Table Of ContentSpringer Tracts in Modern Physics 280
Demetrios Christodoulides · Jianke Yang
Editors
Parity-time
Symmetry
and Its
Applications
Springer Tracts in Modern Physics
Volume 280
Serieseditors
YanChen,DepartmentofPhysics,FudanUniversity,Shanghai,China
AtsushiFujimori,DepartmentofPhysics,UniversityofTokyo,Tokyo,Japan
ThomasMüller,InstfürExperimentelleKernphysik,UniversitätKarlsruhe,
Karlsruhe,Germany
WilliamC.Stwalley,DepartmentofPhysics,UniversityofConnecticut,Storrs,CT,
USA
Jianke Yang, Department of Mathematics and Statistics, University of Vermont,
Burlington,VT,USA
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Demetrios Christodoulides (cid:129) Jianke Yang
Editors
Parity-time Symmetry and Its
Applications
123
Editors
DemetriosChristodoulides JiankeYang
CollegeofOpticsandPhotonics DepartmentofMathematicsandStatistics
UniversityofCentralFlorida UniversityofVermont
Orlando,Florida,USA Burlington,Vermont,USA
ISSN0081-3869 ISSN1615-0430 (electronic)
SpringerTractsinModernPhysics
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Preface
Parity-time (PT) symmetry is nowadays an active area of research, having an
impactinbothscienceandtechnology.Parity-timeconceptsfirstoriginatedwithin
the framework of quantum mechanical formalisms, when Bender and Boettcher
(1998)indicated,forthefirsttime,thatiftheone-dimensionalSchrödingerequation
withacomplexpotential
∂ ∂2
i ψ(x,t)=− ψ(x,t)+V(x)ψ(x,t)
∂t ∂x2
isPT-symmetric,thenthespectrumoftheSchrödingeroperator−∂ +V(x)can
xx
be entirely real. In other words, if this equation is invariant under the combined
actionoftheparityP (x→ −x)andtime-reversalT (t→ −t, i→ −i),then
theenergylevelsEoftheeigenstatesψ(x,t)=u(x)eiEt caninprinciplebeall-real.
Inthiscase,onecanshowthatanecessary(albeitnotsufficient)conditionforthis
complexpotentialtobePT-symmetricis
V(x)=V∗(−x).
WhiletheramificationsofPT symmetryinactualquantumsystemsarestilltobe
assessed,thesameisnottrueinclassicalsettings.Adecadelater,thisfieldstarted
to flourish in earnest, when a series of papers published in the period of 2007–
2008 indicated that optics and photonics can provide a fertile ground where PT-
symmetric ideas can be investigated. In the optical realm, the complex refractive
indexfunctionn(x)=n (x)+in (x)nowplaystheroleofacomplexpotentialV(x),
R I
where n (x) represents the refractive index distribution while n (x) stands for the
R I
gainandlossprofileswithinthemedium.Inthiscase,PT symmetryimpliesthat
n (x)=n (−x), n (x)=−n (−x).
R R I I
Hence, the refractive index function must be even, whereas the gain-loss profile
shouldbeanoddfunctionofposition.WhatmakesopticsanaturalplatformforPT
v
vi Preface
symmetryisthefactthatallthesethreeingredients(refractiveindex,gain,andloss)
canbereadilydeployedinphotonics.Soonafter,thiswasfollowedbyanexplosion
of experimental works, all of which corroborating such possibilities. Symmetry
breaking was first reported in 2009, and full PT symmetry was subsequently
observedinpairsofcoupledwaveguidesin2010.Someveryexcitingapplications
stemmingfromopticalPT symmetryhavealsobeendemonstratedbyanumberof
groups.Theseinclude,forexample,theprospectforunidirectionalinvisibility,ultra-
responsivesensors,unidirectionalelements,single-modemicro-ringPT-symmetric
lasers,andcoherentperfectabsorbers/lasers,tomentionafew.Theimpactofnonlin-
earitiesonopticalPT-symmetricarrangementswasalsoextensivelyinvestigatedin
severalsettings,includingtopologicalphotonics.GeneralizationsofPT symmetry
toaccommodatemoreflexiblegain-lossprofileshavealsobeenproposed.Bynow,
concepts from PT symmetry have permeated several other branches of physics
beyond optics, ranging from nuclear and quantum, to microwave, electronic and
mechanicalsystems.Onecanalsoincludeinthisexpandinglistplasmonics,Bose-
Einstein condensates, acoustics, superconductivity, magnetics, and wireless power
transportsystems.
