Strong field approximation (SFA) is the most important approximation in the analytical theory of intense laser matter interaction. Based on SFA many analytical theories have been developed such that a broad spectrum of strong field physics phenomena can be described. The central idea of SFA-based theories is to approximate the electron propagation in the continuum by the Gordon-Volkov wavefunction - a well studied analytical solution to the time-dependent Schr\"{o}dinger equation where the electron is driven by the laser field only. This approximation captures some of the essential features of strong-field physics, but at the same time causes several problems in the theory. In this thesis a comprehensive study of the SFA has been presented. We introduce the SFA in both the length gauge and the velocity gauge. The adequacy of SFA has been discussed by comparing the theory to the numerical solution to the time-dependent Schr\"{o}dinger equation (TDSE). The numerical method of solving TDSE is presented as a separate chapter. In order to obtain a better understanding of the applicability of SFA-based theory, we tested the major approximations in the theory by using three different models: the zero-range potential, the hydrogen atom and the hydrogen molecular ion. The accuracy of the method of steepest descent (MSD) and other major approximations in the analytical theory have also been examined. Targeting at the generalization of the SFA-based theories, several extensions and improvements of SFA have been proposed. We will review them in detail and bring them into unity. One of the most successful aspect of the SFA-based theories is to describe and decompose electron dynamics into components such that identification of different physical processes becomes possible. For instance, the direct ionization and non-sequential double ionization bear clear definitions only within the SFA-based framework. The physical interpretation becomes more straight forward due to the fact that there is a close connection between the quantum orbital and classical trajectory. The MSD is a mathematical tool to bridge the quantum orbital and the classical trajectory in an SFA-based theory. We will discuss MSD within a systematic framework so that the higher order asymptotic expansion terms can be obtained in a straight forward way. After gaining substantial understanding of the SFA and the MSD we developed a graphic user interface (GUI) software that is capable of calculating strong field ionization rates, photo-electron spectra and high harmonic generation spectra. The software interface and algorithms have been presented in the thesis. Sample calculations were done and compared with the previously obtained results. In the last chapter of the thesis, we further developed the theory to describe a two-laser ionization scheme where one laser is chosen to be resonantly coupled two real states and the other is a strong few-cycle laser pulse. We demonstrate the periodic dependence of the total ionization on the appearance time of the strong few-cycle laser pulse. In the case of few-cycle pulses with lower intensity, we observed side-bands in the photoelectron spectrum, whose intensity vary periodically with the appearance time of the pulse. We show that our extended theory is able to explain these phenomena adequately.

Strong field approximation (SFA) is the most important approximation in the analytical theory of intense laser matter interaction. Based on SFA many analytical theories have been developed such that a broad spectrum of strong field physics phenomena can be described. The central idea of SFA-based theories is to approximate the electron propagation in the continuum by the Gordon-Volkov wavefunction - a well studied analytical solution to the time-dependent Schr\"{o}dinger equation where the electron is driven by the laser field only. This approximation captures some of the essential features of strong-field physics, but at the same time causes several problems in the theory. In this thesis a comprehensive study of the SFA has been presented. We introduce the SFA in both the length gauge and the velocity gauge. The adequacy of SFA has been discussed by comparing the theory to the numerical solution to the time-dependent Schr\"{o}dinger equation (TDSE). The numerical method of solving TDSE is presented as a separate chapter. In order to obtain a better understanding of the applicability of SFA-based theory, we tested the major approximations in the theory by using three different models: the zero-range potential, the hydrogen atom and the hydrogen molecular ion. The accuracy of the method of steepest descent (MSD) and other major approximations in the analytical theory have also been examined. Targeting at the generalization of the SFA-based theories, several extensions and improvements of SFA have been proposed. We will review them in detail and bring them into unity. One of the most successful aspect of the SFA-based theories is to describe and decompose electron dynamics into components such that identification of different physical processes becomes possible. For instance, the direct ionization and non-sequential double ionization bear clear definitions only within the SFA-based framework. The physical interpretation becomes more straight forward due to the fact that there is a close connection between the quantum orbital and classical trajectory. The MSD is a mathematical tool to bridge the quantum orbital and the classical trajectory in an SFA-based theory. We will discuss MSD within a systematic framework so that the higher order asymptotic expansion terms can be obtained in a straight forward way. After gaining substantial understanding of the SFA and the MSD we developed a graphic user interface (GUI) software that is capable of calculating strong field ionization rates, photo-electron spectra and high harmonic generation spectra. The software interface and algorithms have been presented in the thesis. Sample calculations were done and compared with the previously obtained results. In the last chapter of the thesis, we further developed the theory to describe a two-laser ionization scheme where one laser is chosen to be resonantly coupled two real states and the other is a strong few-cycle laser pulse. We demonstrate the periodic dependence of the total ionization on the appearance time of the strong few-cycle laser pulse. In the case of few-cycle pulses with lower intensity, we observed side-bands in the photoelectron spectrum, whose intensity vary periodically with the appearance time of the pulse. We show that our extended theory is able to explain these phenomena adequately.

<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>