1. Introduction

The molecules exhibit exotic dynamic behaviors in intense laser fields depending on the laser intensity [1]. Among these dynamic processes, ionization is fundamental. It underlies most molecular strong-field phenomena such as high-harmonic generation [2], above-threshold ionization [3], laser-induced electron diffraction [4] and non-sequential double ionization [5]. However, due to the complicated molecular structure, it is very difficult to fully understand the ionization process of a molecule in an intense laser field. MO-ADK theory is commonly used to deal with the ionization of some diatomic and triatomic molecules [6,7]. According to this theory, the electron in the highest occupied molecular orbital (HOMO) is considered to be removed because of the exponential decay of the ionization rate on the electron binding energy. The ionization rate is determined by the electronic density of the valence electrons of a molecule. It is largest when the initial electronic cloud is aligned with the laser field direction. Therefore, the angular-dependent ionization reflects the symmetry of the HOMO electron. This prediction has been verified by experimentally measuring the ionic yield as a function of the angle between the molecular axis and the laser polarization [8]. However, there is some discrepancy between theoretic predictions and experimental measurements for some molecules [2,8]. Recent studies have shown that this discrepancy comes from the ionization of multiple valence orbitals in molecules [9–11]. One possible way to identify and separate the contribution of multiple molecular orbitals is the coincident measurement with the state-resolved parent ion [11].

When the laser intensity is further increased, several electrons are removed from molecules irradiated by such intense laser fields. Multiple ionization is complicated for molecules due to the coupling between the nuclear and the electron coordinates. The ionization mechanism can be divided into direction ionization [12,13], electron rescattering [14], charge-resonance enhanced ionization [15–17], and sequential ionization [18,19]. In the case of direct ionization or electron rescattering, the stretch of the internuclear distance is negligible because of the small time interval between two adjacent ionization events. Ionization occurs near the equilibrium internuclear distance R_e. In the case of enhanced ionization, the ionization occurs at the critical distance R_c. R_c is about two folds of R_e for most diatomic molecules. At R_c the appearance intensity of an explosion pathway is lowest and the ionization rate is many orders of magnitude greater than that at either a shorter or longer internuclear distance. Enhanced ionization plays an important role for laser pulses with pulse durations of few hundreds femtoseconds. Recent measurements indicate that ionization occurs at the internuclear distance R with R_e < R < R_c when the pulse duration is few tens femtoseconds [18,19]. During the time interval between two ionization events, the internuclear distance is stretched. This ionization is in a sequential manner and named as sequential ionization or stairstep process.

Single ionization and Coulomb explosion of oxygen have been extensively studied by many groups. One important finding is the ionization suppression of oxygen with respect to its companion xenon that has nearly identical binding energy [20–22]. Some theoretical models are also proposed to explain this observation [6,7,23,24]. Because of the exponential decay of the ionization rate on the electron binding energy, only the HOMO electron has been considered to be stripped away in these models. The electronic configuration of neutral oxygen is (1σ_g)²(1σ_u)²(2σ_g)²(2σ_u)²(3σ_g)²(1π_u)⁴(1π_g)². Its HOMO orbital is π_g, whose density is small in the direction of the molecular axis. According to MO-ADK model, the ionization rate is determined by the electronic cloud in the laser field direction [7]. When the molecule is aligned along the laser field direction, the ionization rate is very small because of the small density of the electron cloud in the laser field direction and explains the ionization suppression of oxygen. Another important finding is the anisotropic angular distribution of the atomic ions generated through Coulomb explosion. This anisotropic angular distribution has been attributed to dynamic alignment or geometric alignment [25–34]. The influence of molecular structure on non-sequential double ionization has been explored by comparing the momentum distributions of electrons produced in the double ionization of nitrogen and oxygen [35]. In this article, we experimentally study the double and triple ionization of oxygen irradiated by intense laser pulses with varied pulse durations and laser intensities. The fragmentation channels are identified by coincident measurement. The KER exhibits definite structures for O²⁺ ₂ → O⁺+ O⁺ independent of the laser parameters. However, the KER shows a single broad peak for O²⁺ ₂ → O²⁺+ O and O³⁺ ₂ → O²⁺+ O⁺. In addition, the KER decreases with increasing the pulse duration. A brief explanation of this phenomenon is also included.

