Abstract

A simple core-shell two-dimensional photonic crystal is studied where the triangular lattice symmetry and the C6 point group symmetry give rich physics in accidental touching points of photonic bands. We systematically evaluate different types of accidental nodal points at the Brillouin zone center for transverse-magnetic harmonic modes when the geometry and permittivity of the core-shell material are continuously tuned. The accidental nodal points can have different dispersions and topological properties (i.e., Berry phases). These accidental nodal points can be the critical states lying between a topological phase and a normal phase of the photonic crystal. They are thus very important for the study of topological photonic states. We show that, without breaking time-reversal symmetry, by tuning the geometry of the core-shell material, a phase transition into the photonic quantum spin Hall insulator can be achieved. Here the “spin” is defined as the orbital angular momentum of a photon. We study the topological phase transition as well as the properties of the edge and bulk states and their application potentials in optics.

© 2016 Optical Society of America

1. Introduction

Since Haldane and Raghu’s seminal work [1] which brought the quantum Hall effect [2] to the realm of photonics, the field of topological photonics attracted significant research interest. The study of topological properties of photonic bands [3–9], polarization states [10,11], as well as their unique physical consequences and applications motivated the field. Recent developments of the field extend to time-reversal symmetric all-dielectric photonic crystals (PhCs) which are easier to fabricate (especially for visible-frequency applications) compared to time-reversal symmetry broken systems [12–17]. In time-reversal symmetric all-dielectric PhCs, because the Kramers theorem does not guarantee double degeneracy for photons, additional symmetry is needed to induce topological effects. These symmetry-protected topological photonic states enrich the study and application values of Berry phases in photonic bands [18]. Many novel properties are to be explored for fundamental physics and applications. Examples include the study of photonic graphene [19], Dirac-like points with zero effective refractive index [20,21], etc.

In spite of much research efforts in the literature, the design and understanding of topological photonic bands remain a challenging task. The existing designs are accidental: there is no clue or guiding principles on how to realize photonic bands with nontrivial topology. One may use Dirac cones as mother states of topologically nontrivial photonic band gap. However, the emergence of Dirac cone is also accidental unless for a few deterministic cases such as in honeycomb lattice photonic crystals where the sub-lattice symmetry dictates Dirac dispersion at the K and K′ points [1,14].

Here we present a systematic study on accidental nodal points in photonic bands in 2D triangle PhCs with core-shell dielectric structures. We show that some of those nodal points are Dirac (and Dirac-like) points. Their dispersion around the Γ point can be easily understood via the k⃗ · P⃗ theory with the assistance of the picture that the Mie resonances can be regarded as atomic orbits for photonic bands [22]. Those atomic orbits can be of s, p, d and f nature and have well-defined parities at the Γ point. The couplings between photonic bands are readily determined by their parity differences. In our core-shell structures, by tuning the inner and outer radii of the core-shell cylinder, we are able to flip the order of the p and d photonic bands, which leads to a phase transition from topologically trivial photonic bands to quantum spin Hall insulator of light: a photonic band gap with nontrivial Z2 topology. The properties of the bulk and edge states of the Z2 topological photonic band gap are studied in details, where essential difference between electronic and photonic systems are emphasized. Differing from realization of Z2 topological photonic band gap using deterministic Dirac cone in honeycomb lattice [14], here the topological phase transition demonstrates rich physics as revealed by our phase diagram and k⃗ · P⃗ analysis. The core-shell triangle photonic crystal structure is simple and mechanically stable. In addition, it is compatible with colloidal self-assembled structure [23]. Our study shows how topological phases of photons can be realized in simple 2D dielectric PhCs and various phases can be induced by tuning the geometry of the dielectric materials, offering guidelines for future studies.

2. 2D triangle lattice photonic crystals

Among common 2D PhCs, triangle lattice PhCs have the highest point group symmetry [24]. As will be shown below the 2D triangle PhCs can host several kinds of topological nodal points. These nodal points are the mother states of different types of photonic topological insulators. In the triangular lattice, the Γ point has two kinds of 2D irreducible representations in the C6 symmetry, which are denoted as the E1 and E2 representations, respectively [21, 24]. The associated Bloch waves are referred as the E1 and E2 modes, respectively. Each representation has two Bloch waves, which can be reorganized into a pair of modes connected by time-reversal operation. Here the relevant modes are the p+ = px + ipy and p = pxipy modes, which are denoted as (pseudo-) spin-up and spin-down states of the p bands [14]. Similarly, the d+ = dx2y2 + idxy and d = dx2y2idxy modes are denoted as the (pseudo-) spin-up and spin-down states for the d bands [14].

The 2D core-shell dielectric structure in triangular lattice is shown in Fig. 1(a). Its outer and inner radii are denoted as R1 and R2, respectively. The dielectric constant of the core-shell material and the background material are ε1 and ε2, respectively. The lattice constant is set as a1 = a2 ≡ 1. We define the inverse structure by the exchange of ε1 and ε2. The inverse structure is hence a core-shell air cylinder in dielectric background. Our architecture is one of the simplest 2D PhCs that can support rich physics of topological nodal points, photonic Z2 topological insulators, and topological phase transitions.

 figure: Fig. 1

Fig. 1 (a) Schematic configuration of a triangle PhC using core-shell dielectric materials. The dielectric constant of the core-shell dielectric material and background material are ε2 and ε1, respectively. R1 and R2 are the outer and inner radii of the core-shell cylinder. a⃗1 and a⃗2 are the two basis vectors of the triangular lattice. We set a1 = a2 ≡ 1 as the lattice constant. The inverse structure is defined by exchange of the permittivity ε1 and ε2. (b) A typical band structure of the core-shell triangle lattice PhC with R1 = 0.16a1, R2 = 0.05a1, ε1 = 1 and ε2 = 12. We use the symbols s, p, d and f to label the modes at the Γ point.

