Optimizing the ionization and energy absorption of laser-irradiated clusters

0.36 MB·English

by  M. Kundu

Optimizing the ionization and energy absorption of laser-irradiated clusters
M. Kundu and D. Bauer
Max-Planck-Institut fu¨r Kernphysik, Postfach 103980, 69029 Heidelberg, Germany
(Dated: February 2, 2008)
It isknownthat rare-gas ormetal clustersabsorb incident laser energy veryefficiently. However,
due to the intricate dependencies on all the laser and cluster parameters it is difficult to predict
under which circumstances ionization and energy absorption is optimal. With the help of three-
dimensionalparticle-in-cellsimulationsofxenonclusters(upto17256atoms)wefindthatforagiven
laserpulseenergyandclusteranoptimumwavelengthexistswhichcorrespondstotheapproximate
8
wavelengthofthetransient,linearMie-resonanceoftheionizingclusteratanearlystageofnegligible
0
expansion. Inasingleultrashortlaserpulse,thelinearresonanceatthisoptimumwavelengthyields
0
muchhigher absorption efficiency than in theconventional, dual-pulsepump-probeset-upof linear
2
resonance duringcluster expansion.

n
a PACSnumbers: 36.40.Gk, 52.25.Os,52.50.Jm
J

8
I. INTRODUCTION ergy is to dope a cluster with atomic/molecular species

] of low ionization potential [23, 24]. An almost two-fold
h
increaseofthehighestchargestateswereobtainedexper-
p The interactionofrare-gasandmetal clusters within-
imentallywithargonclustersdopedwithwatermolecules
- tenselaserlighthasdrawncloseattentionduringthelast
m [25].
tenyears. Reasonsforthislargeinterestare(amongoth-
s ers)thehighchargestates[1,2,3,4,5,6,7,8,9]andthe Experimental results for xenon and silver clusters em-
a highenergiesofbothions[1,2,3,4,6,7,8,9,10,11,12] beddedinheliumdropletswerereportedinRef.[26]. The
l
p and electrons [10, 12, 13, 14, 15] observed (see [16] for a pulse duration and the sign of the chirp of a laser-pulse
s. recent review). also affect ion charge states and ion energies [27]. En-
c The laser-cluster interaction scenario may be qualita- hancedinner ionizationofrare-gasandmetalclusters ir-
i radiatedbyasequenceofduallaserpulseswereobserved
s tively summarizedas follows. Electronsleavetheir “par-
y ent” atoms (“inner ionization”) and absorb further laser [11, 26, 28, 29] experimentally. In these kinds of exper-
h iments one should adjust the delay time between pump
energy while moving in the cluster potential formed by
p
theionicbackground. Someelectronsmayleavetheclus- andprobepulsesuchthattheclusterexpandssufficiently
[
ter (“outer ionization”), leaving behind a net positively to meet the linear resonance ωMie = ωl with the probe

1 charged nanoplasma. The total electric field consists of pulse, where ωMie is the Mie-plasmon frequency and ωl
v the laser plus the space charge field. It may exceed the is the laser frequency. Vlasov simulations [26] and semi-
1 pure laser field in certain spatial regions and thus may classical simulations [30] of a small Xe40 cluster subject
3
liberate further electrons from their parent ions (which to such a pump-probe setup showed an enhancement of
2
1 would remain bound if there was the laser field alone). the ion charge states. An optimum control multi-pulse
. This enhanced inner ionization is called“ionizationigni- simulation has also been performed [31].
1
tion” [17, 18, 19, 20, 21, 22]. Ionization ignition locally Inthisworkweinvestigatetheeffectofthelaserwave-
0
8 continues until the total field drops below the thresh- length by three-dimensional particle-in-cell (PIC) simu-
0 old field required to liberate further electrons. The ionic lations. The goalis to find anoptimum wavelengthfor a
: background expands because of Coulomb repulsion and fixedlaserintensityandagivencluster. Atthisoptimum
v
hydrodynamicpressure,whichultimatelyleadstotheen- wavelength(whichturnsouttobeintheultraviolet(UV)
i
X ergeticionsobservedinexperiments. Becauseofthecom- regime for the Xe clusters under consideration) a single
r plex interplaybetweeninner ionization,outerionization, ultrashortlaser pulse is shownto be much more efficient
a andionicexpansionitis farfromtrivialtopredictquan- than the “conventional” dual-pulse pump-probe setup.
titatively mean or highest charge states, the absorbed Experimental signatures of enhanced x-ray yields and
laser energy, or other observables and their dependence high charge states at short wavelengths [32, 33, 34] indi-
on the laser parameters and the cluster decomposition. cate a clear impact of the laser wavelength on the laser-
Oneofthegoalsinlaser-clusterexperimentsandsimu- cluster interaction. Free electron laser (FEL) cluster ex-
lationsistoconvertasmuchlaserenergyaspossibleinto periments [35] at the DESY facility, Hamburg, and a re-
energetic particles. This can be achieved by optimizing cent x-ray laser-cluster experiment [36] down to wave-
the outer ionization degree, i.e., by removing as many lengths <100 nm also showed enhanced ionization.
