Topological bound modes in anti-PT-symmetric optical waveguide arrays

SHAOLIN KE,1 DONG ZHAO,2,3 JIANXUN LIU,2 QINGJIE LIU,2 QING LIAO,1 BING WANG,2,* AND PEIXIANG LU

1,2,4 1Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan Institute of Technology, Wuhan 430205, China 2School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 3School of Electronics Information and Engineering, Hubei University of Science and Technology, Xianning 437100, China 4lupeixiang@hust.edu.cn *wangbing@hust.edu.cn

Abstract: We investigate the topological bound modes in a binary optical waveguide array with anti-parity-time (PT) symmetry. The anti-PT-symmetric arrays are realized by incorporating additional waveguides to the bare arrays, such that the effective coupling coefficients are imaginary. The systems experience two kinds of phase transition, including global topological order transition and quantum phase transition. As a result, the system supports two kinds of robust bound modes, which are protected by the global topological order and the quantum phase, respectively. The study provides a promising approach to realizing robust light transport by utilizing mediating components.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Topological photonics have attracted considerable attention as they provide additional degree of freedom to reveal new states of light [1–9]. They are also utilized for controlling light flows that is insensitive to disorder [1]. The simplest model to investigate topologically protected modes is the Su–Schrieffer–Heeger (SSH) model, which consists of binary arrays [10,11]. By tuning the intra- and inter- cell coupling, topological order of the array changes when a bandgap closes and then reopens. Topological protected modes appear at the gap of band structure when two structures with different topological orders are interfaced [1,11]. The SSH model is theoretically and experimentally reported in various optical systems, including metallic resonators [9], photonic superlattices [10], silicon waveguides [11], semiconductor micropillars [12], and plasmonic waveguides [13–17].

Studies of topological photonics have so far mostly dealt with Hermitian systems. Recently, a growing interest is paid to investigating topological properties of non-Hermitian systems with gain and loss, especially the Parity-Time (PT) -symmetric systems [18–27]. The PT symmetry is introduced to optics in 2008 for the first time [28], which requires the system Hamiltonian H obeys relation [PT, H] = 0. The system can possess real spectrum in the presence of gain and loss below PT-broken threshold [28–31]. The first experimental observation of PT symmetry is by utilizing two coupled semiconductor waveguides [29], where optical gain is provided though nonlinear two-wave mixing. Many fascinating phenomenon are reveled in PT-symmetric systems [31–35], such as double refraction [28], loss induced transparency [36], non-reciprocal propagation [29], and chiral mode switching [37–39]. Topological phase transition also emerges in PT-symmetric systems, which is closely related to the system degeneracy, known as exceptional point (EP) [19,21,23,40]. The topological bound modes with PT symmetry are firstly observed in coupled waveguide arrays [19], which are Hermitian-like and can be sustained with vanished gain and loss factor. Interestingly, PT-symmetric systems support unique robust bound mode with no Hermitian

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13858

#362178 https://doi.org/10.1364/OE.27.013858 Journal © 2019 Received 12 Mar 2019; accepted 21 Apr 2019; published 29 Apr 2019

counterparts. topological or

Anti-PT-sthe anticommPT symmetrylater studies symmetric as circuits [49], state [45,46,4For exampledemonstrated encircling an anti-PT symmsymmetry [40with C being non-Hermitiandetermining tcoupled systeinduced by ccan be selectiutilized to enhefficiencies, a

In this woarrays with iincorporating coupling strenwaveguides. Wrelated to the the case in Hecan emerge bwhich the imaglobal topolog

Fig. 1array and thcoupli

2. Anti-PT s

We start by incoupling. The

The bound mrder but with dymmetric syst

mutator, have alis in [44], whedemonstrate well [45,47]. optical rings a

48]. Many intre, flat broadb

in double wavexceptional po

metry belongs0,50,51], whichunitary operaton systems contopological prms indicates thharge-conjugaively pumped hance the lasinand robustness rk, we investigimaginary couassistant wave

ngth can be fuWe show the timaginary part

ermitian and Pby tuning the reaginary part ofgical order and

1. Schematic of wto realize the effehe gray and greeing. (c) The geome

symmetry and

nvestigating hoe non-Hermitia

modes appear distinct quantumtems, which oblso attracted muere the refractithe discrete lThe imaginary

