X-RayOptical Diagnostic of Laser Produced … Diagnostic of Laser Produced Plasmas for Nuclear...
Transcript of X-RayOptical Diagnostic of Laser Produced … Diagnostic of Laser Produced Plasmas for Nuclear...
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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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