Ultrafast electron diffraction (UED) is a method of directly imaging system dynamics at the atomic scale with picosecond time resolution. In this thesis I present theoretical analyses of the experimental processes, and construct models in order to better understand UED experiments and to guide future refinements. In particular, I derive a model of electron bunch propagation and a model of electron bunch diffraction, where both models take into account all bunch parameters. To analyse the propagation of electron bunches, I present a mean-field analytic Gaussian (AG) model. I derive a system of ordinary differential equations that are solved quickly and easily to give the bunch dynamics. The AG model is compared to N -body numerical simulations of initially Gaussian bunches, and I demonstrate excellent agreement between the two result sets. I also present a comparison of the AG model with numerical simulations of quasi-Gaussian and non-Gaussian distributions, extending the applicability of the AG model to the propagation of ``real-world'' bunches. During propagation, electron bunches can be shaped by electron-optic devices, which are necessary to attain high brightness, sub-100 fs bunches. I investigate two types of electron-optic devices: one is a magnetic lens used for collimating or focusing bunches, the other is a bunch compressor. I derive bunch parameter transformations for each of the electron-optic devices, and present numerical calculations using these transformations along with the AG model showing the effects of the devices on the evolution of the bunch parameters. To analyse electron bunch diffraction in UED experiments, I present a general scattering formalism. Using single-scattering and far-field approximations, I derive an expression for the diffracted signal that depends on the electron bunch properties just before scattering. Using this expression I identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. I also discuss the effects of different bunch parameters on the measured diffracted flux, and present sample numerical calculations for scattering by nanosize particles based on this model. This simulation demonstrates the cumulative effects of the bunch parameters, and shows the complex interplay of the bunch and target properties on the diffracted signal.

Ultrafast electron diffraction (UED) is a method of directly imaging system dynamics at the atomic scale with picosecond time resolution. In this thesis I present theoretical analyses of the experimental processes, and construct models in order to better understand UED experiments and to guide future refinements. In particular, I derive a model of electron bunch propagation and a model of electron bunch diffraction, where both models take into account all bunch parameters. To analyse the propagation of electron bunches, I present a mean-field analytic Gaussian (AG) model. I derive a system of ordinary differential equations that are solved quickly and easily to give the bunch dynamics. The AG model is compared to N -body numerical simulations of initially Gaussian bunches, and I demonstrate excellent agreement between the two result sets. I also present a comparison of the AG model with numerical simulations of quasi-Gaussian and non-Gaussian distributions, extending the applicability of the AG model to the propagation of ``real-world'' bunches. During propagation, electron bunches can be shaped by electron-optic devices, which are necessary to attain high brightness, sub-100 fs bunches. I investigate two types of electron-optic devices: one is a magnetic lens used for collimating or focusing bunches, the other is a bunch compressor. I derive bunch parameter transformations for each of the electron-optic devices, and present numerical calculations using these transformations along with the AG model showing the effects of the devices on the evolution of the bunch parameters. To analyse electron bunch diffraction in UED experiments, I present a general scattering formalism. Using single-scattering and far-field approximations, I derive an expression for the diffracted signal that depends on the electron bunch properties just before scattering. Using this expression I identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. I also discuss the effects of different bunch parameters on the measured diffracted flux, and present sample numerical calculations for scattering by nanosize particles based on this model. This simulation demonstrates the cumulative effects of the bunch parameters, and shows the complex interplay of the bunch and target properties on the diffracted signal.

