Simulation studies on two-frequency RF gun

Simulation studies on two-frequency RF gun 1 2 first-author 1 1 1 2 1 1 2 1 2 1 corresp Corresponding author, zhaozhentang@sinap.ac.cn Corresponding author, zhaozhentang@sinap.ac.cn zhaozhentang@sinap.ac.cn Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai201800, China University of Chinese Academy of Sciences, Beijing100039, China University of Chinese Academy of Sciences, Beijing100039, China Photocathode RF gun is widely used for particle accelerators as an electron source. When driving an RF electron gun at the fundamental frequency and a higher harmonic frequency simultaneously with proper field ratio and relative phase, it generates electron beams of ultralow emittance and a linear longitudinal phase space distribution. Such a gun provides high quality electron beam with low energy spread, small traverse emittance and high brightness. In this paper, the RF design of a 1.5 cell cavity is presented. Simulation results of beam dynamics for the two-frequency gun and a standard single-frequency RF gun are also shown in this paper. In addition, bunch compression with a two-frequency gun is explored. RF gun Two-frequency Traverse emittance Emittance compensation Energy spread Bunch compression I. INTRODUCTION 1 Generating high quality electron beams is a critical issue for free-electron laser (FEL) facilities. The characteristics of electron beams and reliability of an FEL facility strongly depend on properties of the electron source. To get electron beams of sufficient brightness, photocathode RF guns are widely adopted in most FEL facilities nowadays. Transverse emittance of an RF gun is predominated by space charge forces, RF effects and thermal emittance. As both linear space charge and RF effects can be compensated by solenoid technique [ 1 1 ], the final projected emittance is dominated by space charge oscillation and emittance compensation [ 2 2 , 3 3 ]. Serafini et al . [ 4 4 ] proposed that when an RF gun cavity was operated with an extra harmonic TM 012- π mode, the field could neutralize the emittance blowup induced by RF-time-dependent forces. An RF cavity operated at 1.5 and 4.5 GHz was designed for PSI-XFEL [ 5 5 ]. For future FEL facilities, the final compression is so extreme that non-linear effects such as sinusoidal RF time-curvature may occur easily. A higher harmonic was proposed to compensate the compression transformation, while maintaining the initial temporal bunch profile. X-band RF harmonic compensation for linear bunch compression has been used in the LCLS [ 6 6 ]. Similar schemes have been proposed at DESY [ 7 7 ] and Boeing [ 8 8 ]. With harmonic section, a two-frequency RF gun can provide partial harmonic compensation for bunch compression. In this paper, a two-frequency 1.5 cell gun with optimized RF parameters is designed. The scheme adopted is of easier fabrication than the PSI type. Intensive simulations have been carried out. The emittance compensation process of a two-frequency gun is discussed in detail. The process of bunch compression with a two-frequency gun is performed by the lattice of the Shanghai deep ultraviolet FEL facility (SDUV-FEL) [ 9 9 ] at SINAP. II. RF CAVITY DESIGN 1 SUPERFISH [ 10 10 ], which is an excellent code to simulate electromagnetic field, was used to design the cavity. A convergence optimization was performed to ensure that the meshes are small enough for calculating the field accurately. The 1.5 cells RF cavity is supposed to be operated at 2.998 GHz and 8.994 GHz in TM 010- π and TM 012- π modes, respectively. To make the gun operate at the desired frequencies and expected modes, at least two geometric parameters of the cavity are needed to adjust the frequency. According to simulation results, fundamental frequency is sensitive to the radius of cells, while the third harmonic is sensitive to the length of cells. The working modes are shown in Fig. 1 Fig. 1 . The microwave parameters computed by SUPERFISH are given in Table 1 Table 1 . 