Open Access
Add to BibSonomy
Abstract
Using the quantitative rescattering model, we simulate the correlated two-electron momentum distributions for nonsequential double ionization of helium by 800 nm laser pulses at intensities in the range of (2 − 15) × 1014 W/cm2. The experimentally observed V-shaped structure at high intensities [Phys. Rev. Lett. 99, 263003 (2007) [CrossRef] ] is attributed to the strong forward scattering in laser-induced recollision excitation and the asymmetric momentum distribution of electrons that are tunneling-ionized from the excited states. The final-state electron repulsion also plays an important role in forming the V-shaped structure.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nonsequential double ionization (NSDI) has remained one of the high-interest topics in strong-field physics, even though it has been extensively studied both experimentally and theoretically for more than three decades (for a review, see [1]). The main attractiveness of NSDI stems from the fact that this phenomenon represents a uniquely clean example of electron-electron correlation enforced by an external field. A large number of experiments, from measurements of the total yield of doubly-charged ions versus the peak laser intensity [2–6], especially the ellipticity dependence of the double-ionization yield [7–9], to differential measurements of the recoil-ion momentum distributions [10–12] and the kinematically complete measurements of the correlated two-electron momentum distribution (CMD) [13,14], provide strong evidence that the rescattering mechanism is predominantly responsible for the NSDI process. The classical three-step rescattering model [15] has been widely employed to interpret laser-induced rescattering processes qualitatively. According to the three-step model, the electron in atoms or molecules is first released near the peak of the laser electric field. When the latter field reverses its direction, the electron has a chance to be driven back to the parent ion. In this rescattering picture, NSDI can be attributed to inelastic scattering of the returning electron by the parent ion, including recollision direct ionization (RDI) and recollision excitation with subsequent ionization (RESI) [14].
The characteristics of NSDI depend remarkably strong on the laser intensity, as demonstrated by the measurements on the recoil-ion momentum distributions that evolve from a single-peak into a double-hump structure with increasing field strength [10], and the two-electron momentum spectra that exhibit a transition from correlated to anti-correlated emission as the laser intensity decreases to below the threshold intensity [16]. These novel characteristics reflect detailed microscopic dynamical processes under the recollision mechanism. However, in the NSDI experiments on helium with high resolution and good statistics, performed a decade ago by Staudte et al. [17] and Rudenko et al. [18], respectively, the V-shaped structure observed in the measured CMD at a peak intensity of $15 \times 10^{14}$ W/cm$^2$ bears a strong resemblance to the fingerlike structure at the intensity of $4.5 \times 10^{14}$ W/cm$^2$. Nevertheless, theoretical investigations demonstrate that these similar structures observed at those two intensities are attributed to different mechanisms. The prominent fingerlike structure observed at the relatively low intensity has been intensively investigated by various theoretical models, including the semiclassical quasistatic model [19], the quantitative rescattering (QRS) model [20], and ab initio calculations by solving the time-dependent Schröinger equation (TDSE) [17]. All theoretical studies confirmed that the fingerlike structure is a consequence of the Coulomb interaction between the two emitted electrons.
In contrast, for the relatively high laser intensity, the V-shaped structure has been studied much less. To our knowledge, the only existing theoretical investigation of NSDI of helium in laser pulses with a wavelength of 800 nm at high intensity ($\sim 15 \times 10^{14}$ W/cm$^2$) is based on the classical three-dimensional ensemble model, in which it was argued that the V-shaped structure originates from asymmetric energy sharing at the recollision process, whereas neither the nuclear attraction nor the final-state electron repulsion plays an important role in forming the V-like shape [21]. The lack of theoretical investigations on NSDI at high intensities might be due to the high computational demand, especially when the focal-volume effect is taken into account in the simulations even with a relatively simple numerical method.
In this paper, we present a systematic investigation of the intensity dependence of NSDI by simulating the CMD for double ionization of helium exposed to 800 nm laser pulses at intensities in the range from 2 to $15 \times 10^{14}$ W/cm$^2$ based on the QRS model. Our aim is to unveil the origin of the V-shaped structure from a quantum-mechanical point of view and provide further insight into the role of electron-electron and electron-laser interactions in NSDI.
Atomic units (a.u.) are used in this paper unless otherwise specified.
2. Theoretical model
We employ the QRS model, with focal-volume averaging included, to simulate the CMD for NSDI of helium. The QRS model is based on the factorization formula for rescattering processes originally developed for high-order above-threshold ionization (HATI) [22–24] and high-order harmonic generation [22,25,26]. It was then applied to the description of NSDI including predictions of the total yield of doubly-charged ions [27] and the CMD of rare-gas atoms [28,29]. Recently, the QRS model has been improved by taking into account the lowering of the threshold due to the presence of an electric field at the time of recollision [30–32]. The details of the numerical procedures for simulations of the CMD for NSDI of helium based on the improved QRS model have been presented in Refs. [33–35]. Hence we only give a brief overview here.
The basic philosophy of the QRS model for NSDI is that the CMD can be expressed as a product of the returning-electron wave packet (RWP) and the field-free differential cross section (DCS) for electron impact ionization of the parent ion plus the DCS for electron impact excitation of the parent ion multiplied by the tunneling ionization rate of electrons in the excited states.
