Nonlinear energy chirp compensation with corrugated structures

Nonlinear energy chirp compensation with corrugated structures 1 first-author 1 1 1 1 corresp email: zhangmeng@sinap.ac.cn email: zhangmeng@sinap.ac.cn zhangmeng@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Herein, a feasible method is proposed to compensate the high-order effect during bunch length compression; thereby, enhancing the peak current of a high repetition rate X-ray free-electron laser source. In the proposed method, the corrugated structure is inserted downstream of the high-order harmonic cavities to function as a passive linearizer and enhance the longitudinal profile of the electron beam. Three-dimensional simulations are performed to analyze the evolution of the longitudinal phase space and the results demonstrate that the profile of the electron beam is improved and the peak current can be easily optimized to over 2 kA with a bunch charge of 100 pC. Corrugated structure Nonlinear energy chirp High repetition rate FEL I. Introduction 1 Unlike conventional lasers or synchrotron radiations, a free-electron laser (FEL) can generate ultra-short, ultra-bright, and fully-coherent hard X-ray radiations, which is at the origin of FELs extensive development over the last decade [ 1 1 , 2 2 ]. Nowadays, many X-ray FEL facilities exist or are under-construction around the world, such as LCLS in the USA [ 3 3 ], SACLA in Japan [ 4 4 ], FERMI in Italy [ 5 5 , 6 6 ], Pal-FEL in Korea [ 7 7 ], SXFEL in China [ 8 8 - 12 12 8-12 ], and Swiss-FEL in Switzerland [ 13 13 ]. As these facilities are driven by the common conducting (copper) linear accelerator (linac), their repetition rates are in the range of 1-120 Hz, leading to a relatively low average power. Alternatively, high repetition rate FEL facilities based on superconducting (SC) techniques have been studied for years. For example, FLASH in Germany [ 14 14 ], a soft X-ray user facility based on macro-bunch modes, has accumulated extensive experience on SC techniques. Recently, the European XFEL [ 15 15 ] successfully generated hard X-ray FEL, which is a milestone in the FEL community. LCLS II based in the USA and SHINE in China [ 16 16 ], which have been recently launched, are hard X-ray FEL facilities based on SC linac with continuous wave (CW) modes. These high repetition rates (about 1 MHz) FEL facilities could provide up to a thousand-fold increase in average brightness compared with the common conducting FEL facilities. In order to provide the CW electron beam train, the common conducting very high frequency (VHF) gun [ 17 17 - 19 19 17-19 ], the high-voltage direct current (DC) gun [ 20 20 ], or the SC RF gun [ 21 21 ] are developed as the injector of the CW FEL facilities. As the beam energy from these guns is relatively low compared with that of the usual conducting copper photocathode gun, the high-order terms of the electron longitudinal phase space p lay an important role in the forthcoming electron beam compression process, suffering from strong space charge effects. The peak current of the electron beam at the exit of the SC linac is limited to approximately 1 kA, which is much lower than that of the common conducting linac. This usually leads to a degradation of the FEL performance, more particularly for FEL lasing in hard X-ray regime. Herein, the use of corrugated structures is proposed to compensate for the high-order terms during the compression process, as these structures are usually employed as a beam energy dechirper [ 22 22 - 25 25 22-25 ]. These corrugated structures are inserted upstream of the first bunch compressor chicane, providing a passive wakefield to modulate the longitudinal phase space of the electron beam. Three-dimensional start-to-end simulations are performed to study the evolution of the longitudinal phase space and the results with and without the corrugated pipe are compared. The results demonstrated that the longitudinal phase space is improved and the peak current of the electron beam at the exit of the SC linac can be enhanced to over 2 kA. The principle of the compression process and the basic analytic properties of the corrugated pipe are introduced in section 2 of this manuscript. The start-to-end simulation studies are performed with an optimized corrugated pipe for SHINE in section 3. Finally, a summary and conclusions are briefly stated in section 4. II. Theory 1 In the case of the usual conducting hard X-ray FEL facilities, the nonlinear effects, such as the sinusoidal RF