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[Tetsuyuki Ochiai](https://orcid.org/0000-0003-2933-0014)

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This is an Accepted Manuscript version of the following article, accepted for publication in Waves in Random and Complex Media. Ochiai, T. (2024). Bulk-edge correspondence in open photonic systems. Waves in Random and Complex Media, 34(2), 951–968. https://doi.org/10.1080/17455030.2021.1923858. It is deposited under the terms of the Creative Commons Attribution-NonCommercial License (http://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited.[Creative Commons BY-NC Attribution-NonCommercial 4.0 International](https://creativecommons.org/licenses/by-nc/4.0/)

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[Bulk-edge correspondence in open photonic systems](https://mdr.nims.go.jp/datasets/d04a868f-818b-4f70-a013-f92b8a8b4e23)

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Bulk-edge correspondence in open photonic systemsTetsuyuki OchiaiResearch Center for Functional Materials, National Institute for Materials Science, Tsukuba305-0044, JapanARTICLE HISTORYCompiled August 26, 2020ABSTRACTWe study the bulk-edge correspondence in topological photonic crystals with openboundary. If boundary is open, edge states become leaky inside the light cone, butstill exhibit a chiral and gapless property taking into account the blurring of theirband structure due to the leakage. The so-called bulk-edge correspondence is thusverified. On the other hand, in closed boundaries, edge states exhibit the well-definedband structure without the blurring and show clearly the bulk-edge correspondence.To demonstrate these results, we employ the transfer-matrix formalism and derivereflection matrices of semi-infinite systems. Optical density of states for the systemwith open boundaries is available via the Krein-Friedel-Lloyd formula for the re-flection matrices. The leaky photonic band structure of the edge states is obtainedby following the peaks and widths of the density of states as a function of momen-tum parallel to the boundary. Our derivation of the leaky band structure does notrely on possible effective non-hermitian hamiltonians, but relies on a first-principlescalculation of the Maxwell equation.KEYWORDStopological photonics; edge states; density of states; open quantum systems1. IntroductionRadiation fields are real-valued. This implies that their frequencies are positive-definite, regardless of their spatial profiles, localized or extended. The ”work function”for photons is completely absent between a photonic medium [such as photonic crystal(PhC)] and outer medium (such as air). In this sense, any photonic systems are opento the outer medium, and inversely, an incoming wave in the outer medium can ex-cite internal modes in a photonic system, regardless of frequency, This “open”ness ofphotonic systems shows a striking contrast to that in electronic systems, where inter-nal modes are protected from the mixing with outer modes by finite work functions.The “open”ness of the electronic systems is restricted, only through other kinds ofexcitation modes such as phonons.Suppose that there is a topologically nontrivial photonic system [1,2] with boundary.We expect that the system has gapless boundary states, in accordance with the so-called bulk-edge correspondence [3]. However, such a boundary state is embedded inthe radiation continuum of the outer medium, so that the state becomes leaky unlessit is outside the light cone. We considered this situation in [4], and found the bulk-edgecorrespondence certainly works provided that the blurring (in frequency) of the leakyboundary states is taken into account.Here, we revisit this problem in a more ideal situation than in [4] and show clearlythat the bulk-edge correspondence certainly holds. In this demonstration, the transfer-matrix formalism plays a crucial role. It enables us to study the photonic band struc-tures in the bulk, reflection matrices of semi-infinite systems, skin depth at the surface,and so on [5,6]. The reflection matrix together with a boundary condition determinesthe dispersion relation of possible boundary states, via the