Bright high-order harmonic generation with controllable polarization from a relativistic plasma mirror

1.8 MB·

by  Zi-Yu Chen

Bright high-order harmonic generation with
controllable polarization from a relativistic plasma
mirror
Zi-Yu Chen1,2 and Alexander Pukhov1
6
1
0 1Institutfu¨rTheoretischePhysikI,Heinrich-Heine-Universita¨tDu¨sseldorf,Du¨sseldorfD-40225,Germany
2
2NationalKeyLaboratoryofShockWaveandDetonationPhysics,InstituteofFluidPhysics,ChinaAcademyof

g Engineering Physics,Mianyang621999,China
u
1CorrespondenceandrequestsformaterialsshouldbeaddressedtoZ.Y.C.(email:
A
[email protected])ortoA.P.(email: [email protected]).

7
1
ABSTRACT

]
h
p Ultrafastextremeultraviolet(XUV)sourceswithacontrollablepolarizationstatearepowerfultoolsforinvestigatingthestruc-
- turalandelectronicaswellasthemagneticpropertiesofmaterials. However,suchlightsourcesarestilllimitedtoonlyafew
m
free-electronlaserfacilitiesand,veryrecently,tohigh-orderharmonicgenerationfromnoblegases.Hereweproposeandnu-
s mericallydemonstratealaser-plasmaschemetogeneratebrightXUVpulseswithfullycontrolledpolarization.Inthisscheme,
a
anelliptically-polarizedlaserpulseisobliquelyincidentonaplasmasurface,andthereflectedradiationcontainspulsetrains
l
p and isolated circularly- or highly-elliptically-polarizedattosecond XUV pulses. The harmonic polarizationstate is fully con-
.
s trolledbythe laser-plasmaparameters. The mechanismcan be explainedwithintherelativisticallyoscillatingmirror model.
c ThisschemeopensapracticalandpromisingroutetogeneratebrightattosecondXUVpulseswithdesirableellipticitiesina
i
s straightforwardandefficientwayforanumberofapplications.
y
h
p
[ Ultrafast radiation sources in the extreme ultraviolet (XUV) range have become a major tool to study electronic struc-

turesand thedynamicsof atoms, moleculesand condensedmatter. Because polarizationisa fundamentalpropertyoflight
2
and controlsits interactionwith matter, it is particularlyimportantthatthese lightsourceshave a tunablepolarization. Fur-
v
3 thermore,polarizationcontrolopensa widerrangeofapplications. Forinstance, themagneticaswellastheelectronicand
2 phononic properties of materials can be studied using circularly-polarized (CP) or elliptically-polarized (EP) XUV pulses
9 usingtechniquessuchasmagneticcirculardichroism(MCD)spectroscopy.1 TheMCDtechniquehasproventobeveryuse-
5 fultoprobethespin-resolvedfeaturesinmagneticmaterialsinanelement-specificmanner,2 andthusisofgreatinterestfor
0
understandingcorrelatedsystemsincondensedmatterphysics.Inaddition,CP/EPXUVpulsesalsoenableauniqueprobeof
.
1 chiralmolecules,3,4e.g.,measuringthephotoionizationprocessviaphoto-electroncirculardichroism.5 Assuch,CP/EPXUV
0
pulsesalsofindawiderangeofapplicationsinstudyingchemicalandbiologicalsystems.
6
1 To date, significant efforts have been devoted to generating the ultrafast XUV with a variable polarization state. The
: first free-electron laser facility that was specially designed to produce such a light source, named FERMI (Free Electron
v
laserRadiationforMultidisciplinaryInvestigations,Trieste,Italy),hasbecomeaccessibleveryrecently.6 Polarizationcontrol
i
X
at FERMI is achieved by adjusting the configuration of the undulators. Though powerful, these large-scale facilities are
r expensiveandcomplex,thuslimitingtheirwideaccessibility. Therefore,thereremainsastrongneedforsourcesofcoherent
a
CP/EPXUVradiationatthetable-topscale.
