Returning Electron Wave Packet Research Articles

Enhancing high-order harmonic generation by controlling the diffusion of the electron wave packet

We experimentally study the enhancement of high-order harmonic generation (HHG) driven by synthesized ω − 3 ω laser fields, where we control whether the ionization rate or the electron wave packet’s diffusion is the dominant enhancement mechanism. When minimizing the electron wave packet’s diffusion, the excursion times of the corresponding electron trajectories are reduced by a factor of 2 or more. This result is important for imaging techniques that use the returning electron wave packet to probe the remaining ion. Furthermore, we achieve a 10 × to 3800 × enhancement of the harmonic yield driven by the bichromatic fields relative to that of an optimized single-color field, showing that the bichromatic fields improve HHG’s capability as a light source. We also measure that the two-color field’s harmonics have half the divergence angle compared to their single-color counterpart, suggesting that the “short” electron trajectories play a more prominent role compared to their “long” trajectory counterparts, thus improving the wavefront of the emerging harmonic beam.

Dependence of the returning electron wave packet on the model potential for high-energy plateau photoelectron spectra

SynopsisIn the present work, we study the two-dimensional momentum distributions of high-energy plateau photoelectrons of a model-atom in a linearly polarized few-cycle laser pulse. Based on the improved strong-field approximation (ISFA), it is found that the returning electron wave packet is dependent of the model potential.

Rescattering photoelectron spectroscopy of CO2 molecule with an analytical returning electron wavepacket

SynopsisAngle-resolved photoelectron momentum distributions of rescattering electron generated by intense infrared laser pulses were measured for CO2 molecules. It is shown that the measured distributions along the outermost backward rescattering caustic are well reproduced by calculations based on a newly developed factorization formula with the analytical returning photoelectron wave packet.

Quantitative rescattering theory for nonsequential double ionization**Project supported by the National Natural Science Foundation of China (Grant No. 11274219), the Science and Technology Planning Project of Guangdong Province of China (Grant No. 180917124960522), and the Program for Promotion of Science at Universities in Guangdong Province of China (Grant No. 2018KTSCX062).

We review the recently improved quantitative rescattering theory for nonsequential double ionization, in which the lowering of threshold due to the presence of electric field at the time of recollision has been taken into account. First, we present the basic theoretical tools which are used in the numerical simulations, especially the quantum theories for elastic scattering of electron as well as the processes of electron impact excitation and electron impact ionization. Then, after a brief discussion about the properties of the returning electron wave packet, we provide the numerical procedures for the simulations of the total double ionization yield, the double-to-single ionization ratio, and the correlated two-electron momentum distribution.

Revisiting the recollisional excitation-tunneling process in strong-field nonsequential double ionization of helium

We revisit the excitation-tunneling process in nonsequential double ionization (NSDI) of helium in an 800-nm laser field. The correlated two-electron momentum distributions are calculated by using the improved quantitative rescattering (QRS) model, in which the lowering of the threshold energy due to the presence of an electric field at the instant of recollision is taken into account. In the framework of the QRS model, the correlated two-electron momentum distributions for excitation-tunneling in NSDI can be factorized as a product of the returning-electron wave packet (RWP) and the field-free differential cross section (DCS) for electron impact excitation of the parent ion multiplied by the tunneling ionization rate of electrons in the excited states. The RWPs, which describe the momentum distribution of the returning electrons, are obtained within the strong-field approximation for high-order above-threshold ionization. The DCSs for electron impact excitation of ${\mathrm{He}}^{+}$ are calculated using the state-of-the-art many-electron $R$-matrix theory, and the tunneling ionization rates for electrons in the excited states are evaluated by solving the time-dependent Schr\"odinger equation. The calculated correlated two-electron momentum distribution shows that the fourfold symmetry with regard to the parallel momentum components is broken. This is in contradiction to the prevalent view that the correlation pattern for excitation-tunneling in NSDI is symmetric with respect to the coordinate axes. By including the recollisional ($e,2e$) process, the predicted correlated two-electron momentum distributions are found to be in good qualitative agreement with experiment.

