Effect of pulse duration on resonant heating of laser-irradiated argon and deuterium clusters

Effect of pulse duration on resonant heating of laser-irradiated argon and deuterium clusters

Ayush Gupta and T. M. AntonsenDepartment of Electrical Engineering, University of Maryland, College Park, Maryland 20770

and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20770, USA

T. TaguchiDepartment of Electrical and Electronics Engineering, Setsunan University, Neyagawa, Osaka, Japan

J. PalastroDepartment of Physics, University of Maryland, College Park, Maryland 20770, USA

and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20770, USA�Received 29 June 2006; revised manuscript received 16 August 2006; published 25 October 2006�

We study the effect of pulse duration on the heating of single van der Waals bound argon and deuteriumclusters by a strong laser field using a two-dimensional �2D� electrostatic particle-in-cell �PIC� code in therange of laser-cluster parameters such that kinetic as well as hydrodynamic effects are active. Heating isdominated by a collisionless resonant absorption process that involves energetic electrons transiting throughthe cluster. A size-dependent intensity threshold defines the onset of this resonance �T. Taguchi et al., PhysicalReview Letters, 92, 20 �2004��. It is seen that increasing the laser pulse duration lowers this intensity thresholdand the energetic electrons take multiple laser periods to transit the cluster instead of one laser period. Oursimulations also show that strong electron heating is accompanied by the generation of a high-energy peak inthe ion energy distribution function. We also calculate the yield of thermonuclear fusion neutrons from ex-ploding deuterium clusters using the PIC model with periodic boundary conditions that allows for the inter-action of ions from neighboring clusters.

DOI: 10.1103/PhysRevE.74.046408 PACS number�s�: 52.50.Jm, 36.40.Gk, 52.38.Kd, 52.38.Dx

I. INTRODUCTION

Clusters are nanoscale solid density atomic aggregatesbound by van der Waals forces ranging in size from 102

−106 atoms. Clusters are formed when high-pressure flow ofa cooled gas into vacuum results in adiabatic cooling andexpansion of the gas and particle aggregation �1�. The vol-ume average density of clustered gases is low but the clustersthemselves are at solid density. This enables strong interac-tion of individual clusters with an irradiating laser pulsewhile still allowing propagation of the pulse through theclustered gas. The efficient coupling of laser energy intoclustered gases �2� makes them a unique media for studyingnonlinear laser-matter interaction �3� and leads to many ex-citing applications. The laser-heated clusters generate x-rayand extreme ultraviolet �EUV� radiation, and high-energyions and electrons �4,5�. Collisions between energetic ionsfrom exploding clusters can produce neutrons via thermo-nuclear fusion �5,6�. The dynamics of exploding clustersgives rise to interesting nonlinear optical effects such asharmonic generation �7� and self-focusing �8�. Clusteredgases are also proposed as targets for creating plasma waveguides �9�.

The interaction of clusters with strong laser fields is char-acterized by a large number of tunable parameters such asthe peak intensity, pulse duration, spot size, frequency andfield polarization of the laser pulse, the distribution of clustersizes, fraction of gas present as monomers, and the densityand ionization potentials of the cluster atoms. Various experi-mental �10–13� and theoretical studies �14–19� have ex-plored the role played by some of these parameters withtheoretical studies often focused on the effect of laser inten-

sity and cluster size. Experimental studies show that pulseduration plays an important role in the various aspects oflaser-cluster interaction such as absorption and scattering oflaser pulses �20�, neutron yield of deuterium cluster targets�12�, x-ray generation �13�, and extreme ultraviolet emission�13,21�. In the present work we examine via simulations theeffect of laser pulse duration �full width at half maximum�FWHM� of the laser pulse envelope� on heating of irradi-ated argon and deuterium clusters. We also examine the yieldof fusion neutrons from exploding deuterium clusters for arange of intensities.

The manner in which individual clusters are heated by thelaser pulse and explode has several characteristic regimesbased on the laser intensity and size of cluster. Two relativelydistinct regimes are hydrodynamic expansion and Coulombexplosion. For high intensities or small clusters most elec-trons are removed from the cluster early in the pulse and thecluster “Coulomb explodes” due to the repulsion of the re-maining positive ions. Typically �22� particle models havebeen used to study this regime. In these models the electro-static interaction between particles is treated point wise �par-ticle to particle, also known as molecular dynamics simula-tions� or in the particle in cell �PIC� approximation. Themolecular dynamics method is suited for small clusters andbecomes computationally inefficient for large clusters. ThePIC method is more practical for large clusters. It is, how-ever, relatively more difficult to describe collisions in thePIC framework.

At lower intensities and for large clusters a hydrodynamicapproach is valid. Here, the expansion is driven by the pres-sure of electrons. In this approach, the cluster is modeledusing fluid equations. The first study of hydrodynamic ex-

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pansion of clusters �23� treated the ionized clusters as aspherical ball of uniform “nanoplasma” with no density ortemperature gradients. The electrons are generated by fieldand collisional ionization and heated primarily by theelectron-ion collisions within the cluster. For the brief timeinterval when the electron plasma frequency in the expand-ing cluster satisfies �p0=�3 �0, where �0 is the laser fre-quency, an electrostatic resonance causes the field inside thecluster to rise sharply leading to rapid heating. A more so-phisticated approach allows for nonuniformity of tempera-ture and density during expansion �16�. This model predictsthat the dominant energy absorption occurs in regions wherethe electron density is near the critical density. The mainconsequence of this is that strong absorption occurs over amuch longer duration than that predicted by the uniform den-sity model. In both these models, the response of the clusterto the laser electric field is treated in the cold plasma ap-proximation and is used for calculation of the total electricfield as well as the energy absorbed by the cluster. However,when energetic particles are produced the local cold plasmaapproximation no longer applies. Energetic particles travel adistance comparable to the cluster size in a laser period con-tributing to a nonlocal and nonlinear dielectric response. Thisnecessitates a kinetic treatment of the absorption process.

