We present a theory of ionization of diatomic molecules by a strong laser field. A diatomic molecule is considered as a three-particle system, which consists of two heavy atomic (ionic) centers and an electron. After the separation of the center-of-mass coordinate, the dynamics of this system is reduced to the relative electronic and nuclear coordinates. The exact S-matrix element for ionization is presented in a form in which the laser-molecule interaction is emphasized. This form is useful for application of the molecular strong-field approximation (SFA). We introduced two forms of the molecular SFA, one with the field-free and the other with the field-dressed initial molecular bound state. We relate these two forms of our modified molecular SFA to the standard molecular SFAs, introduced previously using the length gauge and the velocity gauge. Numerical examples of the ionization rates of N{sub 2} and O{sub 2} molecules are shown and compared for all four versions of the molecular SFA and we suggest that our modified molecular SFA should be used instead of the standard molecular SFA.

The role of incoherent scattering in above-threshold ionization (ATI) of atoms and above-threshold detachment of negative ions is investigated. We assume that the ionized (detached) electron may scatter on a target other than its parent ion (atom). We call such a process incoherent in order to distinguish it from the coherent rescattering of an electron on its parent ion (atom). The rescattering process is known to be responsible for high-order ATI (HATI). We show that the contribution to the ionization rate of ATI with a subsequent incoherent scattering can be higher than that of the rescattering-induced HATI if the density of atomic targets is high enough. The spectra of ATI with incoherent scattering have a cutoff-like behaviour, similar to that of ATI and HATI. Our numerical results show that the cutoff energy of the ATI with incoherent scattering is higher than that of the rescattering-induced HATI. These results are supported by classical analysis.

The theoretical description and the experimental methods and results for above-threshold ionization (ATI) by few-cycle pulses are reviewed. A pulse is referred to as a few-cycle pulse if its detailed shape, parametrized by its carrier-envelope phase, affects its interaction with matter. Angular-resolved ATI spectra are analysed with the customary strong-field approximation (SFA) as well as the numerical solution of the time-dependent Schrödinger equation (TDSE). After a general discussion of the characteristics and the description of few-cycle pulses, the behaviour of the ATI spectrum under spatial inversion is related to the shape of the laser field. The ATI spectrum both for the direct and for the rescattered electrons in the context of the SFA is evaluated by numerical integration and by the method of steepest descent (saddle-point integration), and the results are compared. The saddle-point method is modified to avoid the singularity of the dipole transition matrix element at the steepest-descent times. With the help of the saddle-point method and its classical limit, namely the simple-man model, the various features of the ATI spectrum, their behaviour under inversion, the cut-offs and the presence or absence of ATI peaks are analysed as a function of the carrier-envelope phase of the few-cycle laser field. All features observed in the spectra can be explained in terms of a few quantum orbits and their superposition. The validity of the SFA and the concept of quantum orbits are established by comparing the ATI spectra with those obtained numerically from the ab initio solution of the TDSE.

The shape of the field of a few-cycle laser pulse strongly depends on the carrier-envelope phase. For a circularly polarized few-cycle pulse, this phase is correlated with a direction in space. Superposition of two counterrotating circularly polarized few-cycle pulses yields a linearly polarized pulse. High-energy electrons, generated through above-threshold ionization by such a combination of pulses, are emitted in a direction correlated with the carrier-envelope phase. Based on these facts, we propose two schemes for direct measurement and control of the carrier-envelope phase and the phase slip of a pulse train.

Using the example of electron-atom scattering in a strong laser field, it is shown that the oscillatory structure of the scattered electron spectrum can be explained as a consequence of the interference of the real electron trajectories in terms of Feynman's path integral. While in previous work on quantum-orbit theory the complex solutions of the saddle-point equations were considered, we show here that for the electron-atom scattering with much simpler real solutions a satisfactory agreement with the strong-field-approximation results can be achieved. Real solutions are applicable both for the direct (low-energy) and the rescattering (high-energy) plateau in the scattered electron spectrum. In between the plateaus and beyond the rescattering cutoff good results can be obtained using the complex (quantum) solutions and the uniform approximation. The interference of real solutions is related to the recent attosecond double-slit experiment in time.