2.1 Recollision excitation with subsequent ionization
According to the QRS model, for recollision excitation with subsequent ionization in NSDI of atoms in a linearly polarized laser pulse with a peak intensity $I$ and its electric field along the $z$ axis, the CMD for the momentum components $p_1^{||}$ and $p_2^{||}$ of the two outgoing electrons along the laser polarization direction can be expressed as
The parallel momentum distributions $D^{\,\rm exc}_{E_i}(p_1^{||})$ for the returning electron after recollision excitation are obtained by projecting the DCS for electron impact excitation of the parent ion onto the polarization direction to which the parallel momentum $k_1^{||}$ of the projectile electron should be shifted, i.e.,
where $\mathbf {A}_r$ is the vector potential at the instant of recollision. In the QRS model, we take [34] where $k_r$ is the momentum of the returning electron, which is related to the incident electron energy by $k_r=\sqrt {2E_i}$.For the laser-induced recollision process of He$^+$ considered here, only the DCS for the singlet spin channel should be used, since the two electrons involved in the process start in the singlet ground state of He, and their singlet coupling is preserved [36] in a nonrelativistic treatment. The DCS for electron impact excitation of He$^+$ are calculated by using the state-of-the-art multi-electron $B$-spline $R$-matrix (BSR) close-coupling theory [37,38] for incident energies below 100 eV and the distorted-wave Born approximation (DWBA) [39] for higher energies.
The parallel momentum distributions for tunneling ionization of electrons in the excited states $D^{\,\rm tun}_I(p_2^{||})$ are calculated by integrating the 2-dimensional (2D) momentum distributions for single ionization over the momentum component perpendicular to the laser polarization. In the present paper, the 2D momentum distributions for single ionization of He$^+$ are evaluated by solving the TDSE directly for laser intensities below $4.5 \times 10^{14}$ W/cm$^2$ [34,40]. For higher intensities, the strong-field approximation (SFA) [41] is employed. The latter is appropriate for these cases and requires much less computational effort.
The RWPs, which describe the momentum distribution of the returning electron, are evaluated based on the improved SFA model for HATI [23]. We note that an analytical expression of the RWP for HATI at the outermost backward rescattering caustic was recently derived based on the adiabatic theory [24].
2.2 Recollision direct ionization
To simulate the CMD for recollision direct ionization, the singlet triple-differential cross sections (TDCS) for laser-free ($e,2e$) on He$^+$ should be prepared. Here, an approximate three-body scattering wave function that was derived analytically by Brauner, Briggs, and Klar (BBK) [42] is employed to describe the entire system in the final state. The BBK wave function that satisfies the asymptotic three-body Schrödinger equation exactly consists of a product of three Coulomb (3C) wave functions, in which the repulsion between the two outgoing electrons is taken into account. If the interaction between the two outgoing electrons is ignored, the BBK wave function becomes a product of two Coulomb (2C) wave functions. Similar to recollision excitation with subsequent ionization, by projecting the TDCS onto the polarization direction and shifting the parallel momenta of the two outgoing electrons by $-A_r$, we obtain the CMD for recollision direct ionization in a laser pulse with a peak intensity $I$ as
3. Results and discussion
3.1 Total cross sections for electron impact excitation and ionization of He$^+$
According to the rescattering model, the maximum kinetic energy that a returning electron accumulates from the electric field is $3.17 \,U_p$, where $U_p$ is the ponderomotive energy, which is proportional to laser intensity. For the highest intensity of $15 \times 10^{14}$ W/cm$^2$ considered here, the maximum kinetic energy of the returning electron is about 285 eV. Since both RDI and RESI could be involved in NSDI, the total cross sections (TCS) as a function of incident energy for laser-free ($e,2e$) and electron impact excitation of the parent ion should provide an overview of the intensity dependence of the NSDI processes. Therefore, in Fig. 1 we show the singlet TCS for electron impact excitation and electron impact ionization of He$^+$ in the ground state. For electron impact excitation, only the singlet TCSs for excitation to the excited states of $n=2$ and $n=3$ are evaluated, since the excitation to higher states can safely be neglected based on the $n^{-3}$ scaling law. In Figs. 1(a)–1(c), we present the singlet TCS for electron impact excitation of He$^+$ from the ground state to the excited states of $n=2$ and $n=3$ for each angular momentum $l$ with specific magnetic quantum number $m$, respectively. We use the short-hand notation $2p_0 \equiv 2p\,(m\!=\!0)$ and similarly for $2p_1$, $3p_0$, $3p_1$, $3d_0$, $3d_1$, and $3d_2$. As expected, excitations to $2p_0$ and $3p_0$ dominate and exhibit similar energy dependencies, i.e., the TCS increase, reach a maximum around 60 eV, and then decrease with increasing incident energy. For excitations to other excited states, all the TCS (except for $3s$ at a few eV above threshold energy) decrease monotonously as the incident energy increases. Interestingly, for excitations to $2p_1$, $3p_1$, $3d_1$ and $3d_2$, the TCS decrease so slowly that they almost remain constant at energies above 70 eV.
