Topological photonics is a rapidly developing field that leverages geometric and topological concepts to engineer and control the characteristics of light. Currently, the research on topological photonics has expanded from traditional one-dimensional (1D) and two-dimensional (2D) to three-dimensional (3D) and higher-dimensional spaces. However, most reviews on topological photonics focus on 1D and 2D systems, and a review that provides a detailed classification and introduction of 3D and higher-dimensional systems is still missing. Here, we review the photonic topological states in 3D and higher-dimensional systems on different platforms. Moreover, we discuss internal connections between different photonic topological phases and look forward to the future development direction and potential applications of 3D and higher-dimensional systems.

The concept of topology is blossoming into a pervasive principle across a multitude of physics disciplines.1–3 Originating from condensed matter physics, the notion of topology has been expanded to encompass diverse systems, including photonics,4–22 phononics,23–27 mechanics,28,29 and cold atomic gases.30,31 Photonics, in particular, has witnessed a surge in the application of topology driven by three compelling factors. First, by incorporating topology into photonics, robust control of photons at the scale of their wavelengths becomes feasible, enabling advancements in light manipulation. Compared with electrons as information carriers in current communication technology, photons offer superior advantages of higher speed, lower dissipation, and support for more channels. The application of topology in photonics holds great promise for various practical implementations, such as compact waveguides with minimal bending loss, lasers, and cavities.32 Second, photonics provides a tangible and multifaceted platform for the realization and examination of topological phenomena, attributable to the considerably longer wavelengths of photons as opposed to electrons.33–39 Photons, notably within the visible light spectrum, exhibit wavelengths in the hundreds of nanometers, and these lengths extend further for lower frequencies. Conversely, electrons carrying an energy of 1 eV display wavelengths on the nanometer scale. As a result, the photonic band structure and its inherent topological attributes can be crafted without necessitating an exploration into atomic scales. Finally, as electromagnetic waves consist of two vector fields, they exhibit richer physics compared to scalar waves.

So far, photonic topological phases have emerged as a captivating area of research. The topological invariant of a periodic system is defined as the integral of the Berry connection A within the first Brillouin zone,40 and A is given by
A=iunx,qqunx,q,
(1)
where unx,q is the periodic part of the Bloch wave as a function of real space x and momentum space q, satisfying the periodic boundary condition unx,q=unx+a,q. The most prototypical example for 1D topological systems is the Su–Schrieffer–Heeger (SSH) model,41 whose configuration and field profiles of the edge states are shown in Fig. 1(a). In the 1D SSH model, the integral of the Berry connection is defined as the Zak phase,
θnZak=π/Λπ/ΛAdq,
(2)
where the Berry connection in 1D is given by a scale form A=iunx,qqunx,q.42 Moreover, notable examples include investigations in the 1D system on subwavelength topological edge states,43 small-scale topological photonics,44 topological optical waveguiding,45 topological Majorana states in plasmonic systems,46 topologically protected polariton edge states,47 photonic Jackiw–Rebbi states,48 and topological lasing.49 
FIG. 1.

Schematic illustrations of the topological effects and energy spectra of a 1D, 2D, and 3D photonic topological phases. (a) Schematic of 1D SSH model and its energy spectrum (left panel). Upper right panel: configuration of the SSH model. Lower right panel: amplitude Ψ of edge states. (b) Schematic of a quantum Hall insulator with topological chiral edge states propagating unidirectionally along the edges. C denotes the Chern number. (c) Schematic of a quantum spin Hall insulator. The spin-up (red) and spin-down (blue) edge states propagate in opposite directions according to the spin-momentum locking properties. The spin Chern numbers are denoted as C+ and C for spin-up and spin-down edge states, respectively. (d) Domain wall constructed by two valley Hall insulators with distinct valley Chern numbers (denoted by CK and CK). The polarization of edge states (red and blue) is locked to the K and K′ valleys. (e) Schematic of 3D gapless topological phases. The Fermi-arc surface state (blue plane) originates from the Weyl point (green point). (f)–(h) Schematic of 3D gapped topological phases, including 3D strong topological phase with single-Dirac-cone (pink cone) surface states (f), 3D weak topological phase with double-Dirac-cone (pink cones) surface states (g), and 3D Chern insulator with chiral gapless surface states (h), respectively.

FIG. 1.

Schematic illustrations of the topological effects and energy spectra of a 1D, 2D, and 3D photonic topological phases. (a) Schematic of 1D SSH model and its energy spectrum (left panel). Upper right panel: configuration of the SSH model. Lower right panel: amplitude Ψ of edge states. (b) Schematic of a quantum Hall insulator with topological chiral edge states propagating unidirectionally along the edges. C denotes the Chern number. (c) Schematic of a quantum spin Hall insulator. The spin-up (red) and spin-down (blue) edge states propagate in opposite directions according to the spin-momentum locking properties. The spin Chern numbers are denoted as C+ and C for spin-up and spin-down edge states, respectively. (d) Domain wall constructed by two valley Hall insulators with distinct valley Chern numbers (denoted by CK and CK). The polarization of edge states (red and blue) is locked to the K and K′ valleys. (e) Schematic of 3D gapless topological phases. The Fermi-arc surface state (blue plane) originates from the Weyl point (green point). (f)–(h) Schematic of 3D gapped topological phases, including 3D strong topological phase with single-Dirac-cone (pink cone) surface states (f), 3D weak topological phase with double-Dirac-cone (pink cones) surface states (g), and 3D Chern insulator with chiral gapless surface states (h), respectively.

Close modal

Recently, the photonic analogs of the quantum Hall insulator, the quantum spin Hall insulator, and the quantum valley Hall insulator have been realized in 2D photonic topological phases. The photonic counterpart of the quantum Hall effect is proposed in 2D photonic crystals composed of magnetoelectric materials, which can break time-reversal symmetry (T) and result in a nonzero Chern number for a topologically nontrivial bandgap.50,51 The schematic of a quantum Hall photonic insulator is shown in Fig. 1(b). The unidirectional propagation of edge states (indicated by the black arrows) is protected by the nontrivial topological Chern number (C = 1). However, to break T, utilizing external magnetic fields and gyromagnetic materials presents substantial challenges and complexities.

To circumvent the necessities imposed by the external field and gyrotropic materials, many efforts have been made to create photonic pseudospins, mimicking the spins of electrons, to facilitate the realization of the photonic quantum spin Hall effect. The photonic quantum spin Hall effect, shown in Fig. 1(c), is regarded as two copies of the quantum Hall effect for each spin (red and blue). As the spin-up and spin-down are orthogonal with each other and are double degenerate under T, the spin Chern number for spin-up and spin-down is opposite.52 Except for the pseudospin constructed in photonic crystals, the valley has been investigated as a new degree of freedom to realize the so-called quantum valley Hall phase.53,54 When breaking the inversion symmetry of a photonic crystal, the degeneracy of the Dirac point can be lifted at two valleys. The edge states propagate along the domain wall in the opposite direction for (red and blue) K and K′ valleys, as shown in Fig. 1(d).

The expansion of photonic topological phases from 1D and 2D to 3D and higher-dimensional systems proffers a wealth of optical functionalities, exploiting the augmented dimensionality and spatial symmetries. In contrast with their lower-dimensional counterparts, topological states in 3D and higher-dimensional system systems can manifest in bulk on surfaces, interfaces, hinges, and corners. This diversity offers a plethora of opportunities for the innovative design and manipulation of light propagation, thereby boosting the capacity for information transfer.35–39 The added dimensionality enables the realization of glide reflection symmetry with glide planes, screw symmetry with screw axes, and even the stacking of multiple layers or interfaces with different topological properties.35,36 This leads to the creation of strong topological phases, dislocation topological waveguides, and multifunctional devices unique to higher-dimensional systems.

