A Singular Point Waveguide Based on Coupled Topological Interface States

Abstract

The present invention belongs to the field of optical technology, and specifically relates to a singular point waveguide based on coupled topological interface states, including a second-order singular point waveguide with a structure of S1<supgt;8< / supgt;S2<supgt;N< / supgt;S1<supgt;8< / supgt>, a third-order singular point waveguide with a structure of S1<supgt;8< / supgt;S2<supgt;N< / supgt;S1<supgt;N< / supgt>S2<supgt;8< / supgt>, and a fourth-order singular point waveguide with a structure of S1<supgt;8< / supgt;S2<supgt;N1< / supgt>S1<supgt;N2< / supgt>S2<supgt;N1< / supgt>S1<supgt;8< / supgt>. Among them, there are topological interface states between adjacent S1 and S2. The metal patch S2 is obtained by translating the metal patch S1 by half a period along the x direction. The metal patches S1 and S2 have different topological properties within the first photonic bandgap. When electromagnetic waves are incident from the x direction, the interface state mode existing between the metal patch S1 and the metal patch S2 can be excited to regulate the transmission behavior of electromagnetic waves. The present invention solves the problem of PT phase transition, solves the problem of the balance between gain and loss in the microwave system, and realizes the filtering function of a single frequency.

Description

Technical Field

[0001] The present invention belongs to the field of optical technology, and in particular relates to a singular point waveguide based on coupled topological interface states. Background Art

[0002] Optical topological interface states have attracted extensive research and attention due to their excellent transmission properties. For one-dimensional structures with centrosymmetry, the topological invariant of the energy band is typically characterized by the Zak phase, which can be quantized as 0 or π. Based on the bulk-edge correspondence, topological interface states exist at the interface between two structures with different topological properties. Interface state modes composed of one-dimensional topological structures have many advantages, such as high mode localization and a single mode.

[0003] In non-Hermitian optical systems, the system's eigenstates and eigenvectors degenerate at singular points. By precisely designing the system's gain and loss, second-order singular points can be realized in binary coupled systems. A notable characteristic of singular points is their sensitivity to perturbations, and they are commonly used in biomolecule detection, nanoparticle detection, and phase sensing. By extending the binary coupled system to a multi-element system and introducing gain and loss into each of its two resonant modes, singular points of different orders can be realized. Compared to second-order singular points, higher-order singular points are more sensitive to perturbations and can improve the device's detection performance.

[0004] Currently, achieving multi-order singularities in multi-element coupled systems requires precise control of inter-mode coupling and the modal gains and losses. However, in microwave systems, introducing modal gains is challenging. Furthermore, balancing gain and loss in the control system requires extensive experimental and computational resources. Therefore, a simple method for achieving high-order singularities would help advance the development of non-Hermitian devices.

[0005] The purpose of the present invention is to overcome the deficiencies in the prior art and provide a singular point waveguide based on topological interface states.

[0006] In order to achieve the purpose of the present invention, the present invention will be implemented by adopting the following technical solutions.

[0007] A singular point waveguide based on coupled topological interface states, the singular point waveguide comprising at least one of a second-order singular point waveguide, a third-order singular point waveguide, and a fourth-order singular point waveguide, wherein:

[0008] The second-order singular point waveguide is a topological metasurface structure composed of 8 layers of metal patches S1, N layers of metal patches S2 and 8 layers of metal patches S1 arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N S1 8; wherein:

[0009] The metal patch S1 8 and the metal patch S2 N The interface formed has topological interface states with the interface formed by the metal patch S2 N and the metal patch S1 8 ;

[0010] When electromagnetic waves are incident from the x direction, it can excite the interface state modes between the metal patch S1 8 and the metal patch S2 N and between the metal patch S2 N and the metal patch S1 8 to regulate the transmission behavior of electromagnetic waves;

[0011] The third-order exceptional point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N layers, metal patches S1 with N layers, and metal patches S2 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N S1 N S2 8 ; wherein:

[0012] The interface formed by the metal patch S1 8 and the metal patch S2 N , the interface formed by the metal patch S2 N and the metal patch S1 N , and the interface formed by the metal patch S1 N and the metal patch S2 8 all have topological interface states;

[0013] When electromagnetic waves are incident from the x direction, it can excite the interface state modes between the metal patch S1 8 and the metal patch S2 N , between the metal patch S2 N and the metal patch S1 N , and between the metal patch S2 N and the metal patch S1 8 to further regulate the transmission behavior of electromagnetic waves;

