High-order Harmonic Generation Spectra Research Articles

Generation of isolated attosecond pulses in the far field by spatial filtering with an intense few-cycle mid-infrared laser

We report theoretical calculations of high-order harmonic generation (HHG) of Xe with the inclusion of multi-electron effects and macroscopic propagation of the fundamental and harmonic fields in an ionizing medium. By using the time-frequency analysis we show that the reshaping of the fundamental laser field is responsible for the continuum structure in the HHG spectra. We further suggest a method for obtaining an isolated attosecond pulse (IAP) by using a filter centered on axis to select the harmonics in the far field with different divergence. We also discuss the carrier-envelope-phase dependence of an IAP and the possibility to optimize the yield of the IAP. With the intense few-cycle mid-infrared lasers, this offers a possible method for generating isolated attosecond pulses.

High-order harmonic generation enhanced by XUV light

The combination of high-order harmonic generation (HHG) with resonant XUV excitation of a core electron into the transient valence vacancy that is created in the course of the HHG process is investigated theoretically. In this setup, the first electron performs a HHG three-step process, whereas the second electron Rabi flops between the core and the valence vacancy. The modified HHG spectrum due to recombination with the valence and the core is determined and analyzed for krypton on the 3d→4p resonance in the ion. We assume an 800 nm laser with an intensity of about 10(14) W/cm2 and XUV radiation from the Free Electron Laser in Hamburg (FLASH) with an intensity in the range 10(13)-10(16)W cm2. Our prediction opens perspectives for nonlinear XUV physics, attosecond x rays, and HHG-based spectroscopy involving core orbitals.

The effect of light-induced conical intersections on the alignment of diatomic molecules

It has already been shown that dressing of diatomic molecules by standing or by running linearly polarized laser waves gives rise to conical intersections (CIs). Due to the presence of such light-induced CIs (LICI), the rovibronic molecular motions are strongly coupled. Here we investigate an impact of LICI on molecular alignment. We show that the degree of alignment of a diatomic molecule as a function of time as obtained from the rigid rotor calculations (where the LICI is ignored) is very different from the exact calculations (where the LICI is taken into consideration). This claim is valid under an assumption that the initially prepared wavepacket has a significant amplitude at the nuclear geometry where the LICI is located. Moreover, our results clearly show that the LICI increases the electronic excitation of the diatom. In particular for weak laser fields this effect is very significant. Our prediction of the LICI’s effect on the alignment of diatomic molecules can be confirmed by a two-laser experiment. Here, the first laser should have weak intensity and its duration should be sufficiently long as to align the molecules. The wavelength should be tuned in such a way as to ensure that the initial nuclear wavepacket has a significant amplitude at the internuclear distance corresponding to the LICI. The second laser pulse should be strong to produce measurable high-order harmonics. The high-order harmonic generation spectra should be measured as a function of the time delay between the two laser pulses, and are predicted to give an experimental evidence for the LICI effect on the alignment.

Evidence of the2s2p(1P) doubly excited state in the harmonic generation spectrum of helium

By solving the two-active-electron time-dependent Schr\odinger equation in an intense, ultrashort laser field, we investigate evidence of electron correlations in the high-order harmonic generation spectrum of helium. As the frequency of the driving laser pulse varies from 4.6 to 6.6 eV, the 13th, 11th, and 9th harmonics sequentially become resonant with the transition between the ground state and the isolated $2s2p({}^{1}P)$ autoionizing state of helium, which dramatically enhances these harmonics and changes their profiles. When each of the 9th and 13th harmonics are in resonance with this autoionizing state, there is also a low-order multiphoton resonance with a Rydberg state, resulting in a particularly large enhancement of these harmonics relative to neighboring harmonics. When the 11th harmonic is in resonance with the $2s2p({}^{1}P)$ autoionizing state, the 13th harmonic is simultaneously in resonance with numerous higher-energy autoionizing states, resulting in a competition between these two harmonics for intensity. These results demonstrate that even electron correlations occurring over a narrow energy interval can have a significant effect on strong-field processes such as harmonic generation.

