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PHYSICS OF PLASMAS VOLUME 9, NUMBER 7 JULY 2002

Detailed study of nuclear fusion from femtosecond laser-driven explosionsof deuterium clusters

J. Zweiback,a) T. E. Cowan,a) J. H. Hartley, R. Howell, K. B. Wharton,b) J. K. Crane,V. P. Yanovsky,c) and G. HaysLawrence Livermore National Laboratory, L-477, Livermore, California 94550

R. A. SmithImperial College of Science Technology and Medicine, London SW7 2BZ, United Kingdom

T. DitmireLawrence Livermore National Laboratory, L-477, Livermore, California 94550and Department of Physics, University of Texas, Austin, Texas 78712

~Received 26 November 2001; accepted 25 April 2002!

Recent experiments on the interaction of intense, ultrafast pulses with large van der Waals bondedclusters have shown that these clusters can explode with sufficient kinetic energy to drive nuclearfusion. Irradiating deuterium clusters with a 35 fs laser pulse, it is found that the fusion neutron yieldis strongly dependent on such factors as cluster size, laser focal geometry, and deuterium gas jetparameters. Neutron yield is shown to be limited by laser propagation effects as the pulse traversesthe gas plume. From the experiments it is possible to get a detailed understanding of how the laserdeposits its energy and heats the deuterium cluster plasma. The experiments are compared withsimulations. ©2002 American Institute of Physics.@DOI: 10.1063/1.1487382#

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I. INTRODUCTION

A number of experiments have been conductedrecent years examining the interactions of intense femtoond laser pulses with large van der Waals bondclusters.1–15 These experiments have been accompaniedrange of theoretical models.4,16–23 Most of this work hasshown that these interactions can be very energetic, wivariety of experimental manifestations. Very bright x-remissions in the 100–5000 eV range have beobserved;2,3,8,15x-ray emission that was absent when a moatomic gas is irradiated under similar conditions. Absorptexperiments demonstrated that nearly 100% of an intelaser pulse could be absorbed by a modest average degas jet (;1019cm23) within a few millimeters propagationlength when clusters were present in the gas.24,25 This laserabsorption measurement indicated that many kiloelecvolts of energy per atom were being deposited in the clusing gas. The large absorption arises from the fact thatdensity within the clusters is high, nearly that of a solid, acan therefore exhibit the large absorption efficiency usuassociated with solid targets.4

The mechanisms and dynamics of energy depositionfemtosecond pulses in individual clusters have also beenamined by a number of groups. For example, it has bfound that very high charge states are formed in the irration of highZ noble gas clusters.2,5,26This high ionization of

a!Present address: General Atomics, 10240 Flanders Court, San Diego92121.

b!Present address: Department of Physics, San Jose State UniversityJose, CA.

c!Present address: Center for Ultrafast Optical Science, University of Migan, Ann Arbor, MI.

3101070-664X/2002/9(7)/3108/13/$19.00

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the cluster atoms is the result of a number of factors incling enhanced tunnel ionization by charge resonant enhanionization, or so called ‘‘ionization ignition’’16 as well astraditional electron collisional ionization in the laser heatcluster.4 Pump–probe and varying pulse-width experimehave been conducted which show that the clusters disemble on a picosecond time scale.12,13 Most interesting,however, has been the publication by a number of groupsthe fast particles ejected from the explosions of these clusupon intense irradiation. Photoelectron energy measuremindicated that multi-kiloelectron volt electrons are ejectfrom large ~.1000 atom! Xe clusters.27 Perhaps most remarkable has been the discovery that these large cluswhen irradiated at intensity above 1015W/cm2, also ejections with substantial kinetic energy, a fact confirmed bynumber of research groups world wide in recent years.1,7,11,28

This energetic cluster explosion is the result of rapelectron heating within the solid density cluster by the lapulse. In large highZ clusters, space charge forces confithe electrons to the charged ion sphere and a microplasmcreated by field ionization. The laser can heat this electcloud which then drives a rapid expansion of the clusterthe creation of a strong ambipolar electric field. In smalclusters, or lowerZ clusters, such as hydrogen or deuteriuthe laser field is strong enough to remove field ionized eltrons directly from the cluster. In this case, the clusterplodes by the Coulomb repulsion of the closely spaced cter ions. In either case, the accelerating fields in the clucause them to explode ejecting ions with substantial kinenergy. It has been shown that this large release of kinenergy in fast ions can be harnessed to drive nuclear fubetween deuterium ions if deuterium clusters are irradiain a gas of sufficient average density to permit collision b

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3109Phys. Plasmas, Vol. 9, No. 7, July 2002 Detailed study of nuclear fusion . . .

tween ions ejected from different clusters in the gas.29 Inthese experiments, a high intensity, ultrafast laser pulsfocused into a gas jet of deuterium clusters, rapidly ionizthe inertially confined clusters before they can expand. Thclusters subsequently explode, ejecting energetic ions.process creates a plasma filament with a diameter routhat of the laser focus~;100 mm! and a length comparablto the extent of the gas jet plume~;2 mm!. The fast deute-rium ions ejected from the exploding clusters can then clide with ions ejected from other clusters in the plasma. Ifion energy is high enough~greater than a few tens of kiloelectron volts!, nuclear fusion events can occur with higprobability, releasing a characteristic 2.45 MeV neutron frthe fusion reaction, D1D→He31n.30

