Polarization-dependent high-order two-color mixing

Amplitude modulation effect on in-situ temporal characterization of high harmonic attosecond pulses

Ultrafast pulse characterization is one of the fundamental techniques in ultrafast sciences, including the characterization of high harmonic attosecond pulses. The in-situ measurement techniques enable all-optical, spatiotemporal determination of the atto-chirp by applying a perturbing laser field, which is believed to introduce only phase modulations to attosecond pulses. In this paper, we have experimentally revisited in-situ measurement techniques with a collinear or an oblique second-harmonic perturbing laser field. Reduced oscillation contrast ratios of even-order harmonic intensities are observed in the collinear experiment, and the far-field angular profiles of even-order harmonics vary with the perturbing laser phase delay in the oblique one. Both observations confirm the amplitude modulation effect on the in-situ temporal measurement technique of high harmonic attosecond pulses, and such an effect is due to the perturbing laser-induced photoionization rate variations. Finally, we incorporate the amplitude modulation effect into in-situ measurements by utilizing the oscillation contrast ratio information and assuming a constant amplitude-phase modulation delay, correcting the underestimated atto-chirps.

Characterization of photon combination pathways for harmonic generation from indium tin oxide thin films

The harmonic generation in an indium tin oxide (ITO) thin film induced by a ω0 + 2ω0 two-color field (ω0 is the frequency of a fundamental laser field) is investigated based on the numerical solution of the full-wave Maxwell-paradigmatic-Kerr equations. By changing the topological charge number and the amplitude ratio of the ω0 and 2ω0 field components, different photon combination pathways in support of each harmonic generation are distinguished, which are manifested as characteristic tempo-spatial field distributions, doughnut-shaped intensity distributions with different diameters, and spiral phase diagrams with different topological charge numbers. The results here provide a new, to the best of our knowledge characterization method to distinguish the photon combination pathways for each order harmonic generation.

Circularly Polarized High-Harmonic Beams Carrying Self-Torque or Time-Dependent Orbital Angular Momentum

In the rapidly evolving field of structured light, self-torque has been recently defined as an intrinsic property of light beams carrying time-dependent orbital angular momentum. In particular, extreme-ultraviolet (EUV) beams with self-torque, exhibiting a topological charge that continuously varies on the subfemtosecond time scale, are naturally produced in high-order harmonic generation (HHG) when driven by two time-delayed intense infrared vortex beams with different topological charges. Until now, the polarization state of such EUV beams carrying self-torque has been restricted to linear states due to the drastic reduction in the harmonic up-conversion efficiency with increasing the ellipticity of the driving field. In this work, we theoretically demonstrate how to control the polarization state of EUV beams carrying self-torque, from linear to circular. The extremely high sensitivity of HHG to the properties of the driving beam allows us to propose two different driving schemes to circumvent the current limitations to manipulate the polarization state of EUV beams with self-torque. Our advanced numerical simulations are complemented with the derivation of selection rules of angular momentum conservation, which enable precise tunability over the angular momentum properties of the harmonics with self-torque. The resulting high-order harmonic emission, carrying time-dependent orbital angular momentum with a custom polarization state, can expand the applications of ultrafast light-matter interactions, particularly in areas where dichroic or chiral properties are crucial, such as magnetic materials or chiral molecules.

Laser harmonic generation with independent control of frequency and orbital angular momentum

The non-linear optical process of laser harmonic generation (HG) enables the creation of high quality pulses of UV or even X-ray radiation, which have many potential uses at the frontiers of experimental science, ranging from lensless microscopy to ultrafast metrology and chiral science. Although many of the promising applications are enabled by generating harmonic modes with orbital angular momentum (OAM), independent control of the harmonic frequency and OAM level remains elusive. Here we show, through a theoretical approach, validated with 3D simulations, how unique 2-D harmonic progressions can be obtained, with both frequency and OAM level tuned independently, from tailored structured targets in both reflective and transmissive configurations. Through preferential selection of a subset of harmonic modes with a specific OAM value, a controlled frequency comb of circularly polarised harmonics can be produced. Our approach to describe HG, which simplifies both the theoretical predictions and the analysis of the harmonic spectrum, is directly applicable across the full range of HG mechanisms and can be readily applied to investigations of OAM harmonics in other processes, such as OAM cascades in Raman amplification, or the analysis of harmonic progressions in nonlinear optics.

Programmable generation of counterrotating bicircular light pulses in the multi-terahertz frequency range

The manipulation of solid states using intense infrared or terahertz light fields is a pivotal area in contemporary ultrafast photonics research. While conventional circular polarization has been well explored, the potential of counterrotating bicircular light remains widely underexplored, despite growing interest in theory. In the mid-infrared or multi-terahertz region, experimental challenges lie in difficulties in stabilizing the relative phase between two-color lights and the lack of available polarization elements. Here, we successfully generated phase-stable counterrotating bicircular light pulses in the 14-39 THz frequency range circumventing the above problems. Employing spectral broadening, polarization pulse shaping with a spatial light modulator, and intra-pulse difference frequency generation leveraging a distinctive angular-momentum selection rule within the nonlinear crystal, we achieved direct conversion from near-infrared pulses into the designed counterrotating bicircular multi-terahertz pulses. Use of the spatial light modulator enables programmable control over the shape, orientation, rotational symmetry, and helicity of the bicircular light field trajectory. This advancement provides a novel pathway for the programmable manipulation of light fields, and marks a significant step toward understanding and harnessing the impact of tailored light fields on matter, particularly in the context of topological semimetals.