electrons as possible from the cluster in order to gener- Contrary to our findings recent molecular dynamics
ate high charge states so that the asymptotic ion energy simulations [37] concluded that (i) there is no influence
(and thus the total absorbed energy) after Coulomb ex- ofthe laser wavelengthonthe chargingof clusters in the
plosion is largest. regime 100 800 nm for a laser intensity 1016Wcm−2
− ≈
One way to increase the charge states and the ion en- and (ii) that linear resonance plays no role, thus threat-
2
ening the basis of the nanoplasma model [38]. Simi-
lar conclusions were reported by the same authors in Zmax
25
Refs. [39, 40]. Zmin
We consider short laser pulses in this work. Most of Zav
the earlierworkswerereportedfor the long-pulseregime Z)20 Z − Bethe
where linear resonance (LR) absorption [26, 30, 31, 38, es (
41,42,43,44,45]occursduringtheexpandingphaseofa stat15
clusterwhentheMie-plasmafrequencydropssufficiently e
g
so that the laser frequency can be met. At the time of ar10
h
LR the space charge field inside the cluster is strongly C
enhanced, leading to efficient ionization ignition. 5
The paper is organized as follows. In Sec. II we (a)
briefly describe the simulation method and discuss the
0
ionization of a cluster by a short laser pulse in Sec. III. 14 15 16 17 18
In Sec. IV pump-probe simulation results are presented log10[Intensity (W/cm2)]
while Sec. V is devoted to the laser wavelength depen-
dence of the cluster dynamics. A possibility to achieve Zmax
25
100% outer ionization is also discussed in Sec. V before Zmin
we summarize the work in Sec. VI. Unless stated other- Zav
wise we use atomic units (a.u.). s (Z)20 Z − Bethe
e
at15
st
II. DETAILS OF THE SIMULATION e
g
ar10
h
Details of our PIC code are already described in C
Refs. [46, 47]. 5
For the inner ionization we apply the so-called Bethe- (b)
rule or over-the-barrierionization(OBI) model [48]. Ac- 0
cordingto OBIanatomorionisionizedifthe totalfield 14 15 16 17 18
satisfies log10[Intensity (W/cm2)]
|El(t)+Esc(Rj,t)|≥Ip2(Z)/4Z (1) FsoIlGid.),1:mi(nCimoluomr oinolninceh)aMrgaexZimmuinm(tiohnincshoalridge),Zavmearxag(ethiiocnk
at individual ion locations Rj. Here Z is the charge charge Zav (bold black) and the ion charge predicted by
state(afterionization),Ip( )istherespectiveionization OFI alone (dashed) vs peak laser intensity for (a) a Xe2176
potential and Esc(Rj,t) isZthe space charge field. In the cluster of radius R0 ≈ 3.54 nm and (b) a Xe17256 clus-
absence of Esc(Rj,t) ionization is caused by the laser ter of radius R0 ≈ 7 nm in an n = 8 cycle laser pulse
field, known as optical field ionization (OFI). El(t)=E0sin2(ωlt/2n)cos(ωlt) of wavelength λl =800 nm.
In this work we shall vary the wavelength down to
100 nm at an intensity 5 1016Wcm−2, which raises
the questions (i) whether s×uch lasers are available and expected to increase both energy absorption and ioniza-
(ii) whether the Bethe-rule (1) is applicable. With the tion, our results from collisionless PIC calculations may
development of new generation FEL lasers [49, 50] the be considered close to reality for wavelengths ' 400nm
answer to (i) is clearly affirmative. As regards the ion- but slightly underestimating the real charge states and
izationmodel(ii), atshortwavelengthsionizationrather the real absorbed energy for shorter wavelengths.
proceeds via multiphoton ionization than via tunneling
or over-the-barrier ionization so that the Bethe-model
(where the ionization probability switches from zero to III. IONIZATION OF A CLUSTER BY A
unity once a certain threshold field is reached) may not SINGLE SHORT PULSE
yield the precise charging dynamics of the clusters at
short wavelengths. However, the final charge state dis- First we study the response of a xenon cluster in a
tribution should remain unaffected by the details of the linearly polarized n= 8-cycle laser pulse of electric field
ionization model [40] at least qualitatively. strengthE(t)=E sin2(ωt/2n)cos(ωt)andwavelength
l 0 l l
Only collisionless absorption mechanisms are incorpo- λ = 800 nm. Different ionic charge states are self-
l
rated in standard PIC simulations. The neglect of par- consistently produced during the laser pulses according
ticle collisions is an approximation which is the more to the Bethe rule (1).
valid the smaller the clusters, the higher the laser in- Figure 1a shows the maximum charge state , the
max
Z
tensities, and the longer the laser wavelengths are (see, minimumchargestate ,andtheaveragechargestate
min
Z
e.g., [19, 20, 21, 22, 51, 52]). Since collisions can be (defined asthe totalchargeofthe cluster dividedby
av
Z
3
the number of atoms N) and the charge state predicted nism a bigger cluster will capture more electrons (whose
by the OFI (“Z-Bethe” curve)vs peak laser intensity for outer ionizationwould requiremuchhigher laserintensi-
a Xe cluster (N =2176)of initial radiusR 3.54nm tiesthaninthecaseofasmallercluster). Thepresenceof
N 0
≈
afterthepulse(i.e.,after 22fs). Themaximumcharge more electrons in the central region will screen the laser
≈
state variesfrom =5to =26asthelaserinten- fieldmoreefficiently. Asaresultboth aswellas
max av min
sityinZcreasesfrom2.5 Z1014WcmZ−2to7.5 1017Wcm−2. (in Fig. 1b) drop below the corresponZding values forZthe
× ×
The higher value of above the value predicted by smaller cluster (Fig. 1a).
max
Z
theOFIisclearlydue toionizationignition. Thosemax- We conclude that an increasing cluster size (and thus
imum charge states max are mainly acquired by the increasedionizationignitionof, atleast, the ions located
Z
ionsattheclusterperipherywherethespacechargefield close to the cluster boundary) does not always lead to
is highest. Inside the cluster the total field falls be- a higher average charge state. Our aim is to increase
lowthe ionizationthresholdsduetothe decreasingspace not only the highest charge states but also the average
charge produced by the ionic background as well as due ion charge beyond the OFI predicted value through the
to the screening of the laser field by the cluster elec- charging of more ions in the central part of the cluster.
trons. Theionsclosetotheclustercenterhaveminimum In the following sections we study several approaches to
charge states min = 2 20 at laser intensities between achieve this goal.
2.5 1014WcmZ−2 7.5 −1017Wcm−2. Thevalueof
min
× − × Z
remainsmuchlowerthanpredictedbytheOFIforalmost
all laser intensities < 5.0 1017Wcm−2. The average
IV. IONIZATION BY DELAYED PULSES: A
×
charge av remainscloseto(butslightlyhigherthan)the PUMP-PROBE SIMULATION
OFI prZedicted values at intensities <7.5 1015Wcm−2.