High-orderharmonicgeneration(HHG)fromnoblegaseshasbeenexploredextensivelyasaroutetogenerateanultrafast
XUV source.7 This mechanism, however, encountersintrinsic difficulties in generating a CP XUV pulse. This is because
theHHGisbasedonthetunnel-ionization,accelerationandrecombinationofelectronsrippedfromanatominthepresence
of a laser field, which is explainedby the so-called three-step model.8 As a consequence, the emission of HHG decreases
exponentiallywithincreasingthelaserellipticitybecausethelateralmotionofthedetachedelectroninducedbytheellipticity
makestheelectronlesslikelytorecollidewithitsparention. Tobeexact,theelectronneverreturnstotheparentionwitha
CPdrivinglaser. Toovercomethisdrawback,severaltechniqueshavebeenproposedanddemonstratedrecentlytogenerate
quasi-CPorhighly-EPHHG,suchasusingpre-alignedmoleculetargets,9resonantHHGinEPlaserfields,5aco-propagating
bi-chromaticEPorCPdrivinglaserwithoppositehelicity,1,10–12 andaco-propagatingbi-chromaticlinearly-polarized(LP)
drivinglaser with orthogonalpolarization.13 However,owingto theirlow ionizationthresholdsandconversionefficiencies,
thesesourcestypicallysufferfromlowphotonyields.
Tofillthegapbetweenlarge-scalefacilitiesandHHGfromgas,XUVviaHHG14 andothermechanisms15–18 fromlaser-
irradiatedplasmasurfacesoffersapromisingalternativetogenerateanXUVsourcewithhighbrightness. Inprinciple,with
plasma targetsthereis nolimitationontheapplicablelaser intensityandthusthe XUV intensity.14 Severalradiationmech-
anisms have been theoretically and experimentallyidentified as responsible for the HHG process, including coherentwake
emission(dominantintheweaklyrelativisticregime),19–22 relativisticallyoscillatingmirror(ROM;dominantinthestrongly
relativistic regime),23–28 and coherentsynchrotronemission.29–31 It has been demonstratedthat LP HHG can be generated
relativelyefficientlyusinganLPdrivinglaseratobliqueincidence.
IthasbeencommonlyassumeduptonowthattheROMmechanismfailsforaCPdrivinglaser. Moreover,apolarization
gating (aka relativistic coherent control) has been proposed to select a single LP attosecond pulse from a pulse train.32–34
AccordingtotheROMtheory,25theseattosecondpulsesareemittedwhenelectronsattheplasmasurfacearemovingtowards
theobserverandtheirtangentialmomentavanish.ThisisneverthecaseforanEPlaserpulsenormallyincidentattheplasma
surface,whichcausesHHGtobestronglysuppressed. ForanEPlaserpulseatobliqueincidence,twoexperimentalgroups
haveobservedHHGatrelativelysmallanglesofincidence.35,36 However,onlytheharmonicintensitiesweremeasured,with
noinformationabouttheharmonicpolarizationstates.
Inthispaper,weproposeandnumericallydemonstratethegenerationofintenseHHGwithfullycontrolledpolarization
from laser plasmas. We show that this can be achieved using a CP laser obliquely incident onto a plasma surface. Both
pulsestrainsandisolatedcircularorhighlyellipticattosecondXUVpulsescanbeobtained.Bychangingtheincidenceangle,
the harmonicpolarizationstate can be tunedfromquasi-circularthroughelliptical andlinear to an ellipticalpolarizationof
oppositehelicity. Switchingthehelicityoftheincidencelaser,thehandednessoftheharmonicscanbeeasilyreversed. The
schemeworksforawiderangeoflaserandplasmaparameters,andtheefficiencyiscomparabletothatusinganLPlaser. This
very promisingnew procedurethus providesa straightforwardand efficientway to obtain a brightattosecond XUV source
withdesirableellipticities,andholdsthepotentialofmakingaverylargeavenueofresearchmoreaccessibleforanumberof
laserlaboratoriesworldwide.
Results
Scheme
Figure 1(a) shows the scheme of the proposedconfigurationfor the HHG with a desired polarizationstate. The basic idea
is to use a CP relativistic laser pulse obliquely incident on a solid-plasma surface using a radiation mechanism known as
the ROM model. In the ROM model, under the combined action of the laser ponderomotive force of the laser and the
electrostaticrestoringforceresultingfromchargeseparation,thesurfaceelectronsoscillatewithrelativisticspeedsandreflect
thelaserpulselikemirrors. Duringthisnonlinearprocess,harmonicsofthefundamentallaserfrequencyaregeneratedasa
resultofDopplerup-shifting. Exceptforsomespecialcases(i.e.,few-cyclelaserpulseinteractionswithnear-criticaldensity
plasmas37),normal-incidenceCPlaserpulsescannotgenerateharmonicsfortworeasons. First,CPlaserpulseslackthefast
oscillatingcomponentintheponderomotiveforce. Second,drivenbyCPpulses,electronsalwayshavearelativisticallylarge
tangentialmomentum:whenonetangentialcomponentvanishes,theotherreachesitsmaximum. Asaresult,electronsnever
movetowardstheobserveranddonotemithighharmonicsefficiently. ThisdifferencebetweenCPandLPpulsesformsthe
basisofthepolarizationgating,32–34whichisthemethodproposedhereintoobtainanisolatedsingleattosecondpulsefroma
trainofattosecondpulsesofROMharmonics.ThisistrueforaCPlaserpulseatnormalincidence.
