A dual-beam-based finite element optical simulation method and system, and an electronic device

Abstract

The application provides a kind of based on double-beam finite element optical simulation method, system and electronic equipment, comprising: determining multiscale optical model, the propagation region of light is divided into particle characteristic region and wave characteristic region according to the propagation characteristic of light beam;The wave characteristic region includes interference region and / or diffraction region;There are two stable propagation directions of light beam in particle characteristic region;According to the principle that tangential electric field and tangential magnetic field on adjacent region boundary surface are continuously distributed, the boundary condition corresponding to the interface between two different regions is determined;For wave characteristic region, polynomial basis function is used, and for particle characteristic region, double-beam basis function is used;Combined with boundary condition, wave characteristic region is divided into dense grid, and particle characteristic region is divided into sparse grid;The light field distribution of multiscale optical model is obtained by using finite element method simulation. Due to the introduction of double-beam basis function, the number of grids is further reduced, the calculation degree of freedom is reduced, and the solving efficiency of light field distribution is greatly improved.

Description

Technical Field

[0001] This invention belongs to the field of computational electromagnetics, and more specifically, relates to a finite element optical simulation method, system, and electronic device based on dual-beams. Background Technology

[0002] The Finite Element Method (FEM) is an important method in computational mechanics. It emerged in the late 1950s and early 1960s as an interdisciplinary science that integrates applied mathematics, modern mechanics, and computer science. Initially applied in engineering science and technology, the FEM was used to simulate and solve physical problems in engineering mechanics, thermodynamics, and electromagnetism. For problems that were previously unsolvable using analytical methods, and for complex problems with irregular boundary conditions and structural shapes, the FEM is an effective analytical method. The significant mathematical breakthroughs of R. Courant in 1943, and the major breakthroughs in engineering applications by Ogilvy, marked the birth of the FEM. It wasn't until 1960 that Clouph first named the method the finite element method. Since the concept of the finite element was proposed, its theory and applications have developed rapidly.

[0003] In modern optical system modeling and simulation, it is often necessary to consider both the wave and particle properties of light simultaneously, which falls under the category of multi-scale optical analysis. Since the wave nature of light is based on the theory of physical optics, while the particle nature of light is based on the theory of geometrical optics, modeling such multi-scale problems, including the coupled analysis of near-field and far-field effects, has always been a challenging problem in modern optical simulation.

[0004] For multi-scale optical model simulations, the traditional polynomial finite element method generates massive meshes and huge sparse matrices, leading to problems such as insufficient computer memory or excessively long computation times. Simulations using geometric optics theory fail to capture diffraction and interference effects of the device. Although introducing the slowly varying envelope approximation technique into the finite element method can reduce the degrees of freedom in computation, this method can only simulate simple optical fields; it cannot accurately simulate the optical field distribution when the field is complex. Therefore, a technique capable of solving multi-scale finite element problems is currently needed. Summary of the Invention

[0005] To address the shortcomings of existing technologies, the present invention aims to provide a dual-beam finite element optical simulation method, system, and electronic device. This invention addresses the problems of insufficient computer memory or excessive computation time when using polynomial finite element simulation for multi-scale optical models, and the inability to accurately simulate complex optical fields or the failure to simulate the optical field distribution when using slowly varying envelope approximation finite element simulation.

[0006] To achieve the above objectives, in a first aspect, the present invention provides a finite element optical simulation method based on dual-beams, comprising the following steps:

[0007] A multi-scale optical model is determined, and the light propagation region is divided into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the light beam; the wave characteristic region includes an interference region and / or a diffraction region; the light beam has two stable propagation directions within the particle characteristic region;

[0008] Based on the principle of continuous distribution of tangential electric and magnetic fields at the interface between adjacent regions, the boundary conditions corresponding to the interface between two different regions are determined.

[0009] A polynomial basis function is used for the wave characteristic region and a dual-beam basis function is used for the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The light field distribution of the multi-scale optical model is obtained by finite element method simulation.

[0010] In one possible example, during the simulation using the finite element method, it is necessary to interpolate the polynomial basis functions and the dual-beam basis functions to obtain the corresponding weak finite element forms; wherein, the interpolation uses a second-order interpolation function.