Our studies have used a PIC model focused on this inter-mediate range of intensity and cluster sizes where a popula-tion of energetic electrons is produced and the heating andexpansion of the cluster are distinctly nonhydrodynamic.However, the cluster still remains quasineutral as it expands.In this regime energy absorption by electrons is both a ki-netic and a collective effect. The energetic electrons makelarge excursions, comparable to the size of the cluster, andbeing at a high temperature are largely unaffected by colli-sions �refer to the fact that collision frequency is inverselyproportional to temperature�. At the same time, the laser fieldis strongly shielded from the cluster core by the dielectricresponse of the electrons. Other similar particle studies havenot considered ion motion �24� which significantly affectsthe cluster expansion dynamics for long pulse, or do notwork in the regime of free electron excursion comparable tothe cluster diameter for several laser periods �14� which isour regime of interest.

The major result of our previous studies �17,18� was thatthere is a well defined intensity threshold above which ener-getic electrons are created by an absorption process related tothat proposed by Brunel for sharp density gradients �25�.Electrons are first accelerated out from the cluster and thendriven back into it by the combined effects of the laser fieldand the electrostatic field produced by the laser-drivencharge separation. The energetic electrons then pass throughthe cluster and emerge on the other side. If they emerge inphase with the laser field, there is resonant heating, and thecluster quickly absorbs energy. The onset of this resonancecorresponds to the intensity for which the excursion of a freeelectron in the laser field is comparable to the cluster diam-eter. This critical intensity was shown to be

I0 =c

8��me�

2

2�e�2

D20, �1�

where c is the speed of light in vacuum, D0 is the initialcluster diameter, � is the frequency of the laser pulse, and me

and e are the electron mass and charge, respectively.In this paper, we use our PIC model to examine how laser

pulse duration influences the energy absorption by clusters.Our simulations show that increasing the pulse duration en-ables strong heating via a higher order electron transit timeresonance where the electron transit time through the clusterequals multiple laser periods, effectively lowering the inten-sity threshold for strong heating. We modify the scaling lawfor the intensity threshold derived in Ref. �17� to take thehigher order electron transit time resonance into account.

Fusion of ions from exploding deuterium clusters as ob-served in experiments has generated interest in laser irradi-ated clustered gases as a potential source of fusion neutrons.Some past studies �11,12,14,19,26� have investigated fusionneutron yield in the Coulomb explosion regime. In thepresent work, we use the ion energy spectrum from our PICmodel to estimate the fusion yield from deuterium clustersfor a range of pulse intensities and examine the effect ofintensity threshold on fusion yield in the regime where bothkinetic and hydrodynamic effects are active.

The organization of the paper is as follows. The simula-tion model used is described in Sec. II. Simulation resultsshowing the effect of nonlinear resonance on heating of ar-gon and deuterium clusters for a range of intensities at dif-ferent pulse durations are outlined in Sec. III. Modification tothe scaling law for strong heating to incorporate the effectsof increased pulse width is included in this section. SectionIV presents the electron and ion distribution functions forargon and deuterium clusters. The deuterium ion energyspectrum is used in Sec. V to estimate the fusion yield for arange of pulse intensities. Section VI ends the paper withdiscussions and concluding remarks.

II. PIC MODEL OF CLUSTER EXPANSION

To study the effect of laser pulse parameters on clusterheating in the transition regime, we use a two-dimensional�2D� electrostatic particle-in-cell �PIC� model which hasbeen presented in detail in Refs. �17,18�. The laser field E=Ex�t�sin �t is polarized in the x direction with frequency �corresponding to �=800 nm, and time dependent amplitudeEx�t� corresponding to a Gaussian pulse. The electrostaticcalculation is appropriate in the near field limit �D0 /��1�,and for intensities below �5�1017 W/cm2. For higher in-tensities, where the quiver velocity approaches the speed oflight, the Lorentz force becomes important, necessitating anelectromagnetic simulation.

As the intensity rises past a threshold ��1014 W/cm2 forargon atoms� during the pulse neutral cluster atoms are ion-ized by field ionization followed by collisional ionization.This produces a supercritically dense plasma with electrondensity ne�ncr, where ncr is the density at which the electronplasma frequency equals the incident laser frequency. Theplasma shields the core of the cluster from the laser field.The majority of our simulations begin here by modeling theclusters as consisting of preionized atoms and electrons. Thematerial parameters used for argon and deuterium clustersare specified in Table I. The initial temperatures in eithercase are chosen to be 10 eV for electrons and 0 eV for ions.

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This electron temperature is large enough to suppress gridinstabilities inherent to PIC algorithm, but not so large as toaffect results. Electron-ion scattering processes are includedexplicitly in our code through a Monte Carlo method, bution-ion and electron-electron scattering processes are ne-glected. Over the few hundred femtoseconds of the simula-tion, collisional energy relaxation from electrons to ions isnot important, and the ions gain energy dominantly throughacceleration by the space-charge electric field.

The 2D simulations, corresponding to an initially cylin-drical cluster, are carried out on a 1024�1024 square meshof spacing 0.805 nm. In most of our simulations, unless oth-erwise stated, we solve the Poisson equation with openboundary conditions, and particles are allowed to leave thesimulation region. Once a particle leaves, its momentum andposition are still tracked assuming it responds only to thelaser electric field. This is important in calculating the elec-tric dipole moment of the exploding cluster. The distance tothe simulation boundary is sufficiently far from the clustercore that only a small fraction of the electrons leave thesimulation box for most of the pulse duration.

We note that three-dimensional �3D� simulations that wehave carried out earlier and reported in Refs. �17,18� showthe same behavior as found in the 2D simulations reportedthere �17,18� and here. The only difference is a lowering ofthe threshold intensity due to the increased dielectric focus-ing found in 3D geometry. Thus, for the extensive study ofthe effect of pulse duration we restrict ourselves to the lesscomputationally intensive 2D simulations.

