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Abstract
Topological photonics and topological photonic states have opened up a new frontier for optical manipulation and robust light trapping. The topological rainbow can separate different frequencies of topological states into different positions. This work combines a topological photonic crystal waveguide (topological PCW) with the optical cavity. The dipole and quadrupole topological rainbows are realized through increasing cavity size along the coupling interface. The flatted band can be obtained by increasing cavity length due to interaction strength between the optical field and defected region material which is extensively promoted. The light propagation through the coupling interface is built on the evanescent overlapping mode tails of the localized fields between bordering cavities. Thus, the ultra-low group velocity is realized at a cavity length more than the lattice constant, which is appropriate for realizing an accurate and precise topological rainbow. Hence, this is a novel release for strong localization with robust transmission and owns the possibility to realize high-performance optical storage devices.
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1. Introduction
Topological photonics has occupied a wide range of studies. It has become a rapidly developing research topic at present with broad prospects for applications, for instance, topological optical waveguides [1], topological micro-cavities [2,3], and topological lasers [4,5]. It can be argued that topological photonics provides a unique, robust platform to explore new physical phenomena and guidelines for designing the next generation of integrated devices [6,7]. Topological photonics and their topological states have opened up a new frontier for optical manipulation and propagation besides being immune the fabrication errors [8,9]. Due to the topological phase transition, topological edge state (TES) [9] and corner state (TCS) [10] can be generated. The photonic crystal waveguide based on the TES (topological-PCW) supports a topologically protected unidirectional propagation edge state without backscattering [11]. The photonic cavity based on TCS can accomplish a robust field localization without presenting defect structures [12]. The concepts of recently discovered higher-order topological phases [13,14] have shortly been transferred into photonics [15,16], and second-order TCSs have been observed in square lattice PCs [2]. Owing to their capability of confining the field to structures of various dimensions, these topological higher-order phases are promising candidates for topological and laser resonators [17].
Lately, the interest in rainbow phenomena has increased with the advent of topological PCs [18–23], which provide a powerful platform for nanophotonic devices. Posteriorly, the combination of topology and rainbow makes it possible to design desired topologically protected nanophotonic devices [7]. Using a topological rainbow is first to slow down, then trap, and finally separate the topological photonic states of different frequencies into different positions. Hence, potential topological properties can be obtained to eliminate backscattering and immunity to structural disturbances [24–29]. Topological rainbow is a successful and promising frequency-routing technology based on the slow-light property of topological photonic states. It provides a platform to realize a wealth of potential applications related to integrated photonic devices and high-speed information processing chips, such as topological multichannel wavelength multiplexers, topological optical storage, and topological filters [30]. The search for unprecedented and efficient ways to trap and manipulate light using photonic systems is a significant research focus.
Firstly, a topological 2D PC with C4 symmetry is constructed in this work. The different topological properties are the prerequisite for generating topological edge states at the interface of the two valley PCs. The key to realizing valley photonic crystals is breaking the inversion symmetry to open the Dirac points. So, the inversion symmetry can be broken by introducing perturbation on two different sublattices (two different rotation angles in this study). Due to the broken inversion symmetry, a resultant band gap is obtained, and the frequency extremes of bands, i.e., valleys, are formed [31]. Then, TES can be formed at the interface when the two valley PCs with different topological properties are in contact. Finally, the interface rods are replaced by a defected optical cavity along the interface; the light confinement is spatially and temporally obtained with a long photon lifetime. The topological PCW is created with the TES waveguide, and the optical cavity has robust cavity-waveguide interaction. This structure provides high flexibility in controlling the delay time and storage capacity with highly localized light confinement. Topological PCW coupled with a cavity is studied based on cavity length along the x-direction. For $R > a,$ the distance between cavities increases, and the coupling factor decreases. The flatted band can be obtained by increasing the cavity length. The ultra-low group velocity is realized at $R = 3a$ and demonstrated, which is appropriate for realizing a topological rainbow. With gradually increasing the cavity side length ${l_d}$ in the propagation direction, the light will slow down gradually and ultimately approach to stop where the group velocity tends to zero. Therefore, the light of different frequencies is confined at different spatial positions, which is appropriate for realizing accurate and precise dipole and quadrupole topological rainbows.