[0011] In one possible example, a gradient mesh is used for transition at the interface between two regions with different characteristics; the gradient mesh is such that, in a preset region where the particle characteristic region gradually approaches the interface with the wave characteristic region, the mesh density changes from sparse to dense as it approaches the interface; and in a preset region where the particle characteristic region gradually moves away from the interface, the mesh density changes from dense to sparse as it moves away from the interface.

[0012] In one possible example, the boundary condition is:

[0013]

[0014]

[0015] Where, subscript a indicates the region using dual-beam basis functions, and subscript b indicates the region using polynomial basis functions; n a Let n be the normal vector of the interface from region a to region b. b m is the normal vector of the interface from region b to region a; a and m b The matching impedance is the incident plane wave at the interface between the two regions. ∈ ra and ∈ rb The relative permittivity, μ, represents the permittivity of regions a and b, respectively. ra and μrb , respectively, are the relative permeabilities of region a and region b; a Let E be the electric field intensity at the interface of region a. b E represents the electric field strength at the interface of region b. a Using dual-beam basis functions Expand i is the number of beam basis functions, r is the position in space, and k a1 Let k be the direction of the first wave vector in region a. a2 Let e ​​be the direction of the second wave vector in region a. a1i The coefficients of the basis functions corresponding to the first wave vector, e a2i E represents the coefficients of the basis functions corresponding to the second wave vector. b Using polynomial basis functions E bi Expand, E b =∑ i e bi E bi e bi The coefficients of the polynomial basis functions; is the free space wavenumber.

[0016] In one possible example, when using the finite element method for simulation, substituting the boundary conditions into the finite element weak form boundary integral term of the basis functions yields the matrix equation of the multi-scale optical model as follows:

[0017]

[0018] Among them, e Va1 e represents the basis function coefficients corresponding to the first wave vector of a node within region a. Sa1 The basis function coefficients corresponding to the first wave vector of the node on the interface of region a; e Va2 e represents the basis function coefficients corresponding to the second wave vector of a node within region a. Sa2 The basis function coefficients corresponding to the second wave vector of the node at the interface between regions a; e Vb e represents the basis function coefficients of the nodes inside region b. Sb Denotes the basis function coefficients of nodes on the interface of region b; b a1 b a2 and b b Let the initial right-hand vector be ;

[0019]

[0020]

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] Among them, W a1j The test function for the first wave vector of the dual-beam basis functions is taken as... W a2j The test function for the second wave vector of the dual-beam basis functions is taken as follows: W bj Let ∑ be the test function of the polynomial basis functions. j E bj .

[0027] Secondly, the present invention provides a dual-beam finite element optical simulation system, comprising:

[0028] The model determination module is used to determine a multi-scale optical model, dividing the light propagation region into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the light beam; the wave characteristic region includes an interference region and / or a diffraction region; the light beam has two stable propagation directions within the particle characteristic region;

[0029] The boundary condition determination module is used to determine the boundary conditions corresponding to the interface between two different regions according to the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions.

[0030] The light field distribution simulation module is used to apply a polynomial basis function to the wave characteristic region and a dual-beam basis function to the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The light field distribution of the multi-scale optical model is obtained by simulating using the finite element method.

[0031] In one possible example, the optical field distribution simulation module needs to interpolate the polynomial basis functions and dual-beam basis functions to obtain the corresponding weak finite element forms during the simulation using the finite element method; wherein, the interpolation uses a second-order interpolation function.

[0032] In one possible example, the light field distribution simulation module uses a gradient mesh for transition at the interface between two regions with different characteristics. The gradient mesh is as follows: in a preset region where the particle characteristic region gradually approaches the interface with the wave characteristic region, the mesh density changes from sparse to dense as it approaches the interface; in a preset region where the particle characteristic region gradually moves away from the interface, the mesh density changes from dense to sparse as it moves away from the interface.

