A Finite Element Optical Simulation Method and Device Based on Hybrid Beams

Abstract

This invention provides a finite element optical simulation method and apparatus based on hybrid beams, comprising: determining 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 beam; the wave characteristic region includes an interference region and / or a diffraction region; the particle characteristic region includes a single propagation direction region and a dual propagation direction region; determining the boundary conditions corresponding to the interface between two regions with different propagation characteristics according to the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions; applying polynomial basis functions to the wave region and hybrid beam basis functions to the particle region, and dividing the wave region into a dense grid and the particle region into a sparse grid; the hybrid beam basis functions include single beam basis functions and dual beam basis functions; and simulating the light field distribution of the multi-scale optical model using the finite element method by combining the boundary conditions and basis functions. This invention improves the accuracy and speed of optical simulation.

Description

Technical Field

[0001] This invention belongs to the field of computational electromagnetics, and more specifically, relates to a finite element optical simulation method and apparatus based on hybrid 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 Augustus, marked the birth of the FEM. It wasn't until 1960 that Clouph first named the method the FEM. Since the concept of the FEM 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 finite element optical simulation method and apparatus based on hybrid beams. This invention addresses the problems of insufficient computer memory or excessive computation time when using polynomial finite element simulation methods 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 methods.

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

[0007] A multi-scale optical model is determined, and the propagation region of light 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: interference region and / or diffraction region; the particle characteristic region includes: single propagation direction region and dual propagation direction 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 regions with different propagation characteristics are determined.

[0009] A polynomial basis function is used for the wave characteristic region, and a hybrid beam basis function is used for the particle characteristic region. The wave characteristic region is divided into a dense grid, and the particle characteristic region is divided into a sparse grid. The hybrid beam basis function includes a single beam basis function and a dual beam basis function. The single propagation direction region uses a single beam basis function, and the dual propagation direction region uses a dual beam basis function.

[0010] By combining the aforementioned boundary conditions, polynomial basis functions, and hybrid beam basis functions, the optical field distribution of the multi-scale optical model is obtained through simulation using the finite element method.

[0011] In one possible implementation, during the simulation using the finite element method, it is necessary to interpolate the polynomial basis functions, single-beam basis functions, and dual-beam basis functions to obtain the corresponding finite element weak forms; wherein, the interpolation uses a second-order interpolation function.

[0012] In one possible implementation, a gradient mesh is used for transition at the interface between two regions with different propagation characteristics. The gradient mesh is characterized by the following: 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.

[0013] In one possible implementation, the boundary conditions include: a first set of boundary conditions, which corresponds to the interface between the single-beam basis function region and the polynomial basis function region, specifically:

[0014]

[0015]

[0016] Wherein, subscript a indicates the region using single-beam basis functions, and subscript b indicates the region using polynomial basis functions; n abLet n be the normal vector at the interface from region a to region b. ba m is the normal vector at the interface pointing 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 E represents the relative permeability of region a and region b, respectively. 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 single-beam basis functions Expand i is the number of beam basis functions, r is the position in space, and k a Let e ​​be the wave vector direction in region a. ai The coefficients of the single-beam basis function; E 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.

[0017] In one possible implementation, the boundary conditions further include: a second set of boundary conditions, which corresponds to the interface between the dual-beam basis function region and the polynomial basis function region, specifically:

[0018]

[0019]

[0020] Where the subscript c indicates the region where the dual-beam basis function is used; n cb Let n be the normal vector of the interface from region c to region b. bc m is the normal vector of the interface from region b to region c; c Let c be the matching impedance of the plane wave incident on the interface. ∈ rc Let μ be the relative permittivity of region c. rc Let be the relative permeability of region c; c E represents the electric field intensity at the interface of region c. c Using dual-beam basis functions Expand kc1 Let k be the direction of the first wave vector in region c. c2 Let e ​​be the direction of the second wave vector in region c. c1i The coefficients of the basis functions corresponding to the first wave vector, e c2i These are the coefficients of the basis functions corresponding to the second wave vector.