1 1 Parameters 1 1 Fundamental mode 1 1 Triple harmonic mode 1 1 Working mode 1 1 TM 010- π 1 1 TM 012- π 1 1 Shunt impedance (MΩ/m) 1 1 66.093 1 1 3.809 1 1 Frequency (MHz) 1 1 2998 1 1 8994 1 1 Q-value 1 1 15554.3 1 1 17670.4 The field balance [ 11 11 ] of both modes are almost equal to 1, as shown in Fig. 2 Fig. 2 . A total peak flat-top on-axial field is obtained when the two modes are in phase opposition, with the fundamental and the 3 rd harmonic amplitudes being 100 MV/m and 11 MV/m, respectively. III. BEAM DYNAMICS SIMULATIONS 1 Properties of the electron beams generated by the two-frequency RF gun, i.e. beam energy, transverse emittance, the 2 nd order energy chirp and energy spread, were simulated using the codes of ASTRA [ 12 12 ] and ELEGANT [ 13 13 ]. In this paper, the emittance compensation process of a two-frequency RF gun, and all results without the 3 rd harmonic, are presented. To discuss the beam dynamics of an RF gun with a half-cell as the first cell [ 14 14 ], Kim proposed a simple analysis method [ 15 15 ], where the electric field along the axial is assumed as <math display="block" id="M1"><mrow><msub><mi>E</mi><mi>z</mi></msub><mo>=</mo><msub><mi>E</mi><mn>1</mn></msub><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where E 1 is the peak accelerating field, ω is the angular frequency, k = ω / c , and φ 0 is the initial injection phase. The exit normalize energy gain for a 1.5 cell gun is given by <math display="block" id="M2"><mrow><mi>Δ</mi><msub><mi>γ</mi><mn>1</mn></msub><mo>=</mo><mi>α</mi><mo stretchy="false">[</mo><mn>1.5</mn><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>−</mo><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>−</mo><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where α = eE 1 /(2 m 0 c 2 k ). So, an extra mode with frequency tripled to the fundamental mode is imported on the fundamental one, the axial electric field can be written as: <math display="block" id="M3"><mrow><msub><mi>E</mi><mi>z</mi></msub><mo>=</mo><msub><mi>E</mi><mn>1</mn></msub><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi><mn>3</mn></msub><mi>cos</mi><mo stretchy="false">(</mo><mn>3</mn><mi>k</mi><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>3</mn><mi>ω</mi><mi>t</mi><mo>+</mo><msub><mi>φ</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where φ 3 is the initial phase of the 3 rd harmonic, E 3 is the peak accelerating field of the 3 rd harmonic. The normalized energy gain from the 3 rd harmonic is <math display="block" id="M4"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><msub><mi>γ</mi><mn>3</mn></msub><mo>=</mo><mi>α</mi></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mn>3</mn></msub><mo>/</mo><msub><mi>E</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1.5</mn><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>3</mn><mi>ω</mi><mi>t</mi><mo>−</mo><mn>3</mn><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>3</mn><mi>ω</mi><mi>t</mi><mo>−</mo><mn>3</mn><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> From Eqs. (2) and (4), the total energy gain is <math display="block" id="M5"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>Δ</mi><mi>γ</mi><mo>=</mo><mi>α</mi></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">[</mo><mn>1.5</mn><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>−</mo><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo stretchy="false">(</mo><mi>ω</mi><mi>t</mi><mo>−</mo><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mn>3</mn></msub><mo>/</mo><msub><mi>E</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1.5</mn><mo>⋅</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>3</mn><mi>ω</mi><mi>t</mi><mo>−</mo><mn>3</mn><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>3</mn><mi>ω</mi><mi>t</mi><mo>−</mo><mn>3</mn><mi>k</mi><mi>z</mi><mo>+</mo><msub><mi>φ</mi><mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> For