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Photonic Hall effect and helical Zitterbewegung in asynthetic Weyl systemZhang, Shuang

DOI:10.1038/s41377-019-0160-z

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Ye et al. Light: Science & Applications (2019) 8:49 Official journal of the CIOMP 2047-7538https://doi.org/10.1038/s41377-019-0160-z www.nature.com/lsa

ART ICLE Open Ac ce s s

Photonic Hall effect and helicalZitterbewegung in a synthetic Weyl systemWeimin Ye1, Yachao Liu2,3, Jianlong Liu4, Simon A. R. Horsley5, Shuangchun Wen2 and Shuang Zhang3

AbstractSystems supporting Weyl points have gained increasing attention in condensed physics, photonics and acoustics dueto their rich physics, such as Fermi arcs and chiral anomalies. Acting as sources or drains of Berry curvature, Weyl pointsexhibit a singularity of the Berry curvature at their core. It is, therefore, expected that the induced effect of the Berrycurvature can be dramatically enhanced in systems supporting Weyl points. In this work, we construct synthetic Weylpoints in a photonic crystal that consists of a honeycomb array of coupled rods with slowly varying radii along thedirection of propagation. The system possesses photonic Weyl points in the synthetic space of two momenta plus anadditional physical parameter with an enhanced Hall effect resulting from the large Berry curvature in the vicinity ofthe Weyl point. Interestingly, a helical Zitterbewegung (ZB) is observed when the wave packet traverses very close to aWeyl point, which is attributed to the contribution of the non-Abelian Berry connection arising from the neardegenerate eigenstates.

IntroductionSimilar to electrons in solid state materials1, an optical

beam obliquely incident to an interface between twotransparent media experiences a spin-dependent trans-verse shift in the centre of energy (mass) of the reflected/refracted beam. This particular shift is called the spin Halleffect of light (SHEL), which results from the spin−orbitinteractions (SOI) or spin−orbit coupling of photons2–5.Separate from the light-interface interaction, a pure geo-metric spin Hall effect6,7 is observed for a transmittedoptical beam across an oblique polarizer7. SHEL has alsobeen studied for structured surfaces, and it was recentlyshown that a metasurface possessing a linear phase gra-dient could introduce a giant SHEL for an anomalouslyrefracted beam, even when the light was at a normal

incidence8. Moreover, analogous to the quantum spinHall effect of electrons in topological insulators, one-waytransport of the spin-valley-locked edge states has beenobserved in photonic topological insulators9–12. With therapid development in the field of nanophotonics, artifi-cially structured photonic crystals and metamaterialsbring about opportunities to manipulate photonic SOI.Among the reported artificial structures, photonic

honeycomb lattices13–18 have received considerableattention, in part because of their direct analogue tographene, where the electron wave has the dispersion of amassless particle close to the Dirac points. Photonichoneycomb lattices provide two new degrees of freedom,pseudospins and valleys, to describe the state of light. Forin-plane propagation of light, SHEL was proposed in aphotonic analogue of graphene, where the SOI wasinduced by the splitting between transverse-electric (TE)and transverse-magnetic (TM) optical modes13. In astaggered graphene analogue with time-reversal sym-metry14,15, breaking inversion symmetry can also induceSOI and allow a valley Hall effect of photons15. Due to thetime-reversal symmetry, there is an opposite transverseshift of the incident optical beam when the in-plane

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changesweremade. The images or other third partymaterial in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to thematerial. Ifmaterial is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Correspondence: Shuangchun Wen (scwen@hnu.edu.cn) orShuang Zhang (s.zhang@bham.ac.uk)1College of Advanced Interdisciplinary Studies, National University of DefenseTechnology, 410073 Changsha, China2Key Laboratory for Micro/Nano Optoelectronic Devices of Ministry ofEducation, School of Physics and Electronics, Hunan University, 410082Changsha, ChinaFull list of author information is available at the end of the article.These authors contributed equally: Weimin Ye, Yachao Liu

1234

5678

90():,;

1234

5678

90():,;

1234567890():,;

