Design of an rf quadrupole for Landau damping

The recently proposed superconducting quadrupole resonator for Landau damping in accelerators is subjected to a detailed design study. The optimization process of two different cavity types is presented following the requirements of the High Luminosity Large Hadron Collider (HL-LHC) with the main focus on quadrupolar strength, surface peak fields, and impedance. The lower order and higher order mode (LOM and HOM) spectrum of the optimized cavities is investigated and different approaches for their damping are proposed. On the basis of an example the first two higher order multipole errors are calculated. Likewise on this example the required rf power and optimal external quality factor for the input coupler is derived.

Figure 1

Cross section of quadrupolar field profiles providing the transverse kick to the particle according to (1). The force directions corresponds to a negative charge $q$. (a) Pillbox with TM-type mode. (b) Four-Vane-Cavity with TE-type mode.

Figure 2

Profile of elliptical half cells with equator radius $r_{\mathrm{eq}}$, iris radius $r_{\mathrm{ir}}$ and length $l_{\mathrm{hc}}$. The dotted lines subscribe the corresponding ellipses as well as the connecting straight with the concerted tangential angle $\alpha$. (a) Non-reentrant cavity with $\alpha <90\mathrm{deg}$ and (b) reentrant cavity with $\alpha >90\mathrm{deg}$.

Figure 3

Normalized fields distribution of the ${\mathrm{TM}}_{210}$ mode on the surface of an elliptical cavity section.

Figure 4

Quadrupolar strength $b_2$ with respect to the surface peak field $B_{pk}$ (a) and the ratio $B_{pk}/E_{pk}$ (b) of an 800 MHz non-reentrant elliptical cavity. The iris ellipse is fixed to $a_{\mathrm{ir}}=b_{\mathrm{ir}}=10\mathrm{mm}$ while the equator ellipse varies under the condition $a_{\mathrm{eq}}=b_{\mathrm{eq}}$ resulting in a circular profile. Besides, different cavity lengths $l_{\mathrm{hc}}$ are compared to each other showing the maximum in (a) close to $\lambda /2$.

Figure 5

Sketch of the cavity optimization. The optimizer minimizes the quality function $u(\mathbf{x})=B_{pk}/b_2$ where $\mathbf{x}$ is a vector of free design variables such as $a_{\mathrm{ir}}$, $b_{\mathrm{ir}}$, $a_{\mathrm{eq}}$, $a_{\mathrm{eq}}$ and $l_{\mathrm{hc}}$.

Figure 6

Optimal tangential angle as a function of the iris radius $r_{\mathrm{ir}}$. The transition between a reentrant and non-reentrant elliptical cavity is at $r_{\mathrm{ir}}\sim 60\mathrm{mm}$.

Figure 7

Optimized cavity parameters as functions of the iris radius $r_{\mathrm{ir}}$. (a) Cavity length. (b) Ratio between iris and equator ellipse by means of their horizontal half axes. Due to mechanical constraints, this ratio is limited. (c)—(d) Logarithmic ratio between the vertical and horizontal half axis (ellipticity) for the iris and equator ellipse.

Figure 8

(a) Maximal quadrupolar strength of the optimized cavities distinguished by their aperture assuming a magnetic peak field at the surface of $B_{pk}=100\mathrm{mT}$. (b) The corresponding ratio of the surface peak fields $B_{pk}/E_{pk}$.

Figure 9

Effective longitudinal (a) and transverse (b) impedance accumulated over the number of required cavities to satisfy the conditions in Table 1. In solid, the number of cavities is rounded up. In dashed, the number of cavities is not rounded providing smoother relation that shows a shallow minimum at $r_{\mathrm{ir}}=125\mathrm{mm}$ for the longitudinal impedance.

Figure 10

Electric and magnetic field distribution of the quadrupole mode in the transverse plane. The cavity is equipped with ports at the low field regions potentially suitable for damping lower and higher order modes.

Figure 11

Four-vane cavity design. (a) 3D view of the cavity with an open-left segment to depict the unique parts of the longitudinal and transverse cross-section. A quarter of the longitudinal cross section is shown in (b) while (c) and (d) show two different versions of the transverse cross section (only an eighth). The difference is given by a displacement of the circle resulting from a larger iris radius $r_{\mathrm{ir}}$ in (d). The beam is located along the bottom line in (b) or at the lower left corner in (c) and (d), respectively.

Figure 12

Normalized field magnitude of the TE like quadrupole mode on the surface and in the transverse plane at the longitudinal center of a four-vane cavity.

Figure 13

Quadrupolar strength with respect to the magnetic (a) and electric (b) surface peak field as a function of the two half axes of the iris ellipse. In dashed blue, the reference line of a circular profile. In dashed green, regression line of the maximums in ($a_{\mathrm{ir}}$, $b_{\mathrm{ir}}$). The cavity length is optimized for each data set with an iris radius $r_{\mathrm{ir}}=30\mathrm{mm}$, a tapering angle ${\alpha }_{\mathrm{tap}}=30\mathrm{deg}$, and a tangential angle $\alpha =1\mathrm{deg}$. The circle emphasizes the optimized design.

Figure 14

Properties of the optimized 800 MHz four-vane cavities with different iris radius $r_{\mathrm{ir}}$ and tapering angle ${\alpha }_{\mathrm{tap}}$. (a) Maximum achievable quadrupolar strength per cavity. (b) Ratio of surface peak fields. The effective longitudinal (c) and transverse (d) impedance are related to a whole system of cavities to provide the quadrupolar strength in Table 1.

Figure 15

Quarter of the longitudinal cross section of the four-vane cavity with beam pipe enhancement to allow the propagation of the first dipole mode.

Figure 16

Q-factor derived from the transverse wakefield for a design with an iris radius of $r_{\mathrm{ir}}=30\mathrm{mm}$ and a tapering angle ${\alpha }_{\mathrm{tap}}$ of 30 deg (a) or 60 deg (b), respectively. The beam pipe radius $r_{\mathrm{bp}}$ varies in a range where the dominating dipole mode starts to propagate. The tapering angle ${\alpha }_{\mathrm{bp}}$ describing the beam pipe enhancement is applied on the axis of abscissae.

Figure 17

Input power as a function of the external quality factor for a four-vane cavity with an iris radius of $r_{\mathrm{ir}}=30\mathrm{mm}$ and a tapering angle of ${\alpha }_{\mathrm{tap}}=30\mathrm{deg}$. The beam offset is represented by the radial coordinate $\rho$. The bunch form factor $F_{\mathrm{b}}$ at 800 MHz is around 0.5 [11].