X-Ray Optical Diagnosticof Laser ProducedPlasmas

for Nuclear Fusionand X-Ray Lasers

D I S S E R T A T I O N

zur ErlangungdesakademischenGrades

doctorrerumnaturalium(Dr. rer. nat.)

vorgelegt demRatderPhysikalisch-AstronomischenFakultat der

Friedrich-Schiller-Universitat Jena

vonDipl.-Phys.RandolfButzbachgeborenam16.02.1969in Kassel

Gutachter:

1. Prof. Dr. E. FORSTER

2. Prof. Dr. U. SCHUMACHER

3. Prof. Dr. G. DRAGER

Eingereichtam: 23. Oktober2000

Prufungstermine:15. und21. Dezember2000

TagderoffentlichenVerteidigung:4. Januar2001

For myparents

andCecılia

Zusammenfassung

In der vorliegendenArbeit werdenKonzeption,EinsatzundAnwendungvon torisch

gebogenenKristallenfur dierontgenoptischeUntersuchungvonlaserproduziertenPlas-

menzurlasergetriebenenKernfusionsowiezurverstarktenspontanenEmission(”Ront-

genlaser“ ) nahedemsogenanntenWasserfensterdargelegt. Aus dengewonnenenDa-

tenkonnenelementarePlasmaparameterbestimmtwerden.

Der ersteTeil derArbeit stellteinenBeitragzurUntersuchungvonRayleigh-Taylor-

Instablitatendar, diedergewunschtenVerdichtungundZundungdesBrennstoffesder

lasergetriebenenKernfusiondurchTragheitseinschlussentgegenwirken.

Ziel diesesTeils dervorliegendenArbeit war es,eineabbildendeRontgenoptikzu

entwickeln,die denhohenAnspruchenzurBeobachtungderRayleigh-Taylor-Instabi-

lit atengenugt,d.h.einrontgenoptischesSystemzurzweidimensionalenmonochroma-

tischenAbbildung bei gleichzeitighoherLichtstarke, hoherAuflosung(�

3 µm) so-

wie mindestens400µm Scharfentiefe.Die Optik sollteamHochleistungslaserGEK-

KO XII am Instituteof LaserEngineering(ILE) an der Universitat Osaka/Japanim

RahmendesHIPER-Forschungsprogramms(High-IntensityPlasmaExperimentalRe-

search)zumEinsatzkommen.

GrundlagedieserabbildendenOptik ist ein 7 � 7 mm2 großertorischgebogener

QuarzKristall, derbei einerVergroßerungvon zunachst30 � undspater70 � verwen-

detwerdensoll.

Die Simulationder konstruiertenOptik durchStrahlbahnverfolgungsrechnungen

zeigt,daßmittels torischgebogenerKristalle die gefordertenBedingungennicht nur

erreicht,sondernsogarubertroffen werdenkonnen:Die Rechnungergibt einezu er-

wartendeAuflosungvon � 1 µm bei einerKristallaperturvon 3.5 mm undeinerVer-

großerungvon 30 � .

Jedochkonnteder experimentelleNachweisder gefordertenAuflosungaufgrund

iii

von erhohtenAnforderungenan die Justageund technischenProblemenin der zur

VerfugungstehendenStrahlzeitambenotigtenHochleistungslasernochnichterfolgen.

Die bestein dieserArbeit nachgewieseneAuflosungwar6 µm.

Als WeiterfuhrungderArbeit zumNachweisder theoretischzu erwartendenAuf-

losungwird alternativ ein detailliertausgearbeiteterVersuchsvorschlagunterVerwen-

dungeinesRontgengeneratorsvorgestellt.

Der ersteTeil dieserArbeit schließtmit derqualitativenDiskussiondererstener-

haltenenBilder der beobachtetenRayleigh-Taylor-Instabilitaten.Fur einedetaillierte

AuswertungderDatenist einerseitsdie erhalteneDatenmengezugeringundanderer-

seitsdasSignalaufgrundvonProblemenmit demLasersystemzuschwach.

Da einerseitsdie Untersuchungder Rayleigh-Taylor-InstabilitatengroßteWich-

tigkeit fur die Fusionsforschunghat und andererseitsdie hier entwickelte Optik das

”Herzstuck“ desHIPER-Programmsdarstellt,wird dieseArbeit uber dasEndedes

hier gegebenenZeitrahmenshinausweitergefuhrt.Eswird erwartet,daßdie hier kon-

struierteRontgenoptikwichtigeErkenntnissefur dasDesignvon zukunftigenBrenn-

stoffkapselnliefert.

Der zweite Teil dieserArbeit beschaftigt sich mit der spektralund eindimensio-

nal raumlichaufgelostenBeobachtungvon laserproduziertenPlasmenzur verstarkten

spontanenEmission(”Rontgenlaser“ ) nahedemsogenanntenWasserfenster. Ziel der

Arbeit wares,mittelseinesabbildendenRontgenspektrometerseinenZusammenhang

zwischender verstarktenEmissiondesnickelahnlichen4d � 4p–Ubergangsund der

nickelahnlichen4 f � 3d Resonanznachzuweisen.HierbeisolltedasRontgenspektrum

entwederzeitlich odereindimensionalraumlichaufgelost aufgezeichnetwerdenund

ausdengewonnenenDatensolltenweitergehendeplasmaphysikalischeFragestellun-

genwie BestimmungderElektronentemperaturundElektronendichtein Abhangigkeit

derraumlichenPositionim Plasmaerfolgen.

Dazuwurdeein abbildendesSpektrometerauf Basisvon torischgebogenenKri-

stallenentwickelt, welchesentwederin Verbindungmit einerCCD-Kameraraumlich

aufgeloste,aberzeitintegrierteSpektrenlieferteoderin Verbindungmit einerSchmier-

bildkameraeineraumlichintegrierte,aberzeitaufgelosteAufzeichnungder Spektren

ermoglichte.

DasSpektrometerwird fur denverwendetenArbeitsbereichvon 5.7–6.4A voll-

standigdurchStrahlbahnverfolgungsrechnungenin Bezugauf denEinflußderQuell-

iv

große,AuslenkungderQuelleausderBeugungsebenesowie GroßederKristallaper-

tur charakterisiert,gefolgt von dem experimentellenNachweis,daßdie erreichbare

Auflosungim vorliegendenFall durchdieAuflosungderCCD-Kamerabegrenztwird.

DasSpektrometerwurdedurchVergleich der beobachtetenLinienpositionenvon

Al-Plasmenmit derenwohlbekanntenWellenlangenin-situ kalbriert. Aus dem Ver-

gleichkonnendie genauengeometrischenAbstandedesSpektrometerserhaltenwer-

den,durchwelcheumgekehrtdie Dispersionsrelationvollstandigbeschriebenwerden

kann.WennweiterhineinekalibierteCCD-KameraalsDetektorverwendetwird, kann

die absoluteAnzahldervomPlasmaemittiertenPhotonenerhaltenwerden.

Die plasmaphysikalischenExperimentewurdenebenfalls am Hochleistungslaser

GEKKO XII am ILE in Japanisch-Chinesisch-Franzosisch-DeutscherZusammenar-

beit durchgefuhrt. Die ExperimentebestatigendenvermutetenZusammenhangzwi-

schenderEmissionder4 f � 3d-Linie undderverstarktenEmissiondes4d � 4p Uber-

gangs.Als optimalePumplaserintensitatwurdefur Tantal(Ta) � 1 � 5 � 1015 W/cm2 er-

mittelt; einVorpulsvon4% erhohtdie Ionisationin denNi-ahnlichenZustandumdas

bis zu 35–fache.Die weitergehendeAuswertungder zeitlich und raumlichintegrier-

tenSpektrenergibt fur TaeinebeobachteteElektronentemperatur, dieunabhangigvon

derPumplaserintensitat im Bereichvon � 0 � 2 � 2 � 2��� 1015 W/cm2 � 150eV betragt.

MoglicheUrsachenfur diesebeobachtetekonstanteTemperaturwerdendiskutiert.Die

AuswertungderElektronendichtederTa-PlasmenzeigteinenlinearenZusammenhang

zwischenPumplaserintensitat und erreichterDichte bei konstantgehaltenerPlasma-

große.

WeiterhinkonntedieWellenlangeder4 f � 3d-Linien gemessenwerden.Die Uber-

einstimmungdergemessenenWertemit theoretischenLiteraturwertenist beieinerAb-

weichungvon ��� 1 ��� � 10� mA, abhangigvom Autor undderSpektrallinie,sehrgut

bisgut.

Die Auswertungder Datenzeigt, dasseine detaillierteAussageuber die Elek-

tronentemperaturnur in Verbindungmit zeitaufgelostenSpektrenmoglich ist. Diese

konntenjedochin derzurVerfugungstehendenStrahlzeitbishernichtgewonnenwer-

den,sindaberfur die nachsteExperimentseriein voraussichtlichzwei Jahrengeplant.

v

Contents

Zusammenfassung iii

Intr oduction 1

1 Studieson Hydr odynamic Instabilities 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Inertial ConfinementFusion . . . . . . . . . . . . . . . . . . 5

1.1.2 Rayleigh–Taylor Instabilities . . . . . . . . . . . . . . . . . . 8

1.1.3 Theprinciplelayoutof theexperiment. . . . . . . . . . . . . 12

1.2 MonochromaticX-ray imager . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Definitionof resolution. . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Two-dimensionalimagingusingbentcrystals . . . . . . . . . 15

1.2.3 Requirementsof thecamera . . . . . . . . . . . . . . . . . . 19

1.2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.5 Designof theX-ray imager. . . . . . . . . . . . . . . . . . . 27

1.2.6 Calibrationof thegoniometer . . . . . . . . . . . . . . . . . 29

1.3 Imagingtest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3.1 ImagingtestatGEKKO XII . . . . . . . . . . . . . . . . . . 32

1.3.2 Proposalfor animagingtestat anX-ray generator . . . . . . 43

1.4 Experimentalsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.5 Preliminaryresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.6 Concludingremarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2 Diagnosticsof X-Ray Laser Plasmas 50

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2 Determinationof plasmaparameters. . . . . . . . . . . . . . . . . . 53

vi

2.2.1 ElectronTemperature. . . . . . . . . . . . . . . . . . . . . . 53

2.2.2 ElectronDensity . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3 TheSpectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.1 Requirements. . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.2 Theprincipleof theJohann-typespectrometer. . . . . . . . . 60

2.3.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.3.4 Characterizationof theperformanceof thespectrometer . . . 68

2.3.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4 Experimentalsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.5 Processingof theraw data . . . . . . . . . . . . . . . . . . . . . . . 81

2.6 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.6.2 Comparisonof the4d � 4p and4 f � 3d transition . . . . . 86

2.6.3 Studieson tantalumplasmas(point focus) . . . . . . . . . . . 87

2.7 Discussionof thetemperature . . . . . . . . . . . . . . . . . . . . . 91

Summary and Outlook 95

Bibliography 97

Ehrenwortliche Erkl arung der Selbststandigkeit 111

Curriculum Vitae 113

Acknowledgments 115

vii

Intr oduction

High intenselight amplificationby stimulatedemissionof radiationopensthelabora-

tory accessto physicalphenomenaasthey exist in theinteriorof stars.Theinteraction

of high intenselaserlight with mattercancausethegenerationof a practicallytotally

ionizedgaswhich is calledPlasma[1]. Eventhoughthis ‘fourth stateof matter’ is an

exceptionon earth,it is the mostcommonstateof matterin the nature[2]. The sun

andthestarscanbeconsideredasanensembleof plasmas.

Besidesthe pure fundamentalresearchof the behavior of naturein this extreme

region, it is possibleto simulateastrophysicalphenomenain thelaboratoryonamuch

smallerscalelike thethermonuclearfusionprocess[3–8], theprobableenergy source

of the stars,or the explosionof supernovae[9–11]. The former one is particularly

interestingsinceoneexpectsto usethereleasedenergy of thefusionprocessonceasa

terrestrialenergy source.To realizelaser-drivennuclearfusionwith energy gain,huge

lasersystemsare required,which can deliver more than one mega-joulein several

nanoseconds[12].

Another applicationof suchlarge laserfacilities requiredfor the thermonuclear

fusionis thequestfor X-ray lasers[13–17]with emissionline insidethesocalledwater

window, theregionbetweentheK–absorptionedgesof oxygenandcarbon.Thisrange

is particularlyinterestingfor biologicalapplications:Theabsorptionof thewater, or

moreprecisely, theoxygenin thewaterof biologicalsolutionsis relatively smallwhile

thecarbonof theorganicmoleculesabsorbverywell.

In all cases,the main emissionfrom the laserproducedplasmais in the X-ray

regime[18,19]. Therefore,X-ray diagnosticsplayacrucialrole for theunderstanding

of theplasmaphysics.By analyzingtheemissionor absorptionspectraof theplasma,

onecanobtaininformationsuchastheelectrontemperatureor electrondensityof the

plasma[18,19]. Theelectrontemperaturecanbeobtainedfor instanceby measuring

theintensitydistributionof thespectrallinesor by preciselymeasuringthewavelength

1

of theemissionline which is shifteddueto thealteredCoulombpotentialof theatom

[18,19]. The distribution of the ionization stateof atomsin the plasmacausesthe

appearanceof satellitelinesneartheemissionline, from which againonecanobtain

theelectrontemperature[18–22].

The width of the spectralline is directly relatedto the life time of the excited

state[18,19], which in turn is shorteneddueto collisionswith otherparticles: The

higher the densityof the plasma,the more likely is a collision, i.e. the broaderthe

line becomes.Thus,thewidth of the line canbedirectly relatedto thedensityof the

plasma[18,19,23–26].

In order to measurethe line width or shift of the spectrallines, a high spectral

resolutionof about∆λ � λ � 10� 4 � � � 10� 3 is required.

Otherinformationswhichmightbeobtainedfrom thespectraaree.g. thedetection

of hot electronsby observinganun-shiftedKα-line, sincehot electronsarea precon-

dition for theemissionof Kα photonsfrom coldmatter.

Moreover, the plasmadynamicscan be studiedby the spatially and temporally

resolvedobservationof theemissionlinesfrom traceratomsasmarkersof thediffer-

ent partsof the plasmas[27–30]. Thus,ideally the diagnosticinstrumentgiveshigh

spectralresolutioncombinedwith highspatialandhigh temporalresolution[27–31].

Among all imaging and spectralresolvingelementslike e.g. pinholes,gratings

and Kirkpatrick-Baezmicroscopes,two-dimensionallybent crystalsare best-suited

[32–34]. While pinholesare simple devices and can give two-dimensionalimages

of moderate( � 5 � ��� 10µm) spatialresolution[35], theluminosityis verypoorandthe

imagesarepolychromaticandthuscannotprovide informationsaboutthespectrumof

theplasma.

Kirkpatrick-Baezmicroscopes[36] have a muchhigherluminosity thanpinholes

combinedwith a high spatialresolutionof about � 3 µm [37] andan undercut-off

wavelengthdueto theuseof specularreflectionat largebentmirrorswhichcollectthe

emittedphotonsoveralargesolidangle.But thespectralwindow, whoseuppercut-off

wavelengthcanbe obtainedby the useof filters, is still very large andthereforenot

suitedfor spectroscopy. An enhancementof theKirkpatrick-Baezmicroscopecanbe

obtainedby usingmultilayercoatingof the reflectingsurfaceswhich actsasa Bragg

diffractor [35,38]. However, the spectralresolutionis in the order of 1 keV [35],

correspondingto ∆E � E � 0 � 2 anddependsstronglyonslopeerrorsof thesurface[39].

Only two-dimensionallybentcrystalscancombinethe large dispersionandthus

2

high spectralresolutionobtainedat the diffractionof X-rays at crystalswith one-or

two-dimensionalfocusingatlarge( � 0 � 1 ��� � 10msr)aperturesizesandthusverybright

spectraand/orimagingof thesource:Eithertwo-dimensional(2-D) high(�

3µm) spa-

tial resolutionin onenarrow spectralchannel(∆λ � λ � 10� 2 � 10� 4) or 1–D spatial-

andspectral(∆λ � λ � 10� 4) resolutioncanbeobtained.Thetemporalresolutioncanbe

obtainedby usingstreak-andframingcameraswith respectively sub-picosecond[40]

and � 40 ps[31,41] temporalresolutionasa detector, or by a shortpulseplasmaasa

‘speedlight’ for shadowgraphy.

In the presentwork toroidally bentcrystalsystemswereappliedto two different

experiments:Thehighspatiallyresolvedobservationof thegrowth of Rayleigh-Taylor

instabilities,which area harmfuleffect for thecompressionandignition of thefusion

capsulein the inertial confinementfusion process(Chapter1) andthe diagnosticof

the lasingplasmaof a collisional X-ray laserwith the aim of obtaininglasinginside

the so-calledwaterwindow (Chapter2). For both experiments,a particularoptical

instrumentwasdesignedandconstructedwhich fits theparticularrequirementsof the

experiment.

Theseexperimentscanbeonly carriedout at largelasersystemssuchasNOVA in

Livermore/USA,PHEBUSin Limeil/Franceor, likehere,GEKKO XII in Osaka/Japan

in combinationwith toroidally bentcrystals.Thebeamtime at suchlargelaserfacili-

tiesis very limited dueto thelargenumberof researchissuesanda low repetitionrate

of the laserdueto therequiredcooling time: The repetitionrateof GEKKO XII, for

instance,is only oneshotper two hours. Thus,thereareat mostfour shotsper day

possible. The time requiredfor settingup the experimentandalignmentaswell as

frequentproblemswith thelasersystemreducethenumberof availableshotsto effec-

tive 2–3shotsper day. For normally, the quality of the datais influencedby several

problemsandthereis no possibility within two yearsto repeatan experimentunder

improvedconditions.Sinceonly themostimportantresearchissueshave a chanceto

geta secondtime beamtime allocated,thedataobtainedat theselargefacilities (and

presentedin this work) areunique,eventhoughtheir qualitymight bereduced.

Ontheotherhand,only theX-ray opticsgroupattheUniversityof Jenais currently

ableto manufacturetoroidallybentcrystalswith asufficientlyhighquality, i.e. anerror

in thebendingradii of ∆R� R � 10� 3. Theindependentcontroloverbothbendingradii

offers the possibility to designthe optical instrumentaccordingto the experimental

requirements.As a result,ahighspatialresolutionin both,thesagittalandmeridional

3

planeat largeaperturesizes[30,31,34] andthushigh luminosity(cf. Ch.1) might be

obtained.Also, it is possibleto designa highly resolving1–D imagingspectrometer

for theoperationat a distanceof morethan1 m from theplasmasource,which might

be usedin combinationwith a streakcamera(cf. Ch. 2). Sincethesefeaturesarea

preconditionfor theexperimentspresentedin this work aswell astherequirementof

a largelasersystem,theseexperimentsareuniquein theworld.

4

Chapter 1

Studieson Hydr odynamic Instabilities

1.1 Intr oduction

1.1.1 Inertial ConfinementFusion

The fusion of deuterium(D) andtritium (T), two heavier isotopesof hydrogen(H),

producehelium plus a neutronandreleases2 � 85 � 10� 12 J or 17.6 MeV of energy.

Evenif this numberon its own appearsvery small, it is about10000000timesmore

thanthechemicalbindingenergy, and,whenscaledto macroscopicnumbers,theenor-

mousenergy becomesobvious: 1 mg DT releases340MJ, this is equivalentto 85 kg

TNT [12]. Sincein normalwateroneDeuteriumatomcomeswith every5500normal

Hydrogenatoms,onecanobtainfrom oneliter normalwaterthefusionenergy equiv-

alentof 300 liter of gasoline[4]. Thusthe oceanskeepan energy reservoir for 100

billion years(assumingthepresentenergy consumptionandanexclusiveuseof fusion

energy) [42]. Moreover, fusion energy is a ‘clean’ energy: The wasteproductis the

nobelgasHelium.

To startthefusionprocess,onehasto bringthenucleicloseenoughtogethersothat

thestronginteractionof thenucleicanact.However, bothfusionnucleiarepositively

charged,i.e. therearerepulsive forceswhich have to beovercomebeforethenuclear

forcescanact.

The requiredenergy of about1 MeV to passthis ‘Coulomb-wall’ canbe easily

producedwith particleaccelerators.This so-called‘neutrongenerator’is availableat

researchinstitutesasa ‘clean’ alternative to fissionreactorsfor neutronresearch[43].

However, the efficiency of this schemeis by far too low that onecould expectmore

5

Figure 1.1. Theconceptof Inertial ConfinementFusion(LLNL diagram).

releasedenergy from thefusionthanrequiredto startthefusionprocess[4].

A moreefficientscheme,whichwouldallow fusionona largerscale,is to heatthe

fuel to sufficiently high temperatures(10–100keV). At this relatively low energy, the

tunnelprobabilityis alreadysufficiently high. However, theparticleswill bescattered

from eachotherandonly very rarely, oneparticlewill tunnel throughthe Coulomb

wall [4]. Thereforeonehasto increasethetunnelprobabilityby increasingtheparticle

densityn by confinementfor a long confinementtime τ. Theproductnτ is a measure

for thescientificfeasibilityof thereaction,andthethresholdvalueatwhich ignition is

possible,is called‘Lawson-criterium’.For DT, e.g. , nτ � 1014 s/cm3. Insidethestars,

theconfinementis ensuredby thelargegravitational force,which is muchtoo low on

earth.For man-madefusions,onehasto find othermeansof confinement[4].

Oneway is to try to get largeconfinementtimes,i.e. about1 second,in a storage

ring socalledmagneticfusion.Thenucleiareacceleratedin aparticleacceleratorand

circulatein astoragering. On their trajectorythenucleiarefocused,i.e. confined,via

6

magneticfieldsby theLorentz-force.However, to reachsuchhighconfinementtimes,

onehasto correctall higherordertrajectory‘errors’ in thesesocalledtokamaks, which

leadsto complicatedmagneticfieldsandtrajectories[6].

Anotherapproachto fulfill theLawsoncriteriumis to usehigh particledensities,

but low confinementtimes. I.e. onecompressesthe fuel to the requireddensityso

that the fuel ignitesandburnsbeforetheparticlesseparateagain.Sinceoneusesthe

inertia for the confinement,onespeaksof Inertial ConfinementFusion (ICF). On a

largescale,this schemehasbeendemonstratedsuccessfullyin 1950in theHydrogen

Bomb,wherethecompressionis doneby ‘conventional’A-bombs[44].

Immediatelyafter the inventionof the laserby Maymanin 1960,Nuckolls et al.

proposedin 1961[3,6] to uselasersfor therealizationof theICF schemeon a much

smallerscale:A hollow shell of frozenD–T fuel is filled with low densityD–T gas

( � 10 mg/cm3) andcoveredby a plasticlayer, thesocalledablator. Now, oneshoots

with high-powerlasersontotheablator, whichexpandsrapidlyin alow densityplasma

andacceleratesthefuel shellrocket-like to thecenterof thesphere.At thecollisionat

thecenterof thefusioncapsule,thekinetic energy of theimplodingshell is converted

to inner (thermal)energy, which heatsthe fuel to the fusion temperature.The fuel

ignitesin asmallspot,thesocalled‘hot spark’.Theenergy of theα-particlesproduced

in thehotspotwill bedepositedin thefuel andheatsup thefuel further:Thefuel will

burn self-substaining.

To give someroughnumbersfor energy gain conditions[12] in orderto achieve

a hot spot,theshell hasto be acceleratedwithin the capsuleradiusto a velocity, i.e.

a specificenergy, of v � 3 � 107 cm/s. For 3 mg fuel, this correspondsto 135 kJ

kineticenergy of theshell.Assumingacouplingefficiency of 10% betweenthedriver

laserand the shell, the driver laserhasto deliver 1.35 MJ. For an assumedinitial

shell radiusR0 � 1 mm this energy hasto be deliveredin 7 ns, correspondingto an

accelerationof 4 � 1015 cm/s2 and an averagepower deliveredto the fuel of 19 �1012 W, or an averagedriver power of 190 � 1012 W. The new laserfacilities like

the National Ignition Facility (NIF) at LawrenceLivermoreNational Laboratoryin

California/USAor theLaserMegaJouleproject(LMJ) in Francewill beableto deliver

1.8 MJ in 192and240beamsfrom 2002and2012on, respectively [45]. Thusthese

facilitieswill beableto meettherequirementsfor energy gain.

Anothercritical point in orderto achievecompressionis thatonehasto makesure

that the driving forcesarewell balanced,i.e. that the sumof the forceswith respect

7

to the centerof massvanishes.Otherwise,the centerof massis acceleratedandthe

compressionwill not besufficient to ignite thefuel [46,47]. To realizethesymmetric

illuminationof theablator, two schemesarepursued:Direct driveandindirectdrive.

In indirectdrive, thefusioncapsuleis placedinto a cavity (Hohlraum)madefrom

gold for instance.Thelaserbeamsaredirectedto theinnerwalls of thecavity where

they produceX-ray Hohlraumstrahlung.The(ideally) very homogeneousHohlraum-

strahlungthendrives the fusion capsule. The advantageof this schemeis that it is

relatively easyto obtainahomogeneousilluminationof thetargetandthecouplingef-

ficiency for X-raysto theablatoris muchbetterthanfor thedirectlaserlight. However,

the Hohlraumstrahlungis not ideal dueto the entrancewindows for the laserbeams

aswell asobservation windows for diagnostics.Also, the requiredlaserintensity is

higher, sincethelaserenergy hasto befirst convertedinto X-raysbeforeit canbeused

for thedriveof thecapsule.

In direct drive, the fusion capsuleis illuminateddirectly by several laserbeams.

Theadvantageof thisschemeis thattherequiredlaserintensityis lessthanin indirect

drive, but thereare much higher requirementsto the laserbeamoptics: The beam

profile hasto be extremelysmoothin orderto avoid non-uniformitiesin the capsule

surfacedueto laserimprint, whichtriggersRayleigh-Taylor instabilities,whichin turn

canquenchthe ignition. Also, the laserenergy in thedifferentlaserbeamshasto be

verywell balancedin orderto ensureuniformcompression[46,47].

An alternativeto theabovedescribedschemeswasproposedbyTabaketal. in 1994

[48]. In thisso-called‘FastIgnitor’ scheme,thefuel is conventionallycompressedand

thenheatedup furtherby anadditionallaserpulsein orderto ignite thefuel.

1.1.2 Rayleigh–Taylor Instabilities

Oneof themostimportantissuesin ICF researchis hydrodynamicinstability, in par-

ticular Rayleigh–Taylor instabilities,which play an essentialrole in the confinement

processof inertial confinementfusionplasma:

Rayleigh–Taylor instabilities[49,50] (RTI) occurwhena high densityfluid is ac-

celeratedagainsta fluid with a lower density. Classicallyit occur, for examplewhen

onetries to float a layer of wateron top of a layer of lighter fluid suchasoil. Care-

fully doneit might be possible,but slight disturbancein thecontactsurfacebetween

waterandoil will triggeroscillationsof thecontactsurfacethatwill grow until globs

of oil begin to passthroughthewaterto thesurfaceunderbuoyancy forces �Fa driven

8

by gravity g.

In ICF we encountera similar situationduring the accelerationanddeceleration

of theshell: In theaccelerationphase,theexplodinglight ablationplasmapushesthe

denseshell inward andin the stagnationphase,the light centralplasmaslows down

the incomingdensermaterial. In both cases,the inertial force of the shell actslike

the buoyancy force in the exampleabove while the accelerationandde-acceleration,

respectively, actlike thegravitation force.

TheRTI cancausequenchingof the ignition, sinceduring theaccelerationphase

theRTI causesamixing betweentheablatorandthemainfuel while in thestagnation

phasethe RTI causesa mixing of the hot sparkwith the cold main fuel. A good

understandingof theRTI andhow to mitigatethemarethereforemandatory. Thusthe

RTI is for themomentoneof themainresearchissuesin ICF research.