These phenomena represent an interesting possource of pulsed, subnanosecond fast neutrons which cbe used in neutron damage pump probe experiments.31 Be-cause of the need for much higher fusion yields to makeapplication possible, it is desirable to glean a detailed unstanding of these phenomena. Previous work has shsome data on yield scaling with cluster size, along withunexpected limit on neutron yield.32 A better understandingof the factors affecting and limiting the fusion yieldneeded to enable scaling of this source to much higher ntron fluences than demonstrated to date. In this paperpresent a detailed experimental study of the exploding cter physics, neutron yield, and laser energy deposition inclustering gas jet. We find that the fusion neutron yieldstrongly dependent on such factors as cluster size, lasergeometry, and deuterium gas jet parameters. We alsothat energy depletion of the laser pulse as it propagthrough the absorbing cluster gas has a dramatic effecneutron yield and is the ultimate limitation on the fusioyield in our experiments. Our results are consistent witseries of straightforward simulations.

II. SIMPLE MODEL FOR EXPLODING DEUTERIUMCLUSTER FUSION YIELDS

To aid in interpreting the experimental results, we hadeveloped some simple models. While modeling of all dtails of the interaction is extremely complex, we have fouthat straightforward scalings can be ascertained from ssimple assumptions about the laser cluster interactionsfusion process in the deuterium gas.

To model the exploding cluster interaction, it is impotant to assume something about the shape of the ion strum resulting from the explosion of the clusters. The fuscross section of DD fusion is very strongly dependent onenergy, and only exhibits a significant value if the ions ehibit energies of a few to tens of kiloelectron volts. It hbeen shown by a number of groups that large, highZ, ex-ploding clusters exhibit ion spectra that are similar to thata hydrodynamically expanding gas.11,33 This hydrodynamicexplosion, however, is only appropriate when the ion chaof the cluster is very high and a large number of the electrare retained in the cluster via space charge forces. Becdeuterium cannot be ionized to high charge state it is lik


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that for reasonably sized deuterium clusters~thousands ofatoms or less! the laser field will remove most or all of thelectrons during the irradiation of the cluster. This resultsa pure Coulomb explosion, akin to the Coulomb explosionsmall molecules.

The intensity required to drive a pure Coulomb expsion is not easily calculated analytically; it depends ondynamics of the driven electron cloud in the cluster. Theare a few such simulations in the literature.20 These simula-tions suggest that the cluster will be fully stripped of eletrons by the laser field if the laser ponderomotive potentiacomparable to or larger than the surface potential ofcharged cluster. The energy of the Coulomb explosion wbe maximized if the laser field satisfies this condition. If tponderomotive potential is not high enough, the cluster wbe stripped of some fraction of its electrons, raising the sface potential to a value near the ponderomotive potentia

The energy of the Coulomb explosion will be maximuif all electrons are stripped from the cluster on a time scfaster than that of the expansion of the cluster. Full remoof electrons prior to expansion leads to the largest stopotential energy in the cluster and the largest ion energThis fact mandates that the rise time of the laser pulse~to thevalue of the stripping ponderomotive potential! be faster thansome value. This characteristic explosion time can be emated by calculating the time required for a charged clusto expand to twice its initial radius,a. Integration of themotion of a charged deuterium cluster yields for this charteristic explosion time:

tCoul'0.8A4pe0mD

nDe2 , ~1!

wheremD is the mass of a deuteron andnD is the density ofdeuterium atoms in the cluster, roughly 331022cm23 forsolid D2 . Note thattCoul is independent of cluster radius anequals about 20 fs for deuterium clusters. This expanstime assumed an acceleration of a fully charged clustertherefore represents a pessimistic limit on the necessaryrise time. Nonetheless, Eq.~1! seems to suggest that our 3fs pulses are marginally short enough to drive a Couloexplosion at solid density.

If a Coulomb explosion is driven, and it can be assumthat all electrons are removed prior to any ion movement,ion energy spectrum from a single exploding cluster, denof sc(E), can be stated as~when normalized!

f sc~E!5H 32Emax

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whereEmax is the maximum energy of ions ejected and coresponds to those ions at the surface of the cluster.Emax issimply

Emax5e2nDa2

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In deuterium clusters,Emax is about 2.5 keV for 5 nm clus-ters. Equation~3! illustrates that larger clusters are better f


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3110 Phys. Plasmas, Vol. 9, No. 7, July 2002 Zweiback et al.

driving fusion because of the strong dependence of csection with energy and the quadratic dependence of ionergy with cluster radius.

