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 method, the transition amplitude can be expressed as a coherent sum of the partial amplitudes corresponding to different saddle-point (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, 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.

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.

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.