Fig. 1. Singlet total cross sections for electron impact excitation and ionization of He$^+$. Results are shown for excitation of He$^+$ from the ground state to the excited states of (a) $2s$, $2p_0$, and $2p_1$, (b) $3s$, $3p_0$, and $3p_1$, (c) $3d_0$, $3d_1$, and $3d_2$, and (d) $n=2$ and $n=3$. In panel (d), the total cross sections for electron impact excitation to all excited states up to $n=3$ and electron impact ionization of He$^+$ from the ground state are also displayed. For excitation, all results were obtained with the BSR code. For ionization, BSR and BBK results are shown.
In Fig. 1(d), we show the TCS for excitations to $n=2$ and $n=3$, and also their sum. The cross sections for $n=2$ are about three times larger than those for $n=3$ at impact energies above 100 eV, in good accordance with the $n^{-3}$ scaling law. The singlet TCS for electron impact ionization of He$^+$ obtained from the BSR theory and the BBK model are also displayed. It can be seen that excitation dominates for impact energies below 150 eV, corresponding to the maximum energy of the returning electron in an 800 nm laser field at a peak intensity of about $8 \times 10^{14}$ W/cm$^2$. At higher energies, the cross sections for excitation and ionization become comparable.
3.2 Correlated momentum distributions without focal-volume averaging
As indicated in Eq. (1), the CMD for RESI consists of the parallel momentum distribution for the returning electron after the collision, the parallel momentum distribution for the electron ionized from an excited state of the parent ion, and the momentum (energy) distribution of the returning electron before the collision. Each of the three ingredients plays an important role in the simulations of the CMD.
The momentum (energy) distribution of the returning electron before the collision, i.e., the RWP, is calculated within the framework of SFA by employing a hydrogenlike wave function to describe the ground state of He with its depletion taken into account [30]. Calculations were performed for a wide intensity range from 2.0 to $15\times 10^{14}$ W/cm$^2$ with a step of $0.1\times 10^{14}$ W/cm$^2$, using 25 fs pulses with $\cos ^2$ envelope.
The parallel momentum distributions for the returning electron after recollision are obtained by projecting the singlet DCS for laser-free electron impact excitation of He$^+$ onto the polarization direction with the parallel momentum shifted by $-A_r$. Figure 2 displays those parallel momentum distributions for the active electron after recollision for selected excited states. The results shown in Fig. 2 are for the situation in which the recolliding electron returns to the parent ion along the $-\hat {z}$ direction and the vector potential at the moment of recollision is negative ($A_r\;<\;0$). As a result, the smallest parallel momentum $p_1^{||}$ corresponds to rescattering angle $\theta _r=0^{\circ }$. With the increase of rescattering angle, $p_1^{||}$ increases until it reaches its maximum value when $\theta _r=180^{\circ }$. One can see that the momentum distribution probability for excitations to $2s$ and $3s$ first decrease rapidly with increasing parallel momentum and then, due to strong backward scattering, increase again dramatically as the momentum increases further for recollision energies below 100 eV. On the other hand, for excitations to all other states, including the dominant $2p_0$ and $3p_0$ excitation, the large momentum distributions at small momenta exhibit strong forward scattering at all incident energies, except when the energies are very close to threshold.
Fig. 2. Parallel momentum distributions of the active electron after recolliding with the He$^+$ ion and exciting the residual ground-state electron to the excited states of (a) $2s$, (b) $2p_0$, (c) $2p_1$, (d) $3s$, (e) $3p_0$, and (f) $3p_1$ at energies of 60, 100, 150, and 250 eV. The recolliding electron returns along the $-\hat {z}$ direction.
The parallel momentum distribution for the electron tunneling ionized from an excited state of He$^+$ are obtained from TDSE and SFA for laser intensities below and above $4.5\times 10^{14}$ W/cm$^2$, respectively. In the SFA calculations, a decay factor is introduced in the transition amplitude to account for the depletion of the initial state [31]. In the TDSE calculations, this depletion effect is included from first principles. To verify the quality of the SFA model used for laser pulses at high intensities, we show in Fig. 3 the comparison of the parallel momentum distributions from TDSE and SFA for single ionization of He$^+$ from some selected states by an 800 nm laser field at a peak intensity of $4.5\times 10^{14}$ W/cm$^2$ for the recollision process in which the laser-induced recolliding electron returns to the origin along the $-\hat {z}$ direction. The electric field used in the simulations is given explicitly in [34]. It can be seen that the parallel momentum distributions obtained from the SFA are generally narrower than those generated by solving the TDSE. Nevertheless, for excitations to $n=2$, especially for $2p_0$, the SFA results are close to those from the TDSE. Even for excitations to $n=3$, the main feature of the parallel momentum distributions predicted by solving the TDSE can be well reproduced by the SFA model.