It should be noted that the single cluster ion energy dtribution like that of Eq.~2! is not ideal for fusion. It is wellknown that in Maxwellian fusion plasmas of temperatuless than about 10 keV, the majority of the fusion occbetween the small fraction of ions in the hot tail of the Mawellian, owing to the highly nonlinear increase in fusiocross section with ion energy in the 1–100 keV energy ranThe Coulomb explosion ion spectrum has no hot tail, afor equal average ion energies, has a dearth of hot ioncompared to a Maxwellian distribution. This is illustratedFig. 1 where spectra for a Coulomb explosion and a Mwellian are compared, both with 9 keV average ion ene~T156 keV for the Maxwellian andEmax515 keV for theCoulomb explosion!.

In reality, the actual ion energy spectrum will be broaened by any distribution in cluster sizes in the target. Inexperiments, the clusters are produced from a cooled gaand it is well known that clusters formed from such jeexhibit a broad size distribution. This distribution in sizedenotedg(a), can be approximated as

g~a!5bae2~a2a0!2/d2, ~4!

whereb is a normalization factor, andd is a factor charac-terizing the width of the cluster size distribution. In this caassuming that all clusters are completely ionized by the lathe full ion energy distribution can be found by convolvinthe Coulomb explosion distribution of Eq.~2! and the clustersize distribution of Eq.~4!:

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FIG. 1. Ion spectra for a cluster Coulomb explosion and a Maxwellian, ewith a 9 keVaverage ion energy.


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Examples of this ion distribution with average ion energy9 keV are also shown in Fig. 1, where this correspondsa056 nm andd56 nm This comparison shows that the covolved ion distribution closely approximates a Maxwellian

The approximate ion distribution of Eq.~5! is not com-pletely valid in modest laser intensity, If the laser intensitynot sufficient to fully strip the largest clusters, this will leato a cutoff in ion spectrum at high energy. The exact natof this cutoff requires detailed calculations of the laser clter interactions. It is likely, however, that this effect canapproximated by cutting the ion spectrum at an ion enenear Up , the laser peak ponderomotive potential. Thisequivalent to saying that the laser strips the large clusterpotentials ofUp instead ofEmax.

A calculation of the total fusion yield requires a calcultion of all fast ion trajectories in the plasma. The ion mefree path for multi-kiloelectron volt deuterons in gas denties like that used in our experiments (;1019cm23) is of theorder of 2–5 mm. This distance is comparable to or lonthan the characteristic sizes of the plasma filaments crein our experiment. Therefore, it is not appropriate to uspure hydrodynamics calculation of the ion density and fusburn evolution~though such simulations do provide impotant insight!. Instead it is a good approximation to assumthat all ions free steam from the interaction volume afterclusters explode and undergo fusion events as they travthe plasma filament. In this case, the fusion yield will scvery roughly as

Y;ngas2 lV^s&, ~6!

wherengasis the average ion density in the gas jet target,l issome characteristic escape distance~which is more or lessthe dimension of plasma filament!, V is the heated volume ogas, and^s& is some appropriately weighted fusion crosection. This free streaming will occur on a time scale mulonger than the explosions of the clusters. The clustersplode on a 100 fs time scale while the ions free stream acthe plasma on a time scalel /n ion;100 ps for ion velocitiesand plasma sizes of relevance in our experiments. Thusappropriate to think of the cluster explosion and subsequfusion as decoupled events.

To calculates in our simulations, we use the publishescaling values ofs(Etot) for beam target fusion~ion hittingstationary ion! and setting

Etot5E11E212AE1E2 cosu, ~7!

whereu is the angle of incidence between two ions. Thisappropriate for nonrelativistic ions.

We have previously performed calculations of DD fusiyield in our experiment assuming that all ions free streand that the heated plasma is a cylinder, with diameter giby that of the laser focal spot size. These results were plished previously.31 However, this simulation approach wafound cumbersome. It is possible to make additional assutions about the yield. If the cavitation of the plasma filameby escaping ions is ignored, the yield can be calculated fra relation like that of Eq.~6!,

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In the first integral, we find the average escape path lengnumber which will depend on filament shape and aspecttio. In the second integral, we find the weighted fusion crsection averaged over the ion energy distribution of test i(E1) and target ions (E2). The factor of 1/2 accounts fodouble counting of ions. The fusion cross section is aaveraged over all potential incident angles. We have numcally calculated the average ion escape trajectory distancassuming plasma cylinders with initially uniform ion densand isotropic ion velocities and found that this value is eqto between 0.8 and 0.91 times the cylinder radius over a wrange of cylinder aspect ratios~radius/length between 3 an50!. Equation~8! leads to estimates for the yield that awithin 40% of the full ion trajectory calculation of Ref. 31Equation~8! allows an easy calculation of yield scaling wiparameters such as cluster size and laser energy fluencetails of such calculations are presented in Sec. IV.