[0033] In one possible example, the boundary condition determined by the boundary condition determination module is:

[0034]

[0035]

[0036] Where, subscript a indicates the region using dual-beam basis functions, and subscript b indicates the region using polynomial basis functions; n a Let n be the normal vector of the interface from region a to region b. b m is the normal vector of the interface from region b to region a; a and m b The matching impedance is the incident plane wave at the interface between the two regions. ∈ ra and ∈ rb The relative permittivity, μ, represents the permittivity of regions a and b, respectively. ra and μ rb , respectively, are the relative permeabilities of region a and region b; a Let E be the electric field intensity at the interface of region a. b E represents the electric field strength at the interface of region b. a Using dual-beam basis functions Expand i is the number of beam basis functions, r is the position in space, and k a1 Let k be the direction of the first wave vector in region a. a2 Let e ​​be the direction of the second wave vector in region a. a1i The coefficients of the basis functions corresponding to the first wave vector, e a2i E represents the coefficients of the basis functions corresponding to the second wave vector. b Using polynomial basis functions E bi Expand, E b =∑ i e bi E bi e bi The coefficients of the polynomial basis functions; is the free space wavenumber.

[0037] Thirdly, the present invention provides an electronic device comprising: at least one memory for storing a program; and at least one processor for executing the program stored in the memory, wherein when the program stored in the memory is executed, the processor is configured to perform the method described in the first aspect or any possible example of the first aspect.

[0038] Fourthly, the present invention provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to perform the methods described in the first aspect or any possible example of the first aspect.

[0039] Fifthly, the present invention provides a computer program product that, when run on a processor, causes the processor to perform the method described in the first aspect or any possible example of the first aspect.

[0040] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art:

[0041] This invention provides a dual-beam finite element optical simulation method, system, and electronic device. For optical structures with multiple beam propagation directions in space, it adopts a divide-and-conquer approach, combining beam envelope technology and finite element method to propose a dual-beam finite element technique. The computational domain is divided using domain decomposition technology, with different basis functions applied to different regions. Based on the beam propagation characteristics, the light propagation region is divided into a particle characteristic region and a wave characteristic region. The wave characteristic region includes interference and / or diffraction regions; the beam has two stable propagation directions within the particle characteristic region. Boundary conditions are determined at the interface between two different regions based on the principle of continuous distribution of tangential electric and magnetic fields at the interface between adjacent regions. Polynomial basis functions are applied to the wave characteristic region, and dual-beam basis functions are applied to the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense mesh, and the particle characteristic region into a sparse mesh. The finite element method is then used to simulate the light field distribution of the multi-scale optical model. In this invention, the introduction of dual-beam basis functions reduces the number of grids, lowers the computational degrees of freedom, and greatly improves the solution efficiency of the light field distribution. Furthermore, the use of polynomial basis functions in the wave characteristic region ensures the accuracy of the light field simulation. Attached Figure Description

[0042] Figure 1 This is a flowchart of the finite element optical simulation method based on dual beams provided in an embodiment of the present invention;

[0043] Figure 2 This is a schematic diagram of a dual-mirror model provided in an embodiment of the present invention;

[0044] Figure 3 This is a schematic diagram of the simulation area division of the dual-mirror system provided in an embodiment of the present invention;

[0045] Figure 4 These are comparison diagrams of the simulation calculation results of the dual-mirror provided in the embodiments of the present invention; (a) is the electric field mode calculated by the present invention; (b) is the electric field mode calculated using the existing commercial finite element software COMSOL.

[0046] Figure 5 These are comparison diagrams of phase difference between the finite element method and the numerical solution using interpolation functions of different orders, provided in embodiments of the present invention; (a) using a first-order interpolation function; (b) using a second-order interpolation function;

[0047] Figure 6 This is a schematic diagram of a gradient mesh provided in an embodiment of the present invention;

[0048] Figure 7 These are comparison figures of finite element method and numerical solution simulation results using interpolation functions of different orders provided in this embodiment of the invention; (a) using a first-order interpolation function; (b) using a second-order interpolation function.

[0049] Figure 8 This is a diagram of the architecture of a dual-beam finite element optical simulation system provided in an embodiment of the present invention. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0051] In this article, the term "and / or" describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, A and B existing simultaneously, or B existing alone. The symbol " / " in this article indicates that the related objects are in an "or" relationship; for example, A / B means A or B.

[0052] The terms "first" and "second," etc., used in the specification and claims herein are used to distinguish different objects, not to describe a specific order of the objects. For example, "first wave vector" and "second wave vector" are used to distinguish different wave vectors, not to describe a specific order of the wave vectors.

