Quantitative Rescattering Research Articles

Theory of High Harmonic Generation for Probing Time-Resolved Large-Amplitude Molecular Vibrations with Ultrashort Intense Lasers

We present a theory that incorporates the vibrational degrees of freedom in a high-order harmonic generation (HHG) process with ultrashort intense laser pulses. In this model, laser-induced time-dependent transition dipoles for each fixed molecular geometry are added coherently, weighted by the laser-driven time-dependent nuclear wave packet distribution. We show that the nuclear distribution can be strongly modified by the HHG driving laser. The validity of this model is first checked against results from the numerical solution of the time-dependent Schrödinger equation for a simple model system. We show that in combination with the established quantitative rescattering theory this model is able to reproduce the time-resolved pump-probe HHG spectra of N(2)O(4) reported by Li et al. [Science 322, 1207 (2008)].

Imaging ultrafast dynamics of molecules with laser-induced electron diffraction

We introduce a laser-induced electron diffraction method (LIED) for imaging ultrafast dynamics of small molecules with femtosecond mid-infrared lasers. When molecules are placed in an intense laser field, both low- and high-energy photoelectrons are generated. According to quantitative rescattering (QRS) theory, high-energy electrons are produced by a rescattering process where electrons born at the early phase of the laser pulse are driven back to rescatter with the parent ion. From the high-energy electron momentum spectra, field-free elastic electron-ion scattering differential cross sections (DCS), or diffraction images, can be extracted. With mid-infrared lasers as the driving pulses, it is further shown that the DCS can be used to extract atomic positions in a molecule with sub-angstrom spatial resolution, in close analogy to the standard electron diffraction method. Since infrared lasers with pulse duration of a few to several tens of femtoseconds are already available, LIED can be used for imaging dynamics of molecules with sub-angstrom spatial and a few-femtosecond temporal resolution. The first experiment with LIED has shown that the bond length of oxygen molecules shortens by 0.1 Å in five femtoseconds after single ionization. The principle behind LIED and its future outlook as a tool for dynamic imaging of molecules are presented.

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.

Influence of gas pressure on high-order-harmonic generation of Ar and Ne

We study the effect of gas pressure on the generation of high-order harmonics where harmonics due to individual atoms are calculated using the recently developed quantitative rescattering theory, and the propagation of the laser and harmonics in the medium is calculated by solving the Maxwell's wave equation. We illustrate that the simulated spectra are very sensitive to the laser focusing conditions at high laser intensity and high pressure since the fundamental laser field is severely reshaped during the propagation. By comparing the simulated results with several experiments we show that the pressure dependence can be qualitatively explained. The lack of quantitative agreement is tentatively attributed to the failure of the complete knowledge of the experimental conditions.

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.

High-order above-threshold ionisation of atoms and negative ions: channel-closing effects and the low-frequency approximation

The so-called low-frequency approximation (LFA) is an improved version of the strong-field approximation. The LFA describes the rescattering step of high-order above-threshold ionisation by utilising the exact field-free scattering amplitude, which has to be calculated separately. We apply the LFA to high-order above-threshold detachment of the fluorine negative ion. By comparing the so-obtained angle- and energy-resolved spectra with the exact results obtained as solutions of the time-dependent Schrödinger equation we show that the LFA is a more adequate approximation than the so-called improved strong-field approximation used previously. The LFA is also superior to the recently introduced quantitative rescattering (QRS) model. We show this by applying the LFA to the analysis of the intensity-dependent enhancements in above-threshold ionisation spectra of argon atoms. These enhancements can be explained as channel-closing-induced effects. In particular, we have observed and explained the behaviour of these channel-closing enhancements for higher values of the electron emission angle.

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.

Elastic scattering and impact ionization by returning electrons induced in a strong laser field

We study elastic scattering and impact ionization by returning electrons induced in a strong laser field based on the recently developed quantitative rescattering theory. The high-energy angle-resolved photoelectron spectra of molecules in strong fields are attributed to elastic scattering of the returning electrons with the parent ion. We demonstrate that the high-energy photoelectron spectra can be used to retrieve the position of atoms in a molecule. We also investigate the nonsequential double ionization of an atom in intense laser fields which is partly due to the impact ionization of the parent ion by the returning electron.

