Beam performance of the SHINE dechirper

Beam performance of the SHINE dechirper 1 2 first-author https://orcid.org/0000-0002-5995-8382 3 https://orcid.org/0000-0002-1313-0950 1 2 https://orcid.org/0000-0003-2234-3736 3 corresp Corresponding author, guduan@zjlab.org.cn Corresponding author, guduan@zjlab.org.cn guduan@zjlab.org.cn https://orcid.org/0000-0003-2319-5544 1 Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China A corrugated structure is built and tested on many FEL facilities, providing a ’dechirper’ mechanism for eliminating energy spread upstream of the undulator section of X-ray FELs. The wakefield effects are here studied for the beam dechirper at the Shanghai high repetition rate XFEL and extreme light facility (SHINE), and compared with analytical calculations. When properly optimized, the energy spread is well compensated. The transverse wakefield effects are also studied, including the dipole and quadrupole effects. By using two orthogonal dechirpers, we confirm the feasibility of restraining the emittance growth caused by the quadrupole wakefield. A more efficient method is thus proposed involving another pair of orthogonal dechirpers. Corrugated structure Energy spread Wakefield Shanghai high repetition rate XFEL and extreme light facility 1 Introduction 1 Eliminating residual energy chirps is essential for optimizing the beam brightness in the undulator of a free electron laser (FEL). There are currently two traditional ways to eliminate chirps in superconducting linear accelerator (linac)-driven X-ray FEL facilities. One involves exploiting the resistive-wall wakefield induced by the beam pipe. The other option, which involves running the beam ’off-crest’, is inefficient and costly, especially for ultra-short bunches in FEL facilities [ 1 1 ]. Recently, corrugated metallic structures have attracted much interest within the accelerator community, as they use a wakefield to remove linear energy chirps passively before the beam enters the undulator. The idea of using a corrugated structure as a dechirper in an X-ray FEL was first proposed by Karl and Gennady [ 1 1 ]. Several such structrues, XFELs [ 2 2 ], PAL-XFEL [ 3 3 ], pint-sized facilty [ 4 4 , 5 5 ], LCLS [ 6 6 ], and SwissFEL [ 7 7 ] have been built and tested. The feasibility of employing a corrugated structure to precisely control the beam phase space has been demonstrated in various applications. It has also been utilized as a longitudinal beam phase-space linearizer for bunch compression [ 8 8 , 9 9 ], to linearize energy profiles for FEL lasing [ 10 10 ], and as a passive deflector for longitudinal phase-space reconstruction [ 11 11 ]. Meanwhile, many other novel applications of light sources have also been proposed in recent years. Bettoni et al. verified the possibility of using them to generate a two-color beam [ 12 12 ]. A new role for generating fresh-slice multi-color generations in FELs was demonstrated in LCLS [ 13 13 ]. The generation of terahertz waves was also proposed recently [ 14 14 ]. This paper begins by reviewing the dechirper parameters for a small metallic pipe. The wakefield effects are studied with an ultra-short electron bunch in the Shanghai high repetition rate XFEL and extreme light facility (SHINE). Then, the process in dechirper is studied analytically and verified by numerical simulations. The longitudinal wakefield generated by the corrugated structure was adopted as the dechirper was calculated and simulated for SHINE. The following chapter focuses on the transverse wakefield, discussing emittance dilution effects on different arrangements of two orthogonally oriented dechirpers, mainly for the case of an on-axis beam. We also propose using “four-dechirpers” as a novel approach for controlling the beam emittance dilution effect during dechirping, and compare it with the conventional scheme. We then close with a brief conclusion. 