Revisiting the recollisional ( e, 2e ) process in strong-field nonsequential double ionization of helium

We present numerical simulations on the correlated two-electron momentum distributions for the recollisional ($e, 2e$) process in nonsequential double ionization (NSDI) of helium by an 800-nm laser field at an intensity around $4.5\ifmmode\times\else\texttimes\fi{}{10}^{14}\phantom{\rule{0.16em}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$, based on the improved quantitative rescattering model, in which the lowering of the kinetic energy required for the electrons to escape from the parent ion due to the presence of electric field at the time of recollision has been taken into account. According to the QRS model, the correlated two-electron momentum distributions for laser-induced ($e, 2e$) collision in NSDI can be factorized as a product of the returning-electron wave packet (RWP) and the field-free differential cross section (DCS) for ionization of the parent ion by the impact of the laser-induced returning electron. The RWPs which describe the momentum distribution of the returning electrons are obtained within the strong-field approximation for high-order above threshold ionization. In the calculations of the DCSs for electron impact ionization of ${\mathrm{He}}^{+}$, the precollision Coulomb interaction between the incident electron and the parent ion and the postcollision Coulomb interaction between the two outgoing electrons are considered. In addition, the dynamic screening (DS) of the three-body Coulomb interactions in the final state, due to the fact that the strength of the interaction of any two particles is affected by the presence of the third one, has also been taken into account. It has been found that while the postcollision Coulomb interaction is responsible for the observed fingerlike structure, the precollision Coulomb interaction changes the orientation of the two fingers from $\mathsf{V}$-type to parallel, the final state DS reduces the separation of the two fingers, and the lowering of threshold energy shifts the two fingers towards smaller momenta. All these effects improve the agreement of the simulated results with the experimental results.

Ratios of double to single ionization of He and Ne by strong 400-nm laser pulses using the quantitative rescattering theory

We present numerical simulations of the ratio between double and single ionization of He and Ne by intense laser pulses at wavelengths of 390 and 400 nm, respectively. The yields of doubly charged ions due to nonsequential double ionization (NSDI) are obtained by employing the quantitative rescattering (QRS) model. In this model, the NSDI ionization probability is expressed as a product of the returning electron wave packet (RWP) and the total scattering cross sections for laser-free electron impact excitation and electron impact ionization of the parent ion. According to the QRS theory, the same RWP is also responsible for the emission of high-energy above-threshold ionization photoelectrons. To obtain absolute double-ionization yields, the RWP is generated by solving the time-dependent Schr\odinger equation (TDSE) within a one-electron model. The same TDSE results can also be taken to obtain single-ionization yields. By using the TDSE results to calibrate single ionization and the RWP obtained from the strong-field approximation, we further simplify the calculation such that the nonuniform laser intensity distribution in the focused laser beam can be accounted for. In addition, laser-free electron impact excitation and ionization cross sections are calculated using the state-of-the-art many-electron $R$-matrix theory. The simulation results for double-to-single-ionization ratios are found to compare well with experimental data and support the validity of the nonsequential double-ionization mechanism for the covered intensity region.

Temporal and spatial manipulation of the recolliding wave packet in strong-field photoelectron holography

We theoretically demonstrate temporal and spatial manipulation of electron wave packets involved in strong-field photoelectron holography (SFPH) with the orthogonally polarized two-color laser fields. By varying the relative phase of the two-color fields, the recollision time of the returning wave packet can be accurately controlled, which allows us to switch off and on the holographic interference. Moreover, the recollision angles of the returning electron wave packet can be arbitrarily controlled via changing the relative intensity of the two-color fields, and thus the structure information of the target is encoded in the hologram by the recollision electron wave packet from different angles. This makes the SFPH a powerful technique of imaging the molecular structure as well as ultrafast dynamics on an attosecond time scale.

Strong-field approximation and its extension for high-order harmonic generation with mid-infrared lasers

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.