High-order-harmonic generation using gas-phaseH2O molecules

We investigate high-order-harmonic generation of isotropically distributed gas-phase ${\mathrm{H}}_{2}$O molecules exposed to an intense laser field. The induced dipole of each individual molecule by the laser field is first calculated using the recently developed quantitative rescattering theory. In a thin medium, harmonic spectra generated coherently from all the molecules are then calculated by solving Maxwell's equation of propagation. By using accurate transition dipoles of ${\mathrm{H}}_{2}$O, we show that the harmonics in the lower plateau region are quite different from models that employ the simpler strong-field approximation. We also examine the magnitude and phase of the harmonics and their dependence on laser focusing conditions.

Real-time pulse length measurement of few-cycle laser pulses using above-threshold ionization

The pulse lengths of intense few-cycle (4-10 fs) laser pulses at 790 nm are determined in real-time using a stereographic above-threshold ionization (ATI) measurement of Xe, i.e. the same apparatus recently shown to provide a precise, real-time, every-single-shot, carrier-envelope phase measurement of ultrashort laser pulses. The pulse length is calibrated using spectral-phase interferometry for direct electric-field reconstruction (SPIDER) and roughly agrees with calculations done using quantitative rescattering theory (QRS). This stereo-ATI technique provides the information necessary to characterize the waveform of every pulse in a kHz pulse train, within the Gaussian pulse approximation, and relies upon no theoretical assumptions. Moreover, the real-time display is a highly effective tool for tuning and monitoring ultrashort pulse characteristics.

Single-shot carrier-envelope-phase-tagged ion-momentum imaging of nonsequential double ionization of argon in intense 4-fs laser fields

Single-shot carrier-envelope-phase (CEP) tagging is combined with a reaction mircoscope (REMI) to investigate CEP-dependent processes in atoms. Excellent experimental stability and data acquisition longevity are achieved. Using this approach, we study the CEP effects for nonsequential double ionization of argon in 4-fs laser fields at 750 nm and an intensity of $1.6\ifmmode\times\else\texttimes\fi{}{10}^{14}$ W/cm${}^{2}$. The Ar${}^{2+}$ ionization yield shows a pronounced CEP dependence which compares well with recent theoretical predictions employing quantitative rescattering theory [S. Micheau et al., Phys. Rev. A 79, 013417 (2009)]. Furthermore, we find strong CEP influences on the Ar${}^{2+}$ momentum spectra along the laser polarization axis.

Momentum spectra of electrons rescattered from rare-gas targets following their extraction by one- and two-color femtosecond laser pulses

We have used velocity-map imaging to measure the three-dimensional momenta of electrons rescattered from Xe and Ar following the liberation of the electrons from these atoms by 45 fs, 800 nm intense laser pulses. Strong structure in the rescattering region is observed in both angle and energy, and is interpreted in terms of quantitative rescattering (QRS) theory. Momentum images have also been taken with two-color (800 nm $+$ 400 nm) pulses on Xe targets. A strong dependence of the spectra on the relative phase of the two colors is observed in the rescattering region. Interpretation of the phase dependence using both QRS theory and a full solution to the time-dependent Schr\odinger equation shows that the rescattered electrons provide a much more robust method for determining the relative phase of the two colors than do the direct electrons.

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.

Numerical simulation of strong-field electron spectra of Xe in the rescattering region

We investigate the high-energy above-threshold ionization (HATI) plateau for Xe in a strong laser field at peak intensity of about $0.85\ifmmode\times\else\texttimes\fi{}{10}^{14} {\mathrm{W}/\mathrm{cm}}^{2}$ with wavelengths of 800, 1250, 1500, and 2000 nm. Numerical simulations are performed using a recently developed quantitative rescattering (QRS) model. According to the QRS model, the two-dimensional (2D) photoelectron momentum distribution can be treated as a product of the returning electron wave packet and the elastic differential cross section (DCS) for free electrons scattered with the parent ion. From the 2D momentum distributions, the HATI plateau can be obtained. By using different single-active electron potentials in the DCS calculations, we test the potential dependence of the high-energy plateau spectra. Good agreement between the simulated results and the experimental data confirms again the validity of the QRS model.

Complete real-time temporal waveform characterization of single-shot few-cycle laser pulses

A method for complete characterization of the waveform of individual few-cycle laser pulses is presented. By analyzing the ``left'' and ``right'' asymmetries of high-energy photoelectrons along the polarization axis using the recently developed quantitative rescattering theory, we show that the carrier-envelope phase (CEP), pulse duration, and peak intensity of each single-shot pulse can be readily retrieved. By CEP tagging each laser shot, the method permits the study of waveform-dependent processes be extended to relativistic beams and to wavelengths, where CEP stabilization is not yet possible.

