A method for constructing a multi-layer structure plasma mode
By constructing the resonance matrix of multi-layer materials and calculating feature polynomials, the plasma mode parameters are solved, and the structural design problems of multi-layer materials in the prior art are solved, and the effective realization and cost reduction of plasma resonance phenomenon are achieved.
Patent Information
- Application Number
- CN202210088793.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-01-25
- Publication Date
- 2024-12-27
- Estimated Expiration
- 2042-01-25
AI Technical Summary
When designing multi-layer material structures, the material parameters cannot be guided in advance, resulting in limited prior information provided by the experimental method when there are many layers of material, making it difficult to achieve plasma resonance.
By constructing the resonance matrix of multi-layer materials, calculating its characteristic polynomials, solving the roots of the characteristic polynomials to obtain the plasma mode parameters in the material structure, and constructing a multi-layer plasma material model to verify the material parameters to achieve plasma resonance.
This method significantly reduces the eigenvalue calculation time, provides a method to design material parameters, ensures that the multi-layer structure is realized in the plasma resonance phenomenon, and reduces the cost of multi-layer material construction.
Smart Images
Figure CN114420226B_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the field of material construction, and in particular relates to a construction method of a multi-layer structure plasma mode. Background Art
[0002] There are many projects that consider the construction of multi-layer material structures to achieve plasma resonance. The existing technical means mainly use experimental methods to try some specific multi-layer structure materials to determine whether the effect of plasma resonance can be achieved. However, this experimental method is not general and cannot provide advance guidance on how to design material parameters. It can only be tried continuously through experimental means. A big disadvantage of this experimental method is that when there are many layers of materials, the prior information that the experimental method can provide is more limited. Summary of the invention
[0003] The object of the present invention is to provide a method for constructing a plasma mode of a multi-layer structure, which method can design material parameters to achieve plasma resonance when the number of material layers is too large.
[0004] The method for constructing the multi-layer plasma mode provided by the present invention comprises the following steps:
[0005] S1. Construct a resonance matrix of multilayer materials, the order of which is the same as the number of material layers;
[0006] S2. Perform preliminary material construction and use mathematical induction to calculate the final characteristic polynomial based on the resonance matrix;
[0007] S3. Solve the roots of the characteristic polynomial to obtain the parameters corresponding to all plasma modes in the material structure;
[0008] S4. Construct a multilayer plasma material model based on the parameters corresponding to all plasma modes in the material structure, and obtain the final plasma material through verification through the current prior information, and construct a multilayer structure.
[0009] The step S1 comprises constructing a resonance matrix P of a multilayer material having an N-layer structure. N :
[0010]
[0011] Among them, λ j represents the feature parameters of the jth layer; r j represents the radius of the material of the jth layer, j∈1,2,...,N;
[0012]
[0013] Among them, ε j Represents the regular material parameter of the j-th layer material.
[0014] The final characteristic polynomial based on the resonance matrix:
[0015]
[0016] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of ; [·] means rounding down; λ means the original eigenvalue; represents a combination of 2n elements in the set [1,N]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m-1 Indicates the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition α,β∈1,2,...,N; the number of real roots of the final characteristic polynomial is N, and the range of real roots is [-1,2]; q(λ) represents the transformed eigenvalue, q(λ)=λ 2 -λ.
[0017] The step S2 comprises the following steps:
[0018] A1. Perform preliminary material construction including the regular material parameter ε of the j-th layer material j Select and set the feature parameter λ of the jth layer j Perform pre-processing;
[0019] A1. When N≤4, the initial characteristic polynomial is directly calculated;
[0020] A2. Perform recursion to obtain the initial recursive formula;
[0021] A3. Verify the initial recursive formula and obtain the final characteristic polynomial.
[0022] The step A1 includes numbering each layer and calculating the regular material parameter ε of the j-th layer material. j When j is an even number, ε j The value is ε0. When j is an odd number, ε j The value is ε1, ε0 is the parameter of the external background medium, ε1 is the material parameter to be designed; for the j-th layer characteristic parameter λ j Preprocessing is performed. When j is an odd number, λ j Take the value λ, λ represents the original eigenvalue; when j is an even number, λ j The value is the difference between 1 and λ.
