1D Photonic Topological Insulators Composed of Split Ring Resonators: A Mini Review

Abstract

In recent years, topological photonics inspired by electric topological insulators has promoted extensive research on robust electromagnetic (EM) wave manipulation and new wave‐functional devices. Optical resonators can significantly confine EM waves and are the basic building blocks for constructing diverse topological structures under a tight binding mechanism. As an artificial “magnetic atom,” the split‐ring‐resonator (SRR) is one of the most attractive optical resonators. SRRs provide an excellent and flexible platform for constructing various topological structures with complex coupling distributions, uncovering abundant topological properties, and innovating practical devices. Here, the realization and fundamental EM responses of the SRR are briefly introduced. Compared to conventional EM resonance elements, the coupling between SRRs depends not only on the coupling distance but also on the orientation angle of the slits. The recent achievements in various low‐dimensional photonic topological structures composed of SRRs are summarized. Furthermore, this review explains the underlying physical principles and discusses progress in topological devices with SRRs, including wireless power transfer, sensing, and switching. Finally, this review provides an overview of the future of SRR topological structures and their impact on the development of novel topological systems and devices.

Introduction

In recent years, topological insulators, a major research hotspot in physics, have significantly enhanced the understanding of the classification of the state of matter in condensed-matter physics and have made people realize that fully occupied electronic energy bands also have topological properties marked by topological invariants.^[1] Topological insulators also open a new direction for the development of semiconductor devices that are expected to be used in quantum computing and high-fidelity quantum communication.^[2,3] Inspired by the topological physics of electronic energy bands, researchers designed topological photonic energy bands and optical topological states based on artificial photonic microstructures. Optical topological properties must be used to control the movement of photons. Thus, related research content has developed into topological photonics.^[4–9] Some complex topological models in condensed matter physics can be analogized and constructed using artificial photonic structures,^[10–13] which provide the possibility for people to explore the abundant physics properties associated with the topology,^[14–16] such as the high-order topology,^[17,18] non-Hermitian physics,^[19] valley physics,^[20] quasi-period physics,^[21] and non-linear optics.^[22] Moreover, synthetic dimensions^[23,24] and deep learning^[25,26] provide new methods to explore novel topological structures. Simultaneously, the optical topological state can overcome the scattering loss caused by defects and disorders, thus providing a new degree of freedom for controlling electromagnetic (EM) waves and the possibility of designing revolutionary optical devices with topological protection.^[27,28]

Photonic 2D Topological Phases

Photonic Quantum Hall Effect

The most prominent features of topological structures are non-zero topological invariants and associated robust edge states determined by bulk-edge correspondence. The optical analogy of breaking the time-reversal (T-reversal) symmetry to realize the quantum Hall effect was first proposed for a 2D magneto-optical (MO) photonic crystal (PC) under an external magnetic field.^[29,30] For quantum-Hall insulators, the topological curvature F_n = ∇_k × A_n is non-zero, where A_n = ⟨u_n|i∇_k|u_n⟩ denotes the Berry connection and u_n is the Bloch function. Compared with the band structure of conventional trivial optical systems, photonic topological insulators based on MO PCs have an edge mode in the band gap that runs through the upper and lower energy bands. This edge mode with evident topological protection was not affected by the defects during transmission, and this “one-way” propagation free of backscattering was successfully demonstrated in the experiment.^[31] These groundbreaking achievements in photonic quantum-Hall insulators deepen our understanding of Maxwell's theory, reveal analogies with topological insulators for electrons, and offer applications in robust optical interfaces. Subsequently, additional optical topological phenomena breaking the T-reversal symmetry were observed one after another. Topological edge states in graphene-like MO PCs were demonstrated.^[32] In addition, multimode one-way waveguides produced by large Chern numbers,^[33,34] large-area waveguide states,^[35] one-way bulk states,^[36] and topological beam splitting^[37] have recently been explored. Single-surface Dirac cones have been discovered in photonic topological insulators beyond the plane structure.^[38] Although the method of breaking the T-reversal symmetry of the system by applying a magnetic field to MO PCs broadens our way of achieving photonic topological states, the importance of an external magnetic field is a major drawback for the integration of such components on a chip.^[39] In addition, MO-PC-based photonic topological insulators are suitable for the microwave regime, and it is difficult to extend them to the optical band because the MO response is weak in the high-frequency band.

Photonic Quantum Spin Hall (QSH) Effect

Alternative magnetless solutions for photonic topological insulators have also been investigated. After the realization of an actual magnetic field-assistant photonic quantum Hall effect, it was soon discovered that a synthetic magnetic field could exhibit the same effect as an actual magnetic field in producing one-way edge modes.^[40–42] Other topological insulators for electronics including the quantum anomalous Hall effect^[43,44] and QSH effect,^[45–47] naturally overcome the limitation of an external magnetic field. In contrast to the quantum Hall phase, for the QSH phase, the corresponding edge state exhibits an evident spin-locked unidirectional transmission. The Berry curvature of the QSH phase is positive and negative for spin-up and spin-down, respectively, indicating that the spin Chern number of the topological phase is non-zero, but the total Chern number is zero. However, the most significant difference between photons and electrons is that photons do not have intrinsic spins, unlike electrons. Therefore, to realize the QSH effect in optical systems, the first step is to construct a pseudo-spin photon state^[48–50] An optical analogy of the QSH effect protected by T-reversal symmetry using a coupled. resonant ring (CRR) array was realized.^[51] Moreover, photonic CRR arrays under a strong coupling mechanism, which have been theoretically proposed^[52–55] and experimentally demonstrated,^[56,57] can be used to realize topological insulators with a Chern number of zero. For the photonic quantum spin insulators realized by CRR arrays, the roles of pseudo-spins in opposite directions are played by the clockwise and anti-clockwise propagation directions of light.^[51–69] This design was further extended to acoustic systems for topologically protected unidirectional acoustic transmission.^[60,61] High-performance photonic topological devices based on CRR arrays such as optical isolators,^[54] delay lines,^[58,59] and lasers^[68,69] have been proposed as key components of optical communication systems. Importantly, the idea of using degenerate modes to construct photonic pseudospins in a topological CRR array enables the observation of novel optical QSH effects. Based on magneto-electric coupling in split-ring resonators (SRRs) as an effective anisotropic material, a combination of transverse-electric (TE) and transverse-magnetic (TM) polarizations was proposed to construct a photonic pseudospin ψ^±(x_⊥,q) = E_z(x_⊥,q) ± H_z(x_⊥,q) and the topological band structure realized by spin-orbit coupling.^[70] Accompanied by this pioneering work, the magneto-electric coupling-enabled optical QSH effect has been proposed using other configurations and demonstrated in the experiment.^[71–78] A 3D all-dielectric photonic topological insulator with magnetoelectric coupling has been proposed.^[79] Therefore, by tuning the magnetoelectric coupling strength of the permittivity tensor, two types of polarized waves degenerate, and then different pseudospins are constructed using two degenerate polarized lights, which provide a new avenue to observe the optical spin Hall effect directly from the perspective of EM parameters. Recently, many topological properties of other metamaterials with special EM parameters have been discovered, such as photonic topological insulators with zero-index media^[80–84] and hyperbolic media.^[85–93]

Moreover, the photonic QSH effect can be realized in PCs with a special lattice symmetry. A 2D Kekulé-distorted honeycomb PC was proposed^[94] as an explicit photonic example of a topological crystalline insulator.^[95] Mode degeneracy is achieved through band folding in a honeycomb cluster structure composed of all-dielectric media.^[95] In particular, the angular momentum of the wave function plays the role of the pseudospin.^[94] Based on this pioneering theoretical work, extended theoretical analyses^[96–101] and experimental demonstrations^[102–106] have been conducted. Considering the miniaturization and integration of topological devices, particles (finite crystals)^[107] and membrane PCs^[108] have been proposed. Using the local resonance,^[84,109] topological crystalline metamaterials at subwavelength scales have been theoretically proposed and experimentally demonstrated.^[110–112] Importantly, inspired by optical and photonic schemes, the QSH effect realized by 2D Kekulé-distorted honeycomb PCs in acoustics,^[113–118] optomechanics,^[119] exciton-polaritons^[120] and electronics^[121] have been reported recently with desired robust properties. The Kekulé-distorted honeycomb PC not only provides a new paradigm for the study of topological physics from the perspective of photonics pseudospin, but also broadens robust applications in photonic routing,^[122,123] delay line,^[124] waveguides,^[125,126] cavities,^[127] coupler,^[128,129] all-optical logic gates,^[130] and single-mode lasers.^[131–134]