Analysis of effects of macroscopic propagation and multiple molecular orbitals on the minimum in high-order harmonic generation of aligned CO2

We report theoretical calculations on the effect of the multiple orbital contribution in high-order harmonic generation (HHG) from aligned CO$_2$ with inclusion of macroscopic propagation of harmonic fields in the medium. Our results show very good agreements with recent experiments for the dynamics of the minimum in HHG spectra as laser intensity or alignment angle changes. Calculations are carried out to check how the position of the minimum in HHG spectra depends on the degrees of molecular alignment, laser focusing conditions, and the effects of alignment-dependent ionization rates of the different molecular orbitals. These analyses help to explain why the minima observed in different experiments may vary.

High-order-harmonic generation in homonuclear and heteronuclear diatomic molecules: Exploration of multiple orbital contributions

We present a time-dependent density functional theory (TDDFT) approach with proper asymptotic long-range potential for nonperturbative treatment of high-order harmonic generation (HHG) of diatomic molecules with their molecular axis parallel to the laser field polarization. A time-dependent two-center generalized pseudospectral method in prolate spheroidal coordinate system is used for accurate and efficient treatment of the TDDFT equations in space and time. The theory is applied to a detailed all-electron nonperturbative investigation of HHG processes of homonuclear (N${}_{2}$ and ${\mathrm{F}}_{2}$) and heteronuclear (CO, BF, and HF) molecules in intense ultrashort laser pulses with the emphasis on the role of multiple molecular orbitals (MOs). The results reveal intriguing and substantially different nonlinear optical response behaviors for homonuclear and heteronuclear molecules. In particular, we found that the HHG spectrum for homonuclear molecules features a destructive interference of MO contributions while heteronuclear molecules show mostly constructive interference of orbital contributions.

High-order harmonic generation by atoms in a few-cycle laser pulse: Carrier-envelope phase and many-electron effects

Analytic formulas describing high-order harmonic generation (HHG) by atoms in a short laser pulse are obtained quantum mechanically in the tunneling limit. These results provide analytic expressions of the three-step HHG scenario, as well as of the returning electron wave packet, in a few-cycle pulse. Our results agree well with those of numerical solutions of the time-dependent Schr\"odinger equation for the H atom, while for Xe they predict many-electron atomic dynamics features in few-cycle HHG spectra and significant dependence of these features on the carrier-envelope phase of a laser pulse.

Theoretical exploration of laser-parameter effects on the generation of an isolated attosecond pulse from two-color high-order harmonic generation

In the present work, laser-parameter effects on the isolated attosecond pulse generation from two-color high-order harmonic generation (HHG) process are theoretically investigated by use of a wave-packet dynamics method. A 6-fs, 800-nm, $6\ifmmode\times\else\texttimes\fi{}{10}^{14}{\mathrm{W}/\mathrm{cm}}^{2}$, linearly polarized laser pulse serves as the fundamental driving pulse and parallel linearly polarized control pulses at 400 nm (second harmonic) and 1600 nm (half harmonic) are superimposed to create a two-color field. Of the two techniques, we demonstrate that using a half-harmonic control pulse with a large relative strength and zero phase shift relative to the fundamental pulse is a more promising way to generate the shortest attosecond pulses. As a consequence, an isolated 12-as pulse is obtained by Fourier transforming an ultrabroad xuv continuum of 300 eV in the HHG spectrum under half-harmonic control scheme when the relative strength $\sqrt{R}=0.6$ and relative phase $=0$.

High-order harmonics generation by few-cycle and multi-cycle femtosecond laser pulses

This paper investigates experimentally high-order harmonic generation (HHG) of neon gas with 5-fs and 25-fs driving laser pulses. It has been demonstrated that the cutoff energy of the harmonic extreme ultraviolet photons is extended to 131 eV and the HHG spectrum near the cutoff region becomes continuum as the driving laser pulse duration is 5 fs; whereas much lower cutoff photon energy and discrete harmonic spectrum near the cutoff region are presented as the laser pulse duration is 25 fs. The results can be explained by the fact that neutral atoms can be exposed to more intense laser field before they are depleted by ionization because of the extremely short rising time of the few-cycle pulse. The 5-fs driving laser pulse paves the way of generation of coherent x-ray in the water window and single attosecond pulse.