[0023] The step A2 includes when N=4,
[0024]
[0025] Where |P4(λ)| represents the characteristic polynomial of the resonance matrix of the multilayer material with a 4-layer structure; λ represents the original eigenvalue; Represents a combination of 2 elements in the set [1,4]; k represents the value of the first element in the combination of 2 elements, and l represents the value of the second element in the combination of 2 elements; definition α,β∈1,2,...,N.
[0026] The step A3 includes recursively performing a general number of layers to obtain an initial recursive formula:
[0027]
[0028] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of N-1 and λ N The calculation is done using the following formula: λ j represents the j-th layer feature parameter, j∈1,2,...,N; ε j represents the regular material parameter of the j-th layer material; represents the sub-formula formed by the 2nd to N-1th rows and the 2nd to N-1th columns in the resonance matrix; represents the sub-formula formed by the 2nd to the N-2th row and the 2nd to the N-2th column in the resonance matrix; represents the sub-formula formed by the 3rd to N-1th rows and the 3rd to N-1th columns in the resonance matrix; Represents the sub-formula consisting of the 3rd to N-2th rows and the 3rd to N-2th columns in the resonance matrix.
[0029] The step A4 includes assuming that the characteristic polynomials are correct when N≤N0, wherein N represents the number of layers of the multilayer structure, N0 represents the set number of layers, and N0 is a positive integer greater than or equal to 4; using mathematical induction to construct the set characteristic polynomial to verify that N=N0+1 holds; and conducting an in-depth analysis of the number of combinations:
[0030] definition:
[0031]
[0032]
[0033]
[0034]
[0035] Also define:
[0036]
[0037]
[0038]
[0039]
[0040] Wherein, D1 represents the first term of the set characteristic polynomial; D2 represents the second term of the set characteristic polynomial; D3 represents the third term of the set characteristic polynomial; D4 represents the fourth term of the set characteristic polynomial; d1 represents the first term coefficient of the set characteristic polynomial; d2 represents the second term coefficient of the set characteristic polynomial; d3 represents the third term coefficient of the set characteristic polynomial; d4 represents the fourth term coefficient of the set characteristic polynomial; λ represents the original eigenvalue; n represents the summation index; It represents a combination of 2n elements in the set [2,N-1]; It represents a combination of 2n elements in the set [2,N-2]; It represents a combination of 2n elements in the set [3, N-1]; represents a combination of 2n elements in the set [3,N-2]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m-1 Indicates the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition α,β∈1,2,...,N; N0 represents the set number of layers; N represents the number of layers in the multilayer structure; q(λ) represents the transformed eigenvalue, q(λ)=λ 2 -λ; definition α,β∈1,2,...,N; then
[0041] |P N (λ)|=d1D1+d2D2+d3D3+d4D4
[0042] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of ;
[0043] Introduce the following tags to define the first tag a k , second marker b k 、c k The third mark and Fourth mark:
[0044]
[0045]
[0046]
[0047]
[0048] Using the first marker a k , second marker b k 、c k The third mark and The fourth symbol represents the resonance matrix P of the multilayer material with N layers. N The characteristic polynomial of |P N (λ)|:
[0049]
[0050] According to the relationship between the number of combinations, we can get:
[0051]
[0052]
[0053]
[0054] The final characteristic polynomial is obtained through verification.
[0055] The method for constructing the plasma mode of the multilayer structure provided by the present invention provides a resonance matrix corresponding to each multilayer material, and the order thereof is the same as the number of material layers. At the same time, a characteristic polynomial form is provided for the resonance matrix, thereby greatly reducing the calculation time for calculating the characteristic value, and realizing the plasma resonance phenomenon by designing material parameters. BRIEF DESCRIPTION OF THE DRAWINGS
[0056] Figure 1 It is a schematic diagram of the process of the present invention.
[0057] Figure 2 Schematic diagram of the multilayer material structure of the method of the present invention.
