1. Introduction

The interaction of short-pulse high-intensity lasers with condensed-phase matter begins with local ionization and acceleration of the free electrons [1]. The subsequent post-pulse dynamics of the hot electrons play a significant role in the radiative properties and the spatiotemporal behavior of the plasma [2]. The dynamics typically occur on time scales on the order of a few ps. At longer times, energy transfer from the electrons to the ions leads to complex material dynamics [3].

In the work reported here, transmission images of the laser-driven plasma in a 53-μm-diameter water column record the spatial and temporal evolution of the ionized region for times < 0.5 ps. Recent experimental work on fs-pulse interaction with bulk water concentrated on the plasma dynamics for times > 1 ps at lower pulse energies (1 μJ, 100 fs) [4]. The authors present light-scattering measurements that indicate the rapid short-time growth of the plasma. Assuming an electron density above the critical density for the probe light, they infer a radial plasma expansion of 5 μm in 100 fs, corresponding to a short-time growth rate of 0.15c. (c denotes the velocity of light in vacuum.) A second study investigated filaments in bulk water over extended distances (hundreds of microns) at low pulse energies (4 μJ, 120 fs) [5]. The focused pump-beam intensities used by these researchers is four orders of magnitude lower than that used in the present study and led to observed electron densities that were three orders of magnitude lower than those reported here. Probe-beam shadowgraphy was used to monitor the ionization along the pump-beam axis of propagation after its passage. Increased absorption over time of the probe pulse was postulated to be due to multi-photon absorption by solvated electrons. No significant probe-pulse absorption was observed transverse to the pump-propagation axis.

Here, the sequences of transmission images reveal the growth of the lateral edge of the ionized region as a function of delay with respect to the ionizing pump pulse. Analysis of the opacity of the ionized region as a function of incident pump-pulse energy, combined with energy-conservation considerations, indicates electron densities ranging from 1.6 to 2.3 × 10²¹ cm⁻³ and an average electron temperature of 6 - 20 eV. Both the electron density and the average temperature increase with the incident pulse energy. The hot electrons with initial energies ranging from 17 to 100 keV account for only 0.1 - 0.3% of all electrons; however, they carry 80 - 85% of the total electron kinetic energy.

2. Experiment

The target material consisted of a vertical laminar stream of deionized (R = 14 MΩ cm) water that was directed downward through ambient laboratory air. The 53-μm-diameter column was struck by 45-fs pulses from a 1-kHz Ti:sapphire regenerative amplifier (800 nm) at pulse energies ranging from 0.05 to 0.5 mJ. The propagation axis of the laser beam was oriented normal to the axis of the column, with the electric-field vector being parallel to the column axis. The incident beam was focused with a 40-mm-focal-length singlet, and the column was placed upstream of the focus such that at the center of the water column, the beam had a Gaussian radius of 13 μm that was determined through direct imaging. For pulse energies of 0.51 mJ, the peak focused intensity corresponds to 1.9 × 10¹⁵ W/cm². (For this pulse and focusing lens combination, group velocity dispersion in the lens modifies the pulse width by ~12% which we include in calculating the focused intensity [6]. No intensity correction is needed for self focusing since this was not observed when the water column was removed from the pump beam path.) Third-order autocorrelation traces of the pulse train indicated a prepulse leading the main pulse by 8.5 ns with a peak energy that was three orders of magnitude lower. During data collection the transmitted 800-nm light along with the concomitant blue-shifted light was measured as a means of monitoring the amount of incident energy deposited in the target material.

A fraction of the 800-nm light out of the laser was frequency-doubled to yield a 400-nm probe-pulse beam with a pulse energy of ~1 μJ. The probe beam was expanded and collimated to ~1 cm diameter and was delayed with respect to the excitation (pump) pulse using a stepper-motor-driven optical delay with a temporal resolution of ± 15 fs. The probe pulse was directed into the target area at right angles to both the water column and the pump-pulse propagation axis. Transmission images were recorded using a lens pair to collect the transmitted probe light and place a magnified image of the pump-irradiated target area onto a ccd (Foculus, 8-bit, b/w, 1/3”).