2. Experimental Setup

The commercial laser system (Femtolasers, Austria) has been described in our previous paper [36]. It delivers 780 nm, 24 fs laser pulses at a repetition of 3 kHz. For the generation of few-cycle pulses, the spectrum is further broadened in a 1 m long hollow fiber seeded with 2 atm neon. Then the laser is collimated by an f = 1 m concave mirror and compressed by five broadband chirped mirrors. For the compensation of the dispersion in the beam path, another pair of broadband chirped mirrors is used. The attenuation of the laser intensity is achieved by using two thin achromatic film polarizers. The pulse duration is measured to be less than 8 fs. The incident laser is focused by an f = 75 mm concave mirror inside the chamber. The gaseous sample is diffused into the chamber via a leak valve and the pressure is 1 × 10^-9 mbar with the base pressure of 4 × 10^-11 mbar. After the interaction of the laser and the molecules, the ions and electrons are accelerated to opposite direction by the uniform electric and magnetic field. The times and positions that the ions reach the temporal and position-sensitive detector (Roentdek, Germany) are used to measure their momentum vectors. As the leaked gas is warm target, the initial broad velocity distribution will limit the momentum resolution. Here, we use a software cooling method to remove the effect of the initial thermal energy of the gas target. In the following, we introduce the principle of software cooling method with the pathway of O²⁺ ₂ → O⁺+ O⁺. Let us assume that the initial velocity is v⃗ for the warm target O₂ and the mass is 2m. The velocities of the coincident O⁺ generated in the pathway O²⁺ ₂ → O⁺+ O⁺ are v⃗₁ and v⃗₂, respectively. Due to the momentum conservation, we have the formula mv⃗₁ + m⃗v₂ = 2mv⃗. Thus, we can reconstruct the initial momentum of the warm target from the information that we have collected from both ions. In the molecular frame, the velocities of the two atomic ions are:

These expressions demonstrate that we can remove the effect of the initial momentum of warm target. Doppler-free high resolution KERs are obtained with this software cooling method.

3. Results and discussions

Figure 1 shows the kinetic energy release (KER) of O²⁺ ₂ → O⁺+ O⁺ induced by linearly polarized laser pulses with different intensities and pulse durations. The KERs are obtained by measuring the position and arrival time of coincident O⁺. By using the software cooling method, the initial momentum of the warm target is removed and the KER resolution is greatly improved. Owing to the high resolution in the present measurements, clear structures of the KER spectra are observed. These structures look similar independent of the laser intensity and the pulse duration. In comparison with Doppler free KER spectrum obtained by fast electron bombardment [37], we conclude that these structures originate from the dissociation of molecular dications in some definite electronic states. The peaks labeled by a and b result from the dissociation of electronic states, W³Δ_u, B³∑^- _u, 1¹ Δ_u, or 1¹ ∑^- _u. These electronic states are formed through the removal of one HOMO (π_g) electron and one HOMO-1 (π_u) electron. While the peak labeled by c results from the dissociation of electronic state B³∏_g. This state relates to the removal of one HOMO-2 (σ_g) electron and one HOMO electron.

 

Fig. 1. The KER of O²⁺ ₂ → O⁺ + O⁺ induced by linearly polarized (LP) laser pulses with different intensities and pulse durations. (I₀: 10¹⁴ W/cm²).

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Depending on the laser intensity, the molecular dications are formed through nonsequential double ionization or sequential double ionization. In the case of non-sequential double ionization, the electron generated in the tunneling ionization of the neutral molecule returns to the parent ion driven by the oscillating laser field. The electron-ion collision leads to the production of a doubly charged parent ion [14]. However, in the case of sequential double ionization, the tunneling ionization of the neutral molecule is followed some time later by tunneling ionization of the singly charge parent ion [18]. By measuring the ionic yields versus laser peak intensity or comparing the ionic yields between the linearly and circularly polarized laser pulses at equal maximum electric field, the double ionization mechanism can be determined. When the laser intensity is few 10¹⁴ W/cm², non-sequential double ionization dominates for oxygen [32]. Sequential double ionization becomes dominant when the laser intensity is further increased. Under our experimental condition, we believe that the primary mechanism is non-sequential double ionization when the laser intensity is around 2.5 × 10¹⁴ W/cm².

Regardless of the ionization mechanism, the first step of double ionization is a tunneling ionization of the neutral molecule. According to MO-ADK model, the ionization rate depends on the symmetry of the molecular orbital [7]. It is largest when the initial electronic cloud is aligned with the laser field direction. Therefore, the angular dependence of ionization reflects the symmetry of the HOMO electron, which is a π_g for oxygen. The ionization rate should peak near 40° relative to the laser polarization, which has been verified by measuring the ionization probability as a function of alignment angle [8]. Because of the ultrashort dissociation time, it is reasonedly proposed that the angular distribution of the atomic ions represents the molecular axis direction at the formation moment of these molecular dications. Thus the angular distribution is weighted by the angle-dependent ionization if we can neglect the alignment effect of molecules in intense laser fields. However, alignment and ionization are entangled together for molecules in intense laser fields, the angular distributions of atomic ions are influenced by the alignment effect. Both the long pulse and high intensity cause alignment. In order to compare the alignment effect, we measure the angular distribution of coincident O+ under different laser conditions.

