Research Article Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B 2953

Observation of surface mode arcs associated withnodal surfaces in electromagnetic metacrystalsMingli Chang,1 Meng Xiao,2 Jianwen Dong,3 AND C. T. Chan1,*1Department of Physics and Institute for Advanced Study, TheHong KongUniversity of Science and Technology, Hong Kong, China2Key Laboratory of Artificial Micro- andNano-Structures ofMinistry of Education and School of Physics and Technology,WuhanUniversity,Wuhan, China3School of Physics and State Key Laboratory of OptoelectronicMaterials and Technologies, Sun Yat-SenUniversity, Guangzhou 510275, China*Corresponding author: phchan@ust.hk

Received 14 April 2021; revised 27 July 2021; accepted 27 August 2021; posted 30 August 2021 (Doc. ID 427904);published 20 September 2021

In this article, we designed, fabricated, and characterized an electromagnetic metacrystal that topologically carriesnontrivial nodal surface degeneracies. Compared with nodal surfaces observed in an acoustic system, the topologi-cal charge of the nodal surface in our system is compensated by charge-2 Weyl points, and we designed our systemconsidering the rules of symmetry. To demonstrate the existence of the nodal surfaces and their topological proper-ties, we have experimentally observed surface state arcs derived from helicoid sheets of surface states connecting thenodal surface with a charge-2 Weyl point. The surface states support the robust unidirectional transport on the sur-face, and the nodal surface provides more degrees of freedom to engineer the dispersion of surface states. Our sys-tem offers a platform to explore this new class of gapless topological electromagnetic wave systems. © 2021 Optical

Society of America

https://doi.org/10.1364/JOSAB.427904

1. INTRODUCTION

The discovery of topological phases in classical wave systems hasattracted much attention recently as such systems offer a flexibleplatform for realizing various ideas in topological physics. Inparticular, experimental realization of classical wave topologicalsystems and their characterization are typically much easierto achieve than their counterparts in condensed matter phys-ics. Recently, the classical analogues of quantum Hall effect[1], quantum spin Hall effect [2,3], and symmetry-protectedtopological phase [4] have been realized in acoustic [5–7] andphotonic systems [8–13]. The common signatures of topo-logical materials are surface states that exist in a topologicallynontrivial band gap, and these states are protected and guaran-teed to exist by topological invariants that characterize the bulktopological properties [14]. In some cases, these surface statescan propagate in one direction due to the absence of a backwardchannel, and the transport is robust against defects or disorder.Such robustness offers promising application potentials for theone-way transmission of signals and energy.

Topological semimetals possess different kinds of banddegeneracies carrying topological charges. The most commonband degeneracy in topological semimetals is the Weyl point[15–23], where two bands form a conical dispersion along allthree momentum directions. A Weyl point serves as a monopoleof Berry flux in the momentum space, and they appear in pairswith opposite charges in a periodic system [24]. If a crystal

with Weyl points is truncated, the exposed surface will carrygapless surface states that form arcs in momentum space con-necting the Weyl points with opposite charges [19–21,25]. Inaddition to the Weyl point as a zero-dimensional (0D) degen-eracy point possessing a topological charge, nodal lines withone-dimensional (1D) degeneracy and the drumhead surfacestates associated with nodal rings have also been discoveredin metacrystals [26–28]. Similar phenomena have been alsoobserved in acoustics [29]. Nodal surfaces on which the eigen-states are degenerate on a two-dimensional (2D) surface canalso exist in high-symmetry 2D planes, which can also possesstopological charges [30–37]. Gapless surface states and Fermiarc features can also be observed in these systems. As the topo-logically charged nodal surface is a momentum space degeneracythat spans a large area, the dispersion of the gapless surface statethat connects to it can be tuned over a wide range, which is muchmore flexible than the surface state arc that connects the twoWeyl nodal points.

