Imbert-fedorov displacement analysis method and system based on finite thickness topological insulator

Abstract

This invention relates to an Imbert-Fedorov displacement analysis method and system based on finite-thickness topological insulators. The method includes the following steps: Step 1: Establishing a model of a single-layer finite-thickness topological insulator and the media on both sides; Step 2: Determining the electromagnetic properties of topological insulators with infinite and finite surface band gaps; Step 3: Obtaining the reflection and transmission matrices at each interface of the single-layer topological insulator; Step 4: Obtaining the electric and magnetic fields in the single-layer topological insulator and the incident and exiting media; Step 5: Obtaining the Imbert-Fedorov displacement using the energy flow method. This invention calculates the Imbert-Fedorov displacement on finite-thickness topological insulators using the energy flow method, accurately analyzing the Imbert-Fedorov displacement effect characteristics on finite-thickness topological insulators. The analyzed characteristics accurately reflect the changes in Imbert-Fedorov displacement with influence factors such as the surface magnetization direction, surface band gap size, incident angle, layer thickness, and topological magnetoelectric susceptibility of the finite-thickness topological insulator.

Description

Technical Field

[0001] This invention belongs to the field of optical information technology, specifically relating to an analysis method and system for the Imbert-Fedorov displacement based on a single-layer finite-thickness topological insulator. Background Technology

[0002] When an electromagnetic wave is incident at the interface between two media, the center of its reflected wave will shift relative to the center of the incident wave. This shift is divided into lateral displacement parallel to the incident plane, i.e., Goos- The displacement, specifically the lateral displacement perpendicular to the incident plane, is known as the Imbert-Fedorov displacement. According to the energy flux method, the displacement is related to the evanescent wave energy flux in the optically less dense medium. The beam carries energy flux into the optically less dense medium and propagates a certain distance before returning to the optically denser medium; this process generates lateral and transverse displacements. The Imbert-Fedorov transverse displacement has been a focus of research in recent years. In recent years, the Imbert-Fedorov displacements of many special structures and materials, such as PT-symmetric structures, hyperbolic crystals, photonic crystals, disordered materials, and non-integer-dimensional chiral interfaces, have been studied and analyzed, broadening the research direction of beam displacement manipulation.

[0003] Topological insulators represent a novel state of matter, and by opening the surface band gap, they can be made to exhibit topological magnetic polarization. In recent years, the Imbert-Fedorov displacement effect on the surface of topological insulators has received considerable attention. However, studies on the Imbert-Fedorov displacement of topological insulators have largely focused on the surface of infinitely thick topological insulator materials, while research on the Imbert-Fedorov displacement of single-layer topological insulators with finite thickness is relatively limited. Based on this, this invention proposes an analytical method for determining the Imbert-Fedorov displacement characteristics of reflected light beams on finite-thickness topological insulators. Summary of the Invention

[0004] To address the aforementioned issues, this invention provides an analytical method and system for determining the Imbert-Fedorov displacement of a reflected beam on a finite-thickness topological insulator based on the energy flow method. The finite-thickness topological insulator model presented in this invention closely resembles a realistic topological insulator material model, providing a feasible solution for studying the Imbert-Fedorov displacement on topological insulators.

[0005] To achieve the above objectives, the technical solution adopted by this invention is as follows:

[0006] The Imbert-Fedorov displacement analysis method for calculating the reflected beam in a finite-thickness topological insulator based on the energy flow method includes the following steps:

[0007] Step 1: Establish a model of a single-layer topological insulator of finite thickness and the media on both sides;

[0008] Step 2: Determine the electromagnetic properties of topological insulators with infinite and finite surface band gaps;

[0009] Step 3: Calculate the reflection and transmission matrices at each interface of the single-layer topological insulator;

[0010] Step 4: Determine the electric and magnetic fields in the single-layer topological insulator and in the incident and exiting media;

[0011] Step 5: Determine the Imbert-Fedorov displacement using the energy flow method.

