As a landmark of topological photonics, Wang et al. [27] first experimentally demonstrated unidirectional electromagnetic wave featured with backscattering immunity in the 2D magneto-optical square-lattice PhCs that made of yttrium iron garnet (YIG). When an external DC magnetic field is applied, the YIG produces strong magnetic anisotropy, making the PhC a magnetic insulator. The breaking of T via an external applied magnetic field is essential to realize the PCI. Figure 3A presents the band structure of PhCs with T breaking adopted from Wang’s scheme. The primitive unit cell is shown in the inset. The nontrivial band topology can be demonstrated via the first-principle calculation of band Chern number, which can be implemented through the integral of Berry curvature in the discretized Brillouin zone [71] or the Wilson loop approach [63]. Here, we introduce the main idea of the Wilson loop approach to calculate the band Chern number. Note that the Chern number that is related to the Berry phase can be calculated using the following relations [72]: 2 π C n = − ∫ − π π ∫ − π π d k x d k y ( ∂ k x Λ n , k ( y ) − ∂ k y Λ n , k ( x ) ) = ∫ − π π d k y ∂ k y [ ∫ − π π d k x Λ n , k ( x ) ] ≡ ∫ − π π d θ n , k y , (5) where θ n , k y ≡ ∫ − π π d k x Λ n , k ( x ) (6) is the Berry phase for the nth band along the loop k _x ∈ [−π,π] for a fixed k _y which is obtained by the integration over the Berry connection through Eq. 2 for the Wilson loop along k _x . Note that the 2D BZ is equivalent to a torus under the periodic boundary condition for the Bloch states. Therefore, in numerical calculations, the Chern number C _n is obtained by counting the winding phase of when k _y goes from −π to π. The Chern numbers of the first three bands are determined by plotting the Berry phase θ _n,kx (n = 1, 2, and 3) as functions of k _x . For the first photonic band, the Berry phase remains zero for all k _x , leading to a zero Chern number, while for the second (third) photonic band, the Berry phase has a winding number of 1 (-2), corresponding to a Chern number C₂ = 1(C₃ = −2), as shown in Figures 3B–D. The Chern numbers calculated here from the Wilson loop approach agree with the Chern numbers inferred from the chiral edge states in using the bulk-edge correspondence [73]. As shown in Figure 3E, the projected band structure (light blue areas) with a chiral edge state (red line) is calculated in finite systems. A typical field distribution of the chiral edge state is also displayed in Figure 3F. To characterize the chiral edge states, the transmission measurement is implemented. From Figure 3G, it is shown that a strong forward transmission with the second band gap approximately 50 dB greater than the backward transmission at frequencies was observed at mid-gap frequencies. After this milestone of work, a series of work related to the PCIs made of gyromagnetic PhCs are implemented, such as cladding-free guiding of topologically protected edge states [74, 75], steering of multiple edge states along domain walls with large Chern numbers [76, 77], designing of one-way slow-light PhC waveguide [78–81], and antichiral edge states [82, 83]. It is evident that the unidirectional backscattering immunity edge states are expected to have a deep impact on the designing of new optical devices. However, owing to the weak magnetic response in optical materials and the difficulty in device integration, the use of PCIs in optical devices remains a challenge. To date, there is an urgent need to achieve one-way waveguides at optical wavelength.

In 2015, Wu et al. [36] proposed a scheme for achieving PTI by using the all-dielectric PhC, which paves a way for the practical application of PTI. In their seminal work, they start with an expanded cell of photonic honeycomb lattice. Based on the band folding mechanism, the well-known Dirac cone at K and K′ points are folded at Γ point, giving rise to the deterministic double Dirac cone. By either stretching or compressing the expanded unit cell, the double Dirac cone lifted into a trivial or nontrivial band gap. Such a scheme for realizing PTI was later demonstrated by observing the momentum–pseudospin locking in the microwave experiment [84]. Inspired by Wu’s work, a large number of studies of all-dielectric PTI are implemented based on the photonic honeycomb lattice [85–96] and other specific PhC structures, such as core–shell PhCs [97], Stampfli-triangle PhCs [98], and moon-shaped PhCs [99].

