A macroscopic theory of high-order harmonic generation (HHG) is presented, which applies a focal-averaging method based on the integral solution of the wave equation. The macroscopic high-harmonic yield is the coherent superposition of the single-atom contributions of all atoms of the generating medium, which are positioned at different spatial points of the laser focus and exposed to the space-time-dependent laser pulse. The HHG spectrum obtained in our macroscopic simulations is qualitatively different from the one obtained using the microscopic or single-atom theory of HHG. Coherent intensity focal averaging, the simpler and more approximate of two methods we introduced, gives the spectrum which forms a declining plateau with the same cutoff position as that of the microscopic spectrum. The second, more precise method, which we call coherent spatio-temporal focal averaging, shows that it is possible, changing the macroscopic conditions, to obtain an observable peak in the harmonic spectrum at an energy much lower than the microscopic cutoff energy. Generally, the high-harmonic yield appears to be dominated by the contributions of laser-pulse spatio-temporal regions with lower intensities as well as by interference, so that the high-energy plateau and its sharp cutoff are quenched in the theoretical simulation and, presumably, in the experiment. The height and position of this peak strongly depend on the macroscopic conditions. We confirmed these findings by applying our macroscopic theory to simulate two recent experiments with mid-infrared laser fields, one with a linearly polarized field and the other one with a bicircular field.

The differential ionization rate for strong-field ionization by tailored laser fields of atomic systems averaged over the magnetic quantum number satisfies particular inversion and reflection symmetries. The symmetries of the elliptic-dichroism parameter, which is related to the change of sign of the ellipticity of the laser field, are considered in detail, with particular emphasis on high-order above-threshold ionization. The general results are illustrated by the examples of an elliptically polarized laser field and a bi-elliptical orthogonally polarized two-color (BEOTC) field. For the BEOTC field the differential ionization rate and the elliptic-dichroism parameter are investigated for the ω-2ω and ω-3ω field combinations and for various relative phases between the laser-field components. The inversion and reflection symmetries of the photoelectron momentum distribution in the polarization plane of the field depend on the parities of r and s in the rω--sω BEOTC field combination and on the relative phase between the field components. We suggest that, by analyzing the symmetry properties of the measured momentum distribution of the elliptic-dichroism parameter, one can identify the mechanism of strong-field ionization. If the rescattering mechanism is dominant one can use these distributions to obtain information about the atomic and molecular structure and dynamics.

Extreme terahertz (THz) pulses can be generated via interaction of strong infrared ultrashort laser pulses with a suitable target. Inverting this scheme, we propose to use such THz pulses to control strong-laser-field-driven processes. In particular, we show that for THz-pulse-assisted strong-laser-field ionization the electron yield can be increased by one order of magnitude for some energies, and that the maximal emitted photoelectron energy can be a few times higher than that realized with the laser field alone. This can be achieved with the THz field intensity many orders of magnitude lower than that of the ionizing laser field. The only requirement is that the vector potential amplitude of the THz field, which governs the electron dynamics after the ionization by the laser field, be comparable with that of the used laser field. An important control parameter is the time delay between the THz pulse and the laser pulse. Strong-field ionization of Cs atoms is used for an illustration. The numerical results are obtained applying the improved strong-field approximation. For a physical explanation, we use quantum-orbit theory supported by the modified saddle-point method, as well as a classical model.

Generation of an elliptically polarized attosecond pulse train by an orthogonally polarized two-color (OTC) laser field is investigated theoretically and simulated numerically. The OTC field consists of two linearly polarized fields with orthogonal polarizations and frequencies that are integer multiples of the fundamental frequency ω. For the ω−3ω OTC field, the emitted harmonics are elliptically polarized so that they may form an elliptically polarized attosecond pulse train provided that a group of harmonics is phase-locked. This is the case if only one quantum orbit generates the corresponding part of the harmonic spectrum. If so, then two attosecond pulses are emitted per optical cycle due to the dynamical symmetry of the ω−3ω OTC field. Atomic targets with an s ground state only generate attosecond pulses with almost linear polarization. Using, however, targets with a p ground state, attosecond pulses with substantial ellipticity can be produced because ground states with opposite magnetic quantum numbers m=+1 and m=−1 produce harmonics with opposite helicities at different rates. In this case, the harmonic intensity and harmonic ellipticity are different for the ground states with the magnetic quantum number m=±1. These differences are the source of the attosecond pulse ellipticity and can be controlled using the relative phase as a control parameter. In addition, by choosing a particular group of harmonics, one can select the desired ellipticity of the attosecond pulse train.