III. APPARATUS, D 2 CLUSTER TARGET, ANDEXPERIMENTAL CHARACTERIZATION OFTHE INTERACTION

The layout of our experiment with the suite of diagnotics is illustrated in Fig. 2. Our experiment utilized a 10 HTi:sapphire, chirped pulse amplification laser delivering 1mJ of laser energy per pulse with 35 fs pulse width awavelength of 820 nm. This laser was focused into the eof a deuterium gas jet with anf /12 lens. We used a sonic gajet with an orifice of 500mm. This jet produced a broaplume as the gas expanded into the chamber. As theopened the pressure in the chamber would increase untijet closed. Because the van der Waals forces between drium molecules are weak, the gas jet was cryogeniccooled by flowing cooled nitrogen or helium through a jacksurrounding the gas jet body. This encourages cluster fortion. This jacket is similar to that described in Ref. 34. Tcooled jet produced large clusters in the D2 gas. We wereable to cool the jet body to 100 K with liquid nitrogen, and,80 K using liquid helium. This temperature was measuwith a thermocouple mounted near the jet nozzle. We emate that the actual gas temperature, particularly at thercouple readings,100 K was lower than that registered.

The laser spot size within the gas jet varied from roug40 to 200mm depending on the focal spot position, wirespect to the gas jet nozzle. Consequently the peak lintensity in vacuum~at 120 mJ! was between 231016 and431017W/cm2. The laser propagation axis varied from 0mm to roughly 2 mm below the output of the gas jet nozz

To estimate the deuterium cluster size in our gas jet,conducted Rayleigh scattering measurements in the jet35 bypassing low energy~,0.1 mJ! 532 nm pulses from a frequency doubled,Q-switched Nd:YAG laser~pulse width 10ns! through the plume of the gas jet. The Rayleigh scatte


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light was imaged 90° from the laser propagation axis~seeFig. 2!. From the observed onset of Rayleigh scattering,can determine the onset of large cluster formation in the

Using independent measurements of the averagedensity, estimates for the throughput and detection sensity, and the Rayleigh scatter cross section, we are ableestimate the average cluster size in the jet. Estimated clusize as a function of temperature with a 70 atm backpressure is illustrated in Fig. 3. This measurement, basescattered light intensity, yields only a very rough estimatethe average cluster size. It is well known that clustering froan expanding jet results in clusters with a rather broad sdistribution. Our scattering measurement gives us no diinformation on this size spread. From Fig. 3 it is seen thatcan roughly tune the average cluster size~by changing thegas temperature! from clusters of about 10 Å to over 100 Å

The amount of laser energy absorbed in the gas jetmonitored in our experiments. To do this, we measuredtransmitted laser energy collected within anf /3 cone~to col-lect all potentially refracted light from thef /12 cone! with alarge aperture lens and calorimeter. We monitored backstered light with a photodiode~very little was observed inthese experiments!. We also estimated scattered light usingcharge coopted device~CCD! camera and collection lenmonitoring light scattered 90° from the laser propagationrection perpendicular to the laser polarization direction~thedirection of maximum Rayleigh scatter!. Extrapolating into4p we determined that the scattered light was negligible~,1mJ!.

The measured rise in energy absorption as the gastemperature is decreased is illustrated in Fig. 4. As canseen from a comparison of Figs. 3 and 4, the absorpcorrelates quite closely to the growth of cluster diameterthe jet. This absorption difference cannot be explained

FIG. 2. Layout of our experiment with the suite of diagnostics.


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average density change alone and can only be explainecluster formation. Over the range studied here, the gassity changed by approximately a factor of 2 as the jetcooled from room temperature to 100 K, however, absotion increases by a factor of 20.

From interferometry measurements, described in Secwe can make a reasonable estimate for the total energyposited into deuterium ions based on absorption efficienThis analysis36 suggests that roughly 5 keV of energy adeposited per deuterium atom in the plasma. To ascertainion energies more accurately, we conducted ion energy msurement via ion time of flight~TOF!. We installed a flighttube on the interaction chamber and detected ions ejefrom the plasma with a microchannel plate detector. Flidistances of 1.4 and 2.5 m were utilized. The flight timeions ejected from the plasma filament in the gas jet wrespect to the laser irradiation time are measured. This tnique is not an entirely accurate method of measuringenergies given that the ions must traverse a small amoununionized gas~;1 mm! prior to exiting the jet. We estimatethat the mean free path of the kiloelectron volt ions in theis ;1 cm, so the presence of unionized gas will have soaffect on the measurement, particularly for the lower eneions. The measurement is also complicated by space ch

FIG. 3. Estimated cluster size as a function of temperature with a 70backing pressure based on Rayleigh scatter data.

FIG. 4. Laser absorption as a function of gas jet reservoir temperatu


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forces around the high density plasma filament. Nonethelthis measurement yields a rough estimate for the ion enerproduced by the exploding clusters. External magnets winstalled on the flight tube to deflect low energy electrons

TOF spectra for the two flight tube distances are illutrated in Fig. 5~a!. An early time spike in the signal is observed from a flash of fast electrons and UV radiation frothe plasma. The later time signal moves later with longflight tube, confirming that the signal indeed arises from ioand not plasma radiation. An energy spectrum derived frthe 2.5 m flight data is displayed in Fig. 5~b!. This spectrumexhibits deuterons in a broad energy spectrum with enerextending out to nearly 30 keV. The average energy of iofound by averaging over this spectrum is 7 keV.

