1. Introduction

When atoms and molecules are irradiated by intense laser fields, double ionization (DI) may occur [1]. The three-step scenario [2] provides an intuitive picture of the nonsequential DI (NSDI) process in which an electron is freed and then is accelerated by the intense laser field, and it may turn back to near its parent ion when the electric field reverts. When the returning electron collides with the parent ion, the transferred energy makes another electron easier to ionize. Consequently, the probability of the second ionization is greatly enhanced, and a knee structure appears in the curve of ion yield increasing with laser intensity. This process provides a simple prototype of electron correlation that has aroused extensive studies [3–6].

The knee structure is a sign of electron recollision, and is observed frequently in linearly polarized (LP) laser fields, but it is rarely observed in circularly polarized (CP) fields [7]. Recently, the NSDI process in counter-rotating circular two-color (CRTC) fields has attracted much attention [8–11]. The CRTC field is composed of a fundamental frequency and its second harmonic, and both of the two colors are circularly polarized. This field traces out a three-leaf-clover pattern, thus the ionized electron can encounter the parent ion. When the electron revisits its parent ion, it may collide with the parent ion and trigger NSDI. The characterized knee structure was disclosed and NSDI enhancement was identified [8–10]. The CRTC fields are found to be a versatile and well-controlled tool with which to manipulate the electronic behavior on sub-femtosecond timescales [12,13].

In this paper, we propose to enhance the NSDI yield of an O$_{2}$ target by using CRTC laser fields. The NSDI yield is crucial in studying the phenomena related to the electron correlation. However, the NSDI yield is generally quite low due to the large ionization potential. This results in multiple difficulties in experimental observation. One example is the observation of the knee structure by circular polarization. The electron recollision in the CP fields has been disclosed theoretically for noble atoms [14–16], however, no experimental observation has been reported for atoms other than Mg so far [7]. This is due to the lower NSDI yield in the knee region [17]. In order to seize the weak signals, one must use the complicated instruments of high resolution and high stability and establish elaborate experimental programs [18].

Here we propose to enhance the NSDI probability of the O$_{2}$ target by using a CRTC field that is composed of a fundamental wave and its third harmonic. The combined electric field traces out a four-leaf-clover pattern, and has four maxima in one optical cycle. When the ionized electron returns, it has four chances to collide with the remaining electrons in one optical cycle, so the collision probability is greatly increased. Consequently, the NSDI rate is expected to be notably enhanced. This is particularly meaningful for O$_{2}$ molecules, because the single ionization is greatly suppressed owing to the antisymmetric orbital property, which further decreases the NSDI yield [19,20]. Therefore, using the proposed scheme will bring tangible benefits to the study of the electron correlation in O$_{2}$ molecules.

Our study is based on the ensemble method proposed by Haan et al., which have had obtained many successes in treating NSDI of atoms [21]. However, this method cannot fully treat the orbital symmetry, while the latter is crucial to photoionization of molecules [22,23]. Therefore, the study of the molecular NSDI process using this method is limited and has mostly been carried out in the H$_{2}$ molecule [24]. Here, we change the Coulomb potential to treat the orbital symmetry. Our study confirms the collision enhancement, and the NSDI probability is increased as much as 3 orders of magnitude more than that obtained by linear polarization in low intensity region. We further show that both the NSDI yield and the underlying NSDI behavior depend distinctly on the field ratio of two colors. The influence of molecular structure is also discussed.

2. Methods and results

We start from the standard treatment: an initial state of a two-active-electron molecule satisfies the field-free Hamiltonian, and each electron is confined by the Coulomb force of both of the two oxygen nuclei and the other electron. The initial state evolves under the government of the field-free Newton equation until the resulting ensemble maintains a stable position and momentum distributions. The light-free Hamiltonian of a two-active-electron molecule can be written as (in atomic units):

 

Fig. 1. The calculated initial electron distribution of O$_{2}$ (a) and N$_{2}$ (b) molecules. The black circles denote the position of nucleus. The calculated single (c) and double (d) ionization rates of O$_{2}$ and N$_{2}$ molecules are also depicted, respectively. The laser field is of wavelength 800nm and is linearly polarized along x-axis.

