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NATURE PHOTONICS| VOL 8 | NOVEMBER 2014 | www.nature.com/naturephotonics 821

Frequency, wavevector, polarization and phase are degreeso reedom that are ofen used to describe a photonic system.Over the past ew years, topology a property o a photonic

material that characterizes the quantized global behaviour o the

waveunctions on its entire dispersion band has emerged asanother indispensable degree o reedom, thus opening a pathtowards the discovery o undamentally new states o light and pos-sible revolutionary applications. Potential practical applications otopological photonics include photonic circuitry that is less depend-ent on isolators and slow light that is insensitive to disorder.

opological ideas in photonics branch rom exciting develop-ments in solid-state materials, along with the discovery o newphases o matter called topological insulators1,2. opological insula-tors, being insulating in the bulk, conduct electricity on their sur-ace without dissipation or back-scattering, even in the presence olarge impurities. Te first example o this was the integer quantumHall effect, discovered in 1980. In quantum Hall states, two-dimen-sional (2D) electrons in a uniorm magnetic field orm quantized

cyclotron orbits o discrete energies called Landau levels. Whenthe electron energy sits within the energy gap between the Landaulevels, the measured edge conductance remains constant within anaccuracy o around one part in a billion, regardless o sample size,composition and purity. In 1988, Haldane proposed a theoreticalmodel or achieving the same phenomenon in a periodic systemwithout Landau levels3 the quantum anomalous Hall effect.

In 2005, Haldane and Raghu transerred the key eature o thiselectronic model to the realm o photonics4,5. Tey theoreticallyproposed the photonic analogue o the quantum (anomalous) Halleffect in photonic crystals6(the periodic variation o optical mate-rials that affects photons in the same manner as solids modulateelectrons). Tree years later, this idea was confirmed by Wang et al.,who provided realistic material designs7and experimental observa-

tions8

. Tese studies spurred numerous subsequent theoretical913

and experimental investigations1416.In ordinary waveguides, back-reflection is a major source o

unwanted eedback and loss that hinders large-scale optical integra-tion. Te works cited above demonstrate that unidirectional edgewaveguides transmit electromagnetic waves without back-reflectioneven in the presence o arbitrarily large disorder. Tis is an idealtransport property that is unprecedented in photonics. opologicalphotonics promises to offer unique, robust designs and new deviceunctionalities or photonic systems by providing immunity toperormance degradation induced by abrication imperections orenvironmental changes.

Topological photonicsLing Lu*, John D. Joannopoulos and Marin Soljai

The application of topology, the mathematics of conserved properties under continuous deformations, is creating a range ofnew opportunities throughout photonics. This field was inspired by the discovery of topological insulators, in which interfacialelectrons transport without dissipation, even in the presence of impurities. Similarly, the use of carefully designed wavevector-space topologies allows the creation of interfaces that support new states of light with useful and interesting properties. In par-ticular, this suggests unidirectional waveguides that allow light to flow around large imperfections without back-reflection. ThisReview explains the underlying principles and highlights how topological effects can be realized in photonic crystals, coupledresonators, metamaterials and quasicrystals.

In this Review, we present the key concepts, experiments andproposals in the field o topological photonics. Starting with anintroduction to the relevant topological concepts, we introduce the2D quantum Hall phase through the stability o Dirac cones 4,5, ol-

lowed by its realizations in gyromagnetic photonic crystals7,8,13, cou-pled resonators9,10,16and waveguides15, bianisotropic metamaterials11and quasicrystals14. We then extend our discussion to three dimen-sions, wherein we describe the stability o line nodes and Weyl pointsand their associated surace states12. We conclude by considering theoutlook or urther theoretical and technological advances.

Topological phase transitionopology is the branch o mathematics concerned with quantitiesthat are preserved under continuous deormations. For example, thesix objects in Fig. 1a all have different geometries, but there are onlythree different topologies. Te sphere can be continuously changesinto the spoon, so they are topologically equivalent. Te torus andcoffee cup are also topologically equivalent, and so too are the dou-

ble torus and tea pot. Different topologies can be mathematicallycharacterized by integers called topological invariants quanti-ties that remain constant under arbitrary continuous deorma-tions o the system. For the above closed suraces, the topologicalinvariant is its genus, which corresponds to the number o holeswithin a closed surace. Objects with the same topological invari-ant are topologically equivalent; that is, they are in the same topo-logical phase. Only when a hole is created or removed in the objectdoes the topological invariant change. Tis process is known as atopological phase transition.

opologies or material systems in photonics are defined on thedispersion bands in reciprocal (wavevector) space. Te topologicalinvariant o a 2D dispersion band is the Chern number (C, Box 1),a quantity that characterizes the quantized collective behaviour o

the waveunctions on the band. Once a physical observable can bewritten as a topological invariant, it changes only discretely; thus, itwill not respond to continuous small perturbations. Tese perturba-tions can be arbitrary continuous changes in the system parameters.