Fig. 3. Comparison of the parallel momentum distributions from TDSE and SFA for the electron ionized from He$^+$ in the excited states of (a) $2s$, (b) $2p_0$, (c) $2p_1$, (d) $3s$, (e) $3p_0$, and (f) $3p_1$ by an 800 nm laser field with a peak intensity of $4.5\times 10^{14}$ W/cm$^2$.
Since excitations to $n=2$ dominate and the SFA results are expected to be in even better agreement with those from the TDSE at higher intensities, it is reasonable to employ the SFA model to simulate the parallel momentum distributions for the tunneling-ionized electron at intensities above $4.5\times 10^{14}$ W/cm$^2$. Figure 4 shows the SFA results for single ionization of He$^+$ from the excited states $2s$, $2p_0$, $2p_1$, $3s$, $3p_0$, and $3p_1$ by an 800 nm laser field at peak intensities of 6, 8, 10, and $15\times 10^{14}$ W/cm$^2$, respectively. As expected, the momentum distributions spread wider with increasing intensity, since the range of the momentum distribution is roughly within $[-A_0,A_0]$, where $A_0$ is the maximum value of vector potential. In the meantime, the momentum distributions exhibit higher asymmetry for excitations to $n=2$ at higher intensities due to more rapid ionization. For excitations to $n=3$, highly asymmetric distributions are always predicted owing to the smaller ionization potentials compared to the excited states of $n=2$.
Fig. 4. Parallel momentum distributions obtained with the SFA for the electron ionized from He$^+$ in the excited states of (a) $2s$, (b) $2p_0$, (c) $2p_1$, (d) $3s$, (e) $3p_0$, and (f) $3p_1$ by an 800 nm laser field with peak intensities of 6, 8, 10, and $15\times 10^{14}$ W/cm$^2$.
With the well-prepared momentum distributions for the two outgoing electrons and the RWP for the returning electron, it is straightforward to obtain the correlated two-electron momentum distributions for RESI using Eq. (1). Figure 5 displays the CMD for RESI in which the second electron is tunneling-ionized from He$^+$ in the excited $2s$, $2p_0$, $2p_1$, $3s$, $3p_0$, and $3p_1$ states by an 800 nm laser field with a peak intensity of $8.0\times 10^{14}$ W/cm$^2$. It should be noted that the contributions from collisions at all possible incident energies were taken into account. The CMD presented in Fig. 5 imprints the pattern of the momentum distributions of both the returning electron after recollision and the tunneling-ionized electron from excited states of the parent ion. For example, the dense population in the region of small $p_1^{||}$ and large $p_2^{||}$ in Figs. 5(b) and 5(e) reflects strong forward scattering of the projectile electron after the collision and fast tunneling ionization of the electron from excited states. On the other hand, backward scattering is competitive to forward scattering for excitation of $2s$. As a result, two vertical stripes are produced in Fig. 5(a).
Fig. 5. Correlated two-electron parallel momentum distributions for excitation-tunneling from (a) $2s$, (b) $2p_0$, (c) $2p_1$, (d) $3s$, (e) $3p_0$, and (f) $3p_1$ in an 800 nm laser field with a peak intensity of $8.0\times 10^{14}$ W/cm$^2$. The recolliding electron returns along the $-\hat {z}$ direction.
The CMDs for RESI shown in Fig. 5 are only for the situation in which the laser-induced electron returns to the parent ion along the $-\hat {z}$ direction. For the long pulses considered here, the laser-induced electron possesses the same probability for returning to the parent ion along the $+\hat {z}$ direction. Furthermore, since the two outgoing electrons are indistinguishable, the full-space CMD for RSEI should be symmetric with respect to both diagonals $p_1^{||} = \pm p_2^{||}$. Figure 6 depicts the full-space CMD for RESI upon symmetrization of the CMD displayed in Fig. 5. Even though the CMD pattern for different excited states changes like a kaleidoscope, the distributions for excitation-tunneling from $2p_0$ and $2p_1$ dominate and exhibit a V-shaped structure.
Fig. 6. Symmetrized full-space correlated two-electron parallel momentum distributions for excitation-tunneling from the excited states of (a) $2s$, (b) $2p_0$, (c) $2p_1$, (d) $3s$, (e) $3p_0$, and (f) $3p_1$ in an 800 nm laser field with a peak intensity of $8.0\times 10^{14}$ W/cm$^2$.
By summing over all distributions corresponding to the situation in which the initially released electron promotes the second electron in the ground state of He$^+$ to all possible excited states of $n = 2$ and $n = 3$, each of which is specified with angular momentum and magnetic quantum number, we obtain the CMD for RESI of helium in 800 nm laser fields at peak intensities in the range from the threshold intensity to $15.0\times 10^{14}$ W/cm$^2$. The results for some selected intensities are shown in Figs. 7(a)–7(d). One can clearly see that the momentum distributions are not symmetric with respect to the coordinate axes for all the cases considered here. The first and third quadrants accommodate more electrons, and an overall V-shaped structure appears at intensities above $6.0 \times 10^{14}$ W/cm$^2$.
