We experimentally demonstrate straightforward methodologies for generating high harmonics of arbitrary polarization state. Polarization control is realized by adjusting the intensity ratio of the bicircular driving field or by exploiting chirally dependent Cooper minima transitions. OCIS codes: (140.7240) UV, EUV, and X-ray lasers; (020.2649) Strong field laser physics; (320.7110) Ultrafast nonlinear optics The recent re-discovery of circularly polarized high harmonic generation (CPHHG) has resulted in a renaissance of high-energy, polarization-sculpted attosecond light sources, which are capable of interrogating ultrafast, elementspecific chiral dynamics in condensed matter and molecular systems [1]. In particular, CPHHG driven by bichromatic, counter-rotating laser fields—the bicircular field—has been at the forefront of the attosecond polarization resurgence due to the possibility of controlling the high-harmonic spectral, temporal, and polarization properties afforded via the manipulation of these unique two-color fields [1,2]. Practically, direct control over the spectrotemporal structure of the CPHHG emission process is attractive for generating bright, tailor-made highharmonic beamlets for ultrafast magneto-optical chiral spectroscopies and also isolated attosecond pulses of nearly arbitrary polarization. As such, numerous strategies have been proposed to achieve this level of control by exploiting the helicity-dependent aspects of the microscopic and macroscopic response to the external field [1]. Despite sincere efforts, the complexity of many methodologies has limited active polarization control in CPHHG to just a few experimental demonstrations [2, 3]. In this work, we present two straightforward, yet distinct, methodologies for controlling the spectral distribution of leftand right-circularly polarized (LCP/RCP) harmonics in bicircular-driven CPHHG (Fig. 1A), thus allowing for active manipulation of the polarization of the underlying attosecond pulse trains (APTs). First, we find that the intensity ratio of a commensurate bicircular field—where 2w1=w2—serves as a practical, real-time knob for smoothly varying the ellipticity of the APTs, independent of the CPHHG photon energy and bandwidth (Fig. 1B). Second, we show that a non-commensurate nearand short-wave IR (SWIR) field, combined with an effectively phase-matched geometry, can be utilized to extend the CPHHG cut-off to beyond the Cooper minimum in Ar [4], resulting in the natural production of bright, single-helicity harmonic spectra (and thus circularly polarized APTs) spanning the M absorption edges of several magnetic materials (Fig. 1C). Figure 1. (A) Experimental scheme for bicircular-driven circularly polarized high harmonic generation (CPHHG). (B) Theoretical attosecond pulse trains obtained from highly chiral experimental CPHHG spectra generated in an Ar gas jet. (C) Short-wave-IR-driven CPHHG in a phaseHM2A.4.pdf High-brightness Sources and Light-driven Interactions Congress 2018 (HILAS, MICS, EUVXRAY) © OSA 2018 matched waveguide geometry yields a bright spectrum that extends beyond the Cooper minimum in Ar, which generates a single-helicity spectrum covering prominent magneto-optical absorption edges. In the experiment (Fig. 1A), circularly polarized high harmonics are generated via collinear mixing of the fundamental (w, 790-nm, 45-fs, RCP) of a Ti:sapphire amplifier and either its second harmonic (2w, 395-nm, LCP) or a SWIR field (0.4w, 2000-nm, LCP). The combined fields are then focused onto a supersonic gas jet (w-2w) or coupled into a hollow-core waveguide (w-0.4w). In either geometry, bright, circularly polarized high harmonic spectra are produced, which consist of a series of high-harmonic doublets possessing opposite helicities. The harmonics themselves exhibit a high degree of circularity, as indicated by the suppression of spin-forbidden harmonic orders between the circularly polarized doublets (Fig. 1C and Fig. 2A). For commensurate, w-2w, driven CPHHG in an Ar gas jet, the lower (higher) frequency harmonic in each doublet rotates with the w (2w) field, and these individual harmonics can be viewed as being produced by absorption of an additional w (2w) photon from the driving field. The intensity of the RCP (LCP) harmonics in the far-field are directly coupled to the photon density of each color in the two-color driving laser [2]. As such, we can actively control the intensity of either RCP or LCP high-harmonics in the far-field, thus generating highly chiral CPHHG spectra (Fig. 2A). In the time domain, controlling the chirality of the CPHHG process manifests as direct control over the ellipticity of the underlying APTs (Fig. 2B). As the intensity ratio is varied, the x and y components of the bicircular field that correspond to bright CPHHG emission also vary, which imparts ellipticity onto the APT. We confirm the dependence of this attosecond ellipticity on the CPHHG spectral chirality via macroscopic numerical simulations [5] of the upconversion process (Fig 2A, B). Most importantly, this attosecond ellipticity is uncoupled from the polarizations of the driving field, which allows for highly elliptical APTs to be produced without sacrificing the circular polarization of the emitted harmonics [2]. Fig 2. Controlling the ellipticity of APTs in CPHHG. (A) Experimental, w-2w CPHHG spectra in Ar generated at different intensity ratios, I2w/Iw, of the driving field (exact ratios indicated next to spectra). Inset depicts the intensity asymmetry, (IRCP-ILCP)/(IRCP+ILCP), of the CPHHG spectra at different I2w/Iw. (B) Ellipticity of the APTs computed in Ar for experimental intensity ratios. (C) Experimental intensity asymmetries in w-0.4w driven CPHHG as a function of pressure in the hollow-core waveguide. A sharp helicity reversion is observed at the position of the Cooper minimum (~ 50 eV) in Ar. (D) Theoretical simulations confirm the generation of bright, highly chiral CPHHG spectra in the cutoff region. Finally, we show that the generation of highly elliptical APTs can be achieved without manipulation of either the driving laser field or conditions of the generating medium. By driving the CPHHG process with a noncommensurate, w-0.4w field, we generate bright circularly polarized high-harmonics spanning the Cooper minimum in Ar (Fig. 1C). The presence of the Cooper minimum results in a natural suppression of RCP harmonic orders, yielding highly chiral CPHHG spectra in the high energy cut-off region (Fig. 1C and Fig. 2C). Combined with effective phase-matching in the high-pressure waveguide, we generate a single-helicity CPHHG spectrum, which naturally corresponds to circularly polarized APTs [6]. Moreover, this single-helicity region covers many prominent magneto-optical absorption edges, which should allow for spectroscopies of ultrafast magneto-optical phenomenon. In summary, we have shown the spectral chirality, and thus the ellipticity of the APTs, in CPHHG can be actively controlled via two straightforward and simple methodologies. As such, highly elliptical APTs can be readily generated and controlled in real-time, thus greatly extending the applicability of these novel light sources to attosecond spectroscopies of chiral dynamics in a variety of condensed matter and molecular systems. [1] A. D. Bandrauk, J. Guo, K. J. Yuan, “Circularly polarized attosecond pulse generation and applications to ultrafast magnetism,” J. Opt. In Press. https://doi.org/10.1088/2040-8986/aa9673 [2] K. M. Dorney, et al., “Helicity-selective enhancement and polarization control of attosecond high harmonic waveforms driven by bichromatic circularly polarized laser fields,” Phys. Rev. Lett. 119, 063201 (2017). [3] N. Zhavoronkov, M. Ivanov, “Extended ellipticity control for attosecond pulses by high harmonic generation,” Opt. Lett.42, 4720 (2017). [4] D. Baykusheva, et al., “Signatures of Electronic Structure in Bicircular High-Harmonic Spectroscopy,” Phys. Rev. Lett. 119, 203201 (2017). [5] C. Hernández-García, et al., “High-order harmonic propagation in gases within the discrete dipole approximation,” Phys. Rev. A. 82, 033432 (2010). [6] D. B. Milošević. “Low-frequency approximation for high-order harmonic generation by bicircular laser field,” Phys. Rev. A. In Review.