Optical mirrors reflect light o a given requency range, due tothe lack o available optical states inside the mirror. Tus, the re-quency gaps o a mirror are analogous to the energy gaps o aninsulator. Te sum o the Chern numbers o the dispersion bandsbelow the requency gap indicates the topology o the mirror. Tiscan be understood as the total number o twists and untwists othe system up to the gap requency. Ordinary mirrors like air (totalinternal reflection), metal or Bragg reflectors all have zero Chern

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. *e-mail: [email protected]

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numbers, which makes them topologically trivial. Mirrors withnon-zero Chern numbers are topologically non-trivial.

Te most ascinating and peculiar phenomena take place at theinterace between two mirrors with different topological invariants.Te edge waveguide ormed by these two topologically inequiva-lent mirrors (Fig. 1b, right) is topologically distinct rom an ordi-nary waveguide, which is ormed between topologically equivalentmirrors (Fig. 1b, lef). Te distinction lies in the requency spectrao their edge modes inside the bulk requency gap. On the lef oFig. 1c, the two requency bands both have zero Chern numbers,so they can directly connect across the interace without closing

the requency gap. However, when the two mirrors have differentChern numbers, topology does not allow them to connect to eachother directly. A topological phase transition must take place at theinterace: this requires it to close the requency gap, neutralize theChern numbers, then reopen the gap. Tis phase transition (Fig. 1c,right), ensures gapless requency states at the interace: there mustexist edge states at all requencies within the gap o the bulk mir-rors. Te gapless spectra o the edge states are topologically pro-tected; that is, their existence is guaranteed by the difference o thetopologies o the bulk materials on the two sides. In general, thenumber o gapless edge modes equals the difference o the bulktopological invariants across the interace. Tis is known as thebulk-edge correspondence.

Te topological protection o edge waveguides can also be under-stood in reciprocal space. Figure 1d shows the dispersion diagrams

o both ordinary (lef) and gapless (right) waveguides. On the lef, theordinary waveguide dispersion is disconnected rom the bulk bandsand can be continuously moved out o the requency gap into thebulk bands. On the right, however, the gapless waveguide dispersionconnects the bulk requency bands above and below the requencygap. It cannot be moved out o the gap by changing the edge termi-nations. Similar comparisons between the edge band diagrams areshown in Fig. 2. Te only way to alter these connectivities is througha topological phase transition; that is, closing and reopening the bulk

requency gap.Te unidirectionality o the protected waveguide modes can beseen rom the slopes (group velocities) o the waveguide dispersions.An ordinary waveguide (Fig. 1d, lef) supports bidirectional modesbecause it back-scatters at imperections. In contrast, a topologicallyprotected gapless waveguide (Fig. 1d, right) is unidirectional as ithas only positive (or only negative) group velocities. In addition,there are no counter-propagating modes at the same requencies asthe one-way edge modes. Tis enables light to flow around imper-ections with perect transmission the light can only go orwards.Te operating bandwidth o such a one-way waveguide is as large asthe size o the bulk requency gap.

From Dirac cones to quantum Hall topological phase

One effective approach or finding non-trivial mirrors (requencygaps with non-zero Chern numbers) is to identiy the phase tran-sition boundaries o the system in the topological phase diagram,where the bulk requency spectrum is gapless. Correct tuning othe system parameters thus open gaps that belong to different topo-logical phases. In 2D periodic systems, these phase boundaries arepoint-degeneracies in the bandstructure. Te most undamental2D point degeneracy is a pair o Dirac cones with linear disper-sions between two bands. In three dimensions, the degeneraciesinvolve line nodes and Weyl points, which we will discuss later inthis Review.

Dirac cones are protected, in the entire 2D Brillouin zone, byby PT symmetry, which is the product o time-reversal sym-metry (T, Box 2) and parity (P) inversion. Every Dirac cone has a

quantized Berry phase (Box 1) o looped around it17,18

. ProtectedDirac cones generate and annihilate in pairs1923. Te effec-tive Hamiltonian close to a Dirac point in the xy plane can beexpressed by H(k) = vxkxx+ vykyz, where vi are the group veloci-ties and iare the Pauli matrices. Diagonalization leads to the solu-tion (k) = (vx2kx2 + vy2ky2). Although both P and T map theHamiltonian rom k to k, they differ by a complex conjugation:(PT) H(k) (PT)1= H(k)*. PTsymmetry requires the Hamiltonian tobe real and thus absent o y, which is imaginary. A 2D Dirac point-degeneracy can be lifed by any perturbation that is proportionalto y in the Hamiltonian or, equivalently, by any perturbation thatbreaks PT. Tereore, breaking either Por Twill open a bandgapbetween the two bands.