Fig. 7. Normalized correlated two-electron momentum distributions for NSDI of helium in 800 nm laser fields with peak intensities of 6.0, 8.0, 10.0, and $15.0\times 10^{14}$ W/cm$^2$, respectively. (a)-(d): RESI from all possible excited states of $n = 2$ and $n = 3$, (e)-(h): RDI from 2C, (i)-(l): RDI from 3C, (m)-(p) RESI and RDI from 2C, (q)-(t): RESI and RDI from 3C.
The asymmetry of the momentum distributions results from fast tunneling-ionization of electrons from excited states. It has been shown that all the excited He$^+$ ions are ionized within half an optical cycle in the subsequent strong laser fields at intensities above $3.0 \times 10^{14}$ W/cm$^2$ [34]. Since electrons are ionized faster at higher intensities, the momentum distributions become more asymmetric with respect to the coordinate axes as the intensity increases. Besides, the distinct V-shaped structure in the first and third quadrants at high intensities is ascribed to strong forward scattering of the returning electron after recollision.
As demonstrated in Fig. 2, forward scattering dominates in the process of electron impact excitation to all excited states when the incident energy is higher than about 100 eV, which corresponds to the maximum energy of the returning electron in an 800 nm laser field at a peak intensity of $5.3 \times 10^{14}$ W/cm$^2$. In [21], Zhou et al. attribute the off-diagonal feature in the correlated electron momentum distribution to asymmetric energy sharing between the two outgoing electrons. This argument is, indeed, consistent with the present simulations. In addition, Zhou et al. [21] pointed out that the nuclear attraction does not contribute to the V-like shape, which is different from that at the relatively low laser intensity. Again, this point is confirmed by our numerical calculations based on quantum collision theory. When the distorted wave is replaced by a plane wave for the scattering electron, these calculations reveal that strong forward scattering remains and becomes even more pronounced for incident energies above 100 eV.
However, it is still too early to draw a definite conclusion about the origin of the V-shaped structure observed in experiment at a relatively high intensity [18] without considering the contribution from RDI. As stated in the previous section, in the present work, the CMDs for recollision ($e,2e$) are simulated by employing the 2C and 3C models, respectively. Since the BBK model does not reproduce the correct absolute TCS (c.f. Fig. 1), the TDCS obtained from both the 2C and 3C models at each incident energy are renormalized according to the TCS from the BSR theory. The CMDs obtained from the 2C and 3C models are displayed in Figs. 7(e)–7(h) and 7(i)–7(l), respectively. The main patterns of the CMD from both 2C and 3C models almost remain the same when changing the laser intensity. Without taking into account the repulsion between the two outgoing electrons, the 2C model predicts a maximum along the main diagonal. On the contrary, the 3C model always reproduces a minimum in the same region.
Based on the numerical simulations at the quantum-mechanical level [17,20,33], it has been confirmed that the electron-electron repulsion plays a decisive role in forming the finger-like structure in the CMD for NSDI of helium at a relatively low intensity [17]. At high intensity, on the other hand, the influence of the final-state electron repulsion on the V-shaped structure has only been examined by Zhou et al. [21] using a classical three-dimensional ensemble model. In [21], it is demonstrated that the V-shaped structure can also be reproduced in the numerical simulations even when the final-state interaction $V_{ee}=1/r_b$ is replaced by a short-range potential $V_{ee}=\exp (-\lambda r_b)/r_b$, where $\lambda =5.0$ and $r_b=\sqrt {(r_1-r_2)^2+b}$ with $b=0.01$. Based on this observation, it was claimed that the V-shaped structure is not a consequence of the final-state electron repulsion at high intensity [21]. We also note that the laser intensity chosen in the classical simulations was $20\times 10^{14}$ W/cm$^2$, which is much higher than that used in the experiment [18]. In addition, when a short-range potential is used instead of the appropriate long-range potential, the Coulomb repulsion between the two outgoing electrons is not turned off completely, which is different from that in the 2C model used in the present work.
The main focus of the present work is to reexamine the role of the final-state electron repulsion and unveil the origin of the V-shaped structure observed in experiment at a relatively high intensity [18]. For this purpose, we display the entire CMD including both RDI and RSEI for NSDI of helium in 800 nm laser fields at peak intensities from 6.0 to $15.0\times 10^{14}$ W/cm$^2$ in the fourth and fifth rows of Fig. 7, where the 2C and 3C models are employed in the simulations of the CMD for RDI, respectively. It can be seen from Figs. 7(m) and 7(n) that, when the contributions of RDI are added, the gap between the two ends of the “V" in the CMD of RESI at intensities below $8.0 \times 10^{14}$ W/cm$^2$ in Figs. 7(a) and 7(b) is filled by the stripe perpendicular to the main diagonal in the CMD of RDI produced by the 2C model. This is shown in Figs. 7(e) and 7(f), in which the final-state electron repulsion is not taken into account. In contrast, the V-shaped structure remains in Figs. 7(q) and 7(r) due to the fact that the CMD predicted by the 3C model, in which the Coulomb interaction between the two outgoing electrons is retained, is off-diagonal. For intensities higher than $10.0\times 10^{14}$ W/cm$^2$, the NSDI of helium is dominated by RESI such that the pattern of the entire CMD almost remains the same as that of RESI, no matter whether the final-state electron interaction is retained or neglected in RDI. This is displayed in Figs. 7(o), 7(p), 7(s), and 7(t).
