High-order Harmonic Radiation Research Articles

Determining the electronic states that contribute most to solid-state high-order harmonic radiation

Determining the electronic states that contribute most to solid-state high-order harmonic radiation

Critical Laser Intensity of Phase-Matched High-Order Harmonic Generation in Noble Gases

The efficient generation of high-order harmonic radiation has been a challenging task since the early days of strong-field physics. An essential requirement to achieve efficient high-order harmonic generation inside a gas medium is the phase matching of the high-order harmonic radiation and the incident laser pulse. The dominant contribution to the wave–vector mismatch Δk is associated with the ionization probability of the medium. In this work, we derive two analytical formulas to calculate the critical intensity of a general linearly polarized laser pulse that obey the phase-matching condition Δk=0. The analytic formulas are valid in the tunneling regime (ADK model) and the regime of the tunnel and multi-photon ionization (PPT model), respectively. We compare our results to numerical computations and discuss the scaling of the critical intensity depending on the pulse duration and the wavelength of a realistic incident laser pulse. The analytical approach demonstrated in this work is highly accurate and can compete with the existing numerical computational methods by an error of less than 1% and a decrease in the computation time of approximately 4 to 6 orders of magnitude. This enables complex theoretical studies of the efficiency scaling in HHG or to consider the effects of ground state depletion efficiently.

High-order harmonic spectroscopy in an ionized high-density target

We use high-order harmonic spectroscopy to study ionization dynamics in a macroscopic target with tunable density, spanning over six orders of magnitude. In an in situ pump-probe experiment, the target is prepared at different densities with varying degrees of laser-induced ionization. High-order harmonic radiation is generated in the pre-ionized target, and a steepening in the decrease of the harmonic yield is observed for increasing pre-ionization, allowing not only to identify the contributing quantum paths during high-order harmonic generation but also in determining the amount of ionization within the target. The measurements allow probing of ionization dynamics in laser-induced plasma with high spatio-temporal resolution and are specifically of interest for the optimization of the harmonic generation process in high-density targets with number densities of up to 1022 cm−3.

Polarization dependence of atomic high-order harmonic generation: Description using a discrete basis

The optical generation of high-order harmonics is known to have a strong polarization dependence: In contrast to linearly polarized excitation, circularly polarized light induces practically no harmonics. In the current paper we focus on atomic targets, the case when a well-established physical picture explains the effect: For circular polarization, the photoionized electrons never return to their parent nuclei, and the energy they gained while being accelerated by the field is not transferred into high-order harmonic radiation. This is essentially a picture that is based on real-space electron trajectories (or the dynamics of the wave functions). Here, we provide an alternative description that uses the discrete Sturmian basis, and points out how quantum mechanical interference effects and selection rules can explain the polarization dependence of the process. This emphasizes the importance of space-time symmetries during the process of high-order harmonic generation.

Generation of elliptically polarized high-order harmonic radiation with bi-elliptical two-color laser beams

High-harmonic generation (HHG) from an atomic gas target provides a valuable tool to generate high-energy radiation on the tabletop. The polarization properties of this radiation can be manipulated if the HHG process is driven by a bi-elliptical orthogonal two-color laser beam (BEOTC beam). Here we theoretically consider this setup within the so-called strong-field approximation and a quantum-orbit approach. We demonstrate that this approach allows the complete description of the geometry of the outgoing elliptically polarized harmonic radiation. In particular, we identify and classify the relevant quantum orbits that contribute to the process and analyze the harmonic spectrum, ellipticities, and offset angles in detail for $\ensuremath{\omega}\text{\ensuremath{-}}2\ensuremath{\omega}$ BEOTC beams with a fundamental wavelength of $\ensuremath{\lambda}=800\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. For these specific beam parameters, we find that the ellipticity as well as the offset angle of the polarization ellipse with respect to the BEOTC driving beam changes rapidly for consecutive harmonic orders. We show that this oscillatory behavior of the harmonic ellipticity can be explained via the superposition of the dipole radiation emitted from the target atom during different time intervals within one optical cycle of the BEOTC beam. This understanding of the HHG process driven by BEOTC beams will facilitate the generation of highly energetic radiation with complex polarization properties on the tabletop.