If the number density of clusters in the irradiated volumeis high, energetic ions from neighboring clusters are ex-pected to interact especially towards the end of the pulsewhen the cluster has expanded significantly. The effect ofneighboring clusters alters the evolution of the energy spec-trum of ions and electrons. The energy distribution of ions isdirectly used for determining the fusion neutron yield. Inorder to incorporate the effect of ions from neighboring clus-ters on the kinetic energy distribution of ions, we employperiodic boundary conditions to solve the Poisson equationand we allow escaping particles to reenter the simulationdomain. In this case, the size of the simulation box definesthe intercluster distance. We assume that the clusters are dis-tributed uniformly in the irradiated volume. Then the inter-cluster distance is simply �ncl�−1/3, where ncl is the numberdensity of clusters. For our simulations parameters �1024�1024 mesh with mesh spacing of 0.805 nm�, the interclus-ter distance is 824.32 nm which corresponds to ncl=1.78�1012 cm−3.

III. EFFECT OF PULSE DURATION

We have done a series of PIC simulations for a range ofpeak intensities at three different pulse widths for both Argon

and Deuterium. The initial cluster radius was fixed at D0=38 nm. The pulse width defined the full width at half maxi-mum of the Gaussian laser pulse envelope. The peak inten-sity is varied from 1�1013 W/cm2 to 1�1016 W/cm2 withthe pulse widths corresponding to FWHM of 100 fs, 250 fs,and 1 ps for argon and 70 fs, 100 fs, and 500 fs for deute-rium.

In Fig. 1, we plot the average kinetic energy per electronfor cluster electrons �solid� and ions �dotted�, and the elec-trostatic field energy per electron �dashed� in the cluster forD0=38 nm and laser peak intensity of 3�1015 W/cm2 andpulse width of 250 fs. Also shown is the laser intensity pro-file. Initially, the laser electric field pulls out a small fractionof the electrons at the cluster boundary. Accelerated by theelectric field in the cluster potential, these escaped electronsabsorb energy and the mean electron kinetic energy is in-creased. As the laser intensity rises, more electrons are pulledout of the cluster and electron excursion length is also in-creased resulting in stronger heating of the cluster. Themechanism of electron heating is described in detail whenwe discuss phase space plots later in the paper. The chargeimbalance created by the extraction of electrons causes anoutward directed electric field that accelerates the ions. Theelectron energy is thus transferred to ions via the electrostaticfield energy stored in the space-charge field. The cluster ex-pands due to the Coulomb force of the space charge on thecluster ions. Electron extraction and heating decreases laterin the pulse when the laser electric field starts falling and thecluster boundary has expanded. This, accompanied by thecontinued expansion of the cluster core, causes the electronand field energies to fall. Towards the end of the pulse the

TABLE I. Cluster parameters for argon and deuterium clusters.

Clusteramaterial

Initial iondensity �cm−3�

Maximumionized state, Z

Initial electrondensity

Mass of ion�amu�

Argon 1.742�1022 Z=8 ne�80 ncr 40

Deuterium 5.98�1022 Z=1 ne�34 ncr 2

FIG. 1. Electron �solid line with circles�, ion �dashed line withsquares� and electrostatic field energy �solid line with diamonds�per cluster electron as a function of time for an argon cluster withD0=38 nm and laser pulse of peak intensity 3�1015 W/cm2 andFWHM of 250 fs. The laser field profile is shown on the same timeaxis by the dotted line with crosses. The applied field acceleratesthe electrons, creating a charge separation that in turn acceleratesthe ions.

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space charge field energy approaches zero as the expandingions decrease the charge imbalance. The mean ion energysaturates as most of the electron energy is transferred to theions and the electrons are no longer extracting energy fromthe pulse.

Our simulations indicate that there is an intensity thresh-old for strong heating at which there is a sharp increase in thelaser energy coupled to the cluster electrons. This thresholdeffect is seen clearly in Fig. 2 that plots the total absorbedenergy �sum of electron, ion, and electrostatic field energy�per electron at the end of the pulse versus peak laser intensity�scaled to the cluster diameter squared� for different pulsedurations—100 fs, 250 fs, 500 fs, and 1 ps for argon �la-beled as “Ar” in the legend� and 70 fs and 100 fs for deute-rium �labeled with a “D” in the legend�. The curves forshorter pulse durations �100 fs, 250 fs for argon and 70 fs,100 fs for deuterium� exhibit a steep rise in the absorbedenergy indicating that there is a critical intensity beyondwhich the cluster experiences enhanced heating. This thresh-old intensity is different for different cluster material �argon/deuterium� and laser pulse duration. In Ref. �17�, it wasshown for the case of 100 fs pulse duration and argon clus-ters that the threshold for strong absorption is due to a non-linear electron transit time resonance. The critical intensityfor the onset of this resonance was predicted as the intensityfor which energetic electrons take exactly a laser period totransit once through the cluster. The dependence on clustermaterial and laser pulse duration as seen here is not predictedby the scaling law predicted by Taguchi et al. �17� and An-

tonsen et al. �18�. In the following paragraphs we will look atelectron phase space plots to explore this in greater detail.The thresholdlike behavior is less prominent for the longerpulse duration of 500 fs and almost disappears for the evenlonger pulse width of 1 ps. The absence of a clear thresholdfor the 1 ps pulse duration results from strong heating of thecluster over a wider range of peak laser intensities and welook at electron phase space plots at three different intensi-ties for 1 ps pulse duration to understand the process of laserto cluster energy for long pulse durations. Further, we notethat in all cases the average energy absorbed per electronreaches a saturation value at high intensity, and that the satu-ration value does not depend monotonically on pulse dura-tion. In particular, for argon cluster, the saturation value forpulse duration of 250 fs exceeds that for both 100 fs and1 ps pulses. The curve labeled “Ar-Circ” is for a series ofruns for argon clusters irradiated with circularly polarizedlaser light of 100 fs pulse duration.

