The interaction of intense, femtosecond laser pulses with atomic clusters is a field of research that has excited considerable interest over the past ten years. One of the reasons clusters have proved so interesting is the nature of the target - the cluster jet has an overall gaseous density allowing the laser beam to penetrate the jet but within each cluster the density is near-solid. This near-solid density within the cluster leads to a high degree of collisional ionisation and heating when irradiated with a high intensity femtosecond laser pulse and so to an extremely high absorption efficiency of the laser energy. Absorption efficiencies approaching 100 % have been measured in 100 Å Xe clusters [1]. A result of the high absorption efficiency and the Mie resonance that occurs during the cluster heating is the production of multi-keV electrons [2], MeV ions [3] and X-ray yields comparable to that achieved from solid targets [4]. Ions of very high charge state are also created; Xe⁴⁰⁺ has been measured from 65 Å clusters irradiated with a laser pulse of peak intensity just 2×10¹⁶ Wcm^-2 [3]. Experiments conducted with D₂ clusters have shown the deuterons produced during the cluster expansion have sufficient energy to undergo nuclear fusion with deuterons produced by adjacent clusters in a high-density gas jet [5]. Clusters are also promising as an EUV source for photolithography [6] because they provide high EUV yields but are almost debris-free when compared to solid targets.

Experiments and simulations have shown that there exists an optimum laser pulse duration for a cluster of a given size at which maximum absorption of the laser energy is observed [7] and ions of the highest energy are produced. Conversely, there is an optimum cluster size for a given laser pulse duration for which maximum energy ions are produced. This has been shown experimentally by Springate et al. [8]. As well as pulse duration, the optimum cluster size for maximum energy ions is also dependent on the peak laser intensity and the cluster species. Simulations using the nanoplasma model of a cluster [4], which is also used for simulations presented in this paper, have accurately predicted the optimum cluster size observed experimentally [8, 9].

The existence of an optimum laser pulse duration arises from a resonance that occurs during the cluster expansion during which the electric field inside the cluster is enhanced with respect to the external electric field. The resonance occurs when the electron density in the cluster is three times the critical density n_crit(ω) for a planar plasma. The timing of the resonance for a laser pulse of a given duration is dependent upon the laser intensity and the cluster size and species. Small clusters expand more quickly than large clusters and so pass through the resonance early on in the laser pulse while larger clusters expand more slowly and take longer to reach the resonance condition. Resultingly, small clusters have shorter optimum pulse durations than large clusters.

A number of earlier studies have investigated the influence of various laser pulse parameters on the cluster explosion in an intense laser field and the x-ray emission from the irradiated clusters. In particular the effect of the laser pulse duration [6,7,10–12], laser wavelength [13] and polarisation state [2,8,14–16] on such things as laser energy absorption [7,10], X-ray and EUV emission [6,11,12] and electron and ion temperature [2,8,14–16] have been determined. Experiments have also been conducted in which the clusters were irradiated with two laser pulses. In one such experiment the effect of a picosecond prepulse irradiating a cluster prior to the femtosecond main pulse was determined [17] and in another the clusters were irradiated with two laser pulses of differing frequencies delayed with respect to one another [18]. The majority of these investigations were conducted in bulk cluster media, just beneath the nozzle of the gas jet. In contrast, the work reported in Refs. [2,8,14,18] was performed by looking at the fragments from cluster explosions at far lower densities corresponding essentially to isolated clusters, which gives much more insight into the basic physics of the laser-cluster interaction. Here we present results of investigations of the interaction between a laser pulse of variable temporal profile and isolated clusters.

We investigate the effect of the shape of the laser pulse on a cluster explosion by using the fact that when a pulse is frequency chirped the spectral profile is mapped into the temporal domain. The temporal shape can be time-reversed simply by changing the sign of the frequency chirp. Producing a chirped pulse is straight forward in our chirped pulse amplification laser system. Increasing or decreasing the separation of the compressor gratings relative to the position that gives the minimum pulse duration provides a negatively or positively chirped stretched pulse respectively. The pulse shapes we explored have their origin in an asymmetry in the spectrum of the amplified laser pulse in this system. Whilst this has no significant effect on the pulse duration of the optimally compressed pulse, which is limited to ~70 fs by residual higher order phase terms, this spectral asymmetry has provided considerable utility in the current experiments. The spectrum at the output of the laser system has a centre wavelength of 800 nm and a FWHM bandwidth of 20 nm with a small shoulder centred at 752 nm with a bandwidth of ~35 nm at an intensity 0.02 times the intensity of the main laser pulse. Hence, in the time domain, a positively chirped pulse (in which the laser frequency increases with time during the laser pulse) has the shoulder on the trailing edge of the pulse and a negatively chirped pulse (in which the laser frequency decreases with time during the pulse) has the shoulder on the rising edge. Figure 1 shows the temporal intensity profile of a 100 fs negatively chirped pulse from our laser system. The temporal profile was determined from the measured spectral profile through Fourier Transforming the measured spectrum (shown inset) after the appropriate compressor grating phase function had been applied. The temporal profile of the laser pulse can thus be determined for any value of compressor grating separation. The figure shows how the spectral profile of the pulse is mapped into the temporal domain even for small stretch factors (here, 70 fs stretched to 100 fs). A temporal scan made with an autocorrelator either side of the main pulse has not shown the presence of any additional pre/post pulses formed from the higher order phase terms that prevent us from recompressing to the bandwidth limited pulse duration.