Valleytronics in bulk MoS2 with a topologic optical field

The valley degree of freedom1-4 of electrons in materials promises routes towards energy-efficient information storage with enticing prospects for quantum information processing5-7. Current challenges in utilizing valley polarization are symmetry conditions that require monolayer structures8,9 or specific material engineering10-13, non-resonant optical control to avoid energy dissipation and the ability to switch valley polarization at optical speed. We demonstrate all-optical and non-resonant control over valley polarization using bulk MoS2, a centrosymmetric material without Berry curvature at the valleys. Our universal method utilizes spin angular momentum-shaped trefoil optical control pulses14,15 to switch the material's electronic topology and induce valley polarization by transiently breaking time and space inversion symmetry16 through a simple phase rotation. We confirm valley polarization through the transient generation of the second harmonic of a non-collinear optical probe pulse, depending on the trefoil phase rotation. The investigation shows that direct optical control over the valley degree of freedom is not limited to monolayer structures. Indeed, such control is possible for systems with an arbitrary number of layers and for bulk materials. Non-resonant valley control is universal and, at optical speeds, unlocks the possibility of engineering efficient multimaterial valleytronic devices operating on quantum coherent timescales.

Vortex harmonic generation in indium tin oxide thin film irradiated by a two-color field

When a two-color Laguerre-Gaussian laser beam propagates through an indium tin oxide (ITO) material, the spatial distributions of odd- and even-order vortex harmonics carrying orbital angular momentum (OAM) are studied. The origin of vortex harmonics can be directly clarified by investigating their dependence on the incident laser field amplitude and frequency. In addition, it is shown that the spectral intensities of vortex harmonics are sensitive to the epsilon-near-zero nonlinear enhancing effects and the thickness of ITO materials. Thus the vortex harmonics can be conveniently tunable, which provides a wider potential application in optical communications based on high-order OAM coherent vortex beams.

High-order harmonic generation by aligned homonuclear diatomic cations

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.

Enabling elliptically polarized high harmonic generation with short cross polarized laser pulses

Enabling elliptically polarized high-order harmonics overcomes a historical limitation in the generation of this highly nonlinear process in atomic, molecular and optical physics with applications in other branches. Here, we shed new light on a controversy between experimental observations and theoretical predictions on the possibility to generate harmonics with large ellipticity using two bichromatic laser pulses which are linearly polarized in orthogonal directions. Results of numerical calculations confirm the previous experimental data that in short laser pulses even harmonics with large ellipticity can be obtained for the interaction of such cross-polarized laser pulses with atoms initially in a s- or p-state, while odd harmonics have low ellipticity. The amount of the ellipticity can be controlled via the relative carrier-envelope phase of the pulses, their intensity ratio and the duration of the pulses.

High-order harmonic generation from aligned HCN molecules under orthogonally and linearly polarized two-color laser fields

Molecular high-order harmonic generation and molecular orbital ionization probabilities are calculated under orthogonally and linearly polarized two-color laser fields. When a second-harmonic field is applied, the high-order harmonics generated under the linearly polarized two-color laser fields in the antiparallel case are stronger than those generated in the orthogonal polarization case and even stronger than those of the parallel polarization case. The results show that ionization probabilities of various orbitals and harmonic orders are dependent on spatial symmetry of molecular orbitals. It is found that the ionization of low-lying Kohn-Sham molecular orbitals contributes significantly to the ionization and molecular high-order harmonic generation processes. The ionization probability maximum occurs when molecular orbital densities are maximum in the direction of laser field polarization. Furthermore, we show that the degeneracy of π orbitals is broken when the laser-molecule alignment angle deviates from the field axis. Accordingly, we indicated one component of the π orbital is effectively contributed to the ionization and high-order harmonic generation processes. Finally, to confirm the recollision model in the high-order harmonic generation, the quantum time-frequency analysis is used to extract electron paths information on subcycle time scales.

Attosecond spectroscopy for the investigation of ultrafast dynamics in atomic, molecular and solid-state physics

Since the first demonstration of the generation of attosecond pulses (1 as = 10^-18s) in the extreme-ultraviolet spectral region, several measurement techniques have been introduced, at the beginning for the temporal characterization of the pulses, and immediately after for the investigation of electronic and nuclear ultrafast dynamics in atoms, molecules and solids with unprecedented temporal resolution. The attosecond spectroscopic tools established in the last two decades, together with the development of sophisticated theoretical methods for the interpretation of the experimental outcomes, allowed to unravel and investigate physical processes never observed before, such as the delay in photoemission from atoms and solids, the motion of electrons in molecules after prompt ionization which precede any notable nuclear motion, the temporal evolution of the tunneling process in dielectrics, and many others. This review focused on applications of attosecond techniques to the investigation of ultrafast processes in atoms, molecules and solids. Thanks to the introduction and ongoing developments of new spectroscopic techniques, the attosecond science is rapidly moving towards the investigation, understanding and control of coupled electron-nuclear dynamics in increasingly complex systems, with ever more accurate and complete investigation techniques. Here we will review the most common techniques presenting the latest results in atoms, molecules and solids.

