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EEHG-assisted FEL schemes for attosecond X-ray pulses generation
Nuclear Instruments and Methods in Physics Research A 621 (2010) 97–104
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
EEHG-assisted FEL schemes for attosecond X-ray pulses generation Jun Yan, Hai-Xiao Deng n, Dong Wang, Zhi-Min Dai Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, Shanghai 201800, China
a r t i c l e in fo
abstract
Article history: Received 13 March 2010 Received in revised form 6 May 2010 Accepted 11 June 2010 Available online 17 June 2010
In this paper, the schemes of echo-enabled harmonic generation (EEHG) assisted free electron laser (FEL) for generating attosecond soft X-ray pulses are further investigated. We present brief analytical models and three-dimensional simulations for comparison studies of such schemes reported earlier. Moreover, on the basis of these analyses, a more compact and robust EEHG-assisted FEL scheme is proposed for pump-probe experiments using two-color attosecond X-ray pulses. & 2010 Elsevier B.V. All rights reserved.
Keywords: Pump-probe Attosecond Echo-enabled harmonic generation Simulation
1. Introduction Coherent attosecond X-ray sources are urgently needed in modern physical research [1,2]. For example, it can be utilized in high-resolution real-time imaging. Also, attosecond X-ray pulses can be used as probes to investigate complex electron dynamics in materials, molecules and atoms in the so-called pump-probe experiments in which two successive pulses with a specific time interval are needed to initiate and probe the electron dynamics. High-order harmonic generation (HHG) [3–5] of an intense laser is one of the most popular approaches for attosecond X-ray pulses generation, which is widely utilized by the conventional laser community. On the other hand, free electron laser (FEL) plays more and more significant role in the scientific community due to its high brilliance and excellent coherence. Till now, various FEL schemes [6–12] have been proposed to generate coherent attosecond X-ray pulses. To generate attosecond pulses from an intense seed laser via FEL mechanism, the electron beam density has to be modulated on attosecond scale. The coherence of density modulation, known as harmonic bunching, is crucial to the properties (coherence, power, pulse duration, etc) of the generated attosecond pulses. Thus the ever more effective density modulation of the electron beam is of great interest by FEL community [13–15]. Recently a novel FEL scheme with remarkably high harmonic bunching, called echo-enabled harmonic generation (EEHG) [16,17], was proposed, based on which two proposals [18,19] are suggested for the generation of coherent attosecond X-ray pulses.
n
Corresponding author. E-mail address: [email protected] (H.-X. Deng).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.06.099
Fig. 1(a) shows the original EEHG-assisted scheme proposed in Ref. [18] to generate an isolated attosecond X-ray pulse. Due to the contribution of M1, DS1 and M2 (M and DS correspond to modulators and dispersive sections, respectively), the electron beam experiences density modulation and thus forms the energy bands in its phase space. In M3, an intense few-cycle seed laser provides the electron beam an energy chirp, which, with the help from the downstream DS2, converts the energy bands to the longitudinal microbunching. On the other hand, the regions modulated by different cycles of the few-cycle laser have different energy chirps and thus only a small part of the electron beam can be chosen to generate a single attosecond FEL pulse. In order to generate two attosecond X-ray pulses for pump-probe experiments, Ref. [19] proposed an alternative EEHG-assisted scheme in which M2, the modulator to generate a relatively small energy modulation after DS1, is removed and two successive combinations of modulator+dispersive section+ radiator (M +D+ R) is employed after DS1, as shown in Fig. 1(b). In this scheme, the tail part of the electron beam is used to generate the first attosecond pulse in the first M+ D+R and then the fresh part at the beam head generates the second attosecond pulse in the second M+D+ R. In this paper, we show further study on generation of attosecond X-ray pulses using EEHG schemes. By using a simple but extremely useful analytical model, the available maximum harmonic number of the two abovementioned schemes is derived and discussed. Next, their performances are numerically illustrated and compared by three-dimensional (3D) simulation. It is found that for the pump-probe experiments using an infrared seed pulse and a soft X-ray attosecond one, a high-precision control of the time interval between two pulses on the sub-10
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attosecond scale becomes possible. Finally, an EEHG-assisted FEL approach is proposed to generate two-color attosecond pulses for pump-probe experiments, which is more compact and robust compared to the former one [19].
