Strong Field Phenomena in Atoms and Molecules from Near to Midinfrared Laser Fields

Strong Field Phenomena in Atoms and Molecules from near to midinfrared laser fields

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Yu Hang Lai, B.S. Graduate Program in Physics

The Ohio State University 2018

Dissertation Committee: Dr. Louis F. DiMauro, Advisor

Dr. Enam Chowdhury

Dr. Ulrich Heinz

Dr. Ezekiel Johnston-Halperin ⃝c Copyright by

Yu Hang Lai

2018 Abstract

paraStrong field atomic physics is the study of the interaction between an atom andan intense laser pulse such that the field strength is “not-so-small” compared to an atomic unit (50 V/A)˚ and so it could not be treated as just a perturbation to the atomic system.

Depending upon the ionization potential of the target, typical laser intensity required to reach this regime ranging from ∼ 10 to 1000 TW/cm2. In the low frequency limit, the photoionization process can be interpreted as a tunneling process in which the atomic potential is “tilted” by the laser field allowing the electron to escape via quantum tunneling.

The escaped electron wavepacket quivers in the strong laser field whose kinetic energy is characterized by the ponderomotive energy Up which is proportional to the laser intensity and the square of the wavelength. Electron recollision happens when the ionized electron is driven back towards its parent ion by the laser field and it leads to numerous intriguing phenomena such as high-order above-threshold ionization, non-sequential ionization and high-order harmonic generation.

While most of the early experiments were performed in the near-infrared (NIR)(0.8 or

1 µm) wavelengths, an important advance over the last decade has been the emergence of intense mid-infrared (MIR)(∼ 2 − 4 µm) sources. The quadratic scaling of Up with wavelength benefits the study of recollision-driven phenomena and also enables the exploration of strong field interaction deep in the tunneling regime. In addition, the MIR regime isof particular interest for studying molecules due to the presence of vibrational resonances. In this dissertation, we explore several strong field phenomena in atoms and molecules with near and mid infrared fields, including:

ii 1. A comprehensive experimental benchmarking of strong field atomic ionization theories. We performed a comparative study between experiment and theories of the total intensity-dependent ionization yield for different atom species at different laser wavelengths

(0.4 − 4µm) at linear and circular polarizations in order to investigate the applicability of two commonly used strong field ionization theories, PPT and ADK.

2. Ionization and fragmentation of methane in vibrationally resonant MIR fields. We measured the mass spectra of fragments of methane irradiated by MIR fields which cover the resonant frequency range of the C-H bond stretching mode vibration (3.2 − 3.5µm). We observed significant enhancement in ionization and dissociation rate at resonant wavelengths compared with non-resonant wavelengths.

3. Electron recollision in tunnel ionization of C60 fullerenes in MIR fields. From the “soft” recollision for low energy electrons we found an unexpected suppression of the “low- energy structure which might be attributed to the induced dipole field; from the “hard” recollision for high energy electrons we demonstrated the applicability of the “laser-induced electron diffraction technique for imaging macromolecule and observed hint of laser induced deformation of the molecular structure.

4. Strong field double ionization with circularly polarized NIR fields. We searched for the existence of recollision effects in double ionization of magnesium, zinc and calcium at different wavelength in order to examine the validity of the classical interpretation forthe previously observed enhanced double ionization yield in magnesium irradiated by circularly polarized 0.8 µm fields.

iii Dedicated to my parents

iv Acknowledgments

paraI have been very fortunate to have the opportunity to purse my PhD study at The Ohio

State University. First I would like to express my gratitude to my scientific advisors, Prof.

Louis DiMauro and Prof. Pierre Agostini, for offering me the opportunity to be part of their world-class research group. I am grateful that I could spend the last five and a half year working in their state-of-the-art laboratory. Thank Lou and Pierre for their instructions and guidance, I have learned so much from them.

I would like to express my gratitude to my mentors and colleagues in the group. Many of the projects presented in this dissertation would not have been possible without the ideas and guidance by Dr. Cosmin Blaga and Dr. Junliang Xu. I have greatly benefited from working with Cosmin over the last few years. His experience and insight were critical in solving many problems we encountered. I have learned many experimental skills from working with him. In particular, I am glad to have the opportunity to learn how to build an optical parametric amplifier with him. Special thanks go to Junliang, who provided many theoretical ideas and calculations for many projects in the group. I gained a lot of theoretical knowledge from him. I am very grateful for his generous help and advices, and thank you for treating me as a brother. Many thanks go to Dr. Kaikai Zhang, who I worked with during my first two years in the laboratory. I have learned many essential experimental skills from him. Thank Dr. Hyunwook Park for his advice and encouragement. Thank Dr.

Hui Xiong for his generous technical help during my first summer in the laboratory when I didn’t even know how to operate a laser.

I would like to acknowledge the students in the group who worked with me and con-

v tributed to some research projects. Xiaowei Gong contributed to the methane experiments, pulse duration measurements and classical simulations for double ionization. Thanks a lot for your time and hard work. Kent Talbert worked with me on several ionization experiments over the last year. Thank you for helping me to perform the measurements efficiently.

Also, I enjoyed discussions with my officemates, Tim Gorman and Greg Smith.

I would like to acknowledge our collaborators who contributed to some of the results presented in this dissertation or other projects I have participated. Thank Prof. Mathias

Kling’s group for the collaborations on the C60 project, especially I want to thank Dr. Harald Fuest for visiting our laboratory to join our experimental runs and setting up the effusive oven. Thank Prof. Hirohiko Kono and his colleagues for providing important theoretical support. Thank Prof. Jens Biegert for giving me an opportunity to visit his and work in his laboratory in Barcelona.

Special thanks go to Dr. Bruno Schmidt for sharing his phenomenal technique of mid- infrared pulse compression and providing helpful advice on building a dispersion-free SHG

FROG. I enjoyed theoretical discussions with Dr. Elias Diesen and Prof. Jan-Michael Rost on the “low-energy structure, Dr. Fabrice Catoire on strong field ionization of60 C and Dr. Xu Wang on double ionization. I enjoyed collaborations and discussions with Dr. Enam

Chowdhurys group on several projects which utilized our mid-infrared laser.

Also, I want to express my gratitude to the teachers I met in high school and college who inspired me. Thank my high school physics teacher, Dr. Ada Lau, for her teaching and the extra time she spent on answering my physics questions after classes. Thank Prof. Pak

Ming Hui, an excellent teacher I met in college. It was an invaluable experience to work with him on some theoretical problems. Thank Prof. Hin Wing Kui for letting me work in his laboratory during my final year in college. It was my first experience in experimental physics research. I really appreciate their advice and encouragement.

Last but not least, I want to express my deepest gratitude to my dearest mother and father for their endless love and support over all these years. Thank you so much, I can’t express how thankful and grateful I am.

vi Vita

Bachelor of Science, The Chinese University of Hong Kong ...... 2011 Graduate Associate, The Ohio State University ...... 2011 to present

Publications

H. Fuest, Y. H. Lai, C. I. Blaga, K. Suzuki, J. Xu, P. Rupp, H. Li, P. Wnuk, P. Agostini, K. Yamazaki, M. Kanno, H. Kono, M. F. Kling and L. F. DiMauro, “Diffractive imaging of C60 structural deformations induced by intense femtosecond mid-infrared laser fields” (submitted) Y. H. Lai, J. Xu, U. B. Szafruga, B. Talbert, X. Gong, K. Zhang, H. Fuest, M. F. Kling, C. I. Blaga, P. Agostini and L. F. DiMauro “Experimental investigation of strong-field- ionization theories for laser fields from visible to mid-infrared frequencies” Physical Review A 96, 063417 (2017) (Editors’ Suggestion) Z. Wang, H. Park, Y. H. Lai, J. Xu, C. I. Blaga, F. Yang, P. Agostini and L. F. DiMauro “The roles of photo-carrier doping and driving wavelength in high harmonic generation from a semiconductor” Nature Communications 8, 1686 (2017) H. Park, A. Camper, K. R. P. Kafka, B. Ma, Y. H. Lai, C. I. Blaga, L. F. DiMauro and E. A. Chowdhury, “High-order harmonic generation in intense mid-IR fields by cascade 3-wave mixing in a fractal poled LiNbO3 photonic crystal” Optics Letters 42, 4020 (2017) A. Camper, H. Park, Y. H. Lai, H. Kagayama, S. Li, B. Talbert, C. I. Blaga, P. Agostini, T. Ruchon, and L. F. DiMauro “Tunable mid-infrared source of light carrying orbital angular momentum in the femtosecond regime” Optics Letters 42, 3769 (2017) D. R. Austin, K. R. P. Kafka, Y. H. Lai, Z. Wang, K. Zhang, C. I. Blaga, A. Yi, L. F. DiMauro, and E. A. Chowdhury “High spatial frequency laser induced periodic surface structure formation in germanium by mid-IR femtosecond pulses” Journal of Applied Physics 120, 143103 (2016) K. Zhang, Y. H. Lai, E. Diesen, B. E. Schmidt, C. I. Blaga, J. Xu, T. Gorman, F. Legare, U. Saalmann, P. Agostini, J. M. Rost, and L. F. DiMauro “Universal pulse dependence of the low-energy structure in strong-field ionization” Physical Review A 93, 021403(R) (2016)

D. R. Austin, K. R. P. Kafka, S. Trendafilov, G. Shvets, H. Li, A. Y. Yi, U. B. Szafruga, Z. Wang, Y. H. Lai, C. I. Blaga, L. F. DiMauro, and E. A. Chowdhury “Laser induced periodic vii surface structure formation in germanium by strong field mid IR laser solid interaction at oblique incidence” Optics Express 23, 19522 (2015) J. Xu, C. I. Blaga, K. Zhang, Y. H. Lai, C. D. Lin, T. A. Miller, P. Agostini and L. F. DiMauro “Diffraction using laser-driven broadband electron wave packets” Nature Commu- nications. 5, 4635 (2014)

Fields of Study

Major Field: Physics Studies in Strong field Atomic, Molecular and Optical Physics; Nonlinear Optics; Ul- trafast Laser

viii Table of Contents

Page Abstract...... ii Dedication...... iv Acknowledgements...... v Vita...... vii List of Figures ...... xi List of Tables ...... xv

Chapters

1 Introduction ...... 1 1.1 A Brief Historical Overview...... 1 1.2 Basic Theoretical Concepts in Strong-field Physics...... 2 1.2.1 The Keldysh Theory...... 3 1.2.2 Time-dependent Schr¨odingerEquation Simulations...... 7 1.2.3 The Semi-classical Model (Three-step Model)...... 9 1.3 Some Recent Developments and Motivation of this Dissertation ...... 14 1.4 Overview of this Dissertation ...... 16

2 Experimental Apparatus ...... 19 2.1 Laser System...... 19 2.1.1 Mid-infrared Optical Parametric Amplifier...... 19 2.2 High temperature atomic beam source...... 24 2.3 Time-of-flight spectrometer ...... 27 2.3.1 Data acquisition system...... 28 2.3.2 Detection of electrons ...... 30 2.3.3 Detection of ions...... 31 2.3.4 Effect of focal volume averaging...... 33

3 Experimental investigation of Strong field ionization theories from visible to mid-infrared wavelengths ...... 38 3.1 PPT and ADK Formulas ...... 39 3.2 Experiments...... 43 3.2.1 Experimental Data...... 44 3.2.2 Discussions...... 52 3.3 Summary ...... 59

ix 4 Ionization and dissociation of Methane in vibrationally resonant Mid- Infrared Fields ...... 61 4.1 Experiments...... 63 4.1.1 Results at MIR wavelengths...... 64 4.1.2 Results NIR wavelengths ...... 69 4.2 Discussion...... 74

5 C60 Fullerenes in strong mid-infrared fields ...... 76 5.1 Low energy photoelectrons of C60 ...... 78 5.1.1 The low-energy structure in tunnel ionization...... 78 5.1.2 Classical trajectory Monte Carlo simlulations...... 81 5.1.3 Experimental Results and Discussions...... 88 5.2 Self imaging of molecular structure using rescattered electrons ...... 95 5.2.1 Basic Principles...... 95 5.2.2 Results and Discussions...... 100 5.3 Summary ...... 105

6 Nonsequential double ionization of atoms with low ionization potentials 106 6.1 Effect of electron recollision in double ionization ...... 106 6.2 Two-electron classical trajectory simulations...... 110 6.3 Calculations of SDI probabilities with the PPT model ...... 118 6.4 Experimental Results and Discussions ...... 119 6.4.1 Linear Polarization...... 119 6.4.2 Circular Polarization...... 126 6.5 Summary ...... 131

7 Conclusions and Outlook ...... 133

Bibliography ...... 137

Appendices

A Setup of Mid-infrared Optical Parametric Amplifier ...... 149 A.1 White Light Continuum Generation ...... 151 A.2 First Amplification Stage ...... 152 A.3 Second Amplification Stage ...... 153 A.4 Third Amplification Stage...... 154 A.5 Fourth Amplification Stage ...... 155

x List of Figures

Figure Page

1.1 A square potential well under the influence of a strong constant electric field.6 1.2 TDSE simulations of photoelectron energy spectrum from hydrogen atom. 8 1.3 Classical electron trajectories...... 11 1.4 Kinetic energy of rescattering electrons...... 12 1.5 Classical-trajectory Monte Carlo simulation of photoelectron energy spectrum. 13 1.6 Schematic diagram of the three-step model...... 14 1.7 An electron trajectory driven by circularly polarized laser field...... 14

2.1 Schematic diagram of optical parametric amplification...... 20 2.2 The wavelength relation between signal and idler using Eq. 2.1...... 21 2.3 Schematic diagram of the OSU MIR OPA...... 23 2.4 Output pulse energy of the idler beam as a function of wavelength...... 24 2.5 3D drawings (Autodesk Inventor) of the atomic beam source setup...... 25 2.6 Structure of the oven...... 26 2.7 Performance of the oven...... 26 2.8 Schematic diagram of time-of-flight (TOF) spectrometer (top view). . . . . 29 2.9 Conversion from (a) time-of-flight distribution to (b) energy spectrum of photoelectrons...... 31 2.10 Ion mass spectrum of magnesium irradiated by linearly polarized 800 nm pulses...... 32 2.11 Contour plot of intensity distribution of a focused Gaussian beam...... 34 2.12 TDSE simulations of ATI spectrum of argon irradiated by 0.68 µm pulses with a peak intensity 120 TW/cm2...... 35 2.13 Calculated ionization yield of krypton atom as a function of laser intensity using PPT formula...... 36

3.1 Comparison between PPT and ADK ionization yield...... 42 3.2 Ion yields of Ar as a function of intensity ...... 46 3.3 Ion yields of Kr as a function of intensity ...... 47 3.4 Ion yields of Xe as a function of intensity ...... 47 3.5 Ion yields of Ne as a function of intensity ...... 48 3.6 Ion yield of Na as a function of intensity...... 49

xi 3.7 Ion yields of Na and K as a function of intensity ...... 49 3.8 Ion yields of Zn as a function of intensity ...... 51 3.9 Ion yields of Mg as a function of intensity...... 52 3.10 d(log(Y ))/dI of Xe as a function of intensity ...... 53 3.11 The ratio between theoretical (PPT and ADK) and experimental values of Yd at an intensity of 0.8IBSI for different data set...... 54 3.12 Saturation intensity of Mg at 0.8 µm...... 54 3.13 The ratio between measured and theoretically predicted (PPT and ADK) values of saturation intensities as a function of γ...... 55 3.14 The ion yield ratio between CP and LP at different target atoms and wavelengths...... 56 3.15 The ratio between theoretical (PPT and ADK) and experimental values of R ≡ YCP /YLP ...... 58 3.16 Ion yields of Mg as a function of intensity at 0.4 µm...... 59 3.17 Calculated ionization probability of Mg at 0.4 µm using different verisons of PPT...... 60

4.1 MIR absorption spectra of several molecular species in the atmosphere. . . 62 4.2 Mass spectrum of methane irradiated with 3.3 µm pulses at a peak intensity of 70 TW/cm2...... 63 4.3 Total ionization yield of CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm...... 64 4.4 Fragmental ion mass spectrum of CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm...... 65 + + 4.5 Ion ratio CX0−3/CX4 for CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm...... 66 4.6 Mass spectra of methane fragments at different laser wavelengths at the intensity of 70 TW/cm2...... 67 4.7 Mass spectra of methane fragments at different laser wavelengths as a function of intensity. Results of 3.1, 3.3, 3.6 and 3.9 µm are shown in panel (a) to (d) respectively...... 68 + + 4.8 Intensity at which the ion yield ratio CH0−3/CH4 reaches 1 at different wavelengths...... 69 4.9 Ellipticity dependence of ion yields...... 69 4.10 Mass spectrum of methane irradiated with 0.8 µm pulses at a peak intensity of 2 × 1014 W/cm2...... 70 4.11 Ionization yields of methane as a function of laser intensity at 0.8 µm. . . . 71 4.12 Ionization yields of methane as a function of laser intensity 1.1 µm...... 72 + + 4.13 Ion ratio C2X0−5/C2X6 for C2H6 and C2D6 as a function of intensity at (a) 3.1 and (b) 3.5 µm...... 75

5.1 Ionization of C60 at MIR wavelength...... 77 5.2 Photoelectron spectrum of Ar, N2 and H2 irradiated by 2 µm laser pulses. . 79 5.3 Characteristics of the LES...... 80 5.4 Density distribution of initial conditions of electrons as a function of (t0, p0⊥). 82 5.5 Intensity dependence of LES...... 85 5.6 Wavelength dependence (1.4 vs 2.4 µm) of the LES...... 85 xii 5.7 Photelectron spectrum at different emission angles...... 86 5.8 Electron trajectories of different drift K.E. (labelled as E in the plot) calculated from 1D classical model without Coulomb potential...... 87 5.9 Illustration of the Coulomb focusing effect...... 88 5.10 Low energy photoelectron spectrum of Xe and C60 irradiated by 3.1µm laser pulses at 78 TW/cm2...... 89 5.11 Intensity dependence of low energy photoelectron spectrum of (a) N2 and (b) C60 irradiated by 3.1 µm laser pulses...... 90 5.12 Intensity dependence of low energy photoelectron spectrum of (a) Ar and (b) C60 irradiated by 2 µm laser pulses...... 91 5.13 Induced dipole field...... 92 5.14 Illustration of the effect of the induced dipole field...... 93 5.15 Comparison between classical trajectories under the influence of different force components...... 94 5.16 Simulated photoelectron spectra...... 94 5.17 Schematic diagrams of electron diffraction from a diatomic molecule. . . . . 99 5.18 Extraction of molecular interference signal from a photoelectron spectrum. . 101 5.19 Comparison between experimentally retrieved and simulated (IAM) molecular interference signal...... 102 5.20 Graphical illustration of C60 cage field-induced elongation during a laser pulse (red curve)...... 103 5.21 Simulation results...... 104

6.1 Ion yields of single and double ionization of Helium irradiated by linearly polarized, 100 fs, 0.78 µm laser pulses...... 108 6.2 Calculated probability of double ionization as a function of intensity using two-electron classical trajectory simulations...... 109 6.3 Calculated probability of double ionization as a function of intensity for magnesium and helium(inset) with CP field using two-electron classical trajectory simulations...... 111 6.4 Distributions of initial position and KE of the electrons...... 114 6.5 An example of SDI trajectories...... 115 6.6 An example of NSDI trajectories...... 116 6.7 Probability of double ionization as a function of laser intensity for Mg atom irradiated by CP 0.8 µm laser pulses...... 117 6.8 Single and double ionization of Mg with MIR pulses...... 120 6.9 Cross section of electron impact ionization for Mg+ and Xe+ as a function of energy...... 121 6.10 Measured single and double ionization yields of Mg with LP driving fields at (a) 0.8 µm and (b) 1.03 µm...... 122 6.11 Measured single and double ionization yields of Zn with LP driving fields at (a) 0.8 µm and (b) 1.05 µm...... 123 6.12 Double ionization probability of Mg calculated from the classical simulations for LP driving fields at (a) 0.8 µm and (b) 3.6 µm...... 124 6.13 Experimental yields of Xe+ (red symbols) and Xe2+ (blue symbols) versus laser intensity for various wavelengths...... 125

xiii 6.14 Experimental ratios of Xe2+/Xe+ vs laser intensity for various wavelengths. 126 6.15 Double ionization probability of Mg calculated from the classical simulations for CP driving fields at (a) 0.8 µm and (b) 1.03 µm...... 127 6.16 Measured double ionization yields of Mg with CP driving fields at (a) 0.8 µm and (b) 1.03 µm...... 128 6.17 Double ionization of Mg with elliptically polarized pulses...... 129 6.18 Measured ionization yields of Ca (a) and Zn (b) with CP driving fields at 0.8 µm...... 130 6.19 Simulated double ionization probabilities of (a) Ca and (b) Zn as a function of intensity for CP fields at 0.8 µm...... 131 6.20 Phase diagram for NSDI in CP fields...... 132

7.1 Low energy photoelectron spectrum of Xe and C60 irradiated by 3.1µm laser pulses at 78 TW/cm2...... 135

A.1 Full optical layout of the OPA...... 150 A.2 Optical path of the white light continuum generation...... 151 A.3 Optical path of the first OPA stage...... 152 A.4 Optical path of the second OPA stage...... 153 A.5 Optical path of the third OPA stage...... 154 A.6 Optical path of the fourth OPA stage...... 155

xiv List of Tables

Table Page

2.1 Output parameters of the Ti:sapphire system...... 19

3.1 Atomic parameters for PPT and ADK calculations...... 42 3.2 Driving wavelengths λ and intensities ranges (Imin − Imax) of experimental data. The uncertainty of our intensity calibration is about 20%. The γ value at the barrier-suppression intensity and Imin are denoted as γBSI and γmax, respectively...... 45

6.1 Values of Ep and softening parameter a for different atoms in the simulations. The values are in atomic units...... 113

xv Chapter 1 Introduction

1.1 A Brief Historical Overview paraStudies of ionization of atoms using lasers began just in a few years after the first working laser was demonstrated in 1960 [1]. The first photoionization experiment ofa noble gas was performed at The Ohio State University [2] using a pulsed ruby laser. Since the energy of a photon at optical wavelengths is significant smaller than the ionization potential of an atom (e.g., photon energy of the laser used in [2] is 1.79 eV; ionization potential of argon atom is 15.79 eV), ionization of an atom in a laser field is a multiphoton process. In the early days, the most popular theoretical tool in multiphoton ionization was (lowest order) perturbation theory, which predicts that ionization probability has a power-law dependence on the laser intensity with an exponent being the minimum number of photons required to reach the ionization threshold. Although some discrepancies between theory and experimental data had been observed (e.g. [3,4]), the perturbative approach was generally accepted and used by the community. In the late 1970s, the first hint of non-perturbative extension of ionization was observed by Agostini et al [5]. They measured the kinetic energy spectrum of photoelectrons from xenon atoms ionized by a 1 µm pulsed laser and found the existence of multiphoton ionization with one photon more than the minimum requirement. This phenomena is now called “Above-Threshold Ionization” (ATI).

More measurements reported over the few years afterwards revealed many more ATI orders and thus [6,7] provided further evidence for non-perturbative behavior.

Perhaps one of the most striking measurements in the 1980s that demonstrated the

1 breakdown of perturbation theory was the ionization measurements of noble gas atoms using a CO2 laser which has a wavelength of ∼ 10µm [8]. In that case, energetically ∼ 100 photons are required to reach the ionization threshold, but the slope of the measured ionization yields as a function of laser intensity (in log-log scale) was found to be only ∼ 10. The ionization mechanism, as speculated by the authors themselves, could be explained using the “optical tunneling” theory, which is now usually referred as the “Keldysh theory”, proposed by L.

V. Keldysh back in 1964 [9]. In fact, it turned out that Keldysh theory had eventually been recognized to be one of the most important theoretical elements for strong-field physics. As we will introduce in the next section and the subsequent chapters, many strong-field effects could be interpreted using Keldysh theory together with its modifications and extensions that grew out of it.

Thanks to the development of intense femtosecond laser technologies [10–14], a wealth of new results in strong-field physics emerged by the end of 1980s and the early 1990s.In particular, three distinct but in fact related observations, including high-order ATI plateau

[15, 16], non-sequential ionization (NSI) [17, 18] and high order harmonic generation (HHG)

[19], led to the development of a semiclassical model [15, 20] (also known as the “three- step model”) which is a crucial extension to the tunnel ionization theory that gave a more complete view of the intense laser-atom interaction.

Over the last two decades, novel laser and detection techniques and the influx of new ideas have continued driving the field towards new discoveries, some of these recent advances will be highlighted in Sect. 1.3.

1.2 Basic Theoretical Concepts in Strong-field Physics paraThe purpose of the following subsections is to develop the background for the reader of the basic theoretical framework in strong field ionization. The aim is to explain the basic concepts and present without derivation some key formulas and metrics. For comprehensive reviews, see, e.g., [21–23]. Note that all the theories described in this section are under the single-active electron approximation, that is, only one valence electron of the atom is

2 considered to be affected by the laser field.

1.2.1 The Keldysh Theory paraIn the non-relativistic regime, the Hamiltonian of an electron under the influence of atomic potential V (r) and an external laser field F⃗ (t) is given by 1

p⃗2 H(t) = + e⃗r · F⃗ (t) + V (⃗r), (1.1) 2m where e is the magnitude of electron charge. The spatial dependence of F (t) is neglected because the size of an atom is much smaller than the wavelength of an optical field, and this is called the dipole approximation. The transition amplitude for the electron to transit from the initial state ψ0 to a continuum state ψ⃗p of momentum ⃗p is

′ ′ M(⃗p) = lim ⟨ψ⃗p(t)| U(t, t ) |ψ0(t )⟩ , (1.2) t→∞,t′→−∞ where U(t, t′) is the time evolution of H(t). It is a generic expression, the question is how to evaluate M(⃗p). In usual perturbation approach for ionization of an atom, the final state is taken to be an (unbound) stationary state of the atom. It is not the most accurate description because after ionization the electron is still under the influence of the laser field, that is, the ionized electron is being accelerated by a time dependent field. This effect is negligible for weak field but it is a key ingredient in the description of strong field ionization which was first considered by Keldysh in his seminal paper[9]. According to the Keldysh ansatz2, Z ∞ i (V ) ⃗ M(⃗p) = dt0 ⟨ψ⃗p (t)| e⃗r · F (t0) |ψ0(t0)⟩ . (1.3) ~ −∞ (V ) The final state ψ⃗p (t) is called the Volkov function:

1 i m Z t ⟨⃗r|ψ(V )(t)⟩ = exp [m⃗v (t) · ⃗r − ⃗v2(t′)dt′], (1.4) ⃗p 3/2 ⃗p ⃗p (2π~) ~ 2 −∞ where the relation between the kinetic momentum and canonical momentum is

1Equation 1.1 is formulated in the length gauge. In the velocity gauge, H(t) = [⃗p + eA⃗(t)]2/2m + V (⃗r), where A⃗(t) is the vector potential of the laser (F⃗ (t) = −∂A⃗(t)/∂t). For a detailed discussion about gauge transformation in laser-atom interaction, see, e.g., [24]. 2Expression 1.3 is not explicitly presented in [9], but its form can be deduced from Eq. 8 of that paper

3 ⃗ m⃗v⃗p(t) = ⃗p + eA(t), (1.5)

A⃗(t) is the vector potential of the laser (F⃗ (t) = −∂A⃗(t)/∂t). The Volkov state is the eigenstate of a free electron in the field F⃗ (t). Using it as the final state implied that the interaction between the electron and its parent ion after ionization is neglected. In other words, a key difference between perturbative approach and Keldysh approach is that the former one neglected the effect of the laser field for the final state while the latterone neglected the effect of the atomic potential for the final state.In[9] Keldysh presented

R 2 3 the calculations of total ionization rate, that is, w0 = |M| d ⃗p, for a monochromatic field

F0 cos(ωt). Under the approximation that (i) the laser field amplitude is small compared to the atomic field and (ii) the atomic potential is a short-range potential, an analytic expression for w0 was obtained. It was expressed as an infinite sum in which each term can be interpreted as the partial rate due to the absorption of n photons, and the minimum n has to be such that the energy sum of the photons is at least not smaller than the effective ionization potential. That is, each term represents an ATI order. In the expression of w0 there is a crucial parameter γ which is now called the Keldysh parameter: p ω 2mIp γ = , (1.6) eF0 where Ip is the ionization potential and ω is the laser frequency. This dimensionless parameter characterizes the dynamical regime of ionization as we will examine. With exponential accuracy, the ionization rate for the two extreme cases of γ are  F 2⌈K ⌉ ( ) 0 , γ ≫ 1  ω w0 ∝ " 3/2 2 # (1.7) p 1/2 2(2Ip) γ ( IpF ) exp − 1 − , γ ≪ 1  3F 10 where K0 = Ip/~ω and ⌈⌉ is the ceiling function. For γ ≫ 1, ionization is classified to be in the “multiphoton limit”, the ionization rate essentially follows a power law with an exponent being the minimum number of photons required to reach the ionization threshold. The same form is expected if one applies usual perturation theory on nonresonant multiphoton ionization.