In general, photonic topological phases in 3D and higher-dimensional systems can be classified as gapless 3D photonic topological semimetals [Fig. 1(e)] and gapped 3D photonic topological insulators (PTIs) [Figs. 1(f)1(h)].55–59 One key advantage that distinguishes 3D and higher-dimensional system photonic topological phases from their 1D or 2D counterparts is the presence of a 3D band structure. For example, 3D (higher-dimensional) photonic systems have enabled the first observations of several 3D topological band structures that are otherwise challenging in electronic systems, including 3D Chern insulators (CIs).55 Furthermore, the additional dimension confines topological states not only at edges and corners but also on surfaces and hinges, providing opportunities for the manipulation of light propagation across multiple dimensions. These characteristics enable 3D topological photonic systems to accommodate larger information capacities.

For the gapless 3D photonic topological semimetals, the degeneracy of two or more band structures occurs at certain points in the 3D Brillouin zone. The band degeneracy has many interesting properties that cannot have counterparts in 2D. Take Weyl points as examples; they always appear in pairs, whose projections on surfaces are connected by open Fermi arcs [see the blue plane in Fig. 1(e)]. An intriguing characteristic of the Weyl point [see the green point in Fig. 1(e)] is that it acts as a drain or a source of the Berry flux in momentum space, with a nonzero chiral charge defined by the Chern number,
C=12πsFkdS,
(3)
where S is the surface enclosing the Weyl point, and Fk represents the Berry curvature of the lower band structure.

On the other hand, analogous to how gapped 2D topological insulators host 1D gapless edge states, it has been suggested that a 3D bandgap could accommodate 2D gapless surface states [Figs. 1(f)1(h)]. The significance of 3D PTIs lies in their ability to transmit larger information capacities with robustness. These features make them promising candidates for various applications, including low-loss waveguides, advanced imaging and sensing, topological lasers, quantum optics, and photonic circuitry.

Historically, 3D topological insulators (TIs) were originally predicted to exist in condensed matter systems.60–65 Unlike the dominant effect of a single Z2 topological invariant in 2D systems, in 3D systems, there are four Z2 topological invariants v0; (v1 v2 v3) that distinguish 16 phases.60 In particular, v0 determines whether the odd or even numbers of Kramers points are enclosed by a surface Fermi circle. The 3D TIs with Z2 topological invariants can be divided into two categories: strong TIs [v0 = 1, with an odd number of surface Dirac cones; see Fig. 1(f)] and weak TIs [v0 = 0, with an even number of surface Dirac cones; see Fig. 1(g)].60 The strong TIs cannot be constructed by stacking 2D TIs and are extremely robust against disorder, while the weak TIs are like layered 2D quantum spin Hall phases, which can be destroyed by disorders or defects.62,63 On the other hand, the 3D CIs [Fig. 1(h)] correspond to the 3D quantum Hall phase.37 They can be regarded as 3D extended versions of the 2D quantum Hall phases. In particular, the 3D quantum Hall phases have been theoretically predicted in the 3D electron gases under magnetic fields.64 

To understand the aforementioned four Z2 topological invariants v0; (v1 v2 v3) of the strong and weak TIs [see Figs. 1(f) and 1(g)], for a given 3D BZ, the kx-ky plane is determined by fixing a given kz. The Z2 topological invariants are defined in only two momentum planes, kz = π/a and kz = 0, where a represents the lattice constant.37,60 In particular, Z2 topological invariants between two planes (kz = π/a and kz = 0) can be described by the strong topological invariant ν0; if the invariants of the kz = π/a and kz = 0 planes are different (Z2 = 0 and Z2 = 1), then the system represents topologically nontrivial and possesses ν0 = 1 [corresponding to the strong TIs with an odd number of surface Dirac cones, see Fig. 1(f)]; otherwise, if these Z2 topological of the two planes are the same, then the system represents trivial and possesses ν0 = 0 [corresponding to the weak TIs with an even number of surface Dirac cones, see Fig. 1(g)]. On the other hand, the 3D CI systems [Fig. 1(h)] are characterized by a triad of first Chern numbers,
C1(C1x,C1y,C1z),
(4)
where C1x,y,z represent the Chern numbers of the 2D x, y, and z momentum planes,37,55 respectively. Moreover, the gapless surface states in the 3D CI are chiral, whose directionality is determined by the sign of the first Chern number, C1.36 

In photonics, there are several platforms to design 3D (higher-dimensional) photonic topological phases, including 3D photonic crystals, evanescently coupled waveguides, electromagnetic continuum media, and synthetic space, as shown in Fig. 2. Among these available platforms, 3D photonic crystals stand out as the leading choice for designing 3D photonic topological phases. They consist of a periodic arrangement of material elements in 3D [Fig. 2(a)] and possess band structures that share striking similarities with the band structures originally developed in solid-state physics for electrons. Based on 3D photonic crystals, various 3D photonic topological phases have been proposed and experimentally observed. However, the fabrication of 3D structures becomes more intricate in the optical regime due to the reduced length scales, necessitating higher fabrication resolutions. In addition, the magneto-optic effects provided by magnetic materials are so weak in these frequency ranges that most experiments in 3D photonic crystals are limited to the microwave range. Through the evanescently coupled waveguides, however, 3D topological phases have been achieved at optical frequency by incorporating a 2D photonic lattice in the x-y plane and aligning helical waveguides along the z-axis [Fig. 2(b)].

FIG. 2.

Schematic illustrations of photonic platforms for 3D and higher-dimensional topological phases. (a) 3D photonic crystal. (b) Evanescently coupled waveguides. (c) Electromagnetic continuum media platforms. (d) Synthetic space.

FIG. 2.

Schematic illustrations of photonic platforms for 3D and higher-dimensional topological phases. (a) 3D photonic crystal. (b) Evanescently coupled waveguides. (c) Electromagnetic continuum media platforms. (d) Synthetic space.

Close modal

Unlike photonic crystals or evanescently coupled waveguides, electromagnetic continuum media [Fig. 2(c)] represent a uniquely effective medium approach for studying the topological behaviors of electromagnetic waves. In Figs. 2(a)2(c), the topological phases of a photonic system are typically described based on their geometric dimensionality. However, the concept of synthetic dimensions [Fig. 2(d)] enables the exploration of physics in a space with higher-dimensionality. This allows for the investigation of phenomena and properties beyond what is observed solely based on physical dimensions. In photonic systems, there are two approaches to creating a synthetic space: forming a lattice and exploiting the parameter dependency of the system.65 The former approach focuses on increasing the connectivity within the lattice structure. For instance, a 2D lattice can be formed by introducing longer-range coupling into a 1D lattice. On the other hand, the latter approach serves external geometric parameters, excluding spatial dimensions, as extra synthetic dimensions.

Here, we discuss the 3D and higher-dimensional photonic topological phases in the photonic platforms, which have obtained significant developments in experimental implementation. 3D photonic topological phases in the T-breaking system are overviewed in Sec. II. In Sec. III, we introduce the 3D gapless photonic topological semimetals and 3D gapped PTIs in the T-preserved system. We provide an overview of examples of the 3D and higher-dimensional photonic topological phases in synthetic space in Sec. IV. We finish the review by providing a summary in Sec. V.

Weyl semimetals and their classical-wave counterparts represent a fascinating class of topological materials distinguished by band structures that feature two-fold linear crossings in three-dimensional momentum space. These crossing points, termed Weyl points, act as monopoles of Berry flux, marked by their nonzero chiral charges, or Chern numbers.56 3D linear dispersion around a Weyl point can be described as the Weyl Hamiltonian: H = vxkxσx + vykyσy + vzkzσz, where vx,y,z and σx,y,z represent group velocities and Pauli matrices, respectively.