1234

5678

90():,;

momentum matches different valleys in reciprocal space.Recently, increasing focus has been made on waveguidearrays arranged in a honeycomb lattice, which provides apowerful platform for investigating topological photonics,enabling the realization of photonic Floquet topologicalinsulators16, unconventional edge states17, andpseudospin-mediated vortex generation18.Berry curvature underlies many interesting phenomena

in crystalline systems. It is the counterpart of a magneticfield in momentum space and plays an important role inthe motion of a wave packet in both the real space andmomentum space19,20. It is known, for instance, that thewave packet velocity receives an ‘anomalous’ contributionproportional to the Berry curvature in momentumspace1,19. In three-dimensional (3D) systems, the sourcesand drains of Berry curvature are Weyl points, whichoccur at points of double-degeneracy in the 3D bandstructure. Weyl points have been found in solid statesystems of electrons21–23 and 3D photonic crystals24–26,magnetized plasma27, and photonic metamaterials28,29.However, the investigations of Berry curvature effects arenot straightforward in photonic systems possessing Weylpoints in 3D momentum space [kx, ky, kz]. Since the‘anomalous’ contribution to the wave packet velocitydepends on the product of the time derivative of the wavevector and the momentum space Berry curvature1,19,Berry curvature effects cannot be observed in a homo-geneous photonic system, which has an invariant wavevector during the propagation. Recently reported syn-thetic Weyl points30–32 provide a new means to constructphotonic Weyl points in synthetic 3D space. However,similar to homogeneous Weyl systems in 3D momentumspace, a wave packet propagating in the previouslyreported synthetic Weyl systems30–32 has an invariantmomentum in the synthetic space and, therefore, cannotinteract with the Berry curvature.

ResultsIn this Letter we consider a system exhibiting Weyl

points in a synthetic 3D space, where two of the axes aremomentum coordinates kx and ky, and the third axis is thephysical parameter η that adiabatically varies with thespatial coordinate z. A wave packet propagating in the zdirection experiences a variant η and, consequently, a z-dependent Berry curvature generated by the syntheticWeyl points. It is therefore expected that a Berrycurvature-induced Hall effect can be readily observed. Toput it simply, as a wave packet moves through a regionwhere the Berry curvature is large, its velocity will besignificantly modified from the group velocity, and thiseffect will be evident as an additional shift in the finalposition of the packet. Our system consists of an array ofevanescently coupled rods tapered along the z directionand arranged in a honeycomb array. By slowly varying the

diameters of the rods along the direction of propagation(z), the two-dimensional (2D) Dirac points in themomentum space can be turned into synthetic Weylpoints that, hence, generate a distribution of Berry cur-vature in the synthetic space. Based on the semi-classicequations of motion for wave packet propagation in thepresence of Berry curvature1,19,20,33,34, the centre-of-energy velocity of the wave packet contains an ‘anom-alous’ contribution proportional to the Berry curvature ofthe band, and this contribution is responsible for variousHall effects1,19,33. Since each Weyl point is a monopole ofthe Berry curvature in the momentum space, it isexpected that a relatively large Hall effect should beobserved in its vicinity. In addition to this photonic Halleffect, we also observe an interesting helical Zitterbewe-gung (ZB)35–38 arising from the non-Abelian Berry con-nection close to the Weyl point. That is, the centre ofenergy of the optical beam exhibits a helical tremblingmotion around its mean trajectory during thepropagation.Figure 1a shows our proposed array of dielectric rods

with lattice constant a. The radii of rods RA and RB in sub-lattices A and B are designed to vary slowly in the zdirection. At one end of the rod array, the A lattice has alarger diameter than the B lattice, whereas this is reversedon the opposite end of the waveguide array. Somewhere inthe middle of the waveguide array, the A and B latticeshave the same size, closing the band gap at this plane.Figure 1b shows the first Brillouin zone of the honeycomblattice. As will be shown in the following, the Weyl pointsin the 3D synthetic space are located at points denoted byK and K'. The propagation of light in the array of rodsalong z axis can be taken as an adiabatic process where thereflection of light in this direction is negligible. Under thetight-binding approximation, the dynamics of the propa-gation of a Bloch wave with the wave vector (kx, ky) alongthe rods can be cast into the form of a Schrödinger-typeequation for a two-level system (for detailed derivation,see Supplementary information Note 1),

i ddZ juki ¼ H k; ηð Þjuki;H k; ηð Þ ¼ �Re SðkÞ½ �σx þ Im SðkÞ½ �σy � ησz

ð1Þ

where η ¼ ðβA � βBÞ= 2κð Þ, SðkÞ ¼ 1þ 2 cos 12 kxa� �

eiffiffi3

p2 kya.