Actually, to construct all-dielectric PTI, two issues should be highlighted: one is to create the fourfold-degeneracy double Dirac cone, which is the mother state of PTI, the other one is to find out the photonic analog spin–orbit coupling terms. Guided by these two principles, Xu et al. [97] systematically studied accidental band degeneracy in an all-dielectric core–shell PhC (Figure 4A), where the Mie resonance can be regarded as atomic orbits for photonic bands [100]. Those atomic orbits can be of s, p, d, and f nature and have well-defined parities at Γ point [see a typical band structure of the core–shell PhCs in Figure 4B]. Note that due to the C ₆ crystalline symmetry, both the photonic p-orbit and d-orbit are double degenerate. Figure 4C gives the electric field patterns of the p doublets (p _x and p _y ) and d doublets (d _xy and d x ²−y ²). In particular, these four states can be linearly combined into p± = (p _x ± ip _y )/√2 and d± = (d x ²−y ² ±id _xy )/√2, where subscript + (−) refers to pseudospin-up (pseudospin-down). Apparently, the spin DoF here is synthesized by the orbital angular momentum [89]. By tuning the inner and outer radii of the core–shell PhCs, the phase diagram with multiple accidental degeneracies can be acquired (Figure 4D).

Importantly, the double Dirac cone formed by the p and d doublets plays a vital role in the phase transition between PTI and a normal insulator. When the p band is below the d band (Figure 4E), the gap exhibits a trivial phase, while flipping the order of the p and d photonic bands results in a nontrivial band gap (Figure 4F). Such a parity inversion (also known as p-d inversion) picture is at the heart of the quantum spin Hall effect in electronic systems [31]. The topological invariant of PTI can be also acquired via the Wilson loop approach [63]. Considering the Berry phase calculation for multiple bands, the inner product 〈 ψ k i | ψ k i + 1 〉 in Eq. 2 should be replaced by matrix M ^ˆki,ki+1 of which the matrix elements are given. M n n ′ k i , k i + 1 = 〈 ψ n , k i | ψ n ′ , k i + 1 〉 , n , n ′ ∈ 1 … N , (7) where the band indices n and n′ go over all the bands below the concerned band gaps. Then, the Berry phase for a loop in the BZ can be obtained by the matrix product of the Berry connection matrix through the following form, W ^ = ∏ i = 1 N M ^ k i , k i + 1 . (8)

To evaluate the topological invariant, one needs the eigenvalues of the above Berry phase matrix, which can be written as follows: θ n ≡ − ℑ [ log ( w n ) ] , n = 1 , … , N , (9) where w _n is the nth eigenvalues of the matrix W ^ .

Typical Berry phase calculations for a trivial insulator and PTI via the Wilson loop approach are displayed in Figures 4G,H. For the trivial case, the Wilson loop of the first band has a constant value equal to 0 (see left panel in Figure 4G), while that of the set of the second and third bands does not exhibit any winding properties (see right panel in Figure 4G). For the nontrivial case, it is necessary to take the three lowest bands as a set since there is a band crossing between the first and second bands. As shown in Figure 4H, there is no winding but the Wannier centers are localized at the edge of the unit cell, in contrast to that in the trivial case. In the literature, such a topological insulator with non-winding values of Wilson loop is called the photonic obstructed atomic limit [101, 102], which is easily confusing with the concept of fragile topological insulator [96, 99, 102, 103]. Parallel to the topological invariant calculation, a typical calculation of the edge state using two PhCs with distinct topology is presented in Figure 4I. Note that the helical edge states are gapped, which originate from the C ₆ symmetry breaking at the boundary between the two different PhCs. The size of the gap depends on the strength of the perturbation induced by the symmetry breaking. However, the helical feature of the edge states is still clearly demonstrated. As shown in Figures 4J,K, the edge states at the J(K) point are mostly pseudospin-down (pseudospin-up) as recognized from the real space distribution of the Poynting vector with the negative (positive) group velocity.

Utilizing the robust transport properties of the topological PhC interface, many intriguing physical systems are explored, including unidirectional electromagnetic waveguide [84], topological all-optical logic gates [104], topological whispering gallery modes [105], coupled cavity-waveguide system [106], topological converter [95], topological bulk laser [93], Dirac vortex cavity [107], and Dirac vortex fiber [108]. Additionally, all-dielectric PTIs also open an avenue to quantum optics. In 2018, Barik et al. [86] proposed an all-dielectric topological PhC slab with 2D helical edge states confined in a dielectric slab, which is highly desirable to achieve out-of-plane confinement without the use of metal [85, 86, 90, 94]. They also demonstrate the strong interface between the single quantum emitters and topological photonic states [94]. Since all-dielectric PTIs take full advantage of crystalline symmetry and thus go beyond the material limitation, more in-depth research based on all-dielectric PTIs is foreseen.

