Molecular High-order Harmonic Generation Research Articles

Multichannel Molecular High-Order Harmonic Generation from Asymmetric Diatomic Molecules

Multichannel molecular high-order harmonic generation (MHOHG) from a single electron asymmetric molecular ion HeH2+ is investigated numerically. It is found that considerable resonant excitation occurs by laser induced electron transfer (LIET) to neighboring ions and multiple frequency (fractional-order) harmonics are observed from the excited states shifted by some energy Δ from the main Nω energy harmonics. A time series analysis is used to confirm this MHOHG channel which is created by initial ionization from the excited state prepared by LIET and recombination to the neighboring ion at specific field phases, resulting in interference between recombination pathways from ground and excited states.

High-order elliptically polarized harmonic generation in extended molecules with ultrashort intense bichromatic circularly polarized laser pulses

Numerical solutions of the time-dependent Schr\odinger equation (TDSE) for a two-dimensional ${\mathrm{H}}_{2}{}^{+}$ molecule excited by a bichromatic ultrashort intense circularly polarized laser pulse with frequencies ${\ensuremath{\omega}}_{0}$ and $2{\ensuremath{\omega}}_{0}$ and relative carrier envelope phase $\ensuremath{\phi}$ are used to explore the generation of high-order elliptically polarized harmonics as a function of internuclear distance $R$. Optimal values of $\ensuremath{\phi}$ and $R$ for efficient and maximum molecular high-order harmonic generation (MHOHG) are determined from a classical model of collision with neighboring ions and confirmed from the TDSE nonperturbative simulations. Maximum elliptically polarized harmonic energies of ${I}_{p}+13.5{U}_{p}$ are found, where ${I}_{p}$ is the ionization potential and ${U}_{p}={I}_{0}/4{m}_{e}{\ensuremath{\omega}}_{0}^{2}$ is the ponderomotive energy at intensity ${I}_{0}$ and frequency ${\ensuremath{\omega}}_{0}$. The polarization properties of MHOHG, phase difference $\ensuremath{\delta}$, ellipticity $\ensuremath{\varepsilon}$, and orientation angle $\ensuremath{\varphi}$ are presented as well. The high efficiency of the proposed MHOHG scheme should be useful for production of elliptically polarized attosecond extreme ultraviolet pulses.

Revealing molecular structure and dynamics through high-order harmonic generation driven by mid-IR fields

High-order harmonic generation (HHG) from molecules produces spectra that are modulated by interferences that encode both the static structure and the electron dynamics initiated by interaction with the laser field. Using a midinfrared (mid-IR) laser at $1300$ nm, we are able to study the region of the harmonic spectrum containing such interferences in ${\mathrm{CO}}_{2}$ over a wide range of intensities. This allows for isolation and characterization of interference minima arising due to subcycle electronic dynamics triggered by the laser field, which had previously been identified but not systematically separated. Our experimental and theoretical results demonstrate important steps toward combining attosecond temporal and angstrom-scale spatial resolution in molecular HHG imaging.

Molecular high-order harmonic generation with more than one active orbital: Quantum interference effects

We investigate quantum interference effects in high-order harmonic generation in ${\mathrm{N}}_{2}$ and ${{\mathrm{N}}_{2}}^{+}$ beyond the single-active-orbital approximation, with particular emphasis on the role of $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ orbitals in the overall spectra. In the former case, we consider a simplified multielectron wave function which incorporates the $3{\ensuremath{\sigma}}_{g}$ and $1{\ensuremath{\pi}}_{u}$ orbitals, and, in the latter, we assume that the optically active electron is in a coherent superposition of the $3{\ensuremath{\sigma}}_{g}$ and $1{\ensuremath{\pi}}_{g}^{*}$ one-electron states. If the orbitals are energetically close, such as the $3{\ensuremath{\sigma}}_{g}$ and the $1{\ensuremath{\pi}}_{u}$ orbitals of ${\mathrm{N}}_{2}$, we show that the quantum interference patterns observed in the high-order harmonic spectra are predominantly determined by the $3{\ensuremath{\sigma}}_{g}$ orbital. If, on the other hand, there is a significant difference in their binding energies, such as for the $3{\ensuremath{\sigma}}_{g}$ and the $1{\ensuremath{\pi}}_{g}^{*}$ orbitals of ${{\mathrm{N}}_{2}}^{+}$, the most loosely bound orbital will determine the shape of the spectra. Due, however, to the different cutoffs encountered, the more deeply bound orbital will leave an imprint on the high-energy harmonics. This holds both for the situation in which the dynamics of the electron is restricted to the plane ${p}_{x}{p}_{z}$ and for the full three-dimensional case, if the azimuthal angle is integrated over and the degeneracy of the $\ensuremath{\pi}$ orbitals is taken into account.

