ULTRAFAST TIME DOMAIN OPTICS OF SINGLE-CYCLE LASER PULSE INTERACTION WITH MATERIALS

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1 Universiy f Nebraska - Lincln DigialCmmns@Universiy f Nebraska - Lincln Theses, Disserains, and Suden Research frm Elecrical & Cmpuer Engineering Elecrical & Cmpuer Engineering, Deparmen f ULTRAFAST TIME DOMAIN OPTICS OF SINGLE-CYCLE LASER PULSE INTERACTION WITH MATERIALS Ufuk Parali Universiy f Nebraska - Lincln, ufukparali@yah.cm Fllw his and addiinal wrks a: hp://digialcmmns.unl.edu/elecengheses Par f he Elecrical and Cmpuer Engineering Cmmns Parali, Ufuk, "ULTRAFAST TIME DOMAIN OPTICS OF SINGLE-CYCLE LASER PULSE INTERACTION WITH MATERIALS" (010). Theses, Disserains, and Suden Research frm Elecrical & Cmpuer Engineering. 16. hp://digialcmmns.unl.edu/elecengheses/16 This Aricle is brugh yu fr free and pen access by he Elecrical & Cmpuer Engineering, Deparmen f a DigialCmmns@Universiy f Nebraska - Lincln. I has been acceped fr inclusin in Theses, Disserains, and Suden Research frm Elecrical & Cmpuer Engineering by an auhrized adminisrar f DigialCmmns@Universiy f Nebraska - Lincln.

2 ULTRAFAST TIME DOMAIN OPTICS OF SINGLE-CYCLE LASER PULSE INTERACTION WITH MATERIALS by Ufuk Parali A DISSERTATION Presened he Faculy f The Graduae Cllege a he Universiy f Nebraska In Parial Fulfillmen f Requiremens Fr he Degree f Dcr f Philsphy Majr: Engineering (Elecrical Engineering) Under he Supervisin f Prfessr Dennis R. Alexander Lincln, Nebraska December, 010

3 ULTRAFAST TIME DOMAIN OPTICS OF SINGLE-CYCLE LASER PULSE INTERACTION WITH MATERIALS Ufuk Parali, Ph.D. Universiy f Nebraska, 010 Advisr: Dennis R. Alexander In his hesis, ineracin f an ulrashr single-cycle pulse (USCP) wih a bund elecrn wihu inizain is sudied fr he firs ime. Fr a mre realisic mahemaical descripin f USCPs, Hermiian plynmials and cmbinain f Laguerre funcins are used fr w differen single-cycle exciain cases. These single-cycle pulse mdels are used as driving funcins fr he classical apprach mdel he ineracin f a bund elecrn wih an applied field. Tw differen new nvel ime dmain mdificain echniques are develped fr mdifying he classical Lrenz damped scillar mdel in rder make i cmpaible wih he USCP exciain. In he firs echnique, a ime depen mdifier funcin (MF) apprach has been develped ha urns he Lrenz scillar mdel equain in a Hill-like equain wih nn-peridic ime varying damping and spring cefficiens. In he secnd echnique, a ime depen cnvluinal mdifier funcin (CMF) apprach has been develped fr a clse resnance exciain case. This echnique prvides a cninuus updaing f he bund elecrn min under USCP exciain wih CMF ime upgrading f he scillain min fr he bund elecrn. We apply each echnique wih ur w differen driving mdel exciains. Each mdel prvides a

4 quie differen ime respnse f he bund elecrn fr he same applied ime dmain echnique. Differen plarizain respnse will subsequenly resul in relaive differences in he ime depen index f refracin. We shw ha he differences in he w ypes f inpu scillain fields cause subdurain ime regins where he perurbain n he real and imaginary par f he index f refracin dminae successively.

5 iv ACKNOWLEDGMENTS I wuld like express my deepes appreciain all hse wh have exed heir generus help during my Ph.D. prgram and made his wrk pssible. The very firs persn I wuld be graeful f is my advisr Prf. Dennis R. Alexander fr his menrship and advising hrughu my graduae wrk. He has inrigued me a he beginning f my prgram hink n sme f he fundamenals f ulrashr pulse phenmena and ulrashr single-cycle pulse phenmena. I have always learned smehing new frm Prf. Alexander and ver he years I came appreciae his remarkable guidance and suppr. Wihu his experise and encuragemen, his hesis wuld n have been pssible. I wuld als like hank my hesis cmmiee members Prf. Yngfeng Lu, Prf. Ezekiel Bahar and Prf. Crnelis J. Uierwaal fr reading and crrecing he disserain. Our lng discussins wih Prf. Bahar and Prf. Uierwaal were n nly inspirainal bu als exremely helpful fr undersanding he hereical underpinnings f ur wrk. I am hankful fr heir suppr and advice in elecrmagneics and ulrafas pics. I wuld like hank my fris in ur grup fr heir fellwship and suppr: Craig Zuhlke, Nichlas Rwse, Ehan Jacksn, Jhn Bruce III and Dr. Try Andersn. I always enjyed ur discussins and I ruly appreciae heir genersiy and friship. I wuld als like hank my fris in differen research grups: Tanya Gachvska, Yang Ga, Masud Mahjuri Samani, Xuejian Li, Arindra Guha and Ufuk Nalbanglu fr heir suppr and grea friship.

6 v I wuld n have cme his far wihu he lve and suppr f my mher, faher, sisers and brhers. Their cnsan care frm a reme disance was he surce f my energy fr achieving he gal. I cann express my graiude fr heir cninuus encuragemen and selfless sacrifice. This hesis is dedicaed hem.

7 vi TABLE OF CONTENTS ABSTRACT...ii ACKNOWLEDGEMENT...iv NOMENCLATURE...vi 1 INTRODUCTION..1 MODIFIER FUNCTION APPROACH FOR USCP INTERACTION IN TIME DOMAIN WITH A BOUND ELECTRON WITHOUT IONIZATION Mahemaical Mdel. 1. Numerical Resuls and Discussins..3 3 CONVOLUTIONAL MODIFIER FUNCTION APPROACH FOR USCP INTERACTION IN TIME DOMAIN WITH A BOUND ELECTRON WITHOUT IONIZATION Mahemaical Mdel Numerical Resuls and Discussins Numerical sluin f Vlerra inegral equain Numerical resuls and discussins fr VIE sluin.49 4 CONCLUSIONS.56 REFERENCES...58 APPENDIX

8 vii NOMENCLATURE Symbl w 0 Descripin angular frequency c speed f ligh carrier wavelengh Rabi frequency ha prduces Skes sideband 1 Rabi frequency ha prduces ani-skes sideband w deuning frm elecrnic saes w deuning frm Raman sidebands ab D vibrainal ransiin frequency T m repeiin rae phase shif ime delay rder f Laguerre funcin h L m m rder Laguerre funcin z spaial crdinae n z-axis ime scale f he pulse iniial phase phase erm f he elecric field E ulrashr single-cycle elecric field

9 viii p pulse durain n index f refracin permiiviy f free space permeabiliy f free space P pl elecrnic plarizain q e elecrn charge m e elecrn res mass x elecrn scillain field x mdifier funcin k spring cnsan damping cnsan Q ime depen spring cefficien P ime depen damping cefficien w specral bandwidh f rial funcin fr he sluin f Vlerra Inegral Equain phase erm f f f cefficien f i h f i ' x 1 s derivaive f he mdifier funcin x ' ' nd derivaive f he mdifier funcin x V mdifier funcin fund via Vlerra Inegral Equain sluin f V f ha is fund by using x V

