The generation of attosecond-duration (1 as = 10^-18 sec) coherent light through the process of high-order harmonic generation has opened the perspective for probing fundamental processes, such as photoionization, on the natural timescale of electron dynamics in matter. One probing technique is the attosecond streaking method, in which the momentum of the photoelectron is measured as a function of the time delay between the ionizing, attosecond extreme ultraviolet (XUV) pulse, and a weak, femtosecond near-infrared (NIR) pulse which streaks the momentum of the photoelectron, known as a streaking trace. The observed trace contains time information about the photoionization process in the form of a time offset to the vector potential of the streaking field, known as the streaking time delay. Theoretical simulations show that for one-photon ionization this time delay is accumulated by the photoelectron in the continuum when propagating away from the parent ion, whereas for resonant two-photon ionization there exists an additional absorption delay which depends on the properties of the XUV pulse. In this thesis, we use both analytical techniques and numerical simulations to study the contributions of the total time delay observed in streaking, and further explore applications of the streaking time delay to gain insights into the electron dynamics.
We first derive an analytical formula for the streaking time delay in one-photon ionization. The predictions based on the model formula, which can be performed within seconds of computation time, are in good agreement with those of computationally extensive numerical simulations. We demonstrate that the analytical formula not only allows deeper insight into the nature of the time delay, but also offers the opportunity to effectively analyze other theoretical interpretations and potential effects, such as the effect of a chirp in the ionizing attosecond pulse on the time delay measurement. We then apply time-dependent perturbation theory to derive an analytical formula for the absorption delay in resonant two-photon ionization. We use the analytical formula to demonstrate how the absorption delay can be controlled further by the attosecond pulse duration and central frequency in case of an isolated resonance. Furthermore, we show how multiple resonances within the bandwidth of the ionizing pulses as well as the streaking field influence the absorption delay in model systems as well as simple atoms and molecules.