×
Also = 8 and = 3 do not change between
the inZtemnasxities 1015WcZmm−in2 7.5 1015Wcm−2 but Inthissectionweillustratethe“pump-probe”method
− × Zav frequently employedin laser-clusterexperiments. In this
increases slowly as more ions from the cluster center to-
method an initial pump-pulse ionizes the cluster. The
wards the periphery acquire higher charge states 3 8.
→ clusterexpandsfreelybefore,afteradelaytime,aprobe-
The value of remains constant, = 8, due to
the removal ofZamllaxelectrons from the 5sZ2p6 shell of the pulse hits the expanding cluster. The interaction of this
probe pulse with the cluster will sensitively depend on
Xe atoms close to the cluster boundary. As the inten-
the cluster size and thus on the delay time. We revisit
sity 7.5 1015Wcm−2 is approached the laser field is
≈ × such a scenario in our current work since it will allow us
stronglyshieldedfromthecentralpartofthecluster,and
to compare the efficiency of laser energy absorption for
outerionizationaswellasionizationignitiontendtosat-
suchastandardpump-probe method withthe single UV
urate. As a consequence grows slowly between the
intensities 5 1015WcmZa−v2 1016Wcm−2. Unless a pulse scenario which will be introduced in Sec. V.
thresholdin≈tens×ity 1016Wcm−−2 iscrossedfurtherelec- The laser field profile is of the form El(t) =
trons from the clust≈er cannot be removed,which was al- E0sin2(πt/nT)cos(ωlt) for both pump and probe pulse.
The time period T is chosen with respect to the wave-
ready seen in previous model and numerical calculations
[46,53,54,55]. Athigherintensities>1016Wcm−2outer length 800 nm. For, say, n = 4 the product nT deter-
mines the total pulse duration 11 fs. The pulse enve-
ionizationandionizationignitionincreasesagain,leading ≈
lope and intensity are kept the same for all cases under
to an increaseof beyondthe values predicted by the
Zav study, i.e., the laser energy in all pulses is the same too.
OFI due to the strong increase of both and .
Zmax Zmin Figure 2a shows the results for the Xe17256 cluster of
It is commonly believed that ionization ignition be- initial radius R 7 nm at an intensity 5 1016Wcm−2
0
≈ ×
comes increasingly pronounced with increasing cluster when both the pump and the probe pulse have the same
size. Figure 1b shows , , , and the charge wavelength 400 nm. The average charge , the scaled
max min av av
Z Z Z Z
states predicted by the OFI vs the peak laser intensities Mie-frequency ω /ω, the total absorbed energy E
Mie l tot
for a bigger Xe cluster (N = 17256) of initial radius (electrostatic field energy plus the kinetic energy of elec-
N
R 7nm. variesbetween8 26,exceeding again trons and ions), the normalized cluster radius R(t)/R ,
0 max 0
≈ Z −
the charge states predicted by OFI alone. Below the in- and the laser fields are plotted vs time (in units of the
tensity 1017Wcm−2 is higher by a factor of 2 periodT). Duringthefirstfourlasercyclesofthepump-
max
Z ≈
comparedtotheOFIvalue(“Z-Bethe”curve). Although pulse the average charge state rises to 11, the
av
remainsmuchhigher,theaverageioncharge (in frequency ω /ω 2.5 and E 2Z.0 ≈107 while
max av Mie l tot
Z Z ≈ ≈ ×
Fig. 1b) is below the charge states predicted by the OFI the cluster expansion is insignificant. The total energy
for most of the laser intensities. Most of the ions closer E 2.0 107 corresponds to the average energy ab-
tot
≈ ×
to the cluster center acquire chargestates =2 10 sorbedperionE /N 31.4keV.Afterthepump-pulse
min tot
Z − ≈
whichareevenlowerthanforthesmallercluster(Fig.1a) theclusterevolvesfreelyand ,E remainunchanged
av tot
Z
at the corresponding intensities. Hence, ionization igni- but ω /ω drops due to the expansion. Note that the
Mie l
tion is indeed responsible for the highest charge states cluster radius R(t) (defining the cluster boundary) cor-
which increase with the cluster size (as seen in responds to the distance of the most energetic ions from
max
Z
Fig. 1). However, exactly because of the same mecha- theclustercenter. Attheboundary,however,the cluster
4
70 15 (a) up to the value Zav = 14 and Etot ≈ 7.5×107, respec-
1 tively. Such a pump-probe simulation clearly illustrates
E / tot Zav ttheratdythneamlinicesa.rMreosroeneannecregyinidseaebdsoprlabyesd,aleraodleinignttohehicglhuesr-
, 010 chargestates. Theseresultsareinagreementwithhydro-
R
dynamicandVlasovsimulations[11,26]. However,linear
R/
E, l R/R0 rtheseonclaunscteerishmasetalornealydyafetxerpaanrdeeldatisvigenlyifilcoanngtltyim(aeswsheeenn
ω/, Miel 5 ωMie/ωl Etot Ren(etr)g/yR0ab≈so8rpitnionFigin. 2sau)c.h Iaonloizwatidoennsiigtnyitpiolansmanadislaesxer-
ω pected to be less efficient compared to the case where
v, 0 linear resonanceoccurs before the cluster expands signif-
a E
Z l icantly.