10 1 CHAPTER 1 INTRODUCTION The sudies n he generain f ulrashr laser pulses ha cnain nly a few cycles f he elecric field araced remarkable aenin in he scienific cmmuniy [1-37]. Fr he generain f ulrashr pulses, ne requires a wide-bandwidh cheren specrum []. An incheren radiain surce, such as sunligh, a high pressure arc lamp r an amic line emissin lamp cnsiss f many specral cmpnens, all wih randmly varying phases []. The ime srucure frm such a surce is whie nise []. In cnras, a cheren ligh surce has a fixed phase relain amng he specral cmpnens, which inerfere prduce well-defined wavefrms []. Fr mre han w decades and unil very recenly, he shres pical pulses were bained by expanding he specrum f a mde-lcked laser by self-phase mdulain in an pical fiber, and hen cmpensaing fr grup velciy dispersin by using diffracin graing and prism pairs [3]. Fllwing he repr in 1987 f 6 fs pical pulses frm a dye laser sysem [4], ulrashr ligh pulse research has led he creain f laser sysems generaing pulses nly a few cycles in durain [4]. Few-cycle ransiens generain has been bsed by Ti:Sapphire echnlgy [4]. Using sphisicaed inracaviy dispersin cnrl, a pulse durain f 4.4 fs has been achieved direcly wih a resnar [4,5]. Ti:Sapphire amplifiers peraing a reduced repeiin raes enable exreme cmpressin in hllw fibers [4,6,7]. Bradband pical parameric scillars

11 [4,8] and amplifiers [4,9] have prduced pulses as shr as 3.9 fs in he visible [4,10] and 8.5 fs in he near infrared [4,11]. Very recenly, 7.8 fs pulses a a cenral wavelengh f 1. μm were implemened wih erbium-dped fiber echnlgy [4]. All hese resuls crrespnd less han w bu mre han 1.3 scillain cycles f he elecrmagneic field [4]. Since ligh is an elecrmagneic wave, he laser pulses cann be shrer han he carrier wavelengh, λ, which herefre limis he durain f he pulse λ/c, where c is he speed f ligh. T synhesize even shrer pulses, he specra frm femsecnd surces may be shaped in ampliude and phase [4,1] r pulse rains a differen wavelengh may be phase lcked and cmbined [4]. In principle, a sequence f ligh pulses ha are shrer han λ / c can be prduced simply by adding geher waves [Fig. 1.1] ha scillae wih an angular frequency f w + MΔw, where Δw is a fixed shif wih respec he fundamenal laser wave, w = πc / λ, and M is an ineger [13]. Fig 1.1. Hw generae subfemsecnd pulses: The superpsiin f several ligh waves a equidisan frequencies (p) in he ulravile regin can give rise a sequence f subfemsecnd

12 spikes (bm) if he phases f he waves are adjused apprpriaely. The repeiin rae f he spikes is Δν = Δw / π, where Δw is he angular frequency difference beween adjacen cmpnens [13]. 3 The resul is series f inense spikes separaed in ime by 1 / Δν = π / Δw [13]. The durain f hese spikes is inversely prprinal bh he frequency shif, Δν, and he number f waves ha add geher [13]. Cncepually, his echnique is clsely relaed he mde-lcking mehd ha is generally used generae femsecnd pulses in laser resnars [13]. Indeed, Δν mus be s large ha n laser can amplify all hese frequency-shifed waves [13]. The nly way ha hese waves can be prduced is using nnlinear pical echniques ha are n par f he femsecnd laser scillar iself [13]. Fr mre han a decade, laser physiciss and engineers have dreamed f cmbining he upu frm w indepen mde-lcked lasers synhesize single-cycle pulses hrugh cheren inerference [14]. Recenly, he sudies n he generain f ulrashr laser pulses ha cnain nly a few cycles f he elecric field have reached an advanced pin where he ulrashr laser pulses cnain nly a single-cycle f he elecric field [-5,14-4,8-31,33,34]. A single-cycle pulse, he shres pssible wavefrm a a given wavelengh, ccurs when he elecric field wihin he envelpe f an ulrashr laser pulse perfrms jus ne perid befre he pulse s [14]. In he infrared regin a arund 1.5 μm, he durain f ne pical cycle is apprximaely 4 fs [14]. Alhugh he shres pulses achieved s far have durains f less han 100 asecnds, hey are sill mulicycle pulses because he frequency f elecrmagneic radiain is much higher in he exreme ulravile regin han he infrared [14].

13 4 Tday, here are w differen appraches generae single-cycle pulses experimenally. The firs apprach relies n he adiabaic preparain f highly cheren mlecular vibrains r rains in large ensembles f mlecules [3]. The researchers wh prpsed his apprach invesigaed a brad-band Raman ligh surce, which is based n he cllinear generain f wide specrum f equidisan muually cheren Raman sidebands [3,6,7]. Raman scaering ccurs [Fig. 1.] when ligh passes hrugh a gas f mlecules [13]. The ligh can excie vibrainal r rainal energy levels in he mlecules, which subsequenly mdulae he laser Vibrainal Levels (a) (b) (c) (d) Fig. 1.. a) The pump (red) and Skes (dark red) driving lasers drive a mlecular vibrainal ransiin slighly ff-resnance. b) The pump laser mixes wih he mlecular vibrain generae an addiinal ani-skes frequency (brken green line). c) The ani-skes field mixes wih he mlecular vibrain generae he nex ani-skes frequency (brken blue line) d) This prcess cninues generae bh Skes (slid lines he righ f pump pulse) and ani-skes (brken lines he lef f pump pulse). The number f new frequencies deps n he efficiency f he prcess []. radiain [13]. The penial f simulaed Raman scaering fr generaing rains f subfemsecnd pulses has been demnsraed by several sudies [13,4,5,8-34]. In

14 5 hese sudies i has been shwed ha w laser beams whse frequency difference is slighly ffse frm a mlecular ransiin will, fr an apprpriae chice f gas pressure and cell lengh, generae a specrum f Raman sidebands whse Furier ransfrm is a peridic rain f subfemsecnd pulses [8]. The essence f he echnique [Fig. 1.3] is he cncurren generain f a frequency mdulaed wavefrm and he use f grup velciy dispersin emprally cmpress his wavefrm [8]. The cherence f he driven mlecular ransiin is cenral his echnique [8] and i is esablished by deuning [Fig. 1.3(a)] slighly frm he Raman resnance by driving he sysem wih w single-mde laser fields [3,8]. Fig a) Experimenal seup fr empral synhesis and characerizain f single-cycle pulses. b) Mdulain and synchrnizain f he pulse rain wih respec he mlecular scillain. c)

15 Phinizain f xenn fr characerizain f synhesized single-cycle pulse rain [,3,4,13,14,4,8,31,3,35,36]. 6 Classically, his ineracin can be picured [Fig. 1.3(b)] as driving a harmnic scillar near is resnance a he bea ne frequency f he w lasers []. In his manner a very efficien mlecular min mixes wih he w applied fields is prepared prduce new cheren frequencies []. Mlecular min, eiher in phase wih he driving frce (Raman deuning belw resnance) r aniphased (Raman deuning abve resnance) [Fig. 1.3(a)], in urn mdulaes he driving laser frequencies [3]. In is simples erms, mlecular mdulain is very much similar elecr-pic mdulain. The nly difference is ha mlecular mdulain ccurs a he frequencies a 5 rders f magniude larger [36]. Wha he cheren mlecular min des is mdulae he refracive index f he medium. If we cnsider he mlecules f deuerium (D ) [Fig. 1.3(b)], when hey are sreched, hey are easier plarizable [36]. Thus he refracive index f a medium cmpsed f sreched mlecules is larger han he refracive index f a medium cmpsed f cmpressed mlecules [36]. If he mlecules in a sample scillae in unisn, he macrscpic index f refracin f ha medium is mdulaed sinusidal wih he frequency f he mlecular min [35]. S, he mlecular mdulain is essenially due he prducin f Skes and ani-skes Raman sidebands f a special regime f Raman scaering wih maximal cherence [35]. The Raman generar in Fig. 1.3(a) is cnsruced by driving he fundamenal vibrainal ransiin f D by w ransfrm-limied laser pulses, ne frm Nd: Yag laser a μm and he her frm a Ti:Sapphire laser a 807 nm, such ha heir