Photonic Quantum Valley Hall Effect

The original concept of “valley physics” is based on the band structure description of graphene.^[135–137] The two valleys with Dirac cones at K' and K in graphene were constrained by the symmetry of the time inversion. Because the two valleys are separated in the momentum space, they are relatively independent; thus, the valley degree of freedom is also a kind of pseudospin, and can be used to realize various novel devices to control electronic transmission.^[138–142] The Dirac-cone dispersion relationship in a honeycomb PC was experimentally observed.^[143] The Dirac point in graphene originated from the symmetry of the system. Because the band structure has the dispersion relation of the Dirac cone, PCs composed of triangular and honeycomb lattice structures are called “photonic graphene.” In photonic graphene, the coupling strength between two sets of sublattices and the symmetry of the lattice can be flexibly controlled, and some wave behaviors in photonic graphene can be directly observed at the macro scale. Currently, some quantum phenomena and wave propagation characteristics related to the conical dispersion relationship in graphene have been realized in photonic graphene, such as pseudo-diffusion,^[144,145] Zitterbewegung,^[146,147] Klein tunneling,^[148,149] weak anti-localization,^[150] deformation-induced pseudo-magnetic fields,^[151,152] and photonic Landau levels.^[153,154] In addition, by opening the bandgap from the Dirac point, the valley properties have also been further directly explored in the staggered optical lattices, such as the BN and MoS₂ lattices.^[155–157] Importantly, similar to the photonic QSH effect, the topological properties of the photonic quantum valley Hall phase have also been studied theoretically and experimentally.^[158–166] The Berry curvature of the quantum valley Hall phase is positive and negative for K' and K, respectively, indicating that the valley Chern number of the topological phase is not zero.^[160,161] As a type of pseudospin, the valley Chern number can be calculated directly by integrating the Berry curvature into half of the first Brillouin zone. Moreover, the topological valley-edge states can be determined from the bulk–edge correspondence.^[162,163] The topological invariants for two valleys are opposite; therefore, the topological valley-edge states corresponding to K' and K will be transmitted along different directions, that is, the topological valley-edge state corresponds to a one-way transmission with valley-orbit locking.^[164–166] The robust valley-kink states,^[167–170] valley Rabi-like oscillation,^[171] and valley Fano resonance^[172] have been applied in the photonic topological valley structures. In addition, the valley vortex and valley-edge states have also been systematically explored in the acoustic and phononic systems.^[173–181] Valley topology physics was further enriched by considering the synthetic dimension^[182] and non-Hermitian^[183] degrees of freedom. Topological plasmons in graphene have also been proposed.^[184,185] Recently, the hybrid spin-valley Hall effect and the spin-valley coupled-edge states have also been demonstrated.^[186–190] Overall, the charming topological properties of the quantum valley Hall phase have opened up novel capabilities to construct robust photonic devices, including waveguides,^[191,192] delay lines,^[193,194] lasers,^[195,196] beam splitters,^[197] and wireless communications.^[198]

Photonic 3D Topological Weyl Phases

The widely studied 2D topological structures are composed of unit cells arranged periodically along two directions, and gapless edge states with topological protection can be supported in the bandgap. Using a 3D artificial lattice consisting of cells arranged periodically along three spatial directions, the photonic 2D topological insulating phases can be further extended to 3D topological structures.^[199–204] Once the two bands (one above and the other below the bandgap) are topologically distinguished, a topological gapless surface state exists in the bandgap. For the edge states of 2D photonic topological structures, EM waves are confined to the plane; thus, topological protection is effective only for propagation in the plane. In contrast, the surface states of 3D topological insulators are confined to all three spatial directions; thus, the surface states have a stronger topological protection and the EM waves are immune to impurities and defects along any spatial direction, but not in specific planes.^[199–204] For a photonic structure with a 3D quantum Hall phase, a bias magnetic field is applied to break the T-reversal symmetry and a topological surface state is generated in the bandgap. Similarly, considering that T-reversal symmetry remains, the 3D QSH phase can be regarded as a generalization of the 2D QSH phase. The bandgap is not a condition for constructing 3D topological phases. The 3D gapless topological phase, also known as the topological semimetallic phase, is a new topological phase that differs from the topologically insulating phase. The 3D gapless topological phase overturns the conventional view that topological features require a bandgap, and promotes the understanding of photonic topological phases. Unlike 3D gapped topological phases, 3D gapless topological phases do not have 2D counterparts. The characteristic of the 3D gapless topological phase is not bandgap, but Weyl degeneracy, that is, the degeneracy between topological non-equivalent bands.^[205–211] Weyl degeneracy in the 3D gapless topological phase is known as the Weyl point. A synthetic dimension paves an effective way for realizing various 3D topological phases.^[212–216] If the iso-frequency contour (IFC) corresponds to a point at the Weyl frequency, it is called a type-I Weyl point. However, if the band structures are tilted, the corresponding IFCs are two crossed lines and a single line, called type-II and type-III Weyl points, respectively.^[205] Photonic Weyl points have been observed in various 3D systems, such as gyroid PC,^[199] evanescently coupled waveguides,^[205] and metamaterials.^[206,207] To date, a series of promising results has been obtained. Examples include the zero Landau level in an inhomogeneous Weyl system,^[208] topological nodal chains,^[209,210] and helicoidal surface dispersions.^[211] Moreover, the photonic fourfold degenerate Dirac nodal-line semimetals have also been proposed by merging two Weyl points.^[214–219]

Photonic 1D Topological Phases

In addition to the edge states (surface states) in the 2D (3D) systems, the photonic topological states in a 1D system have also attracted extensive attention owing to their rich physics, including invariant topological orders,^[220–228] topological excitations,^[229–232] band inversion,^[233–235] and robust edge states.^[236–238] The 1D topological structure is relatively simple, and its prominent feature is 0D topological end states. The topological phases can be easily constructed by employing lattice modulations.^[239–241] 1D linear Shockley-like surface states in an optically induced semi-infinite photonic superlattice were experimentally demonstrated.^[239] The Su-Schrieffer-Heeger (SSH) model has been carefully studied as a basic geometry for constructing novel photonic topological structures.^[242] In this basic topological model, two topological distinguished phases are determined by the relative magnitude of intra-cell (κ₁) and inter-cell (κ₂) coupling coefficients, and the edge states appear symmetrically at two ends of the chain for the topologically nontrivial phase with κ₁ < κ₂. To date, theoretical and experimental research on the appealing properties of photonic SSH models has also been extended to non-Hermitian,^[243–250] non-linear,^[251–255] and active systems.^[256–259] In contrast to the tight-binding model, 1D PCs based on multiple scattering mechanisms are an important type of photonic topological structure. The topological properties of 1D PCs can be easily affected by the geometrical settings.^[260–262] The topological properties of the bandgaps can be indirectly determined by the Zak phase of the bands in the 1D PC. The Zak phase of each band is used to express the topological invariance. In particular, the reflection phase features of the gaps are related to the Zak phase of the bands.^[262] The electromagnetic response of materials depends on the permittivity (ε) and permeability (μ). In other words, the topological properties of the bandgap can be directly distinguished by the effective electromagnetic parameters.^[260,261] Based on this method, topological edge states in a heterostructure composed of two PCs of different orders have been theoretically proposed and verified through microwave experiments.^[260] The effective EM parameters are related to the topological order of the insulators. When both ε and μ are positive or negative, the material belongs to the conductor of light. However, when one of these values is negative, the material is a light mirror. These are the ε-negative metamaterial (ENG metamaterial, ε < 0, μ > 0) and μ-negative metamaterial (MNG metamaterial, ε > 0, μ < 0). By mapping the 1D Maxwell equation to the Dirac equation, the topological order of the photonic structure can be determined using the effective mass m = ω(ε − μ)/2c associated with the effective EM parameters.^[235] The effective mass of ε-negative metamaterial and μ-negative metamaterial is negative and positive, respectively. Thus, the topological orders of the electric and magnetic mirrors are different. Using the microwave platform, the permittivity and permeability can be flexibly tuned. By choosing different circuit parameters, two mirrors with different topological orders have been constructed.^[263,264] The topological interface state between two PCs with distinct topological gaps was demonstrated and can be used for field enhancement.^[235] To date, abundant topological properties in 1D photonic topological structures have been proposed in PCs,^[260–272] waveguide arrays,^{[229,230,243–249]} resonator lattices,^{[227,238,273–280]} and circuit arrays.^{[253,254,281–283]} In addition, the 1D topological phases realized by the local resonance state combined with multiple scattering mechanisms have also received extensive attention.^{[109,284–286]} Recently, the topological subspace-induced bound state in the continuum have also been theoretically proposed and experimentally demonstrated.^[287–289] In the past decade, topological photonics has experienced vigorous development, and related properties have been extended to acoustics,^{[285,286,290–292]} water waves,^[293] and even thermal diffusion.^[294–301] Research must be conducted to determine whether fascinating topological characteristics can lead to innovations in current optical devices. 1D topological phases with simple designs are easily constructed and have opened up exciting avenues in topological physics and related photonic devices with topological protection against a variety of perturbations and disturbances. However, natural optical resonators have limited optical properties. The ability to tailor the optical properties of meta-atoms flexibly is crucial for the development of novel photonic phases and their applications. Figure 1 shows a schematic of the 2D, 3D, and 1D structures and their corresponding characteristics. For the 1D topological phase, 0D topological end states under the framework of the bulk-edge correspondence (BEC) can be formed at the two ends of the structure. For 2D (3D) topological phases, conventional 1D edge states (2D surface states) and high-order 0D topological corner states (1D topological hinge states and 0D topological corner states) exist in finite structures.

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In this review, we focus on a special photonic meta-atom: the SRR. Compared to disks or spheres, the coupling coefficients between SRRs are not only dependent on the separating distance, but are also related to the relative rotation angle of the slits of two neighboring resonators. 1D topological chains composed of SRRs provide a good platform for studying complex topological models and reveal abundant physical mechanisms and important applications. This study is expected to appeal to a broad audience interested in topological photonics.

