When the strong-field ionization of atoms is induced by an ultrashort pulse instead of a long pulse with a flat envelope, many symmetry properties of the photoelecton momentum distribution are broken. The induced asymmetry is measured using the asymmetry parameter which depends on the values of the driving-pulse parameters and the type of the target. We investigate the driving pulses with two carrier frequencies because in this case the dependence of the asymmetry on the characteristics of the target is more robust. Particular attention is devoted to the pulse which consists of two circularly polarized few-cycle waves and the pulse which has two linearly polarized components with mutually orthogonal polarizations. In the former case, we show that the asymmetry parameter is highly sensitive to the ionization potential and to the structure of the ground state. This is particularly the case for the photoelectron energy just above the value for which the contribution of the electrons which do not interact with the core after the ionization becomes negligible. We explain this sensitivity by investigating the dependence of the short-travel-time saddle-point solutions on the characteristics of the target. On the other hand, for the driving pulse with linearly polarized components, the dependence of the asymmetry parameter on the ionization potential is significant, while the dependence on the structure of the ground state is relatively small. In conclusion, we show that the characteristics of the target are imprinted in the asymmetry parameter and this signature is more pronounced for two-component pulses than for the linearly polarized driving pulse with one carrier frequency. Published by the American Physical Society 2025

The saddle-point solutions for strong-laser-field-induced high-order above-threshold ionization, the complete classification of which was recently presented in , are considered classically. In the limit of vanishing ionization potential the system of saddle-point equations simplifies, allowing a semi-analytical treatment. For a monochromatic field, the analytical nonlinear equations obtained this way allow one to determine the maximum (cutoff) photoelectron energies for the backward- and forward-scattering saddle-point solutions for all values of the multi-indices introduced by our classification scheme. These cutoffs are determined for all photoelectron momenta and it is shown how the backward-scattering solutions from one half of the momentum plane are related to the forward-scattering solutions from the other. The case of a bichromatic linearly polarized field is analyzed in detail. The results are rederived with the help of a simple graphical method, which can be used to qualitatively discuss the effect of varying the field parameters. Published by the American Physical Society 2025

Orbit-based methods are widespread in strong-field laser-matter interaction. They provide a framework in which photoelectron momentum distributions can be interpreted as the quantum interference between different semiclassical pathways the electron can take on its way to the detector, which brings with it great predictive power. The transition amplitude of an electron going from a bound state to a final continuum state is often written as multiple integrals, which can be computed either numerically or by employing the saddle-point method. If one computes the momentum distribution via a saddle-point method, then the obtained distribution is highly dependent on the time window from which the saddle points are selected for inclusion in the “sum over paths.” In many cases, this leads to the distributions not even satisfying the basic symmetry requirements and often containing many more oscillations and interference fringes than their numerically integrated counterparts. Using the strong-field approximation, we find that the manual enforcement of the energy-conservation condition on the momentum distribution calculated via the saddle-point method provides a unique momentum distribution which satisfies the symmetry requirements of the system and which is in a good agreement with the numerical results. We illustrate our findings using the example of the Ar atom ionized by a selection of monochromatic and bichromatic linearly polarized fields. Published by the American Physical Society 2025

Ionization of atoms by a strong laser field can be described using the improved strong-field approximation. The corresponding transition amplitude of high-order above-threshold ionization is presented in the form of a two-dimensional integral over the electron ionization time t0 and the rescattering time t. This integral can be solved using the saddle-point (SP) method and the resulting T-matrix element is expressed as a sum (over the SP times t0 and t) of the partial transition amplitudes. We address the problem of finding the solutions of the system of SP equations for the times t0 and t. For a bichromatic linearly polarized laser field with the frequencies rω and sω (r and s are integers, s>r, and ω is the fundamental frequency) we found that there are 8s2 SP solutions per optical cycle. For one half of them the velocity of the electron emitted in the laser field polarization direction changes the sign at the rescattering time (we call such solutions backward-scattering solutions), while for the other half this velocity remains unchanged (these solutions we call forward-scattering SP solutions). For very short (or even negative) electron travel time we call these solutions backward-like and forward-like scattering SP solutions. For these solutions the imaginary parts of the times t0 and t become large so that the concept of real electron trajectories becomes questionable. Having such a classification, we found additional SP solutions even for the simplest case of a monochromatic linearly polarized laser field. For a bichromatic linearly polarized laser field with s=2 and equal component intensities we presented a detailed analysis of all 32 solutions per optical cycle, showing how the SP times t0 and t and the corresponding differential ionization rates depend on the photoelectron energy. We have also analyzed the case where the intensity of the second component decreases while the sum of the component intensities remains fixed. Published by the American Physical Society 2025

When nonsequential double ionization is treated using the strong-field approximation and the saddle-point (SP) method, the transition amplitude can be expressed as a coherent sum of the partial amplitudes corresponding to different SP solutions. For the case of the recollision excitation with subsequent ionization (RESI) mechanism of the nonsequential double ionization, we examine the partial contributions of the SP solutions which correspond to the electron responsible for the excitation. For a monochromatic linearly polarized laser field, we find that, in addition to the pair of the SP solutions with the shortest travel time, other SP solutions may also make a significant contribution to the photoelectron yield. Moreover, the SP solutions appear in pairs and exhibit notable modifications in comparison to those observed in high-order above-threshold ionization. Furthermore, for a bichromatic linearly polarized driving field, we investigate the intensity range obtained using the simpleman’s model for which the RESI mechanism is dominant. We find that this range must be modified if the photoelectron yield corresponding to the SP solution for which the photoelectron has the highest energy upon return to the parent ion is small. This is particularly the case for the excitation channels involving loosely bound excited states.

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 that 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 (α,β,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 (ν,ρ,μ) 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.