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Abstract
We investigate the double ionization of a model Neon atom in strong middle infrared laser pulses by simulating the classical trajectories of the electron ensemble. After one electron tunnels out from the laser-dressed Coulomb barrier, it might undergo different returning trajectories depending on its initial transverse momentum, which in this wavelength may propagate along or deviate from the polarization direction. This initial transverse momentum determines the rescattering time, and thus some trajectories can have returning time longer than one optical cycle. These late-returning trajectories determine the correlated electron-electron momentum distribution for double ionization and allow us to disentangle each double ionization event from the final momentum distribution. The description of these trajectories allow us also to understand how the nondipole effects modify the correlated electron-electron momentum distribution in double ionization.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nonsequential double ionization (NSDI) driven by strong laser fields is one of the most fascinating and fundamental processes in ultrafast physics. It has attracted a lot of attentions both theoretically and experimentally in past decades [1–3]. Experimentally, many studies of NSDI were performed using Ti:Sapphire lasers in the near-infrared (near-IR) (800 nm) [4–10]. In this wavelength and in the tunneling regime, an electron is ionized at the field maximum where it can gain energy from the field and return to the parent ion [11]. During this revisit, the bound electron may be directly kicked off, if the incoming electron carries enough energy, or jump to an excited state to be ionized later by the field. For each of the two cases, the energy sharing between electrons is different and these two scenarios present distinct final correlated electron-electron momentum distributions (EMDs) [12].
The recent developments of middle infrared (mid-IR) laser technology are now able to produce few-cycle laser pulses with the wavelength longer than 3 micrometers [13]. With these pulses new strong field phenomena have been observed such as high-energy photons in high-harmonic generation [14,15], fork-like structures in photoelectron momentum distributions [16], and the partition of the photon momentum [17,18].
For the case of NSDI the observations show much stronger energy sharing asymmetry in the correlated EMD as compared to the near-IR wavelengths [19,20]. Another important difference with the near-IR case is the importance of the magnetic field induced by the laser field. Since the magnetic field may modify the rescattering and induce the asymmetry in the forward and backward propagation of the laser pulse, the high harmonic generation [21] as well as the high-order above threshold ionization [22] are modified. Dynamics such as the nondipole gating [23] or the unexpectedly high sum momenta in the propagation direction [24] are observed even when the laser intensity is weaker than the previously predicted criteria for the onset of magnetic-field effects, i.e. $\beta _{0}\approx U_{p}/(2\omega c)\approx 1$ [25,26], with $\beta =v/c$, $U_p=\frac {E_0^2}{4\omega ^2}$ the ponderomotive energy, where $\omega$, $E_0$ are the frequency and the electric field amplitude respectively, and $c$ is the light velocity.
Due to the complexity of multiple-electron systems, the fully dimensional time-dependent Schrödinger equation (TDSE), which affords complete and accurate information, is mainly limited to the double ionization induced by EUV and UV fields [27–30]. The simulation of the double ionization driven by mid-IR laser fields is still a too heavy burden for the state-of-the-art computer. Therefore, reduced-dimensionality TDSE models and an extension of the time-dependent surface flux method are developed to qualitatively explain electron-electron correlation [31–34]. Alternatively, the strong field approximation, which neglects the Coulomb action on final states, plays important roles for explaining the correlated EMDs [35–39]. On the other hand, the purely classical method was developed by solving the Newton equation for the classical ensemble in combined laser and Coulomb fields when over-barrier ionization dominates [40–44]. In order to include the tunneling process, a semi-classical method has been proposed, which describes the single ionization by the Ammosov-Delone-Krainov (ADK) theory [45] and treats the rest part purely classically, which reproduces and explains a lot of experimental phenomena, such as the knee structure in the plot of double-ionization yields versus laser intensity [46–48], the double-hump structure of the longitudinal ion-momentum distribution [49], the finger-like structure in the correlated EMD [50,51], and the yield of anticorrelated electrons with high energy [52].
In this paper we use a semi-classical method to study the double ionization of Neon atoms and to describe the differences in the final EMD for the case of near-IR and mid-IR. We discuss the asymmetric energy sharing that is observed in the longer wavelengths and analyze three kinds of trajectories that contribute to different areas in the correlated EMD. Based on this understanding we continue to explain the nondipole effects over these trajectories. In this case, the modification of electron trajectories caused by the magnetic field or a slight ellipticity of the mid-IR laser pulse changes the relative proportion of different rescattering scenarios, which gives the explanation for the variation in the correlated EMD and the reduction of dication yields. The rest of the paper is organized as following. In Sec. II, we introduce the numerical method. The simulation results are shown in Sec. III. The paper ends with a conclusion in Sec. IV.
