Grating modeling method and storage medium

Abstract

The application discloses a grating modeling method based on a periodic finite element, which comprises the following steps: establishing an eigenvalue problem expression about beta, solving a characteristic vector thereof; adding a periodic boundary condition to the eigenvalue problem expression about beta, constructing a mapping matrix P through matrix multiplication, and obtaining an eigenvalue problem expression under the periodic boundary condition by using the mapping matrix P; obtaining an electromagnetic field distribution expression of a grating-air interface close to the air side and expressed as a superposition of plane waves; assuming that reflection coefficients of each mode are and transmission coefficients of each mode are by using a tangential continuity condition equation of a field on the interface; solving the above equation to obtain the reflection coefficients and the transmission coefficients; and establishing a transmission and reflection model of the grating structure. The application can be applied to grating problems of any unit shape, can be applied to various grating materials such as media and metals, meets diversified application requirements, and can guarantee the authenticity of a physical mechanism and the accuracy of calculation.

Description

Technical Field

[0001] This invention relates to the field of microelectronics, and in particular to a grating modeling method based on periodic finite element method. Background Technology

[0002] Gratings have wide applications in micro-optoelectronic systems, semiconductor lasers, and semiconductor optical sensors, and are a key focus in the design of micro-optoelectronic devices. The fabrication cost of gratings in micro-optoelectronic systems is high, and experimental trial-and-error design methods often fail to meet the demands of cost control in research and development. Numerical simulation algorithms are important tools for evaluating grating design schemes. However, the significant differences in the proportions of the three dimensions of the grating structure result in a huge mesh size, which is difficult for general numerical methods to handle. Introducing 2.5D equivalent modeling can effectively reduce the computational load; commonly used methods include analytical methods and rigorous coupled-wave methods. However, accurately modeling gratings with arbitrary unit cell structures remains a major challenge in this field. Summary of the Invention

[0003] The summary of this invention introduces a series of simplified concepts, all of which are simplifications of existing technologies in the field, and will be further explained in detail in the detailed description section. This summary is not intended to limit the key features and essential technical features of the claimed technical solution, nor is it intended to determine the scope of protection of the claimed technical solution.

[0004] The technical problem to be solved by the present invention is to provide a grating modeling method based on finite element method, which is applicable to various grating materials such as media and metals, and can accurately extract the reflection and projection parameters of gratings with arbitrary unit structure.

[0005] To address the aforementioned technical problems, this invention provides a periodic finite element grating modeling method, comprising the following steps:

[0006] Step 1: Use the finite element method to solve the electromagnetic field distribution of the two-dimensional grating structure, establish an expression for the eigenvalue problem of β, and solve for its eigenvectors. Each eigenvector corresponds to an electromagnetic propagation mode.

[0007] Step 2: Add periodic boundary conditions to the expression for the eigenvalue problem of β, construct a mapping matrix P through matrix multiplication, and use the mapping matrix P to transform the ordinary two-dimensional finite element eigenvalue equation.

[0008] A·x=-β 2 B·x is transformed into an expression for the eigenvalue problem under periodic boundary conditions;

[0009] P * |·A·P·x=-β 2 P* ·B·P·x,(3)

[0010] P* is the conjugate transpose of matrix P, which yields the electromagnetic field mode distribution of the periodic structure. The electromagnetic field on the side of the grating-air interface closer to the grating is represented as a weighted superposition of the various modes.

[0011] Step 3: According to Floquet's theorem, the electromagnetic field distribution on the air side of the grating-air interface can be represented as the superposition of plane waves.

[0012] Step 4: Assume the reflection coefficients of each mode are... Transmission coefficient is By applying the tangential continuity condition of the field at the interface, the following equations are established;

[0013]

[0014] Where, E = [E x E y ] T H = [H x H y ] T In the matrix E x E y H x H y The inner product of electric and magnetic field components with the plane wave function

[0015] Each submatrix in the matrix is ​​K y =diag([k y,0 ,...,k y,m ]), K x =diag([k x,0 ,...,k x,n ]), K z =diag([k z,00 ,...,k z,mn ]);

[0016] Step 5: Solve the above equation to obtain the reflection coefficient. Transmission coefficient is Establish transmission and reflection models for the grating structure.

[0017] The expression for the ordinary two-dimensional finite element eigenvalue problem with respect to β is A·x=-β. 2 B·x(1).

[0018] The mapping matrix P is as follows;

[0019]

[0020] Among them, Ψ x Ψ represents the phase difference between the periodic boundaries in the x-direction. y The phase difference between the periodic boundaries in the y-direction is represented by the submatrices corresponding to: internal edge, left boundary edge, right boundary edge, upper boundary edge, lower boundary edge, internal node, left boundary node, right boundary node, upper boundary node, lower boundary node, upper right vertex of boundary, upper left vertex of boundary, lower right vertex of boundary, and lower left vertex of boundary.

