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：色散关系反映了波通过特定材料时，其频率对波矢的依赖关系。它揭示了材料的性质，因此至关重要。然而，色散关系的重构过程耗时且成本高昂。针对这一瓶颈，本文提出了一种基于全局多项式插值的高效色散关系重构方案，用于逼近二维光子能带函数。该方法基于以下事实：在第一布里渊区内，能带函数关于波矢是分段解析的。我们利用第一布里渊区内的适当采样点，求解能带函数计算中涉及的特征值问题，然后采用拉格朗日插值来逼近整个第一布里渊区内的能带函数。数值结果表明，所提方法能够显著提升计算效率。