Statistical Optics and Free Electron Lasers

Transcription

1 Saisical Opics ad Fee leco Lases ialuca eloi uopea XFL Los Ageles UCLA Jauay 5 h 07

2 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 is difficul if o impossible o coceive a eal egieeig poblem i opics ha does o coai some eleme of uceaiy equiig saisical aalysis... [J.W. oodma Saisical Opics] The same siuaio fo Fee-leco Lases High-gai amplifie ca be descibed i a deemiisic way The iiial codiios iheely iclude elemes of uceaiy equiig saisical aalysis A exeal seed lase ca be descibed as a classical deemiisic field A eegy modulaio iduced by modulaos ca also be cosideed deemiisic Deemiisic desiy modulaio is also possible Bu he sho-oise i he eleco beams is always pese ad is vey fudameal some cases i epeses he mai sigal SAS ohes i is deimeal seeded FLs A saisical Opics eame of FLs mus be based o A saisical aalysis of sho-oise A model of he FL pocess

3 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 3 Coes Sho oise as sochasic sigal Deemiisic pa: FL amplificaio The SAS case FL amplifie ad sochasic pocess Coelaio fucios ad figues of mei

4 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 4 Sho-oise as sochasic sigal

5 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 5 Sho-oise descipio i elaivisic eleco beams fom he LNAC Paicle disibuio disceeess of elecic chage: N e - umbe of elecos pe buch C ~ 6e9 elecos p p - adom vaiables: aival imes ad posiio a he eace of he udulao z=0 Oigi of adomess: phooemissio a he cahode Semiclassical heoy based o P ; A = α A x y; Pobabiliy of phooeve i coh. ime A coh. aea phoocahode lase iesiy x y; P > ; A egligible Numbe of phooeves i ad ae saisically idepede

6 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 6 esuls i Poisso space-ime impulse pocess wih pobabiliy of fidig K eves bewee ad + i he aea A give by K is he aveage umbe of phooeves h defies he quaum efficiecy ad is he iegaed iesiy Moe i deail he assumpios fo he semiclassical heoy ae based o elecos as wavefucios evolvig accodig o he Schoedige equaio [see e.g. Madel ad Wolf Opical Coheece ad quaum opics] i m V e m is he eleco mass V he aomic Coulomb poeial c = 0 cos ω kyz he icide field classical deemiisic; hee he dipole appoximaio has bee used ad a calculaio of he asiio pobabiliy yields ou saig assumpios Semiclassical heoy is i ageeme wih fully quaum c

7 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 7 piciple quaum uceaiy may play a ole duig adiaio emissio i ems of: Quaum ecoil Negligible fo a lage aio bewee he FL badwidh ad he eegy of a phoo elaive o he eleco eegy. This efes o he so-called quaum FL paamee ρ = πργmc i he quaum case hω Usually quaum ecoil egligible fo pese pojecs. FL i he quaum egime poposed ad sudied elsewhee by may [e.g. Boifacio Pellegii Piovella Robb Schiavi Schoede ] Quaum diffusio Always pese: i iduces eegy spead i he eleco beam [Saldi Scheidmille Yukov NM A ]: Limis he shoes wavelegh achievable Rece wok [see Aisimov Quaum aue of elecos i classical FLs FL05] wa ha effecs due o he quaum aue of sigle elecos Fee-space dispesio of eleco wavepackes popagaig alog he udulao+quaum aveage may yield o lage-hapeviously-believed quaum effecs fo HXR; ad especially fo hamoic lasig. Hee we eglec hese effecs. helical h e u c e liea mc mc u

8 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 8 Ude hese assumpios we deal wih a classical cue We sick o a classical descipio of he e.m. field ad classical Saisical Opics is ou laguage Summig up sho-oise is a space-ime adom Poissoia pocess ad cue desiy a he eace of he udulao give by Usually N e is ivoked o ea he pocess as aussia. Fo a D cue i he fequecy domai

