Mechanism of laser intensity-dependent below-threshold harmonic generation

We present an ab initio study of the quantum dynamics of high-order harmonic generation (HHG) near the cutoff in intense laser fields. To uncover the subtle dynamical origin of the HHG near the cutoff, we extend the Bohmian mechanics (BM) approach for the treatment of attosecond electronic dynamics of H and Ar atoms in strong laser fields. The time-dependent Schrödinger equation and the self-interaction-free time-dependent density functional theory are numerically solved accurately and efficiently by means of the time-dependent generalized pseudospectral method for nonuniform spatial discretization of the Hamiltonian. We find that the most devoting trajectories calculated by the BM to the plateau harmonics are shorter traveling trajectories, but the contributions of the short trajectories near the cutoff are suppressed in HHG. As a result, the yields of those harmonics in the region near the cutoff are relatively weak. However, for the last few harmonics just above the cutoff, the HHG intensity becomes a little higher. This is because the HHG just above the cutoff arises from those electrons ionized near the peak of the laser pulse, where the ionization rate is the highest. In addition, the longer Bohmian trajectories return to the core with lower energies, these trajectories contribute to the below-threshold harmonics. Our results provide a deeper understanding of the generation of supercontinuum harmonic spectra and attosecond pulses via near cutoff HHG.

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.

The dynamics of high-intense laser fields at coherent Raman self-scattering are investigated by the use of numerical simulation. The processes under investigation take place at the Raman scattering on electronic states in atoms with transition to be forbidden in electro-dipole approximation, on vibrational or rotational states in molecules, on states connected with the spin flip and other. It is found that the threshold intensities exist provided that the pulse duration is smaller than dephasing time of the Raman transition in the medium. Under this, the following processes occur: an effective generation of the high-order harmonics comes up; the plateau forms in the harmonic spectrum. The maximum which arises on this plateau shifts to the higher-harmonic numbers in case of the higher threshold intensities of the pulse. It is shown how the transformation of the initial pulse intensity into high-order harmonics intensity can be considerably increased with the different initial parameters of the pulse and the medium.

We perform a nonperturbative study of the effect of static magnetic fields on the multiple high-order harmonic generation (HHG) processes of atomic H in intense pulsed laser fields. The time-dependent Schrödinger equation in three spatial dimensions can be solved accurately and efficiently by means of the recently developed time-dependent generalized pseudospectral method (Tong X M and Chu S I 1997 Chem. Phys. 217 119). We found that the presence of a static magnetic field perpendicular to the laser field polarization direction can introduce a second HHG plateau, allowing significant extension of the cut-off. The high harmonics in the second plateau reach a maximum enhancement when the cyclotron frequency of the electron is about equal to one half of the laser frequency. Through a detailed time-frequency wavelet analysis of the HHG spectra, we have identified a `synchrotron radiation'-like new mechanism responsible for the production of the second plateau.