and waveguideriguing wave dband light trveguide coupleoint (EP) is res to a more h is first proposor [51]. The ch

nsisting of tworoperty of the hat the topologtion symmetrydue to its spec

ng performanceagainst perturb

gate the topologupling. We sheguides to the

urther controlleopological phat of band strucT-symmetric seal part of refrf refractive indd quantum phas

waveguide arrays ective imaginary cen are for assistaetry of proposed p

d topologica

ow to realize aan coupling em

at the interfam phases [41–4bey {PT, H} =uch attention [ive index of thelattices with iy coupling is res by interactindynamics are fransport and er [45]. The cheported for symfundamental ssed for supercoharge-conjugato sublattices [5

system [40,5gical bound staty [40,52]. Morcial energy spee with improvebation [53,55–gical bound mohow the imagbare array and

ed by tuning thase transition incture rather thasystems. We alractive index, idex is altered. se are also disc

with anti-PT symcoupling. The blueant sites. (b) Theplanar waveguide a

al invariant

nti-PT-symmemerges in coupl

face between 43]. = 0 with the co44–49]. The ore system satisfimaginary courealized in flying the two barfound in anti-P

dispersion-indhiral mode convmmetry-brokensymmetry callonductors and tion symmetry 50], which play50,52–54]. Theates can be genreover, the topectrum [50,54ed single mode

–57]. odes in anti-PT

ginary couplind considering bhe amount of gn anti-PT-sym

an its real part, lso show the quin contrast to PThe robust bo

cussed in detail

mmetry. (a) The pe and red sites repe equivalent arrayarray with anti-PT

etric waveguideled waveguide

two arrays w

ommutator repriginal proposafies n(x) = −n*upling can being atoms [47]re states througPT-symmetric duced dissipa

nversion by dynn modes [48]. led charge-condefined as {CTis a general prys an importane study of asy

nerated via a depological boun], which can be property, hig

T symmetric wng can be reaboth gain and gain and loss i

mmetric arrays iwhich is diffe

uantum phase tPT-symmetric

ound states prol.

proposed sawtoothpresent bare arraysy with imaginary

T symmetry.

e arrays with imes with gain or

with same

placed by al of anti-(−x). The

e anti-PT ], electric gh a third

systems. ation are namically Actually, njugation T, H} = 0 roperty of nt role in ymmetric egeneracy nd modes be further gher slope

waveguide alized by loss. The

in respect is closely rent from transition arrays in

otected by

h s y

maginary loss [58]

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13859

and also in open ring resonator with asymmetric scattering [59]. However, the imaginary part of coupling coefficient is small compared with its real part. Complex coupling is also realized by dynamically modulating refractive index along wave propagation direction [60,61]. However, it increased difficulties for experimental implementation. We follow another approach by incorporating assistant waveguides into the bare array to realize imaginary coupling [45,62,63]. The static on-site complex potentials simplify its practical implementation and the coupling strength can be tuned by the amount of gain and loss. As shown in Fig. 1(a), the proposed weekly coupled array have four waveguides in each unit cell, which are labeled as A, B, C, and D with C and D being the assistant waveguides. The corresponding effective detuning of propagation constants is denoted as δ1, δ2, δ3, and δ4. Only nearest coupling is considered. The coupling is represented by J1 and J2, which are assumed to be real and positive. The wave should evolve according to the coupled mode equation (CMT), which is given by

1 1 2 1

2 1 2

3 1

4 2 1

( )

( )

nn n n

nn n n

nn n n

nn n n

dai a J c J d

dzdb

i b J c J ddzdc

i c J a bdzdd

i d J a bdz

δ

δ

δ

δ

−

+

− = + +

− = + +

− = + +

− = + +

(1)

where an, bn, cn, and dn denote the field amplitudes in respect waveguide in nth unit cell. If the absolute value of detuning |δ3| and |δ4| are much larger than the coupling J1 and J2, the assistant waveguides C and D can be eliminated adiabatically [45], that is,

1 2 1

3 4

( ) ( ),n n n n

n n

J a b J a bc d

δ δ++ +

= − = − (2)

Substituting Eq. (2) into the first two terms of Eq. (1), the corresponding CMT reduces to

2 2 2 21 2 1 2

1 13 4 3 4

2 2 2 21 2 1 2

1 23 4 3 4

( )