However, the bandgaps opened by breaking P24and Tindividu-

ally are topologically inequivalent5,25

, as the bulk bands in these twocases carry different Chern numbers. Te Chern number is the inte-gration o the Berry curvature ((k)in able B1) on a closed suracein wavevector space. (k) is a pseudovector that is odd under Tbuteven under P. In the presence o both Pand T, (k) = 0. When eitherPor Tis broken, the Dirac cones open and each degeneracy-lifingcontributes a Berry flux o magnitude to each o the bulk bands.In the presence o T(Pbroken), (k) = (k). Te Berry flux con-tributed by one pair o Dirac points at kand kare o opposite signs.Integration over the whole 2D Brillouin zone always equals zero,and thus so do the Chern numbers. In contrast, in the presence o P(Tbroken), (k) = (k). Here, the total Berry flux adds up to 2and the Chern number equals one. More pairs o Dirac cones canlead to higher Chern numbers13. Tis T-breaking 2D quantum Halltopological phase is shown in red in the phase diagram o Fig. 2.

Phase transition (gap close)

C = 0

C = 0

Trivial

mirror

b

Waveguide

dispersions

C= +1

C= 1

C= 0

C= 0

x

Topologically protected

waveguide

Ordinary

waveguide

Trivial

mirror

Trivial

mirror

Non-trivial

mirror

C = 1, 2, 3...

Gap Frequency gap Gap

a

Gapless and unidirectional

y

x

ky a

(Air, metal,

Bragg

reflectors...)

Genus = 0 1 2

No transition

C = 0

C = 0

C = 0

C = 0

c

dC= +2

C= 2

C = 1, 2, 3...

Figure 1 |Topological phase transition. a, Six objects of different

geometries can be grouped into three pairs of topologies. Each pair has the

same topological invariant, known as its genus. b, Two waveguides formed

by mirrors of different (right) and same (left) topologies. c, Frequency

bands of different topologies cannot transition into each other without

closing the frequency gap. A topological phase transition takes place on

the right, but not on the left. d, Interfacial states have different connectivity

with the bulk bands, depending on the band topologies of the bulk mirrors.

Here, ais the period of the waveguide propagating along y, and Cis thechange in Chern number between the corresponding bulk bands on the

right and left of the waveguide. The magnitude of Cequals the number

of gapless interfacial modes and the sign of Cindicates the directionof propagation.

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Gyromagnetic photonic crystalsWang et al.were the first to realize the photonic analogue o thequantum Hall effect at microwave requencies8. Teir experimentbroke T by applying a uniorm magnetic field on gyromagneticphotonic crystals, resulting in a single topologically protected edge

mode that propagated around arbitrary disorder without reflection.Such single-mode one-way waveguides can be realized in cou-pled deect cavities26, sel-guide27in ree-standing slabs28and havea robust local density o states29. Tey have enabled novel devicedesigns or tunable delays and phase shifs with unity transmission7,reflectionless waveguide bends and splitters30, signal switches31,directional filters32,33, broadband circulators34and slow-light wave-guides35. Very recently, multimode one-way waveguides o large bulkChern numbers (|C| = 2, 3, 4) have been theoretically discovered byopening gaps o multiple point degeneracies simultaneously13, thusproviding even richer possibilities in terms o device unctionalities.

Wang et al. based their experiments8 on a 2D square latticephotonic crystal comprising an array o gyromagnetic errite rodsconfined vertically between two metallic plates to mimic the 2Dtransverse magnetic (M) modes. Tey also added a metal wall to

the surrounding edges to prevent radiation loss into air (Fig. 3a).Without the external magnetic field, the second and third M bandsare connected by a quadratic point-degeneracy comprising a pairo Dirac cones36. Under a uniorm static magnetic field (0.2 ) thatbreaks T, anti-symmetric imaginary off-diagonal terms develop in

the magnetic permeability tensor (). Te quadratic degeneracybreaks and a complete bandgap orms between the second andthird bands, which both have non-zero Chern numbers. Te reddispersion line in Fig. 3b is the gapless edge state inside the sec-ond bandgap, which has only positive group velocities at around4.5 GHz. Numerical simulation results (Fig. 3c, top) verified that anantenna inside the waveguide can only emit in the orward directionin the bulk requency gap. Te experimental transmission data inFig. 3d shows that the backwards reflection is more than five orderso magnitude smaller than the orwards transmission afer propa-gating over only eight lattice periods. More importantly, there is noincrease in the reflection amplitude even afer the insertion o largemetallic scatterers (Fig. 3c, bottom). Indeed, new one-way edgemodes automatically orm wherever a new interace is created, thusproviding a path or light to circumvent the scatter. Tis is precisely

A closed surace can be smoothly deormed into variousgeometries without cutting and pasting. Te GaussBonnet theo-rem92o equation (1) below, which connects geometry to topology,states that the total Gaussian curvatures ()o a 2D closed suraceis always an integer. Tis topological invariant, named genus (g),characterizes the topology o the surace; that is, the number o

holes within. Examples o suraces with different geni are shownin Fig. 1a.

