Using the strong-field-approximation theory beyond the dipole approximation we investigate above-threshold ionization induced by the monochromatic and bichromatic laser fields. Particular emphasis is on the approach based on the saddle-point method and the quantum-orbit theory which provides an intuitive picture of the underlying process. In particular, we investigate how the solutions of the saddle-point equations and the corresponding quantum orbits and velocities are affected by the nondipole effects. The photoelectron trajectories are two dimensional for linearly polarized field and three dimensional for two-component tailored fields, and the electron motion in the propagation direction appears due to the nondipole corrections. We show that the influence of these corrections is not the same for all contributions of different saddle-point solutions. For a linearly polarized driving field, we focus our attention only on the rescattered electrons. On the other hand, for the tailored driving field, exemplified by the ω–2ω orthogonally polarized two-color field, which is of the current interest in the strong-field community, we devote our attention to both the direct and the rescattered electrons. In this case, we quantitatively investigate the shift which appears in the photoelectron momentum distribution due to the nondipole effects and explain how these corrections affect the quantum orbits and velocities which correspond to the saddle-point solutions. Published by the American Physical Society 2024

The quantum-mechanical transition amplitudes for atomic and molecular processes in strong laser fields are expressed in the form of multidimensional integrals of highly oscillatory functions. Such integrals are ideally suited for the evaluation by asymptotic methods for integrals. Furthermore, using these methods it is possible to identify, in the sense of Feynman's path-integral formalism, the partial contributions of quantum orbits, which are related to particular solutions of the saddle-point equations. This affords insight into the physics of the problem, which would not have been possible by only solving these integrals numerically. We apply the saddle-point method to various quantum processes, which are important in strong-field physics and attoscience. The special case of coalescing or near-coalescing saddle points requires application of the uniform approximation. We also present two modifications of the saddle-point method, for the cases where a singular point of the subintegral function exactly overlaps with a saddle point or is located in its close vicinity. Particular emphasis is on the classification of the saddle-point solutions. This problem is solved for the one-dimensional integral over the ionization time, relevant for above-threshold ionization (ATI), while for two-dimensional integrals a classification by the multi-index $(\alpha,\beta,m)$ is introduced, which is particularly useful for the medium- and high-energy spectrum of high-order harmonic generation (HHG) and backward-scattered electrons (for high-order ATI). For the low-energy structures a classification using the multi-index $(\nu,\rho,\mu)$ is introduced for the forward-scattering quantum orbits. In addition to laser-induced processes such as ATI, HHG and high-order ATI, we consider laser-assisted scattering as an example of laser-assisted processes for which real solutions of the saddle-point equation exist. Particular attention is devoted to the quantum orbits that describe and visualize these processes. We also consider finite laser pulses, the semiclassical approximation, the role of the Coulomb field and the case of laser fields intense enough to lead into the relativistic regime.

Improved molecular strong-field approximation theory is used to calculate the ionisation probability for the high-order above-threshold ionisation process induced by a few-cycle pulse with two carrier frequencies and one envelope. The asymmetry in the photoelectron momentum distribution is due to the ultrashort nature of the driving pulse and due to the relative orientation of the molecule with respect to the laser field. We introduce the generalised asymmetry parameter, which can be used to quantitatively measure the asymmetry between the photoelectron spectra along arbitrarily many selected directions. We investigate the difference between the asymmetry parameters calculated for atomic and molecular targets and show that the contributions to the asymmetry strongly depend on the type of the employed driving pulse. For the driving pulse with components that are linearly polarised with mutually orthogonal polarisations, we find that the main source of the asymmetry, especially in the high-energy part of the spectrum, is the ultrashort nature of the pulse. The relative orientation of the molecule with respect to the laser pulse only affects the low- and medium-energy parts of the spectrum. On the other hand, for the driving pulse with circularly polarised counterrotating components, the asymmetry introduced by molecular orientation is more pronounced. We also analyse the influence of the characteristics of molecular orbitals on the asymmetries using the examples of N2 and O2 molecules.