9 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 m 9 Re T Usually veified..g. ~ 0 9 ~ 80 fs Hz [ ~ 0.m] [ ~ 4m] z Phases uifomly disibued i 0 Aival imes ae saisically idepede of each ohe discussed befoe

10 0 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Fo N e ad i follow aussia disibuios Bu ad i ae acually joily aussia ha is The ca be show o follow he Rayleigh disibuio ad usig he asfomaio we obai wih ha is he usual egaive expoeial disibuio Ad i he ime domai exp i i P exp P P d d P P exp P exp P

11 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Coceig he cue desiy We also have Ay iegal of he cue desiy i ay domai! he follows he amma disibuio wih vaiace equal o he elaive dispesio: moe lae Sho-oise ca be eaed as a space-ime aussia pocess exp j j j j P z z z z

12 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Deemiisic pa: FL amplificaio

13 3 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 As discussed befoe he FL amplifie ca be descibed i a deemiisic way Oe way o descibe he full paicle disibuio is o use he Klimoovich disibuio Hee e is he max volume desiy of elecos ad sadad vaiables ae aleady ioduced i he logiudial diecio: ad The coiuiy equaio holds : Coside oly ieacio of a sigle eleco wih collecive fields fom he beam This meas ha we ae acually ivokig a Vlasov equaio. p N p p p p e p p P P z p P F e ; P c z z k w 0 ; dz z p P df

14 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Helical udulao wih asvese coodiaes as paamees 4 wih θ s = K γ Soluio i he liea egime uses he paicle disibuio backgoud ad a small liea egime peubaio ive he asvese pofile fucio S wih S0= ad 0 he ypical asvese beam size he cue desiy is Vlasov equaio mus be solved ogehe wih Maxwell equaios xpasio i azimuhal hamoics Assume give saig desiy modulaio as iiial codiio Hee ad below follow: Saldi Scheidmille Yukov Op. Comm

15 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 5 The SAS case. FL amplifie ad sochasic pocess

16 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 6 aussia pocess. Ad a liealy fileed aussia adom pocess is also a aussia pocess. Theefoe i he liea egime he field iheis he saisical popeies of he cue paicula he iesiy i he ime ~ o fequecy domai ω ~ ω fixed z follows a egaive expoeial disibuio as well-kow P exp Ad ay iegal of follows a amma disibuio. Fo example give U dd o U d U dd o M M P U M U U M U Wih he amma fucio ad exp M U U M U U - U he ivese elaive dispesio

17 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 7 M is he iepeed as he aveage umbe of modes The meaig of M depeds o he defiiio of he iegal. U dd U d U d modes Ad obviously cosideig U modes hough a moochomao. is he oal eegy i he pulse he M is he oal umbe of modes is he powe he M is he umbe of asvese modes is he eegy desiy a give posiio. The M is he umbe of logiudial dd Degee of asvese ad logiudial coheece is ivese mode umbe oe obais M as he umbe of logiudial Peak Bighess is defied we will comme o his defiiio lae! i ems of logiudial ad asvese modes as B 4 c 3 ave N ph c

18 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Ay iegal of follows a amma disibuio see befoe... 8 Ye logiudial ad asvese modes ae eaed vey diffeely by he amplificaio pocess. Tasvesely diffee self-epoducig modes have diffee gais The well-kow mode-guidig mechaism akes place ad oly oe mode eds o suvive [Saldi Scheidmille ad Yukov Op. Comm ]

19 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 9 Obsevaio Whe oly oe mode eds o suvive oe expecs good asvese coheece This is coec. Howeve he pesece of may logiudial modes limis he max degee achievable due o ieplay bewee logiudial ad asvese modes: asvese modes deped o fequecy

20 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 0 [Saldi Scheidmille ad Yukov New Joual of Physics 00] Behaviou of powe coheece log. & asv. ad bighess LieaNoliea&sauaio Liea egime: wha discussed above holds Sauaio: he saisical popeies of sho-oise ae asfomed by he o-lieaiy of he FL file

21 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Coelaio fucios ad figues of mei