[0012] As a preferred embodiment of the present invention, in step one, the incident medium (medium 1) is a common dielectric, the upper and lower surfaces of the single-layer topological insulator (medium 2) are covered with a thin magnetic layer to open the band gap on the surface of the topological insulator, and the exit medium (medium 3) is a vacuum.

[0013] As a preferred embodiment of the present invention, step two is specifically as follows: In the case of infinite surface band gap, due to the magnetoelectric coupling effect of the topological insulator, the constitutive equation, after being modified by the topological term, is as follows:

[0014]

[0015]

[0016] Where ε TI and μ TI α represents the relative permittivity and relative permeability, α = 1 / 137 is the fine structure constant, and Θ is the topological magnetic polarization.

[0017] In the case of a finite surface bandgap, the electromagnetic field in a topological insulator is described using quantized Lagrange field theory and Maxwell's equations. Combining this with the effective electromagnetic response under the influence of fermions, and substituting these into the standard electromagnetic field equations, we obtain Maxwell's equations:

[0018]

[0019]

[0020]

[0021]

[0022] The integral forms of the dimensionless parameters φ and Φ are:

[0023]

[0024]

[0025] Where m is the surface band gap opened by the magnetic coating of the topological insulator, and sign(m) represents the sign of the surface band gap, corresponding to the sign of the topological magnetic polarization. k0 and k || Let v represent the total wave vector within the topological insulator and the wave vector parallel to the interface, respectively. F It is the Fermi velocity of fermions on the surface of a topological insulator.

[0026] As a preferred embodiment of the present invention, step three is as follows: Simultaneously solving Maxwell's equations, constitutive equations, and boundary conditions, the reflection matrix R from medium k incident on medium l (k, l = 1, 2, 3) is obtained. kl and transmission matrix T kl :

[0027]

[0028]

[0029] For an infinite surface bandgap topological insulator, the above R kl and T kl The matrix elements of a matrix are k kx k kl ε k ε l μ k μ l and ΔΘ kl Decision, where k kx With k kl ε represents the x-component of the wave vector within media k and l. k ε l and μ k μ l ΔΘ represents the permittivity and permeability of media k and l. kl =Θ l -Θ k For a topological insulator with a finite surface bandgap, the above R... kl and T kl The matrix elements of a matrix are k kx k kl ε k ε l μ k μ l , Δφ kl , ΔΦ kl And the band gap m is determined by , where Δφ kl =φ l -φ k , ΔΦkl =Φ l -Φ k .

[0030] When an incident wave enters a monolayer structure, it will be reflected multiple times within the structure. Treating the monolayer structure as a whole, its reflection and transmission matrices can be written in the following form:

[0031]

[0032]

[0033] R and T are the reflection and transmission coefficient matrices of the entire single-layer structure, R kl and T kl It is the reflection and transmission coefficient matrix of each interface in the structure, where the subscripts correspond to the interface from medium k to medium l (k, l = 1, 2, 3), I is the identity matrix, h is the thickness of the single-layer topological insulator, and K2 = jk2 cosθ2.

[0034] As a preferred embodiment of the present invention, step four is as follows: Combining the reflection coefficient matrix and Maxwell's equations, the forms of the electric and magnetic fields of the reflected wave in the incident medium 1 can be obtained as follows:

[0035] Under TE wave incidence,

[0036]

[0037] Under TM wave incident conditions,

[0038]

[0039] Considering that there are various waves with different amplitudes in medium 2, they are divided into ascending waves and descending waves according to their wave vectors. These two types of waves are superimposed and the electric and magnetic fields in medium 2 are represented by undetermined coefficients:

[0040]

[0041]

[0042] Among them, a=k2cosθ2, b=k2sinθ2.