[0053] In embodiments of the present invention, the terms "exemplary" or "for example" are used to indicate that something is an example, illustration, or description. Any embodiment or design described as "exemplary" or "for example" in embodiments of the present invention should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of the terms "exemplary" or "for example" is intended to present the relevant concepts in a specific manner.

[0054] First, the technical terms involved in the embodiments of the present invention will be introduced.

[0055] (1) Dual-beam basis functions

[0056] This invention assumes that there are two directions of light wave propagation in a region, and therefore uses dual-beam basis functions to approximate the slowly varying electromagnetic field. That is... In the two-dimensional case, Substituting into the vector wave equation:

[0057]

[0058] At this point, we can obtain the two sets of weakly form formulas as follows:

[0059]

[0060]

[0061] For a dual-beam basis function, each beam has a set of coefficients, and the relationship between the unknowns corresponding to each beam is obtained by rewriting the boundary conditions, thereby obtaining the coupling terms in the boundary integral terms.

[0062] (2) Polynomial basis functions

[0063] This invention assumes that the optical field is complex in a region, containing interference or diffraction effects, and therefore uses polynomial basis functions to interpolate the electric field. That is, E(x,y,z)=∑ i e i E i (x,y,z).

[0064] In the two-dimensional case, E(x,y,z)=∑ i e i E i (x, y). Substitute into the vector wave equation:

[0065]

[0066] The weak form of the finite element method can be written as:

[0067]

[0068] Next, the technical solutions provided in the embodiments of the present invention will be introduced.

[0069] This invention proposes a dual-beam finite element method (DFBM) to solve the simulation problem of mesoscopic optical structures with two beam propagation directions. Mesoscopic dimensions refer to sizes between the microscopic and macroscopic worlds. Using the visible light wavelength as a reference wavelength, the characteristic dimensions of the mesoscopic structures range from micrometers to millimeters. A novel and efficient algorithm is provided for simulating models such as plane mirrors, beam splitters, and Fabry-Perot resonators. Drawing inspiration from domain decomposition, the DFBM combines beam envelope technology with the beamforming method, using polynomial basis functions and dual-beam basis functions as interpolation functions. This allows for the use of appropriate basis functions for interpolation in different regions.

[0070] It should be noted that when using polynomial basis functions, i.e., traditional finite element method (simulation software COMSOL), simulations can be performed at the microscopic scale. However, due to the relatively dense mesh, the computational load increases with the size of the beam propagation region, resulting in a long computation time. When the beam propagation region size increases to the mesoscopic or macroscopic level, traditional finite element calculations become too time-consuming, making it impossible to obtain simulation results for the light field.

[0071] The polynomial-dual-beam combined optical field simulation method provided by this invention can significantly reduce computational load and improve the speed of microscale simulation when particle-characteristic regions exist in the beam propagation area. It also overcomes the challenge of traditional finite element methods being unable to simulate mesoscale phenomena. For macroscale simulations, when the accuracy of traditional beam basis function simulations is low, using polynomial basis functions for optical field simulation in wave-characteristic regions will greatly improve the accuracy of macroscale optical field simulations.

[0072] This invention divides the overall computational domain into a series of sub-regions according to different field distribution patterns, and employs appropriate mesh sizes and calculation methods in each sub-region. Specifically, a band is used in the computation of field regions with relatively definite propagation directions. Subsequently, first-order impedance-type hybrid boundary conditions are used to couple different regions together on adjacent edges, obtaining the system matrix of the entire system. The electric field intensity of the region is then solved. Furthermore, this invention discovers that finite element methods suffer from dispersion errors when calculating large-scale problems, meaning the solved electric field phase drifts, and the dispersion error increases with the increase of the simulation domain. Therefore, the polynomial basis functions of the dual-beam finite element method must be at least second-order.

[0073] It should be noted that the polynomial-dual-beam finite element simulation method is referred to as the "dual-beam finite element" simulation method in this invention.

[0074] To verify the correctness of this invention, the algorithm was written in Matlab for a dual-mirror model and compared with the results in the numerical simulation software COMSOL. The results show that this invention can achieve the same calculation results as COMSOL with a smaller number of grid cells, significantly reducing computation time and memory usage, thus demonstrating the effectiveness and superiority of this invention.

