research papers
Angular dispersion enhanced prebunch for seeding ultrashort and coherent EUV and soft Xray freeelectron laser in storage rings
^{a}Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China, ^{b}University of Chinese Academy of Sciences, Beijing 100049, China, ^{c}SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA, ^{d}Huazhong University of Science and Technology, Wuhan 430074, China, and ^{e}University of Science and Technology of China, Hefei 230026, China
^{*}Correspondence email: fengchao@sinap.ac.cn, zhaozhentang@sinap.ac.cn
Prebunching is an effective technique to reduce the radiation saturation length and to improve the longitudinal coherence and output stability in storageringbased freeelectron lasers (FELs). A novel technique is proposed which uses angular dispersion to enhance the highharmonic bunching with very small laserinduced energy spread. This technique can effectively reduce the radiation saturation length without significantly reducing the peak power of the FEL. Numerical simulations demonstrate that this technique can be used for the generation of 100 MW scale level, fully temporal coherent femtosecond extremeultraviolet and soft Xray radiation pulses through a 10 mlong undulator based on a diffractionlimited storage ring.
Keywords: angular dispersion; storage rings; freeelectron lasers; extremeultraviolet (EUV); ultrashort; prebunching; highharmonic bunching; radiation saturation length.
1. Introduction
Synchrotron light sources are a mature and reliable technology that can deliver radiation from infrared to soft Xrays with high beam stability, repetition rates and availability of multiple beamlines (Kim et al., 2017; Zhao, 2010). About 10 ps radiation pulses can be generated to study the structure of matter from the molecular to the atomic scale. Linacbased freeelectron lasers (FELs) have been a complementary tool to synchrotron light sources for the generation of radiation pulses with high peak high degree of coherence and ultrafast resolution (Feng & Deng, 2018). Combining these two techniques has been of remarkable interest to the accelerator community (Kim et al., 1985; Murphy & Pellegrini, 1985; Cornacchia et al., 1986; Nuhn et al., 1992; Huang et al., 2008; Dattoli et al., 2012; Cai et al., 2013; Agapov, 2015; Di Mitri & Cornacchia, 2015; Labat et al., 2007, 2009; De Ninno et al., 2008; Khan et al., 2013; Xiang & Wan, 2010).
The selfamplified ; Bonifacio et al., 1984) has been considered for storageringbased FELs (Kim et al., 1985; Murphy & Pellegrini, 1985; Cornacchia et al., 1986; Nuhn et al., 1992; Huang et al., 2008; Dattoli et al., 2012; Cai et al., 2013; Agapov, 2015; Di Mitri & Cornacchia, 2015). The SASE output is typically characterized by excellent spatial but limited temporal coherence. The spontaneous radiation is amplified by instability in the SASE mechanism, leading to large shottoshot fluctuations and a relatively long undulator length (typically 100 m) to reach saturation of the FEL. Long undulators would undoubtedly limit the capacity of the storage rings, as rings usually do not have sufficient space to accommodate very long undulators.
(SASE) mechanism (Kondratenko & Saldin, 1980Prebunching is an effective technique to overcome some of the present limitations of storageringbased SASE FELs. Prebunching means that the electron beam has been prebunched on the scale of the resonant wavelength, prior to the injection of the radiator (Schnitzer & Gover, 1985). Prebunching would not only shorten the saturation length substantially but also improve the longitudinal coherence and output stability (Freund et al., 2012; Jia, 2017). Prebunching normally relies on frequency upconversion schemes, which generally adopt the techniques of opticalscale manipulation of the electronbeam phase space with external coherent laser sources. Massive bunch manipulation techniques have been proposed to produce prebunched electrons, e.g. coherent harmonic generation (CHG) (Girard et al., 1984; Yu, 1991; Yu et al., 2000), echoenabled harmonic generation (EEHG) (Stupakov, 2009; Xiang & Stupakov, 2009) and phasemerging enhanced harmonic generation (PEHG) (Deng & Feng, 2013; Feng et al., 2014).
To illustrate that prebunching has the advantage of miniaturizing facility, here we compare the peak power as a function of propagation distance in the undulator with different initial energy spread for SASE and prebunching cases. Simulations were performed using GENESIS (Reiche, 1999). For the prebunching cases, the initial bunching factors for the entire bunch are directly set in GENESIS. Representative parameters of diffractionlimited storage rings are summarized in Table 1. The bunch is set as 300 A here (Xiang & Wan, 2010; Ding et al., 2013). The radiation wavelength is 13.5 nm and the undulator period is 3 cm.

