Propagation of short laser pulses in plasma channels.pdf

7/30/2019 Propagation of short laser pulses in plasma channels.pdf

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NavalResearchLaboratoryWashington,DC20375-5320

NRL/MR/6790--99-8321

PropagationofShortLaserPulsesin PlasmaChannelsP.S P R A N G L E Beam Physics Branch PlasmaPhysicsDivision

B .H A FI ZIIcarusResearch,Inc.Bethesda,Maryland

P.SERAFIM NortheasternUniversity DepartmentofElectricalEngineeringBoston,Massachusetts

March2,1999

Approvedfo rpublicrelease;distributionsunlimited.

1 9 9 9 0 4 0 14 9


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Interim4. ITLE ANDSUBTITLE

Propagation ofShortLaserPulsesinPlasmaChannels6.AUTHOR(S)

P .Sprangle,B.Hafizi.tan dP .Serafim+7.ERFORMINGORGANIZATION NAME(S)AN D ADDRESS(ES)

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11.SUPPLEMENTARYNOTES tIcarusResearch,Inc.P.O.B ox 30780Bethesda,M D 20824-0780

tNortheastern University DepartmentofElectricalEngineeringBoston,M A 0211512a.DISTRIBUTION/AVAILABILITY STATEMENT

Approvedfor publicrelease;distribution unlimited.12b.DISTRIBUTIONCODE

13 .ABSTRACT (Maximum 200words)Finitepulselengtheffectsareshowntoplayamajorroleinth epropagation,stabilityan dguidingofintenselaserbeamsinplasmas.W epresentthequasiparaxialapproximation(QPA)toth ewave equationthat takesfinitepulselengtheffectsintoaccount.T he QP A is anextension ofth e usualparaxial approximation.The laserfield isshown to be significantlymodifiedfo r pulses less thana few tens ofwavelengthslong.A pair ofcoupledenvelope-powerequationshavingfinitepulselengtheffects,as wellas relativistican datomicelectronnonlinearities,isderivedan danalyzed.Shortlaser pulsespropagatingin plasma channels arefound to undergoan envelopeoscillationinwhichth efrontofth epulseisalwaysdampedwhilethebackinitially grows.The modulationeventually dampsdu etofrequencyspreadphasemixing.naddition,initepulselengtheffectsareshowntosignificantlymodifynonlinearfocusing processes.

14.SUBJECTTERMSIntenselaserpulsesPropagation Plasma channels

Ultra-shortpulsesModulationinstability17.ECURITYCLASSIFICATIONOFREPORT

U NC L AS S I F I E D 18 SECURITY CLASSIFICATIONOF THISPAGE

U N C L A S S I F I E D 19 .SECURITY CLASSIFICATIONOF ABSTRACT

U NC L AS S I F I E D

15.UMBER OFPAGES37

16 . PRICECODE

20. LIMITATIONOF ABSTRACT

UL NSN7540-01-280-5500 StandardForm29 8(Rev.2-89)Prescribedby ANSIStd 239-18 298-102


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CONTENTS

I.ntroduction 1 II .initePulse Length M o d e l 3

a)uasiParaxialApproximation(QPA)to WaveEquation5b)inite PulseLength WaveEquation 7m. ShortPulse Propagation in Vacuum or UniformPlasma 0

a)undamentalTransverse GaussianPulseSolution 0b)roup Velocity 1IV .oupled Envelope-PowerEquations 2

a)ow power,Longpulse 3b)ow power,Shortpulse 3c)ig hpower,Longpulse 4d)ig hpower,Shortpulse 5V .aserEnvelopeModulation 5a)nvelopeOscillation 5b)aser Modulation Mechanism 6

