Phase-merging enhanced harmonic generation free-electron laser

Transcription

1 OPEN ACCESS Phase-merging enhanced harmonic generation free-electron laser To cite this article: Chao Feng et al 014 New J. Phys View the article online for pdates and enhancements. Related content - Flly coherent hard X-ray generation by two-stage phase-merging enhanced harmonic generation * Gang-Lei Wang, Wei-Qing Zhang, Xe- Ming Yang et al. - Triple modlator chicane scheme for seeding sb-nanometer x-ray free-electron lasers Dao Xiang and Gennady Stpakov - Sppression of microbnching instability via a transverse gradient ndlator Chao Feng, Dazhang Hang, Haixiao Deng et al. Recent citations - Phase-merging enhanced harmonic generation free-electron laser with a normal modlator Zhoy Zhao et al - Simplified model for fast optimization of a free-electron laser oscillator Kai Li et al - Generating high-brightness and coherent soft x-ray plses in the water window with a seeded free-electron laser Kaishang Zho et al This content was downloaded from IP address on 09/04/018 at 1:3

2 Phase-merging enhanced harmonic generation free-electron laser Chao Feng, Haixiao Deng, Dong Wang and Zhentang Zhao Shanghai Institte of Applied Physics, Chinese Academy of Sciences, Shanghai 01800, People s Repblic of China denghaixiao@sinap.ac.cn Received 11 December 013, revised 8 Febrary 014 Accepted for pblication 11 March 014 Pblished 3 April 014 New Jornal of Physics 16 (014) doi: / /16/4/04301 Abstract Together with one of its variants, the recently proposed phase-merging enhanced harmonic generation (PEHG) free-electron laser (FEL) is systematically stdied in this paper. Different from a standard high-gain harmonic generation scheme, a transverse gradient ndlator is employed to introdce a phase-merging effect into the transversely dispersed electron beam in PEHG. The analytical theory of the phase-merging effect and the physical mechanism behind the phenomenon are presented. Using a representative set of beam parameters, intensive start-toend simlations for soft x-ray FEL generation are given to illstrate the performance of PEHG. Moreover, some practical isses that may affect the performance of PEHG are also discssed. Keywords: seeded FEL, PEHG, phase-merging effect, transverse gradient 1. Introdction The recent sccess of self-amplified spontaneos emission (SASE)-based x-ray free-electron laser (FEL) facilities [1, ] is enabling ctting-edge science in varios areas. While the radiation from a SASE FEL has excellent transverse coherence, it typically has rather limited temporal coherence as the initial radiation comes from the electron beam shot noise. To overcome this problem, several SASE-based techniqes have been developed, mainly inclding self-seeding [3 5], prified SASE [6], improved SASE [7] and HB SASE [8]. Content from this work may be sed nder the terms of the Creative Commons Attribtion 3.0 licence. Any frther distribtion of this work mst maintain attribtion to the athor(s) and the title of the work, jornal citation and DOI. New Jornal of Physics 16 (014) /14/ $ IOP Pblishing Ltd and Detsche Physikalische Gesellschaft