Krein-Friedel-Lloyd formula[7–9], as we show in this paper.Recently, many important discoveries have been made in non-hermitian [10] and/orPT symmetric [11], and/or open quantum systems [12]. The discoveries include peri-odic tables in topological phases [13], non-hermitian skin effect [14], interaction effects[15]. This paper is closely related to several issues in open quantum systems. However,in studying such a system, possible issues relevant to the light cone are completelyabsent. The light cone is inherent in photonic systems and gives severe constraintson them. For instance, the light cone separates the momentum space into two dis-tinct regions, inside and outside the light cone. Inside the light cone, photons can leakto outer space, whereas outside the light cone, photons are totally internal reflected.The Brillouin-zone folding in PhCs further complexifies the momentum space. Sucha complexity is usually not taken into account in the study of open quantum sys-tems. Therefore, the issues on the light cone may give some insights to open quantumsystems.This paper is organized as follows. In Sec. 2, we present a theoretical framework toinvestigate PhCs with open boundaries. In Sec. 3, we focus on typical topological PhCsvia the time-reversal-symmetry breaking, and verify the bulk-edge correspondence.Section 4 is devoted to present non-topological PhCs with the time reversal symmetryand show that the edge states are clearly gapped. Finally, in Sec. 5, we summarize theresults obtained in this paper.2. Bulk and edge by transfer-matrix formalismIn what follows, we assume two-dimensional (2D) PhCs with the transnational invari-ance in the z direction, and consider the in-plane propagation of light (kz = 0). Thelight polarization is decoupled into the transverse-electric (TE) polarization charac-terized by nonzero Ex, Ey, and Hz and the transverse-magnetic (TM) polarizationcharacterized by nonzero Ez, Hx, and Hy. Here, we assume the TM polarization. Theextension of the following formalism to the TE polarization is straight-forward.Suppose that the bulk PhC is made of a periodic stack of identical monolayers withrelative shift s between two adjacent layers, as depicted in Fig. 1.[Figure 1 about here.]The monolayer consists of circular rods with lattice constant d in the x direction. Thedielectric constant of the background medium is denoted as εb. The stacking directionis taken to be y direction. In the n-th void region between the n-th layer and n+ 1-th2layer, the electric field is expressed asE(n)z (x) =∑h(a+(n)h eiK+h ·(x−xn) + a−(n)h eiK−h ·(x−xn)), (1)K±h = (kx + h)x̂± γhŷ, γh =√q2b − (kx + h)2, (2)h =2πZd, qb =ωc√εb. (3)Here, xn is the reference point of the n-th void region, satisfying xn+1−xn = s, kx isthe Bloch momentum, and ω is the angular frequency of the radiation field concerned.The monolayer S-matrix gives the relation among the plane-wave-expansion (PWE)coefficients a±h of two successive void regions as(a+(n+1)ha−(n)h)=∑h′(S++hh′ S+−hh′S−+hh′ S−−hh′)(a+(n)h′a−(n+1)h′). (4)In a short-hand notation, we rewrite it as(a+(n+1)a−(n))= S(ω, kx)(a+(n)a−(n+1)), (5)S(ω, kx) =(S++ S+−S−+ S−−). (6)The S-matrix is numerically available by the 2D version of the layer Korringa-Kohn-Rostoker method [16–18]. The S-matrix of consecutive two monolayers (S2) is obtainedby the layer-doubling procedure as [19](a+(n+2)ha−(n)h)=∑h′((S2)++hh′ (S2)+−hh′(S2)−+hh′ (S2)−−hh′)(a+(n)h′a−(n+2)h′), (7)S++2 = S++(1− S+−S−+)−1S++, (8)S+−2 = S+− + S++S+−(1− S−+S+−)−1S−−, (9)S−+2 = S−+ + S−−S−+(1− S+−S−+)−1S++, (10)S−−2 = S−−(1− S−+S+−)−1S−−, (11)(12)The successive application of the layer doubling produces the S-matrix of 2m (m:positive integer) layers, and the transmittance and reflectance of the 2m-layer-thickPhC are obtained.The band structure of the corresponding bulk PhC is obtained from the monolayerS-matrix. Between two adjacent voids, the Bloch theorem readsE(n+1)z (x + s) = eik·sE(n)z (x), (13)where x is assumed to be in the n-th void. The lattice vectors of the bulk PhC arespanned by e1 = dx̂ and e2 = s. On the other hand, a±(n+1)h are related to a±(n)h via3the monolayer S-matrix. Therefore, the Bloch theorem reduces to the diagonalizationof the transfer