To detect the production of DD fusion neutrons in theexperiments we employed arrays of neutron sensitive scilators in various configurations. The layout of our detectis shown in Fig. 2. Two classes ofn TOF detectors weredeployed. Detectors with large scintillator pieces were uto measure neutron yield. These detectors had high neudetection efficiency~near unity!. The initial detector of thiskind was a grouping of three plastic scintillator cylinders,cm in diameter and 10 cm long, coupled to two photomuplier tubes~PMTs!. These detectors were located outsidevacuum interaction chamber and were shielded with 6 mmlead. Because of the small solid angle subtended, theseused in a single neutron counting mode. The second clasTOF detectors were thin scintillators closely coupledPMTs, designed for high time resolution.31 These detectorshad subnanosecond response and could be used to exathe time structure of the neutron pulse.

With these neutron detectors we detected the presencsubstantial numbers of 2.45 MeV neutrons when the gasbacked with cooled deuterium.29 We observed no neutron

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FIG. 5. ~a! Time of flight ion spectra for two flight tube distances.~b!Energy spectrum derived from the 2.5 m flight data.


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production with temperatures higher than 150 K, the teperature at which we no longer observe large cluster fortion via Rayleigh scattering.

We were able to estimate the total fusion yield from tcount rates on the large area neutron detectors and assunear unity detection efficiency in the scintillators~an as-sumption that ensures the most pessimistic estimateyield!. The total yield of fusion neutrons varied with the lasand gas jet conditions. Detailed studies of the fusion yiwill be presented in Sec. IV. However, we found that thighest yields attained tended to be around;23104 neu-trons per shot.

Using the thin, high time resolution, neutron detectowe were able to measure the duration of the neutron pulsseveral distances.31 Since the neutron yield is nonlinearldependent on the density@Eq. ~6!#, one would expect thefusion yield to only be significant before the plasma filamehas had time to expand. The density of a multi-kiloelectrvolt deuterium plasma filament with a diameter on the orof the laser focus will rapidly decrease to the point whefusion yield would be negligible. Our models showed ththis time should be less than 1 ns. Figure 6 shows examof the neutron pulse measured at several distances. Thefit was calculated using a Gaussian pulse convolved withtime response of the detector. The increasing pulse wwith distance comes about from Doppler broadening. Etrapolating to zero distance, one finds that initial pulse;500 ps.31 How rapidly the pulse broadens as one movaway from the source gives information on the temperatof the plasma.37 From this, the mean ion temperature is esmated to be about 8 keV, which is similar to the ion TOdata previously discussed.

IV. EXPERIMENTAL MEASUREMENTS OF FUSIONYIELD SCALING

To derive a greater understanding of the nature offusion neutron production in this interaction, we have exained the fusion yield as a function of a wider range of laparameters. These studies were conducted in a numbeways. The primary diagnostic for measuring the fusion yiwas an additional large plastic scintillator detector, placnear the plasma in a reentrant flange to subtend a largeangle. It subtended a solid angle of 0.3 sr and was placecm from the deuterium plasma. Because the time-of-flidata indicate that there is no noticeable x-ray flash andvirtually all of the detected particles are neutrons, we cobtain a shot by shot measure of the fusion yield fromPMT signal. The detection efficiency was calibrated agathe other neutron TOF detectors operating in single partdetection mode. We also calibrated the yield by measuthe average pulse height produced when single neutstruck the detector. Both methods yielded nearly equal vafor the yield calibration.

As discussed in Sec. II, the Coulomb explosion modethe fast deuterium ion production predicts that a strongpendence of fusion yield should exist with increasing avage cluster size. The data of Sec. III indicate that we


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control the average cluster sizes from about 10 to 100 Åvarying the gas jet reservoir temperature. The fusion yimeasured through this technique as a function of clusteris illustrated in Fig. 7. Each data point represents the integtion of five laser shots. The shot to shot spread in the dwas substantial~.50%! owing to the counting statistics fromthe detection of a limited number~,100! of neutrons pershot. This plot represents two data sets, plotted as circlessquares. The circle data points were taken with the lafocus approximately 1 mm before the center of the gasnozzle. This focal point was chosen to optimize the yield ajet temperature of 100 K. The square points were taken wthe laser focus slightly further away from the nozzle cenAs expected, the fusion yield rises rapidly from about 1neutrons per shot with average cluster size of 30 Å and peat around 104 n/shot when the average size in the jet reach55 Å. At higher cluster sizes, corresponding to colder gastemperature, the yield rolls over and falls off. Using Eq.~8!,we have calculated the expected fusion yield as a functiocluster size and compared these predictions with the dThese calculations are shown in Fig. 7 as solid lines. Wethe ion energy distributions of Eq.~5! and have varied thewidth of the cluster size distribution, assuming that the widparameter,d, was equal to a constant factor times the avera

FIG. 6. Neutron pulse width data from the 12312 cm detectors at a varietyof distances.


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cluster diameter. Cluster size distribution widths of 0.7,1.6, and 2.2 times the average cluster diameter are ploThis comparison shows that the data, up to the peak in yat 55 Å, exhibit a yield increase that follows the calculatioquite well, with the best fit being for the calculation coducted with cluster size distribution width equal to the cluter size. This comparison appears to indicate that the trpredicted by the Coulomb explosion model is correct.