In this book, theoretical and experimental progresses in diverse areas of PT
symmetry are reviewed by experts in the field. In Chapter “Linear and Nonlinear
Experiments in PT-Symmetric Photonic Mesh Lattices”, Peschel and collabo-
rators review linear and nonlinear experimental results on light propagation in
PT-symmetric photonic mesh lattices made of fiber components, where Bloch
oscillations and solitons are demonstrated. In Chapter “PT-symmetry on-a-Chip:
Harnessing Optical Loss for Novel Integrated Photonic Functionality”, Feng et al.
discuss experimental results associated with chip-scale PT-symmetric integrated
photonic systems designed for a number of applications. In Chapter “Parity-Time
Symmetry in Scattering Problems”, Alu and colleagues provide an overreview of
PT symmetry in scattering problems, where scattering from open PT-symmetric
systems in coupled waveguide cavity arrangements is analyzed in one and higher
dimensions.InChapter“ScatteringTheoryandPT-symmetry”,Mostafazadehintro-
ducesaone-dimensionalscatteringtheorywithP,T,orPT symmetryandderives
mathematicalconditionsthatencourageorforbidreciprocaltransmission,reciprocal
reflection, and the presence of spectral singularities. In Chapter “Passive PT-sym-
metry in Laser-Written Optical Waveguide Structures”, Szameit and collaborators
discuss how PT-symmetric systems can be implemented in a passive fashion,
without using gain, by employing modulated waveguide structures. In Chapter
“Non-Hermitian Effects Due to Asymmetric Backscattering of Light in Whisper-
ing-GalleryMicrocavities”,Wiersigreviewsprogressonnon-Hermitianeffectsdue
toasymmetricbackscatteringoflightinwhispering-gallerymicrocavitiesandtheir
applications in single-particle detection. In Chapter “Exact Results for a Special
PT-symmetric Optical Potential”, Jones provides exact analytical results of light
propagation inPT-symmetricsinusoidalopticalpotentialsatthephasetransition,
for both transverse and longitudinal configurations. In Chapter “Parity-time-Sym-
metric Optical Lattices in Atomic Configurations”, Xiao and colleagues provide a
roadmapfordesigningandexperimentallyimplementingexactPT-symmetricopti-
Preface vii
callatticeswithgainandlossinatomicvaporsandinvestigatedynamicbehaviorsof
lightpropagatinginsuchinducednon-Hermitianopticallattices.InChapter“Effects
of Exceptional Points in PT-symmetric Waveguides”, Moiseyev et al. discuss
physical effects stemming from exceptional points in PT-symmetric waveguides,
such as a slowdown of light oscillations and possible group-velocity effects. In
Chapter “Higher Order Exceptional Points in Discrete Photonics Platforms”, El-
Ganainy and collegues introduce a systematic approach based on a recursive
bosonicquantizationschemeforgeneratingdiscretephotonicnetworksthatexhibit
exceptional points of any arbitrary order and discuss the spectral properties and
theextremedynamicsnearthesesingularities.InChapter“Non-HermitianOptical
Waveguide Couplers”, Kivshar and collaborators review PT-symmetric effects in
non-Hermitiantwo-corecouplersandtrimersandshowthattheirnonlinearresponse
canbreakPT symmetry.InChapter“Parity-TimeSymmetricPlasmonics”,Dionne
et al. provide an overview of nano-photonic PT devices based on plasmonics,
such as ultra-compact perfect absorber/amplifiers, multiplexers, and polarization
converters with unity-efficiency. In Chapter “PT-symmetry and Non-Hermitian
Wave Transport in Microwaves and RF Circuits”, Kottos and colleagues provide
a review of recent progress in PT symmetry and non-Hermitian wave transport
in microwaves and radio-frequency circuits, where concepts like coherent perfect
absorbers, gain-induced shut-down of lasing, and asymmetric transport naturally
emerge.InChapter“CoupledNonlinearSchrödingerEquationswithGainandLoss:
ModelingPT-symmetry”,KonotopconsiderscouplednonlinearSchrödingerequa-
tions with balanced gain and loss and explores various wave transport phenomena
in nonlinear PT-symmetric settings such as bright and dark solitons and their
interactions with defects, soliton switches, resonant wave interactions, and wave
collapse. In Chapter “Making the PT Symmetry Unbreakable”, Malomed et al.