The exact time-evolution operator of an atom in the presence of a strong laser field is expressed using the phase-space path integral. Presenting this result in the form of a perturbative expansion in the effective interaction of the electron with the rest of the atom enables straightforward derivation of the well-known strong-field approximation and its higher-order corrections. Alternatively, one can use this exact result to obtain a semiclassical approximation by expansion in powers of small fluctuations around the classical trajectories. We present a derivation of such a semiclassical approximation. The obtained result for the momentum-space matrix element of the total time-evolution operator can be useful for studying various processes in strong-field physics. Using the example of above-threshold ionization, it is shown how this approximation can be applied to laser-induced processes. More attention is devoted to the laser-assisted scattering. Using the example of few-cycle laser-pulse-assisted electron-atom potential scattering, we show similarities and differences between the semiclassical and the strong-field approximations. For low energies, the semiclassical scattering cross section is modified and there are trajectories along which the electron is temporarily captured by the atomic potential. Applying stationary-phase method to the integral over the scattering time, we clearly identified relevant semiclassical electron trajectories.

The strong-field-approximation theory of high-order above-threshold ionization of atoms is generalized to include the electron spin. The obtained rescattering amplitude consists of a direct and exchange part. On the examples of excited He atoms as well as ${\mathrm{Li}}^{+}$ and ${\mathrm{Be}}^{++}$ ions, it is shown that the interference of these two amplitudes leads to an observable difference between the photoelectron momentum distributions corresponding to different initial spin states: Pronounced minima appear for singlet states, which are absent for triplet states.

Electron-ion radiative recombination assisted by a bichromatic (two-component) elliptically polarized laser field is analyzed in the frame of the $S\text{-matrix}$ theory. The second Born approximation is applied in the expansion of the $S\text{-matrix}$ element where the first term in the expansion corresponds to the direct recombination of electrons with ionic targets, while the second term corresponds to the recombination preceded by an electron-ion scattering. The latter process is possible in the presence of a laser field. If the electron scatters on an ionic target, it may be subsequently driven back by the laser field and recombine with the same ion. The photon emitted in this process may have a high energy. We have studied the dependence of the energy spectrum on various laser-field and incident electron parameters. The energy spectra obtained show plateaulike structures with abrupt cutoffs. These cutoffs are explained by a classical analysis.

Time evolution of the bound state of a molecular hydrogen cation in an intense, linearly polarized laser field is investigated by solving the full three-dimensional time-dependent Schrödinger equation. Our method is based on the Born-Oppenheimer and dipole approximations, and the wave function is expanded in finite series using B-spline functions and spherical harmonics in prolate spheroidal coordinates. After solving the stationary Schrödinger equation, the initial state is propagated under the influence of the laser field employing the Crank-Nicolson propagator. Using this method we calculate and present high-harmonic photon spectra and above-threshold ionization angle-resolved electron spectra.

Above-threshold ionization (ATI) of atoms by a strong bicircular laser field is investigated using the strong-field approximation and the quantum-orbit theory. The bicircular field consists of two coplanar counterrotating circularly polarized fields with a frequency ratio of 2:1. The velocity map of the angle-resolved ATI spectra, both for direct and rescattered electrons, reflects the shape of a parametric plot of the bicircular field and its symmetries. It is shown that the main characteristics of the ATI spectra can be explained using only a few quantum orbits having short travel times. We also analyze a recently discovered [Phys. Rev. A 93, 052402(R) (2016)] bicircular-field-induced spin asymmetry of the ATI electrons and show that the momentum dependence of the spin-asymmetry parameter is stronger for longer wavelengths.