In this work, we show the first experimental realization of ametacrystal possessing a nodal surface at the Brillouin zone (BZ)boundary with a topological charge 2 at microwave frequencies.Compared with the counterpart in the acoustic wave systems,this metacrystal is designed based on a simple symmetry require-ment that is more general than a tight binding description thatconsiders near-neighbor hopping. The gapless surface states areobserved directly, and the fact that the system carries two surfacemode arcs demonstrates that the charge of the nodal surface is 2.

0740-3224/21/102953-07 Journal © 2021 Optical Society of America

2954 Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B Research Article

The helicoid property of the surface states is also characterized,and such gapless surface states support the one-way transportsurface modes with a specific kz in our system without breakingthe time-reversal symmetry.

2. NODAL SURFACE DEGENERACYPROTECTED BY SYMMETRY

A. Metacrystal with Symmetry-Protected NodalSurface Band Degeneracy

To realize topological nodal surface band degeneracies inelectromagnetic systems, we employ the rule of symmetryto ensure its existence. Figure 1(a) shows the unit cell of themetacrystal projected onto the x − y plane. A unit cell is com-posed of four layers of Y-shaped structures with a width ofw= 0.16a and length d = 0.47a , where a is the lattice constantin the x − y plane. A vertical cylinder with a radius r = 0.18apierces through every layer, and the metacrystals are formedby arranging the unit 2 cell in a hexagonal lattice. The pitchalong the z-axis is h . Here, the distance between the first andthe second, and the third and the fourth layer is h1, while thedistance between the second and the third, and the fourth andthe first layer is h2, and 2(h1+h2)= h . The Y-structures areorientated relative to each other so that the system possessestwo-fold screw symmetry, and the system belongs to the spacegroup P63.

Such a metacrystal can be fabricated at microwave frequen-cies using standard printed circuit board (PCB) technology byetching out the Y-shape slots on metallized PCB. The periodicpatterns of the first and the second layer are formed by etchingout Y-shape slots on each side of the copper surface of the PCBwith the permittivity of ε1 = 4.1 and thickness of h1 = 1.5 mm,

as shown in Fig. S1(a). Similarly, the third and the fourth layerare printed on another PCB with same size and material [seeFig. S1(b)]. These PCBs carrying periodic Y-patterns in thex–y plane are then stacked along the z direction to form a three-dimensional (3D) metacrystal. We insert a dielectric layer asspacer between two layers of PCBs. The spacer has a permittivityof ε2 = 2.2 and thickness of h2 = 3 mm. The overall pitchof unit cell is h = 9 mm in the z direction. By stacking thesePCBs in the z direction and arranging an array of vertical metalrods through the holes [the black circle in Fig. 1(a)], we canbuild a sturdy sample. A 3D model with the unit cell periodi-cally repeated in the x − y direction is shown in Fig. 1(b). Tocharacterize the topological properties of the nodal surface, wecalculate the band structure of the system. The reciprocal spaceof the hexagonal lattice and the high symmetry lines and pointsof the BZ are shown in Fig. 1(c). By taking a specific kz, we candefine a 2D subsystem with a hexagonal 2D BZ as shown inFig. 1(c). The band structure of the metacrystal in Fig. 1(d) indi-cates the bands at the plane of kz = π/h are double degeneratealong the high symmetry lines of the BZ.

To show the band degeneracy at kz = π/h , we focus on thedegeneracy of the fifth and the sixth band (at around 13.2 GHz)and numerically calculate its 3D band structure. The 3D bandstructure at kz = π/h in Fig. 2(a) shows that these two bands(red and blue bands) have the same frequency throughout thewhole BZ, indicating that these two bands form a nodal surfaceat kz = π/h . When kz deviates from π/h , the nodal surfacedegeneracy of these two bands will be lifted. As an example, weconsider kz = 0.75π/h in Fig. 2(b), and the red and blue bandsare completely gapped throughout the whole 2D subsystemBZ, which is consistent with the results shown in Fig. 1(d). Inaddition to the nodal surface at kz = π/h , these two bands meet