matrix asT (ω, kx)(a+(n)a−(n))= eik·s(a+(n)a−(n)), (14)(a+(n+1)a−(n+1))= T (ω, kx)(a+(n)a−(n)), (15)T (ω, kx) =(S++ − S+−(S−−)−1S−+ S+−(S−−)−1−(S−−)−1S−+ (S−−)−1)(16)This is an on-shell band calculation; the momentum in the s direction is obtained asthe outputs, while the frequency and momentum in the x direction are given as theinputs. The transfer matrix T is not unitary, so that the eigen-momenta k · s are notlimited in real value. Among them, the real-valued ones are propagating eigenmodesthat correspond to the photonic band modes in an ordinary (off-shell) band calculation.The complex-valued ones are evanescent eigenmodes which are hidden in the ordinaryband calculation. In a bulk band gap, no propagating eigenmodes are found. Theevanescent eigenmodes having the smallest absolute imaginary part in k · s determinethe penetration depth of the incident wave in the band gap.The transfer matrix is also very important in determining possible surface states inthe PhC. The diagonalization of the transfer matrix leads toT = V ΛV −1, (17)Λ = diag(eik·s) = (Λ+,Λ−), (18)V =(A BC D), V −1 =(A′ B′C ′ D′), (19)Within a bulk band gap, the eigenvalues are either forward evanescent or backwardevanescent. Here, Λ+(−) represents the forward (backward) evanescent eigenmodes. Wenote that the transfer matrix of the N -layer system is simply TN and that ΛN+ → 0 asN →∞. By transforming back to the S-matrix via Eq. (16), the S-matrix of infinitelythick PhC becomesS∞ =(0 R′R 0). (20)Here, R is the reflection matrix of the semi-infinite system for an incident wave fromthe bottom, and is given byR = CA−1 = −D′−1C ′ (21)Similarly, the reflection matrix R′ for an incident wave from the top is given byR′ = BD−1 = −A′−1B′. (22)By truncating into open diffraction channels, these reflection matrices are shown to4be unitary asR̃†R̃ = 1, (23)R̃hoh′o =√γhoRhoh′o1√γh′o, (24)where ho represents an open diffraction channel.From the reflection matrix, we can calculate the eigenfrequency spectrum of possiblesurface states. Inside the light cone, the eigenfrequencies become complex owing to theleakage to the outer space. They have imaginary parts representing the radiation loss.A conventional way to calculate such energy spectra is to diagonalizing effective (andoften non-well-defined) non-hermitian hamiltonians that include the loss. However, wedo not employ such a scheme. Instead, we consider the well-defined transfer matrixand extract the information of the complex energy spectrum from it.The key quantity is the density of states (DOS) of photons. Let us count the numberof possible radiation modes by imposing the Dirichlet boundary condition at far fields.Far away the boundary surface of the PhC, we put the perfect-electric-conductor(PEC) wall parallel to the boundary. The boundary condition is simply given byEz(x,−L) = 0, (25)where we put the PEC wall at y = −L. Then, we havea+(0)hoe−iγhoL + a−(0)hoeiγhoL = 0, (26)where we assume x(0) = 0. Here, we drop unphysical evanescent waves that are expo-nentially growing as y → −∞. Then, we obtain∑h′o(e−2iγhoLδhoh′o +Rhoh′o)a+(0)h′o= 0, (27)in a bulk band gap. Let us consider the case that there is only one open channel. Inthis case, the reflection coefficient Rhohois just a phase factor. Then we have2γhoL+ arg(Rhoho) = (2m+ 1)π (m ∈ Z). (28)The optical DOS per unit angular frequency is then given byρ =∆m∆ω=1π∂γho∂ωL+12π∂∂ωarg(Rhoho). (29)The first term in the right hand side is the DOS in free space with length L, and thesecond term is the contribution of the semi-infinite PhC. Therefore, the increment ofthe DOS due to the semi-infinite PhC structure is∆ρ(ω, kx) =12π∂∂ωarg(Rhoho). (30)Here, arg(Rhohp) is nothing but the scattering phase shift. Therefore, the above equa-tion is the optical analogue of the Krein-Friedel-LLoyd formula, or in other words, the5differential form of the Friedel sum rule [20]. If there are more than one open channels,the above expression is generalized as [21]∆ρ(ω, kx) =12π∂∂ωarg(detR̃). (31)By following the peak frequency ωc of the DOS as a function of kx, we obtain the bandstructure of the leaky surface states.Outside the light cone, there is no open channel. Therefore, the surface states cannotleak to the