0 10 20 30 40 50 While keeping the 400 nm pump as above we now as-
Laser cycles sumeaprobewavelengthof200nmforthepurposeofhit-
70 20 (b) tingthelinearresonanceatanearliertimewhentheclus-
1 terismorecompact. Theenergyintheprobepulseisthe
/ ot sameas inFig.2a. Figure2bshowsthe resultanalogous
Et15 to Fig. 2a. The average charge and the absorbed energy
R, 0 Zav now increase up to Zav =18.5 and Etot ≈9×107 which
E, R/l10 Etot aeonrfeeerhngieigerhsgeybreatihbnasgontrihpnetiFosiangm.in2eaitnahfebteosrtehtchocenadpsersoscbahee.hmiWgehie(tFrhietgffih.ec2ipbeun)lcsiyes
ω, l R/R obvious. The reason is the smaller cluster size at the
/Mie 5 ωMie/ωl 0 time of linear resonance (R(t)/R0 <1.5) and the higher
ω space chargefield relatedto it. Similar findings from ex-
av, 0 E periments havebeen reportedin Ref. [11]. In passing we
Z l note that the average charge av 11 in Fig. 2 due to
Z ≈
0 2 4 6 8 10 the pump (at 400 nm) exceeds 8 in Fig. 1b (at
av
Laser cycles 800 nm) for the same cluster anZd the≈same laser inten-
sity 5.0 1016Wcm−2 despite the higher pulse-energy in
FIG. 2: (Color online) Average ion charge Zav, scaled Mie- Fig. 1b×because of the twice longer pulse.
frequency ωMie/ωl, laser field El (in atomic units), normal- In the following section we study the wavelength de-
ized cluster expansion radius R/R0 and total absorbed en- pendence of the average charge states and the laser en-
ergy Etot (in atomic units) vs time (in 800 nm laser cycles) ergy absorption.
for a Xe17256 cluster of radius R0 ≈ 7 nm. The peak in-
tensity 5×1016Wcm−2 is the same for (a) pump of wave-
length 400 nm (probe, 400 nm) and (b) pump of wavelength
400 nm (probe, 200 nm). The laser field is of the form V. IONIZATION AT DIFFERENT
El(t) = E0sin2(πt/nT)cos(ωlt) with n = 4 and one laser WAVELENGTHS
cycle T corresponding to the wavelength 800 nm. A minor
increase in Etot after the pulses is an artifact of PIC simula- Does the average charge state and the absorbed en-
tions.
ergy for a given cluster increases with decreasing laser
wavelength? One could expect that for a certain wave-
length the linear resonance during the initial ionization
potential is anharmonic. Hence using R(t) for the cal-
stagewhenthe Mie-frequencyrisesfromzerotoits max-
culation of the Mie-frequency ωMie(t) = pN av/R3(t) imum value becomes important. For long wavelengths
Z
the latter is underestimated. Instead we use the defini- this earlyresonanceis passedsoquicklydue to the rapid
tion ωMie(t) =pQb(t)/R03 (as in Ref. [47]) where Qb(t) chargingoftheclusterthatanyindicationofaresonance
is the total ionic charge within the initial cluster radius is washed out.
R0 where the clusterpotentialis close to harmonicatall We assume the same laser field profile El(t) =
times. E sin2(πt/nT)cos(ωt)asinSec.IIIwiththesamepulse
0 l
After 44 laser cycles ω approaches the linear res- duration, pulse energy, and intensity 5.0 1016Wcm−2
Mie
×
onance (dashed line) with respect to the fundamental so that the number of laser cycles in the pulse depends
400 nm, i.e., ω /ω = 1. The probe pulse of wave- onthe wavelength. The laser wavelengthis varied in the
Mie l
length400nmisappliedwithadelayof 42lasercycles range 800 100 nm. Note that in the following we spec-
≈ −
such that the peak of the pulse approximately coincides ify times and pulse durations in units of laser periods at
withtheresonancetime. Duetothelinearresonancethe 800nm (corresponding to T 2.6fs).
≈
average charge and the absorbed energy rises abruptly Figure 3 shows the average charge state and to-
av
Z
5
1.3
25 (a) 1−800 nm−cycles (a) 1−800 nm−cycles
2−800 nm−cycles 2−800 nm−cycles
3−800 nm−cycles 3−800 nm−cycles
20 1.2
4−800 nm−cycles 4−800 nm−cycles
v15 R0
Za (t)/ 1.1
R
10
1
5
0 0.9
0 200 400 600 800 0 200 400 600 800
Wavelength (nm) Wavelength (nm)
35
) (b) 1−800 nm−cycles (b) 1−800 nm−cycles
a.u.30 2−800 nm−cycles 0.8 2−800 nm−cycles
( 3−800 nm−cycles 3−800 nm−cycles
70 25 4−800 nm−cycles 4−800 nm−cycles
1 0.6
y/
rg20
e
n η
d e15 0.4
e
b
r10
o
s 0.2
b
A 5
0 0
0 200 400 600 800 0 200 400 600 800
Wavelength (nm) Wavelength (nm)
FIG. 3: (Color online) Average ion charge Zav (a) and total FIG. 4: (Color online) Normalized cluster expansion radius
absorbed energyEtot (b)vslaser wavelengthafter 1,2,3and R/R0 (a) and outer ionization degree η (b) vs laser wave-
4lasercycles(at800nm)foraXe17256 clusterofradiusR0 ≈ length, corresponding to Fig. 3.
7 nm. Otherparameters as in Fig. 2.
sorption at 125 nm, leading to a marked increase of the
tal absorbed energy E vs the laser wavelength for the average charge up to a value 25 clearly shows
tot av
Z ≈
Xe cluster of radius R 7 nm after t = 1,2,3,4- that wavelengtheffects are undoubtedly important. One
17256 0
≈
laser cycles at 800nm. The value of increases in maycomparetheabsorbedenergyandtheaveragecharge
av
Z
time (in Fig. 3a) for all wavelengths. Ionization mostly with the dual-pulse simulation results in Fig. 2. The ab-
occurs before t = 2 cycles when the peak of the pulse is sorbedenergyE 34 107a.u.andtheaveragecharge
tot
≈ ×
reached. After that the spacechargefieldis highenough 25 are much higher in the present case around the
Z ≈
to generate further charge states between 2 3 cycles. laserwavelength125nm comparedto the respective val-
−
does not change anymore between 3 4 cycles, in- ues E 9 107 and 18.5 in Fig. 2b. The
av tot av
Z − ≈ × Z ≈
dicating a saturation of inner ionization. The average absorptionis 3.78 times higher than in Fig. 2b. More-
≈
charge state increases from = 8 to a maximum over, in the dual-pulse case the total laser-pulse energy
av
Z Z
value 25as the laserwavelengthis decreasedfrom was twice higher. Therefore, the absorption efficiency is
av
Z ≈
theinfrared800nmdowntotheUVwavelength125nm. augmented further by a factor of two.