[0043] In the case of an infinite surface bandgap topological insulator, the undetermined coefficient U can be obtained from the boundary conditions at the interface between medium 2 and medium 3. || U ⊥ V || V ⊥ Specifically, it takes the following form:

[0044] Under TE wave incidence,

[0045]

[0046]

[0047]

[0048]

[0049] Under TM wave incident conditions,

[0050]

[0051]

[0052]

[0053]

[0054] For the case of a topological insulator with a finite surface bandgap, the boundary conditions are:

[0055]

[0056] The undetermined coefficient U is obtained based on the boundary conditions under the finite surface bandgap condition. || U ⊥ V || V ⊥ Specifically, it takes the following form: under TE wave incidence,

[0057]

[0058]

[0059]

[0060]

[0061] Under TM wave incident conditions,

[0062]

[0063]

[0064]

[0065]

[0066] Combining the reflection and transmission coefficient matrix and Maxwell's equations, the forms of the electric and magnetic fields in medium 3 can be obtained as follows: Under the condition of TE wave incidence,

[0067]

[0068] Under TM wave incident conditions,

[0069]

[0070] in

[0071] As a preferred embodiment of this scheme, step five is as follows: The time-averaged Poynting vector x-direction component of the reflected beam in medium 1 is:

[0072]

[0073] The time-averaged Poynting vector y-direction component of the transmitted beam in medium 2 is:

[0074]

[0075] The time-averaged Poynting vector y-direction component of the transmitted beam in medium 3 is:

[0076]

[0077] The specific form of the Imbert-Fedorov displacement obtained by the energy flow method is as follows:

[0078]

[0079] Using the calculation results of each Poynting vector component and equation (39), the Imbert-Fedorov displacement characteristics of a single-layer finite-thickness topological insulator are analyzed.

[0080] This invention also discloses an Imbert-Fedorov displacement analysis system based on finite-thickness topological insulators. This system, based on the aforementioned method, includes the following modules:

[0081] Model building module: Builds a model of a single-layer topological insulator with finite thickness and the media on both sides;

[0082] Electromagnetic property determination module: Determines the electromagnetic properties of topological insulators with infinite and finite surface band gaps;

[0083] Reflection and transmission matrix calculation module: Calculates the reflection and transmission matrices of each interface of a single-layer topological insulator;

[0084] Electric and magnetic field calculation module: Calculates the electric and magnetic fields in a single-layer topological insulator and in the incident and exiting media;

[0085] Imbert-Fedorov displacement determination module: Determines Imbert-Fedorov displacement using the energy flow method.

[0086] The beneficial effects of this invention are:

[0087] 1. This invention calculates the Imbert-Fedorov displacement on a topological insulator of finite thickness using a modified energy flow method, and can accurately analyze the Imbert-Fedorov displacement effect on a topological insulator of finite thickness.

[0088] 2. This invention can accurately reflect the influence of factors such as magnetization direction, layer thickness, surface band gap, topological magnetic polarization, polarization state, and incident angle of a topological insulator with finite thickness on the Imbert-Fedorov displacement. Attached Figure Description

[0089] Figure 1 This is a flowchart of the Imbert-Fedorov displacement analysis method for finite-thickness topological insulators according to an embodiment of the present invention;

[0090] Figure 2 is a schematic diagram of a finite thickness topological insulator model according to an embodiment of the present invention;

[0091] Figure 3 This is a schematic diagram of the system I / O corresponding to the Imbert-Fedorov displacement analysis method on a finite-thickness topological insulator according to an embodiment of the present invention;

[0092] Figure 4 shows the relationship between the Imbert-Fedorov displacement on an infinite surface bandgap topological insulator with finite thickness according to an embodiment of the present invention and the magnetization direction, polarization state, layer thickness, and topological magnetoelectric susceptibility.

[0093] Figure 5 shows the relationship between the Imbert-Fedorov displacement on a topological insulator with finite thickness and finite surface bandgap according to an embodiment of the present invention and the magnetization direction, polarization state, layer thickness, and surface bandgap.