2. Model for the principles of EEHG-assisted attosecond pulses generation In this section, a semi-empirical model, mainly focused on the evolution of the electron beam’s phase space, is developed to help us analyze the EEHG-assisted FEL schemes for attosecond X-ray pulses generation. For clarity, hereafter, the schemes regarding Fig. 1(a) and (b) are called case I and case II, respectively. According to the FEL process and the typical phase space of EEHG, Fig. 2(a) shows three energy bands with p¼0, 7 Dp at the entrance of M3 in case II, where p ¼(E E0)/sE, E0 is the average beam energy and sE is the initial rms energy spread. For simplicity, the energy spread of each energy band and the difference of the spacing between adjacent energy bands are neglected. After being modulated by the large seed power in M3, a linear energy chirp h is induced for the electron beam near y ¼0 as shown in Fig. 2(b), where y is the longitudinal electron phase. Here, Ai ¼ DEi/sE, DEi is the energy modulation in ith modulator, ðiÞ ðiÞ ki sE =E0 , R56 is the dispersive strength of ith DS and ki is the Bi ¼ R56 wave number of the ith seed laser. With the condition of hB2 ¼1, the energy bands are erected as shown in Fig. 2(c). A brief deduction shows that the longitudinal spacing between two adjacent bands in Fig. 2(c) is Dp/h, which indicates a microbunching on the scale of Dp/h. On the other hand, the energy modulation in M3 is A3 sin y whose derivative at y ¼0 is A3 ¼h. Then the relationship between the interested FEL wavelength and A3 can be obtained as
Dp A3
¼
lFEL 2p, lseed
ð1aÞ
or
where lFEL is the concerned harmonic radiation wavelength, lseed the wavelength of the few-cycle seed laser and n is the optimized harmonic number of seed laser. Furthermore, FEL pulse duration can be approximated as
DtFEL ¼
ð1bÞ
with A3 B2 ¼ 1,
ð2Þ
ð3Þ
Once A1 and B1 fixed (i.e. Dp fixed), Eq. (1b) shows that the optimized harmonic number n is proportional to A3 while Eq. (3) shows that FEL pulse duration is inversely proportional to A3. In order to check the linear dependence of n on A3, we numerically integrate the distribution function at the exit of DS2 (Eq. (2) in Ref. [17]) over p, which gives the current profile. Then the optimized harmonic number can be found through the Fourier transform of the integrated current profile. The results are shown in Fig. 3. The coefficient of determination R2 is 0.99991 in a large range, which indicates a perfect linear dependence of n on A3. Although Eq. (1b) indicates that the harmonic number can be pushed up proportionally with increase in A3, Eq. (2) determines the available maximum harmonic number of such kind of scheme. Actually, even if B2 ¼0, the modulator M3 itself has a small intrinsic dispersive strength [20] @y 4p N ¼ , g0 2 @g
ð4Þ
where N is the undulator period number of M3. Now we rewrite Eq. (4) with the normalized notations BM3 ¼ 2p
sE E0
N,
ð5Þ
where BM3 is the dispersive strength of M3 and Eq. (2) becomes A3 ðBM3 þ B2 Þ ¼ 1:
ð6Þ
Combining Eqs. (1b), (5), (6) and B2 Z0, one obtains nr
2p A , n¼ Dp 3
2A1 lFEL A1 lseed 1 ¼ : Dp c pc A3
1
DpNðsE =E0 Þ
,
ð7Þ
which gives the available maximum harmonic number of the EEHG-assisted FEL scheme for attosecond pulses generation. Physically, if A3 is too large, the energy bands are overbunched in M3 and the formed harmonic bunching will be smeared out.
Fig. 1. EEHG-assisted FEL scheme to generate (a) single attosecond X-ray pulse and (b) two attosecond X-ray pulses.
|dp/dθ|=-h
p
Δp/h
Δp Δp
p
p
θ
Δp/h θ
θ
Fig. 2. Longitudinal phase space near y ¼ 0 in case II at (a) the entrance of M3, (b) the exit of M3 and (c) the exit of DS2. The colored lines represent ideal energy bands.
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Table 1 Beam and undulator parameters for generation of single attosecond soft X-ray pulse. Beam energy (GeV) Slice energy spread (keV) Peak current (kA) Normalized emittance (mm mrad) Modulator period length (cm) Modulator period number (M1/M3) Wavelength of seed laser (nm) Undulator period length for radiator 1 (cm) Undulator period number for radiator 1
3 150 1 1 20 6/2 800 4 12
Fig. 3. Dependence of n on A3 obtained from numerical integral of distribution function at the exit of DS2 in caseII, where A1 ¼2, B1 ¼8, B2 ¼1/A3. R42 is the coefficient of determination in linear fitting.
Fig. 4. Scheme to realize pump-probe between an infrared pulse and a soft X-ray FEL pulse.
It is worth mentioning here, typically, for Dp 1, N 10 and sE/E0 10 4, the available maximum harmonic number is about 1000, which is independent of the wavelength of seed laser. In the case involving M2, i.e. case I, the energy modulation after M3 is hy + A2 sin (my + j0), where m is the wavelength ratio of the seed laser in M3 to that in M2 and j0 is the phase difference between the seed laser in M3 and that in M2. Accordingly, for small y, Eq. (1b) becomes n¼
2p
Dp
ðA3 þ mA2 cos j0 Þ:
ð8Þ
From Eq. (8), it is found that the scheme in case I requires a high synchronization between the seed laser in M3 and that in M2 for a steady operation. Moreover, in the case where M2 is involved, the additional energy modulation induced in M2 enlarges the energy spread of each energy band and thus reduces the efficiency of density modulation [15]. Therefore, for a concerned harmonic number, FEL performance in case II will be better than that in case I.
3. Numerical approaches and results In Section 2, we analytically investigated cases I and II to generate attosecond FEL pulses and concluded that the latter should have a better performance than the former. In this section, the scheme without the presence of M2, which is actually the first stage of case II (see Fig. 4), is evaluated through a 3D simulation approach with the same beam and undulator parameters as those reported in Ref. [18] (see Table 1). An intense 800 nm laser is used as a seed to bring energy modulation to the electron beam in modulators. The seed pulse in M3 is operated under few-cycle mode (see Fig. 5), which together with the generated 1 nm FEL pulse can be used for pump-probe experiments.
Fig. 5. Normalized electric field of the few-cycle seed laser.
As for the simulation approach, it benefits from the flexible management of the 6D particle phase space generated by FEL code GENESIS [21]. After the regular time-dependent simulation is carried out in M1 and DS1 by GENESIS, the phase space of the macroparticles is extracted and imported into a 3D tracking code, which serves to simulate the interaction between the electron beam and a few-cycle seed laser in M3. The 3D tracking code is based on the fundamentals of electrodynamics, i.e. the electron’s trajectory is determined by the undulator magnetic field and laser electric field in time domain. Since the harmonic number from M3 to the radiator is as high as 800 and it refers to an ultra-short pulse evolution, at the entrance of the radiator, the whole macroparticle beam is sliced in the scale of the FEL resonance wavelength of the radiator (i.e. 1 nm) and the FEL simulation in the radiator is performed through the modified model of harmonic generation FEL [22,23]. Fig. 6 shows the longitudinal beam phase space at the entrance of the radiator. Only the region shown in the zoom-in figure is erected and thus presents longitudinal microbunching, while other regions in the electron beam are still inclined due to the unmatched A3 and B2. Once such an electron beam passes through an appropriate radiator, only the erected region radiates coherently and generates an intense attosecond radiation pulse. As for a practical consideration, the quantum fluctuation of synchrotron radiation due to the large dispersive strength in DS1 composed of four bending magnets will induce some additional rms energy spread, which can be calculated as [18,19,24] 1=2 r2 1 DsE ¼ 0:662 e y3B g5 E, ð9Þ a L
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Fig. 6. Longitudinal phase space of the electron beam at the entrance of the radiator.