The normalized coordinate Z=Rκdz is an integral of the

effective coupling coefficient κ between the nearest-neighbour rods in sub-lattices A and B. βA and βB arethe propagation constants of the fundamental modesupported by isolated rods with radii RA and RB, respec-tively. Thus, the parameter η is Z dependent when(RA−RB) varies along the propagation direction. Paulimatrices σx, σy and σz are defined by the sub-latticedegrees of freedom. The Hamiltonian H in Eq. 1 is definedin a synthetic 3D space [k, η]. Using the generic

Ye et al. Light: Science & Applications (2019) 8:49 Page 2 of 8

expression of the Berry curvature of a two-level system1,the Berry curvature of the lower-energy state defined inthe synthetic space is found to be

Ω1 ¼ Ωkyη ¼ffiffi3

pa

2β3Zcos 1

2 kxa� �

´ 2 cos 12 kxa� �þ cos

ffiffi3

p2 kya

� �h i

βZ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiη2 þ SðkÞj j2

q ð2aÞ

Ω2 ¼ Ωηkx ¼ a2β3Z

sin 12 kxa� �

sinffiffi3

p2 kya

� �

Ω3 ¼ Ωkxky ¼ � ffiffi3

pa2

4β3Zη sin kxað Þ

ð2bÞ

Figure 1c shows the distribution of the Berry curvature,which becomes singular at the Weyl points located at [±K,0], where the two bands linearly cross each other along alldirections in the 3D synthetic space. The Hamiltonianclose to the Weyl points can be approximately expressedas

H �K þ q; ηð Þ ¼ ±

ffiffiffi3

pa

2qxσx �

ffiffiffi3

pa

2qyσy � ησz

ð3Þ

Furthermore, the Berry curvature (Fig. 1d, e) in its vectorform is

Ωð1Þð�K þ q; ηÞ ¼ ± 3a2

4η2 þ 3q2a2ð Þ3=2qx; qy; η� �

ð4Þ

Due to the presence of time-reversal symmetry, theBerry curvature has an opposite sign at valleys K and K′. Itis straightforward to show that by integrating the Berrycurvature in Eq. 4 over an equi-energy surface enclosing asingle Weyl point, one can obtain a value of ±2π corre-sponding to a quantized Chern number of ±1.Similar to a wave packet propagating in the waveguide

array, the equation of the motion for the centre-of-energycoordinate of the Bloch wave XCk is given by (for detailedderivation, see Supplementary information Note 2)

dXCk

dZ¼ huk j∂kHðk; ηÞjuki

huk juki þ XB0 � XA

0

� � huk jΞðkÞjukihuk juki

ð5Þ

2

a c

b

d e

0

–2

–5

–0.10–0.10

–0.10

–0.05

–0.05

–0.05

0.05

0.05

0.05

0.00

0.00

0.00

0.10

0.10

0.10

0

A

B

y

xz

kxa

qya–0.10

–0.05

0.050.00

0.10

qya

qxa–0.10

–0.05

0.05

0.00

0.10

qxa

ky

kx

M KK ′

K ′

K ′K

K

�

�

�

–0.10

–0.05

0.05

0.00

0.10

�

kya

5–4

–2

0

2

4

Fig. 1 The evanescently coupled rod array arranged in a honeycomb lattice with slowly varying radii along the propagation direction(z-direction). a The schematic illustration. b The first Brillouin zone in the reciprocal space of the honeycomb lattice with the positions of K=−4π

3aexand K′= 4π

3aex. c The distribution of the Berry curvature of the lower-energy state in the synthetic space [k, ƞ]. d, e Local distributions of the Berrycurvature close to the two Weyl points

