Reflection and transmission spectrum analysis method and system for multilayer photonic topological insulators

Abstract

The present invention relates to a reflection and transmission spectrum analysis method and system of a multilayer photon topological insulator, and the method comprises the following steps: S1, establishing a multilayer periodic structure model of a multilayer photon topological insulator medium and a common medium; S2, calculating the electric field and magnetic field in the multilayer photon topological insulator medium layer and the common medium layer; S3, determining the boundary conditions at the interface of the two material layers; S4, calculating the relationship matrix of each interface, and calculating the transmission matrix of the multilayer periodic structure; S5, solving the reflection coefficient and transmission coefficient of the single-layer structure of the photon topological insulator medium and the common medium. The present invention provides an optical method for analyzing multilayer and anisotropic metamaterials based on the transmission matrix calculation method, and the method of rotating the coordinate axis makes the model more universal and produces beneficial technical effects.

Description

Technical Field

[0001] The present invention belongs to the field of optical information technology, and in particular relates to a reflection and transmission spectrum analysis method and system for a multilayer photonic topological insulator. Background Art

[0002] In recent years, inspired by the research on condensed matter topological insulators, topological physics such as integer quantum Hall effect, quantum spin Hall effect, topological semimetal, and high-order topological insulators have been realized in optical systems. Studies have found that photons, phonons, etc. are similar to electrons and have similar band structures, which provides a basis for the research of photonic topological insulators. The boundary states in photonic topological insulators are topologically protected, which means that when light is transmitted along the system boundary, it will not be reflected by impurities or defects, showing the characteristics of unidirectional propagation. This unidirectional transmission characteristic has many potential applications in reflectionless waveguides, topological splitters, and topological lasers. By applying a bias magnetic field to the outside of the gyromagnetic material to realize magnetized plasma, photonic topological insulators are realized, and the dielectric constant of photonic topological insulators is anisotropic. If a photonic topological insulator system with a complex multilayer structure is involved, and the incident angle of the electromagnetic wave and the bias external magnetic field are modulated, its reflection and transmission characteristics can be improved, which can then be used to enhance the polarization deflection effect and regulate the spontaneous emission rate of nearby two-level systems. Therefore, it is very necessary to calculate and analyze the reflection and transmission spectra of multilayer structures of photonic topological insulators.

[0003] In view of the above technical problems, it is necessary to improve them. Summary of the invention

[0004] Based on the above-mentioned deficiencies in the prior art, the present invention provides a method and system for analyzing the reflection and transmission spectrum characteristics of a photonic topological insulator multilayer structure.

[0005] The multilayer structure of the multilayer photonic topological insulator medium and the ordinary medium constructed by the present invention has a broad application prospect as a test model; at the same time, the present invention relates to a transmission matrix calculation method, which provides an optical technical means for analyzing multilayer and anisotropic metamaterial structures.

[0006] In order to achieve the above purpose, the technical solution adopted by the present invention is:

[0007] A reflection-transmission spectrum analysis method for a multilayer photonic topological insulator comprises the following steps:

[0008] S1. Establish a multi-layer periodic structure model composed of multi-layer photonic topological insulator media and ordinary media;

[0009] S2. Calculate the electric and magnetic fields in the multilayer photonic topological insulator medium layer and the ordinary dielectric layer;

[0010] S3, determine the boundary conditions at the interface between the two material layers;

[0011] S4, calculating the relationship matrix of each interface and calculating the transmission matrix of the multi-layer periodic structure;

[0012] S5. Solve the reflection coefficient and transmission coefficient of the single-layer structure constructed by photonic topological insulator medium and ordinary medium.

[0013] Preferably, S1, establishing a multi-layer periodic structure model constructed of multi-layer photonic topological insulator media and ordinary media;

[0014] The structure of the model is a multi-layer periodic structure constructed with photonic topological insulator media and ordinary media. For the sake of convenience, the ordinary medium layer is arranged periodically from left to right, taking the vacuum layer as an example, but the method is still general.

[0015] The constitutive equation of the photonic topological insulator medium is:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022] Where D represents the electric displacement vector, B represents the magnetic induction intensity, E represents the electric field intensity, and H represents the magnetic field intensity. represents the dielectric constant, Represents magnetic permeability. ω p represents the plasma frequency, ω c represents the frequency of the cyclotron, γ is the free carrier damping constant, Γ represents the phonon damping constant, ω L and ω T Represent the longitudinal and transverse optical phonon frequencies respectively. The calculation of the plane wave solution of Maxwell's equations is simplified by using the coordinate system attached to the incident wave on the vacuum side. In order to make the coordinate axis of the incident surface coincide with the optical axis of the material and reduce unnecessary calculations, the coordinate axis angle φ is rotated so that the optical axis of the photonic topological insulator coincides with the coordinate axis of the incident surface, which ensures generality and simplifies subsequent calculations. The new dielectric constant tensor can be obtained:

[0023]

[0024] In Gaussian units, the dielectric constant and magnetic permeability in a vacuum are ε=1, μ=1.

