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Abstract
With a three-dimensional classical ensemble method, we theoretically investigated the correlated electron dynamics in nonsequential double ionization (NSDI) driven by the spatially inhomogeneous fields. Our results show that NSDI in the spatially inhomogeneous fields is more efficient than that in the spatially homogeneous fields at the low laser intensities, while at the high intensities NSDI is suppressed as compared to the homogeneous fields. More interestingly, our results show that the electron pairs from NSDI exhibit a much stronger angular correlation in the spatially inhomogeneous fields, especially at the higher laser intensities. The correlated electron momentum distribution shows that in the inhomogeneous fields the electron pairs favor to achieve the same final momentum, and the distributions dominantly are clustered in the more compact regions. It is shown that the electron’s momentum is focused by the inhomogeneous fields. The underlying dynamics is revealed by back-tracing the classical trajectories.
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1. Introduction
The interaction between an intense femtosecond laser pulse and gas-phases atoms or molecules gives rise to various important physical phenomena, such as high harmonic generation (HHG) [1,2], above-threshold ionization (ATI) [3,4], non-sequential double ionization (NSDI) [5–8], frustrated double ionization (FDI) [9,10], and photoelectron holography [11–16], etc. Among them, NSDI has attracted significant interest due to the highly electron-electron correlation behavior. The widely accepted physical picture of NSDI is the three-step recollision process [17]. Briefly, this process can be summarized listing the subsequent steps: (i) the outermost electron can be launched into the continuum via tunneling when the laser electric field becomes comparable to the binding Coulomb field, (ii) the tunneled electron is accelerated and possibly driven back by the oscillating laser electric field, and (iii) the returned electron collides inelastically with its parent ion core and shares energy with another electron. Depending on the kinetic energy of the returned electron before recollision, the bound electron can be released by recollision impact ionization (RII) channel or recollision excitation with subsequent ionization (RESI) channel [18].
After ionization, the movement of the ionized electron is governed by the laser electric field. Therefore, adjusting the shape of laser electric field is an important way to control the electron dynamics of NSDI and other strong-field processes [19–25]. For instance, the few-cycle elliptically polarized laser field has been employed to control the RII and RESI channels in NSDI through the carrier-envelope phase [19]. The orthogonally polarized two-color field has also been used to control the recollision time of the tunneled electron wave packet with attosecond precision, thus effectively controlling the 2e-correlation in NSDI [20,21]. It has also been shown that the counter-rotating two-color circularly polarized field can induce high-brightness circularly polarized high-order harmonics [22].
Recently, another way to adjust the shape of electric field is to focus a laser pulse on a metal nanostructure, in the vicinity of which there will be a spatially inhomogeneous field due to plasmonic effects [26–37]. In the spatially inhomogeneous field, the electron propagation and acceleration can be characterized by the parameter $\delta = {l_F}/{l_q}$, where ${l_F}$ is the decay length of the electric field and ${l_q} = eE/m{\omega ^2}$ is the electron quiver amplitude [26]. For $\delta \gg 1$, the electron quivers in an almost homogeneous field over multiple cycles, for , the electron rapidly escape from the vicinity of the metal nanostructure, and for $\delta \approx 1$, the electron can be effected to the grea $\delta \ll 1$ test extent within sub-cycle. This unique electron trajectory has led to novel phenomena and applications. For example, the inhomogeneous field can drive the ionized electron with a high energy in the near-keV regime [27]. This high energy leads to a significantly increased high-order harmonic cutoff [28–30]. Besides, the broken potential symmetry results in the simultaneous generation of the odd and even harmonics [29,30]. In addition, it has demonstrated that the higher-energy structure in strong field ionization depends sensitively on the decay length ${l_F}$ [31], and the distribution of the electron momentum is related to the carrier envelope phase of the laser field [32]. Moreover, the inhomogeneous field is a useful tool for splitting the binary and recoil processes in the recollision scenario [33]. The inhomogeneous field has also been applied to the molecules. For H2+, the spatially inhomogeneous field can open a dissociation path: two-step excitation mediated by vibrationally excited states [34], and also can enhance the localization of the electron in one of the two protons [35]. Up to now, there have been very few studies to investigate NSDI in the inhomogeneous field [33].