4 For γ ≪ 1, on the other hand, is called the “tunneling limit”. In the limit ω → 0

3/2 (i.e. γ → 0), the exponential term becomes exp (−2(2Ip) /3F0) which coincides with the expression for the tunnel ionization of an atom by a static field [25, 26]. As a simple illustration of tunnel ionization, Fig. 1.1 shows a narrow square potential well under the influence of a static external field F0. Initially, an electron is in the well with an energy of

−Ip. Due to the external field, the potential is “tilted”, thus a barrier of awidthof Ip/eF0 is formed3. Ionization occurs if the electron tunnels through the barrier. If the external field

is an optical field, it alternates as a function of time and as a result the potential barrier

also varies in time. For every half an optical period the barrier is switched from one side to

the other. In this scenario, one could imagine that a crucial criteria for tunneling to happen

is that the electron has to tunnel through the barrier “fast” enough so that it could escape

before the barrier is “closed”. Rigorous definition and concept of “tunneling time” are not

straightforward (for a review, see [27]) but from the definition that Keldysh used in Ref.

[9] it is simply the time taken for a particle of velocity v to pass through the width of the p barrier. For an electron bound to an atom the average velocity v = 2Ip/m. Thus, the p Keldysh tunneling time is (Ip/eF0)/v = 2mIp/eF0. Tunneling could occur if the ratio tunneling time/optical period is small. In fact, the Keldysh parameter can be interpreted

as such a ratio, since p p ω 2mIp 2mIp/eF0 γ = = . (1.8) eF0 1/ω Now, it is clear why γ ≪ 1 is called the tunneling limit and γ ≫ 1 is called the multiphoton limit. γ can also be expressed as s I γ = p , (1.9) 2Up where e2F 2 U = 0 (1.10) p 4mω2 is called the ponderomotive energy, which is the cycle-averaged quiver kinetic energy of a

free electron in an electromagnetic field. The significance of Up will be illustrated in Sect. 1.2.3. Note that in most of the available strong field experiments γ ∼ 0.5 − 2, which is

3Assuming the width of the potential well is infinitesimally small.

5 −퐼푝 퐼푝/푒퐹0

Figure 1.1: A square potential well under the influence of a strong constant electric field.

neither extremely large nor extremely small. Usually, as long as γ is smaller (greater) than

1 then the ionization process is classified to be in the tunneling (multiphoton) regime. So neither a simple power law nor exponential function could accurately describe the ionization rate in most cases, further discussions will be given in Chap.3.

Soon after the publication of [9], numerous generalizations and corrections appeared.

For a review of the historical development and the details of different theories, see, e.g.,

[22, 23, 28]. Here we will only mention a few of the most notable papers in the literature.

The applicability of using Keldysh’s original formula to compare with experimental data of real atoms is limited because of the disregard of the Coulomb interaction between the photoelectron and the parent ion. Perelomov, Popov and Terent’ev [29, 30] obtained an analytical formula for ionization including a Coulomb correction factor, it is now known as the PPT formula. Ammosov, Delone and Krainov [31] derived a formula (ADK formula) for tunnel ionization of atoms, which is identical to the PPT formula in the limit γ → 0.

PPT and ADK formulas have been often used in calculations of ionization probability. For momentum distribution of photoelectrons, the commonly used theoretical method is called

“strong-field approximation” (SFA), essentially being based on the theory by Keldysh [9],

6 Faisal [32] and Reiss [33] (the central idea is still based on Eq. 1.3, so sometimes SFA is also refered as “KFR theory” 4).

1.2.2 Time-dependent Schr¨odingerEquation Simulations paraSince the late 1980s, ab initio simulations using numerical solutions of the time- dependent Schr¨odingerequation (TDSE) had became available [34–36]. Certainly, the main advantage using TDSE over the theoretical models mentioned above is the accuracy because it includes excited states of the atom and it fully describes the combined effect of the long range Coulomb potential and the laser field. However, one major disadvantage, in addition to long computational time, is the difficulty in identifying the exact origin of certain effects since all the ingredients are included in the calculations. It is also difficult to obtain intuitive physical picture for the electron dynamics unless the relevant physical quantities (e.g.

⟨position⟩,⟨momentum⟩ of the ionized wavepacket) are extracted for every time step in the simulations which is nontrivial and/or impractical to implement in many cases. Neverthe- less, TDSE simulation is a powerful tool and it is often used as a benchmark for testing the accuracies of theoretical models in different scenarios.

Let us have a brief overview on the basic characteristics of photoelectron energy distribution from strong field ionization with two example TDSE simulation results shown in

Fig. 1.2. The simulations were performed using an open-source code Qprop [37]. Panel (a)

shows the case of γ = 1.26 corresponding to the multiphoton regime. The spectrum consists

of a long series of peaks and the energy spacing between two adjacent peaks is equal to the

photon energy corresponding to the central wavelength of the laser (~ω = 1.82 eV). In fact, the energy value of the peaks are given by

En = n~ω − (Ip + Up). (1.11)

The formation of each peak is interpreted as the result of absorption of n photons in which

part of the absorbed energy is consumed to overcome the effective ionization potential Ip+Up and the excess energy becomes the kinetic energy (K.E.) of the electron. In this process,

4The formalism by Faisal [32] and Reiss [33] was based on velocity gauge instead of length gauge

7 −5 10

−10 10 (a) Electron yield (arb. un.) 0 10 20 30 40 50 60 70 80 Electron energy (eV)

−5 10

−10 10 (b) Electron yield (arb. un.) 0 50 100 150 200 250 300 350 400 Electron energy (eV)

Figure 1.2: TDSE simulations of photoelectron energy spectrum from hydrogen atom irradiated by a laser pulse at (a) 680 nm (multiphoton regime) and (b) 1800 nm (tunneling regime). The peak intensity of the laser is 1014 W/cm2. The vertical dashed lines in panel (a) mark the energy values given by Eq. 1.11.

the atom is ionized with more than the energetically required number of photons and so it is called “Above-Threshold Ionization” (ATI), which has been mentioned in Sect. 1.1. Note that the Up term in the effective ionization potential is due to the AC Stark shift ofthe continuum state, whereas the AC Stark shift of the ground state is much less than Up and is neglected. It could also be understood intuitively from the fact that the ionized electron is required to at least have a kinetic energy of Up due to the quiver motion in the field but the K.E. associated with the quiver motion does not contribute to the K.E. at detection.

As for the case of 1800 nm (Fig. 1.2(b)) where the ionization is in the tunneling regime

(γ = 0.47), the photoelectron spectrum consists of two parts with distinct features. For the part of low energy, it has a maximum at zero and decreases exponentially towards a cutoff energy. Beyond this cutoff is the start of the high energy part of the spectrum, itis a long “plateau” like region where the yield is relatively flat and another cutoff exists at the high energy end of the plot. The low energy part of the spectrum is contributed by

8 “direct” electron and the high energy part is contributed by “rescattered electron”. Further illustration will be given using the semi-classical model presented in Sect. 1.2.3.

The two examples above illustrated a key characteristic of the phoelectrons from strong field ionization: the maximum K.E. obtained by the electron is far beyond the energyofa

(or a few) laser photon. In this scenario, the energy scale of the electron is determined by

Up rather than ~ω, and a physical picture to understand this is again tied to the concept of tunneling. A main reason why the concept of tunneling is formidable in strong field physics is because it allows us to treat ionization as a process which responses to the instantaneous field strength of the laser, whereas in the usual perturbation description the ionization rate is usually treated as a cycle-averaged quantity. In other words, the electron emission in tunnel ionization can be approximately treated as an event which happens at a well-defined instant and the probability of emission is strongly correlated to extrema ofthe laser field cycle. These crucial properties are at the origin of the success of the semi-classical model[15, 20] which has become the theoretical foundation of many strong-field phenomena.

1.2.3 The Semi-classical Model (Three-step Model) paraAs we have mentioned, ab initio simulations are important for obtaining accurate results but not quite helpful for getting intuitive physical pictures. In this section, we will describe the semi-classical model, also known as the “three-step model” or “rescattering model”, which has been widely used to interpret and visualize electron dynamics in strong field processes. The first “step” of the model, tunneling, has been introduced inSect. 1.2.1.

After an electron tunnels through the potential barrier, it is still under the influence of the strong laser field. The semi-classical model describes the subsequent dynamics ofthis electron using classical mechanics. Neglecting the Coulomb force due to the parent ion after tunneling, the classical equation of motion for an electron in a linearly polarized laser field is d2x m = −eF cos(ωt), (1.12) dt2 0 in which the following approximations are imposed: (i) relativistic effects are neglected

9 because v ≪ c (c =speed of light) at the laser intensity level of our interest. (ii) The temporal pulse profile of the laser is neglected because the timescale of the recollision dynamic discussed below is sub-optical cycle. (iii) The spatial dependence of the field is neglected5. By integrating Eq. 1.12 the displacement and velocity of the electron are given by

eF x(t; t ) = 0 [cos(ωt) − cos(ωt ) + ωsin(ωt )(t − t )], (1.13a) 0 mω2 0 0 0 eF v(t; t ) = − 0 [sin(ωt) − sin(ωt )], (1.13b) 0 mω 0

where t0 is the instant when the electron just tunneled through the Coulomb potential barrier. The momentum at that instant is assumed to be zero. From Eq. 1.13 it can

be seen that the motion consists of an oscillatory motion and drift motion. Figure 1.3(a)

shows x(t; t0) for different values of t0 and Fig. 1.3(b) shows the amplitude of the field and

vector potential as a function of time. Note that Up (Eq. 1.10) is the optical cycle averaged value of the K.E. due to the first term of Eq. 1.13(b). In practice, the oscillatory motion will vanish once the laser pulse is gone, but the drift momentum of the electron remains and this is the physical quantitiy being measured in photoelectron experiments. The drift momentum depends on t0 and the corresponding K.E. is ranging from 0 (for t0 = 0) to 2Up

6 (for t0 = π/2ω) . As shown in Fig. 1.3(b), the field amplitude (and the ionization rate) is at maximum when t = 0 and minimum when t = π/2ω, so in a photoelectron energy spectrum the signal decreases as a function of K.E. and a cutoff exists at 2Up. The discussion above is under the assumption that the electron never interacts with the parent ion again after tunnel ionization. From Fig. 1.4(a) it can be seen that some electron trajectories cross the horizontal axis at some later times as they return to the parent ion.

The dependence of return energy Er (the kinetic energy when it goes back to x = 0) on t0 is shown in Fig. 1.4(a). The minimum possible Er is zero and the maximum possible Er is

3.17Up (see the dashed line of Fig. 1.4)(a). Except for the case of maximum Er, for each

5Displacement of an electron within a typical laser pulse duration (∼100fs) is ≪ µm which is the length scale of the focused laser spot size. 6 Note: the definition of Up is the average K.E. due to a charged particle oscillating in a laser field, but here we are using it to quantify the K.E. of the drift motion

10 Figure 1.3: Classical electron trajectories. Upper panel: x versus ωt with different values of t0. ωt0 = −0.5 (red); ωt = 0 (blue); ωt0 = 0.5 (green) and ωt0 = 1.4 (orange). Lower panel: amplitude (in arbitrary units) of electric field E(t) and vector potential A(t) vs ωt.

Er there exists two possible values of t0 (one on the left hand side of the dashed line in Fig.

1.4(a) and the other on the right hand side). The trajectory with the smaller t0 is called “long trajectory” and the other one is called “short trajectory”. “Long” and “short” refer

to the time difference tr − t0, where tr is the instant when the electron return to the ion. The returning electron which collides with the parent ion is called electron “recollision”

or “rescattering”, and this is the last step of the three-step model. The outcome of electron

recollision could be categorized into (i) elastic scattering; (ii) inelastic scattering and (iii)

recombination.

For elastic scattering, the direction of the electron momentum changes but not the mag-

nitude. For one dimensional model the only two possible scattering angles are 0◦ (forward

scattering) or 180◦ (back scattering). The effect of back scattering is simply flipping direc-

tion of the electron momentum at t = tr. The motion afterwards is still described by Eq. 1.12. It turns out that the drift momentum of these backscattered trajectories could be

much higher than the ones without scattering, the corresponding K.E. at detection (Ef ) is

ranging from 0 to 10Up. Figure 1.4(b) shows the relation between Er and Ef for long and

11 Figure 1.4: Kinetic energy of rescattering electrons. (a) Return energy of the electron Er as a function of ωt0. (b) Relation between Er and Ef .

short trajectories.

Figure 1.5 shows a simulated energy spectrum of electrons using classical-trajectory monte carlo (CTMC) method. The details of CTMC will be presented in Sect. 5.1.2.

Briefly, the spectrum is obtained from the K.E. distribution of a large number of electron trajectories tunnel ionized at different t0. Both laser field and Coulomb field of the parent ion are included in the calculations so the effect of elastic rescattering is automatically implemented. In the energy range from 0 to 2Up the signal is mostly contributed by “direct” electrons and from 2 to 10Up the signal is from rescattered electrons. During an inelastic scattering, the returning electron transfers energy to the bound electron(s) of the parent ion, which could possibly knock out or excite another or even multiple electrons, this is the mechanism of nonsequential ionization (NSI). NSI will be discussed in Chap.6.

As for recombination, the electron returns to the ground state and due to energy conservation a photon is emitted, this is the mechanism of high-harmonic generation (HHG) at the microscopic level [39]. This process has been used to generating attosecond light pulses

[40] and has led to many applications in the exploration of ultrfast dynamics on the natural timescale of the electron (for a review, see, e.g., [41]). HHG is beyond the scope of this

12 Figure 1.5: Classical-trajectory monte carlo simulation of photoelectron energy spectrum. Laser parameters: wavelength = 2 µm; intensity = 100 TW/cm2.

dissertation.

Figure 1.6 is a visual summary of the three-step model. First, an electron escapes from the atom via tunneling. Secondly, the electron is accelerated and then driven back to its parent ion by the laser field. Finally, it recollides with its parent ion and results inoneof the three possible processes described above.

The discussion above is for the case of linearly polarized laser fields where the field vector is always on a fixed spatial axis. For elliptically polarized fields,

F F⃗ (t) = √ 0 [cos(ωt)ˆx + ϵ sin(ωt)ˆy], (1.14) 1 + ϵ2 where the ellipticity ϵ ranges between 0 (linear polarization) and 1 (circular polarization).

Since the field vector rotates as a function of time, an electron in this field tends tospiral away from the origin and so rescattering is suppressed. The trajectory of an electron in a circularly polarized field is expressed as

eF0 x(t; t0) = √ [cos(ωt) − cos(ωt0) + ωsin(ωt0)(t − t0)], (1.15a) 2mω2 eF0 y(t; t0) = √ [sin(ωt) − sin(ωt0) + ω cos(ωt0)(t − t0)], (1.15b) 2mω2

13 Figure 1.6: Schematic diagram of the three-step model. (a): Step 1: tunnel ionization due to the strong laser field. (b)-(c): Step 2: The electron is accelerated and driven back towards its parent ion by the laser field. (d) Step 3: electron recollides with its parent ion.The three possible outcomes are elastic rescattering, inelastic rescattering and recombination (see text). Reproduced from [38].

Figure 1.7: An electron trajectory driven by circularly polarized laser field. From Eq. 1.15 with t0 = 0.

and it is plotted in Fig. 1.7 (by rotational symmetry, the behavior of the trajectory does not depend on t0). Once the electron starts moving from the origin it never revisit the same point if the atomic potential is neglected. The use of circularly polarized fields in strong field studies have been often treated as a knob to eliminate electron recollision.

1.3 Some Recent Developments and Motivation of this Dis- sertation

14 paraAn important thrust for strong-field physics since the 2000s is the emergence of intense femtosecond laser sources operating at wavelengths longer than the routinely used 0.8 µm

Ti:sapphire laser systems [42]. These sources are usually mainly based on nonlinear optical processes such as optical parametric amplification (OPA), difference frequency generation

(DFG) and optical parametric chirped pulse amplification (OPCPA). The exact wavelength range from different setups varies but overall they provide coverage from1 − 4µm. The major merit of long wavelength is tied to the quadratic scaling of Up with λ. Although laser intensity is the other knob to increase Up but there is always an upper limit of the applied intensity for any atoms due to ground state depletion.

There are two direct consequences due to the drastic increase in Up. First, it leads to small γ values which enabled the exploration of strong field ionization in the deep tunneling regime. Remarkably, it was found, both theoretically [43] and experimentally [44], that the semiclassical model indeed accurately describes the photoelectrons in this regime. The cutoff energies of photoelectrons and high harmonics nicely follows the quadratic scaling in wavelength. But what was even more striking was the discovery of the so-called “low- energy structure” (LES) [45, 46], an unexpected but seemingly universal spike-like feature in the very low energy part (∼ 0.1Up) in photoelectron spectra. LES could be reproduced by TDSE simulations but strong field theories failed to predict its existence. It turns out that the key to understand LES is the effect that most theories neglected: the Coulomb interaction between the tunnel ionized electron and its parent ion. More discussions about

LES will be presented in Chap.5 and in particular we will investigate a special case where

LES is suppressed.

The other consequence of the large Up is the generation of high energy rescattering electrons (Er = 3.17Up, as mentioned), so it benefits the study of recollision-driven phenomena such as NSI [47], HHG [44, 48, 49] and especially “laser-induced electron diffraction” (LIED)

[50]. LIED is an electron diffraction based imaging method using the rescattering electron from the target itself, whereas in conventional electron diffraction (CED) the electrons are prepared by an external source. In LIED analysis, the diffraction pattern embedded in the spectrum of the rescattered photoelectrons is extracted and used to obtain molecular

15 structural information. The advantage of LIED over CED is the capability to achieve a temporal resolution down to ∼ 10 fs. LIED have been proven to work for atoms [51, 52], diatomic molecules[53, 54] and small hydrocarbon molecules [55, 56]. In this dissertation we will test its applicability to a large molecule, C60. Details will be described in Chap. 5 but here we point out why it is important, if not necessary, to employ long wavelength driving lasers for this technique: (i) the spatial resolution is determined by the energies of the rescattering electrons; (ii) interpretation of the process is based on the semiclassical description which is accurate only in the deep tunneling regime.

Another merit of long wavelength drivers is specifically for the studies of molecules. It is because vibrational resonances of various bonds are in mid-infrared region (> 3 µm) such as the stretching modes of N-H (2.86−3.03µm), O-H (2.86−3.13µm), C-H (3.23−3.51µm),

C=O (5.68 − 6.01µm) and C=C (5.95 − 6.1µm) bonds. In particular, we will investigate the effect of vibrational resonance of C-H bonds to tunnel ionization of small hydrocarbon molecules.

1.4 Overview of this Dissertation paraThe topics included in this dissertation covers various aspects in strong field ionization including investigation of ionization under different laser wavelengths and intensities, effect of vibrational resonance in molecules, effect of electron-electron correlations, effect of polarizability, and extending ultrafast molecular imaging to complex molecules. The structure of this dissertation will be as following:

In chapter2, we will introduce the working principles of the experimental apparatus including the laser system, time-of-flight spectrometer for ion and electron detections and a high-temperature atomic beam source.

In chapter3, strong field ionization yield versus laser intensity is investigated for various atomic targets (Ne, Ar, Kr, Xe, Na, K, Mg and Zn) and light polarization from visible to mid-infrared (0.4 − 4µm), from multiphoton to tunneling regimes. The aim is to perform a comprehensive investigation on the applicability of the two most commonly used theoretical

16 models for calculations of ionization rate, PPT and ADK. We found that while PPT is generally satisfactory, ADK validity is much more limited. In addition, our measurements on Na, K, Mg and Zn provide the first mid-infrared tunnel ionization data of these atoms.

In chapter4, we will present our experimental findings of strong field ionization and dissociation of methane molecules. Our main goal is to investigate tunnel ionization when the driving wavelength is “on resonance” with molecular vibrations. The measured ionization and fragmentation yields as a function of laser wavelength. We found that the amount of fragmenting ions relative to the intact molecular ions exhibit a pronounced wavelength dependence near the resonance located at around 3.3 to 3.6 µm. In contrast, the feature is absent in the same measurements with deuterated methane (CD4). The results suggest that the resonance of C-H bond stretching mode plays a significant role in the fragmentation processes. Moreover, by comparing the total ion yields of CH4 with that of CD4, we found that the overall ionization rate of CH4 is also enhanced at around 3.3 to 3.6 µm. This result has important implications in understanding tunnel ionization in the presence of vibrational resonances.

In chapter5, we turn to our study of strong field phenomena of C 60 fullerenes in mid- infrared fields. Our discussions will be focused on two distinct regions: the very lowenergy part and the high energy rescattering plateau of the photoelectron spectra. A surprising feature we observed from the low energy photoelectron spectra is the suppresion of the

LES. We performed classical simulations to compare with our data and proposed that the suppression might be due to the exceptionally large polarizability of C60 which produces a significant induced dipole field that influences the electron trajectories at low energies.As for the high energy rescattered electrons, we applied LIED to probe the molecular structure of C60. It serves as the first test for the applicability of LIED method to study large molecules. Our measured diameter of the C60 cage is found to be a few percent larger than its equilibrium value and we attribute it to molecular vibration induced by the intense laser field.

In chapter6, we will investigate strong field double ionization of metal atoms. We mainly focus on the problem of recollision driven double ionization with circularly polarized

17 driving fields. We performed measurements on magnesium, calcium and zinc at different wavelength and compared with two-electron classical simulations.

18 Chapter 2 Experimental Apparatus

2.1 Laser System paraThe main laser system that we used for the experiments presented in this dissertation consists of two main components: a commercial 0.8 µm Ti:sapphire laser system (Spitfire

ACE PA, Spectra Physics) and a home-built optical parametric amplifier which converts the output wavelength of the Ti:sapphire laser to a tunable range between 3-4 µm.

The output parameters of the Ti:sapphire system is shown in Table 2.1.

Pulse energy 12 mJ Pulse duration (FWHM) 80 fs Central wavelength 800 nm 1/e2 beam diameter 11.5 mm Repetition rate 1 kHz

Table 2.1: Output parameters of the Ti:sapphire system

2.1.1 Mid-infrared Optical Parametric Amplifier paraOptical parametric amplifiers (OPA) have been widely used to generate wavelength- tunable output from a fixed wavelength pump laser (usually Ti:sapphire laser for femtosecond regime). Nowadays, beta barium borate (BBO) crystal based OPAs with high output pulse energy (∼ 1mJ) are routinely used in research in strong field physics. The upper limit 19 of the output wavelength is ∼ 2.5 µm due the absorption of BBO crystal. Other crystals

(see, e.g., [57] for a list) are available for output wavelength > 3 µm. Many of the fem-

tosecond OPA systems working in this regime were built around of one the three crystals

KTiOAsO4 (KTA), KTiOPO4 (KTP), and KNbO3 [58]. In this section we will describe our home-built KTAcrystal based OPA for generating laser pulses in the wavelength range between 3 to 4 µm in sub-millijoule level.

Basic principles of operation

Figure 2.1: Schematic diagram of optical parametric amplification.

Optical parametric amplification is a second order nonlinear optical process which in- volves parametric interaction between three waves of different frequencies in a crystal. Fig- ure 2.1 shows a schematic diagram of the process, two laser beams, one called “pump” of freqency ωp and the other called “signal” of a lower frequency ωs and with much weaker energy, overlap in a nonlinear crystal7. The parametric process results in the pump beam losing energy in favor of the signal beam and a newly generated beam called “idler”. In terms of photon balance, it is interpreted as a pump photon split into a signal photon and an idler photon. By energy conservation,

~ωp = ~ωs + ~ωi. (2.1)

In our case where the wavelength of the pump is 0.8 µm, the relation between the signal and idler wavelength is shown in Fig. 2.2. For the amplification process to be efficient, the

7Optical parametric amplification and difference frequency generation are the same kind of nonlinear process, but the former one usually refer to the case where the lower frequency beam is much weaker than the higher frequency beam

20 Figure 2.2: The wavelength relation between signal and idler using Eq. 2.1. The wavelength of the pump laser is assumed to be 0.8 µm.

phase matching (or momentum conservation) condition

⃗ ⃗ ⃗ ~kp = ~ks + ~ki, (2.2) where ⃗k is the wavevector, must also be satisfied. By the relation k = n(ω)ω/c, where n(ω) is the frequency dependent refractive index of the medium and assuming collinear propagation, Eq. 2.2 can be rewritten as

n(ωp)ωp = n(ωs)ωs + n(ωi)ωi. (2.3)

Therefore, whether phase matching condition can be satisfied depends on material and frequency, and this is a main reason why different type of crystal is required for different wavelength range. In normal dispersion region, where n increases as ω decreases, Eq. 2.3 could not be satisfied in bulk isotropic material. Usually, phase matching is achieved using birefringent8 crystals by aligning the polarizations of the three beams along the ordinary

(o) and extraordinary (e) axes. In our case, where KTA crystal is used, polarization of pump and idler are along e-axis and signal is along o-axis, this is classified as type-IIA phase matching. For the classification of different types of phase matching, see, e.g.,[59].

Regardless, type-IIA phase matching is the only possible configuration for KTA in our

8It means that the material is optically anisotropic. That is, the refractive index depends on the propagation direction and polarization of light.

21 desired wavelength range. Since the refractive index for the beam whose polarization is along the e-axis depends on the orientation of the crystal, so Eq. 2.3 can be satisfied over some range of frequency by adjusting the orientation of the crystal and this is why the output wavelength of an OPA is tunable. The orientation is usually characterized by a set of phase matching angles.

Another key ingredient in an OPA setup, which is unrelated to the optical amplification process itself but is required to initiate the process, is the generation of an initial signal beam

(“seed” beam). Since it is at a different frequency from the pump beam, a nonlinear optical process is required if one wants to provide both pump and seed beam with a single laser system. The most common method is by using white light continuum generation [60–63], which can be achieved by focusing an intense laser beam inside a transparent material such as sapphire or fused silica. The exact process is complicated but essentially the interplay between self-phase modulation and self focusing results in a large spectral broadening (for a review, see, e.g., [64]), then the spectral component of interest is selected and amplified in the nonlinear crystal.

Practical Considerations and Optical Setup

For a given type of material, assuming depletion of the pump beam energy is negligible, the two main factors which determine the amplification efficiency are the pump beamin- tensity and the interaction length of the process. Therefore, to maximize output energy it is favorable to have the pump beam to be as intense as possible without causing material damage. The damage threshold intensity of KTA is ∼ 400 GW/cm2. As for maximizing

the interaction length, one would use a long crystal but ultimately what limits the effec-

tive interaction length is material dispersion: when two laser pulses of different frequencies

propagate in a dispersive material, the group velocity mismatch (GVM) between them tem-

porally separates the two beams, as they separate in time the amplification will diminish

and eventually stop. Another dispersion effect that imposes a limit on the desired crystal

length is the phase matching bandwidth. As mentioned, for a given phase matching angle

of the crystal Eq. 2.3 can be satisfied by a set ofω ( p, ωs, ωi). But since a laser pulse consists

22 of a frequency spectrum9 only the central frequencies satisfy Eq. 2.3 while other frequency components do not due to dispersion. In other words, there is a GVM between the signal and idler waves and as a result they will be temporally separated as they propagate in the crystal. For a detailed discussion about interaction length and phase matching bandwidth, see, e.g., [59].

