Effective Transparency in the XUV: A Pump-Probe Test of Atomistic Laser-Cluster Models
Rishi Pandit, Kasey Barrington, Thomas Teague, Zachary Hartwick, Nicolas Bigaouette, Lora Ramunno, Edward Ackad
EEffective Transparency in the XUV: A Pump-Probe Test of Atomistic Laser-ClusterModels
Rishi Pandit , Kasey Barrington , Thomas Teague ,Zachary Hartwick , Nicolas Bigaouette , Lora Ramunno , Edward Ackad Department of Physics, Southern Illinois University Edwardsville, Edwardsville, Illinois 62026, USA Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada (Dated: September 16, 2018)The effective transparency of rare-gas clusters, post-interaction with an extreme ultraviolet (XUV)pump pulse, is predicted by using an atomistic hybrid quantum-classical molecular dynamics model.We find there is an intensity range for which an XUV probe pulse has no lasting effect on theaverage charge state of a cluster after being saturated by an XUV pump pulse: the cluster iseffectively transparent to the probe pulse. The intensity range for which this phenomena occursincreases with cluster size, and thus is amenable to experimental verification. We present predictionsfor clusters at the peak of the laser pulse profile, as well as the expected experimental time-of-flight signal integrated over the laser profile. Since our model uses only atomic photoionizationrates, significant experimental deviations from our predictions would provide evidence for modifiedionization potentials due to plasma effects.
The extreme ultraviolet (XUV) regime has the simplestinteraction between ultra-intense laser pulses and matter,primarily through photoionization. When a nanoscopicdense clump of matter (cluster) is irradiated, secondaryionization events then take place such as collisional ion-ization. Clusters have solid density but their inter-clusterdistance is so large that clusters do not interact with eachother, thus they bridge the gap between the gas and solidphases of matter.Experimental and theoretical laser-cluster interactionstudies in the XUV regime are simpler to interpret thanat other wavelengths, and it is thus an ideal regime to fur-ther test detailed, atomistic models of laser-cluster inter-actions [1, 2]. At longer wavelengths, even low intensitypulses have efficient processes to transfer energy from thepulse to the free electrons, heating the electron plasma(termed inverse Bremsstrahlung heating, IBH) along theaxis of the laser’s polarization [3, 4]. At shorter wave-lengths, the photoionization that occurs is from the innershell electrons and leads to subsequent Auger ionization[5–9]. Thus, XUV pulses – which through photoioniza-tion only access valence shell electrons, and where IBHis negligible for intensities < W/cm – present theideal regime for experiments to probe the degree to whichthe ionization potential may be modified by the plasmaenvironment [10–15].In this letter, we report on the finding that the ioniza-tion in XUV-cluster interaction can become effectively saturated . In our model, which uses only atomic ioniza-tion potentials, this occurs when a cluster is irradiatedwith an XUV pulse above a saturation intensity. Addi-tional pulses irradiating the cluster leave no net effect onthe ionization or total energy; thus the cluster is effec-tively transparent to the probe pulse.Multiple models of laser-cluster interaction exist, andnew experiments are needed to allow the community todistinguish between the different models [13]. Atomisticcluster models to date fall into two primary categories:those with collisional processes beyond single step colli- sional ionization from the valence shell (atomistic aug-mented collisional model, AACM) and those with en-hanced photoionization processes arising from ionizationpotentials that are lowered below the atomic ionizationpotentials due to the presence of the nanoplasma envi-ronment (atomistic augmented photoionization model,AAPM). Our prediction of effective transparency forpump-probe XUV laser cluster interaction was deter-mined using an AACM, and uses only well-establishedatomic phenomena. Thus an experimental verificationof effective transparency would place an upper bound onthe significance of enhanced photoionization mechanisms.On the other hand, the failure of the AACM to correctlypredict experimental outcomes would be evidence in fa-vor of an enhanced photoionization mechanism (such as,eg, electron screening or barrier suppression). Thus thisletter presents a proposal for an experiment.The schema to distinguish the two models is as follows.A pump pulse irradiates a cluster at an intensity that isabove the saturation intensity predicted by the AACM,where atomic ionization potentials are used. Above thesaturation intensity, the average ion charge state (AICS,the total charge divided by the number of ions since onlyions are detectable) of the cluster in the calculation isindependent of pump intensity. This occurs when the in-tensity is high enough that all possible photoionizationshave occurred, but not high enough for significant mul-tiphoton photoionization or IBH. The AACM predictsthat the cluster would then be effectively saturated, andwould not increase its AICS if subsequently irradiated bya probe pulse.However, if the electromagnetic