harmonic (arb. units)B Figure 510. Variable space. A) All indivi duals of each experiment: maximization of fluorescence (blue squares), maximization of fluorescence efficiency (green triangles), and minimization of fluorescence efficiency (gre y circles). B) Fitness for the best individuals on each generation of each experiment and fitness for individuals with random phases. Looking at Figure 510(B), the da ta presented in this variable space can be separated in three distinctive regions. The center region corresponds to fitness ar ising from pulses with random phases and pulses leading to the optimiza tion of fluorescence alone. Below that region are the best individuals corres ponding to the optimization in which a maximum of fluorescence PAGE 140 140 over second harmonic signal ratio was sought and above all are the individuals corresponding to an evolution in which the goal was the minimiza tion of the ratio. These three regions denote the physical limits of the searched space because each fitness goal is specific ally designed to reach the extremes of this space. In addition, it conf irms that each optimization reaches an independent solution (red stars). The extreme slopes show in Figure 510 are related to the optimization improvement with respect to the experiment with a transform limited pulse. Although this representation shows the correla tion between the variables and the achieved objective it does not reflect either the attained convergence or the generational evolution of the optimization variables. The region explored by eac h optimization appears as a delocalized area where more than one solution for each problem can be found. A new representation is necessary to better understand the evoluti on of the fitness. We plot the ratio of fluorescence over second harmonic signal (fitness) value vers us the difference in the signals. 0.1 0.2 1.2 1.4 1.6 1.8 Ratio (Fluorescence / SHG)(FluorescenceSHG)1 10.05 0.10 0.15 1.15 1.20 Figure 511. Variable space in using ratio and difference of signals. Inset shows a zoom of the left corner of the plot. Individuals co rresponding to the fluorescence efficiency maximization (triangles) a nd minimization (circles). Figure 511 shows the new space for the coordinates of the best individual in each generation (solid shapes) and for the coordina tes of random initial pulses (open shapes). PAGE 141 141 The evolution of the chosen variables for bot h experiments displays different optimization mechanisms. The pathway corresponding to the maximization exhibits a complex optimization route: starting with a random pulse (see label 1 on Figure 511) the data moves horizontally with changes occurring mainly on the difference axis. This is followed by a simultaneous change in both axes reaching an area in which the difference in signals is not pronou nced, but in which the ratio has been improved. In contrast the mini mization shows evolution only on the difference axis while keeping the ratio constant. This im plies that both variables are changing their magnitude approximately by the same amount in the minimization. Although the optimization starts in the same area of the expl ored space (open circles, Figure 511), the convergence zones for each objec tive are clearly delimited in opposite regions. Also, the convergence zone for the maximization of the fluorescence efficiency is localized and the zone reached by the minimization is not. This suggests that the variab le space landscape has local extremum zones with different shapes: the maximization falls into a well while the minimization remains in a more shallow area. Fluorescence Spectrum Control In this thesis we shall try to develop new experiments that can pr ovide insight into the quantum control mechanism. With this goal in mind, we performed an optimization experiment with the objective of controlling the final FranckCondon stat e localization in the first excited state. The proposed experiments optimize sel ectively the low energy region of the emission spectrum. The new fitness function, defined as the ratio of the intensity of the red side fluorescence (Figure 512) over second harmonic, biases the optimization towards those transitions with lower energies The fluorescence spectrum shape is given by the overlap of the wavefunction of the vibrational gr ound state of the first electronic excited state with the various wavefunctions of the vibrational st ates of the electronic ground state. An optimization of only the PAGE 142 142 red side of the emission spectrum could only be produced by an important modification of the potential energy surfaces (strong fi eld control) or by affecting th e wave packet evolution such that the internal vibrational relaxation promotes the excited state to a different region of the excited state inevitably cha nging the FranckCondon coeffici ents and consequently the fluorescence spectrum. To acquire only the red side of the emission spectrum (grey area, Figure 512), a 550 nm longpass filter is placed in front of the photomultiplier. 500 600 700 800 0.0 0.2 0.4 0.6 0.8 1.0 FluorescenceWavelength (nm) Figure 512. Spectrum measured with the cutoff filter (grey area). A B Figure 513. FROG traces for different experiments. A) Maximiza tion of fluorescence efficiency when the whole spectrum is collected. B) Maximization of red si de fluorescence over second harmonic. These experiments produce ratio performances si milar to those optimizations in which the total fluorescence over second harmonic is meas ured. Also, the FROG traces of the optimal pulses have a strong resemblance to their counterpart on the experi ments with the total PAGE 143 143 fluorescence. Therefore, we dis card a control mechanism based on changing the final position of the wave packet or a strong field control. Time (fs)Wavelength (nm) 3000 2000 1000 0 1000 2000 3000 390 392 394 396 398 400 402 404 406 408 410 200 400 600 800 1000 1200 1400 1600 A Time (fs)Wavelength (nm) 3000 2000 1000 0 1000 2000 3000 390 392 394 396 398 400 402 404 406 408 410 500 1000 1500 2000 2500 B Figure 514. FROG traces for differe nt experiments. A) Minimiza tion of fluorescence efficiency minimization when the whole spectrum is collected. B) Minimization of red side fluorescence over second harmonic. Relationship between Objective a nd the Genetic Algorithm Solutions We performed an experiment in which the cost function only includes signals from different regions of the emission spectrum to conf irm the lack of control over the spectrum. The proposed fitness is the intensity of the red side of the emission spectrum over the total fluorescence intensity. 0255075100125150175200 1.06 1.08 1.10 1.12 1.14 FL(w/ cutoff)/(Total FL) (a.u.)Generation 0255075100125150175200 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 FL Intensity (a.u.)Generation Figure 515. A) Fitness evolution of maximization of spectral contro l. B) Evolution of the cost function for minimization of spectral control. PAGE 144 144 Two photomultiplier detectors with different combinations of band pass filter are employed to collect the different signals. Due to the lack of control ov er the spectrum observed in the previous experiment, we expected that the fitness function will oscillate and change randomly without ever converging to any solution. Instead, both experiments present an optimization of the fitness which contradict our initial speculations (Figure 515). Two possible scenario s can be occurring during the optimization. The first involves an actual optimization of the sp ectrum, which is discarded due to the results previously obtained. The second possibility concerns the role of noise and its optimization in an evolution process. The cost function used is red totalF f F, (52) where Fred, Ftotal represent the low energy region of the fluorescence and the overall fluorescence, respectively. The e rror in the cost function is 21red redtotal totaltotalF f FF FF (53) The fitness error function might explain the pseudo op timization achieved in these experiments. In Equation 52 and 53, it is ob served that as the fluorescence is either minimized(maximized), the fitness noise is increased(decreased). Assuming a constant error in all the ranges of fluorescence integrated intensity, this error optimization is due to the inverse relation with the fluorescence va lue, which produces th e above inverse relati onship in the error function. To avoid this problem a function where the noi se does not increase when the magnitude of the signals goes to zero is created. The new cost function is PAGE 145 145 i ij jF fFF F (54) Note that the error function for this fitness function is 2 22 1ii ij jjFF f FF FF. (55) Comparing with the other fitness error (E quation 53), this fitness error is not maximized(minimized) when fluorescence signals decrease(increase). Two experiments are performed with this new noise independent cost function. Here the feedback signals are the blue and red side inte nsities of the emission spectrum. In the first optimization Fi is the blue side and Fj is the red side, and in the second their role is reversed. Since the optimization aims to optimize one side of the spectrum at the expense of the other, both experiments seek the maximizati on of this new cost function. 0255075100125150175200 1.5 2.0 2.5 3.0 3.5 4.0 Fitness (a.u.)GenerationA 0255075100125150175200 0.20 0.18 0.16 0.14 0.12 0.10 0.08 Fitness (a.u.)GenerationB Figure 516. A) Fitness evolution of the cost function (FRFB)FR/FB (black line) and simulated evolution of the cost function (FRFB)FR/FB with the data corresponding to maximization of (FBFR)FB/FR (red line). B) Fitness evolution of minimization of spectral control (black line). Again, the experimental results using th e new fitness function (Equation 54) show optimization, i.e. an increase in the value of the cost function (black lines, Figure 516(A) and PAGE 146 146 (B)). To understand which proce ss is being optimized, we use th e raw data collected in the optimization of (FRFB)FR/FB and use it to simulate the evolution of the function (FBFR)FB/FR. A perfect agreement between both f itness is found (red line, Figure 516(A)). This implies that even though we discard fitne ss noise optimization during the experiments, two different optimizations with opposite objectiv es are producing the same effect. To find out why the fitness is optimized in all these experiments, the raw signal corresponding to the first spectral control (cost functi on intensity of the re d side over the whole spectral intensity) experiment is analyzed (Figure 517). 0 50 100 150 200 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Intensity (a.u.)Generation A) 0 50 100 150 200 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 Intensity (a.u.)GenerationB) Figure 517. Signal of fluorescen ce spectral control experiments. A) Maximization, fluorescence (black line) and second harmonic signals (red line) B) Minimization, fluorescence (black line) and second ha rmonic signals (red line). From Figure 517, it is possible to observe that both signals have exactly the same noise. As expected, these signal trends contradict the noise optimization model because for noise maximization the minimization of both variables is required, and vice versa. Depending on the experiment, second harmonic signal presents higher values than the red side for the maximization, and vice versa. This implies that a crossing in the magnitude of the signals is taking place. PAGE 147 147 A simple mathematical model can be used to describe the optimization effect. The fluorescence signal can be expressed as 2iiiFIabI (56) where I represents the intensity of the excitation laser, a is the response of the detector for zero intensity, and b is a parameter that includes the detector response, the transmission of the color filter, and the twophoton fluorescence quantum yiel d. Using that the intensity is the same for both fluorescence signals (red a nd total), and a mathematical relationship between the fluorescence signals is obtained redtotaltotal totaltotalredred redredabb Fa FabF bb (57) Since the fitness is defined by the ratio of fluorescence intensities (Equation 52), replacing the relation between the de tector signals we obtain red red total redFF f FabF (58) This fitness function depends on the value of the red fluorescence intensity. To understand its behavior the limits of the function can be analyzed. 0lim 0redred F redF abF (59) and 1 limredred F redF abFb (510) These limits show that the fitn ess is modulated by the intens ity of the red fluorescence and the actual cause of the optimization is the a term in the fitness, which is given by the difference in background signals produced by both detectors. This simple model explains the observed PAGE 148 148 trends of the fitness and the cros sing between the signals with differe nt intensities. To avoid this pseudo optimization, it is necessary to make the background signals as close to zero as possible and to maintain both detectors in similar gains. Discussion Physical Interpretation of the Fitness Besides the mathematical description of the fitness of the fluorescence efficiency, we can also try to interpret f itness in terms of the phys ics of the process. Sin ce fluorescence and second harmonic signals depend on the square of the inte nsity, a long temporal pulse not only leads to small fluorescence signals, but also minimizes th e second harmonic signal. When the ratio of these signals is minimized, the changes are co mpensated preventing the system from evolving towards a long structureless pulse. A similar conclusion is observed with the maximization of second harmonic signal and the effect of pulse compression. If no molecular property of the system is involved in the optimization, the fluor escence efficiency cannot be optimized because both fluorescence and second harmonic signal follow the same intensity dependence. This can be observed using the model previously described (E quation 56). Consider a simple case in which the detectors do not have a background signal (a=0) and the measured signals are fluorescence and second harmonic; the fluorescence ove r second harmonic fitness becomes: 22 221FL SHGS abIbI f SabIbIb (511) Thus no matter what type of optimization is pe rformed all the solutions will be located in a line with constant fitness. As observed in the variable space as well as in the power dependence experiments, a slope change in the fluorescen cesecond harmonic optimization is observed when this fitness function (Equation 511) is optimized, confirming the control of molecular properties of Rhodamine 6G. PAGE 149 149 Molecular Interpretation of the Fitness All the experimental evidence points to a contro l of the fluorescence efficiency with pulses specifically adapted to match certain molecular properties of Rhodamine 6G. It is useful to describe fluorescence/second harmon ic fitness in terms of molecular and laser field properties to understand the molecular manipulation produced in the optimization. Using perturbation analysis, a simple model can be derived. Th is perturbation model was first developed by Silberberg et al.42 and later extended by Gerber et al..31 Perturbation Analysis Time dependent perturbation theory can desc ribe the twophoton ab sorption process. A quantum system originally located in the stationary state g can be promoted to the final state f with an electric field interaction at time t0. In the limit of a small perturbation and assuming no resonance with any real intermediate or final state, the amplitude of the final state can be expressed as (see Appendix B for full derivation)42 1 1212 12 21fnngt t itt f natfnngEtEtedtdt (512) Here, ij are the dipole matrix elements between the i and j states, and ij is the corresponding energy difference between those states, ijijEE. (513) In the case of short interaction, the coherent c ontribution of all intermediate levels will also be short, hence it is possible to approximate the summation term in the amplitude (Equation 511) by PAGE 150 150 1 21 21 1 21,2 exp 0, 2.ng n fnng n ngfnngtt iEtt tt (514) The transition probability for the twophoton process is 2 2 2 41 exp() 2TPA fg ngfgfnng p Etitdt (515) The frequency domain representation of the probability is 2 2 41 2TPA fg ngfgfnng pE E d (516) This last expression shows that the twophot on absorption process can be decomposed in two independent factors. 