Considering the constraints described above, it can be calculated10 that the maximum crystal length suitable for our laser and material parameters is about 2.5 mm. With a KTA crystal of such a length pumped by an 800 nm laser at an intensity of 400 GW/cm2, the 1 µm signal beam would be amplified by a factor of ∼ 200 if pump depletion is negligible. Since the amplification factor is a physical limit of the material, to obtain further amplification it is necessary to send the amplified output signal beam to another pumped crystal, and so on. Therefore, in a typical OPA setup the energy of the pump laser is split to supply energy for several crystal stages (one after another) and the number of stages is limited by the total energy of the pump laser. For all the stages, the pump intensity should be kept close to the highest attainable level (∼ 400 GW/cm2 for KTA, as mentioned) but on the other hand the signal should be kept to be much less intense than the pump beam to ensure efficient amplification.

Figure 2.3: Schematic diagram of the OSU MIR OPA.

A schematic diagram of our mid-infrared (MIR) OPA is shown in Fig. 2.3. For a detailed description about the optical componenents and beampaths of the setup, see AppendixA.

9The bandwidth of the spectrum is inversely proportional to the pulse duration. 10SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM

23 In brief, the pump laser beam was split by beamsplitters into five beams and then sent toa sapphire and four KTA crystals, respectively. From the process of white light supercontiuum generation, a broadband seed signal beam whose spectrum extends up to about 1.1 µm is formed after the pump beam has passed through the sapphire. The seed is then amplified through four sequential KTA amplification stages. Note that only a part of the spectrum of the seed is amplified, depending on phase matching angle of the KTA crystals. Inthe last amplification stage, a submillijoule, MIR (3-4 µm) idler beam is generated. The output pulse energy of the idler beam as a function of wavelength is shown in Fig. 2.4.

Figure 2.4: Output pulse energy of the idler beam as a function of wavelength.

2.2 High temperature atomic beam source

We built an effusive oven in collaboration with Prof. M. F. Kling’s group from MaxPlanck

Institute of Quantum Optics (MPQ). It is used for gas phase experiments on metal atoms and C60 fullerences which are solids at room temperature. Initial design and construction were accomplished at MPQ; performance tests and partial redesign and rebuild of the setup were performed at OSU. Figure 2.5 displays 3D drawings of the setup, it is integrated to a 8”

CF flange as it has to be mounted to our time-of-flight spectrometer which will be described

24 Figure 2.5: 3D drawings (Autodesk Inventor) of the atomic beam source setup.

in Sect. 2.3. It is mounted under the interaction region of the spectrometer. The cylindrical structure on top of the flange is a cooling jacket for the oven, which is accomplished by continuous flow of water through the tube wound around the jacket. There is aholeat the top of the setup which is the output for the atom vapor. The actual oven is inside the cooling jacket, as shown in panel (b). The oven is built on a 4 1/2” blank CF flange

(with electrical feedthroughs for heaters and thermocouple pair, not drawn) and is attached to the main flange from the bottom. A schematic diagram of a cross section of thesetup is shown in Fig. 2.6(a) and photos of the oven assembly is shown in Fig. 2.6(b). There are two heaters (HeatWave Labs) in the oven assembly. One is a button shaped heater of

1” diameter sitting on a ceramic plate supported by three legs. The sample reservoir is a stainless steel cup sitting on the heater and a type K thermocouple pair (HeatWave Labs) is in contact with the reservoir. Upon heating, the atom vapor of the sample escapes from the 1 mm aperture on the lid of the reservoir and then passes through the 1mm skimmer aperture which is 2 cm above the reservoir aperture. To prevent clogging of the aperture the skimmer is heated by a donut shaped heater. The outer diameter of the heater is 1” and the inner diameter is 0.4”.

25 (a) (b) Skimmer

Heater

Supporting leg

8” CF flange

4 1/2” CF flange

Figure 2.6: Structure of the oven. Panel (a) shows a schematic diagram and panel (b) shows photos of different parts.

Figure 2.7: Performance of the oven. Panel (a) shows the VI curve of the heater for the reservoir; panel (b) shows the reservoir temperature as a function of input power to the heater.

Figure 2.7(a) shows the voltage-current plot of the heater for the reservoir. The temperature of the reservoir (not the heater) as a function of input power is displayed in Fig.

2.7(b). The temperature measurement is sensitive to the contact between the thermocouple

26 and the reservoir so the uncertainty is tens of degrees. According to the specifications the maximum allowed input power to the button heater is 190 W.

2.3 Time-of-flight spectrometer

Essentially, our experiments of ionization of atoms and molecules are designed to measure (i) ionization probability and (ii) momentum distribution of the photoelectrons. The first one requires counting of the number of ionized atoms due to the laser pulses and the second one is achieved by counting the number of photoelectrons as a function of kinetic energy and emission angle. All measurements (ions or electrons) presented in this dissertation were performed using a home-built time-of-flight spectrometer housed in a ultrahigh vacuum

(UHV) chamber pumped by two turbomolecular pumps (a 345 liter/sec pump (Leybold

Turbovac TMP361) a 360 liter/sec pump (Leybold Turbovac 360 CSVG)). The background pressure of the chamber is ∼ 10−9 torr. A schematic diagram of the setup is shown in Fig.

2.8. The laser beam (labeled as “L” in the figure) passes through a convex lens and enters the chamber through the transparent input window (labeled as “IW” in the figure). The position of the laser focus in the chamber is called the interaction region, in which ionization of atoms (or molecules) takes place. Upon ionization, an ion and an electron are formed at the laser focus. Our spectrometer can detect either ions or electrons and the configurations for the two detection modes will be described in Sect. 2.3.2 and 2.3.3, respectively. In either case, the goal is to measure the distribution of “time-of-flight” ttof of the charged particles, that is, the time taken for the particle to drift along the flight tube from the ionization location to the detector. For a description of the working principle of the detector please see [65, 66]. In brief, when a high voltage is applied across the multichannel plate (MCP), the impact of an ion or electron on its front surface starts a cascade of electrons in the microchannels and a shower of millions of electrons is generated at its back face. These electrons are coupled out capacitively to obtain an electrical pulse.

To avoid the influence of any stray magnetic fields on the electron trajectories, the interaction region and the travel path of the flight tube are surrounded by two grounded,

27 gold-coated mu-metal tubes (labeled as “MM” in the figure), respectively. In addition, a grounded Faraday cage (labeled as “FC” in the figure) is inserted inside the mu-metal tube for the travel path.

There are two ways to deliver the target to the interaction region. For targets which are in gas phase at room temperature, we simply connect a gas reservoir to the chamber and let the gas flow into it via a leak valve which allows a fine control of flow rate. Withthis steady gas inflow and evacuation by the turbomolecular pumps, a constant gas density is achieved for performing experiments. Typical gas pressures ranging between 10−8 and 10−4

torr 11. The lowest attainable pressure is limited by the chamber background pressure and

the highest attainable pressure is limited by space-charge effects and the working pressure

limit of the MCP detector. For metal atoms and C60 fullerences which are solids at room temperature, we used the effusive oven described in Sect. 2.2 to heat up the sample and

produce an atomic vapor beam which intersects the laser beam at the interaction region.

Before we explain the differences between electron and ion detection and how to convert

the time information into the physical quantities of interest, let us have an overview of the

data acquisition system.

2.3.1 Data acquisition system

paraThe data acquisition essentially consists of a “stopwatch” which measures ttof and a controller for data storage and transfer. The “stopwatch” is a high resolution time-to-digital

converter (TDC) (LeCroy TDC4208 or 2228A) which starts counting time upon receiving

a START electrical pulse and stops upon receiving a STOP electrical pulse. Depending

on the TDC model the width of the time bin can be selected between 50 ps, 100 ps, 285

ps or 1 ns. Ionizations occur when the laser pulse arrives at the focus position in the

interaction region. In principle, one might put a photodiode at the laser focus then the

generated photocurrent could be used as the START pulse. However, it is unnecessary

(and impractical) to put the photodiode exactly at the event location. In fact, even doing

that does not give an exact value of the starting time because the electrical pulse will be

11The corresponding ideal gas density is between 3.3 × 108 to 3.3 × 1012 cm−3 at 20 ◦C

28 Figure 2.8: Schematic diagram of time-of-flight (TOF) spectrometer (top view). L: laser; IW: input window; WW: output window; P1,P2,P3: Electrostatic field plates for accelearat- ing ions; MM: mu-metal; FC: Faraday cage; G: grounded grid; MCP: multichannel plates; A: collecting anode. Reproduced from [66].

delayed by the cables and electronic components on its way to the TDC. In other words, the ttof measured by this setup will always have an inherent time offset which is yet to be determined but it is just a constant independent of the physical parameters of the laser and the target. The offset value can be determined unambiguously and will be explained in Sect. 2.3.2. In our setup, an InGaAs photodiode is placed on the side of the optical beam path to the chamber where it receives a small amount of laser energy reflected by a CaF2 pick-off window. The photodiode signal is fed to an amplifier (Ortec 574)and then to a constant-fraction discriminator (Ortec 574). The output of the discriminator is a

NIM-standard 12 fast negative pulse and it is the START pulse for the TDC.

The STOP pulse is provided by the output of the detector. The output is first sent to an amplifier (Ortec VT120C) followed by a pico-timing discriminator (Ortec 9307) which outputs a NIM signal which is finally fed to the TDC. For each laser pulse, multiple par-

12The Nuclear Instrumentation Module standard for logic signal, with -16 mA into a 50Ω load creates a -0.8 V pulse.

29 ticles could be detected which results in multiple STOP pulses. The TDC can record at a maximum of eight STOP pulses for each laser pulse. In addition to the START and STOP pulses, the operation of the TDC requires two additional NIM signal pulses, one is called

“end-of-window” (EOW) pulse and the other is called “clear” (CLR) pulse. For ionization measurement from each laser shot, the TDC receives the signal pulses in this order: START,

STOP (eight of them at max.), EOW, CLR. Once the TDC receives the EOW pulse, no more STOP pulses (if any) will be accepted and the TDC will transfer all the registered time-of-flight values to a controller (CAMAC CC32). After that, a CLR pulse clears allthe existing registered numbers and the TDC is now ready again to take the START and STOP pulses from the next laser pulse. EOW and CLR pulses are generated using a delay/pulse generator (Stanford DG535) synchronized with the 1kHz signal of the laser system.

2.3.2 Detection of electrons paraFrom the ttof of an electron together with the traveled distance L, one could obtain the kinetic energy of the electron simply by

m L E = e ( )2. (2.4) 2 ttof

Typical energy is only hundreds or up to one thousand eV so relativistic effect can be safely neglected (The rest mass energy of an electron is 0.51 MeV). As mentioned earlier, there

′ is a systematic offset between the measured time-of-flight ttof and the actual time-of-flight ′ ′ ttof , that is, ttof = ttof + t0. Electron counts as a function of ttof from ionization of argon atoms by 800 nm pulses is shown in Fig. 2.9(a). The ATI peaks are not equally spaced in time but as explained in Sect. 1.2.2 they are equally spaced in energy with a spacing of the laser photon energy. In fact, this generic property helps us to determine t0 and L. Since the time-to-energy scale conversion depends on t0 and L, so does the spacings of the peaks in the converted spectrum. Thus, the proper t0 and L can be obtained by finding the best fit of the ATI peaks to the photon spacing, as shown in Fig. 2.9(b). Note that the conversion of the signal as a function of time St to the signal as a function of energy SE requires the − 3 − 3 conversion Jacobian dttof /dE ∝ E 2 . From Eq. 2.4 it can be found that dttof /dE ∝ E 2 , 30 Figure 2.9: Conversion from (a) time-of-flight distribution to (b) energy spectrum of photoelectrons. The spectrum is obtained from ionization of argon atoms by linearly polarized 0.8 µm pulses at 1014 W/cm2. The width of the time bin is 1 ns. The vertical dashed lines in panel (b) are shown at increments of the photon energy at 0.8 µm (1.6 eV).

− 3 thus, SE ∝ E 2 St. In this setup only the electrons emitted along the TOF axis (with a finite acceptance angle ±1◦) will reach the detector. For example, the spectrum in Fig. 2.9 was measured with the laser polarization being parallel to the TOF axis so what we measured are the electrons emitted along the laser polarization. To measure photoelectron spectra for another emission angle θ one needs to make the angle between the TOF and the laser polarization to be at

θ. This can be achieved by rotating the laser polarization using a half waveplate.

2.3.3 Detection of ions paraThe interaction between a laser pulse and an atom or molecule does not necessarily produce one electron and one ion. For example, double ionization of a helium atom will give two electrons and one doubly charged ions; double ionization of a carbon monoxide

31 molecule followed by dissociation will give two electrons, one oxygen atom ion and one carbon atom ion. In our setup, measurements of electron energy distribution does not give direct information about the final state of the parent atoms or molecules. For instance, getting two electrons counts with a single laser pulse could be either from a double ionization event from one atom or two single ionization events from two atoms. Although one of the outcomes might be much more probable than the others, experimentally they cannot be distinguished using electron spectroscopy alone. The purpose of measuring ions is to get the charge state distribution of the ionized target atom. For molecules, we can also obtain the fragment distribution due to dissociation assuming that the fragment is charged.

Figure 2.10: Ion mass spectrum of magnesium irradiated by linearly polarized 800 nm pulses.

To perform time-of-flight mass spectrum measurements of ions, static voltages areap- plied to the metal plates P1,P2 and P3 (see Fig. 2.8). The electrostatic fields across the plates make the ions accelerate towards the MCP detector. That is, the main difference between electron and ion measurements is that the velocity gained by the electrons is just

32 from the laser field while the velocity gained by the ions is mostly from the applied static field. Also, unlike photoelectron detection where only the ones emitted along theTOFaxis are collected, essentially the electrostatic field collects all the ions. After passing through the aperture of P3, the ions drift along the field-free region of the flight tube and reach the MCP detector. While P3 is grounded, the voltages on P1 and P2 are positive with

V1 > V2. For optimal resolution of the ion mass spectrum the voltages are set to satisfy the Wiley-McLaren condition [67]. The velocity of an ion of mass m with a charge q at the aperture of P3 is given by r 2qV v = , (2.5) m where V is the electric potential at the position of ionization (the laser focus) and the initial velocity of the ion is assumed to be negligible. Thus, the relation between the mass-to-charge ratio of an ion and its time-of-flight is13

rm t ∝ . (2.6) tof q

As an example, a mass spectrum of magnesium atoms ionized by 0.8 µm pulses is displayed in Fig. 2.10. The three peaks on the right are from the singly charged ions of the three main isotopes and the three peaks on the left are from the singly charged ions. The operational resolution ∆m/m < 10−3 so we could readily resolve single mass differences.

2.3.4 Effect of focal volume averaging paraThe purpose of this section is to explain an averaging effect which is inherent in our detection system. While the number density of the particles within the interaction region can be treated to be uniform to a good approximation, the spatial distribution of the laser intensity is not. Usually, a single mode laser beam is modeled by TEM00 Gaussian beam, assuming z to be the propagation direction and z = 0 to be the focal plane then the intensity distribution is given by

13Again, there is a systematic offset between the measured time-of-flight and the actual time-of-flight due to the reason described in Sec. 2.3.1. But typical time-of-flight for ions is in the order of10 µs which is much large than t0 which is less as 100 ns.

33 Figure 2.11: Contour plot of intensity distribution of a focused Gaussian beam. The beam propagates horizontally on the page. The pair of black dashed curves represent the 1/e2 diameter of the beam as a function of propagation distance.

I0 −2(x2+y2)/w2(z) I(x, y, z) = 2 e , (2.7) 1 + (z/zR)

2 p 2 where zR = πw0/λ is known as the Rayleigh range and w(z) = w0 1 + (z/zR) is the 2 1/e radius as a function of z. I0 is the peak intensity of the distribution. Figure 2.11 is a contour plot of Eq. 2.7 for y = 0 plane. The volume of the region where the intensity is higher than Ii is given by

4 2 4 V (I > I ) = πw2z ( β + β3 − arctan β), (2.8) i 0 R 3 9 3 p where β = I0/Ii − 1. Note that I0 > I1 > I2.... Graphically, it is the volume of the region enclosed by the contour line I = Ii. Therefore, the actual intensity experienced by a particle depends on its position. The measured quantity in experiments Wexp (either electron or ion counts) is the incoherent sum of that quantity for all the intensity values in the focal region, that is, X Wexp = WIi ∆Vi, (2.9) i where ∆Vi(≡ V (I > Ii+1) − V (I > Ii)) is the volume of the iso-intensity shell in which the intensity is between Ii and Ii+1 (i.e. the volume of the region between two neighboring contour lines in Fig. 2.11). The effect of focal volume averaging depends on the sensitivity of

34 the measured features to intenstiy, here we briefly describe two examples which are relevant to our discussions in the subsequent chapters:

ATI spectrum

Figure 2.12: TDSE simulations of ATI spectrum of argon irradiated by 0.68 µm pulses with a peak intensity 120 TW/cm2. Blue line: spectrum from a single intensity (120 TW/cm2). Red line: focal volume averaged spectrum. For visibility, the red spectrum is multiplied by an arbitrary factor relative to the blue spectrum.

Figure 2.12 shows TDSE simulations of photoelectron spectrum for argon atom irradi-

ated by 0.68 µm pulses with a peak intensity 120 TW/cm2. The blue curve is a spectrum

simulated at just one intensity (120 TW/cm2) and the red curve is the focal averaged spec-

trum, that is, the weighted sum of spectra over a wide range of intensities according to

Eq. 2.9. As mentioned in Chap.1 the ATI peaks are at En = n~ω − Ip − Up, so the peak positions shift with intensity. Therefore, focal volume averaging results in broadening and

loss of contrast of the peak structures. It can be seen that some part of the spectrum (e.g.

from 0 to 15 eV) suffers a more severe loss of contrast than some other parts, theexact

mechanism will not be discussed here but essentially it is because the intensity dependence

of photoelectron yields for different energy ranges are different.

35 Ionization yield as a function of intensity

Figure 2.13: Calculated ionization yield of krypton atom as a function of laser intensity using PPT formula. The laser wavelength is 0.8 µm. Blue line: Result without the effect of focal volume averaging. Red line: Result with the effect of focal volume averaging. Black line: I3/2. For visibility, the red curve is multiplied by an arbitrary factor relative to the blue spectrum.

Figure 2.13 shows calculations of ionization yield of krypton atom as a function of laser intensity (experimentally it is measured by counting the number of ions created as a function of intensity) using PPT formula (see Chap.3 for details). The blue curve shows the results without the effect of focal volume averaging. Overall, ionization yield increases with intensity but local minima appear at some particular intensities which satisfies n~ω − Ip − Up = 0. The mechanism of the formation of the minima will be explained in Chap.3 and is not relevant to the discussion here. After focal averaging, the modulations are averaged out and only very shallow modulations remain in the red curve. As intensity goes beyond the so-called saturation intensity Is, population of the atomic ground state is depleted and the ionization probability could not increase anymore, as shown by the blue curve. However, due to the intensity distribution of the laser focus, even when the central part of the region reaches or goes beyond Is, the intensity of some outer region is still lower

36 than Is. As intensity increases, the inner “saturated region” expands and so the yield keep increasing, as shown by the red curve. Regardless of ionization mechanism, this increase which is purely due to the geometry of the Gaussian laser beam has an intensity dependence of I3/2. Such a dependence is simply because the second term of Eq. 2.8 dominates at high intensities. If the laser beam is not a Gaussian or if the ions are not collected from the full focal region (or if the number density of the atoms is not uniform), then the dependence would change.

37 Chapter 3 Experimental investigation of Strong field ionization theories from visible to mid-infrared wavelengths

paraIn Chapter1 we have mentioned that the applicability of Keldysh’s ionization rate formula to compare with experimental data of real atoms is limited since the influence of the atomic core to the photoelectron wavefunction is completely neglected. The prefactor of

Keldysh’s formula was inaccurate. Soon after Keldysh’s publication [9], Perelomov, Popov and Terentev (PPT) presented an improved formula for strong field ionization rate (also see a paper by Smirnov and Chibisov [68]). PPT formula was derived for a short-range potential with a first-order correction to take the long range Coulomb interaction into account sothe asymptotic wavefunction is more accurately described. PPT formula could be applied to different atomic ground state with different l amd m values.

Keldysh theory and other related strong field ionization (SFI) models including PPT

and SFA had been essentially neglected until the 1980s when intense short pulsed lasers

became available. In 1986, Ammosov, Delone and Krainov [31] introduced a formula for

tunnel ionization rate which is now known as the ADK formula. In fact, ADK formula

can be recovered from PPT formula in the limit γ ≪ 1 (although it was not mentioned

by the authors of [31]). It should be stressed that no discrete binding states other than

the ground state is considered in PPT or ADK. In practical application ADK is more

38 commonly used than PPT due to its simplicity. It has been widely used even in the regime of γ ∼ 1. In this chapter, we aim to investigate the applicability of PPT and ADK by a comprehensive comparative study between experiment and theories of the total intensity- dependent ionization yield for different atom species at different laser wavelengths andfor linear and circular polarizations.

3.1 PPT and ADK Formulas paraBefore turning to our experimental results we first recall the theoretical formulas and point out the physical meanings of some of the terms, for derivations the reader is referred to the original papers [29–31]. Same as Keldysh’s result, PPT ionization rate is expressed as a sum in which each term represents the rate due to the ATI process of absorbing q

photons: ∞ X wPPT (F, ω) = wq(F, ω), (3.1)

q≥qmin where F and ω are the amplitude and frequency of the laser field, respectively. The minimum

number of photons required to reach the effective ionization threshold Ip + Up is qmin ≡

⌈(Ip + Up)/ω⌉. The full cycle-averaged PPT expression is [29]:

2n∗−|m|−1 2(2I )3/2 2 2 2 |m|+3/2 − p g(γ) w (F, ω) =c ∗ ∗ f(l, m)I (1 + γ ) 2 A (ω, γ)e 3F , (3.2) PPT n l p F n∗3 m with " # 3 1 p1 + γ2 g(γ) = 1 + sinh−1 γ − , 2γ 2γ2 2γ

2 ∞ 4γ X −α(γ)(q−ν) p Am(F, ω) = √ e wm β(γ)(q − qmin) , 3π|m|!(1 + γ2) q≥qmin

x2|m|+1 Z 1 e−x2tt|m| wm(x) = √ dt, 2 0 1 − t

! γ α(γ) = 2 sinh−1 γ − , p1 + γ2

39 2γ β(γ) = , p1 + γ2

I 1 ν = p 1 + , ω 2γ2 22n∗ (2l + 1)(l + |m|)! where c2 = and f(l, m) = . In above n∗l∗ n∗Γ(n8 + l∗ + 1)Γ(n∗ − l∗) 2|m|(|m|)!(l − |m|)! ∗ p ∗ equations, F is the field amplitude, n = 1/ 2Ip is the effective quantum number, l = n∗ −1 is the effective orbital quantum number,x Γ( ) is the gamma function, and l and m are

orbital and magnetic quantum numbers, respectively, with respect to the quantization axis

defined by the laser polarization direction. The factor/F (2 n∗3)2n∗ in Eq. (3.2) takes long

range coulomb interaction into account and p3F n∗3/π is the result due to cycle averaging.

It is known that the m = 0 orbital dominates the ionization as its electron density is

primarily distributed along the quantization axis, where all nonzero m orbitals present as

nodes which does not favor the ionization of the electron.

For circular polarization, the cycle-averaged PPT rates is given by [69]

2 2n∗ c ∗ ∗ Ip 2 1 ws (F, ω) = √n l (1 + )1/2 PPT 3/2 F n∗3 γ2 4 2πqmin r 2 2 (3.3) ∞ 2 4qmin −1 ζ2+γ2 ζ +γ X 1 + γ − 1+ζ (tanh 2 − 2 ) × (1 + ζ)p(1 − ζ)( )3/4e 1+γ 1+γ , ζ2 + γ2 q≥qmin for s orbitals,

2 2n∗ p 3c ∗ ∗ Ip 2 1 w 0 (F, ω) = √n l (1 + )3/2 PPT 5/2 F n∗3 γ2 16 2πqmin r 2 2 (3.4) ∞ 2 4qmin −1 ζ2+γ2 ζ +γ X 1 + γ − 1+ζ (tanh 2 − 2 ) × (1 − ζ2)p1 − ζ( )5/4e 1+γ 1+γ , ζ2 + γ2 q≥qmin for p orbitals with m = 0, and

40 2 2n∗ p± 3c ∗ ∗ Ip 2 1 w (F, ω) = √ n l (1 + )3/2 PPT 3/2 F n∗3 γ2 8 2πqmin ∞ s X ζ2 + γ2 1 1 + γ2 × ( ∓ ζsign(m))2 √ ( )3/4 (3.5) 1 + γ2 1 − ζ ζ2 + γ2 q≥qmin

r 2 2 2 2 − 4qmin (tanh−1 ζ +γ − ζ +γ ) × e 1+ζ 1+γ2 1+γ2 , for p orbitals with m = ±1. Note that qmin = (2Up + Ip)/ω for circular polarization and

ζ ≡ 2qmin/q − 1. Essentially, Eq. (3.3) and (3.5) are Eq. (88) and (90) in [69], but here we have included the Coulomb correction factor (2/F n∗3)2n∗ [70]. The ionization rate for

m = 0 states is much smaller than the rate for m = ±1 state.

The Ammosov-Delone-Krainov (ADK) ionization rate, on the other hand, is given by

3/2 2 2 2n∗−|m|−1 2(2Ip) w (F ) = c ∗ ∗ f(l, m)I ( ) exp[− ]. (3.6) ADK n l p F n∗3 3F

This ADK formula for instantaneous ionization rate can be derived from eqn.(3.2) by taking the limit γ → 0 (physically, this corresponds to ionization in a static electric field) and dropping the prefactor due to cycle averaging. Since the ADK formula does not contain

ω, it is not able to predict any wavelength dependence of ionization rates. Since it is for calculating instantaneous rate, both CP and LP take the same formula. Alternatively, it can be exactly derived by considering the tunneling rate of an electron through a Coulomb potential tilted by a static field [71]. For the case where the field strength is so strong such that the peak of the potential barrier becomes lower than the atomic ground state energy, the so-called “Barrier Suppression Ionization” (BSI) occurs. The onset intensity for BSI is given by (in atomic units)

4 IBSI = Ip /16. (3.7)

The atomic parameters of the target atoms considered in this chapter are tabulated in Table 3.1. To calculate ionization probability of an atom by a laser pulse we solve the following rate equation dP 0 = −w(F (t))P , (3.8) dt 0 where P0 is the population of the atomic ground state. The initial value of P0 is set to 1

41 ∗ 2 2 Ip (eV) n cn∗l∗ IBSI (TW/cm ) Ne 21.56 0.794 4.244 862 Ar 15.79 0.929 4.116 246 Kr 13.99 0.986 4.025 153 Xe 12.13 1.059 3.882 86 Na 5.14 1.627 2.290 2.8 K 4.34 1.770 1.890 1.4 Mg 7.65 1.334 3.163 13.3 Zn 9.39 1.203 3.532 31

Table 3.1: Atomic parameters for PPT and ADK calculations.

and the system start to evolve as the laser field ramps up. The ionization probability is then given by Z +∞ P = 1 − exp(− w(F (t))dt), (3.9) −∞ where F0(t) is the pulse envelope which is assumed to have a sine-squared shape:  2 2ωt 11nπ 11nπ F0 cos ( ) cos(ωt)ˆz, if − < t < F⃗ (t) = 11n 4ω 4ω (3.10) 0, otherwise where n is the full width of half maximum (FWHM) pulse duration in terms of number of optical cycles.

0 10 PPT ADK

−2 10

−4 Ionization yield (arb. un.) 10

13 14 10 10 Intensity (W/cm2)

Figure 3.1: Comparison between PPT and ADK ionization yield as a function of laser intensity for krypton atom at 0.8 µm driving wavelength.

42 As an example, ionization probability as a function of laser intensity for krypton atom at 0.8 µm driving wavelength calculated by PPT and ADK formulas are shown in Fig. 3.1.

It can be seen that PPT gives a much higher ionization probability than ADK. It is not difficult to understand because in PPT model ionization probability would increase with the photon energy ~ω but on the other hand ADK can be treated as the extreme case of PPT in which ω → 0. Therefore, it is expected that any PPT calculations with a nonzero

ω will give a higher probability than ADK. As mentioned in Sect. 2.3.4, in the PPT curve there are local minima on top of the overall increasing trend. As indicated by the vertical dashed lines, these minima occur at intensities where n~ω −Ip −Up = 0 with nmin being the minimum number of photons required for ionization. This is the so-called channel closure effect: since the effective ionization potential Ip + Up increases with laser intensity, nmin has to increase by one whenever the original energy value nmin~ω is no longer sufficient to ionize the atom. In other words, the first term in expression 3.1 is dropped when the intensity sweeps across a channel closure intensity and that results in a local minimum in ionization probability.