In photonics, Weyl points were theoretically proposed in gyroid photonic crystals [Fig. 3(a)].66 A key advantage of this photonic system compared to the electronic system is the convenience of experimental implementation and characterization. The line node and Weyl point in the double-gyroid photonic crystal system originate from the threefold degeneracy of the band structure in the center of the Brillouin zone by different perturbation mechanisms. The line node is obtained by applying perturbations that hold P and T, and the controlled flat surface dispersions of two pseudo-gaps are displayed in the surface Brillouin zone. The Weyl point is generated by applying perturbations that disrupt P or T and then showing topological phase diagrams based on the number of Weyl points when both P and T are disrupted. At the frequency of the Weyl point, nontrivial surface states connect the opposite-chirality Weyl point and do not close in the surface Brillouin zone. Later on, a proposal is made for a magnetic tetrahedral photonic crystal capable of hosting frequency-isolated Weyl points with opposite Berry fluxes [Fig. 3(b)].67 Compared to a double-gyroid structure, the magnetic tetrahedral design can provide a simple scheme for fabricating photonic crystals with a Weyl point. When T is broken, the threefold quadratic degeneracy points at the corner of the Brillouin zone are lifted, and a single pair of Weyl points appears in the system.

FIG. 3.

Various 3D gapless topological phases in T-breaking systems. (a) Line nodes and Weyl points in gyroid photonic crystals. Reproduced with permission from Lu et al., Nat. Photonics 7, 294–299 (2013). Copyright 2013 Springer Nature. (b) Weyl points in tetrahedral photonic crystals with magnetization. Reproduced with permission from Yang et al., Opt. Express 25, 15772–15777 (2017). Copyright 2017 The Optical Society. (c) Weyl points and Fermi arcs in magnetized plasma. Reproduced with permission from Gao et al., Nat. Commun. 7, 12435 (2016). Copyright 2016 Springer Nature. (d) Twisted magnetized plasma can support stretchable photonic Fermi arcs. Reproduced with permission from Xia et al., Laser Photonics Rev. 12, 1700226 (2018). Copyright 2018 Wiley-VCH. (e) Experimental observation of photonic Weyl points in magnetized plasma. Reproduced with permission from Wang et al., Nat. Phys. 15, 1150–1155 (2019). Copyright 2019 Springer Nature. (f) Topological phase diagram and bulk-edge correspondence in magnetized cold plasma. Reproduced with permission from Fu and Qin, Nat. Commun. 12, 3924 (2021). Copyright 2021 Springer Nature. (g) Topological Fermi nodal disk and topological phase transition in non-Hermitian magnetic plasma. Reproduced with permission from Wang et al., Light: Sci. Appl. 9, 40 (2020). Copyright 2020 Springer Nature. (h) All-angle topological negative refraction in ideal Weyl metamaterials. Reproduced with permission from Liu et al., Light: Sci. Appl. 11, 276 (2022). Copyright 2022 Springer Nature. (i) Topological antichiral surface states in a magnetic 3D photonic crystal. Reproduced with permission from Xi et al., Nat. Commun. 14, 1991 (2023). Copyright 2023 Springer Nature.

FIG. 3.

Various 3D gapless topological phases in T-breaking systems. (a) Line nodes and Weyl points in gyroid photonic crystals. Reproduced with permission from Lu et al., Nat. Photonics 7, 294–299 (2013). Copyright 2013 Springer Nature. (b) Weyl points in tetrahedral photonic crystals with magnetization. Reproduced with permission from Yang et al., Opt. Express 25, 15772–15777 (2017). Copyright 2017 The Optical Society. (c) Weyl points and Fermi arcs in magnetized plasma. Reproduced with permission from Gao et al., Nat. Commun. 7, 12435 (2016). Copyright 2016 Springer Nature. (d) Twisted magnetized plasma can support stretchable photonic Fermi arcs. Reproduced with permission from Xia et al., Laser Photonics Rev. 12, 1700226 (2018). Copyright 2018 Wiley-VCH. (e) Experimental observation of photonic Weyl points in magnetized plasma. Reproduced with permission from Wang et al., Nat. Phys. 15, 1150–1155 (2019). Copyright 2019 Springer Nature. (f) Topological phase diagram and bulk-edge correspondence in magnetized cold plasma. Reproduced with permission from Fu and Qin, Nat. Commun. 12, 3924 (2021). Copyright 2021 Springer Nature. (g) Topological Fermi nodal disk and topological phase transition in non-Hermitian magnetic plasma. Reproduced with permission from Wang et al., Light: Sci. Appl. 9, 40 (2020). Copyright 2020 Springer Nature. (h) All-angle topological negative refraction in ideal Weyl metamaterials. Reproduced with permission from Liu et al., Light: Sci. Appl. 11, 276 (2022). Copyright 2022 Springer Nature. (i) Topological antichiral surface states in a magnetic 3D photonic crystal. Reproduced with permission from Xi et al., Nat. Commun. 14, 1991 (2023). Copyright 2023 Springer Nature.

Close modal

Different from the photonic crystal, the Weyl degeneracy can exist in electromagnetic continua, such as cold magnetized plasma [Fig. 3(c)].68 Under a strong enough magnetic field, the Weyl point arises as a crossing between the transverse helical propagating mode and the purely longitudinal plasma mode. The prominent features of the plasmon Weyl point include the semi-k-plane chirality exhibited in electromagnetic reflection. Unlike artificial metamaterials, as long as the wavelength of interest is much greater than the average distance between charged particles in the plasma, the optical response of the cold magnetized plasma system can be considered local. The photonic Weyl points in the cold magnetized plasma are right at the critical transition between the type-II Weyl point with an open equifrequency surface and the type-I Weyl point with a closed equifrequency surface. Near these Weyl points, the magnetized plasma exhibits exotic parabolic equifrequency surfaces. By varying the plasma density and the strength or direction of the applied static magnetic field, the magnetized plasma system can be reconfigured in real-time. Subsequently, the stretchable photonic Fermi-arc surface state is realized in the twisted, cold magnetized plasma [Fig. 3(d)].69 The length of the Fermi-arc surface state can be arbitrarily stretched by changing the relative orientation and strength of the twisted magnetized plasma system. An unconventional Fermi-arc connection feature is observed, i.e., Fermi-arc connecting two Weyl points with the same chirality but located on different sides of the interface, which brings new insights into the connectivity of Fermi-arc. Weyl points have been studied in other electromagnetic continuums of media, such as gaseous plasmon polariton,70 gyromagnetic metamaterials,71 and bigyrotropic metamaterials.72 

In electromagnetic continua, the experimental observation of photonic Weyl points was reported in a magnetized semiconductor, InSb [Fig. 3(e)].73 It behaves as a cold, magnetized plasma for electromagnetic waves at the terahertz frequency. As it does not involve structure, this represents a simple and adjustable method for implementing photons due to Weyl degeneracy caused by T-breaking. Weyl points and the corresponding photonic Fermi-arc surface states have been demonstrated by changing the magnetic field strength. Moreover, a synthetic gauge potential can be achieved by introducing a spatially variant temperature or magnetic field in the magnetized semiconductor system. A comprehensive picture of topological phases and a study of topological phase transitions are shown in the cold magnetized plasma [Fig. 3(f)].74 The topological phases are systematically mapped out, and the bulk-edge correspondence is established in the magnetized plasma system. For fixed magnetic field B and wave vector kz, only the topological phase transition at the Langmuir wave-cyclotron wave resonance can correspond to topological edge states. Moreover, the topological edge states in the magnetized plasma system exist not only on the vacuum-plasma boundary but also on more general plasma–plasma boundaries.