V I.umericalIllustrations 7VII.onclusions 8

Appendix:ValidityofQuasiParaxialApproximation 0References 2 Figures 7

in


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PROPAGATIONOFSHORTLASERP U L S E S IN PLASMACHANNELS

I. IntroductionAdvancesnasertechnologyhaveresultedinanewclassofcompact,ultrashortpulse

laserswithextremely highintensities[1-3].ntensepulseshavenumerouspotentialapplicationsinareassuchasadvancedlaser-driven accelerators[4-14],harmonicgenerators[15-19],x-rayasers20,21],othershortwavelengthadiationources22],an d fastgnitor"laserusion23-25].aserechnologysno wbeingpushedouc hextremelyhortpulselengthsthatth epulseisonlyafewopticalcyclesinduration.orexample,atawavelengthof0. 8m ,Baity2]asproducedpulsesof4TWpeakpowerwithdurationsof8s~7wavelengths)withplanstoextendtheseresultsto~10fsan d>100TW .nthisregime,finitepulselengtheffectsm ay playanimportantroleinthelaserpulsepropagationdynamics.

Thepropagationdynamicsoflongaserpulseslongcomparedohewavelength)sdescribedyheell-knownaraxialav equation.nhearaxialav equationapproximation,owestorderdiffractioneffectsassociatedwithth elaserbeamar eretainedbut finitepulseengthan dhigherorderdiffractioneffectsareneglected.heolutionsohe paraxialwaveequationinvacuum areth ewell-knownLaguerre-Gaussianfunctionsthatdescribeth edynamicsoflonglaserbeams[26].he nth elaserpulselengthbecomessufficientlyshort,i.e.,lessthan~10'sofwavelengths,finitepulselengtheffectscanplayanimportantrole[27].A nexampleof thisisthepropagationofshortlaserpulsesinaguidingchannel.xtendedlaserpulseropagationn lasmahannel28-40]smportantn umberfapplications,includinghighgradientacceleratorsan dx-raylasers.

Inthispaperaquasiparaxialapproximation(QPA)isintroducedwhichisanextensionofthewell-knownaraxialpproximationoheav eequationoncludeiniteulseengtheffects. EmployingtheQP A ,apairofcoupledenvelope-powerequationssderivedfo rshortManuscriptapprovedDecember2 1,1998.


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laserpulsespropagatinginvacuum,plasmaandchannels.hemodelncludesatomicelectronan drelativisticeffects.heresultscontainedinthispaperinclude:)ananalyticalormulationofshortlaserpulsesusingheQP A,i)aderivationofapairofcoupledaserenvelope-power equations,ii) aserenvelopemodulationwhicheventuallydampsdueorequencypreadphasemixing,iv)demonstrationofsignificantmodificationofnonlinearprocessesdu etofinitepulseengthffects,nd)nalysisfhortulseropagationynamicsnon glasmachannels.heseresultsepresentne weaturesan deffectsassociatedwithhortpulseasers.Shortlaserpulseeffectshavebeenconsiderednumericallyorinon edimensionbyothers41-44],however,th emajorfindingsinthispaperwereno taddressed.orexample,du etofinite pulseengtheffects,herailingedgeofnunmatchedaserpulsepropagatingn plasmachannelsfoundtoundergoanenvelopeoscillation,whileth eleadingedgeisdamped.orasufficientlyhortpulsehemodulationwillodifyhemainbodyofhepulse.heaserenvelopemodulationsshowntobedu etothedependenceofth epulsegroupvelocityonhe spotiz ehroughheulseength.nddition,initeulseengthffectsrehownosignificantlyincreasenonlinearfocusingprocesses.

Theorganizationofthispaperisasfollows.hegeneralwaveequationforth eelectric fieldofainiteengthaserpulseropagatingnacuum,lasmasrhannels,ncludingnonlinear(atomican drelativistic)focusing effects,ispresentedan ddiscussedinSec.II .nSec.II Ithepropagationdynamicsofashortpulseinvacuumoruniformplasmaisderived. InSec.IV ageneralpairofcoupledenvelope-powerequationsderived.heseequationsareusedoanalyzehepropagationofahortaserpulsen preformedplasmachannelwithnonlinearfocusingeffects.ariouspropagationlimitsarediscussedandthelaserenvelopemodulationisanalyzed. TheaserenvelopemodulationsdiscussednSec.V . Numericalllustrationsarc


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presentedinSec.V Ian daconclusionisgiveninSec.VII.ntheAppendixth erangeofvalidity ofth eQPA isobtainedintw olimitingcases.