As explained in Sect. 2.3.4, in our experiments the ions emerge from the full Gaussian laser beam focus, so we need to take the effect of volume averaging into account inour calculations for comparison with experimental data. Unless specified otherwise, effect of focal averaging have been included in the PPT and ADK calculations presented in the following sections.

3.2 Experiments paraIn the past, strong field ionization yields as a function of laser intensity havebeen investigated mainly for noble gas atoms using near-infrared (NIR) wavelengths (≤ 1µm).

Numerous results (e.g. see [18, 72]) have shown deviations from the predictions of the ADK model. Ion yields measurements on alkali and alkaline earth atoms have also been studied in the multiphoton regime [73–76], but no comparison with SFI theories were made. Saturation intensities (relative to xenon) in transition metals have been compared to ADK predictions

43 with strong disagreement factors ranging from 2 to 7[77]. Similar SFI comparisons for organic molecules have resulted in similar disagreement with ADK [78]. The only case of a single electron atom (H) ionization [79, 80] resulted in excellent agreement with TDSE but was not compared to the SFI theories. In our study, ion yield measurements were performed on atomic targets including noble gases (Ne, Ar, Kr, Xe), alkali (Na, K), alkaline earth (Mg) and one transition metal (Zn) with wavelengths ranging from 0.4 to 4 µm.

Table 3.2 summaries the experimental parameters for different target atoms. The laser and detection system have been described in Chapter2. The laser pulse energy is controlled by a half wave plate (HWP) followed by a polarizer, complemented by neutral density filters or pellicle beamspliters. Ellipticity is controlled by the quarter wave plate (QWP) located after these optics. Laser intensities were calibrated by using the 10Up cutoff in photoelectron spectrum of noble gases [53, 81, 82]. The experimental intensities in some cases are slightly scaled (≤ 20% from the calibration) to achieve best fit to the theoretical ion yield curves.

To switch between linear and circular polarization (LP and CP) at a fixed intensity, the √ QWP is rotated without adjusting the laser energy (that the CP field is the LP field / 2).

3.2.1 Experimental Data

Noble Gas Atoms paraNoble gas atoms are the most commonly used targets in strong field studies. The ion yields as a function of laser intensity for Ar, Kr, Xe and Ne at different wavelengths (see captions) are displayed in Fig. 3.2 to 3.5. The symbols are experimental data and the full and dashed curves are the PPT and ADK calculations, respectively. In general, for both LP and CP cases, the yields increase rapidly at low intensities and progressively satu- rate. Beyond saturation the yield follows a I3/2 scaling due to the geometrically expanding

Gaussian focal volume [83], as explained in Sect. 2.3.4.

We first consider the results at visible and NIR wavelengths (0.4, 0.8 and1.3 µm).

For all four target atoms, the agreement between experimental data and PPT results are

44 Table 3.2: Driving wavelengths λ and intensities ranges (Imin − Imax) of experimental data. The uncertainty of our intensity calibration is about 20%. The γ value at the barrier- suppression intensity and Imin are denoted as γBSI and γmax, respectively.

λ Imin Imax γBSI γmax (µm) (TW/cm2) (TW/cm2) Ne 0.8 114 1030 0.46 1.3 Ar 0.4 26 240 1.5 4.5 0.8 24 390 0.73 2.3 1.3 58 230 0.45 0.91 Kr 0.8 16 200 0.88 2.7 3.3 26 140 0.21 0.52 3.6 30 130 0.19 0.44 3.9 29 100 0.18 0.41 Xe 0.8 11 170 1.1 3 3.3 21 89 0.26 0.54 Zn 0.8 21 110 1.6 1.9 1.3 35 150 0.98 0.92 2 23 120 0.64 0.74 3.6 21 120 0.35 0.43 Mg 0.8 6.1 52 2.2 3.2 3.6 9.7 44 0.49 0.57 Na 3.2 1.9 10 0.98 1.2 3.6 2 11 0.87 1 3.7 2.2 9.1 0.85 0.95 4 1.9 12 0.79 0.94 K 3.2 0.83 5.8 1.3 1.7 3.6 0.66 4.8 1.1 1.7 4 0.71 4.7 1 1.4

45 reasonable, while ADK rates always show a more rapid intensity dependence than the experiment data. The deviations between experiments and ADK models are larger for lower laser intensities. Comparing the case of Ar (Ip = 15.76eV) and Kr (Ip = 13.99eV) (Fig. 3.2(b) and 3.3(a)) at the same intensity, the deviation between experimental yield and ADK result is smaller for the case of Kr. Also, it can be seen from the Ar results that the deviation between experiment and ADK are significantly smaller in the case of 1.3 µm than in the

case of 0.4 µm. The observations above are based on the effect of changing three distinct

experimental parameters: ionization potential, intensity and wavelength. Regardless, as

long as the changes lead to smaller values of γ, the deviation between experiment and ADK

results become less. The situation is different when ionization is deep in the tunneling

regime. Figure 3.3(b) and 3.4(b) shows the ion yields of Kr and Xe at MIR wavelengths

with linear polarization. The γ values in these data range from about 0.2 to about 0.5. In

this case PPT and ADK results are very close to each other and are in good agreement with

experimental results over their entire range of intensities. These data illustrate ionization

deep in the tunneling regime where the experiments agree well with both PPT and ADK

model, when γ ≤ 0.5.

0 10 PPT−LP PPT−CP (b) ADK−LP −2 10 Ar(0.8µm) ADK−CP Expt.−LP Expt.−CP −4 10

Ion Yield (arb. un.) −6 Ion Yield (arb. un.) 10

(a) (b) (c) −8 10 1 2 1 2 1 2 10 10 10 10 10 10 Intensity (TW/cm2) Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.2: Ion yields of Ar as a function of intensity at (a) 0.4 µm, (b) 0.8 µm and (c) 1.3 µm for linear and circular polarization. LP: linear polarization; CP: circular polarization.

0 0 PPT−LP Expt.(3.3) 10 10 PPT−CP Expt.(3.6) ADK−LP Expt.(3.9) ADK−CP −2 PPT(3.3) 10 Expt.−LP PPT(3.6) −2 10 Expt.−CP PPT(3.9)

−4 ADK 10

−4 −6 10 10 Ion Yield (arb. units.) −8 10 −6 10 (a) (b) −10 10 1 2 1 2 10 10 10 10 Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.3: Ion yields of Kr as a function of intensity at (a) 0.8 µm and (b) 3.3, 3.6 and 3.9 µm. The three data sets are displaced from each other arbitrarily along the vertical axis for clear illustration.

0 10 (a) (b)

−2 10

−4 10 PPT−LP PPT−CP

Ion Yield (arb. units.) ADK−LP

−6 ADK−CP PPT−LP 10 Expt.−LP ADK−LP Expt.−CP Expt.−LP

1 2 1 2 10 10 10 10 Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.4: Ion yields of Xe as a function of intensity at (a) 0.8 µm and (b) 3.3 µm. LP: linear polarization; CP: circular polarization.

47 0 10

−2 10

−4 10 PPT−LP PPT−CP

Ion Yield (arb. units.) ADK−LP

−6 ADK−CP 10 Expt.−LP Expt.−CP

2 3 10 10 Intensity (TW/cm2)

Figure 3.5: Ion yields of Ne as a function of intensity at 0.8 µm. LP: linear polarization; CP: circular polarization.

Alkali Atoms paraAlkali atoms have much lower ionization potentials than noble gases, so laser intensities have to be kept low to avoid saturation of ionization process. Therefore, long wavelength fields are required to achieve comparable γ values as in the measurements of noble gases.

Also, unlike noble gases, they have a single valence electron (hydrogen-like atoms).

Ionization yields as a function of intensity of Na atom at 3.6 µm (both LP and CP) are shown in Fig. 3. Similar to the data of noble gases in NIR fields, the ionization yields increase rapidly with intensity and eventually approach the I3/2 scaling. Also, the yield for

LP is again about one to two order of magnitude higher than the case of CP. The range of

γ values of this data set is similar to that of the Ne data at 0.8 µm. Also, the ratio between the ionization potential (5.14 eV for Na; 21.56 eV for Ne) and photon energy (0.34 eV for

3.6µm; 1.55 eV for 0.8µm), that is, the degree of nonlinearity, are similar in the two cases.

Other data set of Na and also K at different MIR wavelengths are shown in Fig. 4. Similar to the results of noble gases in NIR fields, all data agree well with the PPT results and deviate from ADK results. At the same intensity, the discrepancy between experimental yield and ADK result is smaller at longer wavelength. Again, this has verified that smaller

48 γ result in smaller deviation between experiment and ADK theory.

PPT−LP 0 10 PPT−CP ADK−LP ADK−CP

−1 Expt.−LP 10 Expt.−CP

−2 10 Ion Yield (arb. units.)

−3 10

0 1 10 10 Intensity (TW/cm2)

Figure 3.6: Ion yield of Na as a function of intensity at 3.6 µm. LP: linear polarization; CP: circular polarization.

0 10 (a) (b)

−2 10

−4 10 Expt.(3.2) Expt.(3.2) Expt.(3.7) Expt.(3.6) Ion yield (arb. units) −6 Expt.(4) Expt.(4) 10 PPT(3.2) PPT(3.2) PPT(3.7) PPT(3.6) PPT(4) PPT(4) −8 ADK ADK 10 0 1 0 1 10 10 10 10 Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.7: Ion yields as a function of intensity of (a) Na at 3.2, 3.7 and 4 µm; (b) K at 3.2, 3.6 and 4 µm. The three data sets in each panel are displaced from each other arbitrarily along the vertical axis for clear illustrations.

49 Although there are big difference between the measurements of noble gases targets at NIR wavelengths and alkali targets at MIR wavelengths in terms of laser intensity, wavelength and ionization potential, the γ values are comparable. As described above, qualitatively there are quite a few of common features in the ionization yields of the two systems. The comparison between the two systems illustrates the universal applicability of the Keldysh parameter.

In fact, the overall good agreement between the experimental data and PPT calculations, which do not take any excited states into account, seems to suggest that the dipole coupling between the ground and the first excited state does not play a significant rolein our experiments. This might seems, at first sight, a bit surprising, but our findings do not contradict with the theoretical studies on ionization of K atom at MIR wavelengths by

Gaarde and Schafer [84]. Their TDSE simulations suggested that the ionization yield would be enhanced around the wavelengths of three and five photon resonance (2.43 and 4.05 µm in field free environment), which are different from the wavelengths used inFig. 3.7(b).

Also, the non-uniform intensity distribution in the laser focal region in experiments may also hinder the visibility of the ionization enhancement as the calculations in [84] has shown that the resonance wavelengths shifts with intensity. To observe such resonance enhancement it is probably necessary to measure the absolute ionization rate at a fixed intensity as a function of wavelength across the resonance. Alternatively, the ratio of ion yields between the case of LP and CP might show similar enhancement feature since the resonance should play no role in ionization for CP since multiphoton transition from s to p state is forbidden.

Zinc and Magnesium

paraIon yields curves of Zn at 3.6, 2, 1.3 and 0.8 µm for both LP and CP are displayed

in Fig. 3.8. Similar to the results of all the other targets presented above, the experiment

agree well with PPT calculations while deviate from the ADK calculations. At the same

intensity, the deviation decreases as wavelength increases, due to the decrease of γ. Also,

the yields from CP fields are closer to the yields from LP fields as the wavelength isgetting

shorter. Figure 3.9 shows the results of Mg at 0.8 and 3.6 µm. The result of 3.6 µm

50 (γ = 0.25 − 0.57) are in good agreement. The data set at 0.8 µm has a much higher γ range

(1.1-3.2). It is not surprising that there are significant deviations between experiment and

ADK calculations. The agreements between PPT results and experiment are good. Note that ionization of Mg using a Ti:sapphire laser had also been studied by Gillen et al [76].

0 (a) (b) 10

−1 10

−2 10 PPT−LP PPT−CP ADK−LP −3 10

Ion Yield (arb. units.) ADK−CP Expt.−CP Expt.−LP −4 10 (c) (d)

0 10

−1 10 Ion Yield (arb. units.)

−2 10

1 2 1 2 10 10 10 10 Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.8: Ion yields of Zn as a function of intensity at (a) 3.6 µm, (b) 2 µm, (c) 1.3 µm, and (d) 0.8 µm. LP: linear polarization; CP: circular polarization.

1 10 PPT−LP PPT−LP 0 PPT−CP 10 ADK−LP ADK−LP Expt.−LP ADK−CP Expt.−LP −1 0 Expt.−CP 10 10

−2 10

−1 10

Ion Yield (arb. units.) −3 10

(a) (b) −2 −4 10 10 1 1 10 10 Intensity (TW/cm2) Intensity (TW/cm2)

Figure 3.9: Ion yields of Mg as a function of intensity at (a) 3.6 µm and (b) 0.8 µm. LP: linear polarization; CP: circular polarization.

3.2.2 Discussions paraIn the following three subsections, we will present absolute comparisons between experiments and theories using the following three quantities which do not require absolute measurements of ionization probabilities. The first is Yd ≡ d(log(Y ))/dI which is the deriva-

−1 tive of log(Ion Yield) with respect to laser intensity I and has a units of I . Note that Yd is also equivalent to (1/Y )dY/dI which is the slope of the ion yield divided by the yield.

The second quantity is the saturation intensity Isat and the last is R ≡ YCP /YLP which is the ion yield ratio between CP and LP, it is dimensionless.

Intensity Dependence of Ion Yields

paraTo compute Yd from discrete experimental data points, a polynomial fit to the data

(log10(Ion yield) vs intensity) is performed (see caption of Fig. 9) from which the derivative is extracted. The data of Xe at 0.8 µm and the fitted curve are shown in Fig. 3.10(a).

The values of Yd calculated from the fitted curve and the theoretical curves are displayed in Fig. 3.10(b). It can be seen that PPT and experiment are in excellent agreement while

52 ADK overestimate. Note that the modulation on the PPT curve is due to channel closures

(condition Ip + Up = n~ω). The same procedure is applied to other data set and the results are compiled in Fig. 3.11, which shows the ratio between theoretical (PPT and ADK) and experimental values of Yd at an intensity of 0.8IBSI . Here γ ranges from 0.2 to 2.4. Note that the closer the ratio is to unity, the better is the agreement between theory and

experiment. In the comparison between experiments and PPT, the ratio is close to one

(ranging between 0.9 and 1.2) in all the data sets, even when γ is greater than 2. For ADK,

good agreement with experiments is observed for small γ values but as it approaches 1 the

deviation becomes significant.

0 10 (a)

−2 10

−4 Expt. 10 Fitting

Ion yield (arb. units) PPT

−6 ADK 10

0 10 (b) Expt.(Fitting) PPT ADK

−1 10 d Y

−2 10

20 40 60 80 100 120 140 160 180 Intensity (TW/cm2)

Figure 3.10: (a) log(Ion yield) of Xe as a function of intensity at 0.8 µm. The fitting function is a polynomial of 1/F up to the fifth order, where F is the field amplitude. The fitting parameters are the coefficients of the polynomial. (b) d(log(Y ))/dI calculated from the fitted curve (green dashed line), PPT (red solid line) and ADK (blue dash-dot line) calculations.

53 1.2 1 Xe Zn Kr Kr (3.3) (3.6) Ar Ne Zn 0.8 (3.6)(3.3) (1.3)(0.8) (2) Na (3.2) (theo.) d 0.6 Ar Kr

/Y (0.8) K (0.8) Xe (3.2) (0.8) Zn (expt.) d 0.4

Y (0.8) Mg Expt./PPT Ar (0.8) Expt./ADK (0.4) PPT/ADK 0.2 0.2 0.4 0.6 0.8 1 2 γ

Figure 3.11: The ratio between theoretical (PPT and ADK) and experimental values of Yd at an intensity of 0.8IBSI for different data set. The bracketed numbers under the atomic symbols represent the driving wavelengths (in µm).

Saturation Intensity

1 10

0 10

−1 10 Expt.

Ion Yield (arb. units) fit(Expt.) fit(Expt.) PPT−LP

−2 ADK−LP 10 1 10 Intensity (TW/cm2)

Figure 3.12: Ion yields of Mg at 0.8 µm. Saturation intensity obtained from experimental data, PPT and ADK calculations are given by the intersection point of the two fitting (black dashed) lines.

54 paraFigure 3.12 shows ion yields of magnesium at 0.8 µm. The two black dashed lines are two linear fits to the data, one is fitted to the data points at low intensities andtheother one is fitted to the data points beyond the saturation intensity which should have aslope close to 1.5. Here Isat is defined to be the intersection of the two linear fits. For thisdata

2 set Isat is determined to be 20 TW/cm . The same fitting procedures are applied to the

ADK and PPT curves to obtained the theoretical predictions of Isat, the values obtained from ADK and PPT are 29 TW/cm2 and 18 TW/cm2 respectively.

Expt./PPT 1.1 Expt./ADK

0.9 Zn (theo.) sat

/I 0.8 (3.6) Na (3.6)

(expt.) sat Ar K

I 0.7 (0.8) (3.2) Xe Mg (0.8) 0.6 (0.8) Zn Ar (0.8) 0.5 (0.4)

0.4 0.5 1 1.5 2 γ (at I(expt.)) sat

Figure 3.13: The ratio between measured and theoretically predicted (PPT and ADK) values of saturation intensities as a function of γ. γ is calculated using the experimental values of the saturation intensities. Each point has an error of 20% in intensity calibration. The bracketed numbers under the atomic symbols represent the driving wavelengths (in µm).

Isat for different data set were obtained using the same method and the results aresum- marized in Fig. 3.13. It shows the ratio between the measured and theoretically predicted

Isat for various targets and driving wavelengths. Overall, PPT agrees well with experiments within 20% uncertainty. Except for small γ, in general ADK overestimates Isat and the de-

viation increases with γ. Note that there is an ambiguity in the determination of Isat using

55 the fitting method described above since the slope of the low intensity part of an ionyield curve varies as a function of intensity. However, since the y-range for the fittings to the experimental and theoretical results are set to be the same, the ratio between the fitted Isat from the two curves indeed represents a real deviation between them.

Linear vs Circular Polarization paraCP and LP lead to very different ATI energy spectra [16, 33, 85] due to the fact that in

CP the photoelectron classical motion never returns to the parent ion. Differences are also expected in the total ionization rates wc and wl. While in the tunneling regime, for small

γ, wl is usually larger than wc, the opposite can be true in the multiphoton domain[86].

0.6 Ar 0.5 Xe 0.4 LP Kr

/Y 0.3 Ne CP

Y 0.2 0.1 (a) 0 0.6 Na 0.5 Zn 0.4 LP Mg

/Y 0.3 CP

Y 0.2 0.1 (b) 0 5 6 7 8 9 10 20 Ip/hω¯

Figure 3.14: The ion yield ratio between CP and LP at different target atoms and wavelengths. Panel (a) shows the results for noble gases and panel (b) shows the results for metal atoms. Blue: 0.4µm; red: 0.8µm; green: 1.3µm; purple: 2µm; black: 3.6µm. Laser intensity is at IBSI .

56 Turning to the experiment, the ratio CP/LP is an interesting quantity which does not imply absolute comparisons. Figure 3.8 shows the results of Zn and Fig. 3.2 shows the results of Ar at different wavelengths for both polarization. One general feature is thatthe yield at LP is larger than the yield at CP at the same laser intensity and the difference increases as the photon energy decreases. It is expected in multiphoton ionization and can be explained using dipole selection rule: for the case of CP there is only one allowed ionization pathway because the angular momentum of the electron is restricted to increase monotonically but for the case of LP there is no such restriction and so multiple ionization pathways exist. The number of possible pathways increases with the number of photons being absorbed to ionize therefore the ratio R ≡ YCP /YLP should be smaller for larger K0. Fig. 3.14 shows the ratio R of different target atoms at different wavelengths and itcanbe observed that the ratio decreases as Ip/~ω increases. In the tunneling regime, it is expected that the yield at LP is much larger than the yield at CP because peak amplitude of the √ field for LP is a factor of 2 larger than that for CP when the intensity is fixed.

To quantify the deviations between experimental results with PPT and ADK, we take the ratio between experimental and calculated (from PPT or ADK) values of R for different

data sets and the results are plotted as a function of γ in Fig. 3.15. All the comparison

are performed approximately at the calculated values of over-the-barrier intensities. Again,

the closer the ratio is to one, the better the agreement between experiment and theory.

The experiment to PPT ratio ranges between 0.8 and 3 and does not show a significant

trend of increase as a function of γ. The experiment to ADK ratio is close to that of PPT for small γ values, but as γ approaches 1 the ratio start to increase significantly and in the multiphoton regime ADK predictions become an order of magnitude larger than the experimental results.

Few-photon Ionization in Large γ Regime paraWhile the data presented in previous sections have demonstrated that PPT formula

(Eq. (3.2)) works well in both multiphoton and tunneling regime, we would like to point out that there is a limit on γ for Eq. (3.2) to be valid. It is due to the fact that the Coulomb

57 8 (a) 6 Ar

Theo. Xe

/R 4 Kr Ne Expt.

R 2

35 (b) 30 Na 25 Zn Theo.

/R 20 Mg 15 Expt.

R 10 5 0 0.4 0.6 0.8 1 2 γ

Figure 3.15: The ratio between theoretical (PPT and ADK) and experimental values of R ≡ YCP /YLP . The filled symbols are the ratio between PPT and experiment andthe open symbols are the ratio betweeen ADK and experiment. Panel (a) shows the results for noble gases and the panel (b) shows the results for metal atoms. The upper panel shows the results for noble gases and the lower panel shows the results for metal atoms. Blue: 0.4 µm; red: 0.8 µm; green: 1.3 µm; purple: 2 µm; black: 3.6 µm. Laser intensity is fixed at the IBSI .

correction (CC) factor (2/F n∗3)2n∗ of Eq. (3.2) was derived under the assumption that √ γ ≪ 2Ip/ F [70]. Figure 3.16 shows ion yields of Mg at 0.4 µm, and intensity from a few to 75 TW/cm2, √ so that γ varies between 1.9 to 8.3. In this case γ > 2Ip/ F for all the data points. Below saturation, the intensity dependence of the data is ∼ I3, as predicted by perturbation theory and consistent with the TDSE results in [87]. However, the PPT calculations (green dashed line in Fig. 3.16) shows that the ionization probability saturates at an intensity much lower than the intensity range of Figure 3.16, so the slope of the curve is just 3/2 due to the expanding focal volume, and even at very low intensities it remains much smaller than 3.

It should be pointed out, that although the short-range potential (with no CC) PPT does predict a power law ∼ IK0 , in agreement with perturbation theory (see Eq. (2.4) in [22]) in the large γ limit , it cannot predict the correct ionization rate. In many cases, it over

58 1 10 Expt. PPT PPT(generalized) 0 10

−1 10 Ion Yield (arb. units)

−2 10

0 1 2 10 10 10 Intensity (TW/cm2)

Figure 3.16: Ion yields of Mg as a function of intensity at 0.4 µm. Squares: Experimental data. The experimental intensities are scaled using the saturation intensity obtained from the TDSE calculations in [87] as a benchmark. Green dashed line: PPT calculations with original version of Coulomb correction factor (Eq. (3.2)). Red solid line: PPT calculations with generalized version of Coulomb correction factor (see text). For visibility, the blue curve is multiplied by 0.01 relative to the red curve.

estimates the saturation intensities by an order of magnitude or more. Popruzhenko et al

[88] derived a new expression for the CC factor of PPT formula, (2/F n∗3)2n∗(1+2γ/e)−2n∗, valid for arbitrary values of γ. With this generalized version of CC factor, good agreement between experiment and PPT calculations (red solid curve in Figure 3.16) is obtained. For clarification, PPT calculations with CC factor/F (2 n∗3)2n∗ (Eq. 3.2), without CC factor and with generalized CC factor (2/F n∗3)2n∗(1 + 2γ/e)−2n∗ are displayed in Fig. 3.17.

3.3 Summary paraIn this chapter, we have presented an experimental study on ionization of atoms in intense laser fields at different wavelengths, intensities, polarizations and types oftargets with the goal of evaluating PPT and ADK models. Our data covers a wide range of γ values.

In particular, we carried out the first experiment on tunnel ionization of alkali and alkaline earth atoms in the mid-infrared as a test of the applicability of Keldysh metric in atoms with very low ionization potentials. The PPT model agrees well with all the experimental 59 0 10

−1 10

−2 10

Ionization probability PPT(generalized) PPT PPT(without CC) −3 10 0 1 2 10 10 10 Intensity (TW/cm2)

Figure 3.17: Calculated ionization probability of Mg at 0.4 µm using PPT without CC factor (black dash-dot line); with CC factor (2/F n∗3)2n∗ (green dashed line); and with generalized CC factor (2/F n∗3)2n∗(1 + 2γ/e)−2n∗ (red solid line). Effect of focal volume averaging is not taken into account. For both solid and dashed lines the slopes are close to K0, but the saturation intensity is different by an order of magnitude.

data presented in this chapter but must include a generalized Coulomb correction factor

[88] in the very large γ regime in which ionization is a few-photon process. The ADK model significantly underestimate the ionization yield except in the deep tunneling regime.

PPT also gives much better predictions for ionization yield ratio between CP and LP than

ADK. ADK underestimates the CP/LP ratio by an order of magnitude when γ is large

(approaches 2).

ADK has also been extended to molecules in a version called molecular ADK (MO-ADK)

[89]. However, MO-ADK fails to give an accurate prediction on the orientation-dependent ionization profile for simple molecules2 CO [90, 91] (and also polar molecules CO [92, 93]) with γ > 1, resulting in a long-term debate. More elaborate models [94–99] have been

attempted with various correction schemes. However, the fact that MO-ADK or ADK

is supposes to be valid only in the regime of γ ≪1, a criterion which is not met in the aforementioned studies, should not be overlooked. Recently, PPT formula has also been generalized to molecules by Zhao et al[100].

60 Chapter 4 Ionization and dissociation of Methane in vibrationally resonant Mid-Infrared Fields

paraWhile a theoretical description for tunnel ionization of atoms could be developed from a simple physical picture in which an electron tunnels through an electric field suppressed

Coulomb potential, modeling the same process for molecules is considerably more complicated. To begin with, the potential of an atom is spherically symmetric while for molecules it is not. Therefore, the potential barrier for the electron to tunnel through actually depends on the molecular orientation with respect to the laser field polarization, thus the ionization rate is orientation dependent. Moreover, the electron wavepacket tunneling from different atomic sites of the molecule might destructively interfere with each other which could lead to an unexpected suppression of the ionization rate [101–103]. Another complication is that strong field ionization of molecules are often accompanied by dissociation, and the induced nuclear displacement could affect the ionization rate [104, 105].

There have been many studies on molecules in strong fields over the last two decades and most of them were performed at the Ti:sapphire laser wavelength (0.8 µm), whereas data at longer wavelengths are rare. Deep in the tunneling regime, one might expect that the ionization process is essentially wavelength independent14. It is important to note,

however, while the use of MIR driving lasers is a prerequisite to reach small γ from an

ionization perspective, vibrational resonances of molecules are also in the MIR region. For

14Recall that the ADK formula Eq. 3.6 does not contain ω.

61 Figure 4.1: MIR absorption spectra of several molecular species in the atmosphere. The region between the two vertical dashed lines range indicate our OPA tuning range. Repro- duced from [106].

example, in the 3-5 µm region, the stretching modes of the following bonds are active: N-H

(2.86−3.03µm), O-H (2.86−3.13µm), C-H (3.23−3.51µm), C=O (5.68−6.01µm) and C=C

(5.95 − 6.1µm)15. Figure 4.1 shows the MIR absorption spectra of several molecular species found in the atmosphere and the region between the two vertical dashed lines range indicate our OPA tuning range. In this chapter, we aim to explore tunnel ionization of molecules in which the driving field is resonant with the molecular vibrational mode with twomain purposes: (i) to test the validity of current tunnel ionization theory and possibly provide hint for the development of a more complete theory including the effect of vibrational degrees of freedom; (ii) explore the potential application of resonant MIR fields in controlling strong field processes. Here, we selected methane4 (CH ) to be the target which is one of the most basic units in organic chemistry.