Fig. 1. A metacrystal possesses the C2z symmetry and has four layers with the unit cell marked by dotted lines, as shown in (a), and the geometricparameters of the Y-shaped slots and the circle rod are labeled. A 3D model of the sample is shown in (b), and the gray color represents the metallic partof the sample. The reciprocal space of the hexagonal lattice is shown in (c), and if a specified kz is chosen, a 2D subsystem BZ can be obtained. Theband structure of this system is shown in (d), and we can find the nodal surface band degeneracy at kz = π/h .

Research Article Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B 2955

to form a Weyl point at 0. This Weyl point exhibits quadraticdispersion along kx and ky , while it has linear dispersion alongkz, indicating that it is a double Weyl point carrying a topo-logical charge of 2 [38]. As the net topological charge of a staticperiodic system should be zero, the nodal surface on the planekz = π/h should also possess topological charge of 2, but itcarries an opposite sign.

We show the two-fold screw symmetry C2z protects thenodal surface degeneracy. Combining the C2z symmetry(C2z : (x , y , z)→ (−x ,−y , z+ h/2)) with time-reversalsymmetry T, we obtain the operator G2z = TC2z, which hasthe symmetry property:

G2z H(kx , ky , kz)G−12z = H(kx , ky ,−kz). (1)

Therefore, the eigenvalue of G2z is (G2z)2= e−ikzh for the

Bloch states, where kx , ky , and kz are the Bloch wave vectorsalong the x , y , and z directions. At kz = π/h , the eigenvalueof G2z is (G2z)

2=−1, which is similar to the time-reversal

symmetry in the fermionic system. Hence, this symmetry canprotect the double degeneracy similar to a “Kramer’s pair” whenkz = π/h , enforcing the bands form nodal surfaces [31]. Oncethe two-fold screw or the time-reversal symmetry is broken,the degeneracy on the plane will either be gapped or becomean isolated nodal point in the band structure (Fig. S1) (SeeSupplement 1).

B. Topological Properties of the Charged NodalSurface

To further illustrate the topological properties of the nodalsurface, we calculate the Chern number of these two bands ofthe 2D systems with specific values kz. While the 3D system istime-reversal symmetric, the bulk bands in 2D subsystems witha non-zero kz have well-defined Chern numbers in the reducedBZ. The Chern number of red (upper) band is −1 for kz > 0(while it is+1 for the kz < 0 subsystems), and the blue band car-ries a Chern number of an opposite sign, as shown in Fig. 2(c).As the kz changes from negative to positive, the Chern numberof the 2D subsystems changes by−2, consistent with the nodalpoint at kz=0 is a charge-2 double Weyl point. Similarly, theChern number of the red band changes by 2 when it passesthrough kz = π/h , which indicates the nodal surface shouldcarry a topological charge 2. The sign change of the Chernnumber indicates the charges of the Weyl point and nodal sur-face are−2 and 2, respectively, because they act as the sink andsource of the Berry flux for the upper band in the momentumspace. When the red band passes through kz = 0, the Chernnumber changes to −2 as the Weyl point offers a sink of Berryflux as −2, and similarly, the nodal surface offers a source ofBerry flux as 2 when the band passes through kz = π/h . Thetopological charges distributions in the momentum space areshown in Fig. 2(d), and the Berry curvature for the upper band isrepresented by the arrows.

Fig. 2. (a) and (b) show the 3D band structure of the fifth and the sixth band in Fig. 1(d) at kz = π/h and kz = 0.75π/h , respectively. The bandsdegenerate at kz = π/h , and they become gapped when kz deviates from π/h . The Chern numbers of these two bands calculated for 2D subsystemswith different kz are presented in (c), and topological charge distributions are presented in (d), showing the Weyl point acts as a sink of the Berry fluxfor the upper band with a topological point charge−2, and the nodal surface acts as a source of Berry flux with a topological surface charge 2. Thearrows represent the Berry flux in the vicinity of the Weyl point and the nodal surface.

