  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.