[0021] Among them, the electric and magnetic field distributions E corresponding to each mode II H II The weighted summation expression is as follows;

[0022]

[0023]

[0024] Among them, the electromagnetic field distribution E I and H I The superposition expression for plane waves is as follows;

[0025]

[0026] Wherein, the plane wave factor is k mn The three components of the wavenumber vector are k mn =(k x,n ,k y,m ,k z,mn The specific expression is k. x,n =2nπ / Λ x k y,m =2nπ / Λ y ,

[0027] e x e y h x h y All of these are coefficients to be determined.

[0028] The present invention provides a computer-readable storage medium having a program stored therein, which, when executed, implements the steps in any of the above-described raster modeling methods.

[0029] The computer-readable medium includes both permanent and non-permanent, removable and non-removable media, which can store information by any method or technology. Information can be computer-readable instructions, data structures, program modules, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include non-transitory computer-readable media, such as modulated data signals and carrier waves.

[0030] The grating modeling method based on periodic finite element method of the present invention has at least the following technical advantages over the prior art:

[0031] 1. The grating modeling method based on periodic finite element method of the present invention discretizes only the two-dimensional cross-section of the grating and only one element, thus resulting in fewer unknowns and higher computational accuracy. Compared with previous analytical methods, it breaks through the limitation of geometric shape and can be applied to grating problems with arbitrary element shapes.

[0032] 2. The grating modeling method based on periodic finite element method of the present invention can be applied to various grating materials such as media and metals, and can meet diverse application needs.

[0033] 3. The grating modeling method based on periodic finite element method of the present invention uses the full-field two-dimensional finite element method for modeling, which ensures the authenticity of the physical mechanism and the accuracy of the calculation. Attached Figure Description

[0034] The accompanying drawings are intended to illustrate the general characteristics of the methods, structures, and / or materials used in specific exemplary embodiments of the invention, supplementing the description in the specification. However, the drawings are schematic diagrams not drawn to scale and may not accurately reflect the precise structural or performance characteristics of any of the given embodiments. The drawings should not be construed as limiting or restricting the range of numerical values ​​or properties covered by exemplary embodiments of the invention. The invention will now be described in further detail with reference to the accompanying drawings and specific embodiments:

[0035] Figure 1 This is a schematic diagram illustrating the verification effect of the reflection and transmission coefficients of a high-contrast dielectric grating.

[0036] Figure 2 This diagram illustrates the computational complexity verification effect of the numerical pattern matching method. Detailed Implementation

[0037] The following specific embodiments illustrate the implementation of the present invention. Those skilled in the art can fully understand other advantages and technical effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through different specific embodiments, and the details in this specification can also be applied based on different viewpoints, with various modifications or changes made without departing from the overall design concept of the invention. It should be noted that, unless otherwise specified, the following embodiments and features can be combined with each other. The following exemplary embodiments of the present invention can be implemented in many different forms and should not be construed as being limited to the specific embodiments set forth herein. It should be understood that these embodiments are provided to make the disclosure of the present invention thorough and complete, and to fully convey the technical solutions of these exemplary embodiments to those skilled in the art.

[0038] This invention provides a grating modeling method based on the finite element method, comprising the following steps:

[0039] Step 1: Use the finite element method to solve the electromagnetic field distribution of the two-dimensional grating structure, establish an expression for the eigenvalue problem of β, and solve for its eigenvectors. Each eigenvector corresponds to an electromagnetic propagation mode.

[0040] Step 2: Add periodic boundary conditions to the expression for the eigenvalue problem of β, construct a mapping matrix P through matrix multiplication, and use the mapping matrix P to transform the ordinary two-dimensional finite element eigenvalue equation.

[0041] A·x=-β 2 B·x is transformed into an expression for the eigenvalue problem under periodic boundary conditions;

[0042] P * |·A·P·x=-β 2 P * ·B·P·x,(3)

[0043] P* is the conjugate transpose of matrix P, which yields the electromagnetic field mode distribution of the periodic structure. The electromagnetic field on the side of the grating-air interface closer to the grating is represented as a weighted superposition of the various modes.

[0044] Step 3: According to Floquet's theorem, the electromagnetic field distribution on the air side of the grating-air interface can be represented as the superposition of plane waves.

[0045] Step 4: Assume the reflection coefficients of each mode are... Transmission coefficient is By applying the tangential continuity condition of the field at the interface, the following equations are established;

[0046]

[0047] Where, E = [E x E y ] T H = [H x H y ] T In the matrix E x E y H x H y The inner product of electric and magnetic field components with the plane wave function

[0048] Each submatrix in the matrix is ​​K y =diag([k y,0 ,...,k y,m ]), K x =diag([k x,0 ,...,k x,n ]), K z =diag([k z,00 ,...,k z,mn ]);

[0049] Step 5: Solve the above equation to obtain the reflection coefficient. Transmission coefficient is Establish transmission and reflection models for the grating structure.