Classically, theRTI in thelin-

Fluid 1

Fluid 2

PSfragreplacements

ξ � t �ρ1

ρ2

x

y

�g � �Fg

�a � �Fa

λRT � 2πk

Figure1.2. Definitionof theRayleigh–Taylor instability.

eargrowthphasecanbedescribed

by a considerationof the vari-

ation of the potentialandkine-

maticalenergy (cf. i.e. [51]) which

leadsto theclassicalstandardmodel

ξ � t ��� ξ0 � eγ t � (1.1)

whereξ is theperturbationam-

plitudefrom theequilibriumpo-

sition andξ0 � ξ � t � 0� i.e. the

initial amplitudeof thenon-uniformitiesat theinterfacebetweentwo liquidsof differ-

entdensities.t is thetimeandγ is thegrowth ratewhichcanbeexpressedby

γ2 � ρ2 � ρ1

ρ2 � ρ1� ��� �A

gk � Agk (1.2)

with thedensitiesρ1 andρ2 of thetwo facingliquids, theaccelerationg andthewave

numberk � 2π � λRT, whereλRT is thewavelengthof theperturbation(cf. Fig. 1.2). A

is calledtheAtwoodnumber.

For ρ2 � ρ1 γ becomespositive andξ � t � grows exponentially:Thesystemis un-

stable.For ρ1 � ρ2 γ2 is negative, i.e. γ is imaginaryandtheξ � t � describesundamped

oscillations.

9

However, in ICF, thesituationis muchmorecomplicatedthantheclassicalmodel

describedby Eq. (1.2). Thefuel is no longerincompressibleandthe(mainfuel) layer

hasa finite thickness.Theablationprocessof theouterlayercausesa mass-andheat

flow aswell asa densitygradient.Also, thethermalconductionby radiationanden-

ergeticparticleshasto betakeninto account.While Eq. (1.2)assumesaflat interface,

in ICF a sphericalconvergenceoccurs. Also, in the simple classicalmodel above,

non-linearmultiple wavelengtheffectslike coupling,mixing andbubblesremainun-

consideredaswell as3-D effects. Thefinite layer thicknessof thefuel causeson the

one handa reductionof the growth rate [51–53] and on the other hand,synergetic

effectsbetweentheRTI invokedat theablationfront andat thecollapse:TheRTI in-

vokedat theablationfront dueto outersurfaceroughnessandnon-uniformitiesin the

laserbeams,which imprintsadrive-beamstructureontothesurface,canfeed-through

thefuel andinterferewith theRTI duringthecollapsefrom theroughnessof theinner

surface.

Nevertheless,all theseeffectscanbe summarizedby a simplefit formula, theso

called‘Takabe-formula’[54,55]

γ � 0 � 9 kg � βkVa (1.3)

with thefit parameterβ � 3 � 4 andtheablationvelocity

Va � m� ρa (1.4)

whereρa is thepeakdensityat theablationfront andm is thearealmassablationrate.

Thedensitygradientat theablationfront is sosharp(ρ1 � 1, ρ2 � 10� 4) thatonemay

assumeA � 1. Thus,thetermAkg reducesto kg.

Equation(1.3) is intendedfor direct-driveexperiments,but canbegeneralizedas

γ � !kg

1 � kL� βkVa � (1.5)

thesocalled‘modifiedTakabeformula’ [56]. Here,L �#" ρ �$� dρ � dz�&% min is thedensity

gradientscalelengthin theablationfront.

More recently, Betti et al. [57] developedanimprovedmodelfor RTI in thepres-

enceof ablationthatobtainsthedensityandvelocity equilibriumprofilesself consis-

tently from amodelof thermalconduction.Thegrowth ratesobtainedfrom thismodel

arein generalcomplex, but have thesamebehavior asEqs.(1.3)and(1.5).

10

Contraryto theclassicalgrowth rate,asdescribedby Eq. (1.2),whereγ increases

with higherperturbationwavenumberk, theablationcausesareductionof thegrowth

rate [54] at higher perturbationwave numbersup to a cut-off wave number, where

γ � 0 andno growth occursasshown in Fig. 1.3.

Cut−off ?

Classical

Takabe/Simulation

Experiment

0

1

2

3

4

5

0 5 10 15 20 25 30 35 40 45 50

Perturbation wavelength/[µm]

R−

T g

row

th r

ate

[1/n

s]

Figure 1.3. The growth rate of the

Rayleigh–Taylor instability. Thedot-

ted line shows the classicalgrowth

rate (Eq. (1.2)), while the solid and

dashedline show thegrowth ratefrom

theTakabeformula(Eq.(1.3)) for the

simulated RTI (ILESTA 1-D code)

and experimentalvalues(Ref. [58]),

respectively.

Experimentalvaluesfor β vary: Remingtonet al. [59] obtainedin 1991with indi-

rectdriveβ � 1 � 2; Nakaietal. [60] obtainedin 1992with directdriveβ � 3. Azechi

et al. [61] obtainedin 1997 at the ILE β � ρa � 2 � 5 cm3/g or β � 4. Shigemoriet

al. [58] observedin 1997amuchstrongerstabilizationeffect thanpredictedby Takabe

etal. [55]: They measuredβ � ρa � 2 � 6 cm3/g whichgivesβ � 6 � 5 � 9 � 1 for asimulated

ablationdensityρa � 3 � 5 � 2 � 5 g/cm3. Comparedto theclassicalgrowth ratefrom Eq.

(1.2) this givesγexp � γclass � 50%. Someof their experimentalvaluesarepresented

in Fig. 1.3 togetherwith theclassicalgrowth rateandtheexpectedgrowth ratefrom

the ILESTA 1-D simulationcode[62]. This stabilizationis interpretedby nonlocal

electronsthat may penetratedeepinto the target, therebyreducingthe target density

via pre-heating,andenhancingablativestabilization[63]. Glendinninget al. [64] ob-

servedγexp � γclass � 50%, too, for verydifferentexperimentalconditions:longerlaser

wavelength,lower laserintensityandhighertargetZ. Comparedto their simulations,

theobservedgrowth rateis about18% lower thanpredicted.

Despiteits greatimportancefor ICF targetdesign,all theseexperimentaldatashow

that theRTI arestill not well understood:Thegrowth ratemeasurementsarestill not

conclusive,thephysicalorigin of thefit parameterβ is still unknown. It couldbeonly

demonstratedthat thereis a stabilizingeffect which is muchstrongerthanexpected.

Also, the predictedcut-off wavelengthcould not yet be observed which is primarily

dueto thetechnicaldifficulty of theexperiment(cf. Sec.1.1.3).

11

BacklighterTarget (Mg−foil)

(CCD−, Streak− or Framing Camera)

Detector

Toroidally bent

Probe Laser(PW−M)

Drive Laser(3+9 Beams)Sample (with

Perturbation)

Quartz Crystal

Figure1.4. Thegenericexperimentalsetupfor theobservationof Rayleigh-Taylor instabilities

in thecaseof thepresentwork.

Theexperimentpresentedherewasintendedto clarify thesituationunderthesim-

ulatedconditionsof futurehigh-gainfusioncapsulesandto observe thecut-off wave-

length,which is expectedto besomewherearound3 � 10 µm.

1.1.3 The principle layout of the experiment

Theprinciple layoutof all experimentsfor theobservationof RTI is asfollows (Fig.

1.4): Oneirradiatesaplanarsampletargetwith laserlight in orderto createtheplasma.

Dependingon theexperimentitself, oneprovokesRTI e.g. by non-uniformitiesin the

incidentlaserbeamor by imposednon-uniformitieson thesurfaceof thesample.An-

other plasmais generated‘behind’ the sampleplasmaas a backlightersource. By

choosinganappropriatewavelengthof thebacklightingX-ray source,togetherwith a

monochromaticimagingoptic, i.e. knowing the wavelengthλ of the backlighterand

thedensityof theplasma,onecanobtaintheoptical thicknessof thesampleby mea-

suringthetransmittedintensityandcompareit with theincidentintensity:A variation

of thethicknessof thesamplewill causeavariationin thetransmittedintensity. By ac-

quiringthetransmittedintensityatdifferenttimesby usingaframing-or streakcamera

or ashortpulsebacklighteratdifferenttimes,onecanobtainthegrowth rateγ.

The backlighterwavelengthλ hasto be chosenasa compromisebetweenthree

constraints:For a preciseobservationof slight variations( � µm) of the thicknessof

the sample,one needsa high absorptioncoefficient and thereforea relatively large

wavelength. However, the initial thicknessof the sampleis very thick comparedto

thethicknessvariationsdueto theRTI sothatmostof theintensityof thebacklighter

12

radiationwill beabsorbedin themainbodyof thesample.Thus,in orderto increase

the amountof transmittedphotons,onewould prefera shorterwavelengthin order

to obtaina smallerabsorptioncoefficient. In additiononeneedsa bright backlighter

source,i.e. one needsto find an intensespectralline in the small rangewherethe

massabsorptioncoefficient is largeenoughto provideapreciseobservationof theRTI

andsmallenoughto ensurethatsufficiently many photonsaretransmittedthroughthe

sample.In otherwords,whendefiningthetransmission

T � Itrans

Iinc� exp �&� µρt � (1.6)

with theincidentandtransmittedintensityIinc andItrans, respectively, themassabsorp-

tion coefficient µ, the densityρ andthe thicknesst, µ shouldbe selectedassmall as

possiblein orderto geta largetransmissionT. However, for large ‘resolution’ of the

thicknessvariations

∆T � T �'� µ � ∆ � ρt � (1.7)

µ shouldbe aslarge aspossible.For an assumedthicknessof the plasticsampleof

50 µm, a thicknessvariationdueto theRayleigh-Taylor instabilitiesof � 3%, andan

absorptioncrosssectionof 2 � 10� 20 cm2 asuitablewavelength,whichis smallenough

to passthesampleandlargeenoughto giveareasonablecontrast,theprobewavelength

hasto bearound9 A. In thiswavelengthregiontheH-like12Mg Lyα emissionline with

a centertransitionwavelengthof the H-like doubletof λ � 8 � 42235A is a suitable

compromisebetweenwavelengthandhigh intensity[63].

Additionally, the imagingoptic hasto have a small energy bandpassin order to

ensurethatonly thedesiredphotonenergy of thebacklighteris detectedandtheself

emissionfrom thesampleis suppressed[65,66].

In summary, thissetshighdemandsontheX-ray imagingsystem:Theopticshould

provideahighspatialresolutionof betterthan � 3 µm in orderto resolve theexpected

cut-off perturbationwavelength,theimagehasto bemonochromaticin orderto havea

well definedabsorptioncoefficient,whichenablesaprecisedeterminationof thethick-

nessvariation,andto suppresstheself-emissionfrom thesample.Sincetheemission

linesof thebacklightingplasmahave fixeddiscretewavelengths,thespectralwindow

of theopticshouldbechosenfreely in orderto matchtheselectedspectralline. More-

over, theapertureof theoptic shouldbeaslargeaspossiblein orderto focusasmany

aspossibleof thefew transmittedphotonson thedetector.

13

Only two-dimensionallybentcrystalscanfulfill all of theabovementionedrequire-

mentsat once[32,33], andarethussuitablefor theX-ray imager.

1.2 Monochromatic X-ray imager

1.2.1 Definition of resolution

Beforedescribingthe basics,design,andperformanceof the constructedimagerfor

the observation of Rayleigh-Taylor instabilities,the (in this work omnipresent)term

resolutionshouldbedefined:

Theimageof apointon thedetectorwill be,dueto imagingerrors,notapoint,but

acomplicatefunction,calledPoint SpreadFunction

PSF� PSF� x � y� (1.8)

wherex andy areorthogonalcoordinatesin asystemwith its origin attheimagecenter.

Usingthepoint spreadfunctionasa startingpoint, we cancalculateotherdescriptive

functionsin commonuse:

TheLineSpreadFunctioncanbecomputedfrom thePSFaccordingto [67]

LSF� x�(�*),+ ∞� ∞PSF� x � y� dy� (1.9)

TheLSF givestheresultof scanning,in x-direction,over the imageof a point source

usinga slit which hasinfinitesimalwidth andis infinitely long in y-direction. It is the

responseof thesystemto a line input [68].

A widely usedform of characterizationis theModulationTransferFunction, given

astheFouriertransformof theLSF [67]:

MTF � k ��� 12π

) + ∞� ∞e� ikx LSF� x� dx (1.10)

wherek is the spatial frequency given by 2π � λ whereλ is the period. The MTF

describesthemodulationof theimageof asinusoidalobjectasafunctionof thespatial

frequency. Oneof its usefulcharacteristicsis thattheMTF of a systemis theproduct

of the MTFs of the seriessystemcomponents[68]. Alternatively, we candefinethe

ContrastTransferFunctionas[69]

CTF� ν ��� � responsemax. � responsemin.�� responsemax.� responsemin.� (1.11)

14

wheretheresponsemaximumandminimumarethosefor thedetectorilluminatedby

a100% contrastbarpatternof spatialfrequency ν line pairsperunit length.TheMTF

maythenbeexpressedas[69]:

MTF � ν �-� π4� CTF� ν � � CTF� 3ν ��� 3 � CTF� 5ν � � 5 � CTF� 7ν � � 7 � � �.�/� (1.12)

In the ideal caseof an optical systemhaving a GaussianPSF, the resolutioncanbe

expressedby theFWHM � x�0� 0 � 44� ν50%, whereFWHM is thefull width athalf max-

imum of theFourier transformof theMTF andν50% is thespatialfrequency in units

of inverselengthx � 1 for which theMTF is 50% [69].

Finally, the Edge SpreadFunction, definedasthe responseof the systemto step

functionillumination, is givenby [67]

ESF� x�(�*),+ ∞� xLSF� x12� dx1 (1.13)

The stepfunction sourceis assumedto have zero intensity for negative x and unit

intensityfor x 3 0 [68].

All of thesefunctionaldescriptionsareinterrelatedmathematically. Thus,if any of

themis measured,thedatacanbeusedto obtaintheothers[68].

In this work, thePSFis obtainedby simulations,andtheLSF is obtainedby the

integrationof the PSFover onedirection. In the following text, the term ‘profile’ is

asa synonym for theLSF. To measuretheresolutionin theexperiment,the imageof

anedgeis considered.The integrationover theedgegivestheESFandits derivative

againtheLSF. Thus,in thiswork, thefull width athalf maximumof theLSFobtained

in thesimulationandtheexperimentis usedfor comparison.Unlessotherwisestated,

this FWHM is usedasa definitionof theresolution.

1.2.2 Two-dimensionalimaging usingbent crystals

For the high-resolutionobservation of the plasmainstabilities,a new X-ray crystal

camera,basedon a toroidally bentcrystal,wasdesignedto fit the requirementsof a

high resolutionmicroscopy.

The imaging at concave mirrors with circular bendingcan be describedby the

well-known lensequation[70]1f� 1

a � 1b

(1.14)

15

where f , a and b are the focal length, the object distanceand the imagedistance,

respectively. In thesagittalplane,thefocal lengthcanbedescribedby [70]

fs � Rs

2sinθ(1.15)

andin themeridionalplaneapproximatelyby [70]

fm � Rm

2sinθ � (1.16)

Here,Rs andRm is the bendingradiusin sagittalandmeridionalplane,respectively,

andθ theangleof incidence,measuredto thesurface.

In orderto obtaina two-dimensionalimage,the focal lengthin meridionalplane

fm hasto beequalto thefocal lengthin sagittalplane fs, which is thecaseif

sinθ � !Rs

Rm� (1.17)

i.e. for an angleof incidence � 904 , both bendingradii have to be different,which

leadsusto thetoroidalshapeof themirror (crystal).

Whenusinga crystalfor ‘reflection’, the angleof incidenceθ hasto be equalto

theBragg-angleθB, which is givenby theBraggequation[71,72]

2dsinθB � nλ (1.18)

with theinterplanarlatticespacingd of thecrystal,theX-ray wavelengthλ andn is an

integernumber.

Thespectralwindow of thecrystalcanbeestimatedby [73]

∆λλ�65555 M � 1

M � 15555 Am

Rmtanθ(1.19)

whereM � b� a is themagnificationof the imageandAm is theaperture(size)of the

crystal in meridionalplane. This estimationexcludesthe influenceof the reflection

curve and the sourcesize. Also, Eq. (1.19) is not valid for the caseM � 1, which

would give a spectralwindow of ∆λ � 0. Thus,by changingtheaperturesizeof the

crystal in meridionalplane,onecancontrol the bandwidth of the reflectedX-rays,

which is importantif oneis interestede.g. in the2-D imagingof asinglespectralline.

16

Spherically bent crystals

The ideal shapeof the mirror (crystal) for aberration-freeimagingof a point source

is an ellipsoid. In all othercases,the aberrationsdependon the angularapertureof

themirror [74]. Aberrationfreeimagingof anextendedsource,however, canonly be

achievedby meansof two mirrors [75]. However, theseschemesarevery difficult to

realize.

A first approximationtowardsaberrationfree imagingis to ensurethat the focal

lengths, i.e. the position of the images,in both, the sagittaland meridionalplane,

coincide: From Eq. (1.17) it is obvious that one hasto control both bendingradii

independentlyand that only for an incident angleof 904 Rs � Rm, i.e. a spherical

crystal,canbeused.

However, sphericalcrystalsarefrequentlyused[76–79] with astonishinglygood

results.Therefore,theuseof sphericalcrystalsat incidentangles� 904 will bebriefly

discussedhere:

Assumingthattheimaginggeometryis well alignedto theimagein themeridional

plane,i.e. at thedistancebm, theimagein thesagittalplanewill lie behindthedetector

planeandthe‘image’ is broadenedaccordingto theopeningangleof thebeamcone,

definedby the imagedistancein sagittalplanebs and the apertureof the crystal in

sagittalplaneAs. In the following discussion,a simplegeometricalapproximationis

applied,wherethesizeof the‘image’ in thesagittalplaneis estimatedby thedeviation

of thedetectorfrom theimagein thesagittalplane

∆b � bs � bm � a fsa � fs

� a fma � fm

� (1.20)

Theestimatedsizeof the‘image’ of apointsourceonthedetector∆X is thengivenby

therelation

∆X � As � ∆bbs

� As 7 1 � fs � a � fs�fm � a � fm�98 � (1.21)

where fs and fm aregivenby Eq.(1.15)andEq.(1.16),respectively.

In orderto estimatethesmalleststructurewhichcanberesolved,onehasto divide

Eq. (1.21)by themagnificationM � b� a, whereb � bm, so that theresolutionin the

objectplane

∆x � ∆XM

� (1.22)

As a furtherstep,onemightsubstitutea by M, i.e.

a � M � 1M

fm (1.23)

17

so that finally, we obtain—aftersubstitutingthe appropriateexpressionsfor fs and

fm—explicitly

∆x � AsM � 1

M � cos2θ � (1.24)

which dependsonly on themagnificationM, theapertureof thecrystalin thesagittal

planeAs, andtheangleof incidenceθ andis independentof theradiusof curvatureof

thecrystalR.

FromEq. (1.24)onecanseethat theobtainableresolutionin thesagittalplaneis

thehigher, thesmallertheapertureAs, the larger themagnificationM andthecloser

theangleof incidenceis to π � 2.

Figure1.5 illustratesEq. (1.24)

Angle of Incidence / [deg]

Res

olut

ion

at 1

mm

Ape

rtur

e / [

µm]

060 65 70 75 80 85 90

100

200

300

400

500

PSfragreplacements

M : 1M ; 10

Figure 1.5. The resolutionin the sagittalplaneat

sphericalcrystals,estimatedby Eq.(1.24)for 1 mm

crystalaperture.

for the caseof an aperturesize in

thesagittalplaneAs � 1 mmandfor

two magnifications,namelyM � 1

and M < 10. For higher magnifi-

cations, the term � M � 1� � M does

notchangeanymoreconsiderablyso

that thesolid curve marksthe limit.

For aBraggangleof 604 , theobtain-

ableresolutionis limited to 250µm

for thehigh magnificationcase,and

in order to reach10 µm, the Bragg

anglehasto belargerthan844 .However, the Bragg anglecannotbe chosenarbitrarily, but dependson the dis-

cretewavelengthof the X-ray source1 andthe discreteinterplanarlattice spacingof

the crystal. Additionally, for plasmadiagnostics,the wavelengthis definedby other

experimentalconditionsso that only the lattice spacingremainsfor the selectionof

theBraggangle.However, thesuitablelatticespacinghasto befoundin anelastically

bendablecrystalsothattheangleof incidencecanbehardlychanged.

The other possibility to increasethe resolutionis to reducethe apertureof the

crystal.Notethatonly theeffectiveaperturesizeis important,not theactualsizeof the

crystal. Aglitskiy et al. [76,77,80] coulddemonstratea spatialresolutionof 1 � 7 µm

usinga crystalsizeof 5 � 10 mm2 at a Braggangleof 81.84 anda magnificationM �1Not consideringherethecontinuousBremsstrahlungandSynchrotronradiation,wherethewave-

lengthcanbefreely chosen

18

3000

4000

5000

6000

7000

8000

9000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Edge Profile

Cou

nts

on C

CD

Distance on CCD / [µm]

Figure1.6. 2-D Imagetestusinga toroidallybentGe(311)crystalanda1000lpi gold mesh.

20. However, sincethe backlightingsourceof 400 µm diameterwasplaced50 mm

behindthetestgrid, while thedistancefrom thegrid to thecrystalwas105mm. Thus,

theeffectiveaperturewasonly about840µm (cf. Section1.3.1).

Even if this high resolutionof 1.7 µm is very impressive as demonstration,its

practicalusemight bedoubtful,sincetheapertureis artificially reduced.For photon

critical experiments,whereoneis interestedin a largeaperture,this artificially small

aperturemighthaveaninsufficientluminosityandmight thereforebeof of limited use.

1.2.3 Requirementsof the camera

For the high-resolutionobservation of the plasmainstabilities,a new X-ray crystal

camerawasdesignedto fit therequirementsof a high resolutionmicroscopy.

In anearlierexperimentin theframeof this work [31,46,47], a spatialresolution

of betterthan6 µm wasalreadyobtainedasshown in Figure1.6, usinga Ge (311)

crystalwith Rm � 200mm andRs � 194� 7 mm bendingradii. Thesizeof thecrystal

was7 � 7 mm anda CCD camerawith a pixel sizeof 22.5µm wasusedasa detector

atadistanceof 1849mm. A testmeshwith 1000linesperinch, i.e. aperiodof 25µm,

and9 µm barwidth wasusedasa testobjectat anobjectdistanceof 104.2mm. The

resultingmagnificationM � 17� 7. As a backlightersource,theM-shell lines from a

gold plasmawereusedwhile the crystalwassetto an angleof incidenceof 80� 634 .Contraryto theabovementioneddemonstrationexperimentby Aglitskiy etal. [76,77],

thefull apertureof thecrystalwasused.

Thebarsareresolved,which demonstratesthatthespatialresolutionis betterthan

19

9 µm. To get morepreciseinformationaboutthe resolution,the grid imagewasan-

alyzedby the following procedure[31]: By Fourier analysisof the imagethe noise

wasremovedby filtering high frequencies.Then,the Fourier dataof the real image

wasdividedby theFourierdataof therealgrid to obtainthepoint spreadfunctionby

backtransformation.By usingRayleighcriteria [70], the obtainedspatialresolution

wasdeterminedto 6.2 µm in theobjectplaneor 110µm in thedetectorplane.These

110µm correspondto fivepixelsof 22.5µm,whichis at thephysicallimit of resolution

of theCCDarray.

Therefore,oneapproachto obtaintherequiredspatialresolutionof lessthan3 µm

in this experimentwasto increasethemagnificationin orderto overcomethelimiting

resolutionby theCCDarray.

In orderto protectthecrystalfrom damageby plasmadebris,theobjectdistance

hasto belargerthan100mm[74]. However, thelargertheobjectdistance,thelargeris

theimagedistance,too,andthesmallerthecoveredsolidanglefromwhichthephotons

arefocused,i.e. the smallerthe luminosity. As a compromise,an objectdistanceof

140mm waschosen,resultingin an imagedistanceof 4200mm for M � 30, which

canbe still realizedin the target chamberhall of GEKKO XII. Thesedistancesare

linkedby a focal lengthof 135.48mm.

Anotherconstraintconcernsthewavelengthof thebacklightersource,which was

selectedaccordingto theboundaryconditionsof a high transmittanceandhigh reso-

lution astheH-like 12Mg Lyα emissionline with acentertransitionwavelengthof the

H-likedoubletof λ � 8 � 42235A (cf. Section1.1.3).

Basedon theseconditions,thecrystallayoutwasdesigned:A suitabled-spacing

wasfoundin theQuartz � 1010� crystallatticeplanewith d � 4 � 254783A, resultingin

aBraggangleof θB � 81� 804 .For the given focal length f � 135� 48 mm and Bragg angle θB � 81� 804 , the

requiredbendingradii of the crystal canbe calculatedby Eqs.(1.15) and(1.16) as

Rs � 268� 2 mmandRm � 273� 8 mm,respectively.

1.2.4 Simulation

In orderto checktheperformanceof theimagerdescribedabove,ray-tracingcalcula-

tionsusingtheT-RAY code[81] werecarriedout.

The original versionof the T-RAY codewasdevelopedparticularly for the Ap-

ple Macintoshplatform,takingadvantageof thegraphicaluserinterface(GUI) of the

20

MacOS.However, for heavy numbercrunching,theGUI is a nuisancesinceit is de-

signedfor the interactionbetweenuserandmachine.But for numbercrunching,one

wantsthemachineto do thejob silently without any furtherinteraction.Additionally,

the GUI is platform dependent.Thusthe original programwill not run underother

machinesexceptfor theMacintosh.

Therefore,all GUI directivesin theT-RAY codewereremovedsothatapureANSI

C/C++ coderemained,which couldbe now compiledandexecutedon any machine.

Thecalculationparametersareprovidedin a simpleASCII file andtheoutputis pro-

videdagainin ASCII andsimplebinaryfiles. For theconvenienceof theuser, a GUI

wasprogrammed,too, in the TCL/Tk scripting language,which is availableon (al-

most)any platform. TheGUI writestheT-RAY input file, startstheT-RAY code,and

displaystheoutputfiles in aconvenientform to theuser. In this format,T-RAY is now

platformindependent.

Thanksto the removal of all interactive partsfrom the T-RAY code,T-RAY can

now berun asa subprocessof any otherprogram.While the interactive characterof

the original T-RAY versionforcesthe userto changea parametermanually, starting

T-RAY, waiting a few minutesfor theresult,analyzetheresultwith anotherprogram,

changea parameter, startingT-RAY again,andso on, it is now possible,for a shell

programto do thejob automatically:Theuserhasjust to write asmallprogram/script

in a scriptinglanguagelike TCL, Perl,IDL, Matlab,etc,which writesor modifiesthe

T-RAY input file, startsthe T-RAY code,analyzesthe calculationresult,writes the

result of the analyzationin a file, andstartsT-RAY again. The advantageis clear:

While at the original version,the usertendsto make calculationsonly for a specific

point(e.g. thebestfocalposition)it is now easyto calculatelongscans,whichcangive

new insightsin thecomplicatedopticalrelationsof thetoroidalcrystal(cf. Ref. [74]).

Figure1.7 shows thepoint spreadfunction(PSF),calculatedby theT-RAY code,

for threedifferentrepresentativepoints.For thecalculations,theactualachievedbend-

ing radii of thecrystalof Rm � 275� 6 mmandRs � 269� 8 mmwereusedinsteadof the

designvaluesmentionedin the lastsection.Thus,thefocal lengthis 136.34mm and

theobjectandimagedistanceare140.88mmand4226.5mm,respectively, in orderto

get a magnificationof 30. The reflectioncurve wascalculatedby the Takagi-Taupin

equation[82–84]andaswidth of theemissionline 16 � 10� 3 A wasassumed.For the

calculation,a pixel sizeof 5 � 0 � 5 � 0 µm2 wasassumedin orderto get a moreinfor-

mative resultthanonewould obtainwith theactualpixel sizeof theCCD array. All

21

0 0

0 0

500 500 1000

500 10

1000

1000 100

0 0

0 0

500 500

500 10

1000 1000

1000 100

Pixel Size:5.0 µm

7.0×7.0 mmAperture Size:

Pixel Size:5.0 µm

7.0×7.0 mmAperture Size:

Pixel Size:0.5 µm

3.5×3.5 mmAperture Size:

Pixel Size:5.0 µm

3.5×3.5 mmAperture Size:

112.0 µm

31.5 µm 16.5 µm

96.0 µm

9.0 µm

27.0 µm

2.3 µm

26.2 µm

(d)

(b)(a)

(c)

Figure 1.7. The point spreadfunction for different aperturesizesand focalizationat 30 =magnification(seetext).