[0025] Preferably, S2, calculate the electric field and magnetic field expressions in the photonic topological insulator medium layer and the ordinary dielectric layer; write the expressions of the incident electric field, reflected electric field, and transmitted electric field of the electromagnetic wave on the interface of the photonic topological insulator medium layer as follows;

[0026] Electric and magnetic fields of incident electromagnetic waves:

[0027]

[0028]

[0029] The electric and magnetic fields of reflected electromagnetic waves:

[0030]

[0031]

[0032] Electric and magnetic fields of transmitted electromagnetic waves:

[0033]

[0034]

[0035] in and represents the electric and magnetic fields of the incident wave, and represents the electric and magnetic fields of the reflected wave, and represents the electric and magnetic fields of the transmitted wave, k x represents the wave vector in the x direction, k z represents the wave vector in the z direction, c represents the speed of light in a vacuum, and ω is the frequency of the electromagnetic wave. x (z), e y (z) and e z (z) represents the magnitude of the electric field components in the x, y, and z directions at z, respectively, and h x (z), h y (z) and h z (z) represents the magnitude of the magnetic field components in the x, y, and z directions at z. According to Maxwell's equations, the wave vector q of the transmitted wave can be obtained: (m) , and find the relationship between the components of the electric and magnetic fields. Note that k x It is conserved at the interface. Substituting the electric and magnetic field equations of the assumed transmitted wave into Maxwell's equations, we get the component in the z direction:

[0036]

[0037]

[0038] For these components, we introduce the vector u 1 =e x ,u 2 =e y ,u 3 =h x , and u 4 =h y By (m) The component single group setting The z contribution of the wave vector in the photonic topological insulator is obtained:

[0039]

[0040]

[0041] The vector u has a unique set of solutions, requiring the determinant to be equal to 0, that is Solving for the dispersion relation:

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] For the four mathematical solutions m=1,2,3,4, where q 1 =-q 3 ,q 2 =-q 4 , when the solution of q is less than 0, it means that the wave propagates in the negative direction. Finally, the expression of the transmission component of the transmission wave E and H parallel to the surface is obtained. In order to facilitate the subsequent calculation, e x , e y ,h x and h y The relationship is simplified:

[0048]

[0049]

[0050]

[0051] The electric and magnetic fields of the electromagnetic waves transmitted from the second interface:

[0052]

[0053]

[0054] The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface:

[0055]

[0056]

[0057] The "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal, ω=2πf, and f is the frequency of the electromagnetic wave.

[0058] Preferably, S3, write out the boundary conditions at the interface between the two material layers; from the electromagnetic field boundary conditions, it is known that at the interface junction, the tangential components of the electric field and the magnetic field are conserved, and after the electromagnetic wave is incident on the photonic topological insulator medium, it is refracted into two beams of electromagnetic waves. Take one interface as an example:

[0059]

[0060]

[0061]

[0062]

[0063] E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y The symbol d represents the length of the photonic topological insulator. After refraction, the electromagnetic wave is transmitted through a length of d and then emitted from the photonic topological insulator. 1 is the angle between the transmitted wave 1 and the normal, θ 2 is the angle between the transmitted wave 2 and the normal, and the continuity of the electric and magnetic fields on the dielectric interface requires Therefore, the refraction angle of the birefringence inside the photonic topological insulator medium can be inferred from the wave vector:

[0064]

[0065] S4. Calculate the relationship matrix of each interface and the transmission matrix of the multilayer periodic structure. Write the transmission matrix according to the boundary conditions when the linear polarization is incident. Write the boundary conditions at the first interface into a matrix form:

[0066]

[0067]

[0068]

[0069] Write the boundary conditions at the second interface in matrix form:

[0070]

[0071] The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation:

[0072]

[0073] Based on the single-layer structure, the transmission matrix is ​​extended to a multi-layer structure:

[0074]

[0075] Where P N0 and P M0 They represent the coefficients before the matrix at the first interface. Since the first interface is taken as the starting point in the calculation, the coefficients are the unit matrix, so they are omitted. Now that the transmission matrix is ​​extended to multiple layers, in order to make the results more regular, they are no longer omitted.