 

Fig. 2. Angular distributions of coincident O⁺ with KER between 6 and 7eV (peak a in Fig. 1). The pulse durations and peak intensities of the laser pulses are (a): 24fs, 2I₀; (b) 8fs, 2.5I₀; and (c) 8fs, 10I₀. (I₀: 10¹⁴ W/cm²).

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Figure 2 shows the angular distribution of coincident O⁺ with KER between 6 and 7 eV (peak a in Fig. 1). In comparison with the theoretical predicated angular-dependent ionization rate [33], we find that the angular distribution in Fig. 2b looks similar to the angular-dependent ionization probability and the peak is near 40° relative to the laser polarization. However, the peak shifts toward the laser polarization direction in Fig. 2a and 2c. The shift originates from the alignment effect of molecules in intense laser field. The long pulse in Fig. 2a results in the dynamic alignment of the neutral molecules before ionization. The high laser intensity in Fig. 2c results in the post ionization alignment during the fragmentation process. Both the dynamic alignment and the post ionization alignment can shift the peak of angular distributions along the laser polarization. Alignment effect can be neglected only for few-cycle laser pulses at low laser intensity. Under this condition, the angular distribution is determined by the angular-dependent ionization and reflects the symmetry of the highest occupied molecular orbital.

 

Fig. 3. Angular distributions of coincident O⁺ induced by 8fs laser pulses with an intensity of 2.5 × 10¹⁴ W/cm². The KERs of coincident O⁺ are shown by (a) peak a in Fig. 1, (b) peak b in Fig. 1, and (c) peak c in Fig. 1.

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The peaks a, b and c in Fig. 1 come from the dissociation of molecular dications in different electronic states. These states are formed through removing two valence electrons from different valence orbitals. The angular distribution of the atomic ions reflects the symmetry of the molecular orbital when the pulse duration is very short and the laser intensity is very low. Therefore, we measure the angular distribution of the coincident O⁺ with different KERs. These atomic ions are produced by 8 fs laser pulse at an intensity of 2.5 × 10¹⁴ W/cm². The results are shown in Fig. 3. The angular distribution looks alike in Fig. 3a and 3b, but exhibits obvious differences in Fig. 3c. According to KERs shown in Fig. 1, we know that the atomic ions are produced through the dissociation of molecular dications in some definite electronic states. The molecular dications are formed through the removal of one π_g electron and one π_u electron in Fig. 3a and 3b, while one π_g electron and one σ_g electron in Fig. 3c. The studies demonstrate that the angular distribution is strongly influenced by the molecular orbitals from which the two electrons are removed. Through coincidentally measuring the two electrons with the state-resolved molecular dications, we can determine the orbital and spin angular momentum quantum numbers of the two electrons. These measurements could provide more precise information about electron-electron correlation in double ionization of molecules.

Figure 4 and Fig. 5 show the KERs of different fragmentation channels of oxygen induced by 8fs and 24fs laser pulses, respectively. The (n, m) represents the fragmentation channel O^(n+m)+ ₂ → Oⁿ⁺ + O^m+. Different from obvious structures observed in the KER spectra of O²⁺ ₂ → O⁺+ O⁺, the KERs show a single broad peak for the pathways O²⁺ ₂ → O²⁺+ O and O³⁺ ₂ → O²⁺+ O⁺. The KER spectrum looks alike independent of the laser polarization and the laser intensity when the pulse duration is fixed. The peak is around 5.8 eV for (2,0) and 18.0 eV for (2,1) when the pulse duration is 8 fs. However, the KER shifts toward lower energy with increasing the pulse duration. The peak is around 2.7 eV for (2,0) and 13.5 eV for (2,1) when the pulse duration is 24 fs. Another big difference between Fig. 4 and Fig. 5 is the presence of (2,2) in 24 fs laser fields and its absence in 8 fs laser fields.

Because of the presence of (2,0) and (2,1) in the circularly polarized laser pulse, we believe that electron rescattering is not the major mechanism of the multiple ionization. The single broad peak observed in the KERs for (2,0) and (2,1) makes us conclude that these fragmentation pathways occur through some repulsive states. When the ionization occurs at small internuclear distance R, the resulting atomic ions share high KER. For oxygen molecule, the equilibrium internuclear distance R_e = 2.30 a.u. and the critical internuclear distance R_c = 5.8 ~7.5 a.u [19]. Here, we assume that the potential energy curve of (2,1) channel can be approximated by Coulomb potential. The KER is ~23.6 eV for direct ionization and ~9.4 eV for enhanced ionization. However, the measured KERs of (2,1) is 18.0 eV for 8 fs laser pulse and 13.5 eV for 24 fs laser pulse. The energies are between the values predicted by direct ionization and enhanced ionization. The KERs indicate that the (2,1) occurs at internuclear distance R with R_e < R < R_c. The mechanisms of direct ionization and enhanced ionization cannot explain the formation of these channels under our experimental conditions.

