Extrapolation methods for Sommerfeld integral tails

A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e., semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (a, infinity) of the integration path, where a>0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points rho and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of rho and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.