3 New J. Phys. 16 (014) An alternative way to significantly improve the temporal coherence of high-gain FELs is to se freqency p-conversion schemes, which generally rely on the techniqes of optical-scale maniplation of the electron beam phase space with external coherent laser sorces. In the highgain harmonic generation (HGHG) scheme [9], typically a seed laser plse is sed to interact with electrons in a short ndlator, called a modlator, to generate a sinsoidal energy modlation in the electron beam on the seed laser wavelength scale. This energy modlation is then transformed into an associated density modlation by a dispersive magnetic chicane, called the dispersion section (DS). Taking advantage of the fact that the density modlation shows Forier components at high harmonics of the seed freqency, intense radiation at shorter wavelengths can be generated. The property of HGHG otpt is a direct map of the seed laser s attribtes, which ensres a high degree of temporal coherence and small plse energy flctations with respect to SASE. These theoretical predictions have been demonstrated in experiments with the HGHG [10 13]. However, significant bnching at higher harmonics is sally needed to strengthen the energy modlation in HGHG, which will reslt in a degradation of the amplification process in the radiator. Ths the reqirement of FEL amplification on the beam energy spread prevents the possibility of reaching a short wavelength in a single-stage HGHG. In order to improve the freqency mltiplication efficiency in a single stage, more complicated phase space maniplation techniqes have been developed, e.g. the echo-enabled harmonic generation [14, 15] techniqe employs two modlators and two dispersion sections, which can be sed to introdce an echo effect into the electron beam phase space for enhancing the freqency mltiplication efficiency with a relatively small energy modlation. The idea of sing a transverse gradient ndlator (TGU) to mitigate the effects of electron beam energy spread in FEL oscillators has been described in [16]. Recently, this idea has been applied to laser-plasma accelerator driven high-gain FELs [17]. Inspired by these early works, a novel phase space maniplation techniqe, originally termed cooled-hghg, has been proposed for significantly improving the freqency p-conversion efficiency of harmonic generation FELs [18]. This techniqe benefits from the transverse-longitdinal phase space copling, while other harmonic generation schemes only maniplate the longitdinal phase space of the electron beam. In comparison with the general idea of sing TGU to compensate for the effects of beam energy spread by making every electron satisfy the resonant condition in the ndlator, this novel scheme tilizes a different operation regime of TGU: when the transversely dispersed electrons pass throgh the TGU modlator, arond the zero-crossing of the seed laser, the electrons with the same energy will merge into the same longitdinal phase, which holds great promise for generating flly coherent short-wavelength radiation at very high harmonics of the seed laser. At first glance, this phase-merging phenomenon is very similar to electron beam energy spread cooling. However, the beam energy spread within the range less than the seed laser wavelength is redced, while the global beam energy spread does not change in sch a process. Therefore, in order to clearly and nambigosly illstrate the physics behind it, we rename sch a scheme as phase-merging enhanced harmonic generation (PEHG), althogh frther stdies demonstrate that this novel techniqe can be tilized for real electron beam energy spread cooling in an x-ray FEL linear accelerator [19]. In this paper, a systematic stdy on the PEHG is presented. The principle of the PEHG is introdced in section. Analytical estimates and one-dimensional simlation reslts are given in section 3 to present the physical mechanism of the phase-merging effect and the possibility of

4 New J. Phys. 16 (014) Figre 1. (a) The original PEHG scheme with a TGU modlator for energy modlation and phase-merging simltaneosly; (b) a PEHG variant with a normal modlator for energy modlation and a TGU for introdcing the phase-merging effect. imprinting ltra-high harmonic microbnching into the electron beam with a relatively small energy spread sing this techniqe. Section 4 gives an optimized design for a soft x-ray FEL with realistic parameters based on the PEHG. Some practical constraints that may deteriorate the performance of PEHG are stdied in section 5. Finally, we conclde in section 6.. Principles of PEHG The initially proposed PEHG consists of a dogleg followed by a HGHG configration with a TGU modlator, as shown in figre 1. Rectanglar coordinates are given in scheme 1, where x and y represent the horizontal and vertical directions, respectively. The dogleg with dispersion η is sed to transversely disperse the electron beam, while the TGU modlator is sed for the beam energy modlation and to precisely maniplate the electrons in the horizontal dimension. It is fond that these two fnctions of the TGU modlator can be separately performed by employing a modified design, as shown in scheme of figre 1. Here, a normal modlator is sed for the energy modlation, and the TGU is responsible only for transverse maniplation of the electrons, a design that will be mch more flexible for practical operation. In principle, the TGU in scheme can be replaced by other kinds of devices with a transverse gradient magnet field e.g. specifically designed wigglers or small chicanes. For convenience of theoretical analysis, we consider scheme first, and then extend the conclsions to scheme 1. Following the notation of [15], we also assme an initial Gassian beam energy distribtion with an average energy γ mc and se the variable p = γ γ σ 0 ( 0) γ for the dimensionless energy deviation of a particle, where σ γ is the rms energy spread. Then the initial longitdinal phase space distribtion shold be f p = N exp p π. We se ( ) ( ) 0 0 ( ) χ = x x0 σx as the dimensionless horizontal position of a particle, where x 0 is the central beam position in the horizontal plane and σ x is the initial horizontal rms beam size. Then the horizontal distribtion of the electron beam can be written as g ( χ) = N exp ( χ ) π 0 0. After the dogleg, χ is changed to χ = χ +Dp, (1) where D = ησ γ σxγ is the dimensionless transverse dispersion of the dogleg, and the horizontal distribtion becomes 3