Generation and complete polarimetry of ultrashort circularly polarized extreme-ultraviolet pulses

The generation of ultrashort circularly polarized pulses in the extreme-ultraviolet spectral range has recently attracted considerable interest for applications in time-resolved circular-dichroism experiments. Here, we demonstrate a simple approach to generate near-circularly polarized femtosecond pulses in the vacuum-ultraviolet. The ellipticity of the generated light can be continuously tuned from linear to near-circular, as demonstrated by detailed polarimetry measurements. Combining optical polarimetry with photoelectron circular-dichroism (PECD) measurements, we demonstrate a novel approach to characterizing the polarization state of light in terms of all four Stokes parameters. For photon energies of 9.3 eV, we obtained S₃ = 0.96 ± 0.02 and a degree of polarization of 97±2%, i.e. the highest values reported from any harmonic-generation source so far. This source is directly applicable to circular-dichroism experiments, also enabling time-resolved PECD in the extreme-ultraviolet, a general approach to probing time-dependent chirality during chemical processes on (sub)-femtosecond time scales.

The High-Order Harmonic Generation from Atom Driven by Co-Rotating Laser Pulses Composed of Fundamental Frequency and High Frequency

By numerically solving the time-dependent Schrödinger equation (TDSE), the harmonic generation process of atoms irradiated by corotating laser pulses composed of a fundamental-frequency and high-frequency field is systematically studied. Compared with the harmonic generated from atoms irradiated by counter-rotating two-color circularly polarized laser pulses, the harmonic efficiency of atoms irradiated by co-rotating two-color circularly polarized (CRTCCP) laser pulses with the same laser parameters is higher. The harmonics are generated by the multiphoton radiation transition after the bound electrons undergo a multiphoton absorption transition to a higher energy level. In addition, the variation of the harmonic efficiency with the field strength of different frequency components in the driving laser pulse is also studied. The circularly polarized harmonics with higher intensity can be obtained by optimizing the field strength of the driving laser field.

Necklace-structured high-harmonic generation for low-divergence, soft x-ray harmonic combs with tunable line spacing

The extreme nonlinear optical process of high-harmonic generation (HHG) makes it possible to map the properties of a laser beam onto a radiating electron wave function and, in turn, onto the emitted x-ray light. Bright HHG beams typically emerge from a longitudinal phased distribution of atomic-scale quantum antennae. Here, we form a transverse necklace-shaped phased array of linearly polarized HHG emitters, where orbital angular momentum conservation allows us to tune the line spacing and divergence properties of extreme ultraviolet and soft x-ray high-harmonic combs. The on-axis HHG emission has extremely low divergence, well below that obtained when using Gaussian driving beams, which further decreases with harmonic order. This work provides a new degree of freedom for the design of harmonic combs-particularly in the soft x-ray regime, where very limited options are available. Such harmonic beams can enable more sensitive probes of the fastest correlated charge and spin dynamics in molecules, nanoparticles, and materials.

Detecting multiple chiral centers in chiral molecules with high harmonic generation

Characterizing chiral is highly important for applications in the pharmaceutical industry, as well as in the study of dynamical chemical and biological systems. However, this task has remained challenging, especially due to the ongoing increasing complexity and size of the molecular structure of drugs and active compounds. In particular, large molecules with many active chiral centers are today ubiquitous, but remain difficult to structurally analyze due to their high number of stereoisomers. Here we theoretically explore the sensitivity of high harmonic generation (HHG) to the chiral of molecules with a varying number of active chiral centers. We find that HHG driven by bi-chromatic non-collinear lasers is a sensitive probe for the stereo-configuration of a chiral molecule. We first show through calculations (from benchmark chiral molecules with up to three chiral centers) that the HHG spectrum is imprinted with information about the handedness of each chiral center in the driven molecule. Next, we show that using both classical- and deep-learning-based reconstruction algorithms, the composition of an unknown mixture of stereoisomers can be reconstructed with high fidelity by a single-shot HHG measurement. Our work illustrates how the combination of non-linear optics and machine learning might open routes for ultra-sensitive sensing in chiral systems.

