The Quantum X-ray Compton Free Electron Laser

Transcription

1 The Quatum X-ray Compto Free Electro Laer Kazuha NAKAJIMA Hgh Eergy Accelerator Reearch Orgazato (KEK) 1-1 Oho, Tukuba, Ibarak, 5-81 Japa Igor V. SMETANIN P. N. Lebedev Phyc Ittute Lek propect 5, Mocow, Rua The quatum operato mode the X-ray Compto free-electro laer (FEL) preeted. Th FEL mode emerge whe the quatum emo exceed the electro beam eergy pread ad FEL homogeeou lewdth, determed by the optcal udulator legth. Such a FEL become a two-level quatum ocllator wth a completely verted actve medum. I the olear teracto regme, vero ca be completely removed a gle pa, aalogy wth π- pule formato coheretly amplfyg medum. Etmate how the quatum FEL provde a promg compact hgh-power hgh-brghte ource of coheret x-ray radato. The deg parameter for the quatum FEL are propoed for a compact hgh brghte coheret X-ray radato ource PACS Code: 41.6.Cr, 4.5.-p, 4.55.Vc, 5.8.Ph, f 1. Itroducto. Sce the demotrato of the frt FEL, lag thee devce ha bee acheved over a broad frequecy rage. It ha bee foud that mot of the expermetal reult ca be explaed by both the quatum ad clacal theore [1,]. It wa accepted that the complete agreemet betwee the reult of the clacal ad quatum approache gfe that a FEL eetally a clacal devce, more ak to covetoal devce of clacal electroc, rather tha a trcally quatum ocllator. I th paper, we ted to how that quatum effect become domat at defte codto the x-ray Compto FEL. The quatum ature of laer-partcle teracto free-electro laer become crucal whe the laer quata exceed the electro beam eergy pread ad FEL homogeeou lewdth, determed by the umber of udulator perod. Such a FEL become a two-level quatum ocllator wth a completely verted actve medum. I the olear teracto regme, vero ca be completely removed a gle pa, aalogy wth π-pule formato coheretly amplfyg medum. Poble tet expermetal cheme baed o a hgh peak power laer ad electro accelerator are dcued.

2 . Clacal ad quatum FEL operato mode The quatum ature ay laer - matter teracto proce revealed whe the tate of materal ytem chage gfcatly a a reult of photo aborpto or emo evet. Or, other word, whe the charactertc eergy cale of materal (atomc, molecular, electro, etc.) ytem of the ame order of magtude a the radato quatum. I free-electro laer, uch a charactertc eergy cale determed by the codto of a effectve teracto of electro beam wth the low poderomotve potetal wave, whch formed by the udulator ad gal feld. I the X-ray Compto FEL, the hgh-tety laer pule cattered by couter propagatg relatvtc electro beam []. The depedece of the emo ad aborpto probablte for hghfrequecy photo FEL o the beam eergy,.e., laer aborpto ad emo le, are thu of reoat ature. The ceter of aborpto (a) ad emo (e) le are at eerge εae, = ε ± η ω /, where the eergy ε = mc ( 1ω / kc) correpod to the ychrom wth the poderomotve wave, of whch frequecy ω = ω ω ad wave umber k = k + k, where ω, ad k, are frequece ad wave umber of pumpg () ad gal () wave repectvely [1]. The quatum properte of a FEL are thu determed by the rato of the eparato ηω betwee the aborpto ad emo le ad ther effectve wdth ε/ ε max{ /, γ / γ, Θ = 14N < > /}, whch caued by the fte legth of teracto rego L = Nλ (homogeeou broadeg, ~1/4N), ad homogeeou broadeg due to the beam eergy pread γ / γ ad dvergece < Θ >, caued by beam emttace. It coveet to troduce the quatum parameter η [4] ω η = η ε (1) I the clacal FEL operato regme η << 1, the emo ad aborpto profle practcally overlap (ee Fg.1a). The tate of electro caot chage gfcatly a the probablte of a drect Compto catterg ad a duced proce for each electro are practcally the ame. The ga coeffcet formula doe ot cota the Plak cotat ad agree wth the formula derved from clacal theory [.5]. I the oppote lmt η >> 1, whch the cae of our partcular teret, the emo ad aborpto le are completely eparated, ee Fg.1b. The tate of electro chage dramatcally wth the cale of ηω, f the electro eergy ear the emo maxmumε e, t caot aborb a quatum ad vce vera. I th regme, the quatum effect become domat. The FEL dyamc reduce to the quatum trato betwee thee two eergy level, ε e ad ε a [4]. A adequate decrpto of lag th regme thu requre the quatum theory ad ca ot be doe wth the clacal electrodyamc. For the cae of the tal laer quatum ηω ~1eV ad Raylegh legth L~1cm, the FEL lewdth determed maly by the beam eergy pread, ad the Compto FEL wll operate the quatum regme at γ > γ / γ, that correpod to the beam electro eergy of ~6MeV for the eergy pread of ~.1%. The emtted quata are thu the X-ray rego, ηω 4γ ηω 4keV.