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By numerically solving the Maxwell equations for transverse magnetic (TM) harmonic modes, of which the electric field is along the core-shell cylinder (i.e., the z direction). Figure 1(b) shows a typical band structure of the PhC. The band structure is calculated by COMSOL based on the finite-element method. In this band structure, the eigen-modes at the Γ point exhibit orbital symmetry of the s, p, d and f waves from low frequency to high frequency. These modes correspond to the A1, E1, E2 and B1 modes of the C6 point group symmetry [24]. The labeling of bands with s, p, d and f modes is only effective at the Γ point which possesses the C6 symmetry. Away from the Γ point these modes couple with each other and their order in frequency may change [14]. Nevertheless, the labels clearly reveal the accidental band degeneracy and band inversion at the Γ point. Besides, the s, p, d and f modes carry physical meanings. The photonic bands can be viewed as derived from transfer (hopping) of local Mie resonances of the core-shell structures between adjacent unit cells (except the lowest photonic band which becomes plane wave in the low frequency limit). Such a tight-binding understanding of the photonic bands has successfully connected the photonic bands with the Mie resonances [22]. Thus the Mie resonances can be regarded as the “atomic orbits” for photonic bands. This picture is useful in our design of photonic Z2 topological insulators and topological nodes.

The s and f (A1 and B1) modes are singlet states, while the E1 and E2 modes are doubly degenerate, according to the C6 symmetry [21]. By continuously changing the radii R1 and R2 or the dielectric constant, different bands cross each other, leading to accidental degeneracy’s at the Γ point. The accidental degeneracy usually results in linear or quadratic dispersion. The scale invariance of the Maxwell equations dictates that the independent variables in our system are the inner and outer radii (divided by the lattice constant) as well as the ratio of the two dielectric constants ε1/ε2. The properties of the photonic bands for R2 = 0 are well-studied in the literature [25], where no topological nodal point or topological band structure is found.

3. Phase diagram

We first discuss the eigen-frequency of the p and d bands at the Γ point. We use the following dimensionless quantity to characterize the relative p-d band-gap size at the Γ point,

Δωpd=2ωdωpωd+ωp,
where ωp and ωd are the egien-frequency’s of the p and d modes at the Γ point, respectively.

By numerically calculating the photonic band structure, we obtain in the R1R2 parameter space with ε1 = 1 and ε2 = 12 (see Fig. 2). The figure clearly demonstrates the p-d band inversion which can result in photonic analog of the Z2 topological insulator. The properties of this photonic Z2 topological states will be discussed in details below. We incorporate in the upper triangle of Fig. 2 the phase diagram for the reversed structure, where the core-shell cylinder of air induces the photonic energy bands. The diagonal line is the homogeneous limit, which can be regarded as no dielectric (for the normal structure) or no air (for the inverse structure). The influence of ε1/ε2 on the phase diagram will be discussed below.

 figure: Fig. 2

Fig. 2 Phase diagram of the p-d-inversion-induced photonic Z2 topological insulator in the R1-R2 parameter space. The diagonal line represents the homogeneous limit which separates the normal and reversed structures in the upper and lower triangular regions. The lower region is the parameter space for core-shell dielectric cylinder in air background (called as the normal structure), while the upper region is the parameter space for core-shell air cylinder in dielectric background. The dielectric constant for air and dielectric are ε1 = 1 and ε2 = 12, respectively. The contour color represents the value of Δωpd. The red region represents that the p bands are lower than the d bands at the Γ point, whereas the blue region stands for the photonic analog of the Z2 topological insulators with the p bands above the d bands at the Γ point. The p-d bands degeneracy at the Γ point is labeled by the black line. The s-p, p-f, s-d, f -d band inversions are also calculated and plotted. Several points in the phase diagram are labeled, which will be used for the discussions in the main text.

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The color scheme in Fig. 2 is as follows: the red region has p bands lower (in frequency) than the d bands at the Γ point, while the blue region has p bands higher than the d bands (i.e., p-d band inversion), at the Γ point. The blue region thus represents the Z2 topological phase (Z2 = 1), while the red region represents the normal phase (topologically trivial, Z2 = 0). The parity inversion at the Γ point for the p-d reversed photonic bands dictate the Z2 topology [26]. At the boundary between the topological phase and the normal phase, the double Dirac-cone dispersion emerges at the Γ point. The properties of the photonic Z2 topological insulator will be discussed in details in next section.

In addition, we use the blue (red) curve in Fig. 2 to denote the crossing between the s and p bands (the f and d bands) in the phase diagram. These two situations lead to Dirac-like cones at the Γ point. The crossing between the s and d bands (the f and p bands) are denoted as the green and purple curves in the phase diagram. These crossings lead to quadratic band touching at the Γ point. All these accidental band degeneracy’s and the properties of the nodal point will be discussed in details below.

