Strong-field Approximation Model Research Articles

Elliptical high-order harmonic generation from current-carrying orbitals of prealigned molecules

The polarization of high harmonics generated from current-carrying state of nitric oxide molecules irradiated by linear laser fields is investigated by numerically solving the three-dimensional time-dependent Schrödinger equation. It is found that the ellipticity of high harmonics is obviously dependent on the polar angle between the driving laser polarization and molecular axis which is interpreted by the strong field approximation model. Moreover, our results also show that the helicity of near-threshold harmonics is opposite to that of plateau harmonics when the molecule is at any polar angle. To analyze this phenomenon, we simulate the dipole matrix element numerically. Finally, the attosecond pulses with opposite helicities can be obtained by synthesizing near-threshold and plateau harmonics. The ellipticity can be tuned by the alignment angle of molecules. Our work may provide a theoretical guiding and detection tool for the electron dynamics of molecular current-carrying states.

Strong-Field Ionization of Hydrogen Atoms with Quantum Light.

We study the strong-field ionization driven by quantum lights. Developing a quantum-optical-corrected strong-field approximation model, we simulate the photoelectron momentum distribution with squeezed-state light, which manifests as notably different interference structures from that with coherent-state (classical) light. With the saddle-point method, we analyze the electron dynamics and reveal that the photon statistics of squeezed-state light fields endows the tunneling electron wave packets with a time-varying phase uncertainty and modulates the photoelectron intracycle and intercycle interferences. Moreover, it is found the fluctuation of quantum light imprints significant influence on the propagation of tunneling electron wave packets, in which the ionization probability of electrons is considerably modified in time domain.

Control of coherent extreme-ultraviolet emission around atomic potential through laser chirp

Substantial neutral atoms can tunnel to excited states in an intense laser field and subsequently generate coherent emission through free induction decay. We experimentally observe an enhanced coherent emission in the harmonic slightly below the threshold, which is consistent with the free induction decay of Rydberg states produced by the frustrated tunnelling ionization (FTI) process. We further find that the intensity of the coherent emission significantly depends on the chirp of laser pulses. The simulations based on the strong field approximation model show that laser chirp affects the probability that the returned electrons recombine to the Rydberg states. Our result shows that coherent emission can be controlled by laser chirp, which facilitates understanding the dynamics of the Rydberg atom and coupling mechanism between the below-threshold harmonics and atomic energy level. In addition, the coherent below-threshold FTI emission we observed has small divergence which is good for EUV light source applications.

Coulomb effects on time-trajectory-resolved high-order harmonic generation

We studied the effects of Coulomb potential on high-order harmonic generation (HHG) numerically and analytically. We focus on the influence of Coulomb potential on emission times of HHG associated with specific electron trajectories. By using a numerical procedure based on a numerical solution of the time-dependent Schrödinger equation (TDSE) in three dimensions, we extract the HHG emission times both for long and short electron trajectories. We compare the TDSE predictions with those of a Coulomb-modified model arising from strong-field approximation (SFA). We show that the Coulomb effect induces earlier HHG emission times than those predicted by the general SFA model without considering the Coulomb potential. In particular, this effect influences long and short electron trajectories differently and is more remarkable for low-energy harmonics than high ones. It also changes the HHG amplitudes for long and short electron trajectories. We validate our discussions with diverse laser parameters and forms of Coulomb potential. Our results strongly support a four-step model of HHG.

Laser-induced deformation of atomic p± orbitals in orthogonally polarized two-color laser fields

We study the photoelectron momentum distribution from strong-field ionization of atomic p ± orbitals in an orthogonally polarized two-color (OTC) laser field. We find that the atomic p ± orbitals are deformed in the OTC field, which significantly varies the left–right asymmetries of the electron yield in the photoelectron momentum distribution. By developing a strong-field approximation model that includes the orbital deformation effect, we show that the orbital deformation has a significant effect on the instantaneous ionization rate, while it plays a negligible role in the subcycle electron interference. By taking the spin-orbit coupling into account, we further show that the orbital deformation effect can be used to produce angular-resolved spin-polarized photoelectrons. Our study paves the way for producing spatially separated high degree spin-polarized electrons.