outer region. These true guided modes are determined asdetB′ = 0. (32)This completes the band calculation of the surface states in open photonic systems.In a closed boundary capped by the PEC wall, the boundary condition is the sameas Eq. (25). However, we have to take account of all the diffraction channels as∑h′(e−2iγhLδhh′ +Rhh′)a+(0)h′ = 0. (33)If the system is covered by the perfect-magnetic-conductor (PMC) wall, the boundarycondition is changed asHx(x,−L) = 0. (34)The secular equation for the surface states is given by∑h′(e−2iγhLδhh′ −Rhh′)a+(0)h′ = 0. (35)In the previous study of Ref. [4], the authors considered the optical DOS for finite-thick PhC with lower and upper open boundaries. Therefore, the contribution of thelower and upper surface states to the DOS is mixed. It is quite often the case thatthe two peaks of the DOS merge to a single peak, so that we cannot separate eachcontribution. Using the semi-infinite system as we do in this paper, we can separateeach contribution. Then, we can verify possible bulk-edge correspondence rule.3. Topological photonic crystal without time-reversal symmetryLet us first consider a typical topological PhC with a magneto-optical medium. In themagneto-optical medium, the permeability tensor in plane is given by←→µ =(µ iκ−iκ µ), (36)assuming the applied magnetic field (or spontaneous magnetization) along the z di-rection. The parameter κ represents the degree of the magneto-optical effect and anonzero κ breaks the time-reversal symmetry. Suppose that the PhC consists of thehoneycomb array of circular cylinders with the magneto-optical effect. Figure 2 showsthe photonic band structures of the PhCs.6[Figure 2 about here.]The photonic band structure without the magneto-optical effect exhibits the Diraccones at the Brillouin zone corners provided that the two sites of the honeycomblattice are equivalent. If the magneto-optical effect is introduced in the cylinder, or ifthe inversion-symmetry breaking is introduced by a difference between the A and Bsites of the honeycomb lattice, the photonic band gap opens around the Dirac cones.In the former case, the relevant photonic bands become topological having the Chernnumbers of ±1. In the latter case, however, the photonic bands are topologically trivialwith vanishing Chern number.Figure 3 shows the photonic band structure of the zigzag and armchair edge statesfor the PEC or PMC boundary. Both the upper and lower edges of the semi-infinitePhC are considered.[Figure 3 about here.]In all the cases, the edge states are clearly gapless; their dispersion curve traverses thebulk band gap.In the zigzag edges, the band structures are quite similar to those of the domain-wall fermion [22]. Namely, the almost linear dispersion is observed in the K and K’valleys. The edge states emerge in the opposite valley with the same chirality betweenthe PEC and PMC boundary. Here, we refer to the chiality as the sign of the averagegroup velocity of the edge states. The chirality is opposite between the upper andlower edges. Such a selective emergence of the chiral edge states are also obtained inthe domain wall introduced in the PhCs [23].The almost-linear dispersion can be understood as follows. Around the K and K’valleys, the k · p effective Hamiltonian is given by [24],HK = v(σ3δkx − σ1δky) + σ2M, (37)HK′ = −v(σ3δkx − σ1δky) + σ2M, (38)Ez(x) = c1E(1)z (x) + c2E(2)z (x). (39)Here, σi (i = 1, 2, 3) is the Pauli matrix, δk is the deviation of the momentum fromthe K or K’ point, and M is the Dirac mass proportional to κ. The Hamiltonian actson the phase space spanned by c1 and c2. The base functions E(1)z and E(2)z are thedoubly degenerate eigenfunctions of the unperturbed honeycomb-lattice system at Kor K’. That is, the modes at the Dirac point of Fig. 2. They form the E representationof C3v, which is the k group at K or K’ of the unperturbed system. We assume thatE(1)z has the even parity with respect to the mirror plane relevant to the K and K’points, and that E(2)z has the odd parity, because the E representation behaves as(x, y). Therefore, E(2)z = 0 on the mirror plane.To study possible edge states within the effective hamiltonian, we replace δky bymomentum operator −i∂/∂y. Then, we