2. Structure design and dispersion analysis
Figure 1 depicts the schematic diagrams of the proposed 2D PC with C4 symmetry of the square lattice. The primitive cell of the lattice with C4 symmetry consists of a four-square Si rod of side-length $l\; = \; 0.2a$ with relative permittivity 11.9, the background material is air, and the lattice constant $a = \; 1\mu m\; $. As shown in Fig. 1(a), the distance from the primitive-cell center to any Si rods is $d$. Symmetry breaking has been proven to be pivotal in realizing topological phases and phase transition [32–35].
Fig. 1. The band structure of the primitive cell of $d = 0.3a$ (a) and $d = 0.2a$ (b), (c) The structure of unit cells at (I) of $\mathrm{\varphi } = {0^0}$, (II) $\mathrm{\varphi } = {45^0}$ and (III)$\mathrm{\varphi } = {225^0}$, (d) the band structures for different angles at (I, II, and III), (e) the electric field ${E_z}$ distribution for band four at the frequency of 0.40658c/a for $\varphi = 45$, and $\varphi = 225,$ and the distribution of the Berry curvatures of the degenerated band, (f) the band structures for different angles at (I, II, and III) along Γ-Y-M-Γdirection.
In general, there are two ways to break the crystal symmetry; one is the displacement by expanding or shrinking the distance between adjacent rods, as shown in Figs. 1(a) and 1(b) respectively. The other is rotating the dielectric rods by an angle $\varphi $ to describe the disordered valley PCs along the axial of symmetry as shown in Figs. 1(c) and 1(d). Without losing generality, the transverse magnetic (TM) mode is only considered throughout this study. Figure 1(a) shows the band structure of the primitive cell of $d = 0.3a$ that is shown in the inset of Fig. 1(a). The M-point of the first Brillouin zone is folded for the first and higher bands. The recognized square lattice PC with nontrivial bulk polarization [2] consists of four equal rods near the corners of the primitive cell, and its topologically trivial counterpart can be obtained by shrinking the four rods (see Fig. 1(b)). Figure 1(b) shows the band structure of $d = 0.2a$ with Dirac degeneracy at the M point. To work for higher bands, the side length of two Si rods in the left corner are increased to be $l = d = \; 0.3a$ as shown in Fig. 1(c). The square lattice of $\mathrm{\varphi } = {0^0}$, the unit cell (I), which is shown in Fig. 1(c), displays accidental Dirac degeneracies; Dirac points away from the high symmetry point can be formed [36,37] as shown by solid black lines in Fig. 1(d). Due to variation of the dimensional of two rods with rotational angle, the topological phase transition appears, such as the extermination of topology-degenerate singularities that carry nonzero Berry curvatures [37]. As a result of the accidental Dirac degeneracies, Dirac points will be formed away from the high symmetry point and appear in the enlarged irreducible Brillouin zone in square valley topological PC in the region (M-Γ-M’) ([38,39] which is similar to our presented case) or in the region (N-Γ-N’) [37]. If the Dirac points are away from the high symmetry points, the Berry curvature $\mathrm{\Omega }$ will have more extremes, and the Chern number will be greater than one [36,40]. Where the gaps with nonzero Berry curvatures are marked with light gray color. Such gaps are generated due to Dirac degeneracy lifting by symmetry breaking from $\mathrm{\varphi } = {0^0}\; $ to $\mathrm{\varphi } = {45^0}$ and $\mathrm{\varphi } = {225^0}$ which shows the same band structure illustrated in Fig. 1(d) using blue solid and red dashed lines, respectively, but different in phase winding of the electric field ${E_z}$ distribution. For $\varphi = {45^0}$, the ${E_z}$ phase increases counterclockwise around the center of the unit cell. In contrast, for $\varphi = {225^0}$, the phase increases clockwise. Thus, the phase winding possesses opposite behavior in the two cases as shown in the upper part of Fig. 1(e).