2. Simulation method
We simulate the double ionization of a Neon atom, in which two electrons are active and all other electrons freeze with the nucleus using a semi-classical approach. In this model, the exit point of the tunneling electron is determined by the effective potential in parabolic coordinates (atomic units are used throughout unless otherwise specified): $-\frac {1}{4\eta }-\frac {1}{8\eta ^{2}}-\frac {E(t)\eta }{8}=-\frac {I_{p1}}{4}$ with the first ionization energy of the atom $I_{p1}=0.8$ a.u. and $E(t)$ the instantaneous electric field. The $\eta$ is the parabolic coordinate relating to the Cartesian coordinate through $\eta =-2x$, with the positive $x$ direction assumed to be the instantaneous polarization direction, and the exit point is $(-\frac {\eta }{2},0,0)$ in Cartesian coordinates [53]. The corresponding Keldysh parameters $\gamma =\sqrt {I_{p1}/2U_{p}}$ are 0.21 (3000 nm) and 0.78 (800 nm) for single ionization of Ne atoms when the laser intensity is $3\times 10^{14}$ W/cm$^{2}$ [54]. The tunneling ionization rate is described by the ADK [45] formula $W(t)=\frac {4(2I_{p1})^2}{|E(t)|}\exp [-\frac {2(2I_{p1})^{3/2}}{3|E(t)|}]$. The longitudinal momentum of the tunneling electron is assumed to be zero, while the transverse momentum distribution is formulated as
3. Simulation results and discussions
When the driving laser wavelength is 800 nm, the correlated EMD of double ionization mainly locates in the first and third quadrants, as shown in Fig. 1(a), and the finger-like structure described in the literature is well reproduced [6,7]. The correlated EMD almost does not change once the magnetic field is included in the simulation, as shown in Fig. 1(b). The correlated EMD driven by the 3000 nm laser pulse, as shown in Fig. 1(c), is quite different from the 800 nm case. The double ionization events locate close to the coordinate axes instead of the diagonal line, and some events invade into the second and fourth quadrants, which is well in agreement with the experimental observations [19,20]. The ponderomotive energy for the driving laser with 3000 nm wavelength ($U_p=9.25$ a.u.) is much higher than 800 nm ($U_p=0.66$ a.u.) when the laser intensities are same ($3\times 10^{14}$ W/cm$^{2}$). The energy sharing between the two electrons is more asymmetric, which gives the explanation that the finger-like structure is still visible but with a very large open angle [44]. Comparing Fig. 1(c) and Fig. 1(d), most electrons are in the first and third quadrants for both panels, however, the near-axis signals are more pronounced in Fig. 1(d). The distributions in each panel have been normalized by their own maxima.
Fig. 1. The correlated EMD when the laser wavelengths are 800 nm (a, b) and 3000 nm (c, d), and magnetic fields of the laser pulses are neglected (a, c) and included (b, d). The laser intensities are $3\times 10^{14}$ W/cm$^{2}$ for 800 nm and 3000 nm laser pulses. Both lasers are linearly polarized.
We first investigate the correlated EMD for the case that the magnetic field is not included. Our analysis of the case of 3000 nm starts with Fig. 2. The upper panel shows the NSDI yields where the first electron is tunneling ionized around $t=0.5$T. The bars show the number of particles which rescatter in each of the three events leading to NSDI, the rescattering time windows are marked as shadow regions. The first rescattering is around $t=1.4$T and corresponds to the usual rescattering trajectory, the second is around $t=1.8$T and the third is around at $t=2.25$T, where the recollision time is defined as the instant when two electrons are closest after the tunneling of the first electron. The short trajectories (rescattering times smaller than 1.2T) are not presented for their low proportion (less than 2.8$\%$). One may see that the multipe rescattering has larger probabilities than the first rescattering, which is general when the driving laser pulse has very long wavelengths [44]. The significance of the late returns has been studied by Shvetsov-Shilovski $et$ $al$. using elliptically polarized laser pulses [55], and later observed by Kang $et$ $al$. in the experiment [56]. In Fig. 2(b), the rescattering energy as a function of the rescattering time is presented. The rescattering time and energy (green solid, dashed and dash-dotted lines in Fig. 2(b)) are obtained by solving the Newtonian equation $\frac {d^{2}\textbf {r}_i}{dt^{2}}=\frac {\partial A(t)}{\partial t}$ ,where $\textbf {r}_i$ denotes the electron displacement, $A$(t) is the laser vector potential, and the Coulomb and the magnetic force are neglected. The shapes of these three curves are quite similar , which indicates a precise map between the ionization time and the rescattering energy. However, the late rescattering events have distinct features compared to the first rescattering. Overlay in Fig. 2(b) is the incoming angle, which is defined as the cross angle between the moving direction of the incoming electron and the polarization axis at the moment $\Delta t$ = 3 a.u. just before recollision [57]. The angle is almost zero for the first rescattering, which means that the initial transverse momentum is also zero and the electrons travel along the laser polarization only. For the second and third rescattering events, the incoming angles are different from zero and change as a function of time. As the parallel momentum component is smaller at the edges of the rescattering time window, the incoming angle is larger. The colorscale in these curves represents the numbers of the rescattering electrons at that instant. From this analysis we observe that electrons collide with different times, angles and energies in these three rescattering time windows. This is an interesting extension to the well-known relation between ionization times and rescattering energy [58].