The process of energy absorption by cluster electrons isrelated to that proposed by Brunel for heating of laser-drivenelectrons at planar solid-vacuum interface where the electronexcursion amplitude is larger than the local density scalelength. This is applicable to clusters where the laser pulseaccelerates energetic electrons out from the cluster core anddrives them back into the shielded cluster. We look at theelectron phase space at peak laser intensity of 5�1015 W/cm2, just beyond the threshold for the case of100 fs pulse duration. Figure 3�a� shows a snapshot of thisphase space at t=188.17 fs �the peak of the pulse occurs at170 fs�. Here, the axes are the x coordinate of position andthe x component of velocity �vx� for all electrons in the nar-row layer y�� along the x axis, where � is the grid size.The high concentration of particles in a band about the centerat small vx represents the core electrons in the cluster. In this

FIG. 2. Total energy absorbed by the cluster �sum of electron,ion and electrostatic field energies� per cluster electron versus thelaser intensity for a range of pulse durations for argon and deute-rium clusters. The curves are labeled in the legend by the pulseduration for that curve and as Ar for argon and D for deuterium. Thecurve labeled 100 fs, Ar-Circ is for a circularly polarized laser pulseof duration 100 fs. All other curves are for linearly polarized laserpulse. The laser intensity has been normalized to the square of theinitial diameter of the cluster. Note the dramatic intensity thresholdfor strong energy absorption for both argon and deuterium at lowerpulse durations. The intensity threshold is lowered as the pulselength is increased, and becomes less prominent for very long pulselengths.

FIG. 3. Phase space �x-vx� plots for electrons near the x axis forArgon cluster. The two plots correspond to peak laser intensity andpulse length of �a� 5�1015 W/cm2, 100 fs and �b� 5�1015 W/cm2, 250 fs. Note that Nres=1 for �a� and Nres=2 for �b�.Increase in the order of resonance leads to lowering of intensitythreshold.

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region, the laser field is strongly shielded. We note that thiscore region is roughly 40 nm in size, indicating that the clus-ter does not expand appreciably before the time of this phaseplot. The less dense bunch of electrons at higher magnitudesof velocity corresponds to energetic electrons in the range of2–10 keV, which have been pulled out from a thin layer atthe cluster surface by the laser electric field. The laser fieldaccelerates these electrons and passing through the clusterthey emerge on the other side in phase with the laser fieldand are further accelerated. The energetic electrons of Fig. 4take one full laser period to travel once around the cluster tocome back to the same position. This matching of electrontransit time to the laser period sets up the resonance condi-tion for irradiating laser pulse of 100 fs duration as describedby Taguchi et al. �17� and Antonsen et al. �18�.

Figure 3�b� plots the phase space for a laser pulse of250 fs FWHM with peak intensity 3�1015 W/cm2 at t=432.4 fs �the peak of the laser pulse occurs at 430.2 fs�. Asseen from Fig. 2, this corresponds to an intensity just beyondthe threshold for strong heating and corresponds to 5�1015 W/cm2 for the 100 fs pulse duration. Here, there aretwo tendrils of energetic electrons, and each bunch moveshalfway around the cluster in a laser period. Thus the ener-getic electron transit time for 250 fs case equals twice thelaser period.

Let the number of laser periods required by the hot elec-tron bunch to transit once around the origin of the phasespace be defined as Nres. Then Nres=1 for a 100 fs pulse, andNres=2 for a 250 fs pulse. In the following paragraph weshow that the electron transit time resonances with highervalues of Nres would lead to a lower intensity threshold.

We assume that the cluster has not expanded appreciablybefore the onset of strong heating. Phase space plots of Fig.3 show that this assumption holds reasonably well for thosecases. For enhanced cluster heating characterized by an elec-tron transit time resonance of order Nres, we can write thecondition for strong heating as

� 2D0

vcluster� = Nres�2�

�� , �2�

where D0 is the initial cluster diameter, cluster is the averagevelocity of an electron transiting the cluster, Nres is the orderof resonance at the time of strong heating, and � is fre-quency of the laser pulse. For the case of electron heating ata planar-solid interface considered by Brunel �25�, the peakvelocity of electrons driven into the planar surface was givenas plane=2eEs /me�, where Es is the laser field at the surface,and the factor of 2 is due to the effective field boost providedby the space charge. The space charge field acts opposite thelaser electric field during extraction, but adds to it when elec-trons are accelerated back into the bulk plasma. For our caseof laser heating in clusters for the 2D geometry, there is anadditional factor of 2 due to the dielectric focusing of theelectric field lines by the cylindrical cluster. The extractedelectrons, however, do not all have the same energy nor arethey at the same phase with respect to the laser field resultingin the typical velocity of electrons transiting the cluster beinglower than the peak velocity. We assume it to be half thepeak velocity, i.e., vcluster=0.5�2vplane�=vplane. Substitutingthis in Eq. �2� we get the scaled critical intensity for strongabsorption as

I0

D20

=c

8��me�

2

2�e�2 1

N2res

. �3�

The threshold intensity is inversely proportional to theorder of resonance as seen in Eq. �3�. Thus the intensitythreshold for 250 fs pulse duration �Nres=2� is lower thanthat for 100 fs case �Nres=1�. It is important to note that thecluster does undergo higher Nres stages even for the case of100 fs FWHM and peak intensity 5�1015 W/cm2 earlier inthe pulse �17�, but the cluster does not absorb significantenergy at these higher resonances. Because of the strongshielding effect of the core electrons, energetic electrons gainenergy only when they are outside the cluster. The higher theorder of the resonance, the smaller the fraction of time anelectron is heated. Consequently, for strong absorption athigher Nres the cluster needs to spend longer time at theresonance. This is possible only with longer pulse durationswhere the temporal variation of the laser electric field islower. For the 250 fs laser pulse, the cluster spends a signifi-cant time in a resonance stage characterized by Nres=2 andexperiences enhanced laser pulse energy absorption at thelower peak intensity of 3�1015 W/cm2. For 100 fs pulsewidth the laser pulse envelope varies more quickly and thepeak laser intensity needs to be as high as 4.7�1015 W/cm2 at which the cluster electrons gain energywith every laser cycle �Nres=1� and experience strongheating.

Figure 4 plots similar phase space plots for deuteriumclusters for a laser pulse of �a� 100 fs FWHM, 3�1015 W/cm2 peak intensity at t=204 fs, and �b� 70 fsFWHM, 5�1015 W/cm2 peak intensity at t=148.9 fs. Wenote that for the 100 fs case, there are two bunches of ener-getic electrons indicating that Nres=2, while Nres=1 for the

FIG. 4. Phase space �x-vx� for electrons near the x axis fordeuterium cluster for �a� 100 fs FWHM, 3�1015 W/cm2 peak in-tensity at t=204.8 fs and �b� 70 fs FWHM, 5�1015 W/cm2 peakintensity at t=148.9 fs. The plots show that Nres=2 for �a� andNres=1 for �b�.