Preface. SILAP 2000: an overview. Part I: Double Ionization of Complex Atoms. Energy distribution of two-electron ionization of helium in an intense field R. Lafon, et al. Double ionization in strong fields: ion momenta and correlated electron momenta A. Staudte, et al. Mechanism of the non sequential double ionization of helium D.G. Lappas, et al. Electron momentum distributions for double ionization in the strong field limit S.P. Goreslavski, S.V. Popruzhenko. S-matrix theory of 'recoil-ion' momentum distribution for double ionization in femtosecond laser fields F.H.M. Faisal, A. Becker. Non-sequential double ionization: a minimal correlation approach R. Kopold, W. Becker. Non-sequential double ionization of atoms in strong fields K. Sacha, B. Eckhardt. Double-electron ionization of two-electron system in strong laser field D.V. Tikhonova. Calculation of double ionization of helium H.G. Muller. The two-electron response in laser driven helium L.R. Moore, et al. The helium atom in strong and short laser pulses: multielectron H. Bachau, R. Hasbani. Dynamics of a two-electron system driven by an ultrashort and intense laser pulse B. Piraux, G. Lagmago-Kamta. Angular distributions for double ionization by an ultrashort, intense laser pulse: the case of Li- G. Lagmago-Kamta, A.F. Starace. Two-and-three electron atoms in strong laser fields P. Lambropoulos, et al. Part II: Ionization and Dissociation of Molecules. Tunnelling ionization and the Franck-Condon principle J.H. Posthumus, et al. Dissociative ionization of few-electron molecules in intense laser fields D. Dundas, et al. One-photon breakup of H+2 in a strong DC field Z. Mulyukov, R. Shakeshaft. Part III: Interaction of Clusters with Very Intense Femtosecond Laser Pulses. Nuclear fusion in gases of deuterium clusters and hot electron generation in droplet sprays under irradiation with an intense femtosecond laser T. Ditmire, et al. The absorption of energy by large atomic clusters from superintense laser pulses V.P. Krainov, M.B. Smirnov. Part IV: Production of Very Intense Femtosecond Laser Pulses. A semi-classical model for high-harmonic generation D.B. Milosevic. Beyond the simple maris model for high harmonic generation M. Brewezyk, K. Rzazewski. Anisotropy induced polarization effects in harmonic generation by an absorptive medium B. Borca, et al. About a new method of high harmonic amplification E.A. Nersesov, et al. X-ray generation via stimulated recombination of electrons and Bohr's correspondence principle A. Jaron, et al. Part V: Stabilization and Relativistic Effects in Super Strong Fields. Interference stabilization: LAMBDA- and V-schemes, dynamics of ionization, initial coherent population of Rydberg levels and quantum phase control of the ionization yield M.V. Fedorov, N.P. Poluektov. Computer experiments on atomic stabilization in a strong laser field A.M. Popov, et al. Quasistationary stabilization of the decay of a weakly-bound level and its breakdown in a strong laser field N.L. Manakov, et al. The strong field limit of atomic stabilization in ultrashort pulses M. Dorr, et al. 3-D numerical calculations of laser atom interactions -- subrelativistic and weakly relativistic regime A. Scrinzi, et al. Momentum space description of hydrogen atom interacting with a low frequency strong laser field A. de Bohan, et al. Relativistic effects in the atomic res

We study the electronic states and nonlinear motion of $\ensuremath{\pi}$ electrons in an armchair nanotube driven by monochromatic intense laser fields with polarization parallel to the nanotube axis. The intensity and frequency (photon energy) of the applied laser fields are varied so their effect on the electrons can be understood. In each case, Floquet-Bloch theory is used to calculate the Floquet-Bloch states, quasienergy band, mean energy band, and electron current density. By summing up the current density of all occupied Floquet-Bloch states, the harmonic generation spectrum can be determined. We demonstrate that the deformation of quasienergy band and mean energy band is related to high-order harmonic generation. Only the states deviating from field-free eigenstates may contribute to high-order harmonic generation.

High-order harmonics generated by a linearly polarized laser field are also linearly polarized. Having in mind that for various application, such as the exploration of magnetic materials, chiral molecules etc., we need circularly polarized high harmonics which serve as coherent soft x-rays, we explore high-order harmonic generation by the so-called bicircular laser field. This field consists of two coplanar counter-rotating circularly polarized fields of different frequencies equal to integer multiples of a fundamental frequency ω. High harmonics generated by such field are circularly polarized with helicity alternating between +1 and −1. Combining a group of such harmonics, instead of obtaining a circularly polarized attosecond pulse train, one obtains a pulse with unusual polarization properties. But, if the harmonics of particular helicity are stronger, i.e., if we have helicity asymmetry in a high-harmonic energy interval, then it is possible to generate an elliptical or even circular pulse train. We theoretically investigated a wide range of bicircular field-component intensities (I1 and I2) and found regions where both the harmonic intensity is high and the helicity asymmetry is large. Particular attention is devoted to the ω−2ω and ω−3ω bicircular fields and atoms having the s and p ground states. In our calculations we use strong-field approximation and quantum-orbit theory. We show that, even in the extreme case of I2 = 8I1, for an ω−3ω bicircular field, high-order harmonic generation is more efficient than in the I2 = I1 case. The obtained results are explained analyzing the relevant electron trajectories and velocities, which follow from the quantum-orbit theory. For the atoms having p ground state the helicity asymmetry parameter is large for a wide range of high-harmonic photon energies, while for the atoms having s ground state the helicity asymmetry parameter can be large only for low harmonics. We confirm this by averaging the obtained results over the intensity distribution in the laser focus.