12

dA= 2(1 g)surface

(1)

A two dimensional Brillouin zone is also a closed suracewith the same topology o a torus due to its periodic bound-ary conditions (Fig. B1). able B1 lists the definitions 93o Berrycurvature and Berry flux with respect to Bloch waveunctionsin the Brillouin zone by comparing them to the amiliar case o

magnetic field and magnetic flux in real space. Integrating theBerry curvature over the torus surace yields the topologicalinvariant known as the Chern number, which gives a measureo the total quantized Berry flux o the 2D surace. Te Chernnumber can be viewed as the number o monopoles o Berryflux inside a closed surace, as illustrated in Fig. B1. An efficient

way to calculate Chern numbers in discretized Brillouin zones isdescribed in re. 94.opological invariants can be arbitrary integers ( )or binary

numbers ( 2, meaning mod 2). Chern numbers are integers(C ) and the sum o the Chern numbers over all bands o agiven system is zero.

Historically, the geometric phase was first discovered in opticsby Pancharatnam95prior to the discovery o the Berry phase96. Tefirst experiments demonstrating the Berry phase were perormedin optical fibres97.

2D Brillouin zone C= 0 C= 1 C= 2 Positive charge of Berry fluxC= 1Negative charge of Berry flux

kxky

kzky

kx

Figure B1 | Chern number as the number of Berry monopoles in momentum space. A 2D Brillouin zone is topologically equivalent to a torus. The

Chern number (C) can be viewed as the number of monopoles (charges) of Berry flux inside a closed 2D surface. The arrows represent Berry curvature

from the positive and negative charges. In a 3D Brillouin zone, these monopoles are Weyl points.

Box 1 | Topological invariant.

Table B1 | Comparison of the Berry phase for Bloch wavefunctions and the AharonovBohm phase.

Vector potential A(r) (k) = u(k)|i |u(k)k Berry connection

AharonovBohm phase A(r) dl (k) dl Berry phase

Magnetic field B(r) = A(r)r (k

) = (k

)k Berry curvatureMagnetic flux B(r) ds (k) ds Berry flux

Magnetic monopoles # = B(r) dseh

C= (k) ds12

Chern number

The Berry connection measures the local change in phase of wavefunctions in momentum space, where i kis a Hermitian operator. Similar to the vector potential and AharonovBohm phase, Berry

connection and Berry phase are gauge dependent (that is, u(k)ei(k)u(k)). The rest of the quantities are gauge-invariant. The Berry phase is defined only up to multiples of 2. The phase and flux can beconnected through Stokes theorem. Here, u(k)is the spatially periodic part of the Bloch function; the inner product of is done in real space. The one-dimensional Berry phase is also known as the Zak phase.

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the topological protection provided by the bulk o a photonic crystalthat contains non-zero Chern numbers.

Tere are other types o one-way waveguides that break T(re. 37), but these are not protected by topology. In general, mag-netic responses are very weak in optical materials. Realization atoptical requencies thereore remains a challenge.

Coupled resonatorsPhotons in an array o coupled resonators are similar to electrons

in an array o atoms in solids. Te photon couplings between the

resonators can be controlled to orm topologically non-trivialrequency gaps with robust edge states. Researchers obtained thephotonic analogues o the integer quantum Hall effect by construct-ing both static and time-harmonic couplings that simulate the elec-trons behaviour in a uniorm magnetic field. When the T-breakingis implemented by accurate time-harmonic modulations, unidi-rectional edge waveguides immune to disorder can be realized atoptical requencies.

In electronic systems, the first quantum Hall effect was observed

in a 2D electron gas subject to an out-o-plane magnetic field. Asillustrated in Fig. 4a, the bulk electrons undergo localized cyclotronmotions, while the unidirectional edge electrons have an extendedwaveunction. Again, the number o gapless edge channels equals theChern number o the system. Here, the physical quantity describingthe magnetic field is the vector potential, which can be written inthe orm A = Byx. An electron accumulates an AharonovBohm(AB) phase o

= A(r) dl

afer a closed loop (see able B1). An electron going against thecyclotron motion acquires a phase o (dotted circle in Fig. 4a), soit has a different energy(rom the electrons moving in solid circles).

Te spin degeneracy o electrons is lifed by Zeeman splitting.Although a photon does not interact with a magnetic field, itdoes acquire a phase change afer passing through a closed loop. Bycareully tuning the propagation and coupling phases, Haezi et al.designed9a lattice o optical resonators in which the photons acquirethe same phase as the AB phase o electrons moving in a uniormmagnetic field. Tis is different rom a true quantum Hall topologicalphase, as Tis not broken in their static and reciprocal resonator array.Tus, back-reflections are allowed because time-reversed channelsalways exist at the same requencies. Nevertheless, Haezi et al. wereable to observe the edge states at near-inrared (1.55 m) wavelengthsin the first set o experiments perormed on a silicon-on-insulatorplatorm16, and in a recent experiment38also showed that robustnessagainst particular types o disorder can still be achieved owing to the

topological eatures o the phase arrangements.