16 7 frequency difference is apprximaely equal he ransiin frequency f 994 cm -1 [4]. The energy and pulse widh f he μm beam are 70 mj and 10 ns. Fr he 807 nm beam, hese quaniies are 60 mj and 15 ns [4]. Bh have a repeiin rae f 10 Hz and are cmbined and lsely fcused in a 50 cm lng D cell [4]. The upu afer he deuerium cell is whie ligh and he generaed specrum, which can be bserved by dispersing he beam wih a prism [], cnsiss f up seveneen sidebands and exs ver many caves f pical bandwidh (frm.95 μm in he infrared 195 nm in he ulravile) [3]. As i is seen in Fig. 1.3(a), nly he seven f hese generaed sidebands are used and he her frequencies are blcked []. The gd muual cherence acrss he spaial and empral prfiles f generaed Raman sidebands by mlecular mdulain, allws hem be recmbined spaially afer he phase adjusmen wih a liquid crysal phase mdular and specral mdificain echniques be used synhesize specified femsecnd ime srucures in a Xe arge cell [see Fig. 1.3(a)] [4]. The desired pulse is synhesized in his fcal regin f verlapping sidebands inside he chamber where fur-wave mixing serves as a pulse shape diagnsic [see Fig. 1.3(c)] []. Fcusing he sidebands in he chamber prduces a very weak UV signal in he range f picjules a several discree frequencies resuling frm he fur-wave mixing nnlinear prcess in Xe []. The magniude f he in signal deps n he inensiy f he synhesized pulse []. The shres pssible pulse ha can be synhesized als has he highes pssible inensiy []. S, he UV signal serves as feedback he spaial ligh mdular fr prviding an adapive phase adjusmen f he seven Raman sidebands []. Fig. 1.3(c) shws he

17 8 crss-crrelain race f he synhesized pulse and he prediced single-cycle elecric field prfile frm his crrelain [,4]. The secnd and he very recen experimenal apprach n he generain f singlecycle f ligh pulses makes use f he erbium-dped fiber laser echnlgy [14]. Fig a) Se-up f a single-cycle fiber laser sysem. OSC: femsecnd erbium-dped fiber scillar, EDFA: erbium-dped fiber amplifier, Si PC: silicn prism cmpressr, HNF: bulk highly nnlinear fiber fr ailred supercninuum generain, F / SF10 PC: pulse cmpressrs wih F and SF10 Brewser prisms, LPF: lw-pass filer (cuff wavelengh 1600 nm), VDL: variable delay line, DBC: dichric beam cmbiner. The divergen upu leaving each HNF face is cllimaed wih ff-axis parablic mirrrs [4]. b) Tempral scillains f he elecric field f w synchrnized ulrashr ligh pulses wih differen cener frequencies [37]. c) The cheren superpsiin f he ransiens. They are cmbined in space and ime such ha he cenral field maxima are exacly in sync wih each her. In

18 his way, hese regins ge amplified. Due he differen frequencies, desrucive inerference ses in already during he scillain cycles befre and afer he cenral maximum [4,14,37,38]. 9 An innvaive way fr he cnsrucin f single-cycle f ligh hrugh he cheren superpsiin f w nn-verlapping specra f separae pulse rains using his echnlgy is repred in [4] [see Fig. 14(a)]. Since he pssibiliy achieve brader bandwidh and shrer pulse durain is cherenly superimpse [see Figs. 1.4(a), 1.4(b)] separaed specra frm indepen bradband lasers a differen cener wavelenghs [39], he cheren inerference beween he upus f w mde-lcked lasers has already been ried be used in several sudies, bu iming jier has always prevened he success [38]. T cmba his drawback, as i is seen in Fig. 1.4, i is he firs ime ha a beam frm a mde-lcked femsecnd erbium-dped fiber scillar peraing a a repeiin rae f 40 MHz is spli in w branches and used as seed pulses fr w differen parallel femsecnd erbium-dped fiber amplifiers [4]. In each branch he average pwer f he femsecnd pulse rain is amplified 330 mw. Using he same scillar as a seed fr deriving bh specra prvides an achievemen in he need reduce he residual iming jier beween he w pulse rains a level f 43 as [14]. In each branch, he pulses are cmpressed pulse durains f 10 fs in a silicn prism sequence [39]. Subsequen supercninuum generain in highly nnlinear fiber assemblies lead ailr cu specra wih cener wavelenghs f 115 nm (dispersive wave) and 1770 nm (slin), respecively [39]. The w cmpnens are hen cmbined wih a dichric mirrr. The empral verlap is aligned wih a piez-cnrlled delay sage in ne branch [39]. A he pimum relaive empral psiin beween he w cmpnens f Δ=0 fs, cnsrucive inerference arises

19 exacly fr he cenral field maxima f each pulse, whereas he res f bh ransiens superimpses desrucively which indicaes he frmain f a single cycle pulse [4]. 10 Fig Fringe-reslved secnd-rder aucrrelains fr w-phn signal frm a GaAs phdide versus differen ime delay Δ beween dispersive wave and slin. A he pimum verlap (Δ=0) he signal feaures an islaed cenral maximum, indicaing he frmain f a single-cycle pulse wih durain 4.3 fs [4]. Due he adven f hese new experimenal sudies, he need fr undersanding he ineracin f a USCP wih he medium hrugh which i is prpagaing in is an impran and imely pic [40,41,4,43,44,45,46]. The ineracin f a laser pulse wih maer invlves he ineracin f he inciden elecric field wih he elecrns f he maerial. Basic physics f he pulse-maer ineracin deps srngly n he rai f he pulse durain and he characerisic respnse ime f he medium (as well

20 11 as n he pulse inensiy and energy). This rai is he key erm in he plarizain respnse f he medium frm a classical pin f view. The gal f his hesis is prvide he mahemaical mdel fr he ineracin dynamics f a USCP wih a bund elecrn wihu inizain fr he firs ime. This sudy is cncerned wih he linear plarizain respnse f dispersive maerials under USCP exciain where he elecric field srengh is lw enugh n prduce inizain. Since he energy is belw he inizain hreshld f he medium, here is n any plasma effec during he ineracin f he applied field wih he maer. Undersanding he linear plarizain respnse is crucial in rder frmulae a realisic field inegral. This realisic field inegral will prvide a mre realisic prpagain mdel f pical pulses hrugh dispersive media [47-67].

21 1 CHAPTER MODIFIER FUNCTION APPROACH FOR USCP INTERACTION IN TIME DOMAIN WITH A BOUND ELECTRON WITHOUT IONIZATION.1 Mahemaical Mdel In rder make an riginal cnribuin fr he analysis f he ineracin f an ulrashr single-cycle pulse (USCP) wih a bund elecrn wihu inizain, firs i is necessary find a realisic mdel fr a USCP. Such pulses have a raher differen srucure frm cnveninal mdulaed quasi-mnchrmaic signals wih a recangular r Gaussian envelpe [40,41,4,43]. Due he fllwing main reasns assciaed wih USCPs, cmbinain f Laguerre funcins and Hermiian plynmials (Mexican Ha) are used in his sudy fr mdeling applied EM field: i) Arbirary ransien seepness: The rising and he falling imes f he signal can be unequal. ii) Varying zer spacing: The disances beween zer-crssing pins may be unequal. iii) Bh he wavefrm envelpe and is firs spaial, secnd spaial and empral derivaives are cninuus. iv) Arbirary envelpe asymmery: USCP wavefrms can be classified cnveninally fr w grups.

22 1) The sharply defined zer-crssing pin a he pulse leading edge as iniial pin (cmbinain f Laguerre funcins). ) The sharply defined narrw maximum agains a backgrund f cmparaively lng ails (Hermiian plynmials Mexican Ha). [40,41,4,43]. Alhugh dela funcin r he Heaviside sep funcin are widely used, hey assume zer signal durain and zer relaxain ime. These assumpins are n suiable fr mdeling he wavefrm f a USCP. There are sme her mre realisic mdels, such as mdulaed Gaussian r recangular ransiens, bu hese mdels assume equally spaced zers which is n suiable fr a USCP, neiher [40,41,4,43]. The cmbinain f Laguerre funcins fr defining he spaiempral prfile f a m USCP is defined as Em BLm Lm where d L x x / / m! exp x a single Laguerre funcin wih rder m and x zc 1 / 0 m 13 m exp x is m dx. Here, B is nrmalizain cnsan, c is he velciy f ligh in vacuum, z is he prpagain direcin and 0 is he ime scale f he pulse. In his sudy, he cmbinain f nd and 4 h rder Laguerre funcins are used define a single USCP: , 4 3 exp 7.5 E (.1) 4 4 where he phase erm is defined as 0 zc 1 /. Here, is he iniial phase, z is he spaial crdinae in he prpagain direcin f he pulse and is he ime