The model calculations indicate that the fusion yield wcontinue to grow as the average cluster size increases be55 Å. The rollover seen at higher average cluster size seethe data is inconsistent with this trend. Clearly, our simmodel will break down when the cluster sizes become laenough that the laser field can no longer completely stripclusters, driving the Coulomb explosion. The rollover wobserved, however, is likely the result of more complicaeffects. The simple model of Sec. II does not account forfact that the real experiment consists of a laser pulse progating into a spatially broad clustering medium. This clead to refraction and absorption of the laser prior toreaching the center of the gas jet plume. We have examthese effects in detail through interferometric imaging oflaser plasma in the jet. These experiments will be descriand analyzed in Sec. V.

To gain a greater insight into the fusion production whave also examined yield as a function of a variety of otparameters. The measured fusion yield as a function ofjet backing pressure is illustrated in Fig. 8. Here the gaswas held at a temperature of 100 K, the point of peak yiseen in Fig. 7. The incident laser energy was 120 mJ.neutrons are observed at backing pressure below 23 atmyield then increases rapidly at pressure above 23 atm.correlates to the point at which large clusters are observethe Rayleigh scattering. These data were taken with the lfocus~measured in vacuum! 1.3 mm before the center of thgas jet.

An important trend observed in our experiments was t

FIG. 7. Measured and calculated fusion yield as a function of cluster s


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the fusion yield was very sensitive to the position of tfocus with respect to the center of the gas jet. We found tthe optimum yield occurred when the laser was focusedfore the center of the gas jet. This effect is illustrated in F9, which shows fusion yield measured as a function of lafocal position with respect to the center of the gas jet fovariety of different gas jet backing pressures. In these dthe laser energy was 50 mJ and the gas jet temperature100 K. First, we see that, at the highest backing pressureyield peaks when the laser is focused;0.75 mm before thegas jet, even through the highest gas density is at jet ceThe yield has dropped by;20% when the laser is focuseright at the center of the gas jet.

Also important to note is the fact that the peak yieposition moves closer to the gas jet center as the bacpressure of the jet is increased. For the lower pressures~,40atm! though the yield is quite low, it peaks when the lasfocus is far from the jet center. The strong dependenceoptimal position with gas jet pressure suggests that this efis the result of some variation in the laser propagatthrough the gas. It should be noted that the observed tren

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FIG. 8. Measured fusion yield as a function of gas jet backing pressu

FIG. 9. Fusion yield measured as a function of laser focal position wrespect to the center of the gas jet for a variety of different gas jet bacpressures.


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counter to a simple explanation based on ionization indurefraction. Refraction will tend to increase the laser size apasses into the jet, so focusing before the jet center wonot counteract this effect.

Similar behavior is observed when the distance betwthe gas jet nozzle and the laser propagation axis are vaData illustrating this are shown in Fig. 10. Because ofdiverging cone of gas, increasing this nozzle–laser distawill decrease the average density and will broaden the spextent of the gas through which the laser propagates. Wethat the yield peak with focal position variation shifts towathe center of the jet as the distance between the nozzlelaser axis decreases.

Data showing laser focalz scans at different incidenlaser energies are shown in Fig. 11. The optimal focal ption, as the laser energy is varied, does not appear to mThe peak yield more or less rests at the same point aslaser energy is decreased.

From these data we can determine the yield scalingfunction of energy. The simple model described in Secsuggests that the yield should increase at least as fast a3/2 power of the laser energy. This scaling results if the laintensity is kept constant and the spot size is allowedincrease as the laser energy grows. Thus, constant inteyields constant ion energy, so the growth in yield resufrom a growth in heated volume~linear with energy! and thegrowth in mean ion transit distance, shown to be nearfocal spot radius in Sec. II for a cylindrical plasma. Sinthis focal spot radius will increase as the square root ofergy, this implies a total yield scaling of laser energy to t1.5 power. Of course this estimate is not valid when the lafocal spot approaches the same dimension as the deposlength. However, in our experiment the focal spot wasways ,200 mm, which is at least an order of magnitudsmaller than the 2 mm deposition length. Nonetheless,estimate likely represents a worse case scaling, since gryield may be derived if the intensity is increased~greatercluster sizes can be fully stripped, for a more energetic C

FIG. 10. Fusion yield measured as a function of laser focal position wrespect to the center of the gas jet for a variety of different separatbetween the jet and the laser propagation axis.


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lomb explosion! or the laser penetrates more effectively inthe gas jet plume.

The measured yield scaling with laser energy is illutrated in Fig. 12. We observe a yield scaling from 25 to 1mJ atE2.2, substantially faster than our simplistic yield scaing estimate. The origin of this fast yield scaling can be,fact, attributed to the way in which more energetic laspulses propagate into the jet. This will be considered in SV.