[0021] In one possible implementation, when using the finite element method for simulation, the first set of boundary conditions and the second set of boundary conditions are substituted into the finite element weak form boundary integral term of the basis function to obtain the matrix equation of the multi-scale optical model:

[0022]

[0023] Among them, e Va e represents the single-beam basis function coefficients of nodes within region a. Sa The coefficients of the single-beam basis functions at the nodes on the interface of region a; e Vb e represents the coefficients of the polynomial basis functions of the nodes within region b. Sb The coefficients of the polynomial basis functions at the nodes on the interface of region b; e Vc1 e represents the basis function coefficients corresponding to the first wave vector of a node within region c. Sc1 The basis function coefficients corresponding to the first wave vector of the node at the interface of region c; e Vc2 e represents the basis function coefficients corresponding to the second wave vector of a node within region c. Sc2 The basis function coefficients corresponding to the second wave vector of the node on the interface of region c; b a b b b c1 and b c2 Let be the initial right-hand vector of the matrix equation;

[0024]

[0025] Among them, W aj The test function for the beam basis function is taken as follows: W bj Let ∑ be the test function of the polynomial basis functions. j E bj ;

[0026]

[0027]

[0028] Among them, W c1j The test function for the first wave vector of the dual-beam basis functions is taken as... W c2j The test function for the second wave vector of the dual-beam basis functions is taken as follows:

[0029] Secondly, the present invention provides a finite element optical simulation device based on hybrid beams, comprising:

[0030] An optical model determination unit 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 particle characteristic region includes: a single propagation direction region and a dual propagation direction region;

[0031] The boundary condition determination unit is used to determine the boundary conditions corresponding to the interface between two regions with different propagation characteristics, based on the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions.

[0032] A basis function determination unit is used to apply polynomial basis functions to the wave characteristic region and hybrid beam basis functions to the particle characteristic region; wherein, the hybrid beam basis functions include single beam basis functions and dual beam basis functions, the single beam basis function is used to the single propagation direction region and the dual beam propagation direction region is used to the dual beam basis function.

[0033] Mesh generation unit is used to divide the wave characteristic region into a dense mesh and the particle characteristic region into a sparse mesh;

[0034] The optical field distribution simulation unit is used to combine the boundary conditions, polynomial basis functions and hybrid beam basis functions to simulate the optical field distribution of a multi-scale optical model using the finite element method.

[0035] In one possible implementation, the optical field distribution simulation unit needs to interpolate the polynomial basis functions, single-beam basis functions, and dual-beam basis functions to obtain the corresponding weak finite element forms during the simulation process using the finite element method; wherein, the interpolation uses a second-order interpolation function.

[0036] In one possible implementation, the mesh division unit uses a gradient mesh for transition at the interface between two regions with different propagation 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.

[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 implementation thereof.

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

[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 implementation thereof.

[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 finite element optical simulation method and apparatus based on hybrid beams. 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 hybrid beam finite element technique. The computational region is divided using domain decomposition technology, and different basis functions are applied to different regions. Based on the beam propagation characteristics, the light propagation region is divided into particle characteristic regions and wave characteristic regions. The wave characteristic region includes interference and / or diffraction regions; within the particle characteristic region, the beam has one stable propagation method and two stable propagation directions. Boundary conditions corresponding to the interface between two different regions are determined according to 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 hybrid beam basis functions are applied 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 into a sparse grid. The finite element method is used to simulate the light field distribution of a multi-scale optical model. In this invention, the introduction of hybrid 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 hybrid beams provided in an embodiment of the present invention;

[0043] Figure 2 This is a schematic diagram of a hybrid refractive optical system model provided in an embodiment of the present invention;

[0044] Figure 3 This is a schematic diagram of the simulation region division of the hybrid refractive optical system provided in an embodiment of the present invention;

[0045] Figure 4These are comparison diagrams of simulation calculation results of the refractive-diffractive hybrid optical system 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 This is a comparison diagram of the electric field intensity at the light screen provided in an embodiment of the present invention;

[0047] Figure 6 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.

[0048] Figure 7 This is a schematic diagram of a gradient mesh provided in an embodiment of the present invention;

[0049] Figure 8 These are comparison diagrams of finite element method and numerical solution simulation results using interpolation functions of different orders provided in this embodiment of the invention; (a) using only the beam envelope basis function; (b) using the first-order interpolation function in this technique; (c) using the second-order interpolation function in this technique;

[0050] Figure 9 This is an architecture diagram of a finite element optical simulation device based on hybrid beams provided in an embodiment of the present invention. Detailed Implementation

[0051] 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.

[0052] 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.

[0053] 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.

[0054] 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.

[0055] First, the technical terms involved in the embodiments of the present invention will be introduced.