[0014] The fourth-order exceptional point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N1 layers, metal patches S1 with N2 layers, metal patches S2 with N1 layers, and metal patches S1 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N1 S1N2 S2 N1 S1 8 ; wherein:

[0015] The metal patch S1 8 and the metal patch S2 N1 form an interface, the metal patch S2 N1 and the metal patch S1 N2 form an interface, and the metal patch S1 N2 and the metal patch S2 N1 form an interface, and the metal patch S2 N1 and the metal patch S1 8 form interfaces that all have topological interface states;

[0016] When electromagnetic waves are incident from the x direction, it can excite the interface state modes between the metal patch S1 8 and the metal patch S2 N1 , between the metal patch S2 N1 and the metal patch S1 N2 , between the metal patch S1 N2 and the metal patch S2 N1 , and between the metal patch S2 N1 and the metal patch S1 8 , and further regulate the transmission behavior of electromagnetic waves;

[0017] Wherein:

[0018] The metal patch S2 is obtained by translating the metal patch S1 by half a period along the x direction;

[0019] The metal patches S1 and S2 have different topological properties within the first photonic band gap.

[0020] As a preferred embodiment of the present invention, the second-order exceptional point waveguide adjusts the number of layers N of the metal patch S2 N to regulate the coupling strength between the topological interface states existing between the metal patch S1 8 and the metal patch S2 N and the topological interface states existing between the metal patch S2 N and the metal patch S1 8 such that the two peaks of the transmission curve will merge into one;

[0021] As a preferred embodiment of the present invention, the third-order exceptional point waveguide adjusts the number of layers N of the metal patch S1 N and the metal patch S2 N to regulate the topological interface states existing between the metal patch S1 8 and the metal patch S2 N , the topological interface states existing between the metal patch S2 Nand the metal patch S1 N The topological interface states existing between and the metal patch S2 N and the metal patch S1 8 The coupling strength between the topological interface states existing between causes the three peaks of the transmission curve to merge into one, and the metal patch S2 N and the metal patch S1 N The electric field strength of the topological interface state existing between is that of the metal patch S1 8 and the metal patch S2 N The topological interface state existing between and the metal patch S2 N and the metal patch S1 8 times that of the electric field strength of the topological interface state existing between; times;

[0022] As a preferred solution of the present invention, the fourth-order exceptional point waveguide adjusts the metal patch S2 N1 , the metal patch S1 N2 and the metal patch S2 N1 to regulate the topological interface states existing between the metal patch S1 8 and the metal patch S2 N1 , the topological interface states existing between the metal patch S2 N1 and the metal patch S1 N2 , the topological interface states existing between the metal patch S1 N2 and the metal patch S2 N1 , and the coupling strength between the topological interface states existing between the metal patch S2 N1 and the metal patch S1 8 such that the three peaks of the transmission curve merge into one and then split into two.

[0023] As a preferred solution of the present invention, the basic unit of the exceptional point waveguide is composed of a top metal plate, a dielectric substrate, and a bottom metal plate, where:

[0024] The length and width of the dielectric substrate are P x = 5 mm, P y = 10 mm, and the thickness is t = 2 mm;

[0025] The length and width of the bottom metal plate are l = 5 mm, w = 3 mm, and the thickness is 0.035 mm;

[0026] The length and width of the top metal plate are l = 5 mm, w = 3 mm, and the thickness is 0.035 mm, and two symmetric rectangular gaps with lengths and widths of l1 = 1.2 mm and l2 = 0.6 mm are provided at the upper and lower boundaries thereof, and the distance b of the rectangular gap part from the center of the structure is used to control the topological characteristics of the structure.

[0027] As a preferred embodiment of the present invention, the topological properties of the metal patches S1 and S2 are characterized by the Zak phase, where the Zak phase is expressed as:

[0028]

[0029] where n represents the n-th energy band, Λ represents the period of the structure, and u k represents the wave function of the system at the wave vector k in the Brillouin zone.

[0030] As a preferred embodiment of the present invention, the topological properties of the metal patch S1 exhibit the characteristics of a conventional insulator, and its topological invariant within the first photonic bandgap is 0;

[0031] The topological properties of the metal patch S2 exhibit the characteristics of a topological insulator, and its topological invariant within the first photonic bandgap is π.

[0032] As a preferred embodiment of the present invention, the exceptional point waveguide is equivalent to a multi-mode coupling model, and the dynamic equations of the interface states are respectively:

[0033]

[0034] where a i (i = 1…N) are the topological modes of different interfaces, ω0 represents the operating frequency of the topological interface state, τ i represents the coupling coefficient from the input port to the left interface state, τ o represents the mode of the output port and the right interface state, κ i represents the mode a i and the mode a i-1 between the coupling strengths.