Ultrafast sub-nanometer matter-wave temporal Talbot effect

The coherent manipulation of the electron wavefunction at the atomic spatial and temporal scales is the fundamental breakthrough underlying far-reaching ultrafast phenomena as high-order harmonic radiation and attosecond pulse generation. In this work, we present a next step in the coherent control of matter waves by translating the concept of Talbot interferometry to the subnanomenter–femtosecond realm. We study the high-harmonic emission from a periodic system irradiated by an intense mid-infrared laser beam at grazing incidence. Our calculations show that Bloch electrons, once ionized, follow a sequence of ultrafast (femtosecond) revivals associated with the temporal Talbot effect. We demonstrate that these revivals leave a distinct signature in the high-frequency harmonic spectrum, in the form of structures extending beyond the main spectral cutoff, toward the x-rays. The reinterpretation of the process of high-order harmonic generation as the temporal realization of a Talbot–Lau interferometer suggests high-harmonic spectroscopy as an appropriate scheme to develop subnanometer ultrafast Talbot interferometry.

Tunable non-integer high-harmonic generation in a topological insulator.

When intense lightwaves accelerate electrons through a solid, the emerging high-order harmonic (HH) radiation offers key insights into the material1-11. Sub-optical-cycle dynamics-such as dynamical Bloch oscillations2-5, quasiparticle collisions6,12, valley pseudospin switching13 and heating of Dirac gases10-leave fingerprints in the HH spectra of conventional solids. Topologically non-trivial matter14,15 with invariants that are robust against imperfections has been predicted to support unconventional HH generation16-20. Here we experimentally demonstrate HH generation in a three-dimensional topological insulator-bismuth telluride. The frequency of the terahertz driving field sharply discriminates between HH generation from the bulk and from the topological surface, where the unique combination of long scattering times owing to spin-momentum locking17 and the quasi-relativistic dispersion enables unusually efficient HH generation. Intriguingly, all observed orders can be continuously shifted to arbitrary non-integer multiples of the driving frequency by varying the carrier-envelope phase of the driving field-in line with quantum theory. The anomalous Berry curvature warranted by the non-trivial topology enforces meandering ballistic trajectories of the Dirac fermions, causing a hallmark polarization pattern of the HH emission. Our study provides a platform to explore topology and relativistic quantum physics in strong-field control, and could lead to non-dissipative topological electronics at infrared frequencies.

Selective high harmonics generated from a carbon nanotube

Selective high-order harmonic radiation can be generated when circularly polarized driving fields interact with single-walled carbon nanotubes. We obtain a gauge-covariant Hamiltonian for electrons confined to a nanotube surface in the presence of a circularly polarized electromagnetic field. Regularized delta-functions are used to approximate the $${\pi }$$ -bonds in the nanotube unit cells, and the electron dynamics, in the presence of the field, is analyzed. The high harmonic radiation occurs for incident field intensities high enough to induce significant nonlinearities and chaos in the electron dynamics. Both (5, 5) and (10, 10) armchair nanotubes are considered and the electron quantum dynamics is modeled using Floquet–Bloch theory. The behavior of the quasienergy spectrum, Floquet states, and the electron current are described for varying intensities of the incident field. We show that, for incident radiation with frequency $${\omega }$$ , the emitted radiation for a (p, p) armchair nanotube will be $$(2p\pm 1){\omega }$$ , $$(4p\pm 1){\omega }$$ , etc.

A multi-terawatt two-color beam for high-power field-controlled nonlinear optics.

Two-color laser beams are instrumental in light-field control and enhancement of high-order harmonic, spectral supercontinuum, and terahertz radiation generated in gases, plasmas, and solids. We demonstrate a multi-terawatt two-color beam produced using a relativistic plasma mirror, with 110 mJ at 800 nm and 30 mJ at 400 nm. Both color components have high spatial quality and can be simultaneously focused, provided that the plasma mirror lies within a Rayleigh range of the driving fundamental beam. Favorable scaling of second-harmonic generation by plasma mirrors at relativistic intensities suggests them as an excellent tool for multi-color waveform synthesis beyond the petawatt level.