2 41 2TPA ngfg f nng g (517) 2 2 fgSEEd (518) While the former represents the properties of the molecule, the later shows the electric field characteristics. The presence of the elect ric field in Equation 518 is one of the most important results of the perturbation mode l because it demonstrates that the twophoton transition probability can be modulated by varying the second order power spectrum of the excitation source, S2. In the limit of very broad inhomogeneous lines the transition probabi lity is the summation over all the possible states63 2 TPATPA p gSd (519) where () is the excited state density of states. PAGE 151 151 Second harmonic intensity as a function of the fundamental electric field can be expressed as110 2 2 2 2 2 2 2 22 111 2s i n c 42SHGLL I kEEd ckvv (520) where (2) is the second order susceptibility, L is the crystal length, is the fundamental frequency between the fundament al and second harmonic fields, k is the phase mismatch, and v1 and v2 are the group velocities of the fundamental and second harmonic wave, respectively. Similarly to the twophoton absorption pr obability, second harmonic generation can be decomposed into two terms: gSHG and S2:31 2 2 2 2 2 2 22 111 sinc 42SHGLL gk ck vv (521) and 2 2 fgSEEd (522) The first term depends on the properties of the nonlinear material while the second term is the integrated second order power spectrum. The si gnals measured in this experiment are then FL mTPAFLINp (523) and SHGSHG I Id (524) where Nm is the total number of molecules present in the irradiated volume, pTPA is the twophoton absorption probability, FL is the quantum yield of tw ophoton induced fluorescence, and ISHG is the second harmonic intensity integrated over the fundamental wave. The fitness becomes PAGE 152 152 2 2 mFLTPA mTPAFL FL SHGSHG SHGNgSd Np I f II gSd (525) Removing all the constants and assuming the second harmonic generation efficiency ( gSHG) to be constant across the laser excitation spectrum, the fitness can be described by 2 2 TPAgSd f Sd (526) This model of the fitness expresses the possibility of controlling the molecular system by changing the second order power spectrum, S2() The validity of this simple model was de monstrated by Joffre and coworkers using a second order interferometric autocorrelator in different dye samples in solution.43 They prove that the twophoton excitation spectrum ( TPAg ) is the main controllable feature within the optimization process. There is experimental evidence that supports a flat twophoton cross section of Rhodamine 6G in the spectral region of our excitation source.135, 137 If this is the case, this optimization step can be discarded from the control steps because it will imply a constant TPAg This suggests that control over the twophoton absorption is no t the critical component. Furthermore, our spectral control experiments confirm the lack of control in the relaxation process in S1, which agrees with the ultrafast internal vibrati onal relaxation times observed in this system. Summary In this work, Rhodamine 6G twophoton induced emission is successfully controlled using phase modulated femtosecond pulses. In the performed experiments, the closed loop optimization produces phase adapted femtosec ond pulses. The adaptation can be driven by stationary, twophoton excitati on spectra, as well as dynami c properties of the system. PAGE 153 153 Interestingly, the pulses that optimize the fluoresce nce efficiency are not necessarily the shortest ones. The presence of a specific time distributi on of the different fre quencies within each excitation pulse might signify the control over th e wave packet dynamics in the excited state. The multipulse structure observed in the optimization of the fluorescence efficiency might be associated with the optimization of a specific deactivation channel (vibrational mode) in the molecular electronic manifold, indicating that when the power dependence is removed, the algorithm finds a new solution which contains mol ecular properties. To confirm this possibility, different studies using pulse sequences are necessary. Finally, spectral control experiments have show n how important the selection of the fitness and the conditioning of the feedback signals are to avoid fake optimizations. PAGE 154 154 CHAPTER 6 COHERENT CONTROL OF ENERGY TRANSFER Introduction Optimal control using pulses with phase, and/ or amplitude, and/or polarization modulation, has been successfully used to govern a variety of processes.24, 51, 77, 85 Many of theses experiments involved the simultaneous measuremen t of two signals arising from independent processes.63 In this work, we explore the optimal c ontrol of energy transf er between donor and acceptor moieties. Moieties with efficient energy tr ansfer must have a strong electronic coupling; thus their states can not be independently m odulated, but one state can be modulated at the expenses of the other. Resonance energy transfer involves Coulombi c coupling between an excited state donor and an acceptor. The relative strength of the coupling between donor and acceptor compared to their spectral overlap (the re sonance condition) defines the ef ficiency and mechanism of the energy transfer process.149 Arbitrary preparation and manipul ation of the donors excited state can then influence the coupling strength, and th erefore manipulate the energy flow. One possible way of influencing the donors excited state is by negatively interfering those relaxations pathways that do not contribute to the energy transfer, indi rectly increasing the coupling efficiencies. Depending on the control objective, optimal control can be used to enhance the systems absorption or to prepare a particular co herent superposition of st ates that will optimize (maximize/minimize) the moieties coupling. Howe ver, molecular systems with energy transfer present other properties that can change the energy transfer efficiency and consequently make its manipulation difficult. One impor tant property is the rigidity of the donoracceptor system. Energy transfer efficiency depends on the donoracceptor distance and their transition dipole moment orientation. Many molecular systems including those generally used for fluorescence PAGE 155 155 resonant energy transfer (FRET) ha ve flexible backbones, thus their energy transfer efficiency is also governed by their molecular dynamics. To prev ent this problem, a molecule with a constant donoracceptor distance should be used. The other important propert y is the acceptor excitation cross section. To avoid direct excitation, the acceptor cannot have a significant one or twophoton cross section in the wavelength region of the excitation source or, if excited, it has to be uncontrollable. This guarantees that the accepto r emission (and its modulation) is arising exclusively from energy transfer. These two proper ties are intrinsic to the molecule, so special care should be taken when choosing a system to perform quantum control. This study is focused on controlling the energy transfer process in 2G2mPer using phase modulated twophoton excitation pulses. We inve stigate the control of the energy transfer process by preparing and controlling those coherent states that lead to the coupling from donor to acceptor purposely avoiding optimization of the ab sorption process. The motivation for these experiments is not only to prove that control over the energy transfer process is feasible, but also to understand the important frequency and time domain parameters that induce the change in the energy transfer process. In addition, we use the re sults from a closed loop experiment to find the crucial parameters to be explored in an open loop experiment. Donor Acceptor System The system under investigation is a phenylene ethynylene dendrimer, 2G2mPer,IV which was specifically designed to mimic light harvesting systems.150 Phenylene ethynylene dendrimers are rigid donor/accep tor macromolecules with an ex tremely high quantum yield for energy transfer.150, 151 2G2mPer architecture is designed su ch that the donor is the dendrimer, IV 2G2mPer a second generation dendrimer with two monodendron on meta positions and a ethynylene perylene trap. PAGE 156 156 and the acceptor is an ethynylene perylene covalently bond to the dendritic backbone (see Figure 61(A)).152 H3CO OCH3 H3CO OCH3 OCH3 OCH3 A B 300350400450500550600 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Emission (arb. units) Absorption (OD)Wavelength (nm) C Figure 61. A)Chemical structur es of phenylene ethynylene dendrim er: 2G2mPer. B) 3D model of the 2G2mPer. C) Absorption (green line) and emission (grey area) spectra of 2G2mPer, and absorption of 2G2mOH (black line) and ethylene perylene (red line) The dendrimer presents a folded spatial arrange ment with all the dendritic branches packed in a bouquetlike 3D spatial arrangement.151 The energy transfer donor consists of two identical and indistinguishable monodendron s that in the ground state ar e electronically decoupled because of their meta connectivity. However, within the monodendron the different length phenylene ethynylene units are bonded in ortho and para substitution creating extended conjugated electronic states w ith broad absorption bands.151 A phenylethynylene perylene acceptor is bonded to the donors backbone in meta position, which crea tes donor and acceptor moieties with only a weak coupling in the ground state. This weak coupling results in an PAGE 157 157 absorption spectrum for the whole molecule wh ich is approximately equal to the sum of absorption spectra of its moieties (Figure 61(C)).151 In 2G2mPer, an initial ex citation is delocalized within each monodendron and the donoracceptor meta connectivity confines the exciton w ithin the monodendrons until it is further transferred to the acceptor. Time resolved spectroscopy previously performed in our group showed that upon single photon excitation the exciton migrates from the in itially excited state through two different pathways to the acceptor (Figure 62). 151 While one pathway consists of an initial step within th e dendrimer backbone and a final transfer to the acceptor, a parallel, direct energy transfer involves direct exciton migration from donor to acceptor. Both pathways are completed in a subpicosencond time scale.151 Indirect energy transfer (a funnel type process) involves an incoherent process, but the direct pathwa y is not understood and it might involve a different initially excited state. This initial state might be a coherent state originated by superposition of spatially close phenylene ethynylene states of the donor. Figure 62. Energy diagram a nd dynamics of 2G2mPer. 1, 2, and 3 are ~0.4ps, ~0.5ps, and ~0.8ps, respectively. FL ~ 2ns FL hS0 Sn 1 2 3 S1 Donor Acceptor FL FL<1ps Sn? PAGE 158 158 The calculated quantum yield for energy transf er, using experimental kinetics times of 2G2mPer is greater than 99 %, where approx imately 60 % and 40 % of the energy transfer occurs via the direct and the in direct processes, respectively.151 Experimental Section Our experimental setup (Fi gure 63) produces the shapi ng of pulses generated by a femtosecond amplified laser. The laser source is a Ti:Sapphire amplified system (Spectra Physics, Spitfire) pumped by a Ti:Sapphire oscillator (Spectra Physics, Tsunami), operated at 1 kHz repetition rate. This source delivers pulses with 25 J energy, Gaussian spectrum centered at 800 nm, a pulse width of 28 nm (FWHM), and temporal duration of 45 fs (FWHM). To avoid energy loss generated by small apertures in some pulse shaper components, a Galilean telescope is used to reduce the beam size to a final 3 mm diameter. Figure 63. Coherent control setup. Arbitrary phase and amplitude modulation is ach ieved with a spatial light modulator based pulse shaper. Our pulse shaper setup is arranged in a folded zerodispersion compressor, and it has been described elsewhere.153 The compressor is constructed in a quasi Littrow configuration 2h PD PMT BS 45 fs Lase r Pulse shaper PAGE 159 159 (46 degrees for 800 nm) with a 300 mm spheri cal lens and an 1800 groves/mm holographic grating. This optical configuration has a reso lution of 0.13 nm/pixel and a maximum spectral window of ca. 84 nm.105 The spatial light modulator is a CRI Inc., SLM640DVN, which has 640 individual pixels, and phase and amplitude modulati on capabilities. Figure 63 shows the experimental configuration. Tailored pulses are split twice: a small amount (<4%) is used to generate twophoton induced current in a GaAsP photodiode and the rest is used either to produce the sample excitation or to characterize the pul ses electric field in a multishot SHGFROG setup. Modulated excitation pul ses with an energy of 330 nJ per pulse are focused on the sample with a 50 mm le ns which yields a experimental ~24 m diameter and a Rayleigh length of 420 m, and produces intensity of 6.8x1011 W/cm2 on the sample. The lens position is carefully set to maximize the emission produced by the sample. The sample is held in a 2 mm optical path qua rtz cell with a magnetic stirrer to ensure fresh sample for each excitation pulse. Since the twophoton cross section is much smaller than the linear cross section, higher sample concentrations (OD=0.4) are used in the region equivalent to one photon excitation (400 nm). The emission signal is collected front face at 45 degrees with respect to the excitation beam with a photomultiplier tube (PMT) to avoid self absorption. A twophoton induced signal is ob tained by focusing the laser beam on a GaAsP photodiode. Since GaAsPs band gap is 680nm, the generated phot ocurrent is a twophot on intensity dependent process similar to second harmonic generation.154 In contrast to the second harmonic generation in doubling crystals, the generate d photocurrent does not have th e constraints present in these nonlinear mediums, such as short pulses bandwidth phase matching and acceptance angles. Fluorescence and twophoton induced currents are processed with a boxcar integrator and acquired with a PC based data acquisition card. PAGE 160 160 A home written genetic algorithm is used to produce the closedloop optimization. Our genetic algorithm is based on previously published algorithms and uses roulette wheel for parent selection, random initial populat ion, and special floating point genetic operators for reproduction.129 Among their various advantages, thes e genetic operators yield a fast convergence with a small number of individuals per generation. We used 80 individuals per generation, from which 16 are sele cted as parents for the next ge neration. Each new generation is originated through crossover (75 individuals) and cloning (5 indivi duals). All the individuals are subject to a possible mutation afte r their creation. Each individual has codified 128 genes (phase parameters) and the 640 phase pixel values are generated through linea r interpolation. The genetic pool is optimized for 200 generations. In the following experiments, the feedback signals are fluorescence and twophoton induced current. The fitness or objective function is defined in the experiments as either the emission or the ratio of emission over twophoton photodiode response. An average of 150 laser shots is used to generate the fitness value of each individual in each generation. Since the noise to signal ratio from our laser system is lowe r than 3 %, no corrections are made for energy fluctuation when fluorescence is used as fitn ess. When the ratio of emission over twophoton photodiode response is used, no corrections for pow er fluctuations are necessary because the twophoton induced current and the fluorescence have the same quadratic intensity dependence. Optimization experiments are repeat ed at least three times to assure experimental reproducibility. During each experiment we monitor the laser st ability by collecting the signals for an extra individual with all its phase