Coulomb Focusing Research Articles

Coulomb focusing in attosecond angular streaking

Angular streaking technique employs a close-to-circularly polarized laser pulse to build a mapping between the instant of maximum ionization and the most probable emission angle in the photoelectron momentum distribution, thereby enabling the probe of laser-induced electron dynamics in atoms and molecules with attosecond temporal resolution. Here, through the jointed experimental observations and improved Coulomb-corrected strong-field approximation statistical simulations, we identify that electrons emitted at different initial ionization times converge to the most probable emission angle due to the previously-unexpected Coulomb focusing triggered by the nonadiabatic laser-induced electron tunneling. We reveal that the Coulomb focusing induces the observed nonintuitive energy-dependent trend in the angular streaking measurements on the nonadiabatic tunneling, and that tunneling dynamics under the classically forbidden barrier can leave fingerprints on the resulting signals. Our findings have significant implications for the decoding of the intricate tunneling dynamics with attosecond angular streaking.

Laser-assisted radiative recombination in a cold hydrogen plasma

Abstract We study the process of laser-assisted radiative recombination of an electron with a proton in a cold hydrogen plasma employing the semiclassical Kramers’ approach which involves calculation of classical trajectories in combined laser and Coulomb fields and the use of the correspondence principle. Due to the Coulomb focusing effect, recombination is the most effective when the initial electron momentum is parallel to the laser polarization. Orders of magnitude enhancement of the cross section, as compared to the laser-free case, is observed in this case. With increasing angle between the electron momentum and polarization, the recombination cross section drops. However, even after averaging over Maxwellian velocity distribution we obtain a substantial enhancement of the recombination rate constant, as compared to the zero-field case. For the field intensities in the range 30–350 MW cm−2, the enhancement occurs in the region of the radiation wavelength from 5 to 20 µm and for the plasma temperature from 20 to 300 K.

Wavelength Dependence of Atomic Excitation for Ar Subject to Intense Midinfrared Laser Pulses

We report experimental and theoretical investigations of wavelength dependence of Rydberg state excitation (RSE) process of Ar subject to intense laser fields. By simultaneously measuring ionization and RSE yields of Ar atoms subject to strong laser fields at a series of wavelengths, we obtain the wavelength scaling law of the ratio of Ar* over Ar+ with respect to the laser intensity, and this result can be well reproduced by a nonadiabatic model, but not by the classical-trajectory Monte Carlo model. Our results indicate that the nonadiabatic corrections of the photoelectron tunneling exit and tunneling probability play a significant role at shorter wavelengths. Analysis shows that the wavelength dependence phenomenon is due to the interplay of the nonadiabatic effect, wave-packet diffusion and Coulomb focusing effect of the liberated electron.

Features of the molecular orbital in the photoelectron momentum distribution using elliptically polarized laser fields

We study the photoelectron momentum distribution from strong-field tunneling ionization of two molecules with similar highest occupied molecular orbitals, i.e., ${\mathrm{O}}_{2}$ and ${\mathrm{CO}}_{2}$. We find that the tiny difference between those two molecular orbitals can be clearly identified from the photoelectron momentum distributions in an elliptically polarized laser field, while it can hardly be identified in a linearly polarized laser field due to the strong Coulomb focusing effect. Furthermore, we show that the momentum distribution in the elliptically polarized laser field can be used to reveal how the molecular orbital affects the ionization time distribution of the photoelectron.

Strong-field approximation with leading-order nondipole corrections

Employing the nondipole Volkov states, i.e., the solutions of the time-dependent Schr\"odinger equation for an electron in a laser field whose spatial dependence is accounted for by including the first-order term in a $1/c$ expansion of the vector potential, we present an analytic approach based on the strong-field approximation that is able to describe ionization and detachment in intense midinfrared laser fields. Our approach includes both the direct transition amplitude and the improved strong-field approximation transition amplitude, accounting for the field-induced rescattering of the liberated electron on its parent core. We apply the theory to strong-field detachment from the ${F}^{\ensuremath{-}}$ ion. Comparing the results to those obtained invoking the dipole approximation, we identify and analyze the effects caused by including the nondipole corrections for the cases of circular and linear polarization. We demonstrate that the introduced nondipole corrections induce asymmetries in photoelectron momentum distributions along the propagation direction. For the case of linear polarization, we report a shift of the low-energy region of the momentum distribution against the propagation direction, a feature previously studied on neutral targets and related to Coulomb focusing effects.