We calculate the Berry curvature to reveal the topological nontriviality of the bandgaps. The Berry curvature of the n-th gap is calculated by summing up the Berry curvature of the bands below the gap. We numerically evaluate the Berry curvature using the presented method [41]. The lower part of Fig. 1(e) shows that the Berry curvatures $\mathrm{\Omega }$ for the degenerated band in the first Brillouin zone have extreme values at specific but not symmetric points with antisymmetric distributions. As we can see from Fig. 1(e), the distributions of the Berry curvatures are antisymmetric for the inversion $({{k_x},\; {k_y}} )\to \; - \; ({ - {k_x},\; - {k_y}} )$. They produce a 180°-rotated Berry curvature profile which are swapped under the rotation, where a nonzero Berry curvature with opposite signs appears at $\mathrm{\varphi } = {45^0}$ and $\mathrm{\varphi } = {225^0}$. Thus, the interfaces of the unit cells of II ($\mathrm{\varphi } = {45^0})\; $ with its inverted counterparts III ($\mathrm{\varphi } = {225^0})$ have valley Chern numbers with different signs. Thereby, topological valley edge states are guaranteed at the geometric boundary between them [42–44]. Figure 1(f) shows the band structures for different angles at (I, II, and III) along the Γ-Y-M-Γ direction. Due to the asymmetry of the four Si rods, the band diagrams from Γ to M via X (Fig. 1(d)) differ from that through the Y direction and are identical in the region from M to Γ. However, the bandgap still exists, which shows that the proposed structure possesses a complete bandgap. The supercell of the proposed structure is shown in Fig. 2(a). Figure 2(b) shows the projected band structure of the supercell of Fig. 2(a). We can see the edge mode inside the bandgap. The projected band structure can be investigated from the dispersion curve by detecting the electric field ${E_z}$ distribution as shown in Fig. 2(c) at the high symmetry point $({{k_x} = 0} )$ in the Brillouin zone. Strong field distribution is observed at the interface where a robust energy exchange exists during the transporting process.
Fig. 2. (a) The supercell of the proposed valley PC, (b) the projected band structure of 2D-PC heterostructure of the supercell in Fig. 2(a) with forming topological edge mode inside the bandgap, (c) the electric field ${E_z}$ distribution at the high symmetry point ($({{k_x} = 0} )$ in the Brillouin zone of the guided mode.
3. Results and discussion
Topological PCW provides a protected unidirectional topologically propagation edge state without backscattering [11,20]. The optical cavity in PC can be presented by removing and/or inserting point and/or line defects; spatially and temporally, light confinement is obtained with a long photon lifetime [45–47]. Consequently, the interaction strength between the optical field and defected region material is extensively promoted. The optical cavity in our proposed structure can be created by replacing the four rods of the interface with defective rods of edge-length ${l_d}$. By selecting appropriate parameters of the defect cavity in the waveguide interface, robust propagation edge mode can be realized for flat-band. Figure 3(a) shows the possible supercells of topological-PCW coupled with cavity-based cavity length $R = na,\; n = 1,2,3$ along the x-direction.
Fig. 3. (a) The possible supercells of topological-PCW coupled with the cavity-based cavity length $R = na,\; n = 1,2,3$ along the x-direction. The cavity was created by replacing the four rods of the interface with defective rods of edge-length ${l_d}$. (b) and (c) the contour map of the dispersion curves and group velocity, Λ is the unit cell period of Λ = 2a, 3a respectively, of the three cases of cavity length R with when edge-length ${l_d}$ varying in a small and extensive range.
Figure 3(b) shows the contour map of the dispersion curves of the three cases of cavity length R with when edge-length ${l_d}$ varying in a small and extensive range. For $R = 1a$, the distance of the adjacent bending neighboring is close enough, the coupling between the two nearest adjacent neighboring occurs due to the evanescent Bloch waves, and the edge mode in the coupling interface is achieved [48]. By only considering the interaction between the nearest neighboring cavities, the sinusoidal dispersion relation of the edge mode can be obtained by [49]: $\omega = {\omega _0}(1 + \kappa \cos {k_x}R)$, $\omega $ is the eigenfrequency of separate cavity, ${\omega _0}$ is the eigenfrequency (at the trapped state), and $\kappa $ is the coupling factor. When n is increased from 1 to 3, the distance between cavities increases based on the above equation at each ${l_d}$ value, the coupling factor $\kappa$ can be determined. For example, at $R = 3a$ and ${l_d} = 0.65a$, $k = 0.0004,$ and at $R = 2a,\; \; k = 0.022$. When n increases from 2 to 3, the distance between the localized states increases, and the coupling factor κ decreases from 0.022 to 0.0004, respectively. Consequently, the flatted band can be obtained by increasing the cavity length due to the interaction strength between the optical field and defected region material which is extensively promoted. The light propagation through the topological PCW and cavity is built on the evanescent overlapping mode tails of the localized fields between bordering cavities; thus, low-group velocity can propagate light as shown in Fig. 3(c), the ultra-low group velocity is realized at $R = 3a$, which is appropriate for realizing topological rainbow. This structure possesses high flexibility in controlling the delay time and storage capacity with highly localized light confinement. Therefore, a unique topological PCW coupled with an optical cavity can be formed if a TES is combined with a cavity. Hence, this configuration for strong localization with robust transmission and owns the possibility to realize high-performance optical storage devices.
