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We experimentally and theoretically study high-order harmonic generation in zinc oxide crystals irradiated by mid-infrared lasers. The trajectories are mapped to the far field spatial distribution of harmonics. The divergence angles of on-axis and off-axis parts exhibit different dependences on the order of the harmonics. This observation can be theoretically reproduced by the coherent interference between the short and long trajectories with dephasing time longer than 0.5 optical cycle. Further, the relative contribution of the short and long trajectories is demonstrated to be accurately controlled by a one-color or two-color laser on the attosecond time scale. This work provides a reliable method to determine the electron dephasing time and demonstrates a versatile control of trajectory interference in the solid high-order harmonic generation.

Various interference effects are known to exist in the process of high harmonic generation (HHG) both at the single atom and macroscopic levels. In particular, the quantum path difference between the long and short trajectories of electron excursion causes the HHG yield to experience interference-based temporal and spectral modulations. In solids, due to additional phenomena such as multi-band superposition and crystal symmetry dependency, the HHG mechanism appears to be more complicated than in gaseous atoms in identifying accompanying interference phenomena. Here, we first report experimental data showing intensity-dependent spectral modulation and broadening of high harmonics observed from bulk sapphire. Then, by adopting theoretical simulation, the extraordinary observation is interpreted as a result of the quantum path interference between the long and short electron/hole trajectories. Specifically, the long trajectory undergoes an intensity-dependent redshift, which coherently combines with the short trajectory to exhibit spectral splitting in an anomalous way of inverse proportion to the driving laser intensity. This quantum interference may be extended to higher harmonics with increasing the laser intensity, underpinning the potential for precise control of the phase matching and modulation even in the extreme ultraviolet and soft X-ray regime. Further, this approach may act as a novel tool for probing arbitrary crystals so as to adjust the electron dynamics of higher harmonics for attosecond spectroscopy.

The mechanism of high harmonic generation (HHG) from condensed matter driven by ultrafast lasers is far from being understood due to the structural diversity and complexity. One of the long-standing open questions is the effect of random disorder in noncrystal HHG. In this work, we identify the critical role of correlations in disordered semiconductor systems and propose them as the criterion to classify the HHG processes. The HHG spectra can be classified into two different categories corresponding to different correlations: short-range correlation and long-range correlation (LRC). Thus correlations build up a solid relationship between the HHG emission and statistical characteristics of a disordered system. In particular, the HHG spectrum from the disordered system with LRC exhibits remarkable consistency with the spectrum from the periodic lattice, revealing that the LRC links HHG processes in disordered and periodic systems. Besides, the mobility edge induced by the long-range correlated disorder enables us to study the behavior of localized- and delocalized-state HHG in one system.

Intraband high-order harmonics in solids reflect the nonlinear motion of carriers in the energy bands and have important applications. In general, multiple energy bands contribute to the intraband harmonics. We reveal the interference mechanism of intraband harmonics between electrons and holes by using the k-resolved semiclassical intraband model. The model gives a quantitative relationship between the intraband radiation yield, the energy band dispersion, and the laser parameters. Based on a preexcitation scheme, we present a method to reconstruct energy bands by using the intraband harmonics. We simulate the intraband harmonics of photo-carriers polarized by a bias field. The variation of harmonic yields as a function of a delay time results from the k-resolved energy band dispersion and reflects the Bloch oscillation of photo-carriers driven by the bias field.

We report the control of tunnel ionization in lithium niobate (${\mathrm{LiNbO}}_{3}$) using phase-controlled two-color laser fields. Through a macroscopic observable of high contrast, we disclose the crucial contribution of the microscopic spontaneous polarization of the ferroelectric material to the ionization rate: as the relative two-color phase is varied, the ablated area of ${\mathrm{LiNbO}}_{3}$ is modulated by 35% when the laser and crystal polarization directions are parallel. Rotating the sample by ${180}^{\ensuremath{\circ}}$ around the laser propagation axis leads to an out-of-phase modulation. We use a two-band model to highlight the key contribution of the material's spontaneous polarization for the symmetry breaking of the ionization rate. Our results open new perspectives for the direct control of ionization dynamics in solids by tailoring the electric field of femtosecond laser pulses.