In this work, we theoretically study NSDI of Ar driven by the spatially inhomogeneous field. The spatial inhomogeneity can be approximated as a linear variation of the laser electric field peak amplitude. The results show that the NSDI yield is obviously higher than that of the homogeneous field at low intensity (<6×1013 W/cm2). With the increase of laser intensity, this enhancement of NSDI yield declines and finally turns into suppression. By back tracing the classical trajectories, we reveal that the multiple-recollision process is greatly suppressed in the inhomogeneous field. In addition, we show that the inhomogeneous fields can induce the angle distribution between the two emitting electrons more and more concentrated near $\textrm{0}^\circ $ as the intensity increases. Moreover, our results indicate that in the inhomogeneous fields, the longitudinal 2e-momentum spectra are substantially located in the third quadrant and cluster along the diagonal.
2. Methods
Due to the huge amount of computation involved in numerically solving the time-dependent Schrödinger equation for two-active-electron system in a strong laser field, an alternative classical ensemble method was proposed by Eberly and coworkers [38,39]. This method provides a high computational efficiency and physically intuitive approach to gain insights into NSDI processes, which has been widely recognized as a reliable and useful method in exploring electron dynamics [38–49]. The general idea of this method is to mimic the evolution of the quantum wavefunction using an ensemble of classically modeled atoms. The evolution of the two-electron system is governed by the Newton’s classical motion equations (atomic units are used unless otherwise stated)
For simplicity, the spatially inhomogeneous field generated by a nanostructure can be approximated as linearly decreasing [27,30–33,36,37]. This approximation is widely used in the study of the electron dynamics driven by the inhomogeneous fields. With this approximation, the laser field can be written as:
Where, ${E_0}$ is the laser electric field peak amplitude, $\omega$= 0.0285 a.u. is the central frequency of the 1600-nm laser field, T is optical cycle period, and $\varepsilon = 1/2{l_F}$ is the parameter characterizing the inhomogeneity strength. Note that we take the electric-field strength to be zero in the region of $x < - 1/2\varepsilon$ to avoid unphysical electric field direction shifts.To obtain the initial values for Eq. (1), the ensemble is populated starting from a classically allowed position for the argon ground-state energy of -1.594 a.u. The available kinetic energy is distributed between the two electrons randomly in momentum space. Then the system is allowed to evolve a sufficient long time (300 a.u.) in the absence of the laser field to obtain stable position and momentum distributions. The ensemble size employed in our calculations is 15 million. Once the initial positions and momenta are obtained, the laser pulse is turned on. We record the energy evolution of the two electrons per 0.01 laser cycle and define a double ionization (DI) event if both electrons achieve positive energies at the end of the laser pulse. The energy of each electron includes the kinetic energy, ion core-electron potential energy, and half of the electron-electron repulsion energy. It is worth while mentioning that the “standard” case for NSDI is helium at 800 nm. For the inhomogeneous fields with the parameter $\delta \gg 1$, the electron quivers in an almost homogeneous field. Therefore, a longer wavelength is needed to highlight the role of inhomogeneous fields. In the current article, we use argon as our target atom which has the advantage of a lower ionization energy and has been well studied experimentally and theoretically [50,51]. Our conclusions and understandings, however, apply equally well to other atoms.
3. Numerical results and discussions
Figure 1 displays the probabilities of double ionization (DI) as a function of laser intensity for the spatially homogeneous ($\varepsilon \textrm{ = }0$) and the plasmonic-enhanced spatially inhomogeneous ($\varepsilon \textrm{ = }0.003$) laser fields, respectively. It is seen that for the inhomogeneous fields, the probability curve exhibit a pronounced “knee” structure, which is the characteristic of NSDI. The global shape of this DI probability curve is similar to that of the homogeneous fields. Clearly, we can see a crossing between the two probability curves around 6×1013 W/cm2, below which DI in the inhomogeneous fields is more efficient and above which DI is more suppressed as compared to the homogeneous fields. In the inset, we show the probability ratios between the inhomogeneous and homogeneous fields. It clearly displays that the ratios decrease gradually with the increase of the laser intensity. Note that the laser intensity of the inhomogeneous field is defined at the position of the atom so that the intensity for the tunneling process of NSDI in the homogeneous and inhomogeneous fields is the same. The difference in Fig. 1 is solely due to the recollision processes of NSDI.