all simulations in this paper, φ 0 and φ 3 stemmed from ASTRA representing the initial phases of fundamental mode and the 3 rd harmonic, corresponding to φ 0 and φ 3 in Eq. (3). All simulations in this section were performed with electron beams of Gaussian profiles and in bunch charges of 1 nC. To demonstrate the two-frequency RF gun for accelerating long bunches, the pre-accelerated beam length was 20 ps, and E 3 =100 MV/m for all cases. According to Serafini et al . [ 4 4 ], an additional harmonic to the RF fundamental field would greatly reduce the RF emittance. A harmonic RF field shall linearize the RF force to the 4 th order, both longitudinally and transversely [ 16 16 ]. The condition can be achieved when the bunch exit phase is 90 and the n -th harmonic field ( E n ) over the fundamental field ( E 1 ) is E n / E 1 = 1/ n 2 . In this case, φ 0 =52.5° is the exit phase for bunch exit phase of 90. However, this set of RF parameters are not applicable for emittance compensation, though the RF emittance is minimized. In an operating BNL type photoinjector, the final projected emittance is dominated by space charge osillation and emittance compensation [ 2 2 ]. In order to minimize chromatic aberration effect, φ 0 is always chosen to minimize energy spread at the gun exit. In an optimal case, φ 0 =47°, φ 3 =4° and E 3 =14 MV/m, by optimizing the rms energy spread at the gun exit. With this set of RF parameters, the 2 nd order energy chirp and the emittance compensation can be optimized. A. Beam energy 2 At φ 0 =47°, the effect of the 3 rd harmonic on beam energy is simulated. According to Eq. (5), the beam energy at the gun exit is decided by the fundamental mode and the 3 rd harmonic. As shown in Fig. 3(a) Fig. 3(a) , with harmonic field amplitude fixed, Δ γ is a sinusoidal function of φ 3 , agreeing with Eq. (5); while with φ 3 fixed ( Fig. 3(b) Fig. 3(b) ), Δ γ 3 varies linearly with E 3 , agreeing with Eq. (4). B. RF emittance, energy spread and the longitudinal phase space 2 To illustrate the RF emittance reduction by harmonic field, the space charge effect is switched off, and φ 0 =52.5° is suggested by the theory [ 16 16 ]. In presence of the 3 rd harmonic, the RF emittance is reduced, compared with the case without the 3 rd harmonic ( Fig. 4(a) Fig. 4(a) ). By optimizing the energy spread at the gun exit at φ 0 =47° under space charge, we found optimal harmonic settings for the two-frequency RF gun: φ 3 =4° and E 3 = 14 MV/m ( Fig. 4(b) Fig. 4(b) ). To discuss the longitudinal phase space distribution, we performed a linear fitting of the relative energy deviation, which is a function of the relative position to a reference particle, it is given by <math display="block" id="M6"><mrow><mi>Δ</mi><mi>E</mi><mo>/</mo><msub><mi>E</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>Z</mi><mo>+</mo><mi>b</mi><msup><mi>Z</mi><mn>2</mn></msup><mn>,</mn></mrow></math> where a and b are the coefficients related to longitudinal phase space, E 0 is energy of the reference particle, Δ E is the relative energy of the electron to the reference particle, and Z is the longitudinal position of the electron with respect to the reference particle. The second derivative of Δ E / E 0 ( Z ) represents the 2 nd order energy chirp, which can be removed at b =0. At φ 0 =47°, simulations of the 2 nd order energy chirp as function of the amplitude of harmonic field was performed in presence of the space charge. In addition, at φ 3 =4°, so that the maximum efficiency for linearizing longitudinal phase space is achieved according to Fig. 4(b) Fig. 4(b) . At φ 0 =47°, E 1 =100 MV/m and φ 3 = 4, coefficient b was -16.78/m 2 , -5.57/m 2 , -0.13/m 2 and 4.34/m 2 at E 3 = 0, 10, 14 and 20 MV/m, respectively. The 2 nd order energy chirp could be almost compensated by the 3 rd harmonic at E 3 =14 MV/m, compared with the case of no 3 rd harmonic. However, if the harmonic