3 . Electro a the quatum aharoc ocllator. To decrbe correctly two-level X-ray FEL dyamc, the Kle - Gordo equato hould be aalyzed aocato wth the wave equato for the X-ray gal ampltude [5]. Takg to accout the mometum ad eergy coervato law, ε( p ) = ( p c + m c ) p = p + k / 4 1, () where p the tal mometum of electro, oe ca repreet the electro wave fucto the form Ψ = mc Ωε( p ) 1 / a( z, t)exp{ [ pz ε( p) t]}, () η whch lead to a et of equato for ampltude of the eergy level gve by Eq. () [5] a = ga 1exp( t) + g+ 1a+ 1exp( + 1t) g = ( e / η) A A ( ε ε ) 1 / 1 (4) Here, A, are the vector potetal ampltude of the pumpg () ad X-ray gal () coheret wave. I fact, Eq. ()-(4) decrbe a aharmoc ocllator, whch characterzed by the detug from the reoace, creag wth the umber of level = ( ε ε 1)/ ηω 4 c ε( p) εe + ηω λ ε (5) At the teracto legth, mc L > 1 λ π ηω γ ~ 497. / γ cm, (6) all the expoet Eq. (4) wth ocllate uffcetly fat o that the exctato of correpodet tate eglgble. A a reult, emo of coheret x-ray photo related to the tmulated trato betwee oly two eergy level =, 1 [4]. 4. Two-level FEL coheret olear dyamc (X-ray π-pule geerato) I the quatum regme the Compto FEL become a two-level quatum ocllator a completely verted actve medum [4]. Such a quatum ytem kow very well the covetoal quatum electroc, ad t atural to apply a correpodet well-elaborated techque to our aaly. I th techque, Eq.(4) the two-level lmt form the Maxwell - Bloch ytem of equato aocato wth the wave equato for the X-ray gal ampltude, the oluto to whch uually eekg term of pule quare [6] z µ / / χ() z = E() z dz cη, µ = ee/ mω ω 1 (7)

4 Here µ a effectve dpole momet of the two-level ytem. The eergy tate ampltude for tally completely verted ytem the are a = co( χ ( z) / ), a 1 =( χ ( z) / ), (8) ad the wave equato become the pedulum equato χ π = α χ, α = 4 z e e E ηωωmc 1 / (9) It eay to ee that α, fact, the lear ga coeffcet. Note that the ga coeffcet formula cota the Plak cotat explct form. It mea that the regme uder coderato ha eetally quatum ature ad qualtatvely dffer from all the covetoal FEL operato regme. The Eq. (9) olear. I aocato wth materal equato (8), t decrbe the tmulated emo of coheret X-ray quata the collectve olear coheret teracto of the laer wth the electro beam. The coherece emerge due to all the electro the ame quatum tate durg the teracto. The Eq. (7)-(9) reveal a well-kow the olear optc effect of π -pule geerato: whe the total quare of the pule reache the value χ( L ) = π, the ytem become the lower eergy tate, ad the complete removal of vero emerge. For the tety of the x-ray gal, we have from Eq. (9) I = d αl I κ, κ ( ) κ = 1/( 1+ I( )/ Ib ) (1) whch the perodcal fucto of the teracto legth L. The maxmum value are I ( L ) = I ( ) + I (11) opt b Here Ib = eηω c the maxmum X-ray tety whch ca be emtted by a buch of two-level electro. The phycal terpretato of Eq. (11) clear a a reult of the collectve coheret laerbeam teracto, every two-level electro emt oe X-ray quata. Note, the maxmum output tety (11) doe ot deped o the pump laer tety, but jut o the dety of electro beam, whch qute dfferet from the covetoal FEL clacal regme. I Fg., the depedece of gal tety o the logtudal coordate how at three dfferet pump laer tete. The calculato are made for the electro beam of the e = 1 15 cm ad eergy γ = 1, ad for 1 tal x-ray photo. Oe ca ee that the mot of X-ray quata are emtted the olear regme whe the collectve teracto become domat. The optmum teracto legth, correpodg to the X-ray π -pule determed by the halfperod of the Jakob d-fucto