Other important information revealed by Fig. 2 is the geometric conditions for the emergence of various nodal points in the photonic spectrum. These nodal points are important for several reasons. First, they are the mother states of topologically nontrivial states of photons. An example has been demonstrated above for the relation between the double Dirac cone and the photonic Z2 topological insulator. The Berry phase of each band for a loop LΓ enclosing the Γ point LΓΨnk|ik|Ψnk (here |Ψ⃗nk⃗〉 denotes the vectorial photonic Bloch wavefunction), is closely related to the formation of the photonic topological states [18,27]. Usually, topological nodal points are the critical states between normal band gaps and topological insulators. By introducing time-reversal/inversion symmetry breaking perturbations, the double-Dirac-cone state can become a photonic quantum anomalous Hall insulator (Z topological insulator), a Z2 topological insulator, or a trivial photonic band gap material, depending on the specific gap opening perturbations [28]. Our study includes the triangle PhCs with dielectric rods as a special limit of R2=0. From the phase diagram, it is clear that the p-d band inversion cannot be attained using dielectric rods with R2=0. Thus the dielectric rod PhCs cannot support double Dirac cone. They can only support Dirac-like cone due to s-p degeneracy or quadratic band touching due to s-d degeneracy.

We plot in Fig. 3 the band structures of different kinds of accidental degeneracy’s marked by the black points in Fig. 2. In Figs. 3(a) and 3(b), the linear dispersion due to the accidental degeneracy of the singlet s mode and the doubly degenerate p modes are plotted, leading to the Dirac-like cone. The Dirac-like cone is associated with a cone and a flat band intersecting at the Dirac point. Its effective Hamiltonian is = v0k⃗ · S⃗ where S⃗ is the pseudo-spin 1 consisting of three modes s, px, and py [20]. v0 is the group velocity around the Dirac point. The upper, lower and flat bands have pseudo-spin along the wave vector direction as 1, −1, and 0, respectively.

 figure: Fig. 3

Fig. 3 Band structures of different kinds of accidental degeneracy. (a) The band structure of the A point in the phase diagram with R1 = 0.1816a1 and R2 = 0. (b) The band structure of the B point in the phase diagram with R1 = 0.17a1 and R2 = 0.0947a1. (c) The band structure of the C point in the phase diagram with R1 = 0.5a1 and R2 = 0.4624a1. (d) The band structure of the D point in phase diagram with R1 = 0.28a1 and R2 = 0.0697a1. (e) The band structure of the E point in the phase diagram with R1 = 0.41a1 and R2 = 0.3844a1. (f) The band structure of the F point in the phase diagram with R1 = 0.45a1 and R2 = 0.2656a1. The geometry of the core-shell material in a unit cell is shown at the left-down corner of each figure. The colored curves labeled with s, p, d, and f represent the modes at the Γ point only (instead of the whole bands). We zoom in the dispersion near the nodal points with a dashed frame.

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The Dirac-like cone can emerge in PhCs with R2 = 0 (cylinder rods) as well as for finite R2 (hollow cylinders). The photonic dispersions for these two situations are illustrated in Figs. 3(a) and 3(b), respectively. Another Dirac-like cone dispersion is shown in Fig. 3(c) due to accidental degeneracy of the singlet f mode and the doubly degenerate d modes. Linear, conical dispersion appears for the s-p and d-f degeneracy’s because the parity of the s and p (d and f) modes are different. According to the k⃗ · P⃗ theory [29,30], the linear in k coupling between two photonic bands originates from the P matrix element which is finite only between bands with different parities. For bands with the same parity, such coupling is quadratic [29, 30]. The quadratic dispersion around the touching point of the s and d bands is shown in Fig. 3(d), while the quadratic band touching around the f -p degeneracy is shown in Fig. 3(e).

The accidental degeneracy of the p doublets and the d doublets [Fig. 3(f)] carries nontrivial topological properties which are the main focus of this work. Since the parity of the p modes and the d modes are different, the dispersion around the accidental degeneracy is conical. The two fold degeneracy’s of the p and d bands result in a double Dirac cone that resembles the dispersion of Dirac’s famous equation for electron and positron with vanishing mass in 2D systems [14,21].

We now exploit the k⃗ · P⃗ theory to derive the effective Hamiltonian for the photonic bands near the nodal points. The Maxwell equations for the TM modes can be written as