[0035] As a further embodiment of the present invention, when τ i = τ o , by adjusting the coupling coefficient between the modes, the system can achieve exceptional points of different orders at the critical point of the PT phase transition.

[0036] As a preferred embodiment of the present invention, the energy coupled from the input end to the interface composed of the metal patch S1 8 and the metal patch S2 N or S1 N1 can be regarded as the gain of the topological interface state between the metal patch S1 8 and the metal patch S2 N or S2 N1 ; the energy coupled out from the topological interface state composed of the metal patch S2 N or S2 N1 and the metal patch S1 8 can be regarded as the metal patch S2N or S2 N1 and the metal patch S1 8 Loss of the topological interface state therebetween.

[0037] As a preferred embodiment of the present invention, the high-order exceptional point can be obtained by measuring the transmission efficiency of the device with a vector network analyzer:

[0038] When the system is in the PT symmetric phase, the transmission curve has multiple transmittance peaks within the energy gap, corresponding to multiple eigenstates of the system;

[0039] When the system is at the PT phase transition point, multiple transmittance peaks of the transmission curve will merge into one, that is, the system supports exceptional points of different orders;

[0040] When the system is in the PT broken phase, the peak of the transmittance of the transmission curve will decrease.

[0041] As a preferred embodiment of the present invention, the exceptional point waveguide is of a metal configuration, and by changing the size and material of the structure of the metal configuration, it can be extended to other working frequency bands.

[0042] Advantageous Effects

[0043] Compared with the prior art, the present invention has the following advantages:

[0044] 1. Invented a high-order exceptional point waveguide based on coupled topological boundary states, solving the problem that traditional exceptional point waveguides need to introduce a specific amount of gain or loss to achieve PT phase transition;

[0045] 2. Using the coupling of the input and output ports with the interface state mode as the gain and loss of the system, solving the problem that it is difficult to control the gain and loss balance in a microwave system;

[0046] 3. The topological waveguide only supports a single interface state mode at the interface of the structure, and can achieve a single-frequency filtering function within a specific frequency range. Description of the Drawings

[0047] Figure 1 In FIG. (a) is the basic unit of the present invention, and this structure is periodically arranged in the x direction, and the lattice constants of the structure are P x and P y respectively, FIG. (b) is the corresponding energy band structure when the structure does not deform, FIG. (c) is the corresponding energy band structure when the structure deforms (b = 1.75 mm), and the electric field distribution at the boundary of the Brillouin zone, and FIG. (d) is the corresponding energy band structure when the structure deforms (b = 0.75 mm), and the electric field distribution at the boundary of the Brillouin zone;

[0048] Figure 2Figure (a) shows the topological waveguide structure with a single interface, the prepared sample, Figure (b) shows the eigenfrequency of the topological waveguide, Figure (c) shows the transmission curve of the topological waveguide obtained by simulation, Figure (d) shows the relationship between the quality factor of the topological interface state and the number of layers of S1 and S2, and Figure (e) shows the equivalent model of the single-interface topological waveguide, the simulation and numerically obtained transmission curves, and Figure (f) shows the transmission curve of the single-interface topological waveguide measured experimentally;

[0049] Figure 3 Figure (a) shows the schematic structure of the high-order exceptional point waveguide based on the coupled topological interface state and the equivalent coupled waveguide model; Figure (b) shows the eigenfrequency diagram of the system when the coupling strength between the input and output ports and the topological interface state is τ = 0.4; Figure (c) shows the relationship between the real part of the system eigenfrequency and τ and κ; Figure (d) shows the relationship between the imaginary part of the system eigenfrequency and τ and κ;

[0050] Figure 4 Figure (a) shows the transmission curves of the topological interface state obtained by simulation under different coupling conditions; Figure (b) shows the transmission curves of the topological interface state measured experimentally under different coupling conditions; Figure (c) shows the electric field distribution diagram at the peak of the transmission curve; Figure (d) shows the electric field intensity distribution diagram at the peak of the transmission curve;

[0051] Figure 5 Figure (a) shows the schematic structure of the third-order exceptional point waveguide based on the coupled topological interface state and the equivalent coupled waveguide model, Figure (b) shows the eigenfrequency diagram of the system when the coupling strength between the input and output ports and the topological interface state is τ = 0.4, Figure (c) shows the relationship between the real part of the system eigenfrequency and τ and κ, and Figure (d) shows the relationship between the imaginary part of the system eigenfrequency and τ and κ;