Numerical simulation of the double-to-single ionization ratio for the helium atom in strong laser fields

We present calculations on the ratio between double and single ionization of helium by a strong laser pulse at a wavelength of 780 nm using the quantitative rescattering (QRS) model. According to this model, the yield for the doubly charged ion $\mathrm{He}^{2+}$ can be obtained by multiplying the returning electron wave packet (RWP) with the total cross sections (TCSs) for electron impact ionization and electron impact excitation of $\mathrm{He}^{+}$ in the singlet spin channel. The singlet constraint was imposed since the interaction of the helium atom with the laser and the recollision processes both preserve the total spin of the system. An $R$-matrix (close-coupling) code is used to obtain accurate TCSs, while the RWPs, according to the QRS, are calculated by the strong-field approximation for high-energy photoelectrons. The laser field, which lowers the required energy for the electron to escape from the nucleus at the time of recollision, is also taken into account. The simulated results are in good agreement with the measured $\mathrm{He}^{2+}\text{/}\mathrm{He}^{+}$ ratio over a broad range of laser intensities. The result demonstrates that the QRS approach based on the rescattering model is fully capable of quantitatively interpreting nonsequential double ionization processes.

Scaling laws for high-order-harmonic generation with midinfrared laser pulses

We derive an analytic expression for the wavelength scaling of the high-order-harmonic generation (HHG) yield induced by midinfrared driving laser fields. It is based on a quasiclassical description of the returning electron wave packet, which is shown to be largely independent of atomic properties. The accuracy of this analytic expression is confirmed by comparison with results of numerical solutions of the time-dependent Schr\odinger equation for wavelengths in the range of $1.4\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}\ensuremath{\leqslant}\ensuremath{\lambda}\ensuremath{\leqslant}4\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$. We verify the wavelength scaling of the HHG yield found numerically for midinfrared laser fields in a recent paper by Le et al. [Phys. Rev. Lett. 113, 033001 (2014)].

Universality of returning electron wave packet in high-order harmonic generation with midinfrared laser pulses.

We show that a returning electron wave packet in high-order harmonic generation (HHG) with midinfrared laser pulses converges to a universal limit for a laser wavelength above about 3 μm. The results are consistent among the different methods: a numerical solution of the time-dependent Schrödinger equation, the strong-field approximation, and the quantum orbits theory. We further analyze how the contribution from different electron "trajectories" survives the macroscopic propagation in the medium. Our result thus provides a new framework for investigating the wavelength scaling law for the HHG yields.

Validity of the quantitative rescattering theory for high-order harmonic generation of atoms in two-color laser pulses

The quantitative rescattering (QRS) theory has been recently developed to study the high-order harmonic generation (HHG) of atoms and molecules exposed to a linearly polarized single-color laser pulse. According to the QRS, the HHG spectra can be expressed as a product of the returning electron wave packet and the photorecombination differential cross section of the laser-free continuum electron back to the initial ground state. In this paper, we extend the QRS to the case of two-color laser fields. The validity of the QRS for two-color pulses is carefully examined by comparing the HHG spectra of several rare-gas atoms obtained from the QRS with those results from the solution of the time-dependent Schrödinger equation (TDSE) based on the single active electron approximation. Our results show that the QRS theory for the HHG of atoms works quite well in the case of two-color pulses.

Attosecond-resolved electron emission in nonsequential double ionization

We have investigated the correlated electron dynamics in nonsequential double ionization (NSDI) of xenon by the orthogonally polarized two-color pulses consisting of 800and 1600-nm laser fields. The two-electron momentum distributions are sensitively dependent on the relative phase of the two pulses. By tracing the history of double ionization trajectories, we find that the revisit time of the returning electron wave packet is controlled with attosecond accuracy. After recollision, one electron is ionized immediately while the other electron is either released immediately or excited with subsequent field ionization. The release time of the excited electron is also steered with attosecond resolution by changing the relative phase of the orthogonal two-color pulses. The attosecond-resolved control of the revisit time of the returning electron wave packet and the release time of the excited electron is responsible for the phase dependence of the correlated behaviors of the two electrons. These results indicate that we can trace the emission of the two electrons in NSDI on attosecond time scales.

Validity of the quantitative rescattering model for two-color strong laser pulses

Recently, we have developed a quantitative rescattering (QRS) model for linearly polarized laser pulses. According to the QRS model, the yield of the high-energy photoelectron spectra can be expressed as a product of the returning electron wave packet and the elastic differential cross section for free electrons scattered with the parent ion. In this paper, the QRS model is applied to the case of two-color strong laser pulses. The validity of the QRS model for two-color pulses is examined by comparing the high-energy photoelectron spectra simulated from the QRS model with the results from solving the time-dependent Schrödinger equation.

