Design of high gradient, high repetition rate damped $C$-band rf structures

The gamma beam system of the European Extreme Light Infrastructure–Nuclear Physics project foresees the use of a multibunch train colliding with a high intensity recirculated laser pulse. The linac energy booster is composed of 12 traveling wave $C$-band structures, 1.8 m long with a field phase advance per cell of $2\pi /3$ and a repetition rate of 100 Hz. Because of the multibunch operation, the structures have been designed with a dipole higher order mode (HOM) damping system to avoid beam breakup (BBU). They are quasiconstant gradient structures with symmetric input couplers and a very effective damping of the HOMs in each cell based on silicon carbide (SiC) rf absorbers coupled to each cell through waveguides. An optimization of the electromagnetic and mechanical design has been done to simplify the fabrication and to reduce the cost of the structures. In the paper, after a review of the beam dynamics issues related to the BBU effects, we discuss the electromagnetic and thermomechanic design criteria of the structures. We also illustrate the criteria to compensate the beam loading and the rf measurements that show the effectiveness of the HOM damping.

Figure 1

Normalized Courant Snyder invariants at the exit of the linac for an initial displacement of all bunches of $200\mu \mathrm{m}$ (in the calculation we have considered an average accelerating field of $28\mathrm{MV}/\mathrm{m}$ that includes the drift spaces, diagnostics etc.).

Figure 2

Maximum value of the normalized Courant Snyder invariants at the end of the $C$-band linac as a function of the first dipole mode quality factor and for different $\beta$-functions (initial displacement of all bunches: $200\mu \mathrm{m}$, perfect transverse wake buildup).

Figure 3

Maximum value of the normalized Courant Snyder invariants at the end of the $C$-band linac as a function of the injection errors and for different Q-factors of the first dipole modes (perfect transverse wake buildup).

Figure 4

hfss single cell geometry with main cell dimensional parameters (table dimensions in mm).

Figure 5

hfss simulated transmission coefficient between two waveguides of a five-cell structure for two different waveguide apertures ${\mathrm{a}}_{\mathrm{damp}}$ in the frequency interval corresponding to the first dipole band (blue curve: ${\mathrm{a}}_{\mathrm{damp}}=7\mathrm{mm}$, red curve: ${\mathrm{a}}_{\mathrm{damp}}=13\mathrm{mm}$).

Figure 6

cst 20-cell simulated structures with perfect matched waveguides (a) and with SiC absorbers (b).

Figure 7

(a) Transverse wake potential per unit length calculated by cst: beam sigma 3 mm, beam off axis 2 mm; (b) corresponding transverse impedance.

Figure 8

hfss simulated structures: (a) module of 12 cells with SiC absorbers; (b) absorbers; (c) main dimensions of the absorber; (d) single cell with absorber (PE means perfect electric boundary condition, PH perfect magnetic boundary condition). Table dimensions are in mm.

Figure 9

Reflection coefficients at the four input ports of Fig. 8, corresponding to the bandwidth of the first dipole modes (hfss simulations, port nomenclature is referred to Fig. 8).

Figure 10

Quality factors of single cell dipole modes as a function of frequency up to 15 GHz (hfss results).

Figure 11

Quality factor of the fundamental accelerating mode as a function of the distance (h) between the SiC absorber and the accelerating cell.

Figure 12

(a) Longitudinal accelerating field profile at ${\mathrm{E}}_{\mathrm{acc}}=33\mathrm{MV}/\mathrm{m}$ and corresponding surface electric (b) and magnetic fields (c).

Figure 13

Single cell parameters as a function of the irises aperture a.

Figure 14

Average accelerating field, irises dimensions, attenuation constant, series impedance and group velocity as a function of the cell number for different types of structures.

Figure 15

(a) input coupler; (b) output coupler; (c) surface magnetic field at the input coupler corresponding to ${\mathrm{E}}_{\mathrm{acc}}=33\text{MV/m}$; (c) surface magnetic field at the output coupler corresponding to ${\mathrm{E}}_{\mathrm{acc}}=33\text{MV/m}$.

Figure 16

Azimuthal magnetic field in the center of the input coupler calculated for three different circle radii (continuous lines: without racetrack; dashed lines: with racetrack).

Figure 17

Multipole magnetic field components in the center of the input/output coupler. The components have been calculated considering the absolute value of the magnetic field $|{\mathrm{B}}_{\theta }|$ and developing it according to Eq. (3).

Figure 18

Longitudinal electric field profile along the whole structure simulated by hfss.

Figure 19

Final mechanical drawing of the single cell and of the 12-cell module.

Figure 20

Temperature distribution along the $C$-band structure.

Figure 21

Radial deformations: (a) last part of the structure (cell number 95); (b) first part of the structure (cell number 4). The probes indicate the deformations on the outer diameter of the cell and on the irises.

Figure 22

Pressure profile along the axis of the structure for two different values of the specific outgassing rate.

Figure 23

Schematic view of the beam loading process in TW structures: (a) traveling beam loading field profile generated by two bunches; (b) steady state condition.

Figure 24

rf pulse shape without shaping (red dashed curve, left scale) and with shaping for beam loading compensation (blue curve, left scale) and bunch energy distribution (right scale) with (blue dots) and without (red dots) compensation.

Figure 25

Picture of a 12-cell module (left); measurements setup (right).

Figure 26

Transmission coefficient between two antennas coupled to a 12-cell module with and without the absorbers (left); detail of the transmission coefficient corresponding to the bandwidth of the accelerating mode (right).

Figure 27

Field attenuation per unit length within the structure (blue); third order polynomial fit (red).

Figure 28

Quality factor as a function of z ($\mathrm{mean}=8860.1$, $\mathrm{sigma}=8.3$).

Figure 29

Inverse equation of motion of the pulse front edge ${\mathit{f}}^{-1}(\mathit{z})$.

Figure 30

Equation of motion of the pulse front edge $\mathit{f}(\mathit{t})$ calculated values (blue), second order polynomial fit (red).

Figure 31

Normalized accelerating field (blue) and rf power (red) in the structure.

Figure 32

Normalized wake potential generated by a charge of 1 pC in the structure W(z).

Figure 33

Wake propagation inside the structure observed at different instants ${\mathrm{t}}_n$.

Figure 34

Total wake potential in steady state beam loading normalized to the bunch charge.

Figure 35

Accelerating field profile along the structure in steady state beam loading (${\mathrm{E}}_{\mathrm{avg}}=33\mathrm{MV}\text{/m}$).

Figure 36

rf input power profile as a function of time for beam loading full compensation.

Figure 37

Input coupler geometry with lines for transverse field calculations.

Figure 38

Equivalent quadrupole gradients due to the E and B fields along the coupler. Integrated gradients ${\mathrm{G}}_E\cong -4\times {10}^{-3}[\mathrm{T}]$, ${\mathrm{G}}_B\cong 4\times {10}^{-2}[\mathrm{T}]$ (without racetrack), ${\mathrm{G}}_E\cong 0$, ${\mathrm{G}}_B\cong 0$ (with racetrack).

Figure 39

Equivalent quadrupole gradients due to the E and B fields with transit time factor for an on-crest particle. Integrated gradients ${\mathrm{G}}_E^{\prime }\cong -1\times {10}^{-3}[\mathrm{T}]$, ${\mathrm{G}}_B^{\prime }\cong -1\times {10}^{-2}[\mathrm{T}]$ (without racetrack), ${\mathrm{G}}_E^{\prime }\cong 0$, ${\mathrm{G}}_B^{\prime }\cong 0$ (with racetrack).