22

distancesin Fig. 1.7 aregiven in micrometersandarerelatedto the detectorplane.

In orderto obtaintheachievableresolution,thesedistanceshave to bedividedby the

magnification,i.e. 30.

Figure 1.7ashows the point spreadfunction for the toroidally bent crystal with

7 � 0 � 7 � 0 mm2 aperturesizeof thecrystalandtheobjectandimagedistancealigned

accordingto Eq. (1.14), i.e. 140.88mm and 4226.5mm, respectively. The image

of the point is broadeneddueto imagingerrorslike focus,astigmatism,coma,field

curvature,andsphericalaberration2, which causethe imageof thepoint to bespread

over a ‘triangle’, i.e. the spreadingoccursin the meridionalplanein onedirection,

only. TheFWHM of theimageis 112.0µm and31.5µm in themeridionalandsagittal

planeandby dividing this sizeby themagnification,oneobtains3.7 µm and1.0 µm

resolution,respectively.

By shifting the detectorslightly ( � 2%) towardsa larger imagedistance(4320

mm), shown in Fig. 1.7b, the detectorwill be in a planewherethe first orderimage

is de-focalized,i.e. broadened,but thehigherorderimages,i.e. the ‘imaging errors’,

are narrowed so that the overall point spreadfunction can reacha minimum. The

sizeof theimagein themeridionalandsagittalplanebecomes96.0µm and16.5µm,

correspondingto a resolutionof 3.2µm and0.6µm, respectively.

Figure1.7cshowsthesameconfigurationasin Fig. 1.7bbut for asmalleraperture

sizeof the crystal,namely3 � 5 � 3 � 5 mm2 insteadof 7 � 0 � 7 � 0 mm2. Sinceonly the

region closeto thecenteris used,theimagingerrorsarereduced,too. FromFig. 1.7c

onecanseethatanimagesizeof thesourceon thedetectorof 27.0µm and9.0µm in

meridionalandsagittalplane,respectively, canbeachieved. Dividing thesenumbers

by themagnificationof 30 � , theachievableresolutionamountsto 0.9µm and0.3µm

in sagittalandmeridionalplanes,respectively.

Finally, Fig. 1.7d shows the calculationfor the sameparametersas in Fig. 1.7c,

but thepixel sizewasassumedto be0.5 µm in orderto geta magnifiedimageof the

point spreadfunctionon theonehandandon theotherhand,to geta higherresolved

numberfor theFWHM of theprofile of thepoint spreadfunctionsincethe9.0µm in

Fig. 1.7ccorrespondto lessthan2 pixels. Thesehigherresolvedcalculationsconfirm

theFWHM valuein meridionalplanebut promiseawidth of thepointspreadfunction

in sagittalplaneof 2.3 µm, which correspondto anachievableresolutionof 0.08µm

2An analytical descriptionof the complicateoptical relation of toroidal surfacesis given by

Dirksmoller in Ref. [74] andshouldnotberepeatedhere.

23

7.0

mm

7.0 mm

4.0 mm

4.0 mm

2.8 mm

2.8 mm

7.0 mm

7.0 mm

4.0

mm

2.8

mm

4.0

mm

2.8

mm

Defocalization / [µm]

Siz

e of

PS

F o

n D

etec

tor

/ [µm

]

Resolution / [µm

]

10.6

2.7

4.0

5.3

6.6

1.3

0.0

7.9

9.3

0

50

100

150

200

250

300

350

400

−600 −400 −200 0 200 400 600

Figure 1.8. The FWHM of the

point spreadfunction in sagittal

( >&>&>�>&> ) andmeridional(—) plane

for 2.8mm,4.0mm and7.0mm

aperturesizeaswell astheresult-

ing resolutionasafunctionof the

de-focalization.

in sagittalplane!

UsingananalyticalapproachtoconsiderdiffractioninsidethecrystalbyChukhovskii

et al. [85], theobtainableresolutionin meridionalandsagittalplanecanbeestimated

as

∆xm � λRm

Am� ∆xs � λRmsinθB

As(1.25)

respectively. For theexampleconsideredhere(Rm � 275� 6 mm, Am � As � 3 � 5 mm,

λ � 8 � 42A, θB � 81� 84 ) ∆xm � 0 � 07µm and∆xs � 0 � 06µm.

However, thesesimulationsassumeaperfecttoroidalshapeof thecrystalandper-

fectcrystallinestructure.In reality, it is verydifficult to achieveaprecisionof 10� 3 in

thebendingradii. Moreover, thecalculationsarebasedon a surfacereflection.In the

nature,theX-rayspenetratetypicallyafew to afew tensof micrometersinto thecrystal

andexit ata locationdifferentfrom theentrancepoint [86]. Thispenetrationdepthde-

pendson thewavelengthof theincidentX-raysand,contraryto thesimulations,there

arealwayshigherorderreflectionspresent,i.e. X-rayswith λ � n n � 2 � 3 � 4 � � ��� with dif-

ferentpenetrationdepthsin thecrystal,which will give superimposedimages.These

effectswill limit theachievableresolution[87] too,andarenottakeninto accounthere.

Thereforetheabovepresentednumbersfor thesub-micrometercaseareoptimistic. In

addition,onehasto considerat thesehigh resolutionsthe limitation by diffraction,

which canbeestimatedfor thecaseconsideredhere(cf. Eq.(1.30))asabout0.03µm.

Nevertheless,it wouldbeworthwhiletestingwhetherasub-micrometerresolutioncan

be obtainedby meansof toroidally bentcrystalsanda large magnification(cf. Sec.

1.3.2).

Another crucial point in the imager, is a large focal depth,sincethe target will

be acceleratedby 5 � 1015 cm/s2 in 4 ns to 2 � 107 cm/s and the Rayleigh-Taylor

instabilitiesshouldbeobservedwithin thefirst 2 ns,correspondingto aflight distance

24

+200−200 0 +200−200 0 +200−200 0

± 0

µm+

500

µm

−50

0 µm

200

300

100

0

200

300

100

0

200

300

100

0

5.3

7.9

2.6

0

5.3

7.9

2.6

0

5.3

7.9

2.6

0

FW

HM

of P

oint

Spr

ead

Fun

ctio

n on

Det

ecto

r / [

µm]

−500 µm ± 0 µm + 500 µm

Dev

iatio

n of

Poi

nt S

oure

from

Opt

ical

Axi

s pe

rpen

dicu

lar

to D

ispe

rsio

n P

lane

Defocalization of Point Source Along the Optical Axis / [µm]

Deviation of Point Soure from Optical Axis in Dispersion Plane

Res

olut

ion

of Im

ager

/ [µ

m]

Meridional Plane

Sagittal Plane

Figure1.9. Thecalculatedwidth of thePSFfor 2.8mmcrystalapertureandanddeviationsof

thepointsourcefrom theopticalaxis.

of ? 400µm.

In orderto checkthefocalcharacteristicsof theimagingsystem,ascriptwaswrit-

tenwhich startsT-RAY for thecalculationof thePSFfor a particularimaginggeom-

etry. TheFWHM of PSFin boththesagittalandmeridionalplanewerecomputedby

anotherprogram,calledPROFILER, whichwaswritten for theevaluationof theT-Ray

output. Then,the objectdistancewasvariedin eachstepby 20 µm andT-RAY was

startedagainuntil thecompletescanwasperformed.

The calculationsassumea toroidally bentcrystalwith Rm @ 275A 6 mm andRs @269A 8mm,asachievedin themanufacturingprocessfor theherepresentedexperiment.

Nominal objectand imagedistanceof 139.9mm and5284 mm, respectively, were

assumedfor thecalculations,sincetheseweretherealizedvaluesof the imagingtest

(cf. below). TheresultingmagnificationM @ 37A 75.

In a first step,thePSFasa functionof thede-focalizationwascalculatedfor three

differentaperturesizes,namely7.0 mm, 4.0 mm and2.8 mm andtheFWHM of the

PSFfor both, the meridionalandsagittalplaneis plottedin Fig. 1.8. The solid line

markstheFWHM in themeridionalplaneandthedottedline theFWHM in thesagittal

plane.While theleft handsideordinategivesthesizeof theimageonthedetector, the

25

right handsideordinategivesthesizeof the imagedividedby themagnification,i.e.

theresolutionof theinstrument.Far away from thetheoreticaloptimalfocal position,

theFWHM in sagittalplanefollowsasimplegeometricalapproximation(cf. Eq.(1.34)

andFig. 1.15)which would becomezerofor thebestfocal position.However, dueto

thehigherorderimagingerrors,thepoint spreadfunctionbecomesvery fastnarrower

andhasaflat bottomaroundthebestfocalposition.Thisflat bottomcanbeexplained

by theimagingerrors,which causethereflectedlight from theoutercrystalregion to

crosstheopticalaxisbeforeor behindthefocalpoint. Thelengthandthewidth of this

‘line focus’alongtheopticalaxisis thelonger, or wider, thelargerthecrystalaperture

is.

In themeridionalplane,for 7 mm aperture,thefocal lengthis about800µm, with

asizeof thepointspreadfunctiononthedetectorof about75µm, correspondingto an

obtainableresolutionof about2 µm, while for 2.8mmaperturesize,theFWHM of the

PSFamountsabout30µm in arangeof 400µm,correspondingto 0.8µm resolution.In

thesagittalplane,thefocal lengthfor all threeaperturesizesamountsaboutto 200µm

with aFWHM of about15 µm, or anobtainableresolutionof 0.4µm.

Sincea largefield of view is required,the focal scanwascalculatedfor thepoint

sourceon the optical axis aswell asfor a deviation of the sourcepoint of B 500 µm

in both themeridionalandthesagittalplane. TheFWHM of the scansareshown in

Fig. 1.9 for eachdeviation in eachplane.ThePSFin Fig. 1.9 is calculatedfor 2.8µm

aperturesize. For larger aperturesizes,the result is similar, taking into accountthe

changesin Fig. 1.8.

For a deviation of thesourcepoint perpendicularto thedispersionplane,thebest

focuspositionin sagittalplaneis shiftedby B 100µm for a deviation of thesourceofB 500µm, while thefocaldepthremainsunaffected.A deviationof thesourcepoint in

dispersionplane,however, hasalmostno influenceon thebehavior of theFWHM in

sagittalplaneof thePSF.

In themeridionalplane,adeviationof thesourcepointperpendicularto thedisper-

sionplaneof B 500µm causesashift of thefocal depthby about B 50 µm, only, while

adeviationof B 500µm in dispersionplanecausesthefocaldepthto bedistorted.The

interestingpoint is thatthedistortionis identicalfor thedeviation in bothdirections.

In summary, a high resolutionfrom thecrystaloptic canbeexpected,with a very

long focal depthevenat suchhigh magnificationsas30 C anda very largeapertureof

7 mm.

26

1.2.5 Designof the X-ray imager

To ensureaproperalignmentof thecrystalconcerningfocusandpositionof theimage

on the detector, a new camera,shown in Fig. 1.10 was designedwhich enablesall

movementsto bemadeby remotecontrolsothatanon-linealignmentis possible.

The main componentof the camerais a toroidally bentQuartz D 1010E crystalof

7 C 7 mm2. Thebendingradii are275.6mm in themeridional(dispersion)planeand

269.8mmin thesagittalplane.Theresulting‘stigmaticangle’θ0 @ 81A 66F . Theinter-

planarlatticespacingis 2d @ 8 A 5096A which givesa BraggangleθB @ 81A 7898F for

thecentralwavelengthof theMg Lyα doubletof λ @ 8 A 42235A, wherethedifference

in thewavelengthamounts5.9mA. For a typical line width of ∆λ G λ ? 10H 3, thewidth

of the doubletcanbe estimatedas ? 14 mA. The spectralwindow of the crystalat

a magnificationof M @ 30 amountsto ? 29 mA, which doesnot changefurther for

highermagnificationssincethe term IJD M K 1E G$D M L 1EMI in Eq. (1.19) for M @ 30 is

alreadyin ‘saturation’.

The crystal is mountedon a mini-goniometerwherethe Bragganglecanbe set

usingastepmotor. Oncethisgoniometeris calibrated(cf. Sec.1.2.6),theBraggangle

canbeeasilysetjustby moving it to thecorrespondingcountervalue.Also, an in-situ

alignmentis possibleby measuringthereflectioncurve.

A debrisshieldingin front of the crystal is necessaryto protectthe crystal from

debriswhich destroy thesurfaceof thecrystalandmake it ‘blind’. ThusPawley et al.

[80] hadto replacetheir crystalafterseveralshots,which is a veryexpensivesolution

of theproblem.The‘standard’solution,namelya thick fix beryllium window in front

of thecrystal,is not possibleheredueto thehigh absorptioncoefficient of Be at this

wavelength.A 50 µm thick Be-window (linearabsorptioncoefficient µ D 8 A 4192A E @357A 0G cm) would absorb97% of the signalsincethe beamhasto passthe window

twice. Laterin theimagingtest,wetesteda10µm thin Be-window, too,whichabsorbs

only 51% of theintensity. But this window wasperforatedafteroneshot.

We thereforechose1.5 µm thin polyesterfoil (‘Mylar’) evaporatedwith 200 nm

aluminum, which transmits63% of the X-rays (total linear absorptioncoefficient

µ @ 1553G cm). The Al coatingis necessarysincethe intensevisible light from the

plasmais suspectedto destroy the glue, makingthe crystalunusable[88]. Sincewe

expectedthefoil to benotstableenoughfor permanentoperation,a ‘shieldingwheel’

wasmountedin front of thecrystal.Thisshieldingwheelconsistsof twelvewindows,

coveredby theshieldingfoil. All windows,apartfrom theonein front of thecrystal,

27

Debris Shielding

Test grid Crystal

Translation Stages

Tiltings

Alignment Needle Pinhole Camera

Backlighter Target

Green He−NeAlignment Laser

Aperture Wheel &

Figure1.10. TheHIPERImagerinsidethetargetchamber.

28

areagainhiddenbehinda thick fix Al-plate. Whenthe debrisdestroys onewindow,

thewheelcanberotatedsothatanothershieldingfoil is in front of thewindow.

For alignmentpurposea motorizedremovablepointer, or ‘needle’,wasintegrated

in thecamera:For thepre-alignmentoutsidethetargetchamber, thetip of theneedle

marksthe positionof the plasmato be imaged. Thusall alignmentscould be done

relative to thetip of theneedle.Whenthecamerais in operation,theneedlecouldbe

removed.Whenthecamerahadto betakenoff thetargetchamber, e.g. for exchanging

theshieldingwheel,theneedlecanbemovedagainto theplasmaposition,thework

canbedoneoutsidethe targetchamberandlateronehasto ensurethat the tip of the

needleis backat thesameposition.

During the constructionphase,it wasnot clearhow many photonscould be ex-

pected,sincethis wasacompletelynew experimentat theILE. Therefore,anaperture

wheelwasmountedin front of thecrystalwith elevendifferentaperturesizes,namely

8.0mm, 5.6mm, 4.0 mm, 2.8mm, 2.0mm, 1.4mm, 1.0mm, 0.7 mm, 0.5mm, 0.35

mm, and0.25 mm in diameter. Thesevalueswerechosenso that the diameterwas

alwaysreducedby a factor N 2, resultingin a reductionof thearea,andthusthetrans-

mitted intensity, by a factortwo. Additionally, therewasa positionwhich hidesthe

crystalcompletelyfrom theX-rays.Thispositioncouldbeusedfor testpurposesor to

protectthecrystalefficiently whenit is not in use.

In orderto aligntheimageonthedetector, thecrystalcouldbeshiftedmotorizedin

three-andtilted in two directions.Thetranslationstagewassetupsothatonedirection

is parallelto theincidentbeam.With this translationdirection,thefocuscouldbeset

properly.

Thedetectorwasmountedat anextensiontubeoutsidethetargetchamber. Close

to the target, a collimator tubewas installedinside the extensiontube,so that only

X-rayswhicharediffractedat thecrystalcouldreachthedetector. In orderto suppress

thescatteredlaserlight, afilter holderwasforseencloseto thedetector.

To make thealignmentusingX-rayspossible,therewasanew so-called‘diagnos-

tics laser’ installedat the ILE with 6 Hz repetitionrate. This lasercould be usedto

createasmallplasmafor thegenerationof X-rays.

1.2.6 Calibration of the goniometer

Oneimportantadvantageof this crystalstageis thestepmotordrivenvariationof the

Braggangle.It is thereforeno longernecessaryfirst to pre-aligntheBraggangleon a

29

separategoniometer, to markthedirectionof incidenceby aneedle,aligningthenthis

needleto themarker of thecenterof a skeleton,which modelsthetargetchamberfor

pre-alignment,thenremoving themarker needlefrom thecrystalholderandaligning

theneedleof thecrystalmanipulatorto thatposition[89].

With this new crystal stage,the integratedmarker needlehasjust to be aligned

to the target chambercenterandall thealignmentcanbedonewithout removing the

crystalor copying theplasmapositionfrom oneinstrumentto another. Thanksto the

stepmotor driven mini-goniometer, the anglecanbe setby moving the goniometer

headto thewell-definedposition.To find therelationbetweenthemotorstepsandthe

angle,themini-goniometerwascalibratedasfollows:

For the calibration, the

Laser

Screen

Mirror

PSfragreplacements

x0

x

2θθ

Figure 1.11. Setupfor thecalibrationof thegoniometer.

set-upasshown in Fig.1.11

wasused.Insteadof thecrys-

tal, a front-sidemirror was

mountedon the mini-goni-

ometer. A laserwaspointed

auto-collimatedto the mir-

ror and the reflectedbeam

wasmonitoredon a screenperpendicularto theauto-collimatedbeamat a distanceof

x0 @ 1000mm. As a ‘screen’,a10mmthick and600mmlongmetalplatewith ahole

in it atonecornerwasused.Now themirror wasturnedonthegoniometerheadby the

angleθ causingthereflectedbeamto bedeviatedfrom theauto-collimatedpositionby

x @ x0 tan2θ A (1.26)

Thustheangleθ canbedeterminedby measuringthedeviationx of thereflectedbeam

on thescreenas

θ @ 12

arctanO xx0 P A (1.27)

The relationsteps–deviation wasmeasuredthreetimes. Themeasuredvaluesare

plottedin Fig.1.12togetherwith thelinearfit andthedeviationof themeasuredvalues

from it.

Thelinearfit wasdeterminedby a least-square-fitas3

θ Q degR @TS 1 A 3206 C 10H 4 B 6 A 7 C 10H 8 U C Steps (1.28)

3Sincetheoff-setangleis arbitraryandthusnot interestingfor thecalibration,it is omittedhere.

30

0

2

4

6

8

10

12

14

0 20000 40000 60000 80000 100000-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Ang

le /

[deg

]

Motorsteps

Dev

iatio

n fr

om th

e fit

/ [d

eg]

Figure 1.12. Calibrationcurve

of the mini-goniometer: The

boxes ( V ) presentthe measured

values togetherwith the linear

fit (straight line; both left ordi-

nate)and the pluses(+) are the

deviation of the measuredval-

uesfrom the linear fit (right or-

dinate). Note the large different

scaleof both ordinates,the er-

ror barsof measuredvaluesare

smallerthantheline width.

or, inverted,

Steps@ D 7572A 3 B 3 A 8EWC θ Q degRXA (1.29)

Besidesthe statisticalscattering,a clearstructureis visible in the deviation from

the linear relation (“ L ” in Fig. 1.12), which seemsto appearfrom the mechanical

precisionof mini-goniometer, which is reasonablefor the manufacturedprecisionof

themini-goniometer[90]: Thisdeviation is reproducibleandall otherinfluences(like

a largedeviation from the90 degreesanglebetweenscreenandincidentlaserbeam)

would givea muchlargerstructure.Also, if it wereastatisticaleffect, it would not be

reproducible.

However, evenif this structureis reproducible,it wasnot takeninto accountin the

calibrationrelationhere,sincethis structureis a functionof thepositionof thecradle

of themini-goniometerandnot of theanglebetweentheincidentandreflectedbeam.

But only the anglebetweenthe incidentandreflectedbeamwasmeasured,which is

interestingfor calibration. A marker at themini-goniometerwould give a possibility

to calibratethemini-goniometerto thepositionof thecradle,but then,onewouldhave

to measurealways relative to the this marker. I.e. when aligning the Bragg angle

onewould first have to measuretheanglebetweennormalincidenceandthis marker

andthento turn—accordingto thecomplicatedcalibrationfunction—thecradleto the

Braggangle. Sinceit is not clear, how preciselythe zeropositionand the angleof

normalincidencecanbedetermined,therewardsof this effort aredoubtful.

Thus,even if the slopeerror of the calibrationfunction is small dueto statistical

31

reasons,the error of a single point is B 0 A 06F , which correspondto an error in the

counterstepsof B 460steps.Anothererrorsourceis thesystematicuncertainty, dueto

themisalignmentof thescreenfrom themirror of ∆x0 @ B 1 mm which resultsin an

uncertaintyof ∆θ @ B 0 A 02F atθ @ 10F . Theplay in thegearwasmeasuredto beabout

7000steps,sothatabacklashof Y 7000stepshasto beconsidered.

In summary, theachievedrangeof confidenceof ∆θ @ B 0 A 06F or B 460stepscov-

ersall influencesof errorsandis sufficiently small for theexperiment,sincetheemis-

sionline doubletunderobservationis reflectedin aBragganglerangeof ∆θB @ B 0 A 2F .1.3 Imaging test

1.3.1 Imaging testat GEKK O XII

Onemajoraimof theexperimentis thedemonstrationof thecut-off perturbationwave-

lengthof theRayleigh–Taylor instability, i.e. theperturbationwavelengthat which no

RTI will occurwhich is expectedto bein theorderof about3 K 10 µm. Thereforethe

demonstrationof theresolvingpropertiesof thediagnosticsis crucial,sinceonehasto

besurethatthenon-observationof RTI below a certainspatialfrequency is dueto the

non-existenceof theRTI andnot dueto thelimited resolutionof thediagnostics.

Therefore,animagingtestwasperformed.Thecrystalandthedetectorshouldbe

set-upsimilarasin therealexperiment,but with agrid asadummysample.Thesetup

is shown schematicallyin Fig. 1.13andthegeometryis summarizedin Tab. 1.1.

Fiveattemptshavebeenmadein orderto carryout thedemonstrationof theperfor-

manceof theimager. However, dueto technicalproblemswith thelasersystem,only

oneexperimentcould be carriedout undervery restrictive conditionsasa so-called

‘backgroundexperiment’at GEKKO XII. I.e. anotherexperiment,called‘foreground

experiment’,hasthefirst priority of laserconditioning,but in thatexperimentnot all

laserbeamsareused.Theunusedlaserbeamsareredirectedto theothertargetchamber

andcantherebeusedparasiticallyby anotherexperiment,thebackgroundexperiment.

However, thebackgroundexperimenthasno influenceon the lasershot;neitherover

thepulseenergy nor thedeliveringtime of theshotnor over thealignmentof the fo-

cal positionbecausere-positioningof the focal point needsoneday, which wasnot

accepted.The latter point wasa crucial point for the imagingtest,sincethe sample

is normallyalignedto thecenterof thetargetchamberandthebacklightingsourceis

32

Crystal Quartz Z 1010[Meridionalbendingradius Rm 275.6mm

Sagittalbendingradius Rs 269.8mm

Stigmaticangle θ0 81.66\Braggangle θB 81.79\Focallength f 136.34mm

Objectdistance a 139.954mm

Imagedistance b 5284mm

DistanceObject–Image c 5150mm

Magnification M 37.75

Apertureof thecrystal A 4 mmdiameter

X-ray wavelength λ 8.42235A

Spectralwindow ∆λ 17mA

Sourcesize S ] 10 mm

Sourcedistance dS ] 1700mm

Effective aperture Aeff ] 0 ^ 8 _ 0 ^ 8 mm2

Pixel sizeof theCCD 22.5µm

Stepwidth of focus 0.04µm/step

Table1.1. Setupof theimagingtest.Distancesaregivenfor theoptimalalignment.

33

GXII Laser

Mirror

Grid

Target Chamber

Detector

Mg−Foil

Lamp

Crystal

PSfragreplacements

S

A

Aeff

Figure 1.13. Theschematiclayoutof theimagingtest.

setwith a certainoffset behindthecenter. Thus,the laserbeamfor the backlighting

sourcehasto bealignedto a positionaway from thecenter. But this is not possiblein

thebackgroundexperiment.Here,thelaserbeamscanonly bealignedto thecenterof

thetargetchamber.

To work aroundthis limitation, thealignmentprocedurewasasfollows: First the

test grid was alignedto the centerof the target chamber, using the target monitors

andthewell controllablestep-motordriventargetpositioner. Then,thealignedtarget

wasshiftedby 2 mm off the centerposition towardsthe crystal,which itself hadto

beshiftedbackby 2 mm,too, takingadvantageof themotorizedtargetpositionerand

focus,sothattheobjectdistanceremainedconstant.

In orderto find thebestfocal position,a focal scanusingvisible light wascarried

out. As an object,the edgeof a grid wassetat the centerof the target chamberand

back-lightedby a tungstencondenserlamp. In orderto placethe light sourceon the

axiswhich links theobjectandthecrystal,a mirror wasmountedinsideat the target

chamberwall, which reflectedthe light from thecondenserlamptowardsthecrystal,

which itself wasplacedoutsidethe target chamberbehinda window port. Thus,the

distanceof thesourcefrom theobjectwasabouttwicetheradiusof thetargetchamber.

34

Table1.2. Theparametersof thefabricatedglassforms.

Glassform “3” “4”

Radiusin meridionalplane Rm 2756 a 0 ` 1 mm 2760 a 0 ` 3 mm

Radiusin sagittalplane Rs 2698 a 0 ` 1 mm 2700 a 0 ` 2 mm

Stigmaticangle θ0 81 66b 81 52bFocallength f 136.34mm 136.49mm

AssociatedCrystalNr. 126 127

As figureof merit, the full width at half maximum(FWHM) of the derivative of the

edgeprofileof theimageof thegrid wasconsidered.

For the imagingtestusingMg Lyα backlightingradiation,a Mg-foil wasaligned

at thecenterof thetargetchamberusingasimpletargetpositionerandthetargetmon-

itors. Laserbeams#07and#08wereused.

However, it wasnotpossibleto obtainsufficiently goodresultsfor thisset-up.The

analysisof the experimentshowed that therewerethreeproblems:The visible light

alignmentwas not possibledue to a too small sourcesize, a crystal with a double

imagewasuseddueto a mistake in the labelingandfinally, dueto a problemin the

stepmotor controller, the crystalwasshiftedjust by thehalf way asintended.After

fixing theseproblems,therewasno morebeamtime availableto getgoodresults.In

thefollowing, theanalysisof theproblemsaredescribedin detail:

Identification of the better crystal

For this imager, a new replicaglassform wasmanufactured.To ensurea sufficiently

high quality toroidalshape,four glassformswerepreparedandtestedasdescribedin

Ref. [91], of which thebestonewasto beselected.

Sincethequality testof thereplicaform couldnotbecarriedoutwith asufficiently

small uncertaintydue to the limited mechanicalprecision,the two bestglassforms

wereusedfor thepreparationsof the crystals.The final imagingtestof the crystals,

usinganX-ray generator, wasto identify thebestcrystal.

