Numerical evaluation of Sommerfeld-type integrals for reflection and transmission of dipole radiation

Abstract

A radiating electric dipole is located near the interface with a layer of material. The electric and magnetic fields reflect off the interface and transmit through the material. The exact solution of Maxwell’s equations can be found in terms of Sommerfeld-type integrals. These integrals have in general a singularity on the integration axis, and the integrands are extremely complicated functions of the parameters in the problem. We present a method for the computation of these integrals, and the corresponding electric and magnetic fields. Key to the solution is the splitting of the incident field in its traveling and evanescent contributions. With a change of variables, the singularities can be transformed away, and the method also greatly improves the accuracy and efficiency of the integration. We illustrate the feasibility of our approach with the computation of the flow lines of electromagnetic energy in the system. For such flow diagrams, a large number of integrals needs to be computed with reasonable accuracy. We show that in our approach even the smallest details in flow diagrams can be revealed. Program summaryProgram titles: CPiP-Auxiliary-1, CPiP-Auxiliary-2, CPiP-Field lines-1, CPiP-Field lines-2.CPC Library link to program files:http://dx.doi.org/10.17632/476n5ffkvv.1Licensing provisions: GPLv3.Programming language: Mathematica.Nature of problem: In near-field optics and nano-photonics, exact solutions of Maxwell’s equations are needed. Of particular interest are the reflected and transmitted electric and magnetic fields of dipole radiation by a layer of material. These solutions involve a large number of integrals, which need to be computed numerically. In the literature, these integrals are known as Sommerfeld-type integrals.Solution method: We split the integration range in two parts. The first part corresponds to traveling dipole waves and the second part results from evanescent dipole waves. In each region we make a (different) change of variables. The result of this transformation is that it removes a possible singularity on the integration axis, and it also has a tendency to smoothen out the integrand. There are 38 different types of integrals. Our method applies to all of them, and is self-contained. There is no need for tweaking of the programs for each of these, and no adjustments need to be made for different values of the parameters. The method is developed for the near field. For large distances to the source (the far field), asymptotic methods are available, and there would be no need for numerical integration.