[0050] The grating modeling method based on periodic finite element method of this invention obtains the matching conditions on the boundary by solving the electromagnetic field distribution of a two-dimensional periodic structure, and establishes equations accordingly. Since there are no requirements for setting the thickness, unit shape, or material of the grating during the modeling process, this method can extract the reflection / transmission coefficients of gratings with various materials, unit structures, and number of layers, demonstrating strong universality in practical grating design applications. Due to its strong modeling universality, this method is applicable to complex grating models with arbitrary cross-sectional shapes, materials, and number of layers. This modeling method can be combined with software to effectively improve the modeling and simulation capabilities of complex gratings, which will contribute to the evaluation and design of products in the fields of micro-optoelectronic systems, semiconductor lasers, and semiconductor optical sensors.

[0051] To verify the accuracy of the grating modeling method based on periodic finite element method of this invention, a model with simulation results from commercial software was selected as the verification example. First, the two-dimensional high-contrast grating was simulated using this invention to solve for its transmission and reflection coefficients. The element spacing along the x and y directions are Λx and Λy, respectively, and the layer thickness of the periodic grating structure is tg. The relative permittivity of the grating region is 10.0. The calculation first verified the accuracy of the method when the layer thickness tg is 0.5 μm. The solution converged after using 30 modes. Comparison with the full-wave finite element method shows that the calculated reflection and transmission coefficients agree well.

[0052] This invention requires relatively little computation and solves quickly. Therefore, it has a significant advantage when optimizing grating thickness parameters. Physically, since changing the grating thickness parameter does not affect the eigenmodes in the periodic structure layer, only one mode solution is needed to calculate the transmission and reflection coefficients for different grating thicknesses. The computational complexity of this invention is verified to be divided into two parts: the time complexity of the finite element matrix filling part is O(N), and the time complexity of the eigenvalue part is O(N^1.5).

[0053] Unless otherwise defined, all terms used herein (including technical and scientific terms) shall have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. It will also be understood that, unless expressly defined herein, terms such as those defined in a general dictionary shall be interpreted as having the meaning consistent with their meaning in the relevant field context, and not as having an idealized or overly formal meaning.

[0054] The present invention has been described in detail above through specific embodiments and examples, but these are not intended to limit the invention. Many modifications and improvements can be made by those skilled in the art without departing from the principles of the invention, and these should also be considered within the scope of protection of the present invention.

Claims

1. A grating modeling method based on periodic finite elements, characterized in that, The method comprises the following steps: Step one, using finite element method to solve electromagnetic field distribution of two-dimensional grating structure, establishing expression of eigenvalue problem about β to solve its eigenvectors, each Eigenvector corresponds to a mode of electromagnetic propagation; Step two, add periodic boundary conditions to the expression of eigenvalue problem about β, construct a mapping matrix P through matrix multiplication, use the mapping matrix P to convert the ordinary two-dimensional finite element eigenvalue equation A·x = -β 2 B·x into the expression of eigenvalue problem under periodic boundary conditions; P * |·A·P·x=-β 2 P * ·B·P·x (3) P* is the conjugate transpose of matrix P, obtaining electromagnetic field mode distribution of periodic structure, electromagnetic field of grating-air interface close to grating side is expressed as weighted superposition of each mode; Step three, according to Floquet theorem, electromagnetic field distribution of grating-air interface close to air side is expressed as superposition of plane waves; Step four, assuming the reflection coefficient of each mode is The transmission coefficient is By the tangential continuity condition of the field on the interface, the following equation is established; where E = [E x ,E y ] T , H = [H x , H y ] T , the inner product of the electric, magnetic field components and the plane wave function in the matrix is x , y , x , y Each sub-matrix in the matrix is K y = diag([k y,0 ,…,k y,m ]), K x = diag([k x,0 ,…,k x,n ]), K z = diag([k z,00 ,…,k z,mn ]) ; Step five, solve the above equation, get the reflection coefficient is The transmission coefficient is Establish the transmission and reflection model of grating structure.

2. The grating modeling method of claim 1, wherein: The mapping matrix P is as follows: wherein Ψ x is the phase difference between the x-direction periodic boundaries, and Ψ y is the phase difference between the y-direction periodic boundaries, the sub-matrices corresponding in turn to: internal edges, left-side boundary edges, right-side boundary edges, upper-side boundary edges, lower-side boundary edges, internal junctions, left-side boundary junctions, right-side boundary junctions, upper-side boundary junctions, lower-side boundary junctions, boundary top-right vertices, boundary top-left vertices, boundary bottom-right vertices, boundary bottom-left vertices.

3. The grating modeling method of claim 1, wherein: Each mode corresponds to a weighted superposition of electric and magnetic field distributions E II , H II The weighted superposition expression is as follows; 4. The grating modeling method of claim 1, wherein: Electromagnetic field distribution E I and H I The representation as a superposition of plane waves is as follows; wherein the plane wave factor is k mn The three components of the wave vector are respectively k mn = (k x,n , k y,m , k z,mn ), and the specific expressions are k x,n = 2nπ / Λ x , k y,m = 2nπ / Λ y , e x , e y , h x , h y are all coefficients to be solved.

5. A computer-readable storage medium, characterized in that: The computer has a program stored therein, and when the program is executed, the steps of the grating modeling method in any one of claims 1-4 are implemented.

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