23 scale f he pulse. Fr Laguerre USCP, 4x10, z 5x10 m and 10 secnds. Here, and z are chsen arbirarily. Wih hese values, we have he phase erm S, we bain he Laguerre USCP [Fig..1(a)] in ime dmain as: E exp (.) Fig..1. (a) Applied Laguerre USCP wih pulse durain τ p =8x10-16 secnds. (b) 1 s derivaive (V/m.sec) f he Laguerre USCP. Fig..1(b) shws he firs derivaive f he applied field and i is seen ha he analyical expressin E in Eq..1 saisfies he cndiins f arbirary ransien seepness and arbirary envelpe asymmery. Frm Fig..1(a), i is als seen ha i saisfies he cndiin f varying zer spacing fr a USCP. In addiin hese, ime prfile f he Laguerre USCP alms saisfies he inegral prpery: 0 d 0. E (.3)

24 15 Fr he Hermiian (Mexican Ha) USCP [Fig..(a)], he fllwing definiin is used: 1 exp /, E (.4) where, 4x10, z 5x10 m and 10 secnds. Wih hese values, we have he phase erm S, we define he Hermiian USCP [Fig..(a)] in ime dmain as: E exp / (.5) Fig..(b) illusraes ha he Hermiian pulse saisfies he abve cncerns. Fig... (a) Applied Hermiian USCP wih pulse durain τ p = 8x secnds. (b) 1 s derivaive (V/m.sec) f he Hermiian USCP. In addiin he quesin hw frmulae ulrashr single cycle ransiens, i is als naural ask hw hese pulses prpagae in pical medium. In his sudy, USCP

25 16 means he smalles pssible single cycle piece (uniy surce) f a wave packe. I is he par f an acual carrier field and des n cnain any her carrier fields in iself. Fr a USCP, i is difficul inrduce he cncep f an envelpe and i is n pssible define a grup velciy. Fr such shr pulses he disincin beween carrier scillains and slwly varying envelpe (SVE), which have w differen empral scales ha are peculiar quasi-mnchrmaic pulses, becmes diffuse r meaningless [47,68,69,70]. Jumping frm many cycle pical waves single cycle pical pulses in dealing wih ligh-maer ineracin, he mahemaical reamens shuld be revised. The radiinal analysis f pulsed EM phenmena is quesinable [40,41,4,43]. If he applied field is a USCP, he shres pssible field as explained abve, hen i is impssible separae he applied surce in pieces find he effec f each par (r piece) by superpsing as being suggesed in he mdels explained in many fundamenal exbks [71]. In rder undersand he USCP-medium ineracin phenmenn, we mus acquire cerain special feaures such as peraing direcly wih Maxwell equains beynd he scpe f Furier represenains [40,41,4,43]. Since he siuains ccur where he ime scale f he pulse is equal r shrer han he relaxain ime f he medium, maerial has n ime esablish is respnse parameers during he essenial par f he pulse cninuance [58,66,7,73,74]. These parameers, which gvern he plarizain respnse f he media, change heir values during he pulse cninuance [58,7]. Thus, sluins f Maxwell equains wih ime-depen cefficiens are required fr he analysis f he wave dynamics [66,74].

26 17 In ur sudy, we cnsider an apprach such ha under a single USCP exciain, he change in he relaive psiin f a bund elecrn is paren am wihu inizain will change he ampliude f he diple in he am and s frh he insananeus plarizain. As a resul f his flucuain in he plarizain, he index f refracin will change in he durain f he single USCP exciain during which he prpagain dynamics f he same applied USCP and he her USCPs cming afer he firs ne will be evaluaed. S physically, we cnsider a case where he medium is including he surce. This is a cmmn siuain especially in pical cmmunicain. In addiin his, we can assciae his apprach sme diagnsic echniques in ulrafas pics such as pump-prbe experimens where bh pump and prbe pulses prpagae and evaluae he ime varying physical parameers f he medium. Bu befre diving in Maxwell equains, we have figure u hw he plarizain respnse f he medium mus be handled fr he ineracin f a USCP EM field wih a bund elecrn. Undersanding he plarizain respnse f he maerial under he exciain f a USCP EM field is ne f he ms impran, n clearly answered ye, cre quesin f day and near fuure ulrafas laser engineering. Plarizain is a crucial physical phenmenn, especially fr pical cmmunicain, since i defines he change in he index f refracin in he maerial due he applied field [58,7,73,75,76]. In erms f permiiviy, we can wrie index f refracin (fr a nnmagneic maerial) as:

27 18 P n 1 pl E 1, (.6) [see Ref. [77], pp: and Ref. [78], pp: fr he jusificain f Eq. (.6)] where is he permiiviy f free space, E is he applied elecric field, and is he elecrnic plarizain. The plarizain respnse f he medium gives he change in he index f refracin. This change r his plarizain respnse affecs he empral and spaial evaluain (Fig..3) f he prpagaing pulse [1,3,79]. P pl Fig..3. Schemaic represenain f self-mdulain (pulse chirping). Alhugh we are ineresed in he lw inensiy applied fields fr linear plarizain in his sudy, empral depence f he inensiy prfile f he applied field can sill cause a empral depence in he refracive index [79]. The saring pin f all hese dynamics is he inhmgeneus wave equain: E z z, 1 Ez, c 0 Ppl, (.7)

28 19 where he plarizain is he surce erm f he gverning differenial equain. In rder find he plarizain, we mus find he scillain field (displacemen) f he bund elecrns. Accrding he Lrenz damped frced scillar mdel: d x me d dx me kx qee, (.8) d x is he ime depen displacemen r he scillain field f a bund elecrn wih respec he applied field E, is he damping cnsan, k is he spring cnsan f he maerial and m e is he mass f elecrn. Fr USCP exciain, unlike he lng pulse exciain fields, he respnse (scillain) f he elecrn mus be handled in a differen manner. Since, bh due he mass f ineria f he elecrn and he shrness f he USCP cmpared he relaxain ime f he medium, he elecrn will n sense he applied field exacly a he leading edge pin f he pulse. The respnse f he elecrn he applied field will increase gradually. During his sense, he elecrn will n fllw he scillain prfile f he applied elecric field. S, he scillain field f he elecrn will n nly have a difference in he phase bu als will have a differen ime prfile (imedepency) wih he applied field. In regular cases, if he applied field is in he frm f jw e ime-depency, hen we assume ha he scillain f he elecrn will be in he same ime-depency frm. In he lieraure, Lrenz scillar mdel is direcly used in jw e ime-depency [80]. Bu fr a USCP exciain, n nly he imedepency jw e is n valid, bu als he scillain field will have a differen

29 wavefrm han he applied field wavefrm (ime-depency). This means ha, he x erm in Eq. (.8), ha is he scillain field f he elecrn, will have a mdified frm f ime-depency wih respec he applied USCP. In rder define he 0 mdified funcin x, we develped a new ime dmain echnique ha we call Mdifier Funcin Apprach. In his apprach, we define he scillain field f he elecrn as he muliplicain f he applied USCP wih he mdifier funcin: x x E, (.9) where x is he mdifier funcin. I has a uni f (meer) /vl which is equivalen culmb*meer/newn. S physically, mdifier funcin defines diple mmen per uni frce. Plugging Eq. (.9) in Eq. (.8), we bain m e d x E dx E d x E q E. me k (.10) e d Afer a few manipulains, we may unie his as Eq. (.13): m e d x d m x e E de d de d E dx me mex k x dx d d E q E, e d m e d E (.11) m E e d x d d E de de dx k me E me E x qee d d d d m (.1) e

30 d x d E de d 1 dx 1 d E de k qe x. d E d E d m (.13) e me We can briefly wrie Eq. (.13) as: where d x d dx q e x, P Q (.14) d me P de, (.15) E d Q 1 E d E d E de d k. m e (.16) I is seen a Eq. (.14) ha i has a similar frm wih a Hill ype equain where fr a regular Hill equain, P and Q erms are peridic and he righ side is zer. A linear equain f his ype ccurs fen when a sysem exhibiing peridic min is perurbed in sme way [81]. This ype f equain was firs derived by G.W. Hill describe he effec f perurbains n he rbi f he Mn, and i ccurs in many her places in physics, including he quanum min f elecrns in a peridic penial f a crysal [81]. The band hery f slids is based n a similar equain, as is he hery f prpagaing elecrmagneic waves in a peridic srucure [81]. Oher applicains include parameric amplifiers. Alhugh P and Q erms are peridic in a Hill equain, in ur case hey are n. S, in ur mdel, Eq. (.14) is a Hill-like