Simulation results are shown in Fig. 1. SASE (ES = 0.05%) means FEL radiation generated from electrons with an energy spread of 0.05%. PreBunch (IB = 0.1) means FEL radiation generated from prebunched electrons with an initial bunching factor of 0.1. As can be seen from Fig. 1, the prebunching cases result in substantial shortening of the saturation length, while keeping the approximate peak power with respect to SASE. It also shows that for either SASE or prebunching cases, in comparison with low energyspread cases, large energyspread cases have lower saturation power and longer saturation length. The energy spread in storage rings is typically 0.1% or larger. With this large energy spread, SASE typically requires a very long undulator to reach saturation. While for prebunched cases, strong coherent at the initial radiation stage speeds up the laserbeam interaction, resulting in a much shorter saturation length.
CHG is the most representative mechanism for prebunching and has been successfully applied to storage rings to generate ultrashort and coherent radiation (Ortega, 1986; Prazeres et al., 1988, 1991; Labat et al., 2007, 2009; De Ninno et al., 2008; Khan et al., 2013). Generating microbunching at the nth harmonic of the seed laser requires the energymodulation amplitude to be approximately n times larger than the initial beam energy spread, whereas large energy modulation is definitely not the best option, as can be seen from Fig. 1. EEHG and PEHG can significantly reduce the required energy modulation for ultra high harmonic generation. However, the laserinduced energy spread seems still too large for highgain FEL in storage rings. Later, a mechanism which can introduce bunching to the electron beam with small energy modulation is proposed (Xiang & Wan, 2010). Compared with previous coherent harmonic generation methods, this scheme relies on the angular modulation instead of the energy modulation. However, the indispensable UV seed laser with TEM_{01} mode is difficult to obtain. Besides, complex lattice design would bring realistic difficulties.
In this work, a promising scheme is proposed, which uses angular dispersion to enhance the prebunching with very small laserinduced energy modulation. This technique can effectively reduce the radiation saturation length without significantly reducing the peak power. Numerical simulations demonstrate that this technique can be used for the generation of 100 MW scale level, fully temporal coherent femtosecond extremeultraviolet (EUV) and soft Xray radiation through a 10 mlong undulator in a storagering freeelectron laser.
The paper is organized as follows: the schematic layout is introduced in Section 2; theoretical derivation and comparison with existing schemes are described in Sections 2.1 and 2.2, respectively; threedimensional simulations for the generation of ringbased EUV radiation with realistic parameters based on the proposed scheme are provided in Section 3; and finally we summarize the results in Section 4.
2. Schematic layout
Fig. 2 illustrates the schematic layout of the proposed scheme and longitudinal beam phasespace evolution within one seedlaser wavelength region. Frenet–Serret coordinates are given here, where x, y and z represent the horizontal, vertical and longitudinal coordinates, respectively. The electron beam is regulated in the z–y plane. The initial longitudinal phase space is shown in Fig. 2(a). Firstly, the electron beam interacts with the wavefront tilted seed laser `seed light' in a short undulator, called a modulator, in which a tilted small sinusoidal energy modulation on the seedlaser wavelength scale will be imprinted onto the phase space of the electron beam. Then the phase space becomes Fig. 2(b). This energy modulation is then transformed into an associated density modulation by a dipole magnet, shown in Fig. 2(c). Taking advantage of the fact that the density modulation reflects as Fourier components at high harmonics of the seed frequency, intense radiation at shorter wavelengths shall be generated at the radiator.
The density modulation of the electron beam that contains highharmonic components is usually quantified by the bunching factor. In the following, we will derive the expression of the bunching factor and analytically compare the proposed scheme with existing schemes.
2.1. Theoretical derivation
To briefly demonstrate the principle of the proposed scheme, we adopt the simplified 4 × 4 dimensional linear beam transport matrix in the y–z plane, i.e. (y, y′, z, δ), where y and z are the vertical and longitudinal coordinates, y′ = dy/dz is the vertical divergence and δ = (γ − γ_{0})/γ_{0} is the dimensionless energy deviation with respect to the reference particle (Chao et al., 2013).