2 Background 1 The layout of the SHINE linac and the beam parameters along the beamline are shown in Fig. 1 Fig. 1 . The electron beam generated by the VHF gun is accelerated to 100 MeV before entering the laser heater, to suppress potential subsequent microbunching by increasing the uncorrelated energy spread of the beam. The overall linac system consists of four accelerating sections (L1, L2, L3, and L4) and two bunch compressors (BC1 and BC2). A higher harmonic linearizer (HL), which works in the deceleration phase, is also placed upstream of BC1 for the purpose of linear compression. The SHINE linac beam specifications are listed in Table 1 Table 1 . After two stages of bunch compressors, the bunch length is shortened to 10●thin;μm, with a time-dependent energy chirp of approximately 0.25% (20 MeV) at the exit of the SHINE linac. Compared with normal conducting RF structures, the wakefield generated by the L-band superconducting structure is relatively weak because of its large aperture [ 15 15 ]. Therefore, it is impossible to compensate the correlated energy spread of the beam by adopting the longitudinal wakefield of the accelerating module, which becomes a key design feature of superconducting linac-driven FELs. In the case of the SHINE linac, the electron bunch length is less than 10●thin;μm after passing through the second bunch compressor. Therefore, the beam energy spread cannot be effectively compensated by chirping the RF phase of the main linac. The SHINE linac adopts the corrugated structure ( Fig. 2 Fig. 2 ) to dechirp the energy spread. This is achieved by deliberately selecting the structural parameters so as to control the wavelength and strength of the field, as verified by beam experiments on many FEL facilities. 1 1 Parameter 1 1 Value 1 1 Energy, E (GeV) 1 1 8 1 1 Charge per bunch, Q (PC) 1 1 100 1 1 Beam current, I (kA) 1 1 1.5 1 1 Bunch length (RMS), σ (μm) 1 1 10 1 1 β x (m) 1 1 60.22 1 1 β y (m) 1 1 43.6 1 1 α x 1 1 1.257 1 1 α y 1 1 1.264 1 1 ϵ nx (mm ●#x22C5; mrad) 1 1 0.29 1 1 ϵ ny (mm ●#x22C5; mrad) 1 1 0.29 3 Longitudinal Wakefield Effect 1 Round pipes and rectangular plates are usually adopted as dechirpers. Round pipes perform better on the longitudinal wakefield by a factor of 16/ π 2 relative to a rectangular plate of comparable dimensions. However, SHINE uses a rectangular plate nonetheless, as its gap can be adjusted by changing the two plates. The corrugated structure is made of aluminum with small periodic sags and crests, with the parameters defined in Fig. 2 Fig. 2 . The surface impedance of a pipe with small, periodic corrugations has been described in detail [ 16 16 - 18 18 16-18 ]. The high-frequency longitudinal impedance for the dechirper can be determined by starting from the general impedance expression. Taking q as the conjugate variable in the Fourier transform, we have <math display="block" id="M1"><mrow><msub><mi>Z</mi><mtext>l</mtext></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>ζ</mi></mrow><mi>c</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi>∞</mi></mrow><mi>∞</mi></msubsup></mrow></mstyle><mtext>d</mtext><mi>q</mi><mi>q</mi><msup><mrow><mtext>csch</mtext></mrow><mn>3</mn></msup><mo stretchy="false">(</mo><mn>2</mn><mi>q</mi><mi>a</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi>q</mi><mi>x</mi></mrow></msup><mn>,</mn></mrow></math> where f ( q )= n/d , and with <math display="block" id="M2"><mtable columnalign="left"><mtr><mtd><mi>n</mi><mo>=</mo><mi>q</mi><mo stretchy="false">[</mo><mi>cosh</mi><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>−</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mn>2</mn><mi>cosh</mi><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><mi>cosh</mi><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>y</mi><mo>+</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mi>i</mi><mi>k</mi><mi>ζ</mi><mo stretchy="false">[</mo><mi>sinh</mi><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>−</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mi>sinh</mi><mo stretchy="false">[</mo><mi>q</mi><mo stretchy="false">(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>y</mi><mo>+</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M3"><mrow><mi>d</mi><mo>=</mo><mo stretchy="false">[</mo><mi>q</mi><mtext>sech</mtext><mo stretchy="false">(</mo><mi>q</mi><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mi>k</mi><mi>ζ</mi><mtext>csch</mtext><mo stretchy="false">(</mo><mi>q</mi><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>q</mi><mtext>csch</mtext><mo stretchy="false">(</mo><mi>q</mi><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mi>k</mi><mi>ζ</mi><mtext>sech</mtext><mo