In this paper we present an analytical solution for the eigenmodes and corresponding electric fields of a composite system made of a nanorod in the vicinity of a plasmonic semi-infinite metallic system. To be specific, we choose Silver as the material for both the nanorod and the semi-infinite metal. The system is composed of two sub-systems with different symmetries: the rod has polar symmetry, while the interface has a rectangular one. Using a boundary integral method, proposed by Eyges, we are able to compute analytically the integrals that sew together the two systems. In the end, the problem is reduced to a one of linear algebra, where all the terms in the system are known analytically. For large distances between the rod and the planar surface, only a few of those integrals are needed and a full analytical solution can be obtained. Our results are important to benchmark other numerical approaches and represent a starting point in the discussion of systems composed of nanorods and two-dimensional materials.

在本文中，我们提出了由纳米棒在等离子体半无限金属系统附近制成的复合系统的本征模式和相应电场的解析解。具体而言，我们选择银作为纳米棒和半无限金属的材料。该系统由两个对称性不同的子系统组成：杆具有极对称性，而界面具有矩形对称性。使用 Eyges 提出的边界积分方法，我们能够分析计算将两个系统缝合在一起的积分。最后，问题被简化为线性代数问题，其中系统中的所有项都是解析已知的。对于杆和平面之间的大距离，只需要其中的几个积分，就可以获得完整的解析解。