However,forobliqueincidenceinteractions,HHGcanbeefficientlygeneratedevenbyCPlaserpulses. Theforceacting
on the plasma surface does contain a fast oscillating component owing to the normal (p-polarization) component of the
laser electric field. Further, the oblique incidence can be reducedto a normalincidence case using Lorentz transformation
to a moving frame of reference where plasma is streaming along the surface (see Methods). In this frame of reference,
electronshaveaninitialtangentialmomentum.Whentheangleofincidenceisadjustedcorrectly,thismomentumcanexactly
compensateforthemomentuminducedbythelaserfieldsothatthereexistmomentswhentheplasmasurfaceelectronsmove
exactlytowardstheobserverandreflecttheCPlaser. However,thedifferentlaserpolarizationsmayhavedifferentphaselags
atthenon-linearreflectionfromtheplasma,whichmakesitnecessarytoperformsimulationstoclarifywhethertheproperties
oftheincidentlaser,suchaspolarizationandcoherence,arepreserved.
2Dsimulationresults
Wefirstcarriedouttwo-dimensional(2D)particle-in-cellsimulationstoshowageneralpictureoftheellipticityofHHGfrom
aCPlaserobliquelyirradiatedonplasmasurfaces.Thelaserandplasmaparametersarechosentomatchrealisticexperiments,
whereinthelaserwithanormalizedamplitudeofa =30isobliquelyincidentatanangleofq =40 ontoaplasmaofdensity
0 ◦
n =100n , where n is the critical density (see Methods). Figure 1(b) presents a snapshotof the electric field component
e c c
E of the reflected pulse in the x y plane at timet =36T , where T is the laser period. The greendashed line marksthe
z 0 0
−
directionofspecularreflectionoftheincidentlaser. Atemporalwaveformoftheradiationinthespecularreflectionisshown
2/10
Figure1. Schemeand2Dsimulationresults. (a)Theproposedexperimentalconfigurationforgenerationofthe
polarization-controlledharmonicsbyacircularly-polarizedlaserpulseobliquelyincidentonaplasmasurface.Theredand
purplearrowsrepresenttheincidentlaserpulseandthereflectedXUVharmonics,respectively.(b)Asnapshotoftheelectric
fieldcomponentE ofthereflectedpulsefromthe2Dsimulationresultsattimet=36T . Thegreendashedlinemarksthe
z 0
specularreflectiondirection.(c)Temporalwaveformand(d)thecorrespondingFourierspectraofthereflectedpulsealong
thespecularreflection.Thegreendashedlineinpanel(d)correspondstothepredictedscalinglawI (n)(cid:181) n 8/3bythe
ROM −
ROMtheory.(e)Thereconstructed3Dimageoftheelectricfieldvectoroftheattosecondpulses(purple),obtainedafter
spectralfilteringbyselectingthe10th-20thharmonicorders(indicatedbytheharmonicswithinthedashedgrayboxinpanel
d). WaveformofthetwoorthogonalelectricfieldcomponentsE (green)andE (blue),aswellastheprojectionofE E
y z y z
−
(gray)arealsoshown.
inFig.1(c),whereboththeE andE componentsaredepicted.ItisapparentthatbothE andE haveanamplitudelevelthe
y z y z
sameasthatoftheincidentlaser.