stretchy="false">(</mo><mi>q</mi><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></math> We use ( x 0 , y 0 ) and ( x , y ) to represent the driving and testing particles, respectively. In Eq. (1), the surface impedance is written as ζ , which is related to the wavenumber k . In Ref. [ 18 18 ], the impedance at large k is expanded, keeping terms to leading order 1/ k and to the next order 1/ k 3/2 . Then, we compare this equation to the round case for a disk-loaded structure in a round geometry, with the same expression in parameters as rectangular plates. The longitudinal impedances at large q for the round and rectangular plates are given by [ 18 18 - 21 21 18-21 ] <math display="block" id="M4"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Z</mi><mtext>r</mtext></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>4</mn><mi>i</mi></mrow><mrow><mi>k</mi><mi>c</mi><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><msup><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mrow><msqrt><mrow><mn>2</mn><mi>k</mi><msub><mi>S</mi><mrow><mtext>0r</mtext></mrow></msub></mrow></msqrt></mrow></mfrac><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Z</mi><mtext>l</mtext></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>4</mn><mi>i</mi></mrow><mrow><mi>k</mi><mi>c</mi><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><msup><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mrow><msqrt><mrow><mn>2</mn><mi>k</mi><msub><mi>S</mi><mrow><mtext>0l</mtext></mrow></msub></mrow></msqrt></mrow></mfrac><mo stretchy="false">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The distance scale factors S 0r and S 0l for the round and flat are strongly influenced by the dechirper parameters: <math display="block" id="M5"><mrow><msub><mi>S</mi><mrow><mtext>0r</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mi>t</mi></mrow><mrow><mn>2</mn><mi>π</mi><msup><mi>α</mi><mn>2</mn></msup><msup><mi>p</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M6"><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mn>0.465</mn><msqrt><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msqrt><mo>−</mo><mn>0.070</mn><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M7"><mrow><msub><mi>S</mi><mrow><mtext>0l</mtext></mrow></msub><mo>=</mo><mn>9</mn><msub><mi>S</mi><mrow><mn>0</mn><mtext>r</mtext></mrow></msub><mo>/</mo><mn>4.</mn></mrow></math> After calculating the inverse Fourier transformation, the distance s between the test and driving particles yields the longitudinal wake at the origin of s =0 + , according to <math id="M8"><mrow><msub><mi>w</mi><mtext>l</mtext></msub><mo>∼</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mtext>0l</mtext></mrow></msub></mrow></msqrt></mrow></msup></mrow></math> . The relationship between the longitudinal point wake and the distance about the test charge behind the driving charge is expressed as Eq. (8) [ 19 19 ]. <math display="block" id="M9"><mrow><msub><mi>w</mi><mtext>l</mtext></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mrow><mi>π</mi><msub><mi>Z</mi><mn>0</mn></msub><mi>c</mi></mrow><mrow><mn>16</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mtext>0l</mtext></mrow></msub></mrow></msqrt></mrow></msup><mn>.</mn></mrow></math> The wakefield of the short bunch is obtained by convoluting the wake with the bunch shape λ ( s ). For a pencil beam, the original wake w 0 on the axis becomes [ 19 19 ] <math display="block" id="M10"><mrow><msub><mi>w</mi><mn>0</mn></msub><mo>=</mo><mfrac><mrow><mi>π</mi><msub><mi>Z</mi><mn>0</mn></msub><mi>c</mi></mrow><mrow><mn>16</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> Eq. (8) shows that the required dechirper length L is determined by the half-gap for a given dechirper strength. The half-gap a = 1●thin;mm was chosen as the baseline for the following reasons. On the one hand, it ensures that a sufficiently large proportion of the beam is included in the clear region for beam propagation, an essential requirement for controlling beam loss in a superconducting linac. On the other hand, the aperture size is constrained by the transverse emittance dilution effect, which is discussed in the following section. In [ 27 27 ], it is assumed that the corrugation