[0075] Furthermore, it can be demonstrated that dual-beam finite element method (DFEM) can be extended to multi-beam finite element method (FEM) to address more complex optical system structures. Currently, optical finite element method (EFEM) lacks a technique capable of solving multi-scale optical simulation problems. Therefore, the algorithm of this invention fills this gap, significantly reducing the computation time and memory usage of optical systems composed of devices with existing large-scale geometrical optical structures and small-scale wave optical structures, thus making a certain contribution to the development of photonics.

[0076] Specifically, beam envelope technology is essentially a multi-scale basis function. In addition to polynomial basis functions, this invention also uses beam basis functions with phase factor terms for interpolation. Beam basis functions are divided into two categories: single-beam, where only one wave vector direction exists in a region, and dual-beam, where two wave vector directions can exist in a region. Dual-beam basis functions are suitable for optical model structures where the beam direction changes during propagation or where there are multiple propagation directions, such as catadioptric phenomena.

[0077] This invention assumes that light waves propagate in two directions within a region and uses a dual-beam basis function to approximate the slowly varying electromagnetic field. Because this invention adds a phase factor term to the dual-beam basis function, making the basis function a form of the electric field envelope, the number of meshes required for computation can be significantly reduced. Traditional finite element methods require very dense meshes to obtain accurate results, while using dual-beam basis functions, since the phase envelope is already included in the basis function, only a sparser mesh is needed to obtain accurate results. Therefore, employing the multi-scale basis function of this invention can effectively reduce the degrees of freedom in computation, thereby reducing computation time and memory usage.

[0078] Understandably, this invention studies a polynomial-dual-beam region coupling technique. This invention decomposes the entire computational domain, using beam basis functions for computation in regions with stable propagation directions, and using polynomial basis functions for computation in regions with complex optical fields, such as interference and diffraction regions. This invention can handle computations in regions containing two stable propagation directions and has scalability to handle regions containing multiple propagation directions. Similarly, this invention can effectively reduce the number of grid cells required for computation, significantly reducing computation time and memory requirements while maintaining computational accuracy.

[0079] The common boundary between different regions uses a first-order impedance-type mixed boundary condition, applied at the interface between regions a and b, as shown in the following equation:

[0080]

[0081]

[0082] Where, subscript a indicates the region using dual-beam basis functions, and subscript b indicates the region using polynomial basis functions; n a Let n be the normal vector of the interface from region a to region b. b m is the normal vector of the interface from region b to region a; a and m b The matching impedance is the incident plane wave at the interface between the two regions. ∈ ra and ∈ rb The relative permittivity, μ, represents the permittivity of regions a and b, respectively. ra and μ rb , respectively, are the relative permeabilities of region a and region b; a Let E be the electric field intensity at the interface of region a. b E represents the electric field strength at the interface of region b. a Using dual-beam basis functions Expand i is the number of beam basis functions, r is the position in space, and k a1 Let k be the direction of the first wave vector in region a. a2 Let e ​​be the direction of the second wave vector in region a. a1i The coefficients of the basis functions corresponding to the first wave vector, e a2i E represents the coefficients of the basis functions corresponding to the second wave vector. b Using polynomial basis functions E bi Expand, E b =∑ i e bi E bi e bi The coefficients of the polynomial basis functions; is the free space wavenumber.

[0083] By substituting this boundary condition into the boundary integral term of the weak form of the finite element method, a complete system equation is formed.

[0084]

[0085] e Va1 e represents the basis function coefficients corresponding to the first wave vector of a node within region a. Sa1 e represents the coefficient of the basis function corresponding to the first wave vector at a node on the discontinuity surface of region a.Va2 e represents the basis function coefficients corresponding to the second wave vector of a node within region a. Sa1 e represents the coefficient of the basis function corresponding to the second wave vector at a node on the discontinuity surface of region a. Vb e represents the basis function coefficients of the nodes within region b. Sb The coefficients of the basis functions at the nodes on the discontinuity of region b are represented.

[0086]

[0087]

[0088]

[0089] Where dl represents the boundary integral and dV represents the area integral over the region.