In order to determine the effect on the cluster expansion of changing the laser pulse shape, the energies of the ions produced were measured using a time-of-flight technique. The apparatus used for this experiment is described elsewhere [19]. In brief, the clusters were produced by the adiabatic expansion of a high-pressure gas through a nozzle into vacuum. A distribution of cluster sizes is produced by the expansion with the mean cluster size dependent on gas backing pressure and temperature and the characteristics of the nozzle. In the experiments detailed here the mean cluster size was estimated using the scaling parameter of Hagena [20]. The ion energies are measured using time-of-flight (TOF) spectrometry in a two stage differentially pumped chamber. The laser-cluster interaction takes part in the lower section of the chamber after the clusters have passed through a skimmer, so reducing the average density in the laser focus and allowing us to study what are effectively single cluster explosions. The ions are detected at the end of a 35 cm field-free flight tube using a microchannel plate detector. A 2 mm acceleration region (-2 kV) in front of the MCP together with a grounded grid at the end of the flight tube compresses the ion energy impact range resulting in a more uniform response of the MCP [21]. The maximum ion energy corresponds to the shortest ion flight time and is determined using a numerical algorithm described in Ref. [8] that locates the point at which the ion signal starts to increase above the noise level. This corresponds to a level of ~5×10^-4 in ion number relative to the peak signal in the ion energy distribution. Each TOF trace recorded was the average of 600 laser shots.

 

Fig. 1. Temporal intensity profile of a 100 fs negatively chirped pulse from our laser system. The temporal profile has been determined by taking a Fourier Transform of the laser spectrum (shown inset) after the application of the grating phase function. A positively chirped pulse of the same duration would be a mirror image of this profile, with the shoulder trailing the main laser pulse in time.

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The laser used in experiments is a Ti:Sapphire laser producing pulses with a minimum FWHM duration of 70 fs and a maximum energy of 60 mJ per pulse. The ion energies were measured as a function of laser pulse duration for both positively and negatively chirped pulses at a constant intensity of 1×10¹⁶ Wcm^-2 within the pulse duration range 70 – 240 fs. The pulse duration and the sign of chirp were varied by changing the separation of the diffraction gratings in the compressor of the laser system. The upper limit for the pulse duration that could be used was imposed by the measurement window of the autocorrelator used for pulse duration measurements. The autocorrelator used was a second harmonic single-shot with a 200 µm thick KDP doubling crystal. At each grating separation, the pulse duration was measured and the laser energy was varied using a waveplate and polariser combination within the laser system to keep the intensity in the laser focus constant. Constant intensity was ensured by using intensity dependent ion appearances in the interaction chamber using a low-pressure backfill of Xe rather than the cluster jet. It was found that for the same energy and pulse duration the peak intensity was independent of the sign of the chirp.

The ion energies from cluster explosions were measured as a function of pulse duration for both positively and negatively chirped laser pulses. The results are shown in Fig. 2. The results shown in Fig. 2(a) are for Xe clusters of mean size N ≈ 12,500 (mean radius 55 Å). The solid squares show the maximum ion energies measured when a laser pulse with the shoulder on the rising edge (negative chirp) was used to irradiate the cluster and the open circles denote the maximum ion energies obtained when a laser pulse with the shoulder on its falling edge (positive chirp) was used. The dotted and dashed lines are polynomial fits to the measured data. It is clear from the data that the pulse with the shoulder on its rising edge gives rise to ion energies that are higher by up to 60% than the pulse with the shoulder on its falling edge. For both signs of laser pulse chirp the maximum ion energy increases with pulse duration in the range measured. The increase is from ~50 keV at a pulse duration of 70 fs to 220 keV and 125 keV at a pulse duration of 220 fs for the pulse with the shoulder on its rising and falling edge respectively. We note that in this experiment, pulse stretching was always accompanied by either positive or negative chirp. Hence, no comparison can be made to an unchirped pulse of varying pulse duration. Such a comparison is possible in simulations as discussed below.

