Open Access
Add to BibSonomy
Abstract
Within the framework of the improved quantitative rescattering (QRS) model, we simulate the correlated two-electron momentum distributions (CMDs) for nonsequential double ionization (NSDI) of Ar by elliptically polarized laser pulses with a wavelength of 788 nm at an intensity of 0.7 × 1014 W/cm2 for the ellipticities ranging from 0 to 0.3. Only the CMDs for recollision excitation with subsequent ionization (RESI) are calculated and the contribution from recollision direct ionization is neglected. According to the QRS model, the CMD for RESI can be factorized as a product of the parallel momentum distribution (PMD) for the first released electron after recollision and the PMD for the second electron ionized from an excited state of the parent ion. The PMD for the first electron is obtained from the laser-free differential cross sections for electron impact excitation of Ar+ calculated using state-of-the-art many-electron R-matrix theory while that for the second electron is evaluated by solving the time-dependent Schrödinger equation. The results show that the CMDs for all the ellipticities considered here exhibit distinct anticorrelated back-to-back emission of the electrons along the major polarization direction, and the anticorrelation is more pronounced with increasing ellipticity. It is found that anticorrelation is attributed to the pattern of the PMD for the second electron ionized from the excited state that, in turn, is caused by the delayed recollision time with respect to the instant of the external field crossing. Our work shows that both the ionization potential of the excited parent ion and the laser intensity play important roles in the process.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As a uniquely clean example of electron-electron correlation enforced by an external field, nonsequential double ionization (NSDI) has attracted wide interest both experimentally and theoretically for about the past four decades and still remains one of the most fundamental and attractive phenomena in strong-field laser physics (for a review, see [1]). Nowadays, it has been widely accepted that the main mechanisms of NSDI are the laser-induced recollisional direct ionization (RDI) and the recollision excitation with subsequent ionization (RESI) [2], which can be qualitatively interpreted by the quasiclassical recollision model [3,4]. In this model, after tunneling through the barrier of the combined atomic and field potential, the first electron may be accelerated and driven back by the oscillating laser field and dislodge a second electron by inelastic scattering with its parent ion. So far the most detailed information about NSDI that experimental measurements can provide are the correlated two-electron parallel momentum distributions (CMDs).
Since the first experiment conducted at the turn of this century, in which the CMDs for NSDI of Ar were measured by Weber et al. [5], a number of kinematically complete measurements have been performed [2,6–13]. Generally, the measurements of CMDs for NSDI by linearly polarized laser light demonstrate that the two electrons have a greater probability of being emitted in the same direction parallel to the laser polarization. However, at intensities below the recollision threshold, the measured CMDs for NSDI [10,11] exhibit anticorrelated behavior, indicating that the two electrons escape opposite to each other along the laser-field direction. To our knowledge, anticorrelation has only been observed in experiment for NSDI of Ar [10–12], whereas it was also predicted theoretically for NSDI of several other atoms including He [14–19].
Classical and semiclassical models have been widely used to unvail the mechanisms responsible for anticorrelation. Multiple recollisions, in the context of RESI, were first put forth to explain the back-to-back emission [10,14,20]. Interestingly, other theoretical investigations, based on the classical ensemble model, disproved that multiple recollisions cause the anticorrelated emission of the two electrons, because the simulated CMDs for the NSDI events, where there is only one recollision, also exhibit distinct anticorrelation [16,17]. In the framework of the classical model, it was also suggested that electron-electron repulsion [15] and Coulomb slingshot motion [18] could be possible mechanisms. In addition, based on the results of the numerical solution of the time-dependent Schrödinger equation (TDSE), Bondar et al. [21] argued that another mechanism, simultaneous electron emission, could also explain the anticorrelation of the electrons. On the other hand, simulations based on $S$-matrix theory showed that the interference between different channels of RESI could give rise to the anticorrelated behavior [22,23].
In light of the long-standing controversial debate on the mechanisms responsible for anticorrelation in NSDI, we recently calculated the CMD for NSDI of He atoms subject to an intense 400 nm linearly polarized laser pulse below the threshold intensity. We employed the improved quantitative rescattering model (QRS) in which the lowering of the threshold due to the presence of the electric field at the instant of recollision is taken into account [19]. The QRS model successfully predicted distinct anticorrelated back-to-back emission of the electrons along the polarization direction and an alternative mechanism responsible for the anticorrelation was suggested [19].
While anticorrelation in NSDI for linearly polarized laser fields has been extensively studied, the issue of elliptical polarization has been addressed much less. In the present paper, we employ the QRS model to simulate the CMDs for NSDI of Ar exposed to elliptically polarized laser pulses with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. Our results show that the CMDs along the major axis of elliptically polarized light with ellipticities ranging from 0 to 0.3 all exhibit clear back-to-back emission of the two electrons. In fact, the anticorrelation is more pronounced as the ellipticity increases. This, in turn, is found to be owing to the decreasing intensity along the major axis of elliptically polarized light.
This paper is organized as follows. In Sec. 2 we introduce the theoretical methods. In Sec. 3 we present and discuss the results. The conclusions are drawn in Sec. 4.
Atomic units ($\hbar =|e|=m=4\pi \epsilon _0=1$) are used unless otherwise specified.
