Modified model for harmonic generation free electron laser

Transcription

1 Nuclear Sciece ad Techiques 1 (010) 1 6 Modified odel for haroic geeratio free electro laser DENG Haixiao * DAI Zhii Shaghai Istitute of Applied Physics, Shaghai 01800, Chia Abstract The logitudial odulatio to the electro bea by a coheret seed laser pulse is widely used for geeratig fully coheret, short wavelegth radiatio i various haroic geeratio free electro laser (FEL) schees. After itroducig the desity odulatio, icrostructures dow to attosecod scale are produced over the distace of oe seed laser wavelegth. I order to take ito accout the icrostructures i the theoretical ad uerical aalysis, i the frae of udulator period averaged approach, a odified odel for haroic geeratio FEL is developed i this paper. With the odified odel, three haroic geeratio FEL exaples are ivestigated by eployig Shaghai soft X-ray FEL (SXFEL) paraeters. I FEL schees with ultra-high haroic geeratio ad ultra-short pulse, the odified odel presets soe iterestig aspects which are helpful for uderstadig of radiatio pulse evolutio, buchig efficiecy ad oise propagatio issues. Key words FEL, Haroic, Model, Attosecod 1 Itroductio Today, various free electro laser (FEL) schees [1 5] based o haroic geeratio of the exteral seed laser are copetitive ad affordable to copact ad fully coheret radiatio sources i the shortwavelegth regio. I haroic geeratio FEL, the exteral seed laser ad the electro bea iteracts with each other i the first udulator, i.e. odulator, ad the the eergy-odulated electro beas eter ito a dispersive sectio where the eergy odulatio is coverted ito spatial odulatio. Fially, the coheret eissio at high haroic of the seed laser geerates i the secod udulator, i.e. radiator. A typical haroic geeratio FEL schee deostrated experietally is high gai haroic geeratio (HGHG) [] i the visible [6] ad ultraviolet regio [7]. Geerally, the stadard HGHG-FEL is liited by a low frequecy ultiplicatio factor to avoid a large odulatio of bea eergy. Thus, soe ovel haroic geeratio FEL schees are recetly proposed so that the haroic uber ca be up to several tes [8 1]. Supported by Shaghai Natural Sciece Foudatio (Grat No. 09JC ). * Correspodig author. E-ail address: deghaixiao@siap.ac.c Received date: FEL radiatio is highly relevat with logitudial distributio of the electro bea, especially i the frae of resoat wavelegth. I the covetioal odel for haroic geeratio FEL, the radiatio evolutios i the odulator ad radiator are deteried by a averaged the electro behaviors over oe seed laser wavelegth. Whe the frequecy ultiplicatio factor is low, such a approach is valid, ad the odel is siple ad easy to solve. However, the logitudial distributio of electro bea is ot uifor over the distace of oe seed laser wavelegth after the desity odulatio. I the radiator, icrostructures dow to attosecod scale are produced, ad this should be take ito accout i the radiatio evolutio of radiator, especially i the case of high frequecy ultiplicatio factor. I this paper, we developed a odified odel for haroic geeratio FEL, based o the udulator period-averaged approach ad the icro-structures i oe seed laser wavelegth. The odified odel is briefly described by eployig a group paraeter of Shaghai soft X-ray FEL (SXFEL) [13], ad is utilized