 

Fig. 2. Experimental results showing the maximum ion energy as a function of pulse duration for a laser pulse with a shoulder on the rising (solid squares) and falling (open circles) edge for (a) 12,500 atom Xe clusters and (b) 2,800 atom Ar clusters. The peak laser intensity was kept constant at 1×10¹⁶ Wcm^-2. The dotted and dashed lines are polynomial fits to the experimental data.

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Measurements of the mean ion energy also show an enhancement when the shoulder is on the rising edge of the pulse compared to when the shoulder is on the trailing edge of the pulse. The enhancement is less than observed for the maximum ion energies however, at most ~30 %. The mean ion energies are 9.5 keV for a 70 fs pulse, increasing to 22 keV and 17 keV at a pulse duration of 220 fs for the shoulder on the rising and falling edge respectively.

Measurements of the maximum and mean ion energies were made for three species of cluster, Xe, Kr and Ar each at three different cluster radii, 30 Å, 55 Å and 80 Å to determine the dependence of the laser pulse shape as a function of cluster species and size [22]. Results show a strong dependence on these variables. Large clusters of a high-Z species (Xe and Kr) showed a large enhancement in ion energy when the shoulder was on the rising edge of the pulse as opposed to the falling edge. This enhancement was as much as 100% for the 80 Å radius Xe clusters. For decreasing Z and cluster radius the enhancement also decreased. For the smallest, lowest Z clusters, 2,800 atom (30 Å) Ar clusters, the results are shown in Fig. 2(b). An enhancement in ion energy was observed when the shoulder was on the falling edge of the pulse. Here the energies of the ions produced by the pulse with the shoulder on the rising edge stay approximately constant as the pulse duration is increased while the energies of the ions produced by a pulse with the shoulder on the falling edge increase slightly from 20–30 keV.

To try and understand the experimental results, simulations were conducted using the nanoplasma model of the laser-cluster interaction [4]. The model was adapted to incorporate a chirped laser pulse. Results of simulations have shown that chirp alone has almost no effect on the ion energies achieved from a cluster-explosion. All the simulations were conducted using a log-normal distribution of cluster sizes in the laser focus, which has previously been shown to be an important consideration for more accurately reproducing experimental data [9].

Simulations were initially conducted for simple, Gaussian pulses with a centre wavelength of 800 nm irradiating 12,500 atom Xe clusters with peak laser intensity equal to that used in experiments, 1×10¹⁶ Wcm^-2. At each pulse duration the maximum ion energies achieved using a pulse that was either unchirped, positively chirped or negatively chirped were determined. Results showed at most a difference in maximum ion energy of 1% between the two signs of chirp for a pulse duration of 220 fs, and no difference in ion energy for shorter pulse durations. It is possible that the chirp could have an effect on the ion energies as the timing of the resonance during the laser pulse depends on the laser frequency through the critical density. The simulations showed, however, that the resonance in the cluster heating occurs on too fast a time scale to be affected by a change in laser frequency over the laser pulse. The laser pulse has insufficient bandwidth for the frequency to change at a high enough rate to track the resonance and so enhance the ion energies significantly in this way. In order to track the resonance, simulations have shown a frequency sweep rate of the order 10¹⁴ Hz/fs is required. Even a transform-limited pulse of duration 3 fs chirped to 4 fs would have a frequency sweep rate a factor of ten too slow. In this case the pulse duration would be too short for the cluster to reach its resonant condition anyway.

Simulations have also been conducted using an asymmetric laser pulse shape. The asymmetric pulse was constructed in the model by combining two Gaussian pulses of different peak intensities delayed in time with respect to one another to create a pulse with a shoulder either before or after the main laser pulse. For each compressor setting the pair of Gaussian pulses were adjusted in relative delay and width so as to provide the best agreement with the estimated experimental laser intensity profile (see Fig. 1). As we will show there is good qualitative agreement with the experimental results, sufficient to support the development of a good physical insight into the mechanisms underlying the process.

Results of the simulations are shown in Fig. 3. These results qualitatively reproduce the trends seen experimentally. For the 12,500 atom Xe clusters (Fig. 3(a)) a laser pulse with the shoulder on its rising edge (dotted line) gives rise to ions of higher energy than a laser pulse with a shoulder on its falling edge (dashed line) for the same peak intensity and FWHM pulse duration. Pulse durations quoted are the FWHM of the main laser pulse. For the shortest duration pulse (70 fs) the pulse was assumed to be unchirped and simulations were conducted for a single Gaussian pulse. The results for a laser pulse with a shoulder on the rising edge appear to predict a shorter optimum pulse duration than is measured experimentally since there is a maximum in the cluster ion energies at a pulse duration ~170 fs. Simulations of the interaction for small, low-Z clusters i.e. 2,800 atom Ar clusters (Fig. 3(b)) show higher ion energies are obtained for a pulse with the shoulder on its falling edge rather than on the rising edge. This agrees with what was measured experimentally (Fig. 2(b)). The simulations predict, however, a steeper increase in ion energy than is measured experimentally. This is not changed if a different distribution of cluster sizes is chosen.