The electron interacts with a wavefront tilted seed laser intersecting at an angle of θ relative to the beampropagation direction in the modulator and obtains an energy change. The electric field of the laser is independent of y. The energy modulation can be simply written as (Feng et al., 2015)
where k_{s} is the wavenumber of the seed light and O = Δγ/γ_{0} is the induced beam energy change by laserbeam interaction. According to the PanofskyWenzel theorem (Panofsky & Wenzel, 1956), there will be an angular modulation,
Considering electrons in one seed wavelength range, the electron beam around the positive zero crossing of the seed laser gets an almost linear energy chirp h = k_{s}O. Therefore the energy change of the electron can be written as δ_{1} ≃ δ_{0} + h(z + θy) and the angular change of the electron can be written as y_{1}′ = y_{0}′ + hθ(z + θy). Then the corresponding transport matrix of the electron beam for this modulation section can be derived as
After the modulator, the energy modulation is converted into density modulation by a dipole magnet. Under the thinlens approximation, the transport matrix of a dipole is
where L_{b} represents R_{12} in the matrix, and b and η are the bending angle and dispersion of the dipole, respectively. Then the transport matrix for the whole beamline is
To enhance the highharmonic bunching, the electrons should merge into the same longitudinal position to the greatest extent, which means the items in the third row of equation (5) should be as small as possible. Therefore, we can obtain the optimization conditions of the proposed scheme:
Once these conditions are satisfied, the longitudinal position of the electron after the density modulation can be written as
It should be emphasized that equations (3)–(7) refer to a small (∼20%) portion of the electron beam. However, the bunching of the entire bunch is primarily determined by the aggregation degree of this portion of the electrons. We can tell from equation (7) that the highharmonic bunching factor will be determined by the product of η and the initial vertical beam divergence . Considering the sinusoidal modulation for the whole beam, a more solid derivation of the proposed scheme gives the straightforward bunching factor as
where J_{n} is the nth order Bessel function.
In order to enhance the bunching factor for a given Δγ in the proposed scheme, one can increase the beta function at the entrance of the proposed scheme to reduce σ′_{y} or increase the incidence angle θ to reduce as required the dispersion strength of the dipole magnet η. Although the balance between quantum excitation and radiation damping results in a relatively large energy spread in the storage ring, it nonetheless provides a beam with quite a low transverse emittance. The proposed technique makes full use of this feature and thus has the potential to significantly enhance the frequency upconversion efficiency.
2.2. Comparison with existing schemes
To illustrate the advantages of the proposed scheme, we compared it with three existing schemes, i.e. CHG, EEHG and PEHG. The bunching factor at the nth harmonic of the CHG is given by (Yu, 1991):
where is the energymodulation amplitude, is the dimensionless strength of the chicane, and k_{s} and R_{56} are the wavenumber of the seed light and the momentum compaction in the chicane, respectively. In the following discussion, the definitions of A and B are the same as those defined here.
The PEHG was proposed to improve the frequency upconversion efficiency of the CHG. The bunching factor at the nth harmonic of the PEHG is given by (Deng & Feng, 2013):
where T is the dimensionless gradient parameter of the transverse gradient undulator (TGU) and is the dimensionless dispersive strength in the horizontal. A previous study (Qi et al., 2017) showed that the intrinsic beam divergence would decrease the phasemerging effect, leading to a lower bunching factor. For simplicity, the influence of intrinsic beam divergence on the bunching factor is ignored here, thus equation (10) indicates the best performance of PEHG.
Another effective way to achieve electronbeam bunching is the EEHG. The bunching factor at the ath harmonic of the EEHG is given by (Stupakov, 2009):
where a = n + mK and C = nB_{1} + (n + mK)B_{2}. Furthermore, A_{1} and A_{2} are the energymodulation amplitudes of the first and second modulator, respectively, and B_{1} and B_{2} are the dimensionless strengths of the first and second chicane, respectively.
From the above discussion, we can plot modulation depth as a function of bunching factor at the 20th harmonic for the above three and proposed schemes based on equations (9)–(11), as shown in Fig. 3. One can see that for either CHG, PEHG or EEHG the energymodulation amplitude should be at least 2 to achieve a bunching factor of 0.1 for the 20th harmonic. This would undoubtedly boost the energy spread in storage rings, resulting in a significant reduction of the peak power. Also, the saturation length will not decrease much. For the proposed scheme, however, the bunching factor is insensitive to the modulation depth. Therefore a very small energymodulation amplitude can generate a large bunching factor. On the whole, the proposed scheme could effectively reduce the saturation length while almost not changing the peak power.