It means that the sub-shells 4s2p6d105s2p6 of almost all
In Fig. 4 we plot the normalized expansion radius
atoms are empty at 125 nm. A further decrease of the
R(t)/R (Fig. 4a) and the outer ionization degree η
0
wavelengthcauses todecreasegraduallytoasmaller
av (number of total electrons outside R(t) divided by the
Z
value 9 at 50 nm.
av totalnumberofelectronsproduced,N (t), inFig.4b)
Z ≈ av
Z
Figure 3b shows a similar qualitative behavior of the vs the laser wavelength corresponding to the results in
absorbed energy both in the time domain and in the Fig. 3. The radius R(t) and the outer ionization de-
wavelength domain. The energy E is maximum at gree η(t) go hand in hand with the absorbed energy
tot
the same wavelength λ = 125 nm. Although the laser- E (t) and the charge (t). After four cycles the
l tot av
Z
pulse energy is the same in all cases the increased ab- cluster has expanded very little, R(4T)/R 1.225 at
0
≈
6
catesthe linearresonance. Chargingofthe clusterstarts
800 nm
around 0.3 cycles for all wavelengths by OFI, leading
400 nm
to an abrupt increase of ω (t)/ω for the longer wave-
Mie l
4 200 nm lengths while for the shorter ones the increase proceeds
150 nm slower. Asaresulttheplasmaisoverdenseduringtheen-
125 nm tire pulse for the long wavelengths but stays close to the
3
ωl 112.5 nm linear resonance for the shorter wavelengths. The more
/e time is spent near the linear resonance,the higher is the
Mi
energy absorption and the average charge state, as seen
ω 2
inFig.3. Atthewavelength125nmtheresonanceismet
at the peak of the pulse so that the energy absorptionis
particularly efficient.
1
We nowdiscuss the time evolutionofthe space charge
field Ex (along the laser polarization) at different posi-
sc
0 tionsinsidetheclustertofurtherillustratetheresonance
0 1 2 3 4 at short wavelengths, leading to efficient ionization igni-
Laser cycles
tion and the generation of high charge states. Figure 6
shows the space charge field Ex and the laser field E
sc l
FIG. 5: (Color online) Scaled Mie-frequency ωMie/ωl vstime vs time at radial positions 0.24R0, 0.48R0, 0.72R0 and
(in 800nm cycles) for wavelengths λl = 800−112.5 nm and 0.96R0forfourdifferentwavelengths. Figure7showsthe
thelaser and cluster parameters of Fig. 3. corresponding phases of Ex with respect to the driving
sc
laser field.
At the long wavelength 800 nm, Ex inside the cluster
sc
λl 125 nm, although the average charge av 25 is at radii 0.24R0, 0.48R0, 0.72R0 mostly oscillates with
≈ Z ≈
very high compared to Fig. 2. With such an insignifi- a phase δ π while δ 0 at the boundary (i.e., at
≈ ≈
cant expansion the space charge field can be considered 0.96R0). This is clearly what one expects from an over-
optimized, leading to maximum ionization ignition. The dense plasma: screening of the laser field in the cluster
ignition field (i.e., the space charge field due to the ionic interior but an opposite behavior outside the electron
background)under the assumption that all electrons are cloud. Theoscillationofthespacechargefieldarisesdue
removedreadsE (t) N (t)/R(t)2. UsingR(t)from totheoscillationsoftheelectronstrappedinsidetheclus-
ig av
≈ Z
Fig. 4a and (t) from Fig. 3a, one obtains at 125 nm ter. These electrons form approximately a sphere which
av
Z
E (2T) 20.0,E (3T) 21.0andE (4T) 16.2,ifall is smaller than the cluster due to outer ionization. If
ig ig ig
≈ ≈ ≈
electronsareremoved(i.e.,η =1). Theexpectedignition the electron cloud was rigid and did not cross the clus-
field 21a.u. is maximum near the pulse peak around ter boundary the phase shouldbe exactly π and 0 inside
≈
2 3-cycles, thereafter decreases to E (4T) 16.2 due andoutside, respectively,if the plasma is overdense,and
ig
− ≈
to an expansion R(4T)/R 1.225 and no further cre- opposite in the underdense case. In reality, the bound
0
≈
ation of charge states. Note that the peak laser field is electron population changes and the electron sphere is
onlyE 1.19. Thereforetheenhancedionizationiscer- neither rigid nor has it a sharp boundary, resulting in
0
≈
tainly due to the ignitionfield. However,atλ 125nm phase distortions and deviations from the idealized case,
l
≈
η 0.6inFig.4b,meaningthat40%oftheelectronsare as seen in Fig. 7a.
≈
still inside the cluster. The presence of these electrons Figure 6 confirms explicitly that the total field at the
lowersEig comparedtothe aboveidealcaseofη =100% boundary is highest and therefore leads to the highest
outer ionization, and one may argue that Eig is not yet ionic charge states while Esxc(t) almost nullifies the laser
optimized. However,evenifη =100%outerionizationis field in the strongly overdense regime. The maximum
achievedfor the above laserfield intensity the maximum valueofthetotalfieldatthepeakofthepulseis 4.0a.u.