Recently, topological phases in non-Hermitian photonic systems have attracted growing interest. The photonic topological Fermi nodal disks and exceptional loops are reported in a non-Hermitian magnetic plasma [Fig. 3(g)].75 In non-ideal magnetohydrodynamic plasma with viscosity dissipations and collisions, the real part of the eigenfrequency of the topologically stable nodal disks degenerates between two band structures. The lossy Fermi-arc surface states at the boundary connect to the middle of the projection of these Fermi nodal disks with nonzero Berry fluxes. Several different topological phase transition processes are studied in the warm magnetic plasma system by varying the dissipation terms and viscosity parameters. Later on, dissipation-induced topological transitions and Weyl exceptional contours in the lossy plasma76 and gyromagnetic metamaterial,77 respectively.

Photonic Weyl metamaterials can achieve all-angle reflectionless negative refraction [Fig. 3(h)].78 The Fermi-arc surface state connecting the two Weyl points takes the form of a half-circle possessing a negative or a positive refractive index by interfacing the Weyl metamaterial with a perfect magnetic conductor (PMC) or a perfect electric conductor (PEC). In the Weyl system, there is no reflection at the boundary between the PMC and PEC-covered regions because of the topological protection. Subsequently, the experimental realization of antichiral surface states is reported in a 3D magnetic Weyl photonic crystal [Fig. 3(i)].79 Two pairs of frequency-shifted Weyl points are achieved by arranging the opposite magnetization in different triangular sublattices of a 3D photonic crystal. By field-mapping measurements, the counterintuitive antichiral surface state of codirectional robust propagation on opposite surfaces is verified. Moreover, the triply degenerate point, Fermi-arc surface state, chiral surface state, and topological transitions have been reported in gyromagnetic/gyroelectric metamaterials, Tellegen metamaterials, gyroelectromagnetic metamaterials, and magnetized plasma.80–96 

3D gapped topological phases can also be realized in T-breaking photonic systems. The 3D strong topological insulator was proposed in a 3D photonic crystal consisting of a body-centered cubic unit cell with four gyroelectric rods in the air.57 The coexistence of T and glide reflections symmetry gives rise to a four-fold degenerate point within the 3D BZ. Following the T-breaking operation that applies alternating magnetizations to the gyroelectric rods, the degeneracy points are lifted, and a gapless surface state appears in the bandgap. The surface state depicted in the top plane in Fig. 4(a) exhibits a single Dirac cone located at the Y point along the M′M line. This prominent phenomenological manifestation is characteristic of 3D strong PTIs. When glide reflection symmetry is also broken in this system, the Dirac cone surface state will open a gap [the bottom plane in Fig. 4(a)], revealing that the surface Dirac cone is protected by the glide symmetries. Moreover, the single Dirac cone surface state is fully robust against arbitrary random disorder, i.e., when the glide symmetry is preserved on average on the surface. Currently, there is no experimental report of the 3D strong PTIs.

FIG. 4.

Various 3D gapped topological phases in T-breaking systems. (a) 3D strong topological insulator in a 3D gyroelectric photonic crystal. Reproduced with permission from Lu et al., Nat. Phys. 12, 337–340 (2016). Copyright 2016 Springer Nature. (b) Theoretical realization of 3D CIs with orientable large Chern vectors in the gyroelectric photonic crystal. Reproduced with permission from Devescovi et al., Nat. Commun. 12, 7330 (2021). Copyright 2021 Springer Nature. (c) Experimental realization of 3D CIs, topological torus loops, and links in the gyromagnetic photonic crystal. Reproduced with permission from Liu et al., Nature 609, 925–930 (2022). Copyright 2022 Springer Nature. (d) One-way fiber of the second Chern number. Reproduced with permission from Lu et al., Nat. Commun. 9, 5384 (2018). Copyright 2018 Springer Nature. (e) Automated discovery and optimization of 3D CIs. Reproduced with permission from Kim et al., ACS Photonics 10, 861–874 (2023). Copyright 2023 American Chemical Society.

FIG. 4.

Various 3D gapped topological phases in T-breaking systems. (a) 3D strong topological insulator in a 3D gyroelectric photonic crystal. Reproduced with permission from Lu et al., Nat. Phys. 12, 337–340 (2016). Copyright 2016 Springer Nature. (b) Theoretical realization of 3D CIs with orientable large Chern vectors in the gyroelectric photonic crystal. Reproduced with permission from Devescovi et al., Nat. Commun. 12, 7330 (2021). Copyright 2021 Springer Nature. (c) Experimental realization of 3D CIs, topological torus loops, and links in the gyromagnetic photonic crystal. Reproduced with permission from Liu et al., Nature 609, 925–930 (2022). Copyright 2022 Springer Nature. (d) One-way fiber of the second Chern number. Reproduced with permission from Lu et al., Nat. Commun. 9, 5384 (2018). Copyright 2018 Springer Nature. (e) Automated discovery and optimization of 3D CIs. Reproduced with permission from Kim et al., ACS Photonics 10, 861–874 (2023). Copyright 2023 American Chemical Society.

Close modal

3D CIs present another class of 3D gapped topological phases and were realized in the gyromagnetic photonic crystal.97 It was theoretically predicted that Chern vectors of any magnitude, sign, or direction could be implemented in 3D cubic photonic crystals. When there is no perturbation, the photonic crystal with four gyroelectric rods in a unit cell possesses a threefold degeneracy at the high symmetry point of the BZ. To achieve 3D CIs with orientable large Chern vectors, T is broken by applying an external magnetic field to the gyroelectric rods, leading to the splitting of the initial threefold degeneracy into a pair of Weyl points. By utilizing the external magnetic field, these Weyl points are then displaced at fractional distances within BZ. Subsequently, through the creation of multi-fold supercells, the Weyl points are folded within the new reduced BZ, resulting in the multi-fold degeneracy. Finally, by adjusting the radius of the cylinders in the designed supercell, the Weyl points are opened, giving rise to a 3D bandgap, as displayed in Fig. 4(b). Depending on the values of the moving distance and the size of the uniaxial supercell, different tailored 3D CIs can be designed, i.e., orientable Chern vectors and tunable larger Chern numbers.98 

Despite the compelling theoretical demonstrations of 3D CIs, their experimental realization remains highly challenging. This difficulty arises from the limited magnitude of the Voigt effect provided by gyroelectric materials as well as the intricate structural designs of photonic crystals, which are insufficient to observe the 3D CIs in experimentation. Through the 3D stacked magnetically tunable photonic crystal, the experimental demonstration of 3D CIs and their topological surface states was performed, marking a significant milestone in the field.55 The left plane in Fig. 4(c) shows the photonic crystal unit cell, consisting of a gyromagnetic rod and a metallic plate perforated with holes. As the degree of inversion symmetry breaking and the external magnetic field strength vary, the photonic crystal exhibits trivial insulators, 3D CIs [the right plane in Fig. 4(c)], and Weyl phases. Using microwave near-field imaging measurements, the topological phase transition from 3D trivial insulators to 3D CIs, mediated by ideal Weyl phases that can host a single Fermi arc, is directly observed. By modulating the system along a spatial axis z, a Chern vector with z-component Cz = 6 is achieved in the system. Furthermore, the isofrequency contours of the chiral surface states in the surface BZ will form torus knots or links with winding numbers of m and n around the median and longitude of the surface BZ when an x–z interface is formed between two 3D CIs with perpendicular Chern vectors mẑ and nx̂. In experimentation, an interface is constructed between two 3D CIs, as shown in Fig. 4(c). As shown in the right plane in Fig. 4(c), the measured surface intensity exhibits two Fermi loops, forming a (2, 2) torus link or Hopf link [Fig. 4(c)], which is topologically distinct from the surface states of other 3D topological insulators. Compared with the 2D CIs, 3D CIs characterized by the Chern vector are weak topological phases whose weak topological indices are defined in a lower dimension.