Strong-field nonsequential double ionization of Ar and Ne

We investigate the nonsequential double ionization (NSDI) of Ar and Ne based on quantitative rescattering theory (QRS). According to QRS theory, each elementary NSDI process can be calculated by multiplying the returning electron wave packet with appropriate differential electron-ion scattering cross sections. We include ($e$, $2e$) and electron-impact excitation cross sections of Ar${}^{+}$ to obtain the correlated electron momentum spectra for the NSDI of Ar by few-cycle pulses to check the dependence of NSDI on the carrier-envelope phase. The results are compared to the ion momentum spectra from the recent experiment of Johnson et al. [Phys. Rev. A 83, 013412 (2011)]. Calculations have also been performed for Ar at another intensity to illustrate the intensity dependence of NSDI and to compare with the earlier data of Feuerstein et al. [Phys. Rev. Lett. 87, 043003 (2001)] and for Ne to illustrate the target dependence. We also address the presence of resonant capture processes in electron-ion collisions in the NSDI spectra.

Polarisation states of high harmonic generation from aligned molecules

We present detailed theoretical calculations based on the recently developed quantitative rescattering theory (QRS) for polarisation and ellipticity of high-order harmonics from aligned N2 and O2 generated with mid-infrared linearly polarised laser pulses. Within the QRS, the two polarisation states of high-order harmonics are due to the high-energy photons emitted with polarisation parallel or perpendicular to the direction of the same returning electron wave packet in the photo-recombination process. The complex transition dipoles responsible for these two components have, in general, difference phases, which results in the ellipticity of emitted harmonics. Our results show clear target-species dependent polarisation states. In particular, we found strong harmonic ellipticity in aligned N2, but weak ellipticity in aligned O2 for harmonics below 100 eV. We further show the factorisation of the macroscopic high harmonic generation under typical experimental conditions. This implies that results of polarisation measurements are quite robust to the macroscopic phase matching conditions.

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.

Medium propagation effects in high-order harmonic generation of Ar andN2

We report theoretical calculations of high harmonic generation (HHG) by intense infrared lasers in atomic and molecular targets taking into account the macroscopic propagation of both fundamental and harmonic fields. On the examples of Ar and N$_2$, we demonstrate that these {\it ab initio} calculations are capable of accurately reproducing available experimental results with isotropic and aligned target media. We further present detailed analysis of HHG intensity and phase, under various experimental conditions, in particular, as the wavelength of the driving laser changes. Most importantly, our results strongly support the factorization of HHG at the macroscopic level into a product of a returning electron wave packet and the photorecombination transition dipole, under typical experimental conditions. This implies that the single-atom/molecule structure information can be retrieved from experimentally measured HHG spectra.

Quantitative rescattering theory of correlated two-electron momentum spectra for strong-field nonsequential double ionization of helium

We study the correlated two-electron momentum spectra of the nonsequential double ionization (NSDI) of an atom in an intense laser field based on the recently developed quantitative rescattering (QRS) theory. According to the rescattering model, an electron that was released earlier in the laser pulse may return to collide again with the target ion. NSDI can occur directly if this electron knocks out another electron in a process similar to the ($e,2e$) process or indirectly by first exciting the target ion to an excited state which is later ionized by the laser field. Using QRS, we obtain the returning electron wave packet. By multiplying this wave packet with standard field-free ($e,2e$) differential cross sections, or with inelastic electron-impact excitation cross sections and the subsequent tunneling ionization, we obtain the correlated two-electron momentum spectra. The calculated spectra agree mostly with the experimental data. However, experimental data show additional features that cannot be accounted for by these two mechanisms only, and other mechanisms for NSDI are suggested. The contributions of these mechanisms to the longitudinal ion momentum distributions are also analyzed. The present quantum mechanical QRS calculation verifies the validity of rescattering model for the NSDI processes at the most fundamental and detailed level.