  As an important figure of merit for characterizing the quantized collective
behaviors of the wavefunction, Chern number is the topological invariant of
quantum Hall insulators. Chern number also identifies the topological
properties of the photonic topological insulators (PTIs), thus it is of crucial
importance in PTI design. In this paper, we develop a first principle
computatioal method for the Chern number of 2D gyrotropic photonic crystals
(PCs), starting from the Maxwell's equations. Firstly, we solve the Hermitian
generalized eigenvalue equation reformulated from the Maxwell's equations by
using the full-wave finite-difference frequency-domain (FDFD) method. Then the
Chern number is obtained by calculating the integral of Berry curvature over
the first Brillouin zone. Numerical examples of both transverse-electric (TE)
and transverse-magnetic (TM) modes are demonstrated, where convergent Chern
numbers can be obtained using rather coarse grids, thus validating the
efficiency and accuracy of the proposed method.