22 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 A saisical sudy of adiaio popeies is bee doe wih he help of coelaio fucios. Fo isace i he ime-domai a fixed z Ad equivalely i he fequecy domai The kowledge of all ode coelaio fucios is eeded o fully chaaceize he sochasic pocess see oodma. he case of a aussia pocess FL i he liea egime he mome heoem applies This meas ha he basic quaiy o sudy i hese cases is O equivalely he same applies i ohe domais fo isace f we ae i he oliea egime i sill makes sese o sudy hese fucios bu hey do o fully chaaceize he pocess * * m m m m * * m m m m p p * *

23 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Logiudial diagosics 3 ive a saisical measueme of he specal coelaio ca we ge he phoo pulse duaio? Based i weighed specal secod ode coelaio fucio Sigle-sho specum assumig give Lie fo he specomee

24 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 4 Use a model fo he amplificaio pocess o deive a aalyical expessio fo Fi he expeimeally measued wih he aalyical expessio o deive he pulse duaio woks icely! The model fo is based o woks also afe he liea egime bu his easoig is sicly ok i he liea egime

25 5 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 f he ivese buch duaio is small compaed o he FL badwidh quasi-saioaiy is a good appoximaio z f f

26 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Wiee-Khichi Theoem Quasi-saioaiy is assumed 6 f 0 f 0 Specal coelaio FT pais Specal desiy esiy 0 f 0 Tempoal coelaio f

27 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Logiudial Coheece 7 his way oe fixes = 0 ad ca sudy logiudial coheece a fixed asvese posiio i he fequecy domai usig 0 f z 0 if quasi saioay O i he ime domai usig 0 f z 0 if quasi saioay Depedece o fixed posiio is sill hee because hee is o quasi-homogeeiy i geeal quasi-homogeeiy is as quasi-saioaiy i he spaial domai Sudy of asvese coheece is doe isead by sudyig slow depedece o if quasisaioay a =0 0 ~ ~ ~ O by sudyig slow depedece o ime if quasi-saioay 0 Tasvese Coheece * * a =0

28 8 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 saioay quasi if f 0 0 Coheece ime Degee of asvese coheece g g d c o z d d d if quasi-saioay 0 * ~ ~ * ca be show [Saldi Scheidmille Yukov Op. Comm ] ha i he liea egime =M

29 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 9 Bighess Radiomey based o geomeical opics he cocep of Radiace is used Radiace is Specal phoo flux pe ui aea pe ui pojecio agle is he phoo flux desiy i phase space Whe Liouville holds o - dissipaive case he desiy of sysem pois ea a give poi evolvig hough phase - space is cosa wih ime The he Bighess is he maximum flux desiy i phase - space ad is he heoeical max coceaio of phoo flux desiy deliveed by a ideal opical imagig sysem

30 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Bighess 30 Beyod geomeical opics oe uses he cocep of Wige disibuio Kwag-Je Kim was he fis o apply his appoach o SR & FLs see Bazaov PRSTAB fo a eview Sa wih coss-specal desiy ~ ~ z z z ca be see as he aalogue of a desiy maix Wige disibuio is he FT of he coss-specal desiy *

31 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Bighess The W disibuio ca obviously be used o expess he degee of asvese coheece i a equivale way as doe wih he coss-specal desiy 3 The W disibuio ca also be used o exed he oio of Bighess i a vey aual way B = max[w Compae wih he pevious defiiio: 4 c 3 ave N ph The lae ca be wie i ems of iegals of Wige fucio B c Ad does o iclude ifomaio o wavefo popeies which is isead icluded i B = max[w hough fo FLs he fudameal mode has ypically a ice wavefo

32 3 Saisical Opics ad Fee leco Lases Theoy ialuca eloi UCLA Los Ageles Jauay 5 h 07 Bighess Noe ha also hee we ae usig he asvese W fucio This migh be geealized usig he full coelaio fucio Ad he We ca sick o he pevious defiiio of Bighess ~ ~ * z z d d i c i A W ] exp[ B = max[w