[0090] Similarly, the compact expression for the system equations corresponding to the second wave vector in region a is:

[0091]

[0092]

[0093]

[0094]

[0095] The compact expression for the system equations in region b can also be written in the following form:

[0096]

[0097]

[0098]

[0099]

[0100] By combining the two regions into a unified system matrix, as shown below, the electric field distribution of the computational region can be obtained by solving the matrix.

[0101]

[0102] This will be illustrated using a dual-mirror model as an example. Figure 2 As shown: A Gaussian beam is incident along the positive y-axis. The beam will be reflected at mirror 1, changing its direction to the positive x-axis. It will also be reflected at mirror 2, changing its direction to the positive y-axis and hitting the screen.

[0103] like Figure 3 The diagram shows the simulation region partitioning for this invention. Regions a and b are the interference regions near the mirror, representing wave characteristics, and are calculated using polynomial basis functions. Region cf is the beam propagation region, representing particle characteristics, with two propagation directions: along the positive y-axis and along the positive x-axis, and is calculated using dual-beam basis functions. Regions a and b are partitioned using a dense grid, and polynomial basis functions are used in regions a and b. Region c is partitioned using a sparse grid, and regions d, f, and e are partitioned using a gradient grid. Region cf is treated as a single region, and dual-beam basis functions are used. Let the wave vector along the positive y-axis be k. a1 The wave vector along the positive x-axis is k. a2 .

[0104] like Figure 4 The figures show the electric field models obtained by the present invention and the commercial finite element simulation software COMSOL. (a) is the electric field model calculated by the method of the present invention; (b) is the electric field model calculated by the existing polynomial finite element commercial software COMSOL. The simulation results of the two are consistent, and the electric field at the interface between the two regions is matched. There is no numerical error caused by the use of different basis functions, which proves the correctness of the present invention.

[0105] Table 1 compares the computational information of the present invention and the traditional finite element method. It can be seen that the mesh count of the present invention is approximately 1 / 10 of that of the traditional finite element method, thus significantly reducing the computation time to approximately 1 / 10 of the traditional finite element method. Therefore, the present invention has significant advantages over the traditional finite element method.

[0106] Table 1 Comparison of Simulation Calculation Information for Dual-Mirror Mirrors

[0107] Number of grids 94252 923714 Degrees of freedom 190750 1850243 Calculation time 2.8s 25s

[0108] It should be noted that the data in Table 1 is for the case where the scale of the light field distribution region is 100 μm. When the scale of the light field distribution increases to the mesoscopic scale, the following conclusions were obtained after verification by the technical personnel of this invention: the traditional finite element method cannot simulate the light field distribution because the simulation freedom is too large and the computer memory cannot meet the calculation needs. The dual-beam finite element method can simulate the light field distribution with 1 / 10 of the degrees of freedom in a finite time.

[0109] It should be noted that in studying the coupling techniques of polynomial and beamforming regions, a wavy, interference-like phenomenon was observed at the interface between the two regions. Further investigation revealed that this is actually caused by dispersion errors in the finite element method. After discretization, the phase velocity determined by the discrete equations of the functional has a certain error compared to the actual phase velocity of the electromagnetic wave in the medium. This error accumulates and increases with the size of the computational target, thus affecting the accuracy of the final field calculation results. Furthermore, the use of different mesh sizes at the interface creates a sudden transition from sparse to dense mesh, which also contributes to the appearance of this wavy pattern.

[0110] Therefore, to solve this problem, this invention proposes two methods: improving the accuracy of the interpolation function and implementing a gradient mesh transition. This invention improves the accuracy of the interpolation function by upgrading the original first-order interpolation function to a second-order interpolation function. The first-order interpolation function is expanded using three basis functions, while the second-order interpolation function is expanded using six basis functions. This improves the accuracy of the interpolation function. Figure 5 As shown, (a) and (b) are comparison diagrams of the phase difference between the finite element method and the numerical solution using the first-order interpolation function and the second-order interpolation function, respectively. It can be clearly seen that when using the first-order interpolation function, the phase of the finite element method and the numerical solution increases with the increase of the propagation distance, while the second-order interpolation function is very effective in optimizing the accuracy of the calculation results.