Figure 2 |Topological phase diagram of the 2D quantum Hall phase.

The top-left image shows a band diagram of edge states in which the bulk

dispersions form a pair of Dirac cones (grey) protected by PTsymmetry.

The green and blue lines represent edge dispersions on the top and bottom

edges. When either Por Tare broken, a bandgap can form in the bulk but

not necessarily on the edges. When T-breaking is dominant, the two bulk

bands split and acquire Chern numbers of 1. Thus, there exists one gapless

edge dispersion on each of the top and bottom interfaces, assuming thebulk is interfaced with topologically trivial mirrors. This T-breaking phase

of non-zero Chern numbers is the quantum Hall phase, plotted in red in the

phase diagram.

Break T

Break

P

Dirac cones

T-breaking

strength

P-

breaking

strength

Phase diagram

C = 0

|C| = 1

kx

kx

kx

Quantum anomalous Hall

C= +1

C= 1

C = 0

C = 0

x

yBulk

x

yBulk

Symmetry considerations are crucial when determining thepossible topological phases o a system. For example, the quantumHall phase requires the breaking o time-reversal symmetry (T). Onthe other hand, in the recently discovered 2D and 3D topologicalinsulators in electronics, T-symmetry is required to protect thesetopological phases characterized by 2topological invariants. Forexample, the 2D topological insulator, also known as the quantumspin Hall effect98, allows the coexistence o counter-propagatingspin-polarized gapless edge states. Without T-symmetry, however,

these edge states can scatter into each other. Te edge energy spec-trum opens a gap and the insulator can continuously connect totrivial insulators, such as the vacuum. A large table o symmetry-protected topological phases have been theoretically classified73,74.Tese systems have robust interacial states that are topologicallyprotected only when the corresponding symmetries are present75.

Here we point out the undamental difference in time-reversalsymmetry between electrons and photons. A photon is a neu-tral non-conserved non-interacting spin-1 Boson that satisfiesMaxwells equations, whereas an electron is a charged conservedinteracting spin-1 Fermion that satisfies Schrdingers equation.Similar to Schrdingers equation, the lossless Maxwells equationsat non-zero requencies can be written as a generalized Hermitianeigenvalue problem:

i = 0

0)) ))E

H ))E

H

))

where = , = and is the bianisotropy term, where isHermitian conjugation.

Te anti-unitary time Toperator is given by:

K))10 01which squares to unity, where Kis complex conjugation (*). When* = , * = and = *, the system is T-invariant. Rotatingthe spin by 2 is the same as applying the T operator twice(T2). But T2has different eigenvalues or photons (T2= +1) andelectrons (T2= 1). It is this minus sign that ensures Kramersdegeneracies or electrons at T-invariant kpoints in the Brillouinzone, thus providing the possibility o gapless connectivity oredge dispersions in the bulk gap. Tis undamental distinctionresults in different topological classifications or photons andelectrons with respect to T. For example, photons do not havethe same topological phases o 2D or 3D topological insulatorsprotected by Tas or electrons.

Box 2 | Time reversal symmetry.



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Figure 4b shows a 2D array o whispering-gallery resonators thatare spatially coupled by the waveguides between. Each resonator hastwo whispering-gallery modes that propagate clockwise (green) andcounter-clockwise (red). Tey are time-reversed pairs and are simi-lar to the spin-up and spin-down degrees o reedom or electrons.Te lengths o the coupling waveguides are careully designed so thatthe total coupling phase between resonators precisely matches theAB phase in Fig. 4a: the vertical couplings have no phase changes,whereas the horizontal couplings have phases that are linear in y.

In each spin space, photons o opposite circulations experienceopposite AB phases (), just like the electrons in Fig. 4a. Teseopposite loops are also illustrated in solid and dashed red lines orthe spin-up photons in Fig. 4b. As a result, the photonic requencyspectrum in this resonator array39,40exhibits both Landau levels andractal patterns known as the Hostadter butterfly, which are the sig-natures o a 2D electron in a uniorm magnetic field the integerquantum Hall effect. However, without breakingT, the two copieso spin spaces are degenerate in requency and couple to each other.Only by assuming that these two spins decouple rom each othercompletely can Chern numbers o the same magnitude but oppositesign be defined and potentially measured41in each spin space. (Tespin-polarized counter-propagating edge modes bear similaritiesto the edge currents in the quantum spin Hall effect or electrons,

but they are undamentally different in symmetry protections andtopological invariants, as discussed in Box 2.) Tese photonic gap-less edge modes are robust against disorder that does not inducespin flips. For example, when a deect edge resonator has a differ-ent size and resonance requency rom the bulk resonators, thenthe edge mode will find another route to pass around this deectresonator. A recent study42,43suggested that the required spatially

varying couplers along ycan be made identical and periodic, butstill achieving the same phenomena. Unortunately, in these recip-rocal schemes, perturbations that induce spin flips are practicallyubiquitous: local abrication imperections on the resonators or thecouplers and even the coupling processes themselves can mix thespins and induce back-scattering.