fields of thenanoplasma sufficiently perturb the atomic ionizationpotentials so that single-photon ionization from deeperstates becomes possible, effective transparency wouldnot be detected in an experiment. The strength of thenanoplasma perturbation would have to be a functionof the plasma density. If a probe pulse irradiates thecluster after only a short delay, while the cluster is still a r X i v : . [ phy s i c s . a t m - c l u s ] J un dense, the different models will strongly disagree. Thenanoplasma perturbation, if large, would allow the clus-ter to further ionize due to the lowered ionization poten-tials. With increasing delay of the probe pulse the clus-ter’s density decreases and so does the nanoplasma’s per-turbation. Thus, the two models predict different trendsas a function of pump-probe delay.This methodology is complimentary to previous pro-posals [2] with the advantage that the effect is enhancedby the cluster size distribution. All previous work in thisarea has neglected the role of collisional excitation whichis known to play a dominant role in the ionization in theXUV [16].Our implementation of an AACM is a hybrid approachwherein the particles are treated as classical charge dis-tributions whose motion is solved by molecular dynamics.The ionization rates are determined from a mix of exper-imental (when available) and theoretical cross-sectionsin the gas phase [17]. During ionization, the pertur-bation on the ions due to the cluster environment (thenanoplasma) has been shown to be well represented byour Local Ionization Threshold (LIT) model [18], whichmaintains the use of atomic ionization potentials. Usingthe LIT model we include single- and multi-photon ion-ization, collisional ionization, augmented collisional ion-ization (ACI) [17], and many-body recombination [19].The model has been successful in reproducing the laser-cluster experimental signals [20], including experimentswhere Auger ionization is dominant [19].In AACMs, collisional ionization beyond a single stepprocess is considered. The standard ionization channelsare augmented to include the possibility of collisional ex-citation, so called augmented collisional ionization (ACI)[17]. A bound electron can first be promoted from theground state to an excited state by a collision of an al-ready ionized electron. Subsequently, this excited elec-tron can be ionized by being promoted from the excitedstate to the continuum through a second collision. Whilethe whole trip can be energetically the same, breaking theprocess up into two steps reduces the energy required foreach transition. This allows an electron with less kineticenergy (compared with single-step ionization) to executethe process. In a nanoplasma, the energy distribution is,on average, Maxwellian and thus there are many moreelectrons with enough energy to excite an atom thanthere are who can ionize an atom directly [20]. Thisionization pathway leads to higher charge states in thecluster and collisionally reduced photoabsorption (CRP)where clusters absorb less photons due to fast collisionalionization removing target ions [20].The current work includes one and two photon ioniza-tion given by the rate, dNdt = (cid:18) IE ph (cid:19) σ (1) + (cid:18) IE ph (cid:19) σ (2) (1)where I is the intensity of the laser, E ph is the photonenergy and σ ( n ) is the n -th order photoionization process.The values of σ (1) = 5 . × − cm [21] and σ (2) = 10 − cm /s (taken as an upper limit from reference [22])were used. The higher σ (2) is, the smaller the range ofintensities in which effective saturation will occur, andthus taking an upper limit gives a conservative estimateof the saturation effect.To show the saturation effect in our AACM model, wesolved the interaction of argon clusters (Ar ) irradiatedby two XUV pulses at λ = 33 nm (37.6 eV) 25-fs apart.Both pulses have a full-width-at-half-maximum of 10 fs.Although short pulses increase the probability of multi-photon ionization (which undermines our signal), theyalso allow the probe pulse to irradiate the cluster whilethe density is still high (which enhances the likelihoodof nanoplasma perturbations which must depend on theplasma density). The number density of the ions at thepeak of the pump pulse is around 3 . × − bohr − (where the distance of the furthest ion is used as the ra-dius of the spherical volume) while at the peak of theprobe pulse the density is 3 . × − bohr − , a percentdifference of about 3.85%. Further, the plasma number-density of the cluster at the same radius is 1 . × − bohr − at the peak of the pump and 1 . × − bohr − at the peak of the probe. Thus, the effects of an enhancedphotoionization mechanism will be most pronounced dur-ing the probe pulse and would decrease as the pulse delayincreases.The specific XUV-wavelength was chosen to be abovethe singly ionized ionization potential for argon (27.6 eV)and below any significant inner-ionization thresholds. Anintensity scan was then performed for the pump pulse.The AICS after 500 fs of the start of the pump pulse isused as a measure of the overall ionization of the cluster,and the cluster is considered to be at the focus of the laserpulse. The average is taken over all ions; ions containingclassically bound electrons