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When the CRTC field acts, we add a dipole term on the field-free Hamiltonian to treat the effect of the laser field. The fundamental wave of the CRTC field is of wavelength 800 nm and has a 3-10-3 trapezoidal pulse envelope, and the third harmonic is of equal pulse duration but opposite helicity. The motion of two electrons is governed by the field-dependent Newton equation, with the initial position and momentum of two electrons being chosen randomly from the initial ensemble. At the end of pulse, a DI event is counted when the energy of each electron is greater than zero. The DI probability is calculated as the number of DI events over that of the initial ensemble. An initial ensemble, including trajectories as much as $6.0\times 10^{7}$, is used in our simulation. The use of a large ensemble recovers some quantum effects of the interaction system [9].

First, we check the validity of our model by calculating the single ionization probability of O$_{2}$ molecules and comparing it with that of N$_{2}$ molecules in linearly polarized (LP) laser fields. Some results are shown in Fig. 1. Generally, for laser intensity higher than $2.0\times 10^{14}$ W/cm$^{2}$, the calculated single ionization probability reaches about unity for both O$_{2}$ and N$_{2}$ molecules, and the distinctive difference appear for the laser intensity below $10^{14}$ W/cm$^{2}$, which shows that the single ionization probability from O$_{2}$ molecues is far lower than that of N$_{2}$ molecules. This difference is caused by the electronic structure in the HOMOs and discloses that our model treats the antibonding electronic structure of O$_{2}$ molecules properly. Then we study the DI probability. For completeness, we calculate the DI probability of aligned O$_{2}$ target in 800 nm LP fields. The molecular ionization depends on the relative orientation with respect to the laser polarization [23]. Owing to the nodal planes in the $1\pi _{g}$ orbital, the single ionization rate from the O$_{2}$ target was found to reach a maximum at an angle of 45$^{\circ }$. This suggests a large dipole momentum in this orientation, which will lead to a high DI rate. The numerical results in low-intensity region ($\leq 10^{14}$ W/cm$^{2}$) confirm this expectation, where the DI probability varies distinctly with the relative orientation and achieves maximum at an angle of 45$^{\circ }$. As laser intensity increases, the nodal planes are distorted, and the electron collision becomes possible for any orientation, so the DI probability increases quickly with laser intensity but depends less on the molecular orientation. When laser intensity is higher than $2.0\times 10^{14}$ W/cm$^{2}$, the electron recollision probability becomes saturated, which reduces the growth of DI rate. In the high-intensity region ($\geqslant 10^{15}$W/cm$^{2}$), sequential DI prevails. Consequently, the DI curve exhibits an apparent knee structure, indicating the nonsequential feature.

Accordingly, we expect that the DI probability in the CRTC field will be notably enhanced in the low-intensity region, induced by frequent electron collisions. The curve for the calculated DI probability increasing with laser intensity is depicted in Fig. 2(a) for the two colors being of equal intensity. The enhancement in NSDI rate is significant, and reaches more than 2 orders of magnitude in the low-intensity region. The knee structure exists and is clear, but is not so evident as that in the LP fields. The knee structure starts from a lower laser intensity and covers a small laser-intensity range. The enhancement in NSDI rate is saturated at a relatively low laser intensity. Owing to the low sequential DI probability below $10^{15}$W/cm$^{2}$, we attribute all of the DI events to the NSDI process.

 

Fig. 2. Curves for calculated DI probabilities increasing with laser intensity are depicted for a variety of conditions. In (a), linear A and C denote that the driving laser field is linearly polarized along with (A) and 45$^{\circ }$ with respect to (C) the molecular axis, respectively. In (b), the electric field ratio is marked. For comparison, curves for the CP fields of frequency of $\omega$ and $3\omega$ are also depicted in (a) and (b), respectively.