Using the strong-field approximation we systematically investigate the selection rules for high-order harmonic generation and the symmetry properties of the angle-resolved photoelectron spectra for various atomic and molecular targets exposed to one-component and two-component laser fields. These include bicircular fields and orthogonally polarized two-color fields. The selection rules are derived directly from the dynamical symmetries of the driving field. Alternatively, we demonstrate that they can be obtained using the conservation of the projection of the total angular momentum on the quantization axis. We discuss how the harmonic spectra of atomic targets depend on the type of the ground state or, for molecular targets, on the pertinent molecular orbital. In addition, we briefly discuss some properties of the high-order harmonic spectra generated by a few-cycle laser field. The symmetry properties of the angle-resolved photoelectron momentum distribution are also determined by the dynamical symmetry of the driving field. We consider the first two terms in a Born series expansion of the T matrix, which describe the direct and the rescattered electrons. Dynamical symmetries involving time translation generate rotational symmetries obeyed by both terms. However, those that involve time reversal generate reflection symmetries that are only observed by the direct electrons. Finally, we explain how the symmetry properties, imposed by the dynamical symmetry of the driving field, are altered for molecular targets.

Scattering of electrons off diatomic molecules in a bichromatic elliptically polarised laser field is considered by applying the S-matrix theory within the second Born approximation. Two characteristic plateaus appear in the energy spectrum of scattered electrons. The higher plateau is observed in the low-energy part of the spectrum and describes the single scattering, while the lower plateau extends to the high-energy part of the spectrum and describes the double scattering. Scattering and rescattering of electrons off molecular targets may occur at any molecular centre and in any order. Interference of contributions results in increasing/decreasing the differential cross section in particular regions of the energy spectrum of scattered electrons. Two contributions for single scattering and four contributions for double scattering exist in the case of diatomic molecules. For some molecular orientations, a sequence of declining maxima may be observed instead of the plateaus in the electron energy spectrum. We have also observed parabolic structures in the angle-resolved energy spectra. Analytical formulas that explain these structures have been provided. We have also explored the impact of the laser-field ellipticity on the scattered-electron energy spectra. The bichromatic ω– and ω– laser fields have been considered. GRAPHICAL ABSTRACT

The molecular strong-field approximation is employed to study high-order harmonic generation by linear and planar polyatomic molecules exposed to an orthogonally polarized two-color laser field, which consists of two orthogonal linearly polarized components with commensurable frequencies. For such a driving field, we find that the harmonic emission rate and the shape of the spectrum strongly depend on the laser-field parameters including the relative phase and the ratio of the intensities of the two components. The values of the relative phase that correspond to the optimal harmonic emission rate, as well as the cutoff position, can be assessed using a classical model. The possible production of an isolated attosecond pulse is investigated. For suitable symmetry of the laser field an attosecond pulse train with only one attosecond pulse per cycle can be generated. Depending on the frequencies of the two field components, the molecular symmetry properties and the orientation of the molecule with respect to the field, the even harmonics can be absent from the spectrum, which can be used to determine the molecular orientation. The emitted harmonics are elliptically polarized and their ellipticity depends on the molecular orientation.

Above-threshold ionization spectra from cesium are measured as a function of the carrier-envelope phase (CEP) using laser pulses centered at 3.1  μm wavelength. The directional asymmetry in the energy spectra of backscattered electrons oscillates three times, rather than once, as the CEP is changed from 0 to 2π. Using the improved strong-field approximation, we show that the unusual behavior arises from the interference of few quantum orbits. We discuss the conditions for observing the high-order CEP dependence, and draw an analogy with time-domain holography with electron wave packets.

Using our theory which is based on the strong-field approximation we analyze high-order above-threshold ionization and high-order harmonic generation processes for the case of the homonuclear diatomic molecules exposed to an orthogonally polarized two-color (OTC) laser field. The OTC field represents a superposition of two linearly polarized fields with mutually orthogonal polarizations and different frequencies. We analyze the photoelectron energy spectra and the harmonic ellipticity as a function of the ratio of the intensities of the OTC laser-field components and the relative phase. Some combinations of the values of these parameters lead to the high-energy electrons, while the harmonic ellipticity depends strongly on the ratio of the intensities of the laser-field components. It is possible to find the value of this ratio for which the ellipticity of the emitted harmonics is large. The signes of ellipticity are opposite for the molecular orientations which are connected through the reflection with respect to the axis along the first OTC field component. This symmetry is explained using the expression which relates the T-matrix element and the harmonic ellipticity.