[0094] Figure 6 This is a block diagram of the Imbert-Fedorov displacement analysis system based on a finite-thickness topological insulator, according to an embodiment of the present invention. Detailed Implementation

[0095] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0096] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0097] like Figure 1 The diagram shows a flowchart of the Imbert-Fedorov displacement analysis method on a topological insulator with a single-layer structure and a finite surface bandgap, according to a preferred embodiment of the present invention. Specifically, the Imbert-Fedorov displacement analysis method on a topological insulator with a finite thickness includes the following five steps:

[0098] Step 1: Establish a model of a single-layer topological insulator of finite thickness and the media on both sides;

[0099] Step 2: Determine the electromagnetic properties of topological insulators with infinite and finite surface band gaps;

[0100] Step 3: Calculate the reflection and transmission matrices at each interface of the single-layer topological insulator;

[0101] Step 4: Determine the electric and magnetic fields in the single-layer topological insulator and in the incident and exiting media;

[0102] Step 5: Determine the Imbert-Fedorov displacement using the energy flow method.

[0103] In step one of this embodiment, the single-layer finite-thickness topological insulator, as shown in Figures 2(a) and 2(b), has an incident medium (medium 1) that is a common dielectric, and a thin magnetic layer covering both the upper and lower surfaces of the single-layer topological insulator (medium 2) to open the band gap on the surface of the topological insulator. The exit medium (medium 3) is a vacuum.

[0104] Step 2: First, determine the electromagnetic properties of the topological insulator with finite thickness. In the case of infinite surface band gap, due to the magnetoelectric coupling effect of the topological insulator, the constitutive equation, after being modified by the topological term, takes the following form:

[0105]

[0106]

[0107] Where, ε TI and μ TI α represents the relative permittivity and relative permeability, α = 1 / 137 is the fine structure constant, and Θ is the topological magnetic polarization.

[0108] In the case of a finite surface bandgap, the electromagnetic field in a topological insulator is described using quantized Lagrange field theory and Maxwell's equations. Combining this with the effective electromagnetic response under the influence of fermions, and substituting these into the standard electromagnetic field equations, we obtain Maxwell's equations:

[0109]

[0110]

[0111]

[0112]

[0113] The integral forms of the dimensionless parameters φ and Φ are:

[0114]

[0115]

[0116] Where m is the surface band gap opened by the magnetic coating of the topological insulator, and sign(m) represents the sign of the surface band gap, corresponding to the sign of the topological magnetic polarization. k0 and k || Let v represent the total wave vector within the topological insulator and the wave vector parallel to the interface, respectively. F It is the Fermi velocity of fermions on the surface of a topological insulator.

[0117] Step 3: Then, using the electromagnetic properties described above, solve the Maxwell's equations, constitutive equations, and boundary conditions simultaneously to obtain the reflection matrix R from medium k incident on medium l (k, l = 1, 2, 3). kl and transmission matrix T kl :

[0118]

[0119]

[0120] For an infinite surface bandgap topological insulator, the above R kl and T kl The matrix elements of a matrix are k kx k kl ε k ε l μ k μ l and ΔΘ kl Decision, where k kx With k kl ε represents the x-component of the wave vector within media k and l. k εl and μ k μ l ΔΘ represents the permittivity and permeability of media k and l. kl =Θ l -Θ k For a topological insulator with a finite surface bandgap, the above R... kl and T kl The matrix elements of a matrix are k kx k kl ε k ε l μ k μ l , Δφ kl , ΔΦ kl And the band gap m is determined by , where Δφ kl =φ l -φ k , ΔΦ kl =Φ l -Φ k .