coefficients set to zero. This individual is not included in the optimization of the population. PAGE 161 161 A multishot SHGFROG is used to fully ch aracterize the optimization pulses (phase modulated pulses) and their electr ic field is retrieved with co mmercially available software, FROG Femtosoft Technoligies.155 Dendrimer samples, 2G2mPer and 2G2mOH, were synthesized by Z. Peng (University of Missouri, KA) and their synthesis has been described elsewhere.152 Perylene sample was purchased from Sigma Aldrich Corporation. All the samples are prepared by making dichloromethane (spectroscopy grade) so lutions at concen trations below 106 M (OD less than 0.2 mm1 at 470nm, acceptors wavelength) to avoid any aggregation or excimer formation.156, 157 The samples photostability is checked by steady st ate absorption spectrum before and after each set of experiments. Results TwoPhoton Cross Section This work is focused on studying the quantum co ntrol of energy transfer using a modulated twophoton excitation. Since the spectrum of the la ser source is in the near IR region (760 nm to 840 nm) and selection rules are different for one and twophoton transition, the accessible electronic states reached after a twophoton excitation could be di fferent from those accessed by a one photon transition with equi valent transition energy (400 nm ). We evaluate the twophoton excitation response using th e twophoton fluorescence over second harmonic technique presented by Montgomery and Damrauer.44 Figure 64(A) shows the twophoton ex citation response of 2G2mPer. The macromolecule presents a flat twophoton ex citation response, which agrees with early experiments performed by Melinger and coworker s on a family of sim ilar phenylene ethynylene dendrimers.150 That study showed that the twophot on cross section for different phenylene ethynylene dendrimers varies with the structure and connectivity of the dendrons. Interestingly, PAGE 162 162 up to the third generation they all present a flat twophoton cross section in the region of our excitation source (760 nm to 840 nm) regardless of th eir internal coupling and conjugation length.150 760780800820840 Wavelength (nm)1 0 1 2 3 4 Two photon cross section (arb. units)A 760 780 800 820 0.00 0.25 0.50 0.75 1.00 Intensity (arb. units)Wavelength (nm)B Figure 64. A) Twophoton cross se ction of 2G2mPer (squares) and excitation laser spectrum (gray area). B) Sprectral windows used to measure the twoph oton cross section. Investigation of the perylene s twophoton cross section sh ows that the trap does not contribute significantly to the systems twophoton cross section, since the integration of its twophoton induced fluorescence (after excitation with a transform limited pulse) is more than nine times smaller than that for 2G2mPer for similar absorptions conditions (OD=0.21 and OD=0.22, respectively, at 470 nm, perylene absorp tion region), which for this decoupled system gives the same concentrations. Closed Loop Experiments Typical experiments compare the optimiza tion of a pure excita tion process (pulse compression) on 2G2mPer with the optimiza tion of the combined excitationdynamicsrelaxation process. Figure 65 presents the fluorescence signal emitted from the energy acceptor in 2G2mPer as a function of the twophoton indu ced current generated by the excitation pulse. Each point corresponds to the best individual in a given generation of the optimization algorithm. PAGE 163 163 When the fitness function is the fluorescen ce intensity, the genetic algorithm will only optimize the excitation process. Since the de ndrimer backbone is excited through twophoton absorption, the process of optimization is e quivalent to compressing the excitation pulse.32, 158 When the fluorescence is plotted against the twophoton induced current signal, the best individuals follow a straight line w ith slope of 1.02. The result of this optimization is a transform limited pulse ( FWHM = 45 fs). 0.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.0 Fluorescence (arb. units)Twophoton induced photocurrent (arb. units)0.2 0.4 0.2 0.4 Figure 65. Best individual evolution for the different op timizations: maximization of fluorescence (red squares), maximization of the ratio (green squares) and minimization of the ratio (blue squares) Inset shows an e xpanded view of the maximization region. We carry out optimizations where we use the fluorescence over th e twophoton dependent GaAsP signal as the fitness f unction to explore the control of processes beyond the twophoton excitation.31 As expected, the plot of the best indivi duals as a function of the twophoton induced current signal reveals a new evol ving pathway (Figure 65, green squares). This new path is significantly different from the first experiment as the evolution of the best individuals departs from the linear response obtained in the previous optimization. The best individual of the last generation has a fluorescence over GaAsP photodi ode signals ratio close 1.17, and for an PAGE 164 164 excitation pulse that generates considerably less fluorescence and twophoton induced current intensities. One way to differentiate an optimization due to pulse compression from the optimization of the overall process is seen in Figure 66. Figure 66 shows the ratio of the fluorescence to GaAsP twophoton signals for optimization ba sed on two different fitness functions: fluorescence only and fluorescence over twophoton induced photodiode signal. 0 50 100 150 200 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 Fitness (arb.units)Generation Figure 66. Fitness evolution for the maximi zation of fluorescence over twophoton induced current. Open squares and full circles repr esent the best indi vidual for the ratio maximization and the calculated fitne ss using the values of the FL only maximization, respectively. Open squares correspond to the ratio of fluorescence ove r twophoton induced current when the objective is chosen to maximize only the fluorescence. After approximately 10 generations, the ratio reaches a saturation point, which is the same ratio obtained for the last generation where the transform li mited pulse is located and the total fluorescence signal is maximized. After the 10th generation, the intensity of fluor escence keeps increasing (see Figure 65). The final fluorescence is five times larger than the intensity in generation 10, but the ratio of fluorescence over twophoton indu ced current signals at generation 200 has the same value as in generation 10. These optimized pulses present almost the same ratio value of their pulse PAGE 165 165 spectral phase. The average and dispersion of the calculated ratio are approximately 1 and 0.05, respectively. A more interesting case is observed when the objective is set to maximize the ratio of fluorescence over twophoton GaAsP signals (full circles). Afte r 200 generations the fitness reaches a plateau zone indicating the existence of an optimum. This optimum represents a 1520 % fitness increase with respect to the previous ex periment. The intensity of fluorescence is about 60 % smaller than the intensity of fluorescence obtained before, but the efficiency of photon emitted per excitation pulse intensity has been improved. This experiment proves that the attained solution cannot be achieved by setting th e genetic algorithm to maximize the emission alone, and that the optimal solution can not be a consequence of noise optimization. FROG traces of critical pulses were measured and their spectral and te mporal electric field profiles reconstructed (Figure 67). Figure 67 shows that the sy stem did not converge to the trivial solution (a transform limited pulse), but to a pulse with a well defined temporal shape. 60 70 80 90 100 110 1.00.50.00.51.01.52.0 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (arb. units)Time (ps) Phase (rad)A10 20 30 760 780 800 820 840 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (arb. units)Wavelength (nm) Phase (rad)B Figure 67. Retrieved (A) Tempor al intensity and phase and (B) Spectrum and spectral phase. This pulse temporal intensity (Figure 67( A)) has an asymmetric profile with two distinguishable main features: a spike component (~50 fs time sc ale) and a long component (~2 ps). The long component is temporally delayed with respect to the short component. In addition, PAGE 166 166 the spectral phase observed in the retrieved electric field presents a step, which is directly related to the control mechanism (see following section). We also performed an optimization in which we tried to minimize the ratio of the fluorescence over twophoton induced current. A new zone of the variable space was explored by the individual pulses (Figure 65, blue squares), but no evolu tion of the optimization process was observed and the optimum zone reached fi ts within the linear dependence region. Comparison of the autocorrelation of the optimal pulses with autocorrelation of the transform limited pulse shows that this optimization leads to a broad and featureless pulse (Figure 68). 6543210123456 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (arb. units)Time (ps) Figure 68. Autocorrelation of the pulse observed after the minimization and of the transform limited pulse. We independently investigate the cohere nt control of acceptor and donoracceptor molecules to further explore which process is being affected by th e phase modulation of excitation pulses. A solution containing the acceptor moiety (perylene) in dichloro methane was excited at the same wavelength and the emissi on and twophoton induced current signals were recorded for excitation with pulses having the same modulated phase as those individuals that lead to the maximization of the emission over th e GaAsP signal for the 2G2mPer molecule in dichloro methane. As stated before, the twophoton cross section of perylene is almost ten times PAGE 167 167 smaller than that of 2G2mPer, thus to obtain a reasonable noise to signa l ratio we average over a larger number of shots. For co mparison, signals are normalized with respect to the perylene fluorescence and twophoton induced signal ob tained with a transform limited pulse. 0.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.0 Fluorescence (arb. units)Twophoton induced photocurrent (arb. units)0.2 0.4 0.2 0.4 Figure 69. Variable space for acceptor and donoracceptor system. Full squares and black stars correspond to the optimization experiment s on the donoracceptor system and using those phase modulated that optimize 2G2mPer to induced excitation on the acceptor molecules alone, respectively. With these excitation pulses, the ratios lie within the linear region determined by the fluorescence maximization. This indicates that the deviation from a unity slope in the optimization of 2G2mPer obtained is not due to a direct excitation of the acceptor, but instead, it involves the excitation of the donor fo llowed by energy transfer to the acceptor. Open Loop Experiments Genetic algorithms are an easy and fast way to optimize an ultrafast laser pulse to achieve a given goal, e.g. molecular energy transfer. The solution obtained in these types of experiments usually is an electric field with very complicated phase features (Figure 67); thus the possibility of gaining any molecular in sight into the dynamics of the process is limited.88 We will show here how certain features of the optimal pulse can be recognized and extracted to be used to simplify the solution with the goal of getting a single knob c ontrol system. Chapter 7 presents a series of PAGE 168 168 statistical analysis used to extract those phase components that most affect the optimal pulse. The experimental retrieved phase leads to Figure 67, while a deeper i nvestigation including statistical analysis is presented in Chapter 7. The spectral phase of the optimal pulse presents a steplike change at 810 nm (Figure 66). A step phase function produces an asymmetric temporal pulse. According to BrumerShapiro asymmetric time pulse profiles have an effect on closed systems interacting with two or more photons.55 To explore the role of asy mmetric electric fields on the excited state dynamics, we investigate the response of 2G 2mPer to excitation with tim e asymmetric pulses (steplike phase) and time symmetric pul ses (quadratic phase). 0.00.20.40.60.81.01.21.41.61.82.0 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 TPIC (arb. units)x.A 760800840 0.00.20.40.60.81.01.21.41.61.82.0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 FL/TPIC (arb. units)x. x.PI Phase (rad) B Figure 610. A) Fluorescence second harmonic ratio versus the step size of the spectral phase for 2G2mPer (stars), 2G2mOH (open triangle) Perylene (crossed circle). Inset shows the applied phase. B) Experimental (s quares) and calculated (line) TPIC. The experiment consists on measuring the ratio of emission over twophoton induced current signals following the twophoton excitation of the different samples. The excitation pulses are modulated with a steplike phase func tion in which the depth of the modulation is continuously varied. The experiments do not provide a feedback si gnal and they are not optimized in any way. They represent the test ing of what we learned from the previous PAGE 169 169 optimization experiments. The experimental result is the raw response of the excited sample to an applied phase. The choice of the phase function (steplike function from 0 to 2 radians) was governed by the result of th e initial retrieved phase. Figure 610(A) exhibits the phase eff ect over the generation of the twophoton photodiode response. The twophoton absorption signal in GaAsP produces the expected symmetric well that can be modeled by calcula ting the twophoton absorption in the GaAsP photodiode of pulses with a step phase f unction plus a residu al cubic phase. Figure 610(B) shows the effect produced by a va riable amplitude step function centered at a fixed wavelength (800 nm) on 2G2mPer (DA) 2G2mOH (D), and perylene (A). The perylene sample shows a symmetric response to the scanning of the applied phase. The emission over twophoton induced current ratio is maximum for an applied phase of 0 (or the equivalent 2 ) and minimum for 0.8 This is the expected response for a system in which the phase only affects the twophoton excitation process. The sma ll shift of the position of the minimum of the response curve (0.8 instead of 1 ) is probably due to solvent effects on the phase of the excitation. Quite different is the response observed fo r the 2G2mOH and 2G2mPer samples. In both cases, the measured ratio does not follow th e expected behavior for a simple twophoton process. The variable step func tion with step size between 1.2 and 1.8 clearly outperforms the response from the transform limited pulse (phase of 0 or 2 For these particular phases, the ratio reaches values close to 1.1 and 1.05 fo r 2G2mPer and 2G2mOH, respectively. These differences represent a 10 and 5 % increase with respect to the transform limited pulse (ratio=1). It is important to highlight that the maximum pos ition is located where the step size is ~4.5 rad PAGE 170 170 (1.4 ) for both samples. This phase position not onl y represents a double pulse with a certain delay, but also determines their relative intensity. 1 0 1 0 2 4 6 0 2 4 6 0 10 20 20 10 0 Intensity (normalized) 2200230024002500 0.0 0.5 1.0 Ang. Frequency (THz) Phase (rad)A Intensity (normalized) 390400410 0.0 0.5 1.0 Wavelength (nm)B Intensity (normalized) 0.50.00.5 0.0 0.5 1.0 Time (ps)C Figure 611. Spectrally modulated pulses. A) Sp ectrum (red line) and spectral phase. B) Second harmonic spectrum (blue line). C) Temporal intensity (black line). From top to bottom: transform limited, 0.4 step, 1.4 step, linearly chirped 2000 fs2, and linearly chirped 2000 fs2 pulses. The signal obtained for perylene and twophot on induced current trends can be explained with the theoretical treatment develop by S ilberbergs group, where the absorption of a multiphoton transition without a resonant intermediate state only depends on the excitation intensity and not on its phase.32 2G2mPer and 2G2mOH do not follow this model, strongly suggesting that a different mechanism must be taking place. This mechanism yields a fluorescence efficiency larger than the one obt ained with a transform limited pulse, and the optimization is phase since the response at x(10.4) is opposite to the signal at x(1+0.4). We also performed an experiment in which double pulses with different phase relationship between the pulses is codified to investigate the phase effect rather than the time ordering of the PAGE 171 171 pulses. In this experiment a doubl e pulse with zero phase relati on and a train of pulses with a phase relation between the two consecutive pulses are examined. 