Analysis of laser acceleration in a semi-infinite space as inverse transition radiation. T. Plettner

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1 Analyss of las acclaon n a sm-nfn spac as nvs anson aaon T. Pln Ths acl calculas h ngy gan of a sngl lavsc lcon nacng wh a sngl gaussan bam ha s mna by a mallc flco a nomal ncnc by wo ffn mhos: h lcc fl ngal along h pah of h lcon an h ovlap ngal of h anson aaon pan fom h conucv fol wh h las bam. I s shown ha fo hs nsanc h wo calculaon mhos yl h sam xpsson fo h xpc ngy chang of h lcon. I. INTRODUCTION A cn poof-of-pncpl xpmn fo las-vn pacl acclaon [] mploy a sngl lnaly pola las bam nacng wh a 3 MV lavsc lcon bam n a sm-nfn vacuum mna by a hn mallc bounay. Th hgh flcv sufac pvns h las bam fom nacng wh h lcon bam n h spac ownsam of h bounay. Fgu llusas h acclao sup us n h poof-of-pncpl xpmn whch s smla o h aangmn ognally popos by nghof an Panll []. ncn las bam bounay lavsc lcon flc las bam FIG : Th las bam lcon bam an bounay confguaon. s h lcc fl of h las an s h fl componn paalll o h lcon bam ajcoy. Fo hs smpl sup h ngy gan fo h lcon bam s mos convnnly calcula by ngang h lcc fl componn of las bam ha s paalll o h lcon bam ( along s ajcoy ov h nacon lngh wh h las. As shown n Fgu hs cospons o h sm-nfn spac upsam of h

2 bounay. Assumng ha h xnal fl causs no sgnfcan chang n h pacl s ajcoy h ngy gan caus by h xnal las fl s b q a ( Th ngy gan obsv n h poof-of-pncpl xpmn was n goo agmn wh h ngy gan xpc by mployng quaon. I was obsv o scal lnaly wh h amplu of h las lcc fl show h xpc las polaaon pnnc an occu only n h psnc of h fl-mnang bounay. Many oh las-vn pacl acclaon schms mploy quaon o calcula h ngy gan. xampls nclu coss gaussan las bams [3] Hm-gaussan las bams [4] an oh abay las bam o lgh ffacon lcc fl pofls [5]. In aon o sm-opn f spac las acclaos oh schms lk Invs-Cnkov acclaos [6] an gu-wav acclaos [7] (opcal o RF also mploy h pah ngal mho of quaon o sma h lcon bam ngy gan. quaon suls fom ngang h Lon foc acng on h lcon ov s pah wh nacs wh h xnal fl. Howv Poynng s Thom offs an alnav mho fo calculang h ngy chang of h lcon. Poynng s Thom [8] sas ha ns a gvn volum V W M S µ ( B ns oal oal wh s h mchancal wok on h lcc chags n h volum V WM s h chang of h lcomagnc ngy so n h volum V an h las m s h ngy flux lavng h volum V hough s bounay S. oal an B oal a h oal lcc an magnc fls a h bounay S. I has bn shown ha fo h cas of an lcon nacng wh an xnal lcomagnc fl (lk a las quaon can b wn o xpss h ngy chang of h lcon as h ovlap ngal of h wak fl of h lcon n h psnc of h sucu o mum wh h las fl [9]. S ( s las wak 3 wh now psns h ngy gan of h chag pacl. Th quans las an wak a h las an lcon fls n h psnc of h sucu []. Fgu llusas a gnal suaon of an lcon nacng wh a sucu o mum n combnaon wh an xnal las fl.

3 wak fl wak fl sucu lcon las wak fl wak fl las FIG : schmac of ovlap ngal of las an wak fl aaon Th ovlap ngal quaon xpsss las acclaon as an nvs-aaon pocss. Howv sval mpoan assumpons a ma n quaon 3 an a mpoan o kp n mn whn makng gnalaons: Th wak fl amplu s much small han h xnal vng amplu ha s wak <<. Thus on can nglc h wak fl aaon conbuon las fom h acclao sucu o mum. Ths can b Cnkov apu anson aaon o vn unulao aaon n a wggl o any combnaon of hs pnng on h spcfc sucu. Th ob of h lcon suffs no appcabl chang n h psnc of h vng fl. Fo lavsc lcons hs s usually a val appoxmaon. Th s no chang n h so lcomagnc ngy n h volum wh h acclaon s calcula hus W M. Th s no ohmc loss n h acclao sucu o mum such ha h ngy flux of h las no h volum s qual o h ngy flux of h las xng h volum an fuhmo h only conbuon o ngy gan of h lcon bam an no fom cuns n h mum. Oh possbl non-aav scang losss a nglc coms fom h Ths assumpons a alsc fo mos las-acclaon xpmns such as h lasacclaon poof-of-pncpl LAP xpmn an hnc s of ns o calcula h ngy chang of h lcon bam by h nvs-aaon pcu scb n quaon 3 an compa o h famla pah-ngal ngy gan calculaon mho. 3