Fig. 1. Probabilities of DI as a function of laser intensity for the homogeneous (blue circles) and inhomogeneous (red squares) laser fields. The inset shows the ratio of yields between the inhomogeneous and homogeneous fields as a function of the laser intensity. The laser wavelengths are both 1600 nm.
The core physical process underlying NSDI is recollision. To show the dynamics of recollision more intuitive in the inhomogeneous fields, Fig. 2(a) displays the schematic diagram of the electron classical trajectories corresponding to electron tunneling ionization to the positive x direction (the red curves) and the negative x direction (the blue curves). The result of the homogeneous field is also shown for comparison [see Fig. 2(b)]. It is seen that for the homogeneous field, the electron can return to the parent ion core many times in a specific electric field phase range, regardless of the direction of electron ionization. However, for the inhomogeneous field, the numbers of return are reduced to one and two for electron ionization to the positive x direction and the negative x direction, respectively.
Fig. 2. Schematic diagrams of the electron classical trajectories corresponding to electron ionization along the positive x direction (red lines) and the negative x direction (blue lines) for the inhomogeneous (a) and homogeneous (b) laser fields, respectively. 1st, 2nd and 3rd in the panels represent the first to third return of the electrons to the parent ion core.
In order to understand the intensity-dependent ratio of the DI yield, we trace back the classical trajectories. Figures 3(a) and 3(b) display NSDI yields as a function of the recollision time for the homogeneous ($\varepsilon \textrm{ = }0$) and inhomogeneous ($\varepsilon \textrm{ = }0.003$) fields at the peak intensities of 0.04 PW/cm2 and 0.07 PW/cm2, respectively. The recollision time is defined as the instant of the closest approach of the two electrons after the first departure of one electron. In Figs. 3(c) and 3(d), we show the kinetic energy of the returning electron vs. the recollision time for the inhomogeneous fields. The results for the homogeneous field are also shown in Figs. 3(e) and 3(f) for comparison. At the low laser intensity of 0.04 PW/cm2, the distributions of the recollision time reveal five pronounced peaks, denoted as P1–P5 in Fig. 3(a). At the peaks P1, P3 and P5, the NSDI yields in the inhomogeneous field are increased by almost three times as compared to the homogeneous field. This is because at the low laser intensity region, the recollision energy is the most important factor for NSDI. In the inhomogeneous laser field, the recollision energies for the peaks P1, P3, P5 are much higher than the case of the homogeneous field, as shown in Figs. 3(c) and 3(e). So, the NSDI yield is much higher.
Fig. 3. (a, b) NSDI yields as a function of the recollision time for the homogeneous and inhomogeneous laser fields. (c, d) The kinetic energy of the returning electron before the recollision vs. the recollision time for the inhomogeneous laser fields. (e, f) are same as (c, d), but for the homogeneous laser fields. The laser intensities are 0.04 PW/cm2 (left column) and 0.07 PW/cm2 (right column), respectively.
When the peak intensity is increased to 0.07 PW/cm2, the distributions of the recollision time reveal six pronounced peaks for the homogeneous and inhomogeneous fields, denoted as P0–P5 in Fig. 3(b). It is shown that the “inhomogeneous” NSDI yields at the peaks P1, P3 and P5 are lower than those of the homogeneous field, though the recollision energies for the inhomogeneous field at these peaks are much higher than those of the homogeneous field. This is because at the high laser intensity region, the returning energy in the homogeneous laser field is already high than the recollision-ionization threshold of Ar. Higher returning energy does not enhance the yield of NSDI. Instead, the inelastic cross section of DI decreases with returning energy. So, the NSDI yields at the peaks P1, P3 and P5 in the inhomogeneous field are less than those of the homogeneous fields. For the peaks P0, P2 and P4, the returning energies for inhomogeneous field are much less than those of the homogeneous field, and thus the NSDI yield is less than that of the homogeneous field. In addition, at this high laser intensity, the recollision energies for the second and third returnings are also large enough to induce the ionization of the second electron, as shown in Fig. 3(f). So, recollision at the second and third returnings also has significant contribution to NSDI. However, the multiple-recollision process is greatly suppressed in the inhomogeneous field, in particular, at the peaks P2 and P4 [see Fig. 3(d)]. This effect leads to further suppression of the NSDI yield in the inhomogeneous field.