It should be stressed that a recently proposed technique (Feng & Zhao, 2017) could also imprint strong coherent microbunching on the electron beam with a small laserinduced energy spread. Mathematically, the energymodulation depths of these two schemes could reach the same level (∼0.2). However, these two schemes adopt different implementations. With only a dipole and a short modulator, the lattice layout for the proposed scheme here is more simplified to achieve similar results.
3. 3D simulations for the generation of intense EUV radiation
To illustrate a possible application with realistic parameters and show the parameteroptimization method of the proposed scheme, threedimensional simulations are carried out based on the beam parameters in Table 1.
The modulation process was simulated with a threedimensional algorithm based on the fundamentals of electrodynamics when considering the appearance of the electromagnetic fields of a wavefront tilted seed laser and the magnetic fields of the modulator. It is assumed that the r.m.s. intrinsic beam size at the entrance of the modulator is σ_{x} = 45 µm and σ_{y} = 25 µm. The modulator is composed of a fourperiod undulator magnet with 2.1 cm period, 7.5 mm gap, and a K value of 1.45. A round laser pulse with a duration of 100 fs FWHM, a central wavelength of 270 nm, a peak power of 700 GW and an incident angle of 5 mrad is adopted. Since the laser is obliquely incident, the center of the laser field gradually deviates from the center of the electron beam as it passes through the undulator. The deviation distance between the seedlaser center and electronbeam center is equal to the product of the modulator length and the seedlaser incident angle, which is 0.42 mm in this case. This parameter is approximately ten times larger than the transverse size of the electron beam. Therefore, the seedlaser spot size is mainly determined by the deviation distance. To fully cover the electron beam, the transverse size of the laser waist is set to 0.84 mm. Interacting with electrons in the modulator, this laser pulse will produce a peak amplitude of energy modulation of ∼0.9 MeV (i.e. 0.2 times of tinitial energy spread). In addition to energy modulation, angular modulation is also induced in the modulator. The energy and angle are both modulated on the scale of the seedlaser wavelength. Besides, the appearances of the energy and divergence distribution after modulation are the same. Here we illustrate the vertical divergence distribution after the laserbeam interaction in Fig. 4. One can see that the divergence distribution is tilted with an inclined angle of 5 mrad and the longitudinal distance between the two modulation peaks is exactly the wavelength of one seed laser. These results are consistent with equation (2).
After the electron beam traverses a dipole magnet or a transport section with a matrix of R_{16} = −0.0312 m, R_{51} = 0.005 and R_{56} = −4 µm, it will enter into the radiator. Introducing R_{56} by the dipole or a chicane magnet would make the electron beam undercompressed, lowering the initial bunching factor while enhancing the final radiation peak power. Simulations for the radiation process were performed by GENESIS (Reiche, 1999). The period length of the radiator is 3 cm with a K value of about 3.7. The FODO structure is well designed to have the following Twiss parameters at the entrance of the FODO cell: β_{x} = 10.5 m, β_{y} = 1.9 m, α_{x} = −1.9, α_{y} = 0.37. With these parameters, the resultant beam size is σ_{x} = 46 µm and σ_{y} = 1.95 µm in the SASE case. In the proposed case, the geometric vertical emittance is boosted to 0.2 nm because of modulation. Then the beam size is σ_{x} = 46 µm and σ_{y} = 20 µm.
The evolutions of the bunching factor and radiation peak power for SASE and the proposed scheme at the 20th harmonic are shown in Fig. 5. The dispersion induced by the undulator becomes apparent for electrons with 0.15% energy spread. Therefore, for the proposed scheme the undercompressed prebunch would subsequently become fully compressed as it propagates in the undulator. This can be seen in Fig. 5(a), where the bunching factor increases from 0.04 to 0.38 within an undulator length of 10 m. Then coherent radiation scales as n_{e}^{2} (where n_{e} is the electronbeam density), which would be emitted by the well prebunched electrons. The radiation reaches the saturation region at an undulator length of 10 m and ∼200 MW of peak power can be generated. However, driven by FEL instability, SASE needs a 60 m or longer undulator to approach power of a comparable level. Hence, compared with SASE, the proposed scheme shows remarkable performance when the undulator is short. It is also shown that the peak power emitted by the electrons scales as n_{e}^{2} again after a undulator length of 20 m for the proposed scheme, which is typical of strong superradiant behavior (Bonifacio et al., 1989), the superradiant regime showing continuous energy extraction from the electron pulse.
