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 [email protected] *[email protected]
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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