[0058] Figure 3 Schematic diagram of characteristic modes and resonance effects of multilayer materials according to an embodiment of the present invention. DETAILED DESCRIPTION
[0059] like Figure 1 It is a schematic diagram of the process of the present invention, such as Figure 2 Schematic diagram of the multilayer material structure of the method of the present invention: The method for constructing the multilayer structure plasma mode provided by the present invention comprises the following steps:
[0060] S1. For a material structure with a selected number of layers and thickness of each layer, the transmission boundary conditions satisfied by the electromagnetic wave propagating in the material structure are used to accurately characterize the impact of changes in its electromagnetic field data, and a metric matrix related to the material electromagnetic parameters, number of layers, and thickness is obtained. The metric matrix is a reversible matrix under the selection of regular material parameters, while the reversibility of the matrix is broken in the plasma mode. We call this matrix a resonance matrix. Each multilayer material corresponds to a resonance matrix, and its order is the same as the number of material layers. The resonance matrix P of a multilayer material with an N-layer structure is constructed. N :
[0061]
[0062] Among them, λ j represents the feature parameters of the jth layer; r j represents the radius of the material of the jth layer, j∈1,2,...,N;
[0063]
[0064] Among them, ε j Represents the regular material parameter of the j-th layer material.
[0065] S2. Perform preliminary material construction and provide a displayed characteristic polynomial form for the resonance matrix, thereby reducing the calculation time for eigenvalues. Use mathematical induction to calculate the final characteristic polynomial based on the resonance matrix:
[0066]
[0067] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of ; [·] means rounding down; λ means the original eigenvalue; represents a combination of 2n elements in the set [1,N]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m-1 Indicates the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition α,β∈1,2,...,N; the number of real roots of the final characteristic polynomial is N, and the range of real roots is [-1,2]; q(λ) represents the transformed eigenvalue, q(λ)=λ 2 -λ.
[0068] S3. Solve the roots of the characteristic polynomial to obtain the parameters corresponding to all plasma modes in the material structure;
[0069] S4. Construct a multilayer plasma material model based on the parameters corresponding to all plasma modes in the material structure, and obtain the final plasma material through verification through the current prior information, and construct a multilayer structure.
[0070] The step S2, constructing the characteristic polynomial comprises:
[0071] A1. Perform preliminary material construction, including numbering each layer as the material construction changes layer by layer, and then calculating the regular material parameter ε of the jth layer material. j When j is an even number, ε j The value is ε0. When j is an odd number, ε j The value is ε1, ε0 is the parameter of the external background medium, ε1 is the material parameter to be designed; for the j-th layer characteristic parameter λ j Preprocessing is performed. When j is an odd number, λ j The same value is denoted as λ, where λ represents the original eigenvalue; when j is an even number, λ j The value is the difference between 1 and λ, that is:
[0072]
[0073] A2. When N≤4, the initial characteristic polynomial is directly calculated: including when N=4,
[0074]
[0075] Where |P4(λ)| represents the characteristic polynomial of the resonance matrix of the multilayer material with a 4-layer structure; λ represents the original eigenvalue; Represents a combination of 2 elements in the set [1,4]; k represents the value of the first element in the combination of 2 elements, and l represents the value of the second element in the combination of 2 elements; definition α,β∈1,2,...,N;
[0076] A3. Perform recursion to obtain the initial recursive formula, including:
[0077] For general layer recursion, the initial recursion formula is obtained:
[0078]
[0079] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of N-1 and λ N The calculation is done using the following formula: λ j represents the j-th layer feature parameter, λ1 represents the first layer feature parameter; λ2 represents the second layer feature parameter; λ N-1 represents the N-1th layer feature parameter; λ N represents the feature parameters of the Nth layer, j∈1,2,...,N; ε j represents the regular material parameter of the j-th layer material; represents the sub-formula formed by the 2nd to N-1th rows and the 2nd to N-1th columns in the resonance matrix; represents the sub-formula formed by the 2nd to the N-2th row and the 2nd to the N-2th column in the resonance matrix; represents the sub-formula formed by the 3rd to N-1th rows and the 3rd to N-1th columns in the resonance matrix; Represents the sub-formula consisting of the 3rd to N-2th rows and the 3rd to N-2th columns in the resonance matrix.