31 equain which has a dc surce n is righ side and a ime-depen damping cefficien (.15) and a ime-depen spring cefficien (.16) in erms f a damped frced scillar mdel. The bjecive f Eq. (.14) is find he mdifier funcin which can be hen used define he scillain field (plarizain respnse) f he maerial. Due he ime-depen damping and spring cefficiens, he mdifier funcin is ally cupled wih he ime depency r ime prfile f he applied field. Eq. (.8) culd als have been slved direcly in he empral dmain, in which case we wuld have ls he analgy wih he Hill-like equain. Bu he apprpriaeness f using he mre cmplicaed apprach wih he mdifier funcin has slid physical reasns. In he case f a USCP exciain, he plarizain respnse f he maerial is n unique all hrugh he pulse cninuance. Due he shrness f he durain f he applied USCP cmparing he relaxain ime f he bund elecrn, he ineracin dynamics and he abiliy f he maerial sense and fllw he applied USCP field during is cninuance will be cmpleely differen han he cnveninal maer-field ineracin apprach. In Eq. (.8), physical parameers (damping and spring cefficiens) are cnsan. Hwever, he ineracin dynamics will n be cnsan during he USCP exciain. S, in rder penerae he effec f he applied field in he scillar mdel via hese physical parameers have a beer undersanding f he scillain respnse f he maerial under USCP exciain, we mus find he definiin f hese physical parameers in erms f he applied field and he physical cnsans f he sysem (maerial). Eq. (.15) and Eq. (.16) are hese definiins. They are being used in Eq. (.14) find he mdifier funcin which has

32 3 been embedded in Eq. (.8). The physical dimensin f he mdifier funcin is a diple mmen per uni frce. I frames he ime depency and he phase delay f he scillain field f he bund elecrn under USCP exciain.. Numerical Resuls and Discussins Fig..4. Bunded elecrn min under Laguerre USCP exciain ((a), (c), (e), (g), (i)) and Hermiian USCP exciain ((b), (d), (f), (h), (j)) fr varius values f spring cnsan ( k ) wih a fixed damping 14 cnsan ( 1x10 Hz).

33 In Fig..4, differen ineracin characerisics f Laguerre and Hermiian pulses are 14 shwn fr a fixed, relaively lw value f damping cnsan ( 1x10 Hz). Due he 4 definiin: k w, ( m 0 m e is he mass f elecrn, k is he spring cnsan fr bund e elecrn), he free scillain frequency f maerial is in UV range fr spring cnsan values f 4 N/m, 9 N/m, 35 N/m, 55 N/m [Figs..4(a),.4(b),.4(c),.4(d),.4(e),.4(f),.4(g),.4(h),.13(a)], 650 N/m [Fig..13(b)] and 750 N/m [Figs..5(b),.13(c)]. Fr spring cnsan values f 1500 N/m [Fig..5(c)], 500 N/m [Figs..4(i),.4(j)] and 7500 N/m [Fig..13(d)], he free scillain frequency is in X-ray range. As i is seen in Fig..4, he Hermiian ineracin has a mre ency scillain han he Laguerre ineracin fr relaively lw values f spring cnsan [see Figs..4(a),.4(b),.4(c),.4(d)]. As he spring cnsan is increased, Laguerre ineracin gains a mre scillary prfile [see Figs..4(e),.4(g)] while he scillain due he Hermiian pulse ineracin sabilizes and is ime prfile seles dwn in he invered phase ime prfile f he exciain pulse (invered Mexican Ha) [see Figs..4(f),.4(h),.4(j)]. Here, he ampliude f scillain r he ampliude f remblinglike min f he elecrn is in he range f 10-0 m 10-1 m which is in he scale f elecrn radius lengh. Finally, as he spring cnsan is increased relaively higher values, he Laguerre ineracin seles dwn in he invered phase ime prfile f he exciain pulse, (invered Laguerre pulse) [see Fig..4(i)]. Fig..4 shws a very clear disincin beween he ineracin characerisics f Laguerre and Hermiian USCPs unil he spring cnsan is 500 N/m (afer his value, we bain nly he invered phase ime prfile f he exciain surce fr he scillain). The scillain characerisics f bund elecrn under differen single USCP surces riginaes frm

34 5 mdifier funcin apprach. The Hill-like equain, which is he resul f he mdificain n he classic Lrenz damped scillar mdel wih he mdifier funcin apprach, causes he ime varying physical parameers cme in play during he ineracin prcess. Since hese physical parameers (ime varying damping and spring cefficiens) are absluely surce depen, hey behave differenly in he pulse durain f each differen USCP surce. As a resul f his, we see differen scillain prfiles fr a bund elecrn under a single Laguerre and Hermiian USCP exciains. Fig..5. Laguerre pulse exciain scillains fr damping cnsan: 16 1x10 Hz. In Fig..5, respnse f a bund elecrn is shwn fr a Laguerre pulse exciain fr varying values f spring cnsan wih a fixed, relaively higher damping cnsan value (1x10 16 ) han he previus case (Fig..4). An ineresing feaure here in Fig..5(a) and Fig..4(g) is ha alhugh hey are a he same spring cnsan value, hey shw differen scillain characerisics. Due a higher dampimg cefficien in Fig..5(a), while he scillain aenuaes quicker a he secnd half cycle f he Laguerre USCP han in Fig..4(g), i his a higher peak a he firs half cycle f he exciain pulse han in Fig..4(g). S, fr a reasnable value f spring cnsan, while relaively higher damping cefficien makes he firs half cycle f he Laguerre USCP mre

35 efficien in he means f ineracin, i makes he secnd half cycle less efficien. In rder cmpare scillain resuls mre deailly beween Figs..5(a) and.4(g), i is 6

36 Fig..6. Laguerre Pulse Exciain physical parameer sluins fr spring cnsan k 55 N/m. (a), 14 (b), (e), (f) and (i) are he sluins f Fig..4(g) (damping cnsan 1x10 Hz). (b) and (f) are he 7 magnified views f (a) and (e) respecively. (c), (d), (g), (h) and (j) are he sluins f Fig..5(a) 16 (damping cnsan 1x10 Hz). (d) and (h) are he magnified views f (c) and (g) respecively. necessary lk a heir physical parameer sluins such as ime varying damping and ime varying spring cefficiens. As i is explained abve, hese ime varying Fig..7. (a) (b): Magnified views f lef wings f Figs..6(a) -.6(c). (c) (d): Magnified views f righ wings f Figs..6(a) -.6(c). parameers cme in play due he naure f Mdifier Funcin Apprach. In Fig..6, ime varying damping cefficien, ime varying spring cefficien and he

37 8 mdifier funcin sluins f Figs..4(g) and.5(a) are shwn respecively fr w differen damping cnsan values wih a fixed spring cnsan a 55 N/m. In Figs..6(a) and.6(c), a sudden jump is seen in he ime varying damping cefficien prfiles a he ime pin where he exciain pulse changes is plarizain direcin. Alhugh hey lk idenical, he magnified views [see Figs..7(a),.7(b),.7(c),.7(d)] f he lef and righ wings f he damping cefficien shw he difference beween w differen damping cnsan cases. Here, he lef wing crrespnds he firs half cycle, righ wing crrespnds he secnd half cycle f he Laguerre exciain pulse. Cmparing he amun f he change n he y-axis wih he ime durain n he x-axis beween Figs..7(a).7(b), and.7(c).7(d), i is easy see he reasnable amun f difference affec he sluin f mdifier funcin [see Figs..6(i),.6(j)]. Fr ime varying spring cefficiens [see Figs..6(e),.6(g)], a significan difference is seen in he ime prfile alhugh he spring cnsan values are he same fr bh cases. The jump in Fig..6(g) his a higher peak han he jump in Fig..6(e). This can be a reasnable explanain fr a relaively lw scillain ency in he secnd half cycle f Fig..5(a) han he Fig..4(g). I can be said ha, due he dissipain f higher energy, his jump causes a lwer scillain prfile fr he bund elecrn during is ineracin wih he secnd half cycle f he Laguerre pulse in Fig..5(a) han in Fig..4(g). In Fig..5(c), as he spring cnsan is increased relaively higher values, same as in Fig..4(i), he scillain prfile seles dwn in he invered ime phase prfile f he exciain pulse. Differen frm Fig..4(i), he scillain seles dwn a a relaivley lwer spring cnsan value. S,