In theprotocolof the laboratoryimagingtest,thetestimagelabeledwith ‘crystal

127’ gave thebetterimage,andaccordingly, crystal127wasselectedfor theHIPER

experiment.As theobtainedimagingqualitywaspoor, bothcrystalsweretestedagain

outsidethe target chamberby a focal scanat visible light. The intentionof this test

35

Crystal 126

−40

−30

−20

−10

0

10

20

30

40

0 50 100 150 2000

50

100

150

200

250

Position on CCD image (pixel)

Grid

pro

file

Der

ivat

ive

of g

rid p

rofil

e

Crystal 127

−40

−30

−20

−10

0

10

20

30

40

0 50 100 150 2000

50

100

150

200

250

Der

ivat

ive

of g

rid p

rofil

e

Grid

pro

file

Position on CCD image (pixel)

(a) (b)

Figure 1.14. Theedgeprofileof thegrid image( c�c�cJc ) andthederivative (—).

wasto identify thebendingradii, labelingand,of course,find thecrystalwith thebest

imagingquality.

Thesetupwasasfollows: A testmeshwassetupatadistanceof about158mmand

illuminatedfrom therearsideby a tungstencondenserlamp. Theimageof theback-

lighted grid wasobserved at a distanceof aboutonemeterby a microscopewith an

attacheddigital camerawith manualcontrolover thefocus(Nikon Coolpix 950). The

objectdistancewasvariedalongtheopticalaxisbyshiftingthegrid usingamicrometer

screw. Theresultingimageswererecordedsothatit waspossibleto comparethebest

objectdistanceof both, crystal126 and127. The analysisof the focal scanshowed

that thebestobjectdistancefor crystal126appearedto beabout0.3mm shorterthan

thatof crystal127.

Sincebothglassformshaveslightly differentradii of curvature,theresultingfocal

length of both preparedcrystalsare slightly different, too (cf. Tab. 1.2). From the

differentradii of curvaturealone,i.e. without assigningthemto a particularcrystal,

onewould expectfrom thecrystalpreparedwith theglassform with Rm @ 275A 6 mm

andRs @ 269A 8 mm, resultingin a focal lengthof f @ 136A 34 mm, thebestimageat

an objectdistanceof 157.7mm. For the crystalwith bendingradii Rm @ 276A 0 mm

andRs @ 270A 0 mm, i.e. f @ 136A 5 mm, the objectdistanceamounts158.1mm, i.e.

0.4mm longerthanthatof theothercrystal.

By comparingthe experimentalresult and the calculateddistances,one would

expect that crystal126 waspreparedwith the glassform with Rm @ 275A 6 mm and

Rs @ 269A 8 mm. Thisconclusionis in agreementwith thelabelingof thecrystalitself

sothatthis resultis evident.

By comparingtheimagesof bothcrystalsby visible impression,theimageof crys-

36

tal 126seemsto bethebetterimage.Also, it wasmucheasierto identify thebestfocal

positionfor thatcrystal.Whenwe look at thederivativeof eachbestfocalizedimage

(solid line in Fig. 1.14),theaverageFWHM of theedgesof thegrid structureamounts

respectively 5.9 and6.4 pixels for crystal126and127,which is in accordanceto the

bettervisible impression.Also, themaximaof thederivative of crystal126aremuch

higher, i.e. theedgesaremuchsteeper, thanthatof crystal127. Moreover, while the

derivativeof theedgesof theimagefrom crystal126consistsof singlegausspeaks,the

derivative of the imagefrom crystal127 shows doublepeaks(arrows in Fig. 1.14b),

whichmightbedueto adoubleimagefrom thecrystalasaresultof thenothomogene

bendingradii. Additionally, theglassform associatedwith crystal127wasmoredif-

ficult to measuresincetheimagingquality of theglassform asa lensseemedto give

a doubleimage,so that this result is confirmed. In summary, the imagingtestof the

crystalsandglassforms are in agreement.Accordingly crystal126 givesthe better

imagethancrystal127.

However, this resultis contraryto thelabelingof thetestprotocol,wherethegood

imageis labeledasto bedueto crystal127andthebadimageasto bedueto crystal

126.Thus,onemightconcludethatthelabelingof theimagetestis wrong. In orderto

obtainfurtherevidenceof this conclusion,thesizesof thetestimagesof theprotocol

were compared. In the laboratoryimaging test, the position of the test object and

the crystal position, i.e. the object distance,were conserved, while the imagewas

focalizedbychangingtheimagedistance.Dueto theslightlydifferentfocallengthsthe

imagesizesareexpectedto beslightly different.For the1 mmlargetestobjectandan

objectdistanceof 182mm,thesizeof theimageshouldbe3.00mm and2.98mm for

136.5mm and136.34mm focal length,respectively. For theabout13 timesenlarged

imagein thetestprotocol,this imagedifferenceshouldamountabout0.2mm. Sucha

smalldifferencecanbeobservedin thetestprotocol,too. However, thelargerimageis

labeledwith “Crystal126” andthesmallerimageis labeledwith “Crystal127”, i.e. the

focal lengthsof crystal127and126wouldbe,respectively, 136.34mmand136.5mm,

thuscontraryto theexpectationandthemeasurementabove. Therefore,thelabelingis

obviouslyexchanged.

Visible light focusingat the target chamber

After installationof the imagingsystemto thetargetchamber, thebestfocal position

shouldbedeterminedby meansof visible light. Thishastheadvantagethatanintense

37

Broadeningat 7 mm (full) aperture

Broadeningat 0.8 mm eff. aperture

Geom. EstimationRay−Tracing

Diffraction limit

0

0.5

1.5

−1.5 −0.5 0 0.5 1.5

1.0

2.0

−1.0 1.0Deviation of object distance from focus / [mm]

Siz

e of

edg

e on

det

ecto

r (F

WH

M o

f der

iv.)

/ [m

m] Figure 1.15. The measuredFWHM of

anedgeasthe functionof the relative de-

viation from the focus togetherwith the

geometricalestimationof the broadening

at full 7 mm ( cdcdc ) and effective 0.8 mm

(——) aperture,respectively, andthelimit

dueto diffraction (- - -). For comparison,

theray-tracingresultfor 7 mm apertureis

plotted,too ZXefc�egc�eh[ .visible light sourceis veryeasyto obtain,andconvenientto use.However, thequality

of thealignmentis limited by thediffractionof thelight at theapertureof thecrystal.

Thus,the final precisealignmentcanbe doneby X-rays only, but the pre-alignment

by visible light limits the rangefor the expensive andtime consumingalignmentby

X-rays.

Figure1.15shows themeasuredFWHM of thederivativeof theimageof anedge,

which wasalignedto thecenterof thetargetchambertogetherwith thegeometrically

estimatedbroadeningof the imageof a point (the edge)at full aperturesizedueto

de-focalization(seebelow) andtheresolutionlimit dueto diffractionof thelight at the

apertureof thecrystal.

Diffraction limit As mentionedabove,thequalityof thealignmentis limited by the

diffraction of the visible light at the apertureof the crystal. The resolutioncan be

describedby [92]

d @ λ fA i 800nm C 136A 3 mm

7 mm @ 16µm (1.30)

wherethe 800 mm wavelengthis arbitrary, sinceno considerationswerecarriedout

concerningtheemissionmaximumof thelamp,which is in thefar-infraredregion,the

transmittivity andreflectivity of thewindows,lenses,mirror, andthesensitivity of the

CCD.For thegivenmagnificationof 37.75theedgewill bebroadenedon thedetector

to

16µm C 37A 75 @ 604µm i 27 pixel ATheobservedminimumwidth of edgeis in very goodagreementwith theprediction.

However, this coincidenceis moreor lessarbitrary, sinceit dependson theunknown

38

effectivewavelengthof thebacklightersource.

Geometricalestimation of the broadening Thede-focalization,i.e. thedistanceof

thecrystalfrom theoptimalobjectdistancecanbeestimatedgeometricallyby consid-

eringthebroadeningof anedge.

Dueto de-focalizationthepoint imagewill belying at a distanceδb in front of or

behindthedetectorposition,i.e. thedesignimagedistanceb0 (cf. Fig. 1.16).Depend-

ing on theaperturesizeA of thelens(crystal),thepointwill bebroadenedto

B @ Ab0 j I δb IkA (1.31)

Thedistanceδb is relatedto a variationof theobjectdistanceδa by thederivative

of thelensequation

b @ f j aa K f(1.32)

(which is equivalentto 1G f @ 1G a L 1G b, where f is thefocal length)as

δb @ K O fa0 K f P 2 j δa A (1.33)

CombiningEqs.(1.31)and(1.33)weobtain

B @ Ab0 j9lllll

O fa0 K f P 2 j δa lllll

A (1.34)

Note,thatthis equationassumesa rectangularapertureof thecrystal,i.e. thatthesize

of thecrystalis constantfor all anglesof incidence.For circularly(elliptically) shaped

crystals,onehasto apply the averagesizeof the aperture,which is reducedby π G 4comparedto thesizeof a quadratic(rectangular)aperture.Also, whenusingX-rays,

Eq.(1.34)canbeappliedin sagittalplaneonly sincein meridionalplane,theeffective

apertureis influencedby thewidth of thespectralline andthewidth of thereflection

curve.

Equation(1.34) is plottedin Fig. 1.15for thesagittalplanetogetherwith the full

ray-tracingcalculationfor comparison.Exceptfor the centerregion, which is domi-

natedby theimagingerrors,Eq. (1.34)is in verygoodagreementwith theray-tracing

result.

For thefull aperturesizeof 7 mm,onewouldexpectfrom Eq.(1.34)abroadening

of the imageasshown by the dashedline in Fig. 1.15. The rangeof uncertaintyin

39

Crystal

Source

Object Point Image Point Detector

PSfragreplacements

A AeffS

dS a

B

b0 δb

Figure 1.16. Theeffective apertureof thecrystal. Despiteof thelargeapertureof thecrystal

A, only asmallfractionof it, Aeff , is usedfor imaging.

thebestfocal positionwould bethereforereducedto about B 0 A 25 mm, i.e. therange,

wherethegeometricalestimationis hiddenby thediffraction.In practice,this rangeis

reducedagain,sinceonewouldexpectthebestfocuspositionin thecenterof thewaist

of thecurve.

However, theobservedFWHM remainedalmostconstantover thefull scanrange

of 2 mm,contraryto Eq.(1.34),whichpredictsachangein thebroadeningof theedge

to morethan2 mm for ade-focalizationof 1 mm.

At thetime of theexperiment,no explanationcouldbefoundfor theflat curve. A

lateranalysisshowedthatthereasonwasthesmallsizeof thesourceatalargedistance

from theobject,asillustratedin Fig. 1.16: DespitethelargeapertureA of thecrystal,

only a small fraction Aeff of it wasusedfor imaging,sincethereis no sourcepoint

lying on theaxiswhich connectstheouterregion of thecrystalwith theobjectpoint

(cf. dottedlines in Fig. 1.16). In consequence,theopeningangleof the raysis much

smaller, anda shift of the imagepoint off the detectorplanewill result in a smaller

broadeningthanexpectedfrom Eq.(1.34)for thefull aperturesize.

In the presentcase,the condenserlamp wasmountedoutsidethe target chamber

and the light was directedto the crystal by meansof a mirror. Thus, the resulting

distanceof thesourcefrom thegrid wasabouttwice theradiusof thetargetchamber,

say1700mm. Whentheapertureof thecondenserlampwould have beenfully open,

thesourcesizewould have beengivenby thesizeof thefilament,say10 mm. In this

case,theeffectiveaperturewouldhavebeenabout(cf. Fig. 1.16)

Aeff @ SdS j a i 10mm

1700mm j 140mm i 0 A 8 mmA (1.35)

40

insteadof thecrystalsizeof 7 mm,asassumed.Theresultingbroadeningof theimage

due to the delocalizationat 0.8 mm aperturesize is shown in Fig. 1.15, too. The

broadeninggrowsveryslowly, asobservedin theexperimentaldata.

However, at the time of the experiment,the apertureof the condenserlamp was

closedto seea sharpshadow of the grid on the crystal in order to ensurethat the

sourcewaslying on the optical axis. Anotherreasonfor the closedaperturewasto

reducetheintensityof thesourceto avoid saturationof theCCD.

In conclusion,onehasto considertheaspectof thesourcesizein futurealignment

procedures.Also, this resultis importantfor thedesignof backlightingexperiments.

Sinceoneis in generalinterestedin auniformbacklightingsource,onelikesto put the

sourcebehindthesampleasfaraspossiblein orderto getablurredimageof thesource

on thedetector. However, whenonelikesto usethefull apertureof thecrystal,too, in

orderto geta high efficiency of theoptic, onehasto make surethat thebacklighting

sourceis largeenoughsothatthefull apertureof thecrystalis really used.

Imaging testusingX-rays

From the visible light focusscanshown in Fig. 1.15, the bestfocal positionwasas-

sumedat the position marked at the ordinatewith zero, which refers to the target

chambercenter.

However, for theX-ray imagingtest,thegrid wasalignedin thecenterof thetarget

chamberasdescriedabove andthenshifted2 mm off the centertowardsthe crystal

in order to placethe backlightersourcein the centerof the target chamberand the

crystal itself had then to be shiftedby these2 mm, too. The laserfocal point was

1 mm behindtheMg foil targetsothatthediameterof thelaserspoton thetargetwas

1/3 mm. This resultingeffective apertureaccordingto Eq. (1.35)was large enough

(21 mm) to ensurethatthefull aperturesizeof thecrystalof 4 mm wasused.

Only at two shotswasthelaserintensityhigh enoughto obtaina reasonableeval-

uation.Thedataof thesetwo shotsaresummarizedin Tab. 1.3.

Basedon the visible light focal scan,the bestfocal position for the centerwas

assumedto beat18000steps,which is definedas0–positionin Fig. 1.15,asdescribed

above. Whenshifting the crystalby 2 mm, or L 50000steps,asit wasdonefor the

grid, the bestfocusshouldbe at 68000,which is beyond the limit of the translation

stage(62703).

Shot#24219wastakenatafocalpositionof 18000steps.TheFWHM of theimage

41

Defocalization / [µm]

Siz

e of

the

PS

F in

Sag

ittal

Pla

ne /

[µm

]

#24223

#24219

Ray−TracingGeom. Est.

0

200

400

600

800

1000

−1000 −500 0 500 1000

Figure 1.17. The resultsof the

X-ray imaging test: The dashed

lines mark the measuredwidths

of theedgeprofile andthe inter-

sectionwith the calculatedsolid

line gives the possiblepositions

of thefocus.

Shot Rel. Focalpos. Width of edge Resol. Est.misalign.

[steps] [mm] [pixel] [µm] [µm] m δa m /[mm]

#24219 18000 0.720 40 900.0 24 0 n 934

#24220 43000 1.720 — — — —

#24223 62703 2.508 29 652.5 17 0 n 683

Table1.3. Summaryof theX-ray imagetest.

of theedgein sagittalplaneis 40pixelsor 900µm. Fromthesimulation(Fig. 1.17)as

well asEq. (1.34)onecanestimatea de-focalizationof B 0 A 937mm. At shot#24223

theFWHM of theimageof theedgeis 29pixelsor 652.5µm, whichwouldcorrespond

to adefocusingof B 0 A 683mm. Betweenbothshots,thefocalpositionwaschangedby

44703stepscorrespondingto 1.789mm. FromEq.(1.34)followsthatthese1.789mm

wouldcauseabroadeningof theedgeof 87pixels.

Whenassumingthatbothimagesarelying on thesamesideof thedetectorplane,

the differencebetweenboth edgewidths shouldbe 87 pixels, which is contraryto

40 K 29 @ 11 pixels.

However, whenassumingthatbothde-focalizedcrystalpositionsarelying on op-

positesidesof the best focal position, one has to add 40 L 29 @ 69 pixels. This

result is in betteragreementwith the estimatedbroadeningof 87 pixels andthe de-

focalizationdistancesof B 0 A 937 mm (#24219)and o 0 A 683 mm (#24223): 0 A 937 L0 A 683 @ 1 A 620mm comparedto theshifted1.789mm.

The measuredvariationof the FWHM of theedgewhencomparingshot#24219

and#24223is 405µm smallerthanonewould expectfrom thesimulationor thedif-

ferencein thede-focalizationis 169µm smallerthanmeasured.Defectsin thecrystal

42

aswell asslopeerrorsof thesurfacewould causea broadeningof theedge[68]; also

theappliedsmoothingof thedatawould causea broadening.Onepossibility for this

narrowing mightbethatthetranslationstagedid notmovetheexpectedway. However,

this is unlikely, sincethetotal distancewascheckedandtheresolutionwasmeasured

as0.040µm/step.Also for thebacklash,this differenceof 169µm is too large. The

backlashwasdeterminedto belessthan1000steps,correspondingto lessthan40µm.

A shotwastakenat a countervalueof 43000(#24220),which is relatively close

to thebestestimatedpositionof 41350steps.However, thesignalin the imageis too

weakto distinguishtheimagefrom thebackgroundaswell asthedamagestructurein

theCCD arrayin orderto measuretheFWHM of theedge.Also, theedgecannotbe

uniquelyassignedto thesagittalbendingradius.However, from thevisualimpression,

thewidth of theedgeof shot#24220is muchbroaderthanonewouldexpect.

Summary

In summary, the detailedanalysisof the experimentaldataof the imaging test has

un-coveredthreemistakes: The testresultsof the crystalswerewrongly marked,no

regard was given to the size of the visible light backlightersource,which madea

pre-alignmentby visible light impossibleand third, not mentionedabove, due to a

softwareerror in the stepmotor controller, the crystalwasshiftedonly half the way

asintendedandassumed.After finding andfixing theseproblems,therewasno more

beamtime available. In this demonstrationexperiment,only a resolutionof 17 µm

couldbeobtainedat ade-focalizationof i 650µm within theavailablebeamtime.

1.3.2 Proposalfor an imaging test at an X-ray generator

Theray-tracingcalculationsin Sec.1.2.4predicta very high achievableresolutionof

assmall as0.08µm in the sagittalplaneusingtoroidally bentcrystals.Even though

thisresultseemsto betoooptimisticdueto thesimplificationsmadefor theray-tracing

calculations(perfectshape,only surfacereflection,neglectingscatteringat small de-

fectsin thecrystal)andunrealisticto achieve, it is interestingto seewhich resolution

canbeactuallyachievedin anexperiment.While theimagingtestusingtheMg-Lyαline asX-ray sourceatGEKKO XII wasnotsuccessfuldueto theverystrictconditions

andshortavailablebeamtime at this high-power laserfacility, anotherimagingtestis

proposedusinganX-ray generatorasanX-ray source.Eventhoughthe imagingtest

43

will becarriedoutat awavelengthwhich is unlikely usedin a plasmaexperiment,the

advantageis thatthis demonstrationexperimentcanbedonein asmall lab.

Besidestheaim of observingthehighestobtainableresolution,anotherconstraint

for thedesignof theexperimentwastheaspectof low costs,i.e. thereshouldbemade

useasmuchaspossibleof thealreadyavailableequipmentonasmalllaboratoryscale.

I.e., asanX-ray source,aconventionalX-ray generatorshouldbeused.Sinceit is very

difficult—andthusvery time consuming—tomanufacturea glasreplicaform with a

sufficiently highquality, i.e. anerrorin theradii of ∆RG R i 10H 3, anexistingglasform

shouldbeused.Together, theboundaryconditionsarevery restrictedsincethereare

only discretewavelengthsof X-Raysavailable,which give togetherwith thediscrete

latticespacingsof thecrystal,a discretenumberof Bragg-angleswhich hasto fulfill

Eq. (1.17)for thediscretesetof availablecombinationsof bendingradii, i.e. radii of

theglasforms.

In orderto solve this problem,a computerprogramwaswritten which calculates

all combinationsof characteristicK-shell linesof all elementsandlatticeplanesup to

the D 101010E reflectionin thecrystal.Anotherboundaryconditionis thatthestructure

factorof thecrystaldoesnotvanishandshouldbeashighaspossiblein orderto geta

high reflectivity of theBraggreflectionandthusa high luminosityof themicroscope.

Also this lattice planehasto be found in an elasticallybendablecrystalof ‘perfect’

quality. Therfore,Si, GeandQuartzweresearchedfor a suitablelatticeplane.For all

availableglasforms,i.e. combinationsof bandingradii, thestigmaticangleθ0 obtained

by Eq.(1.17)wascomparedwith theBraggangle,whereamisfit of 1 A 0F wastolerated.

A suitablecombinationwasfoundin theGe(224)reflectiontogetherwith theCr–

Kα line: Thetransitionwavelengthis λ @ 2 A 289781A, theinterplanarlatticespacing

d @ 1 A 154828,andtheresultingBraggangleθB @ 82A 478F . Thestructurefactor IFH I @144A 6 for theGe(224)reflectionat this wavelength[93].

This fits with the available pair of bendingradii of Rm @ 200A 0 mm and Rs @195A 9 mm with θ0 @ 81A 8F ; i.e. themisfit θB K θ0 @ 0 A 68F correspondingto amisfit in

thewavelengthof λCrKα K λθ0 @ 0 A 00374A. Sincethespectralwindow of thecrystal

∆λ i 0 A 0046A atamagnificationM @ 30andanaperturesizeof thecrystalof 3 mm,

theexperimentcanbecarriedoutat thestigmaticangleθ0 nevertheless.

The correspondingfocal lengthof thesebendingradii is f @ 98A 98 mm so that

a magnificationM @ 30 is obtainedfor an object distancea @ 102A 28 mm and an

imagedistanceb @ 3068mm. Figure1.18shows thesizeof thepoint spreadfunction

44

in meridionalplaneandsagittalplanefor thehereconsideredconfiguration,together

with the resultsfor the sameconfigurationbut a sphericalcrystal. In sagittalplane,

a resolutionof 1 µm can be obtainedalreadyat an apertureof i 9 mm using the

toroidally bentcrystal,while for the sphericalcrystal,the aperturehasto be smaller

than0.1mm in orderto passthe‘1–µm–threshold’.In themeridionalplane,both,the

resultfor thetoroidalandsphericalcrystalarein agreementasexpected,sincein the

meridionalplanetheconfigurationwasnotchanged.The‘1–µm–threshold’is reached

at 3.1mm aperturesize.

Consideringnow theexperimen-

sagittal plane(toroidal crystal)

sagittal plane(spherical crystal)meridional plane

1 µm

Size of Crystal Aperture / [mm]

0

10

20

30

40

50

60

70

80

90

02468100

0.5

1.5

2.5

Res

olut

ion

/ [µm

]

Siz

e of

PS

F o

n de

tect

or /

[µm

]

1.0

2.0

3.0

Figure 1.18. The sizeof the point spreadfunction

(PSF) in sagittaland meridionalplanefor both, a

toroidalcrystal(Rm p 200mm,Rs p 195n 9 mm)and

a sphericalcrystal (R p 200 mm) at 30 _ magnifi-

cation togetherwith the correspondingachievable

resolutionasa function of the aperturesizeof the

crystal.

tal set-up,asimplegeometricalrela-

tion shows that for an X-ray source

sizeof 1 C 1 mm2 andan intended

effective apertureof the crystal of

3 mm, the sourcehasto be placed

25 mm behindthetestgrid, in order

to getafield of view of 1 mm.

The expectedflux from the X-

ray generatorwasestimatedby the

formulaof Honkimaki et al. [94] as

N G 4π i 2 A 1 C 1014 photons/sfor an

excitation tensionU @ 60 kV anda

currentI @ 30 mA. Thus,a 1 mm2

large grid in 25 mm distanceis il-

luminatedby 3 A 4 C 1011 photons/s.

Togetherwith the efficiency of the

crystal,i.e. thenumberof reflectedphotonscomparedto thenumberof emittedpho-

tonsfrom thesource,(calculatedby ray-tracing)of I G I0 @ 7 A 1 C 10H 7, onecanexpect

2 A 4 C 105 photons/son thedetector. And sincetheobjectsizehasanareaof 1 mm2,

whichis 30 C magnified,thereflectedphotonsaredistributedover30 C 30mm2 sothat

theintensityon thedetectoris i 2 A 7 C 10H 4 photons/s/µm2. A typicalvaluefor there-

quiredexposureof anX-ray film is 1 photon/µm2, which leadsto a requiredexposure

timeof i 3700s,or i 1 hour.

A critical point is theabsorptionin air: Sincethe linearabsorptioncoefficient of

air at normal pressure(1013 mbar) µ @ 0 A 0393G cm, the reflectedX-rays would be

45

absorbedon the imagedistanceto I G I0 i 7 C 10H 6. Therefore,the reflectedX-rays

haveto beguidedin vacuum,wherea low vacuumof 1 mbaris sufficient to reducethe

absorptionto 1.2%, i.e. 98.8% of thesignalis transmitted(cf. Ref. e.g. [95] together

with e.g. Ref. [96]).

1.4 Experimental setup

Thesetup of theexperimentis schematicallyshown in Fig. 1.4 (page12). As a sam-

ple,a 25 µm thick CH plasticplatewasusedwith eithera flat surfaceor anintention-

ally imposedsinusoidalsurfacestructure. As the driving laser, all twelve beamsof

GEKKO XII werefocalizedontothesamplein focal spotsizeof 600µm in diameter.

Thetargetwasirradiatedin two steps.In orderto keeptheinitial imprint of the laser

on thetarget(i.e. ξ0 @ ξ D t @ 0E ) assmallaspossible,thetargetwasilluminatedin the

first 2 nsby threebeamsof partialcoherentlight (PCL),eachof 25 J in 2ω mode,i.e.

a wavelengthλ @ 0 A 53 µm, which ensuresa very high uniformity of the laserpulse.

The remainingninebeams,eachof 100J, wereusedfor themain pulsein 3ω mode

(λ @ 0 A 35 µm), which resultsin a muchhigherintensitybut a lower uniformity than

thePCLpulse,“but still betterthananormallaserprofile.” [97] Concretermsnumbers

for theuniformity of thebeamprofilearenotavailablehere.

As a backlightersource,a Mg foil wasplaced10 mm behindthesample.TheMg

foil wasirradiatedwith 4 C 1013 W/cm2 by thePeta-Watt-Module(PW-M) laserasthe

probelaserin a spotsizeof 1400µm diameter. The pulselengthwas160 ps with a

delayof 2 nscomparedto thedriving HIPERbundle.

The transmittedintensitywasimagedby the above describedimageron a streak

camerawith 200pstemporaland100µm spatialresolutionat a distanceof 3674mm,

providing a 25.9C magnifiedimage. The imagerwasalignedby a focal scanusing

visible light. The effective apertureof the crystalwasestimatedto be about3 mm,

whichwassufficiently largeto identify thecenterof thewaistof thecurve(cf. Section

1.3.1).

1.5 Preliminary results

For the observation of the Rayleigh-Taylor growth rate, only aboutsix shotswere

scheduledin this experimentalcampaigndueto the large numberof concurringex-

46

x / [µm]

y / [

µm]

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 800

Figure1.19. Thetime integratedimageof

the laserimprint on a flat target. Thesize

of thespecklesis in theorderof ] 38 µm.

perimentalaimsandproblemswith the lasersystem. The intensityof thesedatais

very low dueto problemswith the probelaser. In consequence,the signal to noise

ratio is very low, whichmakesadetailedanalysisalmostimpossible.Nevertheless,the

obtaineddatawill bebriefly presentedhere.

First we tried to observe the initial imprint of the laserbeamon thetarget. A flat

plastic plate without any initial structurewas usedas a target. For this part of the

experiment,aCCDcamerawasusedasadetectorfor theHIPERimager, whichgavea

2-D spatiallyresolvedbut time-integratedimage.Only oneshotwassuccessfulandthe

resultingimageis shown in Fig. 1.19.Despitethesignalbeingvery weak,a structure

canbe observedafter imageprocessing.The sizeof the dominatingdark andbright

specklesin Fig. 1.19 is in the orderof i 38 µm. As alreadymentioned,rms values

for thenon-uniformityof theincidentlaserbeamarenotavailableheresothatit is not

possibleto comparethe spectralpower densityof the perturbationwavelengthswith

thenon-uniformityof theprofileof thelaserbeam.

Theilluminatedareain Fig. 1.19shouldhave a slightly elliptical shapedueto the

projectionsequence.Thebanana-likeshapeof theilluminatedarea,however, pointsto

problemswith theprobelaser, i.e. thebeamprofilehadnot theintendedcircularshape

but thatbanana-likeshape.A controlimageof thebeamprofile is notavailable.

Next, we tried to observe thetemporalgrowth of theRayleigh-Taylor instabilities.