[0076] Preferably, S5, finding the reflection coefficient and transmission coefficient of a single-layer structure of a photonic topological insulator medium and a common medium, and popularizing the calculation method of the reflection and transmission coefficient to a multi-layer periodic structure;

[0077] When the electromagnetic wave penetrates the last layer of photonic topological insulator medium, there will be no next layer of interface to reflect the transmitted wave back, and we can get and is 0. Reconstruct the transmission matrix:

[0078]

[0079] Take the first two equations and assume that E is and E ip is 0, and the transmission coefficient matrix of a single layer is obtained:

[0080]

[0081] Where Δ = x 11 x 22 -x 12 x 21 At the same time, use E is and E ip express and

[0082]

[0083]

[0084] Substituting this into the last two equations, we obtain the relationship between the incident and reflected waves:

[0085]

[0086]

[0087] Assume that E is and E ip is 0, and the single-layer reflection coefficient matrix is ​​obtained:

[0088]

[0089] After obtaining the single-layer reflection transmission coefficient matrix, simulation can be performed to obtain the transmission spectrum of the single-layer periodic structure constructed by the photonic topological insulator medium and the ordinary medium. On this basis, the transmission matrix is ​​extended to the two-layer or even multi-layer structure to calculate the multi-layer reflection transmission coefficient matrix.

[0090] The present invention also discloses a reflection and transmission spectrum analysis system for a multilayer photonic topological insulator, comprising the following modules:

[0091] Multi-layer periodic structure model building module: build a multi-layer periodic structure model constructed by multi-layer photonic topological insulator media and ordinary media;

[0092] Electric field and magnetic field calculation module: calculate the electric field and magnetic field in multilayer photonic topological insulator medium layer and ordinary dielectric layer;

[0093] Boundary condition determination module: determines the boundary conditions at the interface between two material layers;

[0094] Relationship matrix and transmission matrix calculation module: calculate the relationship matrix of each interface and the transmission matrix of multi-layer periodic structure;

[0095] Reflection coefficient and transmission coefficient solving module: solves the reflection coefficient and transmission coefficient of the single-layer structure constructed by photonic topological insulator medium and ordinary medium.

[0096] The beneficial effects of the present invention are:

[0097] The multilayer structure of the multilayer photonic topological insulator medium and ordinary medium constructed by the present invention has great application prospects as a test model; at the same time, the present invention relates to a transmission matrix calculation method, which provides an optical technical solution for analyzing multilayer and anisotropic metamaterial structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 The figure is a flow chart of the analysis method according to an embodiment of the present invention.

[0099] Figure 2 This is a multilayer structure model of a photonic topological insulator medium and a vacuum layer according to an embodiment of the present invention.

[0100] Figure 3 It is a three-dimensional model diagram of the optical axis and the incident surface coordinate axis of the photonic topological insulator medium according to an embodiment of the present invention.

[0101] Figure 4 This is a system input and output diagram of an embodiment of the present invention.

[0102] Figure 5 This is a graph showing how the transmittance changes with the incident angle for photonic topological insulators with different numbers of layers according to an embodiment of the invention.

[0103] Figure 6 This is a graph showing how reflectivity changes with the incident angle for photonic topological insulators with different numbers of layers according to an embodiment of the invention.

[0104] Figure 7 It is a curve diagram of the transformation of reflectivity with incident angle at different rotation coordinate axis angles of an embodiment of the invention.

[0105] Figure 8 This is a curve diagram of the change of transmittance with the ratio of cyclotron frequency to plasma frequency under photonic topological insulators with different numbers of layers according to an embodiment of the invention.

[0106] Fig. 9 It is a curve diagram of the transformation of reflectivity with the ratio of cyclotron frequency to plasma frequency at different rotation coordinate axis angles of an embodiment of the invention.

[0107] Fig.10 FIG. 4 is a block diagram of an analysis system according to an embodiment of the present invention. DETAILED DESCRIPTION

[0108] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0109] In the description of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "installed", "connected", and "connected" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or it can be indirectly connected through an intermediate medium, or it can be the internal communication of two components. For ordinary technicians in this field, the specific meanings of the above terms in the present invention are understood according to specific circumstances.

[0110] Embodiment 1:

[0111] like Figure 1 As shown, the analysis method of optical topological conversion of multi-layer periodic structure constructed based on photonic topological insulator material and vacuum in the embodiment of the present invention includes the following steps:

[0112] S1. Establish a multi-layer periodic structure model composed of multi-layer photonic topological insulator media and ordinary media;

[0113] The structure of the model is a multi-layer periodic structure constructed with photonic topological insulator media and ordinary media. In order to facilitate the explanation of this method, the ordinary medium layer is arranged periodically from left to right, taking the vacuum layer as an example, but the method is still general.

