Bethe’s theory treats a subwavelength aperture in a metal film as the combination of a parallel magnetic dipole and transverse electric dipole. For linear optics, this gives the usual dipole transmission; however, for nonlinear optics, it is interesting to consider how these two components interact. While many works have studied the nonlinear harmonic generation for metal nanoapertures, here we show that the Lorentz force dominates the second harmonic generation by an order of magnitude at angled incidence where the generation is maximized. The angular dependence matches that of the magnetic and electric dipoles accounting for Lambert’s cosine law. The theoretical analysis and numerical calculations agree well with past experiments.

Bethe的理论将金属膜中的亚波长孔径视为平行磁偶极子和横向电偶极子的组合。对于线性光学器件，这提供了通常的偶极子传输。但是，对于非线性光学，考虑这两个组件之间的相互作用是很有趣的。尽管许多工作研究了金属纳米孔的非线性谐波产生，但在这里我们表明，洛伦兹力在成角的入射角处以最大数量级的数量级主导了二次谐波的产生。角度相关性与考虑朗伯余弦定律的磁偶极子和电偶极子的相匹配。理论分析和数值计算与过去的实验非常吻合。