Therefore,the CCD camerawasreplacedby a streakcamerawith 200 ps temporal

and i 100 µm spatialresolution. In the first step,a flat plasticplatewasusedagain

asa target. However, the obtainedsignal is too weak to be distinguishedfrom the

backgroundnoise.

47

500

2000

2500

3500

0

1500

1000

3000

Integr.

Tim

e / [

ps]

Tim

e / [

ps]

(a) (b)#24388

0 100 200 300 400 500 600 700 800 900 1000

x/[µm]

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500 600 700 800 900

x/[µm]

Figure 1.20. The time resolved growth of theRayleigh-Taylor instabilitiesof a plastictarget

with imposedsinusoidalstructureof 40 µm wavelength.(a) shows theobtainedstreakcamera

imageand(b) shows theprofilesfor differenttimes,integratedover 200ps.

Onereasonfor thelow intensitywasthediameterof theprobelaserbeamprofile,

which wasabouttwice as large as intended,i.e. about1400µm insteadof 700 µm,

so that the intensitywasreducedby a factorfour comparedto thedesignvalue. Ad-

ditionally, thebacklightertargetwassmallerthanthebeamprofile sothatonly about

onefourthof theprobelaserbeamhit thetarget,whichcausedagainareductionin the

numberof emittedphotonsby a factor four. For the next shots,the probelaserwas

re-aligned,but thecontrolimageof thebacklightersource,takenby apinholecamera,

showedthattheproblemremained.

In the next step,we useda 25 µm thick plasticplatewith an imposedsinusoidal

structureon thesurfacewith awavelengthof 40 µm asa targetandtheapertureof the

crystalwasopenedfrom 4 mm to 7 mm in orderto compensatefor the low intensity

of thebacklighter. Theobtainedstreakcameraimageis shown in Fig. 1.20(a)andthe

integratedprofile aswell astheprofilesfor differenttimesareshown in Fig. 1.20(b),

whereaseachprofile is integratedover200ps.Theprofilesarenormalizedto thesame

integratedintensityin oderto compensatefor thevaryingintensityof thebacklighter.

Eventhoughthe intensityis too weakfor a detailedevaluationby a spectralden-

sity analyzationwith a reasonableerror, the growth of the amplitudecanbe clearly

observed.

48

1.6 Concluding remarks

Thelow repetitionratesof thehigh-powerlasersystemsontheonehandandthemany

openresearchquestionson theotherhandlet theprogressof laser-drivennuclearfu-

sion appearto be very slow. The questfor nuclearfusion is a journey of many little

stepsto physicswhichareunknown in thenatureonearth[44]. On thatway, thereare

many “unexpectedsurprises”[12]: As, for instance,DuderstadtandMosesstatedin

1982 [4] that by the end of that decadeall problemswill be solved, the Rayleigh-

Taylor instability was no subject. They mentionedonly that “some scientistsfeel

that Rayleigh-Taylor instability might be an important issuein the future.” Today,

Rayleigh-Taylor instability is a mainresearchissue.Theobservedstabilizationeffect,

which is muchstrongerthan expected(cf. Fig. 1.3), might be due to nonlocalheat

conductionby electrons.Thenonlocalelectronspreheatthemainfuel, whichraisethe

fuel adiabatandmake the fuel lesscompressibleandthusraisingthe ignition thresh-

old. Therefore,it is critically importantto clearify whetherthenonlocaltransportstill

dominatesin future targetsandwhetherthe preheatingof the main fuel by nonlocal

electronscanbeprevented[63].

The presentwork is onelittle stepon the way to answerthis importantquestion:

Eventhoughno high quality experimentaldatacouldbeobtainedwithin theavailable

time for this work, the imageris a main diagnosticof theHIPER researchprojectat

ILE which wasstartedto clearify theabove mentionedquestionfor future high-gain

targets.Theimageris expectedto deliver highly resolveddatain futureexperiments,

which will help considerablyin the understandingof the Rayleigh-Taylor instability

andhow to mitigatethem.

An experimentaldemonstrationof thehereshown theoreticallypossibleresolution

of sub-micrometerswould alsofind many applicationsin biology aswell asmaterial

sciences.

49

Chapter 2

Diagnosticsof X-Ray Laser Plasmas

2.1 Intr oduction

It is a greatchallengeto extendtheadvantagesof the laser, i.e. high coherence,high

naturalcollimation,smallspectralline, andhigh brilliancein theX-ray regime,since

thereit is possibleto resolve atomsandmolecules.Several schemeshave beenpro-

posedfor the pumpingof the X-ray lasingmedium,of which the collisional scheme

is currentlyconsideredas the most reliableandrobust methodfor generatinghigh-

gainamplifiedspontaneousemission(ASE) in theX-UV region [98]. In this method,

a target materialis irradiatedwith a high power optical pumpinglaser, strippingthe

lasing ions to a closed-shellconfiguration,usually the nickel-like or neon-like ions

(Fig. 2.1). For lasingin the Ni-lik e (or Ne-like) configuration,the conditionsin the

plasmashouldoptimizesimultaneouslythe fraction of lasingions andthemonopole

collisional-excitation ratefrom the groundstateinto the 4d (or 3p) manifold so that

populationinversionis formedon a numberof transitionsbetween4d and4p (or 3p

and3s) asthelatteris de-populatedby rapidradiativedecay(cf. Fig. 2.1(b))[98].

Theaim of theexperimentalseriespresentedhereis theextensionof stablelasing

towardstheso-calledwaterwindow [15,99], i.e. therangebetweentheK-absorption

edgesof carbonandoxygenat 2.31nm and4.38nm, respectively. In this region, the

waterof biological solutionsis relatively transparentwhile the carbonatomsof the

biologicalmoleculesabsorbverywell.

Thedifficulty in achieving lasingtowardsshorterwavelengthsis thathigherZ ma-

terialsarerequiredwith a substantialincreasein the requirementsin the plasmapa-

rameters[100]:

50

OpticalPump Laser ~10¹³ W/cm²

~ 200 GW

L ~ 2 cm

~ 100 µm

X−Ray Laser

X−Ray Laser

Target

Plasma

Ground levelof lasing ion

Metastable level

Electron

excitationcollisional Fast radiative

LASING LINE

decay

(a) (b)

Figure 2.1. Thesetup(a) andthe inner-atomicprocesses(b) of thegenericcollisionalX-ray

laserexperiment.

1. Theremustbesufficient ionizationin orderto exceedtheNi-lik e stagebecause

the monopoleexcitation rate from the groundstateof the Ni-lik e ions to the

upper lasing level maximizesat electrontemperatures2–3 times higher than

thatat which themaximumNi-lik e abundanceis realized,whereasa significant

proportionof theionsstill needsto remainin theNi-lik estagesif thelaseris to

function. Thereforea rapidly ionizing plasmais desiredasa gain mediumfor

which thepopulationof theNi-lik e 4d level, which is theupperlasinglevel, is

maximum.

2. A high plasmadensityis requiredfor high-atomic-numberelements[16]. In

theestimationin Fig. 2.2, themaximumdensityfor thepopulationinversionis

restrictedby the thermallimit of the lasingtransitionsandopacityeffectssuch

asthetransitionbetweenthelowerlasinglevel andthegroundstate.Thedesired

electrontemperatureanddensityare1–1.5keV and1 A 5 C 1021 cmH 3 for 70Yb

and1.4–1.8keV and3 A 5 C 1021 cmH 3 for 74W, respectively [16]. The higher

densityprovideshighergain,but thedurationandwidth of thegainaresmaller,

makingpropagationof theX-ray lasermoredifficult [101].

3. Finally, X-ray laserbeampropagationthroughthelasingplasmasmustbecon-

sidered.Thesimplifiedray-tracingcalculationfor X-ray propagationat awave-

lengthof lessthan10nmthroughtheamplifyingmediumshowsthat,in contrast

to thecaseat20nm,theobliquelyincidentX-ray laserbeampenetratesinto and

51

0

500

1000

1500

2000

2500

1019

1020

1021

1022

40 45 50 55 60 65 70 75 80

Exc

itatio

n E

nerg

y (e

V)

Ele

ctro

n T

empe

ratu

re (

eV)

Atomic Number

Excitation Energy

Electron Temperature

Laser transitions

Electron D

ensity (cm¯³)

Electron Density

Figure 2.2. The desiredplasma

parametersfor a Ni-lik e soft X-

ray lasingmediumasa function

of theatomicnumber[16]

is absorbedin the solid-densityplasmaanddoesnot refractbackto the lower

densitygain region becausethe critical densityat 10 nm is q 1 r 1025 cms 3,

which is muchhigherthanthat of the solid-densityplasma.Thereforethe ray

at a 20-nmwavelengthis easilyguidedby a controlledrefractive-index profile

suchasacurvedtarget[102], but suchguidingdoesnotwork well for theshort-

wavelengthX-ray lasers.

Thereforeit is mostimportantto produceahomogeneousplasmawith appropriate

densityρ and scalelength L tvu ρ w$x dρ w dzy&z min asa gain mediumby selectionof a

prepulseandmainpulsewith appropriateindividualpulseduration[16,102–104] and

wavelengthto yield the desiredcrosssectionwith a long effective gain length. The

lasingmaterialsof considerationin this work wereNi-lik e 70Yb, 72Hf, 73Ta and74W,

whoselasinglinesarejust before(70Yb – 73Ta)or inside(74W) thewaterwindow.

Thecontribution of thepresentwork to theseXRL experimentswasthedevelop-

mentandapplianceof a toroidalcrystalspectrometerto thecollisional laserplasmas

with the aim of a betterunderstandingof the spatialand temporalbehavior of the

ionization stateby the detectionof Ni- and Co-like 4 f –3d transitionsin the lasing

plasmas.Theseemissionlines areparticularlyinterestingfor Ni-lik e XRL, sincethe

intensityof theselinesshouldberelatedto theinversionpopulationof thelasingtran-

sitionsincethe4 f level is verycloseto the4d level, theupperstateof the4d–4p lasing

transitionasshown in Fig. 2.3. I.e. the4 f –3d transitioncanserve asanindicatorfor

Ni-lik e4d ions[16].

Moreover, the lifetime of the 4 f –3d transitionis relatively shortso that a time-

resolvedobservationof theionizationstateasa functionof thepump-laserpulsetrain

is possible.In addition,theelectrontemperatureand-densitycanbeobtainedfrom the

intensityratio andwidth of theemissionlines,respectively, of thespectra[25,26].

52

While theXRL wavelengthis around50A, thehereobservedNi- andCo-like4 f –

3d transitionslie atabout6 A, which is roughlyoneorderof magnitudeawayfrom the

lasingtransition.For X-ray optics,this hastheadvantagethat theobservationcanbe

donewith someeasewith e.g. perfectlybentQuartzcrystals.

The organizationof this Chapteris

3d

4d

4f 2098.9

1941.4

0

Ni-like Ta

4p 1664.3

unde

r ob

serv

atio

nXRL

Figure 2.3. Energy level diagramfor Ni-lik e

Ta. (Energiesarein eV.)

as follows: First, the basicsof the de-

terminationof the electrontemperature

and electrondensity is presented.The

requirementsandrealizationof thespec-

trometeris thenpresentedin Section2.3

togetherwith the fundamentalsand the

performanceof the spectrometer. Sec-

tion 2.4 describesthentheexperimental

setup,followed by a descriptionof the

dataprocessingbeforeexperimentalre-

sultsarepresentedanddiscussed.

2.2 Determination of plasmaparameters

2.2.1 Electron Temperature

In thiswork theelectrontemperatureof thelasingplasmais obtainedby acomparison

of the intensityof two spectrallines usingthe ordinaryBoltzmanndistribution: The

intensityratioof theemissionlinesis proportionalto theratioof thepopulationstates,

which is relatedby theBoltzmann-distribution

N1

N0t g1

g0es E1 { E0

kBT (2.1)

whereNi arenumberof particleswith energy Ei in thestatesi t 0 | 1 andgi t'x 2Ji } 1yarethestatisticalweights,i.e. numberof stateswith thesameenergy Ei .

SincetheNi-lik e ion hasa closedshell, the4 f ~ 3d (J t 1 ~ 0) transitionis split

only in threedifferentlines,which canbeclearlydistinguished.Thus,thecalculation

of thetemperatureis very convenient,sinceit ensuresthatthereis only onetransition

in onedetectedline andthat theselineshave thesamegroundlevel, which simplifies

the calculationconsiderably. Otherwise,onewould have to calculatethe theoretical

53

contribution of thedifferenttransitionsto theobserved“line” and/orapply theSaha-

Boltzmann-statisticsandcomparedifferentionizationstageswhichmightbecomevery

difficult or impossible[105].

By comparingtheemittedintensityof two singleemissionlines,wegettherelation

N1

N2t g1

g2es E10 { E20

kBT (2.2)

whereEi0 t E1 ~ E0 is the transitionenergy of the emissionline. This expression

is now independentof the unobservablenumberof particlesin the groundstateN0.

Furthermore,theemittedintensity[106,p. 15f]

Ii0 ∝ NiAi0 � (2.3)

whereAi0 is theEinsteincoefficient for spontaneoustransitionfrom statei to state0.

Insertingthis relationin Eq.(2.2),weobtain

I10

I20t A10g1

A20g2es E10 { E20

kBT � (2.4)

In theliterature,it is morecommonto give theproductgfi j of thestatisticalweightg

andthe oscillatorstrengthfi j , wherethe oscillatorstrengthis relatedto the Einstein

coefficientAi j as[106,p. 34ff]

fi j t meh x 4πε0 yπe2 � ν c3

8πhν3

gi

g j� Ai j t meε0c3

2πe2ν2

gi

g jAi j (2.5)

i.e.

Ai0 ∝ gfi0E2i0 (2.6)

sothatwecanrewrite Eq.(2.4)as

I10

I20t gf10E2

10g1

gf20E220g2

es E10 { E20kBT � (2.7)

IsolatingkBT, wefinally obtain

kBT t�~ E10 ~ E20

ln � gf10E210g1

gf20E220g2

I20I10 � � (2.8)

In our particularcaseg1 t g2 sothatEq. (2.8)reducesagainto

kBT t�~ E10 ~ E20

ln � gf10E210

gf20E220

I20I10 � � (2.9)

54

This is thefinal form which wasusedfor computation.While thevaluesfor I andE

canbedirectlyobtainedfrom thespectra,thegf valueshaveto becalculatedquantum-

mechanically. Table2.1 sumsup somevaluesfrom the literaturetogetherwith the

valueswhich werecalculatedby DongandFritzsche[107] in the frameof this work

by amulti-configurationDirac-Fock code.

Table 2.1. Transitionprobabilitiesof theobserved3d94 f � J � 1� – 3d10 � J � 0� transitionin

Ni–like Yb, Hf, TaandW ions

Ion Transition Ji Jf Type gf a gf b gf c gf d � gf �72Hf44� 3d9

3� 24 f5� 2 � 3d10 1 0 E1 5.71 6.304 5.910 6.14 6 � 02 � 0 � 26

3d95� 24 f7� 2 � 3d10 1 0 E1 1.78 1.755 1.722 1.67 1 � 73 � 0 � 05

73Ta45� 3d93� 24 f5� 2 � 3d10 1 0 E1 6.06 6.248 5.846 6.09 6 � 00 � 0 � 11

3d95� 24 f7� 2 � 3d10 1 0 E1 1.87 1.847 1.72 1.76 1 � 83 � 0 � 11

3p53� 23d104s1� 2 � 3d10 1 0 E1 0.350 0.376 0.368 0.347 0 � 37 � 0 � 01

74W46� 3d93� 24 f5� 2 � 3d10 1 0 E1 6.01 6.189 5.815 6.04 6 � 01 � 0 � 15

3d95� 24 f7� 2 � 3d10 1 0 E1 1.96 1.939 1.900 1.85 1 � 91 � 0 � 05

3p53� 23d104s1� 2 � 3d10 1 0 E1 0.369 0.377 0.368 0.347 0 � 37 � 0 � 01

aRef. [107]: Fritzsche& Dong,1999bRef. [108]: Quinet& Biemont,1991cRef. [109]: Zhang& Sampson,1991dRef. [110]: Zigler et al., 1980

2.2.2 Electron Density

Calculation by the width of the emissionline

Theelectrondensityof theplasmacanbeobtainede.g. from thewidth of theemission

line. Therearethreemaincontributionsto thewidth of theemissionline: Thenatural

linewidth, theDopplerbroadeningandtheresonancebroadening[23,24]:

Natural linewidth: Thenaturalwidthof theemissionline is givenby theHeisenberg

uncertaintyrelation

∆E � ∆t t ∆E � τ �*��w 2 (2.10)

55

Table 2.2. Transitionwavelengthsof theobserved3d94 f � J � 1� – 3d10 � J � 0� transitionin

Ni–like Yb, Hf, TaandW ions

Ion Transition Ji Jf Type λ/[A]a λ/[A]b λ/[A] c λ/[A]d

72Hf44� 3d93� 24 f5� 2 � 3d10 1 0 E1 6.128 6.126 6.122 6.1322(6)

3d95� 24 f7� 2 � 3d10 1 0 E1 6.322 6.323 6.322 6.3225(6)

73Ta45� 3d93� 24 f5� 2 � 3d10 1 0 E1 5.901 5.900 5.895 5.9041(5)

3d95� 24 f7� 2 � 3d10 1 0 E1 6.090 6.092 6.089 6.0892(3)

3p53� 23d104s1� 2 � 3d10 1 0 E1 6.366 6.365 6.366 6.3713(4)

74W46� 3d93� 24 f5� 2 � 3d10 1 0 E1 5.686 5.685 5.682 5.692(3)

3d95� 24 f7� 2 � 3d10 1 0 E1 5.872 5.873 5.871 5.875(4)

3p53� 23d104s1� 2 � 3d10 1 0 E1 6.149 6.149 6.150 6.156(4)

aRef. [107]: Fritzsche& Dong,1999bRef. [108]: Quinet& Biemont,1991cRef. [110]: Zigler et al., 1980dExperiment;thenumberin bracesgivestheuncertaintyin thelastdigit

whereτ is thelife timeof theexcitedstate.Interpretingthe“ � ” as“=” weget

∆E t�� ∆ω t �2τ

(2.11)

or

2∆ω t 1τ

(2.12)

where∆ω is thehalf width at half maximum(HWHM). Thelife-time τ is givenby

1τt,�

j

Ai j (2.13)

the sum of the transitionprobabilitiesof all possiblespontaneousdecaysfrom the

upperstate[25]. Therelativenaturalline width canthenbecalculatedby

∆ωω

tT� j Ai j

2�E � (2.14)

To givearoughnumber, let usassumethattherearenofurtherspontaneousdecays

from the upperstatethan the observed and in Tabs.2.1 and2.2 tabulatedlines. In

thiscase,the � j Ai j t Ai j q 4 r 1014/s for theTa3d93� 24 f5� 2 ~ 3d10 transition,whose

transitionwavelengthλ t 5 � 90A t 2100eV, and

∆ωω

q 6 r 10s 6 �56

This linewidth is broadenedby two maineffects:

Doppler broadening: The Doppler broadeningarisesstraightforwardly from the

Dopplershift causedby thermalparticlemotion. A Maxwellianvelocity distribution

givesriseto a Gaussianline profile [25]

I x ω y�t I x ω0 y exp ��~ x ω ~ ω0 y 2mac2

2ω20kBT � (2.15)

wherema is themassof theatomandkBT the thermalenergy. Theresultingrelative

HWHM is∆ωω

t�� 2ln x 2y kBTmac2 � (2.16)

Insertingtheappropriatevaluefor Ta, i.e. ma t 180� 9 au t 3 � 0 r 10s 19 kg andassum-

ing a thermalenergy of 1 keV, weget

∆ωω

t 9 r 10s 9 |i.e. comparedto a naturalline width which is in theorderof 10s 5 ~ 10s 6 on theone

handandthespectralresolutionof theinstrumentwhich is in theorderof 10s 3 ~ 10s 4

on theotherhand,wecanneglecttheDopplerbroadeningcompletely.

Resonancebroadening: Thedominatingcontribution to theline width is causedby

a shortenof thelife time of theexcitedstatesdueto interactionwith otherions in the

plasma.This broadeningis calledcollisional or Starkbroadeningandis proportional

to thenumberof ionsperunit volume,sincetheprobabilityof acollision increasesthe

moreionsarearound,i.e. thehighertheplasmadensityis.

Again, we get the width of the spectralline dueto collisionsvia the Heisenberg

uncertaintyrelationas

∆ω t 12τ

(2.17)

where∆ω is the half width at half maximumon the ω-scaleandτ is the life-time of

thestate.This time is equalto theaveragenumberof collisionspersecondandcanbe

calculatedby thekinematicalgastheoryas[23,24]

1τt πρ2Nv (2.18)

whereN is thedensityof ions,v is therelativevelocityof thecollision partnersandρis thedistanceof thecollision partnersduringthecollision, i.e. thesumof theradii of

57

thecollision partners.Thisparameteris alsocalledcollision- or Weisskopf-parameter

in theliterature.

It is hard to predict the valueof the Weisskopf parametersincewe don’t know

what happensexactly during the collision. However, one can give estimationsfor

the collision parameter. Weisskopf estimatesρ via the meanfrequency shift of two

classicaloscillatorswhicharecoupledby theirdipoleinteraction.It is [24]

ρ t 12v

c2re

ω0f (2.19)

with ω0 thecenterfrequency andre t e2 w$x x 4πε0 y mec2 yWt 2 � 82 r 10s 15 m is theclas-

sicalelectronradiusand f theoscillatorstrength.

CombiningEquations(2.17),(2.18)and(2.19)weget

∆ω t π2

rec2

ω0f N (2.20)

or

N t 2π

ω0

rec2 f∆ω � (2.21)

Therearemany moresophisticatedways to estimatethe collision parameter(cf.

Ref. [26] andreferencesherein).However, all estimationsassumeneutralatomsand

arenot valid for thepresentcaseof highly ionizedatoms.For the ionizedcase,“the

detailedcalculationsareextremelycomplicatedandhave beendonein detailonly for

a few atoms. . . . Full calculationsgive the constantof proportionality.” [25, p 220].

Therefore,thenumberspresentedhereshouldbeconsideredasrelativenumbers.

Calculation by the intensity ratio of the emissionlines

For completeness,it shouldbementionedthatit is alsopossibleto obtaintheelectron

densityby comparingtheobservedemissionline intensitywith thetheoreticalexpec-

tationby theBoltzmannequation.Dueto self-absorptionor ‘optical thickness’,which

dependson theelectrondensity, temperature,wavelengthandsizeof theplasma,the

observedemissionline intensitywill differ from this expectation.Thus,it is possible

to calculatetheelectrondensityby anappropriatemodelfor theself-absorption:When

knowing theemissionline ratio andtheelectrontemperatureaswell asthesizeof the

plasmaandwavelengthof theemissionline, theelectrondensityis theonly free pa-

rameterin themodel. This methodis describedin moredetailandappliedby Nantel

et al. in Ref. [105].

58

The advantageof this methodis that it is easyto observe the emissionline ratio

which is independentof thebroadeningof theemissionline. However, thetheoretical

effort is considerableandoneneedsto know theelectrontemperatureof theplasmato

beableto computetheBoltzmannequationfor the theoreticalemissionof theX-ray

without absorption.But theelectrontemperatureis alsoobtainedfrom theBoltzmann

distribution. Thus,on theonehand,oneneedstwo spectrallineswith thesameoptical

thicknessfor the calculationof the temperatureand two other emissionlines with

strongdifferentoptical thicknessesfor the calculationof the electrondensity[105].

Eventhoughsimpleformulaefor theestimationof theopacityfor blackbodyradiation

exist [111,112], they areof limited usefor thecharaterizationof theline spectrasince

they areindependentof thewavelength.Thedetailedcalculationof theopacitiesasa

functionof thewavelengthis far beyondtheframeof this work andpublic accessible

codeslikeTOPS[113] arerestrictedto Z � 30.

Sinceon the onehand,the valuesfor the opacitiesarenot accessibleandon the

otherhand,themeasuredtemperaturein this work hasa very high uncertainty, not at

leastdueto theself-absorption,this methodwasnot appliedin this work.

2.3 The Spectrometer

2.3.1 Requirements

For the observationof the 4 f ~ 3d emissionlines,a new spectrometerwasdesigned

which should fulfill the following requirementsat once: The spectralrangeof the

spectrometeris given by the 4 f –3d transitionlines of 70Yb – 74W, which are lying

between5–7A sothatthespectrometershouldcover this range.

Thespectrashouldberecordedeithertimeintegratedbut spatiallyresolvedor time

resolvedbut spatiallyintegrated. In spatialresolutionoperationmode,eithertheho-

mogeneityof theplasmasalongthe lasingaxisor theplasmaparametersin blow-off

directionshouldbeobserved, i.e. at which distancefrom thetargetandat which den-

sity andtemperaturelasingoccurs,while thetemporalresolvedspectrashouldbeused

to observe theinfluenceof thepumpingpulsetrain on theplasmaparameters.

For thespatiallyresolvedoperationmode,aCCD camerawith apixel sizeof 22.5

µm shouldbeused,while the time resolvedspectrashouldbe recordedwith a streak

camerawith 100 µm spatial resolutionfor the recordingof the spectra. The spec-

59

trometershouldprovide a spatialresolutionof q 0 � 1 ~ 0 � 2 mm sincethe sizeof the

expandingplasmais in theorderof 1–2mm. In bothcasesthespectralresolutionof

theinstrumentshouldbe∆λ w λ � 10s 3.

The spectrashouldbe sufficiently bright to give a reasonablestreakcameraim-

agewith ∆t q 20 ps temporalresolution. Due to the useof the streakcamera,the

spectrometerhadto bemountedoutsidethetargetchamber, i.e. theminimumdistance

from thesourceto thespectrometerwasgivenby theradiusof the target chamberas

about � 750 mm. Another importantaspectfor the spectralresolutionis the source

sizewhich is typically 5–30mm in XRL experimentsandwhich cannot beassumed

aspoint sources.

As explainedin thenext Sections,only aspectrometerusingatoroidallybentcrys-

tal canmeetall theserequirementsat once,sinceonecandesignthecrystalaccording

to thegeometricalexperimentalconditionsandthespectralresolutionis almostinde-

pendentof thesourcesize.

2.3.2 The principle of the Johann-typespectrometer

For the diagnosticof the laserproducedplasma,a modified versionof the Johann

spectrometer[114,115] was used,which offers the possibility to recordthe whole

spectrumatonce.Thiscanbedoneby settingthesourceinsideor outsidetheRowland

circle [116]. In theonly consideredcasehere,wherethecrystalis placedoutsidethe

Rowland circle, the X-rays coming from the sourcepassthe Rowland circle over a

largesection.Eachcrossingpointcanbeconsideredasapointsourcewhich is imaged

on theoppositesideof theRowlandcircle (Fig. 2.4).

While thebendingradiusin themeridional(dispersion)planeRm is usedfor spec-

tral focalization,the crystalcanbe bentperpendicularlyto thedispersionplane,too,

whichcanbeusedfor 1-dimensionalimaging:

For imaging,thedistancefrom thesourceto thecrystala andthedistancefrom the

crystalto theimageb is relatedby thewell-known lensequation[70]

1ft 1

a } 1b| (2.22)

where f is the focal lengthof the lensor mirror (crystal). For the bendingradiusin

thesagittalplaneRs, i.e. perpendicularto thedispersionplane,f canbeapproximately

expressedby [70]

fs t Rs

2sinθ(2.23)

60

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Toroidally Bent Crystal(doubly focusing)

ExtendedSourceObject

Spectral

ResolutionRowlandCircle

Rs

Rm

Monochromatic Imagein Sagittal Plane

in Meridional PlanePolychromatic Image

Figure 2.4. Principleof theJohann-typespectrometer

whereθ is the angleof incidence. The distanceb is given asthe distancefrom the

centerof thecrystalto thefocalpoint of thespectralline on theRowlandcircle

b t Rmsinθ � (2.24)

CombiningEqs.(2.22)–(2.24),wefind therelation

Rs t 2Rmsin2 θ � aRmsinθ } a � (2.25)

In orderto enablethecrystalto ‘reflect’ X-rays,θ hasto fulfill theBragg-equation

[71,72]

2dsinθ t nλ | (2.26)

too. Here,d is the interplanarlattice spacing,θ the angleof incidence,λ the wave-

lengthof theincidentX-rays,andn is anintegernumber.