Furthermore, we show the dispersion curve and the group velocity, which can be obtained from the slope of the dispersion curve of the topological modes at $R = 3a$ in Fig. 4. By increasing the cavity side-length ${l_d}$ in two ranges from $0.345a$ to $0.375a$ and from $0.59a$ to $0.65a$, the dipole-like and quadrupole-like distributions can be realized, respectively. With increasing ${l_d}$, the topological modes dispersion curves move to a lower frequency region for the two variation cases. Thus, the forbidden frequency of each topological mode can be trapped in the graded structure by stacking the supercells with continually increasing ${l_d}$ in the propagation direction. Then the mode location happens at the slow-light area of the bands where group velocity tends to be zero; accordingly, the wave is localized at this specific frequency and no longer propagates forward. In Figs. 4(a) and (c), seven certain localized frequencies for dipole and quadrable topological rainbows, respectively. It can be seen that the seven selected operating frequencies in each case are all positioned at the extremely low group velocity, as shown in Fig. 4(b) and (d). Therefore, this feature provides an effective method for spatial separation of topological photonic states of different frequencies and for realizing dipole and quadrupole topological rainbows.
Fig. 4. The dispersion curves and group velocity of the topological modes at $R = 3a$ with increasing the cavity side-length ${l_d}$ in two ranges, (a), (b), from $0.345a$ to $0.375a$ with an increment of $\varDelta {l_d} = 0.005a,\; a = 1\mu m$, respectively, to realize the dipole topological rainbow and (c), (d) from $0.59a$ to $0.65a$, with an increment of $\varDelta {l_d} = 0.01a,\; a = 1\mu m$ to realize quadrupole topological rainbow.
4. Topological rainbow forming
Inspired by the above analysis, the gradient structure can be designed along the x-direction to realize the potential rainbow. It can be obtained by changing the cavity side-length ${l_d}$ to form a topological PCW coupled with an optical cavity transmission waveguide. The structure has 20 periods along the x-axis, and the ultimate structure is composed of the supercells with the gradient dispersion curves designed based on Fig. 5. The parameter ${l_d}$ increases linearly $({l_d} = {l_i} + n \times 0.005 \times a$, ${l_i} = 0.345a,\; n = 1$ for dipole and ${l_i} = 0.59a,\; n = 2$ for quadrupole) in two ranges to realize dipole and quadrupole topological rainbows, as shown in Figs. 5(a) and 5(b), respectively.
Fig. 5. Schematic representation of the topological PCW coupled with the optical cavity transmission waveguide with gradient ${l_d}$ for realizing (a) the dipole and (b) the quadrupole topological rainbows. The parameter ${l_d}$ increases linearly $({l_d} = {l_i} + n \times 0.005 \times a$, ${l_i} = 0.345a,\; n = 1$ for dipole and ${l_i} = 0.59a,\; n = 2\; $ for quadrupole). A multi-frequency plane-wave light source is excited from the port on the left.
A plane wave light source of a multi-frequency is incident from the left port of the structure to realize the topological rainbow. The electric field distribution of ${E_z}$ can be observed using the finite element method (FEM) of COMSOL Multiphysics. Due to group velocity dispersion, different frequencies can be stopped, separated, and trapped at different locations. The primary purpose of the gradient supercells along the x-direction is the spatial separation of each frequency component at a different position contingent on the separation of the dispersion curves. Thus, each section (supercell) reduces the group velocity and focuses the light at a precise frequency, i.e., 0.5476, 0.542, 0.5364, 0.5310, 0.5257, 0.5205, and 0.5155 c / a for dipole topological rainbow and 0.5608, 0.553, 0.5454, 0.5383, 0.5316, 0.5254, and 0.5197 c / a for quadrupole topological rainbow, as shown in Fig. 4 which demonstrated by the plane wave expansion (PWE) method by using the BandSOLVE module of the software Rsoft [45].
