Molecular alignment in degenerated dissociation channels in strong laser fields

In this work, we study strong-field molecular alignment in, for the first time, degenerated channels following the same charged states of molecules. By measuring the angular distribution of dissociation fragments from two degenerated N+ + N+ channels of N2, we observe an opposite angular distribution development in these two channels, one expanding and one contracting, when the laser intensity increases. Our further study shows that the expanding channel comes from a nonsequential transition, while the contracting channel involves a sequential transition. We also study the time sequence of the sequential and nonsequential transitions and find that the opposite angular distribution development is due to the different degrees of molecular alignment in these two degenerated channels.


Results
Polar charts of molecular orientation distribution. The two degenerated N + + N + channels from double-ionization-induced dissociation of N 2 have been identified in our previous studies 22, 23 : a fast N + + N + channel with a higher KER of 7.0 eV [labeled as N(1,1) fast ], and a slow N + + N + channel with a lower KER of 3.8 eV [labeled as N(1,1) slow ] (see Fig. 1(a), which shows the TOF mass spectrum of N + ion peaks obtained with linearly polarized 68-fs pulse at an intensity of 4 × 10 14 W/cm 2 ). Figure 2 shows the polar charts of the measured molecular orientation distribution of N(1,1) slow and N(1,1) fast obtained with 68-fs pulses at different laser intensities, 2I 0 , 3I 0 , and 4I 0 , with I 0 = 10 14 W/cm 2 . To see the change of the distribution profile, we superimpose the curve at 2I 0 to all other curves, shown as the solid dots in Fig. 2. Distinctive difference can be seen between the two channels when the angular distribution profile develops with laser intensity. For N(1,1) slow [ Fig. 2(b),(c) and (d)], the profile at 4I 0 clearly gets contracted compared to the 2I 0 guideline, which can be seen quantitatively from the <cos 2 θ> measure of alignment: 0.83 at 2I 0 and 0.86 at 4I 0 . A contraction in the angular distribution profile with increasing intensity, i.e. an increase in the <cos 2 θ> measure, indicates that molecules are more aligned towards the laser polarization 5,6,11,26 . Therefore, as the laser intensity increases from 2I 0 to 4I 0 , more molecules are aligned towards the laser polarization when N(1,1) slow is created. In contrast, for N(1,1) fast [ Fig. 2(e),(f) and (g)], an expansion is clearly seen in the angular distribution profile when the laser intensity increases from 2I 0 to 4I 0 , with the <cos 2 θ> measure decreasing from 0.77 to 0.73. While an expansion in the angular distribution profile with increasing intensity has not been commonly seen in previous molecular alignment studies, nor explicit mechanisms have been established 5,6 , it certainly does not indicate a higher degree of alignment in this channel.
Note that an isotropic distribution across all θ corresponds to a <cos 2 θ> value of 0.33. However, due to the detection setup of the experiment, the signal measured is not across the entire space but rather is a slice of the entire spatial distribution. For this reason, an isotropic distribution gives a measure of <cos 2 θ> to be 0.5. At the low intensity end of our experiments, both N(1,1) slow and N(1,1) fast show a <cos 2 θ> value larger than 0.5 and this could be due to two possible mechanisms: the ionization rate of the channel is not angularly constant, and/ or a certain degree of alignment that has been achieved when the channel is formed 5,12 . According to previous studies, the angular ionization rate of a molecular fragmentation channel, which is closely related to the molecular orbital characteristics, could significantly impact the angular distribution of fragments 6,11,12,[27][28][29] . In the following studies, we will explore which effect plays a major role leading to the large <cos 2 θ> value in these two channels.
The relationship of the laser intensity, molecular ionization rate, and the angular distribution data. When we rotate the laser polarization and introduce an angle θ between the laser polarization and the TOF axis (i.e. the molecular axis since only ions having their velocity aligned with the TOF axis will be detected), the equivalent E-field strength along the molecular axis is Ecosθ and the equivalent laser intensity along the molecular axis is in proportion to (Ecosθ) 2 . When we vary the angle θ, we vary the equivalent laser intensity proportion along the molecular axis and thus vary the corresponding ion yield rate. Therefore, we fit the angular distribution data by using a cos 2 θ function with an exponential order n, i.e. (cos 2 θ) n , aiming at revealing the correlation between the intensity dependence of the angular distribution data and the intensity dependence of the ion yield rate of the channel. The fitting is shown in Fig. 3(a-d). We can see that N(1,1) fast is fitted by (cos 2 θ) 3 at the high intensity end and (cos 2 θ) 4 at the low intensity end (plus an angle-independent offset that accounts for the weak signal of N(1,1) fast when the laser polarization is perpendicular to the molecular axis, i.e. the components at θ = 90 and 180 degree in Fig. 2), while N(1,1) slow is fitted by (cos 2 θ) 6 at the high intensity end and (cos 2 θ) 5 at the low intensity end.