2. Theoretical methods
The QRS model, based on the factorization formula [24–28], has been widely employed to deal with various laser-induced rescattering processes, including high-order above-threshold ionization (HATI) [29], high-order harmonic generation [30], and NSDI [31,32]. However, since the original QRS model provides the quantitative description based on the classical rescattering model [3,4], it predicts no NSDI events below the threshold intensity. This is not surprising because returning electrons are known to have a maximum classical kinetic energy of about $3.17 U_p$, where $U_p$ denotes the ponderomotive energy. Consequently, a minimum intensity is required for the rescattering electron to have enough energy to excite a core electron. Nevertheless, NSDI was observed to occur at intensities well below the expected threshold [33]. Accordingly, the QRS model was modified by taking into account the lowering of threshold due to the presence of an electric field at the time of the recollision to account for the total yields of doubly charged ions below the threshold intensity [34–36]. The simulated CMDs also showed in better agreement with experiment when calculated with the improved QRS model [37,38].
In this paper, we consider NSDI of Ar induced by elliptically polarized laser pulses given by
For the laser pulses with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ considered here, the intensity is well below the threshold. Since the maximum classical kinetic energy $E_r^{\rm max}$ of the returning electron is about 13 eV, RDI can be ruled out. However, based on the original QRS model, RESI cannot be ignited either since $E_r^{\rm max}$ is still smaller than the energy difference between the ground and first excited state of Ar$^+$. Therefore, one has to consider the lowering of the threshold such that the returning electron can donate more energy to the ionic electron. Practically, in the improved QRS model, we take the prescription proposed in [39] by assuming that the kinetic energy $E_r$ of the laser-induced returning electron with respect to the maximum of the barrier in the combined atomic and electric field potential at a recollision time $t_r$ corresponds to the incoming electron energy $E_i$ in the field-free case, i.e.,
where $\Delta E(t_r)$ is the lowered threshold (see Fig. 1 in [39]). It should be noted, however, that the energy level of an excited state is not lowered. For laser-induced recollision excitation, where $Z(=1)$ is the asymptotic charge of the residual ion seen by the scattered electron, while $F_z(t_r)$ is the electric field at the recollision time along the major polarization.
Fig. 1. (a) Electric fields along the major polarization used in the TDSE calculations with beginning times corresponding to $\omega t_r\,=\,30^{\circ }$ and $60^{\circ }$ marked by points 1 and 2, respectively. The red solid and black dotted lines represent the electric fields with $\omega t_r\,=\,0^{\circ }$ and $60^{\circ }$, respectively. (b, c): 2D photoelectron momentum distributions parallel and perpendicular to the laser polarization direction obtained from the TDSE for single ionization of Ar$^+$ from the excited $3d (m=0)$ sublevel in a 788 nm linearly polarized laser field with a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. (d, e): Parallel momentum distributions for the ionized electron obtained from (b) and (c), respectively.
According to the QRS model, the CMD of RESI for the momentum components $p_1^{||}$ and $p_2^{||}$ of the two outgoing electrons along the major polarization direction can be factorized as a product of the parallel momentum distributions (PMDs) $D^{\,{\rm exc}}(p_1^{||})$ and $D^{\,{\rm tun}}(p_2^{||})$ for the (first) returning electron after recollision and the (second) electron tunneling-ionized from an excited state of the parent ion, respectively. This gives
The PMD $D^{\,{\rm tun}}(p_2^{||})$ for the second electron can be obtained by solving the TDSE for an electron in an excited state of Ar$^+$. Within the dipole approximation and the length gauge, the TDSE reads
The probability amplitude for detecting an electron with momentum $\boldsymbol {p}_2$ at the end of the pulse is given by
where $\Phi _{\boldsymbol {p}_2}^{-}$ is the scattering out eigenstate of the field free Hamiltonian, satisfying the equationFrom the 3D momentum distribution, one obtains the two-dimensional (2D) momentum distribution parallel and perpendicular to the major polarization direction by performing the integral over the azimuthal angle $\phi _2$,
The energy spectra can be obtained by integrating over $\theta _2$ in Eq. (11),
The PMD of the second electron can then be obtained by integrating the 2D momentum distribution over the momentum component perpendicular to the major polarization axis,
The PMD of the first electron can be extracted from the DCS $d\sigma ^{\,{\rm exc}}/d\Omega _1$ for laser-free electron impact excitation of the parent ion. Previously, the first step to this end was to “project” the DCS onto incident direction by converting the DCS as a function of the scattering angle $\theta _1$ at each incident energy $E_i$ into the parallel momentum distribution using $k_{||}=k_f\cos \theta _1$ with an introduced prefactor $2\pi /k_f$, where $k_f=\sqrt {2(E_i-I_p^{\,{\rm exc}})}$ and $I_p^{\,{\rm exc}}$ is the excitation threshold (for details, see [19]). However, this method is inappropriate, since it leads to an unphysical abrupt ending of the parallel momentum distribution [19,38,43].