Ye et al. Light: Science & Applications (2019) 8:49 Page 3 of 8

in which ΞðkÞ ¼ Im SðkÞ½ �σx þ Re SðkÞ½ �σy. XA0 and XB

0 are,respectively, the relative positions of rods located at sub-lattices A and B within one unit cell. The first term on theright-hand side describes the motion of the Bloch wavedetermined by the Hamiltonian. The second term is thevelocity of the centre of energy due to the transitionbetween sub-lattices A and B within each unit cell of thehoneycomb lattice. The Bloch wave can be expressed as asuperposition of the lower- and upper-energy states {Ψ1,Ψ2} of the Hamiltonian in Eq. 1 with the eigenvalues{−βZ, βZ}, i.e., juki ¼ ½jψ1i; jψ2i�ρ; ρ ¼ ½b1; b2�T . Thecolumn vector ρ describes the coefficients of the eigen-states in the Bloch wave. Thus, the Schrödinger-typeequation (Eq. 1) and the centre-of-energy velocity of theBloch wave (Eq. 5) can be rewritten as (for detailedderivation, see Supplementary information Note 3)

iddZ

ρ ¼ Hk � ΛðηÞη′h i

ρ ð6aÞddZ

XCk � ΛðkÞD Eh i

¼ ∂kHk� �� ΩðkηÞ

D Eη′

þV PT � ddZ

ΛðkÞ ð6bÞ

where η′ ¼ dη=dZ;Hk ¼ diag½�βZ; βZ�. The Berry con-nections Λ(k) and Λ(η) defined in the synthetic 3D space

are 2 × 2 matrices with non-zero elements given by ΛðkÞ12 ¼

ihψ1j∂kψ2i and ΛðηÞ12 ¼ ihψ1j∂ηψ2i. In Eq. 6b, �h i refers to

the state average, e.g., ΛðkÞ� � ¼ ρþΛðkÞρ. The right-handside of Eq. 6b consists of four velocity terms. The firstterm is the average group velocity. The second term is theanomalous velocity1, which is proportional to the Berry

curvature ΩðkηÞ ¼ i½ΛðkÞΛðηÞ � ΛðηÞΛðkÞ� in the synthetic

space [ΩðkyηÞ11 ¼ Ω1 in Eq. 2a and ΩðkxηÞ

11 ¼ �Ω2;ΩðkxkyÞ11 ¼

Ω3 in Eq. 2b]. Close to the Weyl points in the syntheticspace studied here, this anomalous velocity is perpendi-cular to the incident plane and therefore leads to thephotonic Hall effect. The third term V PT ¼XB

0 � XA0

� �ΞðkÞh i is the second term in Eq. 5. Because

lattices A and B correspond to the two differentpseudospin states in the honeycomb system, this velocityis therefore named the pseudospin transition velocity(PTV). The fourth term corresponds to the contributionfrom the gradient of Berry connections Λ(k) in Z. Anothercontribution to the motion comes from the displacement

ΛðkÞ� �on the left-hand side of the equation. The

displacement induced by the non-Abelian Berry connec-tion (with nonvanishing off-diagonal elements) plays anessential role in generating the ZB effect, where theoptical beam features a trembling motion around its meantrajectory, which is the photonic analogue of thebehaviour of a free-electron wave packet described bythe Dirac equation35–37. Note that although the Berry

connection Λ(k), the Berry curvature Ω(kη) and the averagegroup velocity ∂kHk are individually gauge dependent,their overall contribution in Eq. 6b is gauge independent.Thus, the centre-of-energy velocity is non-Abelian gaugeinvariant. During the adiabatic evolution, off-diagonalelements of the Berry connection with nondegeneratemodes are usually negligible. An non-Abelian Berryconnection only appears in the synthetic space withdegenerate or near degenerate states, such as in thevicinity of a Weyl point.

To look into the dependence of ZB motion over thedetailed form of the band structure, we consider a rodarray with a uniform radius of rods along the propagationdirection of light, that is η′= 0. A simple expression of thecentre-of-energy position38 can obtained from Eq. 6b

XCk ¼ b�1b2 ei2βZ Z0�Zð Þ � 1� �

ΛðkÞ12 þ c:c:

þ b2j j2� b1j j2� �∂kβZð Þ Z � Z0ð Þ

ð7Þ

where ΛðkÞ12 ¼ ΛðkÞ

12 � eyaSðkÞ=ð2ffiffiffi3

pβZÞ, and c.c. denotes

the complex conjugate. Equation 7 shows that the ZBmotion (first term on the right-hand side) can occur at afrequency equal to the band gap 2βz when the incidentwave is in the superposed state. It is worth noting thatthe ZB motion is proportional to the complex vector Λ