3.3 Focal-volume averaged correlated momentum distributions
So far, all discussions were based on simulated results without taking into account focal-volume averaging. It has been well recognized, however, that the intensity distribution of a focused laser beam is not uniform in space. Therefore, simulated results cannot be used to compare directly with experiment unless the focal-volume effect has been considered, especially for the situation of high intensities considered here. In the present work, we use the same method to do the focal-volume integral as that given in [33]. In the numerical calculations of the integration over intensity, a step size of $0.1\times 10^{14}$ W/cm$^2$ is used.
In Fig. 8 we show the focal-volume-averaged CMD including both recollision ($e,2e$) and excitation-tunneling for NSDI of helium in 800 nm laser fields at peak intensities from 4.0 to $15.0\times 10^{14}$ W/cm$^2$. For each of the intensities considered here, the CMD in the top two rows is quite different from that in the bottom two rows. Figures 8(a)–8(f) show that, without taking into account the final-state electron repulsion, the V-shaped structure cannot be predicted. On the contrary, as displayed in Figs. 8(g)–8(l), when the Coulomb interaction between the two outgoing electrons is considered, the QRS model always successfully reproduces the V-shaped structure at intensities above $6.0\times 10^{14}$ W/cm$^2$. This clearly indicates that, similar to the case for low intensities ($\sim 4.0\times 10^{14}$ W/cm$^2$), the final-state electron-electron interaction also plays an important role in forming the characteristic structure in the CMD of NSDI, at least for helium, at high intensities. Furthermore, as expected, the focal-volume effect is very important at high intensities such that the principal patterns of the CMD exhibited in Fig. 8 for intensities above $10 \times 10^{14}$ W/cm$^2$ differ significantly from the corresponding ones shown in the forth and fifth rows of Fig. 7. We note that the focal-volume effect was not taken into account by Zhou et al. [21] in their numerical calculations.
Fig. 8. Normalized correlated two-electron momentum distributions including both recollision ($e,2e$) and excitation-tunneling for NSDI of helium in 800 nm laser fields with peak intensities of 4.0, 6.0, 8.0, 10.0, 12.0, and $15.0\times 10^{14}$ W/cm$^2$, respectively. The calculations include the averaging over the focal volume of the laser. The momentum distributions for recollision ($e,2e$) in (a)-(f) and (g)-(l) were obtained from the 2C and 3C models, respectively. In (i) and (l), the straight line $p_1^{||}+p_2^{||}=3.0$ perpendicular to the main diagonal indicates the peak position in the recoil-ion momentum distribution.
3.4 Momentum distributions for doubly charged ions
Finally, by projecting the correlated two-electron momentum distribution onto the main diagonal, we obtain the longitudinal momentum distribution of doubly-charged ions. The first differential measurements for the helium double-ionization process were reported by Weber et al. [10] who measured the momentum spectra of He$^{2+}$ ions along the polarization direction of a laser field with a wavelength of 800 nm and a pulse duration of 220 fs at peak intensities of 2.9, 3.8, and $6.6\times 10^{14}$ W/cm$^2$, respectively. With increasing intensity, these spectra evolve from a single-peak into a double-hump structure.
In [18], Rudenko et al. also showed the longitudinal momentum distribution of He$^{2+}$ ions measured in an 800 nm laser field with a pulse duration of 25 fs at an intensity of $15\times 10^{14}$ W/cm$^2$. This distribution exhibits a more pronounced double-hump structure compared to the measurements of Weber et al. [10]. Furthermore, de Jesus et al. [43] presented a comprehensive experimental study on the ion-momentum distributions of He$^{2+}$, Ne$^{2+}$ and Ar$^{2+}$ ions, in which the parallel momentum distributions of the doubly-charged ions were measured for NSDI of He in 23 fs and 795 nm laser pulses at peak intensities of 6.0, 7.0, 10.0, and $12.5\times 10^{14}$ W/cm$^2$, respectively.
In Fig. 9, we compare the present longitudinal He$^{2+}$ momentum spectra with the measurements as well as the existing theoretical results for some selected intensities above $6\times 10^{14}$ W/cm$^2$. It should be noted that although the pulse durations used in experiments are different, only 25 fs laser pulses are used in the present simulations for comparison with all the experimental measurements, since it has recently been demonstrated that the relative counts of NSDI almost do not depend on the pulse duration for 800 nm laser field with pulse durations longer than 25 fs [32]. This direct comparison clearly demonstrates the quality and shortcoming of the theoretical simulations. As expected, all the measured distributions extend up to the upper limits for the kinematically favorable ion momenta of $\pm 4\sqrt {U_p}$, and the double-hump structure is enhanced as the intensity increases. To get the best overall agreement with experiment, a lower intensity had to be used in the present simulations for each of the cases considered here. With the adjusted intensities, overall good agreement with experiment has been achieved, and the trend of the change in the spectra with intensity is also predicted well by the QRS model.