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70 fs case. The intensity threshold is accordingly lower forlaser pulse width of 100 fs FWHM.

We see that the same threshold behavior is operative inboth argon �atomic mass=40� clusters and deuterium �atomicmass=2� clusters. For short pulses the threshold is deter-mined by the resonance condition with Nres=1 and for some-what longer pulses with Nres=2. If we compare the two heat-ing curves in Fig. 2 that correspond to Nres=1 resonance, viz.argon with pulse length 100 fs and deuterium with pulselength 70 fs, we see that they are quite similar, in spite of thefactor of 20 difference in the ion mass. The same can be saidof the two Nres=2 curves: Argon with pulse length 250 fsand deuterium with pulse length 100 fs. Thus, the main dif-ference between argon and deuterium clusters is that thetransition between Nres=1 and Nres=2 occurs for slightlylarger pulse durations in the argon case than in the deuteriumcase. Our simulations seem to indicate that energy absorptionby clusters is much more sensitive to changes in pulse dura-tion rather than scaling of ion mass.

Another feature observed in Fig. 2 is the decreased promi-nence of the intensity threshold for longer laser pulse dura-tions �500 fs and 1 ps runs�. As the pulse duration is in-creased the manner of energy absorption by the clusterchanges significantly. This is seen in Fig. 5 that plots �a� thetotal energy absorbed per cluster electron versus time and �b�root mean squared radius �RMS� of the cluster ions versustime for the peak laser intensities and pulse durations of �5�1015 W/cm2, 100 fs�, �3�1015 W/cm2, 250 fs�, and �5�1014 W/cm2, 1 ps�. These correspond to the intensities be-yond which the total energy absorbed saturates for the par-ticular pulse length. For the 100 fs and 250 fs pulse lengthsthese peak intensities are just above the threshold for strongheating. The plotted RMS radius of the cluster ions is nor-malized to the initial RMS value and is a measure of clusterexpansion. In either figure, time �horizontal axis� is normal-ized to the pulse length. The shape of the pulse envelope isshown as the thin dotted line in Fig. 5�a�. We note that for100 and 250 fs cases energy absorption by the cluster takesplace later in the pulse with fastest absorption taking placejust after the peak has passed. This is due to enhanced heat-ing with the onset of the transit time resonance. For the caseof 1 ps pulse length, however, the cluster starts absorbingenergy much earlier in the pulse when the intensity is low. Inaddition, Fig. 5�b� shows that the cluster starts expandingmuch earlier in the pulse for longer pulse lengths. Specifi-cally the RMS radius of cluster ions increases to roughly 20times the initial value for the case of 1 ps pulse length and toabout 1.5 times the initial value for 250 fs, while it hardlychanges for the shorted pulse length of 100 fs. In the follow-ing paragraph we look at electron phase space plots for the1 ps case.

Figure 6�a� shows the phase space plot at t=1720 fs �peakof pulse is at 1700 fs� for a cluster irradiated with a laserpulse of 1 ps FWHM and peak intensity 5�1014 W/cm2. Inthe phase space plot we can make out 9 distinct bunches ofenergetic electrons. In other words, Nres=9 for the conditionsof Fig. 6�a�. In addition, we note that the cluster core issignificantly enlarged. This is because the cluster absorbedenergy and expanded earlier in the pulse when the intensitywas lower. In Figs. 6�b� and 6�c�, we look at phase space

plots at lower peak intensities 1�1014 W/cm2 �7b� and 5�1013 W/cm2 �7c�. The basic characteristics of these phasespaces are similar to the higher intensity case of 5�1014 W/cm2 but show even higher orders of resonances ofNres=10, and Nres=16, respectively. The phase space studyshows that at pulse length of 1 ps, the cluster can spendmuch longer times at each resonance stage leading to signifi-cant energy absorption at very high orders of resonance andlower peak intensities. As the peak intensity increases it sup-ports lower orders of resonance and absorbs energy moreefficiently, finally leading to saturation in energy absorbedbeyond 5�1014 W/cm2. Absorption of energy at a widerange of Nres leads to gradual increase in energy absorbedwith increasing peak intensity as compared to the 100 fs casewhere there is a sharp threshold, marked by a transition tothe Nres=1 state.

Another feature of Fig. 2 is the variation in the saturationvalue of energy absorbed for varying pulse duration. This isdepicted explicitly in Fig. 7 where we plot the total energyabsorbed per cluster electron versus pulse duration for anargon cluster and peak laser intensity of 1�1016 W/cm2. Wenote that at this peak intensity the energy absorption curves�Fig. 2� for argon for all pulse durations have reached thesaturation value. Figure 7 shows that the saturation value of

FIG. 5. �a� Total energy absorbed per electron and �b� Rootmean squared radius of cluster ions for 100 fs, 5�1015 W/cm2

�solid�, 250 fs, 3�1015 W/cm2 �dashed�, and 1000 fs, 5�1014 W/cm2 �dash-dot�. For the case of 1 ps pulse length, thecluster absorbs energy and expands much earlier in the pulse whenthe electric field is much below the peak.

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absorbed energy increases with pulse duration till about500 fs but decreases for the much longer pulse length of1 ps. The cluster does not disintegrate fully for the shortestpulse lengths, and can thus absorb more energy as the pulselength is increased. However, as the pulse length is increasedto 1 ps the cluster expands and disintegrates earlier in thepulse. Thus, most of the pulse energy is unabsorbed as itpasses through the tenuous plasma formed by the clusterexplosion. Our simulations thus indicate a maximum satura-tion value for 500 fs pulse length �though this observation islimited by the small number of data points on this plot�. Thetrend in Fig. 7 is similar to the trend seen in experiments onstudies of energy absorbed by exploding Argon clusters �Fig.3 in the paper by Zweiback et al. �27��. The experimentalstudy by Zweiback �27� found the energy absorption to peakbetween pulse lengths 200 fs to 300 fs for Argon clustersvarying in diameter from 22 nm–33 nm. The peak intensitywas 1.9�1017 W/cm2 for a 50 fs pulse. Our simulations arelimited in their consideration of a single cluster in a laserfield while in an experiment absorption of laser energy in acluster jet would be affected by additional factors such as thesize distribution of clusters, number density and spatial dis-tribution of clusters and propagation of the laser pulse. How-ever, incorporation of these additional factors should not af-fect the qualitative trend that we observe in the energy

absorbed by clusters, and the experimental results �27� lendsqualitative confirmation to our model.

