# Numerical evaluation of Sommerfeld Integral

1. Aug 6, 2009

### aro

In EM scattering problems in inhomogeneous (layered) media one may encounter Sommerfeld integrals of the form:

$$\int_{0}^{\infty}J_{n}(k_{\rho},\rho)k_{\rho}^{n+1}G(k_{\rho})dk_{\rho}$$

where J is a Bessel function and G is a spectral Green's function, $\rho$ is source-observer distance and in my case $n=0$.
Generally the Green's function results in a branch cut on the real axis and poles either on or slightly off the real axis.
As I see it the integrand contains the mode structure of the field and the integration over wavevectors (k) leads to something like a total 'self-energy' from which dispersion and photonic mode density can be obtained.

I am trying to obtain numerical results for an integral of this type in the context of the problem of decay of an excited molecule above a metal dielectric interface (allowing creation of surface plasmons, hence the pole).

If anyone could give me pointers on how to approach this specific integral or, more generally, how to deal with poles and branch cuts in numerical integration, I would greatly appreciate it.

Thanks,
AR

2. Dec 18, 2009

### amithochman

Hi,

We have recently released the ND-SDP package, which is a free Matlab package for fast and accurate evaluation of 2D Sommerfeld integrals:
http://webee.technion.ac.il/people/leviatan/ndsdp/index.htm

The publication related to the code, "A Numerical Methodology for Efficient Evaluation of 2D Sommerfeld Integrals in the Dielectric Half-Space Problem" and references therein could serve as an introduction to the subject. It will appear soon in IEEE Antennas and Propagation, and can already be viewed at:
http://ieeexplore.ieee.org/xpl/tocpreprint.jsp?isnumber=4907023&punumber=8 [Broken]

HTH
Amit

Last edited by a moderator: May 4, 2017