FromEq.(2.25),it is clearthatRm andRs aredifferent,dependingon theangleof

incidence(Braggangle)andtheobjectdistance.However, someauthors[117–120]use

sphericalcrystalsi.e. R ¨ Rs t Rm for this kind of spectrometer. Thus,theadvantages

anddisadvantagesof sphericalandtoroidallybentcrystalsshouldbebriefly discussed

here:

61

Spherical Crystals: Theadvantageof a sphericalcrystalis that it is easyto manu-

facturethesphericalshapeandto alignsinceazimuthalalignmentis notneeded.

However, accordingto Eq. (2.25)onehasto selectthedistancea from thesource

to thecrystalas

a t Rsinθ2sin2θ ~ 1

t'~ Rsinθcos2θ � (2.27)

which limits theexperimentalsetup:For BragganglesθB © 45ª , a becomesnegative

and focussingis not possible:The raysaredivergent. For 45ª the sagittalraysare

parallel so that only the rangeabove 45ª remainsfor focalization. However, in the

range45ª © θB � 47ª , thedistancea becomesvery large( q 4 m for R t 200mmand

θB t 46ª ) andfor θB � 60ª , thedistancea q R t 50 � � � 300mmsothatthewholespec-

trometerhasto beplacedinsidethevacuumchamber. This might causeinterference

with theincidentlaserbeamsaswell aswith otherdiagnostics.Also, a streakcamera

cannotbeusedinsidethetargetchamber. Moreover onehasto evacuateandventilate

the vacuumchamberfor operationof the spectrometer. Furthermore,the closerthe

sourcecomesto theRowlandcircle, i.e. a ~¬« Rmsinθ0, thesmallerthespectralwin-

dow. If thesourceis placedontheRowlandcircle, thespectralwindow is givenby the

sourcesizeandthewidth of thereflectioncurveonly.

Therefore,the reasonablerangeof Bragg anglesis limited to 47ª –60ª , which

presentsa further limitation of the experiment,sinceonehasto find a combination

of wavelengthandcrystal reflectionwhich givesa Bragganglein this range. This

is only rarely possibledue to the discretecrystal lattice spacingsd, which mustbe

furthermorefoundin anelasticbendablecrystal.

To work aroundthe‘45 ª -problem’mentionedabove,Youngetal. [120] placedthe

detectoroutsidetheRowlandcircle at

b t Ra2asinθ ~ R

(2.28)

so that the detectoris at the imagedistancein the sagittalplane. Dependingon a

andR, thecritical angle,at whichandbelow imagingis notpossible,is now shiftedto

smallervaluesbut theabovementioneddiscussionremainsvalid andtheindependence

of thespectralresolutionfrom thesourcesizeis sacrificedsincethesourceis imaged

2-dimensionallyat thedetectorandtherefore,thespectralresolutionis determinedby

thesizeof the imageof thesource.In thecaseof quasi-pointsources( © 10 µm) this

givesstill good resultsbut is not applicablefor the 5–30mm long targetsusedfor

X-ray laserexperiments.

62

Toroidal Crystals: Theseproblemscanbeavoidedby usinga toroidally bentcrys-

tal, wherebothbendingradii canbedeterminedindependentlyaccordingto theexper-

imentalconditions.Thedistancea canbe freely chosenso that thecrystalis outside

thetargetchamber. This is convenientfor operationof thespectrometerandexcludes

interferencewith otherdiagnosticsor laserbeams.

ThebendingradiusRm hasthento beselectedasa compromisebetweenmagnifi-

cation,spectralwindow andspectralresolution,wherethemagnification

M t bat Rmsinθ

a| (2.29)

thespectralwindow approximately(cf. Sec.1.2.2)

∆λ t Amcosθ � 1a~ 1

Rmsinθ � λ | (2.30)

andthespectraldispersion[114]

dλds ­ λ

Rmtanθ| (2.31)

whereds is thelengthelementalongtheRowlandcircle.

WhenRm anda areselectedandθ is determinedby theBraggangleθB resulting

of thedesiredwavelengthλ andchosencrystalreflection,Rs canbecalculatedby Eq.

(2.25).

Thedisadvantageof the toroidally bentcrystalis that it is muchmoredifficult to

manufacturethetoroidalshape.To ensure∆Rw R � 10s 3 for bothbendingradii in order

to obtaingoodspectralandspatialresolution,theproductionprocessrequiresextensive

and time consumingquality controlsduring manufacturingof the glas forms [91].

Currentlyonly theX-ray opticsgroupat theUniversityof Jenais ableto manufacture

suchcrystals.

In summary, toroidallybentcrystalscanbedesignedaccordingto theexperimental

requirementsbut are more difficult to manufacture,while for sphericalcrystalsthe

experimenthasto bedesignedto fit thegeometricalrequirementsof thecrystal.

2.3.3 Design

An imageof thespectrometeris shown in Fig.2.5andthegeometryof thespectrometer

is summarizedin Table2.4.

63

To Pump

Crystal Stage

To Plasma

To Detector

Crystal

Light Protection

Gate Valve

(a)

(b)

f

xy

θz

StageCrystal

Crystal

χ

Figure2.5. Theoverview of thespectrom-

eter(a) andthe crystalstagein detail (b).

64

As adesignvalue,acentraloperationwavelengthof 6.0A of thespectrometerwas

assumedanda suitablecrystallatticeplanewasfoundin theQuartz x 1010y reflection

with aninterplanardistanced t 4 � 2548174A, resultingin acentralBraggangleθB ® 0 t45ª . 1

For theoperationwith theCCD-camera,thecrystalwasbentto a toroidalshapeas

describedin detail by Forsteret al. [121] with Rm t 200� 0 mm andRs t 177� 7 mm.

For theuseof thespectrometerin connectionwith thestreakcamera,anothercrystal

waspreparedwith Rm t 300� 0 mmandRs t 278� 0 mm. Thelargerbendingradii here

comparedto the crystal for the CCD camerawasnecessarysincethe distancefrom

the connectingflangeto the photocathode,i.e. detectorplane,is larger thanthat for

theCCD cameraon theonehand,andon theotherhandto conserve thehigh spectral

resolution,whichis,amongothers,proportionalto thespatialresolutionof thedetector

dividedby thedistanceto thecrystalat thesameBraggangle.

Thesebendingradii werechosenso that the crystal canbe placedconveniently

outsideat the target chamberwall at a distanceof 1162mm from the sourcewhile

having enoughspaceto mountthedetectoron thespectrometer.

Thecrystalwasmountedon a goniometerwhich againwasmountedon a transla-

tion stageso that thecrystalcanbeshiftedin directionof theplasma,in directionto

thedetectorandperpendicularto this plane.Moreover, thecrystalcanbetilted dueto

a tilting mechanismbetweenthegoniometerandthetranslationstages.

In the centralposition, i.e. at a centralBraggangleof 45ª , the spectralwindow

reachesfrom 5.536A to 6.458A for thecrystalwith Rm t 200mm. In orderto access

the full requiredrangefrom 5–7 A, thecentralBragganglecanbeshiftedaccording

to theusedtargetmaterial.However, whenrotatingthecrystaloff thedesignposition,

thedetectorwill beno longeron theRowlandcircle. Thusthecrystalhasto beshifted

in directionof theopticalaxis,too. Table2.3givestheamountof thenecessaryshift in

eachdirectionrelativeto the45ª –positiontogetherwith theresultingspectralwindow.

I.e. in thecenterpositionthespectralwindow reachesfrom 5.536A to 6.458A. This

window is appropriatefor theTa- andW-spectrum,wherethe4 f ~ 3d emissionlines

reachfrom 5.68 A to 6.37 A. In orderto be ableto observe the Ni-lik e Yb 4 f ~ 3d

emissionlines, which are lying in the rangefrom 6.62 A to 6.92 A, the crystalhas

to beturnedby 7ª sothat thecentralangleof incidenceis 52ª which givesa spectral

1Note,thatit is impossibleto constructsuchanimagingspectrometerby meansof sphericalcrystals,

sincethecentralBraggangleis 45 (cf. Sec.2.3.2).

65

θB ° 0 a ∆a b ∆b θB °min θB °max λmin λmax± ¯X² [mm] [mm] [mm] [mm]± ¯X² ± ¯X² [A] [A]

40 1141.4 � 20� 6 128.6 � 14� 4 35.54 44.41 4.946 5.955

41 1145.1 � 16� 9 131.2 � 11� 1 36.55 45.41 5.068 6.060

42 1149.0 � 13� 0 133.8 � 8 � 23 37.56 46.40 5.187 6.162

43 1153.1 � 8 � 9 136.4 � 5 � 3 38.57 47.39 5.305 6.262

44 1157.5 � 4 � 5 138.9 � 2 � 6 39.58 48.38 5.422 6.362

45 1162.0 � 0 � 0 141.4 � 0 � 0 40.59 49.37 5.536 6.458

46 1166.7 ³ 4 � 7 143.9 ³ 2 � 5 41.60 50.36 5.649 6.553

47 1171.7 ³ 9 � 7 146.3 ³ 4 � 5 42.60 51.36 5.760 6.647

48 1176.7 ³ 14� 7 148.6 ³ 6 � 5 43.61 52.35 5.870 6.738

49 1182.0 ³ 20� 0 150.9 ³ 8 � 2 44.62 53.34 5.977 6.827

50 1187.4 ³ 25� 4 153.2 ³ 9 � 7 45.63 54.34 6.083 6.914

51 1193.0 ³ 31� 0 155.4 ³ 10� 9 46.63 55.33 6.186 7.000

52 1198.7 ³ 36� 7 157.6 ³ 11� 9 47.64 56.33 6.288 7.082

53 1204.5 ³ 42� 4 159.7 ³ 12� 6 48.65 57.32 6.388 7.163

Table 2.3. Thespectralwindow of thespectrometerandthenecessaryshift of thecrystalfor

different centralBragganglesθB ´ 0. Note the differencein ∆b µ� 141¶ 4 · b. ∆b is the shift

parallelto thenormalof theCCD, while b � RmsinθB ´ 0 givesthedistancefrom thecenterof

thecrystalto thedetector.

window from 6.29 A to 7.08 A. In order to ensurethat the detectoris againon the

Rowlandcircle andto obtaina sharpimageof thesource,thecrystalasto beshifted

additionally11.9mm perpendicularto thedetectorplaneaway from thedetectorand

36.7mm away from thesource.

Two 10 µm thin Mylar foils (Mylar: C10H4O8 | ρ t 1 � 39 g/cm3) eachwith 200nm

Al onit weremountedbetweenthecrystalandtheCCDarrayto protecttheCCDfrom

saturationby visible light.

Thetotal throughputof thespectrometer, i.e. thetotal numberof transmittedpho-

tonscomparedto thenumberof emittedphotonsfrom theplasmain 4π, for the35 r10mm2 largecrystalasafunctionof thewavelengthis shown in Fig. 2.6togetherwith

the integratedreflectivity of the crystalandthe transmittivity of the light protection

foil. The integratedreflectivity of thecrystalwascalculatedusingtheTakagi-Taupin

equation[83,84] while the total throughputwasobtainedby the ray-tracingcodeT-

Ray [81] underuseof the transmissioncoefficient of the light protectionfoil which

wascalculatedby the XPower code[122–124] basedon the EvaluatedPhotonData

66

Typeof detector CCD-Array Streak-Camera

Pictureelementsizeof detector 22.5µm 100µm

Crystal Quartz ¸ 1010¹ Quartz ¸ 1010¹Latticespacing d 4.2548174A 4.2548174A

CentralWavelength λ0 º 6 � 0 A º 6 � 0A

CentralBragg-angle θB ° 0 º 45 º 45

Bendingradiusin meridionalplane Rm 200.0mm 300.0mm

Bendingradiusin sagittalplane Rs 177.7mm 278.0mm

Sizein meridionalplane Am 35 mm 30 mm

Sizein sagittalplane As 10 mm 10 mm

Objectdistance a 1162.0mm 1162mm

Imagedistance b 141.4mm 212.1mm

Table2.4. Thedesignparametersof thespectrometer.

Reflectivity

Transmission Total troughput

1

2

3

4

5

6

7

8

9

10

Tot

al th

roug

hput

/ 1e

−10

10

15

20

25

30

35

40

5.5 6.5

0.2

0.4

0.6

0.8

Wavelength [Å]

1.0

0.0

Inte

grat

ed R

efle

ctiv

ity [µ

rad]

Tra

nsm

issi

on o

f lig

ht p

rote

ctio

n

6.0 7.05.0

Figure 2.6. The integratedre-

flectivity of Quartz � 1010� (- - -),

thetransmittivity of thelight pro-

tectionfoil ( »�»�»�»�»J» ) andtheabso-

lute throughputof thespectrom-

eter(——). TheSi K-absorption

edgeis at6.7420A.

67

Library [125]. Knowing the total throughput,it is possibleto measurethe absolute

numberof emittedphotonsfrom thesourcewhenusingacalibrateddetector.

The throughputof the spectrometercan be controlledby changingthe aperture

sizeof the crystal in the sagittalplaneby meansof a slit, which canbe easilymade

by a copperfoil for instanceandmountedon thecrystal. This hastheadvantagethat

the throughputcan be controlled independentlyfrom the spectralsensitivity of the

spectrometercontraryto theuseof filters, wheretheamountof transmissiondepends

on thewavelength.

A critical point of thespectrometermight betheSi K-absorptionedgeat 6.742A

of theQuartzcrystal,wherethereflectivity of thecrystaldecreasesconsiderably. All

herepresenteddata,however, aretakenbelow this absorptionedge.

2.3.4 Characterization of the performanceof the spectrometer

In the caseof the original Johannspectrometerwith a cylindrically bentcrystaland

the sourceon the Rowland circle, the spectralresolutionis, besidethe width of the

reflectioncurve, theverticaldivergence,andthe thicknessof thefilm, limited by the

Johannerroronly, which is almostnegligible [114]. NeglectingtheJohann-error, each

pointof thesourceis imagedontheoppositesideof thecrystal,i.e. thepolychromatic

imageof eachsourcepoint is lying on the oppositesideof the Rowland circle. The

spectralrangeof thispolychromaticimageis givenby thewidth of thereflectioncurve,

only. Whenbendingafilm alongtheRowlandcircle it is ensuredthateachimagepoint

is lying on thedetector.

However, whenplacingthe sourceoutsidethe Rowland circle, the geometrybe-

comesmorecomplicated,anda‘modern’detectorlikeaCCD-or streakcameracannot

bebentalongtheRowlandcircle. Thus,thedetectorhasto beplacede.g. perpendicular

to thecentralbeam.In thefollowing, theanalysisof theinfluenceof thesegeometrical

aspectsis discussed.

General

The simulationsin this sectionarerestrictedto a centralBraggangleof 45ª and

the hereaccessiblespectralrangeof 5.7 A to 6.4 A with the sourceplacedoutside

theRowlandcircle at a distanceof 1162mm away from a Quartz x 1010y crystalwith

bendingradii of Rm t 200mm andRs t 177� 7 mm andthedetectorat a distanceof

68

141.4mm(cf. Tab. 2.4for thecaseof theCCDarray).This restrictionis madebecause

ontheonehand,this is theonly configurationusedin thiswork andontheotherhand,

thecrystalhasto beturnedandshiftedin orderto accessthefull rangefrom 5–7A so

that thediscussionof theresultingnew spectralwindow is analogueto the following

discussionexceptthatthecentralBraggangleis shifted.

Figure 2.7 gives a summaryof

Demagnification

Crystal − Detector

Source − Crystal (−1000 mm)

120

130

140

150

160

170

180

5.4 5.6 5.8 6.2 6.4 6.6 6.8

Dis

tanc

e / m

m

Wavelength / Å6.0

1/10.0

1/9.5

1/9.0

1/8.0

1/7.5

1/7.0

1/8.5

Dem

agni

ficat

ion

Figure 2.7. Distancesof the spectrometerin the

meridionalplaneasafunctionof thewavelengthto-

getherwith theresultingdemagnification.

thedistancesof thespectrometerge-

ometryfor apointsourcein themerid-

ionalplane.Dependingonthewave-

length, the X-rays arediffractedon

differentplacesonthecrystal(cf. Fig.

2.4),i.e. thedistancefromthesource

point to thepointof reflectiononthe

crystalvariesbetween1179mm to

1149mmfor 6.648A to 5.522A, re-

spectively. I.e. thelongerthewave-

length,the longerthedistance.Thedistancefrom thepoint of reflectionto thepoint

of intersectionwith thedetector, however, changesin theoppositeway from 126mm

to 156mm for thesamewavelengthrange.I.e. thelongerthewavelength,theshorter

the distance.Thus, the (de)magnificationof the imagein the sagittalplanechanges

accordinglyfrom 1/9.4to 1/7.4for 6.648A to 5.522A, respectively (Fig. 2.7).

Monochromatic point source

For thecharacterizationof thespectralandspatialresolutionof thespectrometer, the

problemis restrictedfirst to thespectroscopy of amonochromaticpoint source:

Sincethe crystal ‘reflects’ not only X-rays of a singlewavelengthundera single

angle,theabove (Section2.3.2)mentionedconsiderationof theextendedsourceasa

point sourceon theRowlandcircle is not correct.Rather, the imaginedsourceon the

Rowland circle will have a finite width, givenby the angularrangeof acceptanceof

the crystal togetherwith the distancefrom the surfaceof the crystal to the Rowland

circle andthe slopeof the Rowland circle relative to this ‘ray’. Thus, the imageof

theimagined‘point source’on theRowlandcircle will havea finite width, too,which

limits the spectralresolutionof the spectrometer. Even a point sourceoutsidethe

Rowlandcirclehasto beconsideredasanextendedsourceon theRowlandcircle,due

69

150 µm

75 µ

m

5.7 Å 6.0 Å 6.4 Å

+10 mm

+20 mm

+30 mm

± 0 mmS

agitt

al P

lane

Meridional Plane

Figure 2.8. Thecalculatedpoint spreadfunctionof thespectrometerfor selectedwavelengths

anddisplacementsof the sourcein the sagittalplanefor the crystalwith Rm ¼ 200 mm (cf.

Tab. 2.4). Note, that this simulationwascalculatedwith a pixel sizeof 1.5 µm while the real

CCD-arrayhasapixel sizeof 22.5µm.

to this angularrangeof acceptance.

For the caseof a monochromaticpoint source,the imaginedsourcesize on the

Rowlandcirclewill beabout15µm for thehereconsideredcaseat6.0A, correspond-

ing to ∆λ ½ λ ¾ 1 ¿ 1 À 10Á 4 (cf. Fig. 2.8 top middle). This broadeningis directly dueto

thewidth of thereflectioncurve.

However, the spectralresolutiondependson the wavelength,too. For the point

sourcein themeridionalplane,the‘monochromaticfocus’becomessmallerfor longer

wavelengths.Thereasonfor this is thatthecrosssectionof theRowlandcirclethrough

theangularrangeof acceptanceof thecrystal,i.e. theimaginedsourcesize,is smaller

for longerwavelengthsdueto the steeperangleat which the Rowland circle crosses

theopeningangle.In consequence,theimageof theimaginedsourcewill besmaller,

too.

Figure2.8showsrepresentativecalculatedimagesof apointsourceonthedetector

for threewavelengths.The correspondingquantitative numbersfor the full width at

half maximum(FWHM) of thespotsizein themeridionalandthesagittalplanesare

70

10−

4

±30 mm

±20 mm

±10 mm

± 0 mm

±30 mm±0 mm

Meridional Plane

Sagittal Plane

0

5

10

15

20

25

30

35

40

45

5.7 5.8 5.9 6.1 6.2 6.3 6.40

0.5

1.5

2.5

Imag

e S

ize

(FW

HM

) / [

µm]

∆λ/λ

/

Wavelength / [Å]

3.0

2.0

1.0

6.0

Figure 2.9. Resolutionof the spectrome-

ter for apointsource.Thesolid linesshow

the imagesizein themeridionalplanefor

differentdisplacementsperpendicularto it

togetherwith the correspondingspectral

resolution(right ordinate)andthe dashed

lines mark the spatial resolution of the

spectrometerin thesagittalplane.

shown in Fig. 2.9 as a function of the wavelengthtogetherwith the corresponding

spectralresolutionof thespotsizein themeridionalplaneon theright handordinate.

In the sagittalplane,however, the imageis broadenedfor all wavelengthsbeside

thedesignwavelength,sinceonly in thatcase,the imageof thesourceis lying in the

detectorplane.In all othercases,theimageis lying in front of or behindthedetector

plane,whichcausesin eachcaseabroadeningof theimage.

Whenthe point sourceis now shiftedperpendicularto the meridionalplane,the

sizeof thespotonthedetectorincreases,sincethedeviationof theoptimalfocusfrom

the detectorplaneincreases.In the sagittalplane,this broadeningis small, but it is

muchlarger in the meridionalplane,asshown in Figs.2.8 and2.9. For a deviation

of  20 mm away from themeridionalplane,this increasingof thebroadeningof the

imagedueto the de-focalizationcompensatesthe above describednarrowing dueto

the smallerimaginedsourcesize, and for a deviation of  30 mm, this broadening

towardslongerwavelengthsexceedstheabovedescribednarrowing in thiswavelength

direction(Fig. 2.9).

Monochromatic extendedline source

For the next stepof the characterizationof the spectrometer, a monochromaticline

sourcewith theextensionin thediffractionplaneis assumed:Thesourcehasnoexten-

sionin thesagittalplane,sincetherethecrystalis usedin imaginggeometry(cf. Sec.

1.2.2)so that the imageof a point is theappropriatedescriptionof thespatialresolu-

tion. In themeridional(diffraction)plane,theconsiderationof anextendedsourceis

necessarysincethere,theradiationemittedatdifferentsourcepointsincidentsontothe

crystalunderdifferentangles.Thishastwo effectsfor amonochromaticsource:First,

theradiationcomingfrom anotherpointof thesourcecannotbereflectedat thecrystal

71

10−

4

±20 mm

± 0 mm

±40 mm

±30 mm

±50 mmMeridional Plane

Sagittal Plane

(a)

∆λ/λ

/

0

10

20

30

40

50

60

70

5.7 5.8 5.9 6.1 6.2 6.3 6.40

1

2

3

4

5

Imag

e S

ize

(FW

HM

) / [

µm]

Wavelength / [Å]6.0

10−

3

±30 mm

±0 mm

Meridional Plane

(b)

Sagittal Plane

∆λ/λ

/

0

50

100

150

200

250

300

350

5.7 5.8 5.9 6.1 6.2 6.3 6.40

0.5

1.5

Imag

e S

ize

(FW

HM

) / [

µm]

Wavelength / [Å]6.0

1.0

2.0

Figure2.10. Thesizein boththemeridionalandthesagittalplaneof theimageof aline source.

(a) length2 mm,(b) length20mm. Thedifferentlinesreferto differentdeviationof thesource

from themeridionalplane.While thesolid anddottedline refersto a sagittalaperturesizeof

10 mm, thedash-dottedanddashedlines refersto a thesagittalaperturesizeof 1 mm anda

shift of à 0 mmand à 50mm,respectively.

in themeridionalplane,sincetheretheBraggcondition(Eq.(2.26))cannotbefulfilled.

However, second,the Braggconditioncanbe fulfilled off the meridionalplane[22].

Thiscausesaninfluencein the‘image’ on thedetectorin both,themeridionalandthe

sagittalplane.

In otherwords:If thecrystalwouldhave justanextensionin themeridionalplane,

thesourcesizewouldbelimitedby theangularwidthof thereflectioncurveof thecrys-

tal. Consequently, thecaseof themonochromaticpoint sourcewould apply, because

the X-rays, which would comefrom a sourcepoint outsidethis ‘acceptanceangle’,

would not be diffractedat the crystal. For a two dimensionalextendedbentcrystal

however, theX-raysfrom a sourcepoint outsidethe‘openingangle’in themeridional

planecanbe incident in the sagittalplaneunderthe correctBragganglerelative to

the surfaceof the crystal (cf. Ref. [22,73]) andtherebe diffracted. Sincethis point

of intersectionwill not lie perpendicularto theBragg-anglepositionin themeridional

plane,theconsiderationsin Section2.3.2areno longervalid, becausethey assumea

cleardistinctionfor thespectralfocalizationin themeridionalandthe imagingin the

sagittalplane. For the caseof an extendedsourcewhich is diffractedat an extended

crystalin thesagittalplane,this cleardistinctionis notgivenanymore.

The distortionof the imageof a point sourcedueto the extendedcrystalcanbe

observedin Fig. 2.8. However, eventhoughthebroadeningof theimagesizerelative

to thecenterposition(i.e. 6.0 A on theopticalaxis) is largeasshown in Fig. 2.9, it is

smallcomparedto therequirementsandtheresolutionof theCCD array.

72

For anextendedsourcehowever, the resultingFWHM of the ‘image’ sizeon the

detectormight becomea complicatedshape.In Fig. 2.10(a)the FWHM for a 2 mm

long line source,lying in the meridionalplane, is shown for several deviations of

thesourceperpendicularto themeridionalplane. In theX-ray laserexperiment,this

sourcesizeis realizedwhenthelongitudinalextensionof theplasmais perpendicular

to themeridionalplaneasshown in Fig. 2.15. Thesolid lines in Fig. 2.10(a)arethe

resultsof the ray-tracingcalculationfor a thesagittalcrystalapertureof 10 mm. For

‘small’ deviations a parabolicbasicshapeis obtainedfor the FWHM of the image

sizein themeridionalplanewith dipsat ¾ 5 ¿ 8 A and ¾ 6 ¿ 3 A. Thesedipsaredueto

the complicatedbeamcrosssectioncomingfrom the outerregion of the crystal (cf.

Section1.2.4and thereparticularlyFig. 1.8 on page24). An analyticaldescription

of this relation can be found at Dirksmoller [74]. If the sourceis deviated farther

away from the the meridionalplane,the FWHM of the imageon the detectorin the

meridionalplaneis moreandmorebroadened,and the dips aresuperpositionedby

otherinfluences.

However, eventhoughtheinfluencesof thesourcesizeontheimageonthedetector

arelarge,thebroadeningof theimageis still comparableto theresolutionof theCCD

array: The sizeof the imageon the detectorin the meridionalplanevariesbetween

16 µm (at 6.0 A, wherethe detectorintersectswith the Rowland circle, andwithout

deviation from the meridionalplane)and60 µm (6.4 A and50 mm deviation from

themeridionalplane),correspondingto 1–3pixelsof 22.5µm size,or ∆λ ½ λ ¾�Ä 1 ¿ 1 Å4 ¿ 2ÆWÀ 10Á 4. In thesagittalplane,themaximumimagesizeamounts11.6µm, which

is smallerthanthesizeof onepixel so that theresolutionhereis limited by thepixel

sizeonly.

To get someevidencethat the broadeningand complicatedshapeis due to the

sagittalapertureof the crystal,thecalculationswerecarriedout againwith thesame

parametersbut thesagittalapertureof thecrystalwasreducedto 1 mm. Theresulting

FWHM of the imagein themeridionalplaneis shown in 2.10(a),too, for a deviation

of the sourcefrom the meridionalplaneof  0 mm (dash-dottedline) and  50 mm

(dashedline) respectively. As expected,thedipsat ¾ 5 ¿ 8 A and ¾ 6 ¿ 3 A vanish,which

aredueto theouterregionsof thetoroidalshapeof crystalsurface.Also, theimagesize

in themeridionalplaneis reduced,i.e. thespectralresolutionis increased.Moreover,

thedifferencebetweentheFWHM of the imagein the meridionalplanefor the case

with andwithoutadeviation of  50 mm is reduced.

