论文标题
通过不连续的基础在开放光学系统的模式求解器中通过不连续的基础解决吉布斯现象
Resolving the Gibbs phenomenon via a discontinuous basis in a mode solver for open optical systems
论文作者
论文摘要
由于出色的收敛性，经常使用全球基础（例如傅立叶级数）来解决部分微分方程。但是，由于吉布斯现象，当不连续性存在时，收敛会受到损害，从而对模拟速度产生负面影响，并可能产生虚假解决方案。我们通过以固有的不连续的基础来补充平稳的全球基础来解决这一问题，从而结合了不连续性的位置的知识。解决方案的不连续性通过指数收敛复制，加快了模拟。高度约束的不连续基础也消除了产生虚假解决方案的自由。我们采用平滑和不连续的底座来构建一个在开放电磁系统中的谐振器模式的求解器。然后，这些模式可以在没有进一步数字的情况下为任何源配置或发病率的任何散射问题扩展任何散射问题，从而可以访问Green张量的空间变化并进行物理洞察力。解决模式是模态扩展方法中最密集和困难的步骤，因此我们的模式求解器克服了对开放系统使用模态扩展的最后一个主要障碍。
Partial differential equations are frequently solved using a global basis, such as the Fourier series, due to excellent convergence. However, convergence becomes impaired when discontinuities are present due to the Gibbs phenomenon, negatively impacting simulation speed and possibly generating spurious solutions. We resolve this by supplementing the smooth global basis with an inherently discontinuous basis, incorporating knowledge of the location of the discontinuities. The solution's discontinuities are reproduced with exponential convergence, expediting simulations. The highly constrained discontinuous basis also eliminates the freedom to generate spurious solutions. We employ the combined smooth and discontinuous bases to construct a solver for the modes of a resonator in an open electromagnetic system. These modes can then expand any scattering problem for any source configuration or incidence condition without further numerics, enabling ready access and physical insight into the spatial variation of Green's tensor. Solving for the modes is the most numerically intensive and difficult step of modal expansion methods, so our mode solver overcomes the last major impediment to the use of modal expansion for open systems.