5 New J. Phys. 16 (014) N0 g ( p, χ) = ( χ Dp) π exp 1, () 1 where χ now refers to the vale at the entrance of the modlator. After passing throgh the modlator, the electron beam is modlated with an amplitde A = Δγ σγ, where Δγ is the energy modlation depth indced by the seed laser, and the dimensionless energy deviation of the electron beam becomes p = p + Asin( kz s ), where k s is the wave nmber of the seed laser. The two-dimensional distribtion after the seed laser modlation can be written as N0 h ( ζ, p, χ) = ( p A ζ) χ ζ π { D( p A ) } exp sin exp sin, (3) where p refers to the vale after the modlator and ζ = kz s is the phase of the electron beam. When the transversely dispersed electron beam is frther sent throgh a TGU with period length λ, period nmber N, transverse gradient α, and central dimensionless parameter K 0, electrons at different horizontal positions will see different ndlator K vales, where K ( x) = K0 ( 1 + αx) [17]. According to the FEL resonant eqation, different K vales will reslt in different path lengths for a given beam energy, and this converts the longitdinal coordinate z of electrons with different horizontal positions into λ N Kx = + ( ) K 0 LK m 0 1 z z = z + αχσ + ( αχσ ), (4) x x γ γ where Lm = Nλ is the length of the TGU. Considering that the horizontal beam size is sally qite small in FEL, eqation (4) can be re-written as LK m 0 ασx z = z + χ, (5) γ and then the electron beam distribtion after TGU becomes: N0 h ( ζ, p, χ) = ζ χ π { p A ( T ) } exp 1 sin 1 χ ζ χ exp { D p Asin ( T ) }, (6) where ζ now refers to the new phase of the electron after passage throgh the TGU and kl s mk0 ασx T = (7) γ is the dimensionless gradient parameter of the TGU. Finally, after passing throgh the DS with the dispersive strength of R 56, the longitdinal beam distribtion evolves to N0 h ( ζ, p, χ) = ζ χ π { p A ( T Bp) } exp 1 PEHG sin 1 χ ζ χ exp { D p Asin ( T Bp) }, (8) 4

6 New J. Phys. 16 (014) where ζ is the new phase of the electron after passage throgh the DS and B = R 56 k s σγ γ is the dimensionless strength of the DS. Integration of eqation (8) over p and χ gives the beam density N as a fnction of ζ, N ( ζ) = dχ dph ζ χ (, p, ) PEHG, and the bnching factor at the nth harmonic can be written as 1 inp + = χ χ ζ + ζ + b dpe ( TD B) int in f ( p) g ( ) e ( AB sin ) = J [ nab] e ( 1 ) n( TD B) e ( 1 )( nt) n n. (9) 0 0 N 0 For the case withot TGU (i.e. T = 0), eqation (9) redces to the well-known formla for the bnching factor in a standard HGHG FEL. For the harmonics of n > 4, the maximm of the Bessel fnction in eqation (9) is abot 0.67 n 13 and is achieved when its argment is eqal to n n 13. Ths for a given energy modlation amplitde A, the optimal strength of the DS shold be ( ) 13 B = n n na. (10) The maximm of eqation (9) will be achieved when TD = B, which gives an optimized relation of α and η ( ) γ n n αη = nak L K σ s m 0 γ. (11) When we adopt a large A and η, or a small horizontal size σ x, the third term in the righthand side of eqation (9) can be qite close to nity, and the maximal bnching factor of the nth harmonic for PEHG will approach 13 b 0.67 n, (1) n which is mch larger than that of a standard HGHG. For scheme 1 shown in figre 1, the energy modlation process and the phase-merging process are accomplished simltaneosly when the electron beam passes throgh the TGU modlator. The relative phase advance of the electron cased by the gradient of the TGU is the same in both schemes. However, a factor of 1/ shold be introdced in the right-hand side of eqation (7) for scheme 1, becase the energy modlation increases approximately linearly with the modlator period nmber N m, and the phase advance obtained by integration over the modlator length contribtes a factor of 1/. Ths for scheme 1, the optimal dimensionless gradient of the TGU shold become TD = B, and then the relation between α and η becomes: ( ) 4γ n n αη = nak L K σ s m 0 γ This reslt is precisely consistent with the earlier reslts given in [18]. (13) 3. Physical mechanism of PEHG The physical mechanism behind PEHG is transverse-longitdinal phase space copling. The evoltion of the beam longitdinal phase space for scheme is illstrated in figre. For 5