II . FinitePulseLength ModelThe propagationmedium isavacuum,plasmaor preformedplasmachannelconsistingof

freean dboundelectrons,.e.,artiallytrippedplasma.onlinearprocessesrisingro m relativistican datomicpolarizationeffectsareincluded.hewaveequationincludesaplasmacurrentconsistingoffreeelectronsan d polarizationcurrentarisingro mheboundatomicelectronsan disgivenby[31,45,46],

vi+ i -\ dz' d t < Anc

tip a2pn\+ & dt' (1 )whereV^sth etransverse Laplacian,E(r,t)isth eelectricfield,Jpisth eplasmacurrentdensityassociatedwithhere eelectronsan dPshepolarizationieldassociatedwithheboundelectrons. The atomic polarization field consists of a linear an d nonlinear part,P=(l/47r)(?7o-1+2 7 7 o 7 72/E,here j osheinearndex,|2shenonlinearefractiveindexan dI=(c/4jr.)r|oEE)sthetimeaveragedlaserintensity. Inth epresentmodel,he originofhenonlinearndex|2sheanharmonicpotentialnwhichheboundelectronsoscillate.otingthat3J Idt 47t)~lcop(r)Ean dsetting rj0=1 ,th ewaveequationbecomes

vi+ dz' c2t: (02{r) \+/3 E=0, (2)


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whereC D , ,(r)=(47cq2np(r)/m)I/istheplasmafrequencyandn,,(r)istheplasmadensity.nE q.(2 )thecoefficientofthenonlinearterm =, ,+adenotesrelativisticfreeelectioneffectsaswellasnonlinearatomicelectroneffectsarisingfromrj2,

h--I ffay^ 2ylllC j 0) V(3a)=0n2 , b )

wherec op0 (4Ttq2npQIm)112,p0isth edensityonaxis(r =0) ,an dC O isthecharacteristiclaserfrequency.nE q.(3),presultsinacriticalpowerfo rrelativisticfocusing[31,47-50]whileth earesultsnacriticalpowerfo ratomicelectronocusing31,51-53].napartiallytrippedplasma,atomiceffectsca noccuron im ecale~0 " 15ecan dca ndominatere eelectroneffectsinthenonlinearterm[32,45,46].he nonlinearterm,du etorelativisticfreeelectrons,isonlysignificantwhenhelaserpulselengthsongerthan plasmaperiod54].hecriticalpowersfo rrelativisticfocusinginplasmaan dnonlinearfocusinginaga sare,respectively,

Pp=2c(q/re)2(0)/cop)2, 4a )and

Pa=X2I{2KT]2), 4b )wherereistheclassicalelectronradius.ngeneral,whenthelaserpowerexceedseitherof thesecriticalowers,ocusingoccurs31,45,46].heotalonlinearocusingpowerconsistsofcontributionsfrom bothPpan dPaandisgivenby


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The preformedplasmadensitychannelistakentobeafunctionofradialpositioninorderto providefo rguiding ofth elaserpulse.heradialdependence of th eplasmafrequencyisgivenby

(Op(r)= c op0 1 + An2\1/2

(5 )t np0c*

wheren p o +Anisth edensityatth eedgeoftheplasmachannel(r =rc).or guiding,theplasmadensitymustincreaseasafunctionofr,i.e.,A n>0.

Laserinducedplasmawaves,wakefields[55,56],areneglected.hisca nbe justifiedifth elaserpulselengthislessthanaplasmaperiod or ifthelaserintensityissufficientlylow.

a) QuasiParaxialApproximation(QP A)toWaveEquation Thelaser electricfieldisofth eform

E=E0exp(/(fe-0)t))/2+ ex., 6) where0(r,t)sheomplexmplitude, sheavenumberndOsherequency.SubstitutingEqs.(5 )an d(6)intoEq.(2 )givesthewaveequationfo rE0,

Vt +2/ dz c2 & 9zz cLt9E0=0, (7 )whereK =(cOpo/c)(An/npo)1/2isthefocusingparameterassociatedwiththeplasmachannelan d

