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.

一个辐射电偶极子位于与材料层交界的附近。电场和磁场在界面处发生反射并透过材料传播。麦克斯韦方程组的精确解可以表示为索末菲尔德型积分的形式。这些积分通常在积分轴上具有奇点，且积分的被积函数是问题参数的极为复杂的函数。我们提出了一种计算这些积分及其相应电场和磁场的方法。解决问题的关键在于将入射场分解为其行进和衰减贡献。通过变量替换，奇点可以转化为非奇点，且该方法显著提高了积分的准确性和效率。我们通过计算系统中电磁能量流动的流线来说明我们方法的可行性。对于此类流图，需要计算大量积分以确保合理的精度。我们展示，在我们的方法中，即使是流图中最微小的细节也可以被揭示。