From strain engineering of material to photon radiation: a new perspective for investigating crystal microstructure andelectronic structure

Many nonlinear optical scenarios, including the typical high-order harmonic radiation, can be initiated in crystals exposed to ultrashort strong lasers. According to the recent theoretical calculation, the high-order harmonic radiation can be modulated by exerting the strain on a monolayer aluminum nitride. Since the high-order harmonic spectra carry the structural information of the target, inversely, the radiated spectra can be used to extract the microstructure of the target and even the ultrafast electron dynamics.

Above-threshold multiphoton photoemission from noble metal surfaces

Exciting solids with intense femtosecond laser pulses prompts electrons of the interrogated material to respond in a highly nonlinear manner, as is evident in the emission of high-order harmonic radiation and photoelectrons with kinetic energies well above that of the driving photons. Such high-field interactions can be resolved, for example, in above-threshold multiphoton photoemission (ATP) spectroscopy. In this work, we interrogate the nonlinear photoelectric responses of the pristine copper, silver, and gold noble metal surfaces in (111) and (100) crystal orientations in the perturbative regime. Using multiphoton photoemission spectroscopy (mPP) excited by finely tuned optical fields, we characterize enhancement of the mPP and ATP yields from (111) surfaces in selected ${k}_{||}$-momentum ranges when the occupied Shockley surface (SS) states are (near-)resonantly coupled by multiphoton transitions to image potential (IP) intermediate states in the excitation process. The ATP signal from the IP states of (111) surfaces is largely defined by their formation through polarization of SS electrons; this observation is contrasted with ATP experiments from the Ag(100) surface, for which the SS becomes an unoccupied resonance and the IP states can only be excited from bands with significantly more bulk character. In addition, based on the optical power and nonlinear order-dependent mPP spectra, we provide evidence for ATP being a one-step, rather than a sequential process, as previously postulated.

Chirp-controlled high-harmonic and attosecond-pulse generation via coherent-wake plasma emission driven by mid-infrared laser pulses.

Coherent-wake plasma emission induced by ultrashort mid-infrared laser pulses on a solid target is shown to give rise to high-brightness, high-order harmonic radiation, offering a promising source of attosecond pulses and a probe for ultrafast subrelativistic plasma dynamics. With 80-fs, 0.2-TW pulses of 3.9-μm radiation used as a driver, optical harmonics up to the 34th order are detected, with their spectra stretching from the mid-infrared region to the extreme ultraviolet region. The harmonic spectrum is found to be highly sensitive to the chirp of the driver. Particle-in-cell analysis of this effect suggests, in agreement with the generic scenario of coherent-wake emission, that optical harmonics are radiated as trains of extremely short, attosecond ultraviolet pulses with a pulse-to-pulse interval varying over the pulse train. A positive chirp of the driver pulse can partially compensate for this variation in the interpulse separation, allowing harmonics of the highest orders to be generated in the plasma emission spectrum.

Crystal-momentum-resolved contributions to high-order harmonic generation in solids

We analytically and numerically investigate the emission of high-order harmonic radiation from model solids by intense few-cycle midinfrared laser pulses. In single-active-electron approximation, we expand the active electron's wave function in a basis of adiabatic Houston states and describe the solid's electronic band structure in terms of an adjustable Kronig-Penney model potential. For high-order harmonic generation (HHG) from MgO crystals, we examine spectra from two-band and converged multiband numerical calculations. We discuss the characteristics of intra- and interband contributions to the HHG spectrum for computations including initial crystal momenta either from the $\mathrm{\ensuremath{\Gamma}}$ point at the center of the first Brioullin zone (BZ) only or from the entire first BZ. For sufficiently high intensities of the driving laser field, we find relevant contributions to HHG from the entire first BZ. Based on numerically calculated spectra, we scrutinize the cutoff harmonic orders as a function of the laser peak intensity and find good qualitative agreement with our analytical saddle-point-approximation predictions and published theoretical data.