500 250 0 250 500 0.00 0.25 0.50 0.75 1.00 AutocorrelationTime (fs)A 050100150200250300 0.85 0.90 0.95 1.00 1.05 FL/TPIC (arb. units)Time delay (fs) B Figure 612. A) Temporal autocorr elation of the pulse codified with a zero phase difference, where each color represent pulses separate d by a different delay. B) Ratio of fluorescence over twophoton induced current versus the time delay between subpulses for zero phase re lation (black line) and phase relation (red line). Figure 612 presents the results of these experiments. In the case of pulses with zero phase relationship the ratio is increas ed (ratio ~1.07) above the va lue observed for the transformed limited pulse (ratio 1). In cont rast, when the pulses have a phase relationship no improvement of the ratio is observed. It is interesting to not e that the transform limited pulse does not yield the highest response. Also, the delay for the maxi mum response (~70 fs) coincides with the delay between pulses found in the step phase function experiments. Experiments with 2G2mOH yield a similar response. To further prove the uniqueness of the steplike phase function we pe rform an experiment in which a different phase function is scanned, yielding always a symmetri c temporal profile and a second harmonic spectrum similar to that obser ved for a transformed limited pulse (Figure 613). Application of a quadratic phase only produces broadening of the pulse width while maintaining its symmetric temporal shape. We in vestigate the quadratic phase effect over the PAGE 172 172 donoracceptor excited stat e dynamics with pulses with tempor al widths (FWHM) ranging from 42 fs (0 fs2 phase) to almost 600 fs (4600 fs2 phase). Figure 613 shows the results of time symmetric pulses applied to the excitation of 2G2mPer. 54321012345 0.25 0.00 0.25 0.50 0.75 1.00 FL/TPIC (arb. units)Quadratic phase (x1000 fs2) Figure 613. FluorescenceTPIC ratio versus variable quadr atic phase for 2G2mPer. The fluorescence over second harmonic response of 2G2mPer is a nearly symmetric bell. In contrast to the phase respons e observed with the step phase function, the quadratic phase does not yield any improvement in the response with respect to the transfor m limited pulse. The small shift on the position of the maximum is due to the lack of compensation of the phase produced by the solvent. This behavior was predicted by BrumerShapiro theory and it agrees well with Silberbergs twophoton absorption theory and pr eviously published results in Rhodamine B.32, 55 Discussion Closed loop and open loop experiments suggest that control of the energy transfer in 2G2mPer is exerted over the dynamics of the excited state. To confirm if th e system is dynamically controlled, we must analyze all possible processes involved and whether these processes can be controlled or not within our e xperimental conditions. For instance, a mechanism in which the population is dumped from the excited state is not feasible because the respective emission wavelengths (420 nm for the donor and 520 nm for the acceptor) do not match with the spectrum PAGE 173 173 of the excitation laser (near IR, centered at 800 nm). Recen t experiments on Rhodamine B performed in Dantus lab show that if the peak power density is smaller than 1015 W/cm2, 55 twophoton stimulated emission does not play an impor tant role. Our experiments conditions provide a peak power density ~7x1011 W/cm2, ruling out contribution from twophoton stimulated emission. Also, the experiments presented here for linearly chirped pulses present a symmetric response. This response would not be expected if stimulated emission played a role in the mechanism.54 These considerations eliminate the possi bility of producing a control mechanism base on a pump and dump scheme. Another uncontrollable process is the last step of the dynamics: emission from the acceptor. In addition to the difference in energy between excitation and emission (1/2 eV), the population in this state has a decay time of 2 ns; thus all coherences initially created are completely erased by vibrational energy redist ribution, and consequently no control can be exerted on this step. The twophoton absorption process can be mode led in terms of two components: power spectrum of the second harmonic wave and twophot on excitation response (see Chapter 5 for a full description)65 2 2 TPAgSd f Sd (61) where gTPA is the twophoton excitation spectrum of the molecule and S2 is the second harmonic spectrum of the laser. While the component S2 is a property of the laser electric field, the gTPA component is an intrinsic molecular property. Theoretically demonstrated by Silberberg and coworkers and experimentally realized by Gerber and coworkers, at a given energy and power spectrum the twophoton transition probability for molecules without any intermediate resonant PAGE 174 174 state is maximized by the pulse wi th the shortest time duration.32, 63 In addition the experiments presented here for linearly chirped pulses pr esent a symmetric response. We observe this behavior when the optimization is set to maximize solely the 2G2mPer emission. The transform limited pulse produces the highest pop ulation transfer of the donor via twophoton excitation. As a consequence the energy transf er process and acceptor emission are increased, but not due to the control of the excited state dynamics. A smart choice for the fitness function can remove this dependence on the excitation source. In this thesis, it was accomplished by using the ratio of emission over twophoton induced current in a GaAsP photodiode. The optimization process is decoupled from the intensity depe ndence of the twophoton ab sorption and it evolves to a new distinct solution. The new solution is independent of the peak power but it can still depend on the frequency response of the twophoton excitation process ( g(), () see Chapter 5 for a complete description). In experiments involving twophoton emission processes from Damrauers and Joffres groups the maximization/minimizati on of the twophoton excitation response was responsible for most of the control mechanisms.43, 89 If the twophoton exc itation response for a given molecule is not flat, the most effective excitation will be obtained with tailored pulses whose second harmonic spectrum matches the molecular regions of high twophoton cross section and will be minimum or negligible in those spectral regions with lower twophoton absorption cross section. In the closed loop maximi zation experiment presented in this thesis, we do not observe any significant changes in the sp ectrum of the second harmonic of the optimal pulse compared to the transform limited pulse. This lack of spectral selectivity in the excitation agrees well with flat twophot on cross section measured for 2G2mPer (Figure 64). An experiment that confirms the control beyond the tr ivial of spectral control mechanism is shown PAGE 175 175 in Figure 610 and 611. Pulses with identical second harmonic spectra yield very different signal values. For example, application of a 0.6 steplike function to a transform limited pulse generates the same second order spectrum as that with a 1.4 phase function (see Figure 611). However, the measured values for emission over twophoton induced current are strikingly different. The former yields a 0.725 signal wherea s the later yields to optimum value of 1.08. Since different pairs of pulses with identic al second harmonic spectrum have different molecular response, the molecular response mu st be governed by time ordering and relative phase of the double pulse. A step phase function produces a pair of pulses, where the intensity relation between them depends on the si ze of the step. Modulation with a 0.6 step phase creates a double pulse in which the amplitude of the firs t pulse is smaller than the second. Modulation with 1.4 step phase creates the mirror image in time. A similar observation is made on the optimal pulse. The optimum temporal intensity presents a short time component followed by a long time component. The optimal pulse can be explained in terms of two separated and controllable processes occurring du ring the energy transfer: first the population transfer and then control over the excited st ate dynamics. Population transfer is directly controlled with a short early pulse. Control over the molecular energy transfer (represented by the temporal intensitys long time component) strictly depends on the molecu lar properties of the sy stem, e.g. vibrational relaxations channels, coherent states, etc. The single knob experiment clearly demonstrates that the time profile of the laser pulse is the one producing the effect. In a ddition, the double pulse experiments show that not only the time ordering of the pulse is importa nt but also the phase relationship between the pulses. We conclude th at both temporal and phase features of the electric field are the actual knob of the control mechanism. PAGE 176 176 Multipulse structure has been previously observed in other control experiments.51, 82, 85 Many of them have presented strong eviden ce of a control mechanism involving molecular coherences, e.g. vibrational mode selection.50 In our case the dynamics is finished in a couple of hundred femtoseconds leaving the possibility of producing control on the coherences lasting hundreds of femtoseconds. Last but not least, in our c ontrol experiment we rule out the possibility of producing control based on matching the second order power spectrum of the excitation source with the potential energy surface. To our knowledge this is the first experiment which uses the different pulse modulation to rule out this type of control mechanism. Summary We successfully control the energy transfer process of a 2G2mPer in solution. Using closed loop experiments we found that the optimal control mechanism is based on two different components: population transfer control and energy transfer control. Th ese components represent different processes, but they do not have i ndependent evolutions during the optimization, indicating a dependency between them. Moreover, we show that the features of the phase of the pulse leading to maximization of the energy transf er can be extracted and used to simplify the optimal control experiment into an open loop single variable expe riment. The results of this open loop experiment show that the system has coherences which are controlled with specific tailored pulses. The single variable experi ment presents strong indications of coherences present in the donor moiety only. Although the optimal control pulse has been successfully simplified with phase step, a closed inspection of the optimal electric fiel d phase shows two jumps in the phase profile suggesting that an experiment in which the pulse is codified with a double step phase function should produce a higher degree of quantum control of the energy transfer. Hence, a future PAGE 177 177 experiment using a phase function with a double st ep (centered at different wavelengths) could verify the degree of control achieved with the optimal solution in the closed loop optimization experiment. PAGE 178 178 CHAPTER 7 STATISTICAL ANALYSIS OF THE RESU LT S OF THE QUANTUM CONTROL OF ENERGY TRANSFER EXPERIMENTS Introduction Adaptive quantum control, introduced by Judson and Rabitz,26 has been proposed as an effective methodology to control a large number of photoprocesses withou t previous knowledge of the quantum properties of the system. Since its introduction in 1992, many molecular systems have been quantum controlled using this sche me. These breakthroughs range from selective bond cleavage of small molecules in gas phase to en ergy transfer control in complex biological systems in liquid phase.51, 84, 159 Nevertheless, only a few of th ose experimental demonstrations led to results explained in terms of the process under investigation.51, 160 The intricate relationship between the pulse parameterizati on variables and the molecular response, the arbitrary and random manipulation of the pulse parameterization du ring the optimization, and the large number of pulse parameteriza tion variables available make the interpretation of the results difficult and challenge the application of simplifie d schemes, e.g. Taylor phase parameterization. To infer from experimental data the molecu lar properties involved in the optimization, researchers must overcome the inhe rent heuristic character of the optimal control technique. For instance, optimization results contain not only red undancies, e.g. same pulses tested more than once in the closed loop optimization, but al so phase parameters that do not contribute significantly to the optimal outcome. The questio n on how to gather molecular properties from the closed loop optimization data must be acknowle dged, if optimal control is to be implemented to investigate properties of quantum systems. Statistical analysis has been successfully used in many different areas to reduce the dimensionality, to remove unnecessary data and/or noise, and to extract important experimental parameters. Considering the similarities of problems solved by statistical analysis, these PAGE 179 179 exploratory approaches are logi cal methodologies to ex tract the molecular quantum information encoded in the optimal control data. Finding ultr afast pulse parameterization variables directly related to the molecular res ponse opens the possibility of determining unknown molecular characteristics from the quant um control optimization. A couple of examples of these methodologies have been recently implemented in the optimal control field.8890 Using principal component analysis, Bucksbaum and coworkers extracted and explained what components of the phase parameterization are important in controlling stimulated Raman sca ttering in liquid methanol. The authors noted the capabilities of this exploratory analysis to remove extrane ous phase features out of the optimal phase parameterization.90 However, principal components analys is does not link the characteristic features of the experiment with the induced molecular response, e.g. Raman scattering. Damrauer suggested the additional use of a partial least squares methodology to uncover a model that explains the optimization response in terms of the optimal el ectric field variables.88, 89 In their work, they used this statistical met hodology to model the phase variable space in terms of a set of simple collective variables,88 and to reduce a 208 variable optimal control problem into a single control knob experiment.89 Analysis of statistical methodologies and their application to the ener gy transfer efficiency control on a dendritic macromolecule is presented in this chapter. We focus on the understanding of the genetic algorithm evoluti on during the optimization and on th e reduction of the number of laser field variables needed to produce molecular control. In the first part, we present a variable sp ace analysis in conjunction with statistical correlation and multivariate analysis to model the evolution of the genetic algorithm during the optimization. The correlation analysis is used to uncover a characteristic pathway in optimal PAGE 180 180 control experiments and it is followed by partial least squares analysis to extract the principal laser field characteristics affecting the 2G2mPer fluorescence efficiency. The partial least squares method aims at reducing the optim ization variable space dimensionality. Several questions regarding the reduction of the space