outline approaches for extending PT symmetry to very large gain-loss strengths,
whendealingwithsubwavelength-scalewaveguidesandPT-symmetricsolitonsin
one-andtwo-dimensionalmodelshavingself-defocusingnonlinearities.InChapter
“Krein Signature in Hamiltonian and PT -symmetric Systems”, Pelinovsky and
collaborators discuss the concept of Krein signature in Hamiltonian and PT-
symmetric systems such as the one-dimensional Gross–Pitaevskii equation with
a real harmonic potential and a corresponding linear imaginary component. In
Chapter “Integrable Nonlocal PT Symmetric and Reverse Space-Time Nonlinear
Schrödinger Equations”, Musslimani and colleagues overview recent advances in
integrable nonlocal nonlinear Schrödinger equations having PT, reverse-time and
reversespace-timesymmetries,bothincontinuumanddiscretesettings.InChapter
“ConstructionofNon-PT–symmetricComplexPotentialswithAll-RealSpectra”,
Yang reviews the generalization of PT symmetry and shows that, in addition
to PT-symmetric complex potentials, there are also large classes of non-PT-
symmetric complex potentials that allow for more flexible gain-loss profiles and
all-realspectra,thatcanbeconstructedviasymmetry,supersymmetry,andsoliton-
theory methods. In Chapter “Constant-Intensity Waves in Non-Hermitian Media”,
Makris and colleagues systematically discuss how to suppress intensity variations
andreflectionsforwavespropagatingthroughanonuniformpotentiallandscapeby
viii Preface
judiciouslyincorporatinggainandlossinthepotential.InChapter“NonlinearBeam
Propagation in a Class of Complex Non-PT -symmetric Potentials”, Kevrekidis et
al. review nonlinear wave propagation in a class of complex non-PT-symmetric
potentials and show that the departure from a strict PT-symmetric form does not
allowforthenumericalidentificationoftruesolitonsolutions.
The material in this book provides a rather comprehensive survey of recent
progress on the theory and applications of PT symmetry. It could be useful
to scientists, engineers, and graduate students, who wish to further explore and
advancethisactivefield.
Orlando,Florida,USA DemetriosN.Christodoulides
Burlington,Vermont,USA JiankeYang
Contents
LinearandNonlinearExperimentsinPT-SymmetricPhotonic
MeshLattices..................................................................... 1
MartinWimmer,DemetriosChristodoulides,andUlfPeschel
PT-Symmetry on-a-Chip: Harnessing Optical Loss for Novel
IntegratedPhotonicFunctionality ............................................. 33
MingsenPan,PeiMiao,HanZhao,ZhifengZhang,andLiangFeng
Parity-TimeSymmetryinScatteringProblems .............................. 53
Mohammad-AliMiri,RobertS.Duggan,andAndreaAlù
ScatteringTheoryandPT-Symmetry........................................ 75
AliMostafazadeh
PassivePT-SymmetryinLaser-WrittenOpticalWaveguide
Structures......................................................................... 123
T.Eichelkraut,S.Weimann,M.Kremer,M.Ornigotti,andA.Szameit
Non-HermitianEffectsDuetoAsymmetricBackscatteringofLight
inWhispering-GalleryMicrocavities .......................................... 155
JanWiersig
ExactResultsforaSpecialPT-SymmetricOpticalPotential ............... 185
H.F.Jones
Parity-Time-SymmetricOpticalLatticesinAtomicConfigurations....... 215
Zhaoyang Zhang, Yiqi Zhang, Jingliang Feng, Jiteng Sheng,
YanpengZhang,andMinXiao
EffectsofExceptionalPointsinPT-SymmetricWaveguides................ 237
NimrodMoiseyevandAlexeiA.Mailybaev
HigherOrderExceptionalPointsinDiscretePhotonicsPlatforms......... 261
M.H.Teimourpour,Q.Zhong,M.Khajavikhan,andR.El-Ganainy
ix