Recent theoretical predictions have unveiled a novel mode called the one-way fiber mode, which propagates along topological line defects in 3D CIs. The one-way fiber mode, protected by the strong invariant second Chern number in the 4D parameter space, is more robust than the bulk state that gave rise to it.

The fundamental nature of the one-way fiber mode is to compress the surface states, which propagate along the surface of photonic crystals, into an edge state that propagates along line defects within the bulk. This configuration enhances the robustness of electromagnetic waves. The theoretical realization of a one-way fiber mode was reported in the gyromagnetic photonic crystal. In the presence of T breaking, the photonic crystal exhibits two Weyl points, where the unit cell consists of a double magnetic gyroid.99 After a supercell modulation of the crystal in the z-direction, the two Weyl points are superimposed on top of each other to form a 3D Dirac point between four bands. By further modulating the volume fraction of the double magnetic gyroid, the degeneracy points can be opened, resulting in the realization of a 3D CI [Fig. 4(d)]. The one-way fiber mode is obtained by applying a helical modulation to the double magnetic gyroid structure. By increasing the helical winding number, such as double helix, triple helix, and so on, it becomes possible to achieve arbitrary values for the second Chern number and an arbitrary number of one-way modes. To date, experimental evidence for the observation of one-way fiber mode remains lacking. It is worth noting that the design of most 3D PTIs typically relies on manual analysis of their band structures and mode profiles. However, a recent development has introduced a novel approach that combines global and local optimization frameworks for the automated discovery and optimization of 3D topological photonic structures.100 By applying this technique to the optimization of 3D CIs with target Chern vectors C = (0, 0, Cz) for Cz = 1 and Cz = 2, 3D CIs with the largest known complete bandgap are discovered. Within this gap, a pair of chiral surface states exist, which effectively act as a pair of 1D edge states at each kx value, as shown in Fig. 4(e).

The convenience of realization of the T-preserving systems led to the discovery of Weyl points in a T-preserving system, more specifically, a double-gyroid photonic crystal. Through angle-resolved microwave transmission measurements, the photonic Weyl points were experimentally observed [Fig. 5(a)].101 There are two pairs of type-I Weyl points owing to T preserved in the gyroid system. In the experiments, the Weyl points are demonstrated by the two linear dispersion bands touching at four isolated points in the 3D Brillouin zone of the excited bulk states. On this basis, multiple Weyl points and the exploration of the sign change of their topological charges were reported in 3D photonic crystals.102,103 Later on, the experimental observation of type-II Weyl points at optical frequencies and Fermi arc-like surface states is reported in a waveguide array [Fig. 5(b)].104 When the frequency is tuned to the Weyl frequency, the existence of type-II Weyl points is confirmed by conical diffraction along one axis of the 3D waveguide array. By theoretical and numerical simulation, the type-II Weyl point is demonstrated as a critical point of the topological phase transition. Moreover, the existence of the Fermi arc-like surface states associated with the type-II Weyl point is confirmed by the output intensity plots of the 3D photonic crystal structure. Later on, the topological type-II Weyl point was reported in the chiral photonic metamaterial.105 

FIG. 5.

Various 3D gapless photonic topological semimetals in a T-preserving system. (a) Experimental observation of Weyl points in a double-gyroid photonic crystal. Reproduced with permission from Lu et al., Science 349, 622–624 (2015). Copyright 2015 AAAS. (b) Type-II Weyl point and conical diffraction in a waveguide array structure. Reproduced with permission from Noh et al., Nat. Phys. 13, 611–617 (2017). Copyright 2017 Springer Nature. (c) Ideal Weyl points in saddle-shaped metallic inclusion. Reproduced with permission from Yang et al., Science 359, 1013 (2018). Copyright 2018 AAAS. (d) Charge-2 unconventional Weyl point in split ring resonators. Reproduced with permission from Yang et al., Phys. Rev. Lett. 125, 143001 (2020). Copyright 2020 American Physical Society. (e) Maximally charged Weyl point and quadruple-helicoid Fermi arc in the 3D metallic-mesh photonic crystal. Reproduced with permission from Chen et al., Nat. Commun. 13, 7359 (2022). Copyright 2022 Springer Nature. (f) Experimental observation of Dirac points in metamaterial. Reproduced with permission from Guo et al., Phys. Rev. Lett. 122, 203903 (2019). Copyright 2019 American Physical Society. (g) Experimental observation of a 1D nodal line in microwave cut-wire metacrystals. Reproduced with permission from Gao et al., Nat. Commun. 9, 950 (2018). Copyright 2018 Springer Nature. (h) Experimental observation of 1D nodal chains in a metallic-mesh 3D photonic crystal. Reproduced with permission from Yan et al., Nat. Phys. 14, 461–464 (2018). Copyright 2018 Springer Nature. (i) Experimental observation of the Weyl exceptional ring and Fermi arc in a helical waveguide array. Reproduced with permission from Cerjan et al., Nat. Photonics 13, 623 (2019). Copyright 2019 Springer Nature. (j) Landau levels and chiral zero modes in inhomogeneous 3D Weyl metamaterials. Reproduced with permission from Jia et al., Science 363, 148–151 (2019). Copyright 2019 AAAS. (k) Antichiral surface states in a 3D time-reversal-invariant photonic metacrystal. Reproduced with permission from Liu et al., Nat. Commun. 14, 2027 (2023). Copyright 2023 Springer Nature. (l) Real higher-order Weyl point and hinge Fermi arc. Reproduced with permission from Pan et al., Nat. Commun. 14, 6636 (2023). Copyright 2023 Springer Nature.

FIG. 5.

Various 3D gapless photonic topological semimetals in a T-preserving system. (a) Experimental observation of Weyl points in a double-gyroid photonic crystal. Reproduced with permission from Lu et al., Science 349, 622–624 (2015). Copyright 2015 AAAS. (b) Type-II Weyl point and conical diffraction in a waveguide array structure. Reproduced with permission from Noh et al., Nat. Phys. 13, 611–617 (2017). Copyright 2017 Springer Nature. (c) Ideal Weyl points in saddle-shaped metallic inclusion. Reproduced with permission from Yang et al., Science 359, 1013 (2018). Copyright 2018 AAAS. (d) Charge-2 unconventional Weyl point in split ring resonators. Reproduced with permission from Yang et al., Phys. Rev. Lett. 125, 143001 (2020). Copyright 2020 American Physical Society. (e) Maximally charged Weyl point and quadruple-helicoid Fermi arc in the 3D metallic-mesh photonic crystal. Reproduced with permission from Chen et al., Nat. Commun. 13, 7359 (2022). Copyright 2022 Springer Nature. (f) Experimental observation of Dirac points in metamaterial. Reproduced with permission from Guo et al., Phys. Rev. Lett. 122, 203903 (2019). Copyright 2019 American Physical Society. (g) Experimental observation of a 1D nodal line in microwave cut-wire metacrystals. Reproduced with permission from Gao et al., Nat. Commun. 9, 950 (2018). Copyright 2018 Springer Nature. (h) Experimental observation of 1D nodal chains in a metallic-mesh 3D photonic crystal. Reproduced with permission from Yan et al., Nat. Phys. 14, 461–464 (2018). Copyright 2018 Springer Nature. (i) Experimental observation of the Weyl exceptional ring and Fermi arc in a helical waveguide array. Reproduced with permission from Cerjan et al., Nat. Photonics 13, 623 (2019). Copyright 2019 Springer Nature. (j) Landau levels and chiral zero modes in inhomogeneous 3D Weyl metamaterials. Reproduced with permission from Jia et al., Science 363, 148–151 (2019). Copyright 2019 AAAS. (k) Antichiral surface states in a 3D time-reversal-invariant photonic metacrystal. Reproduced with permission from Liu et al., Nat. Commun. 14, 2027 (2023). Copyright 2023 Springer Nature. (l) Real higher-order Weyl point and hinge Fermi arc. Reproduced with permission from Pan et al., Nat. Commun. 14, 6636 (2023). Copyright 2023 Springer Nature.