Another kind of topology-related gapped systems with T is PVHIs [40, 109–135]. Following the idea of valleytronics in the graphene, Chen et al. [119] proposed a PVHI based on the modified honeycomb PhCs, as shown in Figure 5A, where the radii of the two rods are different in a unit cell. Note that most PVHI studies are based on the modified honeycomb lattice [110, 111, 113, 116, 117, 119, 122, 123] since it provides a concise physical picture. Other proposals, including detuning the refractive index [124, 125] and specific geometric designs without P [40, 109, 112, 114, 115, 118, 120, 121, 126], are also adopted to study valley physics. From the point of view of the symmetry, these PVHIs broke the C ₆ symmetry of the structure while preserved the C ₃ symmetry. A typical band structure of PVHI is presented in Figure 5B, where two inequivalent but T valleys (K and K′) with vortex-valley locking (Figures 5C,D) are observed.

Using the valley as a binary DoF, the unidirectional excitation of the valley chirality bulk states can be realized either by sources carrying orbital angular momentum with proper chirality (Figures 5E,F) [119] or by a point-like chiral source based on the azimuthal phase matching condition [120]. Similar to the pseudospin–momentum locking effect in PTIs, there also exist valley pseudospin–momentum locking edge states at the interface of two valley Hall PhCs (Figures 5G,H). Most studies hold that the different valley topological index between two of the valley Hall PhCs gives a nonzero valley Chern number and lead to the emergence of the valley-dependent edge states. However, the valley Chern number cannot be used as a topological invariant because it is not a quantized value. Yang et al. [117] addressed that the chiral vortex-valley locking plays a fundamental role in the emergence of the valley-dependent edge states, rather than the valley Chern number. Thanks to the chiral vortex-valley locking, valley PhCs with valley-dependent edge states exhibit robust transportation against sharp corners (Figure 5I).

For applications, many intriguing photonic devices, including energy beam splitters [127, 131, 132, 135], logic gates [127], switches [132], fiber [134], and filter [132] are created via PVHIs [131, 132]. Based on valley edge states, the electrically pumped terahertz quantum cascade laser is also realized [128]. Xie et al. [133] constructed a topological cavity based on slow-light valley Hall edge states, which exhibit a greatly enhanced Purcell factor. Moreover, it is also desirable to realize PVHIs with a large Chern number, which can support multimode topological transmission [114, 118].

The recent advances in higher-order topology deepen our knowledge of the topology physics. The first HOTI proposed by Benalcazar et al. [41] is a quadrupole topological insulator (QTI), in which the bulk dipole is absent while the quantized, fractional electric multipole moments emerged in the bulk. The key to realize a QTI is to generate both positive and negative nearest-neighbor couplings in a single physical system. In order to meet such requirements, Chen et al. [136] proposed a scheme to realize QTI in plasmon-polaritonic systems by utilizing the sign-reversal mechanism for the coupling between the plasmon-polaritonic cavities. In addition, He et al. [137] extended the idea of QTI from tight-binding models to continuum theories. They demonstrated that the quadrupole topological phase survived in a gyromagnetic PhC, of which the quadrupole moment is quantized by the simultaneous presence of crystalline symmetry and broken T. Since the realization of photonic QTI is limited to some specific systems, most studies focus on the photonic HOTIs with quantized Wannier centers. In 2018, Ezawa [43] constructed a tight-binding model on the breathing kagome lattice, in which both gapped edge states and corner states are observed. In his proposal, the bulk polarization is served as a topological index, which can be defined as the integral of the Berry connection. Intuitively, the bulk polarization characterizes the displacement of the average position of the Wannier center from the center of the unit cell, giving rise to the emergence of corner states in a finite system.