[0111] At the same time, using a gradient mesh at the interface can effectively eliminate this wavy error. For example... Figure 6 As shown, the left side represents a sparse mesh area, the right side a dense mesh area, and the middle area a gradient mesh area. The overall effect is as follows. Figure 7 As shown in the comparison, (a) is the simulation result using first-order basis functions and without a gradient mesh, and (b) is the simulation result using second-order basis functions and a gradient mesh.

[0112] Figure 8 This is a diagram of the finite element optical simulation system architecture based on dual beams provided in an embodiment of the present invention, as shown below. Figure 8 As shown, it includes:

[0113] The model determination module 810 is used to determine a multi-scale optical model, dividing the light propagation region into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the light beam; the wave characteristic region includes an interference region and / or a diffraction region; the light beam has two stable propagation directions within the particle characteristic region;

[0114] The boundary condition determination module 820 is used to determine the boundary conditions corresponding to the interface between two different regions according to the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions.

[0115] The light field distribution simulation module 830 is used to apply a polynomial basis function to the wave characteristic region and a dual-beam basis function to the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The light field distribution of the multi-scale optical model is obtained by simulating using the finite element method.

[0116] The detailed implementation of each module can be found in the description of the aforementioned method embodiments, and will not be repeated here.

[0117] It should be understood that the above system is used to execute the methods in the above embodiments. The corresponding program modules in the system are similar in implementation principle and technical effect to those described in the above methods. The working process of the system can be referred to the corresponding process in the above methods, and will not be repeated here.

[0118] Based on the methods described in the above embodiments, this invention provides an electronic device. The device may include at least one memory for storing a program and at least one processor for executing the program stored in the memory. When the program stored in the memory is executed, the processor performs the methods described in the above embodiments.

[0119] Based on the methods in the above embodiments, this embodiment of the invention provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0120] Based on the methods in the above embodiments, this embodiment of the invention provides a computer program product that, when run on a processor, causes the processor to execute the methods in the above embodiments.

[0121] It is understood that the processor in the embodiments of the present invention can be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. The general-purpose processor can be a microprocessor or any conventional processor.

[0122] The method steps in the embodiments of the present invention can be implemented in hardware or by a processor executing software instructions. The software instructions can consist of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, portable hard disks, CD-ROMs, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor, enabling the processor to read information from and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can reside in an ASIC.

[0123] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of the present invention are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted through the computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state disk (SSD)).

[0124] It is understood that the various numerical designations used in the embodiments of the present invention are merely for descriptive convenience and are not intended to limit the scope of the embodiments of the present invention.

[0125] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A finite element optical simulation method based on dual-beams, characterized in that, Includes the following steps: A multi-scale optical model is determined, and the light propagation region is divided into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the light beam; the wave characteristic region includes an interference region and / or a diffraction region; the light beam has two stable propagation directions within the particle characteristic region; Based on the principle of continuous distribution of tangential electric and magnetic fields at the interface between adjacent regions, the boundary conditions corresponding to the interface between two different regions are determined; the boundary conditions are as follows: ； ； Wherein, subscript a indicates the region using dual-beam basis functions, and subscript b indicates the region using polynomial basis functions; Let be the normal vector of the interface from region a to region b. Let be the normal vector of the interface from region b to region a; and The matching impedance is the incident plane wave at the interface between the two regions. , and Let be the relative permittivity of regions a and b, respectively. and , respectively, are the relative permeabilities of region a and region b; Let be the electric field intensity at the interface of region a. Let be the electric field intensity at the interface of region b. Using dual-beam basis functions Expand , The number of beam basis functions. r Position in space The direction of the first wave vector in region a The direction of the second wave vector in region a and These are polynomial basis functions, and their corresponding wave vector phase terms. and The product is multiplied to form a dual-beam basis function; The coefficients of the basis functions corresponding to the first wave vector are . These are the coefficients of the basis functions corresponding to the second wave vector; Using polynomial basis functions Expand , The coefficients of the polynomial basis functions; For free space wavenumber; A polynomial basis function is used for the wave characteristic region and a dual-beam basis function is used for the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The light field distribution of the multi-scale optical model is obtained by finite element method simulation.

2. The method according to claim 1, characterized in that, In the simulation process using the finite element method, it is necessary to interpolate the polynomial basis functions and the dual-beam basis functions to obtain the corresponding finite element weak forms; wherein, the interpolation uses a second-order interpolation function.