2 DENG Haixiao et al. / Nuclear Sciece ad Techiques 1 (010) 1 6 to study the ultra-high order haroic geeratio, the ode-locked FEL [8] ad echo-eabled haroic geeratio (EEHG) schee [11,1]. The results show that the odified odel is robust for hadlig FEL schees with ultra-high order haroic geeratio ad ultra-short pulse. Modified odel for haroic geeratio FEL Geerally, the electro buch legth is uch larger tha the seed laser wavelegth, ad the bea curret variatio i oe seed laser wavelegth ca be eglected. Therefore, the logitudial distributio of the iitial bea ca be writte as ( 0 ) 1 f(,) e where, σ γ is the RMS eergy spread of the electro bea, ad γ 0 is the ea bea eergy. After passig through various cobiatios of the odulator ad the dispersive sectio, the logitudial distributio of the electro bea ca be writte as g(, ) G[ f(, )] F( ) g(, ) d where, the G fuctio represets all effects to the logitudial phase space before eterig the radiator. For siplicity, here we just cosider the electros over the distace of oe seed laser wavelegth λ s. Fig. 1 The logitudial phase space of electro bea before eterig the radiator i stadard HGHG. (1) () A resoace of radiator at the -th haroic of the seed is assued i the followig discussio. With the G fuctio, the logitudial desity of the electro bea is o ore uifor i [0, λ s ], ad the electros i [0, λ s ] occupy resoat wavelegth of the radiator. As show i Fig.1, the electros are uiforly divided ito slices over the distace of λ s, ad the local curret factor I (k) ad local buchig factor B (k) are defied as, k / F( ) d k / / I ( k) (3) / B ( k) where, k = 1,. k/ k/ / k/ k/ / i F( ) e d F( ) d Fig. Process flow chart of the odified odel for haroic geeratio FEL schee. After passig oe period of radiator, the radiatio slips over the electro buch by the distace of λ s /, ad the electros i each slice are idepedet with other slices. The radiatio field E (k) i each slice is deteried by I (k) ad B (k), thus oe caot average the electros over oe seed laser wavelegth. After passig through periods of radiator, electros i oe seed wavelegth are correlated by the radiatio slippages. However, all the logitudial distributio of the electro bea varies through the radiator due to effects of the radiatio ad dispersio i the radiator. Thus, to appropriately describe a haroic geeratio FEL i the radiator, the particle dyaics should be averaged over oe radiator resoat wavelegth istead of oe seed wavelegth. Therefore, a odified uerical odel for haroic geeratio FEL schee is developed. The (4)

3 DENG Haixiao et al. / Nuclear Sciece ad Techiques 1 (010) bea dyaics i the odulator ad dispersive sectio are calculated by stadard GENESIS [14], as show i Fig.. The steady-state ru of GENESIS is used as a field solver for each bea slice i the radiator, ad all the slippages are artificially processed i exteral of GENESIS after each period of radiator. 3 Exaples ad discussios SXFEL [13] is a two-stage HGHG schee, which is desiged to geerate 9 FEL fro 70 seed laser. The ai paraeters of SXFEL are listed i Table 1. Three haroic geeratio FEL exaples are discussed by usig the odified odel ad the SXFEL paraeters. Table 1 The ai paraeters of SXFEL. Paraeters Values Electro bea eergy E /GeV 0.84 Peak curret I P /A 600 Noralized eittace ε /-rad Local eergy spread σ γ / γ 10 4 Seed laser wavelegth λ S / 70 Modulator period legth λ UM / 58 Modulator udulator paraeter K M 6.83 Modulator udulator periods 16 Radiator resoat wavelegth λ R / 9 Radiator period legth λ UR / 5 Radiator udulator paraeter K R Ultra-high order haroic geeratio I the stadard HGHG, after the dispersive sectio, the logitudial distributio of the electro bea is ( D ) / F( ) 1 e J[ D]cos (5) 1 where, Δγ is the axiu eergy odulatio at the ed of odulator, D is the dispersive stregth cotributed fro the odulator ad dispersive sectio. Fro Eq.(5), oe ay obtai q e p ik k cos si e ik ik si e si b e J [ D ] ( D ) / I the case of haroic geeratio schee with low, whe the radiatio slips forward the electro bea over λ s i radiator, the logitudial distributio of the electro bea varies a little. Thus the local buchig factor ca be averaged over the distace of λ s, ad the Eq.(7) reduces to the uiversal for reported i Ref. [,15]. Fig. 3 Estiated local curret factor ad local buchig factor of the 61 th haroic at the odulator exit. where, I ( k) 1 p b 1 qb 1 B ( k) pb 1 (6) (7) Fig. 4 Siulated local curret factor ad local buchig factor of the 61 th haroic at the odulator exit. I the case of ultra-high haroic, the Bessel fuctio J (x) has the axiu at x 1. With a group of the optiized paraeters, Fig. 3 shows the aalytical results of SXFEL exaple with = 61. Whe a strog eergy odulatio of Δγ = 16.3 is