7 New J. Phys. 16 (014) Figre. The longitdinal phase space evoltion in scheme : (a) the initial phase space after passing throgh the dogleg; (b) the phase space at the exit of the conventional modlator; (c) the phase space at the exit of the TGU; (d) the phase space at the exit of the DS. simplicity, here we assme the horizontal beam size σx = 0 and only show the phase space within one seed laser wavelength region. The energy modlation amplitde is chosen to be A = 3 here, and the optimized condition for the 50th harmonic bnching is B = TD 0.35 according to eqation (10). The initial longitdinal phase space after passing throgh the dogleg is shown in figre (a), where different colors represent different regions of beam energy and also the different horizontal positions of the electrons with respect to the reference electrons with central beam energy. After interaction with the seed laser in the conventional modlator, the longitdinal phase space of the beam evolves to that shown in figre (b). The strong optical field indces a rapid correlated growth of the electron beam energy spread. When the beam travels throgh the TGU, electrons of different colors (different transverse positions) will ndergo different ndlator K vales, ths reslting in the different travel path lengths in the TGU. By properly choosing the gradient of the TGU according to eqation (11), the phase space will evalate to that in figre (c). Dring this process, the electron energy is nchanged. However, arond the zero-crossing of the seed laser, electrons with the same energy will merge into the same longitdinal phase de to the relative phase shift of the electrons in the TGU. This phenomenon is known as the phase-merging effect. After the TGU, the electrons enter the dispersion section where the beam phase space is rotated and the bnching at the desired harmonic is optimized, as shown in figre (d). It has been fond that most of the electrons are 6

8 New J. Phys. 16 (014) Figre 3. Comparison of the bnching factor of PEHG and standard HGHG with different energy modlation amplitdes. The black line is the theoretical prediction of the maximal bnching factor of PEHG. compressed into a small region arond the zero-phase, which indicates that the density modlation has been significantly enhanced for high harmonics. It can be dedced from eqation (9) that the maximal bnching factor of PEHG is mainly determined by the Bessel fnction term, and that it weakly depends on the absolte vale of A when σ x is small or η is qite large. Figre 3 shows the simlation reslts of the maximal bnching factors of PEHG for different energy modlation amplitdes nder the condition of σ x = 0. For comparison, the optimized bnching factors for the standard HGHG with the same energy modlation amplitdes are also shown. One can clearly see that the bnching factor exponentially decreases as the harmonic nmber increases for standard HGHG. However, for PEHG, the bnching factor decreases as n 13 and its maximm fits qite well with the theoretical prediction for n > 4. For a realistic electron beam, the intrinsic horizontal beam size σ x cannot be neglected. It will indce an effective energy spread into the electron beam becase of the transverse field gradient of the TGU. The effective energy spread can be written as [17] σ eff = σ η. (14) x Using the optimized condition of PEHG and plgging eqation (14) into eqation (9), we arrive at [ ] ( ) σ b = J nab e ( 1) nksr56 eff. (15) n n One may find that the bnching factor in eqation (15) redces to the form of a standard HGHG. The only difference is that the initial beam energy spread has been replaced by σ eff. Here, we define an energy spread compression factor C = ησ γ γσx, which can be sed to qantitatively measre the phase-merging effect. The intrinsic beam size is determined by the normalized horizontal emittance ε x and the β fnction. For a relatively short modlator with length L m, it is reasonable to take β L m, and hence σx = εxl m γ. By sing the realistic parameters of the Shanghai soft x-ray FEL (SXFEL) project [0], figre 4 shows the 30th harmonic bnching factor as a fnction of the initial horizontal emittance. The beam energy is 7

