Evolution of topological edge modes from honeycomb photonic crystals to triangular-lattice photonic crystals
aa r X i v : . [ phy s i c s . op ti c s ] D ec Evolution of topological edge modes from honeycomb photonic crystals totriangular-lattice photonic crystals
Jin-Kyu Yang,
1, 2, ∗ Yongsop Hwang, and Sang Soon Oh † Department of Optical Engineering, Kongju National University, Cheonan, 31080, South Korea. Institute of Application and Fusion for Light, Kongju National University, Cheonan, 31080, South Korea. School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom. (Dated: January 1, 2021)The presence of topological edge modes at the interface of two perturbed honeycomb photoniccrystals with C symmetry is often attributed to the different signs of Berry curvature at the K andK ′ valleys. In contrast to the electronic counterpart, the Chern number defined in photonic valleyHall effect is not a quantized quantity but can be tuned to finite values including zero simply bychanging geometrical perturbations. Here, we argue that the edge modes in photonic valley Halleffect can exist even when Berry curvature vanishes. We numerically demonstrate the presence ofthe zero-Berry-curvature edge modes in triangular lattice photonic crystal slab structures in which C symmetry is maintained but inversion symmetry is broken. We investigate the evolution of theBerry curvature from the honeycomb-lattice photonic crystal slab to the triangular-lattice photoniccrystal slab and show that the triangular-lattice photonic crystals still support edge modes in a verywide photonic bandgap. Additionally, we find that the edge modes with zero Berry curvature canpropagate with extremely low bending loss. I. INTRODUCTION
Topological insulators, which are insulating in the bulkpart while conductive along the edge, have been inten-sively studied due to their intriguing physical proper-ties as well as potential applications [1]. As an opticalcounterpart of the topological insulators in condensedmatter physics, photonic topological insulators (PTIs)have been proposed and demonstrated in optical sys-tems [2, 3]. PTIs show unique characteristics, for exam-ple, the guided modes along the edge or interface of PTIswhich are robust against defects and deformations due totopological protection [4–7]. Such robustness has beendemonstrated at telecommunication wavelengths [8].Recently, optical quantum spin-Hall effect (QSHE) andoptical quantum valley-Hall effect (QVHE) have been re-alized by introducing geometrical perturbation in PTIssuch as a honeycomb (HC) photonic crystal (PhC) struc-ture [6, 7, 9]. For the QSHE PTIs [9], a photonic bandgap(PBG) is created at Dirac point by making perturba-tions (extend, shrink) in a way that the C symmetry ismaintained. As a result, the pseudo-time-reversal sym-metry is protected and pseudo-spin channels are main-tained (pseudo-time-reversal operator T is defined to sat-isfy T = − ∗ [email protected] † OhS2@cardiff.ac.uk tices [12], the origin of the edge modes in optical QVHEis not clearly understood because they are not topologi-cally protected as in the Chern insulators (Chern numberis zero for optical QVHE). One way of explaining the ex-istence of edge modes in optical QVHE is the bulk-edgecorrespondence in an extended parameter space ( k , a g )where k is the wavevector and a g is the perturbationstrength [13]. Another most common explanation is thevalley degree of freedom which originates from the differ-ent signs of Berry curvature for different valleys. How-ever, it is not clear whether the different signs of Berrycurvature for different valleys is a necessary condition forthe existence of the edge modes and reflection-less prop-agation at the bending.In this paper, we report that edge modes can be cre-ated even when the Berry curvature vanishes. We demon-strate this by studying the Berry curvature of photonicbands for PhC slabs with the staggered HC lattices in-cluding the HC lattice and the triangular lattice. Weshow the triangular-PhC (Tri-PhC) is an extreme case ofthe honeycomb PhCs (HC-PhCs) which holds their topo-logical characteristics because the reduction of one holein the unit cell to null leads to the Tri-PhCs. This ar-gument is supported by the calculation of photonic bandstructures and Chern numbers and the simulation of one-way propagation. II. PHOTONIC BAND STRUCTURE ANALYSIS
First, we perform photonic band structure analysis tounderstand the characteristics of the guided modes inthe staggered HC-PhC slab structure composed of twoair holes in its unit cell. It is worth noting that thestructure becomes a HC-PhC slab (the inset of Fig. 1(a))if the radii of the two holes are identical and it becomesa Tri-PhC slab (the inset of Fig. 1(b)) if one of the air
