Eigenmode compendium of the third harmonic module of the European X-ray Free Electron Laser

Chains of superconducting radio-frequency resonators are key components of modern particle accelerators such as the European XFEL, which is currently under construction in the north of Germany. In addition to the accelerating mode of the resonators, their beam excited higher order modes are of special interest, because they can harm the beam quality. In contrast to the accelerating mode, these modes are in general not confined within single resonators of the cavity string. For instance, eigenmodes can be localized between adjacent cavities or can be distributed along the entire chain of cavities. Therefore, the full chain has to be considered for a reasonable investigation of its resonant spectra. Accounting for such complex structures is computationally challenging and is therefore often avoided. In this article, the challenge is faced by using the so-called state-space concatenation approach, which is a combination of domain decomposition and model-order reduction. The technique allows for a reduction of the number of degrees of freedom by a factor of $\approx 1.471\times {10}^{-4}$. The method is employed to generate a compendium of eigenmodes in the chain of third harmonic cavities for the European XFEL. The results are discussed in detail and are compared with experimental measurements. The compendium serves as a reference for experiments (inter alia for diagnostics based on higher order modes) at the third harmonic cavity string of the European XFEL, it allows for qualitative understanding of resonant effects appearing in chains of cavities, and it is meant to be a proof of principle of the state-space concatenation approach to handle very long and complex radio-frequency structures. To the authors’ knowledge, it is the first time that a modal compendium of a structure with the given complexity is generated. The article presents geometrical details of the chain, defines quantities relevant to superconducting radio-frequency cavities, and describes the employed computational approach.

Figure 1

(a) Chain of eight superconducting third harmonic cavities. The electron gun is positioned on the right-hand side of the string. Each cavity is equipped with a HOM coupler and a pickup on its left-hand side (downstream) and a HOM coupler with a power coupler on the right-hand side (upstream). Couplers of cavities with an even index are rotated by an angle of 180° around the longitudinal axis. Bellows are mounted between the cavity-coupler combinations. (b) Magnification of the red rectangle in (a), i.e. the first two cavities with couplers. HOM couplers are placed on the left-hand side and HOM couplers with power couplers are mounted on the right-hand side. The HOM couplers are based on the one-leg design [see Fig. 3(a) in 34]. The length $L_{\mathrm{Ca}}$ of the cavity with couplers is 505.92 mm whereas the length of the bellows is 102 mm. Both (a) and (b) are modifications of Fig. 4 in [41].

Figure 2

Segments considered in the framework of this paper: (a) beam pipe with HOM coupler and power coupler, (b) beam pipe with HOM coupler and power coupler 180° rotated, (c) beam pipe with HOM coupler, (d) beam pipe with HOM coupler 180° rotated, (e) bellow, and (f) nine-cell third harmonic cavity. The segments (a)–(f) arise from the decomposition of the chain of cavities (refer to Fig. 1). Table 2 discusses further details of the segments.

Figure 3

Abstract counterpart of the cut plane between the first HOM coupler and the first cavity in the chain. At the cut plane eight 2D port modes are considered. They are accounted for by eight terminals. The subscripts $r$, $p$, $m$ of the modal current $i_{r,p,m}(t)$ and the modal voltage $v_{r,p,m}(t)$ indicate that the quantities refer to the $m$th 2D mode at the $p$th port of the $r$th segment.

Figure 4

Properties of the modes in the chain of eight third harmonic cavities [see Fig. 1]. The orange rectangles denote the location of the monopole bands, the green rectangles the location of the dipole bands, and the purple rectangles the location of the quadrupole bands. The locations of the bands result from an eigenmode computation for a rotational symmetric cavity without couplers [15]. (a) External quality factors, (b) geometrical impedances, and (c) the product of the external quality factors and geometrical impedances for the eigenmodes in the XFEL chain of third harmonic cavities.

Figure 5

Absolute value of the normalized electric fields of the HOM coupler modes. (a) HOM coupler mode with the resonant frequency of approximately 2.60 GHz. The energy of this mode is predominantly localized between the formteil of the coupler and the antenna tip. (b) HOM coupler mode with the resonant frequency of approximately 3.18 GHz. The energy of this mode is localized in the gap between the formteil and the housing as well as in the coaxial connector of the coupler. (c) Legend for the field plots.

Figure 6

Absolute value of the normalized electric fields of a set of several example eigenmodes in the chain. (a) ${\mathrm{TM}}_{01}\text{−}\pi$ mode located in the third cavity: modal index $n=99$, $f_n=3.897618\mathrm{GHz}$, $R_n/Q_n\approx 743\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 6.34\times 10^5$. (b) Dipolelike eigenmode with energy predominantly allocated in the end cells next to the HOM couplers without input couplers: modal index $n=130$, $f_n=4.135416\mathrm{GHz}$, $R_n/Q_n\approx 0\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 2.51\times 10^6$. (c) Dipolelike eigenmode whose energy is distributed along the entire cavity chain (multicavity mode): modal index $n=238$, $f_n=4.582944\mathrm{GHz}$, $R_n/Q_n\approx 0\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 4.68\times 10^2$. (d) Eigenmode with field energy between the cavities (bellow or intercavity mode): modal index $n=359$, $f_n=5.152407\mathrm{GHz}$, $R_n/Q_n\approx 1.44\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 2.17\times 10^4$. (e) Dipolelike eigenmode with field energy distributed along the entire chain: modal index $n=507$, $f_n=5.485864\mathrm{GHz}$, $R_n/Q_n\approx 0\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 5.06\times 10^3$. (f) Eigenmode with energy predominantly allocated at the connection between the first and the second cavity: modal index $n=536$, $f_n=5.556085\mathrm{GHz}$, $R_n/Q_n\approx 0.26\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 4.40\times 10^1$. (g) Quadrupolelike eigenmode which is distributed along the chain: modal index $n=689$, $f_n=6.572043\mathrm{GHz}$, $R_n/Q_n\approx 0\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 8.45\times 10^5$. (h) Eigenmode with field energy between adjacent cavities: modal index $n=741$, $f_n=6.672766\mathrm{GHz}$, $R_n/Q_n\approx 0\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 7.00\times 10^1$. (i) Higher order mode with the largest observed geometrical impedance: modal index $n=1360$, $f_n=7.670048\mathrm{GHz}$, $R_n/Q_n\approx 294\mathrm{\Omega }$, $Q_{\mathrm{ext},n}\approx 4.23\times 10^2$. (j) Legend for the field plots.

Figure 7

Comparison between measured (black curves) and simulated (red curves) scattering transmissions of the chain of eight third harmonic cavities (refer to Fig. 1). The green rectangles indicate the location of the dipole bands of the single third harmonic cavities [15]. (a) Transmission from the left (downstream) HOM coupler to the right (upstream) HOM coupler of cavity 1. The cavity is embedded in the entire chain. (b) Transmission through the entire cavity chain, i.e. from the left (downstream) HOM coupler of cavity 1 to the right (upstream) one of cavity 8.