Fig. 9. Recoil-ion parallel momentum distributions for NSDI of He in laser pulses with peak intensities of (a) 6.6, (b) 7.0, (c) 10.0, and (d) $15\times 10^{14}$ W/cm$^2$. The experimental data are taken from Weber et al. [10], de Jesus et al. [43], and Rudenko et al. [18] (see text for details). The arrows indicate the upper limits for the kinematically favored ion momenta of $\pm 4\sqrt {U_p}$. The dotted line and solid squares in panel (a) are the theoretical results of Chen and Nam [44] and of Becker and Faisal [45], respectively. In the present calculations, focal-volume averaging has been taken into account, and lower intensities were used than those in the corresponding experiments for the best overall agreement.
As demonstrated in Figs. 7 and 8, the double-hump structure is mainly created by the recollision ($e,2e$) process. The enhancement of the double-hump structure is due to the fact that the relative contribution of RDI increases with increasing intensity. Despite the overall agreement, discrepancies between the present model results and the experimental data remain. For example, in Figs. 9(b) and 9(c), the double-hump predicted by the QRS model is located at larger momenta compared to that observed in experiment. Furthermore, it is interesting to see from Fig. 9(d) that at the higher peak intensity of $15\times 10^{14}$ W/cm$^2$ the QRS model reproduces a much deeper minimum at zero momentum, whereas the width of the distribution almost remains the same as that at the lower intensity of $8.0\times 10^{14}$ W/cm$^2$. This can be understood by tracing back to the correlated two-electron momentum distributions. The straight line $p_1^{||}+p_2^{||}=3.0$ drawn in Figs. 8(i) and 8(l) clearly indicates that both the position of the double-hump and the width of the recoil-ion momentum distribution do not change significantly for intensities above $8.0 \times 10^{14}$ W/cm$^2$, although the CMD spreads wider with increasing intensity.
On the other hand, using a semiclassical rescattering model and $S$-matrix theory, Chen and Nam [44] as well as Becker and Faisal [45] also performed numerical calculations for the momentum distributions of recoil ions from laser-induced nonsequential double ionization. Those results for He$^{2+}$ at the intensity of $6.6\times 10^{14}$ W/cm$^2$ are displayed in Fig. 9(a). Both the semiclassical model and the $S$-matrix theory predict a clear double-hump shape with a very deep minimum, which is inconsistent with the experimental findings. In addition, the momentum distribution of the semiclassical rescattering model is much wider than that of experiment. We note again that the focal-volume effect was not considered in the simulations of Chen and Nam [44] or Becker and Faisal [45].
4. Summary and conclusions
We have presented a systematic study of the intensity dependence of nonsequential double ionization of helium by 800 nm laser pulses. The correlated two-electron momentum distributions (CMD) were simulated by employing the quantitative rescattering model (QRS) for intensities in the range of $(2-15) \times 10^{14}$ W/cm$^2$. With the final-state electron repulsion taken into account, the calculated CMD reproduce the experimentally observed V-shaped structure at a relatively high intensity well. For intensities above $6.0 \times 10^{14}$ W/cm$^2$, strong forward scattering occurs in laser-induced recollision excitation due to the high incident energy of the returning electron. The momentum distributions of the tunneling electrons are highly asymmetric owing to fast ionization from excited states. Both effects play a decisive role in forming the V-shaped structure in the CMD for RESI. By adding the contributions from RDI, the V-shaped structure in the CMD for the entire NSDI description only remains if the Coulomb interaction between the two outgoing electron is retained in the simulations of the CMD for RDI. This clearly indicates that the final-state electron repulsion cannot be neglected.
Our calculations also reveal that inclusion of the focal-volume effect could dramatically change the pattern of the simulated CMDs compared to those obtained without focal-volume averaging, especially for high intensities. This implies that discussions of simulated results without taking into account the focal-volume effect could be misleading. Finally, the good agreement between the simulated recoil-ion momentum distributions and the experimental findings once again confirms the quality of the QRS model.
Funding
National Natural Science Foundation of China (11274219); Science and Technology Planning Project of Guangdong Province (180917124960522); Guangdong Province Introduction of Innovative R&D Team (2018KTSCX062); Japan Society for the Promotion of Science (17K05597, 19H00887); National Science Foundation (OAC-1834740, PHY-090031, PHY-1803844).
Disclosures
The authors declare no conflicts of interest.