[0114] The constitutive equation of the photonic topological insulator medium is:

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] Where D represents the electric displacement vector, B represents the magnetic induction intensity, E represents the electric field intensity, and H represents the magnetic field intensity. represents the dielectric constant, Represents magnetic permeability. ω p represents the plasma frequency, ωc represents the frequency of the cyclotron, γ is the free carrier damping constant, Γ represents the phonon damping constant, ω L and ω T Denote the longitudinal and transverse optical phonon frequencies, respectively. Rotating the coordinate axis angle φ makes the optical axis of the photonic topological insulator coincide with the coordinate axis of the incident surface, which ensures the generality and simplifies the subsequent calculations. The new dielectric constant tensor can be obtained:

[0122]

[0123] ε xx and ε zz They are the two optical axes of the photonic topological insulator material; the dielectric constant and magnetic permeability in vacuum in Gaussian units ε=1, μ=1.

[0124] S2. Calculate the electric and magnetic field expressions in the photonic topological insulator medium layer and the ordinary dielectric layer; write the expressions of the incident electric field, reflected electric field, and transmitted electric field of the electromagnetic wave on the interface of the photonic topological insulator medium layer;

[0125] Electric and magnetic fields of incident electromagnetic waves:

[0126]

[0127]

[0128] The electric and magnetic fields of reflected electromagnetic waves:

[0129]

[0130]

[0131] Electric and magnetic fields of transmitted electromagnetic waves:

[0132]

[0133]

[0134] in and represents the electric and magnetic fields of the incident wave, and represents the electric and magnetic fields of the reflected wave, and represents the electric and magnetic fields of the transmitted wave, k x represents the wave vector in the x direction, k z represents the wave vector in the z direction, c represents the speed of light in a vacuum, and ω is the frequency of the electromagnetic wave. x (z), e y (z) and e z(z) represents the magnitude of the electric field components in the x, y, and z directions at z, respectively, and h x (z), h y (z) and h z (z) represents the magnitude of the magnetic field components in the x, y, and z directions at z. According to Maxwell's equations, the wave vector q of the transmitted wave can be obtained: (m) , and find the relationship between the components of the electric and magnetic fields, paying attention to k x It is conserved at the interface. Substituting the electric and magnetic field equations of the assumed transmitted wave into Maxwell's equations, the component in the z direction can be obtained:

[0135]

[0136]

[0137] For these components, we introduce the vector u 1 =e x ,u 2 =e y ,u 3 =h x , and u 4 =h y By (m) The component single group setting The z contribution of the wave vector in the photonic topological insulator is obtained:

[0138]

[0139]

[0140] The vector u has a unique set of solutions, requiring the determinant to be equal to 0, that is Solving for the dispersion relation:

[0141]

[0142]

[0143]

[0144]

[0145]

[0146] For the four mathematical solutions m=1,2,3,4, where q 1 =-q 3 ,q 2 =-q 4, when the solution of q is less than 0, it means that the wave propagates in the negative direction. Finally, the expression of the transmission component of the transmission wave E and H parallel to the surface is obtained. In order to facilitate the subsequent calculation, e x , e y ,h x and h y The relationship is simplified:

[0147]

[0148]

[0149]

[0150] The electric and magnetic fields of the electromagnetic waves transmitted from the second interface:

[0151]

[0152]

[0153] The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface:

[0154]

[0155]

[0156] The "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal, ω=2πf, and f is the frequency of the electromagnetic wave.

[0157] S3. Write down the boundary conditions at the interface between the two material layers. From the electromagnetic field boundary conditions, we know that at the interface, the tangential components of the electric field and magnetic field are conserved. After the electromagnetic wave is incident on the photonic topological insulator medium, it is refracted into two electromagnetic waves.

[0158] Take an interface as an example:

[0159]

[0160]

[0161]

[0162]

[0163] E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, Ey represents the electric field component in the y-axis direction, H y The symbol d represents the length of the photonic topological insulator. After refraction, the electromagnetic wave is transmitted through a length of d and then emitted from the photonic topological insulator. 1 is the angle between the transmitted wave 1 and the normal, θ 2 is the angle between the transmitted wave 2 and the normal, and the continuity of the electric and magnetic fields on the dielectric interface requires Therefore, the refraction angle of birefringence inside the photonic topological insulator medium can be inferred from the wave vector:

[0164]

[0165] S4. Calculate the relationship matrix of each interface and the transmission matrix of the multilayer periodic structure. Write the transmission matrix according to the boundary conditions when the linear polarization is incident. Write the boundary conditions at the first interface into a matrix form:

[0166]

[0167] Write the boundary conditions at the second interface in matrix form:

[0168]

[0169]

[0170]

[0171] The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation:

[0172]

[0173] Based on the single-layer structure, the transmission matrix is ​​extended to a multi-layer structure:

[0174]

[0175] Where P N0 and P M0 They represent the coefficients before the matrix at the first interface. Since the first interface is taken as the starting point in the calculation, the coefficients are the unit matrix, so they are omitted. Now that the transmission matrix is ​​extended to multiple layers, in order to make the results more regular, they are no longer omitted.