We present a non-perturbative theory of a single atom interaction with the multicolor laser field. The theory is based on the use of exact mathematical solutions of the boundary value problem for "an atom in the external field", as a result the case of arbitrary orientation of atomic angular momentum and electromagnetic wave polarization vector can be described in the consistent mathematical form. We also present the results of numerical simulations on silver and argon atoms interaction with single-and two-color laser fields formed by fundamental frequency and second harmonic of Ti:Sapphire laser. The results of numerical calculations show that the efficiency of high-order harmonic generation (HHG) depends strongly on the laser field parameters (especially on the orientation of polarization vectors and chirp of the two-color laser field components). We have demonstrated the saturation of the cut-off frequency in the laser fields of near-atomic strength and investigated the impact of the field phase in the case of ultrashort laser pulses.

High-order harmonic generation (HHG) of bulk crystals in strong laser field is typically investigated with semiconductor Bloch equations (SBEs). However, in the length gauge, it suffers from the divergence for the crystals with a zero band gap, such as graphene, using both Bloch- and Houston-states expansion methods. Here, we present a method of solving the SBEs based on time-dependent Bloch basis, which is equivalent to semiconductor Bloch equations in the velocity gauge. Using this method, we investigate the HHG of a single-layer graphene. It is found that our results for population are in good agreement with the other results. For a initial condition py = 0, we find the electrons just move in single valence band or conduction band, which are in accord with classical results. Our simulations on the HHG dependence of polarization of driving laser pulse confirm that 5th, 7th, and 9th harmonic yields increase to the maximal value when laser ellipticity ε ≈ 0.3. What is more, similar to the case of atoms in the laser field, the total strength of 3rd harmonic decrease monotonically with the increase of ε. In addition, we simulate the dependence of HHG on crystallographic orientation with respect to the polarization direction of linear mid-infrared laser pulse, and the results reveal that for higher harmonics, their radiation along with the change of rotation angle θ reflects exactly the sixfold symmetry of graphene. Our method can be further used to investigate the behaviors of other materials having Dirac points (i.e., surface states of topological insulators) in the strong laser fields.

We perform a dynamical symmetry analysis (DSA) of the high-order harmonic generation (HHG) spectrum of an atom interacting with a bichromatic laser field. Within the framework of the conventional Hermitian quantum mechanics (QM), the HHG spectrum calculated using a single Floquet state, or any finite number of Floquet states, is invariant under the inversion of the relative phase of the two-frequency components, $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$. The asymmetry with respect to the phase inversion seen in the simulated HHG spectra is obtained in the conventional QM only when the Floquet spectrum is continuous and ionization is taken into consideration. However, when the Hamiltonian is complex scaled the description is different. Even a single eigenstate of the complex scaled Floquet operator is enough to describe the breaking of the $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$ symmetry in the HHG spectra. We find that there is a direct correlation between the strength of the asymmetry with respect to the relative phase inversion and the magnitude of the ionization rate. For illustration purposes, the DSA is accompanied by the results obtained for a one-dimensional effective single-electron model Hamiltonian mimicking xenon atom interacting with strong laser field.

In recent years intense mid-infrared lasers with wavelength of a few microns have become the standard tools for research in strong field physics laboratories worldwide. These lasers offer the opportunities to extend the traditional study of high-order harmonics generation and attosecond sciences from the extreme ultraviolet to soft x-rays. In this tutorial we revisit the familiar strong field approximation and its simplification—the quantum orbits theory. We draw special emphasis on the factorization of laser induced dipole moment as the product of a returning electron wave packet with the photo-recombination dipole transition matrix element. The former depends on the laser properties only (up to a normalization constant) while the latter is related to laser-free photoionization transition dipole. The factorization leads to the suggested modification beyond the strong field approximation—the quantitative rescattering theory. In applying these theories to mid-infrared lasers, we analyze the behavior of the returning electron wave packet and its scaling properties vs the wavelength of the laser. A few examples are given to demonstrate how the quantitative rescattering theory is capable of reproducing experimental harmonic spectra under various conditions. Future opportunities in employing harmonics generated by optimized mid-infrared lasers for probing molecular structure and for serving as useful table-top coherent light sources up to the x-ray region are also discussed.