2956 Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B Research Article

Fig. 3. Equi-frequency contours of the surface states for a few frequencies are calculated for a semi-finite strip, as shown in (a)–(c). The gray shadedareas mark the bulk modes of the Weyl point and the nodal surface, and the cyan lines describe the surface states. The surface states connect a Weylpoint to the nodal surface and “rotate” around the Weyl point in a counter-clockwise manner, exhibiting the helicoidal character of these gapless sur-face states.

Meanwhile, we can characterize the topological charges of theWeyl point and nodal surface by counting the number of surfacemode arcs. It is well established that in Weyl point systems,gapless surface states exist on the truncated surfaces, and theyconnect two Weyl points with opposite charges. The fields of thesurface states are localized on one side of the truncated surfaceand have the robust transport characteristics of topologicalsurface modes. When we observe the equi-frequency contoursexperimentally, the surface states usually form an arc connect-ing the bulk modes on the Weyl point cones, and the surfacemode arcs are usually referred as Fermi arcs in condensed mat-ter systems. To observe how this happens in our nodal surfacesystem, we truncated the metacrystal along the y direction [seeFig. 1(b)], and the system remains periodic along the x and zdirections. The truncated surfaces are capped by a perfect metalconductor (PEC) to prevent the surface states from leaking out-side. In Figs. 3(a)–3(c), the equi-frequency contours calculatedfor a few frequencies are shown. The gray shaded areas representthe projected bulk states where the Weyl point locates at the zonecenter, while the projected bulk states near the zone boundarycome from the nodal surface. The two cyan lines are the surfacestates on the truncated surface that connect the charge-2 Weylpoint and charged nodal surface, forming the surface mode arcs.Two surface mode arcs in the equi-frequency contours indicatethat the topological charges of the Weyl point and the nodalsurface are both 2. As the frequency increases, the surface modearcs rotate around the Weyl point in a counterclockwise man-ner. Considering the change of frequencies, these arcs shouldcollectively constitute a helicoid sheet of surface states in the 3Dmomentum space, which is similar to the case of Fermi arcs thatconnect isolated Weyl points [19].

3. EXPERIMENTAL OBSERVATION OF THESURFACE ARCS AND GAPLESS SURFACESTATES

To experimentally realize the nodal surface and to characterizethe associated surface modes, we fabricated the metacrystal sam-ple operating at microwave frequencies using PCBs according tothe blueprint shown in Fig. 1(b). A photo of the real sample canbe seen in Fig. S2(c). The size of sample is 328× 87× 405 mm,

and the experimental setup is shown in Fig. S2(c). An sourceantenna is placed at the center of the upper edge, alignedalong the z direction and another antenna serving as the detec-tor was programmed to scan the x − z plane of the samplewith steps 1x = 4 mm, 1z= 4.5 mm (See Supplement 1).The propagating field distributions [Fig. S2(d)] show that thesurface mode is propagating along the +x direction (with afinite kz), manifesting surface transport along one preferreddirection, although the source antenna itself is symmetricallyplaced and is not biased between the ±x directions. To obtainthe equi-frequency contours of the system, we calculate theFourier transform of the field pattern. We put the source atthe center of four different upper edges, and measure the fieldpatterns four times. By combining these Fourier transformstogether, we can obtain the final result of the equi-frequencypattern.