=< u2Ic1-(02QIC2) Changingvariablesfrom(z ,t)to(z)where=z-9 wehavese tk=( 0 Ic -c opQ 2x1/2 ripCt,andsettingk=r ip C / c ,wherer|p=1-C lp 0 l(Ly11sth elinearon-axisplasmarefractive

index,E q.(7)becomes


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^2)2

3z. E0=0. (8 )The secondtermontheleft-handsideofEq.(8 )representsfirstorderdiffractioneffects,th ethirdtermdenotesfirstorderfinitepulseeffects,thefourthan dfifthtermsdenotehigherorderfinite pulseanddiffractioneffectsespectivelywhileheastwoepresentguidingan donlinear focusing,respectively.

Inth eabsenceofchannelguiding(K c=0)an dnonlinearfocusing( =0)weca nobtain anestimatefo rth eorderofmagnitudeofthevarioustermsinEq.(8).he second,third,fourthan dfifthtermsinE q.(8 )ar eapproximatelyof order

2 T ]p{(0lc)\dldz\ .2'r0 a 2dzdt; ~ r )T O ]p Qr2'0

Mlsi- C O ,3_2X c o U)2 .2'(9a)

(9b)

(9c)r0

a2dz2

( x f2TO]pr ,2 (9d)ID

respectively,wherer0andQrethespotsizean dpulselength. Inobtainingth eestimatesnEq.9)weusedd/d^\ /Qndd/dz\ /ZR,where R=T)0 2/l isheRayleighlengthntheplasmaandXisthevacuumwavelength. Therelativeorderofmagnitudesofthefirst,second,third,fourthan dfifthtermsinEq.(8 )is

1 : 1 mip iQx_ i-nPR _j_

4itilp (00 ' Anup ZK (10)


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Thefirsttw otermsinthewaveequationarecomparablean dleadtoth eparaxialwaveequationapproximation.nth eparaxialapproximationfinitelengtheffectsan dhigherorderdiffraction areneglected,i.e.,th ethird,fourthan dfifthtermsinE q.(8 )areneglected. TheparaxialwaveequationassumesthatX/01 ,1-Vll)(.ZR/0)X/0,an dX /ZRan disgivenby

C O d\Vi+H p-f-0=0r cz (11)Solutionsofth eparaxialwaveequationarethewell-knownLaguerre-Gaussian functions[26].

Whenth eRayleighlengthislargecomparedtoth epulselength,ZR0he higherorderdiffractionterm can beneglectedcomparedtoth efinitepulselengthtermsresultinginth efollowingwaveequation

(V+2 i c d 2 ^ +(l-T]i)-&d2 En=0. (12)Equation(12)containsfirstorderdiffractionan dfinitepulselengtheffectsan dreducestotheparaxialequation,E q. (11),whenth epulselengthismuchlongerthanth ewavelength,0X .

b) FinitePulseLengthW a veEquation Finitepulselengtheffectsarerepresentedbytheterms dzdt;nd3dt;nE q.

(12).hesetermsundercertainconditionsca nbesimplifiedusingth eQ P A ,allowingforth eanalyticalsolutionfo rth efield.oanalyzefinitepulselengtheffectsweassumethatthegeneralsolutionofE q.(12)fo rth efundamentaltransverse Gaussiancomplexamplitudeisgivenby

E0=bexp\i(p-( 1+ie)r2lr}J e, 13 )


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whereb,p ,6an drsarerealunctionsofzand^=z-iPctan d _saunitransversevectordefininghepolarization.nE q.13),bsheamplitude,pshephase,9selatedohewavefrontcurvature,an drsisthespotsizeof thelaserpulse.The centralassumptioninth eQPA isthathemaincontributionromheiniteengthermwillcomero mhe^dependencenhe initial amplitude. Hence, we make the approximations dE0/ , s(3f/j(60)/d)Eo an da2E0/32= l$2n(b0)/d! ;2 (Mn(b0)3)2JE0,hereb0g= b(z 0,).nheAppendixth eQ PA willbeshowntobe wellsatisfiedforabroadrangeof parameters.mploying th eQ P A ,E q.(12),includingth eguidingterman drelativistic/atomic electronnonlinearities, becomes