Next, we compare the angular distribution data with the ionization rate of each channel obtained by ion yield curves 30,31 . We plot the ion yield curves for N(1,1) slow and N(1,1) fast as a function of laser intensity in log scale in Fig. 4. The ion yield curves are obtained with the same TOF spectrometer using linearly polarized laser beam with polarization parallel to the molecular axis. By fitting the ion yield curves of N(1,1) fast with an exponential function, we find that the exponential order is 3.7 and 4.4 at the intensities of 4I 0 and 2I 0 , respectively, as marked in Fig. 4. The exponential order for N(1,1) slow is 2.3 and 3.1 at the high and low intensities, respectively. Note that the exponential order is smaller at the high intensity end than the low intensity end for both channels and this is due to saturation in these channels and/or depletion of this channel by higher charged states when intensity gets higher 22,31 .
The fitting order of the angular distribution data at the high and low intensities, 3 and 4, coincides with the exponential order of the ion yields of N(1,1) fast , 3.7 and 4.4, at the corresponding intensities, as shown in Fig. 4. The small discrepancy could be due to the contribution from that the perpendicular component of the laser field with respect to the molecular axis. However, this component should only play a minor role since we see a very weak signal strength of N(1,1) fast when the laser polarization is perpendicular to the molecular axis (i.e. the horizontal component in Fig. 2). On the other hand, the consistence between the fitting of the angular distribution data and the ion yield curves of N(1,1) fast indicates that the angular distribution of N(1,1) fast is dominantly determined by its ionization rate, or more precisely, angular ionization rate, throughout our intensity range.
In contrast, the angular distribution of N(1,1) slow is fitted by (cos 2 θ) 6 at the high intensity end and (cos 2 θ) 5 at the low intensity end, as shown in Fig. 3(c) and (d). The exponential fitting orders, 6 and 5, significantly diverge from the corresponding fitting orders of the ion yield curve, 2.3 and 3.1, as seen in Fig. 4. This indicates that the angular distribution profile of N(1,1) slow is not solely determined by its angular ionization rate; instead, there must be other mechanisms leading to the substantial divergence between the angular distribution fitting and the ionization rate fitting, which very likely is the molecular alignment effect. The fact that the fitting order of the angular distribution profile is greatly larger than the ionization rate indicates that the molecules (or the fragmented ions) are more aligned towards the laser polarization when N(1,1) slow is formed, resulting in a contracted distribution profile 5,6,11,26 . Furthermore, when the intensity increases from 2I 0 to 4I 0 , the fitting order of ionization rate of N(1,1) slow decreases from 3.1 to 2.3. However, the fitting order of the angular distribution profile increases from 5 to 6. The greater discrepancy between the two fittings as intensity increases indicates that the alignment degree in N(1,1) slow is increased when intensity increases.
Formation dynamics of the degenerated dissociation channels. To understand why we observe different alignment effect in these two degenerated channels, we take a look at their formation dynamics. Our previous studies have shown that N(1,1) fast involves a nonsequential double ionization (NSDI) transition, where the two electrons are removed almost simultaneously when the laser intensity rises sufficiently high 25,32,33 . In contrast, N(1,1) slow is formed through a sequential double ionization (SDI) via enhanced ionization (EI) 23 in a two-step transition: N 2 firstly loses one electron and starts to dissociate, followed by an enhanced ionization of a second electron when the internuclear distance reaches the critical internuclear distance R c in about 43 fs 23,34 . Therefore, N(1,1) fast is mostly formed at the leading edge of the 68-fs pulse, while N(1,1) slow is formed at the trailing edge. This is further confirmed by a time sequence study with varying pulse durations as shown in Fig. 1(a) and (b).  We can see that the N(1,1) slow peaks nearly disappear in the shorter 45-fs pulses and that is because the pulse duration is insufficient for this channel to form; in contrast, N(1,1) fast consistently presents in both 45 and 68-fs pulses, indicating that this channel is formed at the pulse leading edge. A schematic of the formation time sequence of N(1,1) slow and N(1,1) fast is shown in Fig. 1(c).