time-curvature and the second-order momentum compaction terms during the compression process, dominate the final peak current of the electron beam. A harmonic cavity, usually an X-band, is used to linearize the bunch compression process and compensate for the second-order compression effects [ 20 20 ]. Similarly, a third harmonic cavity is used as the linearizer upstream of the first bunch compressor chicane (BC1) in the SC linac-based high repetition rate FEL facilities. The schematic layout of the gun, the linac segments, and the first chicane are presented in Fig. 1 Fig. 1 , with the symbols defined as the voltage ( V ), the RF phase ( φ ), the wavelength ( λ ) of each cryomodule in the injector (L0 CM) and the linac (L1 CM), and the chicane momentum compaction coefficients ( R 56 , T 566 ). Based on the analysis for a high-harmonic cavity [ 26 26 ], the necessary harmonic voltage as can be calculated as follows: <math display="block" id="M1"><mrow><mi>e</mi><msub><mi>V</mi><mn>3</mn></msub><mi>cos</mi><msub><mi>ϕ</mi><mn>3</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mn>0</mn></msub><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msubsup><mi>λ</mi><mn>1</mn><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mfrac><mrow><msub><mi>T</mi><mrow><mn>566</mn></mrow></msub></mrow><mrow><msubsup><mi>R</mi><mrow><mn>56</mn></mrow><mn>3</mn></msubsup></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mi>σ</mi><mi>z</mi></msub></mrow><mrow><msub><mi>σ</mi><mrow><msub><mi>z</mi><mn>0</mn></msub></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mo>−</mo><msub><mi>E</mi><mi>i</mi></msub></mrow><mrow><mn>1</mn><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>λ</mi><mn>1</mn></msub></mrow><mrow><msub><mi>λ</mi><mn>3</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac><mo>,</mo></mrow></math> Where E and σ are the energy and the duration of the electron beam at different locations of the beamline, respectively. However, suffering from the strong space charge effect in the injector, the compression process is dominated by the higher-order terms of the energy chirp. The typical longitudinal phase space of the electron beam at the end of the SC linac is presented in Fig. 2 Fig. 2 . It can be seen that the bunch-length-scale energy modulation leads to a double-horn density distribution and the current in the high-quality central part of the electron beam is approximately 1 kA. In order to improve the peak current of the electron beam, thus the FEL performance, the high-order terms must be compensated during the compression process. The corrugated structure, having a near-maximal possible amplitude for a given aperture, is an attractive solution to compensate for the high-order terms of the energy chirp as the 'additional linearizer' [ 27 27 ]. The geometry of the corrugated structure is presented in Fig.3 Fig.3 . A relativistic electron beam passing on-axis through a periodic corrugated cylindrical pipe with the radius of <math id="M2"><mi>a</mi></math> is considered. The corrugated gap is represented as <math id="M3"><mi>g</mi></math> , the period as <math id="M4"><mi>p</mi></math> and the depth as <math id="M5"><mi>δ</mi></math> , with <math id="M6"><mrow><mi>δ</mi><mo>,</mo><mi>p</mi><mo>≪</mo><mi>a</mi></mrow></math> . At the same time, a "steeply corrugated" structure is assumed, requiring <math id="M7"><mrow><mi>δ</mi><mo>≥</mo><mi>p</mi></mrow></math> . When the relativistic charged beam passes through the structure, the longitudinal point charge wake is approximately expressed as [ 22 22 - 25 25 22-25 , 27 27 ]: <math display="block" id="M8"><mrow><mi>W</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>χ</mi><mi>H</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>cos</mi><mi>k</mi><mi>s</mi><mo>,</mo></mrow></math> where <math id="M9"><mi>χ</mi></math> is the mode loss factor. <math display="block" id="M10"><mrow><mi>χ</mi><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>0</mn></msub><mi>c</mi></mrow><mrow><mn>2</mn><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M11"><mrow><msub><mi>Z</mi><mn>0</mn></msub><mo>=</mo><mn>377</mn><mtext> </mtext><mi>Ω</mi></mrow></math> represents the characteristic impedance of vacuum and <math id="M12"><mi>c</mi></math> is the speed of light. <math id="M13"><mrow><mi>H</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math> is the unit step function defined as: <math