The imaging optics were arranged such that the symmetry plane of the water column normal to the probe beam was in sharp focus. Interference filters were used to reject stray 800-nm light, and neutral-density filters were used to adjust light levels on the CCD. For a range of incident pump-beam energies, transmission images were recorded as a function of pump/probe delay at 30-fs intervals.

Examples of acquired shadowgraph images are shown in Fig. 1 . The alternating light and dark bands parallel to the long axis of the water column are light-diffraction manifestations [7]. In the data presented here, time zero corresponds to the passage of the peak of the pump pulse through the longitudinal axis of the water column. Identification of time zero was done in a two-step process using the shadowgraph images. As the pump pulse propagates it weakly ionizes the air via multi-photon processes. This leaves a small perturbation in the images with a leading edge that can be tracked as a function of delay. So, with the water column translated away from its target location, the images as a function of probe delay were examined to find a delay setting that placed the perturbation at the target location (step 1). After translating the water column back to this location a small adjustment was made in the delay to offset the increased transit time of the water column versus air (step 2).

 

Fig. 1 Transmission images of single-pulse events; pump beam propagates left to right; dashed lines indicate spatial location of Gaussian pump-beam halfwidths. a) Wide view, indicating plasma-formation region in center third of column and unperturbed water above and below. b) Close-up of ionized region at t = 0 fs just as excitation pulse passes center of column. c) Close-up of ionized region 700 fs after arrival of excitation pulse in center of column. Red arrows in b) and c) indicate h(t) – spatial extent of ionized region along centerline of water column.

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As the pump pulse propagates through the water column, it ionizes the neutral material, leading to absorption of the probe pulse by the free electrons. The neutral deionized water exhibits no linear absorption at either 800 or 400 nm, and the third-order susceptibility for water has no imaginary component. Consequently, no intensity-driven third-order absorption will occur [8]. The darkening of the shadowgraph images after passage of the pump pulse is due entirely to absorption of the probe beam by the free electrons. To avoid the complication of separating diffraction effects, the transmission images are analyzed only along the centerline. The red arrows in Figs. 1b) and 1c) indicate the growth of the plasma in time along the centerline. Based on the first appearance of a change in probe-beam transmission at a delay of 1 ps, the threshold for ionization is < 4.5 × 10¹³ W/cm² (an upper limit dictated by the dynamic range of the acquired images). Schaffer et al. [9] found a threshold pulse energy of 0.2 μJ for pure water struck by 100-fs pulses. Using their estimated spot size of 1 μm, this corresponds to a threshold intensity of 6.4 × 10¹³ W/cm². Both measured values are roughly a factor of three larger than that estimated by Noack and Vogel [10].

3. Discussion

Analysis of the pump/probe images proceeded by extracting the measured transmission along the centerline of the water column as a function of time; an example is shown in Fig. 2 for four time delays. The transmission curves are plotted as a function of distance from the beam-propagation axis. Initially, the transmission decreases by 65 - 80% of the pre-pump-pulse value, and over the first ~200 fs the transmission further decreases to final values of 10 - 20% of the initial transmission. No strong correlation was found between the minimum transmission and the incident pump pulse energy. The overall shape of the transmission curves is similar for all delay times and all pump energies. They appear as rounded trapezoids with modulations–presumably due to diffraction–across the transmission minimum. For each curve, an average minimum transmission value was determined across the bottom. Then the width of the full curve was found via curve fitting at half this minimum transmission value. These widths were used to measure the growth of the ionized region as a function of time. Ionized-channel growth for all cases studied was found to be symmetric about the pump-propagation axis.