We therefore conclude that multiple ionization occurs through stairstep process under the condition shown in Fig. 4 and Fig. 5. During the period between two ionization events, the internuclear distance is stretched by the assistance of the laser field. Here, we assume the ionization threshold is I₁ for generating O^(n+m-1)+ ₂ and I₂ for O^(n+m)+ ₂. O^(n+m)+ ₂ originates from the further ionization of O^(n+m-1)+ ₂ and I₂ is larger than I₁. Let us take the laser intensity as $I\left(t\right)=I_0{\sin }^2\left(\genfrac{}{}{0.1ex}{}{\pi t}{2\tau }\right)$ with τ the pulse duration as well as the full width at half maximum. When the laser intensity is increased from I₁ to I₂, the time interval is given by $\mathrm{\Delta t}~\tau ·\left(\arcsin \sqrt{\genfrac{}{}{0.1ex}{}{I_2}{I_0}}-\arcsin \sqrt{\genfrac{}{}{0.1ex}{}{I_1}{I_0}}\right)$. The expression shows that the time interval between two ionization events is proportional to the pulse duration. In consequence, long pulses allow more time for molecular stretching. According to the laser-assisted bond-stretching model [18], the internuclear distance is stretched in intense laser field. The stretching degree depends on the time that the nuclear wavepacket evolves in the laser field. The longer the wavepacket evolves in the laser field, the longer the internuclear distance is stretching. The stretch of the internuclear distance results in the decrease of the KERs. Therefore, the KERs decrease with increasing the pulse duration, which is consistent with our observations.

 

Fig. 4. The KER of different fragmentation channels of oxygen induced by 8fs laser pulses with different intensities and polarizations. (LP: linearly polarized, CP: circularly polarized, I₀: 10¹⁴ W/cm²).

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Fig. 5. The KER of different fragmentation channels of oxygen induced by 24fs laser pulses with different intensities and polarizations. (LP: linearly polarized, CP: circularly polarized, I₀: 10¹⁴ W/cm²).

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The appearance intensity for explosion pathway (n, m), O^(n+m)+ ₂ → Oⁿ⁺ + O^m+, also strongly depends on the transient internuclear distance [38]. It decreases with increasing the internuclear distance when R < R_c. The measured KERs of (2,1) verify that the internuclear distance R is between R_e and R_c upon triple ionization. In addition, R strongly depends on the laser pulse duration, which is smaller in 8 fs laser fields than that in 24 fs laser fields. That means, in order to further ionize O³⁺ ₂, higher laser intensity is required for 8 fs laser pulse than that for 24 fs laser pulses. The R-dependent ionization threshold well explains our observations of the presence of (2,2) in 24 fs laser fields and its absence in 8 fs laser fields with similar laser intensities.

4. Conclusions

We have studied the laser-induced molecular ionization and its subsequent dissociation by applying a coincident three-dimensional momentum imaging method. By measuring the KERs and angular distribution of coincident O⁺ in 8fs and 24 fs laser fields, we conclude that O²⁺ ₂ → O⁺ + O⁺ proceeds through some definite excited electronic states of the molecular dications. The angular distributions are influenced by both the laser intensity and the pulse duration. High laser intensity causes the post ionization alignment during the fragmentation process. Long pulses cause dynamic alignment of the neutral molecules before ionization. These two alignments make the angular distribution peak along the laser polarization. When the laser intensity is 2.5 × 10¹⁴ W/cm² and the pulse duration is 8 fs, the alignment effect can be neglected. The angular distribution is determined by the angular-dependent ionization and reflects the symmetry of the molecular orbital.

We also measure the angular distribution of coincident O⁺ with different KERs, which originate from the dissociation of molecular dications in different electronic states. Depending on the molecular orbitals that the two electrons are removed, molecular dications are formed in different electronic states, whose fragmentation generates different angular distributions of the atomic ions. Through coincident measuring the two electrons with the state-resolved molecular dications, the electron-electron correlation can be studied for the two electrons with selected orbital and spin angular momentum numbers.

Finally, we measure the KERs for the pathways O²⁺ ₂ → O²⁺ + O and O³⁺ ₂ → O²⁺ + O⁺. These KERs show a single broad peak, whose value has no relationship with the laser polarization and the laser intensity when the pulse duration is fixed. KERs indicate that the ionization occurs at internuclear distance R with R_e < R < R_c. Based on these measurements, we conclude that multiple ionization occurs through stairstep process under the current experiment conditions. During the time interval between two ionization events, the internuclear distance is stretched by the assistance of the laser field. The longer the pulse duration is, the longer the internuclear distance is stretching. The stretch of the internuclear distance results in the decrease of the measured KERs with increasing the pulse duration.