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