Fig. 7. Local (a) current profile and (b) bunching factors in the scale of 1 nm.
where yB and L are the bending angle and length of each bending magnet, respectively, re is the classical electron radius, and a E1/137 is the fine structure constant. In order to keep the additional energy spread much smaller than the spacing between two adjacent energy bands, we chose L as 2.5 m, yB as 58.6 mrad and D12(D34), the distance between the first and second magnet (also that between the third and fourth magnet), as 1 m to obtain our required Rð1Þ 56 (i.e. 18.368 mm). According to Eq. (9), the energy spread induced by each bending magnet is 1.2 keV, which is much smaller than the spacing between two adjacent energy bands approximated by Elseed =2Rð1Þ 56 ¼ 65:3 keV. Similarly, the quantum fluctuation of synchrotron radiation in M3 will also induce additional energy spread, which can be
estimated for K 51 as [18,19,25]
DsE ¼ 4:16
re2
a
g2 Nu
eB me c
2 !1=2
E,
ð10Þ
where Nu is the undulator period number, me is the electron rest mass and B is the peak magnetic field in M3. Accordingly, the energy diffusion in M3 is 0.9 keV, which can also be neglected when compared with the spacing between two adjacent energy bands. Besides the synchrotron radiation in DS1 and M3, there are some other detrimental effects, which will possibly destroy the formed microstructure. For example, the effect of energy chirp
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101
Fig. 8. (a) Longitudinal and (b) transverse profile of the FEL pulse at the exit of radiator.
Table 2 Comparison results of cases I and II.
Few-cycle seed pulse
Generated FEL pulse
Peak power (GW) Rayleigh length (m) Waist position (m) Peak power (MW) Pulse duration (as)
Case I
Case II
350 1 0.2 130 20
420 1 0.2 200 20
and curvature along the electron bunch should be of concern. Energy chirp effects on an EEHG FEL have been studied in Ref. [26], which shows that accelerator designs controlling the uniformity of beam longitudinal phase space for high gain harmonic generation [27] FEL are sufficient for EEHG FEL. In our case, since only an 800 nm section of the electron bunch is handled, any reasonable energy variation due to energy chirp effects will be less than the slice energy spread. Furthermore, such kind of small energy chirp will be totally smeared out by the large energy chirp introduced in M3. On the other hand, as the local beam current profile and local bunching factors near the erected beam region are shown in Fig. 7, an extremely high current is modeled before the FEL amplification process. Then the longitudinal space charge force will probably be a big issue to prevent the maintenance of microstructure for such a high current intensity. These potential detrimental effects will be the concentration of our future work. After the bunching process, the electron beam enters the radiator, which is set for 1 nm FEL resonance. The undulator period number of radiator is limited to 12 to prevent the increase of FEL pulse duration caused by slippage. Finally, the FEL pulse with peak power of about 200 MW and pulse duration of about 20 attosecond (FWHM) is obtained (see Fig. 8(a)). For better modulation [17], the rms radius of seed laser is 0.5 mm, 25 times larger than that of the electron beam (i.e. 20 mm), based on which the final transverse distribution of the 1 nm FEL pulse is illustrated in Fig. 8(b). A similar optimization and simulation of case I [18] was also carried out. We set j0 ¼0 in Eq. (8) and thus Eq. (8) becomes
Fig. 9. Relative offset of FEL wavelength versus the time interval shift between the pump pulse and the probe one. For Dt-axis, positive (negative) values correspond to delaying (shortening) the time interval between the pump pulse and the probe pulse.
n¼ 2p/Dp(A3 + mA2), which, compared to Eq. (1b), indicates that to obtain the same harmonic number, a smaller energy chirp is needed for case I due to the contribution from mA2. Here we list the comparison results in Table 2. One can find that the peak power of 1 nm FEL pulse in case II is larger than that in case I, which is consistent with the analytical estimation and statement in Section 2. The 800 nm few-cycle seed pulse and the generated 1 nm FEL pulse can be used for pump-probe experiments, in which the time interval between these two pulses is quite important. Usually, the time interval between two pulses is mainly contributed by the path length difference between the electron beam and the seed pulse when they pass through DS2, which can be calculated as 4L(yB/sin yB 1)+ 2D12(1/cos yB 1). In our case, Rð2Þ 56 ¼ 0:0252 mm; which can be realized by L¼0.8 m, yB ¼3.33 mrad and D12 ¼D34 ¼0.6 m. Then the time interval between two pulses is
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Fig. 10. Scheme to generate two attosecond pulses.
Fig. 11. (a), (b) and (c) shows the longitudinal phase space of the electron beam at the exit of M3, Radiator 1 and DS3, respectively (with its corresponding current profile).