[0176] S5. Calculate the reflection and transmission spectra of a single-layer structure composed of a photonic topological insulator and a common medium, and extend the calculation method of the transmission spectrum to multi-layer periodic structures;

[0177] When the electromagnetic wave penetrates the last layer of photonic topological insulator medium, there will be no next layer of interface to reflect the transmitted wave back, and we can get and is 0. Reconstruct the transmission matrix:

[0178]

[0179] Take the first two equations and assume that E is and E ip is 0, and the transmission coefficient matrix of a single layer is obtained:

[0180]

[0181] Where Δ = x 11 x 22 -x 12 x 21 At the same time, use E is and E ip express and

[0182]

[0183]

[0184] Substituting this into the last two equations, we obtain the relationship between the incident and reflected waves:

[0185]

[0186]

[0187] Assume that E is and E ip is 0, and the reflection coefficient matrix is ​​obtained:

[0188]

[0189] After obtaining the single-layer reflection and transmission coefficient matrix, simulation can be performed to obtain the reflection and transmission spectra of the single-layer periodic structure constructed by the photonic topological insulator medium and the ordinary medium. On this basis, the transmission matrix is ​​extended to the two-layer or even multi-layer structure, the multi-layer reflection and transmission coefficient matrix is ​​calculated, and the multi-layer reflection and transmission spectra are obtained.

[0190] In this embodiment, the system Figure 3As shown, the A end inputs the incident angle parameter, the B end inputs the ratio of the cyclotron frequency and the plasma frequency, the C end inputs the layer number parameter, the D end inputs the rotation coordinate system angle parameter, the E end inputs the material thickness, the F end outputs the relationship between the transmission coefficient and each parameter, and the G end outputs the relationship between the reflection coefficient and each parameter.

[0191] Photonic topological insulators are realized by magnetizing plasmas with an external bias field, such as cyclotron materials described by a dielectric constant. Cyclotron frequency ω c =Bq e / m * , where q e is the electron charge, m * is its atomic mass, n is the free electron density, and γ is the free carrier damping constant. In addition, Γ represents the phonon damping constant, ω L and ω T Represent the longitudinal and transverse optical phonon frequencies respectively. In this embodiment, the setting of the medium parameters of the photonic topological insulator is: ω L =3.62×10 13 rad / s,ω T =3.39×10 13 rad / s,Γ=5.65×10 11 rad / s,γ=3.39×10 12 rad / s,n=1.07×10 17 cm -3 , m * =0.022m e , where m e is the mass of the electron. The dielectric constant in a vacuum is ε = 1, μ = 1, and the thickness of the photon topological insulator medium is d = 10 mm.

[0192] The input angle of incidence parameter is 0-90° at A end, the input ratio of cyclotron frequency to plasma frequency is 200 at B end, the input number of layers parameter is one to three at C end, the input angle parameter of the rotating coordinate system is 60° at D end, and the output curve of the transmittance of photon topological insulator with different number of layers as the incident angle changes is shown in the figure. Figure 5 As shown in the figure, the reflectivity curve of the topological insulator with different layers of photons output at the G end changes with the incident angle. Figure 6As shown. As the number of layers increases, the loss of electromagnetic waves in the medium also increases, and the transmittance under different incident angles decreases; the increase in the number of layers also causes the fluctuation of the reflection coefficient curve to become larger, but as the incident angle increases, the final reflectivity will tend to 1. The incident angle parameter input at the A end is 0~90°, the ratio of the cyclotron frequency and the plasma frequency input at the B end is 200, the number of layers input at the C end is two layers, the rotation coordinate system angle parameters input at D are 30°, 45°, and 60° respectively, and the G end outputs the transformation curve of the reflectivity with the incident angle at different rotation coordinate axis angles as shown in the figure. Figure 7 The transmission curves at different rotation axis angles show obvious differences. The larger the rotation axis angle, the greater the transmittance, and eventually tend to 0.