FIG. 1: Characteristics of the TE photonic band structures.(a) Band diagram of the air-hole HC-PhC slab structure with r a = r b = 0 . a (b) Band diagram of the air-hole Tri-PhCslab structure with r a = 0 . a ( r b = 0) (c) Magnetic fieldprofiles of the normal component ( H z ) at the center of theslab. K , K means the first and the second K-point TEmodes shown in (a, b). (d) Band diagram near the K pointat HC-PhC slab (e) Frequency of the K-point TE modes asa function of r b . (f) Band diagram near the K point at theTri-PhC slab holes is missing. Therefore, the geometrical transitionfrom the HC-PhC to the Tri-PhC can be described bythe change of the radius of the smaller air hole ( r b ) from0 . a to zero.Figure 1(a) shows the band diagram of the HC-PhCslab ( r b = 0 . a ) calculated by the three-dimensional(3-D) plane wave expansion method [14]. Here, the re-fractive index of the slab is set as 3.16, and the radius ofthe large air hole ( r a ) and the thickness of the slab ( t )are fixed as 0 . a and 0 . a , respectively. At the K pointin the band diagram of the HC-PhC, one can observe theDirac cone, where the lowest and the second-lowest TEbands meet with a linear slope. In general, the Diraccone exists when there are C symmetry and the inver-sion symmetry with respect to the mid-point of two airholes in the unit cell (HC lattice) [9].Figure 1 (b) shows the band diagram of the Tri-PhCstructure ( r b = 0). When the inversion symmetry is bro-ken (staggered HC lattice), the degeneracy at the Diraccone is lifted opening a PBG where a one-way propa-gation mode can be introduced at the interface of twoPTIs with different topological invariants such as valleyHall Chern numbers [7]. The evolution of the band di-agram by reducing the radius of one air hole is shownin Figs. 1(d)-(f). As the radius of the one air hole de-creases, the PBG at K point opens because of inversionsymmetry breaking and becomes wider until the smallerair hole is completely removed.The PBG opening also can be numerically understoodby the electromagnetic field profiles of the lowest and thesecond-lowest TE bands . When the inversion symmetryis maintained, i.e. r a = r b , the magnetic fields are local-ized equally at both holes in one unit cell (Fig. 1(c))resulting in a degeneracy as shown in Fig. 1(a). How-ever, the inversion symmetry breaking makes the mag- FIG. 2: (a) The normal magnetic field ( H z ) profiles of thefirst TE mode at the K point with the in-plane electric vectors( E k ) as the structure is gradually changed from the Tri-PhC( r b = 0) to the HC-PhC ( r b = 0 . a ) and (b) the phaseof the in-plane electric vectors. (c) The half Chern numberintegrated in the half of the first Brillouin zone as marked in(d) ∼ (e) with dashed lines. (d-e) Berry curvatures of thefirst TE band (d) r b = 0 (A) (e) r b = 0 . a (B), and (f) r b = 0 . a (C, slightly perturbed HC-PhCs). netic field distribution asymmetric with respect to themidpoint of two holes for the first and band edge modes(K and K ). This is clear because the magnetic field islocalized around larger holes for K in the first band ofstaggered HC-PhC slabs while the magnetic field is local-ized around smaller holes for K in the second band asshown Fig. S1 (see Supplementary Information for moredetails). Remarkably, the overall field distributions ofthe two band edge modes are maintained until r b = 0(Tri-PhC) which has the maximum PBG at the K pointas shown in Fig. 1(c). It is worth to mention the PBGopens even with a very small difference between r a and r b because the PBG originates from the structural inversionsymmetry breaking. III. BERRY CURVATURE AND VALLEYCHERN NUMBER
In order to verify the optical QVHE in the HC-PhCswith non-identical air holes, we investigate the evolutionof the Berry curvature of the first band and its half Chernnumber from r b = 0 to r b = r a . The Berry curvature F ( k ) = ∇ k × i h E k |∇ k E k i is numerically calculated bysumming up the phases of the electric fields E k at the fourpoints of the plaquette in the discretized k space. Thenthe valley Chern number C v is given as C v = C / , K − C / , K ′ where C / , K ( C / , K ′ ) is the half Chern numbercalculated by integrating the Berry curvature over thetriangular area around point K (K ′ ) points [12]. Here, weconsider two-dimensional (2-D) cylindrical air-hole PhCsbecause the field profile of the TE mode calculated by 2-D calculations is the same as the 3-D calculation exceptthe variation along the