5 L K( κ) κα 1 (1) opt = where K(x) the complete ellptc tegral. I Fg, the depedece of optmum teracto legth o the pump laer tety how for the above parameter ad three dfferet electro beam dete ( cm, 1 15 cm, cm ). Dcuo I cocluo, let u dcu ome etmate. The beam eergy pread of.1% lead to the lower lmt of beam eergy of ~5 MeV. The correpodet eergy of x-ray quata are the rego ~4keV (for tal laer wavelegth of ~ 1µm). For the buch of ~1 1 electro (1.6C) the eergy of emtted coheret X-ray radato ~64µJ, or term of tety I,max ~ W/cm at the electro dety e = 1 15 cm. The teracto legth at pump tety of ~ 1 W / cm ~ cm (ee Fg.) whch allow to focu the electro beam ad reach uffcetly hgh beam dete. Suppog the beam dameter of ~1µm, the dvergece of X-ray pule ~.5µrad that lead to the hgh brghte of the quatum x-ray FEL. Really, f we defe the brghte a the photo flux the phercal agle determed by dvergece, we have ~1.6 1 mm mrad ec 1 at the pule durato of 1 fec, whch comparable to SLAC/LCLS X-ray ource. The above parameter, however, may ot be uque ad avalable wth the curret techologe. Ug the hgh-power table-top femtoecod laer, the tete >1 1 W / cm are ow realzed. The race-track mcrotro ca provde electro buche of the charge of 1-5C ad eergy the rego 5-15 MeV at uffcetly low eergy pread (<.1%) ad emttace. Thu, the realzato of the Compto free-electro laer the quatum operato regme ca provde u a compact ource of hgh-brghte coheret X-ray radato. Referece [1]. T.C.Marhall. Free-Electro Laer (Macmlla Publ.Co, 1985). []. C.W.Robero, P.Spragle. Phy Flud B1 (1989) 1. []. P.Dobach, P.Meytre, ad M.O.Scully. IEEE J.Quat.Electro. 19 (198) 181; J.Gea- Baacloche, G.T.Moore, R.R.Schlchter et al. IEEE J.Quat.Electro. (1987) [4]. I. V. Smeta, Laer Phyc, 7, 18 (1997); E.M.Beleov, S.V.Grgor'ev, A.V.Nazark, ad I.V.Smeta. Quatum Electroc (199) 7; JETP 78 (1994) 41. [5]. M.V.Fedorov. Progr. Quat. Electro. 7 (1981) 7; M.V.Fedorov ad J.McIver. JETP 49 (1979) 11. [6]. L. Alle, J. H. Eberly. Optcal reoace ad two-level atom. Wley, 1975.

6 Fgure 1. The depedece of x-ray quatum aborpto ad emo probablte W o the electro eergyε (. e., FEL aborpto ad emo le recprocal poto ) the two lmtg cae: a) the clacal FEL operato regme, η << 1; b) the quatum FEL operato regme, η >> 1. W W a) b) ε ε a ε e ε ε a ε e

7 Fgure. The depedece of the ormalzed x-ray gal tety I ()/ z I o the logtudal coordate z at the pump laer tete (a) 1 19 W/cm, (b) 1 W/cm, (c) 1 W/cm for the beam eergy γ = 1, the beam dety = e 1 15 cm ad the umber of tal X-ray photo - 1. b I (z)/i b 1.8 (c) (b) (a) Iteracto legth z [cm]

8 Fg.. The depedece of the optmum teracto legth L opt o laer tety ( ut 1 19 W/cm ) at the electro beam dete (a) cm, (b) beam eergy γ = 1, ad the umber of tal X-ray photo cm, (c) cm for the L opt (cm) (b) (a) (c) Laer tety [1 19 W/cm ]