×1ε(r)×hn,k(r)=ω2c2hn,k(r),
where n is the band index and h⃗n,k⃗(r⃗) is the Bloch function of the magnetic field of photon. The Bloch function is normalized as u.c.drhn,k*(r)hn,k(r)=δnn with u.c. denoting the unit cell (i.e., integration in a unit cell). The Hermitian operator ×1ε(r)× can be regarded as the photonic Hamiltonian. Expanding the Bloch function h⃗n,k⃗(r⃗) in the basis of the Bloch wavefunctions at the Γ point, h⃗n,0(r⃗), one can establish a k⃗ · P⃗ Hamiltonian,
nn(k)=ωn,02c2δnn+kPnnu.c.drε(r)hn,0*(r)[k×(k×hn,0(r))],
where ωn,0 is the eigen-frequency of the nth band at the Γ point. The matrix element of P⃗ is given by
Pn,n=u.c.drε(r)[hn,0*(r)×(i×hn,0*(r))+(i×hn,0(r))×hn,0*(r)].
We notice that the matrix element of P⃗ is nonzero only when the n and n′ bands are of different parity. Using the above k⃗ · P⃗ theory, to the linear order in k⃗, the effective Hamiltonian of the p and d bands is written in the basis of (p+, p, d+, d)T as,
=(ωp2c20Ak+00ωp2c20A*kA*k0ω2dc200Ak+0ω2dc2),
where k± = kx ± iky, and A is the coupling coefficient. The double Dirac-cone appears at the situations with p-d degeneracy, ωp = ωdω0. The group velocity for the double Dirac-cone dispersion at Γ point is then ±|A|c22ω0 (positive group velocity for bands above the Dirac point, negative group velocity for bands below). The p bands behave as the valence band and the d bands behave as the conduction band in our PhCs. Note that the coupling between the p and d bands are within the same pseudo-spin, i.e., between p+ and d+, or between p and d. The Berry phase for a loop circulating the Dirac point is ±π for spin-up/down bands above the Dirac point. The total Berry phase is zero, in accordance with time-reversal symmetry.

The physics described by Eq. (5) resembles that of the quantum spin Hall effect in electronic systems [18]. The p-d inversion at the Γ point leads to the formation of photonic Z2 topological insulators which have helical edge states. This phenomenon is first discovered in [14]. The phase transition from normal photonic band gaps with trivial topology to the photonic Z2 topological insulator takes place at the black line in Fig. 2, where the double Dirac cone emerges. The key information in Fig. 2 is the appearance of two regions support photonic topological insulators (the two blue regions). This takes place for hollow dielectric cylinders with large outer and inner radii, or for hollow air cylinders with small inner radius. We remark that although Δωpd can be quite large, ∼ 40%, for both normal and reversed structures, the complete photonic band gap is considerably smaller, particularly for the inverse structure. The photonic Z2 topological insulators have helical edge states which can enable unprecedented manipulation of light flow. For example, light propagation can be controlled by the orbital angular momentum. We shall discuss the properties of the edge states below.

It is natural to ask how the phase diagram changes when the permittivity ratio ε1/ε2 is tuned. In most PhCs the photonic band gap increases with the permittivity ratio ε1/ε2 [25]. Here we find that, quite interestingly, the phase boundary between the Z2 topological insulator and the normal photonic band gap changes negligibly for a very broad range of the permittivity ratio. This interesting property is demonstrated in Fig. 4 where we examine the phase boundary along the R2 axis for different permittivity ratio ε1/ε2 for fixed outer radius R1. The critical value of R2 where the double Dirac cone emerges is insensitive to the permittivity ratio ε1/ε2 in a very wide range. Although the calculation is done for R1 = 0.45a1, the observed behavior holds true for other values of the outer radius R1. In fact, we have chosen the R1 in the calculation such that the dependence of the critical value of R2 on the permittivity ratio ε1/ε2 is the strongest. The regions with negative value of ln(ε1/ε2) in Fig. 4 stands for the inverse structure PhC. The horizontal axis represents the case with ε1/ε2 = 1, i.e., the homogenous limit. In the homogeneous limit the photonic band gaps vanish and all the photonic bands become plane waves. Thus they cannot be associated with the s, p, d, and f symmetries. The homogeneous limit is a singular limit for the discussion the topological nodal points and p-d inversion. This explains the discontinuity of, e.g., the s-p band crossing curve for the normal and inverse structure in the phase diagram upon crossing the homogeneous limit. Nonetheless, the p-d band crossing seems to undergo a “continuous transition” from the normal structure to the inverse structure. The possible physical scenario in approaching the homogeneous limit is that the plane-wave component continuously increase to 100%, while the s, p, d, f wave (the local Mie resonances of the hollow cylinder) components gradually vanish in approaching the homogeneous limit.

 figure: Fig. 4

Fig. 4 Relationship between ln(ε1/ε2) and the critical inner radius R2 where the double Dirac cone emerges. The outer radius of the hollow cylinder is fixed at R1 = 0.45a1.

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4. Edge states of photonic Z2 topological insulators

We now discuss the properties of the edge states of the photonic Z2 topological insulators. First, let us consider the G and H points in Fig. 2. The photonic bands p and d are normally ordered (trivial band topology) at the G point, whereas they are reversely ordered (Z2 band topology) at the H point. There is a common complete band gap marked by the cyan region in Figs. 5(a) and 5(b). When these two PhCs are put together, there is topology-induced edge states. The Z2 topology is protected by the pseudo-time-reversal symmetry, of which the operation is defined as 𝒯p = iσ̂y K (K is the complex conjugation operator). Here σ̂y is an operator acting on the pseudo-spin space for both the p and d bands which can be written as a combination of the C6 symmetry operations (see [14]) to ensure that 𝒯p2=1. The Z2 band topology is protected by the double degeneracy at time-reversal invariant wavevectors induced by 𝒯p2=1 [26]. We remark that such anti-unitary operator can always be constructed whenever the double degeneracy of the p and d bands are kept [31]. Therefore, to simulate the Z2 band topology in PhCs, it is crucial to have two doubly degenerate bands with opposite parity (e.g., the p and d bands in our case).

 figure: Fig. 5

Fig. 5 Topology-induced edge states. (a) The normal band structure of the G point in phase diagram with R1 = 0.40a1 and R2 = 0.26a1. (b) The p-d reversed band structure of the H point in the phase diagram (R1 = 0.45a1 and R2 = 0.32a1). The common complete band gap is marked with the cyan ribbon. (c) The projected band structure of two PhCs with oblique line edge. A and B mostly comprise of the pseudo-spin-up and spin-down edge states, respectively. (d) and (e) are the Ez field pattern of A and B, respectively. The time-averaged Poynting vectors S⃗ = Re[E⃗ × H⃗*]/2 near the boundary (between the two dashed lines) are shown by the black arrows.

