Abstract
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.
Highlights
Topology studies the invariant properties of geometry under continuous deformation [1]
In 2005, Haldane and Raghu transferred the key feature of quantum Hall effect in quantum mechanics to classical electromagnetics [5] and soon after, it was numerically and experimentally verified by using photonic crystals (PCs) [6, 7]
The accurate computation of the Chern number is of crucial importance in the photonic topological insulators (PTIs) design [8,9,10]
Summary
Topology studies the invariant properties of geometry under continuous deformation [1]. Chern number in a photonic system is defined on the dispersion bands in wave-vector space. Once the Chern number is calculated, the topological properties of the system can be identified (trivial or non-trivial). The FDFD method is used to compute the band structure of 2D gyrotropic PCs by solving the generalized eigenvalue equations derived from Maxwell’s equations. The formulae for the numerical calculation of the Chern number are derived in the discretized first Brillouin zone. In this work, we will focus on 2D PCs with lossless, non-dispersive, local materials, which is a common practice in computing Chern numbers. The extension of the current method to fully include the effect of material dispersion on the energy bands and Chern number calculations is an interesting direction for future research
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