4 II. TH PATH INTGRAL NRGY GAIN MTHOD A lnaly pola TM gaussan las bam s assum o nac wh a lavsc lcon bam n f spac ov a fn sanc. Th xpsson fo h gaussan bam lcc fl magnu s v by h assumpon ha h las bam pofl vaaon n h ansvs mnson s much small han h vaaon u o h opcal phas n h longunal mnson. Wh hs assumpons an v as a scala h amplu of h ansvs lcc fl componn s [] ansvs w( ( x y cos k η( x y kx R ϕ ( 4 ( Th coonas ( x y a algn o h las bam wh x y a h ansvs coonas whl s h con of popagaon of h las bam. s h pak lcc fl amplu of h las bam ϕ s a phas offs angl an s h fquncy of h las. s h Ralgh ang gvn by w w ( wh w s ( h bam was an w s h bam s loca a R ( ( s h aus of cuvau a an η ( an ( s h Guoy phas shf fo a TM gaussan las bam a. Th focus of h bam n quaon occus a. As shown n Fgu 3 h coonas ( x y a oa by an angl abou h y-axs wh spc o a coona sysm ( x y algn wh h lcon bam. s h wavlngh. w ( w ( ( y x las bam ansvs bounay ( x y ρ x ŷ x ẑ lcon ( FIG 3: Schmac agam of h las bam an h lcon bam ncn on a hn bounay 4

5 Fgu 3 llusas a lnaly pola las bam on a an angl wh spc o h lcon bam an a a polaaon angl ρ. Th lcon avls along h coonas ( ( wh ( c. To calcula h ngy gan of h pacl h fl componn ( ( has o b valua an nga along h -axs. Snc ansvs s a scala quany h onaon of hs fl n h ( x y coonas has o b a by han. Allowng fo h polaaon angl ρ h onaon vco fo h ansvs lcc fl s cos cos ρ n las sn ρ 5 sn cos ρ Thus h xpsson fo h TM gaussan las bam n h ( x y coonas s x y ( cos cos ρ w kx ( x y cos( k η( ( ϕ sn ρ 6 R sn cos ρ ( x y Th coonas ( x y a valua as funcons of al quany. xpss as h al pa of a phaso.. No ha ( s a x y kx ( cos cos ρ ϕ kη ( ( w R ( x y R sn ρ 7 sn cos ρ Wh hs xpsson fo h las lcc fl on can pfom h pah ngal ngy sma n h sp as shown n quaon. Th lcc fl componn paalll o h con of popagaon of h lcon bam an along h lcon bam axs s x y w( kx ( cos η( ϕ sn cos ρ k ( 8 R wh h ( x y coonas cospon o x sn cos 9 5

6 quaon 8 smplfs o sn cos ρ ( cos ψ k cos η sn ( cos cosψ k R sn ϕ ( cos Th ngy chang of h lcon s mos asly oban by numcal ngaon of h lcc fl n quaon. Fo h xpmnal paams n h LAP xpmn h las bam was s w >> an hnc h bam vgnc <<. Fuhmo h las cossng angl s small ( <<. Fgu 4 shows h longunal lcc fl fo a las bam no mna a h bounay. Noc ha h gon of ns wh h las fl has a sgnfcan conbuon s much small han h Raylgh ang <<. Fuhmo noc ha h amplu of oscllaon of h lcc fl has a gaussan-lk nvlop whch s mosly u o h fn ansvs xn of h gaussan bam..8 longunal lcc fl (nomal sanc (/ FIG 4: Th longunal lcc fl fo a sngl gaussan las bam a a shallow angl o h lcon bam On can poc an numcally nga ( o oban h ngy gan howv fo h pupos of vng an analycal xpsson assum ha ajacn lcc fl oscllaons naly cancl xcp fo h vy las fl oscllaon n fon o h mnang bounay. To smplfy quaon assum <<. Thus 6

