propagation of short laser pulses in plasma channels.pdf
TRANSCRIPT
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
1/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
2/37
REPORTDOCUMENTATIONPAGE FormApprovedOMBNo.0704-0188Publicreportingburdenfor thiscollectionofinformationsestimatedtoaverage ou rpe rresponse,includingthe timeforreviewinginstructions,searchingexistingdatasources,gatheringan dmaintainingthe dataneeded,an dcompletingan dreviewingthe collectionofinformation.Sendcommentsregardingthisburdenestimateoranyotheraspectofthiscollectionofinformation,includingsuggestionsforreducingthisburden,toWashingtonHeadquartersServices,DirectorateforInformationOperationsan dReports,1215JeffersonDavisHighway,Suite1204, Arlington,VA 22202-4302,an dtothe Officeof Managementan dBudget,PaperworkReductionProject(0704-0188),Washington,DC 20503,1 .AGENCYUSEONLY{LeaveBlank) 2.EPORT DATE
March2,19993.EPORT TYPE ANDDATESCOVERED
Interim4. ITLE ANDSUBTITLE
Propagation ofShortLaserPulsesinPlasmaChannels6.AUTHOR(S)
P .Sprangle,B.Hafizi.tan dP .Serafim+7.ERFORMINGORGANIZATION NAME(S)AN D ADDRESS(ES)
NavalResearchLaboratory Washington,D C20375-5320 9.PONSORING/MONITORINGAGENCYNAME(S)ANDADDRESS(ES)
OfficeofNavalResearch 80 0NorthQuincyStreetArlington, V A 22217-5660 U.S.Department of EnergyWashington,D C20585
5.UNDINGNUMBERS67-0899-09
8.PERFORMINGORGANIZATION REPORTNUMBERNRL/MR/6790-99-8321
10.SPONSORING/MONITORINGAGENCYREPORTNUMBER
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
3/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
4/37
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.
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
5/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
6/37
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)
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
7/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
8/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
9/37
^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)
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
10/37
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 )
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
11/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
12/37
g) 2 { f i )'o\p*ot (15b)The functionsan dghavemagnitudesmuchlessthanunityinth evicinityof th elaserpulse,i.e.,| |
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
13/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
14/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
15/37
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
16/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
17/37
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()
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
18/37
d) Highpower,Shortpulse(P
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
19/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
20/37
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=^o-8v gtintob~b0exp(-4^ 2/^o).he rateofchangeofth eperturbedamplitudeisdb/dt Sb006vg11\.sing8r=-(ro/b0)8b,w efinddr/dt=-8c0(A/;cr0)2< 5r/^owhichagreeswithth 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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
21/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
22/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
23/37
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)
20
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
24/37
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.
21
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
25/37
References1 ..A .Mourou,C.P.J.Barty,an dM. D.Perry,Phys.Today51,22(1998).2..P.J.Barty,LaserFocusWorld,Juneissue,93(1996).3..D .Perryan dG .Mourou,Science264,917(1994).4..Sprangle,E.Esarey,an dJ.K rall ,Phys.Plasmas3,2183(1996).5..Esarey,P.Sprangle,J.Krall ,an dA .Ting,IEEE Trans.PlasmaSei.24,2 52 (1996).6..A .Edighofferan dR.H.Pantell,J.Appl.Phys.50,6120(1979).7..C .Huang,D.Zheng,W . M .Tulloch,an dR.L.Byer,Appl.Phys.Lett.68 ,75 3(1996);
Y.C.Huangan dR .L.Byer,ibid.69,2175(1996).8..Sprangle,E.Esarey,J.Krall ,an dA .Ting,Opt.Commun.124,69(1996).9..Esarey,P.Sprangle,an dJ.Krall ,Phys.Rev.E 52 ,5443(1995).10..Hafizi,A .Ting,E .Esarey,P.Sprangle,an dJ.Krall ,Phys.Rev.E 55 ,5924(1997);B .
Hafizi,E .Esareyan dP.Sprangle,Phys.Rev.E 55 ,3539(1997).11..Nakajima,D.Fisher,T.Kawakubo,H.Hashaniki,A .Ogata,Y.Kato,Y.K itagawa,R .
Kodama,K .Mima,H.Shiraga,K .Suzuki,K .Yamakawa,T.Zhang,Y.Sakawa,T.Shoji,Y.Nishida,N .Yagami,M .Downer,an dT.Tajima,Phys.Rev.Lett.74 4659(1995).