We have also examined the falloff of neutron yield aschange the intensity though a lengthening of the laser puThis pulse lengthening was achieved by chirping the puthrough a detuning of the pulse compressor. Yield as a fution of the laser pulse width is illustrated in Fig. 13~with allother parameters kept constant!. Instead of the rapid fall seenwhen the laser energy is decreased~Fig. 12! the yield fallsoff slowly as the pulse is broadened from 30 to 300 fs~adrop in intensity by a factor of 10 results in a yield decreaof only a factor of 3.5!. The yield falls off more rapidly atlonger pulse widths. This weak initial scaling of yield witpulse width of course suggests that the strong scaling seeFig. 12 is not attributed simply to laser intensity, but insteis a result of a decrease in laser fluence.

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FIG. 11. Laser focalz scans at different incident laser energies.

FIG. 12. Measured neutron yield scaling as a function of laser energ


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V. LASER PROPAGATION IN THE DEUTERIUM JET

As mentioned in Sec. IV, while we expect the neutryield to increase as the jet temperature falls and the clusize increases, this trend does not hold beyond a certaitemperature. In order to study these effects we performtime resolved interferometry on the laser plasma. The dnostic is illustrated in Fig. 2. A small portion of the lasbeam is split off and sent across the target transverse to mlaser beam. The probe pulse is then analyzed in a Micheinterferometer. By varying the delay of the probe pulsecan watch the evolution of the expanding laser plasma.resulting interferograms can be Abel inverted to find telectron density of the plasma. This ionization informatican then be used to discern where the laser is deposenergy. An example of such raw interferometric data is illutrated in Fig. 14, in which the images are recorded withprobe delayed only 20 ps after the ionizing laser traversesmedium, a time well before any hydrodynamic motion of tions occur. The fringe deviation indicates the area and shof the ionized deuterium ‘‘cigar-shaped’’ filament.

We imaged the plasma at different jet stagnation teperatures to understand how the laser propagation is affeby the gas temperature, and hence cluster size. Figurshows interferograms along the laser propagation axis atferent gas temperature. The three images are for 80, 113155 K. The ionized region appears to move upstream oflaser focus~to the left in the images! as the jet temperaturdecreases. Figure 15 shows electron density along the cof the laser axis as determined from the Abel inversion ofinterferograms in Fig. 14. The maximum measured electdensity is;1.5– 2.031019cm23. This small variation indi-cates that the observed gas temperature effects on yieldnot likely the result of changing average density. The eltron density rises as the pulse propagates toward the centthe jet, illustrating the rise in gas density toward the cente

FIG. 13. Measured yield as a function of the laser pulse width.


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the plume. However, the electron density rolls over andcreases prior to the center of the jet plume, for the dcollected at the lower temperature. At lower temperaturerollover indicates that the laser does not penetrate effectivto the center of the jet. As we move to higher temperatuthe laser can clearly propagate further into the gas plumehigher electron density is reached further before the jet,

FIG. 14. Interferograms along the laser propagation axis approximatelps after the laser ionizes the gas for three measured gas jet temperatur113, and 155 K.

FIG. 15. Deconvolved electron density along the center of the laserdetermined from the interferograms of Fig. 14.


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in the case of the highest temperature, the electron dendoes not roll over. This likely results from there being leenergy absorbed per atom in the smaller clusters, andlaser pulse can travel further before the energy is deplet

The neutron yields in the previous cases were;73103 at 80 K, 43102 at 113 K, and 0 at 155 K. This isremarkable, with over an order of magnitude increase in ntron yield from 113 to 80 K despite the fact that the averaion density is higher atT5113 K. Clearly the laser energmust be absorbed over a smaller number of ions, creatinhigher effective ion temperature, at the colder jet tempeture.

More information about the laser energy absorption aion temperature can be found in looking at the velocity ofblast wave formed as the plasma expands radially. Thelocity of a cylindrically symmetric Tayler–Sedov blast wavwill scale roughly as (El /r0)1/4, whereEl is the energy de-posited per unit length andr0 is the average gas density.38

Thus, faster blast waves indicate larger energy depositionatom is the gas. While it is difficult to compare maximuenergy deposition between two different images becausthe weak quarter power scaling, we can derive some inmation about where the laser deposits its energy alongpropagation path by examining where the later time~fewnanoseconds! blast wave expands with greatest velocity.

This is illustrated by data shown in Fig. 16. These iages are taken 4 ns after the laser pulse ionizes the gtime after the blast wave has developed and expanded odistance many times the initial plasma filament diameter.see that for 80 K the largest radius, and hence greatest enper ion deposition, occurs at the point where the lareaches best focus. The laser will exhibit the highest intenhere and hence be able to strip the electrons from the laclusters and produce the highest energy ions. This inproduces the highest neutron yield for the largest cluseven though the maximum energy deposited at other tperatures occurs at higher densities.