4 4 DENG Haixiao et al. / Nuclear Sciece ad Techiques 1 (010) 1 6 iduced, at the exit of the dispersive sectio, ost electros are odulated to the positio aroud k = 31, where the local bea curret ehaces 14 ties ore tha the uifor curret of the iitial 600 A, ad the local buchig factor is still To validate the aalytical estiatio, threediesioal siulatios were carried out ad the results are show i Fig. 4. With a eergy odulatio of Δγ =1., the local curret factor is about 13.5 ad the local buchig factor is up to 0.39 at the positio aroud k =31. This cosistecy with the estiatio idicates that the relativistic electro bea is odulated o attosecod scale so that it ca produce ultra-high order haroic geeratio of the seed laser withi a very short radiator. 3. The ode-locked FEL The ode-locked FEL [8] was proposed to geerate attosecod radiatio trais fro a FEL aplifier. I the ode-locked FEL, a cob of logitudial odes is sythesized by the optics-free techique applyig a series of teporal shifts betwee the co-propagatig radiatio ad electro buch i a FEL aplifier. The logitudial odes are phase-locked by iducig eergy odulatio fro a seed. Fig. 5 shows a layout of the ode-locked FEL. More details ca be foud i Ref [8,16]. Fig. 5 Scheatic of the ode-locked FEL. I SXFEL, the fial radiatio is the 30 th haroic of the 70 seed laser. If six periods is chose i each seget of radiator ad that a chicae is itroduced to give a electro buch delay of 3 λ R, the total slippage is 9 λ R λ S due to the udulator ad chicae. Uder such circustaces, SXFEL operates as a ode-locked FEL. A eergy odulatio with aplitude about 5 MeV is iduced by seedig the odulator with a 5 GW ad 70 laser. The odified odel for haroic geeratio FEL is used to study the evolutio of 9 radiatio i the ode-locked status of SXFEL. Fig. 6 Peak power growth i the ode-locked SXFEL. Fig.6 illustrates the peak power growth i the radiator of the ode-locked SXFEL, fro which it is see that the peak power goes through a slow growth fro A to B, a draatic growth fro C, ad a fial saturatio at D. I order to uderstad the growth curve, the radiatio pulses at differet radiator positio are give i Fig.7. Accordig to the teporal structures, it is kow that the ode-lockig copetes with self-aplified spotaeous eissio (SASE) fro A to B, doiates at C ad draatically iproves the teporal pulse structure. At the ed of the radiator, 9 radiatio trais with pulse width of 40 as ad peak power of 0 MW level are odeled, ad evely spaced by 0.9 fs (equal to 70 ). For ode-locked FEL, the odified odel is capable for hadlig the details teporally shorter tha oe seed laser cycle, ad it is easy to describe the pulse variatios by usig differet chicae delays ad tapered udulator sectios. But this is beyod the scope of this paper. 3.3 Operatig SXFEL with EEHG schee EEHG schee has a rearkable buchig efficiecy of frequecy up-coversio ad allows for geeratio a high haroic bea desity odulatio with a relatively sall eergy odulatio [11,1]. It has bee show that 9 coheret radiatio with peak power about 400 MW ca be directly geerated fro the 70 seed laser by operatig SXFEL with EEHG schee [17]. This was ivestigated by the odified odel.