[0121] When an incident wave enters a monolayer structure, it will be reflected multiple times within the structure. Treating the monolayer structure as a whole, its reflection and transmission matrices can be written in the following form:

[0122]

[0123]

[0124] R and T are the reflection and transmission coefficient matrices of the entire single-layer structure, R kl and T kl It is the reflection and transmission coefficient matrix of each interface in the structure, where the subscripts correspond to the interface from medium k to medium l (k, l = 1, 2, 3), I is the identity matrix, h is the thickness of the single-layer topological insulator, and K2 = jk2 cosθ2.

[0125] Step four, combining the reflection coefficient matrix and Maxwell's equations, yields the forms of the electric and magnetic fields of the reflected wave in incident medium 1 as follows:

[0126] Under TE wave incidence,

[0127]

[0128] Under TM wave incident conditions,

[0129]

[0130] Considering that there are various waves with different amplitudes in medium 2, they are divided into ascending waves and descending waves according to their wave vectors. These two types of waves are superimposed and the electric and magnetic fields in medium 2 are represented by undetermined coefficients:

[0131]

[0132]

[0133] Among them, a=k2cosθ2, b=k2sinθ2.

[0134] In the case of an infinite surface bandgap topological insulator, the undetermined coefficient U can be obtained from the boundary conditions at the interface between medium 2 and medium 3. || U ⊥ V || V ⊥ Specifically, it takes the following form:

[0135] Under TE wave incidence,

[0136]

[0137]

[0138]

[0139]

[0140] Under TM wave incident conditions,

[0141]

[0142]

[0143]

[0144]

[0145] For the case of a topological insulator with a finite surface bandgap, the boundary conditions are:

[0146]

[0147] The undetermined coefficient U is obtained based on the boundary conditions under the finite surface bandgap condition. || U ⊥ V || V ⊥ Specifically, it takes the following form: under TE wave incidence,

[0148]

[0149]

[0150]

[0151]

[0152] Under TM wave incident conditions,

[0153]

[0154]

[0155]

[0156]

[0157] Combining the reflection and transmission coefficient matrix and Maxwell's equations, the forms of the electric and magnetic fields in medium 3 can be obtained as follows: Under the condition of TE wave incidence,

[0158]

[0159] Under TM wave incident conditions,

[0160]

[0161] in

[0162] Step 5: Finally, the Imbert-Fedorov displacement is calculated using the energy flow method. The time-averaged Poynting vector x-direction component of the reflected beam in medium 1 is:

[0163]

[0164] The time-averaged Poynting vector y-direction component of the transmitted beam in medium 2 is:

[0165]

[0166] The time-averaged Poynting vector y-direction component of the transmitted beam in medium 3 is:

[0167]

[0168] The specific form of the Imbert-Fedorov displacement obtained by the energy flow method is as follows:

[0169]

[0170] Using the calculation results of each Poynting vector component and equation (39), the Imbert-Fedorov displacement characteristics of a single-layer finite-thickness topological insulator are analyzed.

[0171] In this embodiment, as Figure 3As shown, this embodiment of the invention analyzes the Influencing factors such as polarization state, magnetization direction, surface band gap, topological magnetic polarization susceptibility, incident angle, and layer thickness of a topological insulator with finite thickness.

[0172] Input the relevant parameters of the incident medium at port A, including dielectric constant, permeability, and the type of topological insulator. 0 represents an infinite surface bandgap, and 1 represents a finite surface bandgap. Input the magnetization direction of the finite-thickness topological insulator at port B. If the type of the topological insulator input at port A is 0 (infinite surface bandgap), then input the topological magnetic polarization; if the type of the topological insulator input at port A is 1 (finite surface bandgap), then input the size of the surface bandgap. Input the relevant parameters of the incident electromagnetic wave at port C, including the incident angle and polarization state. Input the layer thickness of the topological insulator at port D. Output the Imert-Fedorov displacement of the topological insulator at different layer thicknesses at port E. Output the Imert-Fedorov displacement of the infinite bandgap topological insulator at different topological magnetic polarizations and the finite bandgap topological insulator at different surface bandgap conditions at port F.