Photonic Meta-Atom–Split-Ring-Resonator (SRR)

Resonance Properties

Photonic resonators (PRs) can strongly scatter and confine EM waves and improve the interaction between light and matter. They also play an important role in cavity quantum electrodynamics, non-linear optics, and quantum optics.^[84] The PRs establish a tight-binding regime in which the EM field is mostly confined within the PRs, and the coupling strength is controlled by adjusting the separation between the PRs. The larger the separation, the smaller the coupling strength.^[302] In recent years, an increase in artificial meta-atoms (SRR) has significantly enriched the design and functionality of PRs. The SRR was first proposed for the cavity magnetron of a radar.^[303] Later, it was used as a subwavelength magnetic resonator to study electron spin and nuclear magnetic resonance phenomena.^[304–306] SRRs were first used to design high-Q magnetic resonators in the 200–2000 MHz range to replace the bulky conventional cavities and impractical solenoid coils.^[304] A novel negative refraction of the double-negative metamaterial (DNG metamaterial, ε < 0, μ < 0) was theoretically proposed^[307] However, negative permeability is almost unavailable in natural materials. ENG and DNG. metamaterials have been realized Using artificial SRRs.^[308–310] The realization of metamaterials has rapidly increased the research interest in SRRs. Several types of SRR configurations exist for different research scenarios, as shown in Figure 2 The single-ring configuration of the SRR in Figure 2a is the most basic case, where the directions of the electric and magnetic fields are parallel and perpendicular to the torus, respectively.^[311–314] Rotation is a new degree-of-freedom for the SRR as a PR, as shown in Figure 2b According to Faraday's electromagnetic induction law, the surface of the metal ring stimulates the induced current, whereas at the opening of the SRR, many charges accumulate to form an equivalent capacitance. Thus, an SRR can be regarded as an LCR circuit with a resonant angular frequency $${\omega }_0 = 1/\sqrt {{L}_1{C}_1} $$. Figure 2c shows an effective circuit that can be excited by an external alternating magnetic field passing through the SRR. The effective inductance and capacitance of the SRR can be obtained directly through analytical calculations^[314,315] 1 $$$\begin{equation}{L}_1 = {\mu }_0\left( {r + w/2} \right)\left[ {\ln \frac{{8\left( {r + w/2} \right)}}{{h + w}} - \frac{1}{2}} \right]\end{equation}$$$and 2 $$$\begin{equation}{C}_1 = {\varepsilon }_0\left[ {\frac{{\left( {h + g} \right)\left( {w + g} \right)}}{g} + \frac{{2\left( {h + w} \right)}}{\pi }\ln \frac{{4r}}{g}} \right]\end{equation}$$$respectively. The thickness, height, inner radius, and gap size of the single-ring SRR are w = 1.0 mm, h = 5.0 mm, r = 10 mm, and g = 1.5 mm, respectively.^[238] The resonant frequency of the SRR was f₀ = ω₀/2π = 1.9 GHz. The effective magnetic response of the SRR-based metamaterials can be expressed by the effective permeability^[308–310] 3 $$$\begin{equation}\mu = 1 - \frac{{F\omega _p^2}}{{{\omega }^2 - \omega _0^2 + i\omega {\gamma }_m}}\end{equation}$$$where ω_p and γ_m denote the angular frequency and loss factor of the magnetic plasma, respectively. F is a constant related to the geometry. When the frequency of the external EM wave is lower than the resonant frequency of the SRR, the current excited in the SRR is synchronized with the incident field, the overall material shows a positive response, and the equivalent permeability is positive; if the frequency of the external EM wave is higher than the resonant frequency of the SRR, the current established in the SRR cannot keep up with the alternating incident magnetic field; therefore, it has the effect of lagging the external field. At this time, the material shows a negative overall response, and the effective permeability is negative.^[316] In addition to the microwave regime, the SRR in terahertz and visible bands has also attracted extensive attention with the development of nanotechnology.^[317–322] In metamaterials, many SRR-variant structures based on single-ring configurations have been studied theoretically and experimentally, including double-ring,^[323,324] side coupling,^[325,326] complementary,^[327] and multilayer^[312,328] configurations. In addition to the single-ring SRR, another special configuration considered in this review is the spiral SRR, as shown in Figure 2j. As previously mentioned, the effective inductance of the spiral SRR is formed by a metal strip. When the magnetic field is perpendicular to the spiral SRR, the metal surface also induces a surface current; however, the ring of the spiral SRR is open, and the charges accumulate at the head and tail. Because the charge distribution is uneven throughout the structure, a potential difference between the metal strips of different coils is formed, thus generating an equivalent capacitance. As the most important meta-atom, the transmission characteristics of SRR chains have also been reported successively.^[329–332] In particular, the abundant properties of magnetoinductive waves in SRR chains have been demonstrated,^[333,334] and various waveguide devices have been proposed, such as single-port,^[335,336] multiport,^[337] and ring^[338] waveguides.

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The SRR chains provide a powerful platform for controlling the transportation of EM waves. Owing to the combination of the simplicity of the structure and the exotic electromagnetic response of the topological states, 1D topological modes have been proposed for numerous photonic applications. Here, we briefly discuss the photonic 1D topological phases realized using SRR chains. Depending on band engineering based on tight-binding models in the near-field coupling mechanism regime, topological invariants, edge states, and related applications of various 1D topological models were introduced. The purpose is to provide an up-to-date status of photonic 1D topological phases with novel meta-atoms.^[339,340]

Near-Field Coupling

The EM field was primarily confined to the SRR, and the magnetic field created by the current induced in the element reached its maximum around the opposite part of the split ring. The coupling between two arbitrarily rotated SRRs consists of a positive magnetic and negative electric coupling κ = κ_H + κ_E, as shown in Figure 3a. The rotation angles of the SRRs are φ₁ and φ₂. The coupling strength can be controlled by tuning the angle between the two resonators. They can be modeled using two coupled equivalent LCR circuits, as shown in Figure 3b. Near-field coupling between SRRs can also be obtained through analytical calculations:^[322,324] 4 $$$\begin{equation}\kappa = \frac{{2M}}{L} - \frac{{2C}}{K}\end{equation}$$$where mutual inductance and mutual capacitance can be expressed as 5 $$$\begin{equation}M = - \frac{{{\mu }_0}}{{4\pi }}\int{{\int{{\frac{{{I}_{n1}\left( {{r}_1} \right){I}_{n2}\left( {{r}_2} \right){e}^{ik\left| {{r}_1 - {r}_2} \right|}}}{{\left| {{r}_1 - {r}_2} \right|}}}}}}d{S}_1d{S}_2\end{equation}$$$and 6 $$$\begin{equation}K = {\left[ {\frac{1}{{4\pi {\varepsilon }_0}}\int{{\int{{\frac{{{\rho }_{Ln1}\left( {{r}_1} \right){\rho }_{Ln2}\left( {{r}_2} \right){e}^{ik\left| {{r}_1 - {r}_2} \right|}}}{{\left| {{r}_1 - {r}_2} \right|}}}}}}d{S}_1d{S}_2} \right]}^{ - 1}\end{equation}$$$respectively. According to Equations (4–6), the dependence of the coupling coefficient κ on the relative rotation angle between two SRRs at a fixed separating distance P  =  24 mm is shown in Figure 3c. Both the magnitude and sign of the coupling coefficient can be flexibly tuned by changing the relative rotation angle of the SRRs.

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In addition to the analytical calculation, two types of coupling strengths can be extracted from the reflection spectrum experimentally.^[302] Take two coupling cases with a weak (Case-I: φ₁ = 0° , φ₂ = 0°) and strong (Case-II φ₁ = 180° , φ₂ = 180°) coupling strength for example, which are shown in the center and lower insets of Figure 3c, respectively. When the gaps of neighboring SRRs are adjacent to each other (Case II), the coupling strength is large and negative. However, when the gaps of neighboring SRRs are on opposite sides (Case I), the coupling strength is small and positive. Aotation-controlled energy-level inversion system in a coupled system composed of two SRRs is shown in Figure 4. Schematics of the energy levels for the positive and negative coupling configurations are shown in Figure 4a,d, respectively. The corresponding structures of the coupled SRRs are shown in Figure 4b,e. Figure 4c shows the simulated symmetric (asymmetric) magnetic field distribution of the SRRs with positive hopping at a higher (lower) energy level. By contrast, the symmetry of the modes at different eigenfrequencies is reversed in the case of negative coupling, as shown in Figure 4f.

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To illustrate that the sign of the coupling coefficient between the two SRRs can be flexibly controlled using the rotational degrees of freedom, Figure 5 shows the current distributions of the two coupling configurations. For the case-II (case-I) configuration with positive (negative) coupling, the current distributions in the two coupled SRRs configurations are opposite (the same) at low frequencies, as shown in Figure 5a,c, respectively. Similarly, the current distributions at high frequencies in the two coupled SRR configurations are shown in Figure 5b,d. The magnitude and direction of the current in the SRRs are indicated by the size and direction of the red arrows.