[0052] Figure 6 Figure (a) shows the transmission curves of the topological interface state obtained by simulation under different coupling conditions, Figure (b) shows the transmission curves of the topological interface state measured experimentally under different coupling conditions, Figure (c) shows the electric field distribution diagram at the peak of the transmission curve, and Figure (d) shows the electric field intensity distribution diagram at the peak of the transmission curve;

[0053] Figure 7 Figure (a) shows the schematic structure of the fourth-order exceptional point waveguide based on the coupled topological interface state and the equivalent coupled waveguide model, Figure (b) shows the transmission curves of the electromagnetic waves of the topological interface state under different coupling conditions, Figure (c) shows the real part of the system eigenfrequency under different coupling conditions, and Figure (d) shows the imaginary part of the system eigenfrequency under different coupling conditions;

[0054] Figure 8Figure (a) in [X] is the schematic structural diagram of the second-order exceptional point waveguide of the present invention, Figure (b) is the schematic structural diagram of the third-order exceptional point waveguide of the present invention, and Figure (c) is the schematic structural diagram of the fourth-order exceptional point waveguide of the present invention. Detailed implementation manners

[0055] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0056] As an embodiment of the present invention, as Figure 8 shown, a kind of exceptional point waveguide based on a coupled topological interface state, the exceptional point waveguide includes a second-order exceptional point waveguide, a third-order exceptional point waveguide and a fourth-order exceptional point waveguide, wherein:

[0057] As Figure 8 shown in Figure (a) in [X], the second-order exceptional point waveguide is a topological metasurface structure arranged in a straight line from the input end to the output end, successively including a metal patch S1 with 8 layers, a metal patch S2 with N layers, and a metal patch S1 with 8 layers. The topological metasurface structure is expressed as S1 8 S2 N S1 8 ; wherein:

[0058] The interface formed by the metal patch S1 8 and the metal patch S2 N and the interface formed by the metal patch S2 N and the metal patch S1 8 both have topological interface states;

[0059] When electromagnetic waves are incident from the x direction, the interface state modes existing between the metal patch S1 8 and the metal patch S2 N and between the metal patch S2 N and the metal patch S1 8 can be excited to regulate the transmission behavior of electromagnetic waves;

[0060] As Figure 8 shown in Figure (b) in [X], the third-order exceptional point waveguide is a topological metasurface structure arranged in a straight line from the input end to the output end, successively including a metal patch S1 with 8 layers, a metal patch S2 with N layers, a metal patch S1 with N layers, and a metal patch S2 with 8 layers. The topological metasurface structure is expressed as S1 8 S2 N S1 N S2 8 ; wherein:

[0061] The interface formed by the metal patch S1 8 and the metal patch S2 N The interface formed by...N The interface formed with the metal patch S1 N and the metal patch S1 N The interface formed with the metal patch S2 8 both have topological interface states;

[0062] When electromagnetic waves are incident from the x direction, it can excite the metal patch S1 8 and the metal patch S2 N between them, between the metal patch S2 N and the metal patch S1 N and between the metal patch S2 N and the metal patch S1 8 and the interface state modes between them, thereby regulating the transmission behavior of electromagnetic waves;

[0063] Such as Figure 8 shown in figure (c), the fourth-order exceptional point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N1 layers, metal patches S1 with N2 layers, metal patches S2 with N1 layers, and metal patches S1 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N1 S1 N2 S2 N1 S1 8 ; where:

[0064] The interface formed by the metal patch S1 8 and the metal patch S2 N1 , the interface formed by the metal patch S2 N1 and the metal patch S1 N2 , the interface formed by the metal patch S1 N2 and the metal patch S2 N1 , and the interface formed by the metal patch S2 N1 and the metal patch S1 8 all have topological interface states;

[0065] When electromagnetic waves are incident from the x direction, it can excite the metal patch S1 8 and the metal patch S2 N1 between them, between the metal patch S2 N1 and the metal patch S1 N2 , between the metal patch S1 N2 and the metal patch S2 N1 , and between the metal patch S2 N1 and the metal patch S1 8 and the interface state modes between them, thereby regulating the transmission behavior of electromagnetic waves;

[0066] Where:

[0067] The metal patch S2 is obtained by translating the metal patch S1 in the x direction by half a period;

[0068] The metal patches S1 and S2 have different topological properties within the first photonic band gap.