[0056] (1) Single beam basis function

[0057] This invention assumes that the propagation direction of light waves is almost stable in a certain region, and therefore uses a single-beam basis function to approximate the slowly varying electromagnetic field. That is, E(x,y,z)=∑ i e i E i (,y,z) -ik·r .

[0058] In the two-dimensional case, Substituting into the vector wave equation:

[0059]

[0060] The weak form of the finite element method can be written as:

[0061]

[0062] (2) Dual-beam basis functions

[0063] 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:

[0064]

[0065] At this point, we can obtain the two sets of weakly form formulas as follows:

[0066]

[0067]

[0068] 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.

[0069] (3) Polynomial basis functions

[0070] 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).

[0071] In the two-dimensional case, E(x,y,z)=∑ i e i E i (x, y). Substitute into the vector wave equation:

[0072]

[0073] The weak form of the finite element method can be written as:

[0074]

[0075] Next, the technical solutions provided in the embodiments of the present invention will be introduced.

[0076] Figure 1 This is a flowchart of the finite element optical simulation method based on hybrid beams provided in this embodiment of the invention; as shown... Figure 1 As shown, it includes the following steps:

[0077] S101, Determine the multi-scale optical model, and divide 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: interference region and / or diffraction region; the particle characteristic region includes: single propagation direction region and dual propagation direction region;

[0078] S102, based on the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions, determine the boundary conditions corresponding to the interface between two regions with different propagation characteristics;

[0079] S103, a polynomial basis function is used for the wave characteristic region and a hybrid beam basis function is used for the particle characteristic region. The wave characteristic region is divided into a dense grid and the particle characteristic region is divided into a sparse grid. The hybrid beam basis function includes a single beam basis function and a dual beam basis function. The single propagation direction region uses a single beam basis function and the dual propagation direction region uses a dual beam basis function.

[0080] S104. Combining the aforementioned boundary conditions, polynomial basis functions, and hybrid beam basis functions, the light field distribution of the multi-scale optical model is obtained by finite element method simulation.

[0081] This invention proposes a hybrid beam finite element method to solve the simulation problem of hybrid refractive-diffractive optical systems. Mesoscopic dimensions refer to sizes between the microscopic and macroscopic worlds. Using visible light wavelengths as the reference wavelength, the characteristic dimensions of mesoscopic systems range from micrometers to millimeters. Addressing the challenge of simulating mesoscopic optical systems using neither wave optics nor geometric optics techniques, a novel and efficient algorithm is provided. Addressing the difficulty in simulating hybrid refractive-diffractive optical systems such as diffractive waveguide devices in augmented reality (AR) and virtual reality (VR) glasses, which have received considerable attention in recent years, a divide-and-conquer approach is adopted. This method combines single-beam envelope and double-wave vector envelope techniques with the finite element method to propose a hybrid beam finite element method. The computational domain is divided using domain decomposition techniques. Multi-scale basis functions are used, and beam basis functions with phase factors are employed to calculate field regions with relatively definite propagation directions. Polynomial basis functions are used for regions with complex field distributions. First-order impedance-type hybrid boundary conditions are used to couple adjacent computational domains, thereby reducing the number of meshes, reducing computational degrees of freedom, and significantly improving solution efficiency.

[0082] 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.

[0083] The polynomial-hybrid beamforming 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 beamforming is low, using polynomial basis functions for wave-characteristic regions greatly improves the accuracy of macroscale optical field simulations.

[0084] Drawing inspiration from domain decomposition, this study combines beam envelope technology with the finite element method, employing a multi-scale basis function approach, including polynomial basis functions, single-beam basis functions, and dual-beam basis functions. This allows for the use of appropriate basis functions for interpolation in different regions. Specifically, this includes polynomial-single-beam region coupling techniques, polynomial-dual-beam region coupling techniques, and hybrid region coupling techniques.

[0085] 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 computational methods in each sub-region. Specifically, a band is used in the computation of field regions with relatively definite propagation directions. Subsequently, 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 hybrid beam finite element method must be at least second-order.

[0086] It should be noted that the polynomial-hybrid beam finite element simulation method is referred to as the "hybrid beam finite element" simulation method in this invention.

[0087] To verify the correctness of this technology, the relevant algorithm was written in Matlab, and the same model was calculated in the numerical simulation software COMSOL. The calculation results were then compared, including the field diagram of the complete region and the electric field intensity at the exit surface. The results show that this technology can obtain calculation results consistent with COMSOL with a smaller number of grids, significantly reducing computation time and memory usage. Furthermore, the comparison of the electric field intensity at the exit surface demonstrates that this technology also possesses high computational accuracy.

