4. A review of techniques for evaluation of sommerfeld integrals with applications to multiscale electromagnetic wave propagation
ABSTRACT: Rigorous calculation of electromagnetic wave fields in various multilayer topologies
involve the appropriate layered media Green’s functions, which in turn contain Sommerfeld integrals.
Even for simple topologies such as a single-layer lossy (or lossless) dielectric backed by a perfectly
conducting ground plane, modeling of radiation from microstrip patch antennas, or calculation of crosstalk
between high speed VLSI signal interconnects, estimation of undesired r.f. coupling to signal lines from
external sources, all require the knowledge of appropriate full-wave analysis. The multiscale nature of
such a class of problems becomes evident in the case of source excitations having a wide spectral content -
such as ultrawideband patch antennas or high speed, ultrashort signal pulses along signal lines.
Sommerfeld integrals that have been encountered in various problems over more than a century still pose
computational difficulties because of the nature of its integrand. Fundamentally, these are inverse Fourier
transforms that contain the Bessel function,〖 J〗_(n=0,1,2) (ρξ), an exponential term e^(-jκ|z-z^' | )
in addition to another term that collectively describes the effects of the dielectric backed PEC - generically
known as the “slab function” F(ξ,d). For microstrip patch antenna arrays, the lateral electrical separation
kρ→∞ and z=z^', for calculating mutual coupling between array elements. In case of crosstalk between signal
interconnects, that are densely laid out in a small area, one notes that kρ→0. These two situations for a
fixed lateral separation can arise in wideband signal spectra, which subsequently can result in using large
and small arguments of the Bessel function, respectively. Furthermore, for situations involving mutual
coupling and crosstalk calculations, the convergence of the integrand is severely deteriorated because the
exponential term vanishes as z≅z^'. This happens when sources are located on the interface. The preceding
discussion suggests that in the case of multiscale phenomena, Sommerfeld integrals, at the low and high
frequency ends of the signal spectrum could exhibit different types of convergence behaviors that would
suggest using different types of algorithms for its numerical evaluation.
The slab function F(ξ,d) contains TE, TM surface and leaky wave poles of the Sommerfeld integrand.
The number of these poles, and hence their corresponding residue contributions, increases with electrical
thickness d/λ. Determination of the location of these poles require robust root-finding algorithms and is
thus a computational bottleneck for all full-wave EM algorithms. This observation, in addition to the
preceding convergence issues described above, globally captures the formidable numerical difficulties
in evaluation of Sommerfeld integrals for multiscale electromagnetic problems involving multilayer
There exists several algorithms for evaluating Sommerfeld integrals, with their attendant limitations.
However, examining the various approaches it appears that a significant improvement in the evaluation of
Sommerfeld integrals for multiscale, multilayer problems would be effected if: (a) pole contributions and
(b) Bessel function approximations were obviated. In the presentation recent methods for evaluating
Sommerfeld integrals complying with the stipulations (a) and (b) will be highlighted with applications
to multilayer topologies. Numerical comparisons against existing methods and commercial EM solvers will be
included to demonstrate the more global nature of this algorithm. Possible extensions to the general
multilayer case will be discussed.
 W. C. Chew and L. J. Jiang, “Overview of the Large Scale Computing: The Past, the Present and the Future,” Proceedings of the IEEE, vol. 101, no. 2, pp. 227-241, February 2013
 K. A. Michalski and J. R. Mosig, “Efficient Computation of Sommerfeld Integral Tails - Methods and Algorithms,” Journal of Electromagnetic Waves and Applications, vol. 30, no. 3, pp. 281-317, 2016
 S. Barkeshli, P. H. Pathak and M. Marin, “An Asymptotic, Closed-Form, Microstrip Surface Green’s Function for the Efficient Moment-Method Analysis of Mutual Coupling in Microstrip Antennas,” IEEE Transactions on Antennas and Propagation, vol. 38, no. 9, pp. 1374-1383, September 1990
 D. Chatterjee, S. M. Rao and M. S. Kluskens, “Improved Evaluation of Sommerfeld Integrals for Microstrip Antenna Problems,” Proc. Intl. Symp. Electromag. Theory, (URSI Commission B), pp. 981-984, Hiroshima, Japan, May 2013
BIO: Deb Chatterjee is an associate professor with the Computer Science and Electrical
Engineering Department at the University of Missouri at Kansas City (UMKC), USA. He obtained his B.Tech,
M.Tech, M.A.Sc and PhD degrees in 1982, 1984, 1992 and 1998 respectively. Dr. Chatterjee teaches courses
in electromagnetic theory and antennas. His research interests are asymptotic high-frequency techniques,
Green’s functions, phased arrays, electrically small, pulsed and ultrawideband antennas, characteristic
mode theory, crosstalk modeling in high speed VLSI interconnects, biomedical imaging and r.f. propagation.
Dr. Chatterjee is a Senior Member of the IEEE and URSI Commission B.