Coherence in macroscopic high harmonic generation for spatial focal phase distributions of monochromatic and broadband Gaussian laser pulses

Efficient application of ultrafast laser sources from high harmonic generation requires an understanding of how the spectrum can be controlled - the extent of the highest harmonics and the strength and cleanness of the harmonic lines. We study one important aspect in the coherent build-up of macroscopic high-order harmonic generation, namely the impact of different phase distributions in the focal area on the features of the generated radiation. Specifically, we compare the high harmonic signals for the commonly-used Gouy distribution of a monochromatic beam with those for the phase distribution of a short broadband Gaussian pulse. To this end, we apply a theoretical model in which the microscopic yields are obtained via interpolation of results of the time-dependent Schrödinger equation, which are then used in an individual-emitter approach to determine the macroscopic signals. Regions of poor and good coherent build-up as a function of the position of the gas jet are identified using measures for the strength of the harmonic lines and for the impact of off-harmonic radiation. While the largest extent of the spectra as well as the strongest contribution of off-harmonic radiation is found for positioning the gas jet after the focus for both distributions, the relative strength of the harmonics is overall weaker for the short Gaussian pulse distribution and the spectra differ for a gas jet positioned at the focus. These differences are mainly caused by the additional dependence of the focal phase in the transverse direction for the short Gaussian pulse distribution.

Controlling of the harmonic generation induced by the Berry curvature

High-order harmonic generation in solid state has attracted a lot of attentions. The Berry curvature (BC), a geometrical property of the Bloch energy band, plays an important role for the harmonic generation in crystal. As we all know, the influence of BC on the harmonic emission has been investigated before and BC is simplified as a 1D structure. However, many other materials including MoS₂ are 2D materials. In this work, we extend the investigation for BC to 2D structure and get a generalized equation, which not only gives a new method to control the harmonic emission with BC, but also gives a deeper understanding for the influence of the BC. We show the ability to control the harmonic emission related to the BC using the orthogonal two-color (OTC) laser field. By tuning the delay of OTC laser field, one can steer the trajectory of electrons and modulate the emission of harmonics. This study can provide us a deeper insight into the role of the BC which is difficult to be measured experimentally.

Bright, single helicity, high harmonics driven by mid-infrared bicircular laser fields

High-harmonic generation (HHG) is a unique tabletop light source with femtosecond-to-attosecond pulse duration and tailorable polarization and beam shape. Here, we use counter-rotating femtosecond laser pulses of 0.8 µm and 2.0 μm to extend the photon energy range of circularly polarized high-harmonics and also generate single-helicity HHG spectra. By driving HHG in helium, we produce circularly polarized soft x-ray harmonics beyond 170 eV-the highest photon energy of circularly polarized HHG achieved to date. In an Ar medium, dense spectra at photon energies well beyond the Cooper minimum are generated, with regions composed of a single helicity-consistent with the generation of a train of circularly polarized attosecond pulses. Finally, we show theoretically that circularly polarized HHG photon energies can extend beyond the carbon K edge, extending the range of molecular and materials systems that can be accessed using dynamic HHG chiral spectro-microscopies.

Study on spin angular momentum balance in harmonics generated from counter-rotating two-color laser fields

High-order harmonics generated from Xe driven by counter-rotating two-color driving fields are studied in the frame of a quantum-field scattering theory, and the spin angular momentum transfer is discussed. The driving field is composed by a circularly polarized (CP) mode and an elliptically polarized (EP) mode. We treat the EP mode as a compostition of counter-rotating CP fields of unequal intensity. We use a pair of phased generalized Bessel functions to describe the harmonic generation amplitude, and the conservation of the spin angular momentum during harmonic generation in the two-color field is derived in a solid base and in a straightforward way. The experimentally observed V-type and Λ-type distributions of the harmonic spectra with ellipticity are recovered theoretically. Balance pattern of the spin angular momentum is disclosed substantially.

Elliptically polarized high-harmonic radiation for production of isolated attosecond pulses

Circularly polarized attosecond pulses are powerful tool to study chiral light-matter interaction via chiral electron dynamics. However, access to isolated circularly polarized attosecond pulses enabling straightforward interpretation of measurements, still remains a challenge. In this work, we experimentally demonstrate the generation of highly elliptically polarized high-harmonics in a two-color, bi-circular, collinear laser field. The intensity and shape of the combined few-cycle driving radiation is optimized to produce a broadband continuum with enhanced spectral chirality in the range of 15-55 eV supporting the generation of isolated attosecond pulses with duration down to 150 as. We apply spectrally resolved polarimetry to determine the full Stokes vector of different spectral components of the continuum, yielding a homogenous helicity distribution with ellipticity in the range of 0.8-0.95 and a negligible unpolarized content.

Ellipticity of High-Order Harmonics Generated by Aligned Homonuclear Diatomic Molecules Exposed to an Orthogonal Two-Color Laser Field

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.

Trains of attosecond pulses structured with time-ordered polarization states

Ultrafast laser pulses generated at the attosecond timescale represent a unique tool to explore the fastest dynamics in matter. An accurate control of their properties, such as polarization, is fundamental to shape three-dimensional laser-driven dynamics. We introduce a technique to generate attosecond pulse trains whose polarization state varies from pulse to pulse. This is accomplished by driving high-harmonic generation with two time-delayed bichromatic counter-rotating fields with proper orbital angular momentum (OAM) content. Our simulations show that the evolution of the polarization state along the train can be controlled via OAM, pulse duration, and time delay of the driving fields. We, thus, introduce an additional control into structured attosecond pulses that provides an alternative route to explore ultrafast dynamics with potential applications in chiral and magnetic materials.