[0193] The input angle parameter of incident angle is 60° at A end, the input ratio of cyclotron frequency to plasma frequency is 0 to 200 at B end, the input layer parameter of C end changes from one to three layers, the input rotation coordinate system angle parameter is 60° at D end, and the output curve of transmittance under photon topological insulator with different layers changes with the ratio of cyclotron frequency to plasma frequency is shown in the figure. Figure 8 As shown. As the ratio of cyclotron frequency to plasma frequency increases, the transmittance increases continuously, and decreases as the number of layers increases. The incident angle parameter input at end A is 60°, the ratio of cyclotron frequency to plasma frequency is 0 to 200 at end B, the number of layers input at end C changes by two layers, the rotation coordinate system angle parameters input at end D are 30°, 45°, and 60° respectively, and the reflectivity transformation curve of the ratio of cyclotron frequency to plasma frequency at different rotation coordinate axis angles is output at end G as shown in Fig. 9 The reflection curves at different rotation axis angles show obvious differences. The larger the rotation axis angle, the smaller the direct reflection and the smaller the cross reflectivity.

[0194] Embodiment 2

[0195] like Fig.10 As shown, the reflection and transmission spectrum analysis system of the multilayer photonic topological insulator of this embodiment includes the following modules connected in sequence:

[0196] Multi-layer periodic structure model building module: build a multi-layer periodic structure model constructed by multi-layer photonic topological insulator media and ordinary media;

[0197] Electric field and magnetic field calculation module: calculate the electric field and magnetic field in multilayer photonic topological insulator medium layer and ordinary dielectric layer;

[0198] Boundary condition determination module: determines the boundary conditions at the interface between two material layers;

[0199] Relationship matrix and transmission matrix calculation module: calculate the relationship matrix of each interface and the transmission matrix of multi-layer periodic structure;

[0200] Reflection coefficient and transmission coefficient solution module: solve the reflection coefficient and transmission coefficient of the single-layer structure constructed by the photonic topological insulator medium and the ordinary medium. In summary, the present invention provides a method and system for analyzing the reflection and transmission spectra of multilayer photonic topological insulators based on the transmission matrix. The multilayer structure constructed by the multilayer photonic topological insulator medium and the ordinary medium of the present invention has broad application prospects as a test model; at the same time, the present invention relates to a calculation method based on the transmission matrix, which provides an effective optical technical means for analyzing multilayer and anisotropic metamaterial structures.

[0201] The above description is only a detailed description of the preferred embodiments and analytical calculations of the present invention. For technical researchers in this field, according to the ideas provided by the present invention, the above embodiments can be combined with each other or there can be changes on the basis of the embodiments, and these changes should also be regarded as the protection scope of the present invention.