Vi+2itlp*l fe())|--\r?(\-V2g(0~K?r2/r2 +E0.E* 0/ En=0, (14a)

where4^&)), 14b)c o r ] _ a^

an dS(0:-- 14c)(OT]p d%

Finitepulseengtheffectsareepresentedbyheunctions. ( , )ndg(,). IftheaserpulseamplitudehasaninitialGaussianlongitudinalprofile,~exp(-4


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g) 2 { f i )'o\p*ot (15b)The functionsan dghavemagnitudesmuchlessthanunityinth evicinityof th elaserpulse,i.e.,| |


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HI. ShortPulsePropagationinVacuum or Uniform Plasmaa) FundamentalTransverseGaussianPulseSolution

InhisectionweobtainnddiscussheinitepulseengtholutionoEq.14a)by solvingEqs.16a,b)intheabsenceofguiding(R ,n > < * > )an dnonlinearfocusingeffects(P c-> o).he solutionofE q.(16)inauniformmediumwithrefractiveindexr ]pis

-1,b(Z,Z)= b0(Z)R-l(Z,Z) l+2( )xl/2

1+()(Z+())

(p(Z,Z)=-tan- Z+())+tan"1(())

0(Z,)

ex p (g)()

l+ 2(0Z,

(l+()(Z+()))'f . ,_ -,.,2 V/2

R{Z) 1+(Z+e&Y1 + ( ) (Z+ ( ) )an dth ewavefrontradiusofcurvatureis

RC(Z,)=R2ZR/0(Z,)=(ZR/Z)(l+(Z +(Z,))2)

(17a)

(17b)

(17c)

(17d)

(17e)Inheparaxialimit,e(^) >0,heunctionsb,p ,6,Ran dRenEqs.17)educeohe conventionalexpressionsfo rth efundamentaltransverse Gaussianbeam[26]

b(Z)= b0/R(Z) 18a)

(p(Z)=-tan 'Z 0(Z)=-Z,R(Z)= (\+Z2)'2,RC(Z)= (ZR/Z)(\ ZZ).

(18b)(18c)(18d)(18c)

10


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F or propagationinvacuumwese tr|p= nEqs.(17)an d(18).

b) GroupVelocity

Tocorrectlyobtainhegroupvelocitytsnecessaryousehequasi-paraxialwaveequationnwhichfinitepulselengtheffectsareincluded.heparaxialwaveequationnE q. (11)doesno tcontainfinitepulselengtheffects;itisvalidfo rinfinitelylongbeams.inceE q. (11)isindependentofth evariablewhichdescribesthepulseshape,finitepulselengtheffectsca nbeincludedinanadho cmannerbysimplymultiplyingth eparaxialsolution byanarbitrarypulseenvelope.ngeneral,th egroupvelocityisth evelocitywithwhichth epeakofth epulseenvelopetravels,i.e.,th evelocityfo rwhichdb(Z,t,)/dE,-0.nth eparaxialapproximationthegroupvelocityisfoundtobevg=cr\p= c(ck/a>)wherew ehavetakenth epulseamplitudeto beb0(E , ) -b0exp(-42//Q)an db0sth epeakamplitude.nthisapproximationth eproductofth egroupan dphasevelocitiesisc

Inth equasi-paraxialapproximation,whichincludespulselengtheffects,fromE q.(17a)th epulseenvelopeca nbe writtenas

b{Z^)= b )\-e{ )ZI{\+ Z2)\, 19 )whereb0isth eamplitudeinth eparaxialapproximationan dtheterm proportionaltoeisth eQ PA correctionduetofinitepulselengtheffects.quation(19)iscorrecttoorderendisvalidfo reZ 1 .Inth epresenceoffinitepulselengtheffectsthegroupvelocity isfoundtobe