In the present paper, we take the same methodology for single ionization to simulate the PMD for the first returning electron after rescattering. With the prepared DCS for laser-free electron impact excitation, the total cross section (TCS) at each incident energy is equivalent to the total probability for single ionization at an energy represented by the energy spectra in Eq. (13). Consequently,
To get the 2D momentum spectra for laser-induced recollision excitation, the laser-free momentum distributions should be shifted by $-A_z(t_r)$, where $A_z(t_r)$ is the vector potential along the major polarization at the moment $t_r$ when the recollision takes place. Therefore,
Finally, the PMD for the first electron is obtained by
The RWP actually describes the momentum (energy) distribution of the laser-induced returning electron. Although the RWP cannot be measured directly, it can be obtained by the QRS model for HATI. Since all the laser-induced rescattering processes share the same process in which an electron was first freed by tunnel ionization and then driven back to the parent ion by the laser field, the RWP extracted from HATI can be applied to NSDI. In the QRS model, the RWP is expressed by [29]
where $D(p,\theta )$ is the momentum distribution for HATI photoelectrons, due to elastic scattering of the returning electron with the parent ion, with momentum of magnitude $p$ at a detection angle $\theta$ with respect to the major polarization of the laser field. Furthermore, ${\rm d}\sigma ^{\,{\rm el}}(k_r,\theta _r)/{\rm d}\Omega _r$ is the DCS for laser-free $e$-Ar$^+$ elastic scattering with a momentum of magnitude $k_r$ at an angle $\theta _r$ with respect to the direction of the returning-electron. The detected photoelectron momentum $\boldsymbol {p}$ and the momentum $\boldsymbol {k}_r$ of the rescattered electron are related approximately by withExplicitly, Eq. (22) can be written as [19]
in which the lowering of the threshold has been accounted for. In Eq. (20), the elastic scattering DCS is calculated within the plane-wave first-order Born approximation, and the momentum distribution $D(p,\theta )$ for high-energy photoelectrons is evaluated based on the improved strong-field approximation (SFA) for HATI [29]. In the present work, the SFA model is generalized for the elliptically polarized laser pulses. We note that analytical expressions of the RWP have been derived for elastic [25,27] and inelastic [28] rescattering processes at the caustic, where the long and short trajectories coalesce.3. Results and discussion
We aim to simulate the momentum distributions for the correlated two-electron for NSDI of Ar in 25 fs elliptically polarized laser pulses with ellipticities ranging from 0 to 0.3 at a wavelength of 788 nm and a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. For the laser conditions considered here, only the CMDs for RESI are simulated.
Following the numerical procedures presented in Sec. 2, we first evaluate the PMD for the second electron that is tunnel-ionized from an excited state of Ar$^+$ by solving the TDSE. Here, we assume that tunneling ionization begins at the recollision time $t_r$ immediately after recollision excitation takes place. According to Eq. (3), recollision excitation takes place only when
which indicates that the “effective” recollision time has to be delayed with respect to the field crossing. To control the time at which tunneling ionization begins, in the TDSE calculations we chose the envelope function $a(t)$ to be constant for the first six cycles within $t_r\leq t\leq 6\tau$, where $\tau$ denotes the period, and then ramped it down over four cycles. Specifically,Figure 1(a) displays the electric fields along the major polarization axis used in the TDSE calculations. The returning time $t_r$ is in the range of $0 - \tau /4$, and the carrier envelope phase $\phi$ is set to $\pi /2$ to account for the recollision process in which the laser-induced recolliding electron returns to the origin along the $-\hat {z}$ direction. It should be noted that the condition in Eq. (24) could also be satisfied if the laser-induced electron returns to the parent ion before the field crossing. Here the returning time $t_r$ is chosen to be after the field crossing, since the probability for the first electron to return after the field crossing is much larger than that before the field crossing [37]. This is also consistent with the recent experimental findings [44,45], which reveal that recollision-induced ionization prefers to occur after the field crossing. In Figs. 1(b) and 1(c), we show the 2D photoelectron momentum distributions parallel and perpendicular to the laser polarization direction from the TDSE with $\omega t_r\,=\,30^{\circ }$ and $60^{\circ }$, respectively, for single ionization of Ar$^+$ by linearly polarized laser field from the excited $3d (m=0)$ sublevel. By integrating the 2D momentum distributions over the perpendicular momentum, we obtain the PMD for the tunnel-ionized electron. These are displayed in Figs. 1(d) and 1(e). The important feature of the PMD is that they are not symmetric with respect to $p_2^{||}=0$. The asymmetry of the PMDs can be represented by the parameter $A=P_L/P_R$, where $P_L$ and $P_R$ are the total single-ionization yields of electrons with negative and positive momenta, respectively. As one can see from Figs. 1(d) and 1(e), the asymmetry parameter increases with increasing initial ionization time. The reason for this trend has been analyzed in detail in our previous work [19] by using the Ammosov-Delone-Krainov (ADK) model [46], which provides an intuitive interpretation of the tunnel-ionization process.
In the actual numerical calculations of the PMD for the tunnel-ionized electron, it is a tremendous challenge to consider all the effective recollisions possibly taking place at any time under the conditions given by Eq. (24). Therefore, in order to make the calculations tractable, we choose an “average” recollision (returning) time to account for the lowering of the potential due to the presence of an electric field at the instant of recollision. Here we use an approximate method given in Refs. [19,37] to determine the average returning time for RESI in the laser pulses with different ellipticities.
With the average returning time determined, we calculated the PMD for the electron ionized from Ar$^+$ in the excited states of $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$ for in 788 nm elliptically polarized laser fields at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ by solving the TDSE. For each excited state, the TDSE calculations are carried out for the initial states with magnetic quantum number $m$ with respect to the quantization axis of $x$, and the PMDs are summed over $m$. The results are shown in Fig. 2. Since the intensity of the laser field along the major polarization axis decreases with increasing ellipticity, generally more delayed recollision times are set for larger ellipticities and higher excited states. Again, as demonstrated below, the asymmetry feature of the PMD for the tunnel-ionized electron is of crucial significance in forming the anticorrelated CMD.
Fig. 2. Parallel momentum distributions obtained by solving the TDSE for the electron ionized from Ar$^+$ in the excited states of (a) $3s 3p^6$, (b) $3s^2 3p^4 3d$, (c) $3s^2 3p^4 4s$, and (d) $3s^2 3p^4 4p$ in 788 nm laser fields at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. The results are for the situation where the first electron returns to the parent ion along the $-\hat {z}$ direction with the ellipticity $\epsilon$ and the recollision time corresponding to $\omega t_r$ indicated.
