Laser-plasma interactions have become a rapidly growing area of modern plasma physics and an important subfield of it is laser-plasma acceleration. Using high-intensity lasers, one can drive a plasma structure with electric-field gradients three orders of magnitude higher than the gradients found in traditional, radio-frequency accelerators. This promises to enable great technological advances in medicine, spectroscopy, and experimental particle physics, as well as to open up new avenues of studying matter under extreme conditions.

An important aspect of laser-plasma acceleration is how the transverse electromagnetic field of the laser affects and drives an accelerated particle via longitudinal waves in the plasma. To understand how the laser interacts with the plasma, it is necessary to understand that the transverse characteristics of the laser dictate its longitudinal propagation dynamics. The transverse radiation field of the laser pulse can be described in various ways and decomposed into bases of orthogonal modes. The presence of multiple higher-order modes, copropagating through the plasma, leads to mode beating. Likewise, these modes propagate at different velocities through the plasma and are susceptible to nonlinear interactions with the plasma to varying degrees.

The primary objective of this thesis is to understand how higher-order laser modes interact with the plasma and with one another. In this work, we discuss the detrimental consequences that mode beating may have on a laser-plasma accelerator and how higher-order modes can be filtered out using specially designed plasma structures. Also discussed is how higher-order mode content can be controlled and utilized to shape and control the wakefields. These ideas are extended to the concept of the plasma undulator as a plasma-based light source. Lastly, we discuss how nonlinear effects can excite higher-order mode content as path to understanding laser pulse break up into multiple filaments.

We investigate the statistical properties (e.g., shot-to-shot power fluctuations) of the radiation from a high-gain free-electron laser (FEL) operating in the nonlinear regime. We consider the case of an FEL amplifier reaching saturation whose shot-to-shot fluctuations in input radiation power follow a gamma distribution. We analyze the corresponding output power fluctuations at and beyond first saturation, including beam energy spread effects, and find that there are well-characterized values of undulator length for which the fluctuation level reaches a minimum.

Nuclear isomers impact a broad range of scientific and technical fields, from radio-medicine to stellar nucleosynthesis. Direct manipulation of isomer populations can enable a powerful new technique for mitigating spent-nuclear fuel, as well as enable new approaches to nuclear batteries. This work introduces a novel technique for the direct manipulation of isomer populations utilizing the enhanced nuclear level density (NLD) at high excitation energies, known as the nuclear quasicontinuum. Following excitation into the quasicontinuum, additional coupling of spin can occur through real or virtual photon transfer mediated by nuclear-plasma interactions (NPIs). Laser-plasma accelerators (LPAs) provide energetic, ultra-short pulse electron beams. This work discusses an experimental proof-of-concept study of manipulating isomer populations in Bromine nuclei using LPA-sourced $<100$ fs electron beams. A comparison of bremsstrahlung photon and electron irradiation cases is evaluated to determine the presence of electron-nuclear interaction contributions to isomer populations. Additionally, the potential for LPAs to be used as sensitive probes of NLD models and photon strength functions in an effort to characterize the nuclear quasicontinuum is explored.