Strong-field Approximation Research Articles

Unifying Tunneling Pictures of Strong-Field Ionization with an Improved Attoclock.

We demonstrate a novel attoclock, in which we add a perturbative linearly polarized light field at 400nm to calibrate the attoclock constructed by an intense circularly polarized field at 800nm. This approach can be directly implemented to analyze the recent hot and controversial topics involving strong-field tunneling ionization. The generally accepted picture is that tunneling ionization is instantaneous and that the tunneling probability synchronizes with the laser electric field. Alternatively, recently it was described in the Wigner picture that tunneling ionization would occur with a certain of time delay. We unify the two seemingly opposite viewpoints within one theoretical framework, i.e., the strong-field approximation (SFA). We illustrate that both the instantaneous tunneling picture and the Wigner time delay picture that are derived from the SFA can interpret the measurement well. Our results show that the finite tunneling delay will accompany nonzero exit longitudinal momenta. This is not the case for the instantaneous tunneling picture, where the most probable exit longitudinal momentum would be zero.

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.

General structure of gauge boson propagator and its spectra in a hot magnetized medium

Based on transversality condition of gauge boson self-energy we have systematically constructed the general structure of the gauge boson two-point functions using four linearly independent basis tensors in presence of a nontrivial background i.e. hot magnetized material medium. The hard thermal loop approximation has been used for the heat bath to compute various form factors associated with the gauge boson’s two point functions both in strong and weak field approximation. We have also analyzed the dispersion of a gauge boson (e.g., gluon) using the effective propagator both in strong and weak magnetic field approximation. The formalism is also applicable to QED. The presence of only thermal background leads to a longitudinal (plasmon) mode and a two fold degenerate transverse mode. In presence of a hot magnetized background medium the degeneracy of the two transverse modes is lifted and one gets three quasiparticle modes. In weak field approximation one gets two transverse modes and one plasmon mode. On the other hand, in strong field approximation also one gets the three modes in Lowest Landau Level. The general structure of two-point function may be useful for computing the thermo-magnetic correction of various quantities associated with a gauge boson.

Relativistic effects in the multiphoton ionization of hydrogenlike ions by ultrashort infrared laser pulses

The ionization of hydrogenlike ions in short few-cycle laser pulses of high intensity is investigated using the strong-field approximation. It is found that, for light ions, the Klein-Gordon or a relativistic Schr\odinger treatment agrees well with the results from a full Dirac description when an average over the electron spin is taken.

Contribution of Floquet-Bloch states to high-order harmonic generation in solids

We identify the contribution of Floquet-Bloch states to the high-order harmonic generation (HHG) in solids by numerically solving the time-dependent Schr\odinger equation for both a one-dimensional and a two-dimensional model. Results from the single $k$ point and the full Brillouin zone are compared to each other and the symmetry- breaking effect is discussed. We show that the observed phenomena can be explained under the framework of the Floquet-Bloch theory and the strong-field approximation, respectively. Our results indicate that the total yield of the harmonic radiation increases nonmonotonically with the intensity of the driving pulse. After a rough consideration of the focusing volume effects, we find that the yield of the harmonics shows a steplike structure, which is similar to several recent experimental observations. The present work can contribute to a better understanding of the channel-closing effect in the HHG of solids and may provide a way to detect the Floquet-Bloch bands in a laser field.

Above-threshold ionization of hydrogen atoms in a two-color laser field

By using the method of electron wave packet interference, we investigate the above-threshold ionization (ATI) of hydrogen atoms in a linearly polarized two-color laser pulse (λ1 = 800 nm and λ2 = 400 nm). Firstly, using the strong field approximation (SFA) with the time window function, we obtain the intracycle interference and intercycle interference patterns of hydrogen atoms in a 10 fs laser pulse, and it is found that the general characteristics of two-dimensional photoelectron momentum spectrum are formed by the interplay of the intracycle interference and intercycle interference, and the side rings in two-dimensional photoelectron momentum spectra originate from intercycle interference. Subsequently, we calculate the two-dimensional photoelectron momentum spectra of hydrogen atoms by SFA in full pulse. When the relative phase ϕ = 0, the intracycle interference and intercycle interference fringes in two-dimensional momentum spectrum are symmetrical, and when the relative phase is ϕ=0.5π, the intracycle interference fringes in the two-dimensional momentum spectrum are obviously asymmetrical. The results show that ionization is vitally dependent on relative phase of two laser pulses in two-color laser filed.

Multichannel Interference in Resonancelike Enhancement of High-Order Above-Threshold Ionization.

The intensity-dependent resonancelike enhancement phenomenon in the high-order above-threshold ionization spectrum is a typical quantum effect for atoms or molecules in an intense laser field, which has not been well understood. The calculations of the time-dependent Schrödinger equation (TDSE) are in remarkable agreement with the experimental data, but they cannot clarify the contributions of the bound states. The semiclassical approach of strong field approximation, in which no excited states are involved, can obtain a similar phenomenon, but the laser intensities of enhanced regions predicted by the strong field approximation are inconsistent with the results of the TDSE. In this Letter, a new fully quantum model is established from the TDSE without any excited states. Two types of enhanced structures, unimodal and multimodal structures, are found in the results of the TDSE and model. In addition, the calculations of the model reproduce the key features of the results of the TDSE. It shows that the excited states are not the key factor in the resonancelike enhancements in our calculated system, since there are no excited states in our model. Based on the calculations of our model, we show that such resonancelike enhancements are caused by the constructive interference of different momentum transfer channels. Last, the Fano-like line shapes are also discussed for the features of multichannel interference.

