Significance The new ability to generate circularly polarized coherent (laser-like) beams of short wavelength high harmonics in a tabletop-scale setup is attracting intense interest worldwide. Although predicted in 1995, this capability was demonstrated experimentally only in 2014. However, all work to date (both theory and experiment) studied circularly polarized harmonics only in the extreme UV (EUV) region of the spectrum at wavelengths >18 nm. In this new work done in a broad international collaboration, we demonstrate the first soft X-ray high harmonics with circular polarization to wavelengths λ < 8 nm and the first tabletop soft X-ray magnetic circular dichroism (XMCD) measurements, and also uncover new X-ray light science that will inspire many more studies of circular high-harmonic generation (HHG). We demonstrate, to our knowledge, the first bright circularly polarized high-harmonic beams in the soft X-ray region of the electromagnetic spectrum, and use them to implement X-ray magnetic circular dichroism measurements in a tabletop-scale setup. Using counterrotating circularly polarized laser fields at 1.3 and 0.79 µm, we generate circularly polarized harmonics with photon energies exceeding 160 eV. The harmonic spectra emerge as a sequence of closely spaced pairs of left and right circularly polarized peaks, with energies determined by conservation of energy and spin angular momentum. We explain the single-atom and macroscopic physics by identifying the dominant electron quantum trajectories and optimal phase-matching conditions. The first advanced phase-matched propagation simulations for circularly polarized harmonics reveal the influence of the finite phase-matching temporal window on the spectrum, as well as the unique polarization-shaped attosecond pulse train. Finally, we use, to our knowledge, the first tabletop X-ray magnetic circular dichroism measurements at the N4,5 absorption edges of Gd to validate the high degree of circularity, brightness, and stability of this light source. These results demonstrate the feasibility of manipulating the polarization, spectrum, and temporal shape of high harmonics in the soft X-ray region by manipulating the driving laser waveform.

We present a theory of high-order harmonic generation by a bichromatic elliptically polarized laser field which consists of two coplanar components having the frequencies and (r and s are integers) and is defined in the xy plane. Laser and harmonic fields are decomposed in the components having opposite helicities. Using the conservation laws for energy and projection of the total angular momentum of atom and laser and harmonic photons on the z axis we have derived a general selection rule. This rule reduces to the known result in the case of bicircular field which consists of two counter-rotating circularly polarized fields. We apply our results to explain recent experiment by Fleischer et al 2014 (Nature Photonics 8 543).

Intense-laser-induced above-threshold ionization of a bound electron into continuum states with low energy is investigated in the context of the strong-field approximation that allows for one act of rescattering of the revisiting electron. The quantum orbits for forward and backward scattering are evaluated and generalized to arbitrary scattering angles. The velocity map of the liberated electron exhibits the well-known low-energy structure as well as other features off the polarization axis.

The carrier-envelope phase (CEP) dependence of few-cycle above-threshold ionization (ATI) of Xe is calibrated for use as a reference measurement for determining and controlling the absolute CEP in other interactions. This is achieved by referencing the CEP-dependent ATI measurements of Xe to measurements of atomic H, which are in turn referenced to ab initio calculations for atomic H. This allows for the accurate determination of the absolute CEP dependence of Xe ATI, which enables relatively easy determination of the offset between the relative CEP measured and/or controlled by typical devices and the absolute CEP in the interaction.

The rescattering term of the ionization amplitude in the strong-field approximation is analyzed for very low electron energy and emission in arbitrary direction, first in terms of the classical simple-man model and then in the quantum-mechanical quantum–orbit expansion of the strong-field-approximation amplitude. Particular orbits can be associated with particular patterns in the velocity map. The different roles of forward and backscattering are investigated. In addition to known features such as the LES and the fork, a characteristic and pronounced V structure in the velocity map is identified, which has been observed in recent experiments (2014 Phys. Rev. A 90 063424).

Stimulated by the recent demonstration of the first bright source of circularly polarized high harmonics, we examine the attosecond pulse trains generated by a group of such harmonics. For the s ground state of an atom, the polarization of generated pulses is close to linear, with three different orientations per cycle. However, for the p ground state of the inert gases used in the experiments, the polarization of the attosecond pulses is close to elliptical. We show that this is caused by the different intensities of the high harmonics of the opposite helicity.

We consider high-order above-threshold ionization (HATI) of polyatomic molecules ionized by a strong linearly polarized laser field. Improved molecular strong-field approximation by which the HATI process on polyatomic molecular species can be described is developed. Using this theory we calculate photoelectron angular-energy spectra for different triatomic molecules. Special attention is devoted to the minima that are observed in the calculated high-energy electron spectra of the ozone and carbon dioxide molecules. A key difference between these minima and minima that are observed in the corresponding spectra of diatomic molecules are presented.

High-order above-threshold ionization (ATI) is characterized by the rescattering of the ionized electron off the parent ion. The ATI amplitude can be calculated applying the saddle-point (SP) method and the uniform approximation to a five-dimensional integral over the ionization time, intermediate electron momentum, and rescattering time. The so-obtained amplitude is a coherent sum of amplitudes associated to the two classes of the SP solutions. The low-energy structures in the ATI spectra are determined by the forward-scattering SP solutions, while the high-energy spectra (plateau and cutoff) are determined by the backward-scattering SP solutions. In order to properly describe the intermediate-electron-energy spectra, additional SP solutions should be taken into account. For these solutions, the imaginary part of the rescattering time can be large and, for their explanation, we introduce complex-time quantum orbits.

The low-energy structure (LES) in the energy spectrum of above-threshold ionization of rare-gas atoms is reinvestigated from three different points of view. First, the role of forward rescattering in the completely classical simple-man model (SMM) is considered. Then, the corresponding classical electronic trajectories are retrieved in the quantum-mechanical ionization amplitude derived in the strong-field approximation augmented to allow for rescattering. Third, classical trajectories in the presence of both the laser field and the Coulomb field are scrutinized in order to see how they are related to the LES. It is concluded that the LES is already rooted in the SMM. The Coulomb field enhances the structure so that it can successfully compete with other contributions and become visible in the total spectrum.

S-matrix theory is used to analyze different atomic and molecular processes in a linearly polarized few-cycle laser field. The energy spectra of high-order above-threshold ionization (HATI) are presented. Electron-atom potential scattering assisted by a few-cycle laser pulse is also analyzed. It is shown that the plateau structures in the energy spectra of the electron-atom potential scattering are dependent on the carrier-envelope phase (CEP) of the laser pulse, so that the cutoff positions of the plateaus can be controlled by changing the CEP. Regarding our analysis of the molecular HATI process, the angle-resolved spectra, obtained by different theoretical approaches, are also presented.