SIAM Journal on Applied Mathematics, Volume 80, Issue 2, Page 864-880, January 2020.
A bound state in the continuum (BIC) on a periodic structure sandwiched between two homogeneous media is a guided mode with a frequency and a wavenumber such that propagating plane waves with the same frequency and wavenumber exist in the homogeneous media. Optical BICs are of significant current interest since they have applications in lasing, sensing, filtering, switching, and many light emission processes, but they cannot be excited by incident plane waves when the structure consists of linear materials. In this paper, we study the diffraction of a plane wave by a periodic structure with a second-order nonlinearity, assuming the corresponding linear structure has a BIC and the frequency and wavenumber of the incident wave are one-half of those of the BIC. Based on a scaling analysis and a perturbation theory, we show that the incident wave may induce a very strong second harmonic wave dominated by the BIC and also a fourth harmonic wave that cannot be ignored. The perturbation theory reveals that the amplitude of the BIC is inversely proportional to a small parameter depending on the amplitude of the incident wave and the nonlinear coefficient. In addition, a system of four nonlinearly coupled Helmholtz equations (the four-wave model) is proposed to model the nonlinear process. Numerical solutions of the four-wave model are presented for a periodic array of circular cylinders and used to validate the perturbation results.

中文翻译：

---

通过二次谐波激发连续体中的束缚态

SIAM应用数学杂志，第80卷，第2期，第864-880页，2020年1月。
夹在两个均质介质之间的周期性结构上的连续体（BIC）的束缚状态是一种具有频率和波数的导模，因此均质介质中存在具有相同频率和波数的传播平面波。光学BIC由于在激光，传感，滤波，开关和许多发光过程中都有应用而引起了广泛的关注，但是当结构由线性材料组成时，它们不能被入射平面波激发。在本文中，我们假设具有对应的线性结构具有BIC，并且入射波的频率和波数是BIC的二分之一，我们研究具有二阶非线性的周期性结构对平面波的衍射。基于比例分析和微扰理论，我们表明，入射波可能会诱发由BIC主导的非常强的二次谐波，并且还会引起不可忽略的第四谐波。扰动理论表明，BIC的振幅与小参数成反比，取决于入射波的振幅和非线性系数。另外，提出了一个由四个非线性耦合的亥姆霍兹方程（四波模型）组成的系统，以对非线性过程进行建模。给出了周期性圆柱阵列的四波模型的数值解，并用于验证扰动结果。扰动理论表明，BIC的振幅与小参数成反比，取决于入射波的振幅和非线性系数。另外，提出了一个由四个非线性耦合的亥姆霍兹方程（四波模型）组成的系统，以对非线性过程进行建模。给出了周期性圆柱阵列的四波模型的数值解，并用于验证扰动结果。扰动理论表明，BIC的振幅与小参数成反比，取决于入射波的振幅和非线性系数。另外，提出了一个由四个非线性耦合的亥姆霍兹方程（四波模型）组成的系统，以对非线性过程进行建模。给出了周期性圆柱阵列的四波模型的数值解，并用于验证扰动结果。