plane-normal direction [15]. Inconsideration of the finite thickness of the PhC slab, therefractive index of dielectric materials in 2-D calculationis set as 2.4.Figures 2 (a, b) show the normal magnetic field com-ponent ( H z ) profiles of the first TE mode at the K point,and its phase change. In Fig. 2(a), the arrows indicatethe in-plane electric vectors. At r b = 0 (Tri-PhC), themagnetic field is localized near r a , and this localizationbehavior is preserved unless r b = r a . Another chiralproperty induced by breaking the inversion symmetry arefound in the electric vectors around one of r a .When the inversion symmetry is broken, there are twovortices one at the centre of hexagons and the other atthe centre of smaller holes. In the amplitude plots, theamplitude becomes zero at the two vortices. In the phaseplots, two vortices show different signs. This relation hasbeen proved mathematically [16].Interestingly, at r b = 0 (Tri-PhC), the Berry curvatureis zero, however, two vortices with different signs stillremain, which implies that the topologically protectedmode could exist in the interface or edge within a widespectral range of PBG of the Tri-PhCs (See Figs. S2 andS3 in Supplementary Information). Recently, it was re-ported that inherently, photonic 2D Su-Schrieffer-Heeger(SSH) lattice has C v point group symmetry, and zeroBerry curvature [17, 18]. According to Ref. [17] thenon-trival topological properties with zero Berry curva-ture could be realized by the curl of the magnetic field.The Tri-PhC has also C v point group symmetry andzero Berry curvature that could support the edge statewith one-way propagation because of the chiral propertyof the magnetic field. IV. ONE-WAY PROPAGATION
Given the continuity of the evolution of the band dia-gram, it is obvious that the Tri-PhC is one extreme caseof the PTIs based on the staggered HC-PhCs. Accord-ingly, the Tri-PhCs are expected to have topologicallyprotected edge modes which are the same kind as theones in the staggered HC-PhCs. Hence, we designed astructure of a pair of the HC-PhCs with non-identical airholes, and one of them is vertically flipped and laterallyshifted to form an interface between them as shown inFig. 3. To find guided modes along the interface, we cal-culated a band diagram in Fig. 3(a) showing the existenceof edge modes. The two guided modes with various r b ’sare clearly found in the PBGs which are denoted by thegray arrows at the right side of Fig. 3(a). Figure 3(b)shows the amplitude distributions of the magnetic field ofthe edge modes with different r b ’s marked with red circlesin 3(a). Because of the wide PBG, the edge mode in theTri-PhCs ( r b = 0) is strongly localized at the interface.However, as r b increases, the PBG becomes narrower andthe localization of the edge mode in the interface becomesweaker.We also investigated the one-way propagation proper- FIG. 3: Analysis of photonic topological edge modes (a) Dis-persion properties of the edge modes at the PTIs with various r b . The gray region indicates the bulk bands. The white re-gion (or the gray arrows at the right) indicates the PBG regionof the given r b . (b) H z field profiles of the edge modes at thePTIs with various r b as marked with the red circles at (a).(c) Time-averaged electric field intensity distribution of theedge mode at λ = 1550 nm excited at the center of air holesin the straight interface as shown in the inset with the redmark. (d) Time-averaged electric field intensity distributionin the Ω -shape interface. The marks, ‘ × ’ in the insets of (c)and (d) indicate the position of the excited clock-wise chiralsource. ties of the edge mode in the Tri-PhC by the 3-D Finite-Difference Time-Domain (FDTD) method[19]. In case of r b = 0 (Tri-PhC slab), it is clear that the edge modepropagates unidirectionally along the straight interfaceas shown in Fig. 3(c). Here, the mode is excited by theclockwise (CW) chiral source generated by two dipolesources with π/ λ = 1550 nm . InFig. 