Figure1(d)showstheFourierspectracorrespondingtoFig.1(c).Thegreendashedlinecorrespondstothescalinglawfor
thespectralintensityI asafunctionoftheharmonicordern: I (n)(cid:181) n 8/3,whichisgivenbytheBaeva-Gordienko-
ROM ROM −
Pukhov(BGP)theory25ofROM.Theexcellentagreementofthespectrawiththetheoreticallypredictedpowerlawsuggests
thattheHHGmechanismhereiswithintheROMregime. Harmonicstructuresuptothe20th ordercanbeclearlyobserved
for both the E and E components. Beyond that, however, the spectral lines structure is not periodic, indicating that the
y z
periodicityoftheattosecondpulseschangeswithtime. Itisworthnotingthatthe2DsimulationsforHHGfromsolid-plasma
surfacesarecomputationallyexpensiveandtheresolutionislimited. HereweonlyresolvetheHHGuptothe20th orderfor
demonstrationpurpose,butharmonicspectrawithwell-definedperiodicstructuresuptomuchhigherorderscanbegenerated
andhavebeenobservedexperimentally.Forexample,well-definedharmonicstructuresuptoatleastthe46thorderhavebeen
observedwithalmostthesamelaser(exceptingthatthepolarizationislinear)andplasmaparameters.38 Anotherexperiment
withamuchlowerintensityofa =3.5hasalsodemonstratedthatharmoniccombstructuresuptoaboutthe40th ordercan
0
be observed.39 Therefore, periodic harmonic orders higher than the 20th can be expected. In addition, different harmonic
ordersareappropriatefordifferentapplications. Forexample,theharmonicsofthe7th-20th orders(photonenergiesaround
10-30eV)areofparticularinterestforstudiessuchasmolecularphotoionization,becausethisfrequencyrangeisclosetothe
ionizationthresholdsofmostmolecularsystems.5 Also,harmonicsofthe35th-42ndorders(photonenergiesaround55-65eV)
arerequiredforinvestigatingthemagneticpropertiesofsolids,becausethisfrequencyrangecoverstheMabsorptionedgesof
3/10
themagneticelementsFe,Co,andNi.40 WewillleavetheHHGwithhigherorderstobeinvestigatedwithone-dimensional
(1D)simulationslater.
Applying a band-pass spectral filter that selects harmonics between the 10th-20th orders, we obtain a train of attosec-
ond XUV pulses, as shown in Fig. 1(e). From the helical structures of the electric field contour EH =EH+EH plotted in
y z
this three-dimensional(3D)image, we can see directly thateach attosecondHHG pulseis elliptically polarized. The HHG
pulses reach a peak electric field amplitude of EH =5 (normalized to m w c/e 4 1012 Vm 1), which corresponds to
e 0 −
≈ ×
a dimensional value of EH =2 1013 Vm 1. This clearly demonstrates the potential of the ROM mechanism to obtain a
−
×
bright helical XUV source. The averaged amplitude ratio between the two electric componentsin this frequency range is
e =min EH,EH /max EH,EH =0.96, indicating that a high ellipticity can be reached. The phase shift D f H between
{ y z } { y z }
the two electric field componentsf H and f H is D f H =f H f H=0.36p for the HHG pulse aroundx=30l in Fig. 1(e).
z y z − y 0
The sign of D f H also demonstrates that the HHG pulse generated here has the same helicity as the incident laser pulses
(f L f L=p /2).
z − y
Figure2. Parametricstudyandpolarizationcontrol.(a)-(d)Amplituderatioe andphaseshiftD f Hbetweenthetwo
orthogonalcomponentsoftheharmonicelectricfieldsasafunctionof(a)laseramplitudea ,(b)plasmadensityn ,(c)
0 e
plasmadensityscalelengthL ,and(d)laserincidenceangleq . Theotherparametersare: (a)n =100n ,q =40 ,and
s e c ◦
L =0.2;(b)a =5,q =40 ,andL =0.2;(c)a =5,q =40 ,andn =100n ;(d)a =5,n =200n ,andL =0.1.The
s 0 ◦ s 0 ◦ e c 0 e c s
L valueisnormalizedbyl inthemovingframeforconvenienceinthe1Dsimulations. Thedifferentshadedareasinpanel
s
(d)representharmonicspossessingoppositehelicities. LH:left-handed.RH:right-handed.