( )

nn n n

nn n n

da J J J Ji a b b

dz

db J J J Ji a a b

dz

δδ δ δ δ

δδ δ δ δ

−

+

− = − − − −

− = − − + − − (3)

Therefore, after setting

( ) ( ) 2 21 1 2 2 1 2 3 1 1 4 2 2 / 2, / 2, / , ,/i c c i c c iJ c iJ cδ δ δ δ+= − + − Δ = − + Δ = = (4)

Equation (3) can be written as

1 2 1

1 2 1

,2

.2

nn n n

nn n n

dai a ic b ic b

dzdb

i ic a ic a bdz

−

+

Δ− = − + +

Δ− = + + (5)

The equivalent system described by Eq. (5) is depicted in Fig. 1(b). Each unit cell contains two waveguides A and B with effective detuning of propagation constants being ± ∆/2, respectively. The intra- and inter- coupling coefficients are ic1 and ic2, which are purely imaginary values. In order to realize the imaginary coupling, Eq. (4) indicates that the bare

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13860

arrays should be with gain and the assistant waveguides are with loss. Moreover, the amount of loss significantly exceeds the amount of gain, that is, |δ3|, |δ4| |δ1|, |δ2|. As the adiabatic

elimination requires |δ3|, |δ4| J1, J2, the bare coupling strength should also be much larger

than effective imaginary coupling, that is, J1, J2 c1, c2.

For periodic boundary, the Bloch theorem can be utilized and the system Hamiltonian is given by

1 2

1 2

/ 2 exp( ).

exp( ) / 2

ic ic iH

ic ic i

ϕϕ

−Δ + − = + Δ

(6)

The time-reversal operation T transforms a z-independent operator to its complex conjugate, while the parity operator P exchanges locations of the modes [45]. Thus, the system Hamiltonian fulfills the relation (PT)H(PT)−1 = −H, which implies the system is anti-PT-symmetric. The anti-PT symmetry can be also defined by Pauli matrix σi, which requires σxH(φ)σx = −H*(φ) [23]. Chiral symmetry is also important in determining topological properties of the system, which is defined by σzH(φ)σz = −H(φ). The system possesses chiral symmetry only when the detuning is Δ = 0. This ensures the Berry phase of individual band is quantized. The eigenvalues of Eq. (6) are solved to be

2 2 21 2 1 2/ 4 2 cos .c c c cλ ϕ± = ± Δ − − − (7)

The corresponding right and left eigenvectors are figured out as

cos( / 2)exp( ) sin( / 2) exp( ),

sin( / 2) cos( / 2)

cos( / 2) exp( ) sin( / 2)exp( ),

sin( / 2) cos( / 2)

k k k k

k k

T T

k k k k

k k

i i

i i

φ φλ λ

φ φμ μ

+ −

+ −

Δ − Δ = = Δ Δ

Δ − − Δ − = = Δ Δ

(8)

where Δk = arctan(−2iρk/Δ) and ρ(φ) = c1 + c2exp(−iφ) = ρkexp(iφk). The system has three different phases according to anti-PT symmetry. When |c1 − c2| > Δ/2, the system is in anti-PT-symmetric phase and eigenvalues are purely imaginary across the Brillion zone. When |c1 − c2| < Δ/2, the system is in mixed anti-PT-symmetric and broken phases and eigenvalues become complex-valued at the edge of Brillion zone. When |c1 + c2| < Δ/2, the anti-PT symmetry is fully broken and the eigenvalues develop into purely real values in the entire Brillion zone.

We now investigate the topological invariant and show the topological transition arises in anti-PT-symmetric systems. In non-Hermitian systems, the Berry phase for individual band is

defined by /B ki d dk dkϕ μ λ±

± ±= [41], where ± labels the upper and lower bands and k

represents Bloch momentum. The integration is taken over the 1D Brillouin zone. Substituting Eq. (8) into the expression for φ ± B, the Berry phase is derived as

/ 2 cos( ) / 2B k k kd dϕ φ γ φ± = − ± (9)

When the chiral symmetry holds, that is, Δ = 0, the second term of Eq. (9) is vanished. φ ± B is quantized to be an integer multiple of π, which is φ ±B = π for c1 < c2 and φ ± B = 0 for c1 > c2. In this situation, the Berry phase of individual band can be still regarded as topological invariant and utilized to indicate the topological phase transition. However, as Δ ≠ 0, the second term is not vanished. The Berry phase of a given band is not quantized and becomes complex-valued. Interestingly, the global Berry phase φGB = φB