12..E .Clayton,K . A .Marsh,A .Dyson,M .Everett,A .Lai,W.P.Leemans,R .Williams, an dC.oshi,Phys.Rev.Lett.70,371993);C.E.Clayton,M .J .Everett,A .Lai,D.Gordon,.A .arsh,nd.oshi,hys.lasmas,7531994);.odena,.Najmudin,A . E.Dangor,C.E.Clayton,K . A .Marsh,C.oshi,V .Malka,C.B.Darrow,an dC.Danson,IE E E Trans.PlasmaSei.PS-24,2 89(1996).
13..Umstadter,S.Y.Chen,A .Maksimchuk,G .Mourou,an dR.Wagner,Science273,47 2 (1996).
22
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
26/37
14..Ting,C.I.Moore,K .K rushelnick,C.Manka,E.Esarey,P.Sprangle,R .Hubbard,H .R .Bunis,an dM .aine,Phys.lasmas4,88 91997);C.I.oore,A .Ting,.K rushelnick,E.Esarey,R.F.Hubbard,B .Hafizi,H. R .Bunis,C. Manka,an dP.Sprangle,Phys.Rev.Lett.79 ,3909(1997).
15..Zhou,J.Peatross,M . M .Murnane,H.C.Kapteyn,an dI.P.Christov,Phys.Rev.Lett.76,752(1996).
16..M .Milchberg,CG.DurfeeIII,an dT.J.Macllrath, Phys.Rev.Lett.75,2494(1995).17..Spranglean d E.Esarey,Phys.Rev.Lett.67 2021(1991).18..Esareyan dP. Sprangle,Phys.Rev.A .45 ,5872 (1992).19..Esarey,A .Ting,P.Sprangle,D.Umstadter,an dX .Liu,I EEE Trans.PlasmaSei.2 1, 95 (1993).2 0..C .Eder,P.Amendt,L .B .DaSilva,R .A.London,B .J .M a c G o w a n ,D.L.Matthews,
B . M .enetrante, .D .osen,.C . ilks,T.D.onnelly,R . W .alcone,ndG.L .Strobel, Phys.Plasmas1,1744(1994).
2 1..Suckeweran dC.H.Skinner,CommentsA t.Mol.Phys.30 ,331(1995).2 2 ..E .Lemoff,G.Y .Yin,C.L.GordonIII,C.PJ.Barty,an dS.E.Harris,Phys.Rev.Lett.741574(1995).
2 3..Tabak,.Hammer,M . E .Glinsky,W . L .ruer,S.C.Wilds,.Woodworth,E . M .Campbell,M .C .Perry,an dR.J.Mason,Phys.Plasmas1,1626(1994).
2 4..Deutsch,H.Furukawa,K .Mima,M .Murakami,an dK .Nishihara,Phys.Rev.Lett.77 2483(1996).
2 5..Kodama,K .Takahashi,K . A .Tanaka,M .Tsukamoto,H.Hashimoto,Y.Kato ,an dK .M ima ,Phys.Rev.Lett.77 4906(1996).
23
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
27/37
26..E .Siegman,Lasers(UniversityScienceBooks,MillValley,CA ,1986).2 7..Esarey,P.Sprangle,M .Pilloff,andJ.Krall,J.Opt.Soc.A m.B12 ,1695(1995).28..Borghesi,A.J.McKinnon,L.Barringer,R.Gaillard,L.A.Gizzi,C.Meyer,O.Willi,
A .Pukhov,andJ.Meyer-ter-Vehn,Phys.Rev.Lett78 ,379(1996).29..Borghesi,A.J.Mackinnon,R.Gaillard,O.Willi,andA . A .Offenberger,Phys.Rev.E
57 ,R4899(1998).30..Braun,G .Korn,X .Liu,D.Du, J.Squier,an dG .Mourou,Opt.Lett191544(1995).31..R .ange,G .Grillon,.-F.Ripoche,M . A .ranco,B .amouroux,B .S .Prade,.
Mysyrowicz, E.T.J.Nibbering,andA .Chiron,Opt.Lett.2 3,12 0(1998).32 ..Sprangle,E.Esarey,an dJ.Krall ,Phys.Rev.E 54,4211(1996).33..G .DurfeeIII,T.R.Clark,an dH . M .Milchberg, J.Opt.Soc.A m.B13,59(1996).34..M .ilchberg,.R .lark,.G .urfeeII ,.M .ntonsen,nd.ora,hys.