In Fig. 3 we show the asymmetry parameters of the PMDs displayed in Fig. 2. We see that smaller asymmetry parameters are predicted for higher excited states. For the highest excited state $3s^2 3p^4 4p$ considered here, the asymmetry parameters for all ellipticities are less than 1, corresponding to the CMDs shown in Fig. 2(d) in which the ionization probability for $p_2^{||}<0$ is smaller than that for $p_2^{||}>0$. This can be interpreted in the context of tunnel ionization. Assuming that the electron tunnel ionizes with zero initial velocity, the momentum of the electron after tunnel ionization at the end of laser pulse is determined by $p_2^{||}=-A(t)$. Consequently, the ionization rates before (after) $\tau /4$ with $A(t)\,<\,0$ [$A(t)\,>\,0$] result in momentum distributions with $p_2^{||}\,>\,0$ ($p_2^{||}\,<\,0$) (see Fig. 8 in Ref. [19]). Suppose tunneling begins at $\omega t_r\,=\,0$ in the electric field shown in Fig. 1(a), the total yield before the laser field reaches its maximum value at $\tau /4$ is always larger than that in the subsequent $\tau /4$ due to the effect of depletion. This is also true for ionization of the electron from highly excited states with small ionization potentials, even if the initial ionization time (the recollision time) is delayed. It explains why the asymmetry parameter of the CMDs for tunnel ionization from $3s^2 3p^4 4p$ is always less than 1 even though it increases with increasing ellipticity. However, for ionization from lower excited states with higher ionization potentials, as the initial ionization time is sufficiently delayed, the total yield before the peak of the laser field at $\tau /4$ could be less than that in the subsequent $\tau /4$ interval. As a result, the asymmetry parameter for the PMD becomes greater than 1, as demonstrated in Fig. 3 for ionizations from Ar$^+$ in the excited states of $3s 3p^6$, $3s^2 3p^4 3d$, and $3s^2 3p^4 4s$. In principle, tunneling occurs more slowly for larger ionization potentials and lower intensities. Therefore, it is expected that the asymmetry parameter becomes larger for lower excited states and larger ellipticities. This is true for all excited states considered here, except for the lowest one, $3s 3p^6$. This is because the Keldysh parameter for ionization of Ar$^+$ in the $3s 3p^6$ state in the laser pulses considered here is $\gamma \simeq 1.3$, indicating that the tunneling picture is no longer strictly valid and multiphoton dynamics becomes more significant.
Fig. 3. Asymmetry of the parallel momentum distributions for the electron ionized from Ar$^+$ in the excited states $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$ in 788 nm laser fields at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. See text for details.
Now we turn to simulating the PMD $D^{\,{\rm exc}}(p_1^{||})$ in Eq. (5) for the first returning electron after recollision. To this end, we first evaluate the DCS for laser-free electron impact excitation of Ar$^+$ by using the state-of-the-art multi-electron $B$-spline $R$-matrix close-coupling theory [47,48]. Figure 4 displays the results for electron impact excitation of Ar$^+$ from the ground state to the excited states of $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$ at some selected incident energies below 30 eV. As expected, excitation to $3d$ dominates due to the large overlap of the $3p$ and $3d$ orbitals. While strong backscattering is predicted at low incident energies a few eV above the excitation threshold, forward scattering generally becomes dominant as the incident energy increases.
Fig. 4. Angle-differential cross sections $d \sigma ^{\,{\rm exc}}/d \Omega _1$ at selected incident energies below 30 eV for electron impact excitation of Ar$^+$ from the ground state to (a) $3s 3p^6$, (b) $3s^2 3p^4 3d$, (c) $3s^2 3p^4 4s$, and (d) $3s^2 3p^4 4p$, respectively.
With the prepared DCSs for electron impact excitation of Ar$^+$, by using Eq. (16) we obtain the 2D momentum distributions for the scattered electron after impact excitation of Ar$^+$ from the ground state to $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$, respectively. As displayed in Fig. 5, the 2D momentum distributions provide an overview of the general features regarding the excitation processes. For the energy regime considered here, while the 2D momentum distributions for excitations to $3s 3p^6$ and $3s^2 3p^4 4p$ exhibit more abundant structure, forward scattering dominates excitation to $3s^2 3p^4 4s$ at high energies. Interestingly, the 2D momentum distribution for excitation to $3s^2 3p^4 3d$ is almost uniformly distributed over both scattering angle and energy for $k_f>0.5$.
Fig. 5. Two-dimensional momentum distributions $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ parallel and perpendicular to the incident direction for the scattered electron after impact excitation of Ar$^+$ from the ground state to (a) $3s 3p^6$, (b) $3s^2 3p^4 3d$, (c) $3s^2 3p^4 4s$, and (d) $3s^2 3p^4 4p$, respectively.
Then the 2D momentum distributions $D^{\,{\rm exc}} (p^{\perp }_1,p^{||}_1)$ for the returning electron after recollision in RESI can be obtained from $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ by shifting the parallel momentum by $-A_z(t_r)$ at each of the incident energies using Eq. (17) in which another ingredient left to be evaluated is the RWP.
The relevant RWPs are extracted from the momentum distribution for HATI photoelectrons in elliptically polarized laser fields calculated within the SFA model under the assumption that the laser-induced electron returns to the parent ion along the direction of the major polarization axis. Figure 6 displays the RWP plotted against the kinetic energy of the laser-induced returning electron in elliptically polarized laser pulses with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$. As the ellipticity increases from 0 to 0.3, the RWP drops by an order of magnitude with a cutoff at smaller energy. Whereas the RWPs for all ellipticities exhibit similar structure, each decreases dramatically at low energies. With increasing energy, a plateau follows with some oscillations until a cutoff is reached.