[0069] As an embodiment of the present invention, as shown in Figure 1 (a) of [], it is composed of upper and lower layers of metal patches and a dielectric substrate in the middle. The lattice constant of the photonic crystal is P x = 5 mm, P y = 10 mm, the width of the top metal patch is w = 3 mm, and the length and width of the vacant rectangle at the boundary of the metal patch are l1 = 1.2 mm and l2 = 0.6 mm respectively. The distance b of the rectangular void part from the center of the structure can be used to control the topological properties of the structure. In addition, the thicknesses of the top and bottom metal structures are both 0.035 mm. The thickness and dielectric constant of the middle dielectric substrate are t = 2 mm and ε = 4.2 respectively.

[0070] As an embodiment of the present invention, as shown in Figure 1 (b) of [], it shows the energy bands of the structure when b = 1.25 mm in the range of k x = 0.75π / P x to k x = π / P x . Since the left and right parts of the structure have central and translational symmetries, which is equivalent to doubling the Brillouin zone of the structure, the energy bands of the structure will fold from the boundary of the Brillouin zone towards the center, resulting in the degeneracy of the energy bands at the boundary of the Brillouin zone. By changing the size of b, the translational symmetry on both sides of the structure can be broken, thereby opening the degeneracy points of the energy bands at the boundary of the Brillouin zone and realizing topological phase transition.

[0071] As an embodiment of the present invention, as shown in Figure 1 (c) of [], it shows the energy band structure of the system when b = 1.75 mm. Since the structure has central symmetry, the topological properties of the first energy band can be judged by comparing the eigenstates of the structure at the center and boundary of the Brillouin zone. When the eigenstates of the structure at the center and boundary of the Brillouin zone have the same symmetry distribution, the energy band exhibits topological trivial properties. The eigenstate of the first energy band of S1 at the center of the Brillouin zone has an even symmetry distribution, which is the same as the eigenstate of the structure at the boundary of the Brillouin zone (see Figure 1 (c) of []), so S1 exhibits the characteristics of a conventional insulator within the first energy gap range.

[0072] As an embodiment of the present invention, as shown in Figure 1As shown in Figure (d), the energy band structure of the system is presented when b = 0.75 mm. Since S2 can be obtained by translating S1 along the x - direction by half a period, S1 and S2 have the same energy band structure. However, the eigenstate of the first energy band of S2 at the Brillouin zone boundary shows an odd - symmetric distribution (see Figure (d) in Figure 1 ). Therefore, S2 and S1 exhibit different topological characteristics within the first energy gap range.

[0073] According to the bulk - edge correspondence, topological interface states exist at the interface between two structures with different topological properties. As shown in Figure (a) in Figure 2 , the metasurface structure composed of S1 and S2 (including 9 layers of S1 and 9 layers of S2) and the prepared experimental sample. As shown in Figure (b) in Figure 2 , the eigen - frequency distribution of the system near the energy gap, where the black and red dots represent the bulk state and the interface state modes respectively. The electric - field distributions of the bulk state and the interface state are shown in Figure (b) in Figure 2 . The electric field corresponding to the interface state is mainly concentrated near the interface of the structure and decays rapidly towards both sides of the structure; while the electric field corresponding to the bulk state is distributed on the entire surface of the structure, as shown in Figure (c) in Figure 2 . The transmission curve of the topological interface state near the energy gap is shown in Figure (c) in Figure 2 , where the frequency of the transmission peak is consistent with the characteristic frequency of the interface state in Figure (b) in 3dB . The quality factor (3 - dB bandwidth) of this interface state is Q = f / BW Figure 2 ≈100. In this topological waveguide, the transmittance curve of the system is related to the number of layers of the structure. As shown in Figure (d) in

[0074] , the relationship between the transmittance peak and the quality factor and the number of S1 and S2. The structure shows that the transmission efficiency is inversely proportional to the number of structures, while the quality factor is proportional to the number of structures. Figure 2 This topological interface state can be equivalent to a coupled - waveguide system (see Figure (e) in

[0075]

[0076] ), where the dynamic equation of the topological interface state (resonator) can be expressed as: l where ω0 represents the resonant frequency of the topological interface state, τ i represents the inherent loss of the mode, τ o represents the coupling between the input port and the interface state, and τ

[0077] S - = c s S + + c aa (6)

[0078]

[0079] where c s , c s represent the direct coupling between the output port and the input port and the coupling between the edge state and the output port, respectively. For single-port incidence (S +2 = 0) and neglecting the intrinsic loss of the interface state, the reflection and transmission coefficients of the system can be expressed as:

[0080]

[0081] As shown in the (e) figure of Figure 2 , the transmission efficiencies of the S1 15 S2 15 waveguides are obtained by the electromagnetic simulation software and Equation (9), respectively. As shown in the (f) figure of Figure 2 , the transmission efficiency of the waveguide measured experimentally shows that the topological interface state is well excited within the energy gap.