Parametricstudy
Inthefollowing,weuseaseriesof1DsimulationswithahigherresolutiontostudytheparametricdependenceoftheHHG
ellipticity. FromFig.2(a)wecanseethat,forabroadrangeoflaseramplitudesfroma =5toa =30,boththeamplitude
0 0
ratio e andthephaseshiftD f H betweenthetwo electriccomponentsof HHGchangeinsignificantly. Similarly, e andD f H
exhibitonlya weak dependenceonthe initialplasma densityin the rangeof50-400n (see Fig. 2(b)). Comparedwith the
c
laser amplitude and plasma density, the plasma density scale length L plays a more important role in changing the HHG
s
amplituderatioe , asshowninFig.2(c). Forlaserplasmainteractionsintheultrarelativisticregime(a 1),thedynamics
0
≫
are largely determinedby the dimensionlesssimilarity parameterS=n /a n . Because the reflection mainly occursat the
e 0 c
critical surface with a relativistic density of nRel n a , i.e., at a fixed S=1, the system is expected to be self-similar.41
c ≈ c 0
4/10
Thereforethe dynamicsofharmonicgenerationdonotdependseparatelyupona andn , butinstead uponthescale length
0 e
throughtheelectrondensityprofilene=nRcelexp−√Sx/Ls.38 Apreviousstudyhassuggestedthatthereexistsanopticalscale
lengthwhosevalueisaboutc/w ,38wherecisthelightspeedinvacuumandw isthelaserangularfrequency.Theparametric
0 0
studyhereshowsthatthe helicalHHG existsfora widerangeoflaser andplasmaparametersgiventhatthe scale lengthis
wellcontrolled.Furthermore,thefeasibilityofscalelengthcontrolisconfirmedinexperiments.38
Figure3. AttosecondhelicalXUVpulses. (a)WaveformofaCPXUVattosecondpulsetrainafterspectralfilteringwhere
the15th-20thharmonicordersareselected. HereanEPlaserpulseof30fsdurationisused. (b)WaveformofaCPXUV
isolatedsingleattosecondpulseafterspectralfilteringwherethe35th-42ndharmonicordersareselected. Hereafew-cycleEP
laserpulseof5fsdurationisused. Theotherparametersare:laseramplitudea =5;laserinitialphasef L f L=1.822;
laserincidenceangleq =40 ;plasmadensityn =200n ;plasmascalelength0L =0.1. Inbothpanels,wza−vefyormofthe3D
◦ e c s
electricfieldvector(purple),thetwoorthogonalelectricfieldcomponentsE (green)andE (blue),andtheprojectionof
y z
E E (gray)aredisplayed.
y z
−
Polarizationcontrol
Basedontheparametricstudiesabove,wechooseanoptimalscalelengthofL =0.1,amoderatelaseramplitudeofa =5,
s 0
and a plasma density of n =200n to study polarization state controllability in the HHG pulse. Figure 2(d) shows the
e c
amplituderatioandphaseshiftoftheHHG fieldcomponentsasa functionofthe laserincidenceangleq . TheHHG inthe
frequency range of the 20th-30th orders are selected, except that the 5th-10th orders for q =22.5 and the 15th-20th orders
◦
for q =40 are usedowingto a lowercutoffof well-definedharmonicstructuresatthese relativelysmall incidenceangles.
◦
Nevertheless, we found that, even when using the higher ordersof 35th-42th in the case of q =40 , elliptical HHG pulses
◦
can be generated,thoughwith a smaller ellipticity valuee . Notably, an amplituderatio of e close to unity, togetherwith a
phaseshiftoff H p /2,isobtainedattheangleofq =22.5 . Thisindicatesthatintensequasi-CPHHGpulsesaregenerated
◦
withthissimpleg≈eometry. Moreover,astheangleofincidenceq increases,thephaseshiftD f H changescontinuouslyfrom
+p /2to0,andeventuallyto< p /2.ThissignifiesthatthepolarizationstateoftheHHGpulsesvariesasq increase,froma
polarizationstatethatiscircular−(D f H=p /2)throughonethatiselliptical(0<D f H<p /2)andlinear(D f H=0),andfinally
toonethatisellipticalofoppositehelicity( p /2<D f H<0). Inthisway,therefore,apracticalandstraightforwardmethod
−
to controltheellipticityofthe HHGpulsesisproducedbysimplyadjustingtheincidenceangleofthe laserpulse, whichis
importanttoanumberofapplications.
Circularattosecondpulsesusingellipticlaser
Inadditiontoobtainingquasi-circularorhighlyellipticHHGusingCPlaserpulsesatasmall-angleincidence,hereweshow
CPharmonicsand/orattosecondXUVpulsescanalsobegeneratedusingEPlaserpulsesatanobliqueincidence.Considering
theabovecaseofq =40 inFig.2(d)forexample,theHHGamplituderatioise =0.97andthephaseshiftisD f H=0.42p .For
◦
aperfectCPattosecondXUVpulse,however,thephaseshiftmustbeD f H=p /2;wethereforecompensateforthenecessary
additionalphaseshiftofdf =0.08p byusingEPlaserpulsesthathaveaphaseshiftoff L f L=p /2+df =1.822. The
otherparametersarethesameasthoseusedin thecaseofq =40 inFig. 2(d). Thewavzef−ormyofthegeneratedattosecond
◦
XUVpulsetrainisgiveninFig.3(a),showingapulsetrainwhoseamplituderatioise =1.0andthephaseshiftisD f H=p /2.