dimensionality have not yet been addressed and they can help us gain insight into these statisti cal methodologies. For example, is it necessary to use all the tested phase parameterizations to extract global variables from the experimental data? Do the answers change when different multivariate statistical techniques are used to reduce the complexity of the problem? In the case of phase parameterization, is it the same to analyze the phase in the 0 to 2 space (phase wrapped) as it is without constraints (phase unwrapped)? Experimental This study aims to model the data produced in the optimization of the energy transfer efficiency of 2G2mPer. This molecule harvests light with an extremely efficient energy transfer process (quantum yield >0.98).151, 152 Optimal control experiments were set to optimize the ratio of molecular emission over twophoton induced cu rrent in a semiconductor diode for different spectrally phase modulated pulses with a home written genetic algorithm. The setup used in these experiments is describe d in detail in previous chapters. In brief, ultrafast laser pulses centered at 800 nm and modulated usi ng a 640 pixels spatial light modulator are used to control the efficiency of the energy transf er process in 2G2mPer. Each pulse shape is codified with 128 phase parameters which cover the whole spatial light modulator array. The modulated excitation beam is split by a small amount (>10 %) is used to produce the twophoton induced current on a GaAsP photodiode. Th e rest is used to excite the dendritic sample in solution at room temperature. Th e integrated emission is recorded with a PAGE 181 181 photomultiplier tube. The second harmonic spectrum and autocorrelation of the modulated pulses are recorded to obtain the ultrafast pulse characteristics related to the twophoton process. While the second harmonic spectrum is measured with a spectrometer by gene rating second harmonic in a BBO 100 m crystal, the autocorrelation of the m odulated pulse is determined with a home made multishot SHGFROG. As previously discussed (Chapt er 6), the control of energy tr ansfer in this macromolecule is produced via modulated electric fields w ith phase steps at 805 and 810 nm which correspond to pulses with a short and a long temporal component. Partial Least Squares Regression Closed loop optimization experiments manipulate enormous femtosecond pulse parameterization spaces to induce different mol ecular photoresponses. Partial least squares can link the effective changes in the laser parameterization with the sy stem photoresponse. In partial least squares, the nexus is expressed on a set of la tent variables, i.e. gl obal variables. Latent variables represent the actual c oordinates where the optimization algorithm is evolving because they are selected such that they provide the maximum correlati on with the molecular response. Latent variables are abstracted entities that acquire physical meaning only when they are projected back to their original space. However, the features observed in these global variables do not depend on the projection coefficients. So th ey are useful to correlate the electric field modulation with the molecu lar property unde r control. Optimal Control Theory establishes that the set of independent laser field variables which optimizes a given molecular or laser field property is the optimal solution.90 This optimal solution contains the effective control Hamiltoni an, which has encoded the molecular statistical and/or dynamical proper ties of the system.90 Partial least squares can extract these molecular PAGE 182 182 components from the molecular response produc ed by the system. This methodology not only reduces the variable dimensionality but also removes any data redundancy by selecting the minimum number of latent variables that explains most of the phase parameterization and fitness evolution. The closed loop experiment data can be modeled into independent (X) and dependent (Y) variables Y=A+BX, (71) where A and B represent the coefficients of the linear model. Bold letters express the multidimensional nature of the variables. Implem enting this representation to optimal control experiments, the variables in Equation (71) become ikx X (72) and imyf Y, (73) where f represents the fitness and the phase parameterization. In phase modulated experiments, the phase is a matrix with [generations, phase pa rameters] dimensions and the fitness is a vector of [generations] dimension. The linear model attained with partial least squares is ikkmim k f ba =B+A=, (74) where B contains the model regression coefficients, and A contains the residuals of the linear model. In this expression (Equation 74), the index k represents the number of phase components which are used to produce the line ar regression of the fitness. The phase variables,ik, can be expressed as a linear co mbinations of global variables iaikka ktw=, (75) PAGE 183 183 where wka are the weight of each phase component ik in the global variable, tia. The aim of this linear combination (Equation 75) is to model simultaneously the phase parameterization and fitness; hence the new globa l vectors have to be good estimates of the phase parameterization and fitness ikiaakik atpe = (76) and imiaamim a f tcg =, (77) where pak and cam are coefficients of the linear decomposition and eik and gim are the residuals of the decomposition. Up to this point partial least squares and pr incipal components analysis are the same. The difference between the methodologies arises from the way that the weight vectors ( wka) are found. In principal component an alysis, the weight vectors ( wka) are selected to maximize only the covariance of the phase matrix. In contrast partial least squares calculates the weight coefficients ( wka and cam) such that the covariance of the pha se and the fitness are simultaneously maximized. As a consequence, the new basis set models both the phase va riables and the fitness evolutions. It is important to note that this statistical an alysis can be extended to any problem in which a set of independent and dependent variables are to be linearly modeled. A more complex model can be obtained by purposly including nonlinear term s in the dependent variable rather than by assuming a simple linear model.161 PAGE 184 184 Genetic Algorithm Evolution Analysis Statistical Analysis Maximization of the ratio of fluorescence over second harmonic and maximization of fluorescence directly are analyzed to find out which variables are co ntrolling the energy transfer process. These experiments typically describe the optimization of the combined excitationdynamicsrelaxation on 2G2mPer and the optimization of the excitation process alone. Figure 71 shows all individuals (light and dark grey sq uares) and the individual s with the best fitness (red and green lines) for these experiments in tw o different spaces: fluorescence versus twophoton induced current (Figure 71(A)), and ratio of fluorescence over twophoton induced current versus twophoton induced current (Figure 71(B)). 0.00.51.01.52.02.53.03.54.04.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Fluorescence (arb. units)Two photon induced current (arb. units)A 0.00.51.01.52.02.53.03.54.04.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Ratio of fluorescence/two photon signal (arb. units)Two photon induced current (arb. units)B Figure 71. Variable space evolution and sa mpling for 2G2mPer energy transfer optimal control. A) Typical represen tation of the molecular signa l versus twophoton signal. B) Ratio versus twophoton induced current for each experiment. Red line (grey dots) and green line (dark grey dot s) lines correspond to the best (all) individual for each generation of the ratio of fluorescence ove r twophoton signal and the fluorescence, respectively. The usual representation of the fluorescen ce versus twophoton induced current space (Figure 71(A)) does not provide enough information of the pathway followed by the genetic algorithm to produce the desired objective. Altho ugh the slopes of both experiments are different PAGE 185 185 (see Chapter 5) the change is not that pronounced and the evolution is not clear. The difference in the pathways followed during each optimizat ion is better described when the space is represented in terms of the ra tio of fluorescence over twophot on induced current versus twophoton induced current. In this new representa tion, the fluorescence maximization experiment (when the twophoton signal is measured but no t included in the cost function) evolves by essentially keeping the ratio constant. Although the intensity of the excitation source keeps increasing, the ratio remains constant because the gain in the fluorescence is solely due to the increase in of the number of excited molecules without controlling the excited state dynamics. In contrast, the maximization of en ergy transfer efficiency achiev es its objective by modifying mainly the ratio and only slightly increasing the twophoton induced curr ent. During the first fifteen generations both optimizations share a co mmon pathway although after that they diverge. This behavior suggests that the optimizations ma ke similar changes to th e initial random pulse phase to achieve their respective objectives. After the separation of the pathways, each optimization modifies distinctively the phase parameterization to proceed towards its own optimum region on the surface. The final attained position in this space is unique for each optimization, corroborating the idea that fluores cence and fluorescence over twophoton induced current maximization aim to control ve ry different molecular processes. Analysis of the correlation between fluorescence and twopho ton induced current for all the individual pulses within one generation gives signals which remain highly correlated for all generations (Figure 72, full squares). Similar results are observed whether optimizing fluorescence (black squares) alone or fluorescen ce over twophoton signal (red squares). This strong correlation agrees with the second order intensity de pendence of both signals. A similar correlation study using the ratio and the twopho ton induced current exhibits a completely PAGE 186 186 different behavior (Figure 72, open squares). Initially, th e optimizations have similar correlations, but as the generation number increas es, the correlation between the energy transfer efficiency experiments remains constant (Figure 72, red open squares), while the correlation in the fluorescence maximization spreads in the range of 0.6 to 0.4 and in the last 10 generations has negative values (Figure 72, open black squa res). Again, this agrees with the different pathways followed by each population in the different experiments. 0 50 100 150 200 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Correlation numberGeneration Figure 72. Generational correlation between fluorescence and twophoton induced current (full squares) and between the ratio and twophoton induced current (open squares) for energy transfer efficiency maximization (re d) and fluorescence maximization (black). The pathway for the optimization of the energy transfer efficiency has a clear evolution in the 50 first generation, and after that we do not observe major changes in this space, indicating the presence of a local optimum (F igure 66 and 71). This trend is also observed in the electric field characteristics, i.e. second harmonic spectrum and temporal pulse autocorrelation presented in Figure 73. Spectra of the second harmonic produced by the excitation sour ce and autocorrelations of the fundamental pulses suffer their greatest chan ges during the first 20 generations. After the 20th generation, neither their second harmonic spectr um nor their corresponding autocorrelations show substantial variatio ns of their profiles. The second ha rmonic spectrum remains center at PAGE 187 187 ~403 nm during most of the optimization steps. Th is does not directly correspond with the fitness evolution for this experiment (Figure 72) wher e the fitness suffers the major changes (~40%) in the first 50 generations and later only sm all adjustments are produced. In the 20th generation, the autocorrelation already has the main feature found in the last generation, but the fitness values still show a significant change, going from ~1.03 in the 20th generation to ~1.15 in the 200th generation. 395400405410415 G100 G200 G50 G20 G10 TL G1 395400405410415 0 1 Normalized second harmonic spectrumWavelength (nm)A 3.01.50.01.53.0 G100 G200 G50 G20 G10 TL G1 Normalized Autocorrelation 3.01.50.01.53.0 0 1 Time (ps)B 0100200300400 G100 G200 G50 G20 G10 TL G1 G100 G200 G50 G20 G10 TL G1 Real part of Fourier transform 0100200300400 0.00 0.01 Frequency (cm1)C Figure 73. Pulse characteristics for the fluores cence for generation 1, 10, 20, 50, 100, and 200. A) Second harmonic spectra. B) Autocorrela tions. C) Fourier transforms of the autocorrelations. In contrast to the autocorrelation of the fundamental pulse and the second harmonic spectra, the evolution of the Fourier transform of the autocorrelat ion (Figure 73(C)) exhibits an initial recovery of the missing frequencies and th en a narrowing in the frequency spectrum with some of 300 cm1 contributions disappearing after 100 genera tions. This change indicates that the fine structure of the autocorrelation is still ch anging in the generations where the autocorrelation and second harmonic are not signifi cantly varied, i.e. from the 50th generation and above. A PAGE 188 188 correlation between the fitness of the optimization and the real amplitude of the frequency components (Figure 74) reveals th ree main regions of the freque ncy domain correlated with the evolution of the fitness. The fre quency components between 40 to 130 cm1 and between 190 to 250 cm1 are positively correlated with the fitness func tion. There is also a negatively correlated region with frequencies between 0 to 20 cm1. 0100200300400500600700800900 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Correlation numberFrequency (cm1) Figure 74. Correlation between Fourier frequency amplitude and fitness. Three regions with correlation are observed: between 020 cm1 (negative), 40 to 130 cm1(positive), and 190 to 250 cm1(positive). Partial least squares analysis of the evolution of the sp ectrum of the second harmonic displays one latent variable explaining almo st 100 % of the spectrum and fitness variances (Appendix D). This implies that the optimization can be described within a very good degree of approximation as a system evolving through only one dimension. However, the predicted linear model of the fitness with only this latent variable is very poor (R2= 0.1), and it can be greatly improved with the addition of a second latent variable (R2=0.8). While the first latent variable is a featureless broadband second harmonic spectrum very similar to the transform limited pulse spectrum, the second latent variable is a non symm etric broad spectrum with a 5 nm dip in the center (see Appendix D). A similar result is ob tained when the autocorrelation evolution is statistically modeled with partial least squares. PAGE 189 189 Partial least squares analysis of the phase evolution displays once more an explained variance of almost 100 % using only the first gl obal variable. Figure 75 shows the first and second latent variables. 