Modeling photoelectron spin polarization of xenon atoms in intense circularly polarized pulses

We propose a model for studying the ionization, photoelectron spectra, and spin polarization of xenon atoms under intense circularly polarized (CP) pulses by solving the time-dependent Schrödinger equation (TDSE) with an effective potential for Xe. The ionization mechanism is mainly due to multiphoton absorption in these strong CP fields. The effective potential of a single active electron for Xe consists of asymptotic Coulomb and parametric screening parts and yields low-lying levels that are comparable to those by other models for the total angular momentum J = 3 / 2 of X e + and new for J = 1 / 2 . We show in detail the absorption pathways, above-threshold ionization (ATI) spectra, and spin polarization mechanisms of photoelectrons originated from the initial state 5 p m l of Xe with the magnetic quantum number m l being − 1 , 0, or 1. There exists a substructure with sub-peaks in each of multiple ATI peaks from 5 p − 1 due to pulse effects on the valence electron passing through the absorption pathways. ATI and spin polarization results agree qualitatively with experimental data at the focal volume average to include the spatial effect of CP pulses. We also present the spin polarization results of the widely used strong-field approximation model (SFA), which are inconsistent with those of TDSE and experiments. This shows that TDSE is more accurate with more details than SFA for studying spin polarization in strong CP fields.

Imprint of transient electron localization in using circularly-polarized laser pulse* *Project supported by the National Key Research and Development Program of China (Grant No. 2018YFB1801904), the National Natural Science Foundation of China (Grant No. 12075036), and the Open Research Fund of National Mobile Communications Research Laboratory, Southeast University (Grant No. 2019D14).

Photoelectron momentum distribution of hydrogen molecular ion in a circularly polarized laser pulse is calculated by solving the three-dimensional time-dependent Schrödinger equation (3D-TDSE). At the intermediate internuclear distance, an unusual multi-peak structure is observed in the angular distribution, which is proved to be a signature of the transient localization of the electron upon alternating nucleus. By tracing the time-dependent ionization rate and bound state populations, we provide a clear evidence that the transient electron localization still exists in circularly polarized pulse and the corresponding multiple ionization bursts are directly mapped onto observable angular distributions. In addition, we introduce an intuitive strong-field approximation model which incorporates laser-induced subcycle internal electron dynamics to isolate the effect of the Coulomb potential of the parent ions. In this way, the timing of each ionization burst can be directly read out from the angular distributions. Our results suggest that the ionization time serves as a sensitive tool encoding intramolecular electron dynamics and can be measured using attoclock technique.

Elliptical isolated attosecond-pulse generation from an atom in a linear laser field

We theoretically demonstrate a scheme to produce elliptically polarized isolated attosecond pulses, based on the high-order harmonic generation from the current-carrying state of an atom exposed in linearly polarized laser fields. It is shown that the nonzero ellipticity of the attosecond pulse generated at the single-atom level is attributed to the nonvanishing angular momentum of the state, which is further elucidated by the strong-field approximation model. With specific carrier envelope phases (CEPs), isolated attosecond pulses with large ellipticity can be obtained. Moreover, the ellipticity of high-order harmonics remains almost invariable versus the CEP of the few-cycle driving field. Hence, the temporal profile and ellipticity of the attosecond-pulse radiation can be manipulated separately, which is beneficial to shape the attosecond-pulse radiation without compromising the ellipticity. The obtained elliptical attosecond pulse may serve as a potential tool to explore and manipulate the ultrafast dynamics in magnetic materials and chiral media.

Semiclassical analysis of photoelectron interference in a synthesized two-color laser pulse

We measure the photoelectron energy spectra from strong-field ionization of Kr in a two-color laser pulse consisting of a strong 400-nm field and a weak 800-nm field. The intensities of the main above-threshold ionization (ATI) and sideband peaks in the photoelectron energy spectra oscillate roughly oppositely with respect to the relative phase between the two-color components. We study the photoelectron interferometry in strong-field ATI regime from the view of interference of different electron trajectories in order to extend RABBITT type analysis to the strong-field regime. Based on the strong-field approximation model, we obtain analytical expressions for the oscillations of both ATI and sideband peaks with the relative phase. A phase shift of \pi/4 with respect to the field maximum of the two-color laser pulse is revealed for the interference maximum in the main ATI peak without including the effect of the atomic potential.

Strong-field-approximation model for coherent extreme-ultraviolet emission generated through frustrated tunneling ionization

An atom can be excited through tunneling when it is exposed to a strong laser field. This excitation process is known as frustrated tunneling ionization (FTI). Recently it has been reported that coherent extreme-ultraviolet emission can be achieved through FTI. In the present work we extend the strong-field-approximation (SFA) model developed by Lewenstein et al. to derive a quantum mechanical description of the FTI emission. An analytic expression is derived to describe FTI emission as four steps: tunneling of an electron from the ground state of an atom, acceleration of the electron in the continuum, recombination into the exited state of the parent atom, and free induction decay. The coherence property of FTI emission is studied using the attosecond lighthouse method, in which a spatially chirped laser beam generates FTI emissions in different directions. The spatial distribution of the FTI emission in the far field is analyzed using both the extended SFA model and the time-dependent Schr\odinger equation model. The results obtained from the two models agree well, thereby confirming the validity of our extended SFA model.