have their charges decreasedaccordingly.The solid red curve in Fig. 1 shows the AICS vs pumpintensity when an Ar cluster is irradiated only bya pump pulse. As pump intensity is increased from10 W/cm to 10 W/cm , the AICS starts to in-crease very gradually. At around 10 W/cm theAICS increases dramatically until, at an intensity ofabout 10 W/cm , the AICS becomes saturated aroundAICS=3.5. This is what we call the ”saturation in-tensity”. Further increasing the pump intensity, onlymarginally increases the AICS until after the AICSplateau, around 10 W/cm . This small increase is dueto multiphoton ionization and IBH. At an intensity of10 W/cm AICS reaches about 4.9. Even at this inten-sity, more than 50% of the ionization is due to collisionalionization, almost exclusively through ACI. At the satu-ration intensity ACI accounts for well above 90% of allionizations.To demonstrate effective saturation, a pump probesetup is modeled showing that the probe pulse has al-most no effect on the AICS. The pump pulse is fixed at2 . × W/cm , just above the saturation intensity.The intensity of the subsequent probe pulse (25 fs later) FIG. 1. (Color online) The solid (red) curve shows average ioncharge state (AICS) vs pump intensity for Ar irradiated bya single 10 fs, λ = 33 nm pump pulse (depicted in the lowerleft illustration). The short-dashed (blue) curve shows theAICS vs probe intensity for Ar irradiated by a pump pulsefixed at an intensity of 2 . × W/cm and subsequently(25 fs delay) irradiated by probe pulse of varying intensity(depicted by the lower right illustration). The long-dashed(green) curve shows the AICS vs probe intensity for the samesetup, except where the probe pulse can only photoionize andthus no IBH occurs (depicted by the top right illustration). is scanned from 10 W/cm to 10 W/cm as depictedat the bottom right illustration of Fig. 1. The blue short-dashed curve in Fig. 1) shows AICS versus probe inten-sity, where the AICS is measured 500 fs after the startof the pump pulse. We find that the AICS begins andremains saturated until the intensity of the probe pulseexceeds about 10 W/cm . This is when the probe pulsereaches sufficient intensity for IBH to become significant.Below this intensity, the additional probe pulse does notmeaningfully increase the AICS from what it was afterthe pump; this is the basis for terming the phenomenoneffective transparency, since it is as if the cluster weretransparent to the probe pulse.Why is there a plateau in the AICS? An analysis of thecharge state distribution verses time shows that the irra-diation of the cluster by the pump pulse at the saturationintensity ionizes all possible targets via photoionizationand collisional ionization. Thus, during the pulse thereare no more targets to further photoionize [23]. This isthe high intensity limit to the previously observed CRP[20]. Without any targets the probe pulse does not con-tribute to the AICS. The result is thus the same saturatedAICS, both with and without the probe pulse. Concep-tually, this is where any enhanced photoionization mech-anisms would play a significant role. The probe pulseis irradiating a dense nanoplasma and, according to ion-ization potential (IP) lowering models, would allow forthe photoionization of ions well beyond Ar and thuschange the final AICS significantly [11, 14].Our simulations further show that the effective trans-parency phenomenon is fairly insensitive to the changeof the delay time from 15 fs to around 150 fs. Noticeabledeviations occur only when the delay time is >
200 fs. This insensitivity would be a verifiable trend in the ex-perimental data only if no significant ionization potentiallowering occurs.As the density of the cluster decreases with the clus-ter’s disintegration, one would expect the IP loweringeffect must also decrease. It would tend to zero as thedensity becomes that of a gas, since no IP lowering mech-anism has been observed in gas [11, 13, 24–26]. IP lower-ing effects would thus be sensitive to pump-probe delaytime. If a lack of sensitivity to the delay time (within the15-150 fs range for the aforementioned parameters) werefound experimentally, it would place constraints on howstrong IP lowering contributes to the total ionization.Artificially turning off the probe pulse’s electric field,allowing only direct photoionization the cluster (no IBH),shows that the end of the AICS plateau is due almostexclusively to IBH (short-dashed green curve in figure 1).We now consider calculations that correspond more di-rectly to what an experiment would detect. In any clusterbeam, there is a log-normal distribution of cluster sizes.Thus, we examine the effect of cluster size on saturationintensity, and the intensity range over which the AICSremains constant. In the range of parameters examined,the saturation intensity I sat decreases as the cluster sizeincreases (shown as the red plus signs in Fig. 2 where theline is drawn to aid the eye). This makes intuitive sensesince the larger clusters absorb the same amount of en-ergy per ion as the smaller clusters. However, the amountof energy needed for an electron to escape the cluster re-mains the same [1]. Thus, larger clusters absorb more total energy (than smaller clusters) at the same