ðkÞ12

[≈ΛðkÞ12 in the vicinity of K point because S(K)= 0], which

directly displays the trajectory of the ZB motion projectedonto the x−y plane. Figure 2 shows that the ZB motiondepends on both the momentum relative to K point andthe width of the band gap. In the case that the band gapcloses (η= 0, Fig. 2b), the ZB motion is linear, with azi-muthally oriented linear motion relative to the K point(Fig. 2f). Exactly at the K point, the ZB disappears due tothe degeneracies of the two pseudospin eigenstates.Interestingly, for non-zero η (Fig. 2a, c, d), the ZB motionis elliptical, having an increasing amplitude and a morecircular trajectory when the momentum is closer to the Kpoint (Fig. 2e, g, h). Importantly, the direction of themotion (clockwise or counter-clockwise) depends on thesign of η. Away from the Weyl point (double degenerate)when increasing the parameter η or the distance from theK point, the amplitude of the ZB motion decreasesbecause the non-Abelian Berry connection decreases withthe widened band gap.

Next, we consider a variation of the radii of rods alongthe propagation direction, with a linear dependence ƞ=0.23Z. For a Bloch wave in the lower-energy state withwave vector k= 0.95K at the incident point Z0=−3, Fig.3a shows the evolution of the two mode coefficientsduring the propagation (Eq. 6a). Close to the location ofthe Weyl point (Z= 0) where two eigenmodes are nearlydegenerate, the two coefficients experience rapid changes,and the magnitude of the non-Abelian Berry connection

Ye et al. Light: Science & Applications (2019) 8:49 Page 4 of 8

Λ(η) is at its maximum. The coefficient of the upper-energy state (b2) significantly increases and becomesgreater than that of lower-energy state (b1). When Z isgreater than 2, the Bloch wave turns into a superposedstate with the two relatively stable mode coefficients.Figure 3b shows the evolution of centre-of-energy shift inthe y direction (Hall shift) induced by the displacementhΛðkÞi, the anomalous velocity, and the combined con-tribution of hΛðkÞi, the anomalous velocity, and VPT,respectively. The combined contribution (the last one)agrees well with the exact Hall shift calculated using Eq.6b. It follows that the ZB motion and the transverse tra-jectory of the centre-of-energy of the Bloch wave aremainly determined by hΛðkÞi and the anomalous velocity,whereas VPT is relatively small. Away from the Weyl pointwhen Z is greater than 3, the anomalous velocity becomesvery small due to the nearly vanishing Berry curvature.Similar with the array having a uniform radius in Fig. 2,the motion of the Bloch wave in the real space exhibits aspiral trajectory but has a decreased diameter during thepropagation away from the Weyl point (Fig. 3c). On theother hand, a Bloch wave in the lower-energy state with awave vector k= 0.8K (which is relatively far away fromthe location of the Weyl point), stays mainly in the low-energy state with a very small upper-energy state coeffi-cient during the propagation (Fig. 3d). This is due to thenearly vanishing Berry connection and Berry curvature(Fig. 1c). The Hall shift induced by the displacementhΛðkÞi is approximately zero. Interestingly, the pseudospintransition velocity VPT plays a leading role in the Hall shift

(Fig. 3e). A small wiggling feature at a large Z is a result ofthe contribution by VPT and is shown in Fig. 3f.Furthermore, we investigate the behaviour of the output

waves when a Bloch wave is incident onto the rod arraywith varying in-plane wave vectors in the x direction. Herethe y-direction centre-of-energy shift corresponds to theHall shift. The Z coordinate of the input and outputinterfaces are fixed at −3 and 12, respectively, whichguarantees that the Bloch wave will pass through thelocation of the Weyl point (Z= 0) during its propagation.As the Hall shift depends on the initial states, we studiedthe momentum dependence for two different initial states:the lower-energy state and the pseudospin Sz= 0 state(equal mixing between sub-lattices A and B). For anincident Bloch wave in the lower-energy state, only whenthe wave vector is in the vicinity of the K point, it can beconverted into a combination of states at the output facet(Fig. 4a) due to the effect of the non-Abelian Berry con-nection. However, when the incident Bloch wave withwave vector exactly matching the K point, its energy isonly located at the sub-lattice B during the propagationdue to the conservation of the z component of thepseudospins. As a result, x- and y-directional shifts of itscentre of energy are zero (Fig. 4b, c). In the vicinity of theK point, the Hall shift (Fig. 4c) displays a Fano-like lineshape. On the other hand, for an incident Bloch wave inthe pseudospin Sz= 0 state, the mode coefficient and thelateral shifts of the beam exhibit quite different line shapesfrom that of the lower-energy initial state (Fig. 4a–c) inthe vicinity of the K point, as shown in Fig. 4d–f. A clear