J. Yan et al. / Nuclear Instruments and Methods in Physics Research A 621 (2010) 97–104
about 42 fs. The time interval can be changed by adjusting the optical path of the 800 nm seed pulse, or in a more complicated manner by inserting some additional magnetic elements such as quadrupoles and sextupoles in DS2 to adjust the path length of the electron beam while keeping Rð2Þ 56 unchanged. However, the resolution of such kind of control is presently limited to sub-fs scale. Now we discuss the effects of the carrier-envelope phase (CEP) of the few-cycle seed pulse on output FEL performance. In reality, the electric field is non-periodic in the few-cycle laser pulse, and the density modulation on attosecond scale is highly relevant to the CEP of the few-cycle seed laser pulse. It is found that the arrival time of the attosecond pulse with respect to the infrared seed pulse is apparently proportional to the CEP change of the few-cycle laser. Thus an accurate time interval control can be realized by changing the carrier-envelope phase (CEP) of the 800 nm few-cycle pulse. To our knowledge, the minimum CEP jitter under phase stabilization technology is at the level of 51 [28], which corresponds to 37 as resolution. However, CEP shift will inevitably be accompanied by the offset of the output FEL wavelength. Fig. 9 indicates that within the time interval shifts of several hundreds of attosecond, the offsets of FEL wavelength from the case with CEP¼ 0 are almost negligible compared with the intrinsic bandwidth of an attosecond pulse. Therefore, by locking shot-by-shot CEP [28], the time interval between the infrared pump pulse and the soft X-ray probe can be stably controlled and shifted with the precision of about tens of attosecond. Recently, various single-shot CEP measurement techniques without the need for CEP stabilization were experimentally demonstrated [29,30] in which a precision as high as p/300 can be guaranteed at an optimum measurement point. With the online CEP measurement using such a technology, accurate time interval with 4.5 attosecond resolution can be carried out between the 800 nm few-cycle seed laser and the generated ultra-fast 1 nm pulse, even without the abovementioned CEP stabilization. In Section 2, the theoretical estimation indicates a maximum harmonic number of 1000 for this kind of scheme. Therefore, an alternative case where an intense 200 nm few-cycle seed laser interacts with a 6 GeV electron beam in the scheme shown in Fig. 4 is checked to generate 0.15 nm hard X-ray FEL pulses. The simulation works out a 0.15 nm attosecond pulse with peak power of about 30 MW and pulse duration of about 3 attosecond. However, in this case, the required system parameters are pretty stringent, e.g. a 200 nm laser with several TW peak power is probably unrealistic. Moreover, since Angstrom-scale microbunching is quite vulnerable, the second-order effects (emittance and current) when the beam drifts through the radiator and the chicanes become significant. Therefore, there is still room for further improvements in terms of this case.
region 1 overbunched (see Fig. 11(c)). The different energy chirp of regions 1 and 2 results in their microbunching being optimized at different wavelengths. Radiator 2 is set to satisfy the FEL resonance condition for region 2, which generates the second attosecond FEL pulse. In radiator 2, the FEL amplification in other regions of the electron beam will be suppressed due to their ineffective density modulation. The simulation before DS3 is still based on the beam and undulator parameters listed in Table 1. A new radiator (i.e. radiator 2) with the same undulator period length and undulator period number as that of radiator 1 but whose FEL resonance condition is set for the second FEL pulse is added after DS3. Under these parameters, two attosecond X-ray pulses with the wavelength of 1 and 3.15 nm, respectively, will be obtained using a single 800 nm few-cycle seed pulse. In this case, the quantum diffusion caused by the spontaneous radiation in radiator 1 should also be included. According to Eq. (10) the additional energy spread in radiator 1 is only 0.8 keV, much smaller compared with the spacing between two adjacent energy bands i.e. 65.3 keV. Therefore, the analysis shows that the fine structure in
Fig. 12. Two attosecond FEL radiation profile.
4. Two-color attosecond X-ray pulses generation In Ref. [19], an EEHG-assisted FEL scheme to generate two attosecond soft X-ray pulses was proposed for pump-probe experiments. In this section, we propose a simpler design to generate two attosecond pulses as shown in Fig. 10 where the few-cycle seed pulse only interacts with the electron beam in M3. In this scheme, the principle before DS3 is the same as that in Fig. 4. The beam region modulated by the central cycle of the fewcycle laser (region 1 in Fig. 11(a)) is used to generate the first attosecond FEL pulse in radiator 1. After the beam comes out from radiator 1, DS3 makes the region which interacts with a side cycle of the few-cycle laser (region 2 in Fig. 11(a)) upright while making
103
Fig. 13. Spectrum of two FEL pulses.
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electron beam’s phase space should be able to be preserved before radiator 2. Before performing FEL simulation in two radiators, the time interval between two FEL pulses deserves a detailed discussion. Firstly, region 2 intrinsically advances region 1 by lseed. Secondly, in radiator 1, the first attosecond FEL pulse generated by region 1 catches up region 2 by Nu1l1 (the slippage length), where Nu1 and l1 are the undulator period number and FEL resonance wavelength of radiator 1. Thirdly, when passing through DS3, the path length of region 2 is larger than that of the first attosecond FEL pulse by 4L(yB/sin yB 1)+ 2D12(1/cos yB 1). In our case, Rð3Þ 56 ¼ 0:053 mm; which can be realized by L¼0.8 m, yB ¼4.8 mrad and D12 ¼D34 ¼0.6 m. Then the contribution to the time interval between two pulses from the third factor (about 85 fs) apparently dominates the other two ones. Figs. 12 and 13 show the simulated FEL radiation profile and spectrum from which it can be seen that two FEL pulses with a peak power of a few hundred MW and a pulse duration of about 20 as for the first pulse and about 70 as for the second pulse are obtained. Furthermore, the wavelength of these two pulses can be adjusted by changing the power of the few-cycle seed laser. Besides, the time interval between two attosecond FEL pulses can also be accurately controlled by adjusting the carrier envelope phase of the few-cycle seed laser.
5. Conclusions EEHG-assisted schemes have been suggested to generate coherent attosecond X-ray pulses. In this paper, a semi-empirical, geometrical model is developed to analyze the FEL performances in such kind of schemes and is well validated by three-dimensional simulations. On the basis of these results, a more compact and robust EEHG-assisted FEL scheme is proposed to generate two coherent attosecond soft X-ray pulses for two-color pump-probe experiments. Generally, in such EEHG-assisted FEL scheme for pump-probe experiments, the time interval between the pump and the probe pulses can be adjusted on femtosecond scale by changing the optical path of the few-cycle laser pulse. Studies in this discussion show that changing the carrier envelope phase of the
femtosecond laser may be an alternative way to control the time interval between two pulses at a precision of sub-10 attosecond.
Acknowledgements The authors are grateful to D. Xiang, Z. Huang and G. Stupakov for useful discussions. This work was supported by Shanghai Natural Science Foundation (Grant no. 09JC1416900).