0.2 0.1 0.0 0.1 0.2 760770780790800810820830840 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Wavelength (nm) Phase (a.u.)A 0.2 0.1 0.0 0.1 0.2 0.3 760770780790800810820830840 0.0 0.2 0.4 0.6 0.8 1.0 Intensity (a.u.)Wavelength (nm) Phase (a.u.)B Figure 75. A) Latent variable 1 (black line). B) Latent variable 2(black line). Spectrum of the excitation (grey area) pulse. The first phase latent variable shows two important phase features at 805 and 815 nm, while the second phase latent variable has a sp ectral phase with no re cognizable features are observed. Although both components contribute significantly to the partial least squares model of the fitness, the first phase latent variable represents the process with the greatest influence over the fitness. The second phase latent variable is needed to produce the in itial changes observed in the fitness. The re gression model found by partial leas t squares for the phase agrees with the models that are found with the same methodology to describe second harmonic spectrum and temporal autocorrela tion evolutions. Moreover, the firs t latent variable agrees with the spectral phase retrieved from the experimental optimal pulse (see Chapter 6), corroborating that these pulse features are respons ible for achieving quantum control. Discussion All the statistical analysis results agree with the experimental obser vations presented in Chapter 6. For instance, it has been demonstrated that the process is phas e dependent and that the PAGE 190 190 second order spectrum of the excitation does not play a major role in the energy transfer efficiency optimization. This is also observe d in the evolution of the spectrum of second harmonic of the best pulse, sin ce no shrinking or shifting of the spectrum of the second harmonic is observed during the optimization. In additi on, the Fourier transform analysis of the autocorrelation reveals subtle changes in the te mporal profiles of th e autocorrelation. These changes observed over the autoco rrelation are the actual control features that the genetic algorithm is optimizing, confirming that the achieve control mechanism is not based on modulating the second order power spectrum of the excitation source. While the change in the frequency components of the autocorrelation ar e well correlated to the changes in the experimental fitness, the initial randomness of the optimization does not allow us to confirm whether those frequency componen ts must be optimized to achieve control or whether they are optimized because the initial random population does not have them. In other words, is the critical step just moving a random phase to a well defined smooth function? Would we obtain the same frequency components if we were starting from a nonrandom pulse (e.g. transform limited pulse)? These questions ne ed to be addressed in future work. The trends of the best individual spectrum of second harmonic, applied phase, and autocorrelation statistically modeled with partial least squares have to in clude a second latent variable to produce a correct partial least squares model of the fitness. Since partial least squares produces local models,89 the second latent variable is always necessary to expl ain the outliers, i.e. points far away from the center of the hyperplane defined by the global variables (see Appendix C). In our experiments, these outliers re present the initially random pulse created in the optimization. This implies that the system must be optimizing two di fferent processes that they are not strictly correlated with the process investigated. For instance, in the evolution of the PAGE 191 191 spectrum of the second harmonic, the system fini shes with a second harmonic spectrum with a bandwidth close to that of a transform limited pulse, but because of th e genetic algorithm is initiated with a random set of pulses, the optimi zation must first compress the pulse to produce a second harmonic spectrum close to one of a tran sform limited pulse and then the modifications necessaries to achieve control. This evolution model strongly agrees with the pathway observed in both optimizations, where the optimizations in itially share a common pathway in the variable space and then they separate to accomplish their own objective. Summary We have applied statistics tool to evaluate the results obs erved in the experiments of quantum control of the energy transfer. These results not only provided new insight into the process (e.g. Fourier components of the autocorrelati on) but also confirmed the conclusions obtained from the experimental data. This indicates that statistical analysis is a great tool to be used in the analysis of data pr oduced by closed loop optimizations. Variable Space Reduction To further understand how to utilize statistical data reduction analysis, this section aims to investigate how different effects (e.g. phase unw rapping) affect the predicted model when the multivariate analysis is used. The goal of this se ction is to use multivariate analysis to reproduce the experimental results of the energy transfer optimal control experiments. Variable Space Size Effect In a recent work by Damrauer and coworkers, all the phase parameteri zations tested during a closed loop experiment were used to infer a set of global variables th at control the twophoton induced emission of a ruthenium complex in solution.88, 89 The question that arises from this study is whether the total number of tested phase parameterizations is truly necessary to extract the global variables affecting the experiment. To investigate this effect we apply partial least PAGE 192 192 squares to all the phase encodings (all individuals in all generations) as well as to only the phase parameterization of best individuals of each generation tested during the experiment of maximization of energy transfer efficiency presente d in Chapter 6. The effect of the sampling is inferred by comparing the global variables obtained with each different set of parameterization. The results of this analysis are presented in Figure 76. 0.0 0.1 0.2 Phase (a.u.)020406080100120 0.01 0.00 0.01 ResidualPhase ParameterA 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Phase (a.u.)020406080100120 0.05 0.00 0.05 ResidualPhase ParameterB 0.5 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Phase (a.u.)020406080100120 0.05 0.00 0.05 ResidualPhase ParameterC 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 Phase (a.u.)020406080100120 0.1 0.0 0.1 ResidualPhase ParameterD Figure 76. Latent variables for the best individu als of each of the 200 gene rations (red line) and total population (black line and squares). A) La tent variable 1. B) Latent variable 2. C) Latent variable 3. D) Latent variable 4. Figure 76 presents the first four latent variables. The latent variables obtained with both analyses have the same features, suggesting that th e statistical analysis can be performed directly on the best pulses parameterization of each optimization. PAGE 193 193 Statistical Meth odology Effect The next investigation involves the study of the different statis tical techniques to model the data. As stated before, partial least squares differs from princi pal components analysis because it includes the correlation with the molecular response. It is exp ected that principal component analysis will be outperformed by partial l east squares methodology in data modeling. To investigate whether those two statistical methodologies produce different factorizations, a model produced with only one global component is compared for the two multivariate analysis techniques (partial least squares and principal components). In this test the wrapped phases of the best individuals in each generation are analyzed. The results of this analysis are presented in Figure 78 (A). 020406080100120 0 1 2 3 4 5 6 7 Phase (rad)Phase parameterA 020406080100120 0 1 2 3 4 5 6 7 Phase (rad)Phase parameterB Figure 78. Comparison of the model predicted by different statistical methodologies and the optimal phase. A) Energy transfer optimiza tion. B) Fluorescence maximization. Each graphs shows: original data (black line and squares), the partial least squares model (green line), and the principal component analysis model (red line). Principal components as well as partial least squares model accurately the optimal phase. Surprisingly, the root mean square error is hi gher for partial least squares than for principal components analysis (Table 71). PAGE 194 194 Table 71. Root mean square for th e different statis tical methodologies. Statistic methodology Root mean square error Principal components analysis 4.104 Partial least squares 0.8 To discard a possible effect over the statistical factorization produced by the large data redundancy observed in experiment s with the rapid convergence, such as maximization of the energy transfer efficiency, a similar analysis is performed with the data of the maximization of fluorescence alone. In this particular case the fitne ss changes significantly in at least 130 of the 200 generations. Analysis of this data set displays a similar result, thus discarding an effect due to any rapid convergence which could favor the lo calization of the partia l least squares model. Since the two statistical analys es perfectly match the experimental data, principal component analysis as well as partial least squares can be used to model the data in terms of global variables. To conclude which one produces better phase predictors, it is necessary to perform an experiment where the experimental value of th e fitness corresponding to the predicted phase for both methodologies are measured and compared. Theoretical Phase Unwrapping Effect The last analysis involves the effect of th e unwrapping of the phase before performing the statistical analysis. The closed loop optimization with spatial light modulators explores the effect produced by phases with values in the range from 0 to 2 (see Chapter 3). Unwrapping the phase (so it goes from 0 to n ) produced in the optimization e xperiments causes many problems associated with the statistical factorization. For example if a pixel with negligible contribution to the fitness suddenly produces a jump in the phase bigger than the unwrapping process will take it into consideration and will increase the variability of the phase parameterization even though this points does not contribu te to the fitness. To correc tly analyze the optimal phase obtained by the algorithm we need to consider only those parameters in which the spectral PAGE 195 195 intensity is appreciable, otherwise the phase unw rapping cannot be applied before performing the statistical studies. Most times it is difficult to determine which pixels have sufficient intensity to contribute to the optimization or whether the pixels with low inte nsities have any effect on the fitness or not. We use all the phase variables (128 for the en ergy transfer experime nt) to show how the effect of the unwrapping over the phase parameterizat ion results when the statistical analysis is performed. The model of the phase using only on e global variable is compared to the actual phase of the 200th generation. 0 1 2 3 4 5 6 Phase (radians)020406080100120 2.0 1.0 0.0 1.0 2.0 Residual (rad)Phase ParameterA 0 1 2 3 4 5 6 Phase (rad)020406080100120 2.0 1.0 0.0 1.0 2.0 Residual (rad)Phase ParameterB Figure 79. Comparison between the model of the optimal phase (red line) and the experimental optimal phase (black line and points). A) Without phase unwrapping. B) With phase unwrapping. Figure 79(A) shows that the ph ase is well reproduced with only one global variable when the phase is not unwrapped. In c ontrast, when the phase is unwr apped before the analysis, the model presents very different features compared to the actual phase (Figure 79(B)). The phase unwrapping clearly induces more variability in the sample. A similar behavior is obtained for the phase analysis of the unwrapped and wrapped pha ses even when only the phase variables with more than 5 % of the intensity at the maximu m are considered (Figure 710). Hence, phase PAGE 196 196 unwrapping introduces an unnecessary variability to the phase whic h affects the performance of the statistical analysis. 405060708090100 0 1 2 3 4 5 6 Phase (rad)Phase parameter Figure 710. Comparison among modeling of the optimal phase without unwrapping (red line) and with unwrapping (blue line), and the experimental optimal phase (black line). Experimental Phase Unwrapping Effect We present here an experimental implementa tion to show the effect of the number of components used to reconstruct the data with or without phase unwrapping. First we run an optimization and then we use principal component analysis to extract the principal components of the wrapped or unwrapped experimental phase of th e best individuals of each generation. Choosing the first, the second, or both principal components that explained the phase, we reconstruct the experimental phase da ta. We apply each of the reconstructed optimal phases (phase of the 200th generation) to the phase modulator. The signals obtained by the real optimal phase are compared with those obt ained by using one or two components. Table 72 and Table 73 show raw results and results normalized to the transform limited pulse, respectively. PAGE 197 197 The fitness is perfectly reproduced with m odels containing one or two components when the phase is wrapped. On the contrary, the f itness is poorly predicted when the phase is unwrapped, which agrees with the theoretica l unwrapping effect previously presented. Table 72. Experimental signal and fitn ess values for multiple components. Pulse Signal SHG SHG FL FL FL/SHG FL/SHG Transform limited 3.56 0.06 4.40 0.11 1.24 0.04 Optimal 1.00 0.03 1.41 0.05 1.41 0.07 PC1 unwrapped 0.98 0.03 1.31 0.05 1.34 0.06 PC2 unwrapped 0.57 0.02 0.68 0.04 1.20 0.07 PC1 and PC2 unwrapped 0.47 0.01 0.54 0.03 1.15 0.07 PC1 wrapped 1.12 0.03 1.60 0.06 1.44 0.06 PC2 wrapped 1.10 0.02 1.53 0.05 1.40 0.06 PC1 and PC2 wrapped 0.99 0.02 1.38 0.05 1.39 0.06 Table 73. Experimental normalized fitness values for multiple components. Pulse Response normalized with respect to transform limited pulse FL/SHG Transform limited 1.00 Optimal 1.14 PC1 unwrapped 1.08 PC2 unwrapped 0.96 PC1 and PC2 unwrapped 0.92 PC1 wrapped 1.16 PC2 wrapped 1.12 PC1 and PC2 wrapped 1.12 Discussion In this section, we demonstrated that the dime nsionality reduction is not affected either by considering only the best individuals of the population instead of the total population or by the selected multivariate analysis chosen to reconstruct the optimal phase. Since both methodologies are based on correlating the change on the covari ance matrix, it is not expected that principal components will produce different results than par tial least squares. The reason for this behavior is the heuristic algorithm used for the optimiza tion. Genetic algorithms are based on inheritance and natural selection. Extrapolated to the optimal control experiments, it means that the algorithm is indeed evolving through a space in which the best variables are kept until new and PAGE 198 198 improved variables are found. It is not surprising that the fitness of th e best individual has a monotonous improvement in the spa ce of variables, ultimately causing a matching between the phase optimization directi on and the fitness change. Finally, the phase unwrapping effect is demonstrat ed to have detrimental characteristics if applied before performing the statistical analys is. It is known that pha se wrapping produces errors in the ultrafast generated laser field, wh ich can be exploited by th e genetic algorithm to optimize the photoprocess.55, 162 We conclude that phase unwrapping should not have to be performed before the statistical analysis. Summary We have demonstrated the utility of multivar iate data analysis to extract the global components affecting the evolution of the phase variables during the optim ization. Surprisingly, neither the selected methodology nor the number of individuals used in the analysis affect significantly the result of the global component analysis. Also, we find that the phase unwrapping process not only is unnece ssary but also complicates the statistical factorization of the data. Finally, a very challenging and yet very interesting experiment should be an optimization in which after a certain number of generations without reaching convergence the global variables obtained from a gl obal analysis of the phases are us ed as a basis set to continue the optimization. PAGE 199 199 CHAPTER 8 AZOBENZENE EXCITATED ST ATE DYNAMICS WITH TWOPHOTON EXCITATION Introduction Azobenzene and its derivatives are photochromic molecules exhibiting a trans cis photoisomerization. Their photochromism makes azobenzenelike compounds very attractive to investigate fundamental photochemistry processes a nd to industrial