73

Figure2.10(b)shows the FWHM of the imagesizeof a 20 mm long line source

lying in dispersionplaneasa function of the wavelength. This caseis given, if the

spectrometeris alignedso that the longitudinal extensionof the plasmacolumn is

lying in the dispersionplaneandthe blow-off directionof theplasmais imaged,i.e.

similar to theset-upof Fig. 2.12.

Here,thesizeof thefocal spotin themeridionalplaneis dominatedby thesource

size,which makesthefine structures(dips)in thecurve invisible. Thebehavior of the

curve canbe understoodasthe deviation of the detectorplanefrom Rowland circle

towardsor away from thepolychromaticimageof thesourcein themeridionalplane

(cf. 1.2.2, and thereEq. (1.16)). This polychromaticimageis desiredfor the 2-D

imaging,but herefor thespectrometerratheranuisance.At 6.0A, wherethedetector

intersectswith the Rowland circle, the imagesizeon the detectorhasits minimum,

andideally (for the casethat the FWHM of the reflectioncurve would be zero),the

FWHM shouldbecomezerohereaccordingto Section2.3.2. For longerandshorter

wavelengths,the detectoris deviated from the Rowland circle, wherethe bunch of

diffractedX-rays becomeslarger (cf. Fig. 2.4 on page61). The imagesize in the

meridionalplanevariesbetween35 µm and334µm for 6.0 A and6.4 A, respectively

(cf. Fig. 2.11). Thecorrespondingspectralresolutionamountsto 2 ¿ 5 À 10Á 4 to 2 ¿ 4 À10Á 3, respectively.

Figure2.11shows theinfluenceof thesagittalapertureof thecrystalfor a 20 mm

long line sourcewhich is deviatedby 30 mm off themeridionalplane.Theupperpart

shows theray-tracingresultsfor a sagittalcrystalapertureof 1 mm andthelowerpart

the resultsfor 10 mm sagittalcrystalaperture.The intensityin eachimageis scaled

to its maximum,whereasthemaximumintensityaswell astheintegratedintensityis

reducedby a factortenfor the imagecalculatedfor 1 mm sagittalaperturecompared

to 10 mm sagittalaperture.

At a sagittalapertureof 1 mm, the profilesof the imagein the meridionalplane

for shiftsof  0 mm and  30 mm, areidentical. However, amazingly, if thesagittal

apertureis reducedfrom 10 mm to 1 mm, the FWHM of the imagein sagittalplane

increasesfrom a FWHM of about6 µm for 10 mm sagittalapertureto 66–87µm for

1 mm sagittalaperture,dependingon thewavelength.This broadeningin thesagittal

planeat a smalleraperturesizeis counter-intuitive but canbeunderstoodasfollows.

Dueto thesmallersagittalaperture,thediffractedintensityis decreasedandtheimage

is sharper. However, themain reductionin intensityis in the focusso that themaxi-

74

0 150 300 450 600

0

0

150

150

FWHM: 174 µm

FWHM: 8.4 µm

FWHM: 75 µm

PSfragreplacements

ÇMÈ/ÉËÊmm

ÇMÈÌÉÍÊXÎmm

Ï ÐÒÑÔÓÖÕdÏ ×ØÉÚÙ�Û Ü�ÝÞÊXÎàß ákâÏ ÐÒÑÔÓÖÕdÏ ×ØÉÚÙ�Û ãäÝÞÊXÎ ß ákå

Figure 2.11. The imageof a 20 mm long line sourcefor a sagittalapertureAs ¼ 1 mm and

As ¼ 10 mm. Thewavelengthis 6.2 A, andthedeviation from themeridionalplaneamountsæ30mm. In bothcases,thesameprofile in themeridionalplaneis obtained.

mum intensityis reducedandaccordinglythe full width of the imageat the reduced

maximumintensityis enlarged.

However, evenfor thevery largeplasmasizesexpectedin theX-ray laserexperi-

ments,the influenceof thesourcesizediscussedhereon thespectralandspatialres-

olution is relatively small comparedto the resolutionof the detector, the sizeof the

plasma,and its hugedeviation from the meridionalplane. Even for the very large

20 mm long plasmacolumnin dispersionplanedeviatedby 30 mm,a spectralresolu-

tion of betterthan3 À 10Á 3 canbeobtained,which is in the requiredrangeof about

10Á 3 Å 10Á 4. Thespatialresolution,however, is betterthanthereal resolutionof the

CCD array, which is at leasttwo pixels.

Measurementof the spatial resolution

Figure2.12 presentsthe experimentaldemonstrationof the spatialresolutionof the

spectrometer. Figure2.12(a)shows the raw CCD imageof the Al calibrationspec-

trum, which wastakenwith the CCD arrayat the intendeddistanceandFig. 2.12(b)

showsthespatialprofileof theplasma,integratedoverthefull recordedspectrum:The

incidentlaserbeamcomesfrom the left handside,in which theplasmais expanded.

On theright handsideof theprofile, thesolid targetedgeis visible. Usingthis edge,

75

= 0.39 mm

Incident

Laser Beam FWHM = 2.12 pixel

Heζ Heε Heδ He γLy γ

x

λ

Target

Al

Al

Incident Laser Beam

0 mm -

(a)

(b)0

500

1000

1500

2000

2500

3000

3500

4000

-10 -8 -6 -4 -2 0 2 4 6 8 10-1500

-1000

-500

0

500

1000

Cou

nts

on C

CD

Scale on Object x/[mm]

Der

ivat

ive

of c

ount

s

Figure 2.12. Demonstrationof thespatialresolutionof theinstrument(raw data).Thespikes

in (b) at theleft handsideof thespatialprofilearedueto defectsin theCCD array.

76

therealspatialresolutioncanbemeasuredastheFWHM of thederivativeof theedge

with respectto x (dottedline in Fig. 2.12(b)).Fromthepresenteddatain Fig. 2.12,a

spatialresolutionof ∆x = 2.12pixel = 47.7µm on the CCD arrayor 393 µm on the

target wasobtained,which is in accordancewith thephysicalresolutionof theCCD

arrayof abouttwo pixels.

2.3.5 Calibration

Thespectraldispersionrelation

Figure 2.13. TheAl-spectrumusedfor the in-situ cal-

ibrationof thespectrometer

of thespectrometerwascalculated

by a ray-tracingcode,which was

developedespeciallyfor this pur-

pose:A ray startingat thesource

point hits the crystal, is symmet-

rically reflectedthereandhits the

detector. Thepositionof thepoint

of intersectionon the detectorin

relationto theangleof incidence

andreflectionon thecrystalgives

thedispersionrelation,wherethe

angleof incidenceequalstheBragg

angle(neglectingtherefractioninsidethecrystal).Thecorrespondingwavelengthcan

be obtainedthen from the Bragg angleusing the well-known Bragg equation(Eq.

(2.26)). Thus,thedispersionrelationis well-defined,whenknowing thegeometrical

distancesandanglesof thespectrometer. Theadvantageof this methodcomparedto

a singleanalyticalformula is that this is mucheasierto obtainandit takesautomati-

cally into accountall imagingerrorslike theJohannerroror distortionof the image.

Additionally, this approachis veryeasyto modify in orderto testseveralset-upsor to

applyit to anothergeometry.

In orderto obtaintheprecisevaluesof theessentialparameters,theobservedpo-

sitionsof theH- andHe-like transitionlines of an Al-referencespectrum(Fig. 2.13)

were comparedwith the tabulatedwavelengthsin Ref. [126] using the Levenberg-

Marquardtmethod[127]. Knowing theseparameters,the dispersionrelationis well

defined.Table2.5summarizestheobservedpixel positionon theCCD arraytogether

with the correspondingwavelengthandthe in turn numericallycomputedvaluesλFit

77

He

HeHe

γ

δε

He

He ζη

Ly γ

Heθ

5.6

5.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

0 200 400 600 800 1000 1200

Wav

elen

gth

[Å]

Pixel on CCD

Observed line positionsTheory with fitted parameters

Figure 2.14. Theobserved line positionfrom Fig. 2.13andthedispersionrelationwith fitted

parameters(Tab. 2.6)of theinstrument

from thecalibrateddispersionrelation,which is plottedin Fig. 2.14togetherwith the

usedreferencelines.

As a sideeffect, from thefitted distancesof thespectrometerit couldbeobserved

that the detectorwas placed ¾ 12 mm off the Rowland circle (cf. Tab. 2.6) in the

first experimentalcampaign,dueto a misunderstandingof the distancesof the CCD

camerain the constructionphase.This misalignmentwasleadingto blurredimages

with a spatialresolutionof only 887 µm on the detectorand 6.7 mm in the object

plane.After fixing thisproblemby re-constructingtheconnectorflangebetweenCCD

cameraandspectrometervessel,thespatialresolutionwaslimited by theresolutionof

theCCDarray(cf. Fig. 2.12).

2.4 Experimental setup

The experimentalsetupis shown in Fig. 2.15. The experimentswerecarriedout at

theGEKKO XII laserfacility at theInstituteof LaserEngineering,OsakaUniversity,

in collaborationwith Japanese,French,andChineselaboratoriesaswell astheX-ray

opticsgroupat JenaUniversity. Thewavelengthof the laserwas1.053µm in 100ps

mode. Singleor doubletargetswereused.Thecentersof the targetswereseparated

by 30 mm. Thetargetswereirradiatedby two opposinglaserbeamswith a timedelay

of 100ps for quasi-traveling-wave pumping. The lengthof eachtargetwas8 mm so

78

Line Pixel λRef/A λFit ç A ∆λ/A

Al He γ 303 6.314 6.3141 è 0 é 0001

Al He δ 465 6.175 6.1745 ê 0 é 0005

Al He ε 553 6.100 6.1014 è 0 é 0014

Al He ζ 607 6.059 6.0574 ê 0 é 0016

Al Ly β 614 6.053 6.0517 ê 0 é 0013

Al He η 641 6.0294 6.0300 è 0 é 0006

Al He θ 665 6.0095 6.0109 è 0 é 0014

Al Ly γ 1024 5.739 5.7389 ê 0 é 0001

Table2.5. ThetabulatedwavelengthsλRef. of theAl H- andHe-like transitionfrom Ref.[126],

the observed pixel-positionson the detector(cf. Fig. 2.13)andthe fitted wavelengthsλFit as

well asthedeviation of thefitted wavelengthfrom thereferencewavelength∆λ. Theaverage

deviation ë ∆λ ì ¼ 2 í 5 î 10ï 6.

Parameter Design Fit result

Imagedistance b 141.4mm 153.5mm

CentralBraggangle θB ð 0 45é 0ñ 45é 356ñTilt of CCDarray χ 0 é 0ñ è 0 é 9352ñOffseton CCDarray 12.96mm 13.760mm

Table2.6. Theoutputfrom thefit routineof thefirst experimentalcampaignwith amisplaced

detector. Theotherparametersarekeptfix asof Tab. 2.4.

79

Beam E08GEKKO XII

(Line Focus)

(CCD)Detector

Beam G11GEKKO XII

(Line Focus)

Foil (400 nm Al)Light Protection

Quartz (10.0)Toroidally Bent

of P

lasm

as

1−D Im

age

2 Targets(Yb, Hf, Ta or W)

Spe

ctra

XRL

XRL

Lasing Plasmas

∆T = 100 ps

Figure 2.15. Thesetupof theexperiment

thatthetotal targetlengthsamountedto 16mm.

ThetargetmaterialwasYb, Hf, Ta or W. Thetotal pumplaserenergy wasvaried

between50 J and360J per target in 100pspulseduration.The intensityin the line-

focuswasvariedbetween2 ¿ 4 À 1014 W/cm2 and1 ¿ 7 À 1016 W/cm2 with 0% to102.2%

prepulse.Thetime delaybetweenprepulseandmainpulsewasvariedbetween0.6ns

and3 ns.

A flat-field grazing-incidencespectrometerwasplacedon theX-ray laseraxisas

describedin Ref.[128] in orderto detecttheX-ray lasing.In front of thespectrometer,

100-µm–diameterfiducial wireswereplacedat 0, 4.5,8, and10 mrad,wheretheon-

axisline focuswasdefinedas0 mrad.whichallowedusto determinethedeflectionand

thebeamdivergenceangleof theX-ray laserin thehorizontalplane.No filters were

placedin the spectrometer. However, the first collectingmirror in the spectrometer

wascoveredwith athin carboncontamination;thereforethecarbonK edgewasclearly

visible in theXRL spectra[100].

A varied-spacing2400–lines/mmtoroidal grating focusedthe spectrumonto a

80

flat field perpendicularto the gratingsurface,covering the wavelengthfrom 2 to 13

nm. Time integratedspectrawith anangulardistribution wererecordedwith a back-

illumination-typeCCD camerawhosepixel sizewas24µm À 24µm [100].

Thetoroidallybentcrystalspectrometerwasmountedat1162mmfrom theplasma

at 65ò subtendedby theplanethatincludedtheaxesof thepumplasersandtheX-ray

laserbeam,which wereon the horizontalplane. Thereforethe spectrometerlooked

down ontoahorizontallyexpandingplasmaat 65ò to its expansiondirectionto reduce

theopacitiesfor theresonancelines.A front-sideilluminatedCCDcamera(Princeton

Instruments,modelTEA/CCD – 1242E/3) with a pixel sizeof 22.5µm wasusedasa

detector.

2.5 Processingof the raw data

Theraw experimentaldatawereprocessedasfollowsin orderto obtaintheintensityin

thespectralline andtheline width: All stepswereperformedautomaticallyaccording

to objectivecriteria.

1. The structureof the backgroundfrom the CCD was obtainedby reading-out

theCCD-arrayun-illuminated.This dark-imagewassmoothed(averagingover

3 À 3 pixels)andsubtractedfrom theraw-dataimage.

This subtractiongave a constantbackground,which wasagainsubtracted.The

amountof theuniform backgroundwasdeterminedastheaveragevalueover a

largeun-illuminatedarea.

2. Thepixel valuesweresummedupover thespectraldispersiondirectionin order

to obtainthespatialprofile (“1D image”)of theilluminatedtargets.

3. Sincein the first experimentalcampaign,the CCD arraywasmisalignedby ¾12 mm andthusthe spatialresolutiondecreasedto ¾ 887 µm on the detector,

thedetectedintensitywasintegratedover thespatialdirection:

For each1D-imageof the target, the maximumintensitywassearchedandthe

positionsto wheretheintensityhaddecreasedto 10%.

These10%-positionswereusedasthe boundariesfor the integrationover the

spatialdirectionin orderto obtainthespectra.

81

4. Thedetectedintensityof thespectrawerecorrectedby thecalculatedthroughput

of thespectrometer, whichdependson thewavelengthof theX-rays(Fig. 2.6).

5. Sincethe inner photoeffect wasusedfor the detectionof the X-ray photons,

thedetectedcounts(i.e. photoelectrons)areproportionalto thephotonenergy.

Therefore,the datawere correctedby dividing the count ratesby the photon

energy.

6. The so correctedspectrawere searchedfor the local (i.e. in a certain range

aroundthe expectedposition) maximawhich were assumedto be the wave-

lengthof thetransitionunderconsideration.Comparisonswith Gaussianfits for

randomsampleshaveshown thatthemaximumvalueis in verygoodagreement

with thecenterof thegaussprofile.

7. Onbothsidesof thelocalmaximumthelocalabsoluteminima(i.e. theabsolute

minimum within a specifiedrangearoundthe positionof the local maximum,

markedby arrows in Fig. 2.16(b)–(d))weresearched.Theseminimawereas-

sumedto bethebackgroundof thepseudo-continuum,which againis assumed

to be linearwithin the rangeunderconsideration(areaunderthedottedline in

Fig. 2.16(b)–(d)).

8. The intensityin thespectralline (cf. thehatchedareain Fig. 2.16(b)–(d))was

obtainedby the integral betweenboth local minima over the peak,subtracting

thebackgroundof thepseudo-continuum.

9. Thewidth of theline wasobtainedasthefull width athalf maximum(FWHM);

correctedby thelinearpseudocontinuum.Thehalf-maximumvaluewassearched

from thepeaktowardsthetales.Themeasuredvaluesaroundthehalf-maximum

value were linear interpolatedin order to find a more precisevalue for the

FWHM.

A samplespectrumhasbeendeconvolutedto test the broadeningof the spectral

line due to the instrumentfunction. The instrumentfunction wasgeneratedby the

ray-tracingcodeT-RAY andthedeconvolution wasperformedusingtheFastFourier

Transform. The resultwasthat the non-deconvoluted lines wereabout5% broader

thanthedeconvolutedlines.

82

óôóôóôóôóôóôóôóóôóôóôóôóôóôóôóóôóôóôóôóôóôóôóóôóôóôóôóôóôóôóóôóôóôóôóôóôóôóóôóôóôóôóôóôóôóõôõôõôõôõôõôõôõõôõôõôõôõôõôõôõõôõôõôõôõôõôõôõõôõôõôõôõôõôõôõõôõôõôõôõôõôõôõõôõôõôõôõôõôõôõ

öôöôöôöôöôöôöôöôööôöôöôöôöôöôöôöôööôöôöôöôöôöôöôöôööôöôöôöôöôöôöôöôööôöôöôöôöôöôöôöôö÷ô÷ô÷ô÷ô÷ô÷ô÷ô÷÷ô÷ô÷ô÷ô÷ô÷ô÷ô÷÷ô÷ô÷ô÷ô÷ô÷ô÷ô÷÷ô÷ô÷ô÷ô÷ô÷ô÷ô÷÷ô÷ô÷ô÷ô÷ô÷ô÷ô÷

øôøôøôøôøôøôøôøøôøôøôøôøôøôøôøøôøôøôøôøôøôøôøøôøôøôøôøôøôøôøøôøôøôøôøôøôøôøùôùôùôùôùôùôùôùùôùôùôùôùôùôùôùùôùôùôùôùôùôùôùùôùôùôùôùôùôùôùùôùôùôùôùôùôùôù

(a)

(b) (d)(c)

Co−like

5.6 5.7 5.8 5.9 6.1 6.2 6.3 6.4 6.5Wavelength/Å6.0

0

2

3

Inte

nsity

[a.u

.]

1

5.87 5.89 5.91 5.93 6.37 6.38 6.396.08 6.09 6.10 6.36

PSfragreplacements 3d93ú 24 f5ú 2 û 3d10

3d93ú 24 f5ú 2 û 3d10

3d95ú 24 f7ú 2 û 3d10

3d95ú 24 f7ú 2 û 3d10

3p53ú 23d104s1ú 2 û 3d10

3p53ú 23d104s1ú 2 û 3d10

Figure 2.16. A samplespectrumfrom Ta. Fig. (a) shows thefull recordedspectrumand(b)–

(d) shows thespectrallinesunderconsiderationin detail: Thedottedline markstheassumed

backgroundfrom thepseudocontinuum,whichwasdeterminedby thelocalminimum(arrows

in (b)–(d))andthehatchedareademonstratestheintegratedintensityof thetransition.

Sincetheinstrumentfunctionaffectsonly theshape,i.e. thewidth, of thespectral

line but not theintegratedintensity, andthewidth of theline is only interestingfor the

calculationof thedensity, which canbeestimatedonly very roughlyaspointedout in

Sec.2.2.2,thedetaileddeconvolution for eachspectrumwasomitted.

2.6 Results

2.6.1 General

Figure2.17 shows a typical processedCCD imagefrom the spectrometerwith spa-

tial resolutionalongthe X-Ray lasingaxisof a doubletarget Ta X-ray laserplasma,

togetherwith thecorrespondingprofile plots integratedover bothplasmacolumnsin

the spatialand spectraldirection, respectively. Sincethe CCD was misalignedbyü 12mm,nostructureis visible in thespatialprofile.

Figure2.18shows theNi-lik e Yb, Hf, andTa lasinglinesat 5.0,4.6,and4.5 nm,

respectively, which arecloseto the waterwindow. The upperwavelengthis marked

by theCarbonK-absorptionedge.Thespectrain Fig. 2.18wererecordedby theflat-

field grazing-incidencespectrometerin the X-ray lasingaxis. The spectrumin Fig.

83

λ/Α

020

4060

80

Ta−Spectrum

Co−like

Obj

ect S

cale

[mm

]

E08

G11

5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5

PSfragreplacements3d9

3ú 24 f5ú 2 û 3d10

3d95ú 24 f7ú 2 û 3d10

3p53ú 23d104s1ú 2 û 3d10

Figure 2.17. A representative CCD imageof a doubleTa target shot. The upperpart of the

pictureshows the imageof the target relatedto laserbeamG11 andthe lower part to beam

E08.

)

0

0

0

Carbon K−edge

5.0 4.384.6

Wavelength / [nm]

Inte

nsity

[a.u

.]

Ta

Hf

Yb (a)

(c)

(b)

4.5

Figure 2.18. Line graphsof theX-ray laserspectra.ThecarbonK edgeat 4.38nm represents

theedgeof thewaterwindow. Themaximumcountsfor the lasinglinesare8011,3181,and

159for Yb, Hf, andTa,respectively [98,100].

84

−13.6 cm

−16.6 cm

Pea

k In

tens

ity /

[CC

D C

ount

s]

Hf

Yb

Total double target length / [cm]

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Figure 2.19. The peak X-ray

laser intensity detectedby the

CCD as a function of the to-

tal double target length. The

diamondsand the trianglesrep-

resentthe measuredNi-lik e Yb

andHf lasingintensities,respec-

tively. The gain coefficients

of the Ni-lik e Yb and Hf are

6.6 cmý 1 and3.6 cmý 1, respec-

tively [98,100].

2.18(a)shows the Ni-lik e Yb laserat 5.0 nm with a fairly narrow beamdivergence

of 1.5 mrad [98,100]. Figure 2.18(b)shows the Hf lasing pulseat 4.6 nm with a

similar divergence. Fig. 2.18(c) shows the Ni-lik e Ta lasing line at 4.5 nm which

is at the longerwavelengthedgeof the waterwindow. A relatively weakand large

divergencebeamhasbeenobtained. The atomicnumberscalingseemsto be fairly

strongprobabledueto the1–µm laserpumpingwhich cannotcreatehighdensityand

high temperatureplasmaaspredictedby thehydrodynamicsimulation[100]. Typical

X-ray lasersignalsin theforwardandthebackwarddirectionshowedthatthenarrow-

divergenceandstrongX-ray lasingsignalwasobtainedin theforwarddirection,while

theweaker andmuchhigherdivergencebeamwasobtainedin theoppositedirection

[98,100].

Figure2.19shows thepeakintensityof theX-ray laserasa functionof thelength

of the doubletarget. The slab target at the right handsideactsasan oscillatorand

the target at the left handside actsas an amplifier. The length of the target at the

left-handside waschanged.At the forward traveling wave side, a higher intensity

X-ray laserwasmeasuredthanthat of the oppositeside. The gaincoefficient of the

Ni-lik eYb was6.6cmþ 1 whichcorrespondedto againlengthproductof 11. Thegain

coefficientof Ni-lik eHf laserwas3.6cmþ 1 andthecorrespondinggainlengthproduct

wasapproximately6.

For theHf laser, thegapseparationof thedoubletarget,which wastransverseto

theon-axisof theX-ray laser, waschanged.An intensesoft X-ray laserwith narrow

divergencewasobtainedaroundtheoptimumseparationof 150µm.

85

∆

0

2

4

6

8

10

1.0 1.5 2.00

0.5

1

1.5

2

2.5

Co−like 4f−3dNi−like X−Raylaser

Divergence

∆

Time Delay t(ns)

Bea

m d

iver

genc

e (m

rad)

Inte

nsity

(a.

u.)

4%100 ps

0.5

t Figure 2.20. Intensitiesof the

Yb X-ray laser and the Co-like

Yb af ÿ 3d transitionandtheX-

ray laser beamdivergenceas a

function of the prepulseto main

pulsetimedelay[98,100].

Hf

Peak Intensity of XRL [a.u.]600 1000 1400 1800 2200

5600

5400

5200

5000

Inte

nsity

in 4

f−3d

line

[a.u

.]

Figure 2.21. Correlation be-

tweenthe 4 f ÿ 3d emissionline

intensityand the 4d ÿ 4p -XRL

emission intensity observed in

Hafnium.

2.6.2 Comparisonof the 4d � 4p and 4 f � 3d transition

Figure2.20 shows the intensityof the Yb X-ray laser, its beamdivergenceand the

intensityof the strongestCo-like line asa function of the time interval betweenthe

prepulseandthe main pulse. The strongdependency of the X-ray laserintensityon

the time delaycanbeseen.Theoptimumtime delaywasapproximately1.5 ns,cor-

respondingto themaximumCo-likeabundance,wheretheX-ray laserintensitymaxi-

mizedandtheminimumbeamdivergencewasrealized.

Figure2.21shows thecorrelationof thedetectedNi-lik e 4 f � 3d transitionto the

Hf X-ray lasersignal. Only datawith obtainedlasingareconsidered.Thus,several

parametersarechangedin theherepresenteddata.However, thecorrelationbetween

bothsignalsis linear in thepresenteddataset. This is astonishing,sincethe4 f � 3d

intensitydependsonlyonthepopulationof theionizationstatewhile theXRL intensity

dependsalsoon parameterslike thealignmentof the targets,the lengthof the target,

etc.whichdo notaffect theionizationandthusthe4 f � 3d intensity.

86

Co−like

215

0

1000

2000

3000

4000

5000

6000

0 0.5 1.5 2.5

Inte

nsity

of 4

f−3d

/ [a

.u.]

1.0 2.0Incident Laser Intensity / 10 W/cm

PSfragreplacements

3d93� 24 f5� 2 � 3d10

3d95� 24 f7� 2 � 3d10

3p53 2

3d104s1 2 3d10

Figure 2.22. The optimal pump

laser intensity of the Ta plasma

in dependency of the incident

laser intensity. The maximum

emissionof the Ni-lik e andCo-

like linesappearedatabout1 � 3 �1015 W/cm2

2.6.3 Studieson tantalum plasmas(point focus)

Parallelto theaimof observingamplifiedemissiontowardsthewaterwindow, aseries

of point focusexperimentswerecarriedout in orderto studythe influenceof thepa-

rametersof the incidentlaserbeamon theplasmaparameters.Theexperimentswere

carriedout immediatelyafterthemainX-ray laserexperimentby usingtheremaining

laserbeams.Despitethe reducedsetof parameters,it is possibleto find the optimal

plasmaconditions,beforeoptimizing the set-upof the X-ray laserexperiment,i.e.

finding thebestradiusof curvatureof theslabtarget,thebestgapbetweenthedouble

targets,etc.

ThetargetmaterialwasmainlyTa. Thetotalpumplaserenergy wasvariedbetween

50Jand180Jin 100psmodewhile thefocalspotwasvariedbetween1 � 36 � 10þ 5 cm2

and7 � 85 � 10þ 3 cm2. The so varied resultingincident laserintensitywasbetween

8 � 8 � 1013 W/cm2 and3 � 5 � 1016 W/cm2.

Figure2.22showstheintensityof theNi-lik e4 f � 3d emissionlinesaswell asthe

strongestCo-like line asa functionof theincidentlaserintensityat 4% prepulse.The

maximumemissionwasfoundfor all threelinesatabout1 � 3 � 1015 W/cm2. For lower

intensities,the incident laser is not sufficient to producethis ionization statewhile

at higherlaserintensities,theplasmais muchhigherionized.Thus,onecanconclude

thatat1 � 3 � 1015 W/cm2 theamountof 4p ions,theupperlevel of thelasingtransition,

reachesits maximum,too. However, suchhigh intensitiescouldnotbeachievedin the

line-focusX-ray laserplasmas,which wasonly about3 � 1014 W/cm2, i.e. abouta

factorfour lower.

Figure2.23shows two Ta spectratakenat 1 � 2 � 1015 W/cm2 incidentlaserinten-

sity, i.e. aroundtheoptimumintensity, but differentamountof prepulse.Figure2.23(a)

wastakenwithoutprepulseand(b) wastakenwith 4% prepulse3 nsin advanceof the

87

0

5

10

15

20

25

5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5

35

30

(c)0.0

2.0e+5

4.0e+5

6.0e+5

8.0e+5

1.0e+6

1.2e+6(b)

0.0

2.0e+4

4.0e+4

6.0e+4

8.0e+4

1.0e+5

1.2e+5(a)

Wavelength/ Å

Inte

nsity

[a.u

.]In

tens

ity [a

.u.]