High-order harmonic generation (HHG), a non-perturbative nonlinear light-matter interaction resulting in coherent emission of high-frequency light, has demonstrated promise as an optical probe of carrier dynamics, structural symmetries, and other properties of solids. HHG from bulk solids in transmission geometry, however, is influenced by nonlinear propagation of the driving laser, which leads to spectral skewing and temporal phase variations in the harmonic emission. These effects obscure the microscopic underlying physics, making HHG-based spectroscopy of bulk solids difficult to interpret. HHG in few-to mono-layer materials, however, avoids strong nonlinear propagation effects, and can provide novel material properties for HHG studies. In this work, I compare HHG driven by femtosecond mid-infrared laser pulses in bulk and thin film solids. First, HHG generated from epitaxial ZnO thin films grown using different preparations is compared with HHG from bulk ZnO. I identify spectral signatures that result from nonlinear propagation in bulk samples, while thin films generally yield clean harmonic spectra with features that depend on the crystal growth and preparation. Specifically, I find that as-grown plasma ALD (atomic layer deposition) samples yield monocrystalline polar films, which is modified by annealing. The dependence of the harmonic yield on thickness of the nano-meter scale films was also experimentally measured and found to agree with simulations which incorporated nonlinear conductivity and linear propagation effects. Next, I examine the carrier envelope phase (CEP) dependence of HHG from bulk and thin-film ZnO. I observed a stronger-than-expected sensitivity of the HHG from bulk ZnO to CEP, which results from nonlinear self-compression of the pulse to single-cycle durations. Finally, experimental studies of HHG from novel van der Waals crystals are presented. Together, these results suggest novel frontiers for HHG from few-layer materials.

<p>This dissertation discusses theoretical methods for describing photoionization in different systems in the context of time-dependent and time-independent non-relativistic and time-independent relativistic systems. We introduce a multichannel quantum defect theory (MQDT) model for describing photoionization in the context of pump-probe experiments. The basics of MQDT are introduced and specialized to the argon atom. Two energy regimes are studied in detail and compared to the experiment: (i) a perturbative calculation describing the dynamics of an autoionizing wave packet, (ii) a time-resolved calculation describing the two-photon ionization of a deeply bound wave packet. In both cases, the model accurately describes the relative ionization with respect to the two spin-orbit split thresholds of the ion and the oscillations shown in the delay between the pump and probe. We finalize with a brief presentation, which is primarily pedagogical, of how to use MQDT inside a finite box.</p> <p>Next, we use MQDT to describe the ytterbium atom in different energy regimes and varying degrees of approximation. The motivation behind this lies in the context of quantum information science, but our study is only concerned with calculating atomic properties. We start with a minimal MQDT model to describe the data observed in the experiment, followed by the presentation of an ab initio two-electron model. Both models compare very well to the experiment, and the ab initio method compares favorably with older spectroscopic results. In addition, we show unpublished results that incorporate the hyper-fine effects into the approximate model.</p> <p>Finally, we present an implementation of the two-electron variational R-matrix method for the Dirac equation, including the complete derivation of the solution of the Dirac equation in a central potential. We provide explicit analytic forms for the solutions of the Coulomb potential and use them to derive the generalized quantum defect parameters. A discussion of the variational R-matrix method for the Dirac equation in single and multichannel contexts is presented, with sample calculations for the beryllium and radium atoms. A chapter that summarizes and points to future work for each one of the projects concludes the work.</p>

<p>This dissertation discusses the theoretical methods for quantum scattering in the context of x-ray scattering from electrons and classical scattering in the context of collisions between Rydberg atoms.</p> <p><br></p> <p>A method for describing non-relativistic x-ray scattering from bound electrons is presented. The approach described incorporates the full spatial dependence of the incident x-ray field and is non-perturbative in the incident x-ray field. The x-ray scattering probability obtained by numerical solution for the case of free-electrons is bench-marked with well known analytical free-electron results.</p> <p><br></p> <p>A recent investigation by Fuchs \emph{et al.} [Nat. Phys. 11, 964 (2015)] revealed an anomalous frequency shift of at least 800 eV in non-linear Compton scattering of high-intensity x-rays by electrons in solid beryllium. The x-ray scattering approach described is used to explore the role of binding energy, band structure, electron-electron correlation and a semi-Compton channel in the frequency shift of scattered x-rays for different scattered angles. The results of the calculation do not exhibit an additional redshift for the scattered x-rays beyond the non-linear Compton shift predicted by the free-electron model. </p> <p><br></p> <p>The interference between Compton scattering and nonlinear Compton scattering from a two-color field in the x-ray regime is theoretically analyzed for bound electrons. A discussion of the underlying phase shifts and the dependence of the interference effect on the polarizations of the incident and outgoing fields are presented. </p> <p><br></p> <p>The problem of using x-ray scattering to image the dynamics of an electron in a bound system is examined. Previous work on imaging electronic wave-packet dynamics with x-ray scattering revealed that the scattering patterns deviate substantially from the notion of instantaneous momentum density of the wave packet. Here we show that the scattering patterns can provide clear insights into the electronic wave packet dynamics if the final state of the scattered electron and the scattered photon momentum are determined simultaneously. The scattering probability is shown to be proportional to the modulus square of the Fourier transform of the instantaneous electronic spatial wave function weighted by the final state of the electron.</p> <p><br></p> <p>Collisional ionization between Rydberg atoms is examined. The dependence of the ionization cross section on the magnitude and the direction of orbital angular momentum of the electrons and the direction of the Laplace-Runge-Lenz vector of the electrons is studied. The case of exchange ionization is examined and its dependence on the magnitude of angular momentum of the electrons is discussed.</p> <p><br></p>

