CN115935758B

CN115935758B - Raster modeling methods and storage media

Info

Publication number: CN115935758B
Application number: CN202211714116.2A
Authority: CN
Inventors: 郑宇腾; 蒲菠; 范峻
Original assignee: Ningbo Detu Technology Co ltd
Current assignee: Ningbo Detu Technology Co ltd
Priority date: 2022-12-27
Filing date: 2022-12-27
Publication date: 2026-01-06
Anticipated expiration: 2042-12-27
Also published as: CN115935758A

Abstract

This invention discloses a grating modeling method based on periodic finite element method, comprising: establishing an expression for the eigenvalue problem of β and solving for its eigenvectors; adding periodic boundary conditions to the expression for the eigenvalue problem of β, constructing a mapping matrix P through matrix multiplication, and using the mapping matrix P to obtain the expression for the eigenvalue problem under periodic boundary conditions; representing the electromagnetic field distribution on the air side of the grating-air interface as a superposition of plane waves; assuming the reflection coefficients of each mode are... Transmission coefficient is Solving the above equation using the tangential continuity condition equation of the field at the interface, we obtain the reflection coefficient as follows: Transmission coefficient is A transmission and reflection model for a grating structure is established. This invention is applicable to grating problems with arbitrary unit shapes, and can be used with various grating materials such as dielectrics and metals, meeting diverse application needs while ensuring the authenticity of the physical mechanism and the accuracy of the calculation.

Description

Grating modeling method and storage medium

Technical Field

The invention relates to the field of microelectronics, in particular to a grating modeling method based on periodic finite elements.

Background

The grating has wide application in the fields of micro-photoelectric systems, semiconductor lasers and semiconductor optical sensors, and is also one of the important points of design attention of micro-photoelectric devices. The grating processing cost in the micro-photoelectric system is high, and the requirement for research and development cost control is often difficult to meet through the design mode of experimental trial-and-error. Numerical simulation algorithms are an important tool for evaluating grating designs. However, the large scale difference of three dimensions of the grating structure results in a large grid amount, which is not bearable by the general numerical method. The calculated amount can be effectively reduced by introducing a 2.5-dimensional equivalent modeling mode, and common methods include an analysis method and a strict coupled wave method. How to accurately model gratings of arbitrary cell structures remains a great difficulty in the art.

Disclosure of Invention

In the summary section, a series of simplified form concepts are introduced that are all prior art simplifications in the section, which are described in further detail in the detailed description section. The summary of the invention is not intended to define the key features and essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

The invention aims to provide a grating modeling method which is based on finite elements, can be suitable for various grating materials such as media, metals and the like, and can accurately extract reflection projection parameters of gratings with arbitrary unit structures.

In order to solve the technical problems, the invention provides a modeling method based on a periodic finite element grating, which comprises the following steps:

Step one, solving electromagnetic field distribution of a two-dimensional grating structure by using a finite element method, and establishing an expression of eigenvalue problem about beta to solve eigenvectors of the electromagnetic field distribution, wherein each eigenvector corresponds to an electromagnetic propagation mode;

Step two, adding periodic boundary conditions to the expression of eigenvalue problem about beta, constructing a mapping matrix P by matrix product, using the mapping matrix P to convert the common two-dimensional finite element eigenvalue equation

A.x= -beta ² b.x, into eigenvalue problem expression under cycle boundary conditions;

P^*|·A·P·x＝-β²P^*·B·P·x,(3)

p is the conjugate transpose of the matrix P, the electromagnetic field mode distribution of the periodic structure is obtained, and the electromagnetic field at one side of the grating-air interface close to the grating is expressed as the weighted superposition of each mode;

step three, obtaining electromagnetic field distribution on one side of a grating-air interface close to air according to the Floquet theorem, wherein the electromagnetic field distribution is expressed as superposition of plane waves;

step four, assume that the reflection coefficient of each mode is The transmission coefficient isEstablishing the following equation by tangential continuous conditions of the field at the interface;

Wherein E= [ E _x,E_y]^T,H＝[H_x,H_y]^T ], inner product of E _x,E_y,H_x,H_y electric and magnetic field component and plane wave function in matrix

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]);

Step five, solving the equation to obtain the reflection coefficient asThe transmission coefficient isAnd establishing a transmission and reflection model of the grating structure.