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Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
EEHG-assisted FEL schemes for attosecond X-ray pulses generation Jun Yan, Hai-Xiao Deng n, Dong Wang, Zhi-Min Dai Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, Shanghai 201800, China
a r t i c l e in fo
abstract
Article history: Received 13 March 2010 Received in revised form 6 May 2010 Accepted 11 June 2010 Available online 17 June 2010
In this paper, the schemes of echo-enabled harmonic generation (EEHG) assisted free electron laser (FEL) for generating attosecond soft X-ray pulses are further investigated. We present brief analytical models and three-dimensional simulations for comparison studies of such schemes reported earlier. Moreover, on the basis of these analyses, a more compact and robust EEHG-assisted FEL scheme is proposed for pump-probe experiments using two-color attosecond X-ray pulses. & 2010 Elsevier B.V. All rights reserved.
Keywords: Pump-probe Attosecond Echo-enabled harmonic generation Simulation
1. Introduction Coherent attosecond X-ray sources are urgently needed in modern physical research [1,2]. For example, it can be utilized in high-resolution real-time imaging. Also, attosecond X-ray pulses can be used as probes to investigate complex electron dynamics in materials, molecules and atoms in the so-called pump-probe experiments in which two successive pulses with a specific time interval are needed to initiate and probe the electron dynamics. High-order harmonic generation (HHG) [3–5] of an intense laser is one of the most popular approaches for attosecond X-ray pulses generation, which is widely utilized by the conventional laser community. On the other hand, free electron laser (FEL) plays more and more significant role in the scientific community due to its high brilliance and excellent coherence. Till now, various FEL schemes [6–12] have been proposed to generate coherent attosecond X-ray pulses. To generate attosecond pulses from an intense seed laser via FEL mechanism, the electron beam density has to be modulated on attosecond scale. The coherence of density modulation, known as harmonic bunching, is crucial to the properties (coherence, power, pulse duration, etc) of the generated attosecond pulses. Thus the ever more effective density modulation of the electron beam is of great interest by FEL community [13–15]. Recently a novel FEL scheme with remarkably high harmonic bunching, called echo-enabled harmonic generation (EEHG) [16,17], was proposed, based on which two proposals [18,19] are suggested for the generation of coherent attosecond X-ray pulses.
n
Corresponding author. E-mail address: [email protected] (H.-X. Deng).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.06.099
Fig. 1(a) shows the original EEHG-assisted scheme proposed in Ref. [18] to generate an isolated attosecond X-ray pulse. Due to the contribution of M1, DS1 and M2 (M and DS correspond to modulators and dispersive sections, respectively), the electron beam experiences density modulation and thus forms the energy bands in its phase space. In M3, an intense few-cycle seed laser provides the electron beam an energy chirp, which, with the help from the downstream DS2, converts the energy bands to the longitudinal microbunching. On the other hand, the regions modulated by different cycles of the few-cycle laser have different energy chirps and thus only a small part of the electron beam can be chosen to generate a single attosecond FEL pulse. In order to generate two attosecond X-ray pulses for pump-probe experiments, Ref. [19] proposed an alternative EEHG-assisted scheme in which M2, the modulator to generate a relatively small energy modulation after DS1, is removed and two successive combinations of modulator+dispersive section+ radiator (M +D+ R) is employed after DS1, as shown in Fig. 1(b). In this scheme, the tail part of the electron beam is used to generate the first attosecond pulse in the first M+ D+R and then the fresh part at the beam head generates the second attosecond pulse in the second M+D+ R. In this paper, we show further study on generation of attosecond X-ray pulses using EEHG schemes. By using a simple but extremely useful analytical model, the available maximum harmonic number of the two abovementioned schemes is derived and discussed. Next, their performances are numerically illustrated and compared by three-dimensional (3D) simulation. It is found that for the pump-probe experiments using an infrared seed pulse and a soft X-ray attosecond one, a high-precision control of the time interval between two pulses on the sub-10
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attosecond scale becomes possible. Finally, an EEHG-assisted FEL approach is proposed to generate two-color attosecond pulses for pump-probe experiments, which is more compact and robust compared to the former one [19].