Photoelectron longitudinal momentum distributions containing nondipole effects**Project supported by the National Natural Science Foundation of China (Grant No. 11274149) and the Program of Shenyang Key Laboratory of Optoelectronic Materials and Technology, China (Grant No. F12-254-1-00).

This paper proposes a modified strong field approximation model for evaluating nondipole effects on the ionization of an atom in an intense laser field. The photoelectron longitudinal momentum distributions (PLMD) of a hydrogen-like atom exposed to a mid-infrared laser field is calculated. The theoretical results indicate an obvious asymmetry in the PLMD, and an offset of the PLMD peak appears in the opposite direction of the beam propagation due to nondipole effects. The peak offsets of the PLMD increased with the laser intensity, imposed by the initial state of the hydrogen-like atom.

Angle-Resolved Electron Spectra of Ions by Few-Cycle Laser Pulses**Supported by the National Natural Science Foundation of China under Grant Nos 11647150, 11464026, 11664035 and 11364038, the Young Talents Program of Gansu Province in 2016, the Scientific Research Program of the Higher Education Institutions of Gansu Province under Grant No 2016A-068, and the Doctoral Scientific Research Foundation of Lanzhou City University

The above-threshold detachment of ions induced by a linearly polarized few-cycle laser pulse is investigated theoretically using the strong-field approximation model without considering the rescattering mechanism. We first derive an analytical form of transition amplitude for describing the strong-field photodetachment of ions. The integration over time in transition amplitude can be performed using the numerical integration method or the saddle-point (SP) method of Shearer et al. [Phys. Rev. A 88 (2013) 033415]. The validity of the SP method is carefully examined by comparing the energy spectra and photoelectron angular distributions (PADs) with those obtained from the numerical integration method. By considering the volume effect of a focused laser beam, both the energy spectra and the low-energy PADs calculated by the numerical integration method agree very well with the experimental results.

Ellipticity dependence of neutral Rydberg excitation of atoms in strong laser fields

Rydberg state excitation (RSE) of different atoms in elliptically polarized strong 800 nm laser fields is investigated experimentally, and the results are compared with calculations of the strong-field approximation (SFA) model and the semiclassical model. It is observed that the RSE probability declines with increasing laser ellipticity for all of the He, Ar, and Kr atoms. While the measured ellipticity dependence of He RSE is very consistent with the predictions of both the SFA and semiclassical calculations, the width of the ellipticity dependence for Ar and Kr atoms is wider than that of the SFA model but closer to the semiclassical calculations. Analysis indicates that unlike a tunneling-plus-rescattering process, the decline of the RSE yield with increasing ellipticity can be attributed to a decrease of electrons with low kinetic energy that could be captured in the Rydberg states by the Coulomb potential. It indicates that the atomic RSE process could be related to the very low or near-zero energy structure in the photoelectron spectrum in strong laser fields, which would stimulate further experimental and theoretical studies to reveal their underlying mechanisms.

Broadband XUV supercontinuum generation by time-dependent phase-matching

We investigate the influence of the time-dependent on-axis phase matching on broadband extreme ultraviolet (XUV) supercontinuum generation with an intense laser field. By using the strong-field approximation model and solving Maxwell’s equations we show that time-dependent on-axis phase matching can lead to broadening of the XUV supercontinuum. Moreover, we also demonstrate that the bandwidth of the XUV supercontinuum can be controlled by adjusting the beam waist of the laser field and the position of the gas medium.

Above-threshold ionization and photoelectron spectra in atomic systems driven by strong laser fields

Above-threshold ionization (ATI) results from strong field laser-matter interaction and it is one of the fundamental processes that may be used to extract electron structural and dynamical information about the atomic or molecular target. Moreover, it can also be used to characterize the laser field itself. Here, we develop an analytical description of ATI, which extends the theoretical Strong Field Approximation (SFA), for both the direct and re-scattering transition amplitudes in atoms. From a non-local, but separable potential, the bound-free dipole and the re-scattering transition matrix elements are analytically computed. In comparison with the standard approaches to the ATI process, our analytical derivation of the re-scattering matrix elements allows us to study directly how the re-scattering process depends on the atomic target and laser pulse features -we can turn on and off contributions having different physical origins or corresponding to different physical mechanisms. We compare SFA results with the full numerical solutions of the time-dependent Schroedinger equation (TDSE) within the few-cycle pulse regime. Good agreement between our SFA and TDSE model is found for the ATI spectrum. Our model captures also the strong dependence of the photoelectron spectra on the carrier envelope phase of the laser field.