Two-color XUV+NIR femtosecond photoionization of neon in the near-threshold region

Results of angle-resolved electron spectroscopy of near-threshold photoionization of Ne atoms by combined femtosecond extreme ultraviolet and near infrared fields are presented. The dressed-electron spectra show an energetic distribution into so-called sidebands, being separated by the photon energy of the dressing laser. Surprisingly, for the low kinetic energy (few eV) sidebands, the photoelectron energy varies as a function of the emission angle. Such behavior has not yet been observed in sideband creation and has not been predicted in commonly used theoretical descriptions such as strong field approximation and soft photon approach. Describing the photoionization with a time-dependent Schrödinger equation allows a qualitative description of the observed effect, as well as the prediction of fine structure in the sideband distribution.

Air lasing from singly ionized N2 driven by bicircular two-color fields

Based on the strong-field approximation, we obtain analytical expressions for the initial momentum at the tunnel exit and instantaneous ionization rate of tunneling ionization in elliptically polarized laser fields with arbitrary ellipticity. The tunneling electron reveals a nonzero offset of the initial momentum at the tunnel exit in the elliptically polarized laser field. We find that the transverse and longitudinal components of this momentum offset with respect to the instantaneous field direction are directly related to the time derivatives of the instantaneous laser electric field along the angular and radial directions, respectively. We further show that the nonzero initial momentum at the tunnel exit has a significant influence on the laser phase dependence of the instantaneous ionization rate in the nonadiabatic tunneling regime.

Strong-field ionization dynamics of asymmetric equilateral triatomic model molecules in circularly polarized laser fields

The strong-field-driven ionization dynamics of larger, nondiatomic molecules still constitutes a major challenge for a theoretical and numerical description. While a full numerical treatment of all electronic and nuclear degrees of freedom is computationally prohibitive, several approaches exist to model photoelectron spectra employing certain approximations. Among them are classical trajectory methods or the so-called strong-field approximation (SFA). Here we investigate in detail the strong-field ionization dynamics of simple, noncollinear triatomic model systems. For a detailed characterization of the ionization process of our system, we combine quantum dynamical simulations, modified classical trajectory calculations, and extensions of the SFA in order to analyze the complex pattern of the photoelectron momentum distribution. We aim at disentangling the contributions of excited states and the long-range character of the potential. We show that upon interaction with circularly polarized laser fields, the long-range character merely induces a small shift in the spectra, while the contribution of excited states is in several cases essential: in particular in near-infrared laser fields, compared to mid-infrared drivers, and for systems with larger internuclear distances, when excited electronic states are energetically closer. Key modifications are for our trajectory model to explicitly incorporate the numerically obtained tunnel barrier width directly into the ionization rate, thereby enabling a spatially anisotropic ionization rate, and an approximate treatment of the contribution of excited states via the Stark shift. In contrast, for the SFA, it is important to include electronically excited states for the description of the initial state. We also discuss the effect of the carrier-envelope phase on the ionization dynamics, as well as the influence of molecular averaging.

Tomographic Extraction of the Internuclear Separation Based on Two-Center Interference with Aligned Diatomic Molecules.

We experimentally investigate the two-dimensional photoelectron momentum spectra of aligned diatomic molecules in an intense laser field. Our results reveal a novel prominent valley structure in the molecular alignment dependence of the high-energy photoelectron spectra along the laser polarization. Resorting to the molecular strong-field approximation and a simple semiclassical analysis, we show that this valley structure stems from the destructive two-center interference of the laser-driven rescattered electrons in diatomic molecules. Based on this two-center interference with aligned diatomic molecules, we demonstrate for the first time a tomographic method to extract the molecular internuclear separation, providing a more straightforward approach of molecular imaging, in comparison with, e.g., laser-induced electron diffraction and fixed-angle broadband laser-driven electron scattering.

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.

Strong-field tunneling ionization in the relativistic regime

We study the photoelectron energy and momentum distributions of strong-field tunneling ionization in the relativistic regime. Using the relativistic strong-field approximation, we have derived the position of the tunnel exit, the initial momentum distributions at the tunnel exit, and the instantaneous ionization rate when the electron tunnels through the Coulomb barrier in the relativistic regime. Since the tunneling electron energy is of the order of 1% of the rest of its mass, those nonadiabatic tunneling coordinates are quite different from the nonrelativistic case. We further incorporate the nonadiabatic relativistic tunneling coordinates into the classical trajectory Monte Carlo model and have calculated photoelectron energy and momentum distributions by accurately considering the field distribution in the focus of relativistic femtosecond laser pulses. We show that the nonadiabatic tunneling effect and the focal field distribution are very crucial to modeling strong-field ionization in the relativistic regime because the quiver radius of electrons can be as large as the beam waist.