2. Model for the principles of EEHG-assisted attosecond pulses generation In this section, a semi-empirical model, mainly focused on the evolution of the electron beam’s phase space, is developed to help us analyze the EEHG-assisted FEL schemes for attosecond X-ray pulses generation. For clarity, hereafter, the schemes regarding Fig. 1(a) and (b) are called case I and case II, respectively. According to the FEL process and the typical phase space of EEHG, Fig. 2(a) shows three energy bands with p¼0, 7 Dp at the entrance of M3 in case II, where p ¼(E E0)/sE, E0 is the average beam energy and sE is the initial rms energy spread. For simplicity, the energy spread of each energy band and the difference of the spacing between adjacent energy bands are neglected. After being modulated by the large seed power in M3, a linear energy chirp h is induced for the electron beam near y ¼0 as shown in Fig. 2(b), where y is the longitudinal electron phase. Here, Ai ¼ DEi/sE, DEi is the energy modulation in ith modulator, ðiÞ ðiÞ ki sE =E0 , R56 is the dispersive strength of ith DS and ki is the Bi ¼ R56 wave number of the ith seed laser. With the condition of hB2 ¼1, the energy bands are erected as shown in Fig. 2(c). A brief deduction shows that the longitudinal spacing between two adjacent bands in Fig. 2(c) is Dp/h, which indicates a microbunching on the scale of Dp/h. On the other hand, the energy modulation in M3 is A3 sin y whose derivative at y ¼0 is A3 ¼h. Then the relationship between the interested FEL wavelength and A3 can be obtained as
Dp A3
¼
lFEL 2p, lseed
ð1aÞ
or
where lFEL is the concerned harmonic radiation wavelength, lseed the wavelength of the few-cycle seed laser and n is the optimized harmonic number of seed laser. Furthermore, FEL pulse duration can be approximated as
DtFEL ¼
ð1bÞ
with A3 B2 ¼ 1,
ð2Þ
ð3Þ
Once A1 and B1 fixed (i.e. Dp fixed), Eq. (1b) shows that the optimized harmonic number n is proportional to A3 while Eq. (3) shows that FEL pulse duration is inversely proportional to A3. In order to check the linear dependence of n on A3, we numerically integrate the distribution function at the exit of DS2 (Eq. (2) in Ref. [17]) over p, which gives the current profile. Then the optimized harmonic number can be found through the Fourier transform of the integrated current profile. The results are shown in Fig. 3. The coefficient of determination R2 is 0.99991 in a large range, which indicates a perfect linear dependence of n on A3. Although Eq. (1b) indicates that the harmonic number can be pushed up proportionally with increase in A3, Eq. (2) determines the available maximum harmonic number of such kind of scheme. Actually, even if B2 ¼0, the modulator M3 itself has a small intrinsic dispersive strength [20] @y 4p N ¼ , g0 2 @g
ð4Þ
where N is the undulator period number of M3. Now we rewrite Eq. (4) with the normalized notations BM3 ¼ 2p
sE E0
N,
ð5Þ
where BM3 is the dispersive strength of M3 and Eq. (2) becomes A3 ðBM3 þ B2 Þ ¼ 1:
ð6Þ
Combining Eqs. (1b), (5), (6) and B2 Z0, one obtains nr
2p A , n¼ Dp 3
2A1 lFEL A1 lseed 1 ¼ : Dp c pc A3
1
DpNðsE =E0 Þ
,
ð7Þ
which gives the available maximum harmonic number of the EEHG-assisted FEL scheme for attosecond pulses generation. Physically, if A3 is too large, the energy bands are overbunched in M3 and the formed harmonic bunching will be smeared out.
Fig. 1. EEHG-assisted FEL scheme to generate (a) single attosecond X-ray pulse and (b) two attosecond X-ray pulses.
|dp/dθ|=-h
p
Δp/h
Δp Δp
p
p
θ
Δp/h θ
θ
Fig. 2. Longitudinal phase space near y ¼ 0 in case II at (a) the entrance of M3, (b) the exit of M3 and (c) the exit of DS2. The colored lines represent ideal energy bands.
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Table 1 Beam and undulator parameters for generation of single attosecond soft X-ray pulse. Beam energy (GeV) Slice energy spread (keV) Peak current (kA) Normalized emittance (mm mrad) Modulator period length (cm) Modulator period number (M1/M3) Wavelength of seed laser (nm) Undulator period length for radiator 1 (cm) Undulator period number for radiator 1
3 150 1 1 20 6/2 800 4 12
Fig. 3. Dependence of n on A3 obtained from numerical integral of distribution function at the exit of DS2 in caseII, where A1 ¼2, B1 ¼8, B2 ¼1/A3. R42 is the coefficient of determination in linear fitting.
Fig. 4. Scheme to realize pump-probe between an infrared pulse and a soft X-ray FEL pulse.
It is worth mentioning here, typically, for Dp 1, N 10 and sE/E0 10 4, the available maximum harmonic number is about 1000, which is independent of the wavelength of seed laser. In the case involving M2, i.e. case I, the energy modulation after M3 is hy + A2 sin (my + j0), where m is the wavelength ratio of the seed laser in M3 to that in M2 and j0 is the phase difference between the seed laser in M3 and that in M2. Accordingly, for small y, Eq. (1b) becomes n¼
2p
Dp
ðA3 þ mA2 cos j0 Þ:
ð8Þ
From Eq. (8), it is found that the scheme in case I requires a high synchronization between the seed laser in M3 and that in M2 for a steady operation. Moreover, in the case where M2 is involved, the additional energy modulation induced in M2 enlarges the energy spread of each energy band and thus reduces the efficiency of density modulation [15]. Therefore, for a concerned harmonic number, FEL performance in case II will be better than that in case I.
3. Numerical approaches and results In Section 2, we analytically investigated cases I and II to generate attosecond FEL pulses and concluded that the latter should have a better performance than the former. In this section, the scheme without the presence of M2, which is actually the first stage of case II (see Fig. 4), is evaluated through a 3D simulation approach with the same beam and undulator parameters as those reported in Ref. [18] (see Table 1). An intense 800 nm laser is used as a seed to bring energy modulation to the electron beam in modulators. The seed pulse in M3 is operated under few-cycle mode (see Fig. 5), which together with the generated 1 nm FEL pulse can be used for pump-probe experiments.
Fig. 5. Normalized electric field of the few-cycle seed laser.
As for the simulation approach, it benefits from the flexible management of the 6D particle phase space generated by FEL code GENESIS [21]. After the regular time-dependent simulation is carried out in M1 and DS1 by GENESIS, the phase space of the macroparticles is extracted and imported into a 3D tracking code, which serves to simulate the interaction between the electron beam and a few-cycle seed laser in M3. The 3D tracking code is based on the fundamentals of electrodynamics, i.e. the electron’s trajectory is determined by the undulator magnetic field and laser electric field in time domain. Since the harmonic number from M3 to the radiator is as high as 800 and it refers to an ultra-short pulse evolution, at the entrance of the radiator, the whole macroparticle beam is sliced in the scale of the FEL resonance wavelength of the radiator (i.e. 1 nm) and the FEL simulation in the radiator is performed through the modified model of harmonic generation FEL [22,23]. Fig. 6 shows the longitudinal beam phase space at the entrance of the radiator. Only the region shown in the zoom-in figure is erected and thus presents longitudinal microbunching, while other regions in the electron beam are still inclined due to the unmatched A3 and B2. Once such an electron beam passes through an appropriate radiator, only the erected region radiates coherently and generates an intense attosecond radiation pulse. As for a practical consideration, the quantum fluctuation of synchrotron radiation due to the large dispersive strength in DS1 composed of four bending magnets will induce some additional rms energy spread, which can be calculated as [18,19,24] 1=2 r2 1 DsE ¼ 0:662 e y3B g5 E, ð9Þ a L
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Fig. 6. Longitudinal phase space of the electron beam at the entrance of the radiator.