38 Index f Refracin i can be said ha, fr a higher damping cnsan, a lwer spring cnsan is enugh sabilize he scillain prfile in ime dmain spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m spring cnsan = 55 N/m spring cnsan = 500 N/m (sec) x Fig..8. Time depen index f refracin during he ineracin f a single Laguerre USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan value ( Hz). I is bained frm Eq. (.6) where P Nq x 14 1x10 pl e. Here N 6.0x10 3 and q e is he elecrn charge. Fig..8 shws he perurbain effec f an applied single Laguerre USCP n he index f refracin during is cninuance fr varying spring cnsans wih a fixed damping cnsan value. As i is clearly seen in Fig..8, fr all spring cnsan values excep he relaively higher case (500 N/m), here are hree regins where he perurbain effecs are dminan. These are he railing and leading regins f he pulse and he ime pin where he applied elecric field changes is plarizain sign. The change in he index f refracin arund he railing and leading edges is n as sharp as he change a he pin where he plarizain sign f he field changes. T

39 Index f Refracin Index f Refracin see his sudden effec mre clearly, he zmed view f his regin is shwn in Fig spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m spring cnsan = 55 N/m spring cnsan = 500 N/m (sec) x Fig..9. The jump in he ime depen index f refracin where he elecric field changes is plarizain sign spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m (sec) x Fig..10. Time depen index f refracin during he ineracin f a single Hermiian USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan 14 value ( 1x10 Hz). The same ype f perurbain behavir seen in Fig..8, is seen in he ineracin f a single Hermiian USCP wih a bund elecrn, (see Fig..10). Bh f hese

40 Index f Refracin Index f Refracin figures have he same damping cnsan value. The nly difference in he ime depen perurbain f index f refracin beween hese w cases is ha since spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m (sec) x Fig..11. The jump in he ime depen index f refracin where he elecric field changes is plarizain sign fr single Hermiian USCP ineracin spring cnsan = 55 N/m spring cnsan = 750 N/m spring cnsan = 1500 N/m (sec) x Fig..1. Time depen index f refracin during he ineracin f a single Laguerre USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan 16 value ( 1x10 Hz).

41 here are w pins where he Hermiain USCP field changes is plarizain sign, we have sudden changes in he perurbain f index f refracin wice arund hese pins. The zmed view f hese regins shws he sudden effecs mre clearly in Fig.11. In Fig..1, we see a similar ype f change in he ime depen index f 16 refracin fr damping cnsan 1x10 Hz. 3 Fig..13. Hermiian pulse exciain scillains fr damping cnsan: 1x10 17 Hz. Fr a damping cnsan value f 1x10 17 (Fig.13), very differen scillain behavirs are seen han he previus cases (Fig..4) f Hermiian pulse exciain. The ms prminen feaure in Figs..13(a),.13(b) and.13(c) is he high frequency scillain prfile wih a phase delay wih respec exciain pulse. In Fig..13, he spring cnsan is increased gradually frm.13(a).13(c) while keeping he

42 33 damping value cnsan. Fr a relaively lw value f spring cnsan in Fig..13(a), he main lbe and he railing ail f he exciain pulse have alms n effec n he scillain f he elecrn. The bund elecrn sars sensing he leading ail f he Hermiian exciain afer a phase delay f 5 fs. In Fig..14, he mdifier funcin sluins fr he Hermiian pulse exciain fr Fig..13 is shwn. 17 Fig..14. Hermiian pulse exciain mdifier funcins fr damping cnsan: 1x10 Hz. As i is seen in Fig..14(a), mdifier funcin suppresses he ineracin effec f main lbe and he railing ail f Hermiian funcin. As a resul f his, he bund elecrn sars sensing he exciain pulse wih a phase delay [Fig..13(a)] assciaed wih he mdifier funcin. Same behaviur f he mdifier funcin is seen in Figs..14(b) and.14(c),. As a resul f his, an apprximaely fs phase delay ccurs in Figs.

43 Index f Refracin 34.13(b) and.13(c). In Fig..14(d), he ype f mdifier funcin is seen ha gives a cmpleely phase invered ime prfile f he exciain pulse fr he scillain f he bund elecrn. In Fig..13(d), he sabilized scillain prfile is seen as a resul f his mdifier funcin. In Fig..15, as in he Fig..13, here is a high scillain frequency behaviur in he perurbain effec f he single Hermiian USCP n he index f refracin. Especially, he magniude f he perurbain effec is mre significan arund he main lbe and he railing edge regins han he leading edge regin f he applied field. The effec f he Hermiian USCP n he index f refracin decreases as he spring cnsan increases fr he given fixed damping cnsan value spring cnsan = 55 N/m spring cnsan = 650 N/m spring cnsan = 750 N/m spring cnsan = 7500 N/m (sec) x Fig..15. Time depen index f refracin during he ineracin f a single Hermiian USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan 17 value ( 1x10 Hz).

44 35 CHAPTER 3 CONVOLUTIONAL MODIFIER FUNCTION APPROACH FOR USCP INTERACTION IN TIME DOMAIN WITH A BOUND ELECTRON WITHOUT IONIZATION 3.1 Mahemaical Mdel In secin.1, we explained why he scillain field f he bund elecrn under single USCP expsure mus be defined in erms f he muliplicain f he applied USCP wih a mdifier funcin. In a mre realisic apprximain, we need include a cnsan updaing beween he elecrn min and he ime depen applied field. This is he majr difference beween appraches used in secins.1 and 3.1. Suppse ha we are applying w differen USCPs ranging in differen specral cnen n he same ype f maerial a differen pins. If we assume ha he majriy f he specral cnen f ne f hese USCPs is relaively clser he naural scillain frequency f he bund elecrn f he maerial han he specral cnen f he her USCP (see Fig. 3.1), hen i will n be realisic cnsider exacly he same ype f ime dmain USCP ineracin mechanism (mdifier funcin apprach ha has been explained in secin.1) fr bh f hese w differen USCPs. As i is seen in Fig. 3.1, we ne ha since he majriy f he specral cnen f USCP is clser w han he majriy f he specral cnen f USCP 1, in he cnex f ineracin efficiency he ineracin f USCP will be relaively mre inense han he ineracin

45 36 w w naural scillain frequency f he maerial specral cnen f USCP 1 Specral cnen f USCP w w f Fig Specral cnen f w differen USCPs wih he same pulse durain. They are being applied differen pins n a maerial which has a naural scillain frequency f w. f USCP 1 fr he given specral cnen and fr he given naural scillain frequency. Given he frmulain prvided in secin.1, we are jus direcly masking (muliplying) he mdifier funcin (ha we fund frm Eq..14) n he ime dmain prfile f USCP 1 find he scillain field f he bund elecrn during he cninuance f his pulse. If we fllw he same prcedure calculae he scillain field f he bund elecrn under USCP exciain, his will cause us miss he cumulaive ency due he memry effec f he scillain field f he bund elecrn in ime dmain due he ineracin wih single USCP cmpared he ineracin wih single USCP 1. In rder ake in cnsiderain he cumulaiveness effec under USCP exciain, insead f defining scillain as in Eq..9, we need define he ime depen elecrn min wih a cnvluin perain since a cnvluin can be cnsidered as an perain ha shws he effec f curren and pas inpus he curren upu f a sysem:

46 x x * E. 37 (3.1) If we plug Eq. (3.1) in Eq. (.8), we bain d d d k q (3.) d m m e x * E x * E x * E E. e e Eq. 3. allws us bain he scillain field afer he pulse (wake-field) due he naure f cnvluin perain in Eq The mdifier funcin is a hidden funcin ha mus be evaluaed firs find he scillain field caused by he USCP exciain where he surce durain is much shrer han he relaxain dynamics f he maerial. Due he naure f cnvluin perain in Eq. 3.1, alhugh he USCP acually vanishes a (where is he pulse durain), he mdifier funcin will sill exis afer he f he pulse and ur echnique evaluaes he scillain field afer he pulse durain due he memry effec f he cnvluin perain. In rder find he mdifier funcin in Eq. 3., differen mahemaical sluin echniques can be used. Fr he wrk in his chaper, le us use Eq. 3.1 in he fllwing frm: f x x * E, (3.3 a) f x x E d, (3.3 b) 0