Fig. 6 shows the output radiation pulses, output spectra and transverse distribution of the 20th harmonic radiation pulses for the proposed scheme at z = 10 m. One can see from Fig. 6(a) that ∼60 fs FWHM laser pulses could be generated. Furthermore, one can see from Fig. 6(b) that the spectral bandwidth at 10 m for the proposed scheme is close to the Fourier transformlimited bandwidth. Therefore, the radiation pulse is almost fully coherent. As can be seen from Fig. 6(c), radiation pulses are approximately round in the transverse plane. This is because the geometric vertical emittance is boosted to the value of the geometric horizontal emittance during the modulation process.
In Fig. 6(a), a relatively high spiking shows up in the leading region. This is because there is always a trailing region of the electron beam in which slippage effects give rise to spiking behavior in the head of the radiation pulse. The radiation pulses constantly interact with the fresh bunch in the head region and the energy is extracted from the electrons in a continuous way, leading to a spiking behavior (see Fig. 7). The heading spiking has an approximate 8 fs FWHM temporal structure at z = 40 m, which may be very useful for experiments investigating ultrafast phenomena.
4. Conclusions
In this work, we have proposed a novel technique, which can introduce prebunching to the electron beam. This beammanipulation technique uses a wavefront tilted laser to modulate the electron beam. Relying on angular modulation instead of energy modulation, this technique could enhance highharmonic bunching with a very small laserinduced energy spread. With a simple setup of a short modulator and a dipole magnet, this technique is very dexterous for implementation in existing facilities.
In the threedimensional simulations, 200 MW fully temporal coherent EUV radiation pulses with a duration of 60 fs FWHM are generated through a 10 mlong undulator. This is an unprecedented performance. This technique effectively reduces the radiation saturation length without significantly reducing the peak power with respect to storageringbased SASE FELs. Although EUV radiation pulses are generated in this work, this technique possesses the ability to further extend the output wavelength range towards soft Xrays. The peak power of the seed laser is 700 GW in our case. This would be a primary limitation for the repetition rate in storageringbased FELs. It is found that the energymodulation depth increases as the incident angle decreases. Therefore, under the resonant condition, for a given modulation depth, directly reducing the incidence angle would significantly reduce the intensity of the seed laser. The laser spot size is proportional to the incident angle. Reducing the incidence angle would also lower the laser spot size, which increases the laser intensity. Increasing the modulation length is not working here because the laser spot size is proportional to the modulation length. In addition, we can use an elliptical
seed laser. The laser size in the vertical direction is mainly determined by the deviation distance; however, in the horizontal direction, the laser size is determined by the electronbeam size. The deviation distance is an order of magnitude larger than the beam size. Therefore, the adoption of elliptical rather than circular seed lasers can reduce laser power by one order of magnitude to ∼70 GW. Another possible solution is inserting a quadrupole magnet in front of this scheme. By elaborately setting the parameters of each component, this method could achieve the same bunching factor with a smaller energymodulation depth. Then further lowering the required energymodulation amplitude.The laser–electron interaction would slightly increase the energy spread and boost the vertical emittance of the electron beam. This part of the electron beam may not be conducive to the generation of synchrotron radiation. However, the interaction only happens in 100 fs of the bunch, a tiny fraction of the 10 ps bunch length that is typical in a storage ring. Then the proposed technique may not adversely affect the other beamlines (Schoenlein et al., 2000).
Steady state microbunching (SSMB) is a promising technique to generate high average radiation power in storage rings (Ratner & Chao, 2010; Ratner & Chao, 2011). Preliminary study shows that the proposed technique has the feasibility to be used in reversible SSMB. Further investigations of this topic are ongoing.
Acknowledgements
The authors are deeply grateful to Alex Chao (SLAC) for providing the vision and guidance during the proposal of the scheme. We would also like to thank Bocheng Jiang (Shanghai Institute of Applied Physics – SINAP) and Guanqun Zhou (Institute of High Energy Physics – IHEP) for helpful discussions and useful comments related to the work presented here.
Funding information
Funding for this research was provided by: National Nature Science Foundation of China (grant No. 11475250; grant No. 11605277); National Key Research and Development Program of China (grant No. 2016YFA0401901) and Youth Innovation Promotion Association CAS (grant No. 2015209). The support provided by China Scholarship Council (CSC) during a visit of Xiaofan Wang to SLAC is acknowledged.
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