9 New J. Phys. 16 (014) Figre 4. The 30th harmonic bnching factor of PEHG as a fnction of the horizontal emittance for different energy modlation amplitdes. Figre 5. Comparison of the bnching factor of PEHG and standard HGHG with realistic parameters. 840 MeV with an energy spread of abot 100 kev, the dispersion of the dogleg is η = 1 m, and the average beta fnction in the short modlator is β = m. The energy modlation amplitde has been changed from 1 to 10, and the strength of the DS and the gradient of the TGU have been trned simltaneosly to optimize the 30th harmonic bnching factor according to eqations (10) and (11). The wavelength of the seed laser is 64 nm. One can see from figre 4 that the bnching factor decreases qickly as the horizontal emittance increases when A is smaller than 3. However, the bnching factor is still acceptable for εx = 1μm rad when A is larger than 6. For the case of εx = 1μm rad and A = 6, the optimized bnching factor comparison between PEHG and HGHG is shown in figre 5. The energy spread compression factor is calclated to be C 5.74 for this case, which increases the harmonic p-conversion efficiency approximately six times for PEHG. It shold be pointed ot that the bnching factors are nearly the same for the two schemes in figre 1 when A is mch larger than 1. However, for a relatively small A, the final longitdinal phase space and the bnching factor will be qite different for these two schemes. Figre 6 shows simlation reslts for these two schemes when A = 0.1 and ε = 0. De to the x 8

10 New J. Phys. 16 (014) Figre 6. The longitdinal phase spaces of the electron beams for scheme 1 (a) and scheme (b) at the entrance to the radiators, along with the corresponding bnching factors (c). non-linear effect dring the modlation process for scheme 1, the bnching factor decreases for all harmonics. However, for scheme, the optimized bnching factor for A = 0.1 is nearly the same as for the A = 3 case, which demonstrates the theoretical predictions. 4. Generation of soft x-ray radiation To illstrate a possible application with realistic parameters and show the parameter optimization method of PEHG, we take the nominal parameters of the SXFEL. The SXFEL aims to generate an 8.8 nm FEL from a 64 nm conventional seed laser throgh a two-stage cascaded HGHG. The electron beam energy is 840 MeV, with a slice energy spread of abot 100 kev. The beam peak crrent is over 600 A. As mentioned above, the bnching factor of PEHG is qite sensitive to the beam emittance. The optimized 30th harmonic bnching factor and three-dimensional gain length of the 8.8 nm radiation as a fnction of the initial horizontal emittance are shown in figre 7. From figre 7(a), one can find that the bnching factor decreases qickly as the emittance increases when η is smaller than 0.5 m, and that the bnching factor can be well maintained for η > 1 m. However, when the dispersion η is too large, it will contribte to a FEL gain redction de to the increased beam size. In order to ensre adeqate gain in the radiator, the dispersion indced beam size is reqired to be no larger than the 9

11 New J. Phys. 16 (014) Figre 7. (a) The 30th harmonic bnching factor and (b) three-dimensional gain length of PEHG as a fnction of the horizontal emittance for different transverse dispersion strengths. Figre 8. Simlated parameters at the linac exit: (a) beam energy and crrent distribtion along the electron beam; (b) slice energy spread and normalized emittance distribtion along the electron beam. intrinsic horizontal beam size contribted by the radiator beta fnction. For SXFEL, the beam size in the radiator is abot the 100 μm level, considering the beam energy spread of 100 kev, and the maximm dispersion permitted is abot 1 m. The three-dimensional FEL gain length as a fnction of horizontal emittance with different dispersions is calclated and shown in figre 7(b). The gain length can be well controlled nder m for the 1 μm rad emittance and 1 m dispersion case, which is reasonable for a seeded soft x-ray FEL. Becase the transverse beam size shold be calclated by σ = σx + σ y, where the σ y is nchanged for different η, the three-dimensional gain length will not change too mch (from 1.5 m to 1.8 m) when η is smaller than 1 m for a 1 μm rad emittance. With the above parameters, start-to-end tracking of the electron beam, inclding all the components of SXFEL, has been carried ot. The electron beam dynamics in the photo-injector was simlated with ASTRA [1] to take into accont space-charge effects. ELEGANT [] was then sed for the simlation in the remainder of the linac. The slice parameters at the exit of the linac are smmarized in figre 8. The beam energy in the central part of the electron beam is arond 840 MeV, and the peak crrent is abot 600 A. A constant profile is maintained in an approximately 600 fs wide and over 500 A region. A normalized emittance of approximately 10

