Model calculations of polarization-dependent two-color high-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.

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.

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.

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.

Symmetries and Selection Rules of the Spectra of Photoelectrons and High-Order Harmonics Generated by Field-Driven Atoms and Molecules

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.

Signature of Molecular Orbital Symmetry in High-Order Harmonic Generation by Bichromatic Circularly Polarized Laser Pulses

Molecular orbital symmetry is shown to be an important factor in determining orders and helicities (polarizations) of high-order harmonic generation (HHG) by intense femtosecond counter-rotating bichromatic circularly polarized laser pulses. Numerical solutions of time-dependent Schrödinger equations (TDSE) for the one-electron molecular ions H₂⁺ and H₃²⁺ for different initial electronic states show that harmonic orders and helicities are dependent on orbital symmetries and of the net incident pulse electric field. The numerical results and properties of the harmonics are described by dynamical symmetry theory and time profile analysis of the high-order harmonics, thus confirming that orbital and laser pulse symmetry dependence are generic in HHG of molecules.

Shortcut to study angular momentum transfer of harmonic generation in intense laser fields

High-order harmonics generated from atoms driven by counter-rotating two-color circularly (CRTC) polarized laser fields are studied in the frame of a quantum-field scattering theory. We use a pair of generalized phased Bessel (GPB) functions to describe the harmonic generation amplitude. The use of GPB functions allows us to define the phase of a harmonic channel accurately, by which we obtain the spin angular momentum conservation relation in a straightforward way. The known selection rule of harmonic order in the CRTC field is obtained concisely. Main features of the harmonic spectra are recovered. Our treatment provides a shortcut to study the angular momentum transfer in intense laser fields.

Chiral high-harmonic generation and spectroscopy on solid surfaces using polarization-tailored strong fields

Strong-field methods in solids enable new strategies for ultrafast nonlinear spectroscopy and provide all-optical insights into the electronic properties of condensed matter in reciprocal and real space. Additionally, solid-state media offers unprecedented possibilities to control high-harmonic generation using modified targets or tailored excitation fields. Here we merge these important points and demonstrate circularly-polarized high-harmonic generation with polarization-matched excitation fields for spectroscopy of chiral electronic properties at surfaces. The sensitivity of our approach is demonstrated for structural helicity and termination-mediated ferromagnetic order at the surface of silicon-dioxide and magnesium oxide, respectively. Circularly polarized radiation emanating from a solid sample now allows to add basic symmetry properties as chirality to the arsenal of strong-field spectroscopy in solids. Together with its inherent temporal (femtosecond) resolution and non-resonant broadband spectrum, the polarization control of high harmonics from condensed matter can illuminate ultrafast and strong field dynamics of surfaces, buried layers or thin films.

Strong nonlinearities in solid state materials can lead to interesting applications in photonics. Here the authors study chiral high-harmonic generation at SiO₂ and MgO surfaces using bi-circular two-color driving fields and extract information on crystal properties.

EUV and Hard X-ray Hartmann Wavefront Sensing for Optical Metrology, Alignment and Phase Imaging

For more than 15 years, Imagine Optic have developed Extreme Ultra Violet (EUV) and X-ray Hartmann wavefront sensors for metrology and imaging applications. These sensors are compatible with a wide range of X-ray sources: from synchrotrons, Free Electron Lasers, laser-driven betatron and plasma-based EUV lasers to High Harmonic Generation. In this paper, we first describe the principle of a Hartmann sensor and give some key parameters to design a high-performance sensor. We also present different applications from metrology (for manual or automatic alignment of optics), to soft X-ray source optimization and X-ray imaging.

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.

Reciprocal-Space-Trajectory Perspective on High-Harmonic Generation in Solids

We revisit the mechanism of high-harmonic generation (HHG) from solids by comparing HHG in laser fields with different ellipticities but a constant maximum amplitude. It is shown that the cutoff of HHG is strongly extended in a circularly polarized field. Moreover, the harmonic yield with large ellipticity is comparable to or even higher than that in the linearly polarized field. To understand the underlying physics, we develop a reciprocal-space-trajectory method, which explains HHG in solids by a trajectory ensemble from different ionization times and different initial states in the reciprocal space. We show that the cutoff extension is related to an additional preacceleration step prior to ionization, which has been overlooked in solids. By analyzing the trajectories and the time-frequency spectrogram, we show that the HHG in solids cannot be interpreted in terms of the classical recollision picture alone. Instead, the radiation should be described by the electron-hole interband polarization, which leads to the unusual ellipticity dependence. We propose a new four-step model to understand the mechanism of HHG in solids.