–1.5

0.2

0.1

0

–0.1

–0.21.2 1.1 1 0.9 0.8

0.1

ky (4�/3a)

k y (

4�/3

a)

kx (–4�/3a)1.2 1.1 1 0.9 0.8

kx (–4�/3a)1.2 1.1 1 0.9 0.8

kx (–4�/3a)1.2 1.1 1 0.9 0.8

kx (–4�/3a)

0.2

0.1

0

–0.1

–0.2

k y (

4�/3

a)

0.2

0.1

0

–0.1

–0.2

k y (

4�/3

a)

0.2

0.1

0

–0.1

–0.2

k y (

4�/3

a)

kx (–4�/3a)

0 –0.1

� = –0.3

� = –0.3 � = 0 � = 0.3 � = 0.6

� = 0 � = 0.3 � = 0.6

1.1 1 0.9 0.1

ky (4�/3a) kx (–4�/3a)

0 –0.1 1.1 1 0.9 0.1

ky (4�/3a) kx (–4�/3a)

0 –0.1 1.1 1 0.9 0.1

ky (4�/3a) kx (–4�/3a)

0 –0.1 1.1 1 0.9

–1

–0.5

� z 0

0.5

1

1.5a b c d

e f g h

–1.5

–1

–0.5

� z 0

0.5

1

1.5

–1.5

–1

–0.5

� z 0

0.5

1

1.5

–1.5

–1

–0.5

� z 0

0.5

1

1.5

Fig. 2 Band structure and distribution of the trajectories of the ZB motions projected to the x−y plane in the momentum space (kx, ky)near the K point. a–d Band structures with parameter η equal to −0.3, 0, 0.3 and 0.6, respectively. e–h Corresponding distributions of the trajectoriesof the ZB motions. The ellipse (circle) denoted by +(−) means the trajectory of the ZB is a counter-clockwise (clockwise) ellipse (circle) along thepropagation direction. The geometric sizes of the line, ellipses and circles represent the relative amplitudes of the ZB with different [kx, ky, η]

Ye et al. Light: Science & Applications (2019) 8:49 Page 5 of 8

enhancement of the Hall shift at the K point is evident inFig. 4f. Since the Hall shift in our system originates fromthe pseudospin−orbit interactions (Eq. 1), differentpseudospins lead to different wave motion.

DiscussionTo confirm the above results, we simulate the propa-

gation of Gaussian beam in the rod array (for details, seeSupplementary information Note 4). A hexagonal array(Fig. 1a) with 50 rods arranged on each side (14,554 rodsin total) is used in the calculation. An incident Gaussianbeam with a centre of energy X0, central wave vector kC,and beam waist radius W is selected to be in the lower-energy state of the Hamiltonian H(kC, η(Z0)) at the inci-dent position. We calculate the centre-of-energy shiftsand 3D trajectories of two Gaussian beams with para-meters (X0, kC, W) equal to (0, 0.8K, 50/|K|) and (0,0.95K, 50/|K|), and find that both trajectories agree wellwith those of the Bloch wave obtained by solving Eqs. 6aand 6b, as detailed in the Supplementary information.Thus, the propagation of Gaussian beams with relativelylarge waist radii are well described by Bloch waves.

In conclusion, we have proposed and demonstrated thata coupled rod array arranged in a honeycomb lattice withvariant radii possesses photonic Weyl points in a 3Dsynthetic space, which become Dirac points when pro-jected down to a 2D momentum space. The advantage ofour system is obvious for probing the Berry phase effectsthat arise fromWeyl points. Due to the presence of a localBerry curvature in the synthetic space, the wave packetsexperience an anomalous velocity in the vicinity of Weylpoints, which leads to an enhanced Hall effect. Further-more, because the eigenstates of the system at the Weylpoint are double-degenerate, a non-Abelian Berry con-nection appears in the vicinity of the Weyl point, whichleads to a helical ZB effect. Therefore, our system providesa powerful platform for studying the effects of the Berryphase on the dynamics of photonic wave packets.