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The C6 symmetry breaks down at the boundary between the two different PhCs. Hence the boundary introduces perturbations that gap the helical edge states. The size of the gap depends on the strength of the perturbation. However, since the bulk topology of the two PhCs are distinctive and well-defined, the emergence of edge states is deterministic, although they may not be gapless. This is the essential feature for all bosonic analog of the Z2 topological insulators [32]. A typical calculation of the edge states using 16 unit cells of each type of PhCs (32 unit cells in total) via COMSOL MultiPhysics is presented in Fig. 5(c). The helical edge states are gapped at kx = 0. Away from the kx = 0 point the helical feature of the edge states is clearly demonstrated in Figs. 5(d) and 5(e). The edge state at the B point is mostly spin-down as recognized from the real space distribution of the Poynting vector. Similarly the A point is mostly spin-up and it has positive group velocity, while the B point has negative group velocity. Hence the helical character of the edge states is preserved away from the kx = 0 point.

From symmetry considerations, the boundaries break the pseudo-time-reversal symmetry but still keep the genuine time-reversal symmetry. The genuine time-reversal operator is 𝒯 = σ̂x (since |↑〉 = |px + ipy〉 and |↓〉 = |pxipy〉 for p bands, while ↑〉 = |dx2y2 + idxy〉 and |↓〉 = |dx2y2idxy〉 for the d bands). The spin-operators that are even under time-reversal transformation are σ̂x and σ̂y. Thus there can be two types of “mass terms” that gap the edge states. The general form of the edge Hamiltonian that obeys the (genuine) time-reversal symmetry is edge = vkxσ̂z + mxσ̂x + myσ̂y where v is the group velocity at the Γ point, mx and my are two real quantities. The magnitude of the two masses, mx and my, depend on the specific geometry of the boundary. The energy gap at kx = 0 is 2mx2+my2.

The edge thus has a single branch of massive Dirac spectrum with Chern number ±1/2 (sign depends on the mass terms). This extraordinary property can not be realized in a normal waveguide and is unique to the edge states of the Z2 topological photonic band gap materials. This is a realization of “parity anomaly” in quantum field theory for bosonic particles. If the signs of the mass terms can be modulated, there can be edge solitons emerging at the boundaries [33].

In situations where the two PhCs do not share a common photonic band gap. We can change the ε1 or ε2 of the structure to get the common photonic band gap. Since the p-d band inversion is insensitive to the permittivity (as shown in Fig. 4), such variation of the permittivity does not change the topological properties of the two PhCs. For example, if the PhC with normal band structure has R1 = 0.45a1, R2 = 0.2a1, ε1=9, and ε2=1 [see in Fig. 6(a)], while the PhC with nontrivial Z2 topology has R1 = 0.45a1, R2 = 0.3a1, ε1=12, and ε2=1 [see in Fig. 6(b)]. The two PhCs have a common band gap marked by the cyan ribbon. When those two PhCs are put together the topology-induced edge states are found [see in Fig. 6(c)].

 figure: Fig. 6

Fig. 6 Photonic band structure and topological edge states. (a) The normal band structure with R1 = 0.45a1, R2 = 0.2a1, ε1 = 1, and ε2 = 9. (b) The p-d reversed band structure with R1 = 0.45a1, R2 = 0.3a1, ε1 = 1, and ε2 = 12. They have a common band gap marked with cyan ribbon. (c) Projected band structure of the two PhCs with a line boundary.

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5. Application potentials

One of the applications of the topological nodes, such as Dirac and Dirac-like points, is that they can be used as effective medium with unconventional refractive index. It was first found in [20] that PhCs with a Dirac-like point can be used as effective medium with zero refractive index when the frequency of light is at the Dirac point [20]. We now show that for the double Dirac cone, the effective refractive index of the PhC can be positive, zero, or negative, depending on the frequency of the light. By matching the frequency and wave vector parallel to the boundary, we find, in agreement with [34], that the effective refractive index is frequency dependent,

n(ω)=sin(θ1)sin(θ2)=2ω0|A|c(ωω0ω),
where θ1 and θ2 are the angle of incidence and refraction, respectively. ω0 is the frequency of the Dirac point. The above equation demonstrates that the effective refractive index can be tuned via the frequency. Both positive and negative refractive indices can be achieved, as indicated in Fig. 7. We emphasize that the effective refractive index here are in the range −1 < n(ω) < 1, which are unattainable for natural (lossless) materials. Therefore, the PhC with double Dirac cone can serve as a particular type of lossless metamaterial with unprecedented ability of manipulating light.

 figure: Fig. 7

Fig. 7 Optical property near double Dirac Cone. (a) The band structure near double Dirac cone with R1 = 0.45 and R2 = 0.2656. Four cone-like surfaces touch at ω0 of the Γ point. (b) The iso-frequency surfaces for the frequency of higher (in red) and lower (in blue) than ω0, respectively. In the middle it is the iso-frequency surface of light in air. The refraction law is derived from the conservation of frequency and the wave vector parallel to the interface, kx. Right panel: the property of positive refraction (upper plot) and negative refraction (lower plot).