− + φB+ [41–43], which is

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13861

calculated byutilize global arrays, which

3. Bloch mo

The above thwidely utilize66], we now geometry of px direction anwhich relativInGaAsP is ε0

simplicity. Thuniform, whiwavelength isonly supportscoupled whenhalf maximummode width iWe now invesymmetric w[67,70]. The psimulation sh0.713Δε withand J2 can beJ1 = J2 = J0 =certain amoundetuning of p2(a) and 2(b) Then, the detu10−3, ΔεB = (2strength can b

Fig. 2strengband propainter-l(e) an

y adding the coBerry phase taccurately cap

odes

eory is generaed to demonstrconsider plana

proposed wavend light propage permittivity 0 = 12.25. Thehe width of wich are given s fixed at λ = 1s one propagan the spacing bm of their eigenis 0.31 μm. Thstigate the banaveguide arraypropagation cohows the comph Δε denoting te extracted from= 0.047 μm−1. Tnt of gain and propagation conare Δ/2 = 2 × uning of permi2.8−7i) × 10−3, be properly con

2. The band strucgth. (a), (c), and (e

structure. In all gation constant arlayer coupling vard (f) c2 = 7 × 10−3

omplex Berry to reveal the toptures the topol

al for weekly cate optical anaar InGaAsP w

eguide arrays isgates along zis denoted as

e waveguides awaveguides w

as w = 0.24 1.55 μm. The pating mode unbetween adjacnmode. In thisherefore, the pnd structures anys, which can

onstant of a sinplex propagatithe detuned com a Hermitian Then, the effecloss in respectnstants, effecti10−3 μm−1, c1 =ittivity in waveΔεC = 0.77i, a

ntrolled by tuni

ctures of anti-PTe) The real part of

figures, the intrare fixed at c1 = 4 ×ries with (a) and (bμm−1.

phase in bothopological natlogical transiti

coupled latticealogues of sem

waveguides to es shown in Figaxis. There arεA, εB, εC, and

are assumed toand the interlμm and d0 =

parameters areder TM polar

cent waveguides work, the spaproposed geomnd the field dist be calculated

ngle waveguideion constant oomplex permitdouble waveg

ctive imaginart waveguides aive intra- and

= 4 × 10−3 μm−1

eguides A, B, Cand ΔεD = 3.09iing the detunin

T-symmetric waveband structure. (ba-layer coupling × 10−3 μm−1 and Δb) c2 = 1 × 10−3 μ

h bands, remaiture of anti-PTon point c1 = c

s. As coupled mi-classical elec

exam above thg. 1(c). The strure four wavegud εD. The real o be suspendedlayer spacing

= 0.5 μm, respe chosen such trization. The wes is much laracing is d = d0

metry is in the tributions of B

d by transfer me is figured outof the single wttivity. The baguide coupler, ry coupling canaccording to Eqinter-layer cou

1 and c2 = 1 × C, and D shouli, respectively.ng of imaginary

eguide arrays for b), (d), and (f) The

and the detuningΔ = 4 × 10−3 μm−1,μm−1, (c) and (d) c2

ins quantized. T-symmetric wc2.

waveguides hctron dynamicheoretical analucture is periouides in each part of permid in air with εa

d0 are suppospectively. Thethat a single wwaveguides arrger than the f+ w = 0.74 μmweak coupling

Bloch modes ofmatrix methodt as β0 = 7.76 μwaveguide is

are coupling stwhich is detern be realized bq. (4). For exaupling strength10−3 μm−1, respld be ΔεA = (−. The effective y part of permi

various couplinge imaginary part ofg of real part of, respectively. The2 = 4 × 10−3 μm−1

Here we waveguide

have been s [60,64–lysis. The odic along

unit cell, ittivity of air = 1 for sed to be e incident waveguide

e weakly full width m and the g regime. f anti-PT-d (TMM) μm–1. The kz ≈β0 +

trength J1 rmined as by adding ample, the h in Figs. pectively.

