Resonant Lossy Electromagnetic Structures

Prof. Dr. P. Arbenz, Institute of Computational Science
Dr. B. Oswald, Paul Scherrer Institut

Keywords: Electromagnetic resonators, complex dielectric permittivity and magnetic permeability, dispersive dielectric media, leaking electromagnetic resonators, computational electrodynamics, complex eigenvalue problem, quadratic eigenvalue problem, high performance computing

Funding: This project is supported by the Swiss National Science Foundation by contract #200021-117978/1.


Resonant electromagnetic cavity structures are used in virtually all types of particle accelerators. The X-ray free electron laser currently under study at the Paul Scherrer Institut, is no exception and will consist of a large variety of radio frequency (RF) structures for guiding and accelerating electrons from the photo-cathode through the linear accelerator section, see LEG-Web-Site.

The numerical computation of eigenfrequencies and corresponding eigenmodal fields of large accelerator cavities, based on full-wave, three-dimensional models, has attracted considerable interest in the recent past.

Much work has been invested to compute electromagnetic eigenmodes. In many cases the eigenproblem has been modeled without electromagnetic loss mechanisms. This is a justifiable approximation, especially when saving computational expense is an issue or when the cavity becomes large in terms of the dominant wavelength.

However, the eigenmodal solution is affected considerably by loss. Traditionally, loss has been integrated into the model by computing the quality factor Q. The electromagnetic power dissipated in the cavity boundary is calculated with a perturbation approach using the magnetic field B after the eigenmodes have been computed. While this allows for estimating that important cavity parameter it does not model the effect of loss onto the resonance frequency which is one of the most important cavity parameters. Only if losses are integrated into the eigenvalue problem a priori can we extract thereby affected resonance frequencies from the eigensolution.

We intend to model different loss mechanisms:

  1. Lossy dielectric and/or magnetic materials;
  2. Larger or smaller apertures in the boundary of the resonating structure. These significantly change the character of the eigenvalue problem by introducing loss caused by electromagnetic fields radiating away part of the energy stored in the field;
  3. Finite conductivity cavity wall by the inclusion of a surface impedance boundary condition.

We intend to used the finite element (FE) method because modern cavity designs typically exhibit delicate and detailed geometrical features that must be considered for obtaining accurate results.

There is some work into this direction. We note that often the approaches are relatively small with only a few thousand unknowns, or, losses, introduced either by holes in the aperture or cavity walls with a finite conductivity, are not addressed. Today's accelerator cavities require a computational mesh with tens of millions of unknowns and, ultimately, in the region of 108 unknowns.

We plan to extend an available parallel Jacobi-Davidson-type solver for large-scale real symmetric eigenvalue problems, see Ref [1] into a solver of complex symmetric eigenvalue problems.


  1. P. Arbenz, M. Bečka, R. Geus, U. L. Hetmaniuk, and T. Mengotti: "On a Parallel Multilevel Preconditioned Maxwell Eigensolver". Parallel Computing, 32 (2): 157-165 (2006).
  2. P. Arbenz and O. Chinellato: "On solving complex-symmetric eigenvalue problems arising in the design of axisymmetric VCSEL devices". Appl. Numer. Math. 58 (4): 381-394 (2008), doi:10.1016/j.apnum.2007.01.019
  3. O. Chinellato, P. Arbenz, M. Streiff, and A. Witzig: "Computation of Optical Modes Inside Axisymmetric Open Cavity Resonators". Future Generation Computer Systems 21 (8): 1263-1274 (2005).


Prof. Dr. Peter Arbenz
Institute of Computational Science
Universitätsstrasse 6, CAB G69.3
CH-8092 Zürich
Tel.: +41 44 632 74 32

Dr. Benedikt Oswald
Paul Scherrer Institut
CH-5232 Villigen
Tel.: +41 56 310 32 12

Last update: Aug 14 2008