15The presence of these resonances in a given spectrum allows molecular identification of all alkanes, alcohols or esters. For this reason, the 3-5 µm spectral range is called the functional group region.

62 4.1 Experiments

Figure 4.2: Mass spectrum of methane irradiated with 3.3 µm pulses at a peak intensity of 70 TW/cm2.

paraThe experimental procedures are the same as the ionization studies of atoms de-

scribed in Chap.3, again we mainly focus on measuring the ionization yields as a function

of laser intensity. Laser intensities were calibrated by the 2Up cutoff of photoelectron spectra of noble gases. Also, as described in Sect. 2.3.3, from the measured mass spectrum of ionized molecules the individual yields of the charged fragment ions due to dissociation are monitored. As an example, Fig. 4.7 shows the mass spectrum of charged fragments of CH4 irradiated by 3.3µm laser pulses at an intensity of 70 TW/cm2. The observed fragments

+ + + included CHn (n = 0 − 4), H and H2 ions. Note that there exist neutral fragments which could not be detected by our system. The total sum of all carbon containing fragments

+ (CH0−4), to a good approximation, can be treated as the single ionization yield of CH4. + Although some of the CHn (n = 0 − 3) ions might be due to double ionization followed by 2+ + + + + dissociation (CHn → CHn−1 + H ) and also some of the H2 and H ions might be from + + single ionization (e.g. CH4 → CH3 + H ), they do not cause any significant uncertainty + + + since the ratio of H2 and H counts to the sum of CH0−4 counts is negligible (≪ 0.1) in our experimental intensity range. Unless specified otherwise, “total ionization yield” is referring

63 to the sum of yields of all carbon containing fragment ions for the results presented in this chapter.

4.1.1 Results at MIR wavelengths paraAs mentioned, the C-H bond vibrational resonance is between 3.2 and 3.5 µm and so our investigations will focus on finding the differences between the results within andout of this wavelength range.

7 10 Total yield (CH ) (a) (b) 4 6 Total yield (CD ) 10 4

5 10

4 10 ion yield (arb. un.) 3 10

2 10 30 40 50 60 70 30 40 50 60 70 Intensity TW/cm2 Intensity TW/cm2

Figure 4.3: Total ionization yield of CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm.

First, we consider a back-to-back comparison between CH4 and the deuterated isotopo-

logue, CD4, at 2.9 and 3.3 µm. Note that CD4 has the same Ip and electronic structures

as CH4 but due to the mass difference between hydrogen and deuterium the vibrational √ frequency of C-D bond is lower than that of C-H bond by about a factor of 2, so both

16 wavelengths are off resonant with4 CD but 3.3 µm is resonant with CH4. Figure 4.3 present the total ionization yields for both targets at the two wavelengths. At 2.9 µm, the total ionization yield from CH4 is very similar to CD4, the average difference is only less

16The C-D bond vibrational frequency is longer than 4µm and is beyond the tuning range of our OPA.

64 than 40%. On the other hand, at 3.3 µm, the ionization yield of CH4 is higher than CD4 by about 270% on average.

Figure 4.4: Fragmental ion mass spectrum of CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm. Laser intensity is at 58 TW/cm2.

With such drastic difference between the two wavelengths one could readily conclude that it is related to the vibrational resonance, such small change in wavelength should virtually have no effect on the tunneling process itself. Note that tunnel ionization from vibrationally excited molecules has been studied theoretically using molecular ADK (MO-

ADK) theory [89, 107] for some diatomic and triatomic molecules. For instance, in [107],

Brichta et al calculated tunnel rate as a function of the molecular bond length R and took the average value according to the probability density distribution of the vibrational level.

While the quantitative effect was target dependent, in general the rate from a vibrational excited state was found to be higher than that from the vibrational ground state. We note that these theoretical results have never been verified experimentally and so our results could serve as a first test.

Here we make a remark about the case of nonresonant wavelength (2.9 µm). At first sight, it might be a bit surprising, that the total ionization yield of CH4 is also (slightly) higher than CD4. However, it should be noted that even in a case where vibrational exci-

CH+ /CH+ (a) (b) 0−3 4 CD+ /CD+ 0−3 4

0 10 Ratio of fragments −1 10

30 40 50 60 70 30 40 50 60 70 Intensity TW/cm2 Intensity TW/cm2

+ + Figure 4.5: Ion ratio CX0−3/CX4 for CH4 and CD4 as a function of intensity at (a) 2.9 and (b) 3.3 µm.

tation is completely absent, the nuclei of the molecule are not frozen due to the vibrational

zero-point energy, which is isotope dependent. The difference in tunneling rate between

H2 and D2 due to this reason was theoretically studied [108] and experimentally verified at Ti:sapphire wavelength [109]. In the experiment in [109], it was found that the ionization

rate of H2 is higher than that of D2 by up to 70%. Nevertheless, we would like to clarify that we do not decisively conclude that the difference we observed in4 CH has the same origin as in the results in [109], since the experimental uncertainty such as laser drift and

gas density fluctuation could readily cause an error in the order of 10%. Yet, the drastic

contrast between 2.9 and 3.3 µm was a definitive observation.

We now turn to the details of the mass spectra of the ions since they give more clear

evidence of the vibrational resonance. Figure 4.4 shows the mass spectra of both targets

at the two wavelengths at a fixed intensity (58 TW/cm2), note that all the spectra are

+ normalized to their yields of the parent ions. For CD4, the ratio of the fragments (CD3 + together with very small amount of CD2 ) relative to the parent ion is essentially wavelength

independent, whereas for CH4, fragmentation is much more severe at 3.3 µm than at 2.9 µm. A more quantitative presentation of the data is given in Fig. 4.5, it shows the ratio

of sum of the yields of all carbon containing fragments to the yield of the parent ion as

66 Figure 4.6: Mass spectra of methane fragments at different laser wavelengths at the intensity of 70 TW/cm2. Results of 3, 3.1, 3.3, 3.6 and 3.9 µm are shown in panel (a) to (e) + respectively. All the spectra are normalized to their yields of the parent ion CH4 .

a function of intensity. The ratio for CH4 increased by an order of magnitude when the wavelength is tuned from 2.9 to 3.3 µm.

For completeness, we measured ionization yields of CH4 at several more wavelengths between 3 to 4 µm. Figure 4.6(a)-(e) show the mass spectra at different wavelengths (see

caption) with a fixed intensity 70 TW/cm2. It is clear that there is a pronounced wavelength

dependence of dissociated ions yields relative to the parent ions yields which is peaked at

+ 3.3 µm. The yields of CHn (n = 2 − 4) as a function of intensity at 3.1, 3.3, 3.6 and 3.9 µm + are displayed in Fig. 4.7(a)-(d) respectively. At low intensities, the parent ions CH4 has + + higher yield than that of CH3 . As the intensity increases, the amount of CH3 approaches + and eventually exceeds CH4 . At 3.3 µm, the turnover of the ion yield happens at about 27 TW/cm2, which is much lower than the case of 3.1 µm (70 TW/cm2) and 3.9 µm (50

TW/cm2), as indicated by the vertical dashed lines in Fig. 4.7.

Figure 4.8 serves as a summary of from all of our data run. It shows the “threshold intensity” as a function of wavelength, the threshold intensity is defined to be the intensity

+ at which the amount of detected dissociated ions CH0−3 reaches the amount of detected + 17 parent ions CH4 . a low threshold intensity implies dissociation process is probable, 17Essentially, it is indicated by the vertical black dashed lines in Fig. 4.7 but at a slightly lower value + + + + because now we are considering CH0−3/CH4 instead of CH3 /CH4 .

67 0 10 (a) (b) (c) (d)

10−2

10−4

+ Ion Yields (arb. un.) CH −6 10 CH+ 2 CH+ 3 CH+ −8 4 10 20 100 20 100 20 100 20 100 Intensity (TW/cm2) Intensity (TW/cm2) Intensity (TW/cm2) Intensity (TW/cm2)

Figure 4.7: Mass spectra of methane fragments at different laser wavelengths as a function of intensity. Results of 3.1, 3.3, 3.6 and 3.9 µm are shown in panel (a) to (d) respectively.

vice versa. As seen, a pronounced minimum of the threshold intensity appears within the resonance region.

We also examined the effect of electron recollision on dissociation. Figure 4.9 shows

+ the ion yields of CHn (n = 2 − 4) as a function of laser ellipticity ϵ at an intensity of 78 TW/cm2(3.6 µm). Ion yields of Kr+ under the same laser condition are included for comparison. The yields of all ion species are normalized with respect to their values irradiated by linearly polarized light. Since the ellipticity of the laser is simply controlled by rotating a QWP, the pulse energy remains constant while the peak amplitude of the field drops as the ellipticity increases. As expected, the ionization yields decrease as the laser fields evolve from linear to circular polarization due to the decrease of the field amplitude. However, the

+ + + + ratios CH3 /CH4 and CH2 /CH4 virtually show no dependence on ellipticity over the full range 0 < ϵ < 1. Note that selection rule of vibration transitions has no restriction on laser polarization. This result suggests that electron recollision is not playing a significant role in the dissociation process; otherwise in the case of circular polarization, in which rescattering is mostly suppressed, should exhibit a drop in the ratios.

68 + + Figure 4.8: Intensity at which the ion yield ratio CH0−3/CH4 reaches 1 at different wavelengths (see text). Each point has an error of 20% in intensity calibration.

1 Kr+ CH+ 2 CH+ 0.5 3 CH+ 4 Normalized Yields 0 0 0.2 0.4 0.6 0.8 1

CH+/CH+ 2 3 4 CH+/CH+ Ratios 2 4 1

0 0 0.2 0.4 0.6 0.8 1 Ellipticity

Figure 4.9: Ellipticity dependence of ion yields. Upper panel shows the yields of different + + + + fragments; lower panel shows the ratios CH3 /CH4 and CH2 /CH4 .

4.1.2 Results NIR wavelengths paraWhile strong field ionization of4 CH with vibrationally resonant MIR fields has not 69 Figure 4.10: Mass spectrum of methane irradiated with 0.8 µm pulses at a peak intensity of 2 × 1014 W/cm2. The different species of fragments are labeled.

been considered before to the best of our knowledge, we note that ionization at Ti:sapphire wavelength has been extensively studied at intensities from 1013 W/cm2 up to 1018 W/cm2

[110], so it is beneficial to review and examine the behavior in that regime as well. For instance, Wang et al [111] measured mass spectra of CH4 fragments irradiated by 160 fs pulses at intensity between 1013 to 1014 W/cm2 and proposed a field-assisted dissociation

(FAD) model which described the dissociation of CH4 as a result of distortion of the potential energy surface of the C-H bond by the strong field. When the laser pulse duration is reduced to few cycle regime, Wu et al [112] showed that FAD model disagreed with experimental results and claimed that rescattered electrons are playing the major role in the dissociation processes. Another proposed mechanism of dissociation was based on quasi-equilibrium theory in which the process is interpreted as the redistribution of the energy absorbed from the laser to the internal degrees of freedom of the molecule [113]. There are also studies employing fluorescence spectroscopy to observe neutral dissociation and revealed the roleof superexcited states [114, 115]. Note that none of these proposed mechanisms could explain the wavelength dependence of our results from the previous section.

Similar to the experiments at MIR wavelengths, here we measure ion mass spectrum of CH4 as a function of intensity. Figure 4.10 shows mass spectrum with 0.8 µm pulses at

8 10 (a) (b)

6 10

+ 4 C 10 + CH

ion yield (arb. un.) CH+ 2 CH+ 3 CH+ 2 4 10 + CH + 0−4 CH 4 PPT(13.6eV) CH+ PPT(9.8eV) 0−3

20 40 60 80 100 200 300 20 40 60 80 100 200 300 Intensity TW/cm2 Intensity TW/cm2

Figure 4.11: (a) Ionization yields of methane as a function of laser intensity at 0.8 µm. The + + data consist of the parent ions CH4 and fragments CH0−3. Calculated single ionization yield of methane using atomic PPT model are shown as black solid (Ip=13.6 eV) and 2 dashed curves (Ip=9.8 eV), the two curves are normalized at their values at 300 TW/cm . + + (b) Comparisons between experimental and calculated yields of CH4 and the sum CH0−3. + The calculated yield of CH0−3 is obtained by multiplying P (I) (see text) to the original + + PPT yield; the calculated yield of CH4 is obtained by subtracting the yield of CH0−3 from the original PPT yield. Focal volume averaging is applied as the final step.

14 2 an intensity of 2 × 10 W/cm . Laser intensity calibration was made by using the 10Up cutoff of argon photoelectron spectrum. There are three groups of fragment ions: thefirst

+ group consists of singly charged ions CHn (n = 0 − 4) (the small peak at m/q = 17 is due 13 + 2+ to CH4 ). The second group consists of doubly charged ions CHn (n = 0 − 2), note that 2+ 2+ CH and CH3 are absent from the spectrum because they are unstable. The reason is 2+ 2+ the potential barriers for dissociation are only 0.04 eV for CH and 0.03 eV for CH3 , + + according to the simulations described in [112]. The third group consists of H2 and H . + Intensity-dependent yields of CHn (n = 0 − 4) and the total sum of them are displayed in Fig. 4.11(a).

The intensity dependences of the parent and fragment ions of methane are shown in Fig.

4.11(a). We compared it to calculation using atomic PPT formula (Eq. 3.2), as shown by the black solid curve. In the calculation, Ip is set to be 13.6 eV which is the vertical ionization 71 0 10 (a) (b) 7 10 C+ CH+ 6 CH+ 10 2 CH+ 3 5 10 CH+ 4 CH+ 0−4

4 Ratio 10 ion yield (arb. un.) 3 10 CH+ /CH+(0.8µm) 0−3 4 2 10 CH+ /CH+(1.03µm) 0−3 4 CH+ /CH+(1.1µm) 1 −1 0−3 4 10 10 40 60 80 100 200 40 60 80 100 120 160 200 Intensity TW/cm2 Intensity TW/cm2

Figure 4.12: (a) Ionization yields of methane as a function of laser intensity 1.1 µm. The + + data consist of the parent ions CH4 and fragments CH0−3. Calculated single ionization yield of methane using atomic PPT model are shown as black solid (Ip = 13.6 eV) curves. + (b) Ion ratio CH0−3/CH4 as a function of intensity at three different wavelengths (0.8, 1.03 and 1.1 µm).

potential of CH4, and since it has p-type molecular orbitals we set l = 1. The reasonable agreement between experimental data and atomic PPT theory is consistent with the results in [116], and it seems to imply that ionization occurred before dissociation. If dissociation occurs before ionization, the saturation intensity should have been much lower since the Ip of neutral CH3 is only 9.8 eV, significantly lower than that 4of CH . The black dashed curve is PPT calculation with Ip = 9.8 eV, it can be seen that the saturation intensity is ∼ 20 TW/cm2. In fact, considering the yields of individual fragment ions provides additional evidence. If dissociation occurred before ionization (or at least if it is the main reaction

+ pathway), the saturation intensity of CH3 should have been significantly lower than that + + of CH4 instead of being so similar to each other; also the yield of CH3 should have been + + + higher than CH4 . We also observed that ratio CH0−3/CH4 follows a power law over the entire intensity range (see Fig. 4.12). If dissociation occurs after the laser pulse is gone

+ then the final populations of the ions is related by : sum ofCH0−3 ions = (dissociation probability P (I)) × (PPT ionization yield of CH4), where I is the laser intensity. With this

72 + + simple relation together with an observation that the ratio CH0−3/CH4 follows a power law over the entire intensity range (see Fig. 4.12), one could obtain a simple empirical formula18

n n for the dissociation probability P (I) = (I/I0) /[1 + (I/I0) ], where I0 and n are two free parameters. The two calculated curves in Fig. 4.11(b) are obtained by applying P (I) to

14 the PPT yield (see caption) with I0 = 2 × 10 and n = 0.6 and reasonable agreement are observed.

More measurements at two other NIR wavelengths, 1.1 and 1.03 µm, were performed.

Figure 4.12(a) shows the yields of parent and fragment ions of methane irradiated by 1.1 µm pulses. Similar to the case of 0.8 µm, reasonable agreement between the total ion yield and atomic PPT calculations while the CHn (n = 0 − 4) fragment ions with fewer number of H atoms always have lower yields. However, the dissociation rate for the long wavelength case

+ + is considerably lower. Figure 4.12(b) shows the ratio of fragment ions yields CH0−3/CH4 for three different wavelengths 0.8, 1.03 and 1.1 µm. At low intensities the ratio at 0.8 µm is almost a factor of 2 higher than that at 1.03 µm and 1.1 µm. As intensity increases, the difference reduces. The wavelength dependence of dissociation could not be well explained by the FAD model since potential barrier suppression is wavelength independent. One possible explanation is that the large photon energy (short wavelength) led to more efficient electronic excitations and as a result more efficient coupling to the nuclear vibrations.

Overall, it is obvious to see that the fragmentation observed at NIR wavelengths is much

+ + less significant than at resonant MIR wavelengths. At 0.8 µm, the ion yield ratio CH3 /CH4 is still less than one even when the intensity reaches a few hundred TW/cm2, whereas at 3.3

µm the ratio reaches one when the intensity is only at 27 TW/cm2. Although the results at

NIR wavelengths might not be directly relevant to the focus of our study here because the processes in the NIR cases probably do not involve direct vibrational excitation as in the resonant MIR cases, the contrast between the two wavelength regimes is yet another piece of evidence showing the significance of vibrational resonance in strong field processes.

18 + + −1 + + + + P (I) =(CH0−4/CH0−3) =(CH0−3/CH4 )/[1+(CH0−3/CH4 )]

73 4.2 Discussion paraWhile our results clearly indicate that vibrational resonance plays a significant role in tunnel ionization, the next step will be to try to gain a quantitative and deeper understanding of the process. In our current setup, both vibrational excitation and ionization are induced by the same laser pulse. To deconvolute the two processes, a feasible scheme would employ a pump-probe geometry in which the pump pulse is a weak (non-ionizing) resonant

MIR pulse to induce vibrational excitation and the probe pulse is a strong non-resonant

MIR pulse to promote tunnel ionization. Thus, vibration excitation can be independently controlled by the wavelength and intensity of the pump pulse while the probe pulse would mostly contribute to ionization. However, generating two MIR pulses at different wavelengths simultaneously requires two OPAs pumped by the same laser. With only one OPA, a more practical scheme would use the 0.8 µm pulse (see Fig. 2.1) from the OPA as the probe pulse, although ionization of CH4 at 0.8 µm would not be deep in the tunneling regime.

It should be noted that femtosecond MIR pulses are better suited for vibrational excitation than longer (nanosecond) pulses because short pulses have broad spectral bandwidth

[117]. Since vibrational potentials are anharmonic, which means that the energy difference between two consecutive vibrational levels is getting smaller for higher excited levels, so a driving field with a large spectral bandwidth is required to allow for a wide rangeof resonances with transitions in a stepwise excitation up the vibrational potential. The excitation process could be further optimized by fine tuning the time delays between difference spectral components of the pulse such that the components which are resonant with the transitions between low-lying vibrational states interact with the molecule first and the ones which are resonant with the transitions between higher states arrive later [117]. This type of techniques has been successfully applied in various targets [118? –120]. We believe that

it can also be applied to our further investigations.

As a final remark, we note that the enhanced fragmentation rate observed4 inCH at MIR frequencies should also exist in other molecules which have CH bonds. We verified it by a

74 2 10 C H+ /C H+ 2 (2−5) 2 6 C D+ /C D+ 2 (2−5) 2 6

1 10 Ratio of fragments

0 (a) (b) 10 30 40 50 60 70 30 40 50 60 70 Intensity TW/cm2 Intensity TW/cm2

+ + Figure 4.13: Ion ratio C2X0−5/C2X6 for C2H6 and C2D6 as a function of intensity at (a) 3.1 and (b) 3.5 µm.

comparative measurement between ethane (C2H6) and its deuterated isotopologue (C2D6). Figure 4.13 shows the ratio of the yield of carbon containing fragments to the yield of the parent ion as a function of intensity at 3.1 µm (nonresonant) and 3.5 µm (resonant). The ratio is essentially wavelength independent for C2D6, whereas for C2H6 the ratio increased by an order of magnitude when the wavelength is tuned from 3.1 µm to 3.5 µm.

75 Chapter 5 C60 Fullerenes in strong mid-infrared fields

paraThe C60 fullerene, the third allotrope of carbon next to graphite and diamond, is a soccer ball shaped molecule consisting of 60 carbon atoms bond together with a cage radius of 3.5 A.˚ It has 240 valence electrons forming 180 σ and 60 π orbitals, in which the

HOMO (highest occupied molecular orbital) is a five fold degenerate π orbital. It has a relatively low Ip, 7.6 eV. Due to its approximately spherical symmetric geometry, the electronic structure could be modeled by a jellium model in which the atomic ion cores are treated as a uniform ionic shell. Due to the large number of valence electrons, C60 has a very large polarizability and two plasmon resonances around 20 and 40 eV. In addition to the large electronic degrees of freedom, C60 also has many vibrational modes, therefore the photoionization mechanisms of C60 are complicated. Depending on the timescale of the excitation field, electron-electron couplings and nuclear vibrations influence the ionization process. Campbell et al [121] used a Ti:sapphire laser system to perform a comprehensive study on the laser pulse duration dependence on the photoelectron energy spectrum. For a laser pulse duration of tens of femtoseconds, ATI peaks were observed which implied that the ionization is mainly from a coherent multiphoton process. The contrast of the ATI peaks deteriorates as the pulse duration increases because electron-electron collisions start to destroy the coherence. As the pulse duration was increased to hundreds of femtoseconds, ionization is dominated by the so-called thermoelectron emission which is due to the warm electronic gas formed by electron-electron collisions. In this regime, ATI peaks are com-

76 pletely suppressed and the photoelectron energy spectrum could be modeled by a thermal distribution. For few-picosecond pulses, energy transfer into nuclear degrees of freedom via electron-phonon coupling leads to vibrationally hot molecules. As a result, electron emission can occur from a stochastic vibrational autoionization process. For a review on various types of photoionization of C60, see, e.g., [122].

Figure 5.1: Ionization of C60 at MIR wavelength. (a) Mass spectrum of C60 irradiated with 2 n+ 3.5 µm femtosecond pulses at an intensity of 46 TW/cm . (b) Ionization yields of C60 (n = 1 − 4) as a function of laser intensity, the black line is an atomic PPT calculation with Ip = 7.6 eV and l = 5.

For the ionization of C60 in the femtosecond regime, there have been many studies at the Ti:sapphire wavelength, but experiments at longer wavelengths have been scarce. In

[123], ionization yields of C60 ions versus laser intensity at 1.8 µm revealed two interesting observations. First, there was a lack of fragmentation, in contrast to 0.8 µm femtosecond

experiments or with long pulses. Second, the saturation intensity for ionization was a few

times higher than the prediction of tunnel ionization theory and it was proposed that laser

induced dipole acted as a strong screening which raised the height of the tunneling barrier.

Another proposed explanation argued that the high angular momentum of the valence

electron (l = 5 for HOMO) has an important effect on the electron centrifugal barrier and

so the tunneling rate is suppressed [124]. Both of these explanations are based on tunnel

77 ionization theory and so the suppression effect should persist for even longer wavelength.

It is indeed the case according to our data at 3.5 µm. Figure 5.1(a) shows a C60 ion mass

2 4+ spectrum at 47 TW/cm . The highest charge state detected was C60 and no fragment ions were observed. The ion yields as a function of intensity and atomic PPT calculations for

single ionization (Ip = 7.6 eV; l = 5) are shown in Fig. 5.1(b). While the theory predicted saturation intensity is lower than 20 TW/cm2, the experimental value was at least higher

than 50 TW/cm2.

Although the puzzle of ionization suppression has not yet been totally resolved, the

global features of the electron energy spectrum of C60 are similar to that of an atom: in

the energy range from 0 to 2Up the signal decreases rapidly as a function of energy whereas

from 2 to 10Up the rescattering “plateau” is observed. This implies that the semiclassical

model can still be applied to describe the dynamics of the photoelectons for C60. However,

as one could imagine, the photoelectron spectrum from C60 would not be identical to the spectrum from an atom, and by investigating the differences we wish to acquire a better

understanding of strong field dynamics of60 C . In this chapter, we will discuss the observed

features in the very low energy portion (0-0.1Up) and the high energy portion (2-10Up) of

the photoelectron of C60. The two energy ranges are respectively related to two different types of electron recollision, as we will explain in the followings.

5.1 Low energy photoelectrons of C60

paraIn this first half of the chapter we are going to focus on the low energy partofthe

photoelectron spectrum of C60. In particular, we have found an unexpected suppression of the so called “low-energy structure (LES)” [45], a universal feature observed in tunnel ionization of atoms and molecules. We attributed this phenomenon to induced dipole of

C60 which influences the outgoing electron, an effect greatly magnified60 inC due to its exceptionally large polarizability.

5.1.1 The low-energy structure in tunnel ionization

78 Figure 5.2: Photoelectron spectrum of Ar, N2 and H2 irradiated by 2 µm laser pulses. The right panel shows the full photoelectron spectra and the panel on the left captures the low energy portion of the spectra. Reproduced from [45].

paraFirst, let us illustrate the characteristics of the LES with the data published by Blaga

et al in 2009 [45]. Figure 5.2 (right panel) shows typical photoelectron spectra of different

targets (Ar, N2 and H2) irradiated by 2 µm laser pulses. As introduced in chapter1, the 2 -

10 Up plateau region (∼100-500 eV) is due to rescattered electrons and the direct electrons dominate the signal in the energy range 0 - 2 Up. The left panel of Fig. 5.2 is the low energy portion of the spectra. The LES is the pronounced spike-like feature (yellow shaded

region) peaked at a few percent of Up (typically a few eV). It can be seen that the peak of all three experimental targets are almost identical. Moreover, the LES has been observed

in all noble gases from He to Xe and various diatomic molecules, and we have also observed

the LES in hydrocarbon molecules during the studies presented in chapter4. In fact, it is

now believed that it is a universal feature for tunnel ionization.

The key characteristics of the LES are shown in Fig. 5.3. Panel (a) and (b) illustrate

the wavelength and intensity dependence respectively, it can be seen that the LES extends

to higher energies as wavelength 19 or intensity increases. More importantly, however, the

19LES is not clearly observed at 0.8 µm (black line in panel (a)) because it is in the multiphoton regime and so the spectrum is modulated by ATI peaks.

79 Figure 5.3: Characteristics of the LES. (a) The emergence of the LES as a function of wavelength for Xe at 80 TW/cm2. (b) The progression of the LES in Ar at 2 µm as a function of intensity. (c) The LES in Xe at three different combinations of wavelength and intensity but Up is kept constant (∼ 19 eV). (d) The LES region in Xe for linearly and circularly polarized 2 µm pulses at a fixed intensity. Reproduced from [45].

energy range occupied by the LES is in fact invariant with respect to Up. As shown in panel (c), the three displayed spectra were measured at different wavelengths and intensities but

with the same Up and we can see that the LESs are all peaked at ∼ 1 eV (and has a cutoff

at ∼ 2 eV). In fact, the energy range that the LES occupies is always between 0 − 0.1Up [125], regardless of intensity and wavelength (explanation will be given in 5.1.2). Panel (d)

demonstrates that the LES only exist when the driving field is linearly polarized but not

when it is circularly polarized which suggested that it is related to electron rescattering.

Dubbed as the “ionization surprise” by Faisal[126], the discovery of the LES was con-

sidered to be an important result because the commonly used KFR theory failed to predict

its existence. From the physical picture of tunnel ionization the photoelectron spectrum

should simply decay rapidly but smoothly as a function of energy and the rescattered elec-

tron should only contribute to the high energy “plateau” part of the spectrum. Soon after

the publication of [45] (see also [46]), numerous theoretical studies employing classical and

quantum mechanical simulations have been reported [125, 127–131] and it was found that

80 the origin of the LES was tied to the Coulomb interaction between the parent ion and the outgoing electron, an element that was neglected in the KFR theory. To be more specific, it is the Coulomb focusing effect upon the return of the outgoing electron with its parention which happens at about one and a half optical cycle after tunnel ionization, more details will be illustrated in the coming section.