47 which is called Vlerra Inegral Equain (VIE) f he secnd kind where he surce 38 funcin f and he kernel funcin E are given and x is he unknwn funcin. There are many exising sae f he ar numerical echniques fr slving he VIE in he lieraure [8,83,84,85,86,87,88,89]. Hwever, in his chaper we will fllw a simpler mahemaical prcedure in rder bain physical undersanding and insigh f differences beween cnvluinal mdifier funcin apprach and he mdifier funcin apprach explained in secin.1. Le s define he cnvluin inegral in Eq. 3.3(b) as: 0 x E d f x, (3.4) where f is ging be a reasnable rial funcin ha will be defined fr finding he mdifier funcin in Eq. 3.. By plugging he definiin in Eq. 3.4 in Eq. 3., we bain: d d x d k (3.5) d m x x F, e where F d d k q (3.6) d d m m e f f f E. e e

48 39 While in Eq..14 in secin.1 we are calculaing he mdifier funcin fr ime depen damping and spring cefficiens, in Eq. 3.5 we calculae he mdifier funcin fr cnsan damping and spring cefficiens wih a ime depen surce erm mdified by he rial funcin f. This apprach allws us incrprae he cumulaive ency f he scillain field and memry effec riginaing frm he specral cnen f he USCP and have cnsan damping and spring cefficiens during he pulse cninuance. 3. Numerical Resuls and Discussins Fr ur numerical calculains, we used he fllwing frms as w rial funcins simulaneusly fr he Laguerre USCP exciain case: f 1, (3.7) a f exp Sin 1 f, (3.8) a f exp Sin where 1 zc. a can range frm 1 m accrding he chsen f. S, a he f he calculains, he al scillain field has been evaluaed as: x 1 m m x * E x E i 1 1 i i * (3.9) where 1 is calculaed fr f1 a i and x i is calculaed fr f a i x i frm Eq. 3.5.

49 Fr he Hermiian USCP exciain case, we used he fllwing frm as he rial funcin in he numerical calculains: 40 f a 1 3 f 3 (3.10) and he al scillain field has been evaluaed as: x 1 m m x E i 1 3 i * (3.11) where 3 is calculaed fr f a i x i 3 frm Eq The values f he ampliude cnsans f 1, f, and f 3 are depen n he rial funcins and he number f rial funcins ha are chsen fr he sluin f he mdifier funcin. In Fig. 3., we see sme impran resuls f he cnvluinal mdifier funcin apprach n he scillain field f he bund elecrn under Laguerre and Hermiian USCP exciain and bh have clse specral cnen he naural scillain frequency f he maerial. Alhugh here is n much difference in he scillain frequency cmpared he Fig..4 in secin., here is a significan difference in he scillain ampliude where he cnvluinal mdifier funcin apprach has higher ampliudes. In addiin his (differen han Fig..4), in Fig. 3. we see sme phase delay in he scillain field wih respec he applied USCP fr bh Laguerre and Hermiian exciains (see Figs. 3.(a), 3.(b), 3.(c), 3.(e), 3.(g), 3.(h), 3.(i)

50 41 and 3.(j)). Anher significan resul shwn in Fig. 3., due he naure f he cnvluin perain, we can see he scillain in he wake-field afer he cninuance f he USCP. Fig. 3.. Bunded elecrn min fr he cnvluinal mdifier funcin apprach under Laguerre USCP exciain ((a), (c), (e), (g), (i)) and Hermiian USCP exciain ((b), (d), (f), (h), (j)) fr varius 14 values f spring cnsan ( k ) wih a fixed damping cnsan ( 1x10 Hz).

51 4 Fr Fig. 3.3, we have higher scillain ampliude and alms he same scillain frequency as cmpared Fig..5. Als in Figs. 3.3(a), 3.3(b) and 3.3(c) here is a phase delay which is n seen in Figs..5(a) and.5(b). I is bserved ha cmparing Fig. 3. Fig. 3.3, here is a significan difference in he wake-field scillains which are aenuaed much quicker in Fig. 3.3 afer he f he pulse cninuance. Fig Bunded elecrn min fr he cnvluinal mdifier funcin apprach under Laguerre USCP exciain fr varius values f spring cnsan ( k ) wih a fixed damping cnsan 16 ( 1x10 Hz). In Figs. 3.4 and 3.5, real and imaginary par f he perurbain effec f an applied single Laguerre USCP n he index f refracin f he given k medium. is shwn fr he cnvluinal mdifier funcin apprach. The cmmn behavir ha we ne in Figs..8, 3.4 and 3.5 is ha here is a sudden jump fr real and imaginary pars f he index f refracin a he pin where he USCP field changes is plarizain sign. Anher pin ha we mus ne in Figs 3.4 and 3.5 is ha, when he real par f he perurbain effec vanishes a sme regins f he Laguerre USCP, he imaginary par f he perurbain effec n he index f refracin cmes in play.

52 Imaginary Par f he Index f Refracin Real Par f he Index f Refracin spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m spring cnsan = 55 N/m spring cnsan = 500 N/m (sec) x Fig Real par f he ime depen index f refracin during he ineracin f a single Laguerre USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed 14 damping cnsan ( 1x10 Hz) [see Eq..6] spring cnsan = 4 N/m spring cnsan = 9 N/m spring cnsan = 35 N/m spring cnsan = 55 N/m spring cnsan = 500 N/m (sec) x Fig Imaginary par f he ime depen index f refracin during he ineracin f a single Laguerre USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a 14 fixed damping cnsan ( 1x10 Hz) [see Eq..6].

53 Numerical sluin f Vlerra inegral equain Once he scillain field f he bund elecrn is defined as in Eq. (3.1), VIE f he secnd kind has been uilized in Eq. (3.3b) find a sluin fr he mdifier funcin x. Due he cmmuaive prpery f cnvluin perain, we can wrie Eq. (3.3b) as: f x E, x d, 0 0 1, (3.1) where, E is he applied USCP as he cnvluinal kernel and f is a given surce funcin. Fr a general VIE f he secnd kind wih a cnvluinal kernel, he apprximae clsed frm sluin can be evaluaed by using he Mdified Taylrseries expansin mehd which is defined in [8]. This mehd can be applied a wide class f VIEs f he secnd kind wih smh and weakly singular kernels and i gives an apprximae and explici clsed frm sluin which can be cmpued using symblic cmpuing cdes [8]. Due he smhness f he kernel E,, nly few erms in he Taylr expansin are enugh ge high accuracy [8]. In his wrk, we apply Taylr expansin up he secnd rder. S, if we apply he prcedure explained in [8] n Eq. (3.1), we bain he fllwing equains : f ' " 1 E, d x E, d x E, d x, (3.13)

54 45 f E, d x 0 ' x E, ', (3.14) f '' '' ' x E, x, E, E d x 0, (3.15) where x ' and f ' are he 1 s " derivaive, x and f '' are nd derivaive f hese erms wih respec ime. Here, ur aim is find x, using Eqs. (3.13), (3.14) and (3.15). Perfrming he necessary manipulain, we bain he explici definiin f x as: x 1 I 1 ' '' ' f f I f I 3 E, f I , I E, I E, I E, I I I I (3.16) where I 1 E, d, I E, d, I3 E, d, I 4 E, d 0 and I 5 E, d 0. Here, E E, and E E,,. Fr Laguerre USCP:

55 exp, 3 4 E (3.17) Frm Eq. (3.17), , x E is fund fr Laguerre USCP. Fr Hermiian USCP:. / exp , E (3.18) Frm Eq. (3.18), 0047, 0. E is fund fr Hermiian USCP. Eqs. (3.17) and (3.18) are ging be used in Eq. (3.16) separaely fr Laguerre and Hermiian USCP exciain cases. We als need define f in Eq. (3.16) which is apriri given erm in a VIE sluin f he secnd kind (Eq. (3.1)). In his secin, we will use he same f funcins ha have been used in secin3.1. Frm Eqs. (3.7) and (3.8), we will bain w differen VIE sluins, x 1 and x. Using hese w sluins, we will define he final VIE sluin fr he mdifier funcin as: 1 x x x V. (3.19)