Fig. 7. Local (a) current profile and (b) bunching factors in the scale of 1 nm.
where yB and L are the bending angle and length of each bending magnet, respectively, re is the classical electron radius, and a E1/137 is the fine structure constant. In order to keep the additional energy spread much smaller than the spacing between two adjacent energy bands, we chose L as 2.5 m, yB as 58.6 mrad and D12(D34), the distance between the first and second magnet (also that between the third and fourth magnet), as 1 m to obtain our required Rð1Þ 56 (i.e. 18.368 mm). According to Eq. (9), the energy spread induced by each bending magnet is 1.2 keV, which is much smaller than the spacing between two adjacent energy bands approximated by Elseed =2Rð1Þ 56 ¼ 65:3 keV. Similarly, the quantum fluctuation of synchrotron radiation in M3 will also induce additional energy spread, which can be
estimated for K 51 as [18,19,25]
DsE ¼ 4:16
re2
a
g2 Nu
eB me c
2 !1=2
E,
ð10Þ
where Nu is the undulator period number, me is the electron rest mass and B is the peak magnetic field in M3. Accordingly, the energy diffusion in M3 is 0.9 keV, which can also be neglected when compared with the spacing between two adjacent energy bands. Besides the synchrotron radiation in DS1 and M3, there are some other detrimental effects, which will possibly destroy the formed microstructure. For example, the effect of energy chirp
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Fig. 8. (a) Longitudinal and (b) transverse profile of the FEL pulse at the exit of radiator.
Table 2 Comparison results of cases I and II.
Few-cycle seed pulse
Generated FEL pulse
Peak power (GW) Rayleigh length (m) Waist position (m) Peak power (MW) Pulse duration (as)
Case I
Case II
350 1 0.2 130 20
420 1 0.2 200 20
and curvature along the electron bunch should be of concern. Energy chirp effects on an EEHG FEL have been studied in Ref. [26], which shows that accelerator designs controlling the uniformity of beam longitudinal phase space for high gain harmonic generation [27] FEL are sufficient for EEHG FEL. In our case, since only an 800 nm section of the electron bunch is handled, any reasonable energy variation due to energy chirp effects will be less than the slice energy spread. Furthermore, such kind of small energy chirp will be totally smeared out by the large energy chirp introduced in M3. On the other hand, as the local beam current profile and local bunching factors near the erected beam region are shown in Fig. 7, an extremely high current is modeled before the FEL amplification process. Then the longitudinal space charge force will probably be a big issue to prevent the maintenance of microstructure for such a high current intensity. These potential detrimental effects will be the concentration of our future work. After the bunching process, the electron beam enters the radiator, which is set for 1 nm FEL resonance. The undulator period number of radiator is limited to 12 to prevent the increase of FEL pulse duration caused by slippage. Finally, the FEL pulse with peak power of about 200 MW and pulse duration of about 20 attosecond (FWHM) is obtained (see Fig. 8(a)). For better modulation [17], the rms radius of seed laser is 0.5 mm, 25 times larger than that of the electron beam (i.e. 20 mm), based on which the final transverse distribution of the 1 nm FEL pulse is illustrated in Fig. 8(b). A similar optimization and simulation of case I [18] was also carried out. We set j0 ¼0 in Eq. (8) and thus Eq. (8) becomes
Fig. 9. Relative offset of FEL wavelength versus the time interval shift between the pump pulse and the probe one. For Dt-axis, positive (negative) values correspond to delaying (shortening) the time interval between the pump pulse and the probe pulse.
n¼ 2p/Dp(A3 + mA2), which, compared to Eq. (1b), indicates that to obtain the same harmonic number, a smaller energy chirp is needed for case I due to the contribution from mA2. Here we list the comparison results in Table 2. One can find that the peak power of 1 nm FEL pulse in case II is larger than that in case I, which is consistent with the analytical estimation and statement in Section 2. The 800 nm few-cycle seed pulse and the generated 1 nm FEL pulse can be used for pump-probe experiments, in which the time interval between these two pulses is quite important. Usually, the time interval between two pulses is mainly contributed by the path length difference between the electron beam and the seed pulse when they pass through DS2, which can be calculated as 4L(yB/sin yB 1)+ 2D12(1/cos yB 1). In our case, Rð2Þ 56 ¼ 0:0252 mm; which can be realized by L¼0.8 m, yB ¼3.33 mrad and D12 ¼D34 ¼0.6 m. Then the time interval between two pulses is
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Fig. 10. Scheme to generate two attosecond pulses.
Fig. 11. (a), (b) and (c) shows the longitudinal phase space of the electron beam at the exit of M3, Radiator 1 and DS3, respectively (with its corresponding current profile).