Finally in Fig. 2 we note that the intensity thresholdfor circularly polarized laser pulse is higher than that for thelinearly polarized pulse. Figure 2 shows that the intensitythreshold for circular polarization is just below 1�1016

W/cm2 while the corresponding point for the linear polariza-tion case is 5�1015 W/cm2. For the same peak laser inten-sity, the peak electric field in any one direction for circularlypolarized case is 1 /�2 times the linearly polarized field inthe polarization direction. Thus for the circularly polarizedpulse the threshold peak laser intensity needs to be twice thatfor linearly polarized case for the transit time resonance ofNres=1.

IV. ELECTRON AND ION ENERGY DISTRIBUTIONFUNCTIONS

Figure 8 plots �a� the electron kinetic energy spectra,Fe�E , t�, and �b� the ion kinetic energy spectra, Fi�E , t� for anirradiated argon cluster. The product �Fe,idE� represents thefraction of electrons �ions� in the interval dE at a given ki-netic energy E. The three spectra labeled �a–c� in each plotcorrespond to laser pulses of equal energy for the followingpulse width and peak intensity combinations: �a� 100 fs,1�1016 W/cm2; �b� 250 fs, 4�1015 W/cm2; �c� 1000 fs,1�1015 W/cm2. Each of these cases considered is above thethreshold for strong heating. A quasimonoenergetic peakcharacterizes the energy distribution of the emerging ions. Aselectrons are extracted from the cluster by the laser electricfield they form a halo of energetic electrons around the clus-ter. The ions gain energy as they are pulled out by this spacecharge around the cluster. Initially this leads to an ion energydistribution that is monotonic in energy and radial distance

FIG. 6. Phase space �x-vx� for electrons near the x-axis for argoncluster for 1 ps FWHM and peak intensity of �a� 5�1014 W/cm2,�b� 1�1014 W/cm2, and �c� 5�1013 W/cm2 at t=1720 fs. Theplots show that Nres=9, 15, and 19 for �a�, �b�, and �c�, respectively.

FIG. 7. Total energy absorbed per cluster electron �eV� versuspulse length �fs� for argon cluster and peak laser intensity of1�1016 W/cm2. This peak intensity is much beyond the strongheating threshold for all the pulse durations under consideration andthus the plotted energy absorbed represents the saturation value forthe corresponding pulse length. The trend in the energy absorbed asseen here is similar to that observed in Fig. 3 of the journal articleby Zweiback et al. �27�.

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with the highest energy ions also being at the largest radialdistance from the center of the cluster. As the ions and elec-trons expand, the electrostatic potential changes with time.This results in ions that have been accelerated later gainingmore energy and over taking the ions accelerated earlier. Thebeamlike distribution is associated with this overtaking pro-cess that leads to an increase in the population of ions atenergy slightly below the maximum energy.

Both electron and ion kinetic energy distributions show anincrease in the energy as the pulse width is increased from100 fs to 250 fs and then the energy falls as the pulse widthis further increased to 1 ps. This trend follows that of theaverage energy absorbed in the saturated regime depicted inFig. 3. The effect is more dramatic for ions with the peak ionenergy varying from about 800 keV for laser FWHM of100 fs �a� to 1400 keV for 250 fs FWHM �b� and finallyfalling to 250 keV for 1000 fs �c�. To explain this we notethat for pulse duration of 100 fs, the cluster does not disin-tegrate completely by the time the pulse has passed over it.Thus if the pulse duration is increased from 100 fs the clustercan absorb heat for a longer period of time before it disinte-grates. This leads to higher absorption of energy by the clus-ter electrons and the corresponding higher ion energies.However, as the pulse duration is increased further the clus-ter starts disintegrating earlier in the pulse. Once the clusterhas disintegrated and the electrons respond as if they werefree �that is, the self-field of the cluster is unimportant� heat-

ing stops. This leads to the fall in the peak electron and ionenergies as observed for the 1 ps case. This also explains thehigher saturation value of the total energy absorbed by thecluster for the 250 fs pulse duration as compared to the100 fs and 1 ps pulses.

As a cluster expands it eventually encounters particlesfrom neighboring clusters. The greater the number density ofclusters, the sooner this would happen. Generally when theexpanding clusters encounter one another the result is thatthe accelerating potential resulting due to the expandingelectrons is driven to zero and evolution of the distributionfunctions ceases. Using periodic boundary conditions in thesimulations allows for such intercluster ion and electroninteraction to take place. Figure 9 plots the ion energydistribution functions, f i�E , t� versus energy for a 38 nmdeuterium cluster. The curves labeled �a�–�e� are obtainedusing periodic boundary conditions peak laser intensities of1�1015 W/cm2, 3�1015 W/cm2, 1�1016 W/cm2, 5�1016 W/cm2, and 1�1017 W/cm2, respectively. As in thecase of argon clusters, we see an ion distribution functiondominated by high-energy ions for intensities above thethreshold for strong heating at 3�1015 W/cm2. The spec-trum for the below-threshold peak laser intensity of 1�1015 W/cm2, marked as �a�, does not have this character.The space charge field for such low laser intensities is notenough to accelerate a large number of ions to the boundaryof the electron cloud surrounding the cluster. For intensitiesmuch higher than the threshold intensity the distributionfunction actually increases with increasing energy, reaches amaximum and then finally falls off. Also, the maximum ionenergy is seen to increase monotonically with the peak laserintensity. In the following section we show how this pro-nounced dependence of the ion energy spectrum on the peaklaser intensity would impact the fusion yields for laser-irradiated clusters.