∼ ∼
Asaresult,anattosecondXUVpulsetrainwithnearlyperfectcircularpolarizationhasbeengeneratedusingEPlaserpulses.
Thisapproachisalsoverypromisingandiseasytoimplementexperimentally.
5/10
IsolatedattosecondhelicalXUVpulse
AttosecondHHGpulsetrainshaveproventobeusefulinstudyingultrafastXUVnonlinearprocesses,suchasthephotodis-
sociation of molecules42 and in chiral experiments.1 However, an isolated single attosecond helical XUV pulse holds the
potentialfortime-resolveddichroismmeasurementswithunprecedentedtemporalresolutions.43 Forinstance,importantques-
tionsregardingthetimescaleofmagnetizationdynamicsincorrelatedmaterialsmaybeaddressedwithsuchatool.40 Here
we demonstrate how an isolated single attosecond elliptical/circular HHG pulse can be generated with the present scheme
using a few-cyclelaser pulse. Figure3(b) shows a resultingwaveformof the attosecond HHG pulse after spectral filtering
wherebythe35th-42ndordersareselected.HeretheincidentEPlaserpulsehasadurationof5fsfull-widthathalf-maximum,
an amplitude of a =5, an initial phase of D f L =1.822 and an incidence angle of q =40 , while the plasma density is
0 ◦
n =200n . ItcanbeseenthatanisolatedsingleattosecondXUVpulsewithquasi-circularpolarizationhasbeengenerated.
e c
Thisshowsthepossibilityofapplyingthistechniquetoultrafastdichroismmeasurements.
Figure4. Switchingtheharmonichandedness.Electricfieldwaveformoftheharmonicsusingalaserwithhelicityof(a)
left-handednessand(b)right-handedness.Spectralfilteringisappliedwherethe15th-20thharmonicordersareselected. The
laseramplitudeisa =5andtheincidenceangleisq =40 . Theplasmadensityisn =200n withascalelengthof
0 ◦ e c
L =0.1.
s
SwitchingHHGhandedness
Fordichroismstudyapplications,thedifferenceintheabsorptionofleft-handed(LH)andright-handed(RH)lightismeasured;
i.e.,D A(l )=A (l ) A (l )wherel isthelightwavelength.Thusitisimportanttogeneratehelicallightwithopposite
L LH L RH L L
−
handedness. Thiscan beeasily achievedby changingthe handednessof theincidentlaser pulse in ourschemebecausethe
Vlasov-Maxwellequationsaresymmetricaboutthehandednessofelectromagneticfields. Todemonstratethis,wecompared
the HHG produced by CP laser pulses possessing opposite handedness, where both laser pulses possess an amplitude of
a =5 andareobliquelyincidentattheangleof q =40 ontoa plasmaofdensityn =200n . Figures4(a)-4(b)showthe
0 ◦ e c
HHGwaveformgeneratedbyanLHlaserandanRHlaser,respectively,whereaband-passfilterwasusedtoselectthe15th-
20th harmonicorders. ItisseenthatthetwoHHGpulsesarenearlythesameexceptfortheoppositephaseshiftD f H. This
showsthefeasibilityofreversingtherotationdirectionoftheHHGpulsebysimplyswitchingthehandednessoftheincident
CPlaserpulses.