J. Yan et al. / Nuclear Instruments and Methods in Physics Research A 621 (2010) 97–104
about 42 fs. The time interval can be changed by adjusting the optical path of the 800 nm seed pulse, or in a more complicated manner by inserting some additional magnetic elements such as quadrupoles and sextupoles in DS2 to adjust the path length of the electron beam while keeping Rð2Þ 56 unchanged. However, the resolution of such kind of control is presently limited to sub-fs scale. Now we discuss the effects of the carrier-envelope phase (CEP) of the few-cycle seed pulse on output FEL performance. In reality, the electric field is non-periodic in the few-cycle laser pulse, and the density modulation on attosecond scale is highly relevant to the CEP of the few-cycle seed laser pulse. It is found that the arrival time of the attosecond pulse with respect to the infrared seed pulse is apparently proportional to the CEP change of the few-cycle laser. Thus an accurate time interval control can be realized by changing the carrier-envelope phase (CEP) of the 800 nm few-cycle pulse. To our knowledge, the minimum CEP jitter under phase stabilization technology is at the level of 51 [28], which corresponds to 37 as resolution. However, CEP shift will inevitably be accompanied by the offset of the output FEL wavelength. Fig. 9 indicates that within the time interval shifts of several hundreds of attosecond, the offsets of FEL wavelength from the case with CEP¼ 0 are almost negligible compared with the intrinsic bandwidth of an attosecond pulse. Therefore, by locking shot-by-shot CEP [28], the time interval between the infrared pump pulse and the soft X-ray probe can be stably controlled and shifted with the precision of about tens of attosecond. Recently, various single-shot CEP measurement techniques without the need for CEP stabilization were experimentally demonstrated [29,30] in which a precision as high as p/300 can be guaranteed at an optimum measurement point. With the online CEP measurement using such a technology, accurate time interval with 4.5 attosecond resolution can be carried out between the 800 nm few-cycle seed laser and the generated ultra-fast 1 nm pulse, even without the abovementioned CEP stabilization. In Section 2, the theoretical estimation indicates a maximum harmonic number of 1000 for this kind of scheme. Therefore, an alternative case where an intense 200 nm few-cycle seed laser interacts with a 6 GeV electron beam in the scheme shown in Fig. 4 is checked to generate 0.15 nm hard X-ray FEL pulses. The simulation works out a 0.15 nm attosecond pulse with peak power of about 30 MW and pulse duration of about 3 attosecond. However, in this case, the required system parameters are pretty stringent, e.g. a 200 nm laser with several TW peak power is probably unrealistic. Moreover, since Angstrom-scale microbunching is quite vulnerable, the second-order effects (emittance and current) when the beam drifts through the radiator and the chicanes become significant. Therefore, there is still room for further improvements in terms of this case.
region 1 overbunched (see Fig. 11(c)). The different energy chirp of regions 1 and 2 results in their microbunching being optimized at different wavelengths. Radiator 2 is set to satisfy the FEL resonance condition for region 2, which generates the second attosecond FEL pulse. In radiator 2, the FEL amplification in other regions of the electron beam will be suppressed due to their ineffective density modulation. The simulation before DS3 is still based on the beam and undulator parameters listed in Table 1. A new radiator (i.e. radiator 2) with the same undulator period length and undulator period number as that of radiator 1 but whose FEL resonance condition is set for the second FEL pulse is added after DS3. Under these parameters, two attosecond X-ray pulses with the wavelength of 1 and 3.15 nm, respectively, will be obtained using a single 800 nm few-cycle seed pulse. In this case, the quantum diffusion caused by the spontaneous radiation in radiator 1 should also be included. According to Eq. (10) the additional energy spread in radiator 1 is only 0.8 keV, much smaller compared with the spacing between two adjacent energy bands i.e. 65.3 keV. Therefore, the analysis shows that the fine structure in
Fig. 12. Two attosecond FEL radiation profile.
4. Two-color attosecond X-ray pulses generation In Ref. [19], an EEHG-assisted FEL scheme to generate two attosecond soft X-ray pulses was proposed for pump-probe experiments. In this section, we propose a simpler design to generate two attosecond pulses as shown in Fig. 10 where the few-cycle seed pulse only interacts with the electron beam in M3. In this scheme, the principle before DS3 is the same as that in Fig. 4. The beam region modulated by the central cycle of the fewcycle laser (region 1 in Fig. 11(a)) is used to generate the first attosecond FEL pulse in radiator 1. After the beam comes out from radiator 1, DS3 makes the region which interacts with a side cycle of the few-cycle laser (region 2 in Fig. 11(a)) upright while making
103
Fig. 13. Spectrum of two FEL pulses.
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electron beam’s phase space should be able to be preserved before radiator 2. Before performing FEL simulation in two radiators, the time interval between two FEL pulses deserves a detailed discussion. Firstly, region 2 intrinsically advances region 1 by lseed. Secondly, in radiator 1, the first attosecond FEL pulse generated by region 1 catches up region 2 by Nu1l1 (the slippage length), where Nu1 and l1 are the undulator period number and FEL resonance wavelength of radiator 1. Thirdly, when passing through DS3, the path length of region 2 is larger than that of the first attosecond FEL pulse by 4L(yB/sin yB 1)+ 2D12(1/cos yB 1). In our case, Rð3Þ 56 ¼ 0:053 mm; which can be realized by L¼0.8 m, yB ¼4.8 mrad and D12 ¼D34 ¼0.6 m. Then the contribution to the time interval between two pulses from the third factor (about 85 fs) apparently dominates the other two ones. Figs. 12 and 13 show the simulated FEL radiation profile and spectrum from which it can be seen that two FEL pulses with a peak power of a few hundred MW and a pulse duration of about 20 as for the first pulse and about 70 as for the second pulse are obtained. Furthermore, the wavelength of these two pulses can be adjusted by changing the power of the few-cycle seed laser. Besides, the time interval between two attosecond FEL pulses can also be accurately controlled by adjusting the carrier envelope phase of the few-cycle seed laser.
5. Conclusions EEHG-assisted schemes have been suggested to generate coherent attosecond X-ray pulses. In this paper, a semi-empirical, geometrical model is developed to analyze the FEL performances in such kind of schemes and is well validated by three-dimensional simulations. On the basis of these results, a more compact and robust EEHG-assisted FEL scheme is proposed to generate two coherent attosecond soft X-ray pulses for two-color pump-probe experiments. Generally, in such EEHG-assisted FEL scheme for pump-probe experiments, the time interval between the pump and the probe pulses can be adjusted on femtosecond scale by changing the optical path of the few-cycle laser pulse. Studies in this discussion show that changing the carrier envelope phase of the
femtosecond laser may be an alternative way to control the time interval between two pulses at a precision of sub-10 attosecond.
Acknowledgements The authors are grateful to D. Xiang, Z. Huang and G. Stupakov for useful discussions. This work was supported by Shanghai Natural Science Foundation (Grant no. 09JC1416900).
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