[0080] A4. Verify the initial recursive formula and obtain the final characteristic polynomial:
[0081] Assume that when N≤N0, the characteristic polynomials are correct, where N represents the number of layers of the multi-layer structure, N0 represents the set number of layers, and N0 is a positive integer greater than or equal to 4; use mathematical induction to construct the set characteristic polynomial and verify that N=N0+1 holds; conduct an in-depth analysis of the number of combinations:
[0082] definition:
[0083]
[0084]
[0085]
[0086]
[0087] Also define:
[0088]
[0089]
[0090]
[0091]
[0092] Wherein, D1 represents the first term of the set characteristic polynomial; D2 represents the second term of the set characteristic polynomial; D3 represents the third term of the set characteristic polynomial; D4 represents the fourth term of the set characteristic polynomial; d1 represents the first term coefficient of the set characteristic polynomial; d2 represents the second term coefficient of the set characteristic polynomial; d3 represents the third term coefficient of the set characteristic polynomial; d4 represents the fourth term coefficient of the set characteristic polynomial; λ represents the original eigenvalue; n represents the summation index; It represents a combination of 2n elements in the set [2,N-1]; It represents a combination of 2n elements in the set [2,N-2]; It represents a combination of 2n elements in the set [3, N-1]; represents a combination of 2n elements in the set [3,N-2]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m -1 means the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition α,β∈1,2,...,N; N0 represents the set number of layers; N represents the number of layers in the multilayer structure; q(λ) represents the transformed eigenvalue, q(λ)=λ 2 -λ; definition α,β∈1,2,...,N; then
[0093] |P N (λ)|=d1D1+d2D2+d3D3+d4D4
[0094] Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of ;
[0095] Introduce the following tags to define the first tag a k , second marker b k 、c k The third mark and Fourth mark:
[0096]
[0097]
[0098]
[0099]
[0100] Using the first marker a k , second marker b k 、c k The third mark and The fourth symbol represents the resonance matrix P of the multilayer material with N layers. N The characteristic polynomial of |P N (λ)|:
[0101]
[0102] According to the relationship between the number of combinations, we can get:
[0103]
[0104]
[0105]
[0106] The final characteristic polynomial is obtained through verification.
[0107] like Figure 3 Schematic diagram of characteristic modes and resonance effects of multilayer materials of embodiments of the present invention. The ordinate represents the norm of the polarization tensor (positively correlated with energy). The abscissa represents the wavelength in meters; using the Drew model, experiments were conducted on a 17-layer material structure, and the change in the value of the polarization tensor norm (positively correlated with energy) under wavelength changes was obtained, and the energy was greatly enhanced under the resonance mode. The means of assisting experiments with the theoretical analysis results obtained in this embodiment greatly reduced the cost of constructing multilayer materials.
Claims
1. A method for constructing a multilayer plasma mode, characterized in that The steps include: S1. Constructing a resonance matrix of a multilayer material, the order of which is the same as the number of material layers; including constructing a resonance matrix P of a multilayer material with an N-layer structure N : Among them, λ j represents the feature parameters of the jth layer; r j represents the radius of the material of the jth layer, j∈1,2,...,N; Among them, ε j represents the regular material parameter of the j-th layer material; S2. Perform preliminary material construction and use mathematical induction to calculate the final characteristic polynomial based on the resonance matrix; The final characteristic polynomial based on the resonance matrix: Among them, |P N (λ)| represents the characteristic polynomial of the resonance matrix PN of a multilayer material with an N-layer structure; [·] represents rounding down; λ represents the original eigenvalue; represents a combination of 2n elements in the set [1,N]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m-1 Indicates the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition The number of real roots of the final characteristic polynomial is N, and the range of real roots is [-1,2]; q(λ) represents the transformed eigenvalue, q(λ)=λ 2 -λ; S3. Solve the roots of the characteristic polynomial to obtain the parameters corresponding to all plasma modes in the material structure; S4. Construct a multilayer plasma material model based on the parameters corresponding to all plasma modes in the material structure, and obtain the final plasma material through verification through the current prior information, and construct a multilayer structure.