5.1.2 Classical trajectory Monte Carlo simlulations paraThe classical-trajectory Monte Carlo (CTMC) method has been the main theoretical tool used to investigate the LES. In fact, it is a widely used classical-mechanics based method in various problems in strong field physics. It simulates the momentum distribution ofthe electron which tunnel ionized and propagated away from its parent ion upon the irradiation of a laser pulse by modelling the electron using classical mechanics. In reality, the electron does not ionize from the atom at a definite time with a definite momentum like asingle classical particle due to its quantum mechanical nature; instead these quantities follow a probability distribution given by the ionization formula. CTMC mimics the probabilistic

(quantum mechanical) nature of the tunnel ionization process by modelling the electron wave packet emerged from tunnel ionization as a statistical ensemble of classical particles whose initial conditions follow a probability density function which is derived from the ADK model. After ionization, the subsequent motions of the particles follow Newton’s second law with two forces: one due to the laser field and the other due the Coulomb potential ofthe parent ion. The final momentum of all the particles are then calculated and a momentum distribution is obtained. The advantage of this method is that it is straightforward to trace back the origin of certain features appearing in the final momentum distribution just by inspecting the group of contributing trajectories. It is trivial to implement parallel computing for this method since the particles in the ensemble are independent of each other.

Here we will describe the procedures of CTMC, our main purpose is not just to describe the method itself since it is a commonly used method (see, e.g., [127, 129–133]), rather we aim to provide the reader with a physical picture of the dynamics that caused the formation of the LES with some example calculations. We will also employ this method fo investigate

81 the unexpected results of C60. Initial conditions of electron

Figure 5.4: Density distribution of initial conditions of electrons as a function of (t0, p0⊥). The lower panel shows the waveform of the laser pulse.

The initial conditions of a classical trajectory are specified by initial time, momentum and position. According to ADK model (since we are considering the case of tunneling regime), the dependence of ionization rate on time and initial transverse (perpendicular to the laser polarization) momentum p0⊥ is

3/2 p 2 1 2(2Ip) 2Ipp0⊥ w(t0, p0⊥) ∝ 2 exp[− ] exp[− ], (5.1) F (t0) 3F (t0) F (t0) where F (t0) is the instantaneous field strength at t = t0. The first exponential term is

the same as in Eq. 3.6, but here we ignore the prefactor which has no F (t0) dependence because knowledge of the absolute ionization rate is not needed for obtaining an accurate

“shape” of the momentum distribution. The second exponential term refers to the Gaussian

shape of the electron wavepacket (in momentum space) after it just tunneled out. To obtain

an ensemble of initial conditions one could generate a set of random numbers (t0, p0⊥) by using Eq. 5.1 as the probability density function. Alternatively, one could generate random

82 numbers uniformly on the (t0, p0⊥) plane and assign a “weight” to each sample according to Eq. 5.1. Although the first method makes the simulations more efficient in our case,we adapted the second method because it allows us to easily investigate the robustness of the features in the final momentum distribution with respect to the distribution of the initial conditions simply by changing the weight of each sample points.

The upper panel of Fig. 5.4 shows a typical distribution of initial conditions (ωt, p0⊥). The weight of each point is represented by its color. The lower panel is the laser field as a function of time. As expected, the distribution has the highest density at the peak of the field. In principle, the sample space should cover the entire duration of the laser pulsebutfor the dynamics we are studying, it is sufficient to focus on launching the electron trajectories from just one optical cycle (usually the central cycle of the pulse). It is necessary to include contribution from multiple cycles if, for example, one wants to investigate the interference of trajectories from different optical cycles20. For linearly polarized fields, the system is cylindrically symmetric so emission angles of ⃗p0⊥ is a redundant variable. Therefore, it is sufficient to solve the equation of motion in two dimensions and multiply the initial distribution by a jacobian 2πp0⊥ to take all emission angles into account. The initial

q 2 electron position is set at the tunneling exit x0 = −(Ip + Ip − 4F (t0))/2F (t0)[129]. Classical motion of electron

The classical equation of motion for an electron under the influence of a laser field F⃗ (t)

and the ion Coulomb potential is

d2⃗r 1 = −∇(− ) − F⃗ (t), (5.2) dt2 r where the vector potential of laser field A⃗(t) is assumed to have a cosine squared envelope:  F0 2 2ωt 11nπ 11nπ  cos ( ) cos(ωt)ˆz, if − < t < A⃗(t) = ω 11n 4ω 4ω (5.3) 0, otherwise with the electric field profile given by F⃗ (t) = −∂A⃗(t)/∂t. Here, F0 is the peak field ampli-

20To study interference effects, one also needs to compute the quasiclassical phase of each trajectory, see, e.g. [128]

83 tude; n is the full width of half maximum (FWHM) pulse duration in terms of number of optical cycles. There are two remarks about the modeling of the laser pulse: (i) it would be more convenient to directly define F⃗ (t) as a cosine square pulse just like Eq. 3.10 instead of defining A⃗(t) and then take the time derivative, but the current way ensured that the pulse does not contain any unphysical d.c. component. (ii) We also tested our simulations using Gaussian pulse shape and the results are essentially unchanged, it is just slightly less convenient because the field amplitude does not goes to zero at finite time.

Equation 5.2 is decomposed into a set of first order differential equations and numerically solved using a Runge-Kutta solver in MATLAB. The numerical calculation starts at t = t0

and terminates at the end of the laser pulse t = tf . After the field is off, the motion ofthe electron is only influenced by the ion Coulomb potential. The asymptotic momentum (and

energy) of the electron is then determined analytically with ⃗r(tf ) and ⃗p(tf ) using Kepler

2 orbits (hyperbola). The electrons with energy Ef = ⃗p(tf ) /2 − 1/|⃗r(tf )| < 0 are not ionized and so do not contribute to the photoelectron spectrum.

Electron energy distribution and the formation of the LES

After calculating the asymptotic momentum of all electron trajectories, they are binned

according to energy and emission angle (the angle between laser polarization and the mo-

mentum vector). The electron yield in each energy bin is given by the weighted sum of the

number of electrons. The collection angle is set to be ±1◦ of the specified emission angle.

Unless indicated otherwise, the results presented in this chapter have 0◦ emission angle.

The following discussion focuses on the low energy part of the spectrum but the reader

may go back to Fig. 1.5 to see a typical simulated photoelectron spectrum with full energy

range. Now we are going to introduce the characteristics and the underlying mechanism of

the LES. Figure 5.5(a) shows the dependence of the LES on laser intensity. As shown, the

energy of the LES peak increases with intensity. Figure 5.5(b) shows the same simulations

but the energy scale is in Up, it can be seen that the LES peak is always peaked at slightly

less than 0.1Up.

84 Figure 5.5: Intensity dependence of LES. Both (a) and (b) display the same set of simulation results but in different energy scale (in eV vs in Up). The laser wavelength is 1.8µm. Ip = 15.79eV (argon).

Figure 5.6: Wavelength dependence (1.4 vs 2.4 µm) of the LES. Both (a) and (b) display the same set of simulation results but in different energy scale (in eV vs in Up). The laser 2 intensity is 83 TW/cm . Ip = 15.79eV (argon).

Figure 5.6 shows the LES for two different wavelengths (1.4 and 2.4 µm) at a fixed intensity of 83 TW/cm2. The peak is at a higher energy as the wavelength increases.

Again, the peak is fixed at slightly less thanU 0.1 p. For illustration purposes, the effect of

85 Figure 5.7: Photelectron spectrum at different emission angles. Each spectrum is multiplied by an arbitrary factor for visibility. Ip = 15.79eV (argon).

focal volume averaging21 was not taken into account for the results in Fig. 5.5 and 5.6. From the results above, we can see the CTMC reproduced the experimentally observed intensity and wavelength dependence of the LES and the fact that its peak position is invariant with respect to Up(Fig. 5.3) Another important feature of the LES is revealed from its dependence on the emission angular. As shown in Fig. 5.7, LES is maximized along the laser polarization and the signal drops as a function of emission angle. Note that this property has also been observed [134].

The fact that the peak position of the LES is invariant with respect to Up, just like

the 2Up cutoff of the direct electrons and the10Up cutoff of the backscattered electrons,

indicated that it originates from trajectories ionized within a certain range of t0, regardless of wavelength and intensity. Figure 5.8 shows a few classical trajectories with different final

energies from the simple one dimensional classical model (no Coulomb potential) described

in Sect. 1.2.3. If the drift momentum is low, the electron trajectory will revisit the ion

multiple times before drifting to the detector. Due to the finite values of p0⊥ most of the trajectories do not backscatter even it revisits the ion. However, the Coulomb potential

21Since the peak position shifts with intensity, focal averaging would result in broadening of the peak. So in experiments it is usually easier to determine the cutoff energy of the LES rather than the peak position. 86 Figure 5.8: (a) Electron trajectories of different drift K.E. (labelled as E in the plot) calculated from 1D classical model without Coulomb potential. (b) Amplitude (in arbitrary units) of electric field E(t) and vector potential A(t) vs ωt.

still affects the electron trajectory whenever it gets close to the ion. If the velocity ofthe electron is very low when it returns to the ion, the effective “interaction time” between them is long and as a result the electron trajectory is altered significantly (since ∆⃗p = R F⃗ dt).

This is exactly what happens to the red trajectory in Fig. 5.8 (a), its velocity is almost

zero when it revisits the ion (the circled region). This is why the LES is peaked at ∼ 0.1Up We have just explained why the electron trajectories with low energies are more suscep-

tible to the Coulomb potential than the high energy ones, but we have not discussed how

the effect manifests itself as a peak in the photoelectron spectrum. To briefly illustrate this

we show the comparison between the trajectories with and without Coulomb potential. As

shown in Fig. 5.9(a), the electron trajectory is bended towards the direction of the laser

polarization in the presence of the Coulomb force. This effect is called “Coulomb focus-

22 ing” . Figure 5.9(b) displays a set of trajectories ionized at different ωt0 ranging between

0.06 and 0.2 (see caption). Note that smaller ωt0 would lead to smaller final energy at detection. While all trajectories are bent towards the laser polarization, the bending effect

22The longitudinal component of the Coulomb force also play a role in the formation of the LES, see [125].

87 Figure 5.9: Illustration of the Coulomb focusing effect. (a) Comparison between two classical trajectories with and without the influence of the Coulomb potential. ωt0 = 0.09. (b) Comparison between two sets of classical trajectories with and without the influence of the Coulumb potential; ωt0 = 0.06 + 0.01n, where n = 0, 1, ..., 14. In all the calculations, laser 2 intensity = 67 TW/cm ; wavelength = 3.6 µm; p0⊥ = 0.105.

is particularly severe for the ones with small final energies (circled) and as a result they become very close to 0◦ emission, these are the ones that contributed to the LES. That is, the LES, roughly speaking, is a result of redistribution of electron trajectory population from large emission angles to small emission angles, and this is also why LES does not exist in the photoelectron spectra at large emission angles.

5.1.3 Experimental Results and Discussions paraSince the long range Coulomb potential exist in any ionized targets, so the LES is expected to be an universal feature. It would in fact be a “surprise” if LES does not exist in a specific target. Therefore, our findings60 inC is indeed unexpected and raises some interesting questions.

We start our discussion using our experimental results from a back-to-back comparison between Xe and C60, as shown in Fig. 5.10. The LES can be clearly seen in Xe with a cutoff at ∼ 5 eV. As for C60, although the electron yield also increases as energy decreases, it is

88 Figure 5.10: Low energy photoelectron spectrum of Xe and C60 irradiated by 3.1µm laser pulses at 78 TW/cm2. The two spectra are normalized to each other with respect to the yield at 15eV.

significantly less drastic and it is much more difficult to find a well-defined cutoffenergy.

The data above has definitely shown that the low-energy photoelectrons60 ofC is significantly less abundant than Xe, yet it is insufficient to support a conclusion that the LESis suppressed in C60. It is because while the energy position of the LES should be universal, the magnitude of the structure could vary for different targets, the height of the LES peak in Xe does not serve as a benchmark. As a further test, we included a separate comparison betwen C60 and Xe photoelectron spectra at different laser intensities, as shown in Fig.

5.11. For N2, the LES extends to higher energy when intensity increases, as explained in the previous section (the same trend was observed in Xe and Kr). However, for C60 the shape of the low energy spectrum is essentially unchanged as a function of intensity. In

other words, the low energy photoelectrons of C60 do not seem to exhibit an intensity dependence consistent with the LES has. Another set of measurements were performed at

a wavelength of 2 µm, as shown in Fig. 5.12. The results for Ar again showed that the

LES extends to higher energy when the intensity is increased, as expected. As for C60, no (intensity dependent) pronounced peak structure was observed, similar to the results of 3.1

µm. Additional measurements were performed at one addtional wavelength, 3.6 µm, and

again no evidences of the LES was found. 89 Figure 5.11: Intensity dependence of low energy photoelectron spectrum of (a) N2 and (b) C60 irradiated by 3.1 µm laser pulses. Each spectrum is normalized to its total yield.

Our results suggested that the suppression of the LES is not due to some wavelength sensitive mechanism (resonance), rather it seems to suggest that the dynamics of the low energy photoelectrons of C60 is fundamentally different from atoms and small molecules in some way. One obvious difference between C60 and other previous studied targets is the extensive size of C60 which might alter the interaction between the parent ion and the electron when it returns. However, the impact parameter of electron trajectories contributing to the LES is tens of atomic units (see Fig. 5.9), much larger than the radius of C60 (3.5

A).˚ Another important difference is the polarizability of the system.60 C has a very large polarizability (79 A˚3), while the polarizability of noble gases, diatomic and triatomic gas molecules and small hydrocarbons are usually less than 10 A˚3.

When a polarizable system is in an external electric field, an induced dipole is formed with a potential αF⃗ (t) · ⃗r V (⃗r, t) = − , (5.4) d r3 where α is the polarizability of the system. The field due to the induced dipole is given by the gradient of Eq. 5.4. The spatial dependence of the field direction is shown in Fig.

5.13(a). It can be seen that in some regions this field is opposed to the Coulomb field due to the parent ion which would be always pointing towards the origin. Before trying to

90 Figure 5.12: Intensity dependence of low energy photoelectron spectrum of (a) Ar and (b) C60 irradiated by 2 µm laser pulses. Each spectrum is normalized to its total yield.

understand how the induced dipole field affects electron trajectories, the first question is whether the field strength of the induced dipole is comparable to the Coulomb field strength.

A qualitative inspection is shown in Fig. 5.13(b). It shows the y-component (perpendicular to the external field) of the induced dipole field and the Coulomb field as afunctionof r

(see caption for the definition of r), α is set at 533 a.u. which is the value of C60. The field strength is set at 0.0414 a.u. which is the peak field of a laser with intensity 602 TW/cm .

The dipole field is larger than the Coulomb field when the distance isbelow25 A.˚

Even at 50 A˚ the dipole field strength is still half of the Coulomb field. As discussed in

Sect. 5.1.2, the formation of the LES is due to the transverse component (with respect to the laser polarization) of the Coulomb field which tends to attract the electron trajectories towards the direction of the laser polarization. To oppose this effect the dipole field hasto be in the opposite direction. It is not so trivial to see since the external field changes sign every half an optical cycle.

Figure 5.14 gives a simple illustration: panel (a) shows a low-energy trajectory and panel (b) shows the driving field and the direction of the force due to the induced dipole experienced by the electron (see caption). Once the electron is ionized, it experiences a repelling force from the dipole for about a quarter of an optical cycle. In the subsequent cycles the force changes back and forth depending on the position of the electron and the

91 Figure 5.13: (a) Spatial dependence of the direction of the induced dipole field. The origin is at the center of the plot and the external field is in the x-direction (b) Strength ofa Coulomb field (due to a singly charge ion) and a induced dipole field (duetoEq. 5.4 with α = 533 a.u. and F = 0.0414 a.u.) as a function of r. In this plot, r is defined to be the distance between the origin and a point on the line which is at 45◦ to the external field (the red arrow in panel (a))

direction of the laser field. Full numerical calculations including all the spatial dependence of the forces is required to reveal the overall effect quantitatively but this simple picture suggests that the low-energy electrons should receive a strong initial “push” in the transverse direction during the first quarter of laser cycle which might compete with the Coulomb focusing effect. Figure 5.15 shows three calculated low energy classical trajectories (see caption) in order to illustrate the effect of the induced dipole field. The initial conditions of all three trajectories are identical and are chosen to be relevant to the formation LES. By comparing the red and blue trajectories it can be seen that the effect of the induced dipole field is significant. Note that this single trajectory comparison only serves as anexample and does not necessarily have a direct implication to the photoelectron spectrum as a whole.

To reveal the overall effect of the induced dipole to the low energy photoelectron spectrum, we performed full CTMC simulations following the procedure described in Sect. 5.1.2, which is presented in Fig. 5.16. Results with and without the effect of induced dipole are plotted together (see caption). For both spectra, the potential of the parent ion was simply modeled by a −1/r Coulomb potential and the tunnel ionization was modeled by the same 92 Figure 5.14: Illustration of the effect of the induced dipole field. Panel (a): x-position of an low-energy electron as a function of time. Effect of Coulomb field and induced dipole field are not included. The trajectory is assumed to have anonzero p0⊥. Panel (b): the black line shows the laser field amplitude; the blue dashed line indicates the direction of the dipole force on the electron. Value of 1 means the force is repulsive and -1 means the force is attractive.

ADK formula (Eq. 5.1). In other words, any difference observed between the two results are purely from the effect of the induced dipole field. It can be seen that the LES(thelow energy peak that extends up to about 5 eV) is significantly suppressed when the induced dipole field is imposed. Although the simulation shows a significant suppression oftheLES, it does not fully explain the experimental data where the LES seems completely invisible, which might indicate some other additional factors contributed to the suppression. We note that here the initial conditions were modeled using the ADK formula (Eq. 5.1) developed for atoms. It is likely that the initial momentum (p0⊥) distribution of the electrons from

C60 could not be accurately described by Eq. 5.1. The distribution of p0⊥ is a relevant factor since the LES is formed by the trajectories whose p0⊥ is within a certain range and so it is sensitive to distribution of p0⊥. For atoms and small molecules the p0⊥ distributions are not expected to be drastically different from each other, but it may not be trueforC60 due to its extensive spatial size of the ground state wavefunction. Consequently, the elec-

93 Figure 5.15: Comparison between classical trajectories under the influence of different force components. Black: only laser field is included; blue: only laser field and Coulomb potential are included; red: laser field, Coulomb potential and induced dipole (Eq. 5.4 with α = 100) are included. Initial condition of the electron: ωt0 = 0.09; p0⊥ = 0.105. Laser parameters: intensity = 67 TW/cm2; wavelength = 3.6 µm.

tron wavepacket could tunnel out from different parts of the cage surface and the effective potential barrier for tunneling is probably not uniform along the surface due to the induced dipole field. Future theoretical advances for modeling the tunnel ionization ofC60 could shed light on this question.

0.8 Coulomb field 0.7 Coulomb field+Induced dipole field

0.6

0.5

0.4

0.3

0.2 Normalized electron yield 0.1

0 0 5 10 15 20 25 30 35 40 Energy (eV)

Figure 5.16: Simulated photoelectron spectra. Blue: only laser field and Coulomb potential are included; red: laser field, Coulomb potential and induced dipole (Eq. 5.4 with α = 500) are included. Laser parameters: intensity = 67 TW/cm2; wavelength = 3.6 µm. The effect of focal averaging have been included.

94 5.2 Self imaging of molecular structure using rescattered electrons paraWe will now focus on the high energy electrons with the purpose of testing the applicability of laser-induced electron diffraction (LIED) technique to imaging of large molecules.

LIED is a self-interrogating method which makes use of rescattered electrons in strong field ionization to obtain the structural information of the target itself. That is, the fundamen- tal difference between LIED and conventional electron diffraction (CED) is the waythat scattering electrons are prepared. Certainly, CED is a well-developed technology where pi- cometer spatial resolution and picosecond temporal resolution have been routinely realized

(for a review, see, e.g., [135]). Some recent developments have demonstrated the generation of electron beams with durations down to tens of femtoseconds [136] which suggests that molecular imaging with such a time scale might soon be possible. LIED, on the other hand, has been shown as an alternative route to achieve ultrafast molecular imaging over the past few years. The use of femtosecond laser is certainly a key ingredient for this technique but actually the temporal resolution is not directly limited by the laser pulse duration but by a shorter timescale, the period of one optical cycle. It is because imaging takes place via the three-step process, ionization → propagation → rescattering, all happening within

one optical cycle. For instance, the optical period of a 3 µm laser is 10 fs. Using LIED, ˚ + Blaga et al [53] measured a 0.1 A displacement in the O2 bond length occurring in a time interval of ∼ 5 fs after ionization. In [56], a snapshot of dissociating di-ionized acetylene at ∼ 9 fs after ionization was imaged. These results together with several other studies

[51, 54, 55, 137] have established LIED as a new promising approach for molecular imaging with unprecedented spatial-temporal resolution. However, the targets in all these studies are limited to diatomic and small hydrocarbon molecules. To make one step forward, we apply this technique to a macromolecule, C60.

5.2.1 Basic Principles paraIn Sect. 1.2.3 we have introduced the basic classical kinematics of electron rescattering

95 such as the relation between energy just before rescattering and the final energy at detection.

However, the scattering process itself was not described (it was simply treated as a hard sphere collision). In this section we will describe the properties of electron rescattering and how the rescattered electrons are used to extract molecular structure.

Let us start with some basic elements in the context of CED, which are also applicable in LIED. In CED, a quasi-monochromatic electron beam is prepared and directed to a target sample where they scatter, the spatial distribution of the scattered electrons are then detected. For elastic scattering, the electron momentum before (k⃗0) and after (k⃗1) scattering have the same magnitude but different direction. The scattering angle (θ) is defined as the angle between k⃗0 and k⃗1. The momentum distribution of the scattered electrons after scattering depends on the potential of the target and the initial momentum of the incoming electron, the quantity for characterizing the distribution is the differential cross section σ(k,θ,φ), where k = |k⃗0| and φ is the azimuthal angle. It is defined as the ratio of the number of electrons scattered into direction (θ,φ) per unit time per unit solid angle, divided by the incoming electron flux. For a molecular target, there are multiple atomic sites for the incident electron to scatter. The total scattering amplitude for the incident electron, under the independent atom model (IAM), is the sum of the contributions from all individual atoms: ⃗ ⃗ X i⃗q·Ri F (k;ΩL) = fie , (5.5) i where ΩL is the orientation or alignment angle of the molecule; fi is the scattering amplitude due to the ith atom; momentum transfer ⃗q ≡ k⃗1 − k⃗0; R⃗i is the position vector of the ith atom. In IAM, the redistribution of the atomic electrons due to molecular binding is neglected (so fi is simply the value one would get in a single atom experiment or calculation). Therefore, IAM is an accurate approximation only when the incident electron is sufficiently energetic (∼100 eV or above) such that the valence electrons of the atoms which form molecular bonds do not play any significant role in the scattering process. From Eq. 5.5, the scattering intensity is expressed as

96 ⃗ ⃗ ⃗ 2 X ∗ i⃗q·Rij I(k;ΩL) = |F (k;ΩL)| = IA + fifj e , (5.6) i̸=j

2 where R⃗ij ≡ R⃗i −R⃗j. The first term, IA ≡ Σi|fi| , is an incoherent sum of scattered electron waves from all the individual atomic sites and contains no structural information. The second term is a molecular interference term which contains the structural information of the target molecule. For electron scattering from a sample of randomly oriented molecules, the scattering intensity is obtained by averaging over the angle ΩL:

X ∗ sin(qRij) ⟨I⟩ (k, θ;ΩL) = IA + fifj , (5.7) qRij i̸=j where k ≡ |⃗k| and q ≡ |⃗q| = k sin(θ/2). For the purpose of molecular imaging one wants

to obtain the values of Rij from the electron scattering distribution as a function of q as measured in the experiment. There are two commonly used method to obtain structural

information from the data. One is to take the inverse Fourier transform of the scattering

distribution as a function of ⃗q to obtain a “frequency” spectrum in which each “frequency”

component corresponds to a distance value. The other method is to produce a scattering

distribution from a theoretical simulation which matches the experimental one and thus the

structure information is given by the theoretically constructed molecule.

In CED, scattering electrons and the target are independent of each other. The electron

beam prepared by an external source so its parameters can be directly controlled and

measured. This is not the case in LIED since ionization and rescattering are linked to each

other and so extracting structural information requires additional assumptions and efforts.

The quantitative rescattering theory (QRS) [138] was developed to serve this purpose. It

provides a simple relation between the momentum distribution of the rescattered electrons,

D, and the differential scattering cross section, σ, of the parent molecular ion (which is the

target to be imaged):

D(kf , θf ) = W (kr)σ(kr, θr), (5.8) where the subscripts r and f refer to the value just before rescattering and the final value at detection, respectively. W (kr) is the returning wave packet (RWP), which is the electron

97 flux as a function of momentum. Actually, in the context ofCED,Eq. 5.8 is simply the definition of differential cross section. That is, the central idea of QRS is that laser-induced electron rescattering process can be treated as regular electron scattering. Clearly, it is based on the physical picture of the three-step model. Another main assumption is that the strong field does not have an effect on the differential cross section, it is valid becausethe strong field mostly only affect the valence electrons but not the ion core. To make useofEq.

5.8 to extract the differential scattering cross section σ of the target molecule there are two unknowns have to be identified. First, the detected momenta of rescattered photoelectrons kf are not the same as the momenta just before rescattering kr due to the presence of the laser field. Fortunately, it is straightforward to relate the two quantities classically as shown in Fig. 1.4 (the classical equations can be applied to any scattering angle other than just 180◦ backscattering). Second, the RWP is not directly measurable, and in fact it is not a constant as a function of rescattering momentum according to tunneling theory. If σ of the target is well-known then of course the RWP can be found from the photoelectron distribution using Eq. 5.8. In fact, RWPs for noble gas atoms have been extracted using this method [51] and it was shown that they follow a universal scaling law which is a smooth function of kr and is independent of target species. Nevertheless, extraction of RWP from photoelectron distribution is not generally possible but is, in fact, not necessary.

Our purpose is to extract the molecular structural information from the interference signal embedded in D(kr, θ), so as long as W (kr) is a smooth function which does not affect the frequency content of D(kr, θ) then it can be simply treated as a background that can be removed empirically.

In CED, the knob to vary ⃗q is the scattering angle θr. This knob is also accessible in the laser-induced case, where all that is needed is the angular distribution of the rescattered electrons. This scheme is the angle-swept LIED, which is usually simply referred to as

LIED. The other knob for varying ⃗q is by changing kr. In strong field rescattering, kr is by default broadband since Er is ranged from 0 to 3.17Up. Therefore, structural information can be extracted from the energy distribution of the backscattered electrons. This scheme of using the broadband electron wavepacket at a fixed scattering angle is called “fixed-angle

98 broadband laser-driven electron scattering” (FABLES), it is the method we will employ for the C60 data. The data of angle-swept LIED for C60 has been presented in the dissertation of a former group member [52].

(a) (b)

A 2 A A 1 2 A 1

Figure 5.17: Schematic diagrams of electron diffraction from a diatomic molecule. Assuming the driving laser is polarized horizontally and the electron returns from the right hand side of the molecule. The blue and red curves represent the wavefronts of the electron waves backscattered off from A1 and A2 respectively. The distance between A1 and A2 is l. Panel (a) and (b) show the case in which the molecular axis is parallel and perpendicular to the laser polarization respectively.

Before presenting our experimental results, let us illustrate the working principle and

properties of FABLES with a simple example of diatomic molecule. Figure 5.17(a) is a

schematic diagram showing electron waves backscattered off the two atoms of a diatomic

molecule. Along the backscattering direction (the black dashed line) the wave scattered off

A1 and A2 has a relative phase difference ∆φ = 2kl = ql. Thus, the intensity of detected electron signal oscillates as a function of q due to interference between the waves scattered off A1 and A2. By taking the Fourier transform, a spectrum peaked at l is obtained. From the property of Fourier transform the width of the peak decreases as the q range increases.

Therefore, the larger the q range is, the better the imaging resolution.

If the molecular axis is perpendicular to k⃗r as shown in Fig. 5.17(b), ∆φ = 0 regardless of q and so no interference signal is obtained under this configuration. From this simple example we have shown that FABLES is a 1-dimensional imaging tool which only measures

99 distances with respect to the k⃗r direction. It is an inherent limitation but on the other hand it can also be viewed as an advantage in the sense that it provides a bond selective imaging which essentially only measures the bonds parallel to k⃗r. In principle, tomographic imaging of the molecule can be achieved by taking multiple FABLES signal as a function of molecular alignment angle with respect to the laser polarization [54].