Generation of ellipticity-tunable isolated attosecond pulses from diatomic molecules in intense laser fields

We present low-frequency approximation (LFA) for high-order harmonic generation (HHG) process. LFA represents the lowest-order term of an expansion of the final-state interaction matrix element in powers of the laser-field frequency $\ensuremath{\omega}$. In this approximation the plane-wave recombination matrix element which appears in the strong-field approximation is replaced by the exact laser-free recombination matrix element calculated for the laser-field dressed electron momenta. First, we have shown that the HHG spectra obtained using the LFA agree with those obtained solving the time-dependent Schr\odinger equation. Next, we have applied this LFA to calculate the HHG rate for inert gases exposed to a bicircular field. The bicircular field, which consists of two coplanar counter-rotating fields having different frequencies (usually $\ensuremath{\omega}$ and $2\ensuremath{\omega}$), is presently an important subject of scientific research since it enables efficient generation of circularly polarized high-order harmonics (coherent soft x rays). Analyzing the photorecombination matrix element we have found that the HHG rate can efficiently be calculated using the angular momentum basis with the states oriented in the direction of the bicircular field components. Our numerical results show that the HHG rate for atoms having $p$ ground state, for higher high-order harmonic energies, is larger for circularly polarized harmonics having the helicity $\ensuremath{-}1$. For lower energies the harmonics having helicity $+1$ prevails. The transition between these two harmonic energy regions can appear near the Cooper minimum, which, in the case of Ar atoms, makes the selection of high-order harmonics having the same helicity much easier. This is important for applications (for example, for generation of attosecond pulse trains of circularly polarized harmonics).

Compton scattering is defined as an inelastic scattering process in which the interaction between strong laser fields and electrons in matter leads to photon emission. In recent years, with the rapid development of X-ray free-electron lasers, the intensity of X-ray lasers has steadily increased, and the photon energy in Compton scattering process has risen correspondingly. Previous studies focus on single-photon Compton scattering of free electrons. However, the mechanism of non-relativistic X-ray photon scattering by bound electrons remains to be elucidated. Therefore, we develop a frequency-domain theory based on non-perturbative quantum electrodynamics to investigate single-photon Compton scattering of bound electrons in strong X-ray laser fields. Our results show that the double-differential probability of Compton backscattering decreases with the increase of incident photon energy. This work establishes a relationship between Compton scattering and atomic ionization in high-frequency intense laser fields, thereby providing a platform for studying atomic structure dynamics under high-intensity laser conditions.

We present a detailed study of the multiphoton ionization and high-order harmonic generation (HHG) processes of rare-gas atoms (He, Ne, and Ar) in intense pulsed laser fields by means of a self-interaction-free time-dependent density-functional theory (TDDFT) recently developed. The time-dependent exchange-correlation potential with proper short- and long- range potential is constructed by means of the time-dependent optimized effective potential (TDOEP) method and the incorporation of an explicit self-interaction-correction (SIC) term. The TDOEP-SIC equations are solved accurately and efficiently by the time-dependent generalized pseudospectral technique. In this study, all the valence electrons are treated explicitly and nonperturbatively and their partial contributions to the ionization and HHG are analyzed. The results reveal instructive and qualitatively different behavior from each subshell orbital. Moreover, we found that the HHG yields from Ne and Ar atoms are considerably larger than that of the He atom in strong fields. Three main factors are identified for accounting the observed phenomena: (a) the binding energy of the subshell valence electron, (b) the orientation of the valence electron orbital (with respect to the electric-field polarization), and (c) the effect of multiphoton resonant excitation. In particular, we found that the ${\mathrm{np}}_{0}$ valence electrons (in Ne and Ar) with lowest binding energies and electron orbital orientation parallel to the electric-field direction, make the dominant contributions to both ionization and HHG processes in sufficiently strong fields.