[0082] As shown in the (a) figure of Figure 3 , the second-order exceptional point waveguide based on the coupled topological interface state and the equivalent coupled waveguide model. The dynamic simulation of this system is shown in Equations (2) to (4). Assuming that the system is at the critical point of PT symmetry, the reflection coefficient of the system is zero, i.e., r = 0, and the Hamiltonian of the system can be expressed as:

[0083]

[0084] where the two resonator models are regarded as the resonant modes with gain (ω0 + jτ i ) and the resonant modes with loss (ω0 - jτ o ), respectively. For simplicity, it is considered here that the coupling efficiencies between the output and input ports and the interface state are equal, i.e., τ i = τ o = τ. Figure 3 The (b) figure of Figure 3 shows the eigenfrequencies of the system when τ = 0.4. In addition, the (c) and (d) figures of

[0085] show the relationships between the eigenfrequencies of the system and the coupling coefficients τ and κ, respectively. The results show that when τ = κ, the real part and the imaginary part of the system eigenvalues are equal, i.e., the system has a second-order exceptional point (represented by the black dashed line).

[0085] Figure 4 The (a) figure of 8 shows the transmission efficiency of the two-stage exceptional point waveguide (S1 N S2 8 ). When N = 5, the waveguide has two transmission peaks, corresponding toFigure 3 Two eigenstates with a strong coupling mode in Figure (b) therein. At this time, the system is in the PT-symmetric phase; when N = 16, the two transmission peaks of the system merge into one, corresponding to Figure 3 The critical point of the PT-phase change in Figure (b) therein, where the system supports a second-order exceptional point; when N = 20, the peak value of a single transmission peak will decrease, and at this time the system is in the PT-symmetry-breaking phase. Figure 4 Figure (b) therein is the experimentally measured transmittance curve, showing the change process of the waveguide's transmission peaks merging from 2 to 1. It should be noted that, compared with the simulation results, the frequency of the transmission peaks has shifted, which may be due to the manufacturing error of the sample and the experimental measurement error. Figure 4 Figures (c) and (d) therein respectively show the electric field distributions of the system at the transmittance peak. At the second-order exceptional point, the energies located at the two interfaces are equal.

[0086] Figure 5 Figure (a) therein shows a third-order exceptional point waveguide based on a coupled topological interface state and an equivalent coupled waveguide model. The dynamic simulation of this system is shown in Formulas (2) to (4). Assuming that when the system is at the critical point of PT symmetry, the reflection coefficient of the system is zero, that is, r = 0, the Hamiltonian of this system can be expressed as:

[0087]

[0088] Among them, the three resonator models are respectively regarded as a resonant mode with gain (ω0 + jτ i ), a neutral resonant mode (ω0), and a resonant mode with loss (ω0 - jτ o ). For simplicity, it is considered here that the coupling efficiencies between the input and output ports and the interface state are equal, that is, τ i1 = τ o3 = τ. In addition, the coupling efficiencies between the interface state modes are also equal, that is, κ1 = κ2 = κ. Figure 5 Figure (b) therein represents the eigenfrequencies of the system when τ = 0.4. Figure 5 Figures (c) and (d) therein respectively show the relationships between the eigenfrequencies of this system and the coupling coefficients τ and κ. The results show that when , the real part and the imaginary part of the system eigenvalues are equal, that is, the system has a third-order exceptional point (represented by the black dashed line).

[0089] Figure 6 Figure (a) therein shows the transmission efficiency of a third-order exceptional point waveguide (S1 8 S2 N S1 N S2 8 ). When N = 6, this waveguide has three transmission peaks, corresponding to Figure 5The three eigenstates with strong coupling strength in Figure (b) indicate that the system is in the PT - symmetric phase; when N = 16, the three transmission peaks merge into one, corresponding to the PT phase - transition point of the system. At this time, the system supports a third - order exceptional point; when N = 21, the peak value of the single transmission peak will decrease. At this time, the system is in the PT - symmetry - broken phase. Figure 6 Figure (b) shows the experimentally measured transmittance curve, demonstrating the process of the waveguide's transmission peaks merging from three to one. It should be noted that, compared with the simulation results, the frequencies of the transmission peaks have shifted, which may be due to manufacturing errors of the sample and experimental measurement errors. Figure 6 Figures (c) and (d) show the electric - field distributions of the system at the peak of the transmittance. At the third - order exceptional point, the energy of the interface state in the middle position is about times that of the interface states on the left and right sides, corresponding to the eigenvectors of the third - order exceptional point.