[0088] Currently, there is no technology in optical finite element method that can solve multi-scale optical simulation problems. Therefore, the algorithm of this invention fills this gap, greatly reducing the computation time and memory usage of optical systems composed of existing large-scale geometric optical structures and small-scale wave optical structures, and making a certain contribution to the development of photonics.

[0089] It should be noted that beam envelope technology is essentially a multi-scale basis function. Hybrid beam finite element method can use different basis functions in different regions. 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. Therefore, hybrid beam finite element method uses three types of basis functions simultaneously: polynomial basis functions, single-beam basis functions, and dual-beam basis functions.

[0090] Because this invention adds a phase factor term to the beam basis function, making the basis function take the form of an electric field envelope, the number of grids required for computation can be significantly reduced. Traditional finite element methods require very dense meshes to obtain accurate results, while using 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, the multi-scale basis functions using this technology can effectively reduce the degrees of freedom in computation, thereby reducing computation time and memory usage.

[0091] By applying the aforementioned polynomial region-single-beam region coupling techniques and polynomial region-dual-beam region coupling techniques, along with a series of boundary condition simulation techniques, this invention successfully simulates multi-scale problems in bifolding diffraction hybrid optical systems using hybrid beam finite element technology.

[0092] Specifically, the boundary conditions in this invention include: a first set of boundary conditions and a second set of boundary conditions. The first set of boundary conditions corresponds to the interface between the single-beam basis function region and the polynomial basis function region, specifically as follows:

[0093]

[0094]

[0095] Wherein, subscript a indicates the region using single-beam basis functions, and subscript b indicates the region using polynomial basis functions; n ab Let n be the normal vector at the interface from region a to region b. ba m is the normal vector at the interface pointing 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 E represents the relative permeability of region a and region b, respectively. 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 single-beam basis functions Expand i is the number of beam basis functions, r is the position in space, and k a Let e ​​be the wave vector direction in region a. ai E represents the coefficients of the single-beam basis function. b Using polynomial basis functions E bi Expand, E b =∑ ie bi E bi e bi The coefficients of the polynomial basis functions; is the free space wavenumber.

[0096] The second set of boundary conditions corresponds to the interface between the dual-beam basis function region and the polynomial basis function region, specifically:

[0097]

[0098]

[0099] Where the subscript c indicates the region where the dual-beam basis function is used; n cb Let n be the normal vector of the interface from region c to region b. bc m is the normal vector of the interface from region b to region c; c Let c be the matching impedance of the plane wave incident on the interface. ∈ rc Let μ be the relative permittivity of region c. rc Let be the relative permeability of region c; c E represents the electric field intensity at the interface of region c. c Using dual-beam basis functions Expand k c1 Let k be the direction of the first wave vector in region c. c2 Let e ​​be the direction of the second wave vector in region c. c1i The coefficients of the basis functions corresponding to the first wave vector, e c2i These are the coefficients of the basis functions corresponding to the second wave vector.

[0100] In one possible implementation, when using the finite element method for simulation, the first and second set of boundary conditions are substituted into the finite element weak form boundary integral terms of the basis functions to obtain the matrix equations of the multi-scale optical model:

[0101]

[0102] Among them, e Va e represents the single-beam basis function coefficients of nodes within region a. Sa The coefficients of the single-beam basis functions at the nodes on the interface of region a; e Vb e represents the coefficients of the polynomial basis functions of the nodes within region b. Sb The coefficients of the polynomial basis functions at the nodes on the interface of region b; e Vc1 e represents the basis function coefficients corresponding to the first wave vector of a node within region c. Sc1The basis function coefficients corresponding to the first wave vector of the node at the interface of region c; e Vc2 e represents the basis function coefficients corresponding to the second wave vector of a node within region c. Sc2 The basis function coefficients corresponding to the second wave vector of the node on the interface of region c; b a b b b c1 and b c2 Let be the initial right-hand vector of the matrix equation;

[0103]

[0104] Where dl represents the boundary integral and dV represents the surface integral over the region. aj The test function for the beam basis function is taken as follows: W bj Let ∑ be the test function of the polynomial basis functions. j E bj ;

[0105]

[0106]

[0107] Among them, W c1j The test function for the first wave vector of the dual-beam basis functions is taken as... W c2j The test function for the second wave vector of the dual-beam basis functions is taken as follows:

[0108] It should be noted that this invention does not consider the boundary conditions of single-beam and dual-beam regions in determining the boundary conditions because: 1. Beam direction generally does not change spontaneously, but is altered by the influence of an optical device; 2. If the optical device changes the beam direction, the mechanism of action becomes more complex, and simulation using beam basis functions generally yields larger errors, while simulation using polynomial basis functions is more accurate. Therefore, in this invention, after dividing according to propagation characteristics, there should be no direct boundary between single-beam and dual-beam regions.