and sc ientific applications. Some of the proposed applicati ons include lighttriggered opti cal switches and optical datastorage media.163 Recently, an azobenzene derivative has been successfully used as an ultrafast molecular trigger in protein folding.164 To further develop the applications of azobenzene, its molecular rearrangement pro cess needs to be understood. Azobenzene dynamics have been extensively investigated using many different time resolved as well as steady state spectroscopic techniques.165172 However, the mechanism and dynamics of the molecular rearrangement are stil l controversial. The most debated question in the azobenzene rearrangement is whether the isom erization takes places through an inversion or a rotation pathway.173, 174 New experimental approaches ar e necessary to investigate this molecule and solve the longstanding controversy. Several experiments have probed the possibili ty of coherently controlling isomerization processes.82, 8486 While some of these studies were focused on proof of principle,82, 85 others used the coherent control methodology to ga in insights on the molecular process.84, 86 For instance, using an unmodulated pump and a modulated du mp pulse Gerber and coworkers studied the dynamics of bacteriorhodopsin isomerization in th e excited state potential energy surface close to the conical intersection.86 Another optimal control study focused on the isomerization of cyanine dye has uncovered the molecular motion produced in the excited state, and consequently the understanding of the cyanin e isomerization process.84 The maturity of the coherent control field PAGE 200 200 has reached a point in which we can envision using modulated pulses to further understand molecular processes in which mol ecular rearregement s are involved. The following chapter describes the prelimin ary experiments performed to seek the feasibility of quantum control of the isomerizatio n of azobenzene. It focuses on establishing if a twophoton excitation of azobenzene is possible and on studying possible changes in the excited state dynamics caused by multiphoton excitation. The ultimate goal of this study is to set the necessary foundations of a twophoton exci tation for coherently controlling the photoisomerization of azobenzene with a modulated 800 nm excitation source. Azobenzene Azobenzene has two benzene rings linked via two double bonded nitrogen atoms, the azo group. The molecule presents two possibl e conformations: the thermally stable trans form and the metastable cis form (Figure 81(A)).175 N N N N h h trans azobenzene cis azobenzeneA 200250300350400450500550600650700750800 0.0 0.2 0.4 0.6 0.8 1.0 400 500 0.00 0.05 0.10 Absorption (OD)Wavelength (nm) B Figure 81. A) Isomerization of azobenzene. B) Ground state absorption of azobenzene samples: trans (black line) and cis (red line). Since the azo group presents a l one pair of electrons on the nitrogen atoms, the molecule exhibits two low lying electronic transitions, n and *, found in the UVvisible spectral region. In the trans isomer the two transitions are located at 440 nm (n *) and 330 nm ( *), PAGE 201 201 while for the cis isomer the n transition is observed at the same wavelength and the transition is shifted to the blue (Figure 81(B)). The cause of this spectral shift is the nonplanar structure of the cis isomer which decreases por bital overlap responsible for conjugation. The transition is one photon allowed in both isomers ( cis~104, trans ~3x104 M1cm1) whereas the n transition is one photon forbidden for the trans isomer ( trans~400 M1cm1) and allowed for the cis isomer ( cis~1500 M1cm1).166 Due to the similarities with stilbene two possible mechanisms of isomerization have been proposed for azobenzene: one is based on the inversion around one nitrogen atom while maintaining the rings in the same molecular plan e; the other involves a ro tation of a benzene ring around the nitrogen doub le bond (Figure 82).175 N N N N N N N N Inversion Rotation trans Azobenzene cis Azobenzene Figure 82. Rotation and inversi on pathways of azobenzene isomer ization. Adapted from work by Diau.176 Early steady state measurements of the trans to cis isomerization quantum yield showed an excitation wavelength dependence in which the re arrangement quantum yields are 0.23 and 0.12 for the n and transitions, respectively.172 The wavelength dependence of the quantum yield of the isomerization indicates diverse isomerization mechanisms and/or deactivation pathways taking place on differe nt excited states. Rau and cowo rkers performed a quantum yield PAGE 202 202 study on a rotationally blocked azobenzene. In those experiments the isomerization quantum yields did not vary significantly ( isom(n *) is 0.24 and isom( *) is 0.21).173 Since the bridge blocking the rotation pathways di d not produce significant changes in the electronic structure of the azo derivate, the authors pr oposed a mechanism in which the trans to cis isomerization proceeds through inversion and the rotation coordi nate serves for the excited state relaxation without isomerization. Recently, several timeresolved experiments ha ve been carried out to study the dynamics of this photoisomerization. These experiments focused on the trans to cis isomerization dynamics exciting the molecule in either the n or transitions. Ultrafast studies on the deactivation dynamics after an n excitation showed biexponential decay kinetics in which the time constants were solvent dependent.171 All studies agree on biexponential dynamics with time scales of the subpicoseconds (~0.3 ps in et hanol) and picoseconds (~2.1 ps in ethanol).166 The ultrafast dynamics component is assigned to the initial evoluti on of excited state out of the FranckCondon region and the picos econd component is due to a diffusive motion of the excited state wave function towards the conical intersecti on where the system will transfer back to the electronic ground state.171 In addition, a slower process with a time scale of 5 to 20 ps was observed in some studies177 and this has been assigned to the vibrational cooling of vibrationally hot ground state molecules.171 The dynamics following the excitation have also been investigated.171The general picture of the dynamics includes four different relaxation constants. The first time constant represents the radiationless relaxation process from the second excited state, S2, to the first, S1, in a subpicosecond time scale ( ~120 fs).171 Due to the similar time values ( 2 ~0.5 fs, 3~3 ps and 4~20 ps) all the other constants have been assign ed to processes occurring in the first excited PAGE 203 203 and ground states. The amplitudes of these kinetic steps significantly differ with those observed after the n transition.171 These differences led researches to conclude that th e deactivation of the S2 state involves a popul ation transfer to S1 where the isomerization takes places. In addition the difference in quantum yield is explained w ith the difference in the amplitude values of different processes, which is interpreted as di fferent positioning of the excited state wave function upon relaxation to the first excited state.171 All these studies have shown the time cons tants of the photoprocess but they do not discriminate between an isomerization occurri ng through an inversion or a rotation mechanism. Two experimental studies supporting an inversion pathway are time resolved picosecond Raman168 and steady state spectroscopy usi ng rotationally blocked azobenzenes.173 The first study shows that the NN double bond characte r is maintained during the photoprocess.168 The second study demonstrates the lack of wavele ngth dependence of the isomerization quantum yield when a rotational block azobenzene is excited using ei ther transition, n or *.173 With these experimental precedents, most of the experimental literature considered the isomerization to occur via the inversion path way. However, the isomerization through the rotational mechanism has been gaining support from nume rous theoretical studies.176, 178181 In these studies, the inversion coordina te is challenged as the isomerization coordinate because it is not predicted to be energetically barrierless.180 In addition, the barrierles s rotational coordinate is inferred to lead the excited state close to a conical intersection between S0 and S1, near to the midpoint of the rotation pathway. From the theore tical point of view, this suggests a propitious isomerization via the rotationa l pathway. A time resolved fluorescence study supports this theoretical prediction for solvents with low viscosity and suggests an inversion mechanism in high viscosity solvents.182 PAGE 204 204 Recently, two experiments add more confusi on to the unsolved isomerization problem. Using sub ten femtosecond transient absorption e xperiments with chirped pulses, Kobayashi and coworkers followed the isomerization of an azobenzene derivative (DMAAB).183 The experiments indicate that the NN and CN st retching modes are coupled through at least one vibrational mode. The authors c oncluded that the isomerization mechanism cannot be described either as a pure rotation or pure inversion, but both happening simultaneously. Stollow and coworkers used time resolved photoelectron sp ectroscopy exciting at 330 nm to study the molecule in the gas phase.184 Two photoelectron bands with distinct lifetimes were identified (170 fs and 420 fs) and assigned to degenerate excited states. Following the mechanism proposed by Tajara,169 Stolow suggested that one of thes e degenerated states leads to the deactivation of the S2 to S1 and the other directly re laxes the system from Sn 1,2 to S0, explaining the difference in the isomerization quantum yield. Experimental Transient absorption experiments were performed using a twophoton excitation. Azobenze sample was purchased from Aldrich Corp. (~99% purity) and used without further purification. Azobenzene solutions with a 5.2 mM concentration are prepared with nhexane (Fisher, HPLC grade). The sample is held in a 2 mm optical path quartz cell with magnetic stirring to ensure fresh sample for each excita tion pulse. Since the twophoton cross section is much smaller than the linear cross section,185 high optical densities (OD=0.45 in 2 mm) at the probe wavelength 440 nm were used. In additi on, the solutions were checked for photoproduct build up before and after the experiment s by measuring their absorption spectrum. The experimental setup utilized in this experiment is a femtosecond time resolved transition absorption setup (Figure 83). PAGE 205 205 Figure 83. Experimental setup for transient absorption experime nt probing with UV. nIR pump pulses are obtained from the residual fundamental of the OPA. A chopper wheel is used to compare the signal with and without pump. The laser source is commercially chirped pulse amplifier system (Spitfire, Spectra physics), which delivers sub 45 fs pulses centered at 800 nm with a repetition rate of 1kHz and a pulse energy of 800 J. Approximately 350 J of this laser source are used to pump an optical parametric amplifier (OPA) which generates the probe wavelengths used in this experiment, 440 nm and 480 nm. A 65 J excitation beam is obtained from the leftover fundamental beam of the OPA. An optical chopper is placed in the ex citation pathway to perform double referenced experiments. Chopper Delay line Pump Probe Reference Sample Spitfire laser Amplifier Photodiode BOXCAR OPA Prism compressor PAGE 206 206 The sample is excited with 800 nm and the tr ansient absorption signals are measured with a single color probe and referen ce in two independent photodiode s (Thorlabs 210). To avoid the detection of scattered light from the sample, color filters (BG39) are used. The photocurrent produced in the detector is further processed with a boxcar gated integrator (SRS 250) and acquired with a data acquisition card (National Instruments, PCIMIO16E4). In the data presented herein every point at a fixed delay time is the average of 16000 individual laser shots. The transient absorpti on signal produced by the pu re solvent is later subtracted from the sample signal. Characterization of the instrument response function is achieved using the Raman signal arising from methanol as well as dichloromethane. Results and Discussion The dynamics of the transient absorption signals of azobenzene following twophoton excitation (220 fs FWHM, 800 nm) and probe at 440 nm and 480 nm are investigated. This excitation frequency does not induced one photon absorption: the excitation occurs via twophoton absorption in the spectra l region equivalent to 25000 cm1 which correspond to the blue side of the n transition of trans azobenzene. 101234 0.002 0.000 0.002 0.004 0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 ResidualsTime (ps)Abs (OD) 0.20.00.20.40.6 0.000 0.002 0.004 0.006 0.008 0.010 0.012 A101234 0.004 0.000 0.004 0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 ResidualsTime (ps) Abs (OD) 0.500.250.000.250.500.75 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 B Figure 84. Transient evolution of the absorption changes for azobenzene where open circles and black line represent the data and fitting m odel and the grey plots show the instrument response function. A) Probe at 440 nm. B) Probe at 480 nm. PAGE 207 207 Transient absorption data of trans azobenzene with the two pr obe wavelengths are shown in Figure 84. The signals depicted in these experiments correspond to the excited state absorption following formation of the excited state, S1. At both probe wavelengths, the initial absorption is followed by a fast decay. Photoinduced excited state absorptions completely vanish after 1 ps. The model used to describe this da ta includes two independent decay times convoluted with the instrument response function. () AIRFft (81) 2expIRF IRFIRFAt (82) expii iftAt (83) where A is the transition absorption signal, IRF is the instrument response function, AIRF and IRF are the amplitude and standard deviati on of the instrument response function, Ai and i are the amplitudes and decay times of the model. The time evolution probed at 440 nm has to be modeled with a biexponential func tion; the decay probed at 480 nm is correctly modeled with a single exponential. The results of the modeled dynamics as well as li terature reported values are presented in Table 81. Table 81. Experimental and literature decay times. Excitation exc (nm) prb (nm) 1 (fs) 2 (ps) IRF Solvent Twophoton excitation* 790(395) 440 150 2.5 220 Hexane Twophoton excitation* 790(395) 480 150 200 Hexane One photon166 435 440750 320 2.1 500 Ethanol One photon170 480 370590 340 3.0 80 DMSO One photon167 420 350550 600 2.6 180 Hexane *from this work Our results show good agreement with the literature values indicating that the dynamics are independent of the type of initial excitation used in the experiment (onephoton or two PAGE 208 208 photons). This result suggests a similar localization of the one and twophoton FranckCondon areas which would result in similar dynamics.171 The lack of a biexponential behavior at 480 nm shows that this wavelength only probes the excited state population close to the initial FranckCondon area. In contrast at 440 nm, the excited state is probed during the whole residenc e time in the excited state, which includes the initial FranckCondon region and the wave packet movement towards the conical intersection. This interpretation agrees with the most accepted kinetic model of transazobenzene. Summary We employ twophoton excitation transient abso rption to investigate the excited state dynamics of azobenzene in hexane. The transi ent signal reveals sim ilarities between the dynamics observed with onephoton and twophoton exci tations. This allows us to infer that the system reaches the same potential energy surf aces as well as similar FranckCondon regions for the two different excitations. This opens the pos sibility of using a tw ophoton excitation source to coherently control the isomerization rearrangement in azobenzene. PAGE 209 209 APPENDIX A PULSE SHAPER ALIGNMENT The quality of the m odulation for this apparatus strongly depends on the quality of the alignment. Because of the reflective mode conf iguration, the alignment procedure is highly simplified (i.e. only one grating angle is necessa ry to be aligned). The experiment has two distinctive parts: the polarization splitter and beam steering, and the compressor. The polarization splitter must be setup in a linear arrangement of the following elements: polarizer, Faraday rotator, and /2 waveplate. After these parts ar e aligned, two mirrors are setup to steer the laser into the compressor. Two iris apertures are set up in between these mirrors for the alignment of the compressor. The compressor alignment is a difficult task b ecause it involves the c onstruction of a zero dispersion compressor with a 4 f configuration. Therefore, a r ecipe for the alignment is given below. Define the Fourier plane (mask positi on) and place a translation stage ( S1 ) away from that position at a distance equal to the cylindrical mirror focal length ( f ). Place an aperture (A1 ) at the end point of this stage. 