Wherein the general two-dimensional finite element eigenvalue problem for β has the expression a·x= - β ² b·x (1).

Wherein the mapping matrix P is as follows;

Wherein, ψ _x is the phase difference between the x-direction periodic boundaries, ψ _y is the phase difference between the y-direction periodic boundaries, and the submatrices sequentially correspond to an inner edge, a left boundary edge, a right boundary edge, an upper boundary edge, a lower boundary edge, an inner node, a left boundary node, a right boundary node, an upper boundary node, a lower boundary node, an upper right vertex, an upper left vertex of a boundary, a lower right vertex of a boundary, and a lower left vertex of a boundary.

Wherein, the weighted superposition expression of each mode corresponding to the electric and magnetic field distribution E _II、H_II is as follows;

wherein electromagnetic field distributions E _I and H _I are expressed as superimposed expressions of plane waves as follows;

Wherein the plane wave factor is The three components of the k _mn wave number vector are k _mn＝(k_x,n,k_y,m,k_z,mn respectively), the specific expression is 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.

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

Including both non-transitory and non-transitory, removable and non-removable media, the information storage may be implemented by any method or technology. The information may be computer readable instructions, data structures, modules of a program, or other data. Examples of storage media for a computer 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 technology, compact disc read only memory (CD-ROM), digital Versatile Discs (DVD) or other optical storage, magnetic cassettes, magnetic tape disk storage or other magnetic storage devices, or any other non-transmission medium, which can be used to store information that can be accessed by a computing device. Computer readable media, as defined herein, does not include non-transitory computer readable media (transmission media), such as modulated data signals and carrier waves.

Compared with the prior art, the grating modeling method based on the periodic finite element has at least the following technical effects:

1. The grating modeling method based on the periodic finite element only carries out the discretization on the two-dimensional section of the grating and only carries out the discretization on one unit, so the unknown quantity is less and the calculation accuracy is high. Compared with the prior analysis method, the method breaks through the limitation condition on the geometric shape, and can be suitable for the grating problem of any unit shape.

2. The grating modeling method based on the periodic finite element can be suitable for various grating materials such as media, metals and the like, and meets diversified application requirements.

3. The grating modeling method based on the periodic finite element uses a full-field two-dimensional finite element method to perform modeling, and ensures the authenticity of a physical mechanism and the calculation accuracy.

Drawings

The accompanying drawings are intended to illustrate the general features of methods, structures and/or materials used in accordance with certain exemplary embodiments of the invention, and supplement the description in this specification. The drawings of the present invention, however, are schematic illustrations that are not to scale and, thus, may not be able to accurately reflect the precise structural or performance characteristics of any given embodiment, the present invention should not be construed as limiting or restricting the scope of the numerical values or attributes encompassed by the exemplary embodiments according to the present invention. The invention is described in further detail below with reference to the attached drawings and detailed description:

FIG. 1 is a schematic diagram of the effect of high contrast medium grating reflection and transmission coefficient verification.

FIG. 2 is a schematic diagram of the effect of verification of the computational complexity of a numerical pattern matching method.

Detailed Description

Other advantages and technical effects of the present invention will become more fully apparent to those skilled in the art from the following disclosure, which is a detailed description of the present invention given by way of specific examples. The invention may be practiced or carried out in different embodiments, and details in this description may be applied from different points of view, without departing from the general inventive concept. It should be noted that the following embodiments and features in the embodiments may be combined with each other without conflict. The following exemplary embodiments of the present invention may be embodied in many different forms and should not be construed as limited to the specific embodiments set forth herein. It should be appreciated that these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the technical solution of these exemplary embodiments to those skilled in the art.