Plasmas3, 2149(1996).35..R .Clarkan dH . M .Milchberg,Phys.Rev.Lett.78 ,2773(1997).36..K rushelnick,A .Ting,C.I.Moore,H.R.Burris,E.Esarey,P.Sprangle,an dM .Baine,Phys.Rev.Lett.78 ,4047(1997).37..Zigler,Y.Ehrlich,C.Cohen,.Krall,an dP.Sprangle,.Opt.Soc.A m .B1368
(1996).38 ..Ehrlich,C.Cohen,A .Zigler,J.Krall ,P.Sprangle,an dE.Esarey,Phys.Rev.Lett.77,
4186(1996).39..K aganovich,P.V.Sasarov,Y.Ehrlich,C.Cohen,andA.Zigler,Appl.Phys.Lett.71 ,2295(1997).
24
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
28/37
40 ..hrlich,C.Cohen,D.aganovich,.Zigler,R.F.ubbard,P.prangle,an dE .Esarey,J.Opt.Soc.A m .B15,2416(1998).
41 ..L .Chenan dR . N .Sudan,Phys.FluidsB 5,133(1993).42 ..B .Mori ,CD.Decker,D.E.Hinkel,andT.K atsouleas,Phys.Rev.Lett.72 ,48 2 (1994).43 .D.Decker,W . B .Mori ,andT.K atsouleas,Phys.Rev.E 50 ,R3338(1994).44 ..M o r aan dT . M .Antonsen,Phys.Plasmas4,217(1997).45 ..Sprangle, E .Esarey,an dB .Hafizi,Phys.Rev.Lett.79 ,1046(1997).46 ..Sprangle,E .Esarey,an dB .Hafizi,Phys.Rev.E 56, 5894(1997).47.E.M a x ,J. Arons,an dA . B .Langdon,Phys.Rev.Lett.33 ,2 09 (1974).48 ..G .Litvak,Z h. Eksp.Teor.Fiz.57,629(1969)[Sov.Phys.J E T P 30 ,34 4(1969)].49 ..Sprangle,CM.Tang,an dE .Esarey,EEETrans.PlasmaSei.15 ,145(1987).50..Schmidtan dW .Horton,CommentsPlasmaPhys.Control.Fusion9,85(1985);G.Z.
Sun,E .Ott,Y.CLee,an dP.Guzdar,Phys.Fluids30 ,52 6(1987);A . B .Borisov,A . V .Borovskiy,O .B .Shiryaev,V . V .Korobkin,A . M .Prokhorov,J.CSolem,T.S.Luk,K .Boyer,an dC . K .Rhodes,Phys.Rev.A 45 5830(1992).
51..F .Reintjes,NonlinearOpticalParametricProcessesinLiquidsandGases(Academic,Orlando, FL ,1984).
52 ..R .Shen,T he PrinciplesofNonlinearOptics,(Wiley,N ew York,1984).53..W .Boyd,NonlinearOptics(Academic,S anDiego,1993).54..Sprangle,E .Esarey,andA .Ting,Phys.Rev.Lett.64 ,20111990).55 ..Sprangle, E .Esarey,A .Ting,andG .Joyce,Appl.Phys.Lett.53 ,2146(1988).56..CTzeng,W . B .Mori ,an dT.K atsouleas,Phys.Rev.Lett.79,5258(1997).
25
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
29/37
57. NotethatsimplyequatingpowersofrinE q.(14a)resultsincriticalpowersthataresmall ^byafactorof2. Equation16 )sobtainedbyamoreproperapproachusingthesourcedependentexpansionmethod,P.Sprangle,A .TingandCM.Tang,Phys.Rev.Lett.59,2 02 (1987);Phys,Rev.A36,2773(1987).
26
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
30/37
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
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
31/37
peakofthepulseoccursornegativevaluesof. Inb)initepulseengtheffectsarc Jneglected,.e., =0whilenonlinearfocusingeffectsarcncluded. Parametersarche sameasinFig.3.
28
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
32/37
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
33/37
SpotSizeR(Z)
back yx front
Fig.230
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
34/37
SpotSizeR(Z) front
SpotSizeR(Z) front
Fig.3 31
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
35/37
SpotSizeR(Z)
Z=15
back yx front
Fig.432
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
36/37
PulseAmplitudeb(Z) fron
PulseAmplitudeb(Z) fron
Fig.5 33
-
7/30/2019 Propagation of short laser pulses in plasma channels.pdf
37/37
PowerP(Z)
front yk
(a)
back
PowerP(Z)
(b)