5.2.2 Results and Discussions paraWe now turn to our experimental results. Figure 5.18(a) shows a photoelectron spec-

2 trum of C60 irradiated by 3.6 µm pulses at 84 TW/cm . The data in the backcattering plateau are then selected and the energy variable is converted from Ef to Er using the three step model. As in previous LIED studies [51, 53, 139], it is assumed that the contribution from the “long” trajectories dominates the “short” trajectories due to their difference in tunneling rate. Therefore, the conversion relation is given by the blue curve in Fig. 1.4(b).

The converted backscattered electron spectrum is shown as the blue curve in Fig. 5.18(b).

The next step is to extract the interference signal from the background. To do that we performed a least square fit to the data with an exponential function (the red dashed line in Fig. 5.18(b)) and then subtracted the fitting function from the data. The subtracted signal is shown in Fig. 5.18(c). Unlike the case of a diatomic molecule [54], the signal from C60 contains multiple “frequency” components since there are different characteristic distances between two carbon atoms of the molecule 23. We compared our experimentally retrieved molecular interference signal with theoretical electron scattering simulation with the IAM model. They are not in good agreement if the calculations are performed for C60 which is in its equilibrium geometry. However, if we assume in the calculation that the C60 is elongated along the k⃗r direction such that the length of the cage is 8% larger than the equilibrium value, then the simulated interference signal agrees well with the experimental data, as shown in Fig. 5.19. Elongation of the C60 cage was also inferred from angle-swept LIED analysis [140], the combined averaged elongation value considering both angle-swept

23The shortest distance is the distance between two adjacent carbon atoms (∼ 1.4A)˚ and the longest distance is essentially the diameter of the C60 cage (7.1A).˚

100 0 10 −4 (a) (b) (c) 2 1.5 −2 10 −5 1 0.5 0 −4 10 −6

Log(Yield) −0.5 −1 Electron yield (arb. un.)

−6 Interference signal (arb. un.) −1.5 10 −7 −2

0 500 1000 100 150 200 250 300 6 7 8 9 Detected energy E (eV) Return energy E (eV) Momentum transfer q (a.u.) f r

Figure 5.18: Extraction of molecular interference signal from a photoelectron spectrum. (a) Photoelectron spectrum of C60 along the laser polarization, laser wavelength is 3.6 µm pulses at 84 TW/cm2. (b) Backscattering “plateau” region of the spectrum in (a), the energy is converted from energy at detection Ef to energy at rescattering Er (see text). The red dashed line is an exponential fit to the spectrum. (c) Molecular interference signal as a function of momentum transfer q (q = 2kr for backscattering). It is obtained by subtracting the values of the red line from the values of the blue line in (b), followed by a smoothing procedure using a Gaussian window function.

LIED and FABLES data is found to be (6 ± 3)%.

The observed elongation seems to suggest that the C60 molecules were deformed by the intense laser field. In fact, such effect had been shown in theoretical simulations [141, 142] but has not been verified experimentally. The proposed physical mechanism is impulsive

Raman vibrational excitation [143]. In particular, excitation of the hg(1) mode will induce a periodic deformation of the cage alternating between a prolate and an oblate shape with a period of 125 fs. Figure 5.20 is a schematic diagram for illustrating the process. Initially, the

C60 cage was in its equilibrium geometry, the cage shape started to evolve as the laser field ramped up. At the time when the laser pulse envelope reached its peak the cage elongation was also approaching to its maximum amplitude. Since the detected FABLES spectrum is dominated by the electrons ionized around the peak of the pulse, thus the image result is representative of the degree of elongation around that time.

The theoretical simulations of C60 vibrational dynamics presented in the followings are provided by Prof. H. Kono’s group at Tohoku University, a collaborator for this project.

The computational details can be found in [140–142]. Here, we will just describe the basic 101 Figure 5.19: Comparison between experimentally retrieved and simulated (IAM) molecular interference signal. In the IAM simulation, the diameter of the C60 cage along the laser polarization is elongated by 8%.

principles. In the simulations, the electrons are treated quantum mechanically and the nuclear motion is treated classically. First, the equation for the instantaneous electronic

Hamiltonian24

Hel({ R⃗i } , t)ψ({ R⃗i } , t) = E({ R⃗i } , t)ψ({ R⃗i } , t) (5.9) is solved as a function of the nuclear positions { R⃗i } (i = 1, .., 60) and time t using a density-functional method. The obtained E({ R⃗i } , t) then acts as the adiabatic potential energy surfaces that govern the instantaneous nuclear motion at time t. Note that only

E({ R⃗i } , t) of the ground adiabatic electronic state will be used because it is assumed that the populations of excited adiabatic electronic states are negligible due to the small MIR

photon energy [141].

The forces for the classical equations of nuclear motion are then obtained by taking the

spatial derivatives of E({ R⃗i } , t). To qualitatively understand the vibrational dynamics

first note that E({ R⃗i } , t) can be expanded as

1 E({ R⃗ } , t) = E0({ R⃗ }) − µ({ R⃗ })F (t) − α({ R⃗ })F (t)2 + ..., (5.10) i i i 2 i 24Equation 5.9 is in the form of time-independent Schr¨odingerequation but it contains t so it is called time-dependent adiabatic state approach. 102 Figure 5.20: Graphical illustration of C60 cage field-induced elongation during a laser pulse (red curve). The numbers 1, 2 and 3 represent the three-step model: 1. ionization; 2. propagation; 3. rescattering.

0 where E ({ R⃗i }),µ({ R⃗i }) and α({ R⃗i }) are the field-free ground electronic state potential, permanent electric dipole moment, and polarizability of a molecule, respectively. The first term contains harmonic oscillator-like potentials which describe the vibrational modes of

C60, whereas the subsequent terms are the driving force terms due to the external field. Since the laser frequency is much higher than the vibrational frequencies, the nuclear motion is essentially determined by the optical cycle averaged version of expression 5.10. In other words, the second term can be neglected and so the driving forces mostly depend on the the spatial derivatives of α({ R⃗i }) (this is the origin of Raman excitation). Therefore, this excitation mechanism is essentially independent of laser frequency as long as it is much higher than the vibrational frequency. Basically, the system only “sees” the pulse envelope and so the pulse duration instead of the optical period is the key parameter that influence the excitation.

Figure 5.21(a) shows the elongation of C60 and the laser field amplitude as a function of time. Elongation is defined as d(t)/deq −1, where deq is the equilibrium diameter of C60 and d(t) is the length of the molecular axis along laser polarization as a function of time. The elongation oscillates at a period of the hg(1) mode (∼ 120 fs). The maximum elongation is 1.5 %. More simulations runs were performed at different laser pulse durations (between

40 to 150 fs), wavelengths (3.1 and 3.6 µm) and charge states. The simulated elongation amplitude are shown in Fig. 5.21(b). It can be seen that maximum elongation amplitude

103 Figure 5.21: Simulation results. Panel (a): temporal profile of the elongationd [ (t)/deq] − 1 (red line) of neutral C60 in a mid-IR Gaussian pulse (green line). The parameters of the 2 pulse used are as follows: peak intensity I0=80 TW/cm , pulse duration (FWHM)=60 fs, and wavelength=3.6 µm. Panel (b): Elongation as a function of charge species, wavelength λ, and pulse duration. The horizontal axis indicates the FWHM pulse duration of the laser. Simulations to obtain the values of field-induced distortion are carried out by using the time-dependent adiabatic state approach combined with the SCC-DFTB method (mio- 1-1 parameter set). The time step was chosen to be 0.1 fs. Black squares show values at λ = 3.1µm for neutral C60 and red filled circles at λ = 3.6µm. The upward-pointing triangle + symbol indicates the value for C60 and the downward-pointing symbol indicates the value 2+ for C60 . The value for λ = 3.6 µm is also evaluated by B3LYP/6-31G(d) of DFT, indicated by a green diamond symbol. Simulations performed by Prof. Kono.

is achieved when the applied pulse duration is about 60 fs, which is half of the vibrational

period of the hg(1) mode. We note that although the elongation effect was observed in both experiment and theory,

the agreement is not quantitatively perfect. One possible explanation could stem from the

fact that the experimental measured elongation was inferred from the best fit to IAM

calculation which is not perfectly accurate by itself. Experimental results from CED could

serve as a benchmark for IAM modeling but such data do not exist for our collision geometry.

On the experimental side, one feasible improvement is to perform a comparative mea-

surement between elongated and undisturbed C60. Although it is unavoidable that the molecule would be vibrationally excited by the strong field, it is possible to obtain an image

of the C60 cage close to its equilibrium geometry if one employs a very short laser pulse with a duration down to ∼ 30 fs, which is currently not available in our laboratory but

104 compression of MIR pulses down to few-cycle regime25 has been demonstrated by Fan et al

[144]. It is because it takes at least 30 fs (a quarter of the vibrational period of the hg(1) mode) plus some delay (due to the non-adiabatic response) for the molecule to transit from its equilibrium shape to maximum elongation, so using a short enough pulse could probe the molecule while it is still close to its equilibrium shape. By comparing the frequency components in the measured molecular interference signal between the two pulse durations, it is then possible to determine the relative change in the cage diameter without relying on

IAM modeling.

5.3 Summary paraIn this chapter, we have presented the results about two types of electron recollision phenomena in tunnel ionization of C60. From the “soft” recollision for low energy electrons we found an unexpected suppression of the LES which might be attributed to the induced dipole field; from the “hard” recollision for high energy electrons we demonstrated the applicability of LIED for imaging macromolecule and observed the laser induced deformation of the molecular structure.

25The shortest attainable pulse duration is fundamentally limited by the optical period, which is 12 fs for 3.6 µm 105 Chapter 6 Nonsequential double ionization of atoms with low ionization potentials

paraMost of the phenomena discussed in the previous chapters were restricted to single electron ionization, in this chapter we present our studies on two-electron ionization, with an emphasis on atoms with low ionization potentials irradiated by circularly polarized (CP) fields.

6.1 Effect of electron recollision in double ionization paraWithout considering electron-electron interactions, double ionization can be simply described by a two-step process, where the atom is first singly ionized by the laser field, and then some time later the singly charged ion is further ionized by the same field and a doubly charged ion is formed. This process is known as “sequential double ionization”

(SDI). To model ionization rates of SDI irradiated by intense laser fields, one could again employ the strong field ionization theories discussed in chapter3. Indeed, SDI have been observed in strong-field ionization experiments, but the phenomenon that really motivated the strong-field physics community to perform extensive and in-depth investigations ofdou- ble ionization over the last two decades is the so-called “non-sequential double ionization”

(NSDI). Note that the term “non-sequential” here does not necessarily imply that the emission of the electrons do not follow a temporal sequence, it just means that the two ionization

106 processes are correlated.

Let us first illustrate some of the key features of NSDI with the data of double ionization of helium published by Walker et al in 1994[18], as displayed in Fig. 6.1. As shown, the experimental data of single ionization yield agree well with TDSE simulations under the single-active electron approximation26. The intriguing result was the unexpectedly enormous yield of double ionization. Approximately between the intensity 2 × 1014 W/cm2 to 3 × 1015 W/cm2 the double ionization yields are higher than the theoretical SDI yield by several orders of magnitudes ( the difference is about 6 orders of magnitude at1 × 1015

W/cm2 and would be even larger for lower intensities). The enhancement is attributed to the contribution of NSDI. Only when the intensity is higher than 4 × 1015 W/cm2, then the double ionization yield is dominated by SDI process. Since the two ionization channels (SDI and NSDI) have different intensity dependences and saturation intensities as we will explain in the next section, the resultant yields which is the sum of the two channels exhibited a so-called “knee” structure which is now widely treated as the first signature of NSDI. In strong-field ionization, NSDI have been observed in all noble gas atoms from heliumto xenon and in a number of molecules [145].

As briefly introduced in chapter1, inelastic recollision can induce NSDI if the returning electron has sufficient energy to impact ionize or excite an electron (or even morethan one) of the parent ion. There are more subtle yet important details and questions such as the effect of multiple recollision, the influence of the strong laser field totheeffective ionization potential of the ion, the effect of the Coulomb potential, the effect of recollision induced excitations, etc (for a review, see, e.g., [145]). Moreover, evidence of recollision- unrelated effects which are due to the details of atomic structures have been reported

[146]. Nevertheless, the classical picture of electron recollision plays the central role in the interpretations at least for most of the studies in the tunneling regime. Figure 6.2 shows the probability of double ionization for helium irradiated by 0.8 µm pulse as a function of intensity calculated using two-electron classical trajectory simulations performed by Ho et al [147]. It can be seen that the knee structure is clearly reproduced. Details of this type of

26We have also verified that the data agree well with PPT calculations.

107 Figure 6.1: Ion yields of single and double ionization of Helium irradiated by linearly polarized, 100 fs, 0.78 µm laser pulses. Dots and crosses show different sets of experimental data with different confocal parameters of the laser. The solid line on the left isresult of TDSE calculations (with SAE approximation) for single ionization. The solid line on the right is result of TDSE calculations (with SAE approximation) for sequential double ionization. The dashed line is result of ADK calculations for single ionization. The double ionization yields are higher than the theoretical SDI yield by several orders of magnitudes. Reproduced from [18].

simulation will be described in Sect. 6.2 and will be used to compare with our experimental results.

A puzzle: NSDI with circularly polarized driving fields

There are still unresolved issues in the studies of NSDI (for a review, see, e.g., [145]), one question is whether the process is possible for a circularly polarized (CP) driving field. So far most of the experiments of NSDI focused on linearly polarized (LP) fields. As mentioned in Sect. 1.2.3, the classical trajectory of a free electron in a CP field never revisits the origin after launching, so NSDI is not expected to happen in this case. Indeed, experiments of inert gas atoms [148, 149] showed that the knee structure is absent for the case of CP fields and the double ionization yield agrees well with theoretical predictions of SDI.

108 Figure 6.2: Calculated probability of double ionization as a function of intensity using two- electron classical trajectory simulations (dots). Laser wavelength is 0.8 µm. The dashed curve shows ADK calculations. Effect of focal volume averaging is not taken into account. Reproduced from [147].

However, an experiment on magnesium [76] brought an unexpected result: a knee structure with CP fields. No clear explanation was provided at that time. Almost tenyears later, Mauger et al [150] (see also [151] by Wang and Eberly) proposed, using two-electron classical trajectory simulations, that the knee structure observed in [76] is due to electron recollision. For CP driving fields, the element that makes recollision possible is the Coulomb potential of the ion which tends to pull the outgoing electron back to the ion. The crucial factor that determine whether the Coulomb force can attract the electron back to the ion is the momentum of the outgoing electron. That is, if the electron escapes from the ion with a large enough velocity, it is unlikely that the Coulomb potential can drive it back and cause recollision. For an electron in a CP field, the cycle-averaged drift velocity (which indicates how fast it “escapes” from the ion core) is ∼ F0/ω (since the cycle-averaged kinetic energy

2 2 is Up ≡ eF0 /4mω ), so the probability of recollision decreases when the laser intensity or wavelength increases. Figure 6.3 shows the simulated double ionization probability of

magnesium and helium (inset) presented in [150, 151]. Consistent with the experimental

109 findings [76, 148], the knee structure exists in Mg but not in He. It was attributed to their difference in ionization potential Ip. Since Mg has a much lower Ip than He (Mg: 7.6 eV; He: 24.6 eV), the intensity required to take one electron out from Mg is much lower (about two orders of magnitude) than that for He. As a result, the Up of an electron from He is about two orders of magnitude higher than that from Mg and so recollision happens for Mg but not He.

Further theoretical investigations on this problem have been stimulated by [150], yet most of them relied solely on the comparison with the data of Mg at 0.8 µm in [76] due to the lack of experimental data in the literature. To the best of our knowledge, the data in [76] is the only set of data that shows the knee structure in strong-field atomic double ionization with CP fields (the only other experiment that showed a knee structure was performed on a molecular target NO [149]). Clearly, there are questions that remain unanswered. First, the experiments in [76] were performed well into the multiphoton regime (the max. value of γ ∼ 2), so the validity of using a classical model is questionable27. Second, recollision is a universal phenomenon and so comparing with only one target atom at a fixed wavelength is insufficient for drawing a concrete conclusion. To shed more light on this problem,we have conducted a more comprehensive investigation by performing experiments for various target atoms at different laser wavelengths and compared them with the predictions from classical simulations.

6.2 Two-electron classical trajectory simulations paraThe two-electron classical trajectory simulations, also known as the classical ensemble method, is similar to the CTMC method presented in Sect. 5.1.2, but with two major differences: (i) two electrons are included in the simulations instead of just one; (ii)the ionization processes are treated classically, instead of adapting the ADK tunnel ionization model (Eq. 5.1). The use of this method in the studies of NSDI was first performed by

27According to the classical prediction this type of recollisions only exist for visible to near-infrared wavelengths unless the ionization potential of the target is unrealistically low, as the use of MIR wavelength drivers would increase Up drastically. In other words, investigations for this matter are restricted to the multiphoton regime.

110 Figure 6.3: Calculated probability of double ionization as a function of intensity for magnesium and helium(inset) with CP field using two-electron classical trajectory simulations. Laser wavelength is 0.8 µm. Effect of focal volume averaging is not taken into account. Reproduced from [150].

J. H. Eberly’s group at Rochester in the early 2000s [152]. It has become a widely used method, not only because it has been proven to be able to reproduce some common key features in numerous experiments [145], but also because analysis of electron trajectories is useful for understanding the underlying dynamics. For detailed description of the method, see, e.g., [152, 153].

In this method, the dynamics of the two electrons, including the interaction with the laser field, the interaction with the parent ion and the interaction between the two electrons are simply modeled by classical mechanics. The time dependent Hamiltonian of the system, written in atomic units (~ = m = |e| = 4πϵ0 = 1), is

2 2 ⃗p1 ⃗p2 1 1 1 H = + − − + − ⃗r1 · F⃗ (t) − ⃗r2 · F⃗ (t) (6.1) p 2 2 p 2 2 p 2 2 2 2 ⃗r1 + a ⃗r2 + a |⃗r1 − ⃗r2| + b where ⃗pi and ⃗ri are the momentum and position vectors of the ith electron, respectively. The ion is assumed to remain stationary at the origin of the coordinate system. The first two terms are the kinetic energies of the electrons. The next two terms are the Coulomb potentials between the ion and the electrons. The fifth term is the Coulumb potential

111 between the two electrons and the last two terms are due to the external laser field F⃗ (t).

a and b are softening parameters for the Coulomb interaction. The role of a is to mimic

the ionization potentials of the atomic system. As a increases the “depth” of the Coulomb

potential well due to the ion decreases. That is, large value of a results in a weakly bounded

(low ionization potential) atom and vice versa. Determination of the a values for different

atoms is discussed below. The role of b is to reduce the repulsion of the two electrons to

avoid autoionization. b is set to 1 a.u. (in distance) in all the results presented in this

dissertation.

The simulations proceed in three steps. First, an ensemble of initial conditions for

electrons is set. Next, the equations of motions for the electrons are solved numerically.

Third, the final states of electrons are analyzed.

Initial conditions of electrons

In the classical ensemble method, the initial states of the electrons are mimicked by

an ensemble of initial positions and momenta which are randomly distributed under the

constraint that the total energy of the two electrons has to be equal to the energy required

to ionize the two electrons from the atom Ep, that is,

KE + PE = Ep, (6.2) where the kinetic energy KE is

1 1 KE = (p2 + p2 + p2 ) + (p2 + p2 + p2 ) (6.3) 2 1x 1y 1z 2 2x 2y 2z and the potential energy PE is

1 1 PE = − − px2 + y2 + z2 + a2 px2 + y2 + z2 + a2 1 1 1 2 2 2 (6.4) 1 + . p 2 2 2 2 (x1 − x2) + (y1 − y2) + (z1 − z2) + b

The value of Ep can be obtained by summing the ionization energies of the atom and the singly charge ion. This constraint sets a finite upper limit on the initial distance between the ion and an electron. If the distance is too large, Ep − P E < 0 implying KE < 0 which

112 is unphysical.

The procedures of generating an ensemble of initial conditions are as follows. First, a large set of uniform random numbers in which each sample contains six position coordinates for the two electrons (x1, y1, z1, x2, y2, z2). The corresponding PE for each sample are

calculated, the ones which do not satisfy Ep − P E > 0 are discarded. After calculating the values of KE for the remaining samples, the values of momentum components are

distributed randomly. In order to do that, KE is first split into two partsKE ( 1 and KE2) randomly for the two electrons, where

1 2 2 2 KE1 = (p1x + p1y + p1z), 2 (6.5) 1 KE = (p2 + p2 + p2 ). 2 2 2x 2y 2z Getting random distributions for the momentum components is then equivalent to picking random points on a spherical surface. In the results presented in this dissertation, the size of the ensemble for a simulation at a single intensity ranges from 104 to 105.

Ip (eV) Ip of ion(eV) Ep (a.u.) a 1st excited state Energy of ion (eV)

Ca 6.1 11.9 -0.66 3.7 3.12

Mg 7.6 15 -0.83 3.0 4.2

Zn 9.6 18 -0.94 2.4 6.1

Xe 12.13 22.2 -1.22 2.0 11.27

Table 6.1: Values of Ep and softening parameter a for different atoms in the simulations. The values are in atomic units.

The value of Ep and a for different atoms in our simulations are shown in Table 6.1. In fact, there is a range of valid values of a for each atom. If a is too large, the potential well will be too shallow to make any possible electron configurations to satisfy Ep − P E > 0. On the other hand, if a is too small some electrons in the ensemble will be “autoionized”

113 Figure 6.4: Distributions of initial position and KE of the electrons. (a): Correlation between x1 and x2. (b): Correlation between KE1 and KE2.

even without the presence of any external laser field F⃗ (t).

Using Mg as an example, The correlation between the initial x-coordinates of the two electrons is shown in Fig. 6.4(a). The two electrons tend to separate due to the Coulomb repulsion. Figure 6.4 shows the correlation between KE1 and KE2. Solving equations of motions

After preparing the ensemble of initial conditions, the equations of motions for each

sample are: d2 ⃗r 1 1 1 = −∇(− + ) − F⃗ (t) 2 p 2 p 2 2 dt ⃗r + a2 |⃗r1 − ⃗r2| + b 1 (6.6) d2 ⃗r 1 1 2 = −∇(− + ) − F⃗ (t). 2 p 2 2 p 2 2 dt ⃗r2 + a |⃗r1 − ⃗r2| + b For three dimensional simulations, there are six second order differential equations in total.

The system is then decomposed to twelve first order differential equations and numerically solved using a Runge-Kutta ODE solver in MATLAB. The vector potential of the laser pulse A⃗(t) in our simulations has the form:  F0 2ωt 1 11nπ 11nπ  cos2( )√ [cos(ωt)ˆx + ϵ sin(ωt)ˆy], if − < t <  ω 11n 2 4ω 4ω A⃗(t) = 1 + ϵ (6.7)  0, otherwise

114 Figure 6.5: An example of SDI trajectories. Panel (a): the distance between the electrons and the ion (origin) as a function of time. Panel (b): trajectories in the laser polarization plane. Red and blue represent two different electrons respectively.

then the electric field profile is given by F⃗ (t) = −∂A⃗(t)/∂t. F0 is the peak field amplitude; ϵ is the ellipticity (ϵ = 0 for linear polarization; ϵ = 1 for circular polarization) ; n is the full width of half maximum (FWHM) pulse duration in terms of number of optical cycles. Unless specified otherwise, n is set to be five for all results presented in this chapter. Propagation of electrons under Eq. 6.6 starts at t = −11nπ/4ω and ends at t = 11nπ/4ω + 3T . After that, the final energyKE ( + PE) of each electron are calculated. Ionization occurs if the energy is positive.

Two examples of two-electron trajectories are shown in Fig. 6.5 and 6.6. The calculations were performed for Mg atoms with CP, 0.8 µm laser pulse at an intensity of 240

TW/cm2. Figure 6.5 shows a scenario of SDI. One of the electrons (red trajectory) was driven away from the ion by the laser field and the other electron (blue trajectory) left the ion about four optical periods later. The ionization of the two electrons is uncorrelated, that is, the interaction between them did not play a major role in the process.

115 Figure 6.6: An example of NSDI trajectories. See caption of Fig.6.5.

Figure 6.6 shows a scenario of NSDI. One of the electrons (red trajectory) was first driven away from the ion by the laser field and returned to it after a few optical periods. It collided with the other electron and then both of them left the ion. The ionization of the two electrons is strongly correlated due to the recollision process.

Analysis of electron trajectories

After obtaining all electron trajectories for the entire ensemble and calculating the corresponding final energies, the probability of double ionization is obtained by dividing the number of double ionization events by the size of the ensemble. Each double ionization event is categorized into either SDI or NSDI, based on whether recollision has occurred and whether the energy transfer due to the recollision is significant. To do that, the trajectory of each electron is monitored at each time step in the simulations. As shown in Fig. 6.5(a) and 6.6(a), the black line is defined to be a critical distance r0. For an electron to “escape”,

its distance from the ion r has to exceed r0 at some time (we set r0 = 5×, where <> means ensemble average). For recollision to happen, the distance

r has to be at some later time smaller than r0 again. After recollision, r increases again and

eventually exceed r0 again. Mathematically, if the sign of r − r0 only changed once, that

116 means ionization without recollision occurred. If the sign of r −r0 changed by at least three times or more that means recollision happened. Note that if the number of sign changes is an even number that implies the electron is recaptured by the ion and not considered to be ionized. In our MATLAB code for solving Eq. 6.6, r − r0 is calculated at each time step and the time(s) when it changes sign is recorded.

0 10 NSDI SDI Total −1 10

−2 10 Probability of double ionization −3 10 2 10 Intensity (TW/cm2)

Figure 6.7: Probability of double ionization as a function of laser intensity for Mg atom irradiated by CP 0.8 µm laser pulses. Red: probability due to NSDI process. Blue: probability due to SDI process. Black: total ionization probability, which is the sum of the probabilities due to the two processes.

To determine the gain in energy for the colliding electron, we calculated the total energy of that electron before and after collision. We defined Ebefore to be the time averaged energy from t = t0 − 0.5T to t = t0; and Eafter to be the time averaged energy from t = t0 to t = t0 + 0.5T . The time of recollision t0 is defined to be the instant when |⃗r1 − ⃗r2| is at minimum; T is the laser period. The gain in energy is then given by ∆E = Eafter −Ebefore.

As the final step,E ∆ was compared to a threshold value Eth. If ∆E < Eth or if recollision did not even occur then the event is classified as SDI; otherwise it is NSDI. Weset Eth to

117 be the energy required to excite the ion from ground state to the 1st excited state. For instance, Eth = 4.4 eV for Mg. As an example, double ionization probability as a function of laser intensity for Mg atom irradiated by CP 0.8 µm laser pulses is displayed in Fig.

6.7(black symbols). The contribution of SDI (blue symbols) and NSDI (red symbols) to the total double ionization probability are also shown. At low intensities NSDI is the dominant process and as intensity increases SDI takes over, which results in the formation of a knee structure in the total probability.

6.3 Calculations of SDI probabilities with the PPT model paraAlthough the focus of the chapter is NSDI, it is important to know the contribution of SDI quantitatively. In fact, solely relying on the appearance of the knee structure to determine whether NSDI occurred is not quite sufficient. As illustrated in the previous section, NSDI manifests itself as a knee structure because the dominant ionization mechanism evolves from NSDI at low intensities to SDI at high intensities. However, if NSDI dominated over the whole intensity range then no such transition would be observed. Instead, the measured yields would be simply higher than the predicted yields from SDI process over the whole intensity range. Therefore, a more rigorous criteria to identify the existence of NSDI is from the (dis)agreement between experiment and predicted yield from SDI calculations.