2. The method for constructing a multilayer plasma mode according to claim 1, characterized in that The step S2 comprises the following steps: A1. Perform preliminary material construction including the regular material parameter ε of the j-th layer material j Select the feature parameter λ of the jth layer j Perform pre-processing; A1. When N≤4, the initial characteristic polynomial is directly calculated; A2. Perform recursion to obtain the initial recursive formula; A3. Verify the initial recursive formula and obtain the final characteristic polynomial.
3. The method for constructing a multilayer plasma mode according to claim 2, characterized in that The step A1 includes numbering each layer and calculating the regular material parameter ε of the j-th layer material. j When j is an even number, ε j The value is ε0. When j is an odd number, ε j The value is ε1, ε0 is the parameter of the external background medium, ε1 is the material parameter to be designed; for the j-th layer characteristic parameter λ j Preprocessing is performed. When j is an odd number, λ j Take the value λ, λ represents the original eigenvalue; when j is an even number, λ j The value is the difference between 1 and λ.
4. The method for constructing a multilayer plasma mode according to claim 3, characterized in that The step A2 includes when N=4, Where |P4(λ)| represents the characteristic polynomial of the resonance matrix of the multilayer material with a 4-layer structure; λ represents the original eigenvalue; Represents a combination of 2 elements in the set [1,4]; k represents the value of the first element in the combination of 2 elements, and l represents the value of the second element in the combination of 2 elements; definition 5. The method for constructing a multilayer plasma mode according to claim 4, characterized in that The step A3 includes recursively performing a general number of layers to obtain an initial recursive formula: Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of N-1 and λ N The calculation is done using the following formula: λ j represents the j-th layer feature parameter, j∈1,2,...,N; ε j represents the regular material parameter of the j-th layer material; represents the sub-formula formed by the 2nd to N-1th rows and the 2nd to N-1th columns in the resonance matrix; represents the sub-formula formed by the 2nd to the N-2th row and the 2nd to the N-2th column in the resonance matrix; represents the sub-formula formed by the 3rd to N-1th rows and the 3rd to N-1th columns in the resonance matrix; Represents the sub-formula consisting of the 3rd to N-2th rows and the 3rd to N-2th columns in the resonance matrix.
6. The method for constructing a multilayer plasma mode according to claim 5, characterized in that The step A4 includes assuming that the characteristic polynomials are correct when N≤N0, wherein N represents the number of layers of the multilayer structure, N0 represents the set number of layers, and N0 is a positive integer greater than or equal to 4; using mathematical induction to construct the set characteristic polynomial to verify that N=N0+1 holds; and conducting an in-depth analysis of the number of combinations: definition: Also define: Wherein, D1 represents the first term of the set characteristic polynomial; D2 represents the second term of the set characteristic polynomial; D3 represents the third term of the set characteristic polynomial; D4 represents the fourth term of the set characteristic polynomial; d1 represents the first term coefficient of the set characteristic polynomial; d2 represents the second term coefficient of the set characteristic polynomial; d3 represents the third term coefficient of the set characteristic polynomial; d4 represents the fourth term coefficient of the set characteristic polynomial; λ represents the original eigenvalue; n represents the summation index; It represents a combination of 2n elements in the set [2,N-1]; It represents a combination of 2n elements in the set [2,N-2]; It represents a combination of 2n elements in the set [3, N-1]; represents a combination of 2n elements in the set [3,N-2]; m represents the index value of n elements, m∈1,2,...,n; i 2n is one of the combinations of 2n numbers; i 2m represents the number of the 2mth combination; i 2m-1 Indicates the 2m-1th combination number; is the number of combinations i 2n The symbol indicates that definition N0 represents the set number of layers; N represents the number of layers in the multilayer structure; q(λ) represents the transformed eigenvalue, q(λ) = λ 2 -λ; definition Then |P N (λ)|=d1D1+d2D2+d3D3+d4D4 Among them, |P N (λ)| represents the resonance matrix P of a multilayer material with N layers N The characteristic polynomial of ; Introduce the following tags to define the first tag a k , second marker b k 、c k The third mark and Fourth mark: Using the first marker a k , second marker b k 、c k The third mark and The fourth symbol represents the resonance matrix P of the multilayer material with N layers. N The characteristic polynomial of |P N (λ)|: According to the relationship between the number of combinations, we can get: The final characteristic polynomial is obtained through verification.