3(c), the upper inset shows a schematics of a pairof Tri-PhC slab, and the lower shows the position of thechiral source with the red ‘ × ’ mark. Even in the Ω-shapeinterface with four 120 ◦ bending geometry, the edge modepropagates along the interface without reflection near thesharp corners shown in Fig. 3(d).For the quantitative analysis of chiral coupling of theguided mode along the PTI interface, we calculated one-way coupling efficiency defined as the ratio of the left(or right) propagating flux to both the left and the rightpropagating energy fluxes. Here, the CW chiral sourcewas excited at the center of air holes, and the time-averaged Poynting flux monitored at the left (or right)edge of the photonic crystals was obtained for energyflux. Figure 4(a) shows one-way coupling efficiency asa function of wavelength of the CW chiral source ex-cited at the center of the straight interface. The blueindicates the coupling efficiency of the CW chiral sourceinto the propagating mode toward the left direction. Be-cause of the wide PBG of the Tri-PhCs, high one-waycoupling efficiency over 90% is found in a broad spectralrange from 1490 nm to 1600nm as shown in Fig. S4.For the systematical analysis for the unidirectional cou-pling efficiency of the edge propagation, we defined the FIG. 4: Characteristics of the one-way propagating edgemodes. (a) One-way coupling efficiency of the CW chiralsource into the edge modes in the straight interface with r b = 0 (Tri-PhC slabs). (b) Directionality of the excitededge modes in the straight interface with various r b . (c) Thesource-position dependent directionality with various r b atthe CW chiral source. (d) Directionality of the excited edgemodes in the Ω-shape interface with various r b . directionality as ( P L − P R ) / ( P L + P R ) at the CW chiralsource [10]. Figure 4(b) shows the directionality in theHC-PhCs with non-identical air holes from r b /a = 0 to r b /a = 0 .
20 as a function of the wavelength. In case of r b = 0 (Tri-PhC slabs), the spectral range of directional-ity over 0.90 is about 60 nm. This wide spectral responseof the directionality is very robust to the r b . Especially,at r b /a = 0 .
1, the flat top directionality over 0.98 is about50 nm. The blueshift of the frequency range of high di-rectionality is owing to the blueshift of the PBG range asshown in Fig. 1(e). The pictures in Fig. 4(c) indicate thesource-position dependent directionality with various r b when the CW chiral source was excited near the interface.In the simulation, we fixed the wavelength of the excitedsource as 1500 nm, 1470 nm, and 1440 nm for r b /a = 0,0.1, and 0.15, respectively, for the the highest directional-ity at Fig. 4(b). Here, the white lines indicate the air-holeboundary. When the point-like chiral source lies insidethe upper air hole ( r a ), the unidirectional propagationoccurs along the left, however, the opposite directionalpropagation also observed when the source lies inside thelower air hole ( r ′ a ). The directional propagation occursdominantly when the source is located at the air hole,which is found at different r b . It implies that the unidi-rectionality is related with both the geometry of the largeair hole ( r a ) and the chirality of the source. The asym-metric structural unidirectionality was also reported in the glide PhC waveguide structure [20].To demonstrate the robustness of unidirectional topo-logical transport along the interface without reflection ata sharp bending, we investigated directionality in the Ω-shape interface with various r b . Figure 4(d) shows the di-rectionality as a function of wavelength of the CW chiralsource at the right side of Ω-shape interface. The spec-tral range of directionality over 0.8 is observed over 60 nmwhich is reduced in comparison with the straight inter-face. However, at the optimal wavelength the direction-ality in the Ω-shape interface is almost same as that inthe straight interface. For example, in case of r b = 0 . a ,the directionality at λ = 1470 nm in the straight interfaceand the Ω-shape interface is 0.995, 0.989, respectively.Especially, This indicates that the topological transportis robust with reflection at the sharp bending corner, atleast for the optimal condition. The sudden decrease ofthe directionality at the longer wavelength is due to thebandedge behaviors of the edge mode (see SupplementaryInformation). V. CONCLUSION
We have numerically demonstrated the evolution of thetopological behavior from the perfect HC slab to the Tri-PhC slab. When one of the two air holes in the unit cellof the HC lattice gradually decreases, the Dirac cone atthe K point disappears and the photonic bandgap opensbecause of inversion symmetry breaking. From the sys-tematic investigation of the evolution of the Berry cur-vature from the HC-PhC to the Tri-PhC, we have shownthat the topological behaviors are maintained even withzero Berry curvature at the Tri-PhC. We have numer-ically demonstrated the lossless one-way propagation ofthe edge mode in the straight and bending interfaces withvarious size of the smaller air hole. We believe that ouranalysis of the staggered HC-PhC slab will be useful fordesign of highly efficient platform of lossless photonic in-tegrated circuits.