[0109] Furthermore, if accuracy is not a concern, the boundary between single-beam and dual-beam regions needs to be considered. This invention also addresses this situation: since the entire region does not have only one stable propagation direction, the application of single-beam region-to-single-beam region coupling technology is limited. A common example is the catadioptric model, where the beam direction has three directions: incident, refracted, and reflected. Therefore, this invention develops a dual-beam region-to-dual-beam region coupling technology to solve this problem.

[0110] This explanation uses a catadioptric model as an example. Assume the refractive index of region a on the left is 1.5, and the refractive index of region b on the right is 1.3. A Gaussian beam is incident along the positive x-axis. According to geometric optics, the beam will undergo both refraction and reflection at the interface. Let the wave vector of the incident wave in region a be k. a1 The direction of the wave vector of the reflected wave in region a is k. a2 The wave vector direction of the refracted wave in region b on the right is k. b1 The direction of the wave vector of the virtual wave in region b on the right is k. b2 The reason for having such a virtual wave vector direction is that the phase variable needs to be continuous at the boundary between the two media. The virtual wave vector direction is defined as the direction of the virtual refracted wave calculated through the interface in the opposite direction of the reflected wave; this is necessary to form a continuous phase across the boundary. The first-order impedance-type mixed boundary condition is written in the following form:

[0111]

[0112] 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.

[0113]

[0114] When the catadioptric model is not divided into polynomial regions, the accuracy of the simulation method using beam basis functions is indeed not high, and the simulated beam distribution has ripples and unclear boundaries. Therefore, this invention makes clear divisions for complex optical fields, using polynomial basis functions in complex regions and beam basis functions in regions with clear and simple beam directions. This can further improve the simulation accuracy while ensuring simulation speed, which will be explained in detail below.

[0115] This invention uses simulation as... Figure 2 The illustrated hybrid refractive-diffraction optical system serves as an example. The entire optical system consists of two mirrors and a Fresnel lens. The incident light is a Gaussian beam along the y-axis, which is reflected by the two mirrors and then converged onto the screen by the Fresnel lens. It is well known from experience that the propagation direction of light is relatively stable in the region incident on mirror 1, the transmission region between the two mirrors, the region between mirror 2 and the Fresnel lens, and the light convergence region from the Fresnel lens to the screen. However, diffraction and reflection occur near the Fresnel lens region, resulting in a more complex light field. Therefore, the correct light field distribution cannot be obtained using only the beam envelope basis function.

[0116] like Figure 3The diagram shows the simulation region partitioning for this invention. Regions a and c are interference regions near the mirror, and region d is the diffraction region near the Fresnel lens, representing wave characteristics. Polynomial basis functions are used for calculation. Regions b and fi are beam propagation regions, representing particle characteristics, with two propagation directions: along the positive y-axis and along the positive x-axis. Dual-beam basis functions are used for calculation. Regions e and j are beam propagation regions, representing particle characteristics, with one propagation direction along the positive y-axis. Single-beam basis functions are used for calculation. Regions a, c, and d are partitioned using a dense grid, and polynomial basis functions are used in these regions. Regions b and e are partitioned using a sparse grid, and region fj is partitioned using a gradient grid. Regions b and fi are treated as a single region, and dual-beam basis functions are used. Regions e and j are treated as a single region, and single-beam basis functions are used.

[0117] The mode field distribution calculated by the hybrid beam finite element method provided in this invention is similar to the mode field distribution obtained by the commercial simulation software COMSOL using the traditional finite element method. Figure 4 As shown in Table 1, the simulation results of the two methods are basically consistent, proving the accuracy of the present invention. The number of meshes, degrees of freedom, and computation time required for the two methods are shown in Table 1. It can be seen that the number of meshes required by the present invention is about 1 / 8 of that of the traditional finite element method, and therefore the computation time is also significantly reduced, to about 1 / 8 of that of the traditional finite element method.