field amplitude exceeds certain threshold, the 2 nd order energy chirp would be over compensated. Figure 5(a) Figure 5(a) shows that for 20-ps bunches, the 3 rd harmonic has linearized the longitudinal phase space in absence of the space charge. The 2 nd order energy chirp is removed by 3 rd harmonic at E 3 =14 MV/m and φ 3 =4°. The 3 rd harmonic has improved the longitudinal phase space in presence of the space charge ( Fig. 5(b) Fig. 5(b) ). The full width, correlated energy spread is reduced from 90 keV to 50 keV. C. Emittance compensation of a two-frequency RF gun 2 To illustrate the right set of RF parameters for emittance compensation, a transverse emittance monitoring point is set at the exit of a 1.5 m drift downstream the gun. As shown in Table 2 Table 2 , at φ 0 =52.5° the energy spread at gun exit and the transverse emittance are bigger than at φ 0 =47°. The energy spread at the gun exit is not minimized at φ 0 =52.5°, which indicates that the chromatic aberration effect is not minimized, so that the emittance compensation proceeds incorrectly. At φ 0 =47°, E 3 =14 MV/m and φ 3 =4°, the energy spread is minimized, so that the transverse emittance at the drift exit is optimized. 1 1 φ 0 (°) 1 1 E 3 (MV/m) 1 1 φ 3 (°) 1 1 Solenoid field strength (T) 1 1 Energy spread (keV) 1 1 Transverse emittance (mm mrad) 1 1 52.5 1 1 0 1 1 0 1 1 0.201 1 1 35.5 1 1 3.7 1 1 52.5 1 1 11 1 1 4 1 1 0.178 1 1 48.0 1 1 3.9 1 1 47 1 1 0 1 1 0 1 1 0.198 1 1 17.0 1 1 1.7 1 1 48 1 1 14 1 1 4 1 1 0.177 1 1 6.5 1 1 1.3 IV. BUNCH COMPRESSION WITH A TWO-FREQUENCY GUN 1 As mentioned above, the 3 rd harmonic helps linearize the energy chirp for the process of bunch compression. What is more, by over compensating the 2 nd order energy chirp which imports an extra 2 nd order energy chirp, the two-frequency gun can provide harmonic compensation for bunch compression [ 6 6 ]. As mentioned in Sec. III B, such an extra 2 nd order energy chirp can be produced at E 3 =20 MV/m. The lattice of SDUV-FEL at SINAP, as shown in Fig. 6 Fig. 6 , was used to simulate the process of bunch compression, performed with a 800-pC 20-ps long bunch in presence of the space charge. All settings of the S-band and chicane are the same for the cases with and without the 3 rd harmonic. As shown in Fig. 7(a) Fig. 7(a) , the 3 rd harmonic could improve the beam current distribution at the exit of the bunch compressor. The symmetry bunch distribution of the case with the 3 rd harmonic is better than the case without the 3 rd harmonic. As mentioned in Sec. III B, the bunch shall be over compensated at the gun exit by the 3 rd harmonic at E 3 =20 MV/m. To evaluate its effect on emittance compensation, the transverse emittance is studied in presence of the space charge. As shown in Fig. 7(b) Fig. 7(b) , the transverse emittance evolution is not deteriorated obviously, compared with the optimal case in Table 2 Table 2 . V. CONCLUSION 1 A 1.5 cell two-frequency RF gun, working at 2998 MHz and 8994 MHz, is designed by SUPERFISH. Beam dynamics simulations demonstrate the advantages of a two-frequency RF gun over a single-frequency gun when generating high charge (nC) and high brightness electron beams. Besides, harmonic compensation for bunch compression by manipulating the harmonic component in the two-frequency RF gun is explored. Compared with a single-frequency RF gun without a harmonic linac, simulations show a symmetric bunch compression with comparable transverse emittance. Compared with the case of two-frequency gun without harmonic compensation, the transverse emittance is a bit larger, which indicates the extra 2 nd order energy chirp in the gun for downstream beam energy chirp linearization deteriorates the emittance compensation. References Kim K J . Nucl Instrum Meth A , 1989 , 275 : 201 - 218 . Nucl Instrum Meth A Serafini L and Rosenzweig J B . Phys Rev E , 1997 , 55 : 7565 - 7590 . 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