Laser-assisted charge transfer in positronium collisions with protons and antiprotons

We study the process of laser-assisted charge transfer in collisions of positronium atoms with protons (antiprotons) with formation of Rydberg hydrogen (antihydrogen) atoms by the use of classical trajectories Monte Carlo simulations of Ps-$p$ dynamics in a linearly polarized infrared field. We do not observe a drastic enhancement of the cross sections similar to that predicted before in laser-assisted electron bremsstrahlung and electron recombination since in the present charge-transfer process the Coulomb focusing effect is absent. Still we see a substantial enhancement up to a factor of 3 in the energy range between ${10}^{\ensuremath{-}4}$ and 0.1 eV for the field of intensity between 10 $\mathrm{MW}/{\mathrm{cm}}^{2}$ and 10 $\mathrm{GW}/{\mathrm{cm}}^{2}$. The effect depends weakly on the orientation of the incident Ps velocity relative to the field polarization vector. Rydberg states of the produced hydrogen atoms whose orbits are close to circular can survive against ionization by laser field and decay spontaneously to lower states. This is favorable for spectroscopic studies of antihydrogen atoms.

Enhanced chiral-sensitivity of Coulomb-focused electrons in strong field ionization

Strong-field light–matter interactions initiate a wide range of phenomena in which the quantum paths of electronic wavepackets can be manipulated by tailoring the laser field. Among the electrons released by a strong laser pulse from atomic and molecular targets, some are subsequently driven back to the vicinity of the ionic core by the oscillating laser field. The trajectories of these returning electrons are bent toward the core by the ionic potential, an effect known as Coulomb focusing. This process, studied over the past two decades, has been associated with the long range influence of the Coulomb potential. Here we explore the structural properties of the Coulomb focusing phenomenon. Specifically, we numerically study the sensitivity of the returning electron dynamics to the anisotropy of the ionic potential. We employ orthogonally polarized two-color strong fields and chiral molecules, whose asymmetric features lead to unambiguous fingerprints of the potential on the freed electrons. The Coulomb-focused electrons show an enhanced sensitivity to chirality, related to an asymmetric attoclock-like angular streaking stemming from field-assisted scattering of the electrons onto the chiral ionic potential. Anisotropic features of the ionic potential thus monitor the motion of Coulomb-focused electrons throughout their returning paths, shedding light on the structural properties of the interaction.

Semiclassical theory of laser-assisted dissociative recombination

We study the process of laser-assisted dissociative recombination of an electron with a molecular cation using a semiclassical approach. In the region outside a reaction sphere the electron motion in the combined laser and Coulomb fields is treated classically. Within the sphere the laser-field effects are neglected, and the recombination probability is obtained from quantum-mechanical cross sections calculated for the laser-free process. Specific calculations are performed for dissociative recombination of ${\mathrm{H}}_{2}^{+}$ in the field of the intensity 2.09 GW/${\mathrm{cm}}^{2}$ and the wavelength $22.8\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{m}$. In the energy region above 1 meV the cross section is significantly enhanced compared with the field-free case due to the Coulomb focusing effect. The influence of the indirect process due to electron capture into Rydberg states is also investigated. Although the Rydberg resonances are washed out due to the field effects, they influence significantly the magnitude of the dissociative recombination cross section.

Effect of Coulomb Focusing on the Electron–Atom Bremsstrahlung Cross Section for Tungsten and Iron in Nonthermal Lorentzian Plasmas

The Coulomb focusing effect on the electron–atom bremsstrahlung spectrum is investigated in nonthermal Lorentzian plasmas. The universal expression of the cross section of nonrelativistic electron–atom bremsstrahlung process is obtained by the solution of the Thomas-Fermi equation with the effective atomic charge. The effective Coulomb focusing for the electron–atom bremsstrahlung cross section near the threshold domain is also investigated by adopting the modified Elwert-Sommerfeld factor with the mean effective charge for the bremsstrahlung process. In addition, the bremsstrahlung emission rates are obtained by considering encounters between nonthermal electrons and atoms such as Fe and W atoms. We found that the bremsstrahlung emission rates for nonthermal electron–atoms are lower than those for thermal plasmas. Various nonthermal effects on the bremsstrahlung emission rates in Lorentzian plasmas are also discussed.

Role of nuclei movement in the tunneling ionization of hydrogen molecular ions beyond the dipole approximation

exposed to strong mid-infrared laser pulses undergoes the tunneling ionization, followed by the rescattering of the photoelectron with the parent ion. The two nuclei repel each other during the tunneling and rescattering, which modifies the Coulomb potential experienced by the rescattering electron. We numerically study the role of the nuclear movement in the tunneling ionization of hydrogen molecular ions. The classical trajectory Monte-Carlo simulations give intuitive explanations of photoelectron momentum distributions driven by laser pulses having different wavelengths and different intensities. The competition of the Coulomb attraction and the Lorentz push determines the photoelectron momentum distribution along the laser propagation direction. The nuclear movement modifies the Coulomb focusing, and thus shifts the peaks of the photoelectron momentum distribution along the laser propagation direction.