 

Fig. 2 Relative transmission of 400-nm light along centerline of transmission images for 210-µJ excitation at four time delays. Red arrow indicates h(t = 267 fs).

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The growth of the ionized region as a function of delay and pump-beam energy is shown in Fig. 3 . The y-axis displays the change in channel width after pump-beam arrival in one direction, as indicated by the red arrow in Fig. 2. For each of the five pump-beam energies studied, ionization-channel growth continues for 300 - 400 fs and then ceases. The ionized region persists until a delay of ~10 ps, when large-scale movement of material similar to that reported for solid surfaces begins [3].

 

Fig. 3 Net displacement from incident-beam axis of ionized region as function of delay from arrival of excitation pulse for each of five incident pulse energies.

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By fitting polynomials to the growth-versus-time delay curves and calculating the local derivatives, the speed of the ionization-channel movement as a function of delay was readily determined. Initial values are quite high, ranging from 0.25c to 0.55c for pump energies ranging from 0.05 mJ to 0.5 mJ. It is straightforward to show that for our plasma conditions (see discussion of Table 1 below), the radiant power produced by the associated blackbody radiation or bremsstrahlung is orders of magnitude too weak to produce the observed ionization growth [11]. Consequently, a group of very hot electrons that are initially moving at the speeds indicated above must be responsible for the ionization growth.

Table 1. Summary of Plasma Conditions: n_e (Total Electron Density), Te (Avg. Electron Temperature), Ratio of Hot to Cold Electron Density, Ratio of Calculated Transmission to Measured Transmission at 400 nm, and Ratio of Total Post-pulse Energy to Pulse Energy of Incident Light

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The questions that remain are 1) how do electrons achieve such high initial velocities in the transverse direction, and 2) are they present in sufficient numbers to generate the observed ionization-channel growth. Reviews of many previous reports of experimental measurements of hot electron energies [12] suggest that for pump intensities of 2 × 10¹⁵ W/cm², the anticipated hot electron energy should be on the order of 5 keV, corresponding to a velocity of 0.14c, which is much lower than that observed in the experiments. However, using focused intensities that are five times higher than those reported here, Li et al. [13] observed transverse hot-electron jets in liquid water with energies up to 64 keV, corresponding to peak velocities of 0.46c.

For the peak pump-beam intensity used in this study, the quiver velocity and velocity due to the transverse ponderomotive force are estimated to be 0.02c. Similarly, wakefield acceleration leads to estimated velocities of 10⁻⁶c. Theoretical work by several groups indicates that longitudinal ponderomotive acceleration can be significant if the electron is accelerated by the leading edge of the incident pulse but not decelerated by the trailing edge. Such acceleration is particularly important for electrons that are generated with a low initial energy just before experiencing the high-intensity portion of the leading edge. For example, using very general arguments, Sazegari et al. [14] estimate that leading-edge longitudinal acceleration is capable of accelerating an electron to velocities on the order of the group velocity of the pulse in the target. The analysis of our data discussed below indicates that during post-ionization, the real part of the index of refraction falls from an initial value of 1.34 to 1.2, leading to an estimated pulse group velocity of 0.8c, which is on par with but greater than the highest velocities observed in our measurements.

To evaluate the possibility of generating a hot subgroup of electrons under the stated conditions, the observed degree of probe-beam transmission along with the measured pump-beam energy deposition was examined. The post-pulse total energy was considered to be equal to the sum of the transmitted pump-beam energy, the reflected pump energy, the energy required to ionize water over the volume indicated by the transmission images, the total kinetic energy of the cool electrons, and the total kinetic energy of the hot electrons. The transmitted pump energy was measured directly [15]; all other contributions to the total energy were calculated. We note that bulk water behaves in the same way as a dielectric with a photoionization threshold of 6.5 eV [16]. This value was used in calculating the ionization contribution to the total-energy budget.