Harmonic Generation in Planar Undulators in Single-Pass Free Electron Lasers

The paper presents a theoretical study of the harmonic power evolution in a single-pass free-electron laser (FEL) and a comparison with some of the experimental results. The phenomenological approach which considers the main FEL parameters (current density, the Lorentz factor, electron energy spread, and the beam geometry) provides a description of any undulator. With due regard to the real laser beam parameters, the effect from high-order harmonic radiation and the beam deviation from the undulator axis on the FEL irradiation is investigated. The Bessel coefficients are calculated for the undulator in the presence of the second periodic field. The phenomenological approach provides the free-electron laser simulation experiments at Sorgente Pulsata Auto-amplificata di Radiazione Coerente (SPARC), SPring-8 Angstrom Compact Free-Electron Laser (SACLA) and Linac Coherent Light Source (LCLS). These studies include the high-harmonic generation, a comparison of the emission power evolution with the experimentally obtained data on the harmonic generation including even and odd high-order harmonics, gain and saturation lengths and the respective power. A possible effect from the third harmonic of the planar undulator magnetic field is studied in relation to the FEL irradiation. It is shown that this effect is insignificant even at the magnetic-field harmonic amplitude equaling ~1/10 of the amplitude of the ideal magnetic field in the periodic undulator. It is found that the second-harmonic generation in FEL lasing is rather weak, that is in good agreement with the measurement results as well as the behavior of the high-order harmonics.

Temporal and spectral fingerprints of ultrafast all-coherent spin switching.

Future information technology demands ever-faster, low-loss quantum control. Intense light fields have facilitated milestones along this way, including the induction of novel states of matter1-3, ballistic acceleration of electrons4-7 and coherent flipping of the valley pseudospin8. These dynamics leave unique 'fingerprints', such as characteristic bandgaps or high-order harmonic radiation. The fastest and least dissipative way of switching the technologically most important quantum attribute-the spin-between two states separated by a potential barrier is to trigger an all-coherent precession. Experimental and theoretical studies withpicosecond electric and magnetic fields have suggested this possibility9-11, yet observing the actual spindynamics has remained out of reach. Here we show that terahertz electromagnetic pulses allow coherent steering of spins over a potential barrier, and we report the corresponding temporal and spectral fingerprints. This goal is achieved by coupling spins in antiferromagnetic TmFeO3 (thulium orthoferrite) with the locally enhanced terahertz electric field of custom-tailored antennas. Within their duration of one picosecond, the intense terahertz pulses abruptly change the magnetic anisotropy and trigger a large-amplitude ballistic spin motion. A characteristic phase flip, an asymmetric splitting of the collective spinresonance and a long-lived offset of the Faraday signal are hallmarks of coherent spin switching into adjacent potential minima, in agreement with numerical simulations. The switchable states can be selected by an external magnetic bias. The low dissipation and the antenna's subwavelength spatial definition could facilitate scalable spin devices operating at terahertz rates.

High order harmonic radiation source for multicolor extreme ultraviolet radiography of carbon plumes

Multicolor radiography is used for the characterization of atoms, ions, and molecules of carbon plasma plumes formed by focusing a 200 ps Ti:sapphire laser pulse on a solid graphite target. The radiography of the plume was carried out using a high order harmonic generation based radiation source consisting of the 11th (λ ≈ 72.7 nm) to 21st (λ ≈ 38 nm) odd harmonic orders of the Ti:sapphire laser pulse. The density profile of CI, CII, and carbon dimer molecule (C2) is estimated from the 2D-transmission profile of the harmonics, recorded after passing through the carbon plume. The peak densities of CI, CII, and C2 at a 50 ns delay are estimated to be 8 × 1024 m−3, 4 × 1024 m−3, and 3.5 × 1023 m−3 at distances of 150 μm, 170 μm, and 120 μm away from the target surface, respectively. The expansion speed of the plasma plume front is estimated to be 2 × 104 m/s and the speed of the C2 molecule to be 4 × 103 m/s at a laser intensity of 1011 W cm−2. The present study demonstrates ultrafast multicolor radiography as a simple and versatile tool for a simultaneous estimation of the density profile of neutral atoms, ions, and molecular species of the plasma plume.