Physical review / B 101(21), 214311 (1-6) (2020). doi:10.1103/PhysRevB.101.214311, Laser micromachining techniques using femtosecond laser pulses are employed for the fabrication of a wide range of devices, from optical waveguides and fiber Bragg gratings to microfluidic systems. In this perspective, it is important to characterize how transparent materials respond to the high-intensity laser field in order to fully understand and control the strong-field ionization process at the core of such techniques. Here, we characterize the laser polarization dependence of $y$-cut and $x$-cut ferroelectric lithium niobate (LiNbO$_3$) upon ionization. Using linearly polarized 1800-nm femtosecond pulses, we perform single-shot laser-induced ablation measurements to obtain a macroscopic observable with a large contrast compared to all-optical techniques. In the tunnel ionization regime, the crystal orientation can be correlated with the structural symmetry/asymmetry of the material. This is revealed through a variation of the ablated area when the laser polarization is rotated with respect to the c axis of the crystal. It is further found that ablation is more pronounced when the laser polarization is oriented towards angular regions containing Nb-O bonds of the unit cell, identifying the main chemical bonds contributing to the ionization of the material. The experimental results are supported by numerical simulations based on a two-band model for LiNbO$_3$., Published by Inst., Woodbury, NY

High-order harmonic generation (HHG) in disordered systems is investigated by numerically solving the time-dependent Schrödinger equation. We reveal the vital role of Anderson localization in this situation. It allows us to develop a new and efficient sampling method for the large disordered system, which could significantly reduce the computation cost. It can be applied to both pure liquids and solutions. The structures and cutoffs of the harmonic spectrum are simulated with different field strengths and statistical parameters in the solution. Compared with HHG in pure liquids, the harmonic signal changes little in dilute solutions. HHG from different solutes and concentrations is simulated to confirm this point. This should be the first theoretical study on HHG in solutions. Our results shed light on the way to investigate the ultrafast processes in the solution.

Molecules constituted by different isotopes are different in vibrational modes, making it possible to elucidate the mechanism of a chemical reaction via the kinetic isotope effect. However, the real-time observation of the vibrational motion of isotopic nuclei in molecules is still challenging due to its ultrashort time scale. Here we demonstrate a method to monitor the nuclear vibration of isotopic molecules with the frequency modulation of high-order harmonic generation (HHG) during the laser-molecule interaction. In the proof-of-principle experiment, we report a red shift in HHG from H2 and D2. The red shift is ascribed to dominant HHG from the stretched isotopic molecules at the trailing edge of the laser pulse. By utilizing the observed frequency shift, the laser-driven nuclear vibrations of H2 and D2 are retrieved. These findings pave an accessible route toward monitoring the ultrafast nuclear dynamics and even tracing a chemical reaction in real time., Previous studies on high harmonic generation from molecules have been used to identify the spectral properties and orbital contributions. Here the authors measure the isotopic effects in the energy shift of the HHG spectra caused by the nuclear motion of the molecules.

High harmonic generation (HHG) in solids provides us compact and coherent extreme ultraviolet sources. However, the low yield of HHG is the bottleneck restricting its wide applications. We propose a pump-probe scheme to selectively enhance the yield of HHG. Our simulations illustrate abnormal harmonic yield dependence on the intensity of driving or preexcitation pulses. A multielectron interference model is proposed to reveal the mechanism of the extreme nonlinear optical phenomena. We find that the $k$-resolved intraband harmonic demonstrates different symmetry, phase, and amplitude, which can be observed by using a resonant preexcitation pulse. The position of the interference minimum obtained by solving the semiconductor Bloch equations agrees well with that obtained by our $k$-resolved semiclassical model. This interference model can help us probe the $k$-resolved band structure and optimize the ultrafast electron-hole dynamics for solid harmonic processes.