have the following eigenvalue equation:H(δkx,−i∂∂y)(c1c2)= E(c1c2). (40)We can easily show that the equation around the K point has the edge-state solution7ofE = −vδkx, c1 = 0, c2 ∝ eiδkxx+Mvy, (41)provided that 1) the PEC boundary is imposed on the mirror plane and 2) exp((M/v)y)is decaying away from the boundary. The second condition implies that if M/v > 0,then the upper edge states are allowed to exist. Otherwise, if M/v < 0, the loweredge states are allowed. Similarly, the following solution is also available around theK’ point:E = vδkx, c1 = 0, c2 ∝ eiδkxx−Mvy, (42)provided that 1) the PEC boundary is imposed on the mirror plane and 2)exp(−(M/v)y) is decaying away from the boundary.As for the PMC boundary, possible solutions areE = vδkx, c1 ∝ eiδkxx−Mvy, c2 = 0, (43)around K only for decaying exp(−(M/v)y), andE = −vδkx, c1 ∝ eiδkxx+Mvy, c2 = 0, (44)around K’ only for decaying exp((M/v)y).These four solutions correspond to the four dispersion curves in Fig. 3 (a). Theemergence of the edge states are fully consistent with v > 0 and M < 0, as can benumerically checked from the unperturbed wave functions in the k · p perturbation.In the armchair edge, the dispersion curve is not so simple, and is extended inthe entire surface Brillouin zone. The description by the k · p effective hamiltonian isnot efficient, because the hamiltonian is only valid around the K and K’ valleys. Theedge states still exhibit the chiral and gapless property, namely, the dispersion curvetraverses the band gap with positive average slope for the upper edge and negativeaverage slope for the lower edge. The complexity of the dispersion curve partially comesfrom the fact that the two Dirac cones at K and K’ are overlaid after the projectionto the surface Brillouin zone.In the boundary capped by the PEC or PMC wall, the edge states are well-definedas shown in Fig. 3. They have pure real eigenfrequencies regardless of momentum.In the open boundary, however, the band structure is blurred, as they have nonzeroimaginary part in their eigenfrequencies.Figure 4 shows the photonic band structure of the edge states in the open boundary.The imaginary parts are also plotted as the error bars.[Figure 4 about here.]In the zigzag case, the edge states emerge outside the light cone, so that the bandstructure is well-defined. It exhibits the chiral and gapless property. In the armchaircase, the band structure emerges both inside and outside the light cone. The center ofthe error bar represents the resonant frequency of the edge states, showing the chiraland gapless property. The heights of the error bar represent the inverse lifetimes, whichdecrease toward the light line ω = c|k‖|, and vanishes on it.Figure 5 shows the increment of the DOS at k‖darm/2π = 0.25.8[Figure 5 about here.]By a Lorentzian fit of the DOS via∆ρ(ω, kx) =Γπ(ω − ωc) + Γ2, Q =ωc2Γ, (45)we obtain quality factor Q = 46 for the lower edge state, and Q = 34 for the upperedge state. In the actual calculation of the quality factor, we employ a numerical fittingof the phase shift by the Breit-Wigner formula:detR̃ = e2iδ0 ω − ω0 − iΓω − ω0 + iΓ, (46)from which the Lorentzian form of the DOS is derived.In the PEC/PMC boundary case, the distance between the PhC edge andPEC/PMC wall is crucial. In Fig. 3, we put the PEC/PMC wall at the mirror plane(zigzag edge) or glide plane (armchair edge) of the bulk PhC. In the former case, thesimple analytic solutions of Eqs. (41-44) are available, and describe the numerical so-lutions fairly well. However, if the PEC/PMC wall is put off the mirror/glide plane,such an analytic solution is not available. By increasing the distance L between thePhC edge and PEC/PMC wall, the photonic band structure approaches to that in theopen boundary, while keeping the chiral and gapless property, as shown in Fig. 6.