Spectral interferometry with waveform-dependent relativistic high-order harmonics from plasma surfaces

The interaction of ultra-intense laser pulses with matter opened the way to generate the shortest light pulses available nowadays in the attosecond regime. Ionized solid surfaces, also called plasma mirrors, are promising tools to enhance the potential of attosecond sources in terms of photon energy, photon number and duration especially at relativistic laser intensities. Although the production of isolated attosecond pulses and the understanding of the underlying interactions represent a fundamental step towards the realization of such sources, these are challenging and have not yet been demonstrated. Here, we present laser-waveform-dependent high-order harmonic radiation in the extreme ultraviolet spectral range supporting well-isolated attosecond pulses, and utilize spectral interferometry to understand its relativistic generation mechanism. This unique interpretation of the measured spectra provides access to unrevealed temporal and spatial properties such as spectral phase difference between attosecond pulses and field-driven plasma surface motion during the process.

Control of the helicity of high-order harmonic radiation using bichromatic circularly polarized laser fields

High-harmonic generation in two-colour ($\omega-2\omega$) counter-rotating circularly polarised laser fields opens the path to generate isolated attosecond pulses and attosecond pulse trains with controlled ellipticity. The generated harmonics have alternating helicity, and the ellipticity of the generated attosecond pulse depends sensitively on the relative intensities of two adjacent, counter-rotating harmonic lines. For the $s$-type ground state, such as in Helium, the successive harmonics have nearly equal amplitude, yielding isolated attosecond pulses and attosecond pulse trains with linear polarisation, rotated by 120$^{{\circ}}$ from pulse to pulse. In this work, we suggest a solution to overcome the limitation associated with the $s$-type ground state. It is based on modifying the three propensity rules associated with the three steps of the harmonic generation process: ionisation, propagation, and recombination. We control the first step by seeding high harmonic generation with XUV light tuned well below the ionisation threshold, which generates virtual excitations with the angular momentum co-rotating with the $\omega$-field. We control the propagation step by increasing the intensity of the $\omega$-field relative to the $2\omega$-field, further enhancing the chance of the $\omega$-field being absorbed versus the $2\omega$-field, thus favouring the emission co-rotating with the seed and the $\omega-$field. We demonstrate our proposed control scheme using Helium atom as a target and solving time-dependent Schr{\"o}dinger equation in two and three-dimensions.

Subpetahertz helicity-modulated high-order harmonic radiation

We demonstrate a scheme to produce coherent extreme ultraviolet and soft x-ray radiation with subpetahertz temporal helicity modulation, based on the high harmonic generation from current-carrying orbitals driven by intense linearly polarized laser fields. It is found that the electronic angular momentum of the orbitals oscillates periodically in femtosecond scale in the driving field. This so-far undescribed phenomenon is qualitatively interpreted and attributed to the laser-induced energy shift of orbitals. Consequently, the polarization of the harmonic radiation switches periodically in the temporal domain between left and right elliptical polarizations, and the frequency of the helicity modulation reaches subpetahertz. By varying the intensity of the laser field, the modulation frequency can be continuously controlled. This light source will serve as a potential tool to detect and manipulate the ultrafast dynamics in magnetic materials and chiral media.

Coherent buildup of high-order harmonic radiation: The classical perspective

We present a classical model for high harmonic generation during the propagation of an intense laser pulse through an atomic gas. Numerical simulations of the model show excellent quantitative agreement with the corresponding quantum model for the blueshift and intensity reduction of the propagating laser pulse over experimentally realistic propagation distances. We observe a significant extension of the high-harmonic cutoff due to propagation effects. Phase-space analysis of our classical model uncovers the mechanism behind this extension.