[0173] In this embodiment, the relative permittivity and permeability of medium 1, medium 2, and medium 3 input at port A are: μ1=μ2=μ3=1, ε r1 =1.9, ε r2 =1.45, ε r3 =1, topological insulator type is 0; the input magnetization direction at port B is antiparallel magnetization, and the topological polarizability is |Θ t |=|Θ b |=5π,Θ t Θ is the topological magnetic polarization of the upper surface. b The topological magnetic polarization of the lower surface is given; the input polarization state at port C is TE wave polarization with an incident angle of 30°-60°; the input layer thickness at port D is h = 0.75λ0, 1.5λ0, where λ0 is the wavelength of the incident wave; the output at port E is the graph showing the relationship between the Imbert-Fedorov displacement and the layer thickness, as shown in Figure 4(a).

[0174] The relative permittivity and permeability of medium 1, medium 2, and medium 3 are input at port A as follows: μ1=μ2=μ3=1, ε r1 =1.9, ε r2 =1.45, ε r3 =1, topological insulator type is 0; the input magnetization direction at port B is parallel magnetization, and the topological polarizability is |Θ t |=|Θ b|=π,3π,5π,7π; The input polarization state at port C is TE wave polarization, and the incident angle is 30°-60°; The input layer thickness at port D is h=1.7μm; The output at port F is the relationship between the Imbert-Fedorov displacement and the topological magnetic polarization, as shown in Figure 4(b).

[0175] The relative permittivity and permeability of medium 1, medium 2, and medium 3 are input at port A as follows: μ1=μ2=μ3=1, ε r1 =1.9, ε r2 =1.45, ε r3 =1, topological insulator type is 0; the input magnetization direction at port B is antiparallel magnetization, and the topological polarizability is |Θ t |=|Θ b |=5π;The input polarization state at port C is TM wave polarization, and the incident angle is 30°-60°; the input layer thickness at port D is h=3λ0,6λ0; the output at port E is the relationship between the Imbert-Fedorov displacement and the layer thickness, as shown in Figure 4(c).

[0176] In this embodiment, the relative permittivity and permeability of medium 1, medium 2, and medium 3 input at port A are: μ1=μ2=μ3=1, ε r1 =1.9, ε r2 =1.45, ε r3 =1, topological insulator type is 1; the input magnetization direction at port B is parallel magnetization, and the surface band gap size is |m| = 300ω. R ω R Represented as the resonant frequency ω R =1.1×10 13 The input polarization state at port C is TE wave polarization, with an incident angle of 20°-70°; the input layer thickness at port D is h = 0.75λ0, 1.5λ0, 2.25λ0; the output Imbert-Fedorov displacement at port E is shown in Figure 5(a).

[0177] In this embodiment, the relative permittivity and permeability of medium 1, medium 2, and medium 3 input at port A are: μ1=μ2=μ3=1, ε r1 =1.9, ε r2 =1.45, ε r3 =1, topological insulator type is 1; the input magnetization direction at port B is antiparallel magnetization, and the band gap size is m = 150ω. R 200ω R 250ω R 300ω RThe input polarization state at port C is TM wave polarization, with an incident angle of 20°-70°; the input layer thickness at port D is h = 1.7 μm; and the output at port F is the relationship between the Imbert-Fedorov displacement and the surface bandgap size, as shown in Figure 5(b).

[0178] This invention presents a method for calculating the Imbert-Fedorov displacement of a single-layer finite-surface bandgap topological insulator based on a modified energy flow method. This method can accurately calculate the transmission, reflection, and reflection coefficients of the single-layer finite-surface bandgap topological insulator. Furthermore, it can comprehensively analyze the influence of various parameters on the Imbert-Fedorov displacement based on factors such as the magnetization direction, layer thickness, incident angle, and surface bandgap size. Due to limitations in practical conditions and high material costs, the theoretical model used in this invention closely approximates actual multilayer anisotropic topological insulator materials, making it highly valuable as a theoretical testing model. This method calculates the energy flow in each medium by determining the transmission and reflection coefficients of the single-layer finite-surface bandgap topological insulator, and then tests the Imbert-Fedorov displacement effect of reflected electromagnetic waves. This provides a reference for studying the Imbert-Fedorov displacement effect of single-layer finite-surface bandgap topological insulators and offers a new approach for measuring the topological magnetopolarization properties of topological insulators.