Inspired by Ezawa’s proposal, the studies on the HOTI based on the kagome lattice are later transferred to various classical systems [50, 138–143]. In particular, all-dielectric PhCs provide an excellent platform to study HOTI with quantized Wannier centers and thus have been extensively studied [50, 144–154]. Wang et al. [154] systematically studied the multiple higher-order phase transition in a 2D hexagonal PhC with C ₃ symmetry, where each unit cell consists of three dielectric rods, as illustrated in Figure 6A. By moving the three dielectric rods along three symmetry lines, the PhCs undergo a continuous geometry transformation that includes three triangular lattice configurations and three kagome lattice configurations. Typical band structures for the triangular and kagome lattices are presented in Figures 6B,C. Accompanying with the geometry transformation, the first photonic band experiences multiple phase transitions (Figure 6D). To characterize the higher order topology, one can calculate the bulk polarization (Wannier center positions) via the Wilson loop approach, as depicted in Figures 6E–G, or use the symmetry indicators [155], which can be achieved from C ₃ eigenvalues at high symmetry points.

As a direct manifestation of the higher-order band topology, one may expect that corner states appear in a finite system with C ₃ symmetry. However, the emergence of corner states also depends on the geometric configurations [141, 156]. We remark that often the photonic bands do not have the chiral symmetry and the corner states may shift into the bulk continuum and disappear without the chiral symmetry. To avoid this, it is necessary to place two all-dielectric PhCs with distinct higher topological phases together in a finite system. Along with the geometry transformation, various cases of the calculated supercell are realized, where outside PhCs are of trivial phase, while the insider PhCs are dependent. Several prototype geometries of the calculated supercell are presented in Figure 6H. The eigen solutions are displayed systematically in Figure 6I. Figures 6J,K give the electric field |E _z | distributions of the eigenstates of the corner and edge states. In particular, two types of corner states emerge, as revealed in Refs. 50, 142, 154: type-I corner states due to the nearest neighbor couplings and type-II corner states originating from the next nearest-neighbor coupling.

In addition to the kagome lattice with C ₃ symmetry, the nontrivial bulk polarization and corner states can also appear in the expanded C ₄ symmetric lattice. A typical case is the 2D Su-Schrieffer–Heeger model, which has been extensively studied [151–153]. The 2D Zak phase is employed to characterize the higher-order topology. The photonic HOTIs have been found promising applications in high-quality nanocavities [157], cavity quantum electrodynamics [158], topological nanolasers [159, 160], and multi-channel system fibers [161]. Since HOTIs set up examples with multidimensional topological physics going beyond the bulk-edge correspondence in conventional topological insulators and semimetals, it opens a new avenue toward exploring novel topological phenomena and optical device applications.

Before proceeding, let us comment on the 1D and 2D all-dielectric topological PhCs. Since all-dielectric PhCs are the optical analogs of conventional crystals, it is natural to explore different topological phases as well as to find the potential applications based on all-dielectric PhCs. For 1D topological photonics, the 1D PhC consists of a dielectric AB layered structure that is regarded as a photonic analog of the SSH model. Since the 1D binary PhC supports the Tamm mode, which originates from the lack of translation symmetry, there is a need to clarify the difference as well as a connection between the topological interface states and Tamm modes [162]. For 2D cases, topological phases in all-dielectric PhCs are more diverse, including PCI, PTI, PVHI, and HOTI. We notice that most studies focus on the topological phase transition between trivial and nontrivial phases, while that between two nontrivial phases is very few [163–165]. It is expected that all-dielectric PhCs will become more versatile if two topological phases coexist in a single system. In addition, it is also interesting to introduce other ingredients into the all-dielectric PhCs, such as layer DoF [111, 166].

Generally, 3D topological insulators can be realized in both T-broken and T-invariant system. A typical T-broken case is the 3D quantum Hall phase [also known as a strong topological insulator (STI)], which can be regarded as a 3D extended version of the 2D quantum Hall phase. To the best of our knowledge, Lu et al. [58] first proposed the photonic analog of 3D STI by using PhCs composed of ferrimagnetic materials (Figure 7A). A magnetic field bias breaks T and produces a nontrivial band gap that hosts a single-surface Dirac cone (Figures 7B,C), which is protected by the nonsymmorphic glide reflections. Such a gapless surface state is fully robust against the random disorder of any type. The evolution of Wannier centers is calculated via the Wilson loop approach to characterize the bulk topological invariant (Figure 7D).