The time-dependent Schrödinger equation (TDSE) is usually treated in the real space in the textbook. However, it makes the numerical simulations of strong-field processes difficult due to the wide dispersion and fast oscillation of the electron wave packets under the interaction of intense laser fields. Here we demonstrate that the TDSE can be efficiently solved in the momentum space. The high-order harmonic generation and above-threshold ionization spectra obtained by numerical solutions of TDSE in momentum space agree well with previous studies in real space, but significantly reducing the computation cost.

We introduce a concise methodology to detect the parity of atomic and molecular orbitals based on photoelectron holography, which is more general than the existing schemes. It fully accounts for the Coulomb distortions of electron trajectories, does not require sculpted fields to retrieve phase information and, in principle, is applicable to a broad range of electron momenta. By comparatively measuring the differential photoelectron spectra from strong-field ionization of ${\mathrm{N}}_{2}$ molecules and their companion atoms of Ar, some photoelectron holography patterns are found to be dephased for both targets. This is well reproduced by the full-dimensional time-dependent Schr\"odinger equation and the Coulomb quantum-orbit strong-field approximation (CQSFA) simulation. Using the CQSFA, we trace back our observations to different parities of the $3p$ orbital of Ar and the highest-occupied molecular orbital of ${\mathrm{N}}_{2}$ via interfering Coulomb-distorted quantum orbits carrying different initial phases. This method could in principle be used to extract bound-state phases from any holographic structure, with a wide range of potential applications in recollision physics and spectroscopy.

Intraband high-order harmonics in solids reflect the nonlinear motion of carriers in the energy bands and have important applications. In general, multiple energy bands contribute to the intraband harmonics. We reveal the interference mechanism of intraband harmonics between electrons and holes by using the k-resolved semiclassical intraband model. The model gives a quantitative relationship between the intraband radiation yield, the energy band dispersion, and the laser parameters. Based on a preexcitation scheme, we present a method to reconstruct energy bands by using the intraband harmonics. We simulate the intraband harmonics of photo-carriers polarized by a bias field. The variation of harmonic yields as a function of a delay time results from the k-resolved energy band dispersion and reflects the Bloch oscillation of photo-carriers driven by the bias field.

We studied the high-order harmonic generation (HHG) from 2D solid materials in circularly and bichromatic circularly polarized laser fields numerically by simulating the dynamics of single-active-electron processes in 2D periodic potentials. Contrary to the absence of HHG in the atomic case, circular HHGs below the bandgap with different helicities are produced from intraband transitions in solids with C4 symmetry driven by circularly polarized lasers. Harmonics above the bandgap are elliptically polarized due to the interband transitions. High-order elliptically polarized harmonics can be generated efficiently by both co-rotating and counter-rotating bicircular mid-infrared lasers. The cutoff energy, ellipticity, phase, and intensity of the harmonics can be tuned by the control of the relative phase difference between the 1ω and 2ω fields in bicircularly polarized lasers, which can be utilized as an ultrafast optical tool to image the structure of solids.

We study numerically the Bloch electron wave-packet dynamics in periodic potentials to simulate laser-solid interactions. We introduce an alternative perspective in the coordinate space combined with the motion of the Bloch electron wave packets moving at group and phase velocities under the laser fields. This model interprets the origins of the two contributions (intra- and interband transitions) in the high-order harmonic generation (HHG) processes by investigating the local and global behaviours of the wave packets. It also elucidates the underlying physical picture of the HHG intensity enhancement by means of carrier-envelope phase, chirp, and inhomogeneous fields. It provides a deep insight into the emission of high-order harmonics from solids. This model is instructive for experimental measurements and provides an alternative avenue to distinguish mechanisms of the HHG from solids in different laser fields.

We investigated the high-order harmonic generation (HHG) process of diatomic molecular ion $\mathrm{H}_2^+$ in non-Born-Oppenheimer approximations. The corresponding three-dimensional time-dependent Schrodinger equation is solved with arbitrary alignment angles. It is found that the nuclear motion can lead to spectral modulation of HHG. Redshifts are unique in molecular HHG which decrease with the increase of alignment angles of the molecules and are sensitive to the initial vibrational states. It can be used to extract the ultrafast electron-nuclear dynamics and image molecular structure.