High-resolution spatial and temporal microscopy with intense-laser-induced rescattering electrons

Following the recent developments in intense laser techniques, ultrafast phenomena of the order of a few femtoseconds to attoseconds have been discussed. We are developing new methods for ultrafast imaging for transient atomic and molecular systems. We apply the recently developed quantitative rescattering theory together with a fitting procedure to extract effective potentials as well as elastic differential cross sections of the target ions with free electrons within the single active electron model. Using experimental photoelectron spectra for rare gas atoms of Ne, Ar, Kr, and Xe, we show that the extracted atomic potentials are in good agreement with those obtained theoretically. The current method of retrieval does not require precise knowledge of the peak laser intensities. The results show that accurate charge distribution of target ions indeed can be retrieved from experimental photoelectron spectra generated by lasers, thus paving the way for using infrared laser pulses for dynamic chemical imaging of transient states of molecules with temporal resolution of few femtoseconds.

Quantitative rescattering theory for high-order harmonic generation from molecules

The quantitative rescattering theory (QRS) for high-order harmonic generation (HHG) by intense laser pulses is presented. According to the QRS, HHG spectra can be expressed as a product of a returning electron wave packet and the photorecombination differential cross section of the laser-free continuum electron back to the initial bound state. We show that the shape of the returning electron wave packet is determined mostly by the laser. The returning electron wave packets can be obtained from the strong-field approximation or from the solution of the time-dependent Schr\odinger equation (TDSE) for a reference atom. The validity of the QRS is carefully examined by checking against accurate results for both harmonic magnitude and phase from the solution of the TDSE for atomic targets within the single active electron approximation. Combining with accurate transition dipoles obtained from state-of-the-art molecular photoionization calculations, we further show that available experimental measurements for HHG from partially aligned molecules can be explained by the QRS. Our results show that quantitative description of the HHG from aligned molecules has become possible. Since infrared lasers of pulse durations of a few femtoseconds are easily available in the laboratory, they may be used for dynamic imaging of a transient molecule with femtosecond temporal resolutions.

Retrieval of target photorecombination cross sections from high-order harmonics generated in a macroscopic medium

We investigate high-order harmonic generation (HHG) in a thin macroscopic medium by solving Maxwell's equation using microscopic single-atom induced dipole moment calculated from the recently developed quantitative rescattering (QRS) theory. We show that macroscopic HHG yields calculated from QRS compared well with those obtained from solving the single-atom time-dependent Schr\"odinger equation but with great saving of computer time. We also show that macroscopic HHG can be expressed as a product of a ``macroscopic wave packet'' and the photorecombination cross section of the target gas. The latter enables us to extract target structure from the experimentally measured HHG spectra, thus paves the way to use few-cycle infrared lasers for time-resolved chemical imaging of transient molecules with few-femtosecond temporal resolution.

Retrieval of experimental differential electron–ion elastic scattering cross sections from high-energy ATI spectra of rare gas atoms by infrared lasers

Based on the concept of the recently developed quantitative rescattering theory for the momentum distributions of high-energy photoelectrons generated by infrared lasers, we applied the theory to extract large-angle elastic differential cross sections (DCS) of the target ions with free electrons. Using experimental photoelectron spectra for rare gas atoms of Ne, Ar, Kr and Xe, we showed that the extracted DCS are in good agreement with the DCS calculated theoretically. The current method of retrieval does not require precise knowledge of the peak laser intensities. The results show that accurate DCS between electron–ion scattering indeed can be retrieved from experimental photoelectron spectra generated by lasers, thus paving the way for using infrared laser pulses for dynamic chemical imaging of transient molecules with temporal resolution of few femtoseconds.

Quantitative rescattering theory for laser-induced high-energy plateau photoelectron spectra

A comprehensive quantitative rescattering (QRS) theory for describing the production of high-energy photoelectrons generated by intense laser pulses is presented. According to the QRS, the momentum distributions of these electrons can be expressed as the product of a returning electron wave packet with the elastic differential cross sections (DCS) between free electrons with the target ion. We show that the returning electron wave packets are determined mostly by the lasers only and can be obtained from the strong field approximation. The validity of the QRS model is carefully examined by checking against accurate results from the solution of the time-dependent Schr\"odinger equation for atomic targets within the single active electron approximation. We further show that experimental photoelectron spectra for a wide range of laser intensity and wavelength can be explained by the QRS theory, and that the DCS between electrons and target ions can be extracted from experimental photoelectron spectra. By generalizing the QRS theory to molecular targets, we discuss how few-cycle infrared lasers offer a promising tool for dynamic chemical imaging with temporal resolution of a few femtoseconds.