[0090] Figure 7 Figure (a) shows a fourth - order exceptional - point waveguide (S1 8 S2 N1 S1 N2 S2 N1 S1 8 ) based on coupled topological interface states, and the equivalent coupled - waveguide model. The Hamiltonian of this system can be expressed as:

[0091]

[0092] where the four resonator models are regarded as resonant modes with gain (ω0 + jτ i ), two neutral resonant modes (ω0), and a resonant mode with loss (ω0 - jτ o ). For simplicity, it is considered here that the coupling efficiencies between the input and output ports and the interface states are equal, i.e., τ i = τ o = τ. Figure 7 Figure (b) shows the transmission curve of the fourth - order exceptional - point waveguide. As N2 increases, the peak value of the system transmittance changes from three to one and then to two. Figure 7 Figure (c) shows the change in the real part of the eigenstates of the system under different coupling conditions (the red, black, and blue curves correspond to κ1 = 0.09, κ1 = 0.096, and κ1 = 0.11, respectively). The black curve shows the process of the system changing from PT symmetry to PT - symmetry breaking. Figure 7 Figure (d) shows the relationship between the imaginary part of the eigen - frequency of the system and the coupling of the topological interface states. The results show that at the fourth - order exceptional point, both the real and imaginary parts of the eigen - frequency of the system are degenerate.

[0093] The technical solution of the present invention has been described in detail above in conjunction with the embodiments / attached drawings. However, the present invention is not limited to the above technical solution. For those of ordinary skill in the art, after learning the content recorded in the present invention, without departing from the principle of the present invention, several equivalent transformations and substitutions can still be made, and these equivalent transformations and substitutions should also be regarded as belonging to the protection scope of the present invention.

Claims

1. A singular point waveguide based on a coupled topological interface state, characterized in that, The singularity waveguide includes any one of a second-order singularity waveguide, a third-order singularity waveguide, and a fourth-order singularity waveguide, where: The second-order singular point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N layers, and metal patches S1 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N S1 8 ; where: The metal patch S1 8 and the metal patch S2 N The interface formed by them and the metal patch S2 N and the metal patch S1 8 both have topological interface states; When electromagnetic waves are incident from the x direction, it can excite the metal patch S1 8 and the metal patch S2 N The interfacial state modes existing between and the metal patch S2 N and the metal patch S1 8 are used to regulate the transmission behavior of electromagnetic waves; The third-order singular point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N layers, metal patches S1 with N layers, and metal patches S2 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N S1 N S2 8 ; where: The metal patch S1 8 and the metal patch S2 N form an interface, the metal patch S2 N and the metal patch S1 N form an interface, and the metal patch S1 N and the metal patch S2 8 all have topological interface states; When electromagnetic waves are incident from the x direction, the metal patch S1 can be excited 8 and the metal patch S2 N between them, the metal patch S2 N and the metal patch S1 N between them, and the metal patch S2 N and the metal patch S1 8 between them, the interface state modes can be excited, thereby regulating the transmission behavior of electromagnetic waves; The fourth-order singular point waveguide is a topological metasurface structure composed of metal patches S1 with 8 layers, metal patches S2 with N1 layers, metal patches S1 with N2 layers, metal patches S2 with N1 layers, and metal patches S1 with 8 layers arranged in a straight line from the input end to the output end. The topological metasurface structure is expressed as S1 8 S2 N1 S1 N2 S2 N1 S1 8 ; where: The metal patch S1 8 and the metal patch S2 N1 form an interface, the metal patch S2 N1 and the metal patch S1 N2 form an interface, and the metal patch S1 N2 and the metal patch S2 N1 form an interface, and the metal patch S2 N1 and the metal patch S1 8 form interfaces that all have topological interface states; When electromagnetic waves are incident from the x direction, the metal patch S1 can be excited 8 and the metal patch S2 N1 between them, the metal patch S2 N1 and the metal patch S1 N2 between them, and the metal patch S1 N2 and the metal patch S2 N1 between them, as well as the metal patch S2 N1 and the metal patch S1 8 between them, and the interface state modes between them can be used to regulate the transmission behavior of electromagnetic waves; Where: The metal patch S2 is obtained by translating the metal patch S1 by half a period in the x direction; The metal patches S1 and S2 have different topological properties within the first photonic bandgap; The basic unit of the singularity waveguide is composed of a top metal plate, a dielectric substrate, and a bottom metal plate, where: The length and width of the dielectric substrate are P x = 5 mm, P y = 10 mm, and the thickness is t = 2 mm; The length and width of the bottom metal plate are l = 5 mm and w = 3 mm respectively, and the thickness is 0.035 mm; The length and width of the top metal plate are l = 5 mm and w = 3 mm respectively, and the thickness is 0.035 mm. Two symmetric rectangular gaps with lengths and widths of l1 = 1.2 mm and l2 = 0.6 mm respectively are provided at the upper and lower boundaries thereof. The distance b of the rectangular gap part from the center of the structure is used to control the topological properties of the structure; The topological properties of the metal patches S1 and S2 are characterized by the Zak phase. Among them, the Zak phase is expressed as: Among them, n represents the nth energy band, Λ represents the period of the structure, and u k represents the wave function of the system at the k-point in the Brillouin zone.

