Large-area quantum-spin-Hall waveguide states in a three-layer topological photonic crystal heterostructure Zhihao Lan1Menglin L. N. Chen2Jian Wei You3and Wei E. I. Sha4

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Large-area quantum-spin-Hall waveguide states in a three-layer topological photonic

crystal heterostructure

Zhihao Lan,1Menglin L. N. Chen,2Jian Wei You,3and Wei E. I. Sha4

1Department of Electronic and Electrical Engineering, University College London,

Torrington Place, London, WC1E 7JE, United Kingdom

2Department of Electronic and Information Engineering,

The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

3State Key Laboratory of Milimeter Waves, Institute of Electromagnetic Space, Southeast University, Nanjing, China

4Key Laboratory of Micro-nano Electronic Devices and Smart Systems of Zhejiang Province,

College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China

Topological photonic edge states are conventionally formed at the interface between two domains

of topologically trivial and nontrivial photonic crystals. Recent works exploiting photonic quantum

Hall and quantum valley Hall eﬀects have shown that large-area topological waveguide states could

be created in a three-layer topological heterostructure that consists of a ﬁnite-width domain fea-

turing Dirac cone sandwiched between two domains of photonic crystals with opposite topological

properties. In this work, we show that a new kind of large-area topological waveguide states could

be created employing the photonic analogs of quantum spin Hall eﬀect. Taking the well-used Wu-

Hu model in topological photonics as an example, we show that sandwiching a ﬁnite-width domain

of photonic crystals featuring double Dirac cone between two domains of expanded and shrunken

unit cells could lead to the emergence of large-area topological helical waveguide states distributed

uniformly in the middle domain. Importantly, we unveil a power-law scaling regarding to the size of

the bandgap within which the large-area helical states reside as a function of the width of the middle

domain, which implies that these large-area modes in principle could exist in the middle domain

with arbitrary width. Moreover, pseudospin-momentum locking unidirectional propagations and

robustness of these large-area waveguide modes against sharp bends are explicitly demonstrated.

Our work enlarges the photonic systems and platforms that could be utilized for large-area-mode

enabled topologically waveguiding.

Introduction.— While topological states of matter were

originally discovered in solid state electronic materials

[1,2], the single-particle topological band theory does

not rely on the fermionic nature of electrons and their

Fermi-Dirac statistics. Haldane and Raghu [3,4] made

the ﬁrst eﬀort to extend the quantum Hall physics of two-

dimensional electron gases under strong magnetic ﬁelds

to photons in magneto-optic photonic crystals, where

they predicted the existence of chiral electromagnetic

states whose energy can only propagate in a single direc-

tion. Such unidirectional backscattering-immune topo-

logical electromagnetic states were conﬁrmed in exper-

iments shortly thereafter [5]. As magneto-optical ef-

fects in general are weak in near-infrared and visible

light regions, time-reversal symmetry preserving topolog-

ical photonic systems without magneto-optical materials

have later been proposed based on other members of the

quantum Hall related states, such as quantum spin Hall

[6–10] and quantum valley Hall [11–13] as well as higher-

order topological phenomena [14–16]. The research ﬁeld

of topological photonics [17–20] has not only deepened

our understanding about topological physics in bosonic

systems but also found many interesting applications.

The conventional way to create topological edge states

in photonic systems is based on an interface between two

photonic crystals of diﬀerent topological properties. The

edge states constructed in this way typically are tightly

conﬁned around the interface and decay exponentially

away from it. Moreover, due to the wave nature of

light, tunneling and coupling of topological edge states

at diﬀerent interfaces can also give rise to interesting

physics [21–24]. Recently, interesting large-area topolog-

ical waveguide states were demonstrated in three-layer

heterostructures in both photonic quantum Hall [25,26]

and quantum valley Hall [27] systems. In such topologi-

cal heterostructures, a domain of photonic crystal featur-

ing gapless Dirac cone dispersion is sandwiched between

two domains of photonic crystals with opposite topolog-

ical properties and the resulting topological waveguide

states have ﬁeld amplitudes distributed almost uniformly

in the middle domain. In [25], one-way large-area topo-

logical waveguide states were created in magnetic pho-

tonic crystals with opposite gap Chern numbers and the

optical forces of such waveguide states could be used for

particle sorting and manipulation [26]. In [27], a pho-

tonic crystal with Dirac points was sandwiched between

two valley photonic crystals with opposite valley-Chern

numbers and large-area valley-locked waveguide states

were observed in experiments. Such large-area topolog-

ical waveguide states with a ﬁnite width have a high

capacity for energy transport and in order for them to

exist, the presence of a Dirac cone in the middle do-

main is crucial, which allows the strong coupling and

hybridization of the two topological interface states as-

sociated with the middle domain due to its gapless bulk

spectrum. One could expect that the coupling between

the two interface states would become weaker if the width

of the middle domain increases and consequently, the op-

arXiv:2210.09491v2 [physics.optics] 25 Apr 2023

(a)

(b)

(c)

¯!=!a/2⇡c

1

FIG. 1. Unit cells and band diagrams of the three photonic

crytals for constructing the topological heterostructure. (a)

The expanded unit cell and its topological nontrivial band

diagram with a band inversion between the pand dorbitals.