The localization of the electric field ${E_z}\; $ distribution of the dipole and quadrupole topological rainbows along the coupling interface of the graded ${l_d}$ which is increasing linearly from $0.345a\; \textrm{to}\; 0.375a$ with an increment of $0.005a$ and from $0.59a$ to $0.65a$ with an increment of $0.01a$ are shown in Figs. 6 and 7, respecitvely. Different sections reduce the group velocity and localize the light at a specific frequency matched with the calculations in Fig. 4. Then, multi-topological modes are observed and highly localized along the coupling interface. With increasingly ${l_d}$ in the propagation direction, the light will slow down gradually and ultimately approach to stop where the group velocity tends to zero. Therefore, the light of different frequencies is confined at different spatial positions, which is appropriate for realizing accurate and precise dipole and quadrupole topological rainbows. There is a slight difference in a few frequencies between the two methods, which is principally from the number of plane waves for calculation in the PWE method and the limited discretization of FEM.
Fig. 6. The localization of the electric field ${E_z}$ distribution of the dipole topological rainbow along the coupling interface of the graded ${l_d}$ which is increasing linearly from $0.345a$ to $0.375a$ with an increment of $0.005a$. The light is highly localized at the specific frequencies at ultra-low group velocity is observed when $R = 3a$, which is appropriate for realizing an accurate and precise topological rainbow.
Fig. 7. The localization of the electric field ${E_z}$ distribution of the quadrupole topological rainbow along the coupling interface of the graded ${l_d}$ which is increasing linearly from $0.59a$ to $0.65a$ with an increment of $0.01a$, when $R = 3a$.
The localization of the electric field ${E_z}$ of the topological rainbow phenomenon when $R = 2a$ looks the same as at $R = 3a$. However, the flatted band can be obtained by increasing the cavity length to $R = 3a$. In addition, with increasing the R, the localization increases inside the defect cavity with a longer time for making a hop from one cavity to another, which reflects the ultra-low group velocity as shown in Figs. 4(b and d). The localization of ${E_z}$ distribution when $R = 1a$ of the conventional topological rainbow is observed in two cases. First, when ${l_d}$ is increasing linearly from $0.32a$ to $0.37a$ as shown in Fig. 8(a). The second case is when ${l_d}$ is increasing linearly from $0.59a$ to $0.64a$ as shown in Fig. 8(b). The formed topological rainbow, in this case, is continuously formed, while in the case of $R > 1a$, the formed topological rainbow is isolated, and the energy is accumulated in the cavity. Physically, this can be explained as: at every ${l_d}$ value during the gradually increasing can be considered a defect, and every defect can be considered as a cavity. Thus, a sequence of cavity loads as energy reservoirs of electromagnetic (EM) will be produced in the system. In this case, the wave can be localized in the cavities, and more cavities can be coupled together to form a line of cavities or the cavity waveguide, which form the transmission channel. Thus, each section at every ${l_d}$ value (supercell) reduces the group velocity and focuses the light at a precise frequency, which has a topological mode and can be trapped in the graded structure by stacking the supercells with continually increasing ${l_d}$ in the propagation direction. Then the mode location happens at the slow-light area of the bands where group velocity tends to be zero; accordingly, the wave is localized at this specific frequency and no longer propagates forward. Thus the formed topological rainbow, in this case of $R = 1a$, is continuously formed, due to, the distance between the neighboring cavities is close enough, so the strong coupling occurs due to the evanescent Bloch waves [49], leading to a conventional topological rainbow similar to previously observed. In contrast, in the case of $R > 1a$, the formed topological rainbow is isolated, and the energy is accumulated in the cavity, due to reduction of coupling factor $\kappa $, ultra-low group velocity can be obtained, which is appropriate for realizing a pure topological rainbow.
Fig. 8. The localization of the electric field ${E_z}$ distribution of the conventional topological rainbow in two cases (a) small size of the cavity side-length from $0.32a$ to $0.37a$ and (b) big size of cavity side-length from $0.59a$ to $0.64a$, when $R = 1a$.