The equi-frequency contours at a few frequencies are shownin Figs. 4(a)–4(d). In the experiment, the PEC boundary wasnot applied on the truncated surfaces, and so the light conecan be observed. The blue lines plotted in the figures representthe numerically calculated (using COMSOL) surface modesof a truncated metacrystal exposing the x − z plane with anopen boundary condition. The white ellipse here marks theboundary of the light cone, which engulfs the states of the Weylcone. The shaded areas near the zone boundary at kz = π/hrepresent the projection of the nodal surface modes. The bluelines are the gapless surface states, and they connect the Weylcone states (inside the light cone) and the nodal surface, wherethe bulk topological charges are −2 and +2 respectively forthe upper (6th) band. The background colors of the figuresindicate the Fourier transform of the scanned field obtainedfrom the experiment (See Supplement 1). Here, we can observethe surface state arcs that manifest as the brighter (yellow) color,and the dispersions of the arcs agree quite well with our simu-lation results. In particular, the surface states on one truncatedsurface are chiral as they support transport in one direction. Thehelicoid property of the surface states can also be observed asfrequency increases here. Because we can observe two surface-state arcs as shown by the brighter color, the fact that both theWeyl point and the nodal surface possess a topological charge2 is thus experimentally confirmed. We note that the wave

Research Article Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B 2957

Fig. 4. Comparison of numerical simulation and experimental results of the equal-frequency contour are shown in (a)–(d) at various frequencies.The white ellipse encloses the light cone. The gray shaded areas and the blue lines are the projected bulk bands and surface arc states, respectively. Thebackground color indicates the Fourier transform of the scanned field, and the bright colors indicate the surface states.

transport on the opposite direction can be seen here, although itis much weaker [see the measured real space scan in Fig. S2(d)].This is because the Chern number relies on a well-defined andconserved kz, which is not possible with a finite-sized samplealong the z direction.

In addition to equi-frequency contours, we can also examinethe band structure along some specific k directions. To observethe gapless surface states, we choose the wave vector k alongthe (101) direction. Figure 5(a) shows the band structure alongthe (101) direction. The gray areas indicate the projected bulk

Fig. 5. (a) Numerical and experimental results of band dispersions along the kx = kz line. Here, the white line is the light line, the gray shadedareas are the calculated bulk modes, and the cyan line is the gapless surface state. The background color is obtained from the experiment scanned field.(b) The eigenfields of the surface state at |k| = 0.79π/a marked by the green star in (a) are localized on the boundary between the sample and air. Thesub-figure zooms in the interface region between the sample and air, showing that fields are localized on the upper interface.

2958 Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B Research Article

bands obtained from our simulation, the white line marks thelight line, and the cyan line represents the calculated gaplesssurface state. Figure 5(b) shows the eigen field distributions of|E| at k=

√k2

x + k2z = 0.79π/a as marked by the green star in

Fig. 5(a). The eigen field pattern shows the fields are localized onthe upper boundary between the sample and air and the energytransports along the+x direction, which characterizes the chiralone-way surface states of the system. The background coloris the Fourier transform of the fields measured in the experi-ment at different frequencies stacked along the frequency axis.The bright (yellow) region indicates the surface mode, and theagreement between measured result and the simulation is quitegood.

4. CONCLUSION

In conclusion, we have designed and fabricated, to the best ofour knowledge for the first time, an electromagnetic metacrys-tal exhibiting momentum space topologically charged nodalsurfaces at microwave frequencies, and the nodal surfaces areprotected by a two-fold screw and time-reversal symmetry. Thenodal surfaces have charge 2. Compared with the nodal surfacein acoustic systems, we can fabricate the sample using the PCBtechnique based on a simple symmetry rule, and the gaplesssurface states are identified experimentally. The result of theexperiment clearly shows two surface mode arcs that connect thenodal surface to a charged-2 Weyl point and the equi-frequencycontour of the surface modes exhibits a helicoid character. Suchsurface modes support the unidirectional transport for the EMwaves at specific kz, and they are robust against disorders anddefects. The one-way transport property of the surface modesoffers us a flexible platform to realize the one-way waveguides,optical isolators, and the beam splitter for EM waves.

Funding. Hong Kong University of Science and Technology(N_HKUST608/17); Croucher Foundation (CAS20SC01); ResearchGrants Council, University Grants Committee (16304717).