Development of nonperturbative nonlinear optics models including effects of high order nonlinearities and of free electron plasma: Maxwell–Schrödinger equations coupled with evolution equations for polarization effects, and the SFA-like nonlinear optics model

This paper is dedicated to the exploration of non-conventional nonlinear optics models for intense and short electromagnetic fields propagating in a gas. When an intense field interacts with a gas, usual nonlinear optics models, such as cubic nonlinear Maxwell, wave and Schrödinger equations, derived by perturbation theory may become inaccurate or even irrelevant. As a consequence, and to include in particular the effect of free electrons generated by laser–molecule interaction, several heuristic models, such as UPPE, HOKE models, etc, coupled with Drude-like models [, ], were derived. The goal of this paper is to present alternative approaches based on non-heuristic principles. This work is in particular motivated by the on-going debate in the filamentation community, about the effect of high order nonlinearities versus plasma effects due to free electrons, in pulse defocusing occurring in laser filaments [–]. The motivation of our work goes beyond filamentation modeling, and is more generally related to the interaction of any external intense and (short) pulse with a gas. In this paper, two different strategies are developed. The first one is based on the derivation of an evolution equation on the polarization, in order to determine the response of the medium (polarization) subject to a short and intense electromagnetic field. Then, we derive a combined semi-heuristic model, based on Lewenstein’s strong field approximation model and the usual perturbative modeling in nonlinear optics. The proposed model allows for inclusion of high order nonlinearities as well as free electron plasma effects.

The study of survival for tunneling ionized atom in an elliptically polarized laser field

By using the method of three-dimensional semiclassical electron ensemble simulation, we study the survival mechanism of tunneling ionized atom in an elliptically polarized field. It is found that for atoms with bigger ionization potential such as helium, the unionized atoms are mainly caused by the balancing of electron initial transverse velocity with the drifting velocity. For atoms with smaller ionization potential such as magnesium, Coulomb attraction plays the major role in the survival of atoms. We find a survival window to reflect the effect of Coulomb force. Result from this theory is closer to the three-dimensional semiclassical simulation than the strong-field approximation model. In addition, the distribution of survival yield on ellipticity for atoms with smaller atoms is sensitive to laser intensity which can be used to measure the intensity of super-strong laser pulses.

Selective generation of an intense single harmonic from a long gas cell with loosely focusing optics based on a three-color laser field

The selective generation of an intense single harmonic has been experimentally achieved in argon from a long gas cell with loosely focusing optics using a three-color laser field. When compared with the single harmonic emission from a continuous gas jet, both the intensity and the purity of the selected single harmonic emission from the long gas cell show dramatic improvements; the peak intensity is more intense by as much as 1–2 orders of magnitude, while the contrast ratio (i.e., the spectral purity) is simultaneously increased by several times. The underlying physics of this enhancement can be explained using the strong field approximation model with the propagation effect.

Survival window for atomic tunneling ionization with elliptically polarized laser fields

We find a fraction of atoms remain unionized after the laser pulse when the tunneled electrons are released in a certain window of initial field phase and transverse velocity. The survival window shifts with laser polarization ellipticity and its width varies with respect to laser intensity and atomic ionization potential. Neutral atom yield can be calculated by summing up tunneling probabilities in the window. Our theory can quantitatively reproduce the distribution of the survival yields vs laser ellipticity observed for helium in experiment. For other atom species with smaller ionization potential such as magnesium, our theory predicts a wider distribution than the strong-field approximation model while closer to the three-dimensional semiclassical electron ensemble simulations, indicating the important role of the Coulomb effects.

Elliptical High-Order Harmonic Generation from H+2 in Linearly Polarized Laser Fields

We investigate the elliptical high-order harmonic generation (HHG) from the ground state of H+2 subjected to linearly polarized laser fields, by numerically solving the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) and using the strong-field approximation (SFA) models. Highly elliptical HHG is yielded at intermedial alignment angles from the TDSE, while the standard SFA model fails to predict this. By including the coulomb potential and the stark shift corrections, we yield qualitative agreement results with the TDSE. The comparisons show that in the description of elliptical HHG, both the coulomb potential and the stark-shift are necessary.