Materials and methodsWe discuss the practical realization of the rod array

necessary to reproduce the above findings. Similar to thereported photonic graphene analogues fabricated viafemtosecond direct laser writing in fused silica17,39, we

1.0a

d

b

e

c

f

2.0

1.5

1.0

0.5

0.0

–0.5

y c(a

)

x c(a

)x c

(a)

yc(a)

yc(a)

y c(a

)

–1.0

–1.5

0.6

0.4

0.2

0.0

0.8

0.6

0.4

0.2

0.0

–3 0 3 6

abs(b1)

Exact solution

<Λ(k)>+Anomalous velocity+VPT

<Λ(k)>

Anomalous velocity

Exact solution

<Λ(k)>+Anomalous velocity+VPT

<Λ(k)>

Anomalous velocity

abs(b2)

abs(b1)

abs(b2)

Z

k = 0.95 K

k = 0.8 Kk = 0.8 K

k = 0.95 K k = 0.95 K

k = 0.8 K

Z9 12 –3 0 3 6 9 12

–3 0 3 6

Z Z9 12 –3 0 3 6 9 12

1.0

0.8

0.6

0.4

0.2

0.0

–1.5

12

8

4

0

Z

12

8

4

0

Z

–1.0–0.5

0.00 1 2

–10–8–6–4–2

00.0 0.2 0.4 0.6

Fig. 3 Evolution of the Bloch wave propagating in the array of rods with varying radii along the propagation direction with ƞ= 0.23 Z. Forthe incident Bloch wave with a wave vector k equal to 0.95K in the lower-energy state, a shows the calculated two mode coefficients by solving Eq.6a. b shows the exact centre-of-energy transverse shift (Hall shift, denoted by circles) obtained by solving Eq. 6b, and the calculated centre-of-energy

transverse shifts induced by the combined contribution of hΛðkÞi and anomalous velocity and VPT (denoted by upper triangles), and only the

contribution of the anomalous velocity (denoted by squares), and only the contribution of the displacement hΛðkÞi (denoted by lower triangles),respectively. c Spatial trajectory of the centre of the energy of the wave packet. d–f Corresponding results of the incident Bloch wave with the wavevector k equal to 0.8 K

Ye et al. Light: Science & Applications (2019) 8:49 Page 6 of 8

select the refractive index of the rods Δn= 2 × 10−4

greater than the silica background. At a wavelength of488 nm, the radii R of rods in sub-lattice A (B) linearlyincrease (decrease) from 3.5 μm (4.5 μm) to 6 μm (2 μm)in z direction at the very small ratio ΔR/Δ Z= 0.5 × 10−4,which corresponds to (βA− βB)/2 increasing from −2 to7 cm−1. Thus, the total length of the silica rods array isapproximately 5 cm. Furthermore, with the lattice con-stant a equal to 21 μm, the averaged effective couplingcoefficient κ is approximately 2.9 cm−1 (for details, seeSupplementary information Note 5). It is worth notingthat the incident Bloch wave with the pseudospin Sz= 0 isequivalent to an incident Gaussian wave. So, by graduallyvarying the incident angle of the Gaussian beam, we canmeasure the relative displacements between the centres ofthe input and output beams using a quadrant detector8.Furthermore, the helical ZB can be observed by measur-ing the centre-of-energy displacement of the output beamthrough rod arrays of different lengths.

AcknowledgementsThe work was supported by ERC Consolidator Grant (TOPOLOGICAL) and theNational Science Foundation of China (11374367, 11574079). S.A.R.H. is gratefulfor funding provided by the Royal Society and TATA (RPG-2016-186).

Author details1College of Advanced Interdisciplinary Studies, National University of DefenseTechnology, 410073 Changsha, China. 2Key Laboratory for Micro/NanoOptoelectronic Devices of Ministry of Education, School of Physics andElectronics, Hunan University, 410082 Changsha, China. 3School of Physics &Astronomy, University of Birmingham, Birmingham B15 2TT, UK. 4Departmentof Physics, Harbin Institute of Technology, 150001 Harbin, China. 5Departmentof Physics and Astronomy, University of Exeter, Exeter EX4 4QL, UK

Conflict of interestThe authors declare that they have no conflict of interest.

Supplementary information is available for this paper at https://doi.org/10.1038/s41377-019-0160-z.

Received: 9 December 2018 Revised: 9 April 2019 Accepted: 6 May 2019

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