The invention provides a grating modeling method based on finite elements, which comprises the following steps:

P^*|·A·P·x＝-β²P^*·B·P·x,(3)

According to the grating modeling method based on the periodic finite element, the electromagnetic field distribution of the two-dimensional periodic structure is solved, so that the matching condition on the boundary is obtained, and an equation is established according to the matching condition. The modeling method of the invention has no requirements on the thickness, the unit shape, the material quality and the like of the grating in the modeling process, so that the modeling method of the invention can extract reflection/transmission coefficients of the grating with various materials, unit structures and layer numbers, and has stronger universality in practical grating design application. The modeling method has strong universality on the model, and is suitable for complex grating models with arbitrary cross-section shapes, materials and layers. The modeling method can be combined with software, can effectively improve the modeling simulation capability of the complex grating, and is beneficial to evaluation and design of products in the fields of micro photoelectric systems, semiconductor lasers and semiconductor optical sensors.

In order to verify the accuracy of the periodic finite element-based grating modeling method, a verification example is selected as a model with a commercial software simulation result, and the two-dimensional high-contrast grating is subjected to simulation calculation through the method to solve the transmission coefficient and the reflection coefficient. The cell pitches in the x and y directions are Λx and Λy, respectively, and the layer thickness where the grating periodic structure is located is tg. The relative dielectric constant of the grating region was 10.0. Calculation the accuracy of the calculation of the method was first verified for a layer thickness tg of 0.5 μm. The solution can converge by using 30 modes for expansion. As can be seen from comparison with the full wave finite element method, the calculated reflection coefficient matches the transmission coefficient well.

The invention has small calculation amount and quick solving. Therefore, the invention has obvious advantages when optimizing the grating thickness parameter. In physical analysis, since the change of the grating thickness parameter does not affect the characteristic mode in the periodic structure layer, the transmission coefficient and the reflection coefficient under different grating thicknesses can be calculated only by solving the mode once. The calculation complexity of the invention is divided into two parts in verification, wherein the time complexity of a finite element matrix filling part is O (N), and the time complexity of a characteristic value part is O (N1.5).

Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

The present invention has been described in detail by way of specific embodiments and examples, but these should not be construed as limiting the invention. Many variations and modifications may be made by one skilled in the art without departing from the principles of the invention, which is also considered to be within the scope of the invention.

Claims

1. A grating modeling method, based on periodic finite element method, characterized by comprising the following steps:

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.

Step 2: Add periodic boundary conditions to the expression of the eigenvalue problem with respect to β. Construct a mapping matrix P through matrix multiplication. Use the mapping matrix P to transform the ordinary two-dimensional finite element eigenvalue equation A·x＝-β ² B·x into an expression of the 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, 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.

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.

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;

Where E = [E _x ,E _y ] ^T , H = [H _x ,H _y ] ^T , the inner product of the electric and magnetic field components of E _x ,E _y ,H _x ,H _y with the plane wave function in the matrix;

The submatrices in the matrix are K _y = diag([k _{y,0} ,…,k_y,m ]), K _x = diag([k _{x,0} ,…,k_x,n ]), and K _z = diag([k _{z,00} ,…,k_z,mn]);

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

2. The grating modeling method as described in claim 1, characterized in that: the mapping matrix P is as follows;

Where _Ψx is the phase difference between the periodic boundaries in the x-direction, and _Ψy is the phase difference between the periodic boundaries in the y-direction. The submatrices correspond in sequence 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 the boundary, upper left vertex of the boundary, lower right vertex of the boundary, and lower left vertex of the boundary.

3. The grating modeling method as described in claim 1, characterized in that: the weighted superposition expressions of the electric and magnetic field distributions E _II and H _II corresponding to each mode are as follows;

4. The grating modeling method as described in claim 1, characterized in that: the electromagnetic field distributions E _I and H_I are expressed as the superposition expression of plane waves as follows;

Wherein, the plane wave factor is The three components of the k _mn wavenumber vector are k _mn = (k _x,n ,ky _,m ,k _z,mn ), specifically expressed as k _x,n = 2nπ/Λ _x , k _y,m = 2nπ/Λ _y , _ex , _y , _hx , and _hy are all coefficients to be determined.

5. A computer-readable storage medium, characterized in that: it internally stores a program, which, when executed, implements the steps of the raster modeling method according to any one of claims 1-4.