As we have examined in Ch.3, ionization yield calculations with PPT model are in good agreement with experimental data for various atoms in both tunneling and multiphoton regimes. Here, we apply this model to calculate SDI probability by solving the following set of rate equations:

dP 0 = −w P (6.8a) dt 0 0 dP 1 = w P − w P (6.8b) dt 0 0 1 1 dP 2 = −w P , (6.8c) dt 1 1

where P0, P1 and P2 are the populations of neutral atoms, singly charged ions and doubly

charged ions, respectively (P0 +P1 +P2 = 1). w0 and w1 are PPT ionization rate for neutral

118 atom and ion respectively. P1 and P2 after the laser field are obtained by solving Eq. 6.8 using Euler’s method with initial conditions P0 = 1 and P1 = P2 = 0. The effect of focal volume averaging is taken into account using the same procedure described in chapter3.

6.4 Experimental Results and Discussions

6.4.1 Linear Polarization paraAlthough the main purpose of this chapter is to investigate whether the observed knee structure in the double ionization yield of Mg irradiated by 0.8 µm CP fields [76] is indeed due to recollision, it is beneficial to first examine the cases of LP fields in which recollision is commonly believed to be the main mechanism of NSDI for noble gas atoms. This way we can first justify whether recollision is playing a role in double ionization ofMgbyany means before we move towards the cases of CP fields in which the recollision process, if any, is considered to be even more complicated.

Let us begin the discussion with a data set in the deep tunneling regime, as it is considered to be a “simple” case since the semi-classical description for photoelectrons is known to be accurate in this regime. Figure 6.8(a) displays single and double ionization yields of

Mg irradiated by 3.7 µm pulses. PPT calculations assuming SDI process are also shown

(black solid lines) for comparisons. Qualitatively similar to the data of He in Fig. 6.1, a pronounced knee structure in the double ionization yield curve is shown. Ion yield ratio

Mg2+/Mg+ as a function of intensity is shown in Fig. 6.8(b), which it is almost independent of intensity within the experimental range. In other words, the intensity dependence of double ionization in the non-sequential regime is very similar to that of single ionization.

Such behavior could be understood from the property of electron impact ionization. To understand this, recall the idea of QRS for elastic scattering mentioned in chapter5 that laser-induced rescattering could be modeled as regular electron scattering. Similarly, the probability of NSDI can be treated as the product of the flux of returning electron and the cross section of impact ionization for the parent ion [154]. As shown in Fig. 6.9, the cross section of impact ionization of Mg+ ion has a smooth and gentle dependence on electron

119 Figure 6.8: Single and double ionization of Mg with MIR pulses. (a) Measured single and double ionization yields of Mg with LP driving fields at 3.7 µm, black lines are PPT calculations for single ionization and SDI. (b) Single-to-double ion yield ratio as a function of intensity. The results of Mg is directly obtained from the data in panel (a) and the results of Xe is from the published data of nonsequential ionization of xenon at 3.6 µm by DiChiara et al [47].

energy over the range of rescattering energy for our data (3.17Up ≈ 40−400 eV). As for the flux of the returning electron, according to the ADK formula (seeEq. 5.1) it is inversely √ proportional to I which does not have a rapid dependence neither. Moreover, focal volume averaging tends to flatten out any intensity dependence of ion yields. Considering allthe factors above, it is expected that the double-to-single ion ratio should have a weak intensity dependence. In fact, such trend has been observed in noble gas atoms, for instance, from the study of nonsequential ionization of Xe at 3.6 µm performed by DiChiara et al [47] as shown by the red line in Fig. 6.8(b). They demonstrated that the ratio Xe2+/Xe+ could be well reproduced by simple 1D three-step model classical-trajectory calculations including the effects of impact ionization cross sections and the momentum spread of the returning electrons.

Perhaps a surprising feature in the results of Mg, considering the fact that its impact

120 Figure 6.9: Cross section of electron impact ionization for Mg+ (from [155]) and Xe+ (from [47]) as a function of energy.

ionization cross section is considerably lower than that of Xe+ (see Fig. 6.9), is that the double-to-single ion yield ratio is actually slightly higher than that of Xe, as shown in Fig.

6.8(b). However, it is noteworthy that the cross section of impact excitation to the first excited state for the two atoms over the relevant energy are in fact similar to each other, both are in the order of 102 Mb with less than a factor of two difference. Therefore, if recollision induced excitation followed by strong field ionization also significantly contributed tothe measured NSDI yields then one might expect that the double to single ion ratio for the two targets would not be too different from each other. However, the classical calculations presented by DiChiara et al seems to suggest that recollision excitation did not play any significant role for Xe. In their calculations, if the impact excitation cross section instead of the impact ionization cross section was used, then the results significantly overestimated the observed ion ratio. Interestingly, when the same type of calculation was applied to the case of He irradiated by 0.8 µm pulses (Fig. 6.1) the results underestimated the measured ratio by more than one order of magnitude. Therefore, the effect of recollision excitation should not be ruled out but it is unclear why such effect might be important for some targets

(He,Mg) but not for others (Xe, Ne).

121 0 10 1 1 10 (a) 10 (b) (c) Mg+ Mg+

0 0 10 10

−1 −1 −1 10 10 10

Mg2+ Mg2+ + −2 −2 10 10 /Mg 2+ Mg

−3 −3

Ion Yield (arb. un.) Ion Yield (arb. un.) −2 10 10 10

−4 −4 10 10

−5 −5 10 10 −3 10 13 14 13 14 13 14 10 10 10 10 10 10 Intensity (W/cm2) Intensity (W/cm2) Intensity (W/cm2)

Figure 6.10: Measured single and double ionization yields of Mg with LP driving fields at (a) 0.8 µm and (b) 1.03 µm. Black lines are PPT calculations for single ionization and SDI. Panel (c) shows single-to-double ion yield ratio from the data in panel (a) and (b).

We now turn to the cases of near infrared (NIR) wavelengths (0.8 and 1.03 µm), in which the results are considered to be more complicated to interpret for the following reasons.

First, Up is much smaller than the case of long wavelength. At 3.7 µm, 3.17Up ≈ 40 − 400

2 eV for the intensity range between 10 to 100 TW/cm ; at 0.8 µm, 3.17Up ≈ 2−20 eV for the same intensity range. Second, as mentioned, the accuracy of the semiclassical description in this regime where γ ∼ 1 is questionable. Third, unlike noble gas atoms, Mg (and other alkaline earth atoms) have low lying excited states whose energies are equivalent to just a few near infrared photons, although it is unclear how significant the effects of multiphoton excitation are with our laser wavelengths and how these effects would enhance the double ionization rate.

Figure 6.10(a) displays single and double ionization yields of Mg irradiated by 0.8 µm pulses (red points), PPT calculations are shown as the black solid lines. PPT calculations and experimental data for single ionization yield are in good agreement, as we have presented in chapter3. No knee structure is observed in double ionization but the yield is significantly higher than the theoretical SDI yield over the entire experimental intensity range. Result

122 with a slightly longer wavelength, 1.03 µm, is shown in Fig. 6.10(b). Again, single ionization yields are well predicted by PPT calculations and the double ionization yield is orders of magnitude higher than SDI calculations. The double to single ion ratio for the two wavelengths are shown in Fig. 6.10(c) and the results almost overlap over the entire intensity range. The nearly similar results from the two close but incommensurate NIR wavelengths seems to indicate that excitations to intermediate states may not be the main reason for the observed enhanced yields; otherwise the two results should show some significant “resonant” differences. On the other hand, if it is due to recollision then it is possible that bothcases would give similar results since Up is not drastically different.

−1 −1 10 10 (a) (b) Zn+ Zn+ −2 −2 10 10

−3 −3 10 10

2+ 2+ −4 Zn −4 Zn 10 10 Ion Yield (arb. un.)

13 14 13 14 10 10 10 10 Intensity (W/cm2) Intensity (W/cm2)

Figure 6.11: Measured single and double ionization yields of Zn with LP driving fields at (a) 0.8 µm and (b) 1.05 µm. Black lines are PPT calculations for single ionization and SDI.

We also performed measurements on another target, zinc, at 0.8 and 1.05 µm to see

if a similar enhancement in double ionization is observed. Zinc ([Ar]3d104s2) was chosen

because its Ip (9.4 eV) is not too different from Mg. Also, effects of excited states are expected to be weaker than Mg due to the higher excited states energies. As shown in Fig.

6.11, very strong and similar enhancement was observed for both wavelengths. It seems to

123 again suggest that the enhancement in Mg was not due to its special atomic structure.

Figure 6.12: Double ionization probability of Mg calculated from the classical simulations for LP driving fields at (a) 0.8 µm and (b) 3.6 µm. Red: probability due to NSDI process. Blue: probability due to SDI process. Black: total ionization probability, which is the sum of the probabilities due to the two processes.

If recollision is indeed the main mechanism for NSDI of Mg in both MIR and NIR fields, one question to ask is why the double ionization yield in the latter case is so much more abundant considering the fact that Up is much smaller. Two electron classical simulations might give some hint to this problem. Figure 6.12(a) and (b) are simulated double ionization probabilities as a function of intensity for Mg at 3.6 and 0.8 µm, respectively. As described in Sect. 6.2, a double ionization event is classified as NSDI if the electron near the ion core gained ≥ 4.4 eV (1st excited state energy of Mg+) due to the recolliding electron and then ionized, or else it is classified as SDI. The results showed that NSDI is much more probable in the NIR case. Also, unlike the MIR case, the knee structure is not visible because even at the high intensity end the contribution from SDI did not completely dominate.

Detailed comparisons of electron trajectories for the two wavelengths are not relevant to

124 the goal of this study and so was not pursued, but one obvious and important factor is that the number of returning electron trajectories which “miss” the target increases with wavelength. It is because the time taken for an electron to leave and return to the ion increases with wavelength, so as its transverse displacement upon return (assuming that it has a non-zero initial transverse momentum). Some recent theoretical studies of NSDI for noble gas atoms using classical or semiclassical simulations (e.g., [156, 157]) have also shown that NSDI is more probable for short wavelength.

Figure 6.13: Experimental yields of Xe+ (red symbols) and Xe2+ (blue symbols) vs laser intensity for various wavelengths. Solid lines are PPT calculations for single ionization and SDI. Reproduced from [158]

As further examples, Fig. 6.13 and 6.14 show wavelength dependence studies of NSDI in Xe from two independent groups [157, 158]. We can see from Fig. 6.13 that while a clear knee structure is present at longer wavelengths (1181 and 1579 nm), it disappears at

125 Figure 6.14: Experimental ratios of Xe2+/Xe+ vs laser intensity for various wavelengths. Reproduced from [157].

shorter wavelengths (533 and 571 nm) but the Xe2+ yield is overall higher than the SDI prediction (with PPT formula) over the whole intensity range. Similar trend can be seen from Fig. 6.14. At long wavelengths (1800 and 2200 nm), the Xe2+/Xe+ ratio shows a relatively flat intensity dependence at low intensity (up to about2 100TW/cm ) and this is the region where NSDI dominates. When the intensity goes beyond certain values (slightly higher than 100 TW/cm2), the ratio starts to increase rapidly as SDI takes over. For slightly shorter wavelengths (1200 and 1500 nm), this signature of transition from NSDI and SDI could still be seen, but at a higher intensity (∼ 150 TW/cm2). However, as wavelength becomes even shorter (800 nm), the transition is much less pronounced for observation, it is just that again the ratio is overall higher than the longer wavelength cases.

6.4.2 Circular Polarization paraFrom the results with LP fields we observed some evidence which supported recollision to be the main mechanism of NSDI of Mg, even in the NIR regime. Certainly, recollision under CP fields is considered to be more complicated and subtle since it is required tofully include the long range Coulomb force due to the parent ion in the calculations. On the other hand, complications due to quantum effects should play a less significant role inthe

CP case. It is because the transitions to most of the low lying excited states with NIR fields

126 Figure 6.15: Double ionization probability of Mg calculated from the classical simulations for CP driving fields at (a) 0.8 µm and (b) 1.03 µm.

are forbidden due to the angular momentum selection rule. For Mg+, the allowed states for transition (from ground state) would have to at least have an angular momentum quantum number of 10 and the energy of these states are very close to the ionization threshold [76].

In Sect. 6.1, we have explained how the probability of recollision depends on Ip of the target. In fact, it is also sensitive to the driving wavelength. An increase in wavelength

results in an increase in drift velocity of the photoelectron which would cause recollision

probability to drop. Figure 6.15 are results of classical simulations for Mg at 0.8 µm and 1.03

µm. The knee structure is drastically suppressed when the driving wavelength is increased,

the probability of NSDI dropped by almost two orders of magnitude.

Indeed, similar trend was observed in experiments. Figure 6.16 shows measured single

and double ionization yields of Mg at (a) 0.8 µm and (b) 1.03 µm. Consistent with the data in [76], a knee structure is clearly shown in the case of 0.8 µm. On the other hand, the knee structure is completely suppressed in the case of 1.03 µm, at least within the dynamic range of our measurement. Clearly, the data and the classical simulations follow the same trend that increasing wavelength results in suppression of NSDI. It is possible that the yield

127 Figure 6.16: Measured double ionization yields of Mg with CP driving fields at (a) 0.8 µm and (b) 1.03 µm. Black lines are PPT calculations for single ionization and SDI.

of NSDI is overestimated in simulations so a better experimental dynamic range is required to observed the feature.

Although the complete absence of a knee structure in the case of 1.03 µm might be simply attributed to insufficient experimental dynamic range, it does not completely rule out the possibility that the knee structure observed in 0.8 µm is due to some mechanism other than recollision that works exclusively at that wavelength. To search for the existence of recollision effects at 1.03 µm, we performed measurements with elliptically polarized field.

The ellipticity was set to a large value (0.75) so as to maintain a similar waveform to a CP field and thus would not significantly alter the electron dynamics but only slightly increase the recollision probability. The experimental data shown in Fig. 6.17(a), the CP data from

Fig. 6.16 are also displayed for comparison. As seen, a knee structure is visible and the trend is qualitatively reproduced by classical simulations, as shown in Fig. 6.17(b).

The wavelength and ellipticity dependence of the knee structure in Mg has qualitatively supported the interpretation of recollision, or at the very least these results are not against

128 Figure 6.17: Double ionization of Mg with elliptically polarized pulses. Panel (a): Measured yields for double ionization of Mg using circular and elliptical polarized fields at 1.03 µm. The data of CP fields are from Fig. 6.16(b). The blue line is drawn to guide the eye. Panel (b): Classical simulations for the two cases.

the predictions from classical simulations. If recollision is indeed the mechanism then similar phenomena should be observed in other atoms as well. As explained earlier, recollisions are expected to be suppressed with high Ip so the target atoms for this test need to have comparable Ip as Mg. We selected Zn and Ca as the targets because one has a slightly higher and the other has a slightly lower Ip compared with Mg, as shown in Table 6.1. Moreover, both of them have two valence electrons which match the two-active-electron approximation in the classical model.

Figure 6.18 shows the experimental results of Zn and Ca with 0.8 µm CP driving field.

No knee structure is observed in both cases, but there is a very important difference. For

Zn, the data agrees with the theoretcial SDI yield so it suggested that NSDI is not making any measurable contribution within the dynamic range in the measurements. However, for

Ca the measured double ionization yield is higher than the theoretical SDI yield over the entire intensity range in the experiment, suggesting that there is another ionization channel

129 Figure 6.18: Measured ionization yields of Ca (a) and Zn (b) with CP driving fields at 0.8 µm. Black curves are PPT calculations for single ionization and SDI.

dominating the process (Note that the PPT calculations also overestimates the yield of single ionization of Ca, the reason has not been identified). However, it does not forbid us to conclude that the measured yield is higher than the theoretical SDI yield. In fact, overestimation of single ionization yield in the calculation probably led to an overestimation of SDI yield as well. That is, the deviation between the measured Ca2+ yields and the theoretical SDI yield might be even larger than what is presented in Fig. 6.18.)

The results of classical simulations for Zn and Ca are shown in Fig. 6.19. Although a knee structure appeared in Zn, the signal level is much weaker than Mg at the same driving wavelength (Fig. 6.15(a)). The absence of a knee structure in the measurement might be again attributed to insufficient dynamic range, just like the measurement of Mg at1.03 µm.

Regardless, both experiment and theory shows that higher Ip results in suppression of the knee structure. As for Ca, our simulations shows that the knee structure is absent but it is

because the contribution of NSDI is higher than that of SDI over the whole intensity range.

It might qualitatively explain why the knee structure is absent in the experiment.

130 Figure 6.19: Simulated double ionization probabilities of (a) Ca and (b) Zn as a function of intensity for CP fields at 0.8 µm. Red: probability due to NSDI process. Blue: probability due to SDI process. Black: total ionization probability, which is the sum of the probabilities due to the two processes.

6.5 Summary paraOur main results of CP fields could be summarized by Fig. 6.20. The white points represent the measurements where the double ionization yields agreed with SDI calculations; the black points are the ones which showed deviation. The color map shows the classical predications of NSDI yields as a function of wavelength and Ip. The predicted values are not from full numerical simulations but simply from an approximate analytical expression (2I5])1/4 5/2 √ 2+ + p√ 2 derived by Fu et al [159]. It is expressed as [X ]/[X ] ≃ 10 πω exp[−Ip /2 2ω ]. Overall, we have not found any discerning disagreement between experiments and classical predictions at the qualitative level. Moreover, the appearance of a knee structure in Mg irradiated by 1.03 µm elliptically polarized (close to CP) field provides an important hint of recollison.

The implication of investigating the knee structure of Mg irradiated by CP fields is more than just to understand this seemingly exotic but specific feature. First of all, as 131 mentioned earlier, the validity of classical models in such γ regime is unclear and so this study could serve as a test. It is in fact a bit surprising that our results seem to embrace the classical interpretation. In addition to the concern of large γ, another noteworthy issue is that since the recollision dynamics under CP fields [150] is much more complicated and even chaotic than the simple three-step description for LP fields, how these classical trajectories

“survive” in the quantum reality remain unclear and should be investigated.

9.5 0.03

9 0.025 8.5

0.02 8 [X 2 +

7.5 0.015 /X + ]

7 0.01

6.5

0.005 6 Ca

750 800 850 900 950 1000 1050 1100 wavelength(nm)

Figure 6.20: Phase diagram for NSDI in CP fields (see text).

In addition, the use of CP fields in strong field studies have been often treated asa

“trick” to completely eliminate electron recollision. If the knee structure in Mg is indeed due to recollision, then the assumption that recollision does not exist under CP fields should not be taken for granted, and it is possible that such an effect would become more significant for larger systems.

132 Chapter 7 Conclusions and Outlook

paraIn this dissertation we have investigated atoms and molecules subjected to strong near and mid-infrared pulses. We investigated various problems including investigation of atomic ionization under different laser wavelengths and intensities, effect of vibrational resonance in molecules, effect of electron-electron correlations, effect of polarizability, and extending ultrafast molecular imaging to complex molecules.

In chapter2, we built a sub-millijoule level mid-infrared optical parametric amplifier

(OPA) which is pumped by a Ti:Sapphire laser system. The wavelength of the output is tunable between 3 to 4 µm. It serves as the main laser source for the experiments described in the subsequent chapters.

In chapter3, we measured strong field ionization yield versus intensity is investigated for various atomic targets (Ne, Ar, Kr, Xe, Na, K, Zn and Mg) and light polarization from visible to mid-infrared (0.4 - 4µm), from multiphoton to tunneling regimes. The experimental findings (normalized yield vs intensity, ratio of circular to linear polarization and saturation intensities) are compared to the theoretical models of Perelomov-Popov-Terent’ev (PPT) and Ammosov-Delone-Krainov (ADK). While PPT is generally satisfactory, ADK validity is found, as expected, to be much more limited.

In chapter4, we studied strong field ionization and dissociation of methane molecules.

We found that the amount of fragmenting ions relative to the intact molecular ions exhibit a pronounced wavelength dependence near the resonance located at around 3.3 to 3.6 µm. In contrast, the feature is absent in the same measurements with deuterated methane (CD4).

133 The results suggest that the resonance of C-H bond stretching mode plays a significant role in the fragmentation processes. Moreover, by comparing the total ion yields of CH4

with that of CD4, we found that the overall ionization rate of CH4 is also enhanced when the driving field is on resonant with the vibrational resonance. This result has important

implications in understanding tunnel ionization in the presence of vibrational resonances.

In addition to the total ionization rate, it would be beneficial to find out whether the

resonance also affects the electron wavepacket. Maybe a possible test is to perform acom-

parative FABLES measurement for a hydrocarbon molecule and its deuterated isotopologue,

for example, C2H2 versus C2D2 at 3.5 µm. The molecular interfence signal in both cases are dominatedly by the CC bond and so are identical. Thus, if there is any difference between the rescattering pleateu from the two cases it can be attributed to the difference of electron wavepacket.

In chapter5, we studied two types of electron recollision phenomena in tunnel ionization of C60. From the “soft” recollision for low energy electrons we found an unexpected suppression of the LES which might be attributed to the induced dipole field; from the “hard”

recollision for high energy electrons we demonstrated the applicability of LIED for imaging

macromolecule and observed the laser induced deformation of the molecular structure.

As we have introduced in the begining of chapter5,

134 Figure 7.1: Low energy photoelectron spectrum of Xe and C60 irradiated by 3.1µm laser pulses at 78 TW/cm2.

In chapter6, we searched for the existence of recollision effects in double ionization of magnesium, zinc and calcium at different wavelength in order to examine the validity of the classical interpretation for the previously observed enhanced double ionization yield in magnesium irradiated by circularly polarized 0.8 µm fields. We have not found any discerning disagreement between experiments and classical predictions at the qualitative level. Moreover, the appearance of a knee structure in Mg irradiated by 1.03 µm elliptically polarized (close to CP) field provided an important hint of recollison. Overall, our results seem to embrace the classical interpretation.

A possible further test would be to investigate the pulse duration dependence of the knee structure of Mg at 0.8 µm. From our classical simulations, majority of the recolliding electron trajectories with CP fields have to spend a few optical cycle∼ ( 4) of time before returning to the parent ion, similar to the example trajectory shown in Fig. 6.6 (for further details, see an undergraduate research thesis from our group [160]). Therefore, the knee structure should be suppressed if the laser pulse duration is reduced to ∼ 10 fs. 135 In addition, C60 could potentially be another relevant target to investigate recollision pheonomena with CP fields [161]. Its extended size should lead to a significant higher recollision probability compared with Mg.

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148 Appendix A Setup of Mid-infrared Optical Parametric Amplifier

A full optical layout of the OPA is shown in Fig. A.1. It was built on a 30” by 36” optical breadboard. The pump beam is first reflected by two mirrors M0 and M1toa

90:10 beamsplitter (BS1), at which 90% (10.8 mJ) of the laser energy is reflected and the remaining 10% (1.2 mJ) is transmitted. The transmitted beam is split further to provide light for white light continuum generation and the first three amplification stages; the reflected beam is for pumping the last amplification stage. The lengths of the beampaths were designed such that the signal beam overlaps temporally with the pump beams at the

KTA crystals. The details of the setup will be described in the following paragraphs. We start with the details of the preamplification stages. The 800nm beam passes through BS1 is reflected by M2 and then M3 to reach an 80:20 beamsplitter (BS2). The transmitted beam (∼ 240 µJ) is used to pump the 2nd KTA (C2). The reflected beam is directed to another 80:20 beamsplitter (BS3), the reflection (0.76mJ) is transported to the 3rd KTA

(C3) and the transmission (∼ 190 µJ) is split again by a 95:5 beamsplitter (BS4). The 1st

KTA (C1) takes the reflected part∼ ( 180 µJ) and the remaining ∼ 10 µJ is available for white light continuum generation.

149 Figure A.1: Full optical layout of the OPA.

150 A.1 White Light Continuum Generation

Figure A.2: Optical path of the white light continuum generation.

The optical path of white light continuum generation (WLCG) is shown in Fig. A.2. As described before, WLCG is driven by the beam passing through BS4. The polarization of the beam is rotated by 90 degree using a half waveplate (HWP) so that the polarization of the generated seed signal lies along the e-axis of the KTAs. The beam is focused by a

50 mm plano-convex lens (L1) onto a sapphire plate (S) to generate white light. The laser intensity at the sapphire plate is controlled by an iris aperture placed in front of HWP. It has to be adjusted carefully since the required intensity for WLCG is just slightly lower than the damage threshold of the material. The beam coming out from the sapphire is diverging and has to be collimated by another 50 mm lens (L2). It is then transported by the mirrors (M4→M5→M6→M7) to C1 for the first amplification.

151 A.2 First Amplification Stage

As described before, the 1st KTA stage is pumped by the beam reflected by BS4. To achieve the desired pump intensity, the beam size of the pump is reduced by a factor of 10 using a reflective telescope consisting of a concave mirror CM1 (f = 500mm) and a convex mirror

CM2 (f = -50mm). The pump and the WLC meet at C1 (see Fig. A.3). The estimated peak intensity of the pump at C1 is 430GW/cm2. For overlapping the two beams in time, the optical path length of the WLC is controlled by a delay stage (DS1) to match the path length of the pump beam. After amplification, the remaining energy of the pump is blocked by a beam dump (BD1) and the amplified signal is reflected by M8 to the second KTA

(C2).

Figure A.3: Optical path of the first OPA stage.

152 A.3 Second Amplification Stage

Figure A.4: Optical path of the second OPA stage.

Figure A.4 shows the optical path of the 2nd amplification stage. The KTA is pumped by the beam transmitted through BS2. It is directed by the mirrors

(M9→M10→M11→CM3→CM4) to the second KTA (C2) and overlaps with the signal beam coming from C1. CM3 (f = 400mm) and CM4 (f = -50mm) act as a telescope which reduces the pump beam size by a factor of 8. The estimated peak intensity of the pump at C1 is 370GW/cm2. The temporal overlap between the two beams is controlled by the path length of the pump beam using a delay stage (DS2). The used pump beam is blocked by a beam dump (BD2); the amplified signal beam is reflected by a mirror M12 and transported to the next amplification stage.

153 A.4 Third Amplification Stage

Figure A.5: Optical path of the third OPA stage.

Figure A.5 shows the optical path of the 3rd amplification stage. The signal beam from the 2nd stage is reflected by M12 and then M13 to the KTA (C3). C3 is pumped bythe beam reflected BS2. It is directed by the mirrors→ (M14 M15→M16→M17→M18→M19) to a reflective telescope consists of CM5 (f = 250mm) and CM6 (f = -50mm). Thebeam is then reflected by M19 and DC1 to the KTA (C3). DC1 and DC2 are longpass dichoric mirrors (Thorlabs, DMLP950) which reflect the pump beam and transmits the signal beam.

The use of dichoric mirrors allow the two beams co-propagate in parallel in C3 for optimal phase matching. DS3 controls the time delay between the pump and signal beam. After amplification at C3, the pump and signal beam are separated by DC2. The pump isreflected

154 off and blocked by a beam dump (BD2); the amplified signal beam transmits throughand is reflected by M20 for the next amplification stage. Depending on the wavelength ofthe signal, the pulse energy is ranged between 130 to 180 µJ.

A.5 Fourth Amplification Stage

Figure A.6: Optical path of the fourth OPA stage.

The pump energy involved in the 4th amplification stage is an order of magnitude higher than that in the previous stage. Due to the high intensity, propagation of the pump beam even just in air for a couple meters could be problematic due to nonlinear phase shift caused by Kerr effect. The beam quality could be deteriorated in terms of its temporal profiledue to self phase modulation and spatial profile due to self focusing. The nonlinear phase shiftis R quantified by B integral which is defined as B = (2π/λ) n2Idz, where z is the propagation direction; n2 is the nonlinear refractive index; I is the intensity of the beam. Keeping B integral smaller than 1 is a rule of thumb in ultrafast optical design. The intensity of the

2 −23 −2 final pump beam is about 260 GW/cm and n2 of air is 4 × 10 m /W. Thus, B = 0.82 155 per meter. To improve this, we reduce the intensity by expanding the beam. As shown in

Fig. A.6, the beam reflected by BS1 is expanded by the reflective telescope consists ofCM7

(f = -200mm) and CM8 (f = 500mm). The intensity is then reduced by a factor of 5. It bounced off the mirrors (M23→M24→M25→M26) and then the beam size is reduced to the original size by another telescope consists of CM9 (f = 500mm) and CM10 (f = -200mm).

The beam is then reflected by M27 and DC3 (the same type of dichroic mirror as DC1and

DC2) to the KTA (C4). As for the signal beam from the previous stage, it is reflected by the mirrors (M20→M21→M22) and reaches C4. On its way it is expanded by a telescope consists of L3 and L4 to match the beam size of the pump beam. DS3 controls the time delay between the pump and signal beam.

156