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Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4638

First-principle calculation of Chern number in

gyrotropic photonic crystals

RAN ZHAO,

1,2

GUO-DA XIE,

1,2

MENGLIN L. N. CHEN,

ZHIHAO

LAN,

ZHIXIANG HUANG,

1,2

AND WEI E. I. SHA

5,*

Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, Anhui University,

Hefei 230039, China

Key Laboratory of Electromagnetic Environmental Sensing, Department of Education of Anhui Province,

Hefei 230039, China

Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China

Department of Electronic and Electrical Engineering, University College London, United Kingdom

Key Laboratory of Micro-nano Electronic Devices and Smart Systems of Zhejiang Province, College of

Information Science Electronic Engineering, Zhejiang University, Hangzhou 310027, China

weisha@zju.edu.cn

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.

1. Introduction

Topology studies the invariant properties of geometry under continuous deformation [1]. While

in mathematics, topological invariants are commonly used to classify topological spaces, in

topological physics, topological invariants are explored to distinguish the bulk properties of

materials. If physical observables can be expressed as topological invariants, they can only vary

discretely and will not be aﬀected by small perturbations of system parameters.

With the discovery of the quantum Hall eﬀects [2,3] and recent advances in the study of

topological insulators [4], topological phases of matter have attracted great attention in condensed

matter physics. 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].

Chern number in a photonic system is deﬁned on the dispersion bands in wave-vector space.

For a two-dimensional (2D) periodic system, the Chern number is the integration of the Berry

curvature over the ﬁrst Brillouin zone. Once the Chern number is calculated, the topological

properties of the system can be identiﬁed (trivial or non-trivial). In addition, the Chern number

can be used to explain the phenomenon of “topological protection” of the edge state transmission

in photonic topological insulators (PTIs). Therefore, the accurate computation of the Chern

number is of crucial importance in the PTI design [8–10].

#380077 https://doi.org/10.1364/OE.380077

Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4639

In this paper, we propose a ﬁrst-principle computation method for Chern number calculation,

based on the ﬁnite-diﬀerence frequency-domain (FDFD) method [11–14]. Compared with the

commonly used plane-wave expansion (PWE) method [15–17], the FDFD method is not only

accurate and stable but also has lower computational complexity. Firstly, 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. Then, the formulae for the numerical calculation of

the Chern number are derived in the discretized ﬁrst Brillouin zone. At last, numerical examples

are given to demonstrate the accuracy of the proposed method. Note, 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 eﬀects of material dispersion and loss in a real microwave ferrite

has been discussed in [6]. One remark we would like to make is that as the Chern number is a

global property of the energy band, it does not depend on the local changes of the band caused

by material dispersion as long as the band gap is still open, which also means that the Chern

number has a certain degree of built-in robustness against material dispersion. Nevertheless, 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.

2. Generalized eigenvalue problem using FD method

2.1. Maxwell’s equations in the generalized coordinates

Firstly, the Maxwell’s curl equations in free space in the generalized coordinate (Fig. 1) are

expressed by

∇

H = ık

ˆ(r)

E, ∇

E = −ık

ˆµ(r)

H. (1)

Here, ∇

is the partial diﬀerential operator in the generalized coordinate with three unit vectors

(x, y, z), (q

1, 2, 3).

and

are the normalized vector ﬁelds with



1, 2, 3). Actually, in the numerical implementation, Q

s are the sizes of the discrete grids

along each direction [19]. k

is the free-space wavenumber.

ˆ

and

ˆµ

are the relative permittivity

and permeability in the form of 3 × 3 tensor which are written as

ˆ

(r) = [

¯(r) · g]

· |

; ˆµ

(r) = [

¯µ(r) · g]

· |

; (2)

where

¯

(

) and

¯µ

(

) are the original relative permittivity and permeability in the Cartesian basis.

g is the metric tensor matrix which can be expressed as,

g =













−1

. (3)

With the metric tensor matrix, the length of a vector in the generalized coordinate is calculated

= r

· g

−1

· r. (4)

2.2. Bloch’s theorem

In 2D PCs, due to the translational symmetry, the modes can be written in the Bloch form:

(

n,k

)

(r) = e

−ik

·ρ

−ik

(

n,k

)

(ρ),

(

n,k

)

(r) = e

−ik

·ρ

−ik

(

n,k

)

(ρ).

(5)

Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4640

Fig. 1.

The 2D Yee’s grid in general coordinates. The dashed arrows are where the periodic

boundaries apply.

Here, n is the band number,

is the mapping of vector

in the (u

, u

) plane.

(

) and

(

)

are periodic functions satisfying

(

)

= u

(

ρ + R

(

)

= u

(

ρ + R

), where

stands for any

lattice vectors.

In 2D PCs, we only need to investigate the in-plane propagating modes, i.e. k

0. Hence, the

mode can be rewritten as

E(ρ) = e

−ik

·ρ

(ρ),

H(ρ) = e

−ik

·ρ

(ρ).

(6)

Then, the Bloch’s theorem can be written as

E(ρ + R) = e

−ik

·R

E(ρ),

H(ρ + R) = e

−ik

·R

H(ρ).

(7)

2.3. Discretization of the eigenvalue problems

From the Maxwell’s curl equations (Eq. (1)), we can derive the governing equations for the

modes of the PCs:

∇ × ˆµ

−1

(r)∇ × E(r) = ω

ˆ(r)E(r), (8)

∇ × ˆ

−1

(r)∇ × H(r) = ω

ˆµ(r)H(r). (9)

In 2D PCs, the modes can be separated into two distinct polarizations, transverse-magnetic

(TM) modes with z-polarized electric ﬁelds and transverse-electric (TE) modes with z-polarized

magnetic ﬁelds. The generalized eigenvalue matrix equations for the TM and TE modes of the

2D PCs can be discretized as,





−1

− µ

−1



− U



−1

− µ

−1



= ε

(10)





−1

− ε

−1



− V



−1

− ε

−1



= µ

(11)

Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4641







−1 1

−1 0

−1 1

−1 0

−1







(12)







1 v

−1 1

0 1 v

−1 1

0 1 v

−1







(13)







−1 1

−1

−1 1

−1







(14)

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