Numerical evaluation of Sommerfeld Integral

[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

 

[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

HTH
Amit

 

[charng]

As a fellow scientist, I understand the importance of numerical evaluation in solving complex problems. In this case, the Sommerfeld integral is a crucial component in solving EM scattering problems in inhomogeneous media, and your specific problem of decay of an excited molecule above a metal dielectric interface.

One approach to dealing with poles and branch cuts in numerical integration is to use contour integration techniques. This involves deforming the integration path in the complex plane to avoid the poles and branch cuts, and then evaluating the integral along the new path. This method can be particularly useful when dealing with integrals involving Bessel functions and spectral Green's functions.

Another approach is to use numerical quadrature methods, such as Gaussian quadrature or adaptive quadrature, which can handle integrals with singularities and oscillatory behavior. These methods can also be combined with techniques such as transformation to a finite interval or singularity subtraction to improve accuracy.

In addition, it may be helpful to use specialized software or libraries that are designed for numerical integration of complex integrands, such as the NAG Library or the QUADPACK library.

I hope this information helps in your pursuit of obtaining numerical results for your integral. Good luck with your research!