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 summary Program 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.1 Licensing 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.

中文翻译：

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偶极子辐射反射和透射的索末菲型积分的数值计算

摘要 辐射电偶极子位于与一层材料的界面附近。电场和磁场从界面反射并通过材料传输。麦克斯韦方程组的精确解可以通过 Sommerfeld 型积分找到。这些积分一般在积分轴上有一个奇点，被积函数是问题中参数的极其复杂的函数。我们提出了一种计算这些积分的方法，以及相应的电场和磁场。解决方案的关键是在其行进和渐逝贡献中分裂入射场。随着变量的变化，奇异点可以被变换掉，该方法也大大提高了积分的精度和效率。我们通过计算系统中电磁能量的流线来说明我们方法的可行性。对于此类流程图，需要以合理的精度计算大量积分。我们表明，在我们的方法中，即使是流程图中最小的细节也能被揭示出来。程序概要 程序名称：CPiP-Auxiliary-1、CPiP-Auxiliary-2、CPiP-Field lines-1、CPiP-Field lines-2。CPC 库程序文件链接：http://dx.doi.org/10.17632/476n5ffkvv.1 许可条款：GPLv3。编程语言：Mathematica。问题性质：在近场光学和纳米光子学中，需要麦克斯韦方程组的精确解。特别令人感兴趣的是由一层材料反射和传输的偶极辐射的电场和磁场。这些解决方案涉及大量积分，需要进行数值计算。在文献中，这些积分被称为 Sommerfeld 型积分。解决方法：我们将积分范围分为两部分。第一部分对应于行进偶极波，第二部分来自渐逝偶极波。在每个区域，我们都会对变量进行（不同的）更改。这种变换的结果是它消除了积分轴上可能的奇点，并且还具有平滑被积函数的趋势。有 38 种不同类型的积分。我们的方法适用于所有这些，并且是独立的。不需要为每一个调整程序，也不需要对参数的不同值进行调整。该方法是为近场开发的。对于到源（远场）很远的距离，