HHGefficiency
As is known, the HHG efficiency using a CP laser is low at normal or small-angle incidence.32–34 This result changes
drastically, however, as the angle of incidence increases. Experimental results by Yeung et al. have shown that at 22.5
◦
incidencetheharmonic(13th-28th)efficiencyusingaCPlaserisatleasttwoordersofmagnitudelowerthanthatusinganLP
laser.36 On the otherhand, experimentalresults byEaster etal. have shownthat at35 incidencethe harmonic(13th-19th)
◦
efficiencyusingaCPlaserisjustafactorof3lowerthanthatusinganLPlaserwiththesameharmonicorders.35 Toverifythe
dependenceoftheefficiencyupontheincidenceangle,wecarryoutaseriesofsimulations,asshowninFig.5(a),inwhichthe
laseramplitudeisa =5,theplasmadensityisn =200n andtheharmonicordersare13th-30th.Atincidenceanglesof22.5
0 e c ◦
and35 ,thesimulationresultsareinexcellentagreementwiththeabove-mentionedexperimentalresults.35,36 Inaddition,the
◦
simulationresultspredictthattheharmonicefficiencywiththeCPlaserincreaseswiththeangleofincidence,andreachesthe
6/10
Figure5. Harmonicefficiencyandplasmadynamics.(a)Influenceoftheincidenceangleupontheefficiencyofthe
harmonics(13th-30th)withCPandLPlaserpulses. Thelaseramplitudeisa =5fortheCPlaseranda =7.07fortheLP
0 0
lasertokeeptheintensityandpulseenergythesame. (b)HHGspectraofE component,comparedbetweenthecasesusing
y
CPandLPlaserpulsesat55 incidence.ThegreendashedlinecorrespondstothepredictedscalinglawI (n)(cid:181) n 8/3by
◦ ROM −
theROMtheory.(c)-(d)Spatialdistributionofelectronlongitudinal(p )andtransversemomenta(p and p )forthecaseof
x y z
aCPlaserat55 incidenceattwodifferenttimesof(c)t=28.64T and(d)t=30.40T . Inthesesimulations,theother
◦ 0 0
parametersare: laseramplitudea =5,plasmadensityn =200n ,andscalelengthL =0.1.Themomentaarenormalized
0 e c s
bym c. Thedashedblacklinesinpanels(c)and(d)markthezeromomenta.
e
samevalueaswhenusinganLPlaserat45 incidence. Astheincidenceangleisfurtherincreased,theefficiencieswiththe
◦
CP andthe LP laserstend to stay at the same level. Figure 5(b) showsthe E componentsof the HHG spectrawhen using
y
CPandLPlasersatq =55 incidence,whereitisalsoseenthattheHHGefficiencywiththeCPlaseriscomparabletothat
◦
withtheLPlaser. TheseresultsshowthepotentialofachievingefficienthelicalHHGwiththepresentscheme. Moreover,as
mentioned,intenseHHGcanbegeneratedevenwithalowefficiency,becausetheappliedlaserintensityishigh.
Regimeofvalidity
Figure5(b)againshowsthattheharmonicspectraagreewellwiththeI (n)(cid:181) n 8/3scalinglaw,whichispredictedbythe
ROM −
BGPtheory(alsotermedtheg -spikesmodel)ofROM.25Thistheoryimpliestheexistenceofzerosinthetransversemomenta
ofplasmasurfaceelectrons.44 TofurtherdemonstratethattheHHGmechanismhereiswithintheg -spikesROMregime,we
plotthespatialdistributionoftheelectronlongitudinal(p )andtransversemomenta(p and p )attwodifferenttimesfrom
x y z
thesimulationresultswiththeCPlaserat55 incidence,asshowninFigs.5(c)-5(d). Itcanbeclearlyseenthatmomentsdo
◦
existwhenthebothofthetransversemomentaoftheplasmasurfaceelectronsbecomezerosimultaneously,anditisatthese
momentsthattheharmonicsareefficientlyemitted.Inaddition,theparametricstudiesabovealsoshowthatourschemeworks
wellforabroadrangeoflaseramplitudes(a from5to30)atobliqueincidence. Theseresultssuggestthatthemechanism
0
hereisinaccordancewiththeg -spikesmodelofROM,andthusisrelevanttothestronglyrelativisticregimewitha2 1and
0≫
relativisticallyoverdenseplasmaswithS 1.
≫
In summary, a scheme to generateCP or highly-EPattosecond XUV pulses is proposedand numericallydemonstrated,
whichisbasedonhighharmonicgenerationfromarelativisticplasmamirror.Itisshownthatsuchharmonicscanbeefficiently
generatedwhenthe laser-plasmaparametersaresuitable. Inaddition, the harmonicpolarizationisfully controllablebythe
7/10
laser-plasma parameters. The scheme allows the use of a relativistically intense laser, and thus it is a promisingscheme to
achieveachiralXUVsourcewithhighbrilliance.Thisprovidesanexcitingtoolwithapplicationsinanumberoffields.