Nonlinear time-dependent density-functional-theory study of ionization and harmonic generation in CO2by ultrashort intense laser pulses: Orientational effects

Time-dependent density-functional-theory (TDDFT) methods are used to calculate the orientational dependence of ionization and molecular high-order harmonic generation (MHOHG) in the CO${}_{2}$ molecule as a function of laser intensity ${I}_{0}\ensuremath{\geqslant}{10}^{14}$ W$/$cm${}^{2}$ for few-cycle 800 nm laser pulses. A time-series analysis is used to confirm the recollision model in MHOHG for different density potentials. It is found that at intensities ${I}_{0}g3.5\ifmmode\times\else\texttimes\fi{}{10}^{14}$ W$/$cm${}^{2}$, lower highest occupied molecular orbitals (HOMO's) contribute significantly to ionization and to the MHOHG process. This is due to the symmetry of these orbitals. Even though such lower orbitals have higher ionization potentials (IP), ionization and MHOHG processes occur when orbital densities are maximum with laser polarization direction.

Role of ionization in orientation dependence of molecular high-order harmonic generation

We investigate the orientation dependence of high-order harmonic generation (HHG) from O(2) and CO(2) molecules using the strong-field approximation (SFA). Our simulations reveal the important modulation of the ionization to the HHG orientation dependence, especially at larger orientation angles. By virtue of a simplified model arising from the SFA, we show that this modulation can be read from the harmonic order where the HHG spectra at different orientation angles intersect. These results give suggestions on probing the molecular structure and dynamics using HHG.

Orbital symmetry and interference effects in molecular high-order harmonic generation

We investigate harmonic generation from $\text{H}_{2}{}^{+}$ molecules driven by intense few-cycle laser pulses whose linearly polarization axis makes an arbitrary angle $\ensuremath{\chi}$ with respect to the molecular axis. The $\text{H}_{2}{}^{+}$ molecule is considered initially in various orbitals with nodal planes. It is found that a strong enhancement of high-order harmonics (HOHs) occurs when the laser polarization axis overlaps with major axes of electron distribution in the active orbital, while broad suppression of HOHs occurs when the laser polarization axis is parallel to a nodal plane of the active molecular orbital. We show that this harmonic suppression is enhanced by destructive interferences when the nodal and the laser polarization axes are parallel to the internuclear axis, which leads to a shortening of the harmonic cutoff. It follows that the orientation dependence of HOHs intensities mimics the electronic density in active orbitals through the angular dependence of ionization and recombination processes.

Strong-field approximation for diatomic molecules: Comparison between the length gauge and the velocity gauge

We investigate the ionization and high-order harmonic generation (HHG) of diatomic molecules using the strong-field approximation (SFA) in both the length gauge (LG) and the velocity gauge. We focus on the effect of two-center interference that has an important role in both molecular ionization and HHG. Our simulations reveal the significant disagreement between the two gauges for the description of this effect. For the behavior of molecular ionization rate with respect to the internuclear distance, which is closely related to this effect in ionization, the two gauges give qualitatively conflicting answers in some cases. For the striking hollow structure in the molecular HHG spectrum, where this effect dominates, the predictions of the two gauges also differ significantly. Further comparisons suggest the preference of the LG SFA in describing this interference effect in intense-field--molecule interaction.

Wavelength effect on atomic and molecular high harmonic generation driven by a tunable infrared parametric source

We experimentally investigate the wavelength effect on high-order harmonic generation (HHG) in CH(4) molecules and Xe atoms driven by a tunable infrared parametric source, and observe that the molecular HHG around the vibrational resonance is more sensitive to the driver wavelength than HHG from an atomic gas with comparable ionization potential. The results can be attributed to the light nuclear motion induced by the driving laser field, and it becomes possible to control the proton vibration in the molecular HHG by tuning the infrared wavelength of the driving laser.