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.

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.

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.

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.

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.

Generating circularly polarized radiation in the extreme ultraviolet spectral range at the free-electron laser FLASH

A new device for polarization control at the free electron laser facility FLASH1 at DESY has been commissioned for user operation. The polarizer is based on phase retardation upon reflection off metallic mirrors. Its performance is characterized in three independent measurements and confirms the theoretical predictions of efficient and broadband generation of circularly polarized radiation in the extreme ultraviolet spectral range from 35 eV to 90 eV. The degree of circular polarization reaches up to 90% while maintaining high total transmission values exceeding 30%. The simple design of the device allows straightforward alignment for user operation and rapid switching between left and right circularly polarized radiation.

Phase matching of noncollinear sum and difference frequency high harmonic generation above and below the critical ionization level

We investigate the macroscopic physics of noncollinear high harmonic generation (HHG) at high pressures. We make the first experimental demonstration of phase matching of noncollinear high-order-difference-frequency generation at ionization fractions above the critical ionization level, which normally sets an upper limit on the achievable cutoff photon energies. Additionally, we show that noncollinear high-order-sum-frequency generation requires much higher pressures for phase matching than single-beam HHG does, which mitigates the short interaction region in this geometry. We also dramatically increase the experimentally realized cutoff energy of noncollinear circularly polarized HHG, reaching photon energies of 90 eV. Finally, we achieve complete angular separation of high harmonic orders without the use of a spectrometer.

Selective suppression of high-order harmonics within phase-matched spectral regions

Phase matching in high-harmonic generation leads to enhancement of multiple harmonics. It is sometimes desired to control the spectral structure within the phase-matched spectral region. We propose a scheme for selective suppression of high-order harmonics within the phase-matched spectral region while weakly influencing the other harmonics. The method is based on addition of phase-mismatched segments within a phase-matched medium. We demonstrate the method numerically in two examples. First, we show that one phase-mismatched segment can significantly suppress harmonic orders 9, 15, and 21. Second, we show that two phase-mismatched segments can efficiently suppress circularly polarized harmonics with one helicity over the other when driven by a bi-circular field. The new method may be useful for various applications, including the generation of highly helical bright attosecond pulses.

Controlling Nonsequential Double Ionization in Two-Color Circularly Polarized Femtosecond Laser Fields

Atoms undergoing strong-field ionization in two-color circularly polarized femtosecond laser fields exhibit unique two-dimensional photoelectron trajectories and can emit bright circularly polarized extreme ultraviolet and soft-x-ray beams. In this Letter, we present the first experimental observation of nonsequential double ionization in these tailored laser fields. Moreover, we can enhance or suppress nonsequential double ionization by changing the intensity ratio and helicity of the two driving laser fields to maximize or minimize high-energy electron-ion rescattering. Our experimental results are explained through classical simulations, which also provide insight into how to optimize the generation of circularly polarized high harmonic beams.

Illuminating Molecular Symmetries with Bicircular High-Order-Harmonic Generation

We present a general theory of bicircular high-order-harmonic generation from N-fold rotationally symmetric molecules. Using a rotating frame of reference we predict the complete structure of the high-order-harmonic spectra for arbitrary driving frequency ratios and show how molecular symmetries can be directly identified from the harmonic signal. Our findings reveal that a characteristic fingerprint of rotational molecular symmetries can be universally observed in the ultrafast response of molecules to strong bicircular fields.

Two-color high-harmonic generation in plasmas: efficiency dependence on the generating particle properties

The high-order harmonic generation (HHG) in silver, gold, and zinc plasma plumes irradiated by orthogonally polarized two-color field is studied theoretically and experimentally. We find an increase of the HHG efficiency in comparison with the single-color case, which essentially depends on the plasma species and harmonic order. An increase of more than an order of magnitude is observed for silver plasma, whereas for gold and zinc it is lower; these results are reproduced in our calculations that include both propagation and microscopic response studies. We show that the widely used theoretical approach assuming the 1s ground state of the generating particle fails to reproduce the experimental results; the agreement is achieved in our theory using the actual quantum numbers of the outer electron of the generating particles. Moreover, our theoretical studies highlight the redistribution of the electronic density in the continuum wave packet as an important aspect of the HHG enhancement in the two-color orthogonally polarized fields with comparable intensities: in the single-color field the electronic trajectories with almost zero return energy are the most populated ones; in the two-color case the total field maximum can be shifted in time so that the trajectories with high return energies (in particular, the cut-off trajectory) become the most populated ones.