Acknowledgments
This research was supported by the Basic Science Re-search Program through the National Research Founda-tion of Korea (NRF) funded by the Ministry of Scienceand ICT (2017R1A2B4012181, 2020R1A2C1014498).This work is part-funded by the European Regional De-velopment Fund through the Welsh Government (80762-CU145 (East)). [1] M. Z. Hasan and C. L. Kane,
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Jin-Kyu Yang , , ∗ , Yongsop Hwang , Sang Soon Oh , † Department of Optical Engineering, Kongju National University, Cheonan, 31080, South Korea. Institute of Application and Fusion for Light, Kongju National University, Cheonan, 31080, South Korea. School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom.
I. BAND ANALYSIS
We investigated the field evolution to understand the origin of the photonic topological insulator (PTI) behaviors.Figure S1 shows the normal magnetic field profiles with the in-plane electric field vectors (arrows). The field profileof the first TE mode looks different from that of the second. However, those modes are degenerate, which is causedby different propagation directionality, as shown in Fig. 1(c) (areas in the gray dashed lines). Even in a very smallstructural perturbation regime, for example when r b = 0 . a , the magnetic profile changes dramatically becausethe structural symmetry is broken. This implies that the origin of the PTI behavior at the staggered HC-PhC slabstructure is the symmetry breaking of the structure. Interestingly, the field profiles varies continuously without anyjump in the field values until the extreme case, the Tri-PhC structure ( r b = 0). This evolution behavior is alsoconfirmed by the phase of the in-plane electric vectors as shown in Fig. 2(b). FIG. S1: The normal magnetic field ( H z ) profiles with the in-plane electric vectors ( ~E k ) as the structure is gradually changedfrom the Tri-PhC ( r b = 0) to the HC-PhC ( r b = 0 . a ) (a) The first TE mode at the K point. (b) The second TE mode at theK point. We numerically investigated the effect of the interfacial gap in the HC structure with r b = 0 (Tri-PhC). First, theinterfacial gap is generated by shifting the patterns along the normal direction of the interface, oppositely. FigureS2(a) shows the photonic band structures of the topological edge modes with various gap distances, and Fig. S2(b)shows the schematic view of the shifted structure. As the gap increases, the normalized frequency of the photonicedge mode decreases due to the increase of the dielectric area near the interface. It is worth noting that if the gapincreases further, the structure becomes close to the PhC glider waveguide structure. This means that PhC glidewaveguide can be considered as a PTI structure originated from the symmetry breaking of the HC structure [S1] . FIG. S2: Analysis of photonic topological edge modes at the interfacial gap along the normal direction of the interface in theTri-PhC based topological structure. (a) Dispersion properties of the edge modes with various gap distances. (b) Schematicview of the creation of the interfacial gap. The left inset shows the Brillouin Zone and the symmetric point in reciprocal space.(c) electric energy distributions( ǫE ) and the H z field profiles at the PTIs with various gap distances. FIG. S3: Analysis of photonic topological edge modes with the diagonally-shifted interfacial gap in the Tri-PhC. (a) Dispersionproperties of the edge modes with various gap distances. (b) Schematic view of the creation of the interfacial gap. The leftinset shows the Brillouin Zone and the symmetric point in reciprocal space. (c) electric energy distributions( ǫE ) and the H z field profiles at the PTIs with various gap distances. Second, we also investigated the dispersion of the photonic topological edge mode at the interfacial gap createdby shifting the patterns along the 30 ◦ .-direction of the interface, oppositely. Figure S3(a) shows the dispersioncurves of the photonic topological edge modes with various gap distances, and Fig. S3(b) shows the schematic viewof the 30 ◦ .-directionally shifted structure. As the gap is formed asymmetrically, the mode splitting occurs due to thesymmetry breaking. This implies that topological edge mode still have structural symmetry along the normal to theinterface, which induce the Dirac-point-like band crossing of the two edge modes. However, if the interfacial symmetryis broken, no more Dirac cone exists, but the PBG opens. As the gap increases, PBG becomes wide, however, withfurther increase of the gap, PBG becomes narrow because of the decrease of the frequency of the edge mode. II. FIELD ANALYSIS