Explanation for the smoothness of the phase in molecular high-order harmonic generation

We analyze the orientation dependence of harmonic amplitudes and phases from laser driven ${\text{H}}_{2}^{+}$. We use the Lewenstein model, with and without employing the saddle-point approximation for the summation over electron momenta. This means that the direction of the electron motion is not necessarily restricted to the laser polarization axis in contrast to previous implementations. The model predicts smooth phase jumps by almost $\ensuremath{\pi}$ in the orientation dependence. This demonstrates that the smoothness can be explained without Coulomb effects, but these may be relevant for the size of the phase jumps observed in recent experiments.

Signatures of nuclear motion in molecular high-order harmonics and in the generation of attosecond pulse trains by ultrashort intense laser pulses

Non-Born–Oppenheimer time-dependent Shrödinger equation numerical simulations of the nonlinear nonperturbative response of 1D H2, H+2 molecules (and their isotopes) in few cycle intense 800 nm laser pulses are presented to study the effect of nuclear motion on molecular high-order harmonic generation. A time–frequency analysis is used to identify electron recollision and recombination times responsible for the generation of attosecond pulse trains during the nuclear motion. A very strong signature of nuclear motion is seen in the time profiles of high-order harmonics. In the case of high laser intensity (I ≃ 1015 W cm−2) the nuclear motion shortens the part of the attosecond pulse train originating from the first electron contribution and may enhance the onset of the second electron contribution for longer pulses. Molecular motion thus can act as an important ‘time-gating’ for controlling the length of generated attosecond pulses. The shape of time profiles of harmonics can thus be used for monitoring the nuclear motion. In the case of lower laser intensity, I ≃ 4 × 1014 W cm−2, we also find in time profiles a clear signature of electron excitation due to recollision of the returning electron.

Intensity dependence of intramolecular interference from a full quantum analysis of high-order harmonic generation

We develop a numerical scheme to investigate the high-order harmonic generation (HHG) in intense laser-matter interactions. We represent the time-dependent wave function in the basis of the eigenstates of the field-free Hamiltonian and trace its temporal evolution. Our full quantum simulations reveal that continuum electrons with a broad energy distribution contribute equally to one harmonic and the excited state also plays an important role in the molecular HHG. These results imply a laser-intensity-dependent picture of intramolecular interference in the HHG.

New results in above-threshold ionization and high-order harmonic generation of atomic and molecular systems

Numerical results obtained applying the strong-field approximation to atomic and molecular processes in intense laser fields are presented. In particular, the forward-backward asymmetry in high-order above-threshold ionization (HATI) of Ar atoms by a few-cycle laser pulse is considered. It is shown that the resonantlike enhancements observed in the focal-averaged spectra disappear with the decreasing laser pulse duration. For HATI and high-order harmonic generation (HHG) by molecular systems, we present new results for diatomic molecules. The choice of gauge and the problem of dressing of the initial bound state are discussed. The appearance of the interference minima in molecular HATI and HHG spectra and their role in the characterization of the molecular structure are considered using the example of the Ar2 molecule.

Reading molecular messages from high-order harmonic spectra at different orientation angles

We investigate the orientation dependence of high-order harmonic generation (HHG) from H(2) (+) with different internuclear distances irradiated by intense laser fields both numerically and analytically. The calculated molecular HHG spectra are found to be sensitive to the molecular axis orientation relative to incident laser field polarization and internuclear separation. In particular, our simulations demonstrate that at certain harmonic orders the envelopes of the HHG spectra taken at different orientation angles intersect. Moreover, the position of intersection is largely independent of the laser intensity while strongly dependent on the internuclear separation. This striking "intersection" phenomenon is identified as arising due to intramolecular two-center interference in the HHG and can be used to probe the molecular instantaneous structure.

High-order harmonic generation and molecular orbital tomography: Characteristics of molecular recollision electronic wave packets

We investigate the orientation dependence of molecular high-order harmonic generation (HHG) both numerically and analytically. We show that the molecular recollision electronic wave packets (REWPs) in the HHG are closely related to the ionization potential as well as the particular orbital from which it ionized. As a result, the spectral amplitude of the molecular REWP can be significantly different from its reference atom (i.e., with the same ionization potential as the molecule under study) in some energy regions due to the interference between the atomic cores of the molecules. This finding is important for molecular orbital tomography using HHG [J. Itatani et al., Nature 432, 867 (2004)].

Gauge problem in molecular high-order harmonic generation

The problem that the strong-field approximation (SFA) in the length gauge (LG) cannot give the correct molecular harmonic generation spectrum when the internuclear distance is large is studied. By introducing the first excited state in addition to the ground state in the SFA and a procedure to resum translation invariance, the problem is resolved and the superiority of the LG to the velocity gauge in this case is discussed.