[0179] like Figure 6 As shown, an Imbert-Fedorov displacement analysis system based on finite-thickness topological insulators is disclosed. This system, based on the aforementioned method, includes the following modules:

[0180] Model building module: Builds a model of a single-layer topological insulator with finite thickness and the media on both sides;

[0181] Electromagnetic property determination module: Determines the electromagnetic properties of topological insulators with infinite and finite surface band gaps;

[0182] Reflection and transmission matrix calculation module: Calculates the reflection and transmission matrices of each interface of a single-layer topological insulator;

[0183] Electric and magnetic field calculation module: Calculates the electric and magnetic fields in a single-layer topological insulator and in the incident and exiting media;

[0184] Imbert-Fedorov displacement determination module: Determines Imbert-Fedorov displacement using the energy flow method.

[0185] Other aspects of this embodiment can be found in the above method embodiments.

[0186] This invention calculates the Imbert-Fedorov displacement on a finite-thickness topological insulator using the energy flow method. It can accurately analyze the characteristics of the Imbert-Fedorov displacement effect on a finite-thickness topological insulator. The analyzed characteristics can accurately reflect the changes in the Imbert-Fedorov displacement with the influence of factors such as the surface magnetization direction, surface band gap size, incident angle, layer thickness, and topological magnetoelectric susceptibility of the finite-thickness topological insulator.

[0187] The above description is merely a detailed explanation of preferred embodiments and principles of the present invention. For those skilled in the art, there may be changes in specific implementation methods based on the ideas provided by the present invention, and these changes should also be considered within the scope of protection of the present invention.

Claims

1. An Imbert-Fedorov displacement analysis method based on finite-thickness topological insulators, characterized by: Follow these steps: Step 1: Establish a model of a single-layer topological insulator of finite thickness and the media on both sides; Step 2: Determine the electromagnetic properties of topological insulators with infinite and finite surface band gaps; Step 3: Calculate the reflection matrix and transmission matrix of each interface of the single-layer topological insulator; Step 4: Determine the electric and magnetic fields in the single-layer topological insulator and in the incident and exiting media; Step 5: Determine the Imbert-Fedorov displacement using the energy flow method; The second step is as follows: In the case of infinite surface band gap, due to the magnetoelectric coupling effect of the topological insulator, the constitutive equation, after being modified by the topological term, takes the following form: (1) (2) Where E and D are the electric field strength and electric flux density, and H and B are the magnetic field strength and magnetic flux density. and These are the relative permittivity and relative permeability. and These are the vacuum permittivity and vacuum permeability. It is the fine structure constant. It is the topological magnetic polarizability; In the case of a finite surface bandgap, the electromagnetic field in a topological insulator is described using quantized Lagrange field theory and Maxwell's equations. Combining this with the effective electromagnetic response under fermion interaction, and substituting it into the standard electromagnetic field equations, we obtain Maxwell's equations: (3) (4) (5) (6) Dimensionless parameters and The integral form is: (7) (8) in, It is the surface band gap opened by the magnetic coating of the topological insulator. The symbol for surface band gap corresponds to the symbol for topological magnetic polarization. , and Let represent the total wave vector and the wave vector parallel to the interface within the topological insulator, respectively. It is the Fermi velocity of fermions on the surface of a topological insulator.

2. The method as described in claim 1, characterized in that: In the first step: the incident medium (medium 1) and the top and bottom surfaces of the single-layer finite-thickness topological insulator (medium 2) are covered with magnetic layers to open the band gap on the surface of the topological insulator, and the exit medium (medium 3) is a vacuum.