[0118] Table 1 Comparison of Simulation Calculation Information for Hybrid Refractive and Diffractive Optical Systems

[0119] Number of grids 237300 1639310 Degrees of freedom 476055 3282437 Calculation time 6s 45s

[0120] It should be noted that the data in Table 1 is for the case where the scale of the light field distribution region is 0.1 mm. 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 hybrid beam finite element method can simulate the light field distribution with 1 / 10 of the degrees of freedom in a finite time.

[0121] In addition, this invention compares the electric field intensity distribution at the light screen, such as... Figure 5 As shown, the calculation results of the two methods are basically in agreement. Therefore, this technique has significant advantages over the traditional finite element method.

[0122] In studying polynomial-region and beamforming coupling techniques, 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.

[0123] Therefore, to address this problem, this technology proposes two methods: improving the accuracy of the interpolation function and implementing a gradient mesh transition. This invention upgrades 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 method improves the accuracy of the interpolation function. Figure 6 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.

[0124] At the same time, using a gradient mesh at the interface can effectively eliminate this wavy error. For example... Figure 7 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 8 As shown in the comparison, (a) is the simulation result using only the beam envelope basis function, (b) is the simulation result using the first-order basis function without a gradient mesh, and (c) is the simulation result using the second-order basis function and a gradient mesh. It can be seen that using only the beam envelope basis function results in inaccurate simulations in the catadioptric region. Using the first-order basis function without a gradient mesh produces wavy errors, while using the second-order basis function and a gradient mesh yields an accurate solution.

[0125] Figure 9 This is an architectural diagram of a finite element optical simulation device based on hybrid beams provided in an embodiment of the present invention; as shown below. Figure 9 As shown, it includes:

[0126] The optical model determination unit 910 is used to determine a multi-scale optical model and divide 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 particle characteristic region includes: a single propagation direction region and a dual propagation direction region;

[0127] Boundary condition determination unit 920 is used to determine the boundary conditions corresponding to the interface between two regions with different propagation characteristics according to the principle of continuous distribution of tangential electric field and tangential magnetic field on the interface between adjacent regions.

[0128] The basis function determination unit 930 is used to apply polynomial basis functions to the wave characteristic region and hybrid beam basis functions to the particle characteristic region; wherein, the hybrid beam basis functions include single beam basis functions and dual beam basis functions, the single propagation direction region uses single beam basis functions, and the dual propagation direction region uses dual beam basis functions;

[0129] Mesh generation unit 940 is used to divide the wave characteristic region into a dense mesh and the particle characteristic region into a sparse mesh;

[0130] The optical field distribution simulation unit 950 is used to simulate the optical field distribution of a multi-scale optical model by combining the boundary conditions, polynomial basis functions and hybrid beam basis functions, using the finite element method.

[0131] It should be understood that the above-described device is used to execute the methods in the above embodiments. The implementation principle and technical effect of the corresponding program modules in the device are similar to those described in the above methods. The working process of the device can be referred to the corresponding process in the above methods, and will not be repeated here.

[0132] 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.

[0133] 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.

[0134] 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.

[0135] 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.

[0136] 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.

[0137] 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)).

[0138] 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.