Minimum in the high-order harmonic generation spectrum from molecules: role of excited states

We model the process of high-order harmonic generation by solving the time-dependent Schrödinger equation for H+2 in the fixed nuclei approximation including full 3D electron motion for nonvanishing angles between the nuclear axis and the linear polarization of the driving pulse. We show that the coherent laser coupling induced between the 2Σ+g(1sσg) ground state and the first excited 2Σ+u(2pσu) state leads to two dominating amplitudes for the high-order harmonic generation that may interfere: amplitudes describing recombination back into the σg and σu states, respectively. These two amplitudes may interfere destructively or constructively. The effect of a destructive interference is very clear through the occurrence of a minimum in the high-order harmonic spectrum. We show cases where such a minimum in the spectrum is approximately at the position predicted by the simple two-centre interference formula.

Time-energy properties of an attosecond extreme ultra-violet pulse

The time-energy properties of high-order harmonic generation (HHG) are calculated for a linearly polarized 7-fs laser pulse with different carrier-envelope phases (CEPs). The quantum trajectory paths that contribute to an as (1 as=10−18 s) pulse in HHG are identified. The laser-duration dependence and the CEP dependence of HHG energy property are investigated. The study shows that an as extreme ultra-violet (XUV) pulse can be selected from HHG spectrum near cut-off energy with a bandpass optical filter. The theoretical prediction of the pulse duration is proportional to bandwidth. Analysis suggests that a measured narrowband as XUV pulse may consist of instantaneous shorter pulses each dependent on laser pulse duration, intensity, and CEP. These information can be used as references for producing, selecting, improving and manipulating (timing) as pulses.

Mapping Molecular Orbital Symmetry on High-Order Harmonic Generation Spectrum Using Two-Color Laser Fields

We have measured high-order harmonic generation spectra of D2, N2, and CO2 by mixing orthogonally polarized 800 and 400 nm laser fields. The intensity of the high-harmonic spectrum is modulated as we change the relative phase of the two pulses. For randomly orientated molecules, the phase of the intensity modulation depends on the symmetry of the molecular orbitals from which the high harmonics are emitted. This allows us to identify the symmetry of any orbital that contributes to high-harmonic generation, even without aligning the molecule. Our approach can be a route to imaging dynamical changes in three-dimensional molecular orbitals on a time scale as short as a few hundred attoseconds.

Calculation of high-order harmonic generation spectrum and Floquet resonance states using the dynamical symmetry

The Floquet resonance states of a periodically driven system are obtained using a special weighted power method which forces the dynamical symmetry into regular wave propagation methods. We demonstrate the utility of this method for the calculation of high harmonic generation spectra in rare gases. By using this method the resonance Floquet state which dominates the spectrum is obtained directly at a low computational cost. This may allow the calculation of resonance Floquet states in parameter regimes not accessible before. Implementation involves only slight adjustments to the standard wave propagation codes. The method's unique advantage is explained and demonstrated.

Multiconfiguration time-dependent Hartree approach for electron-nuclear correlation in strong laser fields

The multiconfiguration time-dependent Hartree approach is applied to study the electron-nuclear correlation in the dynamics of molecules subject to strong external laser fields, using the example of a model hydrogen molecular ion. The ground state of the system is well described by as few as two single-particle functions per degree of freedom. A significantly larger but moderate number of configurations is required to predict laser-induced fragmentation probabilities and high-order harmonic generation spectra accurately, showing that the correlation between electronic and nuclear degree of freedom is strongly increased by the presence of the laser field.

Analytic description of high-order harmonic generation by atoms in a two-color laser field

A closed-form analytic formula describing high-order harmonic generation (HHG) in a two-color field of frequencies $\ensuremath{\omega}$ and $2\ensuremath{\omega}$ is derived quantum mechanically in the low-frequency (tunneling) limit for an electron bound by a short-range potential and generalized to the case of an active electron in a neutral atom. The HHG rates are presented as a product of an electron wave packet describing the ionization of an active electron and its propagation in a laser field up to the recombination event and an atom-specific cross section of the electron's photorecombination. In contrast to the case of a monochromatic laser pulse [Frolov et al., Phys. Rev. Lett. 102, 243901 (2009)], the two-color wave packet involves the interference of two terms (involving the Airy function) that describe the emission of harmonics during the first and second half-cycles of the fundamental laser cycle and give rise to the two-plateau structures in the HHG spectra. For the case of the H atom, we show that our analytic results are in good agreement with those obtained from a numerical solution of the three-dimensional time-dependent Schr\odinger equation. The factorization formula is used for describing the dependence of HHG rates for inert gases on the relative phase and intensities of the $\ensuremath{\omega}$ and $2\ensuremath{\omega}$ components of a laser field. It is shown that atomic structure (including electron correlation) effects can modify substantially the two-color HHG spectra of inert gases.