dimensions are no greater than the gap size, i.e., t , p le; a . This ensures that the structure is ‘steeply corrugated’, such that short-range wakes can be neglected. As described in Eq. (7), the distance factor is affected by the dechirper parameters, especially by the ratio t/p . The wakefields induced by the Gaussian bunch with different t/p values are shown in Fig. 3 Fig. 3 . Over the initial 20●thin;μm, all the induced wakefields have the same slope coefficient and differ mainly in terms of the maximal chirp. As t/p increases, the wakefield decreases progressively until it settles when t/p reaches 0.5. Therefore, t/p = 0.5 is selected for SHINE as the dechirper parameter for which deviations are tolerable. Eq. (1) is suitable only for dechirpers with a flat geometry, with corrugations in the y and z directions and with x extending to infinity horizontally. However, in practice, it assumes the presence of a resistive wall in the x direction, as defined by the width w . The wake calculated in the time domain by the wakefield solver ECHO2D [ 23 23 ] is adopted to simulate the actual situation. This is expressed as a sum of discrete modes, with odd mode numbers m corresponding to the horizontal mode wavenumbers kx = mπ/w ( m =1,3,5●#x22C5;s). To obtain the exact simulated wakefield, it has been verified that w a should be satisfied, and that more than one mode contribute to the impedance of the structure [ 17 17 ]. As previously mentioned, t/p = 0.5 was adopted. The longitudinal wakefields corresponding to different widths are shown in the middle subplot of Fig. 3 Fig. 3 . The longitudinal wakefield appears to increase with w , but settles at a maximum value when w = 15●thin;mm. For our calculation, setting a = 1●thin;mm and w = 15●thin;mm yields a sufficiently large ratio w/a = 15. The scenarios in Eq. (1) and ECHO2D can all be regarded as flat geometries. The main parameters chosen for SHINE are summarized in Table 2 Table 2 . 1 1 Parameter 1 1 Value 1 1 Half-gap, a (mm) 1 1 1.0 1 1 Period, p (mm) 1 1 0.5 1 1 Depth, h (mm) 1 1 0.5 1 1 Longitudinal gap, t (mm) 1 1 0.25 1 1 Width, w (mm) 1 1 15.0 1 1 Plate length, L (m) 1 1 10.0 Assuming that the beam goes through an actual periodic structure, the beam entering the finite-length pipe still displays a transient response, characterized by the catch-up distance z = a 2 /2 σ z . Based on the parameters in Table 2 Table 2 , the catch-up distance in SHINE is 50 cm, which is small compared to the structure length, suggesting that the transient response of the structure can be ignored. The effect of a high group velocity in the radiation pulse also merits discussion. At the end of the structure length, the pulse length can be expressed [ 1 1 ] as l p = 2 htL/ap . For the structural parameters of SHINE, we have l p = 5●thin;m, which is much longer than the actual bunch length of the particle. This effect does not have to be taken. During RF acceleration and beam transportation, the electron beam traveling from the VHF gun to the linac is affected by several beam-dynamic processes. These include the space-charge effect in the injector, the wakefield effect from SC structures, coherent synchrotron radiation (CSR), and non-linear effects during bunch compression [ 24 24 ]. Non-linearities in both the accelerating fields and the longitudinal dispersion can distort the longitudinal phase space. The typical final double-horn current distribution after the SHINE linac is shown in Fig. 4 Fig. 4 . The particles are concentrated within the first 20●thin;μm with a linear energy spread. The wakefield for the actual simulated bunch distribution is shown in Fig. 5 Fig. 5 . Compared with the Gaussian and rectangular bunch distributions, as the head-tail wakefield effect of the beam, the double-horn bunch distribution distorts the expected chirp from the Gaussian bunch. The high peak current at the head of the beam complicates the situation further. The maximal chirp generated at different positions is also related to the feature in the bunch distribution. The particles in the double-horn bunch descend steeply over 15●thin;μm, but the particles in the Gaussian and rectangular bunches hold a gender distribution. Nevertheless, the chirp induced by the double-horn beam distribution