Fig. 6. Recolliding wave packet plotted against the kinetic energy of the laser-induced returning electron in elliptically laser pulses with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$.
For the 2D momentum distributions $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ shown in Fig. 5, the incident direction is taken to be along the $+\hat {z}$ direction. Therefore, to get the 2D momentum distributions $D^{\,{\rm exc}} (p^{\perp }_1,p^{||}_1)$ for the scattering electron for the situation where the recolliding electron returns to the origin along the $-\hat {z}$ direction, the 2D momentum distributions $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ should be flipped with respect to $k^{||}_f\,=\,0$ before shifting the parallel momentum by $-A_z(t_r)$ where $A_z(t_r)$ is negative. The results for the ellipticities of 0 and 0.3 are displayed in Fig. 7. In contrast to $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ in Fig. 5, the distributions for different energies of the outgoing electron in $D^{\,{\rm exc}} (p^{\perp }_1,p^{||}_1)$ are no longer concentric, because the $D^{\,{\rm exc}} (k^{\perp }_f,k^{||}_f)$ are shifted by different drift momenta at different incident energies. In addition, forward scattering in Fig. 7, corresponds to the distributions with small $p^{||}_1$. Furthermore, the RWP also plays an important role in $D^{\,{\rm exc}} (p^{\perp }_1,p^{||}_1)$. For example, the roughly uniform distribution for $3s^2 3p^4 3d$ in Fig. 5(b) becomes a rainbow in Figs. 7(b) and 7(f) due to the large probabilities of the electron returning with energies around the cutoff in the RWP displayed in Fig. 6.
Fig. 7. Two-dimensional momentum distributions for the active electron after recolliding with Ar$^+$ and exciting the residual ground-state electron to the excited states of $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$ in elliptically polarized light with ellipticities of 0.0 (top row) and 0.3 (bottom row) at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ with a wavelength of 788 nm. The recolliding electron returns to the origin along the $-\hat {z}$ direction.
By integrating the 2D momentum distributions $D^{\,{\rm exc}} (p^{\perp }_1,p^{||}_1)$ over the $p^{\perp }_1$, we obtain the parallel momentum distributions $D^{\,{\rm exc}} (p^{||}_1)$ which are shown in Fig. 8. Except for $3s 3p^6$, larger probabilities are produced for small momenta, thus indicating strong forward scattering in the recollision excitations.
Fig. 8. Parallel momentum distributions for the active electron after recolliding with the Ar$^+$ ion and exciting the residual ground-state electron to the excited states of (a) $3s 3p^6$, (b) $3s^2 3p^4 3d$, (c) $3s^2 3p^4 4s$, and (d) $3s^2 3p^4 4p$ in elliptically polarized light with ellipticities of 0.0 and 0.25 at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ and a wavelength of 788 nm. The results are for the situation where the first electron returns to the parent ion along the $-\hat {z}$ direction with the ellipticity $\epsilon$ and the recollision time corresponding to $\omega t_r$ indicated.
With the evaluated PMDs for the (second) tunneling electron ionized from excited states and the (first) active electron after recollision displayed in Fig. 2 and Fig. 8, respectively, it is straightforward to simulate the CMD for RESI by using Eq. (5). Figures 9(a)–9(d) display the CMDs for RESI of Ar in a linearly polarized laser field with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ for the situation where the laser-induced electron returns to the parent ion along the direction of $-\hat {z}$. In the CMD, the distribution along $p_2^{||}$ directly reflects the momentum distribution of the tunnel-ionized electron. For example, the denser population in the lower-half plane of the CMD for $3s^2 3p^4 3d$ in Fig. 9(b) corresponds to the higher probability on the left side of the PMD in Fig. 2(b) in the momentum region of $p_2^{||}<0$.
Fig. 9. Correlated two-electron parallel momentum spectra for RESI of Ar in a linearly polarized laser field with a wavelength of 788 nm at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ for the excitation channels of $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$, respectively. Top row: The first electron returns to the parent ion along the $-\hat {z}$ direction; Bottom row: Symmetrized full space.
For relatively long pulses considered here, the laser-induced electron that is born during the other half cycle possesses the same probability to return to the parent ion along the $+\hat {z}$ direction. In addition, due to the indistinguishability of the two outgoing electrons, the CMD for RESI should be symmetric with respect to both diagonals $p_1^{||}=\pm p_2^{||}$. With this taken into account, we obtain the full-space CMD for RESI. The symmetrized full-space CMDs are illustrated in Figs. 9(e)–9(h). Note that the CMDs for the excitation channels of $3s 3p^6$ and $3s^2 3p^4 3d$ exhibit distinct anticorrelation while for $3s^2 3p^4 4p$ the side-by-side emission dominates over back-to-back emission.
Once the CMDs for all the considered excitation channels are calculated, the CMD for RESI can be obtained by summing up the CMD for each individual excitation channel. The CMDs for RESI of Ar including all the excitation channels of $3s 3p^6$, $3s^2 3p^4 3d$, $3s^2 3p^4 4s$, and $3s^2 3p^4 4p$ in a linearly polarized laser pulse is displayed in Fig. 10(a). The CMDs for RESI of Ar in elliptically polarized laser pulses are obtained in the same way. The results for ellipticities of $\epsilon =0.1$, 0.2, and 0.3 are displayed in Figs. 10(b)–10(d), respectively. Despite the slight discrepancies, anticorrelation dominates in all the CMDs for NSDI of Ar in elliptically polarized laser fields with ellipticities between 0.0 and 0.3. It should be noted that the contributions from different excited states are added incoherently.