≈
total field is 22 which is still insufficient to produce (x-componentonly)whichissufficienttoproducecharge
≈
a higher average charge av = 27 (requiring a threshold statesupto 18(alsoseeninFig.1b). Anadditional
Z Z ≈
field ' 24 according to OFI). Hence the average charge contribution (up to a factor √3) to the total field comes
state is optimized. This will be shown explicitly at the from the y and z-components of the space charge field.
end of this section where we actually achieve η 100% At 200 nm the amplitude of Ex around t=1.5 cycles
≈ sc
for this cluster.
at 0.24R increases up to 5 a.u. which, after addition
0
The above results clearly show that there exists a cer- to the laser field, is sufficient to produce charge states
tainwavelengthatwhichthelaser-clustercouplingisvery = 18 even inside the cluster. After 1.75 cycles Ex
Z ≈ sc
efficient. Such a nonlinear dependence of the absorbed at 0.72R behaves similarly to that at 0.96R , i.e., the
0 0
energy and averagecharge state on the laser wavelength laser and space charge fields at 0.96R and 0.72R are
0 0
indicates a resonance around 125 nm in Fig. 3 and 4. now approximately in phase. Figure 6b shows that the
To investigate this further, we plot in Fig. 5 the scaled totalfield 4 8between0.96R and0.72R ,producing
0 0
≈ −
Mie-frequency ω (t)/ω vs time. The dashed line indi- charge states 18 23. However, due to the screen-
Mie l
Z ≈ −
7

0.24R 800 nm 5 0.24R 800 nm (a)
0.48R 0.48R
3
0.72R 0.72R
4
0.96R 0.96R
a.u.) 2 Elaser adian)3
Field ( 01 hase (r2
P
1
−1
(a)
0
0 1 2 3 4 0 0.5 1 1.5 2 2.5 3
Laser cycles Laser cycles
8 0.24R 200 nm 5 0.24R 200 nm (b)
0.48R 0.48R
6
0.72R 0.72R
4
0.96R 0.96R
4 E n)
a.u.) 2 laser adia3
Field ( 0 hase (r2
P
−2
1
−4 (b)
0
0 1 2 3 4 0 0.5 1 1.5 2 2.5 3
Laser cycles Laser cycles

0.24R 125 nm 5 0.24R 125 nm (c)
0.48R 0.48R
10 0.72R 0.72R
4
0.96R 0.96R
E n)
a.u.) 5 laser adia3
Field ( 0 hase (r2
P
1
−5
(c)
0
0 1 2 3 4 0 0.5 1 1.5 2 2.5 3
Laser cycles Laser cycles

6 0.24R 112.5 nm 5 0.24R 112.5 nm (d)
0.48R 0.48R
0.72R 0.72R
4 4
0.96R 0.96R
E n)
a.u.) 2 laser adia3
Field ( 0 hase (r2
P
−2
1
(d)
−4
0
0 1 2 3 4 0 0.5 1 1.5 2 2.5 3
Laser cycles Laser cycles
FIG. 6: (Color online) The x-component of the space charge FIG. 7: (Color online) Phase of the space charge field with
field at radial distances 0.24R0,0.48R0,0.72R0, and 0.96R0 respect to the laser field at different radial distances corre-
inside a Xe17256 cluster of radius R0 ≈ 7 nm and the laser sponding toFig. 6 vs time corresponding to Fig. 6.
field El(t) = E0sin2(πt/nT)cos(ωlt) of peak intensity 5×
1016Wcm−2 vs time (in periods corresponding to 800nm) at
(a) λl = 800, (b) λl = 200, (c) λl = 125, and (d) λl =
112.5 nm.
8
ingofthe laserfieldinsidethe cluster,manyatomsthere
haveonlychargestates <18sothattheaveragecharge 4 η(R)
state is 18 in FigZ. 3. From Fig. 7b it is seen that η(R ) (a)
av 0
pbreofoxriematZte=ly1≈f.o7l5lowcysctlhesepthheaspehaatsteheofsmEasxclleartra0d.7i2iR0.024aRp- R/R0 3 El
and 0.48R0 since the plasma is evolving from under to0 E, l 2 R/R0
overdense. Then, with increasing outer ionization and ),
0
thus shrinking electron sphere, Esxc at 0.72R0 drops and (R 1
approaches the behavior for 0.96R . η
Attheresonantwavelength125n0mviolentoscillations R), 0
(
ofthe electroncloudaredriven,leadingto aparticularly η
hightotalfieldeverywhereinsidetheclusterandanaver-
−1
agechargestate 25inFig.3. Higherchargestates
av
Z ≈
> 26 are not produced because of the high thresh- 0 2 4 6 8 10 12
Zold field 24 necessary to crack the M-shell. One may Laser cycles
≈
argue that the presence of 40% electrons inside the clus-
ter (in Fig. 4b) will deplete the field inside significantly. 70 Zav
) E
However,one should keep in mind that at resonance the u. 60 tot (b)
electroncloud oscillates with a large excursion,exposing a.
(
a substantial part of the naked ionic background, lead- 70 50
1
iwnhgictho acannenphroadnucecde “hdigyhnearmcihcaalrgieonsitzaatteiosnthiganniteioxnp”ec[t1e8d] / ot40
Et
farfotemrtth=e l3asceyrcfileesldEasxlconatetehveenbionusniddeartyhedcrloupsstedru.eFtinoatllhye, av, 30
cluster expansion. As expected, the phases plotted in Z 20
Fig.7c fluctuatearoundπ/2throughouttheclusteronce
10
the resonance condition is met.
At 112.5 nm the plasma remains underdense. Fig- 0

ure6dshowsthatthe spacechargefieldamplitudesdrop 0 2 4 6 8 10 12
Laser cycles
compared to those in Fig. 6c, yielding less ionization ig-
nition and absorbed energy, similar to the 200nm-case.