Close modal

To explore Weyl physics, ideal Weyl points that are well separated from any other nontopological bands in a sufficiently large energy interval are highly desired, which are first realized in an artificial photonic crystal structure [Fig. 5(c)].106 The symmetry operations of the ideal Weyl points are confirmed by experimentally characterizing a microwave photonic crystal of saddle-shaped metallic coils. The intriguing helicoidal structure of topological surface states in the ideal Weyl system is demonstrated by a transmitted near-field scanning scheme. Based on the ideal Weyl system, several topological effects have been studied, such as spontaneous emission,107 Veselago lensing,108 synthetic gravitational field,109 and nonlocal effective medium description.110 

Ideal unconventional Weyl points carrying a topological charge of 2 or more were also experimentally realized [Fig. 5(d)].111 In the ideal charge-2 Weyl metamaterial, two long Fermi-arc surface arcs form a non-contractible loop wrapping around the surface Brillouin zone. The surface states can span a frequency window of around 22.7% relative bandwidth. By field mapping measurements, the surface states are demonstrated to have topological self-collimation and are robust against disorder properties. Moreover, the charge-2 Weyl point was reported in the near-infrared chiral woodpile photonic crystals.112,113 Later on, the experimental realization of the maximally charged (charge-4) Weyl point was reported in a 3D photonic crystal [Fig. 5(e)].114 The maximally charged Weyl point can support quadruple-helicoid Fermi arcs, forming an unprecedented topology of double, non-contractible loops around the surface Brillouin zone. Moreover, the helicoid Fermi arcs exhibit type-II van Hove singularities with saddle point dispersions that can be located in arbitrary momenta.

3D Dirac points, a class of 3D gapless topological phases, can also be realized in photonic metamaterials with electromagnetic duality symmetry [Fig. 5(f)].115 It was found that at the boundary between air and the 3D Dirac metamaterials, a pair of spin-polarized Fermi-arc-like surface states are observed. The surface waves transmit nearly without diffraction due to the linearity of the dispersion surface states, which is significant for imaging applications and information propagation. The eigen reflection fields demonstrate the decoupling process from a Dirac point to two Weyl points with opposite chirality. Moreover, the topological correlation between a Dirac point and vortex or vector beams is confirmed in the photonics system. On this basis, the experimental demonstration of 3D photonic Dirac points and spin-polarized surface arcs was performed in the microwave region with an elaborately designed metamaterial structure.116 Subsequently, the exploration of the giant photonic spin Hall effect,117 nonspecular effects,118 and polarization evolution119 was conducted via the photonic Dirac point. Moreover, different types of Dirac points have been reported in photonic crystals and metamaterials.120,121

Beyond 0D points, the topological degeneracies can also be 1D, known as nodal lines, and 2D, known as nodal surfaces. The surface states linked to these 1D bulk topological phases exhibit a significantly greater degree of complexity compared to the Fermi arc states associated with Weyl/Dirac points. In a microwave cut-wire metacrystal, the experimental demonstration of 1D nodal line degeneracy was performed with a designed negative bulk plasma dispersion [Fig. 5(g)].122 The negative dispersion for the longitudinal bulk plasmon is realized by introducing glide symmetry into the cut-wire metacrystal, which plays a key role in the form of the nodal line degeneracy. When gyroelectric materials are incorporated into the 3D metacrystal, the 1D nodal line degeneracy can transform into the 0D Weyl points with opposite chirality. Later on, the theoretical prediction and experimental observation of nodal chains were conducted in a 3D metallic-mesh photonic crystal [Fig. 5(h)].123 The 3D metallic-mesh structure possesses frequency-isolated linear band-touching rings chained across the entire Brillouin zone. The nodal chains of the metallic-mesh photonic crystal can be protected by mirror symmetry and possess a frequency variation of less than 1%. By performing angle-resolved transmission measurements, the dispersion of the drumhead surface state is mapped out. For non-Hermitian photonic systems, the 1D Weyl exceptional ring was observed in a 3D evanescently coupled bipartite optical waveguide array [Fig. 5(i)].124 The upper and lower bands meet at an exceptional ring, as demonstrated by observing the lack of conical diffraction at the topological phase transition. Moreover, the Fermi-arc surface states connect the projections of the 1D Weyl exceptional rings with opposite Berry fluxes. Subsequently, the Weyl exceptional rings have been explored in chiral photonic metamaterials.125,126

Due to the chirality of Weyl points, the 3D Weyl system can support unidirectional chiral zero modes under strong magnetic fields. It results in the non-conservation of chiral currents, known as chiral anomalies. The chiral zero modes can be realized in inhomogeneous 3D Weyl metamaterials [Fig. 5(j)].127 A gauge field is generated by engineering the individual unit cells of the inhomogeneous Weyl system without breaking T. The observed shift of the Weyl point positions is equivalent to the gauge field in the 3D Weyl system. The dispersion of Landau levels can be derived from the Weyl Hamiltonian of the 3D inhomogeneous Weyl system. The group velocity of the zeroth Landau level is determined by both the chirality of the Weyl point and the artificial magnetic field. The chiral propagation property of the zeroth Landau level in the 3D Weyl systems is demonstrated by measured electric intensities. Moreover, the robust transport of the zero chiral modes against reflection is confirmed by inserting a defect layer of air or dielectric slab into the middle of the 3D Weyl sample and measuring the amplitude of the transmitted bulk states. Later on, the proposal is made for the realization of antichiral surface states in a time-reversal-invariant 3D photonic metacrystal with two asymmetrically dispersed Dirac nodal lines [Fig. 5(k)].128 The nodal lines can be rendered as a pair of offset Dirac points via dimension reduction. Based on microwave experiments, the asymmetric nodal lines and associated twisted ribbon surface states are confirmed in the 3D semimetal system. The antichiral surface transport property is demonstrated by the surface transmission measurements performed. Interestingly, the real higher-order Weyl point and hinge Fermi arc are reported in a 3D photonic crystal, as shown in Fig. 5(l).129 The high Weyl semimetal exhibits the coexistence of surface states, which arise from the nonzero Chern number and the nontrivial generalized real Chern number, respectively, forming a genuine higher-order Weyl photonic crystal. Moreover, the 0D triple degeneracy point, several types of 1D nodal lines, 1D ideal nodal rings, 2D nodal surfaces, and topological transitions have been reported in different optical systems with T.130–165 

The 3D gapped PTIs have revealed extraordinary topological surface states and exhibit potential applications in light robust propagation. In these systems that preserve T, several unique topologically gapless surface states have been observed. In the magneto-electrical coupling metacrystal, the conical Dirac-like dispersion of the surface states can be observed [Fig. 6(a)].166 The theoretical investigation illustrates the intriguing behavior of the interface between two 3D PTIs with different topological invariants. This finding significantly contributes to our understanding of topological transitions and paves the way for controlling topological states in practical 3D photonic devices. Following this, the existence of a nodal ring was experimentally highlighted in a photonic metacrystal. As shown in the right panel in Fig. 6(b), the fully gapped nodal ring can exhibit a Berry curvature vortex with a unique toroidal moment distribution in the momentum space.167 In addition, topological interface states between opposite momentum-space toroidal moments are experimentally observed in the lower panel in Fig. 6(b). This research reveals the importance of momentum space topology and its associated phenomena, presenting a novel direction in the study of topological photonics.

FIG. 6.