56 47 We nw have he explici sluin f he mdifier funcin. Bu his sluin is n cupled wih he physical parameers f he prblem such as spring and damping cnsans. In rder d his, we will g back he prcedure used in secin 3.1 and we will use x V in he prcedure f cnvluinal mdifier funcin apprach fr he sluin f Lrenz damped scillar mdel. This ime, insead f finding a mdifier funcin, we will find a new f funcin and hen we will use his funcin in Eq. (3.16) find he desired mdifier funcin which has already been cupled wih he physical parameers f he prblem. S, if we plug x V in: d d x V d k (3.0) V V d m x x F. e Once we find F explicily in Eq. (0), we can use i in: d d f V d k q (3.1) V V d m m e f f F E, e e where E is he applied USCP and f V is he funcin ha we are ging use i in Eq. (3.16) fr he VIE sluin f he mdifier funcin. Eqs. (3.0) and (3.1) are direcly bained frm secin 3.1 (see Eqs. 3.5 and 3.6). The summary f he prcedure is shwn belw in Fig Same radmap has been fllwed fr he Hermiian USCP exciain where Eq is used fr he iniial f funcin.

57 48 Fig Radmap used in his secin. The flw n he righ wih dark arrws is he prcedure used in secin 3.1. Once we find he mdifier funcin, we can define he scillain field explicily under USCP exciain [see Eq. (3.1)]. This will prvide us see clearly he change f he index f refracin in ime dmain in he perid f ne USCP durain via Eq..6. where P pl is he elecrnic plarizain which is defined as x Nq P e pl. Here, N is he number f elecrns aken in cnsiderain per uni vlume fr he E x x * E m q E x m k E x d d E x d d e e e * * * x f E x * F x m k x d d x d d V e V V E m q F f m k f d d f d d e e V e V V E x x * d x E x f 0, I I I I E I E I E I I I f E I f I f f x V ' 3 '' ' 1, 1, 1, 1, ( x V ) 4 ( f V )

58 ineracin prcess. In his secin, in rder undersand he perurbain effec f he applied USCP n he change f he index f refracin f he medium in ne pulse durain beer, we will use he nrmalized frm f Eq. (.6). This will prvide us see he ime evluin picure f he refracive index in ne USCP durain in a sense f free frm he elecrn densiy effec f he envirnmen. 3.4 Numerical resuls and discussins fr VIE sluin Fig. 3.7 shws he VIE sluin f he cnvluinal mdifier funcin apprach fr he min f a bunded elecrn under Laguerre USCP exciain [see Figs. 3.7(a), 3.7(c), 3.7(e), 3.7(g) and 3.7(i)] and Hermiian USCP exciain [see Figs. 3.7(b), 3.7(d), 3.7(f), 3.7(h) and 3.7(j)] fr varius values f spring cnsan wih a fixed value f damping cnsan. The arrw n he scillain graphs indicae he ime where he durain f he applied USCP field s. Due he naure f cnvluin perain, we can mnir he scillain field f he bund elecrn up where is he acual durain f he USCP. We use exacly he same spring cnsan values fr he same fixed damping cnsan value used in he previus secins. Alhugh i is n very dramaic, here are sme differences beween he resuls f his secin and secin 3.. These differences are n nly seen in he ampliude f he scillain fields bu als seen in he characerisics f he ime prfile. The majr difference fr Laguerre exciain is seen fr he relaively higher values f he spring cnsan. In Fig. 3.7(g), cmparing he scillain field in Fig. 3.(g), which has he same physical parameers used fr Fig. 3.7(g), we see a relaively higher scillain ampliudes in he secnd half cycle f he USCP and in he lae scillain regin jus 49

59 afer he USCP. In Fig. 3.7(i), he difference is mre dramaic cmparing he scillain in Fig. 3.(i). While we see alms exacly he same ype f half cycle 50 Fig Bunded elecrn min fr he Vlerra inegral equain sluin f he cnvluinal mdifier funcin apprach under Laguerre USCP exciain ((a), (c), (e), (g), (i)) and Hermiian USCP exciain ((b), (d), (f), (h), (j)) fr varius values f spring cnsan ( k ) wih a fixed damping cnsan 14 ( 1x10 Hz ). (Arrangemen f he graphs allw ne cmpare early 0 and lae scillains). p p p

60 scillain in he whle perid f he pulse durain in bh Figs. 3.7(i) and 3.(i) wih clsely he same peak value, he lae scillain behavir, which is he scillain afer he ineracin wih he applied USCP, is cmpleely differen. As a lae scillain behavir, we see he invered phase prfile f he applied Laguerre USCP beween he durain f p and p perid. Fr he higher spring cnsan values wih he same fixed damping cnsan, scillain behavir seles dwn in his ime prfile 51 beween 0 and p perid seen in Fig. 3.7(i). Fr he Hermiian ineracin, we see mre dramaic differences beween he Figs 3.7 and 3.. Especially fr he relaively high spring cnsan values, ime phase delay behavir shws impran deviain beween w figures. In Fig. 3.7(f), we have alms ne p phase delay befre he ccurrence f he scillain while here is n phase delay in Fig. 3.(f). On he her hand, in Figs. 3.7(h) and 3.7(j), differen han he scillains in Figs. 3.(h) and 3.(j), we d n see any invered ime phase prfile f he applied USCP and any ime phase delay in he scillain behavir. Fr he higher spring cnsan values, he scillain behavir seles dwn in his ime prfile. given Fig. 3.8 shws he nrmalized value f he change f he refracive index f he k medium in he ineracin durain f he applied Laguerre USCP. As i is shwn in Eq. (.6), he number f elecrns in he uni vlume f he maerial cnribuing he plarizain will effec he change in he index f refracin. Bu mre han he cnribuin f he bund elecrn ppulain, we are ineresed in he effec f he scillain respnse f each single elecrn n he refracive index under a single USCP exciain. S, differen han secin 3., since i is mre inuiive, in his secin we have a nrmalized picure f he prcess in rder undersand he pure

61 5 perurbain effec f he applied USCP bund elecrn ineracin n he ime evluin f he refracive index f he medium. Fr example, if we lk a Fig. 3.8(a), arund he clse prximiy f 7x10-16 secnd, i is seen ha he perurbain effec f he applied USCP n he real par f he refracive index is w imes srnger fr he medium wih k 55N / m han fr he medium k 500N / m. As an her example, fr he maerial k 35N / m in Fig. 3.8(a), i is seen ha he perurbain effec f he Laguerre USCP is apprximaely fur imes higher arund Fig Nrmalized real par perurbain (a) and nrmalized imaginary par perurbain (b) f he ime depen index f refracin during he ineracin f a single Laguerre USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan 14 ( 1x10 Hz). he prximiy f 6x10-16 secnd han he perurbain effec ccurred arund he prximiy f x10-16 secnd fr he same maerial. In Fig. 3.8(b), i is clearly seen ha here are differen incremen and decremen rais a he differen pars f he pulse durain fr he imaginary par as i is seen in he real par. Thus, he

62 real and imaginary pars f he index f refracin are variable during he pulse durain. 53 Fig Nrmalized real par perurbain (a) and nrmalized imaginary par perurbain (b) f he ime depen index f refracin during he ineracin f a single Hermiian USCP wih a bund elecrn wihu inizain fr differen spring cnsan values wih a fixed damping cnsan 14 ( 1x10 Hz). Fig. 3.9 shws he same nrmalized effecs ha are being discussed fr Fig. 3.8, bu his ime i is fr Hermiian USCP exciain case. As i is seen clearly in Figs 3.9(a) and 3.9(b), he nrmalized perurbain effecs are differen han he Laguerre exciain case in Figs. 3.8(a) and 3.8(b). Fr Hermiian case, we see sharper incremens and decremens in he perurbains bh fr he real and imaginary pars. A cmmn and an impran feaure seen (we see he same ype f behavir in secin 3., ) in bh f he Figs. 3.8 and 3.9 is ha, while here is a change in he real par a sme specific par f he applied USCP, he imaginary par is cmpleely suppressed fr bh Laguerre and Hermiian exciain cases. The vice versa f his