Fig. 2. (a) The laser electric field (black solid line) and the laser vector potential (amaranth dashed line). The electron tunnels out during the time interval [0.495T, 0.54T] will rescatter the parent ion in the time interval [1.25T, 1.45T] (the first rescattering), [1.55T, 1.9T] (the second rescattering) and [2.05T, 2,45T] (the third rescattering), and the corresponding double ionization probability are represented by the bars. (b) The incoming angle distribution for three rescattering events at different time. The variation in rescattering energy of these three events is depicted by green solid, dashed and dash-dotted lines. The laser wavelength is $\lambda =3000$ nm, and the laser intensity is $I=3\times 10^{14}$ W/cm$^2$.
To understand the origin of the incoming angle, we analyze the trajectories related to these late rescattering events. The classical trajectory Monte Carlo simulations allow us to trace the electron trajectories and thus view the rescattering details.
Once the electron tunnels out the laser-dressed Coulomb potential, it travels in the combined laser-Coulomb fields. Figure 3 shows the electron trajectories and kinetic momenta along the $z$-axis and $x$-axis for electrons released around $t=0.5$T, and these trajectories will induce double ionization by rescattering. The three columns from left to right are for the events associated with the first, second, and third rescattering, respectively. The first and third rows show the time-dependent displacement $x(t)$ and $z(t)$. The corresponding momenta, $p_x(t)$ and $p_z(t)$, are shown in the second and fourth rows. One may clearly see from the first column that the electron travels along the laser polarization direction with a large excursion ($\sim 800$ a.u.) and rescatters the parent ion at $t=1.44$T. In the transverse direction, the electron trajectories stay close to zero during this time. The electron momentum $p_x(t)$ mirrors the laser vector potential, while $p_z(t)$ keeps constant before rescattering. Therefore, the first rescattering electrons travel along the laser polarization and their incoming angles are close to zero degree.
Fig. 3. The time-evolved electron displacement (the first row) and electron kinetic momentum (the second row) along the laser polarization direction. The third and fourth rows are the time-evolved electron displacement and electron kinetic momentum along the laser propagation direction. The three columns from left to right are for the first, second and third rescattering, respectively.
For the second rescattering shown in the second column, the electron propagates in the laser field and backs to $x=0$ at $t=1.26$T. However, the rescattering does not happen at this moment because the electron is away from the ion in the transverse direction. As the electron passes very quickly through $x=0$ around $t=1.26$T, the Coulomb force reverses the trajectory of the electron in the transverse direction. The sudden change in the transverse momentum can be seen in both the displacement $z(t)$ and $p_z(t)$. The transverse momentum changes signs due to the strong attraction of the Coulomb field. At a later time $t=1.9$T both transverse and parallel coordinates are close to zero ($x(t)=z(t)=0$) and the rescattering occurs.
The trajectories associated with the third rescattering are presented in the right column of Fig. 3. The electron misses the parent ion twice at $t$ = 1.44T and $t$ = 1.56T. Different from the scenario of the second rescattering, the electron is slow when it passes through $x=0$, and thus it may spend more time close to the parent ion. Thereby, the Coulomb field may act longer on this electron and reverse its momentum in the transverse direction even though it is up to $\sim 20$ a.u. away in the $z$ transverse direction. At a later time $t=2.44$T, both the $x(t)$ and $z(t)$ are close to zero and the third rescattering occurs almost two cycles later than the tunnel ionization event. Similar to the second rescattering, the synchronous approaching to zero in both $x$- and $z$-axes explains the rescattering time, energy and incoming angle.