Various 3D gapped PTIs and their surface states. (a) Domain wall between two 3D PTIs in an all-dielectric magneto-electrical coupling metacrystal (left panel). The conical Dirac-like dispersion of the surface states (right panel). Reproduced with permission from Slobozhanyuk et al., Nat. Photonics 11, 130 (2017). Copyright 2017 Springer Nature. (b) Nodal ring in photonic meta-crystal; the Berry curvature vortex shows toroidal moment distribution. Experimentally mapped interface states. Reproduced with permission from Yang et al., Nat. Commun. 12, 1784 (2021). Copyright 2021 Springer Nature. (c) Experimental observation of 3D PTIs and gapless conical Dirac-like surface states in split-ring resonators. Measured total field distributions for the domain wall between two different PTIs. Reproduced with permission from Yang et al., Nature 565, 622 (2019). Copyright 2019 Springer Nature. (d) Self-guided and quadratic dispersion topological surface states in 3D PTIs. Reproduced with permission from Kim et al., Nat. Commun. 13, 3499 (2022). Copyright 2022 Springer Nature. (e) Experimental observation of 3D PTIs in screw dislocation photonic waveguide arrays. Reproduced with permission from Lustig et al., Nature 609, 931–935 (2022). Copyright 2022 Springer Nature.

FIG. 6.

Various 3D gapped PTIs and their surface states. (a) Domain wall between two 3D PTIs in an all-dielectric magneto-electrical coupling metacrystal (left panel). The conical Dirac-like dispersion of the surface states (right panel). Reproduced with permission from Slobozhanyuk et al., Nat. Photonics 11, 130 (2017). Copyright 2017 Springer Nature. (b) Nodal ring in photonic meta-crystal; the Berry curvature vortex shows toroidal moment distribution. Experimentally mapped interface states. Reproduced with permission from Yang et al., Nat. Commun. 12, 1784 (2021). Copyright 2021 Springer Nature. (c) Experimental observation of 3D PTIs and gapless conical Dirac-like surface states in split-ring resonators. Measured total field distributions for the domain wall between two different PTIs. Reproduced with permission from Yang et al., Nature 565, 622 (2019). Copyright 2019 Springer Nature. (d) Self-guided and quadratic dispersion topological surface states in 3D PTIs. Reproduced with permission from Kim et al., Nat. Commun. 13, 3499 (2022). Copyright 2022 Springer Nature. (e) Experimental observation of 3D PTIs in screw dislocation photonic waveguide arrays. Reproduced with permission from Lustig et al., Nature 609, 931–935 (2022). Copyright 2022 Springer Nature.

Close modal

As shown in Fig. 6(c), gapless conical Dirac-like surface states in 3D PTIs that are constructed by split-ring resonators are experimentally observed.58 The total field distributions for the domain wall between two different PTIs are also experimentally observed. Such an experimental breakthrough adds further depth to our understanding of topological surface states in PTIs, showing promise for the exploration of these novel states for practical applications. In a more recent study, self-guided and quadratic dispersion topological surface states were demonstrated in a 3D PTI [Fig. 6(d)].168 The identification of these unique states in a PTI without the need for spin–orbit coupling is significant, as it paves the way for further advancements in topological insulator technology and the design of advanced photonic devices. Finally, experimental observation of a 3D PTI was reported in screw dislocation photonic waveguide arrays [Fig. 6(e)].169 This work illuminates a unique method to achieve PTIs, whereby structural dislocations are used to introduce topological phenomena. The insights gained from this research could prove invaluable for the future engineering of PTIs, highlighting the potential of dislocation-induced topological phases.

In recent years, with the concept of synthetic dimension in photonic systems, it has become possible to explore physics characteristics in the parameter space with a dimensionality that can be higher as compared to the geometrical dimensionality of an actual structure.59 

The photonic Weyl point was proposed in a 2D on-chip ring resonator lattice with a synthetic frequency dimension [Fig. 7(a)].170 Each 2D resonator supports a series of discrete modes equally spaced in its resonant frequency, so a periodic lattice is formed in the synthetic frequency dimension. The one synthetic frequency dimension and two spatial dimensions together form a 3D space. The 2D resonator array system undergoes dynamic modulation of the refractive index. Weyl points with different symmetries can be explored by controlling the phase of modulation [Fig. 7(b)]. The 2D on-chip ring resonator provides a more flexible platform to study and explore a wide range of phase spaces. The nontrivial topology of Weyl points is represented by surface state arcs in the synthesized space, which exhibits unidirectional frequency conversion. Moreover, by varying the dynamic modulation phases, the 2D resonator system can be tuned to support the line node and Weyl point under P or/and T breaking. Subsequently, the Weyl phase transitions are explored in the synthetic frequency space.171 

FIG. 7.

Various 3D and higher-dimensional gapless and gapped photonic topological phases in synthetic space. (a) and (b) Weyl points in the 2D array of ring resonators. Reproduced with permission from Lin et al., Nat. Commun. 7, 13731 (2016). Copyright 2016 Springer Nature. (c) Experimental observation of Weyl points in synthetic parameter space. Reproduced with permission from Wang et al., Phys. Rev. X 7, 031032 (2017). Copyright 2017 American Physical Society. (d) 3D photonic topological insulator in a 2D ring resonator lattice. Reproduced with permission from Lin et al., Sci. Adv. 4, eaat2774 (2018). Copyright 2018 AAAS. (e) Enhanced Hall effect and helical Zitterbewegung effect in an evanescently coupled rod array. Reproduced with permission from Ye et al., Light: Sci. Appl. 8, 49 (2019). Copyright 2019 Springer Nature. (f) 2D topological nodal phase in asymmetric bilayer grating. Reproduced with permission from Lee et al., Phys. Rev. Lett. 128, 053002 (2022). Copyright 2022 American Physical Society. (g) Non-Hermitian Weyl interface physics in curved silicon waveguides. Reproduced with permission from Song et al., Phys. Rev. Lett. 130, 043803 (2023). Copyright 2023 American Physical Society. (h) 4D quantum Hall system in tunable 2D photonic waveguide arrays. Reproduced with permission from Zilberberg et al., Nature 553, 59–62 (2018). Copyright 2018 Springer Nature. (i) Yang monopole and Weyl surface in artificially engineered photonic media. Reproduced with permission from Ma et al., Science 373, 572 (2021). Copyright 2021 AAAS.

FIG. 7.

Various 3D and higher-dimensional gapless and gapped photonic topological phases in synthetic space. (a) and (b) Weyl points in the 2D array of ring resonators. Reproduced with permission from Lin et al., Nat. Commun. 7, 13731 (2016). Copyright 2016 Springer Nature. (c) Experimental observation of Weyl points in synthetic parameter space. Reproduced with permission from Wang et al., Phys. Rev. X 7, 031032 (2017). Copyright 2017 American Physical Society. (d) 3D photonic topological insulator in a 2D ring resonator lattice. Reproduced with permission from Lin et al., Sci. Adv. 4, eaat2774 (2018). Copyright 2018 AAAS. (e) Enhanced Hall effect and helical Zitterbewegung effect in an evanescently coupled rod array. Reproduced with permission from Ye et al., Light: Sci. Appl. 8, 49 (2019). Copyright 2019 Springer Nature. (f) 2D topological nodal phase in asymmetric bilayer grating. Reproduced with permission from Lee et al., Phys. Rev. Lett. 128, 053002 (2022). Copyright 2022 American Physical Society. (g) Non-Hermitian Weyl interface physics in curved silicon waveguides. Reproduced with permission from Song et al., Phys. Rev. Lett. 130, 043803 (2023). Copyright 2023 American Physical Society. (h) 4D quantum Hall system in tunable 2D photonic waveguide arrays. Reproduced with permission from Zilberberg et al., Nature 553, 59–62 (2018). Copyright 2018 Springer Nature. (i) Yang monopole and Weyl surface in artificially engineered photonic media. Reproduced with permission from Ma et al., Science 373, 572 (2021). Copyright 2021 AAAS.