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. A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, “Multiply charged ions formed by multiphoton absorption processes in the continuum,” Phys. Rev. Lett. 48(26), 1814–1817 (1982). [CrossRef]
3. A. L’Huillier, L. A. Lompre, G. Mainfray, and C. Manus, “Multiply charged ions induced by multiphoton absorption in rare gases at 0.53 µm,” Phys. Rev. A 27(5), 2503–2512 (1983). [CrossRef]
4. D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. Kulander, “Observation of nonsequential double ionization of helium with optical tunneling,” Phys. Rev. Lett. 69(18), 2642–2645 (1992). [CrossRef]
5. B. Walker, E. Mevel, B. Yang, P. Breger, J. P. Chambaret, A. Antonetti, L. F. DiMauro, and P. Agostini, “Double ionization in the perturbative and tunneling regimes,” Phys. Rev. A 48(2), R894–R897 (1993). [CrossRef]
6. B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, “Precision measurement of strong field double ionization of helium,” Phys. Rev. Lett. 73(9), 1227–1230 (1994). [CrossRef]
7. 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]
8. D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. Kulander, “Polarization dependence of tunneling ionization of helium and neon by 120-fs pulses at 614 nm,” Phys. Rev. A 49(3), 2174–2177 (1994). [CrossRef]
9. C. Guo and G. N. Gibson, “Ellipticity effects on single and double ionization of diatomic molecules in strong laser fields,” Phys. Rev. A 63(4), 040701 (2001). [CrossRef]
10. T. Weber, M. Weckenbrock, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, F. Afaneh, G. Urbasch, M. Vollmer, H. Giessen, and R. Dörner, “Recoil-ion momentum distributions for single and double ionization of helium in strong laser fields,” Phys. Rev. Lett. 84(3), 443–446 (2000). [CrossRef]
11. R. Moshammer, B. Feuerstein, W. Schmitt, A. Dorn, C. D. Schröter, J. Ullrich, H. Rottke, C. Trump, M. Wittmann, G. Korn, K. Hoffmann, and W. Sandner, “Momentum distributions of Nen+ ions created by an intense ultrashort laser pulse,” Phys. Rev. Lett. 84(3), 447–450 (2000). [CrossRef]
12. A. Rudenko, K. Zrost, B. Feuerstein, V. L. B. deJesus, C. D. Schroter, R. Moshammer, and J. Ullrich, “Correlated multielectron dynamics in ultrafast laser pulse interactions with atoms,” Phys. Rev. Lett. 93(25), 253001 (2004). [CrossRef]
13. 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]
14. 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]
15. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]
16. 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]
17. 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]
18. 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]
19. D. Ye, X. Liu, and J. Liu, “Classical trajectory diagnosis of a fingerlike pattern in the correlated electron momentum distribution in strong field double ionization of helium,” Phys. Rev. Lett. 101(23), 233003 (2008). [CrossRef]
20. 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]
21. Y. Zhou, Q. Liao, and P. Lu, “Asymmetric electron energy sharing in strong-field double ionization of helium,” Phys. Rev. A 82(5), 053402 (2010). [CrossRef]
22. 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]
23. 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]
24. 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]
25. 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]
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. S. Micheau, Z. Chen, A.-T. Le, and C. D. Lin, “Quantitative rescattering theory for nonsequential double ionization of atoms by intense laser pulses,” Phys. Rev. A 79(1), 013417 (2009). [CrossRef]
28. 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]
29. 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]
30. 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]
31. 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]
32. 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]
33. 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]
34. 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]
35. Z. Chen, F. Liu, and H. Wen, “Quantitative rescattering theory for nonsequential double ionization,” Chin. Phys. B 28(12), 123401 (2019). [CrossRef]
36. G. L. Yudin and M. Y. Ivanov, “Correlated multiphoton double ionization of helium: The role of nonadiabatic tunneling and singlet recollision,” Phys. Rev. A 64(3), 035401 (2001). [CrossRef]
37. O. Zatsarinny, “BSR: B-spline atomic R-matrix codes,” Comput. Phys. Commun. 174(4), 273–356 (2006). [CrossRef]
38. 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. Opt. Phys. 46(11), 112001 (2013). [CrossRef]
39. Y. Liang, Z. Chen, D. H. Madison, and C. D. Lin, “Calibration of distorted wave Born approximation for electron impact excitation of Ne and Ar at incident energies below 100 eV,” J. Phys. B: At., Mol. Opt. Phys. 44(8), 085201 (2011). [CrossRef]
40. 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]
41. Z. Chen, T. Morishita, A.-T. Le, and C. D. Lin, “Analysis of two-dimensional high-energy photoelectron momentum distributions in the single ionization of atoms by intense laser pulses,” Phys. Rev. A 76(4), 043402 (2007). [CrossRef]
42. M. Brauner, J. S. Briggs, and H. Klar, “Triply-differential cross sections for ionisation of hydrogen atoms by electrons and positrons,” J. Phys. B: At., Mol. Opt. Phys. 22(14), 2265–2287 (1989). [CrossRef]
43. V. L. B. de Jesus, B. Feuerstein, K. Zrost, D. Fischer, A. Rudenko, F. Afaneh, C. D. Schröter, R. Moshammer, and J. Ullrich, “Atomic structure dependence of nonsequential double ionization of He, Ne and Ar in strong laser pulses,” J. Phys. B: At., Mol. Opt. Phys. 37(8), L161–L167 (2004). [CrossRef]
44. J. Chen and C. H. Nam, “Ion momentum distributions for He single and double ionization in strong laser fields,” Phys. Rev. A 66(5), 053415 (2002). [CrossRef]
45. A. Becker and F. H. M. Faisal, “Interpretation of momentum distribution of recoil ions from laser induced nonsequential double ionization,” Phys. Rev. Lett. 84(16), 3546–3549 (2000). [CrossRef]