[0139] 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 hybrid beams, characterized in that, Includes the following steps: A multi-scale optical model is determined, and the propagation region of light is divided into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the beam. The wave characteristic region includes: the interference region and / or the diffraction region; the particle characteristic region includes: the single propagation direction region and the dual propagation direction 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 regions with different propagation characteristics are determined. The boundary conditions include: a first set of boundary conditions, which corresponds to the interface between the single-beam basis function region and the polynomial basis function region, specifically: in, The symbol represents the imaginary unit. The subscript 'a' indicates the region using single-beam basis functions, and the subscript 'b' indicates the region using polynomial basis functions. Let be the normal vector at the interface from region a to region b. Let be the normal vector at 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 single-beam basis functions Expand , The number of beam basis functions. r Position in space The wave vector direction in region a The coefficients are the basis functions of a single beam. This represents the polynomial basis functions, through which the wave vector direction is characterized by their relationship with this region. Multiply them to construct single-beam basis functions; Using polynomial basis functions Expand , The coefficients of the polynomial basis functions; For free space wavenumber; This represents the wavelength in free space. The boundary conditions also include: a second set of boundary conditions, which correspond to the interface between the dual-beam basis function region and the polynomial basis function region, specifically: Wherein, the subscript c indicates the region where the dual-beam basis function is used; Let be the normal vector of the interface pointing from region c to region b. Let be the normal vector of the interface from region b to region c; Let c be the matching impedance of the plane wave incident on the interface. , Let c be the relative permittivity of region c. Let be the relative permeability of region c; Let be the electric field intensity at the interface of region c; Using dual-beam basis functions Expand , The direction of the first wave vector in region c The direction of the second wave vector in region c The coefficients of the basis functions corresponding to the first wave vector are . The coefficients of the basis functions corresponding to the second wave vector are... and Both represent polynomial basis functions, which, along with this region, characterize the directions of the two wave vectors. and Multiply them separately to construct the dual-beam basis functions; A polynomial basis function is used for the wave characteristic region, and a hybrid beam basis function is used for the particle characteristic region. The wave characteristic region is divided into a dense grid, and the particle characteristic region is divided into a sparse grid. The hybrid beam basis function includes a single beam basis function and a dual beam basis function. The single propagation direction region uses a single beam basis function, and the dual propagation direction region uses a dual beam basis function. By combining the aforementioned boundary conditions, polynomial basis functions, and hybrid beam basis functions, the optical field distribution of the multi-scale optical model is obtained through simulation using the finite element method.

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, single-beam basis functions, and 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 propagation characteristics, a gradient mesh is used for transition. 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.

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

5. A finite element optical simulation device based on hybrid beams, characterized in that, include: The optical model determination unit is used to determine the multi-scale optical model and divide the light propagation region into a particle characteristic region and a wave characteristic region according to the propagation characteristics of the beam. The wave characteristic region includes: the interference region and / or the diffraction region; the particle characteristic region includes: the single propagation direction region and the dual propagation direction region; The boundary condition determination unit is used to determine the boundary conditions corresponding to the interface between two regions with different propagation characteristics, based on the principle of continuous distribution of tangential electric and magnetic fields on the interface between adjacent regions. The boundary conditions include: a first set of boundary conditions, which corresponds to the interface between the single-beam basis function region and the polynomial basis function region, specifically: in, The symbol represents the imaginary unit. The subscript 'a' indicates the region using single-beam basis functions, and the subscript 'b' indicates the region using polynomial basis functions. Let be the normal vector at the interface from region a to region b. Let be the normal vector at 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 single-beam basis functions Expand , The number of beam basis functions. r Position in space The wave vector direction in region a The coefficients are the basis functions of a single beam. This represents the polynomial basis functions, through which the wave vector direction is characterized by their relationship with this region. Multiply them to construct single-beam basis functions; Using polynomial basis functions Expand , The coefficients of the polynomial basis functions; For free space wavenumber; Indicates the wavelength in free space; The boundary conditions also include: a second set of boundary conditions, which correspond to the interface between the dual-beam basis function region and the polynomial basis function region, specifically: Wherein, the subscript c indicates the region where the dual-beam basis function is used; Let be the normal vector of the interface pointing from region c to region b. Let be the normal vector of the interface from region b to region c; Let c be the matching impedance of the plane wave incident on the interface. , Let c be the relative permittivity of region c. Let be the relative permeability of region c; Let be the electric field intensity at the interface of region c; Using dual-beam basis functions Expand , The direction of the first wave vector in region c The direction of the second wave vector in region c The coefficients of the basis functions corresponding to the first wave vector are . The coefficients of the basis functions corresponding to the second wave vector are... and This represents the polynomial basis functions, which, along with this region, characterize the directions of the two wave vectors. Multiply them separately to construct the dual-beam basis functions; A basis function determination unit is used to apply polynomial basis functions to the wave characteristic region and hybrid beam basis functions to the particle characteristic region; wherein, the hybrid beam basis functions include single beam basis functions and dual beam basis functions, the single beam basis function is used to the single propagation direction region and the dual beam propagation direction region is used to the dual beam basis function. Mesh generation unit is used to divide the wave characteristic region into a dense mesh and the particle characteristic region into a sparse mesh; The optical field distribution simulation unit is used to combine the boundary conditions, polynomial basis functions and hybrid beam basis functions to simulate the optical field distribution of a multi-scale optical model using the finite element method.

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

7. The apparatus according to claim 5, characterized in that, The mesh division unit uses a gradient mesh for transition at the interface between two regions with different propagation 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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