3. The method as described in claim 2, characterized in that: The third step is as follows: Solve the simultaneous Maxwell's equations, constitutive equations, and boundary conditions to obtain the equations from the medium. Incident into the medium reflection matrix and transmission matrix : (9) (10) Where k,l=1,2,3; For topological insulators with infinite surface band gaps and The matrix elements of a matrix are composed of , , , , , as well as Decision, among which and This represents the wave vector component in the x-direction within media k and l. , and , Indicates medium and The dielectric constant and permeability, For topological insulators with finite surface band gaps, and The matrix elements of a matrix are composed of , , , , , , , And the band gap m is determined by, where , ; When an incident wave enters a monolayer structure, it will be reflected multiple times within the structure. Treating the monolayer structure as a whole, its reflection and transmission matrices can be written in the following form: (11) (12) R and T are the reflection and transmission coefficient matrices of the entire single-layer structure. and This is the reflection and transmission coefficient matrix of each interface in the structure, where the subscripts correspond to the values ​​from... Media incident The interface of the medium; It is the identity matrix. The thickness of a single-layer topological insulator. .

4. The method as described in claim 3, characterized in that: The fourth step is as follows: Combining the reflection coefficient matrix and Maxwell's equations, the electric and magnetic fields of the reflected wave in the incident medium 1 are obtained as follows: Under TE wave incidence, (13) Under TM wave incident conditions, (14) Considering that there are various waves with different amplitudes in medium 2, they are divided into ascending waves and descending waves according to their wave vectors. These two types of waves are superimposed and the electric and magnetic fields in medium 2 are represented by undetermined coefficients: (15) (16) in , ; In the case of an infinite surface bandgap topological insulator, the undetermined coefficients can be obtained based on the boundary conditions at the interface between medium 2 and medium 3. , , , Specifically, it takes the following form: Under TE wave incidence, (17) (18) (19) (20) Under TM wave incident conditions, (21) (22) (23) (24) For the case of a topological insulator with a finite surface bandgap, the boundary conditions are: (25) The undetermined coefficients are obtained based on the boundary conditions under the finite surface bandgap condition. , , , Specifically, it takes the following form: Under TE wave incidence, (26) (27) (28) (29) Under TM wave incident conditions, (30) (31) (32) (33) Combining the reflection and transmission coefficient matrix and Maxwell's equations, the forms of the electric and magnetic fields in medium 3 are obtained as follows: Under TE wave incidence, (34) Under TM wave incident conditions, (35) in .

5. The method as described in claim 4, characterized in that: The fifth step is as follows: Time-averaged Poynting vector of the reflected beam in medium 1 The directional components are: (36) Time-averaged Poynting vector of the transmitted beam in medium 2 The directional components are: (37) Time-averaged Poynting vector of transmitted beam in medium 3 The directional components are: (38) The specific form of the Imbert-Fedorov displacement obtained by the energy flow method is as follows: (39) Using the calculation results of each Poynting vector component and equation (39), the Imbert-Fedorov displacement characteristics of a single-layer finite-thickness topological insulator are obtained.

6. An Imbert-Fedorov displacement analysis system based on finite-thickness topological insulators, based on the method described in any one of claims 1-5, characterized in that... The system includes the following modules: Model building module: Builds a model of a single-layer topological insulator with finite thickness and the media on both sides; Electromagnetic property determination module: Determines the electromagnetic properties of topological insulators with infinite and finite surface band gaps; Reflection and transmission matrix calculation module: Calculates the reflection and transmission matrices of each interface of a single-layer topological insulator; Electric and magnetic field calculation module: Calculates the electric and magnetic fields in a single-layer topological insulator and in the incident and exiting media; Imbert-Fedorov displacement determination module: Determines Imbert-Fedorov displacement using the energy flow method.

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