A quantum theory of high-order harmonic generation by a strong laser field in the presence of more bound states is formulated. The obtained numerical and analytical results for a two-state hydrogenlike atom model show that the harmonic spectrum consists of two parts: a usual single-state harmonic spectrum of odd harmonics having the energies (2k+1)Omega and a resonant part with the peaks around the excitation energy DeltaOmega. The energy of the harmonics in the resonant part of the spectrum is equal to DeltaOmega +/- Omega, DeltaOmega +/- 3Omega, .... For energies higher than the excitation energy, the resonant part forms a plateau, followed by a cutoff. The emission rate of the harmonics in this resonant plateau is many orders of magnitude higher than that of the harmonics generated in the presence of the ground state alone. The influence of the depletion of the initial states, as well as of the pulse shape and intensity, is analyzed.

Few-cycle above-threshold ionization spectra of atomic hydrogen, calculated via the numerical solution of the time-dependent Schrödinger equation (TDSE), are compared with those predicted by the strong field approximation. Good agreement is obtained for the energetic, rescattered photoelectrons whereas the low-energy part of the electron spectra differ significantly. The latter disagreement is shown to originate from the long-range character of the Coulomb potential. In the second part of the paper a novel quantum distribution function is introduced, which facilitates a direct comparison of the classical electron orbits used in simple man's theory with the exact numerical TDSE result. It is shown that well localized electron wave packets emerge, oriented along the simple man's classical trajectories as the energy resolution in the quantum distribution function is reduced.

We consider high-order harmonic generation by a linearly polarized laser field and a parallel static electric field. We first develop a modified saddle-point method which enables a quantitative analysis of the harmonic spectra even in the presence of Coulomb singularities. We introduce a classification of the saddle-point solutions and show that, in the presence of a static electric field which breaks the inversion symmetry, an additional classification number has to be introduced and that the usual saddle-point approximation and the uniform approximation in the case of the coalescing saddle points have to be modified. The theory developed offers a simple and accurate explanation of the static-field-induced multiplateau structure of the harmonic spectra. The longer quantum orbits are responsible for a long extension of the harmonic plateau, while the larger initial electron velocities are the reason of lower harmonic emission rates.

High-order above-threshold ionization by few-cycle laser pulses is analyzed in terms of quantum orbits. For a given carrier-envelope phase, the number of contributing orbits and their ionization and rescattering times determine the shape of the angle-resolved spectrum in all detail. Conversely, analysis of a given spectrum reveals the carrier-envelope phase and the various interfering pathways from which the electron could choose.

The strong-field approximation (SFA) can be and has been applied in both length gauge and velocity gauge with quantitatively conflicting answers. For ionization of negative ions with a ground state of odd parity, the predictions of the two gauges differ qualitatively: in the envelope of the angular-resolved energy spectrum, dips in one gauge correspond to humps in the other. We show that the length-gauge SFA matches the exact numerical solution of the time-dependent Schr\"odinger equation.

A new scheme for a double-slit experiment in the time domain is presented. Phase-stabilized few-cycle laser pulses open one to two windows (slits) of attosecond duration for photoionization. Fringes in the angle-resolved energy spectrum of varying visibility depending on the degree of which-way information are measured. A situation in which one and the same electron encounters a single and a double slit at the same time is observed. The investigation of the fringes makes possible interferometry on the attosecond time scale. From the number of visible fringes, for example, one derives that the slits are extended over about 500 as.

Potential scattering of electrons in a strong bichromatic laser field is analyzed using the strong-field approximation and the second Born approximation in the scattering potential. In the intermediate step of the scattering process, the electron dynamics is determined only by the laser field and the electron can absorb energy from the laser field, which leads to the appearance of additional plateau and cutoff structures. The features of these structures are analyzed for various incident electron energies and scattering angles, for different laser intensities, and for various relative phases between the bichromatic field components. The electron spectrum strongly depends on this phase, meaning that coherent phase control is possible. For particular values of the relative phase, the rescattering plateau can be much longer than in the monochromatic field case. The results obtained are in agreement with classical estimates.