V. FUSION NEUTRON YIELD

Energetic ions from exploding deuterium clusters can un-dergo thermonuclear fusion producing neutrons as repre-sented by the reaction

FIG. 8. Electron �a� and ion �b� kinetic energy distribution func-tions for �a� 100 fs, 1�1016 W/cm2, �b� 250 fs, 4�1015 W/cm2,and �c� 1 ps, 1�1015 W/cm2. A quasimonoenergetic peak of ionsdominates the ion distribution function. The energy at which thispeak occurs is pulse duration dependent. For both electron and ions,the distribution functions show an increase in energy as the pulselength is increased from 100 fs to 250 fs, but in either case theenergy falls down as the pulse length is further increased to 1 ps.

FIG. 9. Ion kinetic energy distribution functions under periodicboundary conditions for Deuterium ions of initial diameter 38 nmirradiated with laser pulses of 100 fs duration and peak intensitiesof �a� 1�1015 W/cm2, �b� 3�1015 W/cm2, �c� 1�1016 W/cm2,�d� 5�1016 W/cm2, and �e� 1�1017 W/cm2.

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D + D ——→50%

He3�0.82 MeV� + n�2.45 MeV� . �4�

To calculate the rate of fusion reactions we assume that theions at all energies are distributed uniformly in space. This isa reasonable assumption for periodic boundary conditionstowards the end of the pulse when the energetic ions haveexpanded to beyond the simulation boundary. There is alsothe possibility of neutron production via two-step processsuch as

D + D ——→50%

T + p ,

D + T ——→50%

He4 + n , �5�

but for the parameter ranges under consideration the totalyield of products from the first reaction in Eq. �5� is not largeenough to necessitate the consideration of follow-up reac-tion. Hence we only consider Eq. �4� here for calculatingneutron yield. The number of fusion reaction per second perunit volume is then given as

R�t� =1

2nav

2 dE dE�f i�E,t�f i�E�,t���Erel�v − v���,

�6�

where nav is the average density of ions over the simulationregion, f i�E , t� is the kinetic energy distribution of ions givenby, is the reaction cross section, and v is the ion velocitysuch that miv2=2E, mi being the ion mass. The reactioncross section is expressed as a function of the relativeenergy of the interacting ions, Erel=miv−v�2 /2 where v andv� are the velocities of the interacting ions. The angledbrackets denote an averaging over all possible angles �, as-suming that the ion distribution is isotropic. Equation �7�below gives the reaction cross section �28� �in cm2�

�Erel� =A5 + � A4 − A3�Erel�� + 1�−1A2

�Erel��exp A4�Erel�−1/2� − 1�� 10−24, �7�

where �Erel� is in keV. The Duane coefficients Aj have thevalues A1=47.88, A2=482, A3=3.08�10−4, A4=1.177, andA5=0 for the particular reaction in Eq. �4�.

In order to calculate the yield of fusion neutrons we con-sider a cylindrical volume, V= �2�w0��2zr� where w0 is thespot size of the laser beam and zr is the Rayleigh lengthassociated with that spot size. We assume that the interactiontime for ions is limited by the average time required by thehigher energy ion to exit this volume. The total neutron yieldcan then be calculated as

NY =1

2nav

2VDavg dE dE�f i�E,t�f i�E�,t�

���Erel�v − v���

1

max�v, v��, �8�

where Davg is the average distance from the center to theedge of the volume under consideration.

Figure 10 plots reaction rate per unit volume at t=300 fs �solid� and total neutron yield �dotted� for fusion of

energetic ions from exploding laser-irradiated deuteriumclusters with pulse peak intensities ranging from 1�1015 W/cm2 to 1�1017 W/cm2. The threshold intensityfor this set of runs was at 3�1015 W/cm2. The plot shows asharp rise in the reaction rate as the intensity crosses thenonlinear resonance threshold. This is due to the increase inthe number of high-energy ions as the intensity crosses thethreshold. For intensities much above the threshold, the fu-sion rate and neutron yield both saturate as the cluster disin-tegrates earlier in the pulse.

It should be noted that in actual experiments effects suchsize distribution of clusters, absorption and self-focusing ofthe pulse, and spatial distribution of energetic ions wouldplay a significant role in determining the total neutron yield.Thus the neutron yield in an experiment might be differentfrom that based on the fusion reaction rate calculated here.

VI. CONCLUSIONS

We describe the effect of laser pulse width on resonantheating of laser-irradiated clusters. The clusters absorb heatvia a nonlinear resonance mechanism where electrons arefirst accelerated out from the cluster and then driven backinto it by the combined effects of the laser field and theelectrostatic field produced by the laser-driven charge sepa-ration. The energetic electrons then pass through the clusterand emerge on the other side. If they emerge in phase withthe laser field, there is resonant heating, and the clusterquickly absorbs energy. The onset of this “transit time” reso-nance depends on the ratio of laser intensity to cluster sizefor a given pulse duration. Our simulations of argon anddeuterium clusters show that as the pulse duration is in-creased the cluster absorbs energy at higher order resonanceswith the electron transit time equaling more than one laserperiod. This increase in the order of the resonance leads tothe lowering of the intensity threshold for strong heating. Asthe pulse length is increased further our simulations how thatthe threshold becomes less dramatic and almost disappearsfor 1 ps pulse length. This is because at increased pulselengths, even for low peak intensities the cluster can absorbsignificant energy due to longer times in high order resonantstates. The calculation for predicting intensity threshold for a

FIG. 10. Fusion reaction rate per unit volume �left vertical axis�and total neutron yield �right vertical axis� as a function of peaklaser intensity for deuterium clusters with D0=38 nm.

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given cluster diameter as derived in Ref. �17� has been gen-eralized to take the higher order resonances into account. Ourgeneralized formulation predicts intensity thresholds that areinversely proportional to the square of the order of resonanceduring strong heating. However, the predicted threshold in-tensities increasingly deviate from the values obtained fromsimulations as the pulse width is increased to 1000 fs. This isdue to cluster expansion earlier in the pulse for longer pulsewidths, an effect that was not taken into account for calcu-lating the predicted intensity thresholds.