Based on the previous discussion, the trapped frequencies are slightly different when $\textrm{}1a$ and $\textrm{}3a$ with the same value of $\textrm{}{l_d}$; due to the localization being increased inside the cavity with a longer time for making a hop from one cavity to another, which reflects the ultra-low group velocity [50]. Due to the ultra-low group velocity and higher group delay, the ${E_z}$ field is highly localized inside each cavity [51]. Consequently, the interaction strength between the optical field and defective region material is extensively promoted. The ${E_z}\textrm{}$ field shows that the pulse faces different periodic $1a$ and $\textrm{}3a$. Subsequently, the topological photonic states with different frequencies can be slowed and used for temporary information storage. Furthermore, a topological slow-light device can be constructed by coupling the interface modes to output waveguides.
5. Verification of robustness
The critical step is to confirm that the formed topological rainbow is robust and topologically protected from disorders or inserting some defects along the propagation path. Three cases are presented. The first introduces a random variation of the cavity length ${l_d}$. With the same conditions on Figs. 6 and 7, however, the parameter ${l_d}$ is randomly varied according to $({l_d} = {l_i} + n \times 0.01 \times a$, ${l_i} = 0.30a,\; n = 30,\; 1,10,\; 27,\; 4,\; 22)$. The electric field distribution of ${E_z}$ of TM mode is shown in Fig. 9 (a). The frequencies of interface states are localized inside the defect cavity in the waveguide interface with the robust propagation edge mode even with random variation of ${l_d}$ as shown in Fig. 9 (a) (only three localized states are shown for dipole-like and quadrupole-like distributions), which physically can be explained as, each ${l_d}$ value represents an edge mode inside the bandgap. Thus, the forbidden frequency of each topological mode can be trapped when group velocity tends to be zero; accordingly, the wave is localized at this specific frequency and no longer propagates forward.
Fig. 9. Random variations of ${l_d}$ are introduced in the structure along the propagation path with $R = 3a$. The parameter ${l_d}$ is randomly varied according to $({l_d} = {l_i} + n \times 0.01 \times a$, ${l_i} = 0.30a,\; n = 30,\; 1,10,\; 27,\; 4,\; 22).$ The localization of the electric field ${E_z}$ distribution is observed along the waveguide interface (a) without (b) with dielectric obstacles.
To further investigate the robustness of the localized topological states, two dielectric obstacles (rectangular) are inserted in the propagation path with a size of ${l_i}\; \times \; 2{l_i}$ as shown in the framed black boxes of Fig. 9 (b). The frequencies of interface states are affected slightly due to the defects, compared with Fig. 9 (a) in the first two cases near the dielectric obstacles. However, the strong energy concentration occurs and is trapped at the defect cavity along the interface. Obviously, away from dielectric obstacles, the frequency of the interface state is localized at the same value as Fig. 9 (a), which further confirms that this topologically protected proposed configuration.
Finally, some disorders are introduced by changing the positions, rotational angle, and rod length around the interface of the proposed structure, as shown in Fig. 10 (a). The details of perturbations cases are framed in the black box. l of the Si rods marked by black and blue are increased and decreased by $0.05a$ respectively, and the red and yellow rods are shifted in $- x$ and $- y,$ respectively by $0.05a$. The green rod is rotated by an angle $\varphi = 0$ with a missing rod near the interface. The electric field distribution of ${E_z}$ is shown in Fig. 10 (b). The interface frequencies of the formed topological rainbow are affected slightly due to the introduction of the disorders, compared with Fig. 7. However; the topological states are still separated and localized at different positions along the interface. This further achieves that the proposed platform is robust against backscattering from obstacles and defects, which ensures the durability of the designed rainbow device.
Fig. 10. (a) The proposed structure with introducing some disorders around the interface, (b) The localization of the electric field ${E_z}$ distribution of the topological rainbow, even the presence of disorders.
6. Conclusion
In conclusion, dipole and quadrupole topological rainbows have been realized. A topological PCW coupled with the cavity is formed by inserting a gradient-defected cavity along the coupling interface by changing the cavity length. By selecting appropriate parameters of the defect cavity in the waveguide interface, robust propagation edge mode can be realized for flat-band. Accordingly, for $R = 1a$, the conventional topological rainbow is observed. When $R > 1a$, the ultra-low group velocity is attained, which is appropriate for realizing accurate and precise dipole and quadrupole topological rainbows.
Funding
National Natural Science Foundation of China (12274031); Beijing Institute of Technology Research Fund Program for Teli Young Fellows; Beijing Institute of Technology Science and Technology Innovation Plan Innovative Talents Science and Technology Funding Special Plan (2022CX01006).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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