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The edge states can be viewed as unconventional waveguides that correlates the direction of light flow and its group velocity with the angular momentum of light (as demonstrated in Fig. 5). In addition, when the frequency of the light is in the bulk photonic bands, the propagation of light are influenced by the Berry phase in the bulk bands due to the pseudo-spin-orbit coupling, which leads to the spin Hall effect of light [35,36]. The spin-dependent Berry phase can lead to anomalous velocity and shift in a scattering/reflection set-up [36]. Due to the time-reversal symmetry, these anomalous velocity and shift are opposite for opposite pseudo-spins. These properties can be exploited as angular-momentum selective transmission and filtering for advanced photonic circuits.

6. Conclusion and discussions

Using a simple architecture of core-shell triangle PhC with C6 point group symmetry we have systematically studied the evolution of the nodal points in photonic energy bands for various geometry and (isotropic) permittivity parameters. We show that such a simple PhC can support Z2 photonic topological insulators. The edge states of the Z2 topological insulators, unlike for electronic systems, are not ensured to be gapless. Nevertheless, they show properties similar to the helical edge states in electronic systems, such as spin-dependent propagation directions. We also give a full phase diagram for the topological nodal points, Dirac-like cones and double Dirac cones, for various geometric parameters. These topological nodal points are proximate to topologically nontrivial photonic band gaps. The physical origin of the topological nodal points as well as their properties such as Berry phase, (pseudo-)spin-orbit coupling are revealed. Dirac cones at k⃗ = 0 can be exploited to tune the effective refractive index to the range −1 < n(ω) < 1 which are unattainable in natural, lossless materials. These properties, together with the Berry phases of photon, offer great potential for future advanced photonics.

Funding

Soochow University; National Natural Science Foundation of China (NSFC) for Excellent Young Scientists (grant no. 61322504).

Acknowledgments

We thank support from the faculty start-up funding of Soochow University. J.H.J thanks Sajeev John, Xiao Hu, and ZhiHong Hang for helpful discussions.

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4. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013). [CrossRef]  

5. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013); [CrossRef]  

6. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013). [CrossRef]  

7. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014). [CrossRef]  

8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013). [CrossRef]  

9. L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015). [CrossRef]   [PubMed]  

10. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]   [PubMed]  

11. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014). [CrossRef]  

12. W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014). [CrossRef]   [PubMed]  

13. W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014). [CrossRef]  

14. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015). [CrossRef]   [PubMed]  

15. H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016). [CrossRef]  

16. W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

17. L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016). [CrossRef]  

18. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011). [CrossRef]  

19. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007). [CrossRef]  

20. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef]   [PubMed]  

21. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012). [CrossRef]   [PubMed]  

22. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998) [CrossRef]  

23. K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002). [CrossRef]  

24. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).

25. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

26. L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007). [CrossRef]  

27. J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

28. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009). [CrossRef]  

29. Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013). [CrossRef]   [PubMed]  

30. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012). [CrossRef]  

31. J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

32. C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014). [CrossRef]  

33. D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007). [CrossRef]  

34. L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

35. M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011). [CrossRef]   [PubMed]  

36. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004). [CrossRef]   [PubMed]  

References

  • View by:

  1. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
    [Crossref] [PubMed]
  2. K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
    [Crossref]
  3. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
    [Crossref]
  4. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
    [Crossref]
  5. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
    [Crossref]
  6. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
    [Crossref]
  7. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
    [Crossref]
  8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
    [Crossref]
  9. L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
    [Crossref] [PubMed]
  10. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
    [Crossref] [PubMed]
  11. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
    [Crossref]
  12. W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
    [Crossref] [PubMed]
  13. W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
    [Crossref]
  14. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref] [PubMed]
  15. H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
    [Crossref]
  16. W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.
  17. L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
    [Crossref]
  18. X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
    [Crossref]
  19. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
    [Crossref]
  20. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
    [Crossref] [PubMed]
  21. K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
    [Crossref] [PubMed]
  22. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
    [Crossref]
  23. K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
    [Crossref]
  24. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).
  25. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).
  26. L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
    [Crossref]
  27. J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).
  28. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
    [Crossref]
  29. Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
    [Crossref] [PubMed]
  30. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
    [Crossref]
  31. J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.
  32. C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
    [Crossref]
  33. D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
    [Crossref]
  34. L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.
  35. M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
    [Crossref] [PubMed]
  36. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
    [Crossref] [PubMed]

2016 (2)

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

2015 (2)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

2014 (6)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

2013 (6)

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

2012 (2)

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

K. Sakoda, “Dirac cone in two- and three-dimensional metamaterials,” Opt. Express 20(4), 3898–3917 (2012).
[Crossref] [PubMed]

2011 (3)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

2009 (2)

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

2008 (1)

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

2007 (3)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

2004 (1)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

2002 (1)

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

1998 (1)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

1980 (1)

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Barkeshli, M.

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Capasso, F.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Chen, H.-Y.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Chen, W.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Chen, X.