3. The method according to claim 1, characterized in that, At the interface between two regions with different characteristics, a gradient mesh is used for transition; the gradient mesh is as follows: in a preset area where the particle characteristic region gradually approaches the interface with the wave characteristic region, the mesh density changes from sparse to dense as it approaches the interface; in a preset area where the particle characteristic region gradually moves away from the interface, the mesh density changes from dense to sparse as it moves away from the interface.

4. The method according to claim 1, characterized in that, When using the finite element method for simulation, substituting the boundary conditions into the finite element weak form boundary integral term of the basis functions yields the matrix equation of the multi-scale optical model as follows: ； in, This represents the basis function coefficients corresponding to the first wave vector of a node within region a. This represents the basis function coefficients corresponding to the first wave vector of the node at the interface of region a; This represents the basis function coefficients corresponding to the second wave vector of a node within region a. This represents the basis function coefficients corresponding to the second wave vector of the node at the interface of region a; Let represent the basis function coefficients of the nodes inside region b. The basis function coefficients of the nodes at the interface of region b are represented. , and Let the initial right-hand vector be ; ； ； ； ； ； ； ； ； ； in, The test function for the first wave vector of the dual-beam basis functions is taken as... , The test function for the second wave vector of the dual-beam basis functions is taken as follows: , Let be the test function of the polynomial basis functions, taking ; It is a volume integral element used to represent the volume integral within the finite element calculation region; is a line integral infinitesimal element used to represent line integrals on the boundary of the computational domain.

5. A finite element optical simulation system based on dual-beams, characterized in that, include: The model determination module is used to determine a multi-scale optical model, dividing the light propagation region into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the light beam; the wave characteristic region includes an interference region and / or a diffraction region; the light beam has two stable propagation directions within the particle characteristic region; The boundary condition determination module is used to determine the boundary conditions corresponding to the interface between two different regions according to the principle of continuous distribution of tangential electric and magnetic fields on the interface between adjacent regions; the boundary conditions are: ； ； Wherein, subscript a indicates the region using dual-beam basis functions, and subscript b indicates the region using polynomial basis functions; Let be the normal vector of the interface from region a to region b. Let be the normal vector of the interface from region b to region a; and The matching impedance is the incident plane wave at the interface between the two regions. , and Let be the relative permittivity of regions a and b, respectively. and , respectively, are the relative permeabilities of region a and region b; Let be the electric field intensity at the interface of region a. Let be the electric field intensity at the interface of region b. Using dual-beam basis functions Expand , The number of beam basis functions. r Position in space The direction of the first wave vector in region a The direction of the second wave vector in region a and These are polynomial basis functions, and their corresponding wave vector phase terms. and The product is multiplied to form a dual-beam basis function; The coefficients of the basis functions corresponding to the first wave vector are . These are the coefficients of the basis functions corresponding to the second wave vector; Using polynomial basis functions Expand , The coefficients of the polynomial basis functions; For free space wavenumber; The light field distribution simulation module is used to apply a polynomial basis function to the wave characteristic region and a dual-beam basis function to the particle characteristic region. Combined with the boundary conditions, the wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The light field distribution of the multi-scale optical model is obtained by simulating using the finite element method.

6. The system according to claim 5, characterized in that, In the process of simulating using the finite element method, the optical field distribution simulation module needs to interpolate the polynomial basis functions and the dual-beam basis functions to obtain the corresponding finite element weak forms; wherein, the interpolation uses a second-order interpolation function.

7. The system according to claim 5, characterized in that, The light field distribution simulation module uses a gradient mesh for transition at the interface between two regions with different characteristics. The gradient mesh is as follows: in a preset region where the particle characteristic region gradually approaches the interface with the wave characteristic region, the mesh density changes from sparse to dense as it approaches the interface; in a preset region where the particle characteristic region gradually moves away from the interface, the mesh density changes from dense to sparse as it moves away from the interface.

8. An electronic device, characterized in that, include: At least one memory for storing programs; At least one processor is configured to execute a program stored in the memory, wherein when the program stored in the memory is executed, the processor is configured to perform the method as described in any one of claims 1-4.

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