in less than 5% differences in the maximal chirp and show great correspondence on the tail. This proves that the analytical method is also perfectly suitable for the actual bunch in SHINE. According to the middle subplot in Fig. 5 Fig. 5 , the wakefield generated by the same structural parameters in the corrugated structure depends mainly on the shape of the bunch. As shown in the bottom of Fig. 5 Fig. 5 , with the longitudinal wakefield by the actual bunch, the energy chirp in the positive slope after L4 in SHINE can be well compensated. We can conclude that the longitudinal wake generated by the corrugated structure over 10 m is adequate and effective at canceling the energy chirp passively. 4 Transverse Wakefield Effect 1 For the part of the beam near the axis of plates, w y d and w y q are defined as the transverse quadrupole and dipole wakes, where the driving and test particle coordinates y 0 and y a . For a driving particle at ( x 0 , y 0 ) and a test particle at ( x , y ), the transverse wake is given by [ 25 25 ] <math display="block" id="M11"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>w</mi><mi>y</mi></msub><mo>=</mo><msub><mi>y</mi><mn>0</mn></msub><msub><mi>w</mi><mrow><mi>y</mi><mtext>d</mtext></mrow></msub><mo>+</mo><mi>y</mi><msub><mi>w</mi><mrow><mi>y</mi><mtext>q</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>w</mi><mi>x</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>w</mi><mrow><mi>y</mi><mtext>q</mtext></mrow></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Expanding the surface impedance ζ [ 18 18 ] in the first two orders, the short-range vertical dipole and quadrupole wakes near the axis are given by [ 19 19 ] <math display="block" id="M12"><mrow><msub><mi>w</mi><mrow><mi>y</mi><mtext>d</mtext></mrow></msub><mo>≈</mo><mfrac><mrow><msub><mi>Z</mi><mn>0</mn></msub><mtext>c</mtext><msup><mi>π</mi><mn>3</mn></msup></mrow><mrow><mn>64</mn><msup><mi>a</mi><mn>4</mn></msup></mrow></mfrac><msub><mi>s</mi><mrow><mn>0</mn><mtext>d</mtext></mrow></msub><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>d</mtext></mrow></msub></mrow></msqrt><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>d</mtext></mrow></msub></mrow></msqrt></mrow></msup></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M13"><mrow><msub><mi>w</mi><mrow><mi>y</mi><mtext>q</mtext></mrow></msub><mo>≈</mo><mfrac><mrow><msub><mi>Z</mi><mn>0</mn></msub><mtext>c</mtext><msup><mi>π</mi><mn>3</mn></msup></mrow><mrow><mn>64</mn><msup><mi>a</mi><mn>4</mn></msup></mrow></mfrac><msub><mi>s</mi><mrow><mn>0</mn><mtext>q</mtext></mrow></msub><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>q</mtext></mrow></msub></mrow></msqrt><mo stretchy="false">)</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>s</mi><mo>/</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>q</mtext></mrow></msub></mrow></msqrt></mrow></msup></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M14"><mrow><msub><mi>s</mi><mrow><mn>0</mn><mtext>d</mtext></mrow></msub><mo>=</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>r</mtext></mrow></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>15</mn></mrow><mrow><mn>14</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mn>,</mn><msub><mi>s</mi><mrow><mn>0</mn><mtext>q</mtext></mrow></msub><mo>=</mo><msub><mi>s</mi><mrow><mn>0</mn><mtext>r</mtext></mrow></msub><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>15</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mn>.</mn></mrow></math> Figure 6 Figure 6 compares the dipole and quadrupole wakes obtained by convolving with the actual bunch distribution in SHINE and the analytical results verified with the simulated results from the ECHO2D code [ 23 23 ]. Assuming that the beam is close to (and nearly on) the axis, there is good agreement between the numerical and analytical results for the dipole and quadrupole wakes. When the beam is centered off-axis, the emittance growth is generated by the transverse dipole and quadrupole wakefields, leading to a deterioration in the beam brightness. Regardless of whether the beam is at the center, the quadrupole wake focuses in the x direction and defocuses in the y direction, increasingly from the head to the tail. This in turn results in an increase in the