Transverse electron momentum distributions in strong-field ionization: nondipole and Coulomb focusing effects

In strong-field ionization, the combined influence of nondipole and Coulomb focusing effects leave a unique signature on the photoelectron momentum distributions, which may have direct implications on many important strong-field phenomena. We present a comprehensive review of the experimental and theoretical investigations on these effects. We classify these investigations by relativistic and non-relativistic modelling based on different theoretical approaches, e.g. classical, semiclassical, strong-field approximation, time-dependent Schrödinger equation, and time-dependent Dirac equation. Finally, we discuss the latest experimental and theoretical development on this subject that has undergone so far in order to deeply understand the underlying physics.

Semiclassical theory of laser-assisted radiative recombination

We study the process of laser-assisted dissociative recombination of an electron with a molecular cation using a semiclassical approach. In the region outside a reaction sphere the electron motion in the combined laser and Coulomb fields is treated classically. Within the sphere the laser-field effects are neglected, and the recombination probability is obtained from quantum-mechanical cross sections calculated for the laser-free process. Specific calculations are performed for dissociative recombination of ${\mathrm{H}}_{2}^{+}$ in the field of the intensity 2.09 GW/${\mathrm{cm}}^{2}$ and the wavelength $22.8\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{m}$. In the energy region above 1 meV the cross section is significantly enhanced compared with the field-free case due to the Coulomb focusing effect. The influence of the indirect process due to electron capture into Rydberg states is also investigated. Although the Rydberg resonances are washed out due to the field effects, they influence significantly the magnitude of the dissociative recombination cross section.

Gouy's Phase Anomaly in Electron Waves Produced by Strong-Field Ionization.

Ionization of atoms by strong laser fields produces photoelectron momentum distributions that exhibit modulations due to the interference of outgoing electron trajectories. For a faithful modeling, it is essential to include previously overlooked phase shifts occurring when trajectories pass through focal points. Such phase shifts are known as Gouy's phase anomaly in optics or as Maslov phases in semiclassical theory. Because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a focal point as it crosses the symmetry axis in momentum space. In addition, there exist observable Maslov phases already in two dimensions. Clustering algorithms enable us to implement a semiclassical model with the correct preexponential factor that affects both the weight and the phase of each trajectory. We also derive a simple rule to relate two-dimensional and three-dimensional models for linear polarization. It explains the shifted interference fringes and weaker high-energy yield in three dimensions. The results are in excellent agreement with solutions of the time-dependent Schrödinger equation.

Longitudinal photon-momentum transfer in strong-field double ionization of argon atoms

We investigate the longitudinal photon-momentum transfer in nonsequential double ionization of argon atoms exposed to a $2.0\ensuremath{-}\ensuremath{\mu}\mathrm{m}$ linearly polarized strong laser field. The zero momentum of the photoelectron is precisely calibrated using the photoelectron-photoion coincidence spectrum. The sum-longitudinal momentum of two photoelectrons from the double ionization noticeably deflects along the laser propagation direction, which is opposite to and much larger than that of single ionization. By tracking the electron trajectories for double ionization in semiclassical simulations, we find that the Coulomb focusing of the ionic core with the assistance of the Lorentz pushing of the laser field dominates the observed longitudinal momentum deflection. Our results open the possibility to control the photon-momentum transfer by manipulating the waveform of the laser fields and thus the Coulomb and magnetic effects experienced by the photoelectrons.

Classical theory of laser-assisted spontaneous bremsstrahlung

SynopsisAt low energies the classical trajectories for electron motion in a combined Coulomb and ac fields are affected by the Coulomb focusing. This leads to a strong enhancement of the laser-assisted bremsstrahlung cross section as compared to the field-free case.

Coulomb focusing in retrapped ionization with near-circularly polarized laser field.

The full three-dimensional photoelectron momentum distributions of argon are measured in intense near-circularly polarized laser fields. We observed that the transverse momentum distribution of ejected electrons by 410-nm near-circularly polarized field is unexpectedly narrowed with increasing laser intensity, which is contrary to the conventional rules predicted by adiabatic theory. By analyzing the momentum-resolved angular momentum distribution measured experimentally and the corresponding trajectories of ejected electrons semiclassically, the narrowing can be attributed to a temporary trapping and thereby focusing of a photoelectron by the atomic potential in a quasibound state. With the near-circularly polarized laser field, the strong Coulomb interaction with the rescattering electrons is avoided, thus the Coulomb focusing in the retrapped process is highlighted. We believe that these findings will facilitate understanding and steering electron dynamics in the Coulomb coupled system.