Investigation of electron vortices in time-delayed circularly polarized laser pulses with a semiclassical perspective

We theoretically investigate strong-filed electron vortices in time-delayed circularly polarized laser pulses by a generalized quantum-trajectory Monte Carlo (GQTMC) model. Vortex interference patterns in photoelectron momentum distributions (PMDs) with various laser parameters can be well reproduced by the semiclassical simulation. The phase difference responsible for the interference structures is analytically identified through trajectory-based analysis and simple-man theory, which reveal the underlying mechanism of electron vortex phenomena for both co-rotating and counter-rotating component. This semiclassical analysis can also demonstrate the influences of laser intensity and wavelength on the number of arms of vortices. Furthermore, we show the influence of the Coulomb effect on the PMDs. Finally, the controlling of the ionization time intervals in the tens to hundreds of attosecond magnitude is qualitatively discussed.

Attosecond-Scale Streaking Methods for Strong-Field Ionization by Tailored Fields

Streaking with a weak probe field is applied to ionization in a two-dimensional strong field tailored to mimic linear polarization, but without disturbance by recollision or intracycle interference. This facilitates the observation of electron-momentum-resolved times of ionization with few-attosecond precision, as demonstrated by simulations for a model helium atom. Aligning the probe field along the ionizing field provides meaningful ionization times in agreement with the attoclock concept that ionization at maximum field corresponds to the peak of the momentum distribution, which is shifted due to the Coulomb force on the outgoing electron. In contrast, this attoclock shift is invisible in orthogonal streaking. Even without a probe field, streaking happens naturally along the laser propagation direction due to the laser magnetic field. As with an orthogonal probe field, the attoclock shift is not accessible by the magnetic-field scheme. For a polar molecule, the attoclock shift depends on orientation, but this does not imply an orientation dependence in ionization time.

Conservation of Torus-knot Angular Momentum in High-order Harmonic Generation

High-order harmonic generation stands as a unique nonlinear optical up-conversion process, mediated by a laser-driven electron recollision mechanism, which has been shown to conserve energy, linear momentum, and spin and orbital angular momentum. Here, we present theoretical simulations that demonstrate that this process also conserves a mixture of the latter, the torus-knot angular momentum J_{γ}, by producing high-order harmonics with driving pulses that are invariant under coordinated rotations. We demonstrate that the charge J_{γ} of the emitted harmonics scales linearly with the harmonic order, and that this conservation law is imprinted onto the polarization distribution of the emitted spiral of attosecond pulses. We also demonstrate how the nonperturbative physics of high-order harmonic generation affect the torus-knot angular momentum of the harmonics, and we show that this configuration harnesses the spin selection rules to channel the full yield of each harmonic into a single mode of controllable orbital angular momentum.

Transient absorption spectroscopy using high harmonic generation: a review of ultrafast X-ray dynamics in molecules and solids

Attosecond science opened the door to observing nuclear and electronic dynamics in real time and has begun to expand beyond its traditional grounds. Among several spectroscopic techniques, X-ray transient absorption spectroscopy has become key in understanding matter on ultrafast time scales. In this review, we illustrate the capabilities of this unique tool through a number of iconic experiments. We outline how coherent broadband X-ray radiation, emitted in high-harmonic generation, can be used to follow dynamics in increasingly complex systems. Experiments performed in both molecules and solids are discussed at length, on time scales ranging from attoseconds to picoseconds, and in perturbative or strong-field excitation regimes. This article is part of the theme issue 'Measurement of ultrafast electronic and structural dynamics with X-rays'.

Recent advances in ultrafast X-ray sources

Over more than a century, X-rays have transformed our understanding of the fundamental structure of matter and have been an indispensable tool for chemistry, physics, biology, materials science and related fields. Recent advances in ultrafast X-ray sources operating in the femtosecond to attosecond regimes have opened an important new frontier in X-ray science. These advances now enable: (i) sensitive probing of structural dynamics in matter on the fundamental timescales of atomic motion, (ii) element-specific probing of electronic structure and charge dynamics on fundamental timescales of electronic motion, and (iii) powerful new approaches for unravelling the coupling between electronic and atomic structural dynamics that underpin the properties and function of matter. Most notable is the recent realization of X-ray free-electron lasers (XFELs) with numerous new XFEL facilities in operation or under development worldwide. Advances in XFELs are complemented by advances in synchrotron-based and table-top laser-plasma X-ray sources now operating in the femtosecond regime, and laser-based high-order harmonic XUV sources operating in the attosecond regime. This article is part of the theme issue 'Measurement of ultrafast electronic and structural dynamics with X-rays'.

Vectorizing the spatial structure of high-harmonic radiation from gas

Strong field laser physics has primarily been concerned with controlling beams in time while keeping their spatial profiles invariant. In the case of high harmonic generation, the harmonic beam is the result of the coherent superposition of atomic dipole emissions. Therefore, fundamental beams can be tailored in space, and their spatial characteristics will be imparted onto the harmonics. Here we produce high harmonics using a space-varying polarized fundamental laser beam, which we refer to as a vector beam. By exploiting the natural evolution of a vector beam as it propagates, we convert the fundamental beam into high harmonic radiation at its focus where the polarization is primarily linear. This evolution results in circularly polarized high harmonics in the far field. Such beams will be important for ultrafast probing of magnetic materials.