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The enhancement in NSDI rate can be further increased by properly choosing the field ratio of two colors, i.e., $E_{3\omega }:E_{\omega }$. To find an optimized field ratio, we calculate the NSDI probability for several ratios with keeping the total intensity unchanged, see Fig. 3. The enhancement is notable for a broad range of the field ratio, and an optimized ratio exists. The curve is not symmetric about the optimized ratio, and is high on the right side. These features are the same as those in the $\omega -2\omega$ scheme [8–10]. Nevertheless, nontrivial differences also appear. The optimized ratio appears generally at approximately $1.6:1$ and shifts slightly to the $\omega$ side as the intensity increases. The NSDI yield is much high on the $3\omega$ side. These differences are caused by the collision enhancement in high-frequency CP fields. It has been shown that some trajectories still exist in the CP fields by which the ionized electron revisits its parent core and triggers the NSDI process [14]. The knee structure appears and the ion yield in the knee region increases with laser frequency [17]. Thus, the NSDI yield on the $3\omega$ side is generally higher than that on the $\omega$ side. This feature can be also found in the $\omega -2\omega$ CRTC curve [8–10], but is not so evident as in the present $\omega -3\omega$ scheme. As shown in Fig. 2, the DI curve for the $\omega$ field increases monotonously, but the $3\omega$ curve exhibits an apparent knee structure, suggesting that the electron recollision assumes a role.

 

Fig. 3. Variations of the NSDI probability with field ratio $E_{3\omega }/E_{\omega }$ are shown for three intensities. Lissajous curves of electric fields (blue) and vector potentials (red) are shown for field ratios of (d) $0.7:1$, (e) $1:1$, and (f) $1.7:1$, respectively. In each plot, the total intensity keeps constant when the field ratio changes.

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The maximal enhancement can be reached by using the optimized field ratio. The yield curve for the field ratio $E_{3\omega }:E_{\omega }=1.6:1$ is shown in Fig. 2(b). The NSDI probability is further drastically increased. Compared with that obtained by linear polarization, the maximal enhancement exceeds 3 orders of magnitude in the low-intensity region. This is a giant enhancement and brings many advantages to the study of the electron correlation in an O$_{2}$ target.

Second, we study the electronic behavior underlying the NSDI yield. This is done by exploiting the momentum of the recoil ion that equals the minus sum of two electrons’ momenta. The momentum distribution (MD) of the recoil ion varies distinctly with the field ratio, as is shown in the top row of Fig. 4. When the field ratio is small, the MD exhibits a bright region with a four-blade fan-like structure highlighted. As the field ratio increases, the blades shrink to the center, and finally the fan-like structure degenerates into a concentrated region around a reddish center.

 

Fig. 4. MDs of recoil ion (top) and second electron (bottom row) are shown for field ratios of (from left to right colored columns) $1:1$, $1.7:1$, and $3.5:1$, respectively. In each plot, the total intensity keeps constant when the field ratio changes.

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The change of MD with the field ratio discloses different NSDI behaviors underlying the total yield. The NSDI behavior can be further classified into the recollision excitation with subsequent field ionization (RESI) and the recollision impact ionization (RII) processes. Generally, the recoil ion has a small momentum in the RESI process and a large momentum is possible in the RII process. Therefore, we judge that the central peaked MDs reveal the NSDI being dominated by the RESI process, and that the fan-like-structure MDs denote the RII process playing a notable role.

We use the back-trajectory technique [25,26] to identify the electron behavior in detail. The MDs of the second electron are depicted in the bottom row of Fig. 4. In the left two MDs, the inner ring distributes like the vector potential and is formed by the electrons ionized via the RESI process, while the outer four triangle-like regions are formed by electrons ionized via the RII process. Similar to the vector potential, the inner ring shrinks and finally disappears when the field ratio increases. In the MD for a field ratio of $3.5:1$, the bright region is highlighted by four reddish areas. Since the CRTC field is quite closely approaching circular polarization, the NSDI takes place chiefly through special trajectories in the CP field [14], so the MD differs greatly from the prior two.

This further reveals the role of two components in the CRTC field. When the fundamental field is strong, the ionized electron accelerates over a long time and experiences a long excursion trajectory. When it returns, the electron becomes energetic and can transfer more energy to the remaining electrons, so the RII process becomes more likely. In contrast, when the third harmonic is dominant, the first electron experiences a short excursion trajectory and is less energetic when it returns, so the RESI process prevails.