We introduce a quasi-classical model in the k space combined with the energy band structure of solids to understand the mechanisms of high-order harmonic generation (HHG) process occurring in a subcycle timescale. This model interprets the multiple plateau structure in HHG spectra well and the linear dependence of cutoff energies on the amplitude of vector potential A0 of the laser fields. It also predicts the emission time of HHG, which agrees well with the results by solving the time-dependent Schrodinger equation (TDSE). It provides a scheme to reconstruct the energy dispersion relations in Brillouin zone and to control the trajectories of HHG by varying the shape of laser pulses. This model is instructive for experimental measurements.

High-order harmonic generation (HHG) is a nonlinear nonperturbative process in ultrashort intense laser-matter interaction. It is the main source of coherent attosecond (1 as = 10−18 s) laser pulses to investigate ultrafast electron dynamics. HHG has become an important table-top source covering a spectral range from infrared to extreme ultraviolet (XUV). One way to extend the cutoff energy of HHG is to increase the intensity of the laser pulses. A consequence of HHG in such intense short laser fields is the characteristic nonadiabatic red and blue shifts of the spectrum, which are reviewed in the present work. An example of this nonperturbative light-matter interaction is presented for the one-electron nonsymmetric molecular ion HeH2+, as molecular systems allow for the study of the laser-molecule orientation dependence of such new effects including a four-step model of MHOHG (Molecular High-order Harmonic Generation).

We theoretically study high-order harmonic generation (HHG) from solid-phase systems in spatially inhomogeneous strong laser fields originated by resonant plasmons within a metallic nanostructure. The intensity of the second plateau in HHG may be enhanced by two to three orders and be comparable with the intensity of the first plateau. This is due to bigger transition probabilities to higher conduction bands. It provides us with a practical way to increase the conversion efficiency of HHG with laser intensity below the damage threshold. It presents a promising way to triple the range of HHG spectra in experimental measurements. It also allows us to generate intense isolated attosecond pulses from solids driven by few-cycle laser fields.

We use differential holography to overcome the forward scattering problem in strong-field photoelectron holography. Our differential holograms of ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ molecules exhibit a fishbonelike structure, which arises from the backscattered part of the recolliding photoelectron wave packet. We demonstrate that the backscattering hologram can resolve the different nuclear dynamics between ${\mathrm{H}}_{2}$ and ${\mathrm{D}}_{2}$ with subangstrom spatial and subcycle temporal resolution. In addition, we show that attosecond electron dynamics can be resolved. These results open a new avenue for ultrafast studies of molecular dynamics in small molecules.

We investigate the high harmonic generation (HHG) from solids by simulating the dynamics of a single active electron in periodic potentials. The corresponding time-dependent Schr\"odinger equations (TDSE) are solved numerically by using B-spline basis sets in coordinate space. The energy band structure and wave vectors can be directly retrived from the eigenfunctions. The harmonic spectra obtained agree well with the results simulated by TDSE in $k$ space using Bloch states and show a two-plateau structure. Both of the cutoff energies of the two plateaus in the harmonic spectrum scale linearly with the field strength. We also study HHG driven by intense few-cycle laser pulses and find that the cutoff energy of the harmonic spectrum is as sensitive to the changes of the carrier envelope phase, as to HHG from gas samples, which suggests recollision pictures in HHG as found by recent experiments (Nature {\bf 522}, 462 (2015)).

We studied the multi-plateau high-order harmonic generation (HHG) from solids numerically. It is found that the HHG spectra in the second and higher plateaus are redshifted in short laser pulses due to the nonadiabatic effect. The corresponding FWHMs also increase as a function of the harmonic order, suggesting the step-by-step excitation of higher conduction bands in the HHG process. Although the system is symmetric in the coordinate space, even-order harmonics are present. It is due to the fact that the symmetry of electron motions and the population in the higher conduction bands is broken in the k space and time domain based on the indirect step-by-step excitation model. Our numerical results are in good agreement with recent experimental measurements of Ndabashimiye et al. [Nature 534, 520 (2016)].