Ultrafast pulses with controlled parameters are desirable in many applications including probing materials and their interaction with light. Here the authors demonstrate a technique for polarization control of XUV beam using high-harmonic generation and polarization shaping.

Nonsequential double ionization by co-rotating two-color circularly polarized laser fields

Nonsequential double ionization (NSDI) of Ar in co-rotating two-color circularly polarized (TCCP) laser fields is investigated with a three-dimensional classical ensemble model. Our numerical results indicate that co-rotating TCCP fields can induce NSDI by recollision process, while the yield is an order of magnitude lower than counter-rotating case. NSDI yield in co-rotating TCCP fields strongly depends on field ratio of the two colors and achieves its maximum at a ratio of 2.4. In co-rotating TCCP fields, the short recollision trajectory with traveling time smaller than one cycle is dominant. Moreover, the recollision time in co-rotating TCCP laser fields depends on the field ratio, which is mapped to the electron momentum distribution. This provides anavenue to obtain information about recollision time and access the subcycle dynamics of the recollision process.

Floquet group theory and its application to selection rules in harmonic generation

Symmetry is one of the most generic and useful concepts in science, often leading to conservation laws and selection rules. Here we formulate a general group theory for dynamical symmetries (DSs) in time-periodic Floquet systems, and derive their correspondence to observable selection rules. We apply the theory to harmonic generation, deriving closed-form tables linking DSs of the driving laser and medium (gas, liquid, or solid) in (2+1)D and (3+1)D geometries to the allowed and forbidden harmonic orders and their polarizations. We identify symmetries, including time-reversal-based, reflection-based, and elliptical-based DSs, which lead to selection rules that are not explained by currently known conservation laws. We expect the theory to be useful for ultrafast high harmonic symmetry-breaking spectroscopy, as well as in various other systems such as Floquet topological insulators.

Evidence of depolarization and ellipticity of high harmonics driven by ultrashort bichromatic circularly polarized fields

High harmonics generated by counter-rotating laser fields at the fundamental and second harmonic frequencies have raised important interest as a table-top source of circularly polarized ultrashort extreme-ultraviolet light. However, this emission has not yet been fully characterized: in particular it was assumed to be fully polarized, leading to an uncertainty on the effective harmonic ellipticity. Here we show, through simulations, that ultrashort driving fields and ultrafast medium ionization lead to a breaking of the dynamical symmetry of the interaction, and consequently to deviations from perfectly circular and fully polarized harmonics, already at the single-atom level. We perform the complete experimental characterization of the polarization state of high harmonics generated along that scheme, giving direct access to the ellipticity absolute value and sign, as well as the degree of polarization of individual harmonic orders. This study allows defining optimal generation conditions of fully circularly polarized harmonics for advanced studies of ultrafast dichroisms.

Atomic and Molecular Processes in a Strong Bicircular Laser Field

With the development of intense femtosecond laser sources it has become possible to study atomic and molecular processes on their own subfemtosecond time scale. Table-top setups are available that generate intense coherent radiation in the extreme ultraviolet and soft-X-ray regime which have various applications in strong-field physics and attoscience. More recently, the emphasis is moving from the generation of linearly polarized pulses using a linearly polarized driving field to the generation of more complicated elliptically polarized polychromatic ultrashort pulses. The transverse electromagnetic field oscillates in a plane perpendicular to its propagation direction. Therefore, the two dimensions of field polarization plane are available for manipulation and tailoring of these ultrashort pulses. We present a field that allows such a tailoring, the so-called bicircular field. This field is the superposition of two circularly polarized fields with different frequencies that rotate in the same plane in opposite directions. We present results for two processes in a bicircular field: High-order harmonic generation and above-threshold ionization. For a wide range of laser field intensities, we compare high-order harmonic spectra generated by bicircular fields with the spectra generated by a linearly polarized laser field. We also investigate a possibility of introducing spin into attoscience with spin-polarized electrons produced in high-order above-threshold ionization by a bicircular field.

Intensity-dependent two-electron emission dynamics in nonsequential double ionization by counter-rotating two-color circularly polarized laser fields

Nonsequential double ionization of helium in counter-rotating two-color circularly polarized laser fields is investigated with a three-dimensional classical ensemble model. At moderate intensity, the momentum distribution of the two electrons shows a maximum in the middle of each side of the triangle of the negative vector potential. At high intensity, the momentum distribution exhibits a double-triangle structure, which is attributed to the different values of the laser intensity where the two electrons are released after recollision. At low intensity, the momentum distribution shows a shift deviating from the middle of the side of the triangle of the negative vector potential. This is because the first electrons are emitted within a narrow time window after the field maximum. In addition, at low intensity, double-recollision events and NSDI originating from doubly excited states induced by recollision are prevalent.