Strong-field photoelectron holography beyond the electric dipole approximation: A semiclassical analysis

Strong-field photoelectron holography denotes the interference of various electron paths in laser-induced ionization of atoms, leading to interference patterns in the final momentum distribution. For a quantitative description of holography beyond the electric dipole approximation and in the presence of the Coulomb potential, we develop a semiclassical model in which the initial conditions of outgoing electrons are set according to the beyond-dipole strong-field approximation for the ionization step. The phases associated with trajectories are evaluated following the prescription for semiclassical propagators. The validity of the method is confirmed by comparison to the numerical solution of the time-dependent Schr\"odinger equation in two spatial dimensions. The semiclassical model reproduces correctly the backward and forward shifts of the photoelectron momenta along the laser propagation axis that arise from beyond-dipole dynamics. The position of the central holographic interference fringe can be estimated already from a simplified Coulomb-free interference model that provides closed-form expressions for the beyond-dipole shifts. In three dimensions, Coulomb focusing causes a breakdown of the semiclassical model for final momenta with directions close to the polarization axis. We implement a beyond-dipole regularization procedure based on the concept of glory scattering, which was recently used to describe Coulomb focusing in the dipole approximation. While the position of the central maximum and higher-order fringes can already be obtained approximately by simpler semiclassical modeling, this glory model is able to predict the shape of the distribution in the close vicinity of the central maximum. The violation of the dipole approximation in holography should be observable with mid-infrared fields, for which the forward/backward shifts can be comparable with the fringe spacing.

Strong-field ionization of Ar atoms with a 45∘ cross-linearly-polarized two-color laser field

We measure the photoelectron momentum distributions of Ar atoms in cross-linearly-polarized two-color (800 nm+400 nm) laser field with an angle of ${45}^{\ensuremath{\circ}}$. The photoelectron momentum distributions are controlled by tuning the phase of the synthesized two-color fields. With semiclassical numerical simulation, we have analyzed the subcycle dynamics of the tunneling electron wave packets and have studied the Coulomb effect on the evolution of the tunneling electron wave packets. Our investigation indicates that the Coulomb effects are strongly dependent on the trace of the electric field after the tunneling. The Coulomb focusing effect and the Coulomb defocusing effects can be controlled on the subcycle timescale at this interaction geometry. This cross-linearly-polarized field could be widely used in strong-field physics, such as high-order harmonic generation.

Classical theory of laser-assisted spontaneous bremsstrahlung

We study the process of laser-assisted spontaneous electron bremsstrahlung by running classical trajectories in a combined Coulomb and laser (ac) fields. Due to chaotic scattering in the combined Coulomb and ac fields, the radiation probability as a function of the impact parameter and the constant phase of the laser field exhibits fractal structures. However, these structures are smeared out when the cross section is integrated over the impact parameter and averaged over the phase. We analyze the role of different types of orbits, including the trapped orbits, and the dependence of the radiation probability on the impact parameter and the initial phase of the ac field. We show that, at low incident electron kinetic energy, the Coulomb focusing leads to a substantial extension of the range of impact parameters contributing to the bremsstrahlung cross section and results in a substantial increase (by one to two orders of magnitude) of the cross section as compared with the pure Coulomb case. As examples, we discuss the case of relatively high ponderomotive energy ${E}_{p}$ when we obtain an efficient production of photons with frequencies up to $2{E}_{p}$, and the case of low ${E}_{p}$ when only infrared photons are produced. Overall accuracy of the classical approach is estimated to be very good, although it does not describe resonant processes studied previously by quantum-mechanical methods.

Inclusion of Coulomb effects in laser-atom interactions

We investigate the role of the Coulomb interaction in strong field processes. We find that the Coulomb field of the ion makes its presence known even in highly intense laser fields, in contrast to the assumptions of the strong field approximation. The dynamics of the electron after ionization is analyzed with four models for an arbitrary laser polarization: the Hamiltonian model in the dipole approximation, the strong field approximation, the Coulomb-corrected strong field approximation and the guiding center. These models illustrate clearly the Coulomb effects, in particular Coulomb focusing and Coulomb asymmetry. We show that the Coulomb-corrected strong field approximation and the guiding center are complementary, in the sense that the Coulomb-corrected strong field approximation describes well short time scale phenomena (shorter than a laser cycle) while the guiding center is well suited for describing long time scale phenomena (longer than a laser cycle) like Coulomb-driven recollisions and Rydberg state creation.