FIG. 8: (Color online) Variation of (a) normalized cluster
Even in the optimal 125nm case presented so far only
60% of the generated electrons were removed from the radius R/R0, degree of outer ionization η(R),η(R0) within
R and R0, laser field El (in atomic units) (b) average ion
cluster (visible in Fig. 4b). Therefore outer ionization charge Zav and total absorbed energy Etot in time (in units
and ionization ignition was certainly not optimized. We
ofthelaserperiodT correspondingto800nm)fortheXe17256
argued that even if the remaining 40% electrons were
cluster of Fig. 3. The two laser pulses of wavelength 125 and
removed, the average charge state would not be signifi- 415nm, respectively, are included in (a). The peak intensity
cantlyincreasedascomparedtothatshowninFig.3a. To is thesame as in Fig. 3 for both pulses.
prove that, we performed PIC simulations for the same
cluster and the same peak intensity but now employing
two consecutive pulses (shown in Fig. 8). The first pulse 99% within R are achieved, no higher charge states are
0
of resonant wavelength 125nm with respect to the still created. The absorbed energy also does not rise signifi-
compact cluster is ramped up over four 800nm-cycles cantly so that the higher input energy invested into the
and held constant afterwards up to 8 cycles (the details two pulses does not pay off. Hence a single, short UV-
ofhowthepulseisrampeddowndonotmatter;therefore pulseofwavelength125nmturnsouttobe optimalwith
it is simply switchedoff abruptly). Att=8 cyclesa sec- respecttofractionalenergyabsorptionandgenerationof
ond pulse is switched on (over 2 cycles)whose frequency a high averagecharge state under the conditions consid-
is resonantwith the Mie-frequency aroundt=10 cycles. ered.
Afterthe firstpulsetheclusterdoubleditsradius,and
the outer ionization degree amounts to η(R) 0.8 so
≈
that 20% electrons are still inside the cluster of radius VI. SUMMARY
R(t) while 10% are inside a sphere of radius R . The
0
≈
average charge av in Fig. 8b does not change signifi- In summary, we studied the interaction of xenon
Z
cantly compared to Fig. 3a although the pulse energy clusters with intense short laser pulses using a three-
per unit area R8T E2(t)dt is 3 4 times higher. dimensional PIC code. Our aim was to optimize for a
0 l ≈ −
The purpose of the second pulse shown in Fig. 8b is given cluster the laser energy absorption and the gen-
the removal of the residual electrons. Although almost eration of high average charge states. The latter will
95%outer ionizationwithin the expanding radiusR and thenleadto energeticions uponCoulombexplosion. We
9
showed that for a given laser intensity an optimal laser Acknowledgments
wavelengthexiststhat,underthetypicalconditionsstud-
ied in this work, lies in the UV regime. Energy absorp-
tion is optimized when resonance is met during an early
stage of the dynamics when the cluster is still compact. We thank SergeiPopruzhenkofor valuable discussions
The conventional,long-pulselinear resonanceduring the and careful proofreading. This work was supported by
expansion of the cluster is less efficient. the Deutsche Forschungsgemeinschaft.
[1] T. Ditmire, J.W.G. Tisch, E. Springate, M.B. Mason, [24] M. Hohenberger, D.R. Symes, K.W. Madison, A.
N. Hay, J. Marangos, and M.H.R. Hutchinson, Nature Sumeruk, G. Dyer, A. Edens, W. Grigsby, G. Hays, M.
(London) 386, 54 (1997). Teichmann,andT.Ditmire,Phys.Rev.Lett.95,195003
[2] T. Ditmire, R.A. Smith, J.W.G. Tisch, and M.H.R. (2005)
Hutchinson,Phys.Rev.Lett. 78, 3121 (1997). [25] J. Jha, D. Mathur, and M. Krishnamurthy, Appl. Phys.
[3] T. Ditmire, Contemporary Physics, 38, 315 (1997). Lett. 88, 041107 (2006).
[4] T.Ditmire,J.W.G.Tisch,E.Springate,M.B.Mason,N. [26] T. D¨oppner, Th. Fennel, Th. Diederich, J.
Hay,J.P.Marangos,andM.H.R.Hutchinson,Phys.Rev. Tiggesb¨aumker, and K.H. Meiwes-Broer, Phys. Rev.
Lett.78, 2332 (1997). Lett. 94, 013401 (2005).
[5] V. Kumarappan, M. Krishnamurthy, and D. Mathur, [27] Yuji Fukuda, Koichi Yamakawa, Yutaka Akahane,
Phys.Rev.Lett. 87, 85005 (2001). Makoto Aoyama, Norihiro Inoue, Hideki Ueda, and Ya-
[6] V. Kumarappan, M. Krishnamurthy, D. Mathur, and suaki Kishimoto, Phys. Rev.A 67, 061201 (2003).
L.C. Tribedi Phys. Rev.A 63, 023203 (2001). [28] T.D¨oppner,Th.Fennel,P.Radcliffe,J.Tiggesb¨aumker,
[7] M. Krishnamurthy, D. Mathur, and V. Kumarappan, and K.-H. Meiwes-Broer, Phys. Rev. A 73, 031202(R)
Phys.Rev.A 69, 033202 (2004). (2006).
[8] M. Krishnamurthy, J. Jha, D. Mathur, Ch. Jungreuth- [29] Th. Fennel, T. D¨oppner, J. Passig, Ch. Schaal, J.
mayer, L. Ramunno, J. Zanghellini and T. Brabec, J. Tiggesb¨aumker, and K.-H. Meiwes-Broer, Phys. Rev.
Phys.B: At.Mol. Opt.Phys., 39, 625 (2006). Lett. 98, 143401 (2007).
[9] M. Lezius, S. Dobosz, D. Normand, and M. Schmidt [30] C. Siedschlag and J.M. Rost, Phys. Rev. A 71, 031401
Phys.Rev.Lett. 80, 261 (1998). (2005).
[10] V. Kumarappan, M. Krishnamurthy, and D. Mathur, [31] T.Martchenko,Ch.Siedschlag,S.Zamith,H.G.Muller,
Phys.Rev.A 66, 033203 (2002). and M. J. J. Vrakking,Phys.Rev. A 72, 053202 (2005).