Discussion
Previous studies have shown that molecular alignment can be achieved in as fast as tens of femtoseconds for small molecules exposed to intense laser fields 5,7,8 . For example, when N 2 is exposed in 45-fs pulse with a peak intensity of 1.4 × 10 14 W/cm 2 , the highest degree of alignment happens at 67 fs follows the aligning pulse 8 . If an event happens in a similar time scale, it will strongly experience the molecular alignment effect. Since the SDI channel N(1,1) slow is formed at the trailing edge of the laser pulse in about tens of fs 23,34 , similar to the alignment time scale of N 2 , N(1,1) slow will strongly experience an alignment effect (and possibly some post ionization alignment effect 12 ), i.e. the molecules (or molecular ions) that form N(1,1) slow experience stronger alignment and therefore a higher degree of alignment is seen in its angular distribution. As the laser intensity increases, laser field-induced alignment becomes stronger 5,6,8,9,11,35 , and this leads to the contraction as we see in the angular distribution of N(1,1) slow in Fig. 2.
In contrast, the NSDI channel N(1,1) fast is formed at the leading edge of the laser pulse, where both electrons are almost simultaneously removed as fast as the laser intensity rises sufficiently high 25,32,33 , followed by strong Coulomb explosion and molecular bond breaking. Therefore, the NSDI channel only experiences limited alignment effect and/or post ionization alignment 12 . As a result, the angular distribution of N(1,1) fast is less affected by molecular alignment effect, instead, is dominantly determined by its intensity-dependent ionization rate, which leads to the coincidental fitting between the angular ion distribution data and the ionization rate curve as discussed above.
Note that in our current experimental setup we could not exclude the possibility of the post-ionization alignment effect contributing to our experimental results besides the molecular alignment effect. Some thoughts to investigate the relative strength of the post-ionization alignment effect in the SDI channel N(1,1) slow are provided here for future studies: a pump-probe experiment consisting of pump pulse duration being long enough for molecular alignment effect to take place but less than the critical time to trigger the SDI N(1,1) slow channel (e.g. 45 fs, which was reported to well align N 2 molecules [8] but insufficient to produce N(1,1) slow [23]), and probe pulse duration being much shorter (e.g. a few fs) to trigger the SDI N(1,1) slow channel at the critical nuclear distance, R c . The angular distribution of the SDI N(1,1) slow channel from this pump-probe experiment, if compared with single-long-pulse experiment results (e.g. the results shown in this paper), could reveal the relative strength of the post-ionization alignment effect. Here we note on the relative strength and that is because the parental state of the N(1,1) slow channel, i.e. the N 2 + state, may involve both the molecular alignment and the post-ionization alignment effects since single ionization of N 2 could happen immediately once the pump pulse arrives and therefore it may not be easy to completely exclude post-ionization alignment effect in the N 2 + state. In summary, we perform a study on the strong-field molecular alignment effect in two degenerated channels from double ionization-induced dissociation of N 2 . Our previous study identified two N + + N + states, a fast channel with a higher KER and a slow channel with a lower KER; the fast channel is a nonsequential channel and the slow is a sequential channel. By measuring the angular distribution of dissociation fragments, we observe opposite angular distribution development in these two channels as the laser intensity increases, one expanding and one contracting. Our further study shows that the expanding angular distribution comes from the nonsequential channel, whereas the contracting angular distribution comes from the sequential channel. A further analysis of the time sequence of the sequential and nonsequential transitions reveals that the opposite angular distribution development is due to different degrees of strong-field induced molecular alignment in these two degenerated channels.

Methods
Ion detection and collection is realized with a recently modified time-of-flight (TOF) spectrometer 22,23 . The chamber base pressure is less than 5.0 × 10 −10 Torr. We measure the molecular orientation distribution by using our TOF spectrometer. The 2.5-mm pinhole opening on the voltage plates in our TOF spectrometer ensures that only ions having their velocity aligned with the TOF axis will be detected. A half-wave plate (HWP) is used before the focusing lens to control the angle of the laser polarization with respect to the TOF axis. Ion signal of a specific channel is collected as we rotate the HWP, which is equivalent to a scanning of the angular distribution of this channel. Figure 2(a) illustrates the geometric configuration of the experimental setup.
The laser used is a Ti:sapphire system consisting of a regenerative amplifier and a multi-pass amplifier that delivers pulses of 1.0-1.2 mJ/pulse at a 1 kHz repetition rate with the central wavelength at 800 nm. By adjusting the bandwidth of the seed pulse, 25 and 50 nm used in this work, before entering the regenerative amplifier, we manage to generate two pulse durations at 45 and 68 fs, respectively. To minimize the chirp of both pulses, we carefully tune the stretcher and compressor while monitoring the second and third order dispersion with a home-built FROG-like spectrum and phase detection system. This approach allows us to minimize the second and third order dispersion (higher order dispersion may still exist) and therefore achieve the closest-to transform-limited pulses.