Our results indicate a strong pulse duration dependence ofthe kinetic energy distribution of the cluster ions and elec-trons with the maximum ion energy going up from800 keV to 1.4 MeV as the pulse width is increased from100 fs to 250 fs, but then falling off to 250 keV as the pulsewidth is increased further to 1000 fs. The pulse energy was

kept constant for this set of runs. This trend in the maximumenergy of ions mimics the trend in saturation value of energyabsorbed the cluster for different pulse lengths and is consis-tent with the experimental results by Zweiback et al. of laserpulse energy absorption by argon clusters for varying pulselengths �27�. This is of consequence for possible applicationsin laser-based accelerators and for fusion neutron productionfrom clusters.

We have investigated the fusion reaction rate and neutronyield for deuterium clusters for a 100 fs FWHM pulse ofpeak intensities ranging from 1015–1017 W/cm2. We findthat there is a dramatic increase in the reaction rate and neu-tron yield as the intensity crosses the threshold for strongheating at 3�1015 W/cm2. The reaction rate saturates be-yond the peak intensity of 5�1015 W/cm2 as the intensity isincreased to 1�1017 W/cm2.

�1� O. F. Hagena and W. Obert, J. Chem. Phys. 56, 1793 �1972�.�2� T. Ditmire, R. A. Smith, J. W. G. Tisch, and M. H. R. Hutch-

inson, Phys. Rev. Lett. 78, 3121 �1997�.�3� K. Y. Kim, I. Alexeev, V. Kumarappan et al., Phys. Plasmas

11�5�, 2882 �2004�.�4� Y. L. Shao, T. Ditmire, J. W. G. Tisch, E. Springate, J. P.

Marangos, and M. H. R. Hutchinson, Phys. Rev. Lett. 77,3343 �1996�; V. Kumarappan, M. Krishnamurthy, and D.Mathur, ibid. 87, 085005 �2001�.

�5� T. Ditmire, J. Zweiback, V. P. Yanovsky et al., Nature �Lon-don� 398, 489 �1999�.

�6� K. W. Madison, P. K. Patel, M. Allen et al., J. Opt. Soc. Am. B20, 113 �2003�.

�7� T. D. Donnelly, T. Ditmire, K. Neuman, M. D. Perry, and R.W. Falcone, Phys. Rev. Lett. 76, 2472 �1996�.

�8� I. Alexeev, T. M. Antonsen, K. Y. Kim, and H. M. Milchberg,Phys. Rev. Lett. 90, 103402 �2003�.

�9� T. Ditmire, R. A. Smith, and M. H. R. Hutchinson, Opt. Lett.23, 322 �1998�; V. Kumarappan, K. Y. Kim, and H. M. Milch-berg, Phys. Rev. Lett. 94, 205004 �2005�.

�10� V. Kumarappan, M. Krishnamurthy, D. Mathur, and L. C.Tribedi, Phys. Rev. A 63, 023203 �2001�; E. Springate, N.Hay, J. W. G. Tisch, M. B. Mason, T. Ditmire, M. H. R. Hutch-inson, and J. P. Marangos, ibid. 61, 063201 �2000�; T. Ditmire,R. A. Smith, R. S. Marjoribanks et al., Appl. Phys. Lett. 71,166 �1997�.

�11� K. W. Madison, P. K. Patel, D. Price et al., Phys. Plasmas 11,270 �2004�.

�12� K. W. Madison, P. K. Patel, M. Allen, D. Price, R. Fitzpatrick,and T. Ditmire, Phys. Rev. A 70, 053201 �2004�.

�13� E. Parra, I. Alexeev, J. Fan, K. Kim, S. J. McNaught, and H.M. Milchberg, Phys. Rev. E 62, R5931 �2000�.

�14� B. N. Breizman, A. V. Arefiev, and M. V. Fomyts’kyi, Phys.Plasmas 12, 056706 �2005�.

�15� F. Greschik, L. Arndt, and H. J. Kull, Europhys. Lett. 72, 376�2005�.

�16� H. M. Milchberg, S. J. McNaught, and E. Parra, Phys. Rev. E64, 056402 �2001�.

�17� T. Taguchi, T. M. Antonsen, and H. M. Milchberg, Phys. Rev.Lett. 92, 205003 �2004�.

�18� T. M. Antonsen, T. Taguchi, A. Gupta et al., Phys. Plasmas 12,056703 �2005�.

�19� Y. Kishimoto, T. Masaki, and T. Tajima, Phys. Plasmas 9, 589�2002�.

�20� J. Zweiback, T. Ditmire, and M. D. Perry, Opt. Express 6, 236�2000�.

�21� M. Schnurer, S. Ter-Avetisyan, H. Stiel et al., Eur. Phys. J. D14, 331 �2001�.

�22� K. Ishikawa and T. Blenski, Phys. Rev. A 62, 063204 �2000�;G. M. Petrov, J. Davis, A. L. Velikovich, P. Kepple, A. Das-gupta, R. W. Clark, A. B. Borisov, K. Boyer, and C. K.Rhodes, Phys. Rev. E 71, 036411 �2005�; G. M. Petrov, J.Davis, A. L. Velikovich et al., Phys. Plasmas 12, 063103�2005�; M. Eloy, R. Azambuja, J. T. Mendonca et al.,Phys. Scr.T89, 60 �2001�.

�23� T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Falcone, andM. D. Perry, Phys. Rev. A 53, 3379 �1996�.

�24� F. Greschik and H. Kull, Laser Part. Beams 22, 137 �2004�.�25� F. Brunel, Phys. Rev. Lett. 59, 52 �1987�.�26� J. Zweiback, T. E. Cowan, J. H. Hartley et al., Phys. Plasmas

9, 3108 �2002�; J. Zweiback, R. A. Smith, T. E. Cowan, G.Hays, K. B. Wharton, V. P. Yanovsky, and T. Ditmire, Phys.Rev. Lett. 84, 2634 �2000�.

�27� J. Zweiback, T. Ditmire, and M. D. Perry, Phys. Rev. A 59�5�,R3166 �1999�.

�28� G. H. Miley, H. Towner, and N. Ivich, 1974; B. H. Duane,1972.

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