Dispersion relation reflects the dependence of wave frequency on its wave vector when the wave passes through certain material. It demonstrates the properties of this material and thus it is critical. However, dispersion relation reconstruction is very time consuming and expensive. To address this bottleneck, we propose in this paper an efficient dispersion relation reconstruction scheme based on global polynomial interpolation for the approximation of 2D photonic band functions. Our method relies on the fact that the band functions are piecewise analytic with respect to the wave vector in the first Brillouin zone. We utilize suitable sampling points in the first Brillouin zone at which we solve the eigenvalue problem involved in the band function calculation, and then employ Lagrange interpolation to approximate the band functions on the whole first Brillouin zone. Numerical results show that our proposed methods can significantly improve the computational efficiency.

翻译：色散关系反映了波在通过特定材料时，其频率对波矢的依赖关系，体现了该材料的性质，因此至关重要。然而，色散关系的重构非常耗时且成本高昂。为解决这一瓶颈，本文提出了一种基于全局多项式插值的二维光子带函数逼近的高效色散关系重构方案。该方法基于以下事实：带函数在第一布里渊区关于波矢呈分段解析性质。我们利用第一布里渊区中的合适采样点，求解带函数计算中涉及的特征值问题，进而采用拉格朗日插值法在整个第一布里渊区上逼近带函数。数值结果表明，所提方法能显著提升计算效率。