[Figure 6 about here.]Even at L = a/2, the dispersion relation is nearly equal to that in the open boundary.Thus, we can estimate the edge-state dispersion in the open boundary by increasingdistance L.4. Nontopological photonic crystal without space-inversion symmetryFor comparison, we consider a nontopological PhC without the magneto-optical effect.The time-reversal symmetry thus holds and the Chern number vanishes. We considerthe honeycomb-lattice PhC with a difference in the dielectric constants between Aand B rods, yielding the space-inversion-symmetry breaking and the band gap at theDirac points. The effective Hamiltonian around K and K’ are now given by [24]HK = v(σ3δkx − σ1δky) +Mσ2, (47)HK′ = −HK , (48)where M is proportional to εA − εB.Again, we first consider the edge states for the PEC/PMC boundary. Figure 7 showsthe photonic band structure of the edge states with the PEC or PMC boundary.[Figure 7 about here.]In all the cases, the edge states are gapped. However, it is remarkable that, as for thezigzag edge, the upper edge states of the PEC boundary and the lower edge states ofthe PMC boundary are nearly gapless. Moreover, these dispersions are almost lineararound the K and K’ valleys.9These properties are fully understood by the effective hamiltonian. Following asimilar argument as we did in Sec. 3, we can easily find that the results are consistentwith the effective hamiltonian with v > 0 and M < 0. More intuitively, the followingargument also explain the properties. The PEC or PMC wall is like a mirror and themirror-inverted image of the PhC is the PhC with swapped A and B sites. Therefore,the system with the PEC/PMC wall imitates a combined system with a domain wall.As in the domain-wall fermion, the edge states becomes linear and nearly gaplessaround the K and K’ valleys. In the other cases of the upper edge states of the PMCboundary and the lower edge states of the PEC boundary are almost absent. In thesecases, although the domain wall is formed, their masses have the same sign betweenthe two domain, so that the edge states are almost absent. As for the armchair edge,the edge states are clearly gapped regardless of either the PEC or PMC boundary. Bysymmetry, the upper- and lower-edge states are completely degenerate.Next, we consider the open boundary. The dispersion relation of the edge states areshown in Fig. 8.[Figure 8 about here.]Again, the edge states are clearly gapped. As for the armchair edge, the upper- andlower-edge states are completely degenerate. By increasing the distance between thePEC wall and PhC edge, we can see a smooth transition of the edge states to thosein the open boundary, while keeping the gapped property.In this way, the edge states of the nontopological PhC are gapped irrespective ofthe boundary conditions., supporting the bulk-edge correspondence.5. SummaryIn summary, we have presented a detailed theoretical analysis on topological and non-topological PhCs with open boundary. The openness of the system is caused by intrin-sic properties of photons and yields imaginary parts in the eigenfrequency spectrum ofedge states. The transfer-matrix formalism and resulting semi-infinite reflection matri-ces together with the Krein-Friedel-LLoyd formula of the DOS enable us to determinethe leaky and guided edge-state dispersion relations accurately. The leaky edge statesinside the light cone emerge as Lorentzian peaks of the DOS spectrum, whose peakand width in frequency correspond to the eigenfrequency and inverse quality factorof the edge state. Taking the width into account, the edge-state dispersion is shownto be chiral and gapless for topological PhCs with nonzero Chern numbers. The edgestates for nontopological PhCs with vanishing Chern number are shown to be gapped.In this sense, the bulk-edge correspondence certainly holds even for the open photonicsystems.AcknowledgementsThis work was partially supported by KAKENHI Grant No. 17K05507.10References[1] Haldane FDM, Raghu S. Possible Realization of Directional Optical Waveguides in Pho-tonic Crystals with Broken Time-Reversal Symmetry. 