FIG. S4: Time-averaged electric field intensity distributions of the photonic topological edge modes propagating along the leftdirection in the straight interface with r b = 0. The upper is the electric intensity distributions with linear scale and the lower isthe same with logarithmic scale excited by the CW chiral source with (a) 1500 nm, (b)1560 nm, and (c) 1610 nm, respectively. In order to investigate the dispersion properties of the photonic topological edge mode qualitatively, we calculatedthe time-averaged electric field intensity distribution of the edge mode with different wavelength by 3-D FDTDsimulation. Figure S4 shows the xy-cut view of the intensity distribution when the CW chrial source is located atthe center of the upper air hole in the interface. According to Fig. 3(a) and Fig. 4(b), directionality at λ = 1500nmis almost perfect with a moderate group velocity. And, at λ = 1560nm the field intensity is slightly propagating FIG. S5: Time-averaged electric field intensity distribution of the photonic topological edge modes in the Ω-shape interfacewith r b = 0. The upper is the electric intensity distribution with linear scale and the lower is the same with logarithmic scaleat (a) 1500 nm, (b)1560 nm, and (c) 1610 nm.FIG. S6: Directionality of the excited edge mode in the straight and Ω-shape interfaces with various r b . along the right direction so the directionality is still very high about 0.9. However, at λ = 1610 nm the directionalitydecreases further, because of the band edge effect of the Tri-PhC mode. At the lower figures in Fig. S4(c), thefield intensity along the left interface is stronger than that with different wavelength, which indicates that the groupvelocity of the edge mode is smaller than other modes. And also the electric field distribution near the CW chiralsource looks larger than that with different wavelength. Especially, the evanescent field along K-direction is observedwhich means that the edge mode decays exponentially with the field profile of Tri-PhC mode at K-point.We also calculated time-averaged electric field intensity distribution of the edge modes in the Ω-shape interfaces forchecking the bending loss. Figure S5 shows the xy-cut view of the intensity distribution when the CW chrial source islocated at the right-middle of the upper air hole in the interface. At λ = 1500nm, the directionality is almost perfectwhich means no reflection observed as shown in Fig. S5(a). However, as the wavelength increases, the directionalitydecreases gradually and the reflection increases as shown in Fig. S5(b, c).In order to investigate the reflection at the bending of the interface quantitatively, the directionality of the edgemodes in the Ω-shape interfaces with various r b is compared with the directionality in the straight interfaces as shownin Fig. S6. At r b = 0, the directionality of edge mode in the Ω-shape interface is slightly smaller than that in thestraight interface. In view of four 120 ◦ bending structures, the reflection at the single bending is very small. Especially,as r b increases, the directionality of the edge mode increases gradually, and almost the same in the straight interface,which means the no reflection at the bending of the interface. However, with further increasing r b /a over 0.15, thedirectionality becomes small. From this, we expect that there is an optimum r b /a value which has no reflection withhigh directionality. [S1] I. S¨olllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kirˇsansk˙e, T. Pregnolato, H. El-Ella, E. H. Lee, J. D.Song, S. Stobbe, and P. Lodahl, Nat. Nanotechnol.10