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Topological Chains with SRRs

Periodic 1D Structures

Dimer Chain

The SSH model is a typical geometry used to study topological excitations of the organic molecule polyacetylene, where electrons coupled to the domain walls propagate as charged solitons. This photonic system provides a convenient platform for realizing an analogy of the SSH model.^[241] Researchers have demonstrated topological interface states in SSH chains based on dielectric resonators.^[272] They established a tight-binding model, in which the field was mostly confined within the resonators. The coupling strength in the chain was controlled by adjusting the separation between the resonators. Within the tight-binding model, the dynamics of the SSH chain can be described by the Hamiltonian: 7 $$$\begin{equation}H = \sum_{j = 1}^N {\left( {{\kappa }_1c_{A,n}^\dag {c}_{B,n} + {\kappa }_2{c}_{A,n + 1}c_{B,n}^\dag + H.c.} \right)} \end{equation}$$$where κ₁ and κ₂ are the intra- and inter-cell coupling coefficients, respectively. $$c_{i,n}^\dag $$ and c_i,n are the creation and annihilation operators at site i (i = A and B), respectively. The topological properties of the chain can be described by the winding invariant: 8 $$$\begin{equation}\nu = \frac{1}{\pi }\int_{{ - \pi }}^{\pi }{i}\left\langle {{u}_k} \right|\frac{\partial }{{\partial k}}\left| {{u}_k} \right\rangle dk\end{equation}$$$where u_k is the Bloch wavefunction. The topological invariance depends on the relative magnitudes of κ₁ and κ₂. If κ₁ < κ₂, v  =  1, corresponding to a topological nontrivial case. However, if κ₁ > κ₂, v  =  0, corresponding to the trivial topological case. Recently, non-linearresearchers have studied self-induced topological edge states in active SSH chains based on non-linear transmission lines.^[253] The implemented circuit consisted of dimer-type unit cells. Each dimer was composed of two identical LC tanks coupled with linear and non-linear capacitors. The non-linear capacitance was achieved using two series back-to-back variable capacitance diodes (VCDs), and this non-linear capacitance yielded non-linear coupling. At low intensities, the circuit behaved as a topologically trivial chain of dimers with nearly equally excitable resonances. Therefore, the circuit behaved as a topologically trivial chain without an edge state. However, as the intensity increased, the chain became topologically nontrivial, and the self-induced edge state appeared.^[253,254] Photonic parity-time (PT)-symmetric systems that support topological states can be useful for shaping and routing EM waves. Researchers demonstrated that topological interface states can exist in effective PT-symmetric SSH systems.^[244] Using a passive waveguide array, the non-Hermitian PT-symmetry system can be mapped directly onto a system with losses by simply multiplying the wave function by an overall decay factor. The gain–loss profile is globally PT-invariant, and the spectrum exhibits entirely real eigenvalues. Moreover, the topological edge states localized at the interface between two topologically distinct PT-symmetric photonic lattices were demonstrated.^[244] In addition, considerable effort has been made to transplant topological concepts into lasing systems. Using a gain medium, researchers have shown that non-Hermitians can promote single-edge-mode lasing in an active SSH micro-ring array.^[256,257] When uniformly pumped into the SSH laser system, the resulting lasing profile is a complex mixture of all the modes that experience the same gain. However, under the PT-symmetric condition, researchers have shown that only one of the edge modes is favored, whereas all the bulk modes are suppressed. In direct contrast to the uniformly pumped case, the PT-symmetry promotes only one edge state and can be used for single-mode lasers.^[256,257]

In this section, we will mainly introduce the experimental observation of topological invariant and robust edge states in an SSH chain composed of SRRs.^[227,238] The electromagnetic field is mostly confined within the resonators. Consequently, the SRR chain can be treated by the tight-binding model. The dimer chain composed of SRRs can easily mimic an effective SSH model, and the experimental setup is shown in Figure 6a. By exchanging the coupling strength between intra-cell (κ₁) and inter-cell (κ₂), two types of dimer chains with different unit cells (i.e., case-I and case-II configurations) can be constructed, as shown in Figure 6b. Within the tight-binding formalism, the currents distribution of SRRs can be written as: 9 $$$\begin{equation}{p}_n = {a}_n{e}^{i\left( {knd - \omega t} \right)},\;{q}_n = {b}_n{e}^{i\left( {knd - \omega t} \right)}\end{equation}$$$where k and n denote the Bloch wave vector and site number of the unit cell, respectively. Considering the Fourier transforms $${p}_n = \sum_k {{a}_k{e}^{i(knd - \omega t)}} $$ and $${q}_n = \sum_k {{b}_k{e}^{i(knd - \omega t)}} $$, the effective Hamiltonian of the SSH model composed of the SRRs can be described as follows: 10 $$$\begin{equation}H = \left( { \def\eqcellsep{&}\begin{array}{@{}*{2}{c}@{}} {{\omega }_0}&{{\kappa }_1 + {\kappa }_2{e}^{ - ikd}}\\ {{\kappa }_1 + {\kappa }_2{e}^{ikd}}&{{\omega }_0} \end{array} } \right)\end{equation}$$$

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Because the amplitude and phase of the local photonic modes can be accurately measured in the SRR platform, one can obtain direct information on the topological characteristics, compared to other platforms. In particular, the winding number of the bands could be obtained by directly measuring the pseudospin of the SSH chains. First, the wave functions of the unit cells (a_k,b_k) were obtained by measuring the amplitudes and phases of the SRRs. Second, using the discrete Fourier transform, we can obtain the wave function in the momentum space and further derive the pseudospin vectors: 11 $$$\begin{equation}{P}_x = \left\langle {{\sigma }_x} \right\rangle = b_k^*{a}_k + a_k^*{b}_k,\;{P}_y = \left\langle {{\sigma }_y} \right\rangle = ib_k^*{a}_k + ia_k^*{b}_k\end{equation}$$$

The pseudospin angle can then be determined using $${\theta }_k = \arctan ( {{P}_y/{P}_x} )$$. Finally, the winding number of the bands is determined by integrating the pseudospin vectors in the first Brillouin region. 12 $$$\begin{equation}\nu = \frac{1}{{2\pi }}\int_{{ - \pi/d}}^{{\pi/d}}{{d{\theta }_kdk}}\end{equation}$$$

The magnetic field signal above the SRR is proportional to the surface current $${X}_n = {I}_n{e}^{i{\varphi }_n}$$ (X = a or b), where I_n and φ_n denote the amplitude and phase of the current measured by the near-field probe, respectively. Based on near-field detection technology, one of the measured current distributions directly determines the dispersion characteristics using this spectral function. For a finite SSH composed of nine unit cells, the discrete Fourier transform can be expressed as: 13 $$$\begin{equation}{\left| {{C}_{dis}\left( {\omega ,k} \right)} \right|}^2 = {\left| {\sum_{n = 1}^9 {\left[ {{a}_{n,\omega }{e}^{ - ikd\left( {n - 0.5} \right)} + {b}_{n,\omega }{e}^{ - ikdn}} \right]} } \right|}^2\end{equation}$$$

Figure 6c,f show that the measured band dispersions of the trivial (case-II configuration) and topological (case-I configuration) dimer chains are discrete owing to the finite structure. Nevertheless, the band structures could be clearly determined. The experimental results (color spectra) agree well with the theoretical results (theoretical calculations). Moreover, the wave vectors corresponding to different eigenfrequencies can be determined based on the measured band dispersion. From the measured wave function, 14 $$$\begin{equation}\left( { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{a}_k}\\ {{b}_k} \end{array} } \right) = \frac{1}{C}\left( { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\sum_{n = 1}^9 {{a}_{n,\omega }{e}^{ - ikd\left( {n - 0.5} \right)}} }\\ {\sum_{n = 1}^9 {{b}_{n,\omega }{e}^{ - ikdn}} } \end{array} } \right)\end{equation}$$$we obtain the parity of the eigenmodes. This can be used to determine topological characteristics.^[227] In addition, the pseudospin distribution can be obtained using Equation (11). By adding the pseudospin vectors of different eigenmodes to the band dispersion diagrams in Figure 6c,f, the direction of the pseudospin vectors in each band is always the same for the trivial chain, whereas, it will reverse at the band edge in each band for the topological chain. Finally, the winding number, indicated by the closed loops denoted by P_x and P_y as wave vectors spanning the Brillouin zone, can be experimentally observed. The loop of each band winds to zero around the origin for the trivial chain, as shown in Figure 6d,e. However, the loop of each band winds once for the topological chain, as shown in Figure 6g,h. This conclusion is consistent with the theoretical predictions. These attributes make it possible to straightforwardly determine the winding number of a bulk band, as well as other topology-related properties.