Fig. 10. Correlated parallel momentum spectra for RESI of Ar in elliptically polarized light with ellipticities of (a) $\epsilon =0.0$, (b) $\epsilon =0.1$, (c) $\epsilon =0.2$, and (d) $\epsilon =0.3$ at a peak intensity of $0.7\times 10^{14}$ W/cm$^2$ and a wavelength of 788 nm.
To reveal the differences among the CMDs shown in Fig. 10 for various ellipticities, we now introduce another asymmetry parameter $R \equiv (N_2-N_1)/(N_2 +N_1)$, where $N_1$ and $N_2$ are the numbers of double ionization events in the first and second quadrants, respectively. These asymmetry parameters for electron correlation along the $z$ axis, extracted from the CMDs in Fig. 10, are displayed in Fig. 11. Generally, $R$ increases with increasing ellipticity $\varepsilon$, thus indicating that anticorrelation is more pronounced for larger ellipticities. However, the increase of the asymmetry parameter slows down significantly as the ellipticity increases from 0.2 to 0.3. This is partly due to the fact that the asymmetry of the parallel momentum distributions for the electron ionized from Ar$^+$ in the excited state of $3s 3p^6$ decreases as the ellipticity increases from 0.1 to 0.3 (cf. Fig. 3). This might indicate that the analysis based on the tunneling picture alone is no longer valid as the laser intensity decreases to the regime where multiphoton dynamics dominates the ionization of Ar$^+$ in the lowest excited state.
4. Summary and conclusions
We investigated the ellipticity dependence of anticorrelation in NSDI of Ar by elliptically polarized laser pulses with a wavelength of 788 nm at a peak intensity of $0.7 \times 10^{14}$ W/cm$^2$, which is well below the recollision threshold. The improved QRS model, in which the lowering of the threshold due to the presence of the electric field along the major polarization axis at the instance of recollision is taken into account, was employed to simulate the CMDs, which only consist of contributions from RESI. The simulated CMDs exhibit anticorrelation for all the ellipticities considered here. It is demonstrated that anticorrelation is more pronounced with increasing ellipticity.
The merit of the QRS model lies in the fact that the CMD for RESI can be factorized as a product of the PMDs for the two outgoing electrons. This advantage enables us to deal with each of these outgoing electrons individually. The PMD for the first electron is extracted from the DCSs for laser-free electron impact excitation of Ar$^+$, which were evaluated by employing the state-of-the-art multi-electron $B$-spline $R$-matrix close-coupling theory. For the second electron, which is freed by tunnel ionization from an excited state of Ar$^+$, the PMD is obtained by solving the TDSE. Anticorrelation in the CMD for RESI is attributed to the asymmetric distribution of the PMD for the second electron tunnel-ionized from excited states owing to a “delayed” recollision time with respect to the time of zero crossing of the electric field along the major polarization axis. At intensities well below the recollision threshold, the effective recollision is required to be delayed with more time for lower intensities. This results in larger asymmetry parameters for the PMD of the second electron, which, in turn, leads to more pronounced anticorrelation for larger ellipticities, since the intensity of the laser field along the major polarization axis decreases as the ellipticity increases.
Our simulations show that as the ellipticity increases from 0 to 0.3, the RWP drops by at least an order of magnitude, thereby indicating that the probability for recollision significantly decreases with increasing ellipticity. This is in agreement with earlier experimental findings that in elliptically polarized light the nonsequential double ionization rate is greatly reduced [49]. However, it should be noted that decreasing probability for recollision due to the increase of ellipticity does not necessarily lead to double ionization beyond the present recollision picture since the latter strongly depends on the maximum energy of the returning electron rather than the probability for recollision. The obtained RWPs clearly demonstrate that the maximum energy of the returning electron decrease by only about 1 eV when the ellipticity increases from 0 to 0.3. Therefore, the recollision picture remains valid at least for the ellipticities considered here.
Funding
Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515010047); National Science Foundation (OAC-1834740, PHY-090031, PHY-2110023); Japan Society for the Promotion of Science (21K03417, 22H00313); National Natural Science Foundation of China (11274219, 11974380).
