Microwave circuits are used in mobile communications, radio links, and sensors. For special applications in radioastronomy also higher frequencies up to 1 THz are used. Basic elements of microwave integrated circuits are the transmission lines, whose propagation behavior has to be determined accurately. For the numerical treatment, the computational domain has to be truncated by electric or magnetic walls or by a so-called absorbing boundary condition simulating the infinite space. A very efficient formulation for the latter case is the Perfectly Matched Layer (PML) introduced by Sacks *,  which does not require any modification of Maxwell's equation, in contrast to the split-field PML  proposed by Berenger **. The layers are filled with an artificial material with lossy anisotropic material properties, thus permittivity and permeability are complex diagonal tensors. The electromagnetic properties can be calculated by applying Maxwell's equations to the infinitely long homogeneous transmission line structure, which results in an eigenvalue problem for the propagation constants ***. In the presence of losses or absorbing boundary conditions the matrix of the eigenvalue problem becomes complex. The finite volume of the PML introduces additional non physical modes (so-called PML modes) that are not an intrinsic property of the waveguide. The system matrix is sparse and of high order. This requires efficient solvers that preserve sparseness and deliver only the small number of interesting modes out of the complete spectrum. The particular problem is that one has to make sure that all eigenvalues within a certain region in the complex plane are found. Using an estimation for the maximum propagation constant and mapping relations between the planes of eigenvalues and propagation constants an area bounded by a vertical straight line and a parabola is determined containing the eigenvalues which correspond to the desired propagation constants. A new method is presented which finds the eigenvalues of this area solving a sequence of eigenvalue problems with the aid of the invert mode of the Arnoldi method with shifts. In an additional step the non physical PML modes are eliminated.

* Sacks, Z. S., Kingsland, D. M., Lee, R., Lee, J.-F., (1995), "A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition", IEEE Transactions on Antennas and Propagation, Vol. 43 No. 12, pp. 1460--1463.
** Berenger, J.-P., (1994), "Perfectly Matched Layer for the Absorption of Electromagnetic Waves", Journal for Computational Physics, Vol. 114, pp. 185--200.
*** Hebermehl, G., Schlundt, R., Zscheile, H., Heinrich W., (1999) "Improved Numerical Methods for the Simulation of Microwave Circuits", Surveys on Mathematics for Industry, Vol. 9 No. 2, pp. 117--129.