Robust edge states in topological SSH chains composed of SRRs have also been successfully demonstrated.^[237] The 1D dimer chain consists of 32 identical equally spaced SRRs, as shown schematically in Figure 7a. The samples were then placed on a foam substrate and sandwiched between two metallic plates. A near-field probe composed of a non-resonant loop was used to measure the density of states. The density of states spectrum was obtained by averaging the local density-of-state (DOS) spectra for all the sites. The measured DOS distributions in Figure 7b,c show that two isolated bands are separated by a gap in the trivial chain. However, for a non-trivial chain, an additional state exists in the gap, which is the edge state. In addition, the LDOS spectra at all sites were obtained by placing a probe at the center of each SRR. In the topological non-trivial chain without perturbations, the LDOS of the edge state was significantly localized at two ends, whereas the bulk state was mainly distributed, as shown in Figure 7d,e, respectively. Considering the lossy materials inside the SRRs, marked by the perturbation region in Figure 7a, the bulk state is significantly affected, as shown in Figure 7f. However, the edge state is robust against losses that are almost unaffected by the addition of losses to the perturbation region, as shown in Figure 7g. A topological protection of the edge states was further demonstrated by rotating the orientation angle of the slits to introduce structural disorder. Figure 7h,i show that the edge state is nearly unchanged, whereas the bulk state is significantly affected by structural disorders.^[238]

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Trimer Chain

A simple periodic trimer chain is another important 1D topological model with abundant topological phases.^[339–345] Based on the tight-binding mechanism, a schematic model of a trimer chain with N = 10 unit cells is shown in Figure 8a.^[345] The rotation angles of the three SRRs in the unit cell are denoted as φ₁, φ₂, and φ₃. Similar to Equation (7), the dynamics in the trimer chain can be expressed by the Hamiltonian: 15 $$$\begin{equation}H = \sum_{j = 1}^N {\left( {{\kappa }_1c_{A,n}^\dag {c}_{B,n} + {\kappa }_2c_{B,n}^\dag {c}_{C,n} + {\kappa }_3c_{C,n}^\dag {c}_{A,n + 1} + H.c.} \right)} \end{equation}$$$where κ₁ and κ₂ are the intra-cell hopping amplitudes, while κ₃ is the inter-cell hopping amplitude. $$c_{i,n}^\dag $$ and c_i,n are the creation and annihilation operators at site i (i = A, B or C), respectively. The band structures of three topologically distinguished cases with inversion symmetry (κ₁ = κ₂ ≠ κ₃) are shown in Figure 8b–d. The topological transition can occur by changing κ₃. For the P₁ configuration with κ₁ = κ₂ > κ₃ in Figure 8b, there are two open bandgaps at the center and boundary of the first Brillouin zone. The Zak phase of bands, that characterizes different topological phases, can be easily obtained from Equation (8) with ϕ_n,Zak = πν = 0. Therefore, the trimer chain with the P₁ configuration has a trivial phase.Both bandgaps gradually narrow with an increasing κ₃. When κ₁ = κ₃ for the P₂ configuration (κ₁ = κ₂ = κ₃), the two bandgaps are closed, which corresponds to the phase-transition point, as shown in Figure 8c. When κ₃ continues to increase to greater than κ₁ for the P₃ configuration (κ₁ = κ₂ < κ₃), the bandgap reopens and the Zak phase of bands is π, as shown in Figure 8d. The process of the closing and reopening of the bandgap with a continuous change in κ₃ implies that a topological phase transition may occur. A rotation-controlled topological transition and symmetric edge states in the SRR trimer chain are shown in Figure 8e–h. For the trivial trimer SRR chain with the P₁ configuration (φ₁ = 30°, φ₂ = 270°, and φ₃ = 150°), there are no edge states, and the LDOS of the 2N eigenstate is mainly distributed in the bulk state. On the contrary, considering the topological trimer SRR chain with the P₃ configuration (φ₁ = 150°, φ₂ = 270°, and φ₃ = 30°), the LDOS of the 2N eigenstate is symmetrically confined at the two edges of the chain.

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By rotating the SRRs, controllable asymmetric topological edge states can be realized in the trimer chain. For the trimer chain without inversion symmetry (κ₁ ≠ κ₂ ≠ κ₃), the Zak phase is not quantized and the Majorana's stellar representation (MSR) can well characterize the topological phase.^{[289,346–348]} By transforming an n-band eigenstate into n-1 MSs located on a Bloch sphere and finding the roots x_m = tan (θ_m/2)exp (iϕ_m) of the MSR equation, the MS is obtained as follows: 16 $$$\begin{equation}\sum_{l = 0}^{2L} {\frac{{{{\left( { - 1} \right)}}^l{C}_{2L - l + 1}}}{{\sqrt {\left( {2L - l} \right)!l!} }}{x}^{2L - l} = 0} ,\;\;(2L + 1 = n)\end{equation}$$$where C_l(k) represents the component of the eigenvector, and n is the number of sites in each unit cell. The MSs form a closed loop enclosing (excluding) the z-axis in the topological nontrivial (trivial) phase by changing k throughout the Brillouin zone, and the winding number for the MSs is the total winding of their azimuthal angles, given by: 17 $$$\begin{equation}\upsilon = - \frac{1}{{2\pi }}\sum_{m = 1}^2 {\int{{{\partial }_k{\phi }_m\left( k \right)dk}}} \end{equation}$$$which counts the number of times each trajectory winds around the z-axis. The number of edge modes under the open boundary condition is equal to the absolute value of the winding number summed over all energy bands.^[346–348] As k crosses the Brillouin zone, the trajectories of the three Bloch bands in the P₄ configuration (|κ₁| < |κ₃| < |κ₂|) are as shown in Figure 9a–c. The corresponding variation of azimuthal angle ϕ of MSs in the topological trimer chain are shown in Figure 9d–f, respectively. The winding numbers of the first, second, and third bands were 1, 0, and 1, respectively. Therefore, there are two edge states in this topological chain. In the measured DOS spectrum in Figure 9g, the two edge states (E1 and E2) are indicated by red arrows. Figure 9h shows the LDOS spectra of the E1 state for the two configurations after considering the rotation disorder. In particular, the left (right) edge states in the trimer chain with φ₁ = 270°, φ₂ = 30°, and φ₃ = 150°, (φ₁ = 30°, φ₂ = 150°, and φ₃ = 270°) is shown in Figure 9h. Figure 8 and Figure 9 show that the trimer chain has clear advantages in realizing symmetric and asymmetric topological edge states, providing an effective way to observe multiple topological edge states in 1D photonic lattices.^[236] SRRs with rotational degrees of freedom provide a good platform for studying the abundant topological phases of the trimer chain.

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Quasi-Periodic and Quasi-1D Structures

Harper Chain

Quasi-periodic structures with a long-range order are another important class of systems exhibiting topological effects.^{[222,250,284,349–353]} In this section, we will introduce the topological edge state with an aperiodic order based on the Harper model.^[354–358] A schematic of a quasiperiodic Harper chain composed of SRRs is shown in Figure 10a.^[274] Under the tight-binding regime, this Harper chain is defined by the inter-resonator coupling strength:^[358]18 $$$\begin{equation}{\kappa }_n = {\kappa }_0\left[ {1 - \xi \cos \left( {\frac{{2\pi }}{\tau }n + \phi } \right)} \right]\end{equation}$$$where ξ denotes the strength coefficient and $$\tau = ( {\sqrt 5 + 1} )/2$$ is the golden ratio. ϕ is a topological parameter that can be used as a synthetic dimension to construct 2D topological systems with 1D structures.^[23,24] The calculated projected band structure as a function of ϕ is shown in Figure 10b. For the Harper chain with finite resonators, there are three passbands and two bandgaps in the band structure. Moreover, a pair of edge states always exists in the bandgap, independent of the number of resonators. The winding number of the quasiperiodic Harper chain can be expressed as:^[222]19 $$$\begin{equation}\upsilon = \int_{0}^{{2\pi }}{{\frac{{d\delta }}{{d\phi }}}}d\phi \end{equation}$$$where δ denotes the relative spectral position of the edge state within the bandgap. The winding numbers of the first and second gaps are 1 and −1, respectively.

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Considering that the topological parameter is ϕ = 4, a Harper chain composed of 16 identical equally spaced SRRs can be constructed easily. The coupling strength distribution is consistent with the theoretical design. This coupling distribution can be easily realized by tuning the distance between the neighboring SRRs, as shown in Figure 10c. Similar to the analysis of the periodic SSH model, different edge states are marked in the measured DOS spectrum of the Harper chain, as shown in Figure 10d. The measured LDOS distributions of the edge states and one bulk state are shown in Figures 10e–g. The experimental results were in good agreement with the simulated results shown in the inset. The bulk state was mainly distributed in the bulk of the chain, whereas the edge states in the bandgap were strongly localized at the two ends of the chain. Figure 9 shows that in contrast to the periodic topological SSH chain, in which the edge states are symmetry localized at two ends of the structure, the edge states of quasi-period Harper chain are selectively localized to one end of the structure.^[274] This property can be used for selective power transfer. When a source is placed at the center of the chain, for edge states E1 and E2, the energy is mainly transferred to the left and right ends of the chain, respectively. These asymmetric edge states can be used for long-range directional energy transfers, as introduced in Section 4.1.