display="block" id="M14"><mrow><mi>H</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>1</mn><mo>,</mo><mi>s</mi><mo>≥</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn><mo>,</mo><mi>s</mi><mo><</mo><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> and the wave number, <math id="M15"><mi>k</mi></math> , is approximated defined as: <math display="block" id="M16"><mrow><mi>k</mi><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>p</mi></mrow><mrow><mi>a</mi><mi>δ</mi><mi>g</mi></mrow></mfrac></mrow></msqrt></mrow></math> For a bunch with longitudinal distribution <math id="M17"><mrow><mi>λ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></math> , the bunch wake is defined by the convolution: <math display="block" id="M18"><mrow><msub><mi>W</mi><mi>λ</mi></msub><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mo>+</mo><mi>∞</mi></mrow></msubsup><mrow><mi>W</mi><mrow><mo>(</mo><mrow><mi>s</mi><mo>'</mo></mrow><mo>)</mo></mrow><mi>λ</mi><mrow><mo>(</mo><mrow><mi>s</mi><mo>−</mo><mi>s</mi><mo>'</mo></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>s</mi></mrow></mrow></mstyle><mo>'</mo></mrow></math> considering a Gaussian bunch density distribution beam with a bunch length of <math id="M19"><mrow><msub><mi>σ</mi><mi>z</mi></msub></mrow></math> without any current fluctuation. By changing the geometry parameters of the corrugated structure, as presented in Fig. 3 Fig. 3 , thus optimizing the wavelength of the induced energy modulation, the high-order energy chirp of the electron beam can be compensated and the peak current can be improved. It should be noted that, to optimize the parameters of the corrugated structure, the wavelength of the main high-order terms of the bunch energy chirp should be analyzed. Therefore, the proper wavelength of the bunch wake (equal to the wavelength of the high-order terms) can be determined with Eq. (6). For the electron beam in SHINE, the wavelength of the point charge wake is approximately equal to the bunch length. The induced wakefield within the beam is represented in Fig. 4 Fig. 4 , where the quasi-periodic energy modulation along the beam introduced by the wakefield can be observed. III. Nonlinear effect compensation for the SHINE project 1 The SHINE will be the first hard X-ray FEL user facility based on the SC technique in China. The injector consists of a normal conducting electron RF gun (the VHF gun), a 1.3 GHz buncher, an eight 9-cell 1.3 GHz TESLA-like cryomodule (CM) [ 28 28 ], and a laser heater, providing an initial electron beam having a length of approximately 10 ps in full width at half maximum (FWHM), charge of 100 pC, peak current of about 10 A, and bunch rate up to 1 MHz. The main linac consists of three sections: L1 (two CMs with a beam energy of approximately 250 MeV), L2 (18 CMs, 2.1 GeV), and L3 (54 CMs, over 8 GeV), two bunch compressor chicanes BC1 (compressing the beam current from 10 A to 80 A) and BC2 (80 A to 1 kA), and a dechirper line at the end of the linac to compensate the correlated energy spread. Moreover, a 3.9 GHz third harmonic structure is located in L1 acting as a linearizer (harmonic linearizer, HL) to provide extra energy chirp and compensate for the second-order compression effects. The electron beam, after being generated in the injector and accelerated in the linac, with a bunch length of 8 μm in FWHM and a peak current of 1 kA, is finally sent to the downstream switchyard and the undulator systems for X-ray FEL generations. The layout of the injector and the linac of the SHINE are presented in Fig. 5 Fig. 5 . The main beam parameters of the SHINE are listed in Table 1 Table 1 . 1 1 Parameters 1 1 Values 1 1 Energy (GeV) 1 1 8 1 1 Slice energy spread (rms) 1 1 ≤0.01% 1 1 Normalized slice emittance (mm mrad) 1 1 ≤0.4 1 1 Charge (pC) 1 1 100 1 1 Repetition rate (MHz) 1 1 ≤1 1 1 Peak current (kA) 1 1 1 1 1 Bunch length (rms) (μm) 1 1 8 1 1 Transverse beam size (rms) (μm) 1 1 30 In order to illustrate the beam dynamic of the bunch, three-dimensional simulations are performed with all beamline components. The injector simulations are accomplished using the computer tracking codes ASTRA [ 29 29 ] considering the space charge effects. The main linac simulations are performed using ELEGANT [ 30 30 ], in which the coherent synchrotron radiation and the wakefield effects are considered. The longitudinal phase space evolution of the electron beam along the beamline are presented in Fig. 6 Fig. 6 . The peak current of the high-quality part of the electron beam is only of approximately 1 kA, and the double-horn-like current distribution limits the higher