Adapted Contour Integration for Nonlinear Eigenvalue Problems in Waveguide Coupled Resonators

Contour integration methods are claimed to be the methods of choice for computing many (several hundred) eigenvalues of a nonlinear eigenvalue problem inside a closed region of the complex plane. Typically, contour integration methods are designed for circular (or more generally elliptic) shaped contours and rely on the exponential convergence of the trapezoidal rule applied to periodic functions. In this article, the curl–curl eigenvalue problem in a resonator coupled with a waveguide boundary in a way that allows outgoing waves along longitudinally homogeneous waveguide structures is considered. This problem has a square root dependence on the frequency, and thus, adapted integration contours are required to reliably find eigenvalues in the vicinity of branch cuts. The filter function-based analysis of the quadrature rules has been used and improved to reduce the problem to consider the behavior of filter functions on eigenvalues and singular points only. First, conformally mapped circular contours are considered for problems with one branch cut. For problems where there are several branch cuts necessary, the Gauß–Legendre quadrature rules on closed polygonal contours had been analyzed. In both cases, exponential convergence rates were obtained. The estimates are validated numerically using the example of the TESLA cavity.