projected emittance. However, care must be taken that the slice emittance is not affected by the dipole and quadrupole wakes taken by these two orders. For the case of a short uniform bunch near the axis, the quadrupole and dipole inverse focal lengths are given by [ 26 26 ] <math display="block" id="M15"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>f</mi><mtext>q</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>k</mi><mtext>q</mtext><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo>=</mo><mfrac><mrow><msup><mi>π</mi><mn>3</mn></msup></mrow><mrow><mn>256</mn><msup><mi>a</mi><mn>4</mn></msup></mrow></mfrac><msub><mi>Z</mi><mn>0</mn></msub><mi>c</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>e</mi><mi>Q</mi><mi>L</mi></mrow><mrow><mi>E</mi><mi>l</mi></mrow></mfrac></mrow><mo>)</mo></mrow><msup><mi>s</mi><mn>2</mn></msup><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>f</mi><mtext>d</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>k</mi><mtext>d</mtext><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mi>L</mi><mo>=</mo><mfrac><mrow><msup><mi>π</mi><mn>3</mn></msup></mrow><mrow><mn>128</mn><msup><mi>a</mi><mn>4</mn></msup></mrow></mfrac><msub><mi>Z</mi><mn>0</mn></msub><mi>c</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>e</mi><mi>Q</mi><mi>L</mi></mrow><mrow><mi>E</mi><mi>l</mi></mrow></mfrac></mrow><mo>)</mo></mrow><msup><mi>s</mi><mn>2</mn></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> where k q ( s ) is the effective quadrupole strength, which changes with s within the bunch length l . For the case where the beam is near the axis, a short uniformly distributed bunch was deduced in Ref. [ 26 26 ] to calculate the emittance growth after passing through the dechirper. As mentioned above, Eq. (9) is substituted into Eq. (50) of Ref. [ 26 26 ] to completely eliminate 20 MeV from the invariant: <math display="block" id="M16"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>ϵ</mi><mi>y</mi></msub></mrow><mrow><msub><mi>ϵ</mi><mrow><mi>y</mi><mn>0</mn></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><msup><mrow><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msup><mrow><mn>10</mn></mrow><mn>7</mn></msup><msup><mi>π</mi><mn>2</mn></msup><mi>l</mi><msub><mi>β</mi><mi>y</mi></msub></mrow><mrow><mn>6</mn><msqrt><mn>5</mn></msqrt><msup><mi>a</mi><mn>2</mn></msup><mi>E</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>4</mn><msubsup><mi>y</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>σ</mi><mi>y</mi><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Based on the SHINE parameters, Fig. 7 Fig. 7 shows the growth in emittance for different beam offsets. The value of y c is proven to be a significant factor for determining the projected emittance. When the beam is near the axis and the gap a ●#x2A7E; 1 mm, the effect on the growth in emittance induced by the dipole wakefield is tolerable. Hence, as a point of technique in beamline operation, maintaining the beam on-axis is an effective way to restrain emittance growth. We next simply consider the quadrupole wake, where the beam is on-axis ( y c = 0). The transfer matrices for the focusing and defocusing quadrupole are given in Eq. (16), where L is the length of the corrugated structure [ 26 26 ]. <math display="block" id="M17"><mrow><msub><mi>R</mi><mtext>f</mtext></msub><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mrow><mi>cos</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd><mtd><mrow><mfrac><mrow><mi>sin</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow><mrow><msub><mi>k</mi><mtext>q</mtext></msub></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><msub><mi>k</mi><mtext>q</mtext></msub><mi>sin</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd><mtd><mrow><mi>cos</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mn>,</mn><msub><mi>R</mi><mtext>d</mtext></msub><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mrow><mi>cosh</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd><mtd><mrow><mfrac><mrow><mi>sin</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow><mrow><msub><mi>k</mi><mtext>q</mtext></msub></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>k</mi><mtext>q</mtext></msub><mi>sinh</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd><mtd><mrow><mi>cosh</mi><msub><mi>k</mi><mtext>q</mtext></msub><mi>L</mi></mrow></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> One