5 DENG Haixiao et al. / Nuclear Sciece ad Techiques 1 (010) 1 6 Fig. 7 Radiatio pulse structures of the ode-locked FEL. The covetioal odel results a buchig paraeter of 0.08 for the 30th haroic i EEHG operatio of SXFEL. I the odified odel, the local curret factor ad the local buchig factor uevely distributes i oe seed laser wavelegth at the etrace of the radiator, as see i Fig. 8. The axiu of local curret factor ad the local buchig factor are about 1.3 ad 0., respectively. Fig. 8 The local curret factor ad local buchig factor i oe seed wavelegth at the etrace of a 9 radiator, by ruig the SXFEL with EEHG schee, obtaied fro the odified three diesioal siulatios. The peak power growth of the 9 radiatio i the radiator is show i Fig. 9. I the odified odel, the peak power oscillates several ties at the begiig of the radiator. This is cotributed by the local buchig factors with opposite agles i two half-cycle of oe seed wavelegth. However, the saturatio power is cosistet i two differet odels. 5 Fig. 9 The growth of 9 peak power i the radiator. Fig. 10 Teporal ad spectral doai of fial 9 radiatio. Fig. 10 illustrates the fial 9 radiatio pulse ad spectra. The results are very siilar to those of the covetioal odel. The elarged sectio shows ay sall ripples with a period of 70, a pheoeo that caot be observed with a covetioal odel. If the 9 radiatio pulse is itroduced to other haroic geeratio stage for further frequecy ultiplicatio, the ripples will be the iput aplitude oise. This eas that the odified odel is ore copetet for oise propagatio issue i haroic geeratio FEL.

6 6 DENG Haixiao et al. / Nuclear Sciece ad Techiques 1 (010) Coclusios Based o the averagig of oe seed laser wavelegth, the covetioal odel for haroic geeratio FEL sears ad eglects the icrostructures geerated by desity odulatio i oe seed laser wavelegth. Whe oe cosiders that the recetly proposed FEL schees ivolves with the ultra-high haroic geeratio ad ultra-short pulse, these icrostructures are iportat ad should be take ito accout. Based o the averaged wavelegth of the iterested haroic radiatios, a upgraded ad ore realistic odel for haroic geeratio FEL is proposed, ad three haroic geeratio FEL exaples are ivestigated ad discussed by usig SXFEL paraeters. The odified odel shows a robust capability to characterize FEL schees with ultra-high haroic geeratio ad ultra-short pulse, ad presets soe ew results which are helpful for uderstadig of radiatio pulse evolutio, buchig efficiecy ad oise propagatio issues i FEL. Refereces 1 Boifacio R, Salvo L De, Pierii P, et al. Nucl Istru Meth Phys Res A, 1990, 96: Yu L H, Phys Rev A, 1991, 44: Wu J H, Yu L H, Nucl Istru Meth Phys Res A, 001, 475: Golovizi V V, Va-Aersfoort P W, Phys Rev E, 1997, 55: Deg H X, Dai Z M, Chi Phys C, 008, 3: Yu L H, Babzie M, Be-Zvi I, et al. Sciece, 000, 89: Yu L H, DiMauro L, Doyura A, et al. Phys Rev Lett, 003, 91: Thopso N R, McNeil B W J, Phys Rev Lett, 008, 100: Allaria E, Nio G D, Phys Rev Lett, 007, 99: Jia Q K, Appl Phys Lett, 008, 93: Stupakov G, Phys Rev Lett, 009, 10: Xiag D, Stupakov G, Phys Rev ST Accel Beas, 009, 1: Kog X C, Wag J Q, Wag S H, et al., Shaghai Soft X-ray FEL Test Facility Coceptual Desig Report, 008, Shaghai, Chapter : Reiche S, Nucl Istru Meth Phys Res A, 1999, 49: Jia Q K, Phys Rev ST Accel Beas, 005, 8: McNeil B W J, Thopso N R, Duig D J, et al. Proceedig of FEL08, Geyogju, Korea, 008: Xiag D, Stupakov G, SLAC-PUB-13475, 008.