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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 ≈ 1.471 × 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.

Absorbing boundary conditions for waveguide ports in time domain are important elements of transient approaches to treat RF structures. A successful way to implement these termination conditions is the decomposition of the transient fields in the absorbing plane in terms of modal field patterns. The absorbing condition is then accomplished by transferring the wave impedances (or admittances) of the modes to time domain, which leads to convolution operations involving Bessel functions and integrals of Bessel functions. This paper presents a new alternative approach: the convolution operations are approximated by appropriate state-space models whose system responses can be conveniently computed by standard integration schemes. These schemes are indispensable for transient simulations anyhow. Sufficiently far away from the cutoff frequency, a wideband match is achieved.Index Terms-Modal analysis, modal wave absorption, time-domain analysis, waveguide boundary condition.