Back-scattering in the above time-reversal-invariant systems

can be eliminated by breaking T, or example using spatially coher-ent time-domain modulations, as proposed theoretically in re. 10.Fang et al.propose placing two kinds o single-mode resonators inthe lattice shown in Fig. 4c. When the nearest-neighbour coupling isdominant, the two resonators (which have different resonance re-quencies) can couple only through the time-harmonic modulationbetween them. Te vertical coupling phases are zero and the hori-zontal coupling phases increase linearly alongy, thereby producingeffective AB phases rom a uniorm magnetic field. Photons movingin opposite directions have opposite phases, so they have differentrequencies. Floquets theorem in the time domain similar toBlochs theorem in the spatial domain is used to solve this latticesystem o time-periodic modulations. Te resulting Floquet band-structure has the same gapless edge states as that o a static quantum

Hall phase.Achieving accurate and coherent time-harmonic modula-tions or a large number o resonators is experimentally challeng-ing towards optical requencies4446. By translating the modulationrom the time domain to the spatial domain, Rechtsman et al.experimentally demonstrated the photonic analogue o the quan-tum Hall effect using optical photons (633 nm)15. Tese are also thefirst experiments on Floquet topological phases47,48. Starting witha 2D resonator array (Fig. 4d), the researchers extended the cavi-ties along zto give a periodic array o coupled waveguides propa-gating in this direction. In their system, z plays the role o time.More specifically, the paraxial approximation o Maxwells equa-tions results in an equation governing diffraction (propagating inz) that is equivalent to Schrdingers equation evolving in time. Teperiodic helical modulations49,50in zbreak the z-symmetry, which

is equivalent to the time-domain modulations that break T. Tissymmetry-breaking opens up protected band degeneracies in theFloquet bandstructure, thus orming a topologically non-trivialbandgap that contains protected gapless edge modes.

Te idea o creating effective magnetic fields or neutral particles51using synthetic gauge fields was first explored in optical lattices52.Very recently, similar gauge fields have also been studied in optom-echanics53and radiorequency circuits54. Finally, although approxi-mations such as nearest-neighbour in space or rotating-wave intime were adopted through the analysis o the systems described inthis section, these higher order corrections do not undamentallyalter the topological invariants and phenomena demonstrated.

Bianisotropic metamaterialsIn bianisotropic materials ( 0, Box 2)55, the coupling betweenelectric and magnetic fields provides a wider parameter space or therealization o different topological phases. In particular, it has beenshown that bianisotropic photonic crystals can achieve topologicalphases without breaking T(T-invariant); thus, neither magnetism nortime-domain modulations are needed or the topological protectiono edge states. Bianisotropic responses are known as optical activityin chiral molecules and can also be designed in metamaterials.

Bianisotropy acts on photons in a similar way to how spin-orbitcoupling acts on electrons56. In their inspiring theoretical proposal11,Khanikaev et al. enorced polarization (spin) degeneracy or pho-tons by equating to (= ), so that the transverse electric (E) andM modes in two dimensions are exactly degenerate in requencies.

a

Scatterer

Edge waveguide

Metal wall

Antenna

Ferrite rods

Frequency(G

Hz)

0

1

2

3

4

C= 2

C= 1

C= 0

b

Wavevector (2/a)

One-way edge waveguide

Frequency (GHz)

Transmission(dB)

0

20

40

60

4.0

d

B

B

TM bands

c00.5 0.5

B4.5

BackwardsForwards

Bulk

modes

a= 4 cm

Figure 3 |First experimental demonstration of the topologically

protected one-way edge waveguide at microwave frequencies.

a, Experimental set-up for measuring the one-way edge state between the

metal wall and the gyromagnetic photonic crystal confined between the

metallic plates to mimic the 2D TM modes. The inset is a picture of the

ferrite rods that constitute the photonic crystal of lattice period a= 4 cm.

b, The bandstructure of the one-way gapless edge state between the

second and third bands of non-zero Chern numbers. c, Simulated field

propagation of the one-way mode and its topological protection against a

long metallic scatterer. d, The measured robust one-way transmission data

of the edge waveguide.



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When the pseudo-tensor is o the same orm as the gyroelectricor gyromagnetic terms in or , thenacts as a magnetic field oneach polarization with opposite signs, without breaking T. Tis sys-tem can be separated into two independent spin subspaces, in whichquantum anomalous Hall phases exist with opposite Chern numbers.

Very recently, it was suggested that the = condition could poten-tially be relaxed57. Indeed, in their experimental work, Chen et al.58relaxed the material requirements or matching and . Tey alsorealized a broadband effective bianisotropic response by embedding

the /-matched metamaterials in a metallic planar waveguide. Teseadvances enabled them to observe the spin-polarized edge transportat around 3 GHz.