proposal for effectively preventing the growth in emittance caused by the quadrupole wake was to divide the dechirper into two orthogonal dechirpers [ 27 27 ]. This arrangement mode is explored based on beam-optics optimizations in SHINE. First, the entire 10 m length of the dechirper is required and divided into effectively two dechirpers with 5 m intervals. The two sections are oriented orthogonally, one with the plates vertical and the other horizontal. We use ‘V’ and ‘H’ to denote vertical and horizontal plates, respectively. Figure 8 Figure 8 shows the projected emittance growth for four different combinations. The results were verified and compared by simulation using ELEGANT [ 28 28 ]. As expected, while the combinations VV and HH yield a greater projected emittance growth even when the beam is perfectly aligned, the HV and VH combinations preserve the projected emittance effectively after the dechirper section. The different performances are caused by the features in the quadrupole wake. The quadrupole wake holds only one transverse direction in focus, with an equal defocusing strength in the other transverse direction. Therefore, VV and HH degenerate the transverse phase space in one direction, but VH and HV counteract the strength self-consistently. To improve the beam quality in SHINE and maintain the projected emittance, we attempted to divide the dechirper into four sections of uniform length 2.5 m (hereafter named “four-dechirpers”). The two-dechirper and four-dechirper layouts are depicted in Fig. 9 Fig. 9 based on the FODO design. The blue ellipse represents the bunch on-axis. The transverse direction points perpendicular to the page while the black arrow under the e-beam defines the longitudinal direction. The corrugated structures are orthogonal, distributed between the quadrupole magnets. One FODO structure is formed in the two-dechirper and two are formed in the four-dechirper. The hypothesis on the beta functions is validated using a thick-lens calculation. The final transfer matrix is thus expressed as a 2×2 matrix M f , and the original and final Twiss parameters, given by ( α 0 , β 0 , γ 0 ) and ( α , β , γ ), respectively, are related as γ =(1+ α 2 )/ β . As shown in Eq. (17) (where < > denotes the numerical average obtained by integrating over the bunch length), the quadrupole wake transforms exactly like a magnetic quadrupole for any slice position in s . By computing the transfer matrix with the structural parameters, the average of the final Twiss parameter and the emittance growth can be calculated as <math display="block" id="M18"><mrow><msub><mi>ϵ</mi><mtext>f</mtext></msub><mo>/</mo><msub><mi>ϵ</mi><mn>0</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mo>〈</mo><msub><mi>γ</mi><mtext>f</mtext></msub><mo>〉</mo><mo>〈</mo><msub><mi>β</mi><mtext>f</mtext></msub><mo>〉</mo><mo>−</mo><msup><mrow><mo>〈</mo><msub><mi>α</mi><mtext>f</mtext></msub><mo>〉</mo></mrow><mn>2</mn></msup><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mn>,</mn></mrow></math> where the subscripts f(o) represent the final (original) situation. Then, the other plane is also suitable. In contrast to the conventional FODO design, we utilized the two- and four-dechirpers, including the quadrupole wake generated in the corrugated structure. This destroys the periodicity in the Twiss parameters and causes the changes in the projected emittance in the final. The projected emittance growth for different magnet lengths L and magnet strengths K of the quadrupole magnets is demonstrated in Fig. 10 Fig. 10 . However, the minimum projected emittances in the two- and four-dechirpers are 0.0175% and 0.00238%, respectively. These values are comparatively small. The projected four-dechirper emittance still performs better. In the case of the symmetric feature in the FODO structure, the conclusion reached for the x direction does not apply to the y direction. The optimized working point (red dot) is selected for the quadrupole magnets, where the function β x is minimal and shows the best performance in terms of the emittance growth. The above calculation confirms the