peak current due to the high-order terms of beam energy chirp. On the other hand, the bunch length is relatively long in the gun (approximately 30 ps) and the injector (approximately 10 ps after the buncher). This comparable to the wavelength of the wakefield excited by the beam itself. As described above, the field along the bunch is no longer linear (as observed in Figure 4 Figure 4 ). By optimizing the parameters of the corrugated structure, the wavelength and the energy distribution of the structure-induced wakefield based on the given density distribution of the electron beam can be adjusted, and this can be used to compensate for the high-order energy chirp. In order to operate these adjustments, simulations (with LiTrack for 2D and ELEGANT for 3D) are performed to analyze the longitudinal phase space evolution of the electron beam based on the SHINE linac layout and are presented in Fig. 6 Fig. 6 . By plotting the central energy distribution of the electron beam at the exit of BC2 after removal of the linear correlation of energy chirp, the modulation period and amplitude can be calculated. Multiplied by the compression ratio, the modulation period before bunch compressor can be deducted. In order to compensate for the modulation, the gap and the length of corrugated structure can be scanned to adjust the modulation wavelength and amplitude based on the given Gaussian-like current distribution electron beam in the injector. Herein, the compression ratio is approximately 100 and the modulation period is approximately 3 mm. A 0.4 m-long corrugated pipe following the parameters in Table 2 Table 2 is inserted downstream of the 3.9 GHz third harmonic cavities in the L1 section in order to compensate for the high-order terms of energy chirp. The longitudinal phase space evolution of the electron beam along the linac is shown in Fig. 7 Fig. 7 . The double-horn density distribution disappears, and the final current distribution is Gaussian-like with a peak current of approximately 1 kA. Compared with Figure 6 Figure 6 , a distinct improvement of the linearity mainly can be observed in BC2 due to the high-order terms of energy chirp playing a more important role when the compression ratio is higher. 1 1 Parameters 1 1 Values 1 1 Radius, a (mm) 1 1 2.5 1 1 Period, p (mm) 1 1 1.0 1 1 Depth, δ (mm) 1 1 0.4 1 1 Width, g (mm) 1 1 0.2 1 1 Length, L (m) 1 1 0.4 Furthermore, the accelerating phases and the compaction coefficients ( R 56 and T 566 ) of BC1 and BC2 can be optimized to compress the beam to higher peak current, and this is even more critical for FEL operations in a hard X-ray regime. The longitudinal phase space, beam current, and slice energy spread distribution at the exit of the linac are summarized in Figure 8 Figure 8 . The beam current can be easily enhanced to over 2 kA in an approximately 50 fs-wide region. The beam quality has been maintained in this region, where the slice energy spread is approximately 800 keV and the normalized slice emittance is approximately 0.25 mm mrad. IV. Conclusion 1 Due to nonlinear space charge effects, the peak current of the electron beam is limited to approximately 1 kA in high repetition rate free electron laser light source. In order to improve the peak current, the corrugated structure was proposed to be employed as an 'additional linearizer' downstream of the third harmonic cavity to compensate for the high-order terms of energy chirp during the compression process. The optimal parameters of the corrugated structures were determined and the simulation results demonstrated that this novel scheme is feasible. The double-horn current distribution was suppressed and the current distribution profile was optimized based on the SHINE beam. 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Borland , Elegant: A flexible SDDS-compliant code for accelerator simulation. Advanced Photon Source Report No. LS-287 , 2000 Elegant: A flexible SDDS-compliant code for accelerator simulation. Advanced Photon Source Report No. LS-287 The authors thank Prof. Hai-Xiao Deng, Duan Gu, and Zhen-Tang Zhao for the useful discussions. 10.1007/s41365-018-0512-z This work was supported by the National Natural Science Foundation of China (Nos.11675248 and 11775294), and the Youth Innovation Promotion Association CAS (No. 2018300). 2018-07-17 2018-09-19 2018-09-19 2018-11-16 2018-12-01 2021-12-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0512-z http://dx.doi.org/10.1007/s41365-018-0512-z . 2018 Nuclear Science and Techniques 12 29