Methods
PICsimulation
We carried out all simulations using the Virtual Laser Plasma Lab (VLPL) code.45 For 2D simulations, the size of the
simulationboxis45l 70l inthex yplane,withalaserwavelengthofl =800nmandacellsize ofl /200ineach
0 0 0 0
dimension. The laser a×nd plasma para−meters are chosen to match those used in the experiments.38 The laser pulse has a
normalizedamplitudeofa =eE /m w c=30(correspondingtoanintensityof2 1021Wcm 2)andpulsedurationof30
0 0 e 0 −
×
fsfull-widthathalf-maximum,whereE isthelaserelectricfieldamplitude,eistheelementarycharge,andm istheelectron
0 e
mass. The pulse is focused into a Gaussian spot with a diameter of 2 m m, which requires a Ti:sapphire laser system that
candeliverapulseenergyofabout1J.Thelaserpulseisobliquelyincidentatanangleofq =40 ontothetarget,whichis
◦
takentobeafullyionizedplasma. Theplasmaslabhasanelectrondensityofn =100n andathicknessof500nm,where
e c
n =m w 2/4p e2. In the frontof the plasma slab, preplasma exists with an exponentialdensity profile and a density scale
c e 0
lengthofL =0.2l . Tosimulateobliquelaserincidenceinthe1Dsetup,aLorentztransformationfromthelaboratoryframe
s 0
toamovingframeofreferencehasbeenmade.24,46 Assuch,thelaseristransformedtobeatanormalincidenceontoaplasma
slabstreaminginthey-directionparalleltotheplanarsurface. Forallthe1Dsimulations,arelativelyhighspatialresolution
of1000cellsperlaserwavelengthinthemovingframeisused.
Controlofpolarization
ACPlaserpulsecanberepresentedasasuperpositionoftwoLPpulseswithequalamplitudeandaconstantphasedifference
ofp /2:E =E +E withE =E cos(w t)eˆ andE =E sin(w t)eˆ ,whereeˆ andeˆ arerespectivelytheunitvectoralong
CP y z y 0 0 y z 0 0 z y z
theyandzdirections.ThusthecorrespondingvectorpotentialcanbewrittenasA =A sin(w t)eˆ andA = A cos(w t)eˆ .
y 0 0 y z 0 0 z
−
Inthe1Dgeometry,thecanonicalmomentuminthetransversedirectionisconserved: p eA /c=constant,wherep and
A arethetransversemomentumandvectorpotential,respectively. Inthemovingframe⊥o−frefe⊥rence,theinitialmome⊥ntain
th⊥etransversedirectionsarep = m ctanq eˆ andp =0. Thenwecanobtaintheexpressionforthetransversemomentum
y0 − e y z0
as:
p = m ctanq eˆ +eA /c=( m ctanq +eA sin(w t)/c)eˆ (1)
y − e y y − e 0 0 y
p =eA /c= eA cos(w t)/ceˆ (2)
z z − 0 0 z
According to the BGP theory,25 the HHG are emitted when the transverse momentum of the surface electron p reaches
a minimum or vanishes. In the case of a laser normally incident with q =0, we observe that p =eA /c(sin⊥(w t)eˆ
0 0 y
cos(w t)eˆ ), which never vanishes or reaches a minimum, and consequently, no harmonics are g⊥enerated. The situatio−n
0 z
changeswith an obliqueangleof incidenceq , whichmakesitpossible to havep andp simultaneouslyreacha minimum
y z
orvanish. Theincidentangleq thereforeprovidesadegreeoffreedomthatcanbeusedtoadjusttherelativeamplitudeand
phasebetweenthetwocomponentsofthetransversemomentum,andthuscanbeusedtochangethepolarizationstateofthe
harmonicsgenerated.
Data availability
Thedatathatsupportthefindingsofthisstudyareavailablefromthecorrespondingauthorsuponrequest.
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Acknowledgements
Z.Y.C. acknowledgesfinancialsupportfrom the China Scholarship Council (201404890001). This work was supportedby
theDeutscheForschungsgemeinschaftSFBTR18,EUFP7projectEUCARD-2,andtheScienceandTechnologyFundofthe
NationalKeyLaboratoryofShockWaveandDetonationPhysics(China)withprojectNos. 077110and077160.
Author contributions statement
Z.Y.C.conceivedandconductedthesimulations,anddraftedthemanuscript. A.P.developedthecodeandtheory,andsuper-
visedthework.Allauthorsdiscussedtheresultsandreviewedthemanuscript.
Additional information
CompetingfinancialinterestsTheauthorsdeclarenocompetingfinancialinterest.
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