2. The singular point waveguide according to claim 1 based on a coupled topological interface state, characterized in that, The second-order singular point waveguide adjusts the number of layers N of the metal patch S2 N to regulate the topological interface states existing between the metal patch S1 8 and the metal patch S2 N and the coupling strength between the topological interface states existing between the metal patch S2 N and the metal patch S1 8 so that the two peaks of the transmission curve will merge into one; The third-order exceptional point waveguide adjusts the metal patch S1 N and the metal patch S2 N in terms of the number of layers N to regulate the topological interface states existing between the metal patch S1 8 and the metal patch S2 N , the topological interface states existing between the metal patch S2 N and the metal patch S1 N , and the coupling strength between the topological interface states existing between the metal patch S2 N and the metal patch S1 8 , such that the three peaks of the transmission curve merge into one, and the electric field intensity of the topological interface state existing between the metal patch S2 N and the metal patch S1 N is 8 times that of the topological interface state existing between the metal patch S1 N and the metal patch S2 N and the topological interface state existing between the metal patch S2 8 and the metal patch S1; times The fourth-order singular point waveguide adjusts the metal patch S2 N1 , the metal patch S1 N2 and the metal patch S2 N1 layers to control the topological interface states existing between the metal patch S1 8 and the metal patch S2 N1 , the topological interface states existing between the metal patch S2 N1 and the metal patch S1 N2 , the topological interface states existing between the metal patch S1 N2 and the metal patch S2 N1 , and the coupling strength between the topological interface states existing between the metal patch S2 N1 and the metal patch S1 8 , so that the three peaks of the transmission curve will merge into one and then split into two.

3. A singularity waveguide based on a coupled topological interface state according to claim 1, characterized in that: The topological property of the metal patch S1 exhibits the characteristics of a conventional insulator, and its topological invariant within the first photonic bandgap is 0; The topological property of the metal patch S2 exhibits the characteristics of a topological insulator, and its topological invariant within the first photonic bandgap is π.

4. A singularity waveguide based on a coupled topological interface state according to claim 2, characterized in that: The singularity waveguide is equivalent to a multi-mode coupling model, and the dynamic equations of the interface state are respectively: Among them, a i (i = 1…N) topological patterns of different interfaces, ω0 represents the working frequency of the topological interface state, τ i represents the coupling coefficient between the input port and the left interface state, τ o represents the mode of the output port and the right interface state, κ i represents mode a i and mode a i-1 the coupling strength between them.

5. The singular point waveguide according to claim 4 based on a coupled topological interface state, characterized in that: When τ i = τ o , by adjusting the coupling coefficient between modes, the system can achieve exceptional points of different orders at the critical point of the PT phase transition.

6. The singular point waveguide according to claim 4 based on a coupled topological interface state, characterized in that, Coupled from the input end to the metal patch S1 8 and the metal patch S2 N or S1 N1 The energy of the interface formed can be regarded as the gain of the topological interface state between the metal patch S1 8 and the metal patch S2 N or S2 N1 The energy coupled out from the topological interface state between the metal patch S2 N or S2 N1 and the metal patch S1 8 can be regarded as the loss of the topological interface state between the metal patch S2 N or S2 N1 and the metal patch S1 8 and the metal patch S1.

7. The singular point waveguide according to claim 2 based on a coupled topological interface state, characterized in that, The singularity energy can be obtained by measuring the transmission efficiency of the device with a vector network analyzer: When the system is in the PT symmetric phase, the transmission curve has multiple transmittance peaks within the energy gap, corresponding to the number of eigenstates of the system; When the system is at the PT phase transition point, the multiple transmission peaks of the transmission curve will merge into one; When the system is in the PT phase-breaking state, the peak value of the transmittance of the transmission curve will decrease.

8. The singular point waveguide according to claim 1, which is based on a coupled topological interface state, is characterized in that The singularity waveguide is of a metal configuration, and by changing the size and material of the structure of the metal configuration, it can be extended to other working frequency bands.

Images