This work is dedicated to a comprehensive experimental study on the interaction of femtosecond laser pulses with the nonlinear optical medium lithium niobate. The nonlinear optical response in the nanosecond regime was already studied extensively with a variety of techniques, whereas femtosecond pulses were mainly used in transient absorption or transient grating experiments. Naturally, the temporal resolution of these measurements depends on the pulse duration, however, dynamics during the pulse excitation were barely investigated. The motivation of this work is to widen the limits of femtosecond spectroscopy, not only to temporally resolve faster nonlinear optical processes, but further to show a sensitivity to other coupling mechanisms between the pulses and the material. Especially, the role of transient, dynamic holographic gratings is investigated with a careful determination of the pulse duration, bandwidth and frequency chirp. A basis of this work is established in the first part by studying the material response via light-induced absorption before focusing on the main topic, the pulse interaction with elementary (holographic) gratings, both self-induced and static, in the second part. By this detailed study, several features of femtosecond laser pulses, holographic gratings and the ultrafast material response can be revealed: (i) grating recording is feasible even with pulses of different frequencies, provided that their pulse duration is sufficiently short, (ii) grating based pulse coupling causes a pronounced energy transfer even in a common pump-probe setup for transient absorption measurements with (non-)degenerated frequencies, (iii) beyond expectation, oscillations in the phonon frequency range become apparent in different measurements. The presented results point towards appropriate future experiments to obtain a more consistent, microscopic model for the ultrafast response of the crystal, involving the interplay between photo-generated polarons, self-induced gratings, and phonons.

The present work broaches the physics of light-matter interaction, chiefly using nonlinear optical spectroscopy in a newly developed framework termed as Photon Management Concept. This way, existing fragments dealing with specific properties of harmonic and upconversion nanoparticles (HNPs/UCNPs) are consolidated into a full and coherent picture with the primary goal of understanding the underlying physical processes and their impact on the application side, especially in terms of imaging techniques, via suitable experimental and numerical studies. Contemporary optical setups involving contrast-enhancing agents are frequently limited in their excitation and detection configurations owing to a specialization to a select number of markers. As a result, the bandwidth of experimental methods and specimens that may be investigated is severely restricted in a large number of state-of-the-art setups. Here, an alternative approach involving HNPs and UCNPs, respectively, is presented providing an overview from their synthesis to optical characterization and to potential fields of application. Based on their inherent flexibility based on their nonlinear optical response, especially in terms of wavelength and intensity tunability, the PMC alleviates prevalent limitations by dynamically adapting the setup to a sample instead of the preliminary culling to a reduced number of eligible specimens that must not change their optical properties significantly during investigation. The use of HNPs supersedes such concerns due to their nearly instantaneously generated, strongly anti-Stokes shifted, coherent emission capable of producing radiation throughout the visible spectral range, including infrared and ultraviolet wavelengths. This way, HNPs transcend the traditional field of imaging and introduces potential applications in optomanipulation or holographic techniques. Thorough (nonlinear) optical characterization of UCNPs and alkali niobate HNP ensembles is performed to assess the fundamental physical mechanisms interwoven with numerical studies leading to their wide-ranging applicability. Final remarks show that HNPs are ideal candidates for realization of the PMC and yet hold an even further potential beyond current prospects.