Rat

io (

b)/(

a)

Figure2.23. TheTaspectrumwithoutpre-

pulse(a),with 4%prepulse(b) andthera-

tio with prepulseto without prepulse(c).

In the Ni-lik e emission lines, an up to

35 � higherintensitycouldbeobserved in

spectrumtakenwith prepulsecomparedto

the spectrumwithout prepulse,while the

pseudo-continuumis only enlarged by a

factorfour. (Notethedifferentscaleof the

ordinates).

mainpulse.Figure2.23(c)showstheratioof bothspectra.With prepulse,the4 f � 3d

line is about35 times more intensethan without prepulse,indicating that thereare

about35 timesmore4 f ionsand,in conclusion,4p ionsin theplasma.

An interestingfact is thatthereis analmostconstantfactorof aboutfour between

thequasi-continuumof bothspectraabove � 6 � 0 A, while theemissionin thespectral

linesincreasesandthe(Co-like)quasi-continuumof thespectrumtakenwith prepulse

is over-proportionallyenhancedbelow 6.0A.

Figure2.24shows the calculatedtemperaturesof theplasmaasa function of the

incidentlaserintensity. ThetargetwasTa foil andthelaserfocuswaskeptto 333µm

in diameterwhich correspondto 8 � 7 � 10þ 4 cm2 illuminatedareaon thetarget,while

theenergy in thepulsewasvariedbetweenbetween31 J and180J, so thatall shots

aretakenat thesameplasmasize.This is importantsinceotherwise,differentcooling

effectswill influencethe result. The prepulseamountsto 4% of themain pulseand

came3 nsin advance.Thedurationof eachpulsewas100ps.

Thetemperatureof theplasmawasobtainedbyEq.(2.9).Thedifferentplotsin Fig.

88

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

Line 1−2Line 1−3Line 2−3

<T>

gf(Fritzsche & Dong); Lambda: Calc.

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

Line 1−2Line 1−3Line 2−3

<T>

gf(Fritzsche & Dong); Lambda: Exp.

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(Zigler et al); Lambda: Exp.

Line 1−2Line 1−3Line 2−3

<T>

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(Zigler et al): Lambda: Calc.

Line 1−2Line 1−3Line 2−3

<T>

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(Quinet + Biemont); Lambda: Exp

Line 1−2Line 1−3Line 2−3

<T>

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(Quinet + Biemont); Lambda: Calc.

Line 1−2Line 1−3Line 2−3

<T>

0.2 0.6 1.0 1.4 1.8 2.215 2Intensity [10 W/cm ]

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(average); Lambda: Exp.

Line 1−2Line 1−3Line 2−3

<T>

15 2Intensity [10 W/cm ]0.2 0.6 1.0 1.4 1.8 2.2

050

100150200250300350400

Tem

pera

ture

[eV

]

gf(Zhang et al); Lambda: Exp.

Line 1−2Line 1−3Line 2−3

<T>

Figure 2.24. The temperaturesof the Ta plasmasfor different incident laserintensitiesand

differentcalculationof thegf -factorwhile keepingtheexperimentalconditionsconstant,ex-

cept the pulseenergy. The labels‘Line i–j standsfor the calculationof the temperatureby

comparing‘Line i’ with ‘Line j’, where‘Line 1’ standsfor 3d93 24 f5 2 ÿ 3d10, ‘Line 2’ for

3d95 24 f7 2 ÿ 3d10 and‘Line 3’ for 3p5

3 23d104s1 2 ÿ 3d10. Thedottedline markstheaverage

valueof thecomparisonof thethreelines.

89

215

−4E = 31 − 180 J

A = 8.7×10 cm 2

Ele

ctro

n de

nsity

/ [a

.u.]

Incident Laser Intensity / 10 W/cm

7

8

9

10

11

12

13

14

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2.21.0 2.0PSfragreplacements

(a)

E ~ 50 J−5 −3A = 1.4×10 − 7.9×10 cm2

215Incident Laser Intensity / 10 W/cm

7

8

9

10

11

12

13

14

0 5 10 15 20 25 30 35

Ele

ctro

n de

nsity

/ [a

.u.]

PSfragreplacements

(a)

(b)

Figure 2.25. The densityof the Ta plasmain dependency of the incident laserintensityob-

tainedfrom the temporalandspatialintegratedspectra:(a) intensityvariedby changingthe

energy in thepulse( ) togetherwith thelinearfit (—) and(b) intensityvariedby changingthe

focal spot( � ). For comparison,thedataof (a)areplottedin (b), too.

2.24aredueto differentcalculationsof thegf -factors(cf. Tab. 2.1)aswell astheuse

of thecalculatedandexperimentalwavelengths(cf. Tab. 2.2).FromFig. 2.24onesees

thattheaveragetemperatureis almostconstantabout120eV eventhoughtheincident

laserintensitywasvariedby abouta factorof ten. While the obtainedtemperatures

of thecomparisonof thethreelinesis in goodagreementfor thegf –factorsby Zhang

et al. [109] andZigler et al. [110] in conjunctionwith theexperimentalwavelengths,

thegf –factorsfrom Fritzscheetal. [107] andQuinetetal. [108] givelargedifferences

betweenthe threecomparisons.A detaileddiscussionof the temperatureis given in

Sec.2.7.

Figure2.25 shows the calculatedelectrondensityof the plasmaasa function of

the incidentlaserintensity. Theunit of thedensityin Fig. 2.25is arbitrary, sincefor

ionizedgases,the detailedcalculationof the densityis extremelycomplicated,have

beendonein detailonly for afew atoms[25], andis farbeyondtheframeof thiswork.

However, full calculationgive a proportionalrelationto thewidth of thespectralline

(cf. Eq. (2.21)andRef. [25, p 220]). Thus,thedatashown in Fig. 2.25representthe

behavior of thedensity.

Figure2.25(a)shows theelectrondensityfor a constantfocal spotsizebut varied

pulseenergy. Therelationbetweentheelectrondensityandtheincidentlaserintensity

appearsto be linear in theobservedrange.In Fig. 2.25(b)the incidentlaserintensity

wasvariedby keepingtheenergy in thepulseconstantat about50 J but changingthe

spotsizeof thelaserbeam.Here,thedensityis increasingverysteeplyatthebeginning

but goesthenin saturation.Thedatafrom Fig. 2.25(a)areplottedin Fig. 2.25(b),too,

90

for adirectcomparisonof bothcases.

Usingthehighresolutioncharacterof theinstrument,thepositionsof theemission

lines were measured,too. The dataare summarizedin Table2.2 togetherwith the

theoreticalvaluesfrom the literatureandthe calculatedwavelengthsby S. Fritzsche

and Dong which were kindly doneto supportthis work [107]. The calculationof

thewavelengthsweredoneusingtheunpublishedprototypeversionof theGRASP92

code,which is namedGRASP2[129,130]. Theuncertaintyin thelastdigit (standard

error) of the experimentalvaluesis given in bracesin Tab. 2.2. The averageerror

(standarddeviation)is ∆λ � λ 9 � 4 � 10þ 5, ∆λ � λ 6 � 6 � 10þ 5, and∆λ � λ 5 � 9 � 10þ 4

for Hf, Ta, andW, respectively. Thehighererror in theW valuesis dueto thesmall

databaseof only threeshots.

Dependingon the calculationandthe transitionline, the agreementbetweenthe

calculationandthe experimentis excellentto fair. On the otherhand,the measured

wavelengthsarecomparedto thetransitionenergiesof theAl H- andHe-likeions.The

resultingerrorin thedispersionrelationis calculatedas2 � 6 � 10þ 6 A(cf. Tab. 2.5).

2.7 Discussionof the temperature

Thehereappliedwayof calculatingthetemperatureis highly sensitiveto errorsin the

parameters.This is dueto the fact that the temperatureis inverseproportionalto the

logarithmof all parameters(cf. Eq. (2.9)). At high temperatures,the dependency of

the temperatureson the parametersbecomesvery steepso that small changesin the

parameterscauseslargechangesin thecalculatedtemperature.

The error bars in the temperatureplots were estimatedby the following set of

equations:

∂ � kBT �∂E10

� ln � gf10E210I20

gf20E220I10 � E10 � 2E20 � 2E10

ln � gf10E210I20

gf20E220I10 � 2

E10

(2.32a)

∂ � kBT �∂E20

ln � gf10E210I20

gf20E220I10 � E10 � 2E20 � 2E10

ln � gf10E210I20

gf20E220I10 � 2

E20

(2.32b)

∂ � kBT �∂gf10

� E20 � E10

ln � gf10E210I20

gf20E220I10 � 2

gf10

(2.32c)

91

∂ � kBT �∂gf20

E20 � E10

ln � gf10E210I20

gf20E220I10 � 2

gf20

(2.32d)

∂ � kBT �∂I10

E20 � E10

ln � gf10E210I20

gf20E220I10 � 2

I10

(2.32e)

∂ � kBT �∂I20

� E20 � E10

ln � gf10E210I20

gf20E220I10 � 2

I20

(2.32f)

For thecalculation,anuncertaintyof 0.8mA in thetransitionwavelength,� 0 � 1 in

thegf -factorand5 % errorin theintensitywasassumed.

Anotherinfluencein theerrormightbetheopticalthickness,i.e. theself-absorption

of theemittedintensityin theplasma.Thisopticalthicknessdependsbesidethewave-

lengthandthesizeof theplasmaontheelectrondensity, andmightbedifferentfor the

threedifferentemissionlines,aspointedout in Sec.2.2.2. Due to this changein the

self absorption,theemittedspectrummight not representtheBoltzmann-distribution,

asit wasapplied,but hasto becorrectedby theappropriateexpressionfor theoptical

thickness,which might changetheintensityratio of theemissionlinesandwill there-

fore leadto awrongtemperature.Eventhoughthereareformulaefor simpleestimates

for theopacityfor blackbodyradiation[111,112], theseequationsareof limited use

heresincethey are independentof the wavelength. I.e. the obainedopacitycannot

describea variationof the ratio of the intensitiesasa function of the wavelength.A

detailedcalculationhowever is extremelydifficult andcannotbe found in public for

Z � 30[113,131]. However, it is importantto beawarethattheobservedemissionline

rationdependson both,theelectrontemperatureanddensity. Thisalteredline ratioof

theemissionlinescouldexplain that thetemperatureis with � 150eV abouta factor

tensmallerthanexpectedfrom Fig. 2.2.

Also, from Fig. 2.24oneseesthattheaveragetemperatureis almostconstanteven

thoughthe incident laserintensity was variedby abouta factor of ten. This is as-

tonishing(or wrong)sinceoneexpectsa muchhighertemperaturefor a muchhigher

intensity.

One reasonfor the constanttemperaturemight be the time integratedmeasure-

mentof thetemperature.To checkthis possibility, in anotherexperimentalcampaign,

carriedout in May 2000,thespectrashouldberecordedtime resolvedusinga streak

camera.However, dueto technicalproblemswith thelasersystemandthestreakcam-

92

era,no datacouldbeobtainedwithin theavailablebeamtime.

The simulationusing the one-dimensionalMEDUSA code[132] shows, that the

arithmeticmeantemperaturevariesalmostlinear between219keV and589 keV for

Iinc 2 � 1014 W/cm2 andIinc 2 � 1015 W/cm2, respectively. At thetimeof themain

pulse,thetemperatureraisesbriefly to 3.5keV and10 keV for Iinc 2 � 1014 W/cm2

andIinc 2 � 1015 W/cm2, respectively.

Of course,thisarithmeticmeanvaluedoesnot representtheaveragingof thespec-

trometer, but givesanideaof thevariationof thetemperature.For acorrectsimulation

of the averagingof the spectrometer, the absolutenumberof emittedphotonsin the

resonanceshave to becalculatedfor every temperature,timeandspatialdistance.The

generatedphotonshave to besummedup andthencomparedwith eachother. How-

ever, a simulationcodefor the spectrafor suchheavy materialsasTa could not be

foundin public.

In comparisonwith otherworks,Nantelet al. [105] measuredfor thesamepump

laserwavelengthbut Ne-like 30Zn at � 1 � 2��� 1013 W/cm2 in 800psa constanttem-

peratureof 335 � 25 eV, which appearedto be independentfrom the amountof the

prepulse.For comparison,theMEDUSA codewasrunwith thevaluesfrom Ref.[105],

wherethearithmeticmeanvaluefor thetemperatureis 375eV for 2 � 1013 W/cm2.

Anotherinterestingquestionarisesfrom thelargedifferencein theusedgf -values

in the literature,which give very differentagreementsof the threecalculatedvalues

from onespectrum(Fig. 2.24).While temperaturesobtainedfrom thegf -factorspro-

videdby Zhanget al. [109] arein goodconsistence,the temperaturesobtainedfrom

thegf –factorscalculatedby Fritzscheetal. [107] differ by a factorof two. Therefore,

it wouldbeinterestingto measurethegf -factorsfrom theexperimentaldataset.

The idea for basicapproachfor the measurementof the gf -factorswas to use

Equation(2.9) for threedifferent shots j 1 � 2 � 3, i.e. threedifferent temperatures,

takenunderthesameexperimentalconditionsexceptthevariedincidentlaserintensity.

TheenergiesEi andintensitiesI � j �i aretakenfrom thespectra:

T1 T1 � gf10 � gf20 � I � 1�1 � I � 1�2 � E1 � E2 ����� Shot1(2.33a)

T1 T1 � gf10 � gf30 � I � 1�1 � I � 1�3 � E1 � E3 ����� Shot1(2.33b)

T2 T2 � gf10 � gf20 � I � 2�1 � I � 2�2 � E1 � E2 � ��� Shot2(2.33c)

T2 T2 � gf10 � gf30 � I � 2�1 � I � 2�3 � E1 � E3 � ��� Shot2(2.33d)

93

T3 T3 � gf10 � gf20 � I � 3�1 � I � 3�2 � E1 � E2 ����� Shot3(2.33e)

T3 T3 � gf10 � gf30 � I � 3�1 � I � 3�3 � E1 � E3 ����� Shot3(2.33f)

i.e. six equationswith six unknown parameters(T1, T2, T3, gf10, gf20, gf30). Setting

equalthe two equationsfor the sametemperaturethis systemof equationreducesto

threeequationswith thethreeunknown parametersgf10, gf20, andgf30:

T1 � gf10 � gf20 � I � 1�1 � I � 1�2 � E1 � E2 �� T1 � gf10 � gf30 � I � 1�1 � I � 1�3 � E1 � E3 � (2.34a)

T2 � gf10 � gf20 � I � 2�1 � I � 2�2 � E1 � E2 �� T2 � gf10 � gf30 � I � 2�1 � I � 2�3 � E1 � E3 � (2.34b)

T3 � gf10 � gf20 � I � 3�1 � I � 3�2 � E1 � E2 �� T3 � gf10 � gf30 � I � 3�1 � I � 3�3 � E1 � E3 � (2.34c)

However, whentrying to isolatetheexpressionsfor thegf –factors,onefinds,that

this systemof equationshasnosolution.

94

Summary and Outlook

In the presentwork, the conception,designandapplianceof toroidally bentcrystals

for theX-ray opticaldiagnosticsof laserproducedplasmasis discussed.

The first part of this work dealswith thedevelopment,designandcharacterization

of an X-Ray microscopefor the observation of Rayleigh-Taylor instabilities,which

actagainsttheconfinementandignition of thefuel in the inertial confinementfusion

(ICF) process.

Theaim of this work wasthedevelopmentandtestof anX-ray microscopewhich

satisfiesthehighdemandsonthiskind of experiment,i.e. two-dimensionalmonochro-

maticimagingwith high luminosity, high resolution( � 3 µm) aswell as400µm focal

depth.Themicroscopeshouldbeappliedin theframeof theHIPERresearchprogram

at theInstituteof LaserEngineering(ILE) at OsakaUniversity/Japan.

Thebasisof theX-ray microscopeis a toroidally bentquartzcrystalof 7 � 7 mm2

whichshouldbeoperatedat 30 � magnificationandlater70 � magnification.

The simulationof the constructedmicroscopeby ray-tracingcalculationsshows

that therequiredspecificationscannotonly befulfilled but evenexceeded:Thesimu-

lationgivesanexpectedresolutionof lessthan � 1 µm atacrystalapertureof 3.5mm

and30 � magnification.

However, the experimentalevidencecould not be demonstrateddueto technical

problemsandlimited beamtimeat GEKKO XII. Thehighestdemonstratedresolution

in this work was6 µm.

Being aware of the very limited beamtime at ILE and the great interestin the

demonstrationof the sub-micrometerresolution,anotherdemonstrationexperiment

is proposedin detail, which canbe carriedout in a small laboratoryusingan X-ray

generator.

First datacould be obtainedfrom the Rayleigh-Taylor instability experimentand

95

arediscussedqulitatively. For adetailedanalysis,however, theobtaineddatasetis too

smallandthesignaltoo weakto beclearlydistinguishedfrom thenoiseof thestreak-

or CCD-cameradueto focalproblemswith theprobelaser.

Sinceon the onehand,the investigationof the Rayleigh-Taylor instabilitiesare

of greatestimportancein thenuclearfusion researchandon theotherhand,thehere

developedmicroscopeis the main diagnosticfor this kind of experiment,the work

startedherewill becontinuedin thefuturebeyondtheframeof this work.

The aim of the secondpart of the presentwork was the diagnosticof the lasing

mediumfor amplifiedspontaneousemission(ASE)closeto thewaterwindow. For this

purpose,anone-dimensionally(1–D) imagingX-ray spectrometerbasedon toroidally

bentquartzcrystalswasdevelopedfor theobservationof theNi-lik e4 f � 3d transition

of Yb, Hf, Ta,andW ions,whichshouldberelatedto theamplified4d � 4p emission,

sincethe 4 f niveauis very closeto the 4d niveau. Thus,the 4 f � 3d transitioncan

serveasanindicatorfor thepopulationof the4d niveau.

Thespectrometerwasdesignedfor therecordingof thespectraeither1–Dspatially

resolved but time-integratedby usinga CCD camera,or time-resolved but spatially

integratedby usingastreakcamera.

The characteristicsof the spectrometerarediscussedin detail for the usedrange

of operation,i.e. 5.7–6.4A, by usingray-tracingcalculations.It wasfound that the

spectralresolutionof the spectrometeris almostindependentof the sourcesizeand

thedeviation of thesourcefrom thediffractionplane. Evenfor 20 mm large plasma

columnsasthey areusedfor X-Raylaserexperimentsandshiftsof thesourceof 30mm

off thediffractionplane,a spectralresolutionof betterthan3 � 10þ 3 canbeobtained.

Also the spatial resolutionis almost independentof the sourcesize and limited in

thepresentcaseby the resolutionof theCCD array, which couldbedemonstratedin

theexperiment.However, theray-tracingcalculationshows thatthespatialresolution

decreasesthesmallerthesagittalapertureof thecrystalis. This is dueto thedecreased

peakintensityandincreasedimagingerrorsrelative to thepeakreflectivity.

Thespectrometerwascalibratedin-situ by comparingtheobservedline positions

of thewell-known Al plasmawith tabulatedvalues.Fromthecomparison,theactual

distancesof the set-upcan be obtained,which definethe dispersionrelation com-

pletely. Whenusinga calibratedCCD camera,too, the absolutenumberof emitted

photonsfrom theplasmacanbeobtained.

96

Theexperimentswerecarriedout at theGEKKO XII Nd:glas-laser-systemat the

Instituteof LaserEngineeringattheUniversityof Osaka/Japanin aJapanese,Chinese,

French,andGermancollaboration.Theexperimentconfirmstheexpectedrelationbe-

tweentheNi-lik e 4 f � 3d transitionandtheamplified4d � 4p transition.For Ta, the

optimal pump-laserintensitywas found at � 1 � 5 � 1015 W/cm2, a pre-pulseof 4%

increasesthe populationof the 4 f niveauby a factorof up to 35 � . The estimation

of thetemperaturefrom thetime-integratedspectraby usingtheordinaryBoltzmann-

distribution givesan electrontemperatureof � 150eV which wasobservedasinde-

pendentof theincidentpump-laserintensity. Onereasonfor this independencemight

betheaveragingby time integration.Thus,thetime resolvedobservationof thespec-

tra is necessaryin orderto getsomeevidenceof this result,whatcouldnotbedoneyet

in theavailablebeamtime.

Theanalysisof theelectrondensityof theplasmashows a linearrelationbetween

the intensityof the pumplaserandthe electrondensitywhenkeepingthe focal spot

constant. Also, the transitionwavelengthof the 4 f � 3d lines could be measured.

The agreementwith theoreticalvaluesfrom the literatureis at a differenceof �����1 � � � 10� mA, dependingon theauthorandthespectralline, excellentto good.

For thenext experimentalcampaign,thetime-resolvedobservationof the4 f � 3d

transitionis planned,whichshouldgiveanswersto suchimportantquestionsas:What

is the influenceof the pulsetrain on the ionization? At which time of pulsetrain is

the maximumionizationreached?At which time is reachedthe maximumelectron

temperatureand-density?

97

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110

Ehrenwortliche Erkl arung der

Selbststandigkeit

Ich erklarehiermit ehrenwortlich, daßich die vorliegendeArbeit selbststandig,ohne

unzulassigeHilfe Dritter undohneBenutzungandereralsderangegebenenHilfsmittel

undLiteraturangefertigthabe.Die ausanderenQuellendirektoderindirekt ubernom-

menenDatenundKonzeptesindunterAngabederQuellegekennzeichnet.

Bei derAuswahlundAuswertungfolgendenMaterialshabenmir die nachstehend

aufgefuhrtenPersonenin derjeweilsbeschriebenenWeiseunentgeltlichgeholfen:

Eckhart Forster: Themenstellung,AuswahlvonLiteratur, Diskussion,Verbesserungs-

vorschlagezurDissertation

Hir oaki Nishimura: Themenstellung(Rayleigh-TaylorInstabilities),AuswahlvonLi-

teratur, Diskussion

Hir oyuki Daido: Themenstellung(Rontgenlaser),AuswahlvonLiteratur, Diskussion

Yoshiaki Kato: Diskussion

Ingo Uschmann: Einfuhrungin dieThemen,AuswahlvonLiteratur, Diskussion,Hil-

fe bei derKonstruktiondesHIPERKristalls

Manfr ed Larz: KonstruktionderInstrumente

Markus Vollbr echt: KonstruktiondesSpektrometers

JosephNilsen: Literaturhinweise

Kazuhisa Fujita: GemeinsameAuswertungundDiskussion

StephanSebban: Einfuhrungin Rontgenlaser

111

StephanFritzsche: VorschlagundEinfuhrungzur Temperaturbestimmung,Berech-

nungdergf -Faktoren,Diskussion

Fumihir o Koike: Berechnungdergf -Faktoren,Diskussion

Ortrud Wehrhan: Einfuhrungin dieKristallographie,Diskussion

Paul Gibbon: Einfuhrungin denMedusa-Simulationscode,Diskussionder plasma-

physikalischenErgebnisse

WeiterePersonenwarenander inhaltlich-materiellenErstellungdervorliegenden

Arbeit nicht beteiligt. Insbesonderehabeich hierfur nicht die entgeltlicheHilfe von

Vermittlungs-bzw. Beratungsdiensten(Promotionsberateroder anderePersonen)in

Anspruchgenommen.Niemandhatvonmir unmittelbarodermittelbargeldwerteLei-

stungenfur Arbeitenerhalten,die im Zusammenhangmit demInhalt dervorgelegten

Dissertationstehen.DieArbeit wurdebisherwederim In- nochim Auslandin gleicher

oderahnlicherFormeineranderenPrufungsbehordevorgelegt.

DiegeltendePromotionsordnungderPhysikalisch-AstronomischenFakultatistmir

bekannt.

Ich versichereehrenwortlich.daßich nachbestenWissendiereineWahrheitgesagt

undnichtsverschwiegenhabe.

Jena,23.Oktober2000

112

Curriculum Vitae

Name: RandolfButzbach

Birthday, -place: February16,1969 Kassel,Germany

Nationality: German

Personalstatus: single

Schooleducation: 1975–1979 AuefeldschuleKassel

(Grundschule)

1979–1986 Carl-Schomburg-SchuleKassel

(Realschule)

1986–1989 Jacob-Grimm-SchuleKassel

(Gymnasium)

Abitur: May 19,1989

Civil service: 1989–1990

Universityeducation: 1990–1996 Universityof Kassel

Principalsubject:Physics

June27,1996 Degree:Diplom-Physiker

(Dipl.-Phys.)

Diplomarbeit:

(Masterthesis)

“Development of Thin Crystal Monochromators:

The Influenceof Crystal Thicknesson the Tails of

the Rocking Curve” at the EuropeanSynchrotron

RadiationFacility, ESRF, Grenoble/France(on own

initiative)

113

Employment:

1992–1995 Employmentatanengineerofficeasself-responsible

software-engineerfor commercialsoftware

1995–1996 Employmentat theEuropeanSynchrotronRadiation

Facility in Grenoble/France

1997–2000 EmploymentasWiss.Mitarbeiter(ScientificCollab-

orator, PhD Position) at the X-Ray Optics Group

at the Institute of Optics and QuantumElectronics

at the University of Jena/Germany, in collaboration

with the Instituteof LaserEngineering,ILE, at the

Universityof Osaka/Japan.

Staysabroad: – France (18months)

– Japan (11months)

Languages: German,English,French,Modern-Greek– eachin

speechandwriting; somebasicsin Japanese

Awards: ILE-Prizefor thebestscientificwork in 1998

Jena,23. Oktober2000

114

Acknowledgments

First of all, honorto whom honoris due,thanksto the almighty God, the creatorof

heaven and earthand light and atomsand who hasalways guidedmy life, for the

peekin the interrelationsof His wonderfulcreation,which we will never completely

understand.

I like to thankmy parentswho renderedpossiblemy studiesof physicsaswell as

EckhartForster, Hiroaki Nishimura,andHiroyuki Daidowhoproposedthisinteresting

subjectsandfor many fruitful discussions.

The many helpful discussionswith Ingo Uschmann,aswell ashis help in many

ways,wereagreatpleasurefor which I like to expressmy sincerethanks.

Very heartilythanksgo to DiethardKl opfel for his helpin technicalproblemsand

particularlyfor hismoralsupportduringthiswork.

I amgreatfulto MarkusVollbrechtwho undertookwith greatcompetencethede-

signof thespectrometerduringthephaseof writing his thesiswhile I wasa beginner

in this field.

ManfredLarz undertookthenthe mechanicalconstructionof the instrumentsfor

which I amverygreatful.

Theassistancein theexperimentalcampaignof KazuhisaFujitaandJiroFunakura

is greatfullyacknowledged.

I amindebtedto StephanFritzscheandFumihiroKoike who explainedmethere-

quiredatomicphysicsandcalculatedfor methetransitionwavelengthsandgf –factors.

A specialthank goesto the workshopsof the University of Jena,who built the

instrumentsusedin this work in an extremelyshort time. I appreciateit very much

thatthey settheinstrumentsof this work on thehighestpriority while delayingall the

otherordersin orderto preparetherequiredinstrumentsin time.

I alsowould like to thankall the ILE staff for their hospitalityandhelp in many

problems.FurthermoreI liketo thankall thenumerouscolleaguesfrom China,France,

115

Korea,India,andJapan,whocontributedwith their specialskills to theherepresented

experimentalcampaigns.Without their contribution, this work wouldnotbepossible.

Onceagain,I would like to thankOrtrudWehrhan,Elke Andersson,Jorg Tschis-

chgale,Ulrich Wagner, andPaulGibbonwho madethestayin Jenaapleasure.

I also wish to expressmy thanksto the DeutscheForschungsgemeinschaftwho

madethis work possible.

Last but not least,it is a greatpleasurefor me to expressmy sincerethanksto

Cecılia, whogavemethemotivation,strength,andenergy in averydisappointingand

frustratingphaseof thiswork. To sayit with Tang-san’swords(Thankyouverymuch,

Tang-san!):“Take it easythatall yourexperimentsfailed.. . ! At least,youhave found

her. . . ”

116