These data indicate that an important factor in achievhigher neutron yields is to arrange the laser focal positiondeposit the laser energy to the highest density region ofjet before the pulse is depleted. To illustrate this pointaltered the spatial characteristics of the jet plume by altethe timing between the jet opening time and the pointwhich the laser fires. Two shots with different initial jeopening times are shown in Fig. 17. The bottom image wtaken with the gas jet fired 500ms earlier than the uppeimage. The total jet opening time was 800ms. When the jetis opened earlier there is more time for the flow to deveand the size of the plume is larger. This increased plume~of the later opening time, bottom image! leads to laser energy absorption much further upstream than when the jeopened right before the laser arrives~upper image!. The de-convolved axial electron density plots illustrate this,shown in Fig. 18. When the jet is opened early, the electdensity peaks further in front of the center of the jet. Tenergy is depleted prior to the laser focus. The effect ofis shown in Fig. 19 where the blast waves~at 4 ns! of thesesituations are reproduced. The blast waves show that theergy is absorbed much further forward in the jet and far


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front of the laser focus when the jet plume is allowedexpand before the laser arrives~lower image!. This means, ofcourse, that the laser energy is absorbed at lower laser insity in this case. Only small clusters can be fully strippedelectrons and the average explosion energies will be lened. Indeed, the early opening jet~larger plume, bottomimage in Fig. 19! yielded;23103 neutrons while the lateropening jet ~more tightly confined plume! produced;83103 neutrons.

These energy depletion issues in the laser propagacan easily explain the scaling data seen in Fig. 10 whendistance from jet nozzle to laser axis is varied. Figure

FIG. 16. Interferograms at three different gas jet temperatures takenafter the laser irradiates the jet showing the extent of the resulting bwave.

FIG. 17. Interferograms~at 20 ps! of shots with different initial jet openingtimes. The upper image shows the plasma created when the laser awithin close~;100ms! to the initial jet opening. The bottom image is takewhen the laser irradiates the jet well after~.500 ms! the jet opens.


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shows images of the plasma when the laser and gas jeseparated by three different distances. Clearly the laserergy is absorbed further upstream as the distance is increand the effective gas density gradient scale length is lenened. As previously, images of blast waves~at 3 ns!, shownin Fig. 21, confirm that the location of the large energy asorption with respect to the jet center changes with nozzaxis separation. It appears as if the maximum energy abstion occurs very close to best focus at separation of 1.6while it occurs before focus at 3.4 mm and after focus formm. The neutron yield in these experiments was 2.53103

for 0.5 mm separation, 3.53103 for 1.6 mm~best match oflaser focus and energy deposition point! and 13103 for 3.4mm. Once again we see that the maximum neutron yoccurs when the point of highest energy absorption coincwith best laser focus.

The gas jet backing pressure scaling of neutron yipresented in Sec. IV was somewhat unexpected. The hi

FIG. 18. Deconvolved axial electron density plots derived from the tinterferograms in Fig. 17.

FIG. 19. Blast wave profiles determined 4 ns after irradiation of the twoopening times shown in Fig. 17.


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FIG. 20. Interferometric images~at 20 ps! of the plasma when the lasepropagation axis and gas jet are separated by three different distances

FIG. 21. Blast wave profiles~at 3 ns! of the three gas jet-laser axis separtions used in Fig. 20.


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densities at higher pressures should lead to more absorfurther in front of the jet. Intuitively this would lead to thoptimal jet position being further behind the focus, so ththe best focus can be placed where the laser energy is babsorbed. This would be similar to what was observed inheight scaling as the separation is increased. The opptrend is observed in the jet pressure scaling, however, wthe optimal position at 67 atm resting closer to the focus tat 40 atm~Fig. 11!. The interferograms in Fig. 22 clearlshow the reason for this observed trend. The plasma is foing further forward at 40 atm than at 67 atm. Examinationthe electron density shows that, even though the pressuhigher at 67 atm, the density is lower out in front of theplume center than with 40 atm backing, showing thatplume is better collimated at higher backing pressure. This less gas along the path toward the jet center at higbacking pressure and the laser can penetrate further.effect is likely a consequence of changing jet dynamicsthe pressure increases. Finally, this supposition is confirmin the blast waves after 3 ns~Fig. 23!. For data from the blaswave for the 67 atm case, absorption occurs at best fowhile it occurs well ahead of best focus at 40 atm. The ntron yields for these cases were 43103 at 67 atm and 73102 for 40 atm.

VI. CONCLUSION

In this paper we have presented an extended study oscaling of fusion from the irradiation of a gas of deuteriuclusters with an intense 35 fs laser pulse. The various msured scalings indicate that a Coulomb explosion modethe ion energy is a good description of the ion explosion. Tconsequence of this is that larger clusters will clearlyadvantageous in producing faster ions~and higher fusionyields! if high enough laser intensity can be achieved to fustrip the clusters. It is clear from our measurements twhen a sonic nozzle like that utilized in our experiments

FIG. 22. Interferograms~at 20 ps! of the plasma at two different gas jebacking pressures.


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employed, the diffuse cluster plume can lead to substandepletion of the laser during its propagation to the jet cenThis has an important effect on the fusion yield. This initbattery of research now points to a number of improvemeto increase neutron yield in this experiment, namely uswell collimated plumes of large deuterium clusters ahigher energy femtosecond lasers.

ACKNOWLEDGMENTS

We would like to acknowledge the technical assistanof V. Tsai, G. Anderson, and R. Shuttlesworth.

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FIG. 23. Blast wave profiles~at 3 ns! of the plasma at two different gas jebacking pressures used in Fig. 22.


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