Kitaev Chain

This section introduces the design of a complete photonic Kitaev topological model using SRRs as well as its topological phase transitions and edge states. In the condensed state, the Kitaev model contains a Majorana fermion in the topological nontrivial phase. This is one of the simplest models to describe topological superconductivity and has attracted attention from researchers.^[359–365] However, it is difficult to implement this topological model experimentally even in optical systems. Because of the requirement for a complex coupling distribution, it is difficult to construct. Researchers arranged metal disks into a zigzag chain structure and used the coupling between two polarization modes in the metal disk to simulate the edge state of the Kitaev model in the topological nontrivial phase.^[275] This optical state provides a new method of controlling photons. However, owing to the limited adjustable coupling parameters of the disk array structure, a complete Kitaev structure could not be simulated. Therefore, many physical phenomena, including topological phase transitions (i.e., the transition from the topologically trivial phase to the nontrivial phase), are difficult to observe using this simplified photonic Kitaev model. Because the coupling strength of the SRR can be flexibly adjusted by varying the split direction and coupling distance, a complete coupling distribution of the Kitaev chain can be flexibly realized using the SRR. The experimental setup and corresponding tight-binding model of the Kitaev chain composed of SRRs are shown in Figure 11a,b, respectively.^[366] In a double SSH chain composed of SRRs, the diagonal coupling strength between the two SSH chains can be much smaller than the vertical coupling strength between the chains; thus, the Hamiltonian of the system corresponds to the complete Kitaev model. Under a proper linear transformation, the Hamiltonian of the Kitaev model for Bloch modes in momentum-space representation can be expressed as: 20 $$$\begin{equation}{H}^{\prime}\left( k \right) = \left( { \def\eqcellsep{&}\begin{array}{@{}*{4}{c}@{}} 0&{W\left( k \right)}&{{\kappa }_3}&0\\ {{W}^*\left( k \right)}&0&0&{{\kappa }_3}\\ {{\kappa }_3}&0&0&{P\left( k \right)}\\ 0&{{\kappa }_3}&{{P}^*\left( k \right)}&0 \end{array} } \right)\end{equation}$$$where W = κ₁ + κ₂e^−ik and P = κ₂ + κ₁e^−ik. The topological invariant of the Kitaev chain has the form of 21 $$$\begin{equation}\nu = {\mathop{\mathrm sgn}} \left( {\left| {{\kappa }_3} \right| - \left| {{\kappa }_1 + {\kappa }_2} \right|} \right)\end{equation}$$$where ν = 1 (ν = −1) corresponds to a topologically trivial (nontrivial) phase. Because the coupling strength between two SSH chains κ₃ depends on the separation, the topological transition of the photonic Kitaev chain composed of SRRs can be easily controlled by tuning the distance between the two coupled SSH chains. For the Kitaev chain with |κ₁ + κ₂| = 0.13 GHz, the band structures of Kitaev chains with κ₃ = -0.18, -0.13, and -0.08 GHz are shown in Figure 11c–e, respectively. When κ₃ = −0.18 and −0.08 GHz, the band gap corresponded to the topological trivial phase (ν = 1) and nontrivial phase (ν = −1), respectively. For the case |κ₃| = |κ₁ + κ₂| = 0.18 GHz, the band gap was closed, which corresponded to a phase transition point (PTP). Recently, this transition with a critical distance has been called the ‘magnetic distance’ in the moiré superlattice of a bilayer structure.^[367]

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Figure 11 shows that the double-chain structure composed of SRRs effectively mimics the Kitaev model. In an SRR-based Kitaev chain, several basic physical problems must be studied in depth. For example, how to achieve a topological phase transition by adjusting the coupling distance between two chains, and how the field distribution in the structure changes during the phase transition. By avoiding diagonal coupling and delicately tuning the vertical coupling of the two chains, one can study the transition process from a topologically trivial phase to a nontrivial phase, which can be experimentally demonstrated from the phase diagram shown in Figure 12a. As d increases, the Kitaev chain changes from a topologically trivial structure to a nontrivial structure, and topologically bound states appear in the gap. The photonic topological Majorana states in the special nontrivial phase with d = 31.5 mm are marked by M₁ and M₂. The photonic Majorana states in the periodic Kitaev were experimentally observed, as shown in Figure 12c. The topological bound state is confined at two ends of the upper chain. In addition, considering the composite Kitaev chain, in which a trivial chain (d = 24.5 mm) is sandwiched between two topological chains (d = 31.5 mm), the corresponding DOS spectra of the coupled Majorana interface states for different inner trivial chain lengths are shown in Figure 12b. Considering the low-frequency topological bound state of the composite Kitaev chain with N = 1 in Figure 12d, a strong localization exists at the two interfaces between the trivial chain and the topological and Majorana interface state chains, which is the result of the coupling of the two topological main states. The coupling between topologically bound states may be used to explore practical applications for wireless sensing, as introduced in Section 4.2.

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Hofstadter Chain

Based on the Kitaev topological model, another important topological model, the Hofstader model, can be realized by introducing quasi-periodic modulation.^[368] A schematic of the photonic reconfigurable Hofstader chain composed of SRRs is shown in Figure 13a Similar to the Harper chain, considering the quasi-periodic modulation of the coupling coefficient with κ_3i = κ₃₀ + κcos (2πξi), a phase diagram of the finite-sized Hofstadter chain is shown in Figure 13b. A fractal network of spectral gaps can be observed. A standard spectral butterfly (one of the most well-known examples of fractals in condensed matter physics) can be observed in a fractal phase diagram.^[368] Similar to the Hofstader chain with a quasiperiodic modulation of the coupling coefficient in Figure 13b,c shows the phase diagram of the Hofstader chain with finite resonators in a quasiperiodic arrangement controlled by tuning the on-site frequency ω_i = ω₀ + κcos (2πξi). The Hofstader chain constructed by extending the Kitaev chain revealed that inhomogeneities can significantly alter topological phase diagrams. In particular, the quasiperiodic modulation of the onsite resonant frequency of the SRR can be actively controlled by an external bias voltage when a VCD is added to the split of the SRR.^[369,370] Recently, the Kitaev and extended topological models based on the double SSH chain have been successfully realized in acoustic and elastic wave systems.^[371,372]

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Applications

Wireless Power Transfer

In recent years, with the discovery of abundant topological phases and novel edge states with topological protection in 1D topological structures, active efforts have been made to develop robust devices with photonic topological chains. Recently, wireless power transfer (WPT) has triggered immense research interest in a range of practical applications, including mobile phones, robots, medical-implanted devices and electric vehicles.^[373–375] Traditional WPT devices based on magnetic induction are severely limited by the transmission distance. When the distance between the receiving and transmitting coils is large, the transmission efficiency is significantly reduced. An effective solution is magnetic resonance WPT, that is, two coils with the same resonant frequency are used as a transmitter and receiver of the system for magnetic field coupling.^[376–379] Magnetic resonance WPT can effectively improve the transmission distance of WPT, but owing to the exponential attenuation characteristics of the near field, magnetic resonance WPT is also limited to short- and medium-range WPT. Recently, it has been proposed to use relay coils to construct a “domino” structure to realize long-range magnetic resonance WPT.^[380] This scheme effectively solves the problem of the transmission distance of WPT. However, this simple multiresonant coil system has some limitations. First, owing to the near-field coupling effect of multiple resonant coils, the corresponding operating frequency must be adjusted according to the change in the transmission distance. However, the transfer efficiency of the system is sensitive to the arrangement and construction of multiple resonant coils, and construction errors or external disturbances can significantly affect transmission efficiency. Therefore, with the development of WPT devices, efficient long-range and robust WPT is highly desirable, but challenging. The possibility of obtaining photonic topological models that are robust against perturbations by mimicking the topological properties of solid-state systems has had a profound impact on optical sciences. Therefore, extending the concept of optical topological manipulation to the WPT regime is desirable.

Although the scheme of the topological edge state for WPT has been proposed several years ago, its advantages have not been verified in actual systems until recently.^{[238,381–383]} Based on the basic research of photonic 1D SSH topological chains, a non-Hermitian dimer topological system with PT symmetry has been constructed in an actual WPT system, and the topological edge state was applied to the robust WPT.^[384–386] A structural diagram of the SRR dimer chain for the WPT is shown in Figure 14a.^[385] In particular, Figure 14b shows the transmission spectra of the two topologically distinguished dimer chains. At the center frequency (5.62 MHz), the nontrivial chain has an edge state; therefore, its transmission efficiency will be significantly higher than that of the trivial chain. At a working frequency of 5.62 MHz, the transmission efficiency ratio is 44.63, as shown in Figure 14c. Because the topological edge state has the characteristic of topological protection, the implemented long-range WPT is robust against internal disturbances and construction errors of the structure. Moreover, to intuitively demonstrate the long-range WPT realized by the topological edge state in the photonic 1D nontrivial dimer chain, LED lamps were introduced into the system, and the symmetric topological edge state was observed by LED emission, as shown in Figure 14d.^[385]

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Although a non-trivial dimer chain can achieve robust WPT, its idle power loss is evident. When the system is in a standby state, energy can still be input into the system, which not only leads to a waste of energy, but also risks burning the circuit owing to excessive no-load power. In addition, in the equivalent second-order PT symmetric system realized by topological edge states, the frequency of stable transmission energy changes with different loads. This load-dependent characteristic requires frequency tracking in practical applications, which significantly increases the complexity of the devices. To overcome these challenges, researchers have proposed combining topological edge and interface states to construct non-Hermitian WPT systems with higher-order PT symmetry and achieve efficient and stable long-range WPT.^[382] In contrast to the second-order PT symmetric system, the third-order PT symmetric system always has a pure real eigenvalue at the working frequency for the change in load power, which can be used to achieve a stable WPT, as shown in Figure 15. Finally, the idle power loss of the equivalent third-order PT symmetric non-Hermitian topological chain was analyzed. When the system was operating, the reflectivity of the system was low, and the LED lights at both ends of the chain were lit. However, when the system was in the standby state, the reflectivity of the system was high at the working frequency, and the LED lights at both ends of the chain were not lit, which proves that the idle power loss of the dimerized topology chain with effective third-order PT symmetry is small. Inspired by long-range WPT with topological edge states, the use of more complex topological structures is expected to enable energy transmission with more functions. A high-performance multi-load WPT can be realized for the corner and hinge states in the recent widely concerned high-order topological structures.^[387–394]