vg{t)~cr\t 1 X i-cVizl),*VJl + cV/z^)2 crip X nripr (20)1 1


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an dR =rs/r0,Z =z/ Z R.nobtainingEqs.(21)higherorderfinitelengtheffectscontainedina,whichareofordere2 ,havebeenneglected.heoupledenvelope-powerequationsorhe guidedpulsegivenbyEqs.2 1)arenonlinearunctionsofspotizean dcontaininitepulselengtheffects.everallimitingcases,dependingonthelaserpoweran dpulsedurationwillbe discussed.nheollowing,owaserpowerefersopowersmuchesshanhenonlinearfocusingpower,PLPc-

a)ow power,Long pulse(P 1,8 =0) ,In th elow power,long pulselimit,th eenvelopeequationreducesto

dz2 mTheecondtermnE q.2 2)denotesplasmachannelocusingwhileheas ttermepresentsdiffraction.nthislowpower,longpulselimit,thebeam ha san equilibrium spotsizeR ,=Rm.

b)ow power,Shortpulse(P


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andth epulsegroupvelocitysgivenbytheexpressionnE q.20). Theeffectoffinitepulselength(e 0)istomakethepeakofthepulseshiftbackwardasthepulsepropagates.

c) Highpower,Longpulse(P 24b)< peqG,Z)=-(l-Pq/)Z/R?q, 24c)

whereP^=Po()=(Pmax/Pc)(&n()^o)Smeinitialnormalized laserpowerasafunctionof^. Inthislimitthelaserpowerdoesno tevolvewithdistance,i.e.,P e q =Po()isindependentofZ.F or peaklaserpowerslessthanth ecriticalpower,Po()


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d) Highpower,Shortpulse(P


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where8R0=8 R (Z=0) ,dWdZ=0atZ=0and(/=0)salwaysdampedwhilenheback2^|^|/(3^p/l).

b) LaserModulationMechanism Themechanism fo rth eenvelopemodulationca nbe understoodby notingth e

relationships betweenth egroupvelocity, spotsizean dpowerofapulsepropagating inaplasmachannel.hegroupvelocityofapulseinaplasmachannelca nbe writtenasvg=V g o +5vg,whereth emeangroupvelocityvgo isgivenby E q.(20),theperturbed groupvelocityis8vg=c ( A / 7 r , r o ) 2r/roan d8risth eperturbed spotsize.olowestorder,conservationofpowerimplies8b =-bo8r/ro,where8ban dboareth eperturbedan dunperturbed laserfieldamplitudes,respectively.igure (a,b)showsth eamplitudeandspotsizeofafinitepulselengthlaserinareferenceframemoving withth emeangroupvelocityvg0.he solidcurveshowstheequilibrium amplitudeandspotsizeasafunctionofz-vg0t.f thespotsizeisuniformlyincreased (8r>0)alongth epulse,thegroupvelocityincreasesbytheamount8vg.he amplitudeinfrontof theunperturbedpulseincreases(S b>0) whiletheamplitudeinbackoftheunperturbedpulsedecreases(8 b< 0) .onservationofpowerindicatesthattheperturbedspot

16


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sizeinbackofthepulseisfurtherincreasedsince8r=-r08b/b0>0an ddecreasedinfrontofthepulsesince8r=-r08b/b0< 0.f,instead,initiallythespotsizewereuniformlydecreased,th espotsizeatthebackwouldbe furtherdecreasedwhileinth efrontitwouldincrease.ence,th eperturbedspotsizeisdampedinfrontofthepulsean disunstableinth eback.ubstituting =z-vgt=z -vg0t-5vgt=ô-8v gtintob~b0exp(-4^ 2/ô).he rateofchangeofth eperturbedamplitudeisdb/dt Sb006vg11\.sing8r=-(ro/b0)8b,w efinddr/dt=-8c0(A/;cr0)2< 5r/ôwhichagreeswithth egrowthterm inE q.(29).nherentto afinitelaserpulseisafrequencyspreadgivenby S O D ~d0.enceth eenvelopemodulationfrequencyA=2c/Z RacquiresaspreadSQ-Oeco/)~Q.e(X/27il0).his envelopefrequencyspreadresultsinphasemixing ofth emodulationinadistanceZ d(normalizedto ZR).