Some typical trajectories are depicted in Fig. 4 in the three-dimensional space. The trajectories for tunnel-ionized electrons with the initial transverse momenta equal to zero ($p_z=p_y=0$), the first circle and the second circle in Fig. 5(a) produce rotationally symmetric (in the dipole case) surfaces corresponding to the first, second and third rescattering. On each surface, a typical trajectory determined by the Coulomb focusing and laser driving is plotted by a solid line. These trajectories also identified as glory trajectories [59] are the only trajectories that can rescatter in the combined effect of the laser field and the Coulomb potential. As pointed by Huang $et$ $al$., even higher-order rescattering may happen using a multiple-cycle laser pulse [44]. Here, the fourth rescattering is truncated because a few-cycle laser pulse is used. We may parenthetically point that the high-order returns are overestimated in this semi-classical model since the wave packet expansion is not comprised.
Fig. 4. The typical trajectories for the first (green), second (red) and third (blue) rescattering without considering the magnetic field. The laser wavelength is $3000$ nm, and the intensity is $3\times 10^{14}$ W/cm$^2$. Surfaces are formed by rotating each trajectories with respect to the $x$-axis. The dashed curve is the trajectory starting with the identical displacement and momentum of the first rescattering trajectory but including the magnetic effect.
Fig. 5. (a) The initial transverse momentum distribution of the tunneling electron associated with the later double ionization. The magnetic field is not included. (b), (c) and (d): The correlated EMDs associated with the initial transverse momenta around 0, in the first and second circles, respectively. The laser wavelength is 3000 nm, and the intensity is $3\times 10^{14}$ W/cm$^{2}$. The laser is linearly polarized.
The identification of the mapping between the initial transverse momentum to the rescattering time, energy and incoming angle allow us to disentangle all the rescattering events in the final correlated EMD. The three types of rescattering trajectories with well defined rescattering times can now help us to understand the formation of the correlated EMDs shown in Figs. 5(b), (c) and (d).
According to the Simpleman model [60], where the Coulomb potential and magnetic fields are neglected, the first rescattering occurs at the moment that the vector potential A is maximum, and the rescattering energy is up to 3.17$U_p$. The corresponding kinetic momentum $p=\sqrt {2\times 3.17U_p }=1.26A_0$ [61], where $A_0$ is the amplitude of the laser vector potential. For the mid-IR laser field, the rescattering energy is much larger than the Ne$^+$ ionization potential $I_{p2}=1.5$ a.u., which means the pitcher (the tunneling electron) only transfers very little energy to the catcher (the bound electron). Just after the rescattering, the pitcher and catcher move in the same direction. In the later propagation, the low energetic catcher is first reversed and then accelerated to high speed by the remaining laser field. However, the pitcher with the kinetic momentum up to 1.26$A_0$ is substantially decelerated but does not reverse its moving direction, and finally contribute to the double ionization events in the second and fourth quadrants. For the less energetic pitcher whose momentum just after the rescattering is less than A will finally reverse its direction, and the corresponding double ionization events locate in the first or third quadrants. According to the classical estimation, the critical laser vector potential for producing rescattering double ionization events in the second and fourth quadrants is $A=\sqrt {3.4I_{p2}}$. Differently, the second rescattering brings the rescattering energy less than $1.54U_p$ (the corresponding maximum kinetic momentum is 0.88$A_0$). Therefore, both the pitcher and catcher will be driven to the same direction by the laser field, as showed in Fig. 5(c). For the third rescattering, the maximum rescattering energy is 2.4$U_p$(the corresponding kinetic momentum is 1.1$A_0$), the pitcher moving direction is very likely to be reversed, thereby the double ionization events locating in the second and fourth quadrants are less than that of the first rescattering.
After having understood the rescattering details within the dipole approximation, we are ready to analyze how the magnetic field changes the rescattering processes and modifies the correlated EMD. When the magnetic field is take into consideration, the Lorentz force would affect the rescattering process of electrons and some interesting phenomena have been presented for double ionization [23,24]. The dashed line in Fig. 4 represents the trajectory of the electron starting with the identical initial position and momentum with those of the green solid line for the first rescattering, and now the electron is pushed to the positive $z$ axis by the Lorentz force and the rescattering is suppressed.
Figure 6(a) shows the initial transverse momentum distribution of the tunneling electron that will ultimately induce rescattering double ionization driven by a 3000 nm, $3\times 10^{14}$ W/cm$^2$ linearly polarized laser pulse by including the action of the magnetic field. For compensating the push of the Lorentz force, the initial transverse momentum distribution has been pushed oppositely to the laser propagation direction in order to meet the nucleus, which has been pointed by Walser $et$ $al$ and Daněk $et$ $al$ [62,63]. Besides the central point shifts to $p_z=-0.08$ a.u., the circles changes into ellipses, which is caused by the break of the cylindrical symmetry of the rescattering trajectories.
Fig. 6. The initial transverse momentum distributions of the tunneling electron that will lead to double ionization when (a) the laser field is linearly polarized and the magnetic field is included, (b) the laser field is elliptically polarized (with the ellipticity 0.005) and the magnetic field is not included, (c) same as (b) but the magnetic filed is included. (d) The correlated EMD for the laser condition of (c). The laser wavelength is 3000 nm, and the intensity is $3\times 10^{14}$ W/cm$^2$.
Since the initial transverse momentum distribution of the tunnel-ionized electron has a gaussian shape centered at zero, as formulated by Eq. (1), the magnetic deviation will change the proportion of the first, second, and third rescattering, as well as the correlated EMD. In Fig. 1(c), the relative proportions for the first, second, and third rescattering are $6\%$, $45\%$ and $49\%$. In the nondipole case as showed in Fig. 1(d), however, the relative proportions are $9\%$, $37\%$ and $54\%$ for the three rescattering scenarios respectively. Consistently, one may notice that the events in the first and third quardrants in Fig. 1(c), mainly contributed by the second rescattering, are more than those in the first and third quardrants in Fig. 1(d).
If the driving laser pulse is not purely linearly polarized, which is the usual case in experiment, the electric component $E_y$ of the laser pulse will modify the electron trajectory in the $y$ direction. Accordingly, the initial transverse momentum distribution of the tunnel-ionized electron and the later on trajectory are modified, so does the correlated EMD. Figure 6(b) presents the initial transverse momentum distribution of the tunneling electron associated with the rescattering double ionization when the magnetic field is neglected and the ellipticity is $0.005$. Because of the drag along y-axis by the tiny electric field component $E_y$, the concentric circles presented in Fig. 5(a) are pulled apart and two set of broken ellipses are observed as expected. The relative proportions of different rescattering scenarios are also modified, i.e. $17\%$ for the first rescattering, $30\%$ for the second rescattering, and $53\%$ for the third rescattering, and we find (not shown) the corresponding correlated EMD is reshaped as predicted. When the unavoidable magnetic field is further introduced in the simulation, the initial transverse momentum distribution moves downward, as displayed in Fig. 6(c). The corresponding correlated EMD is shown in Fig. 6(d). The relative proportions for the first, second, and third rescattering are $26\%$, $30\%$ and $44\%$ respectively in Fig. 6(d). Since more double ionization events happen through the first and third rescattering, the main structure in the correlated EMD locates in the near-axis area and the pitcher and catcher have quite different momenta. Such correlated EMD looks similar to the experimental measurements [19,20].
Besides reshaping the correlated EMDs, the magnetic field also modifies the dication yields via altering the trajectories of rescattering electrons. Figure 7 shows the dication yield ratio of the nondipole and the dipole approximation cases as a function of laser wavelengths when the intensity is fixed at $3\times 10^{14}$ W/cm$^{2}$. The movement of the initial transverse momentum distribution away from the center of the gaussian shape given by Eq. (1) weakens the rescattering double ionization, which reduces the yield of dications. The magnetic field must be included though it is still far from relativistic regimes.
Fig. 7. The ratio of the dication yields for the nondipole and dipole approximation cases as a function of the laser wavelength. The driving lasers are linearly polarized and have the fixed intensity $3\times 10^{14}$ W/cm$^{2}$.
4. Conclusions
To summarize, by following the electron trajectory in the combined Coulomb field and laser field of the driving mid-infrared laser, we found the first released electron may rescatter the parent ion within one optical cycle, or around 1.25 cycles, or 1.75 cycles after its tunneling. These three different rescattering scenarios are mainly governed by the initial transverse momentum at tunneling. Different rescattering events lead to well defined regions on the correlated EMD and help to understand the asymmetric energy sharing leading to double ionization. The inclusion of the unavoidable magnetic field and the slight ellipticity of the mid-IR laser pulse modifies the electron trajectories, weakens the rescattering double ionization, and reshapes the correlated EMD. Physically, due to the limitation of the numerical model, for example, the missing of the wave packet expansion during the electron propagation, the second and third rescattering events are overestimated. Nevertheless, our study provides qualitative understanding of a set of new electron trajectories leading to double ionization driven by mid-IR laser pulses, which are now available in advanced laboratories.
Funding
National Key Research and Development Program of China (2018YFA0404802); National Natural Science Foundation of China (11574205, 11721091, 91850203); Shanghai Municipal Education Commission (2017-01-07-0002-E00034); Ministerio de Economía y Competitividad (FIS2016-75652-P).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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