12 New J. Phys. 16 (014) Figre 9. Comparisons of the horizontal beam size (a) and beam central position changes (b) before and after the dogleg in the simlation. Figre 10. FEL performance of PEHG at 8.8 nm. (a) The radiation peak power as a fnction of ndlator distance, and (b) the radiation spectrm at satration μm-rad and a slice energy spread abot 100 kev are observed in figre 8(b). Figre 9 shows the transverse beam center and beam size changes after passing throgh the dogleg. The average vale of the horizontal beam size σ x is increased from abot 60 μm to abot 70 μm, which will not significantly affect FEL performance. However, as shown in figre 9(b), the horizontal beam position changes a lot de to the large energy chirp in the electron beam. The FEL performance of PEHG was simlated by the pgraded three-dimensional FEL code GENESIS [3] based on the otpt of ELEGANT. A 64 nm seed plse with the longitdinal plse length mch longer than the bnch length is sed in the simlation. The length of the TGU modlator is abot 1 m, with a period length of 80 mm and K vale of arond 5.8. To maximize the bnching factor at the 30th harmonic of the seed laser, the 1 optimized parameters are set to be A = 6, B = 0.18, α = 0 m, η = 0.5 m. The maximal bnching factor along the electron beam at the entrance of the radiator exceeds 5%, which fits qite well with the theoretical prediction of figre 7(a). The period length of the radiator is 5 mm with a K vale of abot 1.3. The evoltion of the radiation peak power is shown in figre 10. The large bnching factor at the entrance to the radiator offered by the PEHG scheme is responsible for the initially steep qadratic growth in power. The significant enhancement in performance sing the PEHG is clearly seen in figre 10(a), where the peak power of the 30th harmonic radiation exceeds 400 MW, which is qite close to the otpt peak power of the original design of SXFEL with two-stage HGHG. Moreover, the 8.8 nm radiation satrates 11

13 New J. Phys. 16 (014) within the 15 m long ndlator, which is in the range of the original design of SXFEL. As mentioned above, the optimized condition for the bnching factor of PEHG is TD = B, which indicates that the beam energy chirp effect indced by the dispersions of the TGU and the DS will conteract each other. Ths, the otpt wavelength of PEHG will be immne to the large residal beam energy chirp in figre 8(a). The single-shot radiation spectrm at satration is shown in figre 10(b), from which one can find that the bandwidth of the radiation at satration is qite close to Forier-transform-limited. 5. Some practical isses The niqe featre of PEHG is the tilization of an ndlator with a transverse field gradient of α. We will discss some practical isses that may affect the performance of PEHG in this section. Unlike a conventional planar modlator ndlator in standard HGHG, the TGU in PEHG will introdce an additional kick in the transverse dimension de to the gradient field, which will reslt in a deviation of the electron trajectory in the horizontal plane. For a TGU, the magnetic field distribtion can be written as B ( x, z) = B ( 1 + αx) sin k z, (16) y 0 where B 0 is the reference peak magnetic field. The electric field of the seed laser can be simply represented as ( ) ( s ) E z = E sin kz + φ, (17) x 0 0 where E 0 is the peak electric field and φ 0 is the carrier envelop phase of the seed laser. According to Maxwell s eqations, the trajectory eqations of the electron with initial horizontal position x 0 in the modlator can be written as dx dz = dx cdt, (18) dx 1 dx e dx = = B ( 1 + αx) sin k z. (19) 0 dz c dt γmc dt Considering that the electron wiggles in the ndlator, the horizontal position of the electron beam in the modlator can be written as K0 x( z) = x0 + sin kz. (0) k γ Then eqation (19) is changed to dx dz e dx K 0 = B 1 + α x + sinkz sinkz γmc dz kγ. (1) e dx αk 0 = B0 ( 1 + αx0) sinkz + sin kz γmc dz k γ 0 0 1