Acknowledgment. We acknowledge the help of Dr. Qinghua Guo, Dr.Xiaodong Chen, and Dr. Fulong Shi in the experimental measurements.

Disclosures. The authors declare no conflicts of interest.

Data Availability. Data underlying the results presented in this paper arenot publicly available at this time but may be obtained from the authors uponreasonable request.

Supplemental document. See Supplement 1 for supporting content.

REFERENCES1. K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-

accuracy determination of the fine-structure constant based onquantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).

2. C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,”Phys. Rev. Lett. 95, 226801 (2005).

3. B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Halleffect and topological phase transition in HgTe quantum wells,”Science 314, 1757–1761 (2006).

4. X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protectedtopological orders and the group cohomology of their symmetrygroup,” Phys. Rev. B 87, 155114 (2013).

5. X. Wen, C. Qiu, Y. Qi, L. Ye, M. Ke, F. Zhang, and Z. Liu, “AcousticLandau quantization and quantum-Hall-like edge states,” Nat. Phys.15, 352–356 (2019).

6. C. He, X. Ni, H. Ge, X.-C. Sun, Y.-B. Chen, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “Acoustic topological insulator and robust one-way soundtransport,” Nat. Phys. 12, 1124–1129 (2016).

7. J. Mei, Z. Chen, and Y. Wu, “Pseudo-time-reversal symmetry andtopological edge states in two-dimensional acoustic crystals,” Sci.Rep. 6, 32752 (2016).

8. F. D. M. Haldane and S. Raghu, “Possible realization of directionaloptical waveguides in photonic crystals with broken time-reversalsymmetry,” Phys. Rev. Lett. 100, 013904 (2008).

9. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic,“Observation of unidirectional backscattering-immune topologicalelectromagnetic states,” Nature 461, 772–775 (2009).

10. A. B. Khanikaev, S. H. Mousavi, W.-K. Tse, M. Kargarian, A. H.MacDonald, and G. Shvets, “Photonic topological insulators,” Nat.Mater. 12, 233–239 (2013).

11. W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, andC. T. Chan, “Experimental realization of photonic topological insulatorin a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).

12. L.-H. Wu and X. Hu, “Scheme for achieving a topological photoniccrystal by using dielectric material,” Phys. Rev. Lett. 114, 223901(2015).

13. L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,”Nat. Photonics 8, 821–829 (2014).

14. Y. Hatsugai, “Edge states in the integer quantum Hall effect andthe Riemann surface of the Bloch function,” Phys. Rev. B 48,11851–11862 (1993).

15. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, “Weyl points andline nodes in gyroid photonic crystals,” Nat. Photonics 7, 294–299(2013).

16. L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M.Soljacic, “Experimental observation of Weyl points,” Science 349,622 (2015).

17. M. Xiao, W.-J. Chen, W.-Y. He, and C. T. Chan, “Synthetic gauge fluxand Weyl points in acoustic systems,” Nat. Phys. 11, 920–924 (2015).

18. W.-J. Chen, M. Xiao, and C. T. Chan, “Photonic crystals possessingmultiple Weyl points and the experimental observation of robust sur-face states,” Nat. Commun. 7, 13038 (2016).

19. B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan, W. Gao, H. Liu,Y. Xiang, J. Chen, C. Fang, A. Hibbins, L. Lu, and S. Zhang, “IdealWeyl points and helicoid surface states in artificial photonic crystalstructures,” Science 359, 1013 (2018).

20. J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen, and M.Rechtsman, “Experimental observation of optical Weyl points andFermi arc-like surface states,” Nat. Phys. 13, 611–617 (2017).

21. F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, “Weyl points and Fermi arcs ina chiral phononic crystal,” Nat. Phys. 14, 30–34 (2018).