Close modal

The experimental observation of generalized optical Weyl points was reported in a 1D photonic crystal based on the concept of synthetic dimensions [Fig. 7(c)].172 The reflection on the boundary of truncated photonic crystals exhibits phase vortices because of the synthesized Weyl points, which in turn ensures the existence of optical interface states between any reflective substrate and the 1D photonic crystal. The existence of the optical interface states is protected by the topological characteristic of Weyl points. The trajectory of the interface states in the parameter space is similar to the trajectory of the Weyl semimetal Fermi-arc surface state in wave vector space. The reflection phase vortices are demonstrated to serve as an experimental feature of the generalized Weyl points. By tuning the third geometric parameter in the 1D photonic crystal, the topological transition from Weyl semimetals to nodal line semimetals is further confirmed in 4D space. Later on, the tunable THz generalized Weyl points and topological interface states are explored in the 1D photonic crystal.173,174 On this basis, the photonic Dirac nodal-line semimetals were reported in a 1D hypercrystal.175 

A weak 3D PTI with minimal interlayer coupling can be achieved in a 2D ring resonator lattice by incorporating a synthetic frequency dimension [Fig. 7(d)].176 The 2D resonator system bypasses the difficulty of studying the 3D topological effects of constructing complex 3D optical structures. The screw dislocation along the frequency axis is created by modulating a few of the resonators, which supports robust one-way transmission of photons along the frequency axis. The dislocation can be implemented with only a few modulated rings in the 2D resonator system. It substantially reduces the experimental challenge of exploring band topology in the synthetic frequency dimension.

The photonic Hall effect and helical Zitterbewegung were reported in a synthetic Weyl system [Fig. 7(e)].177 In a honeycomb lattice, the evanescently coupled rod array is arranged along the transport direction with slowly varying radii. The system has photonic Weyl points in the synthetic space of a physical parameter and two momenta, and the enhanced Hall effect is due to the large Berry curvature in the vicinity of the synthetic Weyl point. When the wave packet traverses very close to the Weyl point, the helical Zitterbewegung can be observed because of the contribution of the non-Abelian Berry connection. Later on, the exploration of the 2D topological nodal phase is conducted in a bilayer resonant grating structure [Fig. 7(f)].178 The interlayer shift can simulate an extra momentum dimension for creating the 2D topological nodal phase by performing the mathematical analogy between the vertically asymmetric photonic lattice and the topological semimetal. The numerical results confirm the complex gapless dispersion of the topological edge modes, which connect the two 1D Dirac points in the topologically nontrivial area of the synthetic dimension. For non-Hermitian photonics systems, the observation of Weyl interface states is reported in the near-infrared wavelength of a silicon waveguide array [Fig. 7(g)].179 The Weyl interface in the silicon waveguides with distinct topological origins is experimentally realized by establishing the non-Hermitian Hamiltonian in the synthetic parameter space.

Recently, the scheme of synthetic dimensions has expanded the field of topological physics to higher-dimensional (such as 4D and 5D) space. The dynamically generated 4D quantum Hall system is experimentally realized using tunable 2D arrays of photonic waveguides [Fig. 7(h)].180 The transport of light through the waveguide array over momenta in two additional synthetic dimensions can achieve a 2D topological pump. The quantized bulk Hall response with 4D symmetry can be supported due to the band structure having 4D topological invariance. When the synthetic momenta are modulated, the 4D topological bulk response can be carried by the localized edge states in a finite-sized system. Later on, the observation of Weyl surfaces and Yang monopoles was made in 5D space (with two synthetic dimensions) [Fig. 7(i)].181 Via selected 3D subspaces, several intriguing bulk and surface phenomena are explored, such as the linking of Weyl surfaces and surface Weyl arcs. In the higher-dimensional system, the 1D Weyl arcs on a 3D Fermi hypersurface of the 5D system possessing Yang monopoles or Weyl surfaces are highlighted to demonstrate the distinctive phenomenon protected by C2. Moreover, exceptional cones182 and chiral zero modes183 are reported in 4D parameter space and 5D Yang monopole metamaterials, respectively.

In summary, the marriage of topology and photonics, driven by the unique advantages of photonics and the rich physics of electromagnetic waves, opens up new avenues for research and technological advancements. Unprecedented control over light at the nanoscale can be unlocked by leveraging topological principles, paving the way for transformative applications in communication, optical devices, and beyond. In the review, we delve into the realm of topological photonics in 3D and higher-dimensional systems, focusing on prominent examples of gapless 3D photonic topological semimetals and gapped 3D PTIs. These remarkable structures are realized through the ingenious utilization of photonic crystals, evanescently coupled waveguides, electromagnetic continuum media, and synthetic space. By exploring the world of synthetic space, we transcend the physical dimensions of these structures, enabling us to investigate phenomena and properties that extend beyond conventional limitations.59 In addition, we establish connections between diverse types of 3D and higher-dimensional photonic topological phases, unveiling the underlying interplay and shared characteristics among these extraordinary photonic systems.

The exploration of 3D and higher-dimensional photonic topological phases has paved the way for groundbreaking advancements in the field of topological photonics. These significant contributions not only hold immense potential for practical applications in photonics but also serve as a wellspring of inspiration for further research endeavors. Exciting possibilities await as we delve deeper into the fascinating realm of 3D and higher-dimensional photonic topological phases, unearthing novel aspects and expanding the frontiers of their applications in the domain of topological photonics. For example, given the prominent role of nonlinear phenomena in the functionality of many photonic devices, including modulators, switches, light sources, and lasers, exploring how 3D and higher-dimensional photonic topological phases can be combined with nonlinearity to achieve robust responses and enhance light–matter interactions has become a natural issue, whether in classical or quantum applications.39,184–202 In addition, the emergence of 3D and higher-dimensional photonic topological phases establishes an ideal photonic platform for the exploration of captivating physical phenomena, including non-Abelian band topology,203–206 non-Hermitian skin effects,207–212 and adiabatic trapping potentials.213–215 These avenues of inquiry hold tremendous promise for future scientific breakthroughs, propelling the field of topological photonics to new heights of discovery and innovation.

The work is sponsored by the Key Research and Development Program of the Ministry of Science and Technology under Grant Nos. 2022YFA1405200 (Y.Y.), 2022YFA1404704 (H.C.), 2022YFA1404902 (H.C.), and 2022YFA1404900 (Y.Y.); the National Natural Science Foundation of China (NNSFC) under Grant Nos. 62175215 (Y.Y.), and 61975176 (H.C.); the Key Research and Development Program of Zhejiang Province under Grant No. 2022C01036 (H.C.); the Fundamental Research Funds for the Central Universities (Grant No. 2021FZZX001-19) (Y.Y.); the Excellent Young Scientists Fund Program (Overseas) of China (Y.Y.);, the National Natural Science Foundation of China under Grant Nos. 12104211, 62375118, and 6231101016 (Z.G.); Shenzhen Science and Technology Innovation Commission under Grant No. 20220815111105001 (Z.G.); SUSTech under Grant Nos. Y01236148 (Z.G.) and Y01236248 (Z.G.); and the National Natural Science Foundation of China under Grant No. 12304484 (Y.M.).

The authors have no conflicts to disclose.

N.H., X.X., and Y.M. contributed equally to this work.

Ning Han: Investigation (equal); Writing – original draft (equal). Xiang Xi: Investigation (equal); Writing – original draft (equal). Yan Meng: Investigation (equal); Writing – original draft (equal). Hongsheng Chen: Conceptualization (lead); Supervision (lead); Writing – review & editing (lead). Zhen Gao: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). Yihao Yang: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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