−2.8−7i) × coupling ittivity.

g f f e ,

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13862

Figure 2 coupling, whiremain unchaBloch modes.dots representthe condition phase, the imaclosed. As theIn Figs. 2(c) acloses at the E−Δ/2, as showremains closewhich indicatwaveguide arphase of mattmodified. In real part of bphase transitistructure is cltrivial as c1 > imaginary parband indices lsystem is a pr

Here we wthat is, the mothe amplified input beam. Tof Gaussian benergy of Gauis excited, thesubject to mo

Fig. 3respecaddedJ2 = J

plots the banile the intra-la

anged. There a. The lines stant the simulatioc1 − c2 > Δ/2 aaginary part ofe coupling satiand 2(d), the coEP near the edwn in Figs. 2(eed. The simulates the structurrray is realizedter changes whthe Hermitian band structureion relates to losed all the timc2. Therefore, rt of band strulose their meanroperty of the ewant to explainode have negatband structure

The imaginary beam Im(kz) = ussian beam ane wave should dified Bessel f

3. (a) and (b) are tctively. (c) The fied in assistant waveJ0, c1 = c2 = c0 = 4 ×

nd structures iayer coupling aare two bands nd for the resuons of TMM. Iand then the syf band structuresfies |c1 − c2| <oupling is c1 = ge of Brillion e) and 2(f), thation and the e is surly in th

d. The general hen a bandgap

and PT-symm. Here in antithe imaginary

me. The systemour study enri

ucture is closening at a band entire Hamiltonn the role of Imtive Im(kz), woue can be extractpart of band st−Ln(Eout/Ein)/

nd L presents tevolve analog

function [68].

the mode profiles eld fidelity as a funeguides C and D fo× 10−3 μm−1.

in the first Brand detuning oin the system

ults calculated bIn Figs. 2(a) anystem should be is gaped whil< Δ/2, the imag

c2. One can sezone. As c2 is e imaginary ptheoretical re

he weak couplintopological bacloses and the

metric systemsi-PT-symmetriy part of bandm is topologicches the knowd in the entirecrossing. Ther

nian, not indivim(kz) under prould remain visited from the wtructure determ/2L, where Eou

the total propaggously to discr

of Bloch modes fnction of coupling

for different coupli

rillouin zone of real part of

m, representing by CMT accornd 2(b), the co

be in anti-PT-syle the real part

ginary part of bee the imaginarfurther modifiart of band re

esults coincide ing regime andand theory staen reopens whe, the topologic

ic systems, wed structure as c nontrivial as

wledge of topoloe broken regiorefore, the topoidual bands [23opagation. Theible over a long

wave propagatiomines the ampliut and Ein denogation distancerete diffraction

for φ = 0 in upperg strength J0. (d) Ting strength. In th

for various inf propagation two different

rding to Eq. (7oupling strengtymmetric phas

t of the band stband structure ry part of bandied and fulfills opens and the well with ea

d the anti-PT-syates that the toen the system cal phase relate show the to the real partc1 < c2 and toogical transitio

on (|c1 − c2| < ological invari3]. e more amplifiger distance. Mon for a broad ification of tot

ote the output ae. If a single wn with amplifie

r and lower bands,The amount of losshe calculation, J1 =

nter-layer constants kinds of

7) and the th fulfills se. In this tructure is is closed.

d structure c1 − c2 <

real part ach other, ymmetric

opological is further tes to the

opological t of band opological on. As the Δ/2), the ant in the

ied mode, Moreover,

Gaussian tal energy and input

waveguide ed energy

, s =

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13863

Figures 3(zone correspoThe mode in waveguide is not ideal as aorder to quansymmetric wa

|λ+,m and |λ’+

arrays with elimination anin arrays withfunction of thmode 1 at uppof mode 2 atunity as J0/c0 waveguides ipropagation crapidly growsloss in experTherefore, theFigs. 3(a) andmodes reachewaveguides m

4. Normal to

In the above sits topologicainterface betwNow we study

Fig. 4(b) Thtopoloμm−1.