Similar to the T-invariant resonator arrays in Fig. 4b andres 9,16,38,42,43, the above metamaterial realizations also requirestrict conditions in order to decouple the two copies o spins. Inthese cases, the requirements are on the accurate realization o theconstitutive parameters during metamaterial manuacturing. Telack o intrinsic T-protected quantum spin Hall topological phaseis one o the most undamental differences between electronic andphotonic systems, as discussed in Box 2. Finally, gapless suracestates were also proposed to exist in a bulk hyperbolic metamaterialthat exhibits bianisotropic responses59.

QuasicrystalsQuasicrystals are aperiodic structures that possess spatial order.Tey also have requency gaps and interacial states. Quasicrystalscan be constructed rom the projections o periodic crystals in

higher dimensions. Krauss et al.14projected the 2D quantum Hallphase onto a 1D quasicrystal model containing a tunable parameterthat is equivalent to the Bloch wavevector lost during the projection.Scanning this periodic parameter reproduced the ull gapless re-quency spectrum o the 2D quantum Hall phase; that is, the 0D edgemode requency o the 1D quasicrystal continuously swept throughthe 1D bulk gap. Te researchers abricated 1D optical waveguidearrays to be spatially varying along z according to the continuoustuning o this parameter. In their system, zplays the role o time.

Tey observed the edge state start rom one edge o the waveguidearray, merge into the bulk modes, then switch to the other edge othe array. Tus, light is adiabatically transerred in space rom edgeto edge. Going a step urther, they proposed the potential realizationo the quantum Hall phase in 4D using 2D quasicrystals60.

Weyl points and line nodes: Towards 3D topological phases2D Dirac points are the key bandstructures that led to the first pro-posal and experiments o the photonic analogue o the quantumHall effect. For 3D61,62 topological phases, the key bandstructuresare line nodes63, 3D Dirac points64and, more undamentally, Weylpoints65. However, Weyl points have not yet been realized in nature.Recently, Lu et al.theoretically proposed12the use o germanium orhigh-index glass or achieving both line nodes and Weyl points in

gyroid photonic crystals at inrared wavelengths.A line node63is a linear line-degeneracy: two bands touch at aclosed loop (Fig. 5a) while being linearly dispersed in the other twodirections, thus making it the extension o Dirac cone dispersionsin three dimensions. For example, H(k) = vxkxx+ vykyzdescribes aline node along kz. PTthereore protects both Dirac cones and linenodes. Te line node bandstructure in Fig. 5b is ound in a doublegyroid (DG) photonic crystal with both Pand T. Te surace disper-sions o a line-node photonic crystal can be flat bands in controlledareas o the 2D surace Brillouin zone. When PTis broken, a linenode can either open up a gap or split into Weyl points. Figure 5cshows a phase diagram o the DG photonic crystals, where theline node splits into one or two pairs o Weyl points under Tor Pbreaking, respectively.

A Weyl point65

is a linear point-degeneracy: two bands touch ata single point (Fig. 5d) while being linearly dispersed in all threedirections. Te low-requency Hamiltonian o a Weyl point isH(k) = vxkxx+ vykyy+ vzkzz. Diagonalization leads to the solution(k) = (vx2kx2+ vy2ky2+ vz2kz2). Because all three Pauli matricesare used in the Hamiltonian, the solution cannot have a requencygap. Te existence o the imaginary yterm means that breaking PTis a necessary condition or obtaining Weyl points. Weyl points aremonopoles o Berry flux (Fig. 5d): a closed surace in a 3D Brillouinzone containing a single Weyl point has a non-zero Chern numbero 1. Tis means a single Weyl point is absolutely robust in 3Dmomentum space, as Weyl points must be generated and annihi-lated pairwise with opposite Chern numbers. When only Pis bro-ken (Tpreserved), the minimum number o pairs o Weyl points is

two because Tmaps a Weyl point at k to k without changing itsChern number. When only Tis broken (Ppreserved), the minimumnumber o pairs o Weyl points is one. Te bandstructure in Fig. 5e,which contains the minimum o our Weyl points, is realized in aDG photonic crystal under P-breaking. Note that a Dirac point inthree dimensions64is a linear point-degeneracy between our bands,consisting o two Weyl points o opposite Chern numbers sitting ontop o each other in requency.

A photonic crystal that contains requency-isolated Weyl pointshas gapless surace states. Consider the brown plane in the bulkBrillouin zone o Fig. 5: it encloses either the top red Weyl point(C= +1) or the lower three Weyl points, depending on the choice odirection. Either way, this plane has a non-zero Chern number, simi-lar to the 2D Brillouin zone in the quantum Hall case. Tus, any sur-ace state with this particular fixed k

y

is gapless and unidirectional.