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Unlike the symmetric topological edge states in periodic SSH chains, the asymmetric topological edge states in the quasiperiodic Harper chain described in Sec. III-B.1 can also be used for long-range WPT with topological protection. The edge states in the Harper chain are localized at the left or right ends of the chain and can be used for directional WPT, as shown schematically in Figure 16a.^[258] A nonresonant coil was placed at the center of the Harper chain as the transmitting coil, and two nonresonant coils were placed at the left and right ends of the Harper chain as the receiving coils. Based on the near-field detection technology, the DOS spectrum of the 1D Harper chain was obtained, as shown in Figure 16b. The left edge state (f = 5.26 MHz) and right edge state (f = 5.45 MHz) are visible in the bandgap. Figure 16c shows the transmittance spectrum of the Harper chain in two directions. The green (red) line represents the ratio of the left (right) transmission to the right (left) transmission S_L/S_R (S_R/S_L). When f = 5.26 MHz (5.45 MHz), S_L/S_R is significantly higher (lower) than S_R/S_L, indicating that the topological boundary state is selectively localized at the left or right end of the quasi-periodic chain at different frequencies. Because the Harper chain is an asymmetric structure, it not only leads to different distributions of the left and edge states, but also leads to different transmission efficiencies on the left and right sides. Consequently, the transmission ratios S_L/S_R and S_R/S_L are not equal, as shown in Figure 16c. Therefore, directed long-range WPT was realized based on the asymmetric edge state in the Harper chain. Similar to Figure 14d, two Chinese characters composed of LED lights were connected to the non-resonant coil at the left and right ends of the Harper chain. Once the magnetic field in the resonant coil was sufficiently strong, the LED lights could be turned on to explain the directional transmission characteristics of the Harper chain more intuitively. To visually display the WPT direction, the non-resonant coil at the left (right) end of the disturbed Harper chain was connected with the Chinese character “Tong” (“Ji”) composed of LED lights. Figure 16d,e show that under the working frequency of the left (right) edge state at f = 5.26 MHz (5.45 MHz), the Chinese character “Tong” at the left end of the chain is bright (dark), while the Chinese character “Ji” at the right end of the chain remains dark (right).^[258]

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Implementing active control or coding photonic topological insulators in long-range WPT will significantly improve the flexibility of devices.^[395–398] Therefore, non-linearity appears as a convenient tool enabling the tuning of the properties of excitations in topological systems.^[399–407] A non-linear resonant unit based on the active control of the external circuit can realize directional long-range WPT with active control and robustness. The experimental device for actively controlling directional WPT measurements is shown in Figure 17.^[258] The ratio of the transmission distance to the coil radius in the active-control Harper chain is 13.6. To realize active control of the WPT, all the SRR coils were connected in parallel with the DC source. The signal was first generated by the vector network analyzer, and then, it was input to the nonresonant coil, which was used as the excitation source of the system. In addition, another non-resonant coil was placed at both ends of the Harper chain as the receiving coil, and the transmission coefficients of the left and right sides were measured.

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In the experiment, the active-controlled SRR was composed of a basic LC resonant structure, VCD, and protective elements. The capacitance of the VCD decreased with an increase in the applied DC voltage, thus realizing the regulation of the resonant frequency. Figure 18a shows a photograph of an actively controlled SRR for directional topological WPT.^[258] The corresponding equivalent-circuit model is shown in Figure 18b. The structure of the experiment is similar to that shown in Figure 16. The signal was generated by the vector network analyzer and then input into the non-resonant coil as the transmitting coil. A nonresonant coil was placed at both ends of the chain as the receiving coil, and the transmission spectra of the left and right sides were measured. The relationship between the applied voltage and resonant frequency of the nonlinear SRR was obtained through experiments, as shown in Figure 18c. With increasing applied voltage, the frequency of the nonlinear SRR increased. Figure 18d shows the coupling characteristics of the two resonant SRRs. The coupling strength of the SRR decreased exponentially with increasing distance. The coupling strength between the resonant coils was almost independent of the applied external voltage. Consequently, the frequency of the edge state could be easily adjusted by changing the external voltage without changing the structure. When the applied voltage was U = 0 V, the working frequency of the right edge state was 38. 4 MHz, as shown in Figure 18e. At this frequency, energy was transmitted to the right side of the chain. When the voltage applied to the VCD was increased to U = 4 V, the resonant frequency of the coil increased. At this time, the spectrum blue-shifted and the 38.4 MHz corresponded to the left edge state in Figure 18e. Therefore, by adjusting the external voltage of the system, the system changed from the right-edge state to the left-edge state at a fixed frequency in the non-linear Harper chain, that is, the system energy was transferred from the right side to the left side.^[258] This feature can be used to actively control the opening and closing of the WPT equipment in a specific direction. Moreover, the novel topological skin effect with asymmetric coupling,^[408–416] meta-source with artificial emission control,^[417] and the anomalous non-reciprocal topological edge states^[418,419] can be expected in the robust directional WPT devices.

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Sensing

Currently, non-Hermitian topological photonics is an interesting research topic in topological physics.^[420–433] Research on non-Hermitian photons based on PT symmetry has led to new design methods for photonic topological insulators.^[434,435] The eigenvalues of open optical non-Hermitian systems are complex. PT symmetric structures with pure real eigenvalues belong to a special class of non-Hermitian systems. Degenerate points exist on the Riemannian surface in the parameter space of a non-Hermitian system. At this time, the eigenvalues and corresponding eigenvectors coalesced simultaneously. These non-Hermitian degenerates are also called exceptional points (EPs).^[436,437] An EP is the origin of many abnormal phenomena, such as mode merging,^[438] dynamic energy transmission,^[376,439] chiral inversion,^[440] and topological mode transfer.^[441–443] EPs provide a new method to design a new type of high-sensitivity sensor that exceeds the linear response, and its sensitivity performance can be further improved by increasing the order of Eps.^[444–449] Non-Hermitian topological chains exhibit new phenomena that are difficult to observe using ordinary Hermitian topologies. As one of the simplest topological structures, 1D dimer chains have been widely used to study photonic topological excitations. Researchers have extensively studied the unique topological order, phase transition, and edge states of non-Hermitian 1D dimer chains.^[243,273] The topological non-Hermitian system provides a new way to study topological photonics related to EPs and design new functional optical devices.^[450–453]

Near-field mode coupling is a basic physical effect that plays an important role in controlling EM waves. Researchers have found many interesting phenomena in topological edge-state near-field coupling, such as the robust topological Fano resonance^{[172,260,454,455]} and Rabi splitting.^[235] However, in a finite non-Hermitian dimer waveguide array, the coupling effect of the edge state causes the edge state to deviate from the topological zero mode, weakening the robustness of the edge state.^[247] Mode splitting caused by near-field coupling can be eliminated by increasing the chain length. To restore topological protection, the coupling of the two edge states must be reduced by increasing the chain length such that the split edge states can return to zero energy. However, the splitting frequency can be reduced again at the EP by directly changing the gain or loss strength, while keeping the chain length constant.^[385] The topological edge state is the result of a nonlocal response based on the bulk boundary correspondence, which is typically robust to structural disturbances. In contrast, EPs are often used to obtain sensors that are highly sensitive to small environmental changes. Therefore, whether topological edge states can be combined with EP points to design new highly sensitivite sensors must be solved.^[456–458] Figure 19a shows a topologically coupled non-Hermitian system comprising the Kitaev chain in Sec. III-B-2, which can be used to study topological sensors when losses and gains are introduced into the system.^[365] In this PT symmetric Kitaev chain, the real and imaginary eigenfrequencies ω_± are shown in Figure 19b,c, respectively. Considering the frequency-detuning perturbation ε on the left Kitaev chain, the real parts of the eigenfrequencies are shown in Figure 19d. Moreover, the dependence of the frequency-splitting $$\Delta \omega = {\mathop{\mathrm Re}\nolimits} ( {{\omega }_ + - {\omega }_ - } )$$ on the strength of the perturbation at the EP is shown in Figure 19e. The results are presented on a logarithmic scale. The coupled edge states at the EP exhibit a slope of 1/2 for a small perturbation, as shown by the blue dots.

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Recently, topological circuits have been widely used as a multi-functional platform for studying topological physics.^[459–462] Recently, the sensitivity characteristics of the EP in a dimer chain based on finite non-Hermitian topological circuits were discovered on a circuit platform.^[457] Considering a finite non-Hermitian topological dimer chain consisting of subwavelength SRRs composed of a basic LC resonant unit, negative impedance converter (NIC) components, and adjustable resistors, the near-field coupling between the two edge states is closely related to the realization of the equivalent second-order EP of the system. The modulation of the gain and loss in the composite resonant cell is realized by the NIC module and adjustable resistor element, respectively, and a metal-oxide semiconductor field-effect transistor (MOSFET) is used to provide an effective gain. By adding the loss and gain at both ends of the dimer chain, a non-Hermitian topological chain satisfying PT symmetry was obtained. The circuit model and NIC components are shown in Figure 20a. The complete circuit model of the composite SRR is shown in Figure 20b. Figure 20c shows the experimental setup in which the dimer chain is composed of 10 SRRs. The gain-, neutral-, and loss-resonant units are denoted as G, N and L, respectively. The signal was input from the left end and used to measure the reflection spectrum of the chain. By fixing the gain, the coupling of the edge states was modulated by adjusting the loss. Because of the near-field coupling between the edge states, the two split-edge states have different eigenfrequencies, corresponding to A in the reflection spectrum in Figure 20d. Then, the loss gradually increases, and the degeneracy point of the eigenfrequency corresponds to the EP of the edge state in the non-Hermitian dimer chain. This corresponds to B in the reflection spectrum in Figure 20d. The resistance is further reduced, that is, the loss of the SRR is further increased. At this time, the edge state remains degenerate, which corresponds to C in the reflection spectrum, as shown in Figure 20d. As shown in Figure 20d, by adjusting the resistance of the resistor, that is, the loss of the SRR, the non-Hermitian topological dimer chain can easily realize the phase transition process related to the EP. The EP is realized by increasing the loss of the system, and the sensitivity of the topological edge state to environmental disturbances near the EP has been studied. In the experiment, a low-frequency perturbation was added to the lossy SRR by changing the lumped capacitance. A logarithmic diagram of the relationship between the frequency disturbance of the lossy SRR and capacitance disturbance is shown in Figure 20e. The edge state at Point B in Figure 20d was used to realize the EP sensor, and the slope was 1/2 in the case of a small disturbance, as indicated by the circle in Figure 20e. At Point A in Figure 20d, the non-Hermitian chain is in the split region with a small loss. The slope of the sensor realized by the edge state was one when there was a small disturbance, as shown by the pentagram in Figure 20e. At Point C in Figure 20d, the non-Hermitian chain belongs to the degenerate region, and the edge state is barely affected by external disturbances, as indicated by the triangle in Figure 20e. A comparison of the three phases revealed that the edge state at the EP was sensitive to the disturbance at the end of the chain.

