AI 中文总结
该研究针对克尔型非线性高折射率介质谐振器的三维波散射问题，通过非线性模态降阶方法扩展线性模态分解至非线性波散射，结合数值框架验证了相关理论并分析了谐振器的共振特性。
AI 中文摘要
我们通过非线性利普曼-施温格方程，研究在平面波入射下具有克尔型非线性的高折射率介质谐振器的三维波散射，以及相关的非线性介质散射共振。利用牛顿势的简单本征模附近的李雅普诺夫-施密特降阶，我们将散射波局部分解为共振贡献和受控余项，并针对足够小的对比度参数、局部小的共振振幅及入射场，推导出共振模态系数的显式非线性方程。第二次降阶适用于一阶入射场和弱非线性的情况。我们的结果首次将线性模态分解扩展到非线性波散射问题。我们通过开发基于奈斯特罗姆离散化、实牛顿迭代和伪弧长延拓的数值框架来补充这些结果。针对多种几何形状的单个谐振器，我们的计算证实了预测的高对比度共振标度和非线性模态近似，并展示了多值入射波响应。对于镜面对称的谐振器二聚体，我们在宽间距范围内追踪对称、反对称及破缺对称的共振分支。我们的计算表明，对称破缺阈值随谐振器间隙减小而升高，这与主导阶分岔理论一致。
英文摘要
We study three-dimensional wave scattering by high-index dielectric resonators with Kerr-type nonlinearity under plane-wave incidence, together with the associated nonlinear dielectric scattering resonances, through a nonlinear Lippmann-Schwinger equation. Using a Lyapunov-Schmidt reduction near a simple eigenmode of the Newtonian potential, we obtain a local decomposition of the scattered wave into a resonant contribution and a controlled remainder and derive an explicit nonlinear equation for the resonant modal coefficient for sufficiently small contrast parameter and locally small resonant amplitude and incident field. A second reduction covers the regime of incident fields of order one and weak nonlinearity. Our results extend for the first time the linear modal decomposition to nonlinear wave-scattering problems. We complement them by developing a numerical framework based on Nyström discretization, real Newton iteration, and pseudo-arclength continuation. For single resonators of several geometries, our computations confirm the predicted high-contrast resonance scaling and the nonlinear modal approximation, and exhibit the multivalued incident-wave response. For a mirror-symmetric resonator dimer, we track symmetric, antisymmetric, and symmetry-broken resonance branches over a broad range of separations. Our computations show that the symmetry-breaking threshold increases as the gap between the resonators decreases, in agreement with the leading-order bifurcation theory.
Comments32 pages, 17 figures