(a) and 3(b) ponding to Fig. upper band, asalmost vanish

a small amounntitatively comaveguide array

+,m denote theassistant wavnd Eq. (4), the h assistant wavhe ratio of bareper band is almt lower band in

→ ∞. Howevs serious for l

constants as a s with the increriment and it ie ratio of coupld 3(b), the raties as high as

match well with

opological bo

sections, we haal invariant. Onween topologicy the topologic

4. Topological bouhe propagation coogical edge modesThe detuning of r

lot mode profi2(c). Most of es shown in Fig

hed. However, nt of energy of

mpare the simils, we define th

F =

e normalized eveguides, respe

field fidelity sveguides. Figue coupling stre

most perfect as ncreases as th

ver, at the samlarge ratio. Fifunction of th

ease of couplinis also hard toling strength shio of coupling s 0.9998 and h the anti-PT-sy

ound modes

ave discussed tne of most remcally inequivalecal bound mode

und modes in finiteonstants for all sus. The effective coureal propagation co

files (|Hy|) of Benergy is config. 3(a), is almothe mode in lo

f is concentratlarity of the ei

he field fidelity

* ',

A,B

.m mm

λ λ+=

=

eigenvectors ofectively. Consshould relate toure 3(c) plots tength to effectthe field fidelie ratio of coup

me time, the amgure 3(d) plot

he ratio J0/c0. ng strength. It o maintain thehould be carefustrength is J0/0.9856, whic

ymmetric array

s

the band structmarkably featurent media necees in anti-PT-s

e waveguide arrayupermodes. (c) andupling strength is onstant is Δ = 4 ×

Bloch modes afined at the barost perfect as tower band, as ted at the assiigenmode with

y, which is give

f ideal anti-PTsidering the o J1 and J2, thethe fidelity of tive imaginary ity approximatpling strength

mount of loss ats the detuningThe detuning is difficult to

e real part of fully designed. /c0 = 11.75. T

ch implies theys.

ture of periodicre of topologicessarily hosts rsymmetric array

ys. (a) The geometd (d) are the modc1 = 1 × 10−3 μm−

10−3.

at the center ofre waveguides the field in the

present in Figstant waveguidh that in idealen by

T-symmetric arcondition of e bare couplingtwo Bloch mocoupling stren

tes to unity. Thincreases and

added into theg of imaginarin assistant wrealize huge apropagation cFor the mode

The field fidelite arrays with

c waveguide acal photonics irobust boundarys.

try of finite latticede profiles for the1 and c2 = 4 × 10−3

f Brillion A and B.

e assistant g. 3(b), is des C. In l anti-PT-

(10)

rrays and adiabatic

g strength odes as a ngth. The he fidelity d tends to e assistant ry part of waveguide amount of constants. shown in ty of two assistant

arrays and is that the ry modes.

. e 3

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13864

Figure 4(acorrespondingwith assistant10−3 μm−1 andis topologicalstructures witand the topospectrum of pnumber of bargap of imaginequal while thin Figs. 4(c) other is confinthe dots are confining to indication. Inexhibits vanidistributed in topological prmodes under

On the oththe system is strength is awaveguides. Tsolely by gainprevious stud(an', bn')

T = (a

Both inter- anAs Δ = 0, the zero, that is, energy of topo

Fig. 5(c) an

a) depicts the g actual structut waveguides. Td the detuning l nontrivial. Tth different toplogical trivial propagation core waveguides nary part of baheir real parts aand 4(d). One ned at the left. theoretical resthe assistant

n Fig. 4(c), theshing amplituwaveguide B

rotection on tha continuous dher hand, as thtopological tr

ctually controTherefore, ourn and loss con

dy of SSH-likean, ibn)

T. After t

H

nd intra- couplichiral symmetkz = 0. The n

ological mode

5. Propagation of td (d) are for a sing

ideal finite wure with assistThe effective cof propagationhe end of the

pological order,air. Therefore

onstants of the is N = 30. One

and. The imagiare different. Tmode is locatThe lines repr

sults for the idwaveguides,

e field is mainlude. In contras. The vanishine edge mode a

deformation of he inter-layer rivial without colled by tuninr results also introl, not the He model with athe basis chang

1 2

/

exH

c c

−Δ= − −

ing becomes atry is fulfilled,nonvanished ois not remaine

opological edge mgle waveguide exc

waveguide arratant waveguidecoupling strengn constants is Δ

structure can , that is, the tope, it must posfinite wavegu

e can see thereinary part of prThe mode profted at the rightresent the simudeal model. Dthe simulatio

ly concentratedst, as plotted ng amplitudes as it prevents ththe system’s pspace d0 is unconsidering gag the imaginaindicate the top

Hermitian factoasymmetric coge, the system H

1 22

xp( )

c c

iϕ+

Δ

asymmetric and which retains

onsite potentiaed at zero anym

modes. In (a) and (citation from the ri

ays with imagies. The actual gth is c1 = 1 × Δ = 4 × 10−3. A

be regarded apological non-ssess topologic

uide arrays is se is a pair of edropagation con

files of two edgt termination o

ulation results cDespite of a smon agrees weld on waveguidin Fig. 4(d), at waveguide

he edge mode fparameters. niform betweenain and loss. Tary part of ppological phas

ors [27]. Our moupling [21,40Hamiltonian ta

exp( ).