7 ( ( ( ϕ η ψ R k k Inoucng h fnons of w ( η an ( R ( ( ( ϕ ψ ϕ ψ ϕ ψ 3 3 an an an k w w k k k k snc w assum ha << h nvs-angn can b appoxma by an x x x 3 hus h opcal phas m bcoms ( ϕ ψ 3 k 4 h m vaabl can b lmna by assumng h lcon avls a a consan vlocy o v c v. Thus... w w k k k c k γ γ γ 5 Wh hs n mn h opcal phas m n quaon 4 bcoms ϕ γ ψ 6 Wh h pvously sa assumpons ha >> an hus << h phas angl slps by n a sanc much small ha h Raylgh ang. In hs gon wh << h phas m can b appoxma by 7

8 ψ ϕ γ 7 Noc ha h opcal phas m fo coss las bams v by Spangl say an Kall conans nsa of. Ths ffnc s u o h ncluson of h nnsc longunal lcc fl of a gaussan bam n h vaon ha s nglc n hs scusson whch s val fo h psn suaon wh >>. Snc w a analyng h lm wh h gaussan bam s a na-plan wav <<. Hnc hs ffnc of has no sgnfcanc. Th lcc fl n quaon bcoms cos ρ cosψ 8 Wh h assumpon ha h las s a na-plan wav an ha all xcp fo h vy las half lcc fl oscllaon cancl h ngal xpsson of quaon bcoms slppag q ( 9 Th slppag sanc s gvn by h conon wh h opcal phas avancs by. slppag ψ γ hus slppag γ γ Snc w a assumng ha h amplu faco has no appcabl chang ov ha s ( xp n h gon of ns h ngy gan s appoxmaly slppag slppag ϕ q cos ρ cos ϕ slppag 8

9 q q cos ρ cos ρ slppag slppag cosϕ ( cosϕ cosuu snϕ snuu Thus h ngy chang fo an lcon sma by h pah ngal mho s q cos ρ cosϕ γ Fom quaon w can s ha h ngy chang pns on h opcal phas ϕ ha scals lnaly wh h lcc fl amplu of h las ha follows a cosn pnnc wh h polaaon angl ρ an ha scals lnaly wh h las wavlngh. Fuhmo shows a pnnc wh h las cossng angl ha scals as f ( ( γ ha has an opmum angl of max γ. Fgu 5 shows ha fo h LAP xpmn las an lcon bam paams h analycal appoxmaon of quaon s n goo agmn wh h numcal pah ngal of h longunal lcc fl of h las bam. 5 ngy gan (kv V (k n g a y g n 5 numcal analycal las cossng angl (ma las cossng angl (ma FIG 5: compason bwn analycal appoxmaon an numcal ngal of h longunal lcc fl 9

10 III. TH INVRS TRANSITION RADIATION PICTUR W poc o calcula h lcc a fl of an lcon appoachng an nfn conucv bounay a nomal ncnc an mol as a supposon of a chag an an mag chag appoachng ach oh a unfom vlocy an soppng abuply a h bounay. Fgu 6 llusas h suaon wh h lcc fl a h obsvaon R φ s o b calcula. pon O wh coonas ( O bounay ( Rφ q -q chag mag chag FIG 6: anson aaon calculaon by h mho of mags Sang fom h a scala an vco ponals fo a pon chag A Φ ( x ( x µ q 4 q 4ε vδ V δ ( R c R( x ; ( R c R( x ; an usng h lcc an magnc fl xpssons B A Φ A 3 4 Th a lcc fl of an acclang chag can b foun o b a ( n q n KR 5 4 K

11 In h fquncy spcum hs cospons o q n ( R c ( x n 6 4R K Assum h collson m s nfnly sho hnc h m ngaon m wh changs valu. q n q n ( x n n 4R K 4 K ( R c s consan ov h 7 h oal fl caus by h chag plus h mag chag s ( x q n n n 4R K K 8 whch smplfs o q n ( ( n x 9 R n ( no ha h fl s aally pola cos cosφ n ( n cos snφ 3 sn hnc n pola coonas h anson aaon fl s ( R φ q sn R cos cos cosφ cos snφ sn 3 Th compac xpsson of h lcc fl n aal coonas mpls ha h ovlap ngal calculaon bwn h las fl las an h anson aaon fl ( φ s mos convnnly ca ou n h ( R φ coona sysm. Thus las ns o b xpss as a supposon of plan wavs n ( φ spac. Wn n phaso noaon h scala pa of h las fl s