Destructive and Constructive Interference of High-Order Harmonic Generation in Mixture of He and Ne Gases

Using the single-atom induced dipole moment under strong field approximation as a source, we suggest a model to simulate the macroscopic high-order harmonic generation (HHG) from the mixed gases (He and Ne) interacting with intense infrared laser by solving the three-dimensional Maxwell's equation of the harmonic field. Regular destructive interference (DI) and constructive interference (CI) are observed in the macroscopic HHG spectra when the gas jet is put at a good phase-matching position. A semiclassical model of short and long electron trajectories is applied to interpret the DI and CI of HHG qualitatively.

Reliability of soft-core approximations in theoretical studies of molecules in intense laser fields

The popular approach of softening the Coulomb singularity in models with reduced dimensionality leads to uncontrollable discrepancies with numerical results obtained using exact potentials. This is demonstrated for ionization and dissociation probabilities as well as for high-order harmonic generation spectra even of the simple hydrogen molecular ion in aligned geometry. Soft-core models perform better in full dimensions, thereby losing one of their advantages, namely the gain in computation time that is achieved by going to reduced dimensionality. However, the much simpler numerical implementation of soft-core potentials carries over to full dimensions. Moreover, a generalization of the soft-core potential given in previous work [M. Lein, P. P. Corso, J. P. Marangos, and P. L. Knight, Phys. Rev. A 67, 023819 (2003)] is proposed, leading to a tremendous improvement toward the exact results, which is very promising for future calculations without cylindrical symmetry.

Quantum path interferences of electron trajectories in two-center molecules

We report a new quantum path interference effect of electron trajectories in high-order harmonic generation (HHG) from two-center molecules, in which the interference minima are mainly located in the high-energy portion of HHG spectrum. The quantum calculations of the time-frequency analyses and the classical results of the electron trajectories demonstrate very good agreement and reveal that the positions of the interference minima are associated with the cutoff of various kinds of molecular electron trajectories. The interference fringes within a half optical cycle can be clearly seen in the time-frequency analysis spectrum. Moreover, the characteristics of both the HHG in frequency domain and the corresponding attosecond pulse generation in time domain permit tracing back the interference information of these electron trajectories. These interference phenomena offer new possibilities for getting insight into the attosecond electronic dynamics in molecules.

Effect of the relative phase between two-colour pump pulses on structure of harmonic spectra

We numerically calculate the high-order harmonic generation (HHG) power spectra from a one-dimensional model atom irradiated by linearly polarised 12 fs two-colour laser pulses composed of a fundamental pulse from Ti:sapphire laser and its second harmonic. It is found that a distinct double plateau structure appears when the relative phase of the two pulses is set as π/8, 2π/8 or 3π/8, and the double plateau structure disappears when the relative phase is set as 4π/8, 5π/8, 6π/8 or 7π/8. The relative-phase-dependent plateau structure is explained by the temporal profile of the synthesised electric fields as well as the semi-classical “three-step” model. Moreover, our numerical result shows that cut-off frequencies of the two-colour pulse HHG spectra can be exactly predicted by use of the semi-classical “three-step” model.

Effect of laser polarization direction on high order harmonic generation of oxygen molecule——A simulation via TDDFT

The high order harmonic generation (HHG) of O2 radiated by ultra-strong femto-second pulsed laser beam is simulated by the time dependent density functional theory (TDDFT) method with the general gradient approximation (GGA) of the exchange-correlation energy. The effect of different laser polarization directions on HHG is discussed. The results indicate that the HHG spectrum of O2 has the similar characteristic as the typical atom HHG spectrum. The HHG yields peaks when the molecules are aligned at about 45° from the polarization axis. Good agreements are found with the experiment.