Acknowledgments
The authors gratefully acknowledge the invaluable contributions by the late Dr. Oleg Zatsarinny who carried out the $B$-spline $R$-matrix calculations used in parts of this manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. W. Becker, X. J. Liu, P. J. Ho, and J. H. Eberly, “Theories of photoelectron correlation in laser-driven multiple atomic ionization,” Rev. Mod. Phys. 84(3), 1011–1043 (2012). [CrossRef]
2. B. Feuerstein, R. Moshammer, D. Fischer, A. Dorn, C. D. Schröter, J. Deipenwisch, J. R. C. Lopez-Urrutia, C. Höhr, P. Neumayer, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, and W. Sandner, “Separation of recollision mechanisms in nonsequential strong field double ionization of Ar: The role of excitation tunneling,” Phys. Rev. Lett. 87(4), 043003 (2001). [CrossRef]
3. J. Krause, K. Schafer, and K. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68(24), 3535–3538 (1992). [CrossRef]
4. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]
5. T. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature 405(6787), 658–661 (2000). [CrossRef]
6. R. Moshammer, J. Ullrich, B. Feuerstein, D. Fischer, A. Dorn, C. D. Schröter, J. R. Crespo López-Urrutia, C. Höhr, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Strongly directed electron emission in non-sequential double ionization of Ne by intense laser pulses,” J. Phys. B: At. Mol. Phys. 36(6), L113–L119 (2003). [CrossRef]
7. M. Weckenbrock, D. Zeidler, A. Staudte, Th. Weber, M. Schöffler, M. Meckel, S. Kammer, M. Smolarski, O. Jagutzki, V. R. Bhardwaj, D. M. Rayner, D. M. Villeneuve, P. B. Corkum, and R. Dörner, “Fully Differential Rates for Femtosecond Multiphoton Double Ionization of Neon,” Phys. Rev. Lett. 92(21), 213002 (2004). [CrossRef]
8. A. Staudte, C. Ruiz, M. Schöffler, S. Schössler, D. Zeidler, T. Weber, M. Meckel, D. M. Villeneuve, P. B. Corkum, A. Becker, and R. Dörner, “Binary and recoil collisions in strong field double ionization of helium,” Phys. Rev. Lett. 99(26), 263002 (2007). [CrossRef]
9. A. Rudenko, V. L. B. de Jesus, T. Ergler, K. Zrost, B. Feuerstein, C. D. Schröter, R. Moshammer, and J. Ullrich, “Correlated two-electron momentum spectra for strong-field nonsequential double ionization of He at 800 nm,” Phys. Rev. Lett. 99(26), 263003 (2007). [CrossRef]
10. Y. Liu, S. Tschuch, A. Rudenko, M. Dürr, M. Siegel, U. Morgner, R. Moshammer, and J. Ullrich, “Strong-field double ionization of Ar below the recollision threshold,” Phys. Rev. Lett. 101(5), 053001 (2008). [CrossRef]
11. Y. Liu, D. Ye, J. Liu, A. Rudenko, S. Tschuch, M. Dürr, M. Siegel, U. Morgner, Q. Gong, R. Moshammer, and J. Ullrich, “Multiphoton Double Ionization of Ar and Ne Close to Threshold,” Phys. Rev. Lett. 104(17), 173002 (2010). [CrossRef]
12. M. Kübel, K. J. Betsch, N. G. Kling, A. S. Alnaser, J. Schmidt, U. Kleineberg, Y. Deng, I. Ben-Itzhak, G. G. Paulus, T. Pfeifer, J. Ullrich, R. Moshammer, M. F. Kling, and B. Bergues, “Non-sequential double ionization of Ar: from the single- to the many-cycle regime,” New J. Phys. 16(3), 033008 (2014). [CrossRef]
13. K. Henrichs, S. Eckart, A. Hartung, D. Trabert, K. Fehre, J. Rist, H. Sann, M. Pitzer, M. Richter, H. Kang, M. S. Schöffler, M. Kunitski, T. Jahnke, and R. Dörner, “Multiphoton double ionization of helium at 394 nm: A fully differential experiment,” Phys. Rev. A 98(4), 043405 (2018). [CrossRef]
14. S. L. Haan, Z. S. Smith, K. N. Shomsky, and P. W. Planting, “Anticorrelated electrons from weak recollisions in nonsequential double ionization,” J. Phys. B: At. Mol. Phys. 41(21), 211002 (2008). [CrossRef]
15. Y. Zhou, C. Huang, and P. Lu, “Coulomb-tail effect of electron-electron interaction on nonsequential double ionization,” Phys. Rev. A 84(2), 023405 (2011). [CrossRef]
16. X. Ma, Y. Zhou, and P. Lu, “Multiple recollisions in strong-field nonsequential double ionization,” Phys. Rev. A 93(1), 013425 (2016). [CrossRef]
17. S. Dong, X. Chen, J. Zhang, and X. Ren, “Pulse-duration effect in nonsequential double ionization of Ar atoms,” Phys. Rev. A 93(5), 053410 (2016). [CrossRef]
18. G. P. Katsoulis, A. Hadjipittas, B. Bergues, M. F. Kling, and A. Emmanouilidou, “Slingshot nonsequential double ionization as a gate to anticorrelated two-electron escape,” Phys. Rev. Lett. 121(26), 263203 (2018). [CrossRef]
19. Z. Chen, A. Zhou, T. Morishita, Y. Bai, X. Hao, O. Zatsarinny, and K. Bartschat, “Anticorrelation in nonsequential double ionization of helium,” Phys. Rev. A 103(5), 053102 (2021). [CrossRef]
20. D. F. Ye and J. Liu, “Strong-field double ionization at the transition to below the recollision threshold,” Phys. Rev. A 81(4), 043402 (2010). [CrossRef]
21. D. I. Bondar, G. L. Yudin, W.-K. Liu, M. Yu. Ivanov, and A. D. Bandrauk, “Nonsequential double ionization below laser-intensity threshold: Anticorrelation of electrons without excitation of parent ion,” Phys. Rev. A 83(1), 013420 (2011). [CrossRef]
22. X. L. Hao, J. Chen, W. D. Li, B. Wang, X. Wang, and W. Becker, “Quantum Effects in Double Ionization of Argon below the Threshold Intensity,” Phys. Rev. Lett. 112(7), 073002 (2014). [CrossRef]
23. A. S. Maxwell and C. Figueira de Morisson Faria, “Controlling Below-Threshold Nonsequential Double Ionization via Quantum Interference,” Phys. Rev. Lett. 116(14), 143001 (2016). [CrossRef]