VI. NumericalIllustrationsFigure2isaplotofth enormalizedspotsizeR(Z)inE q.(17d)asafunctionoffo r

variousvaluesofZ =0,1 ,2 ,3.nthisfigureth elaserpulsepropagatesinvacuuman dth epulselengthis0=6X .nth eabsenceof finitelengtheffectsth espotsizewouldbeindependentof.igure2indicatesthatthetailofth epulseflaresou tmorethanth efrontofthepulse,leading toa"trumpet"pulseshape.

InFigs.3- 6th elaserpulseparametersareX = im ,c O p / c o 10 2 0\im(6 7fs)an dthepeakpowerisP p e ak=0.56Pc. Thetotalnonlinearfocusingpower,Pc,isgivenbyEq.4c )andconsistsofcontributionsfromfreean datomicelectrons. Inallth efiguresthereisaninitial

1 7


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mismatchn(liespotsizecomparedtotheequilibrium spotsize,i.e.,R ()=R(0,0)= an dR ,=1.15.igure3(a)showsthespotsizeR(Z,,)asafunctionofZ=z/ Z Randt/X,withfinitepulselengtheffectse- t -0)ncluded.orcomparisonFig.3(b)howsheam eplotexceptnhe absenceoffinitelengtheffects(e=0) .he laserenvelopemodulationisclearlyseeninFig.3(a)wherethespotsizeoscillationsatthefrontofthepulse( ,>0)aredampedandinth eback( ,< 0)grow.initepulseeffectsnotonlyesultnanenvelopemodulationbutalsoignificantly enhancenonlinearfocusing.hisisshowninFig.4whereth espotsizewithfinitepulselengtheffects(solidcurve)approacheszeroatIjX= -3fo rZ =15.hespotsizewithoutfinitelengtheffects(dottedcurve)showslessthana10% decreaseat ,~0fo rZ =15.igures5(a)an d(b)showth elaserpulseamplitudeb(Z,^)asafunctionofZan dt/Xwithan dwithoutfinitepulselengtheffects,respectively.saresultoftheenhancednonlinearfocusingduetoth efinitepuleslength,Fig.5(a)showsasignificantincreaseinthepulseamplitudeatZ =15comparedtoFig.5(b).igure6showsth epulsepowerasafunctionoft/Xan dradialcoordinater/roatZ=15.FinitelengtheffectsinFig.6(a)resultinanincreaseinthepeakpoweraswellasadistortionofthepulsecomparedtoFig.6(b),wherefinitelengtheffectsareabsent.nFig.6(a),finitelengtheffectsreduceth epulsepropagationvelocity,i.e.,peakofth epulseoccursatnegativevaluesof.nFig.6(b),nonlinearocusingeffectsarencludedwhileinitepulseengtheffectsareneglected,8=0.or e=0thepulsevelocityiscan dnonlinearfocusing issubstantially reduced.

VII. ConclusionsThencreasingus eofultrahortaserpulsesnanypplicationsequireshathe

paraxialwaveequationbeextendedtoincludefinitepulselengtheffects.W epresentthequasi18


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paraxialapproximationQP A )ohewaveequation.heQ PAsnextensionofth eusualparaxialapproximationan dakesinitepulseengtheffectsntoaccount. pairofcoupledenvelope-powerequationsisderivedfo rshortlaserpulsespropagatinginvacuum,plasmasan dpreformedplasmachannels.hemodelncludesatomicelectronan delativisticeffects.efindthatfinitelengtheffectsca nsignificantlymodifythelaserfield.henewresultsinclude:i)annalyticalormulationfhortaserpulses,i)derivationfapairofcoupledaserenvelope-powerequations,ii)aaserenvelopemodulation,v)demonstrationof ignificantmodificationofnonlinearprocessesbyfinitepulselengtheffects,an dv)analysisofshortpulse propagationdynamicsinlongplasmachannels.