14 New J. Phys. 16 (014) Figre 11. The electron trajectory in the horizontal dimension: (a) TGU modlator; (b) TGU with correction magnetic field. Figre 1. The 30th harmonic bnching factors at varios shots for a flctating amplitde of the seed laser power. Integration of this formla over z gives the total x-deviation of the electron after passage throgh the TGU modlator α Δx = K Nmλ. () 0 4γ According to the parameters sed in section 5, Δx is calclated to be abot 96 μm, which will not significantly affect the performance of PEHG. Moreover, it can be derived from eqation (1) that the x-deviation can be compensated by introdcing an external magnetic field: B external αk0 = B0. (3) k γ To illstrate the particle trajectory in the TGU and check the simlation reslts of GENESIS, we developed a three-dimensional algorithm based on the fndamentals of electrodynamics when considering the appearance of a gradient ndlator magnetic field and laser electric field in the time domain [4]. The simlation reslts are shown in figre 11. It can 13

15 New J. Phys. 16 (014) be fond from figre 11(a) that the deviation of the electron trajectory in the horizontal dimension at the exit of the modlator is abot 95 μm, which fits qite well with the theoretical calclation. As the seed laser size is mch larger than the beam size i.e. abot 1000 μm (rms) in the simlation the horizontal deviation will not significantly affect the modlation process. For the case shown in figre 11(a), the maximal bnching factor decreases from 5.6% to abot 5.4%, and this deviation can be easily compensated by introdcing an external magnetic field of abot 5.6 Gs, as shown in figre 11(b). The sensitivity of the bnching factor to the shot-to-shot flctations of the laser power has also been stdied by introdcing random flctations of the laser power in the modlator. The reslting 1000 shots of the flctations of the 30th harmonic bnching factor are shown in figre 1. One can find that, with ±5% tolerance on the seed laser peak power, the bnching factor of PEHG can be well maintained over 5%. 6. Conclsion In smmary, intensive analytical and nmerical investigations on PEHG schemes have been performed. The reslts demonstrate the potential of generating ltra-high harmonic radiation with a relatively small energy modlation by a single-stage PEHG. It is fond that the optimized nth harmonic bnching factor of PEHG is almost determined by the maximal vale of the nthorder Bessel fnction, which decreases as n 13. The transverse dispersion-indced increase in beam size will not degrade the FEL performance when the system parameters are properly set. For a PEHG FEL operated at 8.8 nm directly from a 64 nm conventional seed laser, the nmerical example demonstrates an otpt peak power exceeding 400 MW, which is comparable with that of the original two-stage HGHG design. Considering the ability to exploit the fll electron bnch in the PEHG, the otpt bandwidth and the plse energy will be significantly improved, ths leading to a FEL average brightness two orders of magnitde higher than the two-stage HGHG baseline. In addition to the generation of flly coherent radiation, the concept of the phase-merging effect also offers a novel method for flexible beam control, which may be sefl for cooling the electron beam energy spread, and for ltra-intense and ltra-short FEL plse generation. Acknowledgments The athors wold like to thank B Li, T Zhang, G Stpakov, Y Ding, D Xiang and Z Hang for helpfl discssions and sefl comments. This work is spported by the Major State Basic Research Development Program of China (011CB808300) and the National Natral Science Fondation of China ( , and ). References [1] Emma P et al 010 Nat. Photon [] Ishikawa T et al 01 Nat. Photon [3] Feldhas J et al 1997 Opt. Commn [4] Geloni G, Kocharyan V and Saldin E 011 J. Mod. Opt

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