Control of Harmonic Generation by the Time Delay Between Two-Color, Bicircular Few-Cycle Mid-IR Laser Pulses

We study control of high-order harmonic generation (HHG) driven by time-delayed, few-cycle ω and 2ω counterrotating mid-IR pulses. Our numerical and analytical study shows that the time delay between the two-color pulses allows control of the harmonic positions, both those allowed by angular momentum conservation and those seemingly forbidden by it. Moreover, the helicity of any particular harmonic is tunable from left to right circular without changing the driving pulse helicity. The highest HHG yield occurs for a time delay comparable to the fundamental period T=2π/ω.

Helicity asymmetry in strong-field ionization of atoms by a bicircular laser field

Ionization of atoms by an intense bicircular laser field is considered, which consists of two coplanar corotating or counterrotating circularly polarized field components with frequencies that are integer multiples of a fundamental frequency. Emphasis is on the effect of a reversal of the helicities of the two field components on the photoelectron spectra. The velocity maps of the liberated electrons are calculated using the direct strong-field approximation (SFA) and its improved version (ISFA), which takes into account rescattering off the parent ion. Under the SFA all symmetries of the driving field are preserved in the velocity map while the ISFA violates certain reflection symmetries. This allows one to assess the significance of rescattering in actual data obtained from an experiment or a numerical simulation.

Collective Effects in Second-Harmonic Generation from Plasmonic Oligomers

We investigate collective effects in plasmonic oligomers of different symmetries using second-harmonic generation (SHG) microscopy with cylindrical vector beams (CVBs). The oligomers consist of gold nanorods that have a longitudinal plasmon resonance close to the fundamental wavelength that is used for SHG excitation and whose long axes are arranged locally such that they follow the distribution of the transverse component of the electric field of radially or azimuthally polarized CVBs in the focal plane. We observe that SHG from such rotationally symmetric oligomers is strongly modified by the interplay between the polarization properties of the CVB and interparticle coupling. We find that the oligomers with radially oriented nanorods exhibit small coupling effects. In contrast, we find that the oligomers with azimuthally oriented nanorods exhibit large coupling effects that lead to silencing of SHG from the whole structure. Our experimental results are in very good agreement with numerical calculations based on the boundary element method. The work describes a new route for studying coupling effects in complex arrangements of nano-objects and thereby for tailoring the efficiency of nonlinear optical effects in such structures.

Spectral control of high harmonics from relativistic plasmas using bicircular fields

We introduce two-color counterrotating circularly polarized laser fields as a way to spectrally control high harmonic generation (HHG) from relativistic plasma mirrors. Through particle-in-cell simulations, we show that only a selected group of harmonic orders can appear owing to the symmetry of the laser fields and the related conservation laws. By adjusting the intensity ratio of the two driving field components, we demonstrate the overall HHG efficiency, the relative intensity of allowed neighboring harmonic orders, and that the polarization state of the harmonic source can be tuned. The HHG efficiency of this scheme can be as high as that driven by a linearly polarized laser field.

Optical Chirality in Nonlinear Optics: Application to High Harmonic Generation

Optical chirality (OC)-one of the fundamental quantities of electromagnetic fields-corresponds to the instantaneous chirality of light. It has been utilized for exploring chiral light-matter interactions in linear optics, but has not yet been applied to nonlinear processes. Motivated to explore the role of OC in the generation of helically polarized high-order harmonics and attosecond pulses, we first separate the OC of transversal and paraxial beams to polarization and orbital terms. We find that the polarization-associated OC of attosecond pulses corresponds approximately to that of the pump in the quasimonochromatic case, but not in the multichromatic pump cases. We associate this discrepancy with the fact that the polarization OC of multichromatic pumps vary rapidly in time along the optical cycle. Thus, we propose new quantities, noninstantaneous polarization-associated OC, and time-scale-weighted polarization-associated OC, and show that these quantities link the chirality of multichromatic pumps and their generated attosecond pulses. The presented extension to OC theory should be useful for exploring various nonlinear chiral light-matter interactions. For example, it stimulates us to propose a tricircular pump for generation of highly elliptical attosecond pulses with a tunable ellipticity.

Multiple recollisions in nonsequential double ionization by counter-rotating two-color circularly polarized laser fields

With the three-dimensional (3D) classical ensemble method, we theoretically investigate the recollision dynamics in strong-field nonsequential double ionization (NSDI) of Ar by counter-rotating two-color circularly polarized laser fields. With the analysis of the NSDI trajectories, we find that not only multiple-recollision but also single-recollision processes occur in the double ionization events. Furthermore, the multiple-recollision and single-recollision processes both undergo the recollision-induced excitation with subsequent ionization (RESI) and recollision-induced ionization (RII). The angle between the momentum and the force of the laser field at the recollision moment can affect the times of the recollision.

Controlling the polarization and vortex charge of attosecond high-harmonic beams via simultaneous spin-orbit momentum conservation

Optical interactions are governed by both spin and angular momentum conservation laws, which serve as a tool for controlling light-matter interactions or elucidating electron dynamics and structure of complex systems. Here, we uncover a form of simultaneous spin and orbital angular momentum conservation and show, theoretically and experimentally, that this phenomenon allows for unprecedented control over the divergence and polarization of extreme-ultraviolet vortex beams. High harmonics with spin and orbital angular momenta are produced, opening a novel regime of angular momentum conservation that allows for manipulation of the polarization of attosecond pulses-from linear to circular-and for the generation of circularly polarized vortices with tailored orbital angular momentum, including harmonic vortices with the same topological charge as the driving laser beam. Our work paves the way to ultrafast studies of chiral systems using high-harmonic beams with designer spin and orbital angular momentum.