+

2

+2

3+3

+

2

+2

3+3

+

Reflectionlessedge

mod

es when spins decouple

yx

z

Protectedone-way

edge

modemoving along z+

+

+2+2

A B

+3 +3

3 3

Protectedone-wayedgem

ode

under timemodulations

b

c d

aInteger quantum Hall effect

Protectedone

-w

ayedge current

+

+

+

22

Vector potential A= Byx

AB phase = Adl

Spin

Spin

Figure 4 |Quantum Hall phase of electrons in a magnetic field and of

photons in coupled resonators exhibiting an effective magnetic field.

a, Cyclotron motions of electrons in a static magnetic field (Bz). The vector

potential increases linearly in y. b, A 2D lattice of photonic whispering-

gallery resonators coupled through static waveguides. The horizontal

coupling phases increase linearly in y. The two spins of the whispering-

gallery resonators are degenerate in the effective magnetic field. c, A 2D

lattice of photonic resonators consisting of two types of single-mode

cavities. The nearest neighbours are coupled through time-domain

modulations, with horizontal phases increasing linearly in y; this breaks T.

d, An array of helical photonic waveguides, breaking zsymmetry, induces

harmonic modulations on any photons propagating through it.



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Plotted on the lef o Fig. 5 is an example o such surace states orthe P-broken DG photonic crystal.

OutlookTe field o topological photonics has grown exponentially in recentyears. Non-trivial topological effects have been proposed and real-ized across a variety o photonic systems at different wavelengthsand in all three spatial dimensions. Tis Review has introduced the

main concepts, experiments and proposals o topological photon-ics, ocusing on 2D and 3D realizations. 1D examples are discussedin res 6672.

Over the coming years, we expect the discovery o new topologi-cal mirrors, phases and invariants, which could be classified withrespect to different symmetries7377. Te topological phases o inter-acting photons7880could be explored by considering nonlinearity81and entanglement. Various topologically protected interacial statesbetween different topological mirrors will be studied. Te immu-nity to disorder and Anderson localization o those interacial statesmust be addressed. Moreover, the concepts and realizations o topo-logical photonics can be translated to other bosonic systems suchas surace plasmons70,71, excitons82, excitonpolaritons83,84, phon-ons85,86and magnons87. Certain other robust wave phenomena canbe explained through topological interpretations88.

echnologically, the exploitation o topological effects coulddramatically improve the robustness o photonic devices in thepresence o imperections. As a result, it will become easier todesign robust devices. For example, designers will soon worry lessabout insertion loss and FabryProt noise due to back-reflections.opologically protected transport could solve the key limitationrom disorder and localization in slow light35and in coupled resona-tor optical waveguides38. Unidirectional waveguides could decreasethe power requirements o classical signals and improve coherence

in quantum links89,90. One-way edge states o T-breaking topo-logical phases could be used as compact optical isolators91. Edgestates o T-invariant topological phases11,58do not have reflectioneven when the system is reciprocal; thus, isolators may be unnec-essary or photonic circuits comprising T-invariant topologicalphases. Te realization o practical, topologically protected unidi-rectional waveguides at optical requencies is currently the mainchallenge o this emerging field. Much like the field o topologicalinsulators in electronics, topological photonics promises an enor-mous variety o breakthroughs in both undamental physics andtechnological outcomes.

Received 24 May 2014; accepted 18 September 2014;published online 26 October 2014

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Gapless surface modes

Wavevector

c Weyl point

|C| = 1

P-breaking

strength

T-

breaking

strength

Phase diagram

1 pair

2 pairs

k

Line node (PT)

Line node

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Chern numbers. The red and blue colours indicate the opposite Chern

numbers of the Weyl points. Left: the surface states associated with this

plane have protected gapless dispersions.



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AcknowledgementsL.L. thanks L. Fu, C. Wang and A. Khanikaev or discussions. Te authors thankP. Rebusco and C.W. Hsu or critical reading and editing o the manuscript.J.J. was supported in part by the U.S.A.R.O. through the ISN, under contractW911NF-07-D-0004. L.L. was supported in part by the MRSEC Program o the NSF

under award DMR-0819762. M.S. and L.L. were supported in part by the MI S3ECEFRC o DOE under grant DE-SC0001299.

Author contributionsAll authors contributed equally to this work.

Additional informationReprints and permissions inormation is available online at www.nature.com/reprints .Correspondence and requests or materials should be addressed to L.L.

Competing financial interestsTe authors declare no competing financial interests.

http://lanl.arxiv.org/abs/1408.0237http://lanl.arxiv.org/abs/1408.0237http://www.nature.com/reprintshttp://www.nature.com/doifinder/10.1038/nphoton.2014.248http://www.nature.com/doifinder/10.1038/nphoton.2014.248http://www.nature.com/reprintshttp://lanl.arxiv.org/abs/1408.0237http://lanl.arxiv.org/abs/1408.0237

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