1 9


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Appendix:alidityoftheQuasiParaxialApproximation fTheapproximation,whicheadsoheimplifiedwaveequationnEq.14a),equires

that

En N ,e+ T ]pD% Er (Al)whereEoan dearegivenby E q.(13)and(14b).he shortpulseapproximationrequiresthatEq. (Al)besatisfied.

a) ApproximationinVacuumFor pulsepropagationinvacuum,T \P= ,th einequalityinE q.(Al)ca nberewrittenan d

th eapproximationisshowntobe validif

t \In 1 +i(Z+e) 1+ (r/r0Y1 +i(Z+ e) ( A 2 )wherewehaveassumed|e|1.hefinitepulseengthapproximationnvacuum,usedoreplaceBEQIB!;ith-{aIC)E{^)EQnE q.(14a),iswellsatisfiedeverywhereexceptwithinasmallfunctionofawavelengthof thepulse'scenter.

b) ApproximationinGuidingChannelTodeterminehevalidityofhehortpulseapproximationnguidinghannele

considerthelowpower,shortpulselimitofE q.21),.e.,imitb)inSec.V .hefieldinEq.(13)fo rtheequilibriumsolutiongiveninEqs.(23)is

Eeq=bo(0 p[-Z/Rl-ir2/r0Rm)2\ A 3)

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ubstitutingeqor0nq.Al)ieldsheonditionorhePAoealid. The approximationsvalidprovided |e|c/corjp)\dzld^\ziR}, whichor pulsehavingGaussianlongitudinalprofileis

ii- z. A 4 ,TheQ PA approximationisno tvalidfo rlongpropagationdistances.

Acknowledgment Theuthorscknowledgeiscussionsith.F .ubbard,.ing,.iglernd

acknowledgeth eassistanceofJ.Penanoinpreparingthecomputergraphics.hisworkw as supportedbyth eOfficeofNavalResearchan dth eDepartmentofEnergy.

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FigureCaptions

Fig. Illustrationofthephysicalmechanismforthelaserenvelopemodulation.Theamplitudeisshownna)an dth espotsizein(b),asafunctionof$=z-V g o t ,wherevg0isth emeangroupvelocity.he solidcurvescorrespondtotheequilibrium.he dashedcurvesshowth eamplitudean dspotsizefo rthecasewithgroupvelocitylargerthanvg0(1 )an dth ecasewithgroupvelocitysmallerthanvg0(2).

Fig.2lo tofnormalizedspotizeR(Z)asafunctionofforapulseoflength0=6Xpropagatinginfreespace.hefourcurvescorrespondtonormalizedaxialpointsZ=0(lowestcurve),1,2,an d3.

Fig.3 SurfaceplotsofspotsizeRsafunctionofB,IXndpropagationdistanceZ= z/ZRwitha)initepulselengtheffectse*0)an d(b)finitepulselengtheffectsneglected(e= 0).he parametersareX = l]im,>0=20/im, p e a k = 0.56P.

Fig.4 Plotofpotizeas unctionof/X afterapropagationdistancequalo5RayleighengthsZ=15 ) Theoliddotted)urvencludesneglects)initepulselengtheffects.arametersareth esameasinFig.3.

Fig.5 SurfaceplotsoflaserpulseamplitudebasafunctionofE,lXndpropagationdistanceZ= z/ZRwith(a)finitepulselengtheffectse*0)an d(b)finitepulselengtheffectsneglectede=0).arametersarethesameasinFig.3.

Fig.6 SurfaceplotsoflaserpulsepowerPasafunctionoft/Xan dradialcoordinater/ r0afterapropagationdistanceequalo5RayleighengthsZ=5). Ina)initepulesength

|ffectsareincludedan dshowenhancedfocusingan ddecreasedpropagationvelocity,i.e.,27


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peakofthepulseoccursornegativevaluesof. Inb)initepulseengtheffectsarc Jneglected,.e., =0whilenonlinearfocusingeffectsarcncluded. Parametersarche sameasinFig.3.

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SpotSizeR(Z)

back yx front

Fig.230


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SpotSizeR(Z) front

SpotSizeR(Z) front

Fig.3 31


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SpotSizeR(Z)

Z=15

back yx front

Fig.432


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PulseAmplitudeb(Z) fron

PulseAmplitudeb(Z) fron

Fig.5 33


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PowerP(Z)

front yk

(a)

back

PowerP(Z)

(b)

Propagation of short laser pulses in plasma channels.pdf

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