Molecular high order harmonic generation by laser induced electron transfer with intense ultrashort pulses

We present a study of the nonlinear molecular high harmonic order generation (MHOHG) process in the linear one-electron triatomic ion [ H + – H 2 + ] under a two cycle intense pulse ( I > 10 14 W cm −2) and λ = 800 nm. We have solved numerically the 3-D time dependent Schrödinger equation (TDSE) describing the motion of the electron in the presence of such short pulses. We investigate MHOHG for the internuclear distance R = α, πα/2 between the H + and H 2 + , and for different internuclear distance r of H 2 + , where α = E 0 ω 2 is the ponderomotive radius. We show that, by varying the carrier-envelope phase (CEP) of the laser pulse, one can control the direction of the electron transfer which we call Laser Induced Electron Transfer (LIET). We have focused on the case the transfer occurs from H + to H 2 + ( 1 s σ g ) / ( 1 s σ u ) . We showed that the collision of the electron with H 2 + generates harmonics exceeding the atomic recollision law I p + 3.17 U p. MHOHG spectra we obtained show a series of peaks with maxima and minima. The minima are shown to be independent of the symmetry of the orbital, whereas there are dependent on the pondermotive electron energy U p = E 0 2 4 ω 2 , and the electron collision time with the H 2 + .

Attosecond localization of electrons in molecules

AbstractNumerical solutions of the time‐dependent Schrödinger equation for a 1D model non‐Born–Oppenheimer H are used to illustrate the nonlinear nonperturbative response of molecules to intense (I ≥ 1013 W/cm2), ultrashort (t < 10 fs) laser pulses. Molecular high‐order harmonic generation (MHOHG) is shown to be an example of such response and the resulting nonlinear photon emission spectrum is shown to lead to the synthesis of single attosecond (10−18 s) pulses. Application of such ultrashort pulses to the H system results in localized electron wavepackets whose motion can be detected by asymmetry in the photoelectron spectrum generated by a subsequent probe attosecond pulse, thus leading to measurement of electron motion in molecules on the attosecond time scale. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Attosecond molecular spectroscopy – The one-electron H2+ system

Numerical solutions of the time-dependent Schrödinger equation for a 1-D model non-Born–Oppenheimer H2+ are used to illustrate the nonlinear, nonperturbative response of molecules to intense (I ≥ 1013 W/cm2), ultrashort (t < 10 fs) laser pulses. Molecular high-order harmonic generation (MHOHG) is shown to be an example of such response, and the resulting nonlinear photon emission spectrum is shown to lead to the synthesis of single attosecond (10–18 s) pulses. Application of such ultrashort pulses to the H2+ system results in localized electron wave packets whose motion can be detected by asymmetry in the photoelectron spectrum generated by a subsequent probe attosecond pulse, thus leading to measurement of electron motion in molecules on an attosecond time scale. Key words: attosecond spectroscopy, attosecond photoionization.

Impact of electron ionization on the generation of high-order harmonics from molecules

When the laser frequency is tuned to be equal to the molecular electronic excitation, high-order harmonics are generated due to the electronic dipole transitions between the corresponding two potential-energy surfaces (PES). A natural, often taken, choice is the PES of the field-free molecular system. In this special choice the ionization phenomenon is not considered. Only the effect of the dissociation is considered. The method we developed enables one to remain within the framework of the 2-PES approximation and yet to include also the ionization effect in the calculations of molecular high-order harmonic generation spectra. In this approach the coupling between the electronic and nuclear motions is taken into consideration by using coupled complex adiabatic PES. As an illustrative numerical example, we calculated the high harmonic generation (HHG) spectra of ${\mathrm{H}}_{2}^{+}$ in a 730-nm laser with the intensity of $8.77\ifmmode\times\else\texttimes\fi{}{10}^{13}{\mathrm{W}/\mathrm{c}\mathrm{m}}^{2}.$ The inclusion of the ionization in our approach not only enables the electrons to tunnel through the effective static potential barrier, but also apply an asymmetric force which accelerates the electron before ionization takes place. Therefore, indirectly the inclusion of the ionization by the laser field may lead eventually to an enhanced HHG spectra in comparison with the calculated one when the ``natural'' choice of the field-free 2PES is taken.