validity of dividing the dechirpers into four. The β functions for both models are plotted in Fig. 9 Fig. 9 . In [ 29 29 ], the emittance growth caused by the quadrupole wakefield is fully compensated only if β x = β y . In practice, however, the beta functions always fluctuate, and the beam suffers from the residual quadrupole wakefield. For a period FODO cell, the difference between the maximum and minimum β values is given by Eq. (18) [ 30 30 ], where K and L denote the quadrupole strength and length separately, and l the length for each separated dechirper (5 and 2.5 m for the two- and four-dechirpers, respectively). <math display="block" id="M19"><mrow><msub><mi>β</mi><mrow><mtext>max</mtext></mrow></msub><mo>−</mo><msub><mi>β</mi><mrow><mtext>min</mtext></mrow></msub><mo>=</mo><mn>4</mn><mi>l</mi><mo>/</mo><msqrt><mrow><mn>64</mn><mo>−</mo><msup><mi>K</mi><mn>2</mn></msup><msup><mi>L</mi><mn>2</mn></msup><msup><mi>l</mi><mn>2</mn></msup></mrow></msqrt><mn>.</mn></mrow></math> Figure 11 Figure 11 compares the β functions for two- and four-dechirpers by scanning the magnet strength K and the magnet length L . The minimum differences are 2.50 m and 1.25 for the two- and four-dechirpers, respectively. This implies a better compensation of the quadrupole wakefield-induced emittance growth in the four-dechirper schemes. In practical engineering applications, four-dechirpers are considered appropriate to simplify the system. Finally, the projected emittance changes for the two- and four-dechirpers were simulated separately in the actual bunch with the working point, as optimized. We also compared both schemes with the ELEGANT code using the actual bunch distribution with the optimized working point. The results are summarized in Table 3 Table 3 . The transverse phase space is shown in Fig. 12 Fig. 12 . Because of the mismatch in the actual bunch when it goes through either the two- or four-dechirper scheme, the actual bunch hardly maintains the projected emittance as analyzed. The x and x’ in the Gaussian bunch are almost invariant, but the mismatch in the actual bunch cannot be ignored and must be taken into account. As a result, the actual bunch has a lower emittance in the four-dechirpers scheme. This therefore makes the four-dechirpers a more feasible and efficient scheme for preserving the emittance for SHINE. 1 1   4 1 Gaussian bunch 2 1 Actual bunch 1 1   2 1 Analytical results 2 1 Simulation 2 1 Simulation 1 1   1 1 Two-dechirpers 1 1 Four-dechirpers 1 1 Two-dechirpers 1 1 Four-dechirpers 1 1 Two-dechirpers 1 1 Four-dechirpers 1 1 δϵ x / ϵ x 1 1 0.0175% 1 1 0.00238% 1 1 0.209% 1 1 0.008% 1 1 0.647% 1 1 -0.33 1 1 δϵ y / ϵ y 1 1 0.0031% 1 1 0.0006% 1 1 0.195% 1 1 0.006% 1 1 1.282% 1 1 -0.36 % 5 Brief conclusion and Discussion 1 This study systematically investigated the effectiveness of using a corrugated structure as a passive device to remove residual beam chirp in the SHINE project. We simulated the application of the dechirper to the SHINE beam and studied the transverse and longitudinal wakefield effects. A detailed parameter optimization of the corrugated structure was carried out using analytic formulas. 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Johnsen , Theoretical aspects of the behavior of beams in accelerators and storage rings , p 34 ( 1977 ). https://cds.cern.ch/record/118362/files/CERN-77-13.pdf doi: 10.5170/CERN-1977-013 http://doi.org/10.5170/CERN-1977-013 Theoretical aspects of the behavior of beams in accelerators and storage rings All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by You-Wei Gong, Meng Zhang and Duan Gu. The first draft of the manuscript was written by You-Wei Gong and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-021-00860-8 This work was supported by the Youth Innovation Promotion Association CAS (No. 2018300), the National Key Research and Development Program of China (No. 2018YFE0103100), and the National Natural Science Foundation of China (No. 11935020). 2020-08-14 2021-01-11 2021-01-13 2021-03-17 2021-03-01 2021-03-31 © 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. 2021 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/s41365-021-00860-8 http://dx.doi.org/10.1007/s41365-021-00860-8 . 2021 Nuclear Science and Techniques 03 32