[11] E. Springate, N. Hay, J.W.G. Tisch, M.B. Mason, T. [32] K. Kondo, A. B. Borisov, C. Jordan, A. McPherson, W.
Ditmire, M.H.R. Hutchinson, and J.P. Marangos, Phys. A. Schroeder, K. Boyer, and C. K. Rhodes, J. Phys. B
Rev.A 61, 063201 (2000). 30, 2707 (1997).
[12] E. Springate, S.A. Aseyev, S. Zamith, and M.J.J. [33] W. A. Schroeder, F. G. Omenetto, A. B. Borisov, J.
Vrakking,Phys.Rev.A 68, 053201 (2003). W. Longworth, A. McPherson, C. Jordan, K. Boyer, K.
[13] Y.L. Shao,T. Ditmire, J. W.G. Tisch, E. Springate, J. Kondo, and C. K. Rhodes,J. Phys.B 31, 5031 (1998).
P.Marangos,andM.H.R.Hutchinson,Phys.Rev.Lett. [34] A.McPherson,B.D.Thompson,A.B.Borisov,K.Boyer,
77, 3343 (1996). and C. K. Rhodes, Nature(London) 370, 631 (1994).
[14] V. Kumarappan, M. Krishnamurthy, and D. Mathur, [35] H. Wabnitzet al., Nature(London) 420, 482 (2002).
Phys.Rev.A 67, 043204 (2003). [36] S. Namba, N. Hasegawa, M. Nishikino, T. Kawachi,
[15] L.M. Chen, J.J. Park, K.H. Hong, I.W. Choi, J.L. Kim, M. Kishimoto, K. Sukegawa, M. Tanaka, Y. Ochi, K.
J. Zhang and C.H. Nam, Phys. Plasmas 9, 3595 (2002). Takiyama, and K. Nagashima, Phys. Rev. Lett. 99,
[16] U. Saalmann and J.M. Rost, J. Phys. B: At. Mol. Opt. 043004 (2007).
Phys.39, R39-R77(2006). [37] G.M. Petrov and J. Davis, Phys. Plasmas 13, 033106
[17] C. Rose-Petruck, K.J. Schafer, K.R. Wilson, and C.P.J. (2006).
Barty,Phys. Rev.A 55, 1182 (1997). [38] T.Ditmire,T.Donnelly,A.M.Rubenchik,R.W.Falcone,
[18] D.BauerandA.Macchi,Phys.Rev.A68,033201(2003). and M.D. Perry,Phys. Rev.A 53, 3379 (1996).
[19] K. Ishikawa and T. Blenski, Phys. Rev. A 62, 063204 [39] J.Davis,G.M.Petrov,andA.Velikovich,Phys.Plasmas
(2000). 14, 060701 (2007).
[20] F. Megi, M. Belkacem, M.A. Bouchene, E. Suraud, and [40] G.M. Petrov, J. Davis, A.L. Velikovich, P.C. Kepple, A.
G. Zwicknagel, J. Phys. B: At. Mol. Opt. Phys. 36, 273 Dasgupta,R.W.Clark,A.B.Borisov,K.Boyer,andC.K.
(2003). Rhodes, Phys. Rev.E 71, 036411 (2005).
[21] C. Jungreuthmayer, M. Geissler, J. Zanghellini, and T. [41] L. K¨oller, M. Schumacher, J. K¨ohn, S. Teuber, J.
Brabec, Phys.Rev.Lett. 92, 133401 (2004). Tiggesb¨aumker, and K.H. Meiwes-Broer, Phys. Rev.
[22] D. Bauer, J. Phys. B: At. Mol. Opt. Phys. 37, 3085 Lett. 82, 3783 (1999).
(2004). [42] S. Zamith, T. Martchenko, Y. Ni, S.A. Aseyev,
[23] I. Last and J. Jortner, Phys. Rev. Lett. 87, 033401 H.G. Muller, and M.J.J. Vrakking, Phys. Rev. A 70,
(2001), I. Last and J. Jortner, J. Phys. Chem. A 106, 011201(R) (2004).
10877(2002),I.LastandJ.Jortner,J.Chem.Phys.121, [43] I. Last and J. Jortner, Phys.Rev.A 60, 2215 (1999).
8329 (2004). [44] Ulf Saalmann and Jan-Michael Rost, Phys. Rev. Lett.
10
91, 223401 (2003). [52] C. Jungreuthmayer, L. Ramunno, J. Zanghellini and T.
[45] Th. Fennel, G.F. Bertsch, and K.H. Meiwes-Broer, Eur. Brabec,J.Phys.B:At.Mol.Opt.Phys.38,3029,(2005).
Phys.J. D 29, 367 (2004). [53] M. Kundu and D. Bauer, Phys. Rev. Lett. 96, 123401
[46] M. Kundu and D. Bauer, Phys. Rev. A 74, 063202 (2006).
(2006). [54] P. Mulser and M. Kanapathipillai, Phys. Rev. A 71,
[47] M.Kundu,S.V.Popruzhenko,andD.Bauer,Phys.Rev. 063201 (2005); P. Mulser, M. Kanapathipillai, and
A 76, 033201 (2007). D.H.H. Hoffmann,Phys. Rev.Lett. 95, 103401 (2005).
[48] H.A. Bethe and E.E. Salpeter, Quantum mechanics of [55] T. Taguchi, T.M. Antonsen, Jr., and H.M. Milchberg,
one- and two-electron atoms(PlenumPublishingCorpo- Phys. Rev. Lett. 92, 205003 (2004); Thomas M. Anton-
ration, New York,1977). sen,Jr. ToshihiroTaguchi,AyushGupta,JohnPalastro,
[49] J. Andruszkowet al.,Phys. Rev.Lett. 85, 3825 (2000). and Howard M. Milchberg, Phys. Plasmas 12, 056703
[50] V.Ayvazyanet al., Phys.Rev.Lett. 88, 104802 (2002). (2005).
[51] C.SiedschlagandJ.M.Rost,Phys.Rev.Lett.93,043402
(2004).