Claims

1. A reflection-transmission spectrum analysis method for a multilayer photonic topological insulator, characterized in that: The following steps are involved: S1. Establish a multi-layer periodic structure model composed of multi-layer photonic topological insulator media and ordinary media; S2. Calculate the electric and magnetic fields in the multilayer photonic topological insulator medium layer and the ordinary dielectric layer; S3, determine the boundary conditions at the interface between the two material layers; S4, calculating the relationship matrix of each interface and calculating the transmission matrix of the multi-layer periodic structure; S5. Solve the reflection coefficient and transmission coefficient of the single-layer structure constructed by the photonic topological insulator medium and the ordinary medium; Step S1 specifically includes: The constitutive equation of the photonic topological insulator medium is: Where D represents the electric displacement vector, B represents the magnetic induction intensity, E represents the electric field intensity, and H represents the magnetic field intensity. represents the dielectric constant, represents magnetic permeability; ω p represents the plasma frequency, ω c represents the frequency of the cyclotron, γ is the free carrier damping constant, Γ represents the phonon damping constant, ω L and ω T represent the longitudinal and transverse optical phonon frequencies respectively; by rotating the coordinate axis angle φ, the optical axis of the photonic topological insulator coincides with the coordinate axis of the incident surface, and a new dielectric constant tensor is obtained: ε xx and ε zz are the two optical axes of the photonic topological insulator material; the dielectric constant and magnetic permeability in a vacuum in Gaussian units are ε=1, μ=1; Step S2 specifically includes: determining the expressions of the incident electric field, reflected electric field, and transmitted electric field of the electromagnetic wave at the interface of the photonic topological insulator medium layer; the electric field and magnetic field of the incident electromagnetic wave: The electric and magnetic fields of reflected electromagnetic waves: Electric and magnetic fields of transmitted electromagnetic waves: in, and represents the electric and magnetic fields of the incident wave, and represents the electric and magnetic fields of the reflected wave, and represents the electric and magnetic fields of the transmitted wave, k x represents the wave vector in the x direction, k z represents the wave vector in the z direction, c represents the speed of light in a vacuum, and ω is the frequency of the electromagnetic wave; e x (z), e y (z) and e z (z) represents the magnitude of the electric field components in the x, y, and z directions at z, respectively, and h x (z), h y (z) and h z (z) represents the magnitude of the magnetic field components in the x, y, and z directions at z, respectively; according to Maxwell's equations, the wave vector q of the transmitted wave is obtained: (m) , and find the relationship between the components of the electric field and magnetic field, k x It is conserved at the interface. Substituting the electric and magnetic field equations of the assumed transmitted wave into Maxwell's equations, we get the component in the z direction: For these components, we introduce the vector u1=e x ,u2=e y ,u3=h x , and u4=h y ; By (m) The component single group setting The z contribution of the wave vector in the photonic topological insulator is obtained: Solution The dispersion relation is obtained: For the four mathematical solutions m = 1, 2, 3, 4, where q1 = -q3, q2 = -q4, when the solution of q is less than 0, it means that the wave propagates in the negative direction; finally, the expression of the transmission component of the transmission wave E and H parallel to the surface is obtained. First, e x , e y ,h x and h y The relationship is simplified: The electric and magnetic fields of the electromagnetic waves transmitted from the second interface: The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface: Among them, "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal, ω = 2πf, f is the frequency of the electromagnetic wave; Step S3 specifically includes: It can be known from the electromagnetic field boundary conditions that at the interface junction, the tangential components of the electric field and the magnetic field are conserved. Assume an interface: E x : E y ： H x ： H y ： E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y represents the magnetic field component in the y-axis direction; the symbol d represents the length of the photonic topological insulator. After refraction, the electromagnetic wave is transmitted through a length of d and emitted from the photonic topological insulator; θ1 is the angle between the transmitted wave 1 and the normal line, and θ2 is the angle between the transmitted wave 2 and the normal line. The continuity requirements of the electric field and magnetic field on the dielectric interface are Therefore, the refraction angle of the birefringence inside the photonic topological insulator medium can be inferred from the wave vector: Step S4 specifically includes: determining the transmission matrix according to the boundary conditions through the transmission matrix at the time of linear polarization incidence, and writing the boundary conditions at the first interface into a matrix form: Write the boundary conditions at the second interface in matrix form: The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation: Based on the single-layer structure, the transmission matrix is ​​extended to a multi-layer structure: Where P N0 and P M0 They represent the coefficients before the matrix at the first interface. Since the first interface is taken as the starting point in the calculation, the coefficients are the unit matrix, so they are omitted. Step S5 specifically includes: When the electromagnetic wave penetrates the last layer of photonic topological insulator medium, there is no next layer of interface to reflect the transmitted wave back, and we get and is 0, and the transmission matrix is ​​reconstructed: Take the first two equations and assume that E is and E ip is 0, and the transmission coefficient matrix of a single layer is obtained: Where Δ = x 11 x 22 -x 12 x 21 At the same time, use E is and E ip express and Substituting this into the last two equations, we obtain the relationship between the incident and reflected waves: Assume that E is and E ip is 0, and the reflection coefficient matrix is ​​obtained: After obtaining the single-layer reflection and transmission coefficient matrix, simulation is performed to obtain the reflection and transmission spectra of the single-layer periodic structure constructed of photonic topological insulator medium and ordinary medium.