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Phys Rev B.2012;86:075152.11x0x1x2x3x41st layer2nd layer3rd layer4th layer5th layer1st void2nd void3rd void4th voidboundaryLdsABFigure 1. Schematic illustration of the honeycomb-lattice photonic crystal presented in this paper. It consistsof a periodic stack of identical monolayers with relative shift s. The monolayer is a periodic arrangement ofcircular cylinders with period d in the x direction.12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Γ M K Γωa/2πcM K 0.23 0.24 0.25 0.26 0.27 0.28 0.29ΓMKK'Figure 2. Bulk photonic band structures of the honeycomb-lattice PhCs made of circular rods. The rodshave dielectric constant εA(B), diagonal permeability µA(B), imaginary off-diagonal permeability κA(B), andradius rA(B) for the A(B) sites of the honeycomb lattice. The background medium is air (ε = µ = 1, κ = 0).Three configurations are considered: 1) εA = εB = 12 and κA = κB = 0 (solid line), 2) εA = εB = 12 andκA = κB = 0.2 (dotted line), and 3) εA = 13, εB = 11 and κA = κB = 0 (dashed line). The other parametersare kept fixed as µA = µB = 1 and rA = rB = 0.2a, where a is the lattice constant of the honeycomb lattice.The upper right inset shows the first Brillouin zone of the honeycomb lattice. Points of high symmetry arenoted.13 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||dzig/2πPEC, lowerPMC, lowerPEC, upperPMC, upper(a) 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||darm/2πPEC, lowerPMC, lowerPEC, upperPMC, upper(b)Figure 3. Photonic band structures of (a) the zigzag and (b) armchair edge states for the PEC or PMCboundary. The following PhC parameters are assumed: εA = εB = 12, µA = µB = 1, κA = κB = 0.2, andrA = rB = 0.2a. The shaded region represents the bulk photonic bands and the blank region is the photonicband gap. We consider the upper and lower edges of the semi-infinite PhC in the perpendicular direction tothe edge. The symbol dzig(arm) represents the spatial periodicity in the zigzag (armchair) edge and is givenby dzig = a and darm =√3a. The distance L to the PEC/PMC boundary (See Fig. 1) is taken to be zero,assuming the reference point xn on the mirror plane for the zigzag edge, and a glide planes for the armchairedge.14 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||dzig/2πlowerupper(a) 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||darm/2πlowerupper(b)Figure 4. Photonic band structures of (a) the zigzag and (b) armchair edge states for the open boundary.Both the upper and lower edges of the semi-infinite PhC are considered. The PhC parameters are the sameas in Fig. 3. The dash-dotted line is the light cone ω = c|k‖|. Inside the light cone ω > c|k‖|, the edge statesbecome leaky having finite lifetimes. The error bar length represents the inverse lifetime Γ of the edge states.15 0 50 100 150 200 250 300 350 0.23  0.24  0.25  0.26  0.27  0.28  0.29Δρ2π2 c/aωa/2πclowerupperFigure 5. Increment of the optical density of states ∆ρ due to the presence of the semi-infinite PhC with theopen armchair edge. Both the upper and lower edges are considered. The PhC parameters are the same as inFig. 3. The momentum parallel to the boundary is taken to be k‖darm/2π = 0.25.16 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||dzig/2πL=0L=a/16L=a/8L=a/4L=a/2openFigure 6. Dependence of the edge-state dispersion on the distance L from the PhC edge to the PEC wall.The lower zigzag edge is considered. The PhC parameters are the same as in Fig. 3.17 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||dzig/2πPEC, lowerPMC, lowerPEC, upperPMC, upper(a) 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||darm/2πPEC, lower & upperPMC, lower & upper(b)Figure 7. Photonic band structures of (a) the zigzag and (b) armchair edge states in the honeycomb latticePhC with the time-reversal symmetry but without the space-inversion symmetry. The PhC parameters are asfollows: εA = 13, εB = 11, µA = µB = 1, κA = κB = 0, and rA = rB = 0.2a. Both the upper and lower edgesof the semi-infinite PhC are considered. In the armchair case, the upper and lower edge states are completelydegenerate by symmetry. The boundary is capped by the PEC/PMC wall with distance L = 0.18 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||dzig/2πlowerupper(a) 0.23 0.24 0.25 0.26 0.27 0.28 0.29-0.5  0  0.5ωa/2πck||darm/2πlower & upper(b)Figure 8. Photonic band structures of the edge states in the honeycomb lattice PhC with the time-reversalsymmetry but without the space-inversion symmetry. The PhC parameters are the same as in Fig. 7. Theboundary is open to air. The dash-dotted line is the light cone ω = c|k‖|. The error bar length represents theinverse lifetime Γ of the edge states.19