Modeling of the transmission of the probe was conducted following the approach of Theobald et al. [17] who considered the imaginary part of the complex refractive index of water to be due solely to the free electrons. Analysis of the pump-produced plasma proceeded by first estimating the values of the total electron density (n_e) and the average electron temperature (Te) and then self-consistently calculating a probe-transmission value (T) along with a total energy (E) value. This procedure was iterated by adjusting the values for n_e and Te until the calculated probe transmission agreed with the measured transmission and the total energy agreed with the incident pump-beam energy. The calculations were made for the transmission conditions at the arrival of the peak of the pump beam in the center of the water column–probe delay equal to 0 fs. The kinetic energy of the hot electrons was calculated using the initial hot-electron velocities found from the time derivatives of the curves in Fig. 3. The fraction of electrons with these high initial velocities was adjusted such that the initial total kinetic energy of the hot electrons would be sufficient to ionize via binary collisions the volume of water indicated by the growth shown in Fig. 3. The results of these calculations are summarized in Table 1 for four pump-beam energies. Columns 4 and 5 display the degree to which the calculated and observed transmission values agree and how well energy conservation is achieved. As shown, the electron density varies by a factor of approximately two, while the average electron temperature varies by a factor of approximately four for incident pulse energies that are changing by a factor of five. The mean electron density of 2 × 10²¹ cm⁻³ (over all four cases) indicates an ionization fraction of 6%, assuming single ionization of each parent water molecule. The ratio of hot electrons to cold electrons varies by a factor of approximately three and is < 0.3% for all cases examined.

Examination of the intermediate values of n_e and Te during the iteration process indicates that the optimal values for these parameters are rather tightly constrained in this model. For example, a ± 20% change in the value of n_e leads to a calculated optical transmission that deviates by ± 40% from the measured transmission. Interestingly, the ratio of hot to cold electrons is not very sensitive to changes in the total electron density or average electron temperature. This is likely an indirect reflection of the mechanism by which they gain their initial kinetic energy via longitudinal acceleration from the incident pulse. This process would largely be independent of variations in the local plasma environment.

The above considerations of energy conservation and probe-beam transmission clearly indicate that the available pump-beam energy was sufficient to produce an ensemble of hot electrons of sufficient density to yield the observed ionization growth–both in spatial extent and growth rate. Since the hot electrons are initially generated with velocities parallel to the pump-beam propagation, collisions must be responsible for redirecting them into trajectories with a significant transverse component. Recently, Date et al. [18] used Monte Carlo simulations to study collisions between energetic electrons (up to 10 keV) and water molecules in liquid water. Their work shows that the hot-electron trajectories rapidly gain significant transverse velocity components via collisions–both ionizing and excitation collisions. These authors also note that the incident hot electrons carry away most of the kinetic energy post-collision. Such electron dynamics would explain the high growth rate of the ionized water volume observed in our measurements. There have been numerous studies of the energy loss range of a free electron in liquid water, both experimental and computational, as a function of incident electron energy (see for example [19,20] and references therein). Such studies do not include a background of positive ions mixed in with the neutral molecules and therefore miss the drag on the particles due to Coulomb forces present in our plasma environment. However, the lateral ionization growth observed in this study is of the same order as that seen in the electron stopping studies – just a little smaller as it should be.

4. Conclusion

In summary, pump/probe-transmission images of a 53-μm water column following the arrival of the 45-fs pump pulse reveal the short-time spatial and temporal growth of the ionized region. Following initial photoionization, the rapid ionization transverse to the pump-propagation axis is driven by hot electrons with unusually high initial velocities (energies) of 0.55c (~100 keV) for a pump intensity of 1.9 × 10¹⁵ W/cm². These hot electrons are accelerated by the leading edge of the pump pulse via the longitudinal ponderomotive force and then acquire trajectories with a significant transverse component through ionization and excitation collisions with neutral water. Retaining most of their pre-collision kinetic energy, these hot electrons rapidly ionize the region radially surrounding the pump-pulse propagation axis.