International Physics Conference in Bosnia and Herzegovina (PHYCONBA 2020) Organizer: Physical Society in Federation of Bosnia and Herzegovina with support of the Academy of Sciences and Arts of Bosnia and Herzegovina Date: October 19, 2020 Venue: Premises of The Academy of Sciences and Arts of Bosnia and Herzegovina, 7 Bistrik street, Sarajevo, Bosnia and Herzegovina. Memebers of Organizing committee: 1. Maja Đekić (Faculty of Science, University of Sarajevo), chairwoman, 2. Mirza Hadžimehmedović (Faculty of Science, University of Tuzla), member, 3. Rifat Omerović (Faculty of Science, University of Tuzla), member, 4. Ena Žunić-Ćejvanović, member, 5. Amra Salčinović Fetić, (Faculty of Science, University of Sarajevo), Technical secretary, 6. Benjamin Fetić (Faculty of Science, University of Sarajevo), member Members of Scientific committee: 1. Dejan Milošević (Faculty of Science, University of Sarajevo, Academy of Sciences and Arts of BiH), chairman, 2. Dijana Dujak (Faculty of Electrical Engineering, University of Sarajevo), member, 3. Hedim Osmanović (Faculty of Science, University of Tuzla), member, 4. Siniša Ignjatović (Faculty of Natural Sciences and Mathematics, University of Banja Luka), member.

High-harmonic generation by aligned diatomic molecules in orthogonally po-larized two-color laser fields is considered using the molecular strong-field approximation. Regions of the parameter space with large harmonic ellipticity are identified.

We investigate emission rate and ellipticity of high-order harmonics generated exposing a homonuclear diatomic molecule, aligned in the laser-field polarization plane, to a strong orthogonally polarized two-color (OTC) laser field. The linearly polarized OTC-field components have frequencies rω and sω, where r and s are integers. Using the molecular strong-field approximation with dressed initial state and undressed final state, we calculate the harmonic emission rate and harmonic ellipticity for frequency ratios 1:2 and 1:3. The obtained quantities depend strongly on the relative phase between the laser-field components. We show that with the OTC field it is possible to generate elliptically polarized high-energy harmonics with high emission rate. To estimate the relative phase for which the emission rate is maximal we use the simple man’s model. In the harmonic spectra as a function of the molecular orientation there are two types of minima, one connected with the symmetry of the molecular orbital and the other one due to destructive interference between different contributions to the recombination matrix element, where we take into account that the electron can be ionized and recombine at the same or different atomic centers. We derive a condition for the interference minima. These minima are blurred in the OTC field except in the cases where the highest occupied molecular orbital is modeled using only s or only p orbitals in the linear combination of the atomic orbitals. This allows us to use the interference minima to assess which atomic orbitals are dominant in a particular molecular orbital. Finally, we show that the harmonic ellipticity, presented in false colors in the molecular-orientation angle vs. harmonic-order plane, can be large in particular regions of this plane. These regions are bounded by the curves determined by the condition that the harmonic ellipticity is approximately zero, which is determined by the minima of the T-matrix contributions parallel and perpendicular to the fundamental component of the OTC field.

We consider the problem of the choice of gauge in nonrelativistic strong-laser-field physics. For this purpose, we use the phase-space path-integral formalism to obtain the momentum-space matrix element of the exact time-evolution operator. With the assumption that the physical transition amplitude corresponds to transitions between eigenstates of the physical energy operator rather than the unperturbed Hamiltonian H0=(−i∂/∂r)2/2+V(r), we prove that the aforementioned momentum-space matrix elements obtained in velocity gauge and length gauge are equal. These results are applied to laser-assisted electron-ion radiative recombination (LAR). The transition amplitude comes out identical in length gauge and velocity gauge, and the expression agrees with the one conventionally obtained in length gauge. In addition to the strong-field approximation (SFA), which is the zeroth-order term of our expansion, we present explicit results for the first-order and the second-order terms, which correspond to LAR preceded by single and double scattering, respectively. Our general conclusion is that in applications to atomic processes in strong-field physics the length-gauge version of the SFA (and its higher-order corrections) should be used. Using the energy operator as the basis-defining Hamiltonian, we have shown that the resulting transition amplitude is gauge invariant and agrees with the form commonly derived in length gauge.