1. Place the gratings rotation stage ( RS ) close to S1. To meet the 4 f configuration, a second translation stage ( S2) must be set such that the distance from the grating to this stage plus the distance from S2 to the cylindrical mirror is approximately f Place another aperture ( A2 ) on one side of S2. 2. With the help of a HeNe la ser place two more apertures ( A3 and A4 ) in the back of the mask (one close to the back of the mask and th e second one as far away as possible) position and check that the aperture height is the same for a ll of them (beam path parallel to the table). PAGE 210 210 3. Place the steering mirrors to bring the fe mtosecond laser beam into the compressor and align the laser beam through A1 and A2 4. Place the grating in the RS and check that the grating surface and the rulings are perpendicular to the plane of the optical table. This can be achiev ed by setting the height of all the grating diffraction orders to the same height at different distances (parallel to the optical table). Align the angle of the grating with A2 and then remove A2 5. Place the folding mirror in S2 without changing the rotation angle of RS tilt the vertical angle of the grating such that the beam is reflected in this mirror. 6. Place the cylindrical mirror in S1. Use aperture A3 and A4 to align the folding mirror and the cylindrical mirror. To check the position of the cylindrical mirror, place a regular mirror where the back of the spatial light modulat or will be placed, and make the incoming and outgoing beams to be the same size. Having the incoming and outgoing beams collimated and with the same size guarantees that the distan ce between the cylindrical mirror and the masks back mirror is equal to f 7. Block the beam (be careful, it is focusing at the mask position) and place the spatial light modulator in the Fourier plane. Before allowing th e beam in the pulse shaper rotate the grating to the Littrow angle (this angle can be calculated using the grating e quation). Unblock the beam and place the first diffraction order close to the middle of the mask array. Rotate the mask array until the outgoing beam goes through the entrance aperture ( A5 ). 8. With an autocorrelator or other technique (i.e. FROG), slowly move the first stage to minimize the pulse width. If the pulse presents a lot of temporal structure, minimize the pulse width and then rotate the grating. This rotation will minimize the temporal distortion. Repeat this step until the shortest structureless pulse is obtained. PAGE 211 211 9. After this alignment, the pulse must be analyzed spatially to verify that the beam does not have spatial chirp. If spatial chirp is present, rotate the optical axis of the cylindrical mirror until the effect disappears. Figure A1. Scheme of the setup with the apertu res and translation stages. S1 and S2: translation stages; RS: rotation stage; A1, A2, A3, A4, and A5: iris apertures; M1, M2 steering mirrors; FM, folding mirror. S2 S1 A3 G A5 A2 A1 RS A4 PAGE 212 212 APPENDIX B TIME DEPENDENCE PERTURBATION THEORY A quantum system in the absence of any interaction can be described by the timeindependent Hamiltonian 0000kkkHE (B1) where E0 k and 0 k are the eigenvalues (energies) and eigenfunctions (wave functions) of H0. When a timedependent perturbation is applied to the system, the time dependence wave function evolution is expressed by the ti medependent Schrdinger equation H it (B2) Here, H is the total Hamiltonian and contains both the unperturbed and the perturbation operators. Thus, 0 HHH (B3) Assuming no perturbation, the timed ependent Schrdinger equation is 0 H it (B4) 0 kk kc (B5) Since can always be expressed in the complete basis set of 0 k 0 0000kk k kkkkk kkc HcEc it (B6) Projecting this equation in the basis set of 0 k 00 000 kkk k kkkk kc Ec it (B7) PAGE 213 213 0 k k kk kc Ec it (B8) This differential equation can be solved for each coefficient individually 0expkkkctciEt (B9) Replacing this result in equation B5, we obtain 00 0expkkkkk kktctciEt (B10) Now the Schrdinger equation with the perturbation is 00 00exp expkkk k kkk kctiEt HctiEt it (B11) Note that because the Hamiltonian is timede pendent, the state function will evolve with time and the coefficients of the linear combination have to evolve with time, too. Equation B11 can be simplified to 00 00 expexpk kkkkk kkct iEtctiEtH it (B12) Again projecting in the complete basis set 0 k 00 0 0 expexpk kmkkkmk kkct iEtctiEtH it (B13) Since mk is 1 when k is equal to m the sum of the left side reduces to 0000 expm km kmk kct i ctiEEtH t (B14) Thus the Nth order solution is 1 0000 expN N m mm kmk kct i ctiEEtH t (B15) PAGE 214 214 Assuming that the system is unperturbed in a state 0 i before the perturbation, the solution of these equations are, 0 mm kct (B16) 1 0000 expt mm k m ki ctiEEtHtdt (B17) 2 1 0000 expt mmnmnn ni ct iEEtHtctdt (B18) Then the second order time dependent perturbation theory coefficient is 2 21 exp exptt mmnmnnknk nct itHtitHtdtdt (B19) where 00ij i jHtHt (B20) 00 ijijEE (B21) For an ultrafast laser field, th e perturbation is expressed as iiHtQxEt (B22) Thus, replacing the perturbation on the second order coefficient, the following expression is obtained 2 21 expexptt mfnngmnnk nct EtEtititdtdt (B23) PAGE 215 215 APPENDIX C PARTIAL LEAST SQUARES Partial least squares, PL S, is a multidimensional linear regression. Like any linear regression, partial leas t squares describes the relationship between the variable predictor, X, and variable response, Y, so that, YXB+F (C1) where B represents the matrix of regression coefficients, and F the residual matrix. Like principal component analysis, partial least squa res is a matrix orthogonal linear transformation, which relies on diagonalizing the X matrix, or its covariance ma trix, to extract a new orthogonal basis set based on its variance. E ach eigenvector (latent variable) in the basis set is a linear combination of the independent variables, xi. As any basis set, the la tent variables do not have correlation among them. Therefore, X is decomposed in, T=XW (C2) Here, T and W are matrices containing the coll ective variable coefficients and their weight, respectively. T elements are selected such that they represent good approximations of X and Y. XTP+E (C3) YTC+F (C4) Here, the regression coefficients of P and C are calculated so that they make the matrices E and F small. Y=XWC'+F=XB+F (C5) However, unlike other multilinear regressi on methods such as principal component regression, PLS finds the mini mum number of inde pendent variable se ts by including the covariance structure between X and Y in the calculation of the weight matrix, W. In other words, PAGE 216 216 the criteria for selecting latent variables not only includes the maximi zation of the explained variance in the independent variable data, X, but also the maximum possible correlation with the dependent variables, Y. Hence, partial least squares as multivariate data analysis models dependent variables, Y, in terms of a set of independent variables, xi, with the maximum correlation with Y. Topologically, this represen ts a projection of the nthdimension of the laser parameterization to a hyperplane with N dimensions (or N latent variables) in such a way that the projection coordinates are good approximations of the molecular response. In other words, latent variables are a new set of coordinates in the independent variable space, where the projection of the actual xi components points to the direction with maximum correlation with the values of Y (Figure C1, d vector). Figure C1. Geometrical representati on of partial least squares model. t1t2x1x2d=c1t1+c2t2 PAGE 217 217 APPENDIX D PARTIAL LEAST SQUARES ANALYSIS RESULTS Partial Least Squares Regression for The Autocorrelation Evolution 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Variance Explained X Y A 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Latent vector #Percentage of Variance Explained X Y B Figure D1. A) Variance explained by each late nt variable. B) Cumulative variance explained. Autocorrelation model (Blue bars). Fitness model (red bars). 1 2 3 4 5 6 7 8 9 10 0 200 400 600 800 1000 1200 Latent vector #RMSE X Y A 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #R square X Y B Figure D2. A) RMSE error a nd B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Autocorrelation model (Blue bars). Fitness model (red bars). 1 2 3 4 5 6 7 8 9 10 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Latent vector #Y and Ypred Linear Correlation A 1 2 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 0.12 Latent vector #Y Chi Square B Figure D3. Error of the fitness model: A) Linear correlation and B) Chi square. PAGE 218 218 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 105 Latent vector #X Chi Square Figure D4. Chi square error of the autocorrelation model. 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 Time (fs)Intensity A 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 600 400 200 0 200 400 600 800 1000 Time (fs)Intensity B Figure D5. Autocorrelation latent va riables: A) First and B) Second. 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 200 0 200 400 600 800 1000 1200 Time (fs)Intensity B 5000 4000 3000 2000 1000 0 1000 2000 3000 4000 5000 500 400 300 200 100 0 100 200 Time (fs)Intensity C Figure D6. Autocorrelation latent variables: A)Third and B) Forth. PAGE 219 219 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) A 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) B Figure D7. Fitness (blue line) and PLS linear model (green line). A) With one latent variable. B) With two latent variables. 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) Figure D8. Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) With four latent variables. Partial Least Squares Regression for the Spectral Evolution 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Variance Explained X Y A 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Latent vector #Percentage of Variance Explained X Y B Figure D9. A) Variance explained by each late nt variable. B) Cumulative variance explained. Spectral model (Blue bars). Fitness model (red bars). PAGE 220 220 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 Latent vector #RMSE X Y A 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #R square X Y B Figure D10. A) RMSE error a nd B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Spectral model (Blue bars). Fitness model (red bars). 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Y and Ypred Linear Correlation A 1 2 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 0.12 Latent vector #Y Chi Square B Figure D11. Error of the fitness model: A) Linear correlation and B) Chi square. 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 Latent vector #X Chi Square Figure D12. Chi square erro r of the spectral model. PAGE 221 221 395 400 405 410 415 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Wavelength (nm)Intensity A 395 400 405 410 415 10 5 0 5 10 15 20 Wavelength (nm)Intensity B Figure D13. Spectral latent variab les: A) First and B) Second. 395 400 405 410 415 70 60 50 40 30 20 10 0 10 Wavelength (nm)Intensity A 395 400 405 410 415 10 0 10 20 30 40 50 60 70 Wavelength (nm)Intensity B Figure D14. Spectral latent variab les: A) Third and B) Forth. 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) A 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) B Figure D15. Fitness (blue line) and PLS linear model (green line). A) With one latent variable. B) With two latent variables. PAGE 222 222 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) A 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 GenerationFitness (arb. units) B Figure D16. Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) With four latent variables. Partial Least Squares Regression for the Phase Evolution 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Variance Explained X Y A 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 80 90 100 Latent vector #Percentage of Variance Explained X Y B Figure D17. A) Variance explai ned by each latent variable. B) Cumulative variance explained. Phase model (Blue bars). F itness model (red bars). 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Latent vector #RMSE X Y A 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #R square X Y B Figure D18. A) RMSE error a nd B) Linear correlation (R2) for the fitness model versus the number of latent variables used in the model. Phase model (Blue bars). Fitness model (red bars). PAGE 223 223 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Y and Ypred Linear Correlation A 1 2 3 4 5 6 7 8 9 10 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 Latent vector #X Chi Square B Figure D19. Error of the fitness model: A) Linear correlation and B) Chi square. 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Latent vector #Y Chi Square Figure D20. Chi square e rror of the phase model. 20 40 60 80 100 120 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 Phase parameter #Phase A 20 40 60 80 100 120 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 Phase parameter #Phase B Figure D21. Phase latent variab les: A) First and B) Second. 20 40 60 80 100 120 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 Phase parameter #Phase A 20 40 60 80 100 120 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 Phase parameter #Phase B PAGE 224 224 Figure D22. Phase latent variab les: A) Third and B) Forth. 20 40 60 80 100 120 140 160 180 200 0.7 0.8 0.9 1 1.1 1.2 GenerationFitness (arb. units) A 20 40 60 80 100 120 140 160 180 200 0.7 0.8 0.9 1 1.1 1.2 1.3 GenerationFitness (arb. units) B Figure D23. Fitness (blue line) and PLS linear model (green line). A) With one latent variable. B) With two latent variables. 20 40 60 80 100 120 140 160 180 200 0.7 0.8 0.9 1 1.1 1.2 GenerationFitness (arb. units) A 20 40 60 80 100 120 140 160 180 200 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 GenerationFitness (arb. units) B Figure D24. Fitness (blue line) and PLS linear model (green line). A) With three latent variables. B) With four latent variables. PAGE 225 225 APPENDIX E SPATIAL LIGHT MODULATOR CHARACTERIZATION Figure E1. Spatial light modulator mask pixel inhomegeneity for mask 1 at level 4095 and mask 2 and level 600. PAGE 226 226 Figure E2. Spatial light modulator mask pixel inhomegeneity for mask 1 at level 600 and mask 2 and level 4095. 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After attending Don Zeno School for 5 years, he be gan his undergraduate studies in the spring of 1996 at Universidad de Buenos Aire s, in Buenos Aires city, Argen tina. With an intense physical chemistry background and a special interest in quantum mechanic s and chemical kinetics, he came to the University of Florida, Department of Chemistry in the August of 2002 to begin doctoral studies under the supervision of Profes sor Valeria Kleiman in the area of coherent control of light matter interactions. His profe ssional career as a Ph.D. will continue in Philadelphia where he plans to take a post doctoral position. 