12 x y k η ( kx R ϕ w( ( ( A ( 3 Usng h Fou ansfomaon pa ha has h fom ( ( ( ( 33 Thus h spcum of h (monochomac las lcc fl s 34 ( δ ( A( Dfnng h plan wav spcum as A u kxu kyv ( v A( x y xy 35 a h amplu faco of h lcc fl s A w( ( x y x y ϕ k η ( kx R( 36 Assumng ha h cossng angl s small a h bounay h aus of cuvau an Guoy phas shf ms can b nglc. Fuhmo h las spo s s w ( w. Wh hs assumpons fo A ( x y h plan wav spcum A ( u v bcoms (S appnx A u x y w kxu kyv ( ϕ k ( v xy 37 usng h coona ansfomaons n h small angl appoxmaon x xcos x ' xsn x y' y 38

13 A u x y ϕ w kx kxu kyv ( v xy 39 valuang h ngal n quaon 39 A u ( v w kwv kw ( u ϕ 4 Th faco kw can b plac by h las bam vgnc angl w ( u v ( u v w ϕ A 4 wh ( u v a h fa-fl paaxal angls. No ha h xponnal s quckly (assumng <<. Assumng ha h anson aaon fl sbuon has a nglgbl chang ov hs h angula ang h gaussan faco n quaon 4 can b appoxma by a la funcon ( u v δ ( u v 4 Ths s h quvaln assumpon ma n h pah-ngal calculaon ha h las bam s a na-plan wav. Thus ( v A u ( u ( u v v w ϕ ϕ δ ( u- v δ ( u- v 43 No ha h fa-fl paaxal angls ( v u fom a cangula coona sysm ha xpss n h aal angl coonas s u sn cosφ v sn snφ 44 hus h las plan wav spcum of quaon 35 bcoms ( v ( u v A u ϕ δ ( - φ 45 sn 3

14 h faco sn s a sul of h coona sysm ansfomaon. Whou h ngal of h la funcon woul gv h xa sn m. Ang h las polaaon vco fo ach of h wo plan wavs n quaon 45 las ( φ sn ( u v o ϕ δ ( - φ δ ( cos cos ρ sn ρ sn cos ρ 46 Ths s h xpsson of h las fl avlng away fom h focus n h fa fl an n h absnc of a flcv scn. Th s anoh m n ( φ coonas fo h ncomng las bam avlng no h focus. Ths ncn fl s loca a h coonas ( φ. No ha h vcos a wn n Casan coonas. Snc h onaon of h vco of h ncn las bam s h sam as h xng bam h xpsson h x-componn pcks up a ngav sgn whn h aal φ φ coonas ungo h ansfomaon ( las ( φ sn ( u v ϕ δ ( - φ δ ( cos cos ρ sn ρ sn cos ρ 47 In h psnc of h bounay h a wo mo ms on ansmsson m ha s φ an ha maks h oal fl bhn h bounay o an a qual o ( o las flcon m a coonas ( φ. las ( φ sn ( u v ϕ δ ( - φ δ ( cos cos ρ sn ρ sn cos ρ 48 Thus h oal las fl n h psnc of h bounay s las 49 ( φ las ( φ las ( φ s Thus las ϕ sn ( φ δ ( s { n δ ( ( φ n δ ( ( φ } 5 wh 4

15 cos cos ρ cos cos ρ n sn ρ n sn ρ 5 sn cos ρ sn cos ρ A agam of h fl componns s shown n Fgu 7 ( φ las ( ( las las ( bounay FIG 7: Th las fl n h psnc of h flcv bounay No ha las ( φ an las ( φ a qual n magnu. Bas on h physcal assumpon ha h ncn bam s nng h volum of nacon whl h flc bam s xng h sam volum h fa-fl ovlap ngal of h las fl wh slf s o fo a hgh flco bounay. Ω ( φ ( φ Ω ( φ ( φ Ω las las las las Ω 5 Ths wll sasfy h conon ha fo a losslss bounay Ω las ( φ ( φ s las s 53 Assumng no lcomagnc ngy s so n h volum of nacon an no ohmc losss Poynngs Thom pcs an ngy chang fo h lcon 5

16 T T v v S T wh s assum ha ( φ n ( φ na ( φ ( φ v T a. Ths xpsson can b xpan o 54 T T T ( ( ( ( ( las ( n Ω las ( ( ( Ω ( ( Ω ( ( Ω las las T las 55 I hav sablsh ha fo h las fl assumng a losslss bounay Ω las ( ( las 56 Fuhmo followng h assumpon ha h las fl magnu s much lag han h anson aaon fl magnu w can nglc h anson aaon ngy loss m an a lf wh ( ( T las ( ( Ω 57 T Ths s val fo al fls. Fo fls wn as phasos quaon 57 bcoms [] T T T T T T R R R ( ( R ( ( Ω ( ( ( ( ( las Ω T ( ( ( Ω R ( ( las las las T ( las Ω 58 Th scon m n 58 has ms wh posv fquncs only an hfo avags o o fo long ngaon ms. Thfo T ( ( ( R las Ω 59 T Now w n o conv hs m pnn xpsson no a fquncy pnn xpsson. Th nga ponng vco magnu s popoonal o 6