2. A reflection-transmission spectrum analysis system for multilayer photonic topological insulators, characterized by: Includes the following modules: Multi-layer periodic structure model building module: Establish a multi-layer periodic structure model composed of multi-layer photonic topological insulator media and ordinary media; specifically: the constitutive equation of the photonic topological insulator media is: Where D represents the electric displacement vector, B represents the magnetic induction intensity, E represents the electric field intensity, and H represents the magnetic field intensity. represents the dielectric constant, represents magnetic permeability; ω p represents the plasma frequency, ω c represents the frequency of the cyclotron, γ is the free carrier damping constant, Γ represents the phonon damping constant, ω L and ω T represent the longitudinal and transverse optical phonon frequencies respectively; by rotating the coordinate axis angle φ, the optical axis of the photonic topological insulator coincides with the coordinate axis of the incident surface, and a new dielectric constant tensor is obtained: ε xx and ε zz are the two optical axes of the photonic topological insulator material; the dielectric constant and magnetic permeability in a vacuum in Gaussian units are ε=1, μ=1; Electric field and magnetic field calculation module: calculate the electric field and magnetic field in multilayer photonic topological insulator medium layers and ordinary dielectric layers; specifically: determine the expressions of the incident electric field, reflected electric field, and transmitted electric field of electromagnetic waves on the interface of photonic topological insulator medium layers; Electric and magnetic fields of incident electromagnetic waves: The electric and magnetic fields of reflected electromagnetic waves: Electric and magnetic fields of transmitted electromagnetic waves: in, and represents the electric and magnetic fields of the incident wave, and represents the electric and magnetic fields of the reflected wave, and represents the electric and magnetic fields of the transmitted wave, k x represents the wave vector in the x direction, k z represents the wave vector in the z direction, c represents the speed of light in a vacuum, and ω is the frequency of the electromagnetic wave; e x (z), e y (z) and e z (z) represents the magnitude of the electric field components in the x, y, and z directions at z, respectively, and h x (z), h y (z) and h z (z) represents the magnitude of the magnetic field components in the x, y, and z directions at z, respectively; according to Maxwell's equations, the wave vector q of the transmitted wave is obtained: (m) , and find the relationship between the components of the electric field and magnetic field, k x It is conserved at the interface. Substituting the electric and magnetic field equations of the assumed transmitted wave into Maxwell's equations, we get the component in the z direction: For these components, we introduce the vector u1=e x ,u2=e y ,u3=h x , and u4=h y ; By (m) The component single group setting The z contribution of the wave vector in the photonic topological insulator is obtained: Solution The dispersion relation is obtained: For the four mathematical solutions m = 1, 2, 3, 4, where q1 = -q3, q2 = -q4, when the solution of q is less than 0, it means that the wave propagates in the negative direction; finally, the expression of the transmission component of the transmission wave E and H parallel to the surface is obtained. First, e x , e y ,h x and h y The relationship is simplified: The electric and magnetic fields of the electromagnetic waves transmitted from the second interface: The electric and magnetic fields of the electromagnetic wave reflected from the third interface to the second interface: Among them, "+" and "-" represent the direction of the component of the electromagnetic wave propagation direction on the z-axis. When the component direction is the same as the z-direction, it is "+", and the opposite is "-". θ is the angle between the incident electromagnetic wave and the normal, ω = 2πf, f is the frequency of the electromagnetic wave; Boundary condition determination module: Determine the boundary conditions at the interface between two material layers; Specifically: From the electromagnetic field boundary conditions, it can be seen that at the interface junction, the tangential components of the electric field and the magnetic field are conserved. Suppose an interface: E x ： E y ： H x ： H y ： E x represents the electric field component in the x-axis direction, H x represents the magnetic field component in the x-axis direction, E y represents the electric field component in the y-axis direction, H y represents the magnetic field component in the y-axis direction; the symbol d represents the length of the photonic topological insulator. After refraction, the electromagnetic wave is transmitted through a length of d and emitted from the photonic topological insulator; θ1 is the angle between the transmitted wave 1 and the normal line, and θ2 is the angle between the transmitted wave 2 and the normal line. The continuity requirements of the electric field and magnetic field on the dielectric interface are Therefore, the refraction angle of the birefringence inside the photonic topological insulator medium can be inferred from the wave vector: Relation matrix and transmission matrix calculation module: calculate the relationship matrix of each interface and calculate the transmission matrix of the multi-layer periodic structure; specifically: determine the transmission matrix according to the boundary conditions through the transmission matrix at linear polarization incidence, and write the boundary conditions at the first interface into a matrix form: Write the boundary conditions at the second interface in matrix form: The intermediate variable Eliminate and get E is 、E ip 、E rs 、E rp and relation: Based on the single-layer structure, the transmission matrix is ​​extended to a multi-layer structure: Where P N0 and P M0 They represent the coefficients before the matrix at the first interface. Since the first interface is taken as the starting point in the calculation, the coefficients are the unit matrix, so they are omitted. Reflection coefficient and transmission coefficient solution module: solve the reflection coefficient and transmission coefficient of the single-layer structure constructed by the photonic topological insulator medium and the ordinary medium; specifically: when the electromagnetic wave is transmitted to the last layer of the photonic topological insulator medium, there is no next layer of interface to reflect the transmitted wave back, and we get and is 0, and the transmission matrix is ​​reconstructed: Take the first two equations and assume that E is and E ip is 0, and the transmission coefficient matrix of a single layer is obtained: Where Δ = x 11 x 22 -x 12 x 21 At the same time, use E is and E ip express and Substituting this into the last two equations, we obtain the relationship between the incident and reflected waves: Assume that E is and E ip is 0, and the reflection coefficient matrix is ​​obtained: After obtaining the single-layer reflection and transmission coefficient matrix, simulation is performed to obtain the reflection and transmission spectra of the single-layer periodic structure constructed of photonic topological insulator medium and ordinary medium.

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