24. T. Morishita, A.-T. Le, Z. Chen, and C. D. Lin, “Accurate retrieval of structural information from laser-induced photoelectron and high-order harmonic spectra by few-cycle laser pulses,” Phys. Rev. Lett. 100(1), 013903 (2008). [CrossRef]
25. T. Morishita and O. I. Tolstikhin, “Adiabatic theory of strong-field photoelectron momentum distributions near a backward rescattering caustic,” Phys. Rev. A 96(5), 053416 (2017). [CrossRef]
26. Y. Okajima, O. I. Tolstikhin, and T. Morishita, “Adiabatic theory of high-order harmonic generation: One-dimensional zero-range-potential model,” Phys. Rev. A 85(6), 063406 (2012). [CrossRef]
27. O. I. Tolstikhin and T. Morishita, “Strong-field ionization, rescattering, and target structure imaging with vortex electrons,” Phys. Rev. A 99(6), 063415 (2019). [CrossRef]
28. J. Svensmark, O. I. Tolstikhin, and T. Morishita, “Adiabatic theory of strong-field ionization of molecules including nuclear motion: Rescattering,” Phys. Rev. A 104(6), 063115 (2021). [CrossRef]
29. Z. Chen, A.-T. Le, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for laser-induced high-energy plateau photoelectron spectra,” Phys. Rev. A 79(3), 033409 (2009). [CrossRef]
30. A.-T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for high-order harmonic generation from molecules,” Phys. Rev. A 80(1), 013401 (2009). [CrossRef]
31. Z. Chen, Y. Liang, and C. D. Lin, “Quantum theory of recollisional (e, 2e) process in strong field nonsequential double ionization of helium,” Phys. Rev. Lett. 104(25), 253201 (2010). [CrossRef]
32. Z. Chen, Y. Liang, D. H. Madison, and C. D. Lin, “Strong-field nonsequential double ionization of Ar and Ne,” Phys. Rev. A 84(2), 023414 (2011). [CrossRef]
33. B. Sheehy, R. Lafon, M. Widmer, B. Walker, and L. F. DiMauro, “Single- and multiple-electron dynamics in the strong-field tunneling limit,” Phys. Rev. A 58(5), 3942–3952 (1998). [CrossRef]
34. Z. Chen, Y. Zheng, W. Yang, X. Song, J. Xu, L. F. DiMauro, O. Zatsarinny, K. Bartschat, T. Morishita, S.-F. Zhao, and C. D. Lin, “Numerical simulation of the double-to-single ionization ratio for the helium atom in strong laser fields,” Phys. Rev. A 92(6), 063427 (2015). [CrossRef]
35. Z. Chen, X. Li, O. Zatsarinny, K. Bartschat, and C. D. Lin, “Ratios of double to single ionization of He and Ne by strong 400-nm laser pulses using the quantitative rescattering theory,” Phys. Rev. A 97(1), 013425 (2018). [CrossRef]
36. Z. Chen, L. Zhang, Y. Wang, O. Zatsarinny, K. Bartschat, T. Morishita, and C. D. Lin, “Pulse-duration dependence of the double-to-single ionization ratio of Ne by intense 780-nm and 800-nm laser fields: Comparison of simulations with experiments,” Phys. Rev. A 99(4), 043408 (2019). [CrossRef]
37. Z. Chen, Y. Wang, L. Zhang, and X. Jia, “Revisiting the recollisional (e, 2e) process in strong-field nonsequential double ionization of helium,” Phys. Rev. A 99(3), 033401 (2019). [CrossRef]
38. Z. Chen, Y. Wang, T. Morishita, X. Hao, J. Chen, O. Zatsarinny, and K. Bartschat, “Revisiting the recollisional excitation-tunneling process in strong-field nonsequential double ionization of helium,” Phys. Rev. A 100(2), 023405 (2019). [CrossRef]
39. H. W. van der Hart and K. Burnett, “Recollision model for double ionization of atoms in strong laser fields,” Phys. Rev. A 62(1), 013407 (2000). [CrossRef]
40. X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At. Mol. Phys. 38(15), 2593–2600 (2005). [CrossRef]
41. T. Morishita, Z. Chen, S. Watanabe, and C. D. Lin, “Two-dimensional electron momentum spectra of argon ionized by short intense lasers: Comparison of theory with experiment,” Phys. Rev. A 75(2), 023407 (2007). [CrossRef]
42. V. N. T. Pham, O. I. Tolstikhin, and T. Morishita, “Images of molecular orbitals in strong-field photoelectron momentum distributions generated by circularly polarized pulses,” Phys. Rev. A 99(1), 013428 (2019). [CrossRef]
43. Z. Chen, Y. Liang, and C. D. Lin, “Quantitative rescattering theory of correlated two-electron momentum spectra for strong-field nonsequential double ionization of helium,” Phys. Rev. A 82(6), 063417 (2010). [CrossRef]
44. H. Kang, K. Henrichs, M. Kunitski, Y. Wang, X. Hao, F. Fehre, A. Czasch, S. Eckart, L. Ph. H. Schmidt, M. Schöffler, T. Jahnke, X. Liu, and R. Dörner, “Timing recollision in nonsequential double ionization by intense elliptically polarized laser pulses,” Phys. Rev. A 120(22), 223204 (2018). [CrossRef]
45. H. Kang, K. Henrichs, Y. Wang, X. Hao, S. Eckart, M. Kunitski, M. Schöffler, T. Jahnke, X. Liu, and R. Dörner, “Double ionization of neon in elliptically polarized femtosecond laser fields,” Phys. Rev. A 97(6), 063403 (2018). [CrossRef]
46. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64(6), 1191–1194 (1986).
47. O. Zatsarinny, “BSR: B-spline atomic R-matrix codes,” Comput. Phys. Commun. 174(4), 273–356 (2006). [CrossRef]
48. O. Zatsarinny and K. Bartschat, “The B-spline R-matrix method for atomic processes: application to atomic structure, electron collisions and photoionization,” J. Phys. B: At. Mol. Phys. 46(11), 112001 (2013). [CrossRef]
49. P. Dietrich, N. H. Burnett, M. Ivanov, and P. B. Corkum, “High-harmonic generation and correlated two-electron multiphoton ionization with elliptically polarized light,” Phys. Rev. A 50(5), R3585–R3588 (1994). [CrossRef]