The variation of MD with field ratio also depends on the peak intensity of the CRTC field. In the low-intensity region, the fan-like structure is not obvious, because the final momentum of the ion is small and the blades are invisible. A full filled reddish center is highlighted. As laser intensity increases, the fan-like structure becomes easy to be identified and is general, as shown in Fig. 5(b) for 3.2$\times 10^{14}$ W/cm$^{2}$. The timing information indicates that the second ionization happens almost equally in each sub-cycle. When laser intensity is further increased, the second ionization occurs unequally in the four sub-cycles. Only one or two blades are highlighted with their orientation varying with the laser intensity, so the MD resembles a rotating fan. This is clearly shown in the left-hand column of Fig. 5. The temporal statistics on the second ionization confirms that the RII electron contributes much to the fan-structured MDs, as shown in the right-hand column of Fig. 5. The RII electron is released near the crossing point of the electric field, and forms the small peaks located at around the crossing points. The variation of these small peaks in height and location supports our judgement.

 

Fig. 5. Left-hand column, MDs of the recoil O$_{2}^{2+}$; right-hand column, time statistics of the second ionization. Laser intensities form top to bottom rows are 2.0, 3.2, and 4.8$\times 10^{14}$ W/cm$^{2}$, respectively, and the field ratio is 1:1. In each plot, the total intensity keeps constant when the field ratio changes.

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Finally, we study the role of molecular structure for further analyzing the collision enhancement. The molecular structure of the O$_{2}$ target manifests itself by the dominance of the RESI process. Owing to the four-lobe electronic distribution, the first electron has more chance to collide with the remaining electrons. This shortens its excursion in the CRTC field and makes it less energetic. Moreover, when it meets the remaining electrons, an oblique impact is general in the CRTC fields. Consequently, the electron collision occurs more frequently but the energy transferred to the remaining electrons becomes small, so the second ionized electron tends to be excited first.

The above statements can be deduced by comparing the MDs of the recoil O$_{2}^{2+}$ and Xe$^{2+}$ ions. Several MDs at the knee region are depicted in Fig. 6, the left-hand column of which shows the MDs for a field ratio of $1:1$. The MD of the Xe$^{2+}$ ion exhibits four highlighted reddish branches around a light-colored center, while that of the O$_{2}^{2+}$ ion shows a fan-like structure in which the center is also highlighted. This indicates that the RESI process comprises a prevailing portion in the O$_{2}$ target. A similar difference exists for other field ratios, as shown in the right-hand column of Fig. 6 for a field ratio of $1.6:1$. In this case, two MDs are peaked at the center, indicating that the RESI process is prevailing. However, the MD of the Xe$^{2+}$ ion has four branches while that of the O$_{2}^{2+}$ ion does not. This suggests that more RII processes occur for Xe atoms. That is to say, the RESI process is more likely to occur in the NSDI of molecules. For the energetic returning electron, it has more opportunity to knock out the second electron, so the RII process is also altered by the molecular structure. This can be found in the MDs of the second electron, as shown in Fig. 4(e), in which the outer ring of the MD exhibits more bright twigs.

 

Fig. 6. MDs of Xe$^{2+}/$O$_{2}^{2+}$ ions are shown in top/bottom rows, respectively. Left- and right-hand columns are for field ratios of $1:1$ and $1.6:1$, respectively. The CRTC field driving O$_{2}$ molecules is of fundamental wavelength 800 nm and combined intensity $6.0\times 10^{14}$ W/cm $^{2}$, and that driving Xe atoms is of 910 nm and $4.08\times 10^{14}$ W/cm$^{2}$. Such a choice ensure that the electronic dynamics in the two CRTC fields are equivalent [27], so the differences indicate the influence of molecular structure.

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3. Conclusions

The NSDI probability of molecules can be enhanced significantly by using an $\omega -3\omega$ CRTC driving field. The CRTC field increases the electron collision probability in O$_{2}$ molecules, which results in the NSDI probability being enhanced significantly in low-intensity region. The enhancement in NSDI rate stems two factors: one is high-frequency component of the CRTC field and the other is the electronic structure of $1\pi _{g}$ state of O$_{2}$ molecules. The enhance effect is notable in low-intensity region. The enhancement in NSDI rate appears in a broad range of the relative intensity of two colors and is optimized at the field ratio of approximately $1.6:1.$ In addition, both the MD of the O$_{2}^{2+}$ ion and the electronic behavior vary distinctly with field ratio. Owing to the frequent collision, the RESI mechanism dominates the molecular NSDI process. The enhancement in NSDI rate in the CRTC scheme depends less on the molecular alignment, so the molecules do not to be pre-aligned, which brings many advantages in experimental observations.