/ 2

iϕ− Δ

d the onsite po the energy of

al Δ breaks chmore, as shown

(b), the input fieldight and left termin

inary couplingstructure is te10−3 μm−1 and

As a result, the as the interfac-trivial wavegucal bound mo

shown in Fig. 4dge modes locanstants of two ge modes are iof the structurecalculated by Tmall amount oll with the thde A and wav

the energy iA or B is the

from merging w

n different wavThe imaginary ermittivity in se transition ismodel can also0,52] by a basiakes the form

otential is −Δ/2f topological mhiral symmetryn in Fig. 4(b).

ds are eigenmodesnation.

g and the erminated d c2 = 4 ×

structure ce of two uide array odes. The 4(b). The

ated at the modes is

illustrated e and the

TMM and of energy heoretical

veguide B is mainly

result of with bulk

veguides, coupling assistant

s induced o relate to is change

(11)

2 and Δ/2. mode to be y and the

.

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13865

Figure 5 pis simulated u5(b), the incidconfined at thdifferent propresults shownis zero. The sand 5(d). Theunderstood. Wmodes are exmodes, the buthe edge modmodes are mo

5. Abnorma

In spite of thesymmetric arrefers to an aquantum phas

Fig. 6Berry effect

Figures 6(> 1. The resuEq. (9) is vanindividual banresults can besymmetric. Ththat is, c1− c2

complex. In pand the systemthree phases transition denfrom one to unusual robus

presents the prusing Rigoroudent fields are he ends of wavpagation distann in Fig. 4(b) asituation is diffe wave diffuse

When wave is lxcited simultanulk modes domdes, one may postly confined a

al topologica

e general topolrays also expeabrupt change se transition em

6. The Berry phasephase. The red

ive coupling stren

(a) and 6(b) ploults are numericnished and the Bnd is not quant

e divided into thhe eigenvalues

2 < Δ/2 < c1 + phase III, that im is in fully ahave distinct

notes the transitanother. Ther

st bound modes

ropagation of ts coupled-wavthe eigenmodeeguide arrays d

nce z is almost as the imaginarferent when a ses into the strlaunched from neously. As sominated after epump the two wat the terminati

l bound mod

logical phase trerience quantuof Berry pha

merges in the an

e as a function of dand blue curves rgth is c1 = 4 × 10−

ot the complexcally calculateBerry phase is tized and contihree phases. Ins are purely imc2, the eigenvs, Δ/2 > c1 + c

anti-PT-brokenboundary as

tion threshold efore, our syss appear at the

two topologicave analysis (RCes shown in Fiduring the propneither decayery part of propsingle waveguiructure during

the single wavome bulk modnough long prwaveguides at ion waveguide

des

ransition, the pum phase transse of individunti-PT-symme

detuning Δ. (a) Rerepresent upper a

−3 μm−1 and c2 = 1 ×

x Berry phase wd by Eq. (9). Iquantized. As inuously varyi

n phase I, that imaginary and φB

values become 2, the eigenvalu

n phase. The Bthe Berry ph

where the syststem exhibits t

Phase I to III t

al edge modes.CWA) methodigs. 4(c) and 4pagation. The ed nor amplifiepagation constaide is excited. the propagati

veguide, the budes are more aropagation dist

the structure tes.

previous study sition. The qu

ual band [41–4etric waveguide

eal part and (b) imaand lower bands, × 10−3 μm−1.

with increasingIn the limit Δ = Δ is not vanising by increasiis, c1− c2 < Δ/2B ± are imaginacomplex and ues are real in