Nanoscale magnetic imaging using circularly polarized high-harmonic radiation

This work demonstrates nanoscale magnetic imaging using bright circularly polarized high-harmonic radiation. We utilize the magneto-optical contrast of worm-like magnetic domains in a Co/Pd multilayer structure, obtaining quantitative amplitude and phase maps by lensless imaging. A diffraction-limited spatial resolution of 49 nm is achieved with iterative phase reconstruction enhanced by a holographic mask. Harnessing the exceptional coherence of high harmonics, this approach will facilitate quantitative, element-specific, and spatially resolved studies of ultrafast magnetization dynamics, advancing both fundamental and applied aspects of nanoscale magnetism.

Signatures of Electronic Structure in Bicircular High-Harmonic Spectroscopy

High-harmonic spectroscopy driven by circularly polarized laser pulses and their counterrotating second harmonic is a new branch of attosecond science which currently lacks quantitative interpretations. We extend this technique to the midinfrared regime and record detailed high-harmonic spectra of several rare-gas atoms. These results are compared with the solution of the Schrödinger equation in three dimensions and calculations based on the strong-field approximation that incorporate accurate scattering-wave recombination matrix elements. A quantum-orbit analysis of these results provides a transparent interpretation of the measured intensity ratios of symmetry-allowed neighboring harmonics in terms of (i) a set of propensity rules related to the angular momentum of the atomic orbitals, (ii) atom-specific matrix elements related to their electronic structure, and (iii) the interference of the emissions associated with electrons in orbitals corotating or counterrotating with the laser fields. These results provide the foundation for a quantitative understanding of bicircular high-harmonic spectroscopy.

Extended ellipticity control for attosecond pulses by high harmonic generation

Attosecond (1  as=10^-18  s) pulses produced through high harmonic generation (HHG) are a basis for studies of electron dynamics during light-matter interaction on an electron's natural time scale. Extensively exploited HHG technology has, however, a few unsolved problems, where producing of circularly polarized or chiral attosecond pulses belongs to them. We have demonstrated experimentally a way to control the ellipticity of attosecond pulse trains produced via HHG in two-color, bi-circular laser fields. We show that the combination of a nonlinear medium position and the intensities of the two-color driving laser fields create an effective helicity-dependent filter. Based on this approach, we report generation of chiral spectra providing potential to produce attosecond pulses with polarization tuned from the rotating, but linear to highly elliptic, with ellipticity as much as ϵ=0.75. This new way to create a chiral-sensitive element offers a simple and practical knob to control polarization for a combined harmonics field in a smooth and predictable manner.

Time-resolved high harmonic spectroscopy of dynamical symmetry breaking in bi-circular laser fields: the role of Rydberg states

The bi-circular scheme for high harmonic generation, which combines two counter-rotating circular fields with frequency ratio 2:1, has recently permitted to generate high harmonics with essentially circular polarization, opening the way for ultrafast chiral studies. This scheme produces harmonic lines at 3N + 1 and 3N + 2 multiples of the fundamental driving frequency, while the 3N lines are forbidden owing to the three-fold symmetry of the field. It is generally established that the routinely observed signals at these forbidden harmonic lines come from a slight ellipticity in the driving fields, which breaks the three-fold symmetry. We find that this is neither the only nor it is the dominant mechanism responsible. The forbidden lines can be observed even for perfectly circular, long driving pulses. We show that they encode rich information on the sub-cycle electronic dynamics that occur during the generation process. By varying the time delay and relative intensity between the two drivers, we demonstrate that when the second harmonic either precedes or is more intense than the fundamental field, the weak effects of dynamical symmetry breaking caused by finite pulse duration are amplified by electrons trapped in Rydberg orbits (i.e., Freeman resonances), and that the forbidden harmonic lines are a witness of this.

Helicity-Selective Enhancement and Polarization Control of Attosecond High Harmonic Waveforms Driven by Bichromatic Circularly Polarized Laser Fields

High harmonics driven by two-color counterrotating circularly polarized laser fields are a unique source of bright, circularly polarized, extreme ultraviolet, and soft x-ray beams, where the individual harmonics themselves are completely circularly polarized. Here, we demonstrate the ability to preferentially select either the right or left circularly polarized harmonics simply by adjusting the relative intensity ratio of the bichromatic circularly polarized driving laser field. In the frequency domain, this significantly enhances the harmonic orders that rotate in the same direction as the higher-intensity driving laser. In the time domain, this helicity-dependent enhancement corresponds to control over the polarization of the resulting attosecond waveforms. This helicity control enables the generation of circularly polarized high harmonics with a user-defined polarization of the underlying attosecond bursts. In the future, this technique should allow for the production of bright highly elliptical harmonic supercontinua as well as the generation of isolated elliptically polarized attosecond pulses.