In our experiments, the shortest pulse duration we could use was 70 fs, with no chirp. Reproducing this constraint in our simulations for 2,800 atom Ar clusters irradiated with negatively chirped pulses (shoulder on rising edge) gives a peak ion energy at a pulse duration of 100 fs. However, from other simulations, it is clear that higher ion energies could be obtained with even shorter negatively chirped pulses. For example, simulations assuming the 70 fs duration pulse was asymmetric and negatively chirped (shoulder on rising edge) show ion energies of 50 keV.

These simulations show the sensitivity of the cluster expansion to variations in the pulse shape on the rising edge of the pulse that determines when the cluster starts to expand and the velocity at which it does so. A difference in the pulse shape between the exact experimental pulse shape used and the pulse shape used in could lead to the differences observed between the experimental and simulated data presented here.

 

Fig. 3. Numerical simulations conducted using an asymmetric pulse shape for (a) 12,500 atom Xe clusters and (b) 2,800 atom Ar clusters for a peak laser intensity of 1×10¹⁶ Wcm^-2. The results show the maximum ion energy as a function of pulse duration for a laser pulse with a shoulder on the rising edge (dotted line), on the falling edge (dashed line) and a Gaussian laser pulse (solid line).

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For a cluster of a given size and species the laser pulse with the shoulder on the rising edge always has a shorter optimum pulse duration than a laser pulse with the shoulder on the trailing edge. The reason for this is that when the shoulder is on the rising edge of the pulse the cluster expansion is initiated at an earlier time than with the main pulse alone. This reduces the time needed to reach the resonance point in relation to the start of the main laser pulse, which in turn results in a shorter optimum pulse duration. When the shoulder comes after the main laser pulse no pre-expansion of the cluster can occur. This picture is confirmed by simulations using a single Gaussian laser pulse of the same duration as the asymmetric pulse used in experiments (Fig. 3(a)). In this case the predicted ion energies vary with pulse duration in a similar way to the asymmetric pulse with the shoulder on the trailing edge but are of slightly lower energy. Although the shoulder on the trailing edge of the pulse does not affect the optimum pulse duration it can still impact on the energies of the ions produced by the cluster expansion and so gives rise to slightly higher energy ions than a Gaussian pulse of the same duration.

Simulations show that the ion energies increase quite rapidly with pulse duration up to their maximum value at the optimum pulse duration and then fall off quickly again as the pulse duration extends beyond the optimum. For the 12,500 atom Xe clusters, the optimum pulse duration determined through simulations for a Gaussian pulse irradiating the largest cluster within the distribution is greater than 220 fs, as shown in Fig. 3(a). This is beyond the longest pulse duration investigated experimentally. The effect of the shoulder on the rising edge of the laser pulse is to reduce this optimum pulse duration. The shortening of the optimum pulse duration means that experimentally we are recording ion energies closer to their maximum value than we would for a Gaussian laser pulse or a pulse with the shoulder on its falling edge.

For small, low Z clusters the optimum pulse duration for a Gaussian pulse is short, ~120 fs for the 2,800 atom Ar clusters. The pulse with the shoulder on the rising edge has an optimum pulse duration shorter than this, meaning that most of the measurements made are for pulse durations greater than the optimum value and so are in the region where the ion energy falls off with increasing pulse duration. For a pulse with the shoulder on its trailing edge the position of the optimum pulse duration is unaffected and ion energy measurements are in fact made close to this optimum value. This results in higher ion energies measured for the pulse with the shoulder on the falling edge than the rising edge.

In conclusion, we have shown that the shape of the laser pulse irradiating a cluster plays an important role in the cluster expansion dynamics. Large differences in ion energy, up to a factor of 2, have been achieved by varying the shape of the laser pulse. The measured difference depends strongly on cluster size and species with the most favourable pulse shape determined by whether we have large, high-Z or small, low-Z clusters. Simulations suggest that the ion energies produced using an asymmetric laser pulse are higher than could be achieved using a Gaussian pulse of the same FWHM duration and peak intensity. Finally, we note while our experimental technique only allowed simple pulse shapes to be investigated, future investigations are well suited to adaptive pulse shaping under the control of evolutionary algorithms.