Nondipole strong-field approximation for spatially structured laser fields

The strong-field approximation (SFA) is widely used to theoretically describe the ionization of atoms and molecules in intense laser fields. We here propose an extension of the SFA to incorporate nondipole contributions in the interaction between the photoelectron and the driving laser field. To this end, we derive Volkov-type continuum wave functions of an electron propagating in a laser field of arbitrary spatial dependence. Based on previous work by L. Rosenberg and F. Zhou [Phys. Rev. A 47, 2146 (1993)], we show how to construct such Volkov-type solutions to the Schr\odinger equation for an electron in a vector potential that can be written as an integral superposition of plane waves. These solutions are therefore not restricted to plane waves but are also appropriate to deal with more complex laser fields like twisted Bessel or Laguerre-Gaussian beams, where the magnetic field plays an important role. As an example, we compute photoelectron spectra in the above-threshold ionization of atoms with a single-mode plane-wave laser field of midinfrared wavelength. Especially, we demonstrate how peak offsets in the ${p}_{z}$ direction can be extracted that result from the nondipole nature of the interaction. Here, we find good agreement with previous theoretical and experimental studies for circular polarization and discuss differences for linear polarization.

Fast retrieval of temporal characteristics of FEL pulses using streaking by THz field.

A simple and very fast method for reconstruction of temporal structure of linearly polarized FEL pulses from the THz streaking spectra of photoelectrons is suggested. The method is based on a quantum mechanical approach within the strong-field approximation. The method is suitable for online retrieval of the temporal characteristics of the FEL pulses. It can be applied for any photon frequency in a broad range of FEL pulse duration with a proper selection of the streaking field. To enhance its accuracy, it is suggested to simultaneously analyze the streaking spectra for several emission angles.

Atom-Volkov strong-field approximation for above-threshold ionization

We introduce an approximation for strong-laser-field-induced ionization of atoms, which we call atom-Volkov strong-field approximation (aV SFA). For the final state, instead of a plane wave (used in the standard well-known SFA) or a Coulomb wave (used in the Coulomb-Volkov SFA), in the aV SFA we use the exact atom-specific scattering states. We first introduce the aV SFA as an ansatz and show that it yields results that are closer to the exact results than those obtained using the SFA. After that, we systematically derive an iterative theory whose zeroth order is equal to the introduced aV SFA. We analyze its first-order correction, which corresponds to electron rescattering off the parent ion and show that, in the case of forward scattering, this correction leads to the low-energy structures in the photoelectron spectra which are observed in the experiments.

The influence of the driving-bicircular-field component intensities on the helicities of emitted high-order harmonics

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.

Coulomb-Corrected Strong Field Approximation without Singularities and Branching Points

The domain of validity of the Coulomb-Corrected Strong-Field Approximation (CCSFA) is going to be analyzed in relation to the semi-classical dynamics of electrons during ionization of hydrogen-like targets. Our analysis is limited to ionization driven by Ti-sapphire laser pulses with intensities up to roughly 1014 Wcm−2. For such parameters, the effects related to radiation pressure are small and the laser field can be described in the dipole approximation. By applying the Magnus expansion for the exact retarded electron propagator we obtain an effective action which is free of Coulomb singularities and branch points when the complex-time trajectories are used. Furthermore, we show that the classical action is exactly recovered as the asymptotic limit of its effective counterpart. The applicability of such limit is also discussed.

Electron Bessel States in High-Energy Ionization

The creation of electron Bessel states of large topological charge is discussed in the context of high-energy photoionization. We show that, for short (three cycle) driving pulses of circular polarization and intensities of 5 × 1016 Wcm−2, it is possible to obtain electron vortex states with orbital angular momentum of few thousands units of ℏ. Our analysis is based on the quasi-relativistic strong-field approximation where an additional term, the relativistic mass correction, is accounted for. The effect of this modification, together with the recoil corrections, over the energy spectra of photoelectrons is also analyzed.

Strong-Field Ionization of Linear Molecules by a Bichromatic Elliptically Polarized Laser Field with Coplanar Counterrotating or Corotating Components of Different Frequencies

We investigate strong-field ionization of linear molecules by a two-color laser field of frequencies rω and sω having coplanar counterrotating or corotating elliptically polarized components (ω is the fundamental laser field frequency and r and s are integers). Using the improved molecular strong-field approximation we analyze direct above-threshold ionization (ATI) and high-order ATI (HATI) spectra. More precisely, reflection and rotational symmetries of these spectra for linear molecules aligned in the laser-field polarization plane are considered. The reflection symmetries for particular molecular orientations, known to be valid for a bicircular field (this is the field with circularly polarized counterrotating components), are valid also for arbitrary component ellipticities. However, specific rotational symmetries that are satisfied for HATI by a bicircular field, are violated for an arbitrary elliptically polarized field with counterrotating components. For the corotating case and the N2 molecule we analyze molecular-orientation-dependent interferences and plateau structures for various ellipticities.