We perform a systematic comparison between photoelectron momentum distributions computed with the rescattered-quantum orbit strong-field approximation (RQSFA) and the Coulomb-quantum orbit strong-field approximation (CQSFA). We exclude direct, hybrid, and multiple scattered CQSFA trajectories, and focus on the contributions of trajectories that undergo a single act of rescattering. For this orbit subset, one may establish a one-to-one correspondence between the RQSFA and CQSFA contributions for backscattered and forward-scattered trajectory pairs. We assess the influence of the Coulomb potential on the ionization and rescattering times of specific trajectory pairs, kinematic constraints determined by rescattering, and quantum interference between specific pairs of trajectories. We analyze how the Coulomb potential alters their ionization and return times, and their interference in photoelectron momentum distributions. We show that Coulomb effects are not significant for high or medium photoelectron energies and shorter orbits, while, for lower momentum ranges or longer electron excursion times in the continuum, the residual Coulomb potential is more important. We also assess the agreement of both theories for different field parameters, and show that it improves with the increase of the wavelength.

We introduce the theory of high-order harmonic generation by aligned homonuclear diatomic cations using a strong-field approximation. The target cation is represented as a system which consists of two atomic (ionic) centres and one active electron, while the driving field is either a monochromatic or bichromatic field. For a linearly polarised driving field, we investigate the differences between the harmonic spectra obtained with a neutral molecule and the corresponding molecular cation. Due to the larger ionisation potential, the molecular cations can withstand much higher laser-field intensity than the corresponding neutral molecule before the saturation effects become significant. This allows one to produce high-order harmonics with energy in the water-window interval or beyond. Also, the harmonic spectrum provides information about the structure of the highest-occupied molecular orbital. In order to obtain elliptically polarised harmonics, we suggest that an orthogonally polarised two-colour field is employed as a driving field. In this case, we analyse the harmonic ellipticity as a function of the relative orientation of the cation in the laser field. We show that the regions with large harmonic ellipticity in the harmonic energy-orientation angle plane are the broadest for cations whose molecular orbital does not have a nodal plane. Finally, we show that the molecular cations exposed to an orthogonally polarised two-colour field represent an excellent setup for the production of elliptically polarised attosecond pulses with a duration shorter than 100 as.

Using a strong-field-approximation theory, we investigate the high-order above-threshold ionization of diatomic molecules exposed to the monochromatic and bichromatic elliptically polarized fields. We devote particular attention to the difference between the photoelectron momentum distributions obtained with fields with opposite helicity. This difference is quantified using the elliptic-dichroism parameter, which represents the normalized difference between the differential ionization rates calculated with driving fields with opposite helicity. We find that this parameter strongly depends on the molecular orientation with respect to the laser field. In addition, this dependence is different for molecules with different types of highest-occupied molecular orbital. In other words, we show that the molecular structure is imprinted onto the elliptic-dichroism parameter for both monochromatic and bichromatic driving fields. This is explained by analyzing the interferences between various partial contributions to the differential ionization rate. In this way, elliptic dichroism also serves as a tool to analyze the electron dynamics. Finally, for heteronuclear diatomic molecules, we show that the elliptic dichroism is different from zero even for the direct electrons, i.e., the electrons that after liberation go directly to the detector. In this case, the dependence on the molecular orientation is far more pronounced for a bichromatic driving field.

The contributions of two energetically highest molecular orbitals to the harmonic emission rate are analysed for a two-component laser field. For diatomic molecules exposed to the elliptically polarised field, the emission from the highest-occupied molecular orbital (HOMO) is dominant for various molecular orientations with respect to the laser field. However, the contribution of the lower molecular orbital (HOMO-1) can become significant or even dominant for some molecular orientations. We introduce the ratio of the coherent over the incoherent sum of the HOMO and HOMO-1 contributions as a quantitative measure of the significance of the particular molecular orbital. Also, the gaseous medium response is different for the left and right elliptically polarised light and the molecular characteristics are imprinted into this difference. Moreover, for the orthogonally polarised two-colour (OTC) laser field the relative contributions of HOMO and HOMO-1 depend to a great extent on the relative phase between the field components. The importance of the HOMO-1 can be assessed by the relative error which is made if the harmonic spectra are obtained only with the HOMO contribution. Finally, we investigate the interference of the contributions of two highest molecular orbitals. We show that, for the OTC field, the destructive interference depends linearly on the intensity of the field components. Also, the interference minima shift towards the higher energies with the increase of the component wavelength.