17 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( δ 58 Thus ( ( Ω U las R 59 Insng h xpsson fo an las ( ( ( ( ( ( Ω Ω φ φ φ φ n q n q U las las las las cos sn R 4 cos sn R 6 Insng h xpsson of n quaon 6 las ( ( ( ( ( { } Ω n n n q U cos sn sn R 4 φ δ φ δ δ ϕ whch can b wn ( ( ( ( ( Ω Ω n n n n q U φ δ φ δ ϕ cos R Ingang h la funcon xpssons 7

18 ( { n ( n ( } n n ( q cosϕ sn cos 6 Th vco o poucs n quaon 6 a n n n n ( ( cos cos ρ cos cosφ sn ρ cos snφ sn cos ρ sn ( cos sn cos ρ cos cos ρ cos cosφ sn ρ cos snφ sn cos ρ sn ( cos sn cos ρ cos ρ cos ρ ( ( 6 Thus q cosϕ sn cos ( { cos ρ ( cos ρ } ( q q cosϕ sn cos ( ( cosϕ sn cos ρ cos cos ρ 63 Assumng << q cosϕ cos ρ 64 Assumng q cosϕ cos ρ 65 γ Compang o quaon on can conclu ha fo a las bam mna by a conucv bounay a nomal ncnc o h lcon bam h ngy gan xpc 8

19 fom h pah ngal (quaon s h sam as h ngy gan calcula by h nvs-aaon ovlap ngal mho. 9

20 APPNDIX Dfn h plan wav composon of a fl sbuon ( x y kxu kyv ( u v C f ( x y xy f as Ψ A wh ( u v a angula coonas an C s an abay consan o b mn by consvaon of oal ngy flux. Th oal ngy flux of f x y has h fom ( Φ Ρ ( x y f ( x y f xy A Th consan C s chosn such ha Φ Ρ Ψ ( u v Ψ ( u v uv A4 Thus fom quaon A Φ Ρ ΡCC ΡCC ΡCC ΡCC C ΡCC f f kxu kyv ( x y xyc f ( x y f f f f ( x y f ( x y k ( x x u k ( y y ( x y f ( x y k ( x x u k ( y y ( x y f ( x y ( x y f ( x y ( δ ( x x δ ( y y ( x y f ( x yxy k v v kx u ky v kxu kyv kx u ky v x y xyuv uvx y xy uvx y xy x y xy x y uv A5 Thus C. Hnc h plan wav composon s Ψ kxu kyv ( u v f ( x y xy A6

21 RFRNCS. T. Pln al o appa n Phys. Rv. L. (5. R.H. Panll M.A. Psup F-lcon momnum moulaon by mans of lm nacon lngh wh lgh Appl Physcs Ls 3 no. p ( say P. Spangl J. Kall Las Acclaon of lcons n Vacuum Physcal Rvw 5 p ( J. Bochov G.T. Moo M.O. Scully Acclaon of pacls by an asymmc Hm-Gaussan las bam Physcal Rvw A Vol 46 No. p (99 5. J.A. nghof R.H. Panll ngy xchang bwn f lcons an lgh n vacuum Jounal of Appl Physcs 5 p 6-6 ( W. D. Kmua G. H. Km R. D. Roma L. C. Snhau I. V. Pogolsky K. P. Kusch R. C. Fnow X. Wang Y. Lu Las Acclaon of Rlavsc lcons Usng h Invs Chnkov ffc Phys. Rv. L ( X.. Ln Phoonc ban gap fb acclao Phys. Rv. ST Accl. Bams 4 53 ( 8. S fo xampl Jackson Classcal lcoynamcs n on chap 6 p M. X Pocngs of h 3 Pacl Acclao Confnc (3.. Huang G. Supakov an M. oloov Calculaon an Opmaon of Las Acclaon n Vacuum Phys. Rv. Spcal Topcs - Acclaos an BamsVol. 7 3 (4. A. Yav Opcal lconcs 4 h on Sauns Collg Publshng p. 48. D.K. Chng Fl an Wav lcomagncs n on Ason-Wsly Publshng Company p. 383