Berry phase in hase is not cotem-associatedthree differenttransition inter

The wave prod [69]. In Fig. 4(d). The fieldsenergy of two ed. This agreesants of two edgAs shown in F

ion. The reasoulk modes and

amplified than tance. In ordertermination as

has shown thauantum phase t43]. Here we e arrays as wel

aginary part of therespectively. The

g the detuning Δ= 0, the seconshed, the Berrying the detunin2, the system isary as well. In the Berry phathe entire Brilthis region is ontinuous. An

d quantum phast quantum pharface.

opagation 5(a) and

s are well modes at

s with the ge modes Figs. 5(c)

on can be d the edge

the edge r to select the edge

at the PT-transition show the

ll.

e e

Δ as c1/c2 d term of

y phase of ng Δ. The s anti-PT-

n phase II, se is also llion zone real. The

ny abrupt se transits ases. The

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13866

Fig. 7spectrof boucalcul

Figure 7(waveguides w> c2 such thatof propagationthe calculatioand Δ2 = 22 Therefore, thethe same. Hosymmetric phFigure 7(b) pwaveguides aspectrum, whthe mode prosimulated moof energy resipropagation opropagation.

The bounddisorder. To vthe structure. c1−δc and c2 8(b). The pergeometric parbound modes waveguide, inFig. 8(d), the modes can is results of TM

7. Bound mode indrum of eigenmodesund mode. The linlated by the couple

(a) depicts twith different qt the end of then constants of n, the paramet× 10−3 μm−1,

e two arrays arowever, their hase (phase I) presents the pare set to be Nhich is the robufile of the boude profile agreides at the assiof the bound m

d mode inducevalidate its robFigure 8(a) de+ δc. The corrrmittivity of crameters remaifor different δ

ndicating its rolocal couplingnot destroyed

M and CMT, r

duced by quantum s. The green dot dnes present the simed mode theory. (d

the structures quantum phasese structure wiltwo semi-arra

ters are chosenwhich satisfiere under the saquantum phasand the right

ropagation coN = 30. There iust mode generund mode, whicees well with thstant waveguid

mode. The energ

ed by the quanbustness, we inepicts the ideal responding arrcenter five wain unchanged. δc. The interfacobustness agaig strength is tod, showing largrespectively. T

phase transition. (enotes the robust b

mulated results andd) Propagation of b

under consis are interfacedl not support t

ays are differenn as c1 = 4 × 1es the relation ame topologicses are differet part is in fulnstants of all is an eigenmodrated by quantuch energy is whe theoretical pdes. Figure 8(dgy is well conf

ntum phase trantroduce a pertarrays as coup

ray with assistaveguides in thFigures 8(c) a

ce modes persiinst local topootally reversed ge fault-toleranhey are well ac

(a) The proposed sbound modes. (c) d the dots are the bound mode.

ideration, whd. The couplingtopological edgnt, which are d0−3 μm−1, c2 =Δ1/2 < |c1− cal order since ent. The left plly anti-PT-bro

eigenmodes ade located at tum phase trans

well concentratprediction, excd) shows the sifined to the cen

ansition is robrturbation of copling near the itant waveguidhe dotted box

and 8(d) plot thist with field coological perturb

(c1 < c2). In thnce. The linesccordant with e

structures. (b) TheThe mode profilestheoretical results

here two semg strength is sege modes. The

denoted as Δ1 a= 1 × 10−3 μm−

c2| and Δ2/2 > global Berry p

part is in the oken phase (phas the numberthe bandgap osition. Figure 7ted at the intercept that a smalimulation resulnter interface d

bust against tooupling strenginterface is alte

des is illustratex is changed whe mode profiloncentrated at

rbation. Remarhis situation, t

s and dots staneach other.

e s s

mi-infinite et to be c1 e real part and Δ2. In 1, Δ1 = 0, |c1 + c2|.

phase are anti-PT-

hase III). r of bare f the real 7(c) plots face. The ll amount lts for the during the

opological th δc into ered to be ed in Fig. while the les of the the same

rkably, in the bound nd for the

Vol. 27, No. 10 | 13 May 2019 | OPTICS EXPRESS 13867

Fig. 8tuningwavegstreng

6. Summary

In conclusiontopological boThe imaginarwaveguides akinds of topquantum phaswhich are prothe coupling topological paddition, the qin contrast tmodulated. Tsystems and components.

Funding

National NatuDistinguishedHubei, ChinaInstitute of Te

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