Figure 4 shows the statistical distribution of the angle between the two emitting electrons at the end of the laser pulses for the inhomogeneous (the upper row) and homogeneous (the lower row) laser fields at three different laser intensities, namely 0.04, 0.05 and 0.07 PW/cm2. For the homogeneous fields, most of the distributions of angle are concentrated around $\textrm{0}^\circ $, and there is only a very small part of the distribution is focused near $\textrm{180}^\circ $ meaning that the two electrons are most likely to be emitted into the same direction. This is another characteristic of NSDI. Interestingly, for the inhomogeneous fields the two electrons are dominantly emitted into the same direction, while the distribution near 180° is completely suppressed. With the increase of laser intensity, the angle distribution becomes more and more concentrated to $\textrm{0}^\circ $.
Fig. 4. Distribution of the angle between the two emitting electrons at the end of the laser pulses for the inhomogeneous (upper row) and homogeneous (lower row) laser fields. The laser intensities are 0.04 PW/cm2 (left column), 0.05 PW/cm2 (middle column) and 0.07 PW/cm2 (right column), respectively.
This behave can be more intuitively seen in the correlated electron momentum distributions. Figure 5 shows the correlated longitudinal momentum spectra between the two emitted electrons for the inhomogeneous (the upper row) and homogeneous (the lower row) laser fields. The laser intensities are 0.04 PW/cm2 (left column), 0.05 PW/cm2 (middle column) and 0.07 PW/cm2 (right column), respectively. The spectra are plotted in units of $\sqrt {{U_p}} $, where ${U_p} = {E^2}/4{\omega ^2}$ is the ponderomotive energy. One can see that for the homogeneous fields shown in Figs. 5(d)–5(f), the populations are mainly distributed in the first and third quadrants, meaning that the two electrons are emitted into the same direction. There is also a small part of events distribution in the second and fourth quadrants, responsible for the minor distribution around 180° in Fig. 4. In the case of inhomogeneous fields, as shown in Figs. 5(a)–5(c), all of the spectra substantially shift to the third quadrant and closely cluster along the main diagonal. It seems that the momentum distributions of the correlated electrons are “focused” by the inhomogeneous fields.
Fig. 5. The correlated momentum spectra between the two electrons along the polarization direction for the inhomogeneous (upper row) and homogeneous (lower row) laser fields. The laser intensities are 0.04 PW/cm2 (left column), 0.05 PW/cm2 (middle column) and 0.07 PW/cm2 (right column), respectively.
Previous studies have shown that in the homogeneous fields with high returning energies, asymmetric energy sharing (AES) between the electron pair in the recollision process is prevail and this AES results in the different final momentum of the two electrons and thus the off-diagonal distribution [42–44]. In the inhomogeneous field, the returning energy is much higher than the recollision-ionization threshold (as shown in Fig. 3). It is strange why the momentum distributions are mainly along the main diagonal and focused in very narrow regions.
To answer this question, we focus our attention on the NSDI events correspond to the peaks P1, P3 and P5 in Fig. 3(b), which have higher returning energies. The results of the statistical distributions of energy difference ($\Delta E$) at the instant of 0.03 o.c. after recollision for the peaks P1, P3 and P5 are depicted in Fig. 6(a). The value of $\Delta E$ is an indication of the degree of asymmetric energy sharing of the recollision process. In Fig. 6(b), we show the statistical distributions of the final momentum differences of the two electrons (Px1-Px2) of the three peaks.
Fig. 6. (a) The statistical distributions of energy difference ($\Delta E$) at the instant of 0.03 o.c. after recollision for the peaks P1, P3 and P5 in Fig. 3(b). (b) The statistical distributions of the final momentum differences of the electron pairs for the peaks P1, P3 and P5. (c-e) The correlated momentum spectra between the two electrons along the polarization direction for the peaks P1, P3 and P5, respectively.