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Dong, J.-W.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Dorda, G.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Fang, C.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

Fu, L.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

Hafezi, M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

He, W.-Y.

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

Hou, J.-M.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

Hsu, C. W.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Hu, X.

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Jian, S.-K.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Jiang, J.-H.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Jiang, S.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Joannopoulos, J. D.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
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L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Johnson, S. G.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Kane, C. L.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

Kargarian, M.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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Khanikaev, A. B.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Klitzing, K. v.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Lee, D.-H.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

Li, Y.

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Lu, L.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Lubensky, T. C.

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

Lumer, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

MacDonald, A. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Mei, J.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Mei., J.

Migdall, A.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Mittal, S.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Moroz, A.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Mousavi, S. H.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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Murakami, S.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Nagaosa, N.

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
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Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

Pepper, M.

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

Plotnik, Y.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Podolsky, D.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Qi, X.-L.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

Ran, L.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Sakoda, K.

Segev, M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Shvets, G.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Soljacic, M.

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Stone, A. D.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Taylor, J. M.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Tse, W. K.

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

van Blaaderen, A.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Velikov, K. P.

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

Wang, H.-X.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Wang, L.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Wang, Z.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Wu, L. H.

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

Wu, S.

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Wu, Y.

Y. Li, Y. Wu, X. Chen, and J. Mei., “Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals,” Opt. Express 21(6), 7699–7711 (2013).
[Crossref] [PubMed]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Xiang, T.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
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Xiao, M.

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

Xu, L.

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Yao, H.

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

Ye, D.

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Zeuner, J. M.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Zhang, G.-M.

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

Zhang, S.-C.

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

Zhen, B.

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

K. P. Velikov, A. Moroz, and A. van Blaaderen, “Photonic crystals of core-shell colloidal particles,” Appl. Phys. Lett. 80(1), 49–51 (2002).
[Crossref]

J. Phys.: Condens. Matter (1)

J.-H. Jiang and S. Wu, “Non-Abelian topological superconductors from topological semimetals and related systems under the superconducting proximity effect,” J. Phys.: Condens. Matter 25(5), 055701 (2013).

Nat. Commun. (1)

W.-J. Chen, S.-J. Jiang, X.-D. Chen, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Nat. Mater. (3)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Nat. Photon. (3)

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photon. 7(4), 294–299 (2013).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photon. 8(11), 821–829 (2014).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photon. 7(12), 1001–1005 (2013).
[Crossref]

Nat. Phys. (2)

L. Lu, C. Fang, L. Fu, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Symmetry-protected topological photonic crystal in three dimensions,” Nat. Phys. 12(4), 337–340 (2016).
[Crossref]

C. L. Kane and T. C. Lubensky, “Topological boundary modes in isostatic lattices,” Nat. Phys. 10(1), 39–45 (2014).
[Crossref]

Nature (London) (2)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature (London) 496(7444), 196–200 (2013);
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature (London) 461(7265), 772–775 (2009).
[Crossref]

Opt. Express (2)

Phys. Rev. A (1)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75(6), 063813 (2007).
[Crossref]

Phys. Rev. B (4)

L. Fu and C. L. Kane, “Topological insulators with inversion symmetry,” Phys. Rev. B 76(4), 045302 (2007).
[Crossref]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, “First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals,” Phys. Rev. B 86(3), 035141 (2012).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80(15), 155103 (2009).
[Crossref]

H.-X. Wang, L. Xu, H.-Y. Chen, and J.-H. Jiang, “Three-dimensional photonic Dirac points stabilized by point group symmetry,” Phys. Rev. B 93, 235155 (2016).
[Crossref]

Phys. Rev. Lett. (8)

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljačić, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45(6), 494–497 (1980).
[Crossref]

D.-H. Lee, G.-M. Zhang, and T. Xiang, “Edge solitons of topological insulators and fractionalized quasiparticles in two dimensions,” Phys. Rev. Lett. 99(19), 196805 (2007).
[Crossref]

M. Barkeshli and X.-L. Qi, “Topological response theory of doped topological insulators,” Phys. Rev. Lett. 107(20), 206602 (2011).
[Crossref] [PubMed]

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004).
[Crossref] [PubMed]

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81(7), 1405–1408 (1998)
[Crossref]

Rev. Mod. Phys. (1)

X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83(4), 1057–1110 (2011).
[Crossref]

Sci. Rep. (1)

W.-Y. He and C. T. Chan, “The emergence of Dirac points in photonic crystals with mirror symmetry,” Sci. Rep. 5, 8186 (2014).
[Crossref]

Science (1)

L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos, and M. Soljačić, “Experimental observation of Weyl points,” Science 349(6248), 622–624 (2015).
[Crossref] [PubMed]

Other (5)

W.-J. Chen, M. Xiao, and C. T. Chan, “Experimental observation of robust surface states on photonic crystals possessing single and double Weyl points,” arXiv:1512.04681.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2005).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

L. Wang, S.-K. Jian, and H. Yao, “Topological photonic crystal with equifrequency Weyl points,” arXiv:1511.09282.

J.-M. Hou and W. Chen, “Accidental degeneracy and hidden antiunitary symmetry protection: a novel Weyl semimetal phase as an example,” arXiv:1507.02024.