Similarly, the 3D quantum spin Hall phase [also known as a weak topological insulator (WTI)] can be viewed as a generalization of the 2D quantum spin Hall phase and corresponds to the T-invariant case. In 2011, Yannopapas [167] proposed a scheme for realizing a 3D photonic analog of WTI. A tetragonal lattice of uniaxial dielectric cavities in a lossless metallic host was investigated using a coupled dipole method, which is the photonic counterpart of topological crystalline insulators in an electronic system [168]. This system with T and point-group symmetry exhibit a complete 3D band gap and gapless topological surface states. Topological photonics with 3D band gaps also have been proposed by using 3D bianisotropic structures [59, 169]. As shown in Figure 7E a stacked layer of triangular arrays of mirror-symmetry-broken dielectric rods (Figure 7F) supports a conical dispersion of topological surface states (Figures 7G,H) and backscattering immunity propagation of the surface modes. The scheme for achieving 3D photonic WTI paves a way to various optical devices, such as topological lasers and circuits in previously inaccessible 3D geometries.

As one of the earliest studies of topological semimetals, Lu et al. theoretically [60] and experimentally [61] demonstrated the existence of Weyl points in gyroid PhCs. The Weyl point refers to the 3D linear point between two bands, of which the dispersions are governed by the Weyl Hamiltonian. Unlike the 2D Dirac point, which is protected by PT symmetry, the 3D Weyl point only exists on the systems that lack T or P or both [170 –176]. Such a character makes a single Weyl point absolutely robust to any perturbations. The only way to eliminate and create Weyl points is through pair annihilations and pair generations of Weyl points with opposite chirality. It seems the emergence of Weyl points is somehow accidental, nevertheless, one can have a 3D topological phase with symmetry protection first and then have the Weyl points by symmetry reduction (Figure 8A) [170–173]. From this point of view, Wang et al. [170, 171] systematically studied the 3D Z ₂ Dirac point, which can be viewed as a pair of Weyl points with opposite chirality, based on all-dielectric PhCs. Usually, the 3D Dirac points are unstable when two Weyl points annihilate each other and form a gap. Nevertheless, Wang et al. [170, 171] pointed out that 3D Z ₂ Dirac can survive stably via certain crystalline symmetry, and split into the Weyl points when the P is broken.

As an illustration, Figure 8B presents a kind of 3D all-dielectric PhCs with nonsymmorphic symmetry. In each unit cell, there are two dielectric blocks of the same shape and permittivity, which are connected via screw symmetry S _x and S _y (Figure 8C). To realize the Z ₂ photonic Dirac point, both Kramers double degeneracy and parity-inversion should be synthesized, in which the crystalline symmetry plays a key role in realizing these two elements. On the one hand, anti-unitary operators that combine the screw symmetry with T symmetry are created to simulate the photonic Kramers degenerate pairs. On the other hand, the eigenvalue of a two-fold rotation symmetry operator is employed to define the parity of photonic states. The distribution of 3D Z ₂ Dirac points in the BZ is displayed in Figure 8D.

The 3D Dirac point emerges from the band crossing of two doublets with distinct parities (Figures 8E,F). The spit–orbit physics of the Dirac points can be understood via a symmetry-based ^∼ k⋅p ∼ theory. When the space symmetry is reduced, it is expected that the Dirac point will split into Weyl points. As shown in Figures 8G,H, by transforming the dielectric blocks into other shapes, namely, breaking the screw symmetry while keeping the two-fold rotation symmetry, the crossing between the band with distinct parities (i.e., p and d bands) result in Weyl points (Figure 8I). Following this idea, the Weyl point is also realized in the metasurface [172] and twisted 1D dielectric PhCs [173]. Because the 3D Dirac point acts as the mother state of the Weyl points, it is interesting to explore the properties of surface states according to the bulk-edge correspondence principle. Figure 8J shows a gapless surface band traversing the projected photonic band gap. The topological surface states carry finite total angular momentum as indicated in Figure 8K by the winding profile of the Poynting vectors. The sign of the photonic total angular moment is changed when the wavevector is reversed. This property is similar to the “spin–wavevector locking” on the edges of topological insulators. Based on such a salient feature, one can further study frequency-, angle-, wavevector-, and angular momentum-selective transmission in Weyl/Dirac PhCs. In addition, note that the topological surface states are below the light-line and hence can form cavity states on the PhC–air interfaces with no need for additional cladding. Last but not least, the conical dispersions in Dirac/Weyl all-dielectric PhCs provide a new mechanism to realize unconventional optical properties, such as anomalous refraction [171].