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Although the topological edge state is sensitive to disturbances at the end of the chain, it is robust to disturbances in the middle of the chain. Therefore, this sensor, based on the edge-state EP in the non-Hermitian topological dimer chain, is highly sensitive to disturbances in the frequency of the end node of the structure caused by the external environment, but it is still topologically protected internally; that is, it is immune to internal disturbances caused by changes in coupling strength. The experimental results shown in Figure 20f verify the robustness of the non-Hermitian dimer chain, which originates from the recovery of the topological zero mode. As shown in Figure 20f, with an increase in the disorder intensity, the fluctuation of the edge state in the splitting area increases significantly, and when the edge states coalesce, the influence of the same disturbance on the edge state is significantly reduced. Therefore, the recovered topological zero mode was more robust.

Switching

Based on the reconfigurable topological dimer chain composed of non-linear SRRs, active tuning of the state from a bulk state to an edge state, or vice versa, can be realized. Recently, a VDC was mounted inside the gap of an SRR to construct a magnetic resonator with Kerr-type non-linearity.^[255] For a 1D dimer chain with an odd number of resonators, only one topological edge state exists in the bandgap, which avoids the near-field coupling effect of the edge state. A schematic of the external homogeneous pump-controlled topological edge state in a non-linear SRR dimer chain is shown in Figure 21a. The spectral shifts of the resonant frequencies for different non-linear SRRs under strong external pumps are shown in Figure 21b. The calculated and measured probe reflection spectra at each SRR for a dimer chain with seven resonators in the linear regime are shown in Figure 21c,d, respectively. An edge state near 1500 MHz is localized at the center of the spectrum. Considering the external pumps applied to this dimer chain in the non-linear regime, the resonant frequency of the edge state was extracted from the probe spectra as a function of the pump power, as shown in Figure 21e. The calculated and measured results are marked with black curves and stars, respectively. As shown in Figure 21e, the edge state can be tuned dynamically using an external pump. With an increase in the pump power, the edge state was blue-shifted. In particular, the corresponding ratio of the edge-state amplitude to that of the bulk state as a function of pump power was observed, as shown in Figure 21f. At higher pump intensities, the edge state became less evident.^[255] Recently, signal processing,^[463] signal filtering,^[464] and magnetic resonance imaging^[465] were demonstrated in photonic dimer chains.

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Conclusion and Outlook

The emerging field of meta-atoms with unusual optical properties is promising for significantly impacting photonic technology. However, despite active efforts in implanting photonic devices with topological protection in a simple dimer chain with SRRs, applications of other topological structures remain almost uncharted. Recently, research has been stimulated by theoretical predictions and experimental observations of novel effects such as topological defects,^[466,467] Möbius insulators,^[468,469] non-Euclidean topology,^[470,471] vortex topological modes,^[472,473] and topological rainbows.^[474,475] In the coming years, we expect to discover the new topological structures and novel topological phenomena with SRRs. The study of photonic topological structures is a frontier field at the intersection of optical and condensed-matter physics. As highlighted in the introduction to this review, topological edge states have now penetrated several sub-disciplines in physics. The photonic design of SRR topological structures not only enriches the research scope of the current photonic topological field,^[476] but also extends the results to other classical systems.^[477]

Tremendous challenges still exist that must be addressed. First, to apply SRRs to high-frequency topological photonics construction, a high-precision micro/nanoprocessing technology is required. In addition, an active regulation in the outfield and non-linear topological regulation of SRR-based topological structures still requires further study in high-frequency regimes. Second, SRRs have a strong local resonance, and the photonic topological structure constructed by SRRs has greater intrinsic (dissipative) losses, compared to all-dielectric photonic topological structures. Reducing the intrinsic loss of SRR-based photonic topological structures is a challenge. Third, the coupling strength between two neighboring SRRs depends not only on the spacing, but also on the relative rotation angle. Therefore, high-dimensional (2D and 3D) topological structures constructed using SRRs exhibit evident anisotropic characteristics. Consequently, when constructing high-dimensional topological systems using SRRs, next-nearest neighbor coupling and anisotropic coupling must be considered, and their corresponding richer and more complex topological characteristics must be studied. Fourth, currently, the topological structures composed of SRRs mainly focus on the near-field coupling mechanism, while far-field coupling and even hybrid coupling (with both near-field and far-field couplings)-enabled topological phases remain elusive. Finally, the applications of photonic devices in various frequency spectra (including microwave,^{[77,122,124,191,192,263,384–386, 457,462,463,465,475,481,483,489,490,494,496–499]} terahertz^{[123,198,478,482,484,491,495]} infrared,^{[58,126,128,129,163,164,170,184,474,479,482,485,487,488]} visible,^{[68,69,125, 130–134,195,196,234,256,257,480,482,492,493]} and X-ray^[235]) are summarized in Table 1. The applications of SRR-based topological structures are limited to WPT, wireless sensing, wireless communications, and magnetic resonance imaging. More practical and robust devices such as antennas and optical switches are expected to be explored with the help of photonic topological structures.

Table 1 Summary of exemplary applications of topological structures that have been demonstrated.

	Microwave	Terahertz	Infrared	Visible	X-ray
Monochromators	[263]	[478]	[479]	[480]	[235]
Rainbow trapping	[475, 481]	[482]	[474, 482]	[482]
Switch	[483]	[484]	[485]	[130]
Waveguides	[191, 192]	[198, 486]	[126, 184, 487, 488]	[125]
Coupler	[489, 490]	[478]	[128, 129]	[234]
Routing	[122]	[123, 491]	[164]	[492]
Wave-division multiplexing	[77]	[478]	[170]	[482]
Delay lines	[124]	[478]	[58, 163]	[493]
Antenna	[494]	[495]
Fiber	[496, 497]
Wireless communications	[498]	[198]
Wireless power transfer	[384–386]
Magnetic resonance imaging	[465]
Wireless sensing	[457, 499]
Signal Filtering	[463, 464]
Lasers				[68, 69, 131–134, 195, 196, 256, 257]

Several potential opportunities exist in this field. First, the negative coupling efficiency realized by the SRRs enables an effective construction of an artificial gauge field, which provides a flexible platform for studying the novel topological skin effect beyond non-reciprocal coupling and complex topological bands, such as Mobius rings and Klein bottles. Second, based on non-linear SRRs with variable capacitance diodes, more complex photonic topological models, such as quasi-crystals and disordered crystals, are expected to be explored. Third, SRR-based topological non-Hermitian systems provide an effective avenue for studying the intriguing properties of topological photonics involving EPs, novel skin effects, and the development of new functional devices. Fourth, tuning the topological properties of metasurfaces composed of SRRs has attracted significant interest in recent years because of its usefulness in designing planar devices that are easier to integrate and have smaller losses. Finally, the topological structure constructed by SRRs and the emerging interdisciplinary novel and counterintuitive physical properties are worth investigating, including: twist and the Moiré physics of multi-layer structures, non-Euclidean hyperbolic lattices, phonon polaritons in natural 2D materials, hyperbolic and zero-index metamaterials, bound states in the continuum, machine learning, and quantum optical phenomena.

In this review, we systematically discussed the fundamentals and applications of photonic 1D topological chains that comprise SRRs. Using the rotation degree of freedom, the main concepts (such as topological invariant, excitation, and edge states) have been introduced for various topological chains, including dimers, trimer, quasi-periodic Harper, composite Kitaev, and Hofstadter chains. In addition, by combining topological characteristics with non-linear and non-Hermitian characteristics, new functional photonic devices such as WPT, sensing, and switching have been introduced, which will promote the practical application of photonic topological insulators. Overall, the fascinating optical properties and functionalities of the SRR chains open an unprecedented path for the realization of novel devices with topological protection that can find widespread applicability in various photonic systems.

Acknowledgements

This study was funded by the National Key Research and Development Program of China (Grant Nos. 2021YFA1400602, 2023YFA1407600), the National Natural Science Foundation of China (NSFC; Grant No. 12004284, 12374294), the Fundamental Research Funds for the Central Universities (Grant No. 22120210579), and the Chenguang Program of Shanghai Education Development Foundation and Shanghai Municipal Education Commission (Grant No.21CGA22).

Conflict of Interest

The authors declare no conflict of interest.

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