(b) The unit cell that gives a gapless band diagram with a

double Dirac cone at Γ. (c) The shrunken unit cell and its

trivial band diagram. The band inversion is indicated by the

red arrowed lines. As labeled in (b), the lattice constant,

radius of the dielectric cylinder, and distance of the cylinder

to the unit cell center are a,rand drespectively. Here, r=

0.12a,d= 0.36a, a/3,0.3afor (a-c) and the dielectric constant

of the cylinder = 12.

erational bandwidth of the topological waveguide states

would become smaller. However, the scaling laws of the

operational bandwidth as a function of the width of the

middle domain in these systems are not known.

As photonic quantum spin Hall eﬀect is an impor-

tant class of phenomena in topological photonics, it is

natural to ask whether large-area topological waveguide

states could also be created utilizing this eﬀect. In this

work, taking the Wu-Hu model [9] widely used for cre-

ating pseudospin-momentum-locked helical electromag-

netic edge states, we show that inserting a domain of

photonic crystal with a double Dirac cone dispersion into

two domains of photonic crystals with expanded and

shrunken unit cells, a pair of large-area helical waveg-

uide modes emerges within a ﬁnite bandgap after gap-

ping out the double Dirac cone in the middle domain.

Importantly, we show that the operational bandwidth of

these large-area modes decays as a power law with re-

spect to the width of the middle domain and pseudospin-

momentum locking unidirectional propagations as well as

robustness of these large-area waveguide modes against

sharp bends are further explicitly demonstrated. Con-

sidering that the Hu-Wu model has already found many

interesting applications, such as coupling with quantum

emitters [28], reconﬁgurable devices [29,30], all-optical

logic gates [31], third-harmonic generation [32], topo-

logical lasing [33,34], and bound topological edge state

in the continuum [35], the demonstration of large-area

waveguide modes in this setup could oﬀer new opportuni-

ties beneﬁtting from the additional width degree of free-

dom, e.g., high-capacity topological transport and easy-

interfacing with conventional waveguides and devices.

Three photonic crystals for constructing the topologi-

-0.2

0.2

0.43

0.44

0.45

0.46

0.47

0.48

k=ka/⇡

¯!=!a/2⇡c

III

(a)

(b)

(c)

III

min

max

-0.2

0.2

0.43

0.44

0.45

0.46

0.47

0.48

k=ka/⇡

-0.2

0.2

0.43

0.44

0.45

0.46

0.47

0.48

k=ka/⇡

FIG. 2. (a) Projected band diagram of the three-layer topo-

logical heterostructure I/II/III, calculated by imposing peri-

odic boundary conditions along the vertical direction whereas

scattering boundary conditions along the horizontal direction

of the supercell. The three domains I, II and III are con-

structed from the unit cells of (a), (b) and (c) in Fig.(1) re-

spectively, and the numbers of the unit cells along the horizon-

tal direction in the three domains are NI=NII =NIII = 10.

The red lines are the large-area helical waveguide modes

within a bandgap indicated by the green region, whereas the

blue region shows the common bulk bandgap of domains I

and III. The mode proﬁles of two modes labeled by A and B

are also shown. (b) and (c) Projected band diagrams similar

to (a) but for two ordinary three-layer heterostructures with

conﬁguration of I/II/I and III/II/III, which can not support

the large-area helical waveguide modes as in (a).

cal heterostructure.— We begin by brieﬂy discussing the

Wu-Hu model [9] for emulating pseudospin-momentum-

locked helical photonic edge states based on the C6vcrys-

talline symmetry. The model considers photonic crystals

with six dielectric cylinders in a hexagon unit cell (see

Fig.1). When the distance of the cylinders to the unit cell

center d=a/3, where ais the lattice constant, the cylin-

ders reduce to a honeycomb array and due to the band-

folding mechanism, the original Dirac points at K/K0as-

sociated with the transverse magnetic modes of the hon-

eycomb lattice are folded into a double Dirac point at the

Γ point (see Fig.1(b)). Starting from this photonic crys-

tal with a double Dirac point, a topological nontrivial

(or trivial) photonic crystal with a ﬁnite bandgap could

be created by expanding (or shrinking) the six cylinders

away from (or towards) the unit cell center, see Fig.1(a)

(or Fig.1(c)). Due to the C6vsymmetry of the cylin-

ders, the {px, py}and {dxy, dx2−y2}orbitals associated

with the two two-dimensional irreducible representations

of the C6vpoint group at the Γ point form two double-

degenerate pairs and a band inversion could be induced

by this shrinking-expanding operation (see the red arrows

in the band diagrams of Fig.1). Especially, the porbitals

will be higher in frequency than the dorbitals in the

expanded unit cell, resulting in a topological nontrivial

bandgap whereas the shrunken unit cell gives a topolog-

ical trivial bandgap. While the original proposal of the

Wu-Hu model only uses two photonic crystals made of ex-

panded and shrunken unit cells for creating helical edge

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Large-areaquantum-spin-Hallwaveguidestatesinathree-layertopologicalphotoniccrystalheterostructureZhihaoLan,1MenglinL.N.Chen,2JianWeiYou,3andWeiE.I.Sha41DepartmentofElectronicandElectricalEngineering,UniversityCollegeLondon,TorringtonPlace,London,WC1E7JE,UnitedKingdom2DepartmentofElectronicandInforma...

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Large-area quantum-spin-Hall waveguide states in a three-layer topological photonic crystal heterostructure Zhihao Lan1Menglin L. N. Chen2Jian Wei You3and Wei E. I. Sha4

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