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic configuration of a triangle PhC using core-shell dielectric materials. The dielectric constant of the core-shell dielectric material and background material are ε2 and ε1, respectively. R1 and R2 are the outer and inner radii of the core-shell cylinder. a⃗1 and a⃗2 are the two basis vectors of the triangular lattice. We set a1 = a2 ≡ 1 as the lattice constant. The inverse structure is defined by exchange of the permittivity ε1 and ε2. (b) A typical band structure of the core-shell triangle lattice PhC with R1 = 0.16a1, R2 = 0.05a1, ε1 = 1 and ε2 = 12. We use the symbols s, p, d and f to label the modes at the Γ point.
Fig. 2
Fig. 2 Phase diagram of the p-d-inversion-induced photonic Z2 topological insulator in the R1-R2 parameter space. The diagonal line represents the homogeneous limit which separates the normal and reversed structures in the upper and lower triangular regions. The lower region is the parameter space for core-shell dielectric cylinder in air background (called as the normal structure), while the upper region is the parameter space for core-shell air cylinder in dielectric background. The dielectric constant for air and dielectric are ε1 = 1 and ε2 = 12, respectively. The contour color represents the value of Δωpd. The red region represents that the p bands are lower than the d bands at the Γ point, whereas the blue region stands for the photonic analog of the Z2 topological insulators with the p bands above the d bands at the Γ point. The p-d bands degeneracy at the Γ point is labeled by the black line. The s-p, p-f, s-d, f -d band inversions are also calculated and plotted. Several points in the phase diagram are labeled, which will be used for the discussions in the main text.
Fig. 3
Fig. 3 Band structures of different kinds of accidental degeneracy. (a) The band structure of the A point in the phase diagram with R1 = 0.1816a1 and R2 = 0. (b) The band structure of the B point in the phase diagram with R1 = 0.17a1 and R2 = 0.0947a1. (c) The band structure of the C point in the phase diagram with R1 = 0.5a1 and R2 = 0.4624a1. (d) The band structure of the D point in phase diagram with R1 = 0.28a1 and R2 = 0.0697a1. (e) The band structure of the E point in the phase diagram with R1 = 0.41a1 and R2 = 0.3844a1. (f) The band structure of the F point in the phase diagram with R1 = 0.45a1 and R2 = 0.2656a1. The geometry of the core-shell material in a unit cell is shown at the left-down corner of each figure. The colored curves labeled with s, p, d, and f represent the modes at the Γ point only (instead of the whole bands). We zoom in the dispersion near the nodal points with a dashed frame.
Fig. 4
Fig. 4 Relationship between ln(ε1/ε2) and the critical inner radius R2 where the double Dirac cone emerges. The outer radius of the hollow cylinder is fixed at R1 = 0.45a1.
Fig. 5
Fig. 5 Topology-induced edge states. (a) The normal band structure of the G point in phase diagram with R1 = 0.40a1 and R2 = 0.26a1. (b) The p-d reversed band structure of the H point in the phase diagram (R1 = 0.45a1 and R2 = 0.32a1). The common complete band gap is marked with the cyan ribbon. (c) The projected band structure of two PhCs with oblique line edge. A and B mostly comprise of the pseudo-spin-up and spin-down edge states, respectively. (d) and (e) are the Ez field pattern of A and B, respectively. The time-averaged Poynting vectors S⃗ = Re[E⃗ × H⃗*]/2 near the boundary (between the two dashed lines) are shown by the black arrows.
Fig. 6
Fig. 6 Photonic band structure and topological edge states. (a) The normal band structure with R1 = 0.45a1, R2 = 0.2a1, ε1 = 1, and ε2 = 9. (b) The p-d reversed band structure with R1 = 0.45a1, R2 = 0.3a1, ε1 = 1, and ε2 = 12. They have a common band gap marked with cyan ribbon. (c) Projected band structure of the two PhCs with a line boundary.
Fig. 7
Fig. 7 Optical property near double Dirac Cone. (a) The band structure near double Dirac cone with R1 = 0.45 and R2 = 0.2656. Four cone-like surfaces touch at ω0 of the Γ point. (b) The iso-frequency surfaces for the frequency of higher (in red) and lower (in blue) than ω0, respectively. In the middle it is the iso-frequency surface of light in air. The refraction law is derived from the conservation of frequency and the wave vector parallel to the interface, kx. Right panel: the property of positive refraction (upper plot) and negative refraction (lower plot).

Equations (6)

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Δ ω p d = 2 ω d ω p ω d + ω p ,
× 1 ε ( r ) × h n , k ( r ) = ω 2 c 2 h n , k ( r ) ,
n n ( k ) = ω n , 0 2 c 2 δ n n + k P n n u . c . d r ε ( r ) h n , 0 * ( r ) [ k × ( k × h n , 0 ( r ) ) ] ,
P n , n = u . c . d r ε ( r ) [ h n , 0 * ( r ) × ( i × h n , 0 * ( r ) ) + ( i × h n , 0 ( r ) ) × h n , 0 * ( r ) ] .
= ( ω p 2 c 2 0 A k + 0 0 ω p 2 c 2 0 A * k A * k 0 ω 2 d c 2 0 0 A k + 0 ω 2 d c 2 ) ,
n ( ω ) = sin ( θ 1 ) sin ( θ 2 ) = 2 ω 0 | A | c ( ω ω 0 ω ) ,

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