inten-sity. It thus takes less intensity to effectively saturate thecluster’s ionization channels. It should be noted that thetrend ends once the cluster’s size becomes large enoughthat the pulse is significantly depleted by the photoab-sorption ( N ≥ FIG. 2. (Color Online) The saturation intensity (minimumintensity needed to saturate the cluster) I sat as a function ofthe cluster size is shown as the (red) pluses for pump pulseduration of 10 fs at λ = 33 nm. The intensity range (rightvertical axis) as a function of cluster size over which the probepulse has a negligible effect on the average ion charge state isshown as the (blue) x’s. FIG. 3. (Color Online) Time-of-flight signal for Ar irradi-ated only by a 10 fs pump pulse of I = 2 . × W/cm at λ = 33 nm shown as the dashed red line. The blue boxesshow the time-of-flight signal when the pump pulse is fol-lowed by an identical probe pulse. Inset: The average ioncharge state as a function of the pump pulse’s intensity (solidred) integrated over the full laser pulse’s spatial distribution.The dashed (blue) curve is the average ion charge state for apump pulse fixed at 2 . × W/cm as a function of theprobe pulse’s intensity, integrated over the spatial profiles ofthe two pulses. The range of intensities over which the cluster is effec-tively transparent ( I high − I low ) to the probe pulse also in-creases with the size of the cluster (shown as the blue x’sin Fig. 2). This indicates that the effective saturation ismore pronounced in all measures in larger clusters. It wasfurther found that AICS ≈ α ln( βN ), where α = 0 . β = 35457 . N is the cluster size less than 2057 forargon [23]. The cluster size distribution will change theAICS but not by much due to the logarithmic relation-ship between AICS and N . Thus, experiments with thecluster size peaked at a few hundred atoms would havetheir signals enhanced by the cluster beam’s size distri-bution.Thus far the results have been for the spatial peak ofthe laser pulse(s) and may be achievable if the beam ismasked to reduce the wings of the pulse as in Ref. [5].Otherwise, we now consider what an experiment woulddetect due to the spatial distribution of the pulse. Whilea small subset of clusters will be irradiated by both pulsesat the peak intensities, many clusters spatially located inthe wings of the pulse will be irradiated by a pump pulseof insufficient intensity to saturate the ionization chan-nel. The probe pulse will then increase their ionization.In the pump-probe setup, clusters were assumed to inter-act with the same intensity region of both pulses, i.e., thepulses were assumed to be spatially identical and focusedat the same location. The resulting time-of-flight (TOF)signal for the pump pulse alone at I = 2 . × W/cm is shown as the dashed red line in Fig. 3. It was calcu- lated using the methodology from reference [19], wherethe signal is integrated over the intensities of the pulse fora single cluster size and using the TOF setup describedin reference [6]). It shows that the signal will containprimarily singly charged ions with an almost linear de-crease in the higher charge states. The signal only seesa small change when an identical probe pulse is included(shown as the blue boxes in Fig. 3). If the probe pulse isbelow the saturation intensity (but with the same spatio-temporal profile), the TOF is quite close to the pump-only signal. However, increasing the probe pulse to be-yond the saturation intensity results in some increase inthe signal from the multiply-charged states. This is tobe expected as now more clusters will fall into a spatialregion where they will be saturated by the probe pulse.To illuminate this effect and show the trends an ex-periment would see in the absence of IP lowering, theAICS is calculated over the entire spatial distribution ofthe pulse. The saturation of the AICS is not observablefor a single pulse (red solid curve in the inset of Fig. 3)due to the spatial wings of the pulse. However, satura-tion is observable when the clusters are further irradiatedby a probe pulse. Fixing the pump pulse’s peak inten-sity at 2 . × W/cm and changing the intensity ofthe probe pulse shows the saturation of the AICS. Asthe probe pulse’s intensity increases from 10 to about10 W/cm , the AICS remains constant at about 1.5(blue dashed curve in the inset of Fig. 3). Further in-creases in the intensity of the probe pulse increase theAICS as more clusters in the wings of the pulse becomesaturated. This result is again constant with delay timesless than about 200 fs. IP lowering models would showmuch sharper increases in the AICS as a function of theintensity. This would result in the AICS increasing evenfor a low intensity probe pulse since the additionallyphotoionized electrons, allowed by IP lowering, wouldbe cluster bound causing additional collisional ionizationevents.In conclusion, we have shown that atomic-based laser-cluster interaction models predict that it is possible toinduce effective transparency in the XUV using a pump-pulse setup. This effect is insensitive to the delay betweenthe pulses, and thus insensitive to nanoplasma density;this is in contrast to what IP lowering models would pre-dict. 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