Intense circularly polarized attosecond pulse generation from relativistic laser plasmas using few-cycle laser pulses

We have investigated the polarization of attosecond light pulses generated from relativistic few-cycle laser pulse interaction with the surface of overdense plasmas using particle-in-cell simulation. Under suitable conditions, a desired polarization state of the generated attosecond pulse can be achieved by controlling the polarization of the incident laser. In particular, an elliptically polarized laser pulse of suitable ellipticity can generate an almost circularly polarized attosecond pulse without compromising the harmonic generation efficiency. The process is thus applicable as a new tabletop circularly-polarized XUV radiation source for probing attosecond phenomena with high temporal resolution.

Bicircular High-Harmonic Spectroscopy Reveals Dynamical Symmetries of Atoms and Molecules

We introduce bicircular high-harmonic spectroscopy as a new method to probe dynamical symmetries of atoms and molecules and their evolution in time. Our approach is based on combining a circularly polarized femtosecond fundamental field of frequency ω with its counterrotating second harmonic 2ω. We demonstrate the ability of bicircular high-harmonic spectroscopy to characterize the orbital angular momentum symmetry of atomic orbitals. We further show that breaking the threefold rotational symmetry of the generating medium-at the level of either the ensemble or that of a single molecule-results in the emission of the otherwise parity-forbidden frequencies 3qω  (q∈N), which provide a background-free probe of dynamical molecular symmetries.

Electron rescattering in a bicircular laser field

Above-threshold ionization of rare-gas atoms by a bicircular field with its two components counterrotating is theoretically investigated by means of the improved strong-field approximation. Both direct and rescattered electrons are considered and the quantum orbits that lead into a specific final state are calculated and depicted. The angle-dependent spectrum reflects the discrete rotational symmetry of the bicircular field. The backward-scattering contributions are very similar to those generated by a linearly polarized field; several such contributions are rotated one versus the other by the symmetry angle of the discrete rotational symmetry. The forward-scattering contributions dramatically affect the velocity map at comparatively low momenta. The direct-electron spectrum observes reflection symmetry about several symmetry axes determined by the field symmetry. This is broken by rescattering.

Tomographic reconstruction of circularly polarized high-harmonic fields: 3D attosecond metrology

Bright, circularly polarized, extreme ultraviolet (EUV) and soft x-ray high-harmonic beams can now be produced using counter-rotating circularly polarized driving laser fields. Although the resulting circularly polarized harmonics consist of relatively simple pairs of peaks in the spectral domain, in the time domain, the field is predicted to emerge as a complex series of rotating linearly polarized bursts, varying rapidly in amplitude, frequency, and polarization. We extend attosecond metrology techniques to circularly polarized light by simultaneously irradiating a copper surface with circularly polarized high-harmonic and linearly polarized infrared laser fields. The resulting temporal modulation of the photoelectron spectra carries essential phase information about the EUV field. Utilizing the polarization selectivity of the solid surface and by rotating the circularly polarized EUV field in space, we fully retrieve the amplitude and phase of the circularly polarized harmonics, allowing us to reconstruct one of the most complex coherent light fields produced to date.

Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism

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.

Generating Isolated Elliptically Polarized Attosecond Pulses Using Bichromatic Counterrotating Circularly Polarized Laser Fields

We theoretically demonstrate the possibility to generate both trains and isolated attosecond pulses with high ellipticity in a practical experimental setup. The scheme uses circularly polarized, counterrotating two-color driving pulses carried at the fundamental and its second harmonic. Using a model Ne atom, we numerically show that highly elliptic attosecond pulses are generated already at the single-atom level. Isolated pulses are produced by using few-cycle drivers with controlled time delay between them.

Generation of elliptically polarized attosecond pulse trains

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.

Generation of high-order harmonics with controllable elliptical polarization

We propose and numerically demonstrate a method for obtaining high-harmonic radiation with desirable elliptical polarization. Atoms are shined by a combination of a strong linearly-polarized laser field and an additional weak field, which is elliptically polarized in a plane perpendicular to the polarization direction of the strong field. The strong driver ionizes and recollides electrons with their parent ion, while the weak field perturbatively drives the electrons away from "head-on" collision. Upon recombination, new elliptically polarized harmonics with same ellipticity as the weak driver are emitted at efficiency which linearly depends on the intensity of the weak beam, but is independent of its elilipticity.