22. H. He, C. Qiu, L. Ye, X. Cai, X. Fan, M. Ke, F. Zhang, and Z. Liu,“Topological negative refraction of surface acoustic waves in a Weylphononic crystal,” Nature 560, 61–64 (2018).

23. A. A. Burkov, “Topological semimetals,” Nat. Mater. 15, 1145–1148(2016).

24. H. Weyl, “Elektron und Gravitation. I,” Z. Phys. 56, 330–352 (1929).25. B. Yang, Q. Guo, B. Tremain, L. E. Barr, W. Gao, H. Liu, B. Béri, Y.

Xiang, D. Fan, A. P. Hibbins, and S. Zhang, “Direct observation oftopological surface-state arcs in photonic metamaterials,” Nat.Commun. 8, 97 (2017).

26. Q. Yan, R. Liu, Z. Yan, B. Liu, H. Chen, Z. Wang, and L. Lu,“Experimental discovery of nodal chains,” Nat. Phys. 14, 461–464(2018).

27. W. Gao, B. Yang, B. Tremain, H. Liu, Q. Guo, L. Xia, A. P. Hibbins,and S. Zhang, “Experimental observation of photonic nodal linedegeneracies in metacrystals,” Nat. Commun. 9, 950 (2018).

28. L. Xia, Q. Guo, B. Yang, J. Han, C.-X. Liu, W. Zhang, and S. Zhang,“Observation of hourglass nodal lines in photonics,” Phys. Rev. Lett.122, 103903 (2019).

29. W. Deng, J. Lu, F. Li, X. Huang, M. Yan, J. Ma, and Z. Liu, “Nodal ringsand drumhead surface states in phononic crystals,” Nat. Commun.10, 1769 (2019).

Research Article Vol. 38, No. 10 / October 2021 / Journal of the Optical Society of America B 2959

30. W. Wu, Y. Liu, S. Li, C. Zhong, Z.-M. Yu, X.-L. Sheng, Y. X. Zhao,and S. A. Yang, “Nodal surface semimetals: theory and materialrealization,” Phys. Rev. B 97, 115125 (2018).

31. Q.-F. Liang, J. Zhou, R. Yu, Z. Wang, and H. Weng, “Node-surfaceand node-line fermions from nonsymmorphic lattice symmetries,”Phys. Rev. B 93, 085427 (2016).

32. X. Zhang, Z.-M. Yu, Z. Zhu, W. Wu, S.-S. Wang, X.-L. Sheng, and S.A. Yang, “Nodal loop and nodal surface states in the Ti3Al family ofmaterials,” Phys. Rev. B 97, 235150 (2018).

33. C. Zhong, Y. Chen, Y. Xie, S. A. Yang, M. L. Cohen, and S. B. Zhang,“Towards three-dimensional Weyl-surface semimetals in graphenenetworks,” Nanoscale 8, 7232–7239 (2016).

34. T. Bzdušek and M. Sigrist, “Robust doubly charged nodal linesand nodal surfaces in centrosymmetric systems,” Phys. Rev. B 96,155105 (2017).

35. M. Xiao, L. Ye, C. Qiu, H. He, Z. Liu, and S. Fan, “Experimentaldemonstration of acoustic semimetal with topologically chargednodal surface,” Sci. Adv. 6, eaav2360 (2020).

36. Y. Yang, J.-P. Xia, H.-X. Sun, Y. Ge, D. Jia, S.-Q. Yuan, S. A. Yang,Y. Chong, and B. Zhang, “Observation of a topological nodal sur-face and its surface-state arcs in an artificial acoustic crystal,” Nat.Commun. 10, 5185 (2019).

37. M. Kim, D. Lee, D. Lee, and J. Rho, “Topologically nontrivial photonicnodal surface in a photonic metamaterial,” Phys. Rev. B 99, 235423(2019).

38. C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, “Multi-Weyl topo-logical semimetals stabilized by point group symmetry,” Phys. Rev.Lett. 108, 266802 (2012).