As shown in Fig. 6(a), the distributions of the energy difference extend to the value as large as about 5 a.u. for all of the three peaks P1, P3 and P5, indicating the extreme AES during the recollision. For the peak P1, as a result of this AES, one can see that the distribution of the final momentum differences shows a double-hump structure in Fig. 6(b). However, for P3 and P5, the distributions of the final momentum differences exhibit a single-hump structure with a very narrow width although AES is even more prevalent [Fig. 6(a)]. To make it more intuitive, the 2e-momentum spectra corresponding to the peaks P1, P3 and P5 are shown in Figs. 6(c)–6(e), respectively. Obviously, the off-diagonal momentum distribution only exists for P1, while the 2e-momentum spectra of P3 and P5 are concentrated in the narrow regions along the diagonal. These results indicate that for the DI events of peaks P3 and P5, although the initial momenta of the correlated electrons just after recollision are very different, they could achieve the same final momentum due to the post-collision acceleration by the inhomogeneous laser field. Note that the soft-Coulomb potential underestimates the Coulomb interaction to some extent. The returning energies of the first electron are much larger than those of the homogeneous field. Thus, the first electron can quickly pass though the vicinity of the parent ion core and this underestimate can be ignored.
In order to understand this issue, we analyze the evolution of the electron in the inhomogeneous fields. In Figs. 7(a)–7(c), we show the classical momentum evolution of the electrons respectively released at time ranges around P1, P3 and P5 with four different initial momenta (0, -0.4 a.u., -0.8 a.u., -1.2 a.u.). Here, we only consider the electric field of the laser field while the coulomb interaction between the ion core and the electron has been neglected. It is shown that for P1, the magnitude of the final momentum obviously depend on the initial momentum, i.e. the electrons launching with different initial momenta can achieve different final momenta. However, for P3 and P5, the final momenta are insensitive to the initial momenta, and are concentrated around 1.5 $\sqrt {{U_\textrm{p}}} $ and 0.7 $\sqrt {{U_\textrm{p}}} $, respectively. It means that the inhomogeneous field focuses the momenta of the ionized electrons. This “focusing effect” can also results in electron energy concentration in strong field ionization, and the emergence of a sharp high energy peak can be observed [31,37]. Figure 7 shows that the “focusing effect” depend on the release time of the electron. For the electron released at the falling edge of the laser envelope (5T-10 T in Fig. 7), the focusing effect is strong, and the electrons released during different 1/8th cycles are clustered at difference momentum. This focusing effect is weak when electrons ionize at the rising edge of the laser envelope.
Fig. 7. (a-c) The classical momentum evolution of the electrons ionized with different momentum around the peaks P1, P3 and P5 in Fig. 3(b). The dotted line shows the envelope of the pulse and the dashed lines shows the oscillating laser electric field at the origin.
Finally, one issue should be noted that the yields of the peaks P0, P2 and P4 are not negligible as shown in Fig. 3(b). In particular, the peak P2 is shown to have a larger yield than peak P5. In Fig. 8(a), we show the statistical distributions of energy difference ($\Delta E$) at the instant of 0.03 o.c. after recollision for the peaks P0, P2 and P4. One sees that the energy difference distributions of those three peaks all converge within about 1 a.u.. It means that the AES does rarely take place for these peaks due to the lower returning energy of the first electron. As a result, the 2e-momentum spectrum of P2 is concentrated along the diagonal as shown in Fig. 8(b).
Fig. 8. (a) The statistical distributions of energy difference ($\Delta E$) at the instant of 0.03 o.c. after recollision for the peaks P0, P2 and P4 in Fig. 3(b). (b) The correlated momentum spectrum between the two electrons along the polarization direction for the peaks P2.
4. Conclusion
In summary, using the three-dimensional classical ensemble method, we have theoretically investigated non-sequential double ionization of Ar in the inhomogeneous fields. Our results show that compared with the spatially homogeneous fields NSDI is more efficiently at the low laser intensities. This enhancement of NSDI yield decreases with laser intensity and finally the NSDI yield becomes lower than that of the homogeneous field at the high intensities. This intensity-dependent ratio of the NSDI yields between the homogeneous and inhomogeneous fields is traced back to the recollision energy of the tunneled electron and the recollision at its multiple returnings. More interestingly, we show that the inhomogeneous fields exhibit a focusing effect on the correlated electrons in NSDI. The electrons’ final momentum are clustered in a very narrow region. It thus offers a novel way to control the electron angular correlation in NSDI.
Funding
National Key Research and Development Program of China (2019YFA0308300); National Natural Science Foundation of China (12074329, 12